New User Special Price Expires in

Let's log you in.

Sign in with Facebook


Don't have a StudySoup account? Create one here!


Create a StudySoup account

Be part of our community, it's free to join!

Sign up with Facebook


Create your account
By creating an account you agree to StudySoup's terms and conditions and privacy policy

Already have a StudySoup account? Login here

Gen Econ TheoryMacroeconomics

by: Alan Doyle

Gen Econ TheoryMacroeconomics ECON 510

Marketplace > Yale University > Economcs > ECON 510 > Gen Econ TheoryMacroeconomics
Alan Doyle
GPA 3.89


Almost Ready


These notes were just uploaded, and will be ready to view shortly.

Purchase these notes here, or revisit this page.

Either way, we'll remind you when they're ready :)

Preview These Notes for FREE

Get a free preview of these Notes, just enter your email below.

Unlock Preview
Unlock Preview

Preview these materials now for free

Why put in your email? Get access to more of this material and other relevant free materials for your school

View Preview

About this Document

Class Notes
25 ?




Popular in Course

Popular in Economcs

This 237 page Class Notes was uploaded by Alan Doyle on Thursday October 29, 2015. The Class Notes belongs to ECON 510 at Yale University taught by Staff in Fall. Since its upload, it has received 8 views. For similar materials see /class/231029/econ-510-yale-university in Economcs at Yale University.


Reviews for Gen Econ TheoryMacroeconomics


Report this Material


What is Karma?


Karma is the currency of StudySoup.

You can buy or earn more Karma at anytime and redeem it for class notes, study guides, flashcards, and more!

Date Created: 10/29/15
Lecture notes for Macroeconomics I 2004 Per Krusell Please do NOT distribute Without permission Comments and suggestions are welcome Chapter 1 Introduction These lecture notes cover a one semester course The overriding goal of the course is to begin provide methodological tools for advanced research in macroeconomics The emphasis is on theory although data guides the theoretical explorations We build en tirely on models with microfoundations ie models where behavior is derived from basic assumptions on consumers7 preferences production technologies information and so on Behavior is always assumed to be rational given the restrictions imposed by the primi tives all actors in the economic models are assumed to maximize their objectives Macroeconomic studies emphasize decisions with a time dimension such as various forms of investments Moreover it is often useful to assume that the time horizon is in nite This makes dynamic optimization a necessary part of the tools we need to cover and the rst signi cant fraction of the course goes through in turn sequential maximization and dynamic programming We assume throughout that time is discrete since it leads to simpler and more intuitive mathematics The baseline macroeconomic model we use is based on the assumption of perfect com petition Current research often departs from this assumption in various ways but it is important to understand the baseline in order to fully understand the extensions There fore we also spend signi cant time on the concepts of dynamic competitive equilibrium both expressed in the sequence form and recursively using dynamic programming In this context the welfare properties of our dynamic equilibria are studied In nite horizon models can employ different assumptions about the time horizon of each economic actor We study two extreme cases all consumers really dynasties live forever the in nitely lived agent model and ii consumers have nite and deterministic lifetimes but there are consumers of different generations living at any point in time the overlapping generations model These two cases share many features but also have important differences Most of the course material is built on in nitely lived agents but we also study the overlapping generations model in some depth Finally many macroeconomic issues involve uncertainty Therefore we spend some time on how to introduce it into our models both mathematically and in terms of eco nomic concepts The second part of the course notes goes over some important macroeconomic topics These involve growth and business cycle analysis asset pricing scal policy monetary economics unemployment and inequality Here few new tools are introduced we instead simply apply the tools from the rst part of the course Chapter 2 Motivation Solow s growth model Most modern dynamic models of macroeconomics build on the framework described in Solow7s 1956 paper1 To motivate what is to follow7 we start with a brief description of the Solow model This model was set up to study a closed economy7 and we will assume that there is a constant population 21 The model The model consists of some simple equations Ot1t Yt FKtL 21 It Kt1 1 i 6 Kt 22 It 5F Kt7 L 23 The equalities in 21 are accounting identities7 saying that total resources are either consumed or invested7 and that total resources are given by the output of a production function with capital and labor as inputs We take labor input to be constant at this point7 whereas the other variables are allowed to vary over time The accounting identity can also be interpreted in terms of technology this is a one good7 or one sector7 economy7 where the only good can be used both for consumption and as capital investment Equation 22 describes capital accumulation the output good7 in the form of investment7 is used to accumulate the capital input7 and capital depreciates geometrically a constant fraction 6 E 01 disintegrates every period Equation 23 is a behavioral equation Unlike in the rest of the course7 behavior here is assumed directly a constant fraction 5 E 01 of output is saved7 independently of what the level of output is These equations together form a complete dynamic system an equation system de n ing how its variables evolve over time for some given F That is7 we know7 in principle7 what Kt110 and K70t71t0 will be7 given any initial capital value K0 In order to analyze the dynamics7 we now make some assumptions 1No attempt is made here to properly assign credit to the inventors of each model For example7 the Solow model could also be called the Swan model7 although usually it is not F 07 L 0 6 FK 07 gt klirnsFKKL17 6 lt1 F is strictly concave in K and strictly increasing in K An example of a function satisfying these assurnptions7 and that will be used repeat edly in the course7 is FK7 L AKu Ll D with 0 lt 04 lt 1 This production function is called Cobb Douglas function Here A is a productivity pararneter7 and Oz and 1 7 Oz denote the capital and labor share7 respectively Why they are called shares will be the subject of the discussion later on The law of motion equation for capital may be rewritten as KML 17 6 K 5F Kt7 L Mapping K into Kt1 graphically7 this can be pictured as in Figure 21 k1 Figure 21 Convergence in the Solow model The intersection of the 450 line with the savings function determines the stationary point It can be veri ed that the system exhibits global convergence77 to the unique strictly positive steady state7 Ki that satis es K 176K5FKLor 6K 5F K7 L there is a unique positive solution Given this inforrnation7 we have Theorem 1 3K gt 0 VKO gt 07 Kt 7 K Proof outline 1 Find a K candidate show it is unique 2 1f K0 gt K show that K lt Kt lt K Vt 2 0 using Kt 7 K 5F KhL 7 HQ 1f K0 lt K show that K gt Kt gt Kt Vt gt 0 3 We have concluded that K is a monotonic sequence7 and that it is also bounded Now use a math theorem a monotone bounded sequence has a limit I The proof ofthis theorem establishes not only global convergence but also that conver gence is monotonic The result is rather special in that it holds only under quite restrictive circumstances for example7 a one sector model is a key part of the restriction 22 Applications 221 Growth The Solow growth model is an important part of many more complicated models setups in modern macroeconomic analysis Its rst and main use is that of understanding why output grows in the long run and what forms that growth takes We will spend considerable time with that topic later This involves discussing what features of the production technology are important for long run growth and analyzing the endogenous determination of productivity in a technological sense Consider7 for example7 a simple Cobb Douglas case In that case7 a the capital share determines the shape of the law of motion function for capital accumulation If a is close to one the law of motion is close to being linear in capital if it is close to zero but not exactly zero7 the law of motion is quite nonlinear in capital In terms of Figure 217 an a close to zero will make the steady state lower7 and the convergence to the steady state will be quite rapid from a given initial capital stock7 few periods are necessary to get close to the steady state 1f7 on the other hand7 a is close to one7 the steady state is far to the right in the gure7 and convergence will be slow When the production function is linear in capital when a equals one we have no positive steady state2 Suppose that 5A176 exceeds one Then over time output would keep growing7 and it would grow at precisely rate 5A 1 7 6 Output and consumption would grow at that rate too The Ak production technology is the simplest tech nology allowing endogenous growth 7 ie the growth rate in the model is nontrivially determined7 at least in the sense that different types of behavior correspond to different growth rates Savings rates that are very low will even make the economy shrink if 5A 1 7 6 goes below one Keeping in mind that savings rates are probably in uenced by government policy7 such as taxation7 this means that there would be a choice both by individuals and government7 of whether or not to grow The Ak model of growth emphasizes physical capital accumulation as the driving force of prosperity It is not the only way to think about growth7 however For example7 2This statement is true unless 3A 1 7 6 happens to equal 1 Figure 22 Random productivity in the Solow model one could model A more carefully and be speci c about how productivity is enhanced over time via explicit decisions to accumulate RampD capital or human capital learning We will return to these different alternatives later In the context of understanding the growth of output7 Solow also developed the methodology of growth accounting 7 which is a way of breaking down the total growth of an economy into components input growth and technology growth We will discuss this later too growth accounting remains a central tool for analyzing output and productivity growth over time and also for understanding differences between different economies in the cross section 222 Business Cycles Many modern studies of business cycles also rely fundamentally on the Solow model This includes real as well as monetary models How can Solow7s framework turn into a business cycle setup Assume that the production technology will exhibit a stochastic component affecting the productivity of factors For example7 assume it is of the form FAtFKtL where A is stochastic7 for instance taking on two values AH7 AL Retaining the assump tion that savings rates are constant7 we have what is depicted in Figure 22 It is clear from studying this graph that as productivity realizations are high or low7 output and total savings uctuate Will there be convergence to a steady state In the sense of constancy of capital and other variables7 steady states will clearly not be feasible here However7 another aspect of the convergence in deterministic model is inherited here over time7 initial conditions the initial capital stock lose in uence and eventually after an in nite number of time periods77 the stochastic process for the endogenous 8 variables will settle down and become stationary Stationarity here is a statistical term one that we will not develop in great detail in this course although we will de ne it and use it for much simpler stochastic processes in the context of asset pricing One element of stationarity in this case is that there will be a smallest compact set of capital stocks such that once the capital stock is in this set it never leaves the set the ergodic set In the gure this set is determined by the two intersections with the 4501ine 223 Other topics In other macroeconomic topics such as monetary economics labor scal policy and asset pricing the Solow model is also commonly used Then other aspects need to be added to the framework but Solow7s one sector approach is still very useful for talking about the macroeconomic aggregates 23 Where next The model presented has the problem of relying on an exogenously determined savings rate We saw that the savings rate in particular did not depend on the level of capital or output nor on the productivity level As stated in the introduction this course aims to develop microfoundations We would therefore like the savings behavior to be an outcome rather than an input into the model To this end the following chapters will introduce decision making consumers into our economy We will rst cover decision making with a nite time horizon and then decision making when the time horizon is in nite The decision problems will be phrased generally as well as applied to the Solow growth environment and other environments that will be of interest later Chapter 3 Dynamic optimization There are two common approaches to modelling real life individuals they live a nite number of periods and ii they live forever The latter is the most common approach7 but the former requires less mathematical sophistication in the decision problem We will start with nite life models and then consider in nite horizons We will also study two alternative ways of solving dynamic optimization problems using sequential methods and using recursive methods Sequential methods involve maxi mizing over sequences Recursive methods also labelled dynamic programming methods involve functional equations We begin with sequential methods and then move to re cursive methods 31 Sequential methods 311 A nite horizon Consider a consumer having to decide on a consumption stream for T periods Con sumer7s preference ordering of the consumption streams can be represented with the utility function U0075177CT39 A standard assumption is that this function exhibits additive separability 7 with stationary discounting weights T UcoclcT Z tuct t0 Notice that the per period or instantaneous utility index 22 does not depend on time Nevertheless if instead we had 22 the utility function U 0001 CT would still be additively separable The powers of B are the discounting weights They are called stationary because the ratio between the weights of any two different dates t 2 and t j gt 2 only depends on the number of periods elapsed between 2 and j and not on the values of2 or j 11 The standard assumption is 0 lt B lt 17 which corresponds to the observations that hu man beings seem to deem consumption at an early time more valuable than consumption further off in the future We now state the dynamic optimization problem associated with the neoclassical growth model in nite time T max Z tu ct Ckt1o t0 st 0 kt f kt E F kt7 N 1 7 6 tht 07 T 0 2 07W 07 T kt 2 07 07 T k0 gt 0 given This is a consumption savings decision problem It is7 in this case7 a planning prob lem there is no market where the individual might obtain an interest income from his savings7 but rather savings yield production following the transformation rule f The assumptions we will make on the production technology are the same as before With respect to u7 we will assume that it is strictly increasing Whats the implication of this Notice that our resource constraint 0 kt S f kt allows for throwing goods away7 since strict inequality is allowed But the assumption that u is strictly increasing will imply that goods will not actually be thrown away7 because they are valuable We know in advance that the resource constraint will need to bind at our solution to this problem The solution method we will employ is straight out of standard optimization theory for nite dimensional problems In particular7 we will make ample use of the Kuhn Tucker theorem The Kuhn Tucker conditions are necessary for an optimum7 provided a constraint quali cation is met we do not worry about it here A A x are suf cient if the objective function is concave in the choice vector and the con straint set is convex We now characterize the solution further It is useful to assume the following lin u c 00 This implies that 0 0 at any It cannot be optimal7 so we can ig H nore the non negativity constraint on consumption we know in advance that it will not bind in our solution to this problem We write down the Lagrangian function T L Z t lu Ct t Ct kH1 f Mtkt1l7 A t0 where we introduced the LagrangeKuhn Tucker multipliers t and 6 for our con straints The next step involves taking derivatives with respect to the decision variables 0 and kt and stating the complete Kuhn Tucker conditions Before proceeding7 however7 let us take a look at an alternative formulation for this problem 12 T L 2 u f kt kt1l mm B t0 Notice that we have made use of our knowledge ofthe fact that the resource constraint will be binding in our solution to get rid of the multiplier t The two formulations are equivalent under the stated assumption on u However7 eliminating the multiplier 6 might simplify the algebra The multiplier may sometimes prove an ef cient way of condensing information at the time of actually working out the solution We now solve the problem using formulation A The rst order conditions are 6L 5 tuct7t0 t0T t 6L t t t1 aktH 5 At5 t5 t1fkt107 t077T 1 For period T7 6L T 7 A T 0 akT 5 T 5 MT The rst order condition under formulation B are 6L t t t1 aktH 5 u Ct 5 Mt 5 Ct1fkt1 07 t 077T 1 6L akT 5Tu CT 5TMT 0 Finally7 the Kuhn Tucker conditions also include Mtkt1 07 t 07 AtZO t0T km 2 0 t 0T Ht 2 013 07T These conditions the rst of which is usually referred to as the complementary slackness condition are the same for formulations A and B To see this7 we use 1 ct to replace in formulation A At in the derivative 6kt Now noting that u c gt 0 V0 we conclude that MT gt 0 in particular This comes from the derivative of the Lagrangian with respect to kT1 76W CT BTW 0 But then this implies that kT1 0 the consumer leaves no capital for after the last period7 since he receives no utility from that capital and would rather use it for consump tion during his lifetime Of course7 this is a trivial result7 but its derivation is useful and will have an in nite horizon counterpart that is less trivial 13 The summary statement of the rst order conditions is then the Euler equation 1 lf kt kt1l u lfltkt1l kt2l f kt1v t 07 WT 1 0 Ewen kT1 07 where the capital sequence is what we need to solve for The Euler equation is sometimes referred to as a variational condition as part of calculus of variation given to boundary conditions kt and kHz it represents the idea of varying the intermediate value kt so as to achieve the best outcome Combining these variational conditions7 we notice that there are a total of T 2 equations and T 2 unknowns the unknowns are a sequence of capital stocks with an initial and a terminal condition This is called a dz erence equation in the capital sequence It is a second order difference equation because there are two lags of capital in the equation Since the number of unknowns is equal to the number of equations7 the difference equation system will typically have a solution7 and under appropriate assumptions on primitives7 there will be only one such solution We will now brie y look at the conditions under which there is only one solution to the rst order conditions or7 alternatively7 under which the rst order conditions are suf cient What we need to assume is that u is concave Then7 using formulation A7 we know that U Zuct is concave in the vector ct since the sum of concave functions is t0 concave Moreover7 the constraint set is convex in ct7 kt17 provided that we assume concavity of f this can easily be checked using the de nitions of a convex set and a concave function So7 concavity of the functions u and f makes the overall objective concave and the choice set convex7 and thus the rst order conditions are suf cient Alternatively7 using formulation l37 since ufkt 7 kt is concave in kmrt which follows from the fact that u is concave and increasing and that f is concave7 the objective is concave in kt1 The constraint set in formulation B is clearly convex7 since all it requires is kt 2 0 for all t Finally7 a unique solution to the problem as such as well as to the rst order con ditions is obtained if the objective is strictly concave7 which we have if u is strictly concave To interpret the key equation for optimization7 the Euler equation7 it is useful to break it down in three components 1 Ct u CH1 39 f kt1 V bvd W Utility lost if you Utility increase Return on the invest one more next period per invested unit by how unit a marginal unit of increase in Ct1 many units next period s cost of saving c can increase Thus7 because of the concavity of u7 equalizing the marginal cost of saving to the marginal bene t of saving is a condition for an optimum How do the primitives affect savings behavior We can identify three component determinants of saving the concavity of utility7 the discounting7 and the return to saving Their effects are described in turn i Consumption smoothing if the utility function is strictly concave the individual prefers a smooth consumption stream Eccample Suppose that technology is linear ie fk Rh and that RB 1 Then 5 kt1 5R 1 i u 0 1 CH1 i C Ct1 if u is strictly concave lmpatience via 6 we see that a low 6 a low discount factor or a high discount A x rate B 7 1 will tend to be associated with low ct17s and high ct7s iii The return to savings f kt1 clearly also affects behavior but its effect on con sumption cannot be signed unless we make more speci c assumptions Moreover kt is endogenous so when f nontrivially depends on it we cannot vary the return independently The case when f is a constant such as in the Ah growth model is more convenient We will return to it below To gain some more detailed understanding of the determinants of savings let us study some examples Example 1 Logarithmic utility Let the utility indep be u c log c and the production technology be represented by the function f Rh Notice that this amounts to a linear function with eccogenous marginal return R on in uestment The Euler equation becomes 1 Ct 5 39 1 CH1 39 f kt1 7R l 7 Ct CH17 and so Ct1 BR ct 31 The optimal path has consumption growing at the rate BR and it is constant between any two periods From the resource constraint recall that it binds CO k1 Rko 01 k2 Rkl CT kT1 RkT kT1 0 15 With repeated substitutions we obtain the consolidated or quotintertemporal budget con straint 1 1 1 CO Cl 02 WET The left hand side is the present value of the consumption stream and the right hand side is the present value of income Using the optimal consumption growth rule Ct1 5R Ct 1 1 1 Co EBB Co 5sz Co oTRT c0 Rko c0 1oozoT Rko This implies Rko C0 1o62 T39 We are now able to study the e ects of changes in the marginal return on savings R on the consumer s behavior An increase in R will cause a rise in consumption in all periods Crucial to this result is the chosen form for the utility function Logarithmic utility has the property that income and substitution e ects when they go in opposite directions emactly o set each other Changes in B have two components a change in relative prices of consumption in di erent periods and a change in present value income Rho With logarithmic utility a relative price change between two goods will make the consumption of the favored good go up whereas the consumption of other good will remain at the same level The unfavored good will not be consumed in a lower amount since there is a positive income e ect of the other good being cheaper and that e ect will be spread over both goods Thus the period 0 good will be unfavored in our eccample since all other goods have lower price relative to good 0 ifR goes up and its consumption level will not decrease The consumption of good 0 will in fact increase because total present value income is multiplicative in R Necct assume that the sequence of interest rates is not constant but that instead we have Pablo with R di erent at each t The consolidated budget constraint now reads cic cc c kR 0 R11 R1R22 131112219133 R1RTT 0 039 Plugging in the optimal path Ct1 Rcht analogous to 31 one obtains c01 62 T kORO from which C i koRO 0 1 62 T C koRoR16 1 1 62 T 7 kORORt t ct i 16 Now note the following comparatiue statics R T i c0c1 ct1 are una ected savings at 0 t7 1 are una ected In the logarithmic utility case if the return betweent and t1 changes consumption and savings remain unaltered until t 7 1 Example 2 A slightly more general utility function Let us introduce the most commonly used additiuely separable utility function in macroeconomics the CES con stant elasticity of substitution function This function has as special cases 0 0 linear utility o gt 0 strictly concaue utility o 1 logarithmic utility o 00 not possible but this is usually referred to as Leontief utility function Let us de ne the intertemporal elasticity of substitution IE3 d Ctk Ct ES 2 thtk Rtk We will show that all the special cases of the CES function have constant intertemporal elasticity of substitution equal to 7 We begin with the Euler equation 0 u 0 u CH1 Rt1 Replacing repeatedly we have 1 Ct kuCtk1t1Rt2Rtk E Rttk u c c i c kCfthtk l 1 5k quot Rawk Ct This means that our elasticity measure becomes d dlog 7 Ct 7 thtk i d10gRtgttk i o Rttk 17 When 0 1 eccpenditure shares do not change this is the logarithmic case When 0 gt 1 an increase in Rt k would lead ct to go up and savings to go down the income e ect leading to smoothing across all goods is larger than substitution e ect Finally when o lt 1 the substitution e ect is stronger sauings go up whenever Rt k goes up When 0 0 the elasticity is in nite and savings respond discontinuously to Raw 312 In nite horizon Why should macroeconomists study the case of an in nite time horizon There are at least two reasons 1 Altruism People do not live forever but they may care about their offspring Let u ct denote the utility ow to generation t We can then interpret t as the weight an individual attaches to the utility enjoyed by his descendants t generations down 00 the family tree His total joy is given by Z Btu ct A B lt 1 thus implies that the t0 individual cares more about himself than about his descendants lf generations were overlapping the utility function would look similar 00 Z t lu0yt 5UCotl t0 utility ow to generation t The existence of bequests indicates that there is altruism However bequests can also be of an entirely sel sh precautionary nature when the life time is unknown as it is in practice bequests would then be accidental and simply re ect the remaining buffer the individual kept for the possible remainder of his life An argument for why bequests may not be entirely accidental is that annuity markets are not used very much Annuity markets allow you to effectively insure against living too long and would thus make bequests disappear all your wealth would be put into annuities and disappear upon death It is important to point out that the time horizon for an individual only becomes truly in nite if the altruism takes the form of caring about the utility of the descen dants lf instead utility is derived from the act of giving itself without reference to how the gift in uences others7 welfare the individuals problem again becomes nite Thus if I live for one period and care about how much I give my utility function might be uc 1109 where 1 measures how much I enjoy giving bequests b Although b subsequently shows up in another agent7s budget and in uences his choices and welfare those effects are irrelevant for the decision of the present agent and we have a simple static framework This model is usually referred to as the warm glow77 model the giver feels a warm glow from giving For a variation think of an individual or a dynasty that if still alive each period dies with probability 7139 Its expected lifetime utility from a consumption stream ct0 is then given by Z tntu ct t0 18 This framework the perpetual youth model or perhaps better the sudden death77 model is sometimes used in applied contexts Analytically it looks like the in nite life model only with the difference that the discount factor is 6W These models are thus the same on the individual level On the aggregate level they are not since the sudden death model carries with it the assumption that a de ceased dynasty is replaced with a new one it is formally speaking an overlapping generations model see more on this below and as such it is different in certain key respects Finally one can also study explicit games between players of different generations We may assume that parents care about their children that sons care about their parents as well and that each of their activities is in part motivated by this altru ism leading to intergenerational gifts as well as bequests Since such models lead us into game theory rather quickly and therefore typically to more complicated characterizations we will assume that altruism is unidirectional E0 Simplicity Many macroeconomic models with a long time horizon tend to show very similar results to in nite horizon models ifthe horizon is long enough ln nite horizon models are stationary in nature the remaining time horizon does not change as we move forward in time and their characterization can therefore often be obtained more easily than when the time horizon changes over time The similarity in results between long and in nite horizon setups is is not present in all models in economics For example in the dynamic game theory the Folk Theorem means that the extension from a long but nite to an in nite horizon introduces a qualitative change in the model results The typical example of this discontinuity at in nity77 is the prisoners dilemma repeated a nite number of times leading to a unique noncooperative outcome versus the same game repeated an in nite number of times leading to a large set of equilibria Models with an in nite time horizon demand more advanced mathematical tools Consumers in our models are now choosing in nite sequences These are no longer ele ments of Euclidean space 3 which was used for our nite horizon case A basic question is when solutions to a given problem exist Suppose we are seeking to maximize a function U z z E S If U is a continuous function then we can invoke Weierstrass7s theorem provided that the set S meets the appropriate conditions S needs to be nonempty and compact For S C 3 compactness simply means closedness and boundedness In the case of nite horizon recall that x was a consumption vector of the form 01 CT from a subset S of 3 In these cases it was usually easy to check compactness But now we have to deal with larger spaces we are dealing with in nite dimensional sequences kt0 Several issues arise How do we de ne continuity in this setup What is an open set What does compactness mean We will not answer these questions here but we will bring up some speci c examples of situations when maximization problems are ill de ned that is when they have no solution A Examples where utility may be unbounded Continuity of the objective requires boundedness When will U be bounded If two consumption streams yield in nite utility it is not clear how to compare them The 19 device chosen to represent preference rankings over consumption streams is thus failing But is it possible to get unbounded utility How can we avoid this pitfall Utility may become unbounded for many reasons Although these reasons interact let us consider each one independently Preference requirements Consider a plan specifying equal amounts of consumption goods for each period throughout eternity Ct0 320 Then the value of this consumption stream according to the chosen time separable utility function representation is computed by U Z tu ct Z tu E What is a necessary condition for U to take on a nite value in this case The answer is B lt 1 under this parameter speci cation the series 20311 E is convergent and has a nite limit If u has the CES parametric form then the answer to the question of convergence will involve not only 6 but also 039 Alternatively consider a constantly increasing consumption stream Ct0 CO391 7t0 ls U 20311 0 20 Btu co 1 Wt bounded Notice that the argument in the instantaneous utility index u is increasing without bound while for B lt 1 t is decreasing to 0 This seems to hint that the key to having a convergent series this time lies in the form of u and in how it processes the increase in the value of its argument In the case of CES utility representation the relationship between B 039 and y is thus the key to boundedness In particular boundedness requires 6 1 170 lt 1 Two other issues are involved in the question of boundedness of utility One is tech nological and the other may be called institutional Technological considerations Technological restrictions are obviously necessary in some cases as illustrated indi rectly above Let the technological constraints facing the consumer be represented by the budget constraint Ct kH1 R kt kt Z 0 This constraint needs to hold for all time periods t this is just the Ak case already mentioned This implies that consumption can grow by at most a rate of R A given rate B may thus be so high that it leads to unbounded utility as shown above Institutional framework Some things simply cannot happen in an organized society One of these is so dear to analysts modelling in nite horizon economies that it has a name of its own It expresses 20 the fact that if an individual announces that he plans to borrow and never pay back7 then he will not be able to nd a lender The requirement that no Ponzi games are allowed77 therefore represents this institutional assumption7 and it sometimes needs to be added formally to the budget constraints of a consumer To see why this condition is necessary7 consider a candidate solution to consumers maximization problem 00 and let 0 S EVt ie7 the consumption is bounded for every t Suppose we endow a consumer with a given initial amount of net assets7 10 These represent real claims against other agents The constraint set is assumed to be 0 at1 RatVt 2 0 Here at lt 0 represents borrowing by the agent Absent no PonZi game condition7 the agent could improve on 00 as follows 1 Put 60 03 17 thus making 31 171 2 For every t 2 1 leave 6 c by setting 311 a 7 Rt With strictly monotone utility function7 the agent will be strictly better off under this alternative consumption allocation7 and it also satis es budget constraint period by period Because this sort of improvement is possible for any candidate solution7 the maximum of the lifetime utility will not exist However7 observe that there is something wrong with the suggested improvement7 as the agents debt is growing without bound at rate R7 and it is never repaid This situation when the agent never repays his debt or7 equivalently7 postpones repayment inde nitely is ruled out by imposing the no PonZi game nPg condition7 by explicitly adding the restriction that a i t 2 039 lntuitively7 this means that in present value terms7 the agent cannot engage in borrowing and lending so that his terminal asset holdings77 are negative7 since this means that he would borrow and not pay back Can we use the nPg condition to simplify7 or consolidate the sequence of budget constraints By repeatedly replacing T times7 we obtain T 1 a T1 goCt l RT SQO R By the nPg condition7 we then have T 1 a T 1 a i T1 i i i T1 T1320 gay RT Thuga wr 712T 00 1 a i i T1 Za wmm 7 t0 and since the inequality is valid for every T7 and we assume nPg condition to hold7 1 ZCtE 00R t0 21 This is the consolidated budget constraint In practice we will often use a version of nPg with equality Example 3 We will now consider a simple eccample that will illustrate the use of nPg condition in in nite horizon optimization Let the period utility of the agent u c logo and suppose that there is one asset in the economy that pays a net interest rate of r Assume also that the agent lives forever Then his optimization problem is 00 max 2 6 log ct Ctgtat1o t0 st ctat1at1rVt20 a0 given nPg condition To solve this problem replace the period budget constraints with a consolidated one as we have done before The consolidated budget constraint reads 0 0 a01r t0 With this simpli cation the rst order conditions are 1 1 x Vt20 ct 17 where A is the Lagrange multiplier associated with the consolidated budget constraint From the rst order conditions it follows that ct B 1 rtc0Vt 2 1 Substituting this eppression into the consolidated budget constraint we obtain i5t1rtco ao 1 7 t0 1 r c0233j a01r t0 From here c0 a0 1 7 B 1 r and consumption in the periods t 2 1 can be recovered from ct B 1 rt Co B Su cient conditions Maximization of utility under an in nite horizon will mostly involve the same math ematical techniques as in the nite horizon case In particular we will make use of Kuhn Tucker rst order conditions barring corner constraints we will choose a path such that the marginal effect of any choice variable on utility is zero In particular con sider the sequences that the consumer chooses for his consumption and accumulation of 22 capital The rst order conditions will then lead to an Euler equation which is de ned for any path for capital beginning with an initial value he In the case of nite time horizon it did not make sense for the agent to invest in the nal period T since no utility would be enjoyed from consuming goods at time T1 when the economy is inactive This nal zero capital condition was key to determining the optimal path of capital it provided us with a terminal condition for a difference equation system In the case of in nite time horizon there is no such nal T the economy will continue forever Therefore the dif ference equation that characterizes the rst order condition may have an in nite number of solutions We will need some other way of pinning down the consumers choice and it turns out that the missing condition is analogous to the requirement that the capital stock be zero at T 1 for else the consumer could increase his utility The missing condition which we will now discuss in detail is called the transversality condition It is typically a necessary condition for an optimum and it expresses the following simple idea it cannot be optimal for the consumer to choose a capital sequence such that in present value utility terms the shadow value of kt remains positive as t goes to in nity This could not be optimal because it would represent saving too much a reduction in saving would still be feasible and would increase utility We will not prove the necessity of the transversality condition here We will however provide a suf ciency condition Suppose that we have a convex maximization problem utility is concave and the constraint set convex and a sequence ht11 satisfying the Kuhn Tucker rst order conditions for a given ho ls kt11 a maximum We did not formally prove a similar proposition in the nite horizon case we merely referred to math texts but we will here and the proof can also be used for nite horizon setups Sequences satisfying the Euler equations that do not maximize the programming problem come up quite often We would like to have a systematic way of distinguishing between maxima and other critical points in 3 that are not the solution we are looking for Fortunately the transversality condition helps us here if a sequence kt11 satis es both the Euler equations and the transversality condition then it maximizes the objective function Formally we have the following Proposition 1 Consider the programming problem ma 2 t F kt kt kt1t0 t0 st kH1 2 0 Vt An eccample is Fzy u f i If k10 3220 satisfy 239 kit 2 0 W ii Euler Equation F2 hi hi 5171 kiln 12 f 0 W m o 2 0 o kt11 2 0 W iv tlirglo tFi k 51 kl 0 and Fzy is concave in zy and increasing in its rst argument then kj10 mapimizes the objective 23 Proof Consider any alternative feasible sequence k E kt10 Feasibility is tan tamount to kt 2 0 Vt We want to show that for any such sequence T TlLIEOZBt F k kg 7 Fkt 1mg 2 0 t0 De ne T ATk E 26 F k kg 7 F kt 1mg t0 We will to show that as T goes to in nity ATk is bounded below by zero By concavity of F t F1 k kill Wk kt F2 k kill kill kt1l Ma AT k 2 n H 0 Now notice that for each It kt1 shows up twice in the summation Hence we can rearrange the expression to read T71 AT k 2 EH 1 7 km F2 k m m m km t0 F1 1637 kl 39 k3 k0 TFz k kin 39 kin kT1 Some information contained in the rst order conditions will now be useful 112k 51 5111 kirk 52 va together with kg 7 k0 0 k0 can only take on one feasible value allows us to derive a 71 AT 0 2 Bill 39 kt 51 BTFZ If kf kin kTH t H 0 Next we use the complementary slackness conditions and the implication of the Kuhn Tucker conditions that M kt1 Z 0 to conclude that Mkt1 7 kg 2 0 In addition F2 kid 7BF1 kid k rz 7 MT so we obtain Ma AT k 2 t kt krill 6T l Fl kirk ka TH kTH 341 39 n H 0 Since we know that Mal kt1 7 kg 2 0 the value of the summation will not increase if we suppress nonnegative terms AT 0 Z BTHFI kirk kfu kTH kf Z BTHFI kf v kf2 39k139 24 In the nite horizon case It would have been the level of capital left out for the day after the perfectly foreseen end of the world a requirement for an optimum in that case is clearly It 0 ln present value utility terms one might alternatively require k1 TT 0 where 6 is the present value utility evaluation of an additional unit of resources in period t As T goes to in nity the right hand side of the last inequality goes to zero by the transversality condition That is we have shown that the utility implied by the candidate path must be higher than that implied by the alternative I The transversality condition can be given this interpretation F1ktkt1 is the marginal addition of utils in period t from increasing capital in that period so the transversality condition simply says that the value discounted into present value utils of each additional unit of capital at in nity times the actual amount of capital has to be zero If this requirement were not met we are now incidentally making a heuristic argument for necessity it would pay for the consumer to modify such a capital path and increase consumption for an overall increase in utility without violating feasibility1 The no PonZi game and the transversality conditions play very similar roles in dy namic optimization in a purely mechanical sense at least if the nPg condition is inter preted with equality In fact they can typically be shown to be the same condition if one also assumes that the rst order condition is satis ed However the two conditions are conceptually very different The nPg condition is a restriction on the choices of the agent In contrast the transversality condition is a prescription how to behave optimally given a choice set 32 Dynamic Programming The models we are concerned with consist of a more or less involved dynamic optimization problem and a resulting optimal consumption plan that solves it Our approach up to now has been to look for a sequence of real numbers kill 0 that generates an optimal consumption plan In principle this involved searching for a solution to an in nite sequence of equations a difference equation the Euler equation The search for a sequence is sometimes impractical and not always intuitive An alternative approach is often available however one which is useful conceptually as well as for computation both analytical and especially numerical computation It is called dynamic programming We will now go over the basics ofthis approach The focus will be on concepts as opposed to on the mathematical aspects or on the formal proofs Key to dynamic programming is to think of dynamic decisions as being made not once and for all but recursively time period by time period The savings between t and t 1 are thus decided on at t and not at 0 We will call a problem stationary whenever the structure of the choice problem that a decision maker faces is identical at every point in time As an illustration in the examples that we have seen so far we posited a consumer placed at the beginning of time choosing his in nite future consumption stream given an initial capital stock k0 As a result out came a sequence of real numbers kj10 indicating the level of capital that the agent will choose to hold in each period But 1This necessity argument clearly requires utility to be strictly increasing in capitali 25 once he has chosen a capital path suppose that we let the consumer abide it for say T periods At t T he will nd then himself with the k decided on initially If at that moment we told the consumer to forget about his initial plan and asked him to decide on his consumption stream again from then onwards using as new initial level of capital k0 k what sequence of capital would he choose If the problem is stationary then for any two periods t 31 5 kt 5 gt ktj k5j for all j larger than zero That is he would not change his mind if he could decide all over again This means that if a problem is stationary we can think of a function that for every period t assigns to each possible initial level of capital kt an optimal level for next period7s capital kt and therefore an optimal level of current period consumption kt g Stationarity means that the function g has no other argument than current capital In particular the function does not vary with time We will refer to g as the decision rule We have de ned stationarity above in terms of decisions in terms of properties of the solution to a dynamic problem What types of dynamic problems are stationary lntuitively a dynamic problem is stationary if one can capture all relevant information for the decision maker in a way that does not involve time In our neoclassical growth framework with a nite horizon time is important and the problem is not stationary it matters how many periods are left the decision problem changes character as time passes With an in nite time horizon however the remaining horizon is the same at each point in time The only changing feature of the consumers problem in the in nite horizon neoclassical growth economy is his initial capital stock hence his decisions will not depend on anything but this capital stock Whatever is the relevant information for a consumer solving a dynamic problem we will refer to it as his state variable So the state variable for the planner in the one sector neoclassical growth context is the current capital stock The heuristic information above can be expressed more formally as follows The simple mathematical idea that max fzy maxymaxw fzy if each of the max operators is well de ned allows us to maximize in steps rst over z given y and then the remainder where we can think of x as a function of y over y If we do this over time the idea would be to maximize over k51t rst by choice of k51t1 conditional on kt and then to choose kt That is we would choose savings at t and later the rest Let us denote by Vkt the value of the optimal program from period t for an initial condition kt oo ZBSquotFkSkS1 st k1 e l ksVs 2 t st Vkt E max ks1sgtot where Hist represents the feasible choice set for kt given ktz That is V is an indirect utility function with kt representing the parameter governing the choices and resulting 2The onesector growth model example would mean that Fzy 7 y and that lquotz 0 the latter restricting consumption to be nonnegative and capital to be nonnegative 26 utility Then using the maximization by steps idea7 we can write 00 max Fkt7kt1k max 2 BSTtFkS7kS1 S39t39 ks1 E 2 t1 k161quotk s1 t1 St1 which in turn can be rewritten as Fltktktlgt k max 2 s Fksks1 at k91E Pltksgtvs 2 ms s max kt1 FW 1sgt0t1 Ft But by de nition of V this equals F k k V k Wrinea xkt t7 t1 t1 So we have Vkt max Fktkt1 vkt1 kt1 Fkt This is the dynamic programming formulation The derivation was completed for a given value of kt on the left hand side of the equation On the right hand side7 however7 we need to know V evaluated at any value for kt1 in order to be able to perform the maximization lf7 in other words7 we nd a V that7 using k to denote current capital and k next period7s capital7 satis es Vk max Fk7 76 Vk k eI k for any value of k then all the maximizations on the right hand side are well de ned This equation is called the Bellman equation7 and it is afunctz39zmal equation the unknown is a function We use the function g alluded to above to denote the arg max in the functional equation gk arg max Fk7 k 6Vk k eI k or the decision rule for k k This notation presumes that a maximum exists and is unique otherwise7 9 would not be a well de ned function This is close to a formal derivation of the equivalence between the sequential formu lation of the dynamic optimization and its recursive7 Bellman formulation What remains to be done mathematically is to make sure that all the operations above are well de ned Mathematically7 one would want to establish o If a function represents the value of solving the sequential problem for any initial condition7 then this function solves the dynamic programming equation DPE o If a function solves the DPE7 then it gives the value of the optimal program in the sequential formulation o If a sequence solves the sequential program7 it can be expressed as a decision rule that solves the maximization problem associated with the DPE o If we have a decision rule for a DPE7 it generates sequences that solve the sequential problem 27 These four facts can be proved7 under appropriate assumptions3 We omit discussion of details here One issue is useful to touch on before proceeding to the practical implementation of dynamic programming since the maximization that needs to be done in the DPE is nite dimensional7 ordinary Kuhn Tucker methods can be used7 without reference to extra conditions7 such as the transversality condition How come we do not need a transversality condition here The answer is subtle and mathematical in nature In the statements and proofs of equivalence between the sequential and the recursive methods7 it is necessary to impose conditions on the function V not any function is allowed Uniqueness of solutions to the DPE7 for example7 only follows by restricting V to lie in a restricted space of functions This or other7 related7 restrictions play the role of ensuring that the transversality condition is met We will make use of some important results regarding dynamic programming They are summarized in the following Facts Suppose that F is continuously differentiable in its two arguments7 that it is strictly increasing in its rst argument and decreasing in the second7 strictly concave7 and bounded Suppose that P is a nonempty7 compact valued7 monotone7 and continuous correspondence with a convex graph Finally7 suppose that B E 07 1 Then 1 There exists a function that solves the Bellman equation This solution is unique 2 It is possible to nd V by the following iterative process i Pick any initial Vb function7 for example Vb 0 Vk ii Find Vn17 for any value of k by evaluating the right hand side of using n The outcome of this process is a sequence of functions which converges to 3 V is strictly concave 4 V is strictly increasing 5 V is differentiable 6 Optimal behavior can be characterized by a function 97 with k gk7 that is increasing so long as F2 is increasing in k The proof of the existence and uniqueness part follow by showing that the functional equation7s right hand side is a contraction mapping7 and using the contraction mapping theorem The algorithm for nding V also uses the contraction property The assump tions needed for these characterizations do not rely on properties of F other than its continuity and boundedness That is7 these results are quite general 3See Stokey and Lucas 1989 28 In order to prove that V is increasing it is necessary to assume that F is increasing and that P is monotone In order to show that V is strictly concave it is necessary to assume that F is strictly concave and that P has a convex graph Both these results use the iterative algorithm They essentially require showing that if the initial guess on V Vb satis es the required property such as being increasing then so is any subsequent Vn These proofs are straightforward Differentiability of V requires F to be continuously differentiable and concave and the proof is somewhat more involved Finally optimal policy is a function when F is strictly concave and P is convex valued under these assumptions it is also easy to show using the rst order condition in the maximization that g is increasing This condition reads 7F2kk Vk The left hand side of this equality is clearly increasing in k since F is strictly concave and the right hand side is strictly decreasing in k since V is strictly concave under the stated assumptions Furthermore since the right hand side is independent of k but the left hand side is decreasing in k the optimal choice of k is increasing in k The proofs of all these results can be found in Stokey and Lucas with Prescott 1989 Connection with nitehorizon problems Consider the nite horizon problem T max 2 Btu ct Ctlio t0 st kt 0 F kt Although we discussed how to solve this problem in the previous sections dynamic pro gramming offers us a new solution method Let Vn denote the present value utility derived from having a current capital stock of k and behaving optimally if there are n periods left until the end of the world Then we can solve the problem recursively or by backward induction as follows If there are no periods left that is if we are at t T then the present value of utility next period will be 0 no matter how much capital is chosen to be saved Vb 0 Vk Then once he reaches t T the consumer will face the following problem v1 k mkam f k 7 M m 1 Since Vb k 0 this reduces to V1 mkax u f 7 The solution is clearly k 0 note that this is consistent with the result kT1 0 that showed up in nite horizon problems when the formulation was sequential As a result the update is V1k u f We can iterate in the same fashion T times all the way to VT by successively plugging in the updates V This will yield the solution to our problem In this solution of the nite horizon problem we have obtained an interpretation of the iterative solution method for the in nite horizon problem the iterative solution is like solving a nite horizon problem backwards for an increasing time horizon The statement that the limit function converges says that the value function of the in nite horizon problem is the limit ofthe time zero value functions ofthe nite horizon problems 29 as the horizon increases to in nity This also means that the behavior at time zero in a nite horizon problem becomes increasingly similar to in nite horizon behavior as the horizon increases Finally notice that we used dynamic programming to describe how to solve a non stationary problem This may be confusing as we stated early on that dynamic pro gramming builds on stationarity However if time is viewed as a state variable as we actually did view it now the problem can be viewed as stationary That is if we increase the state variable from not just including h but t as well or the number of periods left then dynamic programming can again be used Example 4 Solving a parametric dynamic programming problem In this ecc ample we will illustrate how to solve dynamic programming problem by nding a corre sponding value function Consider the following functional equation V ma logc 6V sit c Ah 7 k The budget constraint is written as an equality constraint because we know that prefer ences represented by the logarithmic utility function emhibit strict monotonicity goods are always valuable so they will not be thrown away by an optimizing decision maker The production technology is represented by a Cobb Douglass function and there is full depreciation of the capital stock in every period F k 1 1 7 6k W Aka117a 0 A more compact erpression can be derived by substitutions into the Bellman equation V mag og Aka 7 M 6V We will solve the problem by iterating on the value function The procedure will be similar to that of solving a T problem backwards We begin with an initial guess Vb 0 that is a function that is zero valued everywhere V1 k 7 Iggg oe AW 7 k l 3 14 m g og Aka 7 M B 0 ggg oe AW 7 k l This is mamimized by taking k 0 Then V1logA alogk Going to the necct step in the iteration V2 k 7 5333 109 AW 7 M 6V1 k riglag log Aka 7 M 7l7 B logA 04 log 14 30 The rst order condition now reads 1 75744 7046Ak Akaik ik 1ao39 We can interpret the resulting eppression for k as the rule that determines how much it would be optimal to save if we were at period T71 in the nite horizon model Substitution implies 046Ak0 1 046 a 0426 logk log A i log Akai 5 logA0410g 7 046A 1046 046Ak0 1 045 logA 04610g a A 104639 We could now use V2 again in the algorithm to obtain a V3 k and so on We know by the characterizations above that this procedure would make the sequence of value functions converge to some V However there is a more direct approach using a pattern that appeared already in our iteration Let 046A 1 046 046A 1046 aEIOgA7 logA04610g and b E 04 0426 Then V2 ab log k Recall that V1 logA0410g k ie in the second step what we did was plug in a function V1 a1 b1 log k and out came a function V2 a2 b2 log k This clearly suggests that if we continue using our iterative procedure the outcomes V3 V4 Vn will be of the form Vn an bn logk for all n Therefore we may already guess that the function to which this sequence is conuerging has to be of the form Vk ablogk So let us guess that the value function solving the Bellman has this form and determine the corresponding parameters a b V a blogk rig1 10g Aka 7 k 6a blog k Vk Our task is to nd the values ofa and b such that this equality holds for all possible values of k If we obtain these values the functional equation will be solved The rst order condition reads 1 i b b 77 7Ak Akaik 14g 16 We can interpret as a savings rate Therefore in this setup the optimal policy b 1 5b will be to save a constant fraction out of each period s income De ne LHS E a b log k 31 and RHS E 113 10g Aka 7 k 61 b log k Plugging the epppession for k into the RHS we obtain i 0 0 0 RHS 7 log Ak 71 bAk a b 10gltr bf1k 7 55 on 55 a 7 10g 1 1 b Ak a b 10glt1 bAk 1b logAloglt a b 10glt aa b10gk 1 ob 1b 16 Setting LHSRHS we produce a1b lo A10 a b 10 i g g 1 b6 g 1 6b baa b which amounts to two equations in two unknowns The solutions will be i a i 1 7 046 and using this nding 1 a 1 b 10gA b 10g b6 7 1 b6 10g 1 b so that 71 1 11Alt1 m1 lt1 B 61 61 aili 17016 0g 04 0g 04 04 oga Going back to the savings decision rule we haue 55 k iAk 1 b6 k a Ak If we let y denote income that is y E Aka then k a y This means that the optimal solution to the path for consumption and capital is to save a constant fraction 046 of income This setting we have now shown provides a micpoeconomic justi cation to a constant savings rate like the one assumed by Solow It is a very special setup however one that is quite restrictive in tepms offunctionalfopms Solow s assumption cannot be shown to hold generally We can also visualize the dynamic behauioiquot of capital 32 Figure 31 The decision rule in our parameterized model Example 5 A more complex example We will now look at a slightly di erent growth model and try to put it in recursive terms Our new problem is 00 max 6 u ct Cto t0 st ct it F kt and subject to the assumption is that capital depreciates fully in M periods and does not depreciate at all before that Then the law of motion for capital given a sequence of investment it0 is given by kt 271 2372 Then k L1 L2 there are two initial conditions L1 and L2 The recursive formulation for this problem is V L1 L2 Hiaix V i7 i1 87 C f l1 23972 7 Notice that there are two state variables in this problem That is unavoidable here there is no way of summarizing what one needs to know at a point in time with only one variable For eaample the total capital stock in the current period is not informative enough because in order to know the capital stock neat period we need to know how much of the current stock will disappear between this period and the neat Both L1 and L2 are natural state variables they are predetermined they a ect outcomes and utility and neither is redundant the information they contain cannot be summarized in a simpler way 33 33 The functional Euler equation In the sequentially formulated maximization problem7 the Euler equation turned out to be a crucial part of characterizing the solution With the recursive strategy7 an Euler equation can be derived as well Consider again V kHelf fli F k7 k 5V As already pointed out7 under suitable assumptions7 this problem will result in a function k gk that we call decision rule7 or policy function By de nition7 then7 we have v k F k M 9v 9 km 32 Moreover7 gk satis es the rst order condition F2 k 14gt WW 0 assuming an interior solution Evaluating at the optimum7 ie7 at k gk7 we have F2 7 9 6V 9 0 This equation governs the intertemporal tradeoff One problem in our characterization is that V is not known in the recursive strategy7 it is part of what we are searching for However7 although it is not possible in general to write in terms of primitives7 one can nd its derivative Using the equation 32 above7 one can differentiate both sides with respect to k since the equation holds for all k and7 again under some assumptions stated earlier7 is differentiable We obtain V F1 k7 9 9 F2 k7 9 BV 9l indirect effect through optimal choice of k From the rst order condition7 this reduces to V F1 k7 9 which again holds for all values of k The indirect effect thus disappears this is an application of a general result known as the envelope theorem Updating7 we know that V F1gk7 gg also has to hold The rst order condition can now be rewritten as follows F2 k7 9 F197 9 9l 0 Vk 33 This is the Euler equation stated as a functional equation it does not contain the un knowns kt kt and kHz Recall our previous Euler equation formulation F2 km kt1l Fi kt17 kt2l 07Vt7 where the unknown was the sequence kt1 Now instead7 the unknown is the function 9 That is7 under the recursive formulation7 the Euler Equation turned into a functional equation 34 The previous discussion suggests that a third way of searching for a solution to the dynamic problem is to consider the functional Euler equation and solve it for the function g We have previously seen that we can look for sequences solving a nonlinear difference equation plus a transversality condition or ii we can solve a Bellman functional equation for a value function The functional Euler equation approach is in some sense somewhere in between the two previous approaches It is based on an equation expressing an intertemporal tradeoff but it applies more structure than our previous Euler equation There a transversality condition needed to be invoked in order to nd a solution Here we can see that the recursive approach provides some extra structure it tells us that the optimal sequence of capital stocks needs to be connected using a stationary function One problem is that the functional Euler equation does not in general have a unique solution for g It might for example have two solutions This multiplicity is less severe however than the multiplicity in a second order difference equation without a transver sality condition there there are in nitely many solutions The functional Euler equation approach is often used in practice in solving dynamic problems numerically We will return to this equation below Example 6 In this example we will apply functional Euler equation described above to the model given in Example 4 First we need to translate the model into quotV F language With full depreciation and strictly monotone utility function the function F has the form F la6 u fk 900 Then the respective derivatives are F1 k k u fk 14 f k F2 k k u fk 14 In the particular parametric example becomes 1 aA9k 1 Am 7 M A 900 7 g M 0M This is a functional equation in g Guess that g sAk ie the savings are a constant fraction of output Substituting this guess into functional Euler equation delivers 1 046A sAku f Tl 17 5 Ah 7 A SAWY 7 5A SA60 As can be seen h cancels out and the remaining equation can be solved for 5 Collecting terms and factoring out 5 we get 5 046 This is exactly the answer that we got in Example 4 35 36 Chapter 4 Steady states and dynamics under optimal growth We will now study in more detail the model where there is only one type of good that is only one production sector the one sector optimal growth model This means that we will revisit the Solow model under the assumption that savings are chosen optimally Will as in Solow7s model output and all other variables converge to a steady state It turns out that the one sector optimal growth model does produce global convergence under fairly general conditions which can be proven analytically If the number of sectors increases however global convergence may not occur However in practical applications where the parameters describing different sectors are chosen so as to match data it has proven dif cult to nd examples where global convergence does not apply We thus consider preferences of the type 2 t u Ct t0 and production given by Ct kt1 where fkt F kt7 N 1 7 6 kt for some choice of N and 6 which are exogenous in the setup we are looking at Under standard assumptions namely strict concavity B lt 1 and conditions ensuring interior solutions we obtain the Euler equation 1 Ct 5 u Ct1 f kt1 A steady state is a constant solution kt WVt ct 0Vt This constant sequence ct0 00 will have to satisfy 1 0 39u 0 WW 37 Here 1 0 gt 0 is assumed7 so this reduces to 5 39 f V 1 This is the key condition for a steady state in the one sector growth model It requires that the gross marginal productivity of capital equal the gross discount rate 16 Suppose k0 If We rst have to ask whether kt I Vt a solution to the steady state equation will solve the maximization problem The answer is clearly yes7 provided that both the rst order and the transversality conditions are met The rst order conditions are met by construction7 with consumption de ned by 0 fk 7 If The transversality condition requires tlimo t F1k 7 kt kt 0 Evaluated at the proposed sequence7 this condition becomes 3306 F1 W W k 0 and since F1 W7 W I is a nite number7 with B lt 17 the limit clearly is zero and the condition is met Therefore we can conclude that the stationary solution kt If Vt does maximize the objective function If f is strictly concave7 then kt I is the unique strictly positive solution for k0 If It remains to verify that there is indeed one solution We will get back to this in a moment Graphically7 concavity of fk implies that B f k will be a positive7 decreasing function of k and it will intersect the horizontal line going through 1 only once as can be seen in Figure 41 Figure 41 The determination of steady state 38 41 Global convergence In order to derive precise results for global convergence7 we now make the following assumptions on the primitives i u and f are strictly increasing7 strictly concave7 and continuously differentiable ii f0 07 ll iinijr 007 and kliamof Ur E b lt11 iii lightc 00 iv 3 E 07 1 We have the following problem Wk u Wk W 6Vk max k 6l0gtfkl leading to k gk satisfying the rst order condition 1 Wk k l WWW Notice that we are assuming an interior solution This assumption is valid since assump tions ii and iii guarantee interiority In order to prove global convergence to a unique steady state7 we will rst state and re derive some properties of the policy function g and of steady states We will then use these properties to rule out anything but global convergence Properties of gk Property 1 gk is single valued for all k This follows from strict concavity of u and V recall the theorem we stated previ ously by the Theorem of the Maximum under convexity Property 2 gk is strictly increasing We argued this informally in the previous section The formal argument is as follows Proof Consider the rst order condition u Wk k l V k V is decreasing7 since is strictly concave due to the assumptions on u and f De ne LHSkk u fkik RHSW BV W 1It is not necessary for the following arguments to assume that IliIIEfk ooi They would work a even if the limit were strictly greater than ll 39 Let I gt k Then 7 k gt fk 7 k Strict concavity of u implies that u 7 k lt u 7 k Hence we have that gtk LHS 3ltLHSwy As a consequence7 the RHS k must decrease to satisfy the rst order condition gince V is decreasing7 this will happen only if k increases This shows that k gt k gk gt gk The above result can also be viewed as an application of the implicit function theorem De ne Hkk E u 7 k 7 V k 0 Then 6Hk k 6L4 L 6k 6Hk k 614 wum7w w 7a W 7 M 7 6V k r HA u lfk7k lf k gt0 WUW7MMWW 7 where the sign follows from the fact that since u and V are strictly concave and f is strictly increasing7 both the numerator and the denominator of this expression have negative signs This derivation is heuristic since we have assumed here that V is twice continuously differentiable It turns out that there is a theorem telling us that under some side conditions that we will not state here V will indeed be twice continuously differentiable7 given that u and f are both twice differentiable7 but it is beyond the scope of the present analysis to discuss this theorem in greater detail I The economic intuition behind 9 being increasing is simple There is an underlying presumption of normal goods behind our assumptions strict concavity and addi tivity of the different consumption goods over time amounts to assuming that the different goods are normal goods Speci cally7 consumption in the future is a normal good Therefore7 a larger initial wealth commands larger savings Property 3 gk is continuous This property7 just as Property 17 follows from the Theorem of the Maximum under convexity Property 4 90 0 This follows from the fact that fk 7 k 2 0 and f0 0 40 Property 5 There exists 1 st gk S I for all k lt I Moreover7 exceeds f 11 The rst part follows from feasibility because consumption cannot be negative7 k cannot exceed Our assumptions on f then guarantee that fk lt k for high enough values of k the slope of f approaches a number less than 1 as k goes to in nity So gk lt k follows The characterization of 1 follows from noting that I must be above the value that maximizes fk 7 k since fk is above k for very small values of k and f is strictly concave and ii that therefore 9 gtf 11gtf 11 We saw before that a necessary and suf cient condition for a steady state is that B f k 1 Now let us consider different possibilities for the decision rule Figure 42 shows three decision rules which are all increasing and continuous Line 1 quot Line 2 1 quot Line 3 3 Line 1 kquot Line 1 kquot Line 1 kquot Line 2 kquot Figure 42 Different decision rule candidates All candidate decision rules start at the origin7 as required by the assumption that f0 0 and that investment is nonnegative They all stay in the bounded set 07 Line 1 has three different solutions to the steady state condition k k line 2 has only one steady state and line 3 has no positive steady state We do not graph decision rules with more than three steady states Now7 we need to investigate further the number of steady states that our model will allow Let us now have a look at Figure 42 As we derived earlier7 a single steady state must obtain7 due to strict concavity of u This rules out line 1 in Figure 427 and it also rules out any decision rule with more than one positive crossing of the 450 line Line 37 with no positive steady state7 can be ruled out by noting that the steady state requirement 6 f k 17 together with Property 5 mean that there will be a strictly positive steady state in the interval 07 Therefore7 the only possibility is line 2 We can see that line 2 is above the 45 0 line to the left of If and below to the right This implies that the model dynamics exhibit global convergence 41 The convergence will not occur in nite time For it to occur in that manner the decision rule would have to be at at the steady state point This however cannot be since we have established that gk is strictly increasing Property 2 42 Dynamics the speed of convergence What can we say about the time it takes to reach the steady state The speed of global convergence will depend on the shape of gk as Figure 43 shows kl Figure 43 Different speeds of convergence Capital will approach the steady state level more rapidly ie in a smaller number of steps along trajectory number 2 where it will have a faster speed of convergence There is no simple way to summarize in a quantitative way the speed of convergence for a general decision rule However for a limited class of decision rules the linear or af ne rules it can be measured simply by looking at the slope This is an important case for it can be used locally to approximate the speed of convergence around the steady state If The argument for this is simple the accumulation path will spend in nite time arbi trarily close to the steady state and in a very small region a continuous function can be arbitrarily well approximated by a linear function using the rst order Taylor expansion of the function That is for any capital accumulation path we will be able to approxi mate the speed of convergence arbitrarily well as time passes If the starting point is far from the steady state we will make mistakes that might be large initially but these mis takes will become smaller and smaller and eventually become unimportant Moreover if one uses parameter values that are in some sense realistic it turns out that the resulting decision rule will be quite close to a linear one In this section we will state a general theorem with properties for dynamic systems of a general size To be more precise we will be much more general than the one sector 42 growth model With the methods we describe here it is actually possible to obtain the key information about local dynamics for any dynamic system The global convergence theorem in contrast applies only for the one sector growth model The rst order Taylor series expansion of the decision rule gives 1 gltkgt wrkwkw k k i If 7 t 9 k k k Next periodls gap Current gap This shows that we may interpret g k as a measure of the rate of convergence or rather its inverse lf g k is very close to zero convergence is fast and the gap decreases signi cantly each period 421 Linearization for a general dynamic system The task is now to nd g k by linearization We will use the Euler equation and linearize it This will lead to a difference equation in ht One of the solutions to this difference equation will be the one we are looking for Two natural questions arise 1 How many convergent solutions are there around If 2 For the convergent solutions is it valid to analyze a linear difference equation as a proxy for their convergence speed properties The rst of these questions is the key to the general characterization of dynamics The second question is a mathematical one and related to the approximation precision Both questions are addressed by the following theorem which applies to a general dynamic system ie not only those coming from economic models Theorem 2 Let wt 6 3 Given xi hzt with a stationary point i i If 1 h is continuously di erentiable with Jacobian around i 2 I 7 is non singular then there is a set of initial conditions 0 of dimension equal to the number of eigenvalues of that are less than 1 in absolute value for which xi a i We will describe how to use this theorem with a few examples Example 7 n 1 There is only one eigenvalue A h i Z W 2 1 i no initial condition leads to wt converging to i In this case only for do i will the system stay in i 2 W lt1 x a i for any value ofxo Example 8 n 2 There are two eigenvalues A1 and A2 1 Rd Hal 2 1 i No initial condition 0 leads to convergence 43 2 M11 lt 17 M21 2 1 i Dimension of ms leading to convergence is 1 This is called saddle path stability 3 M11 M21 lt 1 i Dimension of ms leading to convergence is 2 xi a i for any value of do The examples describe how a general dynamic system behaves It does not yet7 however7 quite settle the issue of convergence ln particular7 the set of initial conditions leading to convergence must be given an economic meaning ls any initial condition possible in a given economic model Typically no for example7 the initial capital stock in an economy may be given7 and thus we have to restrict the set of initial conditions to those respecting the initial capital stock We will show below that an economic model has dynamics that can be reduced to a vector difference equation of the form of the one described in the above theorem In this description7 the vector will have a subset of true state variables eg capital while the remainder of the vector consists of various control7 or other7 variables that are there in order that the system can be put into rst order form More formally7 let the number of eigenvalues less than 1 in absolute value be denoted by m This is the dimension of the set of initial zo7s leading to i We may interpret m as the degrees offreedom Let the number of economic restrictions on initial conditions be denoted by 7 These are the restrictions emanating from physical and perhaps other conditions in our economic model Notice that an interpretation of this is that we have 7 equations and m unknowns Then the issue of convergence boils down to the following cases 1 m m i there is a unique convergent solution to the difference equation system 2 m lt m i No convergent solution obtains 3 m gt m i There is indeterminacy ie many solutions how many dim mim 422 Solving for the speed of convergence We now describe in detail how the linearization procedure works The example comes from the one sector growth model7 but the general outline is the same for all economic models 1 Derive the Euler equation Fktkt1kt2 0 1 WW kt1l 5 39 u lfkt1 kt2l f ktH 0 Clearly7 Iii is a steady state gt F If7 ki ki 0 2 Linearize the Euler equation De ne lit kt 7 Iii and using rst order Taylor ap proximation derive amd17 and a2 such that aZ gt2al kt1QO It0 44 3 r 9 Write the Euler equation as a rst order system A difference equation of any order can be written as a rst order difference equation by using vector notation De ne x lt k5 and then kt xi H zt Find the solution to the rst order system Find the unknowns in t t xtclA1vl02A2vg 41 where 01 and 02 are constants to be determined A1 and A2 are distinct eigenvalues of H and 01 and 112 are eigenvectors associated with these eigenvalues Determine the constants Use the information about state variables and initial con ditions to nd 01 and 02 In this case x consists of one state variable and one lagged state variable the latter used only for the reformulation of the dynamic system Therefore we have one initial condition for the system given by k0 this amounts to one restriction on the two constants The set of initial conditions for 0 in our economic model has therefore been reduced to one dimension Finally we are looking for convergent solutions If one of the two eigenvalues is greater than one in absolute value this means that we need to set the corresponding constant to zero Consequently since not only k0 but also k1 are now determined ie both elements of 0 and our system is fully determined all future values of k or s can be obtained If both eigenvalues are larger than one the dynamics will not have convergence to the steady state only if the system starts at the steady state will it remain there If both eigenvalues are less than one we have no way of pinning down the remain ing constant and the set of converging paths will remain of one dimension Such indeterminacy effectively an in nite number of solutions to the system will not occur in our social planning problem because under strict concavity it is guaran teed that the set of solutions is a singleton However in equilibrium systems that are not derived from a planning problem perhaps because the equilibrium is not Pareto optimal as we shall see below it is possible to end up with indeterminacy The typical outcome in our one sector growth model is 0 lt A1 lt 1 and A2 gt 1 which implies m 1 saddle path stability Then the convergent solution has 02 0 In other words the economics of our model dictate that the number of restrictions we have on the initial conditions is one namely the given initial level of capital k0 ie 7 1 Therefore m t so there is a unique convergent path for each k0 close to 1 Then 01 is determined by setting 02 0 so that the path is convergent and solving equation 41 for the value of 01 such that if t 0 then kt is equal to the given level of initial capital k0 We now implement these steps in detail for a one sector optimal growth model First we need to solve for H Let us go back to u kt1l 5 39U lfltkt1 kt2l f kt1 0 45 In order to linearize it7 we take derivatives of this expression with respect to kt kt and kHz and evaluate them at If We obtain 611 f k 152 7 mm B u ltcgt Wm B u ltcgt f ltkgt I1 u c mm 1 0 Using the steady state fact that B f k 17 we simplify this expression to My 45127 ii10 nil u c um fImrl mm 1 t1 1u cIt 0 Dividing through by u c7 we arrive at W A 1 u A 1 A kt2 1 u kt1B kt0 6 WWW Then I I l l 3 With 1 W WW 1 H WW39W 3 1 0 This is a second order difference equation Notice that the second row of H delivers kt kt so the vector representation of the system is correct Now we need to look for the eigenvalues of H7 from the characteristic polynomial given by lHiAIl 0 As an interlude before solving for the eigenvalues7 let us now motivate the general solution to the linear system above with an explicit derivation from basic principles Using spectral decomposition7 we can decompose H as follows A1 0 HVAV 1 A 0 A2 where A1 and A2 are eigenvalues of H and V is a matrix of eigenvectors of H Recall that st H zt A change of variables will help us get the solution to this system First premultiply both sides by V l V l xt1 V l Hzt V 1VAV 1zt A V71 wt 46 Let 2 E V lxt and 211 E V lxt Then7 since A is a diagonal matrix Zt1 A 2t 2 At 20 21 Cl 39 210 39 22 220 39 We can go back to Xt by prernultiplying Z by V x V2t V V 7 t 11 t 12 quot A1m A2w g3 kt 39 The solution7 therefore must be of the form kt 61 1 62 where 31 and 32 are to be determined from initial conditions and values of A1 and A2 Let us now go back to our example To nd the eigenvalues in our speci c setting7 we use lH 7 All 0 to obtain 1 u f 1 1 i 777 7 BW 6 0 1 7A 1 u 1 A271BJAB07 4 where u 7 u 7 f 7 f denote the corresponding derivatives evaluated at 1 Let A27 1l3f 43 F 6 w W B This is a continuous function of A and F0 gt0 F1 72 lt0 Therefore7 the mean value theorern implies that HA1 6 01 F A1 0 That is7 one of the eigenvalues is positive and smaller than one Since Alirn F A 00 gt 07 the other eigenvalue A2 must also be positive and larger than 1 47 We see that a convergent solution to the system requires 02 0 The remaining constant 01 will be determined from 1 61x 10 E kOA Elk0Ak The solution therefore is ktk ikoik Recall that kt 7 If gI kt 7 If Analogously in the linearized system kt1ik1ktik It can thus be seen that the eigenvalue A1 has a particular meaning it measures the inverse of the rate of convergence to the steady state As a different illustration suppose we were looking at the larger system kt 61A 02A 03A 04AL k0 given That is some economic model with a single state variable leads to a third order difference equation If only one eigenvalue A1 has M11 lt 1 then there is a unique convergent path leading to the steady state This means that 02 03 04 will need to be equal to zero choosing the subscript 1 to denote the eigenvalue smaller than 1 in absolute value is arbitrary of course In contrast if there were for example two eigenvalues A1 A2 with M11 M21 lt 1 then we would have m 2 two degrees of freedom But there is only one economic restriction namely k0 given That is 7 1 lt m Then there would be many convergent paths satisfying the sole economic restriction on initial conditions and the system would be indeterminate 423 Alternative solution to the speed of convergence There is another way to solve for the speed of convergence It is related to the argument that we have local convergence around I ifthe slope ofthe gk schedule satis es g k 6 711 The starting point is the functional Euler equation u lfk 9kl u lf9k 99klf 9k7 Vk Differentiating with respect to k yields gtlt A m A Pr V V h q A m A Pr V l m A m A V Pr V i z A m A Pr V V A Pr V lt Pr 48 Evaluating at the steady state and noting that gk If we get U Clf k 9 kl Bu clf k9 k 9 k2lf k u cf k9 k This equation is a quadratic equation in g k Reshuf ing the terms and noting that B f k 17 we are lead back to equation 42 from before with the difference that we have now g k instead of A Using the same assumptions on and f7 we can easily prove that for one of the solutions gik 6 711 The nal step is the construction of gk using a linear approximation around k 49 50 Chapter 5 Competitive Equilibrium in Dynamic Models It is now time to leave pure maximization setups where there is a planner making all decisions and move on to market economies What economic arrangement or what al location mechanism will be used in the model economy to talk about decentralized or at least less centralized behavior Of course different physical environments may call for different arrangements Although many argue that the modern market economy is not well described by well functioning markets due to the presence of various frictions incomplete information externalities market power and so on it still seems a good idea to build the frictionless economy rst and use it as a benchmark from which exten sions can be systematically built and evaluated For a frictionless economy competitive equilibrium analysis therefore seems suitable One issue is what the population structure will be We will rst look at the in nite horizon dynastic setup The generalization to models with overlapping generations of consumers will come later on Moreover we will whenever we use the competitive equilibrium paradigm assume that there is a representative consumer That is to say we think of it that there are a large truly in nite perhaps number of consumers in the economy who are all identical Prices of commodities will then have to adjust so that markets clear this will typically mean under appropriate strict concavity assumptions that prices will make all these consumers make the same decisions prices will have to adjust so that consumers do not interact For example the dynamic model without production gives a trivial allocation outcome the consumer consumes the endowment of every product The competitive mechanism ensures that this outcome is achieved by prices being set so that the consumer when viewing prices as beyond his control chooses to consume no more and no less than his endowments For a brief introduction imagine that the production factors capital and labor were owned by many individual households and that the technology to transform those factors into consumption goods was operated by rms Then households7 decisions would consist of the amount of factors to provide to rms and the amount of consumption goods to purchase from them while rms would have to choose their production volume and factor demand The device by which sellers and buyers of factors and of consumption goods are driven together is the market which clearly brings with it the associated concept of 51 prices By equilibrium we mean a situation such that for some given prices individual households7 and rms7 decisions show an aggregate consistency7 ie the amount of factors that suppliers are willing to supply equals the amount that producers are willing to take7 and the same for consumption goods we say that markets clear The word competi tive indicates that we are looking at the perfect competition paradigm7 as opposed to economies in which rms might have some sort of market power Somewhat more formally7 a competitive equilibrium is a vector of prices and quantities that satisfy certain properties related to the aggregate consistency of individual decisions mentioned above These properties are 1 Households choose quantities so as to maximize the level of utility attained given their wealth factor ownership evaluated at the given prices When making decisions7 households take prices as given parameters The maximum monetary value of goods that households are able to purchase given their wealth is called the budget constraint E0 The quantity choice is feasible By this we mean that the aggregate amount of commodities that individual decision makers have chosen to demand can be pro duced with the available technology using the amount of factors that suppliers are willing to supply Notice that this supply is in turn determined by the remunera tion to factors7 ie their price Therefore this second condition is nothing but the requirement that markets clear 3 Firms chose the production volume that maximizes their pro ts at the given prices For dynamic economic setups7 we need to specify how trade takes place over time are the economic agents using assets and7 if so7 what kinds of assets Often7 it will be possible to think of several different economic arrangements for the same physical environment that all give rise to the same nal allocations It will be illustrative to consider7 for example7 both the case when rms rent their inputs from consumers every period7 and thus do not need an intertemporal perspective and hence assets to ful ll their pro t maximization objective7 and the case when they buy and own the long lived capital they use in production7 and hence need to consider the relative values of pro ts in different periods Also7 in dynamic competitive equilibrium models7 as in the maximization sections above7 mathematically there are two alternative procedures equilibria can be de ned and analyzed in terms of in nite sequences7 or they can be expressed recursively7 us ing functions We will look at both7 starting with the former For each approach7 we will consider different speci c arrangements7 and we will proceed using examples we will typically consider an example without production endowment economy and the neoclassical growth model Later applied chapters will feature many examples of other setups 51 Sequential competitive equilibrium The central question is the one of determining the set of commodities that are traded The most straightforward extension of standard competitive analysis to dynamic models 52 is perhaps the conceptually most abstract one simply let goods be dated so that for example in a one good per date context there is an in nite sequence of commodities consumption at t 0 consumption at t 1 etc and like in a static model let the trade in all these commodities take place once and for all We will call this setup the date 0 or Arrow Debreu McKenZie arrangement In this arrangement there is no need for assets If for example a consumer needs to consume both in periods 0 and in future periods the consumer would buy rights to future consumption goods at the beginning of time perhaps in exchange for current labor services or promises of future labor services Any activity in the future would then be a mechanical carrying out of all the promises made at time zero An alternative setup is one with assets we will refer to this case as one with sequential trade In such a case assets are used by one or more agents and assets are traded every period In such a case there are nontrivial decisions made in every future period unlike in the model with date 0 trade We will now in turn consider a series of example economies and for each one de ne equilibrium in a detailed way 511 An endowment economy with date0 trade Let the economy have only one consumer with in nite life There is no production but the consumer is endowed with wt 6 3 units of the single consumption good at each date It Notice that the absence of a production technology implies that the consumer is unable to move consumption goods across time he must consume all his endowment in each period or dispose of any balance An economy without a production technology is called an ewehange economy since the only economic activity besides consumption that agents can undertake is trading Let the consumers utility from any given consumption path ct0 be given by 2311 The allocation problem in this economy is trivial But imagine that we deceived the consumer into making him believe that he could actually engage in transactions to buy and sell consumption goods Then since in truth there is no other agent who could act as his counterpart market clearing would require that prices are such that the consumer is willing to have exactly wt at every t We can see that this requires a speci c price for consumption goods at each different point in time ie the commodities here are consumption goods at different dates and each commodity has its own price pt We can normaliZe 100 1 so that the prices will be relative to t 0 consumption goods a consumption good at It will cost pt units of consumption goods at t 0 Given these prices the value of the consumers endowment is given by 00 E Pt Wt t0 53 The value of his expenditures is 00 10 Ct t0 and the budget constraint requires that 00 00 10 0 S 10 Wt t0 t0 Notice that this assumes that trading in all commodities takes place at the same time purchases and sales of consumption goods for every period are carried out at t 0 This market structure is called an Arrow Debreu McKenZie or date 0 market as opposed to a sequential market structure in which trading for each period7s consumption good is undertaken in the corresponding period Therefore in this example we have the following De nition 1 A competitive equilibrium is a vector of prices pt0 and a vector of quantities ci0 such that Z c0 arg max 6 uct t0 000 00 00 St 10 0 S 10 39Wt t0 t0 ct 2 0 Vt 2 c wt Vt market clearing constraint Notice as mentioned earlier that in this trivial case market clearing condition 2 re quires that the agent consumes exactly his endowment in each period and this determines equilibrium prices Quantities are trivially determined here but prices are not To nd the price sequence that supports the quantities as a competitive equilibrium simply use the rst order con ditions from the consumers problem These are BtuWt Apt W7 where we have used the fact that equilibrium consumption ct equals wt and where A de notes the Lagrange multiplier for the budget constraint The multiplier can be eliminated to solve for any relative price such as i AM Pt1 uWtJrl This equation states that the relative price of today7s consumption in terms of tomorrow7s consumption the de nition of the gross real interest rate has to equal the marginal rate of substitution between these two goods which in this case is inversely proportional to the discount rate and to the ratio of period marginal utilities This price is expressed in terms of primitives and with it we have a complete solution for the competitive equilibrium for this economy remember our normalization p0 1 54 512 The same endowment economy with sequential trade Let us look at the same eccchange economy but with a sequential markets structure We allow l period loans which carry an interest rate of Rt E l 7quot V V gross rate net rate on a loan between periods t7 l and t Let at denote the net asset position of the agent at time t ie the net amount saved lent from last period Now we are allowing the agent to transfer wealth from one period to the next by lending l period loans to other agents However this is just a ction as before in the sense that since there is only one agent in the economy there cannot actually be any loans outstanding since lending requires both a lender and a borrower Therefore the asset market will only clear if a 0 Vt ie if the planned net asset holding is zero for every period With the new market structure the agent faces not a single but a sequence of budget constraints His budget constraint in period t is given by Ct at1 at RWt7 W W uses of funds sources of funds where R denotes the equilibrium interest rate that the agent takes as given With this in hand we have the following De nition 2 A competitive equilibrium is a set of sequences ci0 a10 RHZO such that 00 Z cfaf10 argmax tuct f0tgtat1o t0 st ct a L at R wt Vt thOVta00 71 00 tlim at1 Rig 0 no Ponzi game condition 2 Feasibility constraint a 0 Vt asset market clearing 3 c wt Vt goods market cleaiing Notice that the third condition necessarily follows from the rst and second ones by Walras7s law if n 7 l markets clear in each period then the nth one will clear as well To determine quantities is as trivial here with the same result as in the date 0 world Prices ie interest rates are again available from the rst order condition for saving the consumers Euler equation evaluated at c wt uWt 5 uWHI 39Rerlv so that 1 u wt Rl1 39 739 5 U Wt1 Not surprisingly this expression coincides with the real interest rate in the date 0 econ only 55 513 The neoclassical growth model with date0 trade Next we will look at an application of the de nition of competitive equilibrium to the neoclassical growth model We will rst look at the de nition of competitive equilibrium with a date 0 market structure7 and then at the sequential markets structure The assumptions in our version of the neoclassical growth model are as follows H 3 9 4 U The consumer is endowed with 1 unit of time each period7 which he can allocate between labor and leisure The utility derived from the consumption and leisure stream ct7 l 7 nt0 is given by U071 710 26 u6t That is7 we assume for the moment that leisure is not valued equivalently7 labor supply bears no utility cost We also assume that is strictly increasing and strictly concave The consumer owns the capital7 which he rents to rms in exchange for rt units of the consumption good at t per unit of capital rented Capital depreciates at rate 6 each period The consumer rents his labor services at t to the rm for a unit rental or wage rate of wt The production function of the consumptioninvestment good is FKn F is strictly increasing in each argument7 concave7 and homogeneous of degree 1 The following are the prices involved in this market structure Price of consumption good at every t pt pt intertemporal relative prices if p0 17 then pt is the price of consumption goods at t relative to in terms of consumption goods at t 0 Price of capital services at t pt rt 73 rental rate price of capital services at t relative to in terms of consumption goods at t Price of labor pt wt wt wage rate price of labor at t relative to in terms of consumption goods at t De nition 3 A competitive equilibrium is a set of sequences Prices 100 73 Bio wi fio Quantities 00 K 10 710 such that 56 Z ci0 K10 710 solve the consumer s problem 00 of7 K31 710 argmaX t uct 0tgtKt1nO t0 517 Zpl lctwLKt l ZpirfKt1 76 Ktnt t0 t0 ct 2 0 Vt so given At every period t capital is quoted in the same price as the consumption good As for labor recall that we have assumed that it has no utility cost Therefore wt gt 0 will imply that the consumer supplies all his time endowment to the labor market wtgt0 n1Vt 2 Ki0 710 solve the rms problem Vt K31 argmaXlp FltKt7ntgt 10 K ipiwi mt Kant The rm s decision problem involves just a one period choice it is not of a dynam ical nature for epample we could imagine that rms live for just one period All of the model s dynamics come from the consumer s capital accumulation problem This condition may equivalently be edpressed as follows Vt r wf satisfy 2 FK K31 51 w Fn KS7 1 Notice that this shows that if the production function F K771 is increasing in n then n 1 follows 3 Feasibility market clearing ciKf1FKf1176Kf This is known as the one sector neoclassical growth model since only one type of goods is produced that can be used either for consumption in the current period or as capital in the following There is also a vast literature on multi sector neoclassical growth models in which each type of physical good is produced with a di erent production technology and capital accumulation is speci c to each technology Let us now characterize the equilibrium We rst study the consumers problem by deriving his intertemporal rst order conditions Differentiating with respect to ct we obtain Ct 6 WW Pl39bi where 3 is the Lagrange multiplier corresponding to the budget constraint Since the market structure that we have assumed consists of date 0 markets7 there is only one budget and hence a unique multiplier 57 Consumption at t 1 obeys Ct1 i t139uC1 1044 AT39 Combining the two we arrive at p 1 u C 7 7 52 1031 B M 011 p v 1 u C i We can7 as before7 interpret as the real interest rate7 and 7 i as the marginal t t1 rate of substitution of consumption goods between t and t 1 Differentiating with respect to capital7 one sees that Kt133910 T 101 731 1 6 Therefore7 i i F 1 7 6 i t 1 1044 Using condition 517 we also nd that p i 1 FKKH11 176 53 10t1 The expression FK Kf11 1 7 6 is the marginal return on capital the marginal rate of technical substitution transformation between 0 and ct Combining expres sions 52 and 537 we see that uc uc1 FKK11 17 6 54 Notice now that 54 is nothing but the Euler Equation from the planners problem Therefore a competitive equilibrium allocation satis es the optimality conditions for the centralized economy the competitive equilibrium is optimal You may recognize this as the First Welfare Theorem We have assumed that there is a single consumer7 so in this case Pareto optimality just means utility maximization ln addition7 as we will see later7 with the appropriate assumptions on FKn namely7 non increasing returns to scale7 an optimum can be supported as a competitive equilibrium7 which is the result of the Second Welfare Theorem 514 The neoclassical growth model with sequential trade The following are the prices involved in this market structure Capital services at t Kt 7 R Rt rental rate price of capital services at t relative to in terms of consumption goods at It Just for the sake of variety7 we will now assume that R is the return on capital net of the depreciation costs That is7 with the notation used before7 R E rt 1 7 6 58 Labor nt a wt wt wage rate price of labor at t relative to in terms of consumption goods at t De nition 4 A competitive equilibrium is a sequence R7 w c Kill ni0 such that 00 7 Z ct7 KH17 nt t0 solves the consumer s problem 00 cg7 Kf1n0 argmaX 2J3 uct CKt1mo st ctKt1 KtR nt 10 so given and a no Ponzi game condition Note that accumulating Kt is analogous to lending at t 2 Kg17 n 0 solves the rms problem Vt K31 argmaXFKtnt 7 R Kt 176 Kt 7w nt Kn 3 Market clearing feasibility Vtc K 1 FK 1176K The way that the rental rate has been presented now can be interpreted as saying that the rm manages the capital stock7 funded by loans provided by the consumers However7 the capital accumulation decisions are still in the hands of the consumer this might also be modeled in a different way7 as we shall see later Let us solve for the equilibrium elements As before7 we start with the consumers problem ct t uc t With the current market structure7 the consumer faces a sequence of budget constraints7 and hence a sequence of Lagrange multipliers Ai0 We also have CH1 3 5H1 39uC1 5H1 Alu Then A 7 0 55 31 u 031 Differentiation with respect to capital yields Kt1 t A 5H1 39R1 Al17 so that At 7 Pi 5 6 At 7 a H1 lt gt t1 59 Combining expressions 55 and 56 we obtain We 7 m 751 57 From Condition 2 of the de nition of competitive equilibrium R FkK117 6 58 Therefore combining 57 and 58 we obtain WC 6 W 0111 le K31 1 5l This again is identical to the planners Euler equation This shows that the sequential market equilibrium is the same as the Arrow Debreu McKenzie date 0 equilibrium and both are Pareto optimal 52 Recursive competitive equilibrium Recursive competitive equilibrium uses the recursive concept of treating all maximization problems as split into decisions concerning today versus the entire future As such this concept thus has no room for the idea of date 0 trading it requires sequential trading Instead of having sequences or vectors a recursive competitive equilibrium is a set of functions quantities utility levels and prices as functions ofthe state the relevant initial condition As in dynamic programming these functions allow us to say what will happen in the economy for every speci c consumer given an arbitrary choice of the initial state As above we will state the de nitions and then discuss their rami cations in the context of a series of examples beginning with a treatment of the neoclassical growth model 521 The neoclassical growth model Let us assume again that the time endowment is equal to 1 and that leisure is not valued Recall the central planner7s problem that we analyzed before V K V K lt gt we 6 lt gt stcK FK1176K In the decentralized recursive economy the individuals budget constraint will no longer be expressed in terms of physical units but in terms of sources and uses of funds at the going market prices In the sequential formulation of the decentralized problem these take the form of sequences of factor remunerations Rh wt0 with the equilibrium levels given by R FKK1176 w FnK 71 60 Notice that both are a function of the aggregate level of capital with aggregate labor supply normalized to 1 ln dynamic programming terminology what we have is a law of motion for factor remunerations as a function of the aggregate level of capital in the economy If R denotes the current aggregate capital stock then Rmm w Therefore the budget constraint in the decentralized dynamic programming problem reads 0KRKMm am The previous point implies that when making decisions two variables are key to the agent his own level of capital K and the aggregate level of capital K which will de termine his income So the correct syntax for writing down the dynamic programming problem is VKK V K K 510 lt Cg 0uc lt lt gt where the state variables for the consumer are K and We already have the objective function that needs to be maximized and one of the restrictions namely the budget constraint Only K is left to be speci ed The economic interpretation of this is that we must determine the agents perceived law of motion of aggregate capital We assume that he will perceive this law of motion as a function of the aggregate level of capital Furthermore his perception will be rational it will correctly correspond to the actual law of motion K GK 511 where G is a result ofthe economy7s that is the representative agent7s equilibrium capital accumulation decisions Putting 59 510 and 511 together we write down the consumers complete dynamic problem in the decentralized economy VKK cma 0uc VKK 512 Si 0 K RU Kwf KGK 512 is the recursive competitive equilibrium functional equation The solution will yield a policy function for the individuals law of motion for capital K gKK argmaX uRKKwKiK hL WKCK H Ke0RKKwK st 1 em We can now address the object of our study De nition 5 A recursive competitive equilibrium is a set offurietioris 61 Quantities GUT 9K K Lifetime utility level V K K Prices RK wK such that Z V KJ solves 512 and gKK is the associated policy function 2 Prices are competitively determined Ru FKK1176 wK FAR 1 In the recursive formulation prices are stationary functions rather than sequences 3 Consistency is satis ed 7 7 7 7 GK gKK VK The third condition is the distinctive feature of the recursive formulation of competi tive equilibrium The requirement is that whenever the individual consumer is endowed with a level of capital equal to the aggregate level for example only one single agent in the economy owns all the capital or there is a measure one of agents his own individual behavior will exactly mimic the aggregate behavior The term consistency points out the fact that the aggregate law of motion perceived by the agent must be consistent with the actual behavior of individuals Consistency in the recursive framework corresponds to the idea in the sequential framework that consumers7 chosen sequences of say capital have to satisfy their rst order conditions given prices that are determined from rms7 rst order conditions evaluated using the same sequences of capital None of the three conditions de ning a recursive competitive equilibrium mentions market clearing Will markets clear That is will the following equality hold 5K FK1176K where E denotes aggregate consumption To answer this question we may make use of the Euler Theorem If the production technology exhibits constant returns to scale that is if the production function is homogeneous of degree 1 then that theorem delivers FK 1 1 76KRK KwK ln economic terms there are zero pro ts the product gets exhausted in factor payment This equation together with the consumers budget constraint evaluated in equilibrium K K implies market clearing Completely solving for a recursive competitive equilibrium involves more work than solving for a sequential equilibrium since it involves solving for the functions V and g which specify off equilibrium behavior what the agent would do if he were different from the representative agent This calculation is important in the sense that in order to justify the equilibrium behavior we need to see that the postulated chosen path is not worse than any other path VKK precisely allows you to evaluate the future conse quences for these behavioral alternatives thought of as one period deviations lmplicitly 62 this is done with the sequential approach also although in that approach one typically simply derives the rst order Euler equation and imposes K K there Knowing that the FCC is suf cient one does not need to look explicitly at alternatives The known parametric cases of recursive competitive equilibria that can be solved fully include the following ones logarithmic utility additive logarithms of consumption and leisure if leisure is valued Cobb Douglas production and 100 depreciation ii isoelastic utility and linear production and iii quadratic utility and linear production It is also possible to show that when utility is isoelastic and no matter what form the production function takes one obtains decision rules of the form gK K AKK MK where the two functions A and u satisfy a pair of functional equations whose solution depends on the technology and on the preference parameters That is the individual decision rules are linear in K the agents own holdings of capital More in the spirit of solving for sequential equilibria one can solve for recursive competitive equilibrium less than fully by ignoring V and g and only solve for G using the competitive equilibrium version of the functional Euler equation It is straightforward to show using the envelope theorem as above in the section on dynamic programming that this functional equation reads Using the Euler Theorem and consistency K K we now see that this functional equation becomes u F011 lt1 7 M 7 GU 7 W ltFltGltKgt1gt lt1 7 6gtGltKgt7 7GGK F1GK 1 1 i 6 VK which corresponds exactly to the functional Euler equation in the planning problem We have thus shown that the recursive competitive equilibrium produces optimal behavior 522 The endowment economy with one agent Let the endowment process be stationary wt w Vt The agent is allowed to save in the form of loans or assets His net asset position at the beginning of the period is given by 1 Asset positions need to cancel out in the aggregate 6 0 since for every lender there must be a borrower The de nition of a recursive equilibrium is now as follows De nition 6 A recursive competitive equilibrium is a set offimetwhs Va 9a R such that Z Va solves the consumer s functional equation Va moaxl B Va stca aRw 2 Consistency 90 0 63 The consistency condition in this case takes the form of requiring that the agent that has a null initial asset position keep this null balance Clearly7 since there is a unique agent then asset market clearing requires a 0 This condition will determine R as the return on assets needed to sustain this equilibrium Notice also that R is not really a function it is a constant7 since the aggregate net asset position is zero Using the functional Euler equation7 which of course can be derived here as well7 it is straightforward to see that R has to satisfy 1 R 6 since the 1 terms cancel This value induces agents to save zero7 if they start with zero assets Obviously7 the result is the same as derived using the sequential equilibrium de nition 523 An endowment economy with two agents Assume that the economy is composed of two agents who live forever Agent 239 derives utility from a given consumption stream 40 as given in the following formula Ul ul i 12 t0 Endowments are stationary I I u w1 Vt7i12 Total resource use in this economy must obey 0 0 w1w2 Vt Clearing of the asset market requires that 6 Eaga0Vt Notice this implies a fag that is7 at any point in time it suf ces to know the asset position of one of the agents to know the asset position of the other one as well Denote A1 E 11 This is the relevant aggregate state variable in this economy the time subscript is dropped to adjust to dynamic programming notation Claiming that it is a state variable amounts to saying that the distribution of asset holdings will matter for prices This claim is true except in special cases as we shall see below7 because whenever marginal propensities to save out of wealth are not the same across the two agents either because they have different utility functions or because their common utility function makes the propensity depend on the wealth level7 different prices are required to make total savings be zero7 as equilibrium requires Finally7 let q denote the current price of a one period bond q Also7 in what t1 follows7 subscript denotes the type of agent We are now ready to state the following De nition 7 A recursive competitive equilibrium of the two agent economy is a set offunetions 64 Quantities 91 a17 A1 7 92 a27 A1 7 GU11 Lifetime utility leuels V1 a17 A1 7 V2 a27 A1 Prices q A1 such that 1 V7 a77 A1 is the solution to consumeri s problem Vi ltin M m ltcigt vi a2 An 02 st cl a qA1 a7 wi A GU11 a perceived law of motion for A1 The solution to this functional equation delivers the policy function gl a77 A1 2 Consistency G 91A17 A1 VAl 7G 927A17 A1 The second condition implies asset market clearing 91A17 A1 92iA17 A1 GU11 GU11 0 Also note that q is the variable that will adjust for consistency to hold For this economy7 it is not easy to nd analytical solutions7 except for special para metric assumptions We will turn to those now We will7 in particular7 consider the following question under what conditions will q be constant that is7 independent of the wealth distribution characterized by A1 The answer is7 as one can show7 that it will be constant if the following two conditions hold 1 51 52 11 ul u27 an t ese ut1 1ty 1n ices e ong to t e orman c ass d h l d b l h G l W V lfcl where fc a b c quadratic i exponential V is 6170 7 1 CRRA This proposition is also valid for models with production7 and can be extended to uncertainty and to the case of valued leisure The underlying intuition is that q will be independent of the distribution of wealth if consumers7 utility functions induce constant marginal propensities to save7 and these propensities are identical across consumers And it is a well known result in microeconomics that these utility functions deliver just that 65 524 Neoclassical production again with capital accumulation by rms Unlike in the previous examples recall the discussion of competitive equilibrium with the sequential and recursive formulations7 we will now assume that rms are the ones that make capital accumulation decisions in this economy The single consumer owns stock in the rms In addition7 instead of labor7 we will have land77 as the second factor of production Land will be owned by the rm The functions involved in this model are the dynamic programs of both the consumer and the rm K GK aggregate law of motion for capital qK current price of next period7s consumption V0 1 K consumer7s indirect utility as function of K and his wealth a a go a K policy rule associated with V0 1 K Vf K7 K market value in consumption goods7 of a rm with K units of initial capital7 when the aggregate stock of capital is K K gf K7 K policy rule associated with Vf a K The dynamic programs of the different agents are as follows 1 The consumer Vca K E10auc Vca K 513 511 cqK aa K c K The solution to this dynamic program produces the policy rule a go a K 2 The rm 7 7 7 7 7 7 Vf K K 7 mIaXFK 1 1 6 K K qK Vf K K 514 sf K c K The solution to this dynamic program produces the policy rule K W K K We are now ready for the equilibrium de nition De nition 8 A recursive competitive equilibrium is a set of functions 66 Quantities go a K gf K K G Lifetime utility levels values V0 a K Vf K K Prices q such that 1 V0 a K and go a K are the value and policy functions respectively solving 513 2 Vf K K and 9f K K are the value and policy functions respectively solving 514 3 Consistency 1 9f K K G foiquot all K 4 Consistency 2 go Vf K K K Vf G G VK The consistency conditions can be understood as follows The last condition requires that the consumer ends up owning 100 of the rm next period whenever he started up owning 100 of it Notice that if the consumer starts the period owning the whole rm then the value of a his wealth is equal to the market value of the rm given by Vf That is aVfK K 515 The value of the rm next period is given by VfK K To assess this value we need K and K But these come from the respective laws of motion 7 7 7 VfltK K gt Vf MK K GltKgtl Now requiring that the consumer owns 100 of the rm in the next period amounts to requiring that his desired asset accumulation a coincide with the value of the rm next period 7 7 a Vf MK K GltKgtl But a follows the policy rule gca A substitution then yields Maj Vf MK 061170 516 Using 515 to replace a in 516 we obtain 9c lVf K K K Vfl9fK7 0611701 517 The consistency conditioniis then imposed with K K in 517 and using the Con sistency 177 condition gf KK G yielding go Vf K K K Vf GK GK To show that the allocation resulting from this de nition of equilibrium coincides with the allocation we have seen earlier eg the planning allocation one would have to derive functional Euler equations for both the consumer and the rm and simplify them We leave it as an exercise to verify that the outcome is indeed the optimal one 67 68 Chapter 6 Uncertainty Our program of study will comprise the following three topics 1 Examples of common stochastic processes in macroeconomics 2 Maximization under uncertainty 3 Competitive equilibrium under uncertainty The rst one is closely related to time series analysis The second and the third one are a generalization of the tools we have already introduced to the case where the decision makers face uncertainty Before proceeding with this chapter7 it may be advisable to review the basic notation and terminology associated with stochastic processes presented in the appendix 61 Examples of common stochastic processes in macroe conomics The two main types of modelling techniques that macroeconomists make use of are 0 Markov chains 0 Linear stochastic difference equations 611 Markov chains De nition 9 Let x E X where X 2112 in is a nite set of values A station ary Markov chain is a stochastic process z fio de ned by X a transition matricc P and an initial probability distribution 7T0 for 0 the rst element in the stochastic an 1Xn process The elements of P represent the following probabilities VLX VL PM Prt1 fjlit 69 Notice that these probabilities are independent oftime We also have that the probability two periods ahead is given by Przt2 Ej it Z 39 ij h1 E lPZlmv where 132 denotes the i jth entry of the matrix P2 Given 7T0 7T1 is the probability distribution of alas of time t 0 and it is given by 7T1 7T0 39 Analogously 7T2 7T0 39 P2 if 7T0 Pt and also in 7Tt P De nition 10 A stationary or invariant distribution for P is a probability vector 7139 such that 7139 7T P A stationary distribution then satis es 7T I 7T P where I is identity matrix and 7T77TP 0 7TIP 0 That is 7139 is an eigenvector of P associated with the eigenvalue A 1 Example 9 7 3 7 3 ZPlt396 394gt7T1 7T27T1 7T2lt396 394 Youshouldverifythat QC H QC HQ V 1 H A p t twp A 11 09 i 7139 1 0 The rst state is said to be absorbing 70 ivP continuum of invariant distributions i 7139 a 17a a 6 01 In this last case there is a The question now is whether in converges in some sense to a number WOO as t a 00 which would mean that WOO WOO P and if so whether WOO depends on the initial condition 7T0 lfthe answers to these two questions are Yes77 and No respectively then the stochastic process is said to be asymptotically stationary with a unique invariant distribution Fortunately we can borrow the following result for suf cient conditions for asymptotic stationarity Theorem 3 P has a unique invariant distribution and is asymptotically stationary if 612 Linear stochastic difference equations AR1 ARIMA Let at 6 3 wt E Wquot t1 A 39t O wt1 We normally assume E wt1l Etlwt1lwt7 1047 0 E wHL w 1 I Example 10 AR1 process Let yt1P39yt t1b and assume Et 8H1 0 E 83 oz E l tk tk1l 0 Even if yo is known the yt0 process will not be stationary in general However the process may become stationary as t a 00 By repeated substitution we get b EolytlPt39yo lipt b lPl lt1 tILTglOEO lytl Then the process will be stationary if lpl lt 1 Similarly the autocovariance function is given by Wt k E E0 11 7 E lytl 1H 7 E with 02 p 7 i 02 lPllt1 tILI 07t7 k fpz The process is asymptotically weakly stationary if lpl lt 1 71 We can also regard we or yo in the case of an AR1 process as drawn from a distribution with mean no and covariance E zo 7 no 0 7 no E To Then the following are suf cient conditions for 882 to be weakly stationary process i no is the eigenvector associated to the eigenvalue A1 1 of A 6 6 1 ii All other eigenvalues of A are smaller than 1 in absolute value Mil lt1 i 2 ii To see this notice that condition implies that t1 M0 A 1 0 O 39 wt1 Then r0 r0 E E xt filo z lmy AP0 A OO and P E E tk M0 1 Moll Ak P 0 This is the matrix version of the autocovariance function y t k presented above Notice that we drop t as a variable in this function Example 11 Let x gt 6 3B A p C 02 and wt a we are accommodating the ARZ process seen before to this notation We can do the following change of variables at b A o 1 39yt l 0 wt1 1 A 7 P yt1 7 0 W A Then using the previous results and ignoring the constant we get P0 pZP002 02 P017p 2 62 Maximization under uncertainty We will approach this topic by illustrating with examples Let us begin with a simple 2 period model where an agent faces a decision problem in which he needs to make the following choices 1 Consume and save in period 0 2 Consume and work in period 1 72 The uncertainty arises in the income of period 1 through the stochasticity ofthe wage We will assume that there are 71 possible states of the world in period 1 ie LUZ 6 W17 quot7 Wn7 where 7T E Pr LUZ 411 for 239 1 n The consumers utility function has the von Neumann Morgenstern type ie he is an expected utility maximizer Leisure in the second period is valued U in u co 01 E E u 00 01 i1 Speci cally the utility function is assumed to have the form U we 6 in who 71 mlgt17 i1 where 1 lt 0 Market structure incomplete markets We will assume that there is a risk free77 asset denoted by a and priced q such that every unit of a purchased in period 0 pays 1 unit in period 1 whatever the state of the world The consumer faces the following budget restriction in the rst period 00 a q I At each realization of the random state of the world his budget is given by 01awini z391 n The consumers problem is therefore u 00 B 39 7U U 01139 1 max 00 11 01in1i1 st 00 a q I 01awini 239 1 n The rst order conditions are V L co ucoZ R i1 1 where RE 7 q 01139 3 5 39 7U 39U 01139 w39 711139 3 76m 3911 w39 wi 73 i 71 cu w v uco ZmucliR E BEuc1R The interpretation of the last expression is both straightforward and intuitive on the margin the consumers marginal utility from consumption at period 0 is equated to the discounted eppected marginal utility from consuming R units in period 1 170 7 1 Example 12 Let uc belong to the CES class that is uc This is a a common assumption in the literature Recall that o is the coe cient of relative risk aversion the higher a the less variability in consumption across states the consumer is willing to su er and its inverse is the elasticity of intertemporal substitution the higher a the less willing the consumer is to epperience the fluctuations of consumption over time In particular let a 1 then uc logc Assume also that vn log1 7 Replacing in the rst order conditions these assumptions yield CM LUl39 39 7 and using the budget constraint at i we get awi CMT Therefore q 7 n 2 Ieaq Z7 aw i1 From this equation we get a unique solution even if not epplicit for the amount of savings given the price q Finally notice that we do not have complete insurance in this model why Market structure complete markets We will now modify the market structure in the previous example Instead of a risk free asset yielding the same payout in each state we will allow for Arrow securities77 state contingent claims n assets are traded in period 0 and each unit of asset i purchased pays off 1 unit if the realized state is i and 0 otherwise The new budget constraint in period 0 is 71 00ZQi39ai 1 i1 In the second period if the realized state is i then the consumers budget constraint is 01139 ai71i wi 74 Notice that a risk free asset can be constructed by purchasing one unit of each 1 Assume that the total price paid for such a portfolio is the same as before ie 71 q 2 i1 The question is whether the consumer will be better or worse off with this market structure than before lntuitively we can see that the structure of wealth transfer across periods that was available before namely the risk free asset is also available now at the same cost Therefore the agent could not be worse off Moreover the market structure now allows the wealth transfer across periods to be state speci c not only can the consumer reallocate his income between periods 0 and 1 but also move his wealth across states of the world Conceptually this added ability to move income across states will lead to a welfare improvement if the ws are nontrivially random and if preferences show risk aversion ie if the utility index u is strictly concave Solving for a in the period 1 budget constraints and replacing in the period 0 con straint we get V L V L 00ZQi39Cii I l ZQi wi ni i1 i1 We can interpret this expression in the following way q is the price in terms of co of consumption goods in period 1 if the realized state is 239 q wi is the remuneration to labor if the realized state is 239 measured in term of co remember that budget consolidation only makes sense if all expenditures and income are measured in the same unit of account in this case it is a monetary unit where the price of co has been normalized to 1 and q is the resulting level of relative prices Notice that we have thus reduced the n 1 constraints to 1 whereas in the previous problem we could only eliminate one and reduce them to n This budget consolidation is a consequence of the free reallocation of wealth across states The rst order conditions are 00 uco 01139 3 B39Wi uCii39 711i 3 B Wi Uni wr i iucliwiv uCOB7T Qi The rst condition intra state consumption leisure choice is the same as with in complete markets The second condition re ects the added exibility in allocation of consumption the agent now not only makes consumption saving decision in period 0 but also chooses consumption pattern across states of the world Under this equilibrium allocation the marginal rates of substitution between consump tion in period 0 and consumption in period 1 for any realization of the state of the world is given by i u 01 239 1n MRS 007 01139 In 75 and the marginal rates of substitution across states are Qi Cu 61739 i 1739 Example 13 Using the utility function from the previous epample the rst order condi tions together with consolidated budget constraint can be rewritten as 1 71 71 1 l 7U Qi 01139 01 3900 W The second condition says that consumption in each period is proportional to con sumption in CO This proportionality is a function of the cost of insurance the higher q in relation to in the lower the wealth transfer into state i 621 Stochastic neoclassical growth model Notation We introduce uncertainty into the neoclassical growth model through a stochastic shock affecting factor productivity A very usual assumption is that of a neutral shock affect ing total factor productivity TFP Under certain assumptions for example Cobb Douglas y AKu nl D production technology a productivity shock is always neutral even if it is modelled as affecting a speci c component capital K labor n technology A Speci cally a neoclassical constant returns to scale aggregate production function subject to a TFP shock has the form Eltkt71zt fkt7 where z is a stochastic process and the realizations 2 are drawn from a set Z 2 E Z Vt Let Z denote a t times Cartesian product of Z We will assume throughout that Z is a countable set a generalization of this assumption only requires to generalize the summations into integration however this brings in additional technical complexities which are beyond the scope of this course Let 2 denote a history of realizations a t component vector keeping track of the previous values taken by the 27 for all periods j from 0 to t 2t 27 2717 m7 20 Notice that 20 20 and we can write 2 2t zt l Let 7TZt denote the probability of occurrence of the event 2 2t1 20 Under this notation a rst order Markov process has 7r 2t1 2t zt 77l2t17 2 lztl care must be taken of the objects to which probability is assigned 76 Sequential formulation The planning problem in sequential form in this economy requires to maximize the func tion 2 Z t7rzt u ct E E 236 uct t0 z eZt Notice that as t increases7 the dimension of the space of events Zt increases The choice variables in this problem are the consumption and investment amounts at each date and for each possible realization of the sequence of shocks as of that date The consumer has to choose a stochastic process for 0 and another one for kt 0 2 V2 Vt kt 2t Vzt Vt Notice that now there is only one kind of asset kt1 available at each date Let It7 2 denote a realization of the sequence of shocks 2 as of date It The bud get constraint in this problem requires that the consumer chooses a consumption and investment amount that is feasible at each It7 2 0 2t kt12t S 2 f kt 71 17 6ktzt 1 You may observe that this restriction is consistent with the fact that the agents infor mation at the moment of choosing is 2 Assuming that the utility index u is strictly increasing7 we may as well write the restriction in terms of equality Then the consumer solves t t t awg zwzogzgf HZ MONT SP at at lt2tgt 2tfktlt2quot1gtlt17 6gt kt lt2 v a 2t k0 given Substituting the expression for 0 2 from budget constraint7 the rst order condition with respect to kt1 2 is W 2 39u Ct t Z 5 2t17 2t 39u CH1 2t17 N X zt16Zt1 gtlt 2H1 39 f kt1 17 t t 7TZt17zt i Alternatively7 if we denote 7T 2t1 z 2 W then we can write a at w Z M m 21 If CH1 2w x Zt16Ztl gtlt 2t1f kH1 2 17 6 SEE 77 E zt uCt1zt17 2t Rt1 7 where Rt E 2111 kt1 1 7 6 is the marginal return on capital realized for each 2t1 SEE is a nonlinear stochastic difference equation In general we will not be able to solve it analytically so numerical methods or linearization techniques will be necessary Recursive formulation The planner7s problem in recursive version is V k z mkax u 7 k17 516 B Z 7TZ Vh 23 z eZ where we have used a rst order Markov assumption on the process 2t0 The solution to this problem involves the policy rule k g k z If we additionally assume that Z is not only countable but nite ie Z 21 2 then the problem can also be written as V mkax u 7 14 1 7 6 h xingV7 j1 where m denotes the probability of moving from state i into state j ie m E 7139 211 2712 Stationary stochastic process for kz Let us suppose that we have gk 2 we will show later how to obtain it by linearization What we are interested is what will happen in the long run We are looking for what is called stationary process for kz ie probability distribution over values of kz which is preserved at t 1 if applied at time t It is analogous to the stationary or invariant distribution of a Markov process Example 14 Let us have a look at a simpli ed stochastic version where the shock vari able 2 takes on only two values 2 E 212h An eccample of this kind of process is graphically represented in Figure 6 Following the set up we get two sets of possible values of capital which are of signif icance for stationary stochastic distribution of The rst one is the transient set which denotes a set of values of capital which cannot occur in the long run It is depicted in Figure 6 The probability of leaving the transient set is equal to the probability of 78 high shock capital line low shock capital line transient set A ergodic set B transient set Figure 61 An example of kz stochastic process when 2 E 21 2h capital reaching a value higher or equal to A which is possible only with a high shock This probability is non zero and the capital will therefore get beyond A at least once in the long run Thereafter the capital will be in the ergodic set which is a set that the capital will never leave once it is there Clearly the interval between A and B is an ergodic set since there is no value of capital from this interval and a shock which would cause the capital to take a value outside of this interval in the nept period Also there is a transient set to the right ofB Let Phz denote the joint density which is preserved over time As the stochastic process has only two possible states it can be represented by the density function Ph z From the above discussion it is clear to see that the density will be non zero only for those values of capital that are in the ergodic set The following are the required properties of Ph z 1 fPk Phdk 1 f 131kldk 7T1 where in and in are invariant probabilities of the low and high states 3 Prob h S hz 2h fkag Phhdh fwmgkmmdk mm fwmgmmdk mh Prob h S hz 21 fkq Hhdh fkghkSEPhkdk m fkglkgEPlkdk ml 79 Solving the model linearization of the Euler equation Both the recursive and the sequential formulation lead to the Stochastic Euler Equation 1 Ct 5 39 Ezt lu Ct1 l2t1 fkt11 5ll SEED Our strategy to solve this equation will be to use a linear approximation of it around the deterministic steady state We will guess a linear policy function and replace the choice variables with it Finally we will solve for the coef cients of this linear guess We rewrite SEE in terms of capital and using dynamic programming notation we get u zfk176kik Ezu z fk176k ik gtlt gtlt ifH 1 7 6 SEE2 Denote LHS E u zfk176kik RHS E Ezuzfk 1fork7W 2 f k 176 Let h be the steady state associated with the realization 2t0 that has 2 E for all but a nite number of periods t That is E is the long run value of 2 Example 15 Suppose that 2th follows art AR1 process 2H1 P Zt 1 P 3 t17 where p lt1 IfE8t 0 02 lt 00 and Eet 8tjl 0 VjZ 1 thert by the Law of Large Numbers we get that plz39m 2t 3 Having the long run value of 2t the associated steady state level of capital h is solved from the usual deterministic Euler equation WE u 53fh15l 1 B g gs 73076 25 anyiaE 176 N v v denote the variables expressed as deviations from their steady state values Using this notation we write down a rst order Taylor expansion of SEE2 around the long run values as LHS LLHSEaLEbLhcLh dL RHS LRHSEEZaRE bRh cRh dR 80 where the coef cients IL 13 bL etc are the derivatives of the expressions LHS and RHS with respect to the corresponding variables evaluated at the steady state for example aL u E In addition LLHS LRHS needs to hold for E 2 E P Z 0 the steady state and therefore dL dR Next we introduce our linear policy function guess in terms of deviations with respect to the steady state as A A kgk kgz392 The coef cients gk 91 are unknown Substituting this guess into the linearized stochastic Euler equation we get LLHS 0L35LECL399k39ECL39gz393dL LRHS E2aR El l bR gk z l bR gz E l CR gk E l CR gz EidR E2QR El l bR gk z l bR gz E l CR gi z l CR gk gz E l CR399239339ldR aR EzlgllbR gk g l bR gz g l cR gg g l CR gk gz E l CR399239Ezl3 ldR and our equation is LLHSLRHs amp Notice that dL dR will simplify away Using the assumed form of the stochastic process 2t0 we can replace E1 by p E The system LS needs to hold for all values of Z and 3 Given the values of the coef cients 1 bi c for 239 L R the task is to nd the values of 9 91 that solve the system Rearranging LS can be written as 3AE m3 0q where A aL l CL gz bR gz CR gk gz B aR CR gz O bL l CL gk bR gk CR gi As O is a second order polynomial in gk the solution will involve two roots We know that the value smaller than one in absolute value will be the stable solution to the system Example 16 Let 2t0 follow an AR1 process as m the previous emample Zt1P Zt1 P E t1 Then 339 E 27 pz17p387E p27 6 81 Itfollows that EAE p2 artd LRHS0R P393bR399k39EbR3992393CR39gi39ECR39gk39gz393CR39gz39P393dR We cart rearrartge LS to where A aLCL gz70R P bR gz CR gk gz CR gz p B bL i CL gk bR gk CR gg The solution to LS requires A 0 B 0 Therefore the procedure is to solve rst for gk from B picking the value less thart one artd thert use this value to solve for 91 from A Simulation and impulse response Once we have solved for the coef cients gk 917 we can simulate the model by drawing values of 2330 from the assumed distribution7 and an arbitrary k0 This will yield a stochastic path for capital from the policy rule kt1 gk39kngz39Et We may also be interested in observing the effect on the capital accumulation path in an economy if there is a one time productivity shock E which is the essence of impulse response The usual procedure for this analysis is to set so 0 that is7 we begin from the steady state capital stock associated with the long run value 3 and 30 to some arbitrary number The values of 3 for t gt 0 are then derived by eliminating the stochastic component in the 20 process For example7 let 2t0 be an AR1 process as in the previous examples7 then 3H1 P 3t 8 Let 30 A and set at 0 for all t Using the policy function7 we obtain the following path for capital E0 0 k1 gz A k2 gkgzAgzpAgkgzgzpA ks 9 gzgkgzpgz92 A kt 92 192 2p gkpquot2pquot1 gzA 82 and A lgkl lt1 amp lpl lt1 tlimkt 0 The capital stock converges back to its steady state value if lgkl lt 1 and lpl lt 1 z39 Figure 62 An example of an impulse response plot7 using 91 087 gk 097 p 7075 References and comments on the linearquadratic setup You can nd most of the material we have discussed on the neoclassical growth model in King7 Plosser and Rebelo 1988 Hansen and Sargent 1988 discuss the model in a linear quadratic environment7 which assumes that the production technology is linear in z and k and u is quadratic yzk ayzbyk uc iaucicu2bu This set up leads to a linear Euler equation7 and therefore the linear policy function guess is exact ln addition7 the linear quadratic model has a property called certainty equivalence 7 which means that gk and 91 do not depend on second or higher order moments of the shock 8 and it is possible to solve the problem7 at all t by replacing sz with E szl and thus transform it into a deterministic problem This approach provides an alternative to lineariZing the stochastic Euler equation We can solve the problem by replacing the return function with a quadratic approximation7 and the technological constraint by a linear function Then we solve the resulting linear quadratic problem Zettwmwufatem1 Return function 83 The approximation of the return function can be done by taking a second order Tay lor series expansion around the steady state This will yield the same results as the linearization Finally7 the following shortfalls of the linear quadratic setup must be kept in mind The quadratic return function leads to satiation there will be a consumption level with zero marginal utility Non negativity constraints may cause problems In practice7 the method requires such constraints not to bind Otherwise7 the Euler equation will involve Lagrange mul tipliers7 for a signi cant increase in the complexity of the solution A linear production function implies a constant marginal product technology7 which may not be consistent with economic intuition Recursive formulation issue There is one more issue to discuss in this section and it involves the choice of state variable in recursive formulation Let us consider the following problem of the consumer Z 6 WM u0t2 mtax E MW lo t0 zteZt 2t thtil 3 Ct2t Qht2t aht12t Qlt2t alt12t wtzt a1 ztil 2t thztil 3 Ct2t Qht2t 39aht12t Qlt2t alt12t wtzt ahtzt 1 Vt Vzt and no PonZi game condition7 where 2 follows a rst order Markov process and even more speci cally7 we only have two states7 ie 2 6 2mm As can be seen7 we have two budget constraints7 depending on the state at time t Let us now consider the recursive formulation of the above given problem To simplify matters7 suppose that 2 21 wtzt u 2 zh wtzt wh What are our state variables going to be Clearly7 2 has to be one of our state variables The other will be wealth w differentiate from the endowment w which we can de ne as a sum of the endowment and the income from asset holdings 2t 21 wtzt u altzt 1 2t 2h wtzt 4 ahtzt 1 The recursive formulation is now Vwzi E 84 1513 w Qih 3909 Iii 39 02 5 l ih 39 VhWh 09 7711 39V1Wi ID l 7 apah W W wk wl where the policy rules are now all 9ihw a gilwilh Could we use a as a state variable instead of it Yes we could but that would actually imply two state variables a and a1 Since the state variable is to be a variable which expresses the relevant information as succinctly as possible it is it that we should use 63 Competitive equilibrium under uncertainty The welfare properties of competitive equilibrium are affected by the introduction of un certainty through the market structure The relevant distinction is whether such structure involves complete or incomplete markets lntuitively a complete markets structure allows trading in each single commodity Recall our previous discussion of the neoclassical growth model under uncertainty where commodities are de ned as consumption goods indexed by time and state of the world For example if z and 2 denote two different realizations of the random sequence zj0 then a unit of the physical good c consumed in period t if the state of the world is 2 denoted by C is a commodity different from ct A complete markets structure will allow contracts between parties to specify the delivery of physical good c in different amounts at t than at t 2 and for a different price In an incomplete markets structure such a contract might be impossible to enforce and the parties might be unable to sign a legal contract that makes the delivery amount contingent on the realization of the random shock A usual incomplete markets structure is one where agents may only agree to the delivery of goods on a date basis regardless of the shock In short a contract specifying ct 31 ct is not enforceable in such an economy You may notice that the structure of markets is an assumption of an institutional nature and nothing should prevent in theory the market structure to be complete How ever markets are incomplete in the real world and this seems to play a key role in the economy for example in the distribution of wealth in the business cycle perhaps even in the equity premium puzzle that we will discuss in due time Before embarking on the study ofthe subject it is worth mentioning that the structure of markets need not be explicit For example the accumulation of capital may supply the role of transferring wealth across states of the world not just across time But allowing for the transfer of wealth across states is one of the functions speci c to markets therefore if these are incomplete then capital accumulation can to some extent perform this missing function An extreme example is the deterministic model in which there is only one state of the world and only transfers of wealth across time are relevant The possibility of accumulating capital is enough to ensure that markets are complete and allowing agents also to engage in trade of dated commodities is redundant Another 85 example shows up in real business cycle models which we shall analyze later on in this course A usual result in the real business cycle literature consistent with actual economic data is that agents choose to accumulate more capital whenever there is a good77 realization of the productivity shock An intuitive interpretation is that savings play the role of a buffer77 used to smooth out the consumption path which is a function that markets could perform Hence you may correctly suspect that whenever we talk about market completeness or incompleteness we are in fact referring not to the actual explicit contracts that agents are allowed to sign but to the degree to which they are able to transfer wealth across states of the world This ability will depend on the institutional framework assumed for the economy 631 The neoclassical growth model with complete markets We will begin by analyzing the neoclassical growth model in an uncertain environment We assume that given a stochastic process 2t0 there is a market for each consumption commodity 0 2 as well as for capital and labor services at each date and state of the world There are two alternative setups Arrow Debreu date 0 trading and sequential trading ArrowDebreu date0 trading The Arrow Debreu date 0 competitive equilibrium is Ct2t7 kt12t7 lt2t710t2t7 732 wt2t o such that 1 Consumers problem to nd Ct2t kt1zt ltzt 0 which solve max tWzt uc 2t17lzt CtZtgtkt11tltzt0 gzgt tlt tlt st 2 Z 1942 Cd2 kt12 l E Z Z 1942 lltrtzt1 5 X t0 ztezt t0 ztezt gtlt kt zt l 10 2 lt 2 First order conditions from rm7s problem are rtzt ztFkktzt 1ltzt wtzt ztFlktzt 1ltzt 3 Market clearing is Ct2t kt1zt 17 6 ktzt 1 2t Fktzt 1ltzt Vt Vzt 86 You should be able to show that the Euler equation in this problem is identical to the Euler equation in the planners problem In this context it is of interest to mention the so called no arbitrage condition which can be derived from the above given setup First we step inside the budget constraint and retrieve those terms which relate to kt1zt 0 From the LHS ptzt kt1zt 0 From the RHS Zz lp z hzt rt12t1zt 1 i 5 kt1zt The no arbitrage condition is the equality of these two expressions and it says that in equilibrium the price of a unit of capital must equal the sum of future values of a unit of capital summed across all possible states Formally it is kt12t 1091 Zpt12t172t 39 l7t12t172t 1 i 5 0 Zt1 What would happen if the no arbitrage condition did not hold Assuming kt1zt 2 0 the term in the brackets would have to be non zero If this term were greater then zero we could make in nite pro t by setting kt1zt foo Similarly if the term were less than zero setting kt1zt 00 would do the job As neither of these can happen in equilibrium the term in the brackets must equal zero which means that the no arbitrage condition must hold in equilibrium Sequential trade In order to allow wealth transfers across dates agents must be able to borrow and lend lt suf ces to have one period assets even with an in nite time horizon We will assume the existence of these one period assets and for simplicity that Z is a nite set with 71 possible shock values as is illustrated in Figure 63 Z 1216 Z zm 22 e Z zm 23 e Z t zm 2n 6 2 Figure 63 The shock 2 can take 71 possible values which belong to Z 87 Assume that there are q assets with asset j paying off 77 consumption units in t 1 if the realized state is 2 The following matrix shows the payoff of each asset for every realization of 2t1 a1 a2 aq 21 T11 T12 39 39 39 Tiq 22 T21 T22 39 39 39 Ta 23 T31 T32 39 39 39 Tag E R Zn Tn1 Tn2 39 39 39 an Then the portfolio 1 a1 a2 aq pays p in terms of consumption goods at t 1 where p R a 7 ngtlt1 an qgtlt1 q and each component p Z n aj is the amount of consumption goods obtained in state 239 from holding portfolio 1 What restrictions must we impose on B so that any arbitrary payoff combination p 6 3 can be generated by the appropriate portfolio choice Based on matrix algebra the answer is that we must have 1 q 2 n 2 rankR n If R satis es condition number 2 which presupposes the validity of the rst one then the market structure is complete The whole space 3 is spanned by R and we say that there is spanning It is useful to mention Arrow securities which were mentioned before Arrow security 239 pays off 1 unit if the realized state is z and 0 otherwise If there are q lt n different Arrow securities then the payoff matrix is 11 12 rag 21 1 0 0 22 0 1 0 23 0 0 0 z 0 0 1 2n 0 0 0 632 General equilibrium under uncertainty the case of two agent types in a twoperiod setting First we compare the outcome of the neoclassical growth model with uncertainty and one representative agent with the two different market structures 88 Only sequential trade in capital is allowed There is no spanning in this setup as there is only one asset for n states Spanning either with Arrow Debreu date 07 or sequential trading Will equilibria look different with these structures The answer is no7 and the reason is that there is a single agent Clearly7 every loan needs a borrower and a lender7 which means that the total borrowing and lending in such an economy will be zero This translates into the fact that different asset structures do not yield different equilibria Let us turn to the case where the economy is populated by more than one agent to analyze the validity of such a result We will compare the equilibrium allocation of this economy under the market structures 1 and 2 mentioned above Assumptions Random shock We assume there are n states of the world corresponding to n different values of the shock to technology to be described as 2 E 217 227 7 Zn njPrzz Let Edenote the expected value of z V L 3 Z 7Tj 39 2739 j1 Tastes Agents derive utility from consumption only not from leisure Preferences satisfy the axioms of expected utility and are represented by the utility index u Speci cally7 we assume that Ul ui 06 6 an z3912 j1 where ul x and uz is strictly concave gt 07 ug lt 0 We also assume that limju z 00 In this fashion7 agents7 preferences exhibit different attitudes ma towards risk Agent 1 is risk neutral and Agent 2 is risk averse Endowments Each agent is endowed with we consumption goods in period 07 and with one unit of labor in period 1 which will be supplied inelastically since leisure is not valued Technology Consumption goods are produced in period 1 with a constant returns to scale technology represented by the Cobb Douglass production function n 170 Kaltgt y 27 2 89 where K n denote the aggregate supply of capital and labor services in period 1 respectively We know that n 2 so yj 27 K Therefore the remunerations to factors in period 1 if state j is realized are given by 73 zjozKWl wj 27 g K Structure 1 one asset Capital is the only asset that is traded in this setup With K denoting the aggregate capital stock 1 denotes the capital stock held by agent 239 and therefore the asset market clearing requires that 11 12 The budget constraints for each agent is given by 63 a we 0 airjwj To solve this problem we proceed to maximize each consumer7s utility subject to his budget constraint Agent 1 The maximized utility function and the constraints are linear in this case We therefore use the arbitrage condition to express optimality 1 Z7rjrj j1 For a not to be in nite which would violate the market clearing condition that part of the arbitrage condition which is in brackets must equal zero Replacing for 73 we get then 1BZ7rjazjKDquot1 EEl j1 V L 1a K 1Z7rjzj j1 Therefore the optimal choice of K from Agent 17s preferences is given by i K 30131704 39 90 Notice that only the average value of the random shock matters for Agent 17 consis tently with this agent being risk neutral Agent 2 The Euler equation for Agent 2 is 12 woiag anu 2 ag Tw EE2 11 Given K from Agent 17s problem7 we have the values of r and w for each realization j Therefore7 Agent 27s Euler equation EE2 is one equation in one unknown 12 Since limju z 007 there exists a unique solution Let a be the solution to Then the ma values of the remaining choice variables are 7 7 aliKaZ i 7 CO 7 woiai More importantly7 Agent 2 will face a stochastic consumption prospect for period 17 which is 0arw where r and w are stochastic This implies that Agent 1 has Lot provided full insurance to Agent 2 Structure 2 Arrow securities It is allowed to trade in n different Arrow securities in this setup In this case7 these securities are contingent claims on the total remuneration to capital you could think of them as rights to collect future dividends in a company7 according to the realized state of the world Notice that this implies spanning ie markets are complete Let aj denote the Arrow security paying off one unit if the realized state is 27 and zero otherwise Let qj denote the price of 17 In this economy7 agents save by accumulating contingent claims they save by buying future dividends in a company Total savings are thus given by M S E qja1j 127 1 x H lnvestment is the accumulation of physical capital7 K Then clearing of the savings investment market requires that qu01j02 K 51 j1 Constant returns to scale imply that the total remuneration to capital services in state j will be given by K 77 by Euler Theorem Therefore7 the contingent claims that 91 get activated when this state is realized must exactly match this amount each unit of dividends that the company will pay out must have an owner but the total claims can not exceed the actual amount of dividends to be paid out In other words clearing of all of the Arrow security markets requires that 11739 12739 K 39 Tj 1 71 If we multiply both sides of ASMC by qj for each j and then sum up over j7s we get 2 aljfaz K qu rj j1 7391 But using S I to replace total savings by total investment V L K K 39 Z 1739 39 Tj j1 Therefore the equilibrium condition is that V L Egg471 EC j1 The equation EC can be interpreted as a no arbitrage condition in the following way The left hand side 22 qj 73 is the total price in terms of foregone consumption units of the marginal unit of a portfolio yielding the same expected marginal return as physical capital investment And the right hand side is the price also in consumption units of a marginal unit of capital investment First suppose that 22 qj 77 gt 1 An agent could in principle make unbounded pro ts by selling an in nite amount of units of such a portfolio and using the proceeds from this sale to nance an unbounded physical capital investment In fact since no agent would be willing to be on the buy side of such a deal no trade would actually occur But there would be an in nite supply of such a portfolio and an in nite demand of physical capital units In other words asset markets would not be in equilibrium A similar reasoning would lead to the conclusion that 22 qj 77 lt 1 could not be an equilibrium either With the equilibrium conditions at hand we are able to solve the model With this market structure the budget constraint of each Agent 239 is V L 023qu aij Wo j1 c 17 10 Using the rst order conditions of Agent 17s problem the equilibrium prices are 11 5 7939 You should also check that 1 K 3045 7 92 as in the previous problem Therefore Agent 1 is as well off with the current market structure as in the previous setup Agent 27s problem yields the Euler equation M03 Aq1 rju 203 Replacing for the equilibrium prices derived from Agent 17s problem this simpli es to 12 cg uZ j1 n Therefore with the new market structure Agent 2 is able to obtain full insurance from Agent 1 From the First Welfare Theorem which requires completeness of markets we know that the allocation prevailing under market Structure 2 is a Pareto optimal allocation It is your task to determine whether the allocation resulting from Structure 1 was Pareto optimal as well or not 633 General equilibrium under uncertainty multipleperiod model with two agent types How does the case of in nite number of periods differ from the two period case In general the conclusions are the same and the only difference is the additional complexity added through extending the problem We shortly summarize both structures As before Agent 1 is risk neutral and Agent 2 is risk averse Structure 1 one asset Agent 1 Agent 17s problem is max Z 3 7TZt ctzt zteZt t0 st 0112t alt12t 732 01t2t71 1042 Firm7s problem yields using Cobb Douglas production function rtzt ztakf 1zt 11i 6 1 7 04 My z 2 k zt l Market clearing condition is 01t12t 02 1ltZtgt kt12t First order condition wrt a1gtt1zt gives us Zt v 2t t 1 5 12 My Tt12t17 Z t1 gt1 Ezt1 zTt1 93 Using the formula for n1 frorn rrn7s rst order conditions we get 1 527TltZt1lztgt 2H1 39 04 39 k53121 1 i 5 n1 a kff11ztlz7r2t1lzt Zt1 51 5 n1 EZtllzt 1 16 39 KEQ gt kt1ltzt aE2t1 zt Agent 2 Agent 27s utility function is u02tzt and his rst order conditions yield u02tzt Eztmz u02t12t1 1 7 6 a 2H1 Using the above given Euler equation and KEQ together with Agent 27s budget con straint we can solve for 022t and a2gtt1zt Subsequently using the market clearing condition gives us the solution for 012t The conclusion is the same as in the two period case Agent 2 does not insure fully and his consurnption across states will vary Structure 2 Arrow securities Agent 1 The problem is very similar to the one in Structure 1 except for the budget constraint which is now Civ Z 1712 wind alt2t 1 wt2t j1 As we have more than one asset the no arbitrage condition has to hold It can expressed as 2 112 39ajt1zt kt1zt j1 ajt1zt 11 5 Tt12j7ztl kt121 i 1 Eggff 17 6 rt1zjzt j1 Solving the rst order condition of Agent 1 wrt alt1zt yields 273 2 7TZt 94 173 339 5 M271 APQ which is the formula for prices of the Arrow securities Agent 2 The rst order condition wrt ai zt yields 0 73412 41 39uCt22t 5H1 WW7 39uCEHW 2t Substituting APQ gives us 7139 2zt 0 5 2 5 u ltcltz gtgt 6 m 2 u c12j72 uC zth 0319 2t i 032 0119 2t This result follows from the assumption that gt 0 and u lt 07 and yields the same conclusion as in the two period case7 ie Agent 2 insures completely and his consumption does not vary across states 634 Recursive formulation The setup is the same as before Agent 1 is risk neutral and Agent 2 is risk averse We de note the agent type by superscript and the state of the world by subscript The stochastic process is a rst order Markov process Using the recursive formulation knowledge from before7 we use wealth denoted by w to differentiate it from wage7 which is denoted by w as the state variable More concretely7 there are three state variables individual wealth Lu7 the average wealth of risk neutral agents 17 and the average wealth of risk averse agents 2 The risk averse consumer7s problem is then W1ww1w2nggxu1w Z qijw1wza 6 Zn V a mi 7391 739 739 f wjGiwl7wz7ijwhwz wjGiwl7wz7 DiZjthz wjGiwl7wzl7 VF1 where a dfjw7w17w2 DMthz dljw17w17w27 V 7j7w17w2 Djw17w2 djw27w17w27 V 7j7w17w2 wahwz Zqijw1w2Dijw1w2 Dw1w2 V msz 739 From the rms problem7 we get the rst order conditions specifying the wage and the interest rate as wjk 2739 Fzk717V M 770 i 2739 39 j 95 Also the market clearing requires that Mm DigEma lt17 6 nGiwlmzv Gil61 1 The same holds for Agent 2 when we change the superscript in VFl from 1 to 2 The formulation is very similar to our previous formulation of recursive competitive equilibrium with some new unfamiliar notation showing up Clearly d is the law of motion for the asset holdings in state i of an asset paying off one unit of consumption good in future state j The aggregate law of motion for the asset holdings is denoted by ng for the risk neutral agents and D3 for the risk averse agents The average capital is denoted by Gi The following are the unknown functions V d j Digj gigJ Gib wj rj It is left as an exercise to identify the elements from the recursive formulation with elements from the sequential formulation 64 Appendix basic concepts in stochastic processes We will introduce the basic elements with which uncertain events are modelled The main mathematical notion underlying the concept of uncertainty is that of a probability space De nition 11 A probability space is a mathematical object consisting of three elements I a set 9 of possible outcomes w 2 a collection fof subsets of 9 that constitute the events to which probability is assigned a o algebra and 3 a set function P that assigns probability values to those events A probability space is denoted by 9 f P De nition 12 A o algebra f is a special kind offamily of subsets of a space 9 that satisfy three properties I Q E f 2 f is closed under complementation E E f i E0 E f 3 f is closed under countable union if is a sequence of sets such that E E f Vi then UfilEi E f De nition 13 A random variable is afunction whose domain is the set of events 9 and whose image is the real numbers or a subset thereof z Q a 3 For any real number 04 de ne the set Eawzw lt04 De nition 14 A function x is said to be measurable with respect to the o algebra f or f measurable if the following property is satis ed VaE EEa f 96 Conceptually if z is f rneasurable then we can assign probability to the event z lt 04 for any real number a We may equivalently have used gt S or 2 for the de nition of rneasurability but that is beyond the scope of this course You only need to know that if z is f rneasurable then we can sensibly talk about the probability of z taking values in virtually any subset of the real line you can think of the Borel sets Now de ne a sequence of o algebras as E 1 1 g E g g f Conceptually each U algebra ft re nes77 EA in the sense that distinguishes in a probabilistic sense between more events than the previous one Finally let a sequence of random variables pt be ft rneasurable for each t which models a stochastic process Consider an to E Q and choose an 04 6 3 Then for each t the set Em E w x w lt 04 will be a set included in the collection the o algebra ft Since E Q f for all t Em also belongs to f Hence we can assign probability to Eat using the set function P and P Em is well de ned Example 17 Consider the probability space 9 f P where o 9 0 1 o f B the Borel sets restricted to 0 o P A the length of an interval Aa bl b 7 a Consider the following collections of sets 39 39 1 2 2t 71 ill vliv ll 2t 2t 2 70 For every t let E be the minimum U algebra containing At Denote by a At the col lection of all possible unions of the sets in A notice that Q E 0At Then ft 0 At a At you should check that this is a o algebra For epample A1071l7 7 07 it i7 1 XE l071l707 0 7 A2 07 it i7 7 B72 e0012 l07 7 07 2 i7 2 i7 1l7 l 1l7 07 EU 7 2U um i u a 1i 07 i u a 1i 07 a u a 1 7 ii a u a 1 7 07 ll Now consider the eccperiment of repeated fair coin flips ct E 0 1 The in nite sequence ct0 is a stochastic process that can be modeled with the probability space and associated sequence ofo alqebras that we have de ned above Each sequence ct0 is an outcome represented by a numberw E 9 NH 97 For every t let yt cgBiL this will be a t dimensional vector of zeros and ones and to each possible con guration of yt there are 2 possible ones associate a distinct interval in At For epample for t 1 and t 2 let 1110 1 0 O u n n u n n u 19 MH NH O Nib A A H O O H H O 999 H O VVVVVV H Munch NH H NIH t In 7 Fort 3 we will have a three coordinate vector and we will have the following restrictions on 3 T 40 40 H HOHO VVVV T T T A A A A H O u Muncme H MmMHtMH I and so on for the following t Then a numberw E 9 implies a sequence of intervals It0 that represents for every t the partial outcome realized that far Finally the stochastic process will be modeled by a function it that for each t and for each w E Q associates a real number such that it is ft measurable For epample take w 7 and w 8 then 1 1 E 1 that is the rst element of the respective sequences cg c is a 1 say quotHeads It holds that we must have 1 w 1 LON E b We are now ready to answer the following question What is the probability that the rst toss in the eccperiment is Heads Or in our model what is the probability that z1w b To answer this question we look at measure of the set ofw that will produce the value 1 w b Ewz1wb EA Efl The probability of the event E 1 is calculated using P 1 A 1 5 Th at is the probability that the event ct1 to be drawn produces a Head as its rst toss is De nition 15 Let B E 7 Then the joint probability of the events xt tn E B is given by Pt1yvvtnB P w E Q t1w7 tn E B De nition 16 A stochastic process is stationary if BHVVWHAB is independent of t Vt Vn VB 98 Conceptually7 if a stochastic process is stationary7 then the joint probability distribu tion for any xt tn is independent of time Given an observed realization of the sequence xiE in the last 5 periods pkg xi at757 at7 the conditional probability of the event xt tn E B is denoted by Pt1tn B lxtis 0797 7 1 atl De nition 17 A rst order Markov Process is a stochastic process with the property that Pt1tn B lxtis 0797 7 1 ail Pt1tn B l t atl De nition 18 A stochastic process is weakly stationary or covariance stationary if the rst two moments of the joint distribution of zt17 tn are independent of time A usual assumption in macroeconomics is that the exogenous randornness affecting the economy can be modelled as a weakly stationary stochastic process The task then is to look for stochastic processes for the endogenous variables capital7 output7 etc that are stationary This stochastic stationarity is the analogue to the steady state in deterrninistic rnodels Example 18 Suppose that productivity is subject to a two state shock y 2Fk z E 2L72H Imagine for eccample that the 2 s are iid with Pr 2 2H Pr 2 2L Vt The policy function will now be afunction of both the initial capital stock K and the realization of the shock z ie 9 k7 z E 9 k7 2L 7 9 k7 VK We need to nd the functions 9 k7 Notice that they will determine a stochastic process for capitalie the trajectory of capital in this economy will be subject to a random shock The Figure 64 shows an eccample of such a trajectory 99 k high shock capital line low shock capital line I k ergodic set 3 Figure 64 Stochastic levels of capital The interval AB is the ergodic set once the level of capital enters this set7 it will not leave it again The capital stock will follow a stationary stochastic process within the limits of the ergodic set 100 Chapter 7 Overlappinggenerations model 71 Welfare in models with multiple agents We are interested in analyzing the neoclassical growth model in an environment with mul tiple agents Our objective will be to study the ef ciency properties of competitive equi librium under such setups Will the introduction of several7 possibly non homogeneous7 agents substantially modify the welfare properties of equilibria Heterogeneity among agents can be modelled by assuming agents with different pref erences7 or by endowing them with varying amounts of initial wealth Another way of thinking of a non homogeneous population is by using the overlapping generations7 ap proach7 as opposed to the dynastic approach we have been dealing with so far The dynastic model has all its agents born in period t 07 thereafter to live during all the periods modelled possibly in nite time On the contrary7 the overlapping generations scheme assumes that new agents are born into the economy each period7 and these indi viduals7 life spans are shorter than the economy7s time horizon The overlapping generations approach seems more realistic7 and raises issues that the dynastic model does not address7 such as why agents would die leaving non consumed savings behind ln fact7 a vast literature on the inter generational game has been devel oped7 trying to explain the motives for bequests7 and studying the impact of this game on the dynamic properties of the economy 711 De nitions and notation In what follows7 we will introduce some general de nitions By assuming that there is a nite set H of consumers and7 abusing notation slightly7 let H be an index set7 such that H E cardU O7 we can index individuals by a subscript h l7 H So H agents are born each period t and they all die in the end of period t l Therefore7 in each period t the young generation born at t lives together with the old people born at t 7 1 Let c t 239 denote consumption at date t of agent h born at It usually we say of generation It 7 and we have the following De nition 19 A consumption allocation is a sequence a ltc lttgtcfltt1gtgtham t0 U 3110h6H 101 A consumption allocation de nes consumption of agents of all generations from t 0 onwards7 including consumption of the initial old7 in the economy Let ct E ZheH cft clLlt denote total consumption at period t7 composed of the amount c t consumed by the young agents born at t7 and the consumption cf1t enjoyed by the old agents born at t 7 1 Then we have the following De nition 20 A consumption allocation is feasible if ct S Yt Vt Example 19 Storage economy Assume there is quotintertemporal production mod elled as a storage technology whereby investing one unit at t yields 7 units at t 1 In this case the application of the previous 1 f 39 39 reads r feasible in this economy if there eccists a sequence Ktf0 such that g is Ct Kt 1 S Ya K007 Vt where Yt is an endowment process Example 20 Neoclassical growth model Let Lt be total labor supply at t and the neoclassical function Yt represent production technology Wt F KW Ltl Capital is accumulated according to the following law of motion Kt 1 1 76 It Then in this case regardless of whether this is a dynastic or an overlapping generations setup we have that a consumption allocation is feasible if there epists a sequence It0 such that ct t S F Kt7 Lt Vt The de nitions introduced so far are of physical nature they refer only to the material possibility to attain a given consumption allocation We may also want to openjudgement on the desirability of a given allocation Economists have some notions to accommodate this need7 and to that end we introduce the following de nition De nition 21 A feasible consumption allocation c is ef cient if there is no alternative feasible allocation E such that at 2 ct Vt7 and at gt ct for some t An allocation is thus deemed ef cient if resources are not wasted that is7 if there is no way of increasing the total amount consumed in some period without decreasing consumption in the remaining periods The previous de nition7 then7 provides a tool for judging the desirability of an allocation according to the aggregate consumption pattern The following two de ni tions allow an extension of economists7 ability to assess this desirability to the actual distribution of goods among agents 102 De nition 22 A feasible consumption allocation cA is Pareto superior to cB or cA quotPareto dominates CB if 1 No agent strictly prefers the consumption path speci ed by cB to that speci ed by CA CA Eh CB E HVt 2 At least one agent strictly prefers the allocation cA to cB 39 E Hi CA gtijB Notice that this general notation allows each agent7s preferences to be de ned on other agents7 consumption as well as on his own However in the overlapping generations model that we will study the agents will be assumed to obtain utility or disutility only from their own consumption Then condition for Pareto domination may be further speci ed De ne cf cft c t 1 if t 2 0 and cf cllt1 otherwise Pareto domination condition reads 1 No agent strictly prefers hisher consumption path implied by cB to that implied by cA 0A a CB Vh e H Vt 2 At least one agent strictly prefers the allocation cA to cB E Ht 3 CA gtijBtl Whenever CE is implemented the existence of cA implies that a welfare improvement is feasible by modifying the allocation Notice that a welfare improvement in this context means that it is possible to provide at least one agent and potentially many ofthem with a consumption pattern that he will nd preferable to the status quo while the remaining agents will nd the new allocation at least as good as the previously prevailing one Building on the previous de nition we can introduce one of economists7 most usual notions of the most desirable allocation that can be achieved in an economy De nition 23 A consumption allocation c is Pareto optimal if39 1 It is feasible 2 There is no other feasible allocation 37 c that Pareto dominates c Even though we accommodated the notation to suit the overlapping generations framework the previous de nitions are also applicable to the dynastic setup In what follows we will restrict our attention to the overlapping generations model to study the ef ciency and optimality properties of competitive equilibria You may suspect that the fact that agents7 life spans are shorter than the economy7s horizon might lead to a differ ent level of capital accumulation than if agents lived forever In fact a quite general result is that economies in which generations overlap lead to an overaccumulation of capital This is a form of dynamic inef ciency since an overaccumulation of capital implies that the same consumption pattern could have been achieved with less capital investment 7 hence more goods could have been freed up77 to be consumed In what follows we will extend the concept of competitive equilibrium to the overlap ping generations setup We will start by considering endowment economies then extend the analysis to production economies and nally to the neoclassical growth model 103 712 Endowment economy We continue to assume that agents of every generation are indexed by the index set H Let w t i denote the endowment of goods at ti of agent h born at t Then the total endowment process is given by h h Wt 2 wt t waxw heH We will assume throughout that preferences are strongly monotone which means that all inequality constraints on consumption will bind Sequential markets We assume that contracts between agents specifying one period loans are enforceable and we let Rt denote the gross interest rate for loans granted at period t and maturing at t 1 Then each agent h born at t 2 0 must solve max a C1 c2 71 0102 c1 l S aunt 62 3 am 1 1 Rt and generation 71 trivially solves h h 0 72 Milled l C210 3 120 Unlike the dynastic case there is no need for a no Ponzi game restriction In the dynastic model agents could keep on building debt forever unless prevented to do so But now they must repay their loans before dying which happens in nite timel De nition 24 A competitive equilibrium with sequential markets is a consump tion allocation c and a sequence R E Rt0 such that Z c t c t 1 solve generationt s agent h 71 problem and cljl0 solves pro 2 Market clearing is satis ed E ectiuely we need only to require the credit market to be cleared and Walras law will do the rest due to feasibility of c thh0Vt0oo heH In the initial setup of the model the agents were assumed to live for two periods Because of this no intergenerational loan can be ever paid back either a borrower or a lender is simply not there next period Therefore there is no intergenerational borrowing in the endowment economy 1Notice that in fact both the no Ponzi game and this paybefore you die restrictions are of an institutional nature and they play a key role in the existence of an intertemporal market 7 the credit market 104 ArrowDebreu date0 markets In this setup we assume that all future generations get together at date t 71 in a futures market and arrange delivery of consumption goods for the periods when they will live2 The futures market to be held at t 71 will produce a price sequence pt0 of future consumption goods Then each consumer knowing in advance the date when he will be reborn to enjoy consumption solves max a c1 c2 73 0102 st p ap heSp iw p Dwd D whenever his next life will take place at t 2 0 and the ones to be born at t 71 will solve max ugc 74 st pWWCSP Ww wl De nition 25 A competitive equilibrium with ArrowDebreu date0 markets is a consumption allocation c and a sequence p E 10tt0 such that Z cft cft 1 solve generationt s agent h problem and cljl0 solves 74 problem 2 Resource feasibility is satis ed markets clear Claim 1 The de nitions of equilibrium with sequential markets and with Arrow Debreu date 0 trading are equivalent Moreover if c p is an Arrow Debreu date 1 trading equilibrium then c R is a sequential markets equilibrium where Rt 1 75 Proof Recall the sequential markets budget constraint of an agent born at t 7 h c1 l 7 wt t c2 w t 1 l Rt where we use the strong monotonicity of preferences to replace the inequalities by equal ities Solving for l and replacing we obtain h c2 h wt t 1 c 7 w t 2You may assume that they all sign their trading contracts at t 71 thereafter to die immediately and be reborn in their respective periods 7 the institutional framework in this economy allows enforcement of contracts signed in previous livesi 105 Next recall the Arrow Debreu date 0 trading budget constraint of the same agent W 61 pt 1 62 pt WNW pt 1 wlt 1 Dividing through by pt we get W 1 h N 1 h ic2w t7w t1 W t W As can be seen with the interest rate given by 75 the two budget sets are identical Hence comes the equivalence of the equilibrium allocations l 01 An identical argument shows that if c R is a sequential markets equilibrium then c p is an Arrow Debreu date 0 trading equilibrium where prices pt are determined by normalizing p0 p0 usual normalization is p0 1 and deriving the remaining ones recursively from p t pt 1 7 R Remark 1 The equivalence of the two equilibrium de nitions requires that the amount of loans that can be drawn l be unrestricted that is that agents face no borrowing constraints other than the ability to repay their debts The reason is that we can switch from c1 l wtht 62 wft1lRt w m 1 C c1 i who R Mt only in the absence of any such restrictions Suppose instead that we had the added requirement that l 2 b for some number b such h wt t 1 1W 76 that b gt 7 In this case I and 76 would not be identical any more3 Application Endowment economy with one agent per generation We will assume that H 1 therefore agents are now in fact indexed only by their birth dates and that for every generation t 2 0 preferences are represented by the following utility function ut Cy c0 log cy log c0 Similarly the preferences of generation t 71 are represented by utility function 1L1 c logo 3 wt 1 lf b 7R7 then this is Just the paybefore you die restriction implemented in fact by h t 1 nonnegativity of consumption Also if b lt 7 then l 2 b would never bind for the same reason 106 The endowment processes are given by wtt wtt 1 wy 4 for all t Trading is sequential and there are no borrowing constraints other than solvency Agent t 2 0 now solves max log Cy log 00 Cygt00 We can substitute for co to transform the agents problem into cy wy mCaX log Cy log wy 7 Cy Rt Taking rst order conditions yields 1 Rt T w 07 y Lug K 7 Cy Rt Cy wy K 7 Cy Then from rst order condition and budget constraint we get ww Wy 1W W0 39 Cy lewlw co Market clearing and strong monotonicity of preferences require that the initial old consume exactly their endowment 010 4 Therefore using the feasibility constraint for period t 0 that reads C00 0710 Wy W07 follows 000 LUZ4 Repeating the market clearing argument for the remaining t since 000 wy will imply 001 4 we obtain the following equilibrium allocation Vt Mt Ctt 1 4Notice that the same result follows from clearing of the loans market at t 0 lo 0 This together with 000 10 my implies the same period 0 allocationi My 4 107 Given this allocation we solve for the prices Rt that support it You may check that these are Rt 21 This constant sequence supports the equilibrium where agents do not trade they just consume their initial endowments Let us now use speci c numbers to analyze a quantitative example Let wy 3 wo 1 This implies the gross interest rate of Rt The net interest rate is negative 2 rt E Rt71 g The natural question hence is whether the outcome Rt is a ef cient and b optimal a Ef ciency Total consumption under the proposed allocation is Ct 4 which is equal to the total endowment It is not possible to increase consumption in any period because there is no waste of resources Therefore the allocation is e cient b Optimality To check whether the allocation is optimal consider the following alternative allocation 340 2 am 2 t 1 2 That is the allocation Eis obtained from a chain of intergenerational good transfers that consists of the young in every period giving a unit of their endowment to the old in that period Notice that for all generations t 2 0 this is just a modi cation of the timing in their consumption since total goods consumed throughout their lifetime remain at 4 For the initial old this is an increase from 1 to 2 units of consumption when old It is clear then that the initial old strictly prefer Eto c We need to check what the remaining generations think about the change It is clear that since utility is concave the log function is concave this even split of the same total amount will yield a higher utility value In fact ut log2 log2 2 log2 log4 gtlog3log1log3 U 0 Therefore EPareto dominates c which means that c can not be Pareto optimal Suppose instead that the endowment process is reversed in the following way LUZ1 w03 108 There is the same total endowment in the economy each period but the relative assign ments of young and old are reversed From the formula that we have derived above this implies Rt 3 The no trade77 equilibrium where each agent consumes his own endowment each period is ef cient again since no goods are wasted ls it Pareto optimal This seems a dif cult issue to address since we need to compare the prevailing allocation with all other possible allocations We already know that an allo cation having 2 2 will be preferred to 1 3 given the log utility assumption However is it possible to start a sequence of intergenerational transfers achieving consumption of Cy co from some t 2 0 onwards while keeping the constraints that all generations receive at least log3 units of utility throughout their lifetime some generation is strictly better off and the initial old consume at least 3 units If any of these constraints is vi olated the allocation thus obtained will not Pareto dominate the no trade77 allocation It is left to you to check that the answer to this question is No Notice that in analyzing Pareto optimality we have restricted our attention to sta tionary allocations Let us introduce a more formal de nition of this term De nition 26 Stationary allocation A feasible allocation c is called stationary if Vt 0N 027 ctt1 00 With this de nition at hand we can pose the question of whether there is any station ary allocation that Pareto dominates 2 2 Figure 71 shows the resource constraint of the economy plotted together with the utility level curve corresponding to the allocation 2 2 Figure 71 Pareto optimality of 2 2 allocation 109 The shaded area is the feasible set7 its frontier given by the line Cy 00 4 It is clear from the tangency at 27 2 that it is not possible to nd an alternative allocation that Pareto dominates this one However7 what happens if we widen our admissible range of allocations and think about non stationary ones Could there be a non stationary allocation dominating 27 2 In order to implement such a non stationary allocation7 a chain of inter generational transfers would require a transfer from young to old at some arbitrary point in time t These agents giving away endowment units in their youth would have to be compen sated when old The question is how many units of goods would be required for this compensation Figure 72 lmpossibility of Pareto improvement over 27 2 allocation Figure 72 illustrates that7 given an initial transfer 81 from young to old at t the transfer 82 required to compensate generation It must be larger than 61 given the concave utility assumption This in turn will command a still larger 63 and so on ls the sequence at0 thus formed feasible An intuitive answer can be seen in the chart no such transfer scheme is feasible in the long run with stationary endowment process Therefore7 for this type of preferences the stationary allocation 27 2 is the Pareto optimal allocation Any proposed non stationary allocation that Pareto dominates 27 2 becomes unfeasible at some point in time Somewhat more formally7 let us try to use the First Welfare Theorem to prove Pareto optimality Notice that our model satis es the following two key assumptions 1 Preferences exhibit local non satiation since u is strictly increasing 2 The market value of all goods is nite we will come back to this Proof Pareto optimality of competitive equilibrium Let an economy7s pop ulation be indexed by a countable set I possibly in nite7 and consider a competitive equilibrium allocation x that assigns xi to each agent 239 might be multi dimensional 110 If x is not Pareto optimal then there exists E that Pareto dominates s that is a feasible allocation that satis es l ii 3j61 j gtj7x Then we can use local non satiation to show that 10 39 ii 2 10 39 13 p f gt p must hold Summing up over all agents we get Zp39fi gt przv id 61 61 61 The last inequality violates the market clearing condition since the market value of goods with local non satiation must be equal to the market value of endowments in an equilibrium You may observe that this proof is in fact an application of the separating hyperplane theorem I This proof is quite general In the speci c case of overlapping generations we have the following two peculiarities 1 p and z are in nite dimensional vectors 2 There is an in nity of consumers Therefore the series 2p f and 2p s might take in nite value in which case the l I 1 last comparison in the6 proof mightenot hold We need to specify further conditions to ensure that the rst welfare theorem will hold even with the correct77 assumptions on preferences To this effect let us assume that the following conditions are met by the economy H Regularity conditions on utility and endowments E0 Restrictions on the curvature of the utility function 7 that has to be somewhat77 curved but not too much An example of curvature measure is one over the elasticity of intertemporal substitution f z f 96 5This ratio is also called the coef cient of relative risk aversion whenever the environment involves uncertainty 1n the expected utility framework the same ratio measures two aspects of preferences intertemporal comparison and degree of aversion to stochastic variability of consumption 111 3 Other technical details that you may nd in Balasko and Shell 1980 but that are beyond the scope of this course Then we have the following Theorem 4 Balasko and Shell Journal of Economic Theory 1980 A compet itive equilibrium in an endowment economy populated by overlapping generations of agents is Pareto optimal if and only if M8 i700 19007 7 t H o where pt denote Arrow Debreu prices for goods delivered at time t Recall our example The allocation 2 2 implied Rt 1 and from the equivalence of sequential and Arrow Debreu date 0 trading equilibria we have that which implies t0 t1 In the case of 3 1 we have 19 t 3 p0 Then 00 00 t 00 1 3 1 1 1 gm gm gt 720 2 pm lt 0 And nally for 1 3 1 7 31 2 pm pltogt Therefore by applying the theorem we conclude that 2 2 and 1 3 are Pareto optimal allocations whereas 3 1 can be improved upon which is the same conclusion we had reached before So what if the economy in question can be represented as 3 1 type of situation How can a Pareto improvement be implemented Should the government step in and if so how A possible answer to this question is a pay as you go type of social security system that is used in many economies worldwide But a distinct drawback of such a solution is the forced nature of payments when social security becomes social coercion ls there any way to implement Pareto superior allocation with the help of the market One of the solutions would be to endow the initial old with intrinsically useless pieces of paper called money lntuitively if the initial old can make the young in period t 0 believe that at time t 1 the next young will be willing to trade valuable goods for these pieces of paper a Pareto improvement can be achieved relying solely on the market forces We will examine this issue in the following section in greater detail 112 713 Economies with interternporal assets In the previous section we have looked at overlapping generations economies in which only consumption goods are traded A young agent selling part of his endowment to an old one obviously needs something which serves the purpose of a storage of value so that the proceeds from the sale performed at time t can be used to purchase goods at t 1 A unit of account is therefore implicit in the framework of the previous section which is obvious from the moment that such thing as prices77 are mentioned However notice that such units of account are not money they exist only for convenience of quoting relative prices for goods in different periods We will now introduce intertemporal assets into the economy We will consider in turn at money and real assets Economies with at money In this section we introduce at77 money to the economy To this end any paper with a number printed on it will ful ll the need of value storage provided that everybody agrees on which are the valid papers and no forgery occurs We have assumed away these details agents are honest As before consider an overlapping generations economy with agents who live for two periods one agent per generation An endowment process is given by wttwtt 1 wyw0 W The preferences will once again be assumed to be logarithmic ut Cy co log Cy log 00 W In contrast to the previous setup let the initial old be endowed with M units of at currency A natural question to address is whether money can have value in this economy A bit of notation let 10m denote a value of a unit of money at time t in terms of consumption goods at time t Also let pt E 7 be price level77 at time t that is the mt price of a unit of consumption goods at time t in terms of money Notice the difference between pt in this model and Arrow Debreu date 0 prices denoted pt Assume for the moment that pt lt 00 Then the maximization problem of generation It agent is max log Cy log 00 77 0ycoMt1 Mm Cy 7 Wyv Pt Mt1 C0 W0 7 Pt1 Mt1 Z 0 And the agent of generation 71 trivially solves 1 0 0g61 113 M 010 wo A 100 The meaning of the last constraint in 77 is that agents cannot issue money7 or7 budget constraint of an agent born at period t is alternatively to sell it short Combining the constraints from 7 7 the consolidated co 4 Cy Pt wy 10 7 Pt1 10t1 real return on money is yicyZO The budget set under these constraints is presented in Figure 73 As can be seen7 the t T 2 77311 for small values of 7Tt1 Co Here 7311 denotes in ation rate From rst 1 1 7Tt1 order Taylor approximation it follows that net return on one dollar invested in money is Figure 73 Budget set in the economy with at money Momentarily ignore Aug 7 Cy 2 0 Then the solution to 77 is 1 Pt1gt 7 w w07 7 2lt y Pt 1 ltwyw0 Pt1gt Pt Pt 10t139 Having found Cy we can recover the real demand for money of the young at t M11 1 Pt1 7 7 c 7w in 7 Pt y y 2 y 2 0 10 lmposing market clearing condition on the money market7 Mt1 MV 12 114 we can recover the law of motion for prices in this economy M 1 1 Pt1 741 70 7w77w7 Pt y y 2y 20 10 Aug 2M 10t110t 7 wo wo Consider the following three cases u oilgt1 wo u oil1 wo u offlt1 W0 Solution to this rst order difference equation is presented graphically on the Figure 74 Figure 74 Dynamics of price level As can be seen7 the only case consistent with positive and nite values of pt is the rst one7 when wy gt 4 The following solutions can be identi ed 1 If wy gt wo we can observe the following there exists a solution pt 16 gt 0 So7 money can have real value a Money can overcome suboptimality77 when wy gt wo and consumption level is w in constant Cy co L since 1 implies that MRS 17 and the resulting allocation is Pareto optimal by Balasko Shell criterion 115 b There is no equilibrium with p0 lt 1 which means that one unit of money at t 0 has value at most 10 c If p0 gt 1 there is an equilibrium7 which is the solution to Aug 2M 10t1 i 10 i 7 wo wo with p0 given In this equilibrium7 pt a 00 pm a 07 and E increases Pt monotonically to E This is an equilibrium with hyperin ation Money loses value in the limit 0 d pmo 0 pt 00 is also an equilibrium So7 there is a continuum of equilibria The fact that money has value may be seen as a rational bubble what people are willing to pay77 for money today depends on what they expect others will pay77 for it tomorrow The role of money here is to mitigate the suboptimality present in the economy It is the suboptimality that gives money positive value If we add borrowing and lending opportunity7 we get from arbitrage condition and loans market clearing Pt R 7 Pt1 So7 real interest rate is non positive7 and real money holdings are still present 1t 0 Vt E0 If Aug 3 wo there is no equilibrium with pt lt oo However7 autarky7 pt 007 is still an equilibrium Economies with real assets In this subsection we will consider the assets that are real claims7 rather than at money That is7 they will be actual rights to receive goods in the following periods Two different kinds of assets are of interest A tree that produces a given fruit yield dividend each period Capital7 that can be used to produce goods with a given technology A tree economy We assume that the economy is populated by one agent per generation7 and that each agent lives for two periods Preferences are represented by a logarithmic utility function as in previous examples it t 7 t if U Cy co 7 logcy logco Agents are endowed with LUZ7100 consumption units fruits when young and old7 respectively7 and there is also a tree that produces a fruit yield of d units each period Therefore total resources in the economy each period are given by Yt wyw0d 116 Ownership of a given share in the tree gives the right to collect such share out of the yearly fruit produce Trading of property rights on the tree is enforceable7 so any agent that nds himself owning any part of the tree when old will be able to sell it to the young in exchange for consumption goods The initial old owns 100 of the tree Let 1HL denote the share of the tree purchased by the young generation at t7 and pt denotes the price of the tree at t It is clear that asset market clearing requires 1HL 1 for all t Generation t consumer solves max log ct log cf ct ct y y 0 Pt at1 C Wm cf we at1pt1 d Notice that the returns on savings are given by Pt1 d Pt 39 The rst order conditions yield 1 Pt ct7 w7wu0gt7 y 2 y Pt1d which implies that generation t7s savings satisfy p a 7141 pt w t tlii 739 0 2 y Pt1d lmposing the market clearing condition and rearranging we get the law of motion for prices W0 Pt1 T 7 d 7 2 Pt This is a rst order non linear difference equation in pt Figure 75 shows that it has two xed points7 a stable negative one and an unstable positive one What is the equilibrium pt1 sequence It must be a constant sequence since any deviation from the positive xed point leads directly into the negative one or creates a bubble that eventually collapses due to infeasibility So7 pt 10 Vt where 10 is the positive solution to in P1 at d6 p 6Notice that for the case d 0 we are back in at money economy7 and the constant positive value fch1i of money is once again pmt P 117 Figure 75 Fixed points for price of the tree ls this competitive equilibrium Pareto optimal We can answer this question by checking whether the Balasko Shell criterion is satis ed First notice that if we multiply ti1 t72 10 by p plt plt plt we can write pt 1 pt 2 p1 p0 1 Jae1gtpltte2gtuplt1gtpltogtH m7 Wpt1p1p0 igRs Hh where 100 E 1 and RSJH denotes the interest rate between periods 5 and s 1 198 RSS E H ps L W But we already know that the return on savings is given by Pt1 d Pt 39 Therefore the interest rate for each period using equilibrium prices is 19 d Rss1 pak Replacing for i we get that W 00 1 00 d t gmp01E The limit of this series is in nity for any d 2 0 The Balasko Shell criterion is met hence the competitive equilibrium allocation supported by these prices is Pareto optimal 118 Finally notice that the optimality of the result was proven regardless of the actual endowment process therefore it generalizes for any such process Now consider two cases of economies with production a simple model with CRS technology that uses only capital and a more complicated neoclassical growth model Storage economy We will assume the simplest form of production namely constant marginal returns on capital Such a technology represented by a linear function of capital is what we have called storage77 technology whenever no labor inputs are needed in the production pro cess Let the yield obtained from storing one unit be equal to one That is keeping goods for future consumption involves no physical depreciation nor does it increase the physical worth of the stored goods Let the marginal rates of substitution between consumption when old and when young be captured by a logarithmic function as before and assume that the endowment process is Aug 4 3 1 Generation t7s problem is therefore max log ct log cf ct ct y y o t 5tcyiwy t coistw0 Ct l 411 y 2 y Rt39 The return on storage is one B 1 So using the values assumed for the endowment process this collapses to The rst order conditions yield t Cy 2 t 7 co 7 2 st 1 Notice that the allocation corresponds to what we have found to be the Pareto optimal allocation before 2 2 is consumed by every agent In the previous case where no real intertemporal assets existed in the economy such an allocation was achieved by a chain of intergenerational transfers enforced if you like by the exchange in each period of those pieces of paper dubbed at money Now however agent buries his potato when young and consumes it when old ls the current allocation Pareto optimal The answer is clearly no since to achieve the consumption pattern 2 2 the potato must always be buried on the ground The people who are born at t 0 set aside one unit of their endowment to consume when old and thereafter all their descendance mimic this behavior for a resulting allocation 6 1 u 2 2o 119 However7 the following improvement could be implemented Suppose that instead of storing one7 the rst generation It 0 consumed its three units when young In the following period the new young would give them their own spare unit7 instead of storing it7 thereafter to continue this chain of intergenerational transfers through in nity and beyond The resulting allocation would be a 1 u 3 2 U 2 217 a Pareto improvement on 0 ln fact7 Eis not only a Pareto improvement on c but simply the same allocation 0 plus one additional consumption unit enjoyed by generation 0 Since the total endowment of goods is the same7 this must mean that one unit was being wasted under allocation 0 This problem is called overaccumulation of capital The equilibrium outcome is dynamically inef cient Neoclassical growth model The production technology is now modelled by a neoclassical production function Capital is owned by the old7 who put it to production and then sell it to the young each period Agents have a labor endowment of Aug when young and wo when old Assuming that leisure is not valued7 generation t7s utility maximization problem is t t ma ut Cy co C C y o t 7 cyst flag10 Co 5t TH1 Wo wt1 If the utility function is strictly quasiconcave7 the savings correspondence that solves this problem is single valued 5 h wt7 Tt17 wt1l The asset market clearing condition is St Kt1 We require the young at tto save enough to purchase next period7s capital stock7 which is measured in terms of consumption goods the price of capital in terms of consumption goods is 1 The rm operates production technology that is represented by the function FK7 Market clearing condition for labor is 71 wy w0 From the rms rst order conditions of maximization7 we have that factor remuner ations are determined by 7 F1Kt7wywo7 wt F2Ktwyw0 120 If we assume that the technology exhibits constant returns to scale we may write K FltKngt 77117 71 K K where f E Fi 1 Replacing in the expressions for factor prices 71 n r f Kt t wyw0 7 w f Kt i K f Kt t wyw0 wyw0 wyw0 39 denote the capitallabor ratio If we normalize wyw0 1 we have Kt Let kt E wy wo that K kt Then 7 fkt7 wt fltkt kt fktl Substituting in the savings function and imposing asset market equilibrium km h fkt 7 kt fkt7 fkt7 fltkt1 7 kt1 fkt1l We have obtained a w order difference equation Recall that the dynastic model lead to a second order equation instead However proving convergence to a steady state is usually more dif cult in the overlapping generations setup Recall that the steady state condition with the dynastic scheme was of the form 5 f V 1 ln this case steady state requires that W 7 h WW 7 W f k7 f k7 f0 7 W f kl 714 Dynamic ef ciency in models with multiple agents We have analyzed the welfare properties of consumption allocations arising from a mul tiple agent environment under the form of a population consisting of overlapping genera tions of individuals The purpose of this section is to generalize the study of the dynamic ef ciency of an economy to a wider range of modelling assumptions In particular we will present a theorem valid for any form of one sector growth model We assume that the technology is represented by a neoclassical production function that satis es the following properties f0 0 f gt 0 121 f lt 0 f 6 OZ 02 denotes the space of twice continuously differentiable functions7 lirr f m 00 lim f 0 Notice that since we de ne fx E F1 1 7 6 x the last assumption is not consistent with the case of 6 lt 1 This assumption is implicit in what follows Then we can show the following Theorem 5 A steady state I is e cient if and only if P E f k 2 1 lntuitively7 the steady state consumption is 0 fk 7 1 Figure 76 shows the attainable levels of steady state capital stock and consumption If7 07 given the as sumptions on f The k0 CG locus corresponds to the golden rule77 level of steady state capital and consumption7 that maximize CG 1 031 f klt1 7 Ef cient Inef cient I W k Figure 76 Ef ciency of the steady state Proof i Pi lt 1 is inef cient Assume that I is such that f k lt 1 Let 0 denote the corresponding level of steady state consumption7 let co 0 Now consider a change in the consumption path7 whereby k1 is set to k1 I 7 8 instead of k1 1 Notice this implies an increase in 00 Let kt k1 Vt 2 1 We have that 0170 fk1k1 fkk E fk7e7k8fkk 122 Notice that strict concavity of f implies that fk lt fk 8 W i W i 8l f k 8 for 8 E 0 k 7 kc and we have that f k 7 8 lt 1 Therefore fk lt fk 7 8 k 7 k 7 8 This implies that 01 7 0 gt 0 which shows that a permanent increase in consumption is feasible ii R 2 1 is ef cient Suppose not then we could decrease the capital stock at some point in time and achieve a permanent increase in consumption or at least increase consumption at some date without decreasing consumption in the future Let the initial situation be a steady state level of capital k0 k such that f k 2 1 Let the initial 00 be the corresponding steady state consumption 00 0 fk 7 k Since we suppose that k is inef cient consider a decrease of capital accumulation at time 0 k1 k 7 81 thereby increasing 00 We need to maintain the previous consumption pro le 0 for all t 2 1 0t 2 0 This requires that 61 7 Wu 7 kg 2 mm 7 1a C7 k2 S 1 7 fk k k2 7 W S fk1 7 fk Concavity of f implies that fk1 f0 lt WW 39 61 7 Vl T Notice that 82 E kg 7 k lt 0 Therefore since f k 2 1 by assumption we have that 1821 gt 1811 The size of the decrease in capital accumulation is increasing By induction QB is a decreasing sequence of negative terms Since it is bounded below by 7k we know from real analysis that it must have a limit point 800 E 7k 0 Consequently the consumption sequence converges as well 000fk76007k7800 It is straightforward to show using concavity of f that 000 lt 0 Then the initial increase in consumption is not feasible if the restriction is to maintain at least 0 as the consumption level for all the remaining periods of time I We now generalize the theorem dropping the assumption that the economy is in steady state 123 Theorem 6 Dynamic e iciency With possibly nonstationary allocations Let both kt0 and the associated sequence Rt kt E f ktf0 be uniformly bounded above and below away from zero Let 0 lt a 3 7ft kt S M lt 00 Vt th Then kt0 is e cient if and only if 2 R5 1 00 t0 91 Recall that 00 t 00 1 Z 1112 as e t0 91 t0 pt The Balasko Shell criterion discussed when studying overlapping generations is then a special case of the theorem just presented 72 Welfare theorems in dynastic and CG models From our discussion so far we can draw the following summary conclusions on the ap plicability of the rst and second welfare theorems to the dynamic economy model First Welfare Theorem H Overlapping generations Competitive equilibrium is Lot always Pareto optimal Sometimes it is not even ef cient E0 Dynastic model Only local non satiation of preferences and standard assumption 6 lt 1 are required for competitive equilibrium to be Pareto optimal Second Welfare Theorem H Overlapping generations In general there is no applicability of the Second Welfare Theorem E0 Dynastic model Only convexity assumptions are required for any Pareto optimal allocation to be implementable as a competitive equilibrium Therefore with the adequate assumptions on preferences and on the production tech nology the dynastic model yields an equivalence between competitive equilibrium and Pareto optimal allocations Of course the restrictions placed on the economy for the Second Welfare Theorem to apply are much stronger than those required for the First one to hold Local non satiation is almost not an assumption in economics but virtually the de ning characteristic of our object of study recall that phrase talking about scarce resources etcetera In what follows we will study the Second Welfare Theorem in the dynastic model To that effect we rst study a 1 agent economy and after that a 2 agents one 124 721 The second welfare theorem in a 1agent economy We assume that the consumers preferences over in nite consumption sequences and leisure are represented by a utility function with the following form U Ctv lt0l Z t 0th where 0 lt B lt l and the utility index n is strictly increasing and strictly concave For simplicity leisure is not valued This is a one sector economy in which the relative price of capital in terms of con sumption good is 1 Production technology is represented by a concave homogeneous of degree one function of the capital and labor inputs Yt F Kt 71 Then the central planner7s problem is V K max t u c lt 0 CbKWdZO y m Ct Kt1 F Km nt 7Vt The solutions to this problem are the Pareto optimal allocations Then suppose we have an allocation cg Kill nt0 solving this planner7s problem and we want to support it as a competitive equilibrium Then we need to show that there exist sequences 10 307 R107 1030 such that i c Kj1nt0 maximizes consumer7s utility subject to the budget constraint determined by pi Rf wf0 ii K in 0 maximize rm7s ro ts t v 70 p iii Markets clear the allocation ca Kj10 is resource feasible Remark 2 Even though nt can be treated as a parameter for the consumer s problem this is not the case for the rms These actually choose their amount of labor input each period Therefore we must make the sequence nt part of the competitive equilibrium and require that the wage level for each t support this as rms equilibrium labor demand A straightforward way of showing that the sequences 1930 1330 1030 exist is directly by nding their value Notice that from concavity of F R F1K7nt7 w F2K7nt will ensure that rms maximize pro ts or if you like that the labor and capital services markets clear each period In addition homogeneity of degree 1 implies that these 125 factor payments completely exhaust production7 so that the consumer ends up receiving the whole product obtained from his factor supply Then the values of p remain to be derived Recall the rst order conditions in the planners problem t 4110 A A F1K17nt1 39 A2117 which lead to the centralized Euler equation WC 6 WELD F1 K1317 711m Now7 since A is the marginal value of relaxing the planners problem resource con straint at time t it seems natural that prices in a competitive equilibrium must re ect this marginal value as well That is7 p A seems to re ect the marginal value of the scarce resources at t Replacing in the planners Euler equation7 we get that p F1K1nt1 10t1 Replacing by R this reduces to P 1031 It is straightforward to check that 78 is the market Euler equation that obtains from the consumers rst order conditions in the decentralized problem you should check this Therefore these prices seem to lead to identical consumption and capital choices in both versions of the model We need to check7 however7 that the desired consumption and capital paths induced by these prices are feasible that is7 that these are market clearing prices To that effect7 recall the planners resource constraint which binds due to local non satiation R 78 c Kill FK1nt1 W The equality remains unaltered if we premultiply both sides by 10 10 c KAI 10 FK17 nt1gt Vt And summing up over t we get 00 00 10 Cf Kiri 10 F Kirk 1 t0 t0 Finally7 homogeneity of degree 1 of F 7 and the way we have constructed R and w imply that 00 00 1 0 K 1l 1 lR K w Ml t0 t0 Therefore the budget constraint in the market economy is satis ed if the sequence 0 Kf10 is chosen when the prevailing prices are 107 w R ZO Next we need to check whether the conditions for 0 Kg 71 10 w R 0 to be a competitive equilibrium are satis ed or not 126 i Utility maximization subject to budget constraint We have seen that the budget constraint is met To check whether this is in fact a utility maximizing consumption capital path we should take rst order conditions But it is straightforward that these conditions lead to the Euler equation 78 which is met by the planners i 00 optimal path K1t0 ii Firms maximization By construction ofthe factor services prices and concavity of the production function we have that K nt0 are the rms7 pro t maximizing levels of factor inputs iii Market cleaning We have discussed before that the input markets clear And we have seen that if the consumers decentralized budget constraint is met this implies that the planners problem resource constraint is met for the corresponding consumption and capital sequences Therefore the proposed allocation is resource feasible Recall we mentioned convexity as a necessary assumption for the Second Welfare Theorem to hold Convexity of preferences entered our proof in that the rst order conditions were deemed suf cient to identify a utility maximizing consumption bundle Convexity of the consumption possibilities set took the form of a homogeneous of degree one jointly concave function F Concavity was used to establish the levels of factor remunerations Rf it that support K and in as the equilibrium factor demand by taking rst order conditions on F And homogeneity of degree one ensured that with R and it thus determined the total product would get exhausted in factor payment an application of the Euler Theorem 722 The second welfare theorem in a 2agent economy We now assume an economy with the same production technology and inhabited by two agents Each agent has preferences on in nite dimensional consumption vectors represented by the function Ui Cit0l 5 39W Cit 217 27 t0 where B E 0 1 and u is strictly increasing concave for both i 1 2 For some arbitrary weights 1 2 we de ne the following welfare function W 010537 CZtol M1 39 U1 Clt0l 2 39 U2 CZtol Then the following welfare maximization problem can be de ned 1 39 5 39U101t 2 39 5 39 U2 020 t0 t0 VK0 7 max 01t02K10 01 621 Kt1 S F Km 71 NR 127 where 71 711 712 denotes the aggregate labor endowrnent which is fully utilized for production since leisure is not valued If we restrict p1 and 2 to be nonnegative and to add up to 1 then W is a convex combination of the Us we have the Negishi characterization By varying the vector 1 2 all the Pareto optimal allocations in this economy can be obtained from the solution of the problem VK0 That is for every pair 1 2 such that 1412 2 0 1 2 1 we obtain a Pareto optimal allocation by solving VK0 Now given any such allocation 0 03 Kj10 is it possible to decentralize the problem VK0 so as to obtain that allocation as a competitive equilibriurn outcorne Will the price sequences necessary to support this as a competitive equilibriurn exist In order to analyze this problem we proceed as before We look for the values of p Bi 1030 and we guess them using the same procedure p A R F1K7nt7 w F2K 7m The planner7s problem rst order conditions yield 1 39Bl Chi tv 2 5 12 Ca Atv A t1 Fl Kt17 71t1 Does the solution to these centralized rst order conditions also solve the consumers decentralized problem The answer is yes and we can verify it by using pt At to replace in the previous expression for consumer 1 identical procedure would be valid for consumer 2 M1 395 Cit 107 1 3955 01t1 Pt1 So dividing we obtain i p 11101 7 31 u101t1 39 Pt1 This is the decentralized Euler equation notice that the multiplier M1 cancels out Next we turn to the budget constraint We have the aggregate expenditure incorne equation 00 00 10 39 Cit02t Kt1l 10 39 Rt Kt l wt ntl t0 t0 By homogeneity of degree 1 of F the factor rernunerations de ned above irn ply that if the central planner7s resource constraint is satis ed for a 01h 02 Kt110 sequence then this aggregate budget constraint will also be satis ed for that chosen consurnption capital accurnulation path 128 However satisfaction of the aggregate budget constraint is not all We have an addi tional dilemma how to split it into two different individual budget constraints Clearly we need to split the property of the initial capital between the two agents kio kzo K0 Does km contain enough information to solve the dilemma First notice that from the central planner7s rst order condition A t1 Fl Kt17 71t1gt we can use the pricing guesses Rt F1 Kt 71 pt At and replace to get Pt 10t1 39Rt1 Therefore we can simplify in the aggregate budget constraint 10 39 Kt1 Pt139Rt1 39 Kt1 for all t Then we can rewrite 00 00 10 39 Clt 02 100 39 R0 39 kio kzo 10 39 wt 39 71 t0 t0 And the individual budgets where the labor endowment is assigned to each individual read Zpt39Cit PO39RO39k10ZPt wt nlh 79 t0 t0 Zpt CZtpO RO kZO l Zpt wt nZt 710 t0 t0 Notice that none of them include the capital sequence directly only indirectly via wt Recall the central planner7s optimal consumption sequence for Agent 1 BBB the one we wish to implement and the price guesses u F2 Ki nt0 and 10 A ZO lnserting these into 79 we have 00 00 1 62 pE R3k10zp WU 411 t0 t0 The left hand side 2010 03 is the present market value of planned consumption path for Agent 1 The right hand side is composed by his nancial wealth p3 R3 km and his human wealth77 endowment 2010 11 71 The variable km is the adjust ment factor that we can manipulate to induce the consumer into the consumption capital accumulation path that we want to implement Therefore km contains enough information there is a one to one relation between the weight M and the initial capital level equivalently the nancial wealth of each consumer The Pareto optimal allocation characterized by that weight can be implemented with the price guesses de ned above and the appropriate wealth distribution determined by km This is the Second Welfare theorem 129 Notes on Macroeconomic Theory Steve Williamson Dept of Economics University of Iowa Iowa City7 IA 52242 August 1999 Chapter 1 Simple Representative Agent Models This chapter deals with the most simple kind of macroeconomic model which abstracts from all issues of heterogeneity and distribution among economic agents Here we study an economy consisting of a represen tative rm and a representative consumer As we will show this is equivalent under some circumstances to studying an economy with many identical rms and many identical consumers Here as in all the models we will study economic agents optimize ie they maximize some objective subject to the constraints they face The preferences of consumers the technology available to rms and the endowments of resources available to consumers and rms combined with optimizing behavior and some notion of equilibrium allow us to use the model to make predictions Here the equilibrium concept we will use is competi tive equilibrium ie all economic agents are assumed to be price takers 11 A Static Model 111 Preferences endowments and technology There is one period and N consumers who each have preferences given by the utility function uc where c is consumption and E is leisure Here u is strictly increasing in each argument strictly concave and 2 CHAPTER 1 SIMPLE REPRESENTATIVE AGENT MODELS twice di erentiable Also assume that limcno u1cE 00 E gt 0 and liming u2cE 00 c gt 0 Here ulCJ is the partial derivative with respect to argument i of ucE Each consumer is endowed with one unit of time which can be allocated between work and leisure Each consumer also owns 11 units of capital which can be rented to rms There are M rms which each have a technology for producing consumption goods according to y Zfk7nv where y is output k is the capital input n is the labor input and z is a parameter representing total factor productivity Here the function f is strictly increasing in both arguments strictly quasiconcave twice di erentiable and homogeneous of degree one That is produc tion is constant returns to scale so that Ayzfkn 11 for A gt 0 Also assume that limkqo f1kn 00 llmkqoo f1 n 0 limnno f2kn 00 and limHOG f2kn 0 112 Optimization In a competitive equilibrium we can at most determine all relative prices so the price of one good can arbitrarily be set to l with no loss of generality We call this good the numeraire We will follow convention here by treating the consumption good as the numeraire There are markets in three objects consumption leisure and the rental services of capital The price of leisure in units of consumption is w and the rental rate on capital again in units of consumption is 7 Consumer s Problem Each consumer treats w as being xed and maximizes utility subject to hisher constraints That is each solves WWW 11 A STATICMODEL 3 subject to c 7vz17 rkS 12 160 lt lt 0 7 CS 7 N 13 0 g E g 1 14 c 2 0 15 Here k5 is the quantity of capital that the consumer rents to firms 12 is the budget constraint 13 states that the quantity of capital rented must be positive and cannot exceed what the consumer is endowed with 14 is a similar condition for leisure and 15 is a nonnegativity constraint on consumption Now given that utility is increasing in consumption more is pre ferred to less we must have k5 1 and 12 will hold with equality Our restrictions on the utility function assure that the nonnegativity constraints on consumption and leisure will not be binding and in equi librium we will never have I 1 as then nothing would be produced so we can safely ignore this case The optimization problem for the con sumer is therefore much simpli ed and we can write down the following Lagrangian for the problem 190 I uc Mw TN 7 w 7 c where p is a Lagrange multiplier Our restrictions on the utility func tion assure that there is a unique optimum which is characterized by the following first order conditions 81 EU1MO all a U2 7 mu 0 Z wrk 7wf7c0 Here 11 is the partial derivative of u with respect to argument i The above first order conditions can be used to solve out for p and c to obtain k k wu1wr o 774 7u2wr o 7wEE 0 16 4 CHAPTER 1 SIMPLE REPRESENTATIVE AGENT MODELS which solves for the desired quantity of leisure E in terms of w 7 and 1 Equation 16 can be rewritten as U2 U1 7 7 ie the marginal rate of substitution of leisure for consumption equals the wage rate Diagrammatically in Figure 11 the consumer s budget constraint is ABD and heshe maximizes utility at E where the budget constraint which has slope 7w is tangent to the highest indi erence curve where an indi erence curve has slope 7 Firm s Problem Each rm chooses inputs of labor and capital to maximize pro ts treat ing w and 7 as being xed That is a rm solves naxzfkn 7 rk 7 um and the rst order conditions for an optimum are the marginal product conditions zfl 7 17 ng w 18 where denotes the partial derivative of with respect to argu ment i Now given that the function is homogeneous of degree one Euler s law holds That is differentiating 11 with respect to A and setting A 1 we get zfkn zf1kzf2n 19 Equations 17 18 and 19 then imply that maximized pro ts equal zero This has two important consequences The rst is that we do not need to be concerned with how the rm s pro ts are distributed through shares owned by consumers for example Secondly suppose 16 and 71 are optimal choices for the factor inputs then we must have zfkn irkiwn0 110 11 A STATIC MODEL Fxgurell Figure 1 1 6 CHAPTER 1 SIMPLE REPRESENTATIVE AGENT MODELS for k 16 and n 11 But since 110 also holds for k AW and n m for any A gt 0 due to the constant returns to scale assumption the optimal scale of operation of the rm is indeterminate It therefore makes no difference for our analysis to simply consider the case M 1 a single representative rm as the number of rms will be irrelevant for determining the competitive equilibrium 113 Competitive Equilibrium A competitive equilibrium is a set of quantities c E n k and prices w and 7 which satisfy the following properties 1 Each consumer chooses c and E optimally given w and 7 2 The representative rm chooses n and k optimally given w and 7 3 Markets clear Here there are three markets the labor market the market for consumption goods and the market for rental services of capital In a competitive equilibrium given 3 the following conditions then hold N17E n 111 y NC 112 kg k 113 That is supply equals demand in each market given prices Now the total value of excess demand across markets is Nciywn7N17Erk7k0 but from the consumer s budget constraint and the fact that pro t maximization implies zero pro ts we have Nciywn7N17Erkik00 114 Note that 114 would hold even if pro ts were not zero and were dis tributed lump sum to consumers But now if any 2 of 111 112 11 A STATICMODEL 7 and 113 hold then 114 implies that the third market clearing con dition holds Equation 114 is simply Walras7 law for this model Walras7 law states that the value of excess demand across markets is always zero and this then implies that if there are M markets and M 7 1 of those markets are in equilibrium then the additional mar ket is also in equilibrium We can therefore drop one market clearing condition in determining competitive equilibrium prices and quantities Here we eliminate 112 The competitive equilibrium is then the solution to 16 17 18 111 and 113 These are ve equations in the ve unknowns E n k w and 7 and we can solve for c using the consumer s budget constraint It should be apparent here that the number of consumers N is virtually irrelevant to the equilibrium solution so for convenience we can set N 1 and simply analyze an economy with a single repre sentative consumer Competitive equilibrium might seem inappropriate when there is one consumer and one rm but as we have shown in this context our results would not be any di erent if there were many rms and many consumers We can substitute in equation 16 to obtain an equation which solves for equilibrium E MFR9071 Eu1zfk0717 111 11291009071 1 0 115 Given the solution for Z we then substitute in the following equations to obtain solutions for 7 w n k and 0 21010901 71 7 116 2120911 71 w 117 n 1 71 k ko c zfk01 71 118 It is not immediately apparent that the competitive equilibrium exists and is unique but we will show this later 8 CHAPTER 1 SIMPLE REPRESENTATIVE AGENT MODELS 1 14 Pareto Optimality A Pareto optimum generally is de ned to be some allocation an al location being a production plan and a distribution of goods across economic agents such that there is no other allocation which some agents strictly prefer which does not make any agents worse off Here since we have a single agent we do not have to worry about the allo cation of goods across agents It helps to think in terms of a ctitious social planner who can dictate inputs to production by the representa tive firm can force the consumer to supply the appropriate quantity of labor and then distributes consumption goods to the consumer all in a way that makes the consumer as well off as possible The social planner determines a Pareto optimum by solving the following problem max uc E 015 subject to czfk01i 119 Given the restrictions on the utility function we can simply substitute using the constraint in the objective function and differentiate with respect to E to obtain the following first order condition for an optimum 2090717 01111210060 1 7 EM 7 112121116071 Ml 0 120 Note that 115 and 120 are identical and the solution we get for c from the social planner s problem by substituting in the constraint will yield the same solution as from 118 That is the competitive equilibrium and the Pareto optimum are identical here Further since u is strictly concave and is strictly quasiconcave there is a unique Pareto optimum and the competitive equilibrium is also unique Note that we can rewrite 120 as U2 2f 2 7 U1 where the left side of the equation is the marginal rate of transforma tion and the right side is the marginal rate of substitution of consump tion for leisure In Figure 12 AB is equation 119 and the Pareto 11 A STATICMODEL 9 optimum is at D where the highest indifference curve is tangent to the production possibilities frontier In a competitive equilibrium the representative consumer faces budget constraint AFG and maximizes at point D where the slope of the budget line 7w is equal to 7 In more general settings it is true under some restrictions that the following hold 1 A competitive equilibrium is Pareto optimal First Welfare The orem 2 Any Pareto optimum can be supported as a competitive equilib rium with an appropriate choice of endowments Second Welfare Theorem The non technical assumptions required for l and 2 to go through include the absence of externalities completeness of markets and ab sence of distorting taxes eg income taxes and sales taxes The First Welfare Theorem is quite powerful and the general idea goes back as far as Adam Smith s Wealth of Nations ln macroeconomics if we can successfully explain particular phenomena eg business cycles using a competitive equilibrium model in which the First Welfare Theorem holds we can then argue that the existence of such phenomena is not grounds for government intervention In addition to policy implications the equivalence of competitive equilibria and Pareto optima in representative agent models is useful for computational purposes That is it can be much easier to obtain com petitive equilibria by rst solving the social planner s problem to obtain competitive equilibrium quantities and then solving for prices rather than solving simultaneously for prices and quantities using market clearing conditions For example in the above example a competitive equilibrium could be obtained by rst solving for c and E from the social planner s problem and then nding w and r from the appropriate mar ginal conditions 116 and 117 Using this approach does not make much difference here but in computing numerical solutions in dynamic models it can make a huge difference in the computational burden 10 CHAPTER 1 SIMPLE REPRESENTATIVE AGENT MODELS FxgurelZ 1 lasure Figure 1 2 11 A STATICMODEL 11 115 Example Consider the following specific functional forms For the utility func tion we use 01 uc E 1 7 1 where y gt 0 measures the degree of curvature in the utility function with respect to consumption this is a constant relative risk aversion utility function Note that BUM oss 1 0 1 01quot 7 1 i 11 1m 7 1m 7 P1 177 771 log 0 using L Hospital s Rule For the production technology use 10097 kanl ay where 0 lt CZ lt 1 That is the production function is Cobb Douglas The social planner s problem here is then on 7 170 174 7 maxzk01 1 1M 4 1 7 y and the solution to this problem is z 1 7 1 7 azk 0 1 1 a 121 As in the general case above this is also the competitive equilibrium solution Solving for c from 119 we get 0 717 a1 azk3m 122 and from 117 we have u 7 1 7 a1azkgm9w 123 From 122 and 123 clearly c and w are increasing in z and 1 That is increases in productivity and in the capital stock increase aggregate consumption and real wages However from equation 121 the effects 12 CHAPTER 1 SIMPLE REPRESENTATIVE AGENT MODELS on the quantity of leisure and therefore on employment are ambigu ous Which way the e ect goes depends on whether 7 lt l or y gt 1 With 7 lt 1 an increase in z or in Co will result in a decrease in leisure and an increase in employment but the effects are just the opposite if y gt 1 If we want to treat this as a simple model of the business cycle where uctuations are driven by technology shocks changes in 2 these results are troubling In the data aggregate output aggregate consumption and aggregate employment are mutually positively corre lated However this model can deliver the result that employment and output move in opposite directions Note however that the real wage will be procyclical it goes up when output goes up as is the case in the data 116 Linear Technology Comparative Statics This section illustrates the use of comparative statics and shows in a somewhat more general sense than the above example why a produc tivity shock might give a decrease or an increase in employment To make things clearer we consider a simpli ed technology 112717 ie we eliminate capital but still consider a constant returns to scale technology with labor being the only input The social planner s prob lem for this economy is then mlaxuz17 1 and the rst order condition for a maximum is 7zu1z17 DJ u2z17 1 O 124 Here in contrast to the example we cannot solve explicitly for E but note that the equilibrium real wage is 8y 7 an 7 so that an increase in productivity 2 corresponds to an increase in the real wage faced by the consumer To determine the e ect of an increase U 2 7 11 A STATICMODEL 13 in 2 on E apply the implicit function theorem and totally differentiate 124 to get 7u1 7 217 Eu11 U2117 Edz 22U11 221112 1 U22d We then have 1 KU1Z1 U11U211 I dz 221111 7 22u12 U22 Now concavity of the utility function implies that the denominator in 125 is negative but we cannot sign the numerator In fact it is easy to construct examples where g gt 0 and where lt 0 The ambiguity here arises from opposing income and substitution effects In Figure 13 AB denotes the resource constraint faced by the social planner c 211 7 E and ED is the resource constraint with a higher level of productivity 22 gt 21 As shown the social optimum also the competitive equilibrium is at E initially and at F after the increase in productivity with no change in E but higher 0 Effectively the repre sentative consumer faces a higher real wage and hisher response can be decomposed into a substitution effect E to G and an income effect G to Algebraically we can determine the substitution effect on leisure by changing prices and compensating the consumer to hold utility con stant ie uc h 126 where h is a constant and 7zu1cf u2cf 0 127 Totally differentiating 126 and 127 with respect to c and E and us ing 127 to simplify we can solve for the substitution effect subst as follows E U1 b t lt 0 dz su 8 221111 7 2Zu12 U22 From 125 then the income effect inc is just the remainder d i 1111 U211 inc gt 0 dz 221111 7 2Zu12 U22 14 CHAPTER 1 SIMPLE REPRESENTATIVE AGENT MODELS D FxgurEIE 1 leisure Figure 1 3 12 GOVERNMENT 15 provided I is a normal good Therefore in order for a model like this one to be consistent with observation we require a substitution effect that is large relative to the income effect That is a productivity shock which increases the real wage and output must result in a decrease in leisure in order for employment to be procyclical as it is in the data In general preferences and substitution effects are very important in equilibrium theories of the business cycle as we will see later 12 Government So that we can analyze some simple scal policy issues we introduce a government sector into our simple static model in the following man ner The government makes purchases of consumption goods and fi nances these purchases through lump sum taxes on the representative consumer Let g be the quantity of government purchases which is treated as being exogenous and let T be total taxes The government budget must balance ie g 739 128 We assume here that the government destroys the goods it purchases This is clearly unrealistic in most cases but it simpli es matters and does not make much difference for the analysis unless we wish to consider the optimal determination of government purchases For example we could allow government spending to enter the consumer s utility function in the following way Ma a 1401 119 Given that utility is separable in this fashion and g is exogenous this would make no difference for the analysis Given this we can assume 719 0 As in the previous section labor is the only factor of production ie assume a technology of the form 1 2n Here the consumer s optimization problem is max uc E 015 16 CHAPTER 1 SIMPLE REPRESENTATIVE AGENT MODELS subject to c w1 7 E 7 739 and the rst order condition for an optimum is 7wu1 U2 O The representative rm s pro t maximization problem is m7axz 7 wn Therefore the rm s demand for labor is in nitely elastic at w z A competitive equilibrium consists of quantities c E n and 739 and a price w which satisfy the following conditions H The representative consumer chooses c and E to maximize utility given w and 739 N The representative rm chooses n to maximize pro ts given w 9quot Markets for consumption goods and labor clear 4 The government budget constraint 128 is satis ed The competitive equilibrium and the Pareto optimum are equivalent here as in the version of the model without government The social planner s problem is max uc 015 subject to c g 21 7 E Substituting for c in the objective function and maximizing with re spect to E the rst order condition for this problem yields an equation which solves for E 7zu1217 E 7 91 u2217 E 7 gl 0 129 In Figure 14 the economy s resource constraint is AB and the Pareto optimum competitive equilibrium is D Note that the slope of the resource constraint is 72 7w 12 GOVERNMENT 17 Ly lexsure Figure 1 4 18 CHAPTER 1 SIMPLE REPRESENTATIVE AGENT MODELS We can now ask what the e ect of a change in government expen ditures would be on consumption and employment In Figure 15 9 increases from 91 to 92 shifting in the resource constraint Given the government budget constraint there is an increase in taxes which rep resents a pure income effect for the consumer Given that leisure and consumption are normal goods quantities of both goods will decrease Thus there is crowding out of private consumption but note that the decrease in consumption is smaller than the increase in government purchases so that output increases Algebraically totally di erentiate 129 and the equation 0 21 7 E 7 g and solve to obtain d5 7 Zun U12 7 lt 0 d9 221111 i 221112 U22 E M lt 0 130 119 221111 i 221112 U22 Here the inequalities hold provided that 7zu11 U12 gt 0 and man 7 U22 gt 0 ie if leisure and consumption are respectively normal goods Note that 130 also implies that 2 lt l ie the balanced budget multiplier is less than 1 13 A Dynamic Economy We will introduce some simple dynamics to our model in this section The dynamics are restricted to the government s nancing decisions there are really no dynamic elements in terms of real resource alloca tion ie the social planner s problem will break down into a series of static optimization problems This model will be useful for studying the e ects of changes in the timing of taxes Here we deal with an in nite horizon economy where the represen tative consumer maximizes time separable utility Z tuct7 t7 t0 where is the discount factor 0 lt lt l Letting 6 denote the dis count rate we have where 6 gt 0 Each period the con sumer is endowed with one unit of time There is a representative rm 13 A DYNAMIC ECONOMY 19 zrgl Fxgure 1 5 17g2z lrglz 1mm Figure 1 5 20 CHAPTER 1 SIMPLE REPRESENTATIVE AGENT MODELS which produces output according to the production function yt ztnt The government purchases gt units of consumption goods in period t t 0 1 2 and these purchases are destroyed Government purchases are nanced through lump sum taxation and by issuing one period gov ernment bonds The government budget constraint is 9t 1 73 Ttbt17 131 t 01 2 where bi is the number of one period bonds issued by the government in period t 7 1 A bond issued in period t is a claim to 1rt1 units of consumption in period t1 where n1 is the one period interest rate Equation 131 states that government purchases plus principal and interest on the government debt is equal to tax revenues plus new bond issues Here 0 0 The optimization problem solved by the representative consumer is gt0 E140 gt 0 max 511Czlz 0 t subject to at wt17 ft 7 73 7 5H1 1 73 132 t 012 50 0 where 5H1 is the quantity of bonds purchased by the consumer in period t which come due in period t 1 Here we permit the representative consumer to issue private bonds which are perfect substitutes for government bonds We will assume that 4320 1130 n 3933 which states that the quantity of debt discounted to t 0 must equal zero in the limit This condition rules out in nite borrowing or Ponzi schemes and implies that we can write the sequence of budget con straints 132 as a single intertemporal budget constraint Repeated substitution using 132 gives gt0 gt0 1 7 COZ1EOTOZM t1 134 111T 74139 t1 211 1 74139 13 A DYNAMIC ECONOMY 21 Now7 maximizing utility subject to the above intertemporal budget constraint7 we obtain the following rst order conditions A t z 7 0t123 u10t7 t 211Ti 7 7 7 7 Au u c1 7 0t123 27 7 241 U1ltCO7 0 A 0 U2ltCO7 0 A100 0 Here7 A is the Lagrange multiplier associated with the consumer s in tertemporal budget constraint We then obtain U2Ct7 t U1Ct7 t ie the marginal rate of substitution of leisure for consumption in any period equals the wage rate7 and u1ct17 t1 1 U1Ct7 t 1 17 wt 135 136 ie the intertemporal marginal rate of substitution of consumption equals the inverse of one plus the interest rate The representative rm simply maximizes pro ts in each period7 ie it solves nilaxzt 7 100m and labor demand7 m is perfectly elastic at wt 2t A competitive equilibrium consists of quantities7 ct7 ft m 5H1 bi 7380 and prices wt7rt10 satisfying the following conditions 1 Consumers choose ct7 It 5H1 Bio optimally given 73 and wt7 rt1f0 2 Firms choose min optimally given w fio 3 Given g fim bt1739tf0 satis es the sequence of government budget constraints 131 22 CHAPTER 1 SIMPLE REPRESENTATIVE AGENT MODELS 4 Markets for consumption goods labor and bonds clear Wal ras7 law permits us to drop the consumption goods market from consideration giving us two market clearing conditions 5t1 bt17t0717277 137 and 1Et ntt012 Now 133 and 137 imply that we can write the sequence of government budget constraints as a single intertemporal government budget constraint through repeated substitution Tt 90 73901L 7 138 11241 72 210 n ie the present discounted value of government purchases equals the present discounted value of tax revenues Now since the government budget constraint must hold in equilibrium we can use 138 to sub stitute in 134 to obtain ct wt1ift igt C w 1 7 E 7 139 0 Z 1711 74139 O 0 90 Z 1117L 74139 Now suppose that wtrt10 are competitive equilibrium prices Then 139 implies that the optimizing choices given those prices re main optimal given any sequence yiio satisfying 138 Also the representative firm s choices are invariant That is all that is relevant for the determination of consumption leisure and prices is the present discounted value of government purchases and the timing of taxes is irrelevant This is a version of the Ricardizm Equivalence Theorem For example holding the path of government purchases constant if the representative consumer receives a tax cut today heshe knows that the government will have to make this up with higher future taxes The government issues more debt today to nance an increase in the government de cit and private saving increases by an equal amount since the representative consumer saves more to pay the higher taxes in the future 13 A DYNAMIC ECONOMY 23 Another way to show the Ricardian equivalence result here comes from computing the competitive equilibrium as the solution to a social planner s problem ie max i tum 7 5 gtvgtl 111020 t0 This breaks down into a series of static problems and the rst order conditions for an optimum are Ztu1lzt1 t gtvztlu2lzt1 t gtv tl 07 140 t 0 1 2 Here 140 solves for It t 0 1 2 and we can solve for at from at zt1 7 It Then 135 and 136 determine prices Here it is clear that the timing of taxes is irrelevant to determining the competitive equilibrium though Ricardian equivalence holds in much more general settings where competitive equilibria are not Pareto op timal and where the dynamics are more complicated Some assumptions which are critical to the Ricardian equivalence result are 1 Taxes are lump sum 2 Consumers are in nite lived 3 Capital markets are perfect ie the interest rate at which private agents can borrow and lend is the same as the interest rate at which the government borrows and lends 4 There are no distributional e ects of taxation That is the present discounted value of each individual s tax burden is una ected by changes in the timing of aggregate taxation 24 CHAPTER 1 SIMPLE REPRESENTATIVE AGENT MODELS Chapter 2 Growth With Overlapping Generations This chapter will serve as an introduction to neoclassical growth theory and to the overlapping generations model The particular model intro duced in this chapter was developed by Diamond 1965 building on the overlapping generations construct introduced by Samuelson 1956 Samuelson s paper was a semi serious meaning that Samuelson did not take it too seriously attempt to model money but it has also proved to be a useful vehicle for studying public nance issues such as gov ernment debt policy and the e ects of social security systems There was a resurgence in interest in the overlapping generations model as a monetary paradigm in the late seventies and early eighties particularly at the University of Minnesota see for example Kareken and Wallace 1980 A key feature of the overlapping generations model is that mar kets are incomplete in a sense in that economic agents are nite lived and agents currently alive cannot trade with the unborn As a re sult competitive equilibria need not be Pareto optimal and Ricardian equivalence does not hold Thus the timing of taxes and the size of the government debt matters Without government intervention resources may not be allocated optimally among generations and capital accu mulation may be suboptimal However government debt policy can be used as a vehicle for redistributing wealth among generations and inducing optimal savings behavior 25 26 CHAPTER 2 GROWTH WITH OVERLAPPIN G GENERATIONS 21 The Model This is an in nite horizon model where time is indexed by t 0 l 2 00 Each period Lt two period lived consumers are born and each is en dowed with one unit of labor in the rst period of life and zero units in the second period The population evolves according to L L01n 21 where L0 is given and n gt 0 is the population growth rate In period 0 there are some old consumers alive who live for one period and are col lectively endowed with K0 units of capital Preferences for a consumer born in period t t 012 are given by 1401217 Cf17 where 01 denotes the consumption of a young consumer in period t and of is the consumption of an old consumer Assume that u is strictly increasing in both arguments strictly concave and de ning 7 a 9 001 7 00 a 7 ac assume that limcynovcyco 00 for cquot gt 0 and limcanovcyco 0 for Cy gt 0 These last two conditions on the marginal rate of substitu tion will imply that each consumer will always wish to consume positive amounts when young and when old The initial old seek to maximize consumption in period 0 The investment technology works as follows Consumption goods can be converted one for one into capital and vice versa Capital con structed in period t does not become productive until period t l and there is no depreciation Young agents sell their labor to rms and save in the form of capi tal accumulation and old agents rent capital to rms and then convert the capital into consumption goods which they consume The repre sentative rm maximizes pro ts by producing consumption goods and renting capital and hiring labor as inputs The technology is given by Y FKt7Lt7 22 OPTIMAL ALLOCATIONS 27 where Yt is output and Kt and Lt are the capital and labor inputs respectively Assume that the production function F is strictly in creasing strictly quasi concave twice differentiable and homogeneous of degree one 22 Optimal Allocations As a benchmark we will rst consider the allocations that can be achieved by a social planner who has control over production capi tal accumulation and the distribution of consumption goods between the young and the old We will con ne attention to allocations where all young agents in a given period are treated identically and all old agents in a given period receive the same consumption The resource constraint faced by the social planner in period t is FKtth Kt Kt1CLtCfItilv 22 where the left hand side of 22 is the quantity of goods available in period t ie consumption goods produced plus the capital that is left after production takes place The right hand side is the capital which will become productive in period t 1 plus the consumption of the young plus consumption of the old In the long run this model will have the property that per capita quantities converge to constants Thus it proves to be convenient to express everything here in per capita terms using lower case letters De ne kt E 15 the capitallabor ratio or per capita capital stock and fkt E Fkt 1 We can then use 21 to rewrite 22 as 0 t fUW f kt 1 kt1 f 0121 23 1 n De nition 1 A Pareto optimal allocation is a sequence 0 cf kt1f0 satisfying and the property that there exists no other allocation aiagkt go which satis es and of 2 cf 621 2 140 021 for all t 0 1 2 3 with strict inequality in at least one instance 28 CHAPTER 2 GROWTH WITH OVERLAPPIN G GENERATIONS That is a Pareto optimal allocation is a feasible allocation such that there is no other feasible allocation for which all consumers are at least as well off and some consumer is better off While Pareto optimality is the appropriate notion of social optimality for this model it is somewhat complicated for our purposes to derive Pareto optimal allocations here We will take a shortcut by focusing attention on steady states where kt k 01 Cy and of cquot where k Cy and cquot are constants We need to be aware of two potential problems here First there may not be a feasible path which leads from Co to a particular steady state Second one steady state may dominate another in terms of the welfare of consumers once the steady state is achieved but the two allocations may be Pareto non comparable along the path to the steady state The problem for the social planner is to maximize the utility of each consumer in the steady state given the feasibility condition 22 That is the planner chooses Cy co and k to solve max ucy co subject to M ink cy10n 24 Substituting for cquot in the objective function using 24 we then solve the following nlyagmcy 1 nlfk k Cyl The first order conditions for an optimum are then ul 7 1 nu2 0 or U1 1 25 W M lt gt intertemporal marginal rate of substitution equal to l n and Ne n 26 marginal product of capital equal to Note that the planner s prob lem splits into two separate components First the planner nds the 23 COMPETITIVE EQUILIBRIUM 29 capital labor ratio which maximizes the steady state quantity of re sources from 26 and then allocates consumption between the young and the old according to 25 In Figure 21 k is chosen to maximize the size of the budget set for the consumer in the steady state and then consumption is allocated between the young and the old to achieve the tangency between the aggregate resource constraint and an indifference curve at point A 23 Competitive Equilibrium In this section we wish to determine the properties of a competitive equilibrium and to ask whether a competitive equilibrium achieves the steady state social optimum characterized in the previous section 231 Young Consumer s Problem A consumer born in period t solves the following problem 9 0 ymax uctvct1 weal subject to 013 wt 7 st 27 0544 5t1 73H 28 Here wt is the wage rate 73 is the capital rental rate and st is saving when young Note that the capital rental rate plays the role of an in terest rate here The consumer chooses savings and consumption when young and old treating prices wt and n1 as being xed At time t the consumer is assumed to know n1 Equivalently we can think of this as a rational expectations or perfect foresight equilibrium where each consumer forecasts future prices and optimizes based on those forecasts In equilibrium forecasts are correct ie no one makes sys tematic forecasting errors Since there is no uncertainty here forecasts cannot be incorrect in equilibrium if agents have rational expectations SUCHAPTER 2 GROWTH WITH OVERLAPPING GENERATIONS cons ofold Hg 2 1 lmxfrkrnk momk cons Figure 2 1 23 COMPETITIVE EQUILIBRIUM 31 Substituting for c and of in the above objective function using 27 and 28 to obtain a maximization problem with one choice vari able7 st the rst order condition for an optimum is then 7u1wt 7 stst1 rt1 u2wt 7 stst1 rt11rt1 0 29 which determines st ie we can determine optimal savings as a function of prices st swtrt1 210 Note that 29 can also be rewritten as 3 l n1 ie the in tertemporal marginal rate of substitution equals one plus the interest rate Given that consumption when young and consumption when old are both normal goods7 we have 821 gt 07 however the sign of indeterminate due to opposing income and substitution e ects s is 3Tz1 232 Representative Firm s Problem The rm solves a static pro t maximization problem maxFKt7 L 7 tht 7 rth KuLz The rst order conditions for a maximum are the usual marginal con ditions F1Kt7Lt Ti 07 F2KtLt 7 wt 0 Since F7 is homogeneous of degree 17 we can rewrite these marginal conditions as fact 73 07 211 fkt 7 ktf kt 7 wt 0 212 233 Competitive Equilibrium De nition 2 A competitive equilibrium is a sequence of quantities kWh stf0 and a sequence ofprices whrt im which satisfy con sumer optimization rm optimization market clearing in each period t 017 27 given the initial capitallabor ratio k0 32 CHAPTER 2 GROWTH WITH OVERLAPPIN G GENERATIONS Here we have three markets for labor capital rental and con sumption goods and Walras7 law tells us that we can drop one market clearing condition It will be convenient here to drop the consumption goods market from consideration Consumer optimization is summa rized by equation 210 which essentially determines the supply of capital as period t savings is equal to the capital that will be rented in period t1 The supply of labor by consumers is inelastic The demands for capital and labor are determined implicitly by equations 211 and 212 The equilibrium condition for the capital rental market is then kt11 5wtvrt1v 213 and we can substitute in 213 for wt and n1 from 211 and 212 to get kt11 71 SHOW kf kt7f kt1 214 Here 214 is a nonlinear rst order di erence equation which given 1 solves for k fir Once we have the equilibrium sequence of capital labor ratios we can solve for prices from 211 and 212 We can then solve for s tio from 210 and in turn for consumption allocations 24 An Example Let ucycquot lncy lncquot and FK L VK ILl a where gt 0 y gt 0 and 0 lt CZ lt 1 Here a young agent solves mgxllnwt 5t lnl Tt15tlv and solving this problem we obtain the optimal savings function 5t Given the Cobb Douglass production function we have 760 and yczka l Therefore from 211 and 212 the rst order conditions from the rm s optimization problem give 73 39yczk l 216 24 AN EXAMPLE 33 wt 717 Czkf 217 Then using 214 215 and 217 we get kt11 n 717 czkf 218 i 1 Now equation 218 determines a unique sequence k til given k0 see Figure 2m which converges in the limit to 16 the unique steady state capital labor ratio which we can determine from 218 by setting kt kt 16 and solving to get 7 wow k 7 1n1 219 Now given the steady state capital labor ratio from 219 we can solve for steady state prices from 216 and 217 that is a1n1 M1 Cl 7 M0 7 Ct 10 1 7 Cr llt1 mm a We can then solve for steady state consumption allocations w cyw7 w 1 1 co w1 74 In the long run this economy converges to a steady state where the capital labor ratio consumption allocations the wage rate and the rental rate on capital are constant Since the capital labor ratio is constant in the steady state and the labor input is growing at the rate n the growth rate of the aggregate capital stock is also n in the steady state In turn aggregate output also grows at the rate n Now note that the socially optimal steady state capital stock 1 is determined by 26 that is yaka l n 34 CHAPTER 2 GROWTH WITH OVERLAPPIN G GENERATIONS 01 l A my a k 7 7 220 Note that in general from 219 and 220 16 a 1 ie the competi tive equilibrium steady state is in general not socially optimal so this economy suffers from a dynamic inef ciency There may be too little or too much capital in the steady state depending on parameter values That is suppose l and n 3 Then if CZ lt 103 16 gt 1 and if a gt 103 then 19 lt 1 25 Discussion The above example illustrates the dynamic inef ciency that can result in this economy in a competitive equilibrium There are essentially two problems here The rst is that there is either too little or too much capital in the steady state so that the quantity of resources available to allocate between the young and the old is not optimal Second the steady state interest rate is not equal to n ie consumers face the wrong interest rate and therefore misallocate consumption goods over time there is either too much or too little saving in a competitive equilibrium The root of the dynamic inef ciency is a form of market incomplete ness in that agents currently alive cannot trade with the unborn To correct this inef ciency it is necessary to have some mechanism which permits transfers between the old and the young 26 Government Debt One means to introduce intergenerational transfers into this economy is through government debt Here the government acts as a kind of nancial intermediary which issues debt to young agents transfers the proceeds to young agents and then taxes the young of the next gener ation in order to pay the interest and principal on the debt Let BHl denote the quantity of one period bonds issued by the government in period t Each of these bonds is a promise to pay ln1 2 6 GOVERNMENT DEBT 35 units of consumption goods in period t 1 Note that the interest rate on government bonds is the same as the rental rate on capital as must be the case in equilibrium for agents to be willing to hold both capital and government bonds We will assume that BH1 bLt 221 where b is a constant That is the quantity of government debt is xed in per capita terms The government s budget constraint is Bt1 T 1 7403 222 ie the revenues from new bond issues and taxes in period t Tt equals the payments of interest and principal on government bonds issued in period t 7 1 Taxes are levied lump sum on young agents and we will let 7 denote the tax per young agent We then have T nLt 223 A young agent solves msaxuwt 7 st 7 73 1 rt1st where st is savings taking the form of acquisitions of capital and gov ernment bonds which are perfect substitutes as assets Optimal savings for a young agent is now given by 5t 5wt 7377344 224 As before pro t maximization by the rm implies 211 and 212 A competitive equilibrium is de ned as above adding to the de ni tion that there be a sequence of taxes 7380 satisfying the government budget constraint From 221 222 and 223 we get 73711 b 225 Ti 1n l The asset market equilibrium condition is now k11n b sw 7mn17 226 36 CHAPTER 2 GROWTH WITH OVERLAPPIN G GENERATIONS that is per capita asset supplies equals savings per capita Substituting in 226 for wt 73 and n1 from 211 we get kt11nb 7 s ltfkt 7 kf k 7 b mm 227 We can then determine the steady state capital labor ratio kb by setting kb kt kt in 227 to get f kb n WwwmsOWDWWfWD 1 awwww 228 Now suppose that we wish to nd the debt policy determined by b which yields a competitive equilibrium steady state which is socially optimal ie we want to nd I such that Now given that n from 228 we can solve for I as follows 13 7k1nsf 71mm 229 In 229 note that i may be positive or negative If lt 0 then debt is negative ie the government makes loans to young agents which are nanced by taxation Note that from 225 73 0 in the steady state with b i so that the size of the government debt increases at a rate just suf cient to pay the interest and principal on previously issued debt That is the debt increases at the rate n which is equal to the interest rate Here at the optimum government debt policy simply transfers wealth from the young to the old if the debt is positive or from the old to the young if the debt is negative 261 Example Consider the same example as above but adding government debt That is ucycquot lncy lncquot and FK L VK ILl a where gt 0 y gt 0 and 0 lt CZ lt 1 Optimal savings for a young agent is s 7 107 n 230 2 7 REFERENCES 37 Then from 216 217 227 and 230 the equilibrium sequence k tio is determined by i 1 and the steady state capital labor ratio kb is the solution to kb1nb 17a7kba 7W k11n b lt 17 we 7 1n 15 1n Then from 229 the optimal quantity of per capita debt is 13 7 5 17a m 7m 1n i 1 7 n n 7 CgOi 1 7 cz 7 Cr 7 l n 1 n I Here note that given 7 n and 13 lt 0 for CM suf ciently large and b gt 0 for CM suf ciently small 262 Discussion The competitive equilibrium here is in general suboptimal for reasons discussed above But for those same reasons government debt mat ters That is Ricardian equivalence does not hold here in general because the taxes required to pay 011 the currently issued debt are not levied on the agents who receive the current tax bene ts from a higher level of debt today Government debt policy is a means for executing the intergenerational transfers that are required to achieve optimality However note that there are other intergenerational transfer mecha nisms like social security which can accomplish the same thing in this model 2 7 References Diamond P 1965 National Debt in a Neoclassical Growth Model American Economic Review 55 1126 1150 38 CHAPTER 2 GROWTH WITH OVERLAPPING GENERATIONS Blanchard O and Fischer S 1989 Lectures on Macroeconomics Chapter 3 Kaieken7 J and Wallace7 N 1980 Models of Monetary Economies Federal Reserve Bank of Minneapolis Minneapolis MN Chapter 3 Neoclassical Growth and Dynamic Programming Early work on growth theory particularly that of Solow 1956 was carried out using models with essentially no intertemporal optimizing behavior That is these were theories of growth and capital accu mulation in which consumers were assumed to simply save a constant fraction of their income Later Cass 1965 and Koopmans 1965 de veloped the first optimizing models of economic growth often called optimal growth models as they are usually solved as an optimal growth path chosen by a social planner Optimal growth models have much the same long run implications as Solow s growth model with the added bene t that optimizing behavior permits us to use these models to draw normative conclusions ie make statements about welfare This class of optimal growth models led to the development of stochas tic growth models Brock and Mirman 1972 which in turn were the basis for real business cycle models Here we will present a simple growth model which illustrates some of the important characteristics of this class of models Growth mod el will be something of a misnomer in this case as the model will not exhibit long run growth One objective of this chapter will be to introduce and illustrate the use of discrete time dynamic programming methods which are useful in solving many dynamic models 39 40 CHAPTER 3 NEOCLASSICAL GROWTH AND DYNAMIC PROGRAMMING 31 Preferences Endowments and Tech nology There is a representative infinitely lived consumer with preferences given by 0 Z um t0 where 0 lt lt l and at is consumption The period utility function is continuously differentiable strictly increasing strictly concave and bounded Assume that limcho u c 00 Each period the con sumer is endowed with one unit of time which can be supplied as labor The production technology is given by 1 Fktnt 31 where yt is output kt is the capital input and m is the labor input The production function F is continuously differentiable strictly increasing in both arguments homogeneous of degree one and strictly quasiconcave Assume that F0n 0 limkho F1kl 00 and The capital stock obeys the law of motion km 17 6m it 32 where it is investment and 6 is the depreciation rate with 0 S 6 S l and k0 is the initial capital stock which is given The resource constraints for the economy are Ct 1 3 11m 33 and m S 1 34 32 Social Planner s Problem There are several ways to specify the organization of markets and pro duction in this economy all of which will give the same competitive equilibrium allocation One speci cation is to endow consumers with 32 SOCIAL PLANNER S PROBLEM 41 the initial capital stock and have them accumulate capital and rent it to rms each period Firms then purchase capital inputs labor and capital services from consumers in competitive markets each period and maximize per period pro ts Given this it is a standard result that the competitive equilibrium is unique and that the rst and sec ond welfare theorems hold here That is the competitive equilibrium allocation is the Pareto optimum We can then solve for the competitive equilibrium quantities by solving the social planner s problem which is max i0 tuct ctantaitaktlltoo t0 subject to at 1 3 Fkt m 35 kt 1 6kt 1 36 m S 1 37 t 012 and k0 given Here we have used 31 and 32 to substitute for yt to get 35 Now since uc is strictly increasing in c 35 will be satis ed with equality As there is no disutility from labor if 37 does not hold with equality then m and at could be increased holding constant the path of the capital stock and increasing utility Therefore 37 will hold with equality at the optimum Now substitute for it in 35 using 36 and de ne E Fk l as m l for all t Then the problem can be reformulated as max 2 tuct 01k11oo t0 subject to Ct f kt1 fkt1 519 t 012 k0 given This problem appears formidable particularly as the choice set is in nite dimensional However suppose that we solve the optimization problem sequentially as follows At the beginning of any period t the utility that the social planner can deliver to the consumer depends only on kt the quantity of capital available at the beginning of the period Therefore it is natural to think of kt as a state 42 CHAPTER 3 NEOCLASSICAL GROWTH AND DYNAMIC PROGRAMMING variable for the problem Within the period the choice variables or control variables are at and kHl ln period 0 if we know the maximum utility that the social planner can deliver to the consumer as a function of k1 beginning in period 1 say vk1 it is straightforward to solve the problem for the rst period That is in period 0 the social planner solves gifwco MUCH subject to CO k1 i This is a simple constrained optimization problem which in principle can be solved for decision rules k1 gk0 where is some function and CO fk0 l 7 5160 7 gk0 Since the maximization problem is identical for the social planner in every period we can write wee Igolg mco mm subject to Co k1 fko17 51907 or more generally 110 61113ch f vkt1l 38 subject to at kt 1 7 6kt 39 Equation 38 is a functional equation or Bellman equation Our pri mary aim here is to solve for or at least to characterize the optimal decision rules kt gkt and at l 7 6kt 7 gkt Of course we cannot solve the above problem unless we know the value function In general is unknown but the Bellman equation can be used to nd it In most of the cases we will deal with the Bellman equation satis es a contraction mapping theorem which implies that 1 There is a unique function which satis es the Bellman equa tion 32 SOCIAL PLANNER S PROBLEM 43 2 If we begin with any initial function U006 and de ne 12141 by woe nde WW subject to ck fk176k for i 0 1 2 then liming 114106 vk The above two implications give us two alternative means of un covering the value function First given implication 1 above if we are fortunate enough to correctly guess the value function v then we can simply plug L t1 into the right side of 38 and then verify that 00 solves the Bellman equation This procedure only works in a few cases in particular those which are amenable to judicious guessing Second implication 2 above is useful for doing numerical work One approach is to nd an approximation to the value function in the following manner First allow the capital stock to take on only a nite number of values ie form a grid for the capital stock k E k1k2km S where m is nite and 16 lt cl1 Next guess an initial value function that is m values v6 v0kii l 2 m Then iterate on these values determining the value function at the jth iteration from the Bellman equation that is v nygxlmc vfill subject to 0 kl f 1 51913 lteration occurs until the value function converges Here the accu racy of the approximation depends on how ne the grid is That is if 16 7 cl1 y i 2m then the approximation gets better the smaller is y and the larger is m This procedure is not too computa tionally burdensome in this case where we have only one state variable However the computational burden increases exponentially as we add state variables For example if we choose a grid with m values for each state variable then if there are n state variables the search for a maximum on the right side of the Bellman equation occurs over mquot grid points This problem of computational burden as n gets large is sometimes referred to as the curse of dimensionality 44 CHAPTER 3 NEOCLASSICAL GROWTH AND DYNAMIC PROGRAMMING 321 Example of Guess and Verify Suppose that Fktnt k ni a lt cz lt 1 uct lnct and 6 1 ie 100 depreciation Then substituting for the constraint 39 in the objective function on the right side of 38 we can write the Bellman equation as we 1333111133 7 km v ml 310 Now guess that the value function takes the form vkt ABlnkt 311 where A and B are undetermined constants Next substitute using 311 on the left and right sides of 310 to get A B lnkt maxlnkf 7 kt1 A B lnkt1 312 11 Now solve the optimization problem on the right side of 312 which gives Bk 1 B and substituting for the optimal kt in 312 using 313 and col lecting terms yields kH1 313 ABlnkt Bln B 7 1 133 ln1 133 A 1 BCzln kt 33914 We can now equate coef cients on either side of 314 to get two equa tions determining A and B A Bln B71 Bln1 B A 315 B 1 BCz 316 Here we can solve 316 for B to get B 317 7 a 717C10 32 SOCIAL PLANNER S PROBLEM 45 Then we can use 315 to solve for A though we only need B to determine the optimal decision rules At this point we have veri ed that our guess concerning the form of the value function is correct Next substitute for B in 313 using 317 to get the optimal decision rule for 16t1 16t1 a kf 318 Since at 165 7 16t1 we have at 17 a 16 That is consumption and investment which is equal to 16t1 given 100 depreciation are each constant fractions of output Equation 318 gives a law of motion for the capital stock ie a first order nonlinear difference equation in kit shown in Figure 31 The steady state for the capital stock 16 is determined by substituting kit 16t1 16 in 318 and solving for 16 to get H am Given 318 we can show algebraically and in Figure 1 that kit con verges monotonically to 16 with kit increasing if 160 lt 16 and kit decreas ing if 160 gt 16 Figure 31 shows a dynamic path for kit where the initial capital stock is lower than the steady state This economy does not exhibit long run growth but settles down to a steady state where the capital stock consumption and output are constant Steady state con sumption is 0 1 7 a 160 and steady state output is y 322 Characterization of Solutions When the Value Function is Differentiable Benveniste and Scheinkman 1979 establish conditions under which the value function is differentiable in dynamic programming problems Supposing that the value function is differentiable and concave in 38 we can characterize the solution to the social planner s problem using first order conditions Substituting in the objective function for at using in the constraint we have we 1233mm lt1 7 61kt 1mm mm 319 46CHAP IER 3 NEOCLASSICAL GROWTH AND DYNAMIC PROGRAMMING Fxgure3l kw k0 Figure 3 1 32 SOCIAL PLANNER S PROBLEM 47 Then the rst order condition for the optimization problem on the right side of 38 after substituting using the constraint in the objective function is Ull kt f 1 6kt kt1l f lmCH1 0 320 The problem here is that without knowing v we do not know v However from 319 we can differentiate on both sides of the Bellman equation with respect to kt and apply the envelope theorem to obtain WWW Ull kt f 1 6W kt1llf kt 1 Sly or updating one period Wkt Ullfkt11 6kt1 kt2llflkt11 6l 321 Now substitute in 320 for v kt1 using 321 to get Ull kt f 1 6kt kt1l wu tml lt1 e 6m e mumcut 1 e 6 0 33922 UTCt f ulct1lflkt11 6l 07 The rst term is the bene t at the margin to the consumer of consum ing one unit less of the consumption good in period t and the second term is the bene t obtained in period t l discounted to period t from investing the foregone consumption in capital At the optimum the net bene t must be zero We can use 322 to solve for the steady state capital stock by setting kt kt kt2 16 to get f k 71 6 323 ie one plus the net marginal product of capital is equal to the inverse of the discount factor Therefore the steady state capital stock depends only on the discount factor and the depreciation rate 48 CHAPTER 3 NEOCLASSICAL GROWTH AND DYNAMIC PROGRAMMING 323 Competitive Equilibrium Here l will simply assert that the there is a unique Pareto optimum that is also the competitive equilibrium in this model While the most straightforward way to determine competitive equilibrium quantities in this dynamic model is to solve the social planner s problem to nd the Pareto optimum to determine equilibrium prices we need some infor mation from the solutions to the consumer s and firm s optimization problems Consumer s Problem Consumers store capital and invest Le their wealth takes the form of capital and each period they rent capital to rms and sell labor Labor supply will be 1 no matter what the wage rate as consumers receive no disutility from labor The consumer then solves the following intertemporal optimization problem 0 t max u c Ctakt1oo t subject to at kt wt 73kt l 7 316 324 t 012 k0 given where wt is the wage rate and n is the rental rate on capital lf we simply substitute in the objective function using 324 then we can reformulate the consumer s problem as 0 max Z tUOJR f 731 1 6kt kt1 kt1100 t0 subject to kt 2 0 for all t and k0 given lgnoring the nonnegativity constraints on capital in equilibrium prices will be such that the con sumer will always choose kt gt 0 the first order conditions for an optimum are tUIUUt f 731 f 1 6kt kt1 1u wt1 Tt1kt1 f 1 6kt1 kt27 t1 1 6 0 325 32 SOCIAL PLANNER S PROBLEM 49 Using 324 to substitute in 325 and simplifying we get u c 1 M 326 u ct 1 n1 7 6 that is the intertemporal marginal rate of substitution is equal to the inverse of one plus the net rate of return on capital ie one plus the interest rate Firm s Problem The rm simply maximizes pro ts each period ie it solves maxF t7m wtnt Ttktlv and the first order conditions for a maximum are F1ktm n 327 FZUWJ H wt 328 Competitive Equilibrium Prices The optimal decision rule kt gkt which is determined from the dynamic programming problem 38 allows a solution for the compet itive equilibrium sequence of capital stocks k til given 1 We can then solve for 820 using 39 Now it is straightforward to solve for competitive equilibrium prices from the first order conditions for the firm s and consumer s optimization problems The prices we need to solve for are whn im the sequence of factor prices To solve for the real wage plug equilibrium quantities into 328 to get F206 wt To obtain the capital rental rate either 326 or 327 can be used Note that n 7 6 f kt 7 6 is the real interest rate and that in the steady state from 326 or 323 we have 1 7 7 6 or if we let where 77 is the rate of time preference then 7 7 6 77 ie the real interest rate is equal to the rate of time preference 50 CHAPTER 3 NEOCLASSICAL GROWTH AND DYNAMIC PROGRAMMING Note that when the consumer solves her optimization problem she knows the whole sequence of prices wh rtf0 That is this a rational expectations or perfect foresight equilibrium where each period the consumer makes forecasts of future prices and optimizes based on those forecasts and in equilibrium the forecasts are correct In an economy with uncertainty a rational expectations equilibrium has the property that consumers and rms may make errors but those errors are not systematic 33 References Benveniste L and Scheinkman J 1979 On the Differentiability of the Value Function in Dynamic Models of Economics Econo metrica 47 727 732 Brock W and Mirman L 1972 Optimal Economic Growth and Uncertainty The Discounted Case Journal of Economic Theory 4 479 513 Cass D 1965 Optimum Growth in an Aggregative Model of Capital Accumulation Review of Economic Studies 32 233 240 Koopmans T 1965 On the Concept of Optimal Growth in The Econometric Approach to Development Planning Chicago Rand McNally Chapter 4 Endogenous Growth This chapter considers a class of endogenous growth models closely related to the ones in Lucas 1988 Here we use discrete time models so that the dynamic programming methods introduced in Chapter 2 can be applied Lucas s models are in continuous time Macroeconomists are ultimately interested in economic growth be cause the welfare consequences of government policies affecting growth rates of GDP are potentially very large In fact one might argue as in Lucas 1987 that the welfare gains from government policies which smooth out business cycle fluctuations are very small compared to the gains from growth enhancing policies Before we can hope to evaluate the ef cacy of government policy in a growth context we need to have growth models which can successfully confront the data Some basic facts of economic growth as much as we can tell from the short history in available data are the following 1 There exist persistent differences in per capita income across countries 2 There are persistent differences in growth rates of per capita in come across countries 3 The correlation between the growth rate of income and the level of income across countries is low 4 Among rich countries there is stability over time in growth rates 51 52 CHAPTER 4 ENDOGENOUS GROWTH of per capita income and there is little diversity across countries in growth rates 5 Among poor countries growth is unstable and there is a wide diversity in growth experience Here we first construct a version of the optimal growth model in Chapter 2 with exogenous growth in population and in technology and we ask whether this model can successfully explain the above growth facts This neoclassical growth model can successfully account for growth experience in the United States and it offers some insights with regard to the growth process but it does very poorly in accounting for the pattern of growth among countries Next we consider a class of endogenous growth models and show that these models can potentially do a better job of explaining the facts of economic growth 41 A Neoclassical Growth Model Exoge nous Growth The representative household has preferences given by gt0 02 Z tNt 7 41 t0 7 where 0 lt lt l y lt l at is per capita consumption and Ni is population where N 1 n N0 42 n constant and N0 given That is there is a dynastic household which gives equal weight to the discounted utility of each member of the household at each date Each household member has one unit of time in each period when they are alive which is supplied inelastically as labor The production technology is given by Yt KNtAt1Ta7 43 where Yt is aggregate output Kt is the aggregate capital stock and At is a labor augmenting technology factor where 4 1 a A0 44 41 A NEOCLASSICAL GROWTH MODEL EXOGENOUS GROWTH53 with a constant and A0 given We have 0 lt CZ lt l and the initial capital stock K0 is given The resource constraint for this economy is Ntc KH1 K 45 Note here that there is 100 depreciation of the capital stock each period for simplicity To determine a competitive equilibrium for this economy we can solve the social planner s problem as the competitive equilibrium and the Pareto optimum are identical The social planner s problem is to maximize 41 subject to 42 45 So that we can use dynamic programming methods and so that we can easily characterize long run growth paths it is convenient to set up this optimization problem with a change of variables That is use lower case variables to de ne quantities normalized by ef ciency units of labor for example yt E Also let xi E it With substitution in 41 and 45 using 42 44 the social planner s problem is then max 1 n 1 1 7 t t mmm subject to xt1n1akt1 kf t012 46 This optimization problem can then be formulated as a dynamic pro gram with state variable kt and choice variables xi and kt1 That is given the value function vkt the Bellman equation is nk venm j umamww subject to 46 Note here that we require the discount factor for the problem to be less than one that is l nl 17 lt 1 Substituting in the objective function for xi using 46 we have vkt max kz1 W kt11 n1 COW aumumww j 47 54 CHAPTER 4 ENDOGENOUS GROWTH The rst order condition for the optimization problem on the right side of 47 is 71 n1 1002 1 1 n1 aquotv kt1 0 48 and we have the following envelope condition v kt ak lel 49 Using 49 in 48 and simplifying we get 1 17795371 akl l l 0 410 Now we will characterize balanced growth paths that is steady states where xi f and kt 16 where f and 16 are constants Since 410 must hold on a balanced growth path we can use this to solve for 16 that is 411 Then 46 can be used to solve for f to get 7 1 n 1 a 412 1 a 50 lt gtlt gt lt gt Also since yt kf then on the balanced growth path the level of output per ef ciency unit of labor is x m Maw a ya k4ltoz In addition the saVings rate is Kt1 kt11 0 a 5t Y k so that on the balanced growth path the saVings rate is 5 k1quot11 n1 a 41 A NEOCLASSICAL GROWTH MODEL EXOGENOUS GROWTH55 Therefore using 411 we get 5 a1 n1 11 414 Here we focus on the balanced growth path since it is known that this economy will converge to this path given any initial capital stock K0 gt 0 Since 16 f and y are all constant on the balanced growth path it then follows that the aggregate capital stock Kt aggregate consumption Ntct and aggregate output Yt all grow approximately at the common rate a n and that per capita consumption and out put grow at the rate 1 Thus long run growth rates in aggregate vari ables are determined entirely by exogenous growth in the labor force and exogenous technological change and growth in per capita income and consumption is determined solely by the rate of technical change Changes in any of the parameters C1 or 7 have no effect on long run growth Note in particular that an increase in any one of Cl or 7 results in an increase in the long run savings rate from 414 But even though the savings rate is higher in each of these cases growth rates remain unaffected This is a counterintuitive result as one might anticipate that a country with a high savings rate would tend to grow faster Changes in any of Cl or y do however produce level effects For example an increase in which causes the representative household to discount the future at a lower rate results in an increase in the savings rate from 414 and increases in 16 and y from 411 and 413 We can also show that 1 n1 11 lt 1 implies that an increase in steady state 16 will result in an increase in steady state f Therefore an increase in leads to an increase in x Therefore the increase in yields increases in the level of output consumption and capital in the long run Suppose that we consider a number of closed economies which all look like the one modelled here Then the model tells us that given the same technology and it is hard to argue that in terms of the logic of the model all countries would not have access to At all countries will converge to a balanced growth path where per capita output and consumption grow at the same rate From 413 the differences in the level of per capita income across countries would have to be ex plained by differences in CM or 7 But if all countries have access 56 CHAPTER 4 ENDOGENOUS GROWTH to the same technology then CZ cannot vary across countries and this leaves an explanation of differences in income levels due to differences in preferences This seems like no explanation at all While neoclassical growth models were used successfully to account for long run growth patterns in the United States the above analysis indicates that they are not useful for accounting for growth experience across countries The evidence we have seems to indicate that growth rates and levels of output across countries are not converging in con trast to what the model predicts 42 A Simple Endogenous Growth Model In attempting to build a model which can account for the principal facts concerning growth experience across countries it would seem necessary to incorporate an endogenous growth mechanism to permit economic factors to determine long run growth rates One way to do this is to introduce human capital accumulation We will construct a model which abstracts from physical capital accumulation to focus on the essential mechanism at work and introduce physical capital in the next section Here preferences are as in 41 and each agent has one unit of time which can be allocated between time in producing consumption goods and time spent in human capital accumulation The production technology is given by Yt ClhtUtNm where C1 gt 0 Yt is output ht is the human capital possessed by each agent at time t and ut is time devoted by each agent to production That is the production function is linear in quality adjusted labor in put Human capital is produced using the technology ht1 WHO W 415 where 6 gt 0 l 7 ut is the time devoted by each agent to human capital accumulation ie education and acquisition of skills and he is given Here we will use lower case letters to denote variables in per capita terms for example yt E The social planner s problem can then 42 A SIMPLE ENDOGENOUS GROWTH MODEL 57 be formulated as a dynamic programming problem where the state variable is ht and the choice variables are ct ht and ut That is the Bellman equation for the social planner s problem is C Y Walt max 7t M1 n ht1 31711 h1 subject to at czhtut 416 and 415 Then the Lagrangian for the optimization problem on the right side of the Bellman equation is C Y 3 7t 1 nvht1 tahtut Ct Mtl6ht1 ut ht1lv where At and m are Lagrange multipliers Two rst order conditions for an optimum are then 81 a 03 1 7 A 0 417 81 W WW 7 m o 418 ah 415 and 416 In addition the rst derivative of the Lagrangian with respect to ut is all Ataht 7 Mt ht But Now if 3 gt 0 then ut 1 But then from 415 and 416 we have hs CS 0 for s t1t 2 But since the marginal utility of consumption goes to in nity as consumption goes to zero this could not be an optimal path Therefore 88 S 0 If 88 lt 0 then ut 0 and at 0 from 416 Again this could not be optimal so we must have 81 8 1 Ataht 7 Mt ht 0 419 at the optimum 58 CHAPTER 4 ENDOGENOUS GROWTH We have the following envelope condition v ht gut Ata1 7 ut or using 417 v ht ac 420 From 417 420 we then get 51 M643 7 a 0 421 Therefore we can rewrite 421 as an equation determining the equi librium growth rate of consumption W1 nm 422 Ct Then using 415 416 and 422 we obtain W1 We W W Ut1 1 u 7 t 423 Now 423 is a first order difference equation in ut depicted in Figure 41 for the case where 111 16 1 lt 1 a condition we will assume holds Any path 1 in satisfying 423 which is not stationary a stationary path is ut u a constant for all t has the property that limtnoo ut 0 which cannot be an optimum as the representative consumer would be spending all available time accumulating human capital which is never used to produce in the future Thus the only solution from 423 is u u 17 1 nm for all t Therefore substituting in 415 we get ht1 7 T i 151 61 7 43 ENDOGENOUS GROWTH WITHPHYSICAL CAPITAL AND HUMAN CAPITAL59 and human capital grows at the same rate as consumption per capita lf gt 1 which will hold for 6 suf ciently large then growth rates are positive There are two important results here The first is that equilibrium growth rates depend on more than the growth rates of exogenous factors Here even if there is no growth in popu lation n 0 and given no technological change this economy can exhibit unbounded growth Growth rates depend in particular on the discount factor growth increases if the future is discounted at a lower rate and 6 which is a technology parameter in the human capital accu mulation function if more human capital is produced for given inputs the economy grows at a higher rate Second the level of per capita income equal to per capita consumption here is dependent on initial conditions That is since growth rates are constant from for all t the level of income is determined by ho the initial stock of human capital Therefore countries which are initially relatively rich poor will tend to stay relatively rich poor The lack of convergence of levels of income across countries which this model predicts is consistent with the data The fact that other factors besides exogenous technological change can affect growth rates in this type of model opens up the possibility that differences in growth across countries could be explained in more complicated models by factors including tax policy educational policy and savings behavior 43 Endogenous Growth With Physical Cap ital and Human Capital The approach here follows closely the model in Lucas 1988 except that we omit his treatment of human capital externalities The model is identical to the one in the previous section except that the production technology is given by Yt KNthtut1ia7 where Kt is physical capital and 0 lt CZ lt l and the economy s resource constraint is NtCt Kt1 K NthtuN a mm CHAPTER 4 Fxgure41 Figure 4 1 ENDOGENO US GRD WTH 45 deg me u tanonary soln 0 43 ENDOGENOUS GROWTH WITHPHYSICAL CAPITAL AND HUMAN CAPITAL61 As previously we use lower case letters to denote per capita quantities In the dynamic program associated with the social planner s optimiza tion problem there are two state variables kt and h and four choice variables ut ct ht1 and kt The Bellman equation for this dynamic program is C Y 110 ht max t 51 nvkt17 ht1 Ctautaktlahtl 7 subject to c 1 mic1 7 khtut1quotquot 424 hH1 7 ma 7 ut 425 The Lagrangian for the constrained optimization problem on the right side of the Bellman equation is then 07 I 7t 1 kt1v ht1tlkhtm1 0 Cr1nkt1lm6ht17mhml The first order conditions for an optimum are then 8E 8 0 7 oz 17 A 7 o 426 8 1 7 Mi 7 akf hquotquotuquot 7 Mb 7 0 427 But all M1 WKCH1 ht1 Mt 07 428 aht1 8E 7At1 n 1 nvlkt1ht1 0 429 8kt1 424 and 425 We also have the following envelope conditions Ulktht Atak 1htut1 a 430 122kt ht At1 7 czkf h 1u a Mt61 7 ut 431 Next use 430 and 431 to substitute in 429 and 428 respec tively then use 426 and 427 to substitute for At and m in 428 and 429 After simplifying we obtain the following two equations 70271 c fczk llwt ut 1 0 0 432 62 CHAPTER 4 ENDOGENOUS GROWTH 70341921qu 6 1 maggfkglhmugfl 0 433 Now we wish to use 424 425 432 and 433 to characterize a balanced growth path along which physical capital human capital and consumption grow at constant rates Let Mk p4 and MC denote the growth rates of physical capital human capital and consumption respectively on the balanced growth path From 425 we then have 1Mh 61 Ut7 which implies that 1 Mn 6 7 a constant along the balanced growth path Therefore substituting for ut qu and growth rates in 433 and simplifying we get ut17 lt1mgt13lt1mgtwlt1uhgta 6 1n 434 Next dividing 424 through by kt we have Ct k 1 int 1 k31hu1a 435 k k Then rearranging 432 and backdating by one period we get 1 T M01T Y 047 70 T kt 1htut1 436 Equations 435 and 436 then imply that Ct 1 T Mc177 l l ktlt ngtlt m 50 But then 2 is a constant on the balanced growth path which implies that Ms Mk Also from 436 since ut is a constant it must be the case that We p4 Thus per capita physical capital human capital and per capita consumption all grow at the same rate along the balanced growth path and we can determine this common rate from 434 ie 1M01Mk1Mh1p6 1n 437 4 4 REFERENCES 63 Note that the growth rate on the balanced growth path in this model is identical to what it was in the model of the previous section The savings rate in this model is i Kt1 kt11n Sti Yt i ktk 1htUt1 a Using 436 and 437 on the balanced growth path we then get 5 a mu mn 438 In general then from 437 and 438 factors which cause the savings rate to increase increases in n or 6 also cause the growth rate of per capita consumption and income to increase 44 References Lucas RE 1987 Models of Business Cycles Basil Blackwell New York Lucas RE 1988 On the Mechanics of Economic Development Journal of Monetary Economics 22 3 42 64 CHAPTER 4 ENDOGENOUS GROWTH Chapter 5 Choice Under Uncertainty In this chapter we will introduce the most commonly used approach to the study of choice under uncertainty expected utility theory Expected utility maximization by economic agents permits the use of stochastic dynamic programming methods in solving for competitive equilibria We will first provide an outline of expected utility theory and then illustrate the use of stochastic dynamic programming in a neoclassical growth model with random disturbances to technology This stochastic growth model is the basis for real business cycle theory 51 Expected Utility Theory In a deterministic world we describe consumer preferences in terms of the ranking of consumption bundles However if there is uncertainty then preferences are de ned in terms of how consumers rank lotteries over consumption bundles The axioms of expected utility theory im ply a ranking of lotteries in terms of the expected value of utility they yield for the consumer For example suppose a world with a single con sumption good where a consumer s preferences over certain quantities of consumption goods are described by the function uc where c is consumption Now suppose two lotteries over consumption Lottery i gives the consumer 01 units of consumption with probability pi and 02 units of consumption with probability 1 ipi where 0 lt p lt l i l 2 65 66 CHAPTER 5 CHOICE UNDER UN CERTAIN TY Then the expected utility the consumer receives from lottery i is 13144011 1 13314012 and the consumer would strictly prefer lottery l to lottery 2 if P1UCl 1 130140 gt 132140 1 1391405 would strictly prefer lottery 2 to lottery 1 if P1UCl 1 130140 lt 132140 1 1391405 and would be indifferent if P1UCl 1 130140 132140 1 1391405 Many aspects of observed behavior toward risk for example the obser vation that consumers buy insurance is consistent with risk aversion An expected utility maximizing consumer will be risk averse with re spect to all consumption lotteries if the utility function is strictly con cave lf uc is strictly concave this implies Jensen s inequality that is Elu0l S UEl0l7 51 where E is the expectation operator This states that the consumer prefers the expected value of the lottery with certainty to the lottery itself That is a risk averse consumer would pay to avoid risk If the consumer receives constant consumption 5 with certainty then clearly 51 holds with equality In the case where consumption is random we can show that 51 holds as a strict inequality That is take a tangent to the function uc at the point see Figure 1 This tangent is described by the function 90 Ct W 5 where Cl and are constants and we have ex EM 53 Now since uc is strictly concave we have as in Figure 1 Cr e 2 uc 54 51 EXPECTED UTILITY THEORY 67 for c 2 0 with strict inequality if c a Since the expectation operator is a linear operator we can take expectations through 54 and given that c is random we have C1 Elcl gt EMCH or using 53 UEl0l gt Elu0l As an example consider a consumption lottery which yields 01 units of consumption with probability p and Cg units with probability 1 7 p where 0 lt p lt l and Cg gt 01 In this case 51 takes the form PU01 1 PUC2 lt U P01 1 PM In Figure 2 the difference UP01 1 PC2 PU01 1 PUC2l is given by DE The line AB is given by the function a 02ucl 7 clu02 u02 7 1401 C Cg Cl Cg 01 A point on the line AB denotes the expected utility the agent receives for a particular value of p for example 1 0 yields expected utility ucl or point A and B implies p l Jensen s inequality is reflected in the fact that AB lies below the function Note that the distance DE is the disutility associated with risk and that this distance will increase as we introduce more curvature in the utility function ie as the consumer becomes more risk averse 511 Anomalies in Observed Behavior Towards Risk While expected utility maximization and a strictly concave utility func tion are consistent with the observation that people buy insurance some observed behavior is clearly inconsistent with this For exam ple many individuals engage in lotteries with small stakes where the expected payoff is negative 68 CHAPTER 5 CHOICE UNDER UN CERTAIN TY Another anomaly is the Allais Paradox Here suppose that there are four lotteries which a person can enter at zero cost Lottery 1 involves a payoff of 1 million with certainty lottery 2 yields a payoff of 5 million with probability 1 1 million with probability 89 and 0 with probability 01 lottery 3 yields 1 million with probability 11 and 0 with probability 89 lottery 4 yields 5 million with probability 1 and 0 with probability 9 Experiments show that most people prefer lottery 1 to lottery 2 and lottery 4 to lottery 3 But this is inconsistent with expected utility theory whether the person is risk averse or not is irrelevant That is if is an agent s utility function and they maximize expected utility then a preference for lottery 1 over lottery 2 gives u1 gt 1u5 89u1 01u0 or 11u1 gt 1u5 01u0 55 Similarly a preference for lottery 4 over lottery 3 gives 11u1 89u0 lt 1u5 9u0 OI 11u1 lt 1u5 9u0 56 and clearly 55 is inconsistent with 56 Though there appear to be some obvious violations of expected util ity theory this is still the standard approach used in most economic problems which involve choice under uncertainty Expected utility the ory has proved extremely useful in the study of insurance markets the pricing of risky assets and in modern macroeconomics as we will show 512 Measures of Risk Aversion With expected utility maximization choices made under uncertainty are invariant with respect to af ne transformations of the utility func tion That is suppose a utility function W M We 51 EXPECTED UTILITY THEORY 69 where Cl and are constants with gt 0 Then we have EWCH C1 EMCH since the expectation operator is a linear operator Thus lotteries are ranked in the same manner with 120 or uc as the utility function Any measure of risk aversion should clearly involve u c since risk aversion increases as curvature in the utility function increases However note that for the function 00 that we have 1 0 u c ie the second derivative is not invariant to af ne transformations which have no effect on behavior A measure of risk aversion which is invariant to af ne transformations is the measure of absolute risk aversion ARAc 350 A utility function which has the property that ARAC is constant for all c is uc 76quot C1 gt 0 For this function we have 7 2 70w ARAC 7L cz Cleft It can be shown through Taylor series expansion arguments that the measure of absolute risk aversion is twice the maximum amount that the consumer would be willing to pay to avoid one unit of variance for small risks An alternative is the relative risk aversion measure We 39 A utility function for which RPM10 is constant for all c is RRAC 70 01quot 7 1 uc 1 Y 7 where y 2 0 Here 7 41M RRAC foile 7 0 7 70 CHAPTER 5 CHOICE UNDER UN CERTAIN TY Note that the utility function uc lnc has RPM10 l The measure of relative risk aversion can be shown to be twice the maximum amount per unit of variance that the consumer would be willing to pay to avoid a lottery if both this maximum amount and the lottery are expressed as proportions of an initial certain level of consumption A consumer is risk neutral if they have a utility function which is linear in consumption that is uc e where gt 0 We then have E ucl Elcl so that the consumer cares only about the expected value of consump tion Since u c 0 and u c we have ARAC RRAC O 52 Stochastic Dynamic Programming We will introduce stochastic dynamic programming here by way of an example which is essentially the stochastic optimal growth model studied by Brock and Mirman 1972 The representative consumer has preferences given by gt0 E0 2 tuct t0 where 0 lt lt 1 at is consumption is strictly increasing strictly concave and twice differentiable and E0 is the expectation operator conditional on information at t 0 Note here that in general at will be random The representative consumer has 1 unit of labor available in each period which is supplied inelastically The production technology is given by M ZtFkt7 07 where F is strictly quasiconcave homogeneous of degree one and increasing in both argument Here kt is the capital input m is the labor input and 2t is a random technology disturbance That is z fio is a sequence of independent and identically distributed iid random variables each period 2t is an independent draw from a xed probability distribution In each period the current realization 2t is learned 52 STOCHASTIC DYNAMIC PROGRAMMING 71 at the beginning of the period before decisions are made The law of motion for the capital stock is kt1 it 1 519 where it is investment and 6 is the depreciation rate with 0 lt 6 lt l The resource constraint for this economy is Ct 12 yt 521 Competitive Equilibrium In this stochastic economy there are two very different ways in which markets could be organized both of which yield the same unique Pareto optimal allocation The rst is to follow the approach of Arrow and Debreu see Arrow 1983 or Debreu 1983 The representative consumer accumulates capital over time by saving and in each period heshe rents capital and sells labor to the representative rm However the contracts which specify how much labor and capital services are to be delivered at each date are written at date t 0 At t 0 the representative rm and the representative consumer get together and trade contingent claims on competitive markets A contingent claim is a promise to deliver a speci ed number of units of a particular object in this case labor or capital services at a particular date say date T conditional on a particular realization of the sequence of technology shocks 202122 2T In a competitive equilibrium all contingent claims markets and there are potentially very many of these clear at t 0 and as information is revealed over time contracts are executed according to the promises made at t 0 Showing that the competitive equilibrium is Pareto optimal here is a straightforward extension of general equilibrium theory with many state contingent commodities The second approach is to have spot market trading with rational expectations That is in period t labor is sold at the wage rate wt and capital is rented at the rate 73 At each date the consumer rents capital and sells labor at market prices and makes an optimal sav ings decision given his her beliefs about the probability distribution of future prices In equilibrium markets clear at every date t for every 72 CHAPTER 5 CHOICE UNDER UN CERTAIN TY possible realization of the random shocks 20 2122 2t ln equilib rium expectations are rational in the sense that agents7 beliefs about the probability distributions of future prices are the same as the ac tual probability distributions In equilibrium agents can be surprised in that realizations of 2t may occur which may have seemed ex ante to be small probability events However agents are not systematically fooled since they make ef cient use of available information In this representative agent environment a rational expectations equilibrium is equivalent to the Arrow Debreu equilibrium but this will not be true in models with heterogeneous agents In those models complete markets in contingent claims are necessary to support Pareto optima as competitive equilibria as complete markets are required for ef cient risk sharing 522 Social Planner s Problem Since the unique competitive equilibrium is the Pareto optimum for this economy we can simply solve the social planner s problem to determine competitive equilibrium quantities The social planner s problem is 0 max E0 2 tuct fetaktllzoio t0 subject to Ct kt1 thUW f 1 lktv where E FUC 1 Setting up the above problem as a dynamic program is a fairly straightforward generalization of discrete dynamic programming with certainty In the problem given the nature of uncer tainty the relevant state variables are kt and 2t where kt is determined by past decisions and 2t is given by nature and known when decisions are made concerning the choice variables at and kHl The Bellman equation is written as Ukt72t 0133 lUCt EtUkt172t1l subject to Ct kt1 thkt f 1 lkt 52 STOCHASTIC DYNAMIC PROGRAMMING 73 Here 1 is the value function and Et is the expectation operator conditional on information in period t Note that in period t at is known but CHI 139 l 23 is unknown That is the value of the problem at the beginning of period t l the expected utility of the representative agent at the beginning of period t l is uncertain as of the beginning of period t What we wish to determine in the above problem are the value function v and optimal decision rules for the choice variables ie kt 90 2t and at zthct l 7 6kt 7 gkt 2t 523 Example Let Fkn mania with 0 lt a lt 1 uc lnct 6 1 and Eln 2t M Guess that the value function takes the form vktzt A BlnktDlnzt The Bellman equation for the social planner s problem after substi tuting for the resource constraint and given that m l for all t is then AB lnktD lnzt k ax mmcf 7 n1 EJA B lnkt1 Dln 2H1 11 or AB ln ktD lnzt max mmcf 7 n1 A B lnkt1 DM 11 57 Solving the optimization problem on the right hand side of the above equation gives B a CH4 mztkt Then substituting for the optimal kt in 57 we get 2k 1 B thcf 1 B ABlnktDlnztlnlt gt A Blnlt gt DM 59 74 CHAPTER 5 CHOICE UNDER UN CERTAIN TY Our guess concerning the value function is veri ed if there exists a solution for A B and D Equating co ef cients on either side of equation 59 gives 1 B Bczcz B 511 D 1 B 512 Then solving 510 512 for A B and D gives CZ km 1 km 7 1 w W Ai li ln1ia 1ia lna 1ia We have now shown that our conjecture concerning the value function is correct Substituting for B in 58 gives the optimal decision rule kt a ztkf 513 and since at 2k 7 kt1 the optimal decision rule for at is at 1 7 a ztkf 514 Here 513 and 514 determine the behavior of time series for at and kt where kt is investment in period t Note that the economy will not converge to a steady state here as technology disturbances will cause persistent fluctuations in output consumption and investment How ever there will be convergence to a stochastic steady state ie some joint probability distribution for output consumption and investment This model is easy to simulate on the computer To do this simply assume some initial 1 determine a sequence able using a random number generator and xing T and then use 513 and 514 to de termine time series for consumption and investment These time series 5 3 REFERENCES 75 will have properties that look something like the properties of post war detrended US time series though there will be obvious ways in which this model does not t the data For example employment is constant here while it is variable in the data Also given that output yt ztkf if we take logs through 513 and 514 we get 1nkt1 lncz 1n yt and lnct ln1 7 a 1n yt We therefore have uarlnkt1 uarlnct uarlnyt But in the data the log of investment is much more variable about trend than is the log of output and the log of output is more variable than the log of consumption Real business cycle RB C analysis is essentially an exercise in mod ifying this basic stochastic growth model to t the post war US time series data The basic approach is to choose functional forms for util ity functions and production functions and then to choose parameters based on long run evidence and econometric studies Following that the model is run on the computer and the output matched to the ac tual data to judge the t The tted model can then be used given that the right amount of detail is included in the model to analyze the effects of changes in government policies For an overview of this literature see Prescott 1986 and Cooley 1995 5 3 References Arrow K 1983 Collected Papers of Kenneth J Arrow Vol 2 General Equilibrium Harvard University Press Cambridge MA Brock WA and Mirman L 1972 Optimal Economic Growth and Uncertainty the Discounted Case Journal of Economic Theory 4 479 513 Cooley T 1995 Frontiers of Business Cycle Research Princeton University Press Princeton NJ 76 CHAPTER 5 CHOICE UNDER UN CERTAIN TY Debreu G 1983 Mathematical Economics Twenty Papers of Gerard Debreu Cambridge University Press Cambridge Prescott E 1986 Theory Ahead of Business Cycle Measurement Federal Reserve Bank of Minneapolis Quarterly Review Fall 9 22 Stokey N and Lucas R 1989 Recursive Methods m Economic Dy namics Harvard University Press Cambridge MA Chapter 6 Consumption and Asset Pricing In this chapter we will examine the theory of consumption behavior and asset pricing in dynamic representative agent models These two topics are treated together because there is a close relationship between the behavior of consumption and asset prices in this class of models That is consumption theory typically treats asset prices as being exogenous and determines optimal consumption savings decisions for a consumer However asset pricing theory typically treats aggregate consumption as exogenous while determining equilibrium asset prices The stochastic implications of consumption theory and asset pricing theory captured in the stochastic Euler equations from the representative consumer s problem look quite similar 61 Consumption The main feature of the data that consumption theory aims to explain is that aggregate consumption is smooth relative to aggregate income Traditional theories of consumption which explain this fact are Fried man s permanent income hypothesis and the life cycle hypothesis of Modigliani and Brumberg Friedman s and Modigliani and Brumberg s ideas can all be exposited in a rigorous way in the context of the class of representative agent models we have been examining 77 78 CHAPTER 6 CONSUMPTION AND ASSET PRICING 611 Consumption Behavior Under Certainty The model we introduce here captures the essentials of consumption smoothing behavior which are important in explaining why consump tion is smoother than income Consider a consumer with initial assets A0 and preferences tMct 61 where 0 lt lt 1 at is consumption and is increasing strictly concave and twice di erentiable The consumer s budget constraint is At1 1 At 7 Ct wtb 62 for t 0 1 2 where 7 is the one period interest rate assumed con stant over time and wt is income in period t where income is exoge nous We also assume the no Ponzi scheme condition At hm taco 1 TV This condition and 62 gives the intertemporal budget constraint for the consumer lt1 at A0 1 mt 6393 The consumer s problem is to choose ctAt1f0 to maximize 61 subject to 62 Formulating this problem as a dynamic program with the value function 0At assumed to be concave and di erentiable the Bellman equation is Am A 13 WW W 571mm The rst order condition for the optimization problem on the right hand side of the Bellman equation is A u w A 7 1 5mm 0 64 1 1 7 and the envelope theorem gives A tut u w A 7 65 61 CONSUMPTION 79 Therefore substituting in 64 using 65 and 62 gives MW 5 Ct1 That is the intertemporal marginal rate of substitution is equal to one plus the interest rate at the optimum Now consider some special cases lf 1 7 1 7 66 ie if the interest rate is equal to the discount rate then 66 gives Ct Ct1 0 for all t where from 63 we get c1 A0 67 Here consumption in each period is just a constant fraction of dis counted lifetime wealth or permanent income The income stream given by 71in could be highly variable but the consumer is able to smooth consumption perfectly by borrowing and lending in a perfect capital market Also note that 67 implies that the response of con sumption to an increase in permanent income is very small That is suppose a period is a quarter and take 7 01 an interest rate of approximately 4 per annum Then 67 implies that a 1 increase in current income gives an increase in current consumption of 0099 This is an important implication of the permanent income hypothe sis because consumers smooth consumption over time the impact on consumption of a temporary increase in income is very small Another example permits the discount factor to be different from the interest rate but assumes a particular utility function in this case where y gt 0 Now from 66 we get W1 mi 68 Ct 80 CHAPTER 6 CONSUMPTION AND ASSET PRICING so that consumption grows at a constant rate for all t Again the consumption path is smooth From 68 we have Ct 00W1 7 and solving for 00 using 63 we obtain co 17 1r1ij A0g 1 134 612 Consumption Behavior Under Uncertainty Friedman s permanent income hypothesis was a stochastic theory aimed at explaining the regularities in short run and long run consumption behavior but Friedman did not develop his theory in the context of an optimizing model with uncertainty This was later done by Hall 1978 and the following is essentially Hall s model Consider a consumer with preferences given by E0 2 tuct where has the same properties as in the previous section The consumer s budget constraint is given by 62 but now the consumer s income wt is a random variable which becomes known at the beginning of period t Given a value function UAt wt for the consumer s problem the Bellman equation associated with the consumer s problem is A 39UAt7wt max UAt wt H1 Et39UAt17wt1 7 Az1 l 7 and the first order condition for the maximization problem on the right hand side of the Bellman equation is Am 1 4 Etvi At17wt1 69 u At wt 7 71 We also have the following envelope condition Am A A 7 610 Uilt tth U twt 17 61 CONSUMPTION 81 Therefore from 62 69 and 610 we obtain Etu ct1 u ct 611 1 1 7 Here 611 is a stochastic Euler equation which captures the stochas tic implications of the permanent income hypothesis for consumption Essentially 611 states that u ct is a martingale with drift How ever without knowing the utility function this does not tell us much about the path for consumption 1f we suppose that is quadratic ie uct 75 7 ct2 where E gt 0 is a constant 611 gives 51 r 71 EC c t t lt1m t so that consumption is a martingale with drift That is consumption is smooth in the sense that the only information required to predict future consumption is current consumption A large body of empirical work summarized in Hall 1989 comes to the conclusion that 611 does not t the data well Basically the problem is that consumption is too variable in the data relative to what the theory predicts in practice consumers respond more strongly to changes in current income than theory predicts they should There are at least two explanations for the inability of the perma nent income model to t the data The rst is that much of the work on testing the permanent income hypothesis is done using aggregate data But in the aggregate the ability of consumers to smooth con sumption is limited by the investment technology In a real business cycle model for example asset prices move in such a way as to induce the representative consumer to consume what is produced in the cur rent period That is interest rates are not exogenous or constant as in Hall s model in general equilibrium In a real business cycle model the representative consumer has an incentive to smooth consumption and these models t the properties of aggregate consumption well A second possible explanation which has been explored by many authors see Hall 1989 is that capital markets are imperfect in prac tice That is the interest rates at which consumers can borrow are typically much higher than the interest rates at which they can lend 82 CHAPTER 6 CONSUMPTION AND ASSET PRICING and sometimes consumers cannot borrow on any terms This limits the ability of consumers to smooth consumption and makes consumption more sensitive to changes in current income 62 Asset Pricing In this section we will study a model of asset prices developed by Lucas 1978 which treats consumption as being exogenous and asset prices as endogenous This asset pricing model is sometimes referred to as the lCAPM intertemporal capital asset pricing model or the consumption based capital asset pricing model This is a representative agent economy where the representative consumer has preferences given by gt0 EOZ tMct 612 t0 where 0 lt lt l and is strictly increasing strictly concave and twice di erentiable Output is produced on n productive units where yit is the quantity of output produced on productive unit i in period t Here yit is random We can think of each productive unit as a fruit tree which drops a random amount of fruit each period It is clear that the equilibrium quantities in this model are simply ct Zyit 613 i1 but our interest here is in determining competitive equilibrium prices However what prices are depends on the market structure We will suppose an stock market economy where the representative consumer receives an endowment of 1 share in each productive unit at t 0 and the stock of shares remains constant over time Each period the output on each productive unit the dividend is distributed to the shareholders in proportion to their share holdings and then shares are traded on competitive markets Letting pit denote the price of a share in productive unit i in terms of the consumption good and Zn the 62 ASSET PRICING 83 quantity of shares in productive unit i held at the beginning of period t the representative consumer s budget constraint is given by ZPitzit1 Ct 2 2110111 yit7 614 i1 i1 fort 0 1 2 The consumer s problem is to maximize 612 subject to 614 Letting pt 2t and yt denote the price vector the vector of share holdings and the output vector for example yt ylt ygt ym we can specify a value function for the consumer vztpt yt and write the Bellman equation associated with the consumer s problem as Uzt7pt7yt max UCt Et39UZt17pt17yt1l Cuzz subject to 614 Lucas 1978 shows that the value function is differen tiable and concave and we can substitute using 614 in the objective function to obtain n 02t7Pt7 1h U 12110111 1111 pitzit1l Et39UZt17pt17Ut1 i1 Now the first order conditions for the optimization problem on the right hand side of the above Bellman equation are quot 8v Pitul 2 12110111 1111 Pitzit1l Et 07 615 1 aZit1 for i 1 2 n We have the following envelope conditions 81 n 82 pit 111011 2 12110111 1111 Pitzit1l 616 it i1 Substituting in 615 using 613 614 and 616 then gives Pitul 1m Et pit1 yit1ul yit1 07 617 i1 i1 for i 12n or M 0 618 pit Et pit1 yit1 u at 84 CHAPTER 6 CONSUMPTION AND ASSET PRICING That is the current price of a share is equal to the expectation of the product of the future payoff on that share with the intertemporal marginal rate of substitution Perhaps more revealing is to let 7m denote the gross rate of return on share i between period t and period t l ie pit1 yit1 7 pit and let mt denote the intertemporal marginal rate of substitution 5 Ct1 mt U Ct Then we can rewrite equation 618 as Et Tritmt 17 or using the fact that for any two random variables X and Y covX Y EXY EXEY7 covt7ritmt Et 7 Et mt 1 Therefore shares with high expected returns are those for which the covariance of the asset s return with the intertemporal marginal rate of substitution is low That is the representative consumer will pay a high price for an asset which is likely to have high payoffs when aggre gate consumption is low We can also rewrite 618 using repeated substitution and the law of iterated expectations which states that for a random variable xi Et Et xt r Etxt r 5 2 s 2 0 to get 0 situCS t t um That is we can write the current share price for any asset as the ex pected present discounted value of future dividends where the discount factors are intertemporal marginal rates of substitution Note here that the discount factor is not constant but varies over time since consump tion is variable 62 ASSET PRICING 85 Examples Equation 617 can be used to solve for prices and we will show here how this can be done in some special cases First suppose that yt is an iid random variable Then it must also be true that pt is iid This then implies that Et pit1 yit1U WHO Au 620 i1 for i l 2 n where A gt 0 is a constant That is the expression inside the expectation operator in 620 is a function of pm and yit1i l 2 n each of which is unpredictable given information in period t therefore the function is unpredictable given information in period t Given 617 and 620 we get p A it u 21 1111 Therefore if aggregate output which is equal to aggregate consumption here is high then the marginal utility of consumption is low and the current price of the asset is high That is if current dividends on assets are high the representative consumer will want to consume more today but will also wish to save by buying more assets so as to smooth consumption However in the aggregate the representative consumer must be induced to consume aggregate output or equivalently to hold the supply of available assets and so asset prices must rise A second special case is where there is risk neutrality that is uc c From 617 we then have pit Etpit1 yit17 ie the current price is the discount value of the expected price plus the dividend for next period or pit1 yit1 pit 1 E 71 621 t pit Equation 621 states that the rate of return on each asset is unpre dictable given current information which is sometimes taken in the 86 CHAPTER 6 CONSUMPTION AND ASSET PRICING Finance literature as an implication of the ef cient markets hypothe sis Note here however that 621 holds only in the case where the representative consumer is risk neutral Also 619 gives 0 pit Et 2 5571157 st1 or the current price is the expected present discounted value of divi dends A third example considers the case where 110 lnc and n 1 that is there is only one asset which is simply a share in aggregate output Also we will suppose that output takes on only two values yt y1 y2 with yl gt yg and that yt is iid with Pryt 111 7r 0 lt 7T lt 1 Let 1 denote the price of a share when yt 1 for i 12 Then from 617 we obtain two equations which solve for p1 and p2 P1 5WampP1 1111 702012 112 111 112 p2 a 921 y1lt1 7 702022 112 111 112 Since the above two equations are linear in p1 and p2 it is straightfor ward to solve obtaining 7 3111 P17 17 7 3112 P2717 Note here that p1 gt p2 that is the price of the asset is high in the state when aggregate output is high Alternative Assets and the Equity Premium Puzzle Since this is a representative agent model implying that there can be no trade in equilibrium and because output and consumption are exogenous it is straightforward to price a wide variety of assets in this type of model For example suppose we allow the representative agent to borrow and lend That is there is a risk free asset which trades on 62 ASSET PRICING 87 a competitive market at each date This is a one period risk free bond which is a promise to pay one unit of consumption in the following period Let le denote the quantity of risk free bonds acquired in period t by the representative agent note that le can be negative the representative agent can issue bonds and let qt denote the price of a bond in terms of the consumption good in period t The representative agent s budget constraint is then 7L 7L pitzit1 Ct tht1 22110311 1111 bt i1 i1 In equilibrium we will have bi 0 ie there is a zero net supply of bonds and prices need to be such that the bond market clears We wish to determine qt and this can be done by re solving the consumer s problem but it is more straightforward to simply use equa tion 617 setting pm 0 since these are one period bonds they have no value at the end of period t l and yit1 l to get Ullt t1l E 622 q a uw lt gt The one period risk free interest rate is then 1 r71 623 It If the representative agent is risk neutral then qt and n 7 1 that is the interest rate is equal to the discount rate Mehra and Prescott 1985 consider a version of the above model where n l and there are two assets an equity share which is a claim to aggregate output and a one period risk free asset as discussed above They consider preferences of the form 01quot 7 1 110 In the data set which Mehra and Prescott examine which includes annual data on risk free interest rates and the rate of return implied by aggregate dividends and a stock price index the average rate of return 88 CHAPTER 6 CONSUMPTION AND ASSET PRICING on equity is approximately 6 higher than the average rate of return on risk free debt That is the average equity premium is about 6 Mehra and Prescott show that this equity premium cannot be accounted for by Lucas s asset pricing model Mehra and Prescott construct a version of Lucas s model which incorporates consumption growth but we will illustrate their ideas here in a model where consumption does not grow over time The Mehra Prescott argument goes as follows Suppose that output can take on two values yl and y2 with yl gt yg Further suppose that yt follows a two state Markov process that is Prlyt1 11339 l M yil 7n We will assume that 71quot p for i 12 where 0 lt p lt 1 Here we want to solve for the asset prices 1 pi i 12 where qt 11 and pt p when yt 1 for i 12 From 617 we have ply7 7 p031 11011 1 7 Wm 1191137 7 624 132167 7 p032 My 1 7 mm y yfl 625 Also 622 implies that q1 7 p 1 7 p 626 q2 7 p 1 7 p 627 Now 624 and 625 are two linear equations in the two unknowns p1 and p2 so 624 627 give us solutions to the four asset prices Now to determine risk premia we rst need to determine expected returns In any period t the return on the risk free asset is certain and given by n in 623 Let n 7 when yt 1 for i 12 For the equity share the expected return denoted Rt is given by Rt E ltPt1 yt1 pt I Pt 62 ASSET PRICING 89 Therefore letting R denote the expected rate of return on the equity share when yt yi we get R1pltp1y11ip ltP2112gt71 P1 p1 R2 pltp2y2 17plt131y1gt 71 P2 p2 Now what we are interested in is the average equity premium that would be observed in the data produced by this model over a long period of time Given the transition probabilities between output states the unconditional long run probability of being in either state is here Therefore the average equity premium is 1 6 mp7y17y2 R1 7 m 5 R2 7 r2 1 5 Mehra and Prescott s approach is to set p yl and y2 so as to replicate the observed properties of aggregate consumption in terms of serial correlation and variability then to nd parameters and 7 such that e y p y1 y2 E 06 What they nd is that 7 must be very large and much outside of the range of estimates for this parameter which have been obtained in other empirical work To understand these results it helps to highlight the roles played by y in this model First 7 determines the intertemporal elasticity of substitution which is critical in determining the risk free rate of inter est n That is the higher is y the lower is the intertemporal elasticity of substitution and the greater is the tendency of the representative consumer to smooth consumption over time Thus a higher 7 tends to cause an increase in the average risk free interest rate Second the value of y captures risk aversion which is a primary determinant of the expected return on equity That is the higher is y the larger is the expected return on equity as the representative agent must be com pensated more for bearing risk The problem in terms of tting the model is that there is not enough variability in aggregate consumption to produce a large enough risk premium given plausible levels of risk aversion 90 CHAPTER 6 CONSUMPTION AND ASSET PRICING 63 References Hall R 1978 Stochastic Implications of the Life Cycle Permanent lncome Hypothesis Journal of Political Economy 86 971 987 Hall R 1989 Consumption in Modern Business Cycle Theory R Barro ed Harvard University Press Cambridge MA Lucas R 1978 Asset Prices in an Exchange Economy Economet rica 46 1426 1445 Mehra R and Prescott E 1985 The Equity Premium A Puzzle Journal of Monetary Economics 15 145 162 Chapter 7 Search and Unemployment Unemployment is measured as the number of persons actively seeking work Clearly there is no counterpart to this concept in standard rep resentative agent neoclassical growth models lf we want to understand the behavior of the labor market explain why unemployment uctu ates and how it is correlated with other key macroeconomic aggregates and evaluate the ef cacy of policies affecting the labor market we need another set of models These models need heterogeneity as we want to study equilibria where agents engage in different activities ie job search employment and possibly leisure not in the labor force Fur ther there must be frictions which imply that it takes time for an agent to transit between unemployment and employment Search models have these characteristics Some early approaches to search and unemployment are in McCall 1970 and Phelps et al 1970 These are models of one sided search which are partial equilibrium in nature Unemployed work ers face a distribution of wage offers which is assumed to be xed Later Mortensen and Pissarides developed two sided search models for a summary see Pissarides 1990 in which workers and rms match in general equilibrium and the wage distribution is endogenous Pe ter Diamond 1982 also developed search models which could exhibit multiple equilibria or coordination failures Diamond s was probably the rst attempt to rigorously explain Keynesian unemployment phe nomena Search theory is a useful application of dynamic programming and 91 92 CHAPTER 7 SEARCH AND UNEMPLOYMENT it also allows us to introduce some basic concepts in the theory of games We will first study a one sided search model similar to the one studied by McCall 1970 and then look at a version of Diamond s coordination failure model 71 A OneSided Search Model Suppose a continuum of agents with unit mass each having preferences given by 0 E0 2 tuct t0 where 0 lt lt l and is concave and strictly increasing Let 1 1T where 7 is the discount rate Note that we assume that there is no disutility from labor effort on the job or from effort in search ing for a job There are many different jobs in this economy which differ according to the wage w which the worker receives From the point of view of an unemployed agent the distribution of wage offers she can receive in any period is given by the probability distribution function F w which has associated with it a probability density func tion Assume that w 6 0747 ie the set 010 is the support of the distribution If an agent is employed receiving wage w assume that each job requires the input of one unit of labor each period then her consumption is also w as we assume that the worker has no op portunities to save At the end of the period there is a probability 6 that an employed worker will become unemployed The parameter 6 is referred to as the separation rate An unemployed worker receives an unemployment bene t I from the government at the beginning of the period and then receives a wage offer that she may accept or decline Let Vu and V5042 denote respectively the value of being unem ployed and the value of being employed at wage w as of the end of the period These values are determined by two Bellman equations v a m fm mmm mm 71 V500 Mw 6Vu 1 5V5wl 72 71 A ONE SIDED SEARCH MODEL 93 In 71 the unemployed agent receives the unemployment insurance bene t I at the beginning of the period consumes it and then receives a wage offer from the distribution The wage offer is accepted if V510 2 Vu and declined otherwise The integral in 71 is the expected utility of sampling from the wage distribution In 72 the employed agent receives the wage 10 consumes it and then either suffers a separation or will continue to work at the wage 10 next period Note that an employed agent will choose to remain employed if she does not experience a separation because V510 2 Vu otherwise she would not have accepted the job in the rst place In search models a useful simpli cation of the Bellman equations is obtained as follows For 71 divide both sides by substitute T1 and subtract Vu from both sides to obtain TV 1110m max mm 7 V7 0 f10d10 73 On the right hand side of 73 is the ow return when unemployed plus the expected net increase in expected utility from the unemployed state Similarly 72 can be simpli ed to obtain MW uw 6M 7 V500 74 We now want to determine what wage offers an agent will accept when unemployed From 74 we obtain u10 6V lle 16 Therefore since u10 is strictly increasing in 10 so is Thus there is some 10 such that V510 2 Vu for 10 2 10 and V510 S Vu for 10 S 10 see Figure 71 The value 10 is denoted the resemation wage That is an unemployed agent will accept any wage offer of 10 or more and decline anything else Now let yt denote the fraction of agents who are employed in period t The ow of agents into employment is just the fraction of unemployed agents multiplied by the probability that an individual agent transits from unemployment to employment 1 7 m 1 7 Further the CHAPTER 7 SEARCH AND UNEMPLOYMENT Fxgure 7 1 Vew Figure 7 1 71 A ONE SIDED SEARCH MODEL 95 ow of agents out of employment to unemployment is the number of separations yt6 Therefore the law of motion for yt is Yt1 Yt 1 Yt 1 Yt6 75 17 Fw Yt 7 6 Since l Fw 7 6 llt l yt converges to a constant 7 which is deter mined by setting 7H1 yt y in 75 and solving to get 17Fw m 76 Y Therefore the number of employed decreases as the separation rate increases and as the reservation wage increases The unemployment rate in the steady state is given by l 7 7 so that if the separation rate increases increasing the ow from employment to unemployment the steady state unemployment rate goes up Similarly if unemployed agents are more choosy about the job offers they accept so that 10 increases this decreases the ow from unemployment to employment and the steady state unemployment rate increases One policy experiment we could conduct in this model is to ask what the effects are of an increase in b interpreted as an increase in the generosity of the unemployment insurance program One can show from 73 and 74 that V5042 7 Vu decreases for each u when 1 increases Therefore 10 increases with b and from 76 the number of employed agents falls and the unemployment rate rises That is an increase in unemployment insurance bene ts acts to make unemployed agents more choosy concerning the jobs they will accept decreasing the ow of agents from the unemployment pool to employment and the unemployment rate therefore increases in the steady state 711 An Example Suppose that there are only two possible wage offers An unemployed agent receives a wage offer of 4i with probability 7T and an offer of zero with probability 1 7 7T where 0 lt 7T lt 1 Suppose first that 0 lt b lt 1 Here in contrast to the general case above the agent knows that when she receives the high wage offer there is no potentially higher offer that 96 CHAPTER 7 SEARCH AND UNEMPLOYMENT she foregoes by accepting so high wage o ers are always accepted Low wage o ers are not accepted because collecting unemployment bene ts is always preferable and the agent cannot search on the job Letting V denote the value of employment at wage if the Bellman equations can then be written as Wu W 7TVe Vu7 We HOD 5Vu Vat and we can solve these two equations in the unknowns V and Vu to obtain 7 7ruw 6ub KB rr67r 7 M Vui rr67r Note that V 7 Vu depends critically on the di erence between 1 and b and on the discount rate 7 The number of employed agents in the steady state is given by 7139 Y my so that the number employed unemployed increases decreases as 7T increases and falls rises as 6 increases Now for any I gt 1 clearly we will have 7 0 as no o ers of employment will be accepted due to the fact that collecting unemploy ment insurance dominates all alternatives However if b 1 then an unemployed agent will be indi erent between accepting and declining a high wage o er It will then be optimal for her to follow a mixed strat egy whereby she accepts a high wage offer with probability 77 Then the number of employed agents in the steady state is n 7 Y i 6 wry which is increasing in 77 This is a rather stark example where changes in the Ul bene t have no effect before some threshold level but increasing bene ts above this level causes everyone to turn down all job o ers 72 DIAMOND S COORDINATION FAILURE MODEL 97 712 Discussion The partial equilibrium approach above has neglected some important factors in particular the fact that if job vacancies are posted by rms then if a wage offer is accepted this will in general affect the wage offer distribution faced by the unemployed Two sided search models as in Pissarides 1990 capture this endogeneity in the wage distribution In addition we do not take account of the fact that the government must somehow nance the payment of unemployment insurance bene ts A simple nancing scheme in general equilibrium would be to have Ul bene ts funded from lump sum taxes on employed agents Note in the example above that it would be infeasible to have b gt w in general equilibrium 72 Diamond s Coordination Failure Model Diamond s model is probably the rst account of a coordination failure problem interpreted as a Keynesian type low employment steady state where there is a possibility of welfare improving government interven tion A simple coordination failure model is constructed by Bryant 1983 and Cooper and John 1988 captures the general features of coordination failure models The key feature of Diamond s model is that there are multiple steady state equilibria which can be Pareto ranked That is if the agents living in the model could somehow choose among steady states they would all agree on which one was most preferred However the problem is that in equilibrium the outcome may not be this most preferred steady state In principle there might be a role for government in helping the private economy coordinate on the right equilibrium though search models do not typically permit government intervention in a nice way The idea that a coordination failure problem might imply a stabiliz ing role for government can be found in Keynes 1936 though Keynes was thinking in terms of nominal wage rigidity and the coordinating role of the central bank as follows Except in a socialised community where wage policy is set tled by decree there is no means of securing uniform wage 98 CHAPTER 7 SEARCH AND UNEMPLOYMENT reductions for every class of labor The result can only be brought about by a series of gradual irregular changes justi able on no criterion of social justice or economic ex pedience and probably completed only after wasteful and disastrous struggles where those in the weakest bargaining position will suffer relatively to the rest A change in the quantity of money on the other hand is already within the power of most governments by open market policy or anal ogous measures Having regard to human nature and our institutions it can only be a foolish person who would pre fer a exible wage policy to a exible money policy pp 267268 721 The Model There is a continuum of agents with unit mass each with preferences given by E0 0 MCt atlv where 0 lt lt l is strictly increasing with u0 0 at is con sumption and at is production effort At the end of a given period an agent is either unemployed or employed lf unemployed the agent is searching for a production opportunity With probability 0 one production opportunity arrives during the period and with prob ability 1 7 0 no production opportunities arrive A given production opportunity is de ned by its production cost Cz Given that a produc tion opportunity arrives it is a draw from a probability distribution F a with corresponding probability density function f a Assume that Cl 6 goo where ggt 0 On receiving a production opportunity the unemployed agent must decide whether to take it or not If she does not take it then she continues to search for a production opportunity in the next period and if she does take the opportunity she produces 1 units of an indivisible consumption good and is then deemed employed The consumption good has the property that the agent who produces it can not consume it but any other agent can Thus an agent with a consumption good must search for a trading partner For any em 72 DIAMOND S COORDINATION FAILURE MODEL 99 ployed agent the probability of nding a trading partner in the current period is b y where is a strictly increasing function with b0 0 and bl S l and y is the fraction of employed agents in the popula tion Thus the probability of nding a trading partner increases with the number of would be trading partners in the population Once two employed agents meet they trade consumption goods since the goods are indivisible and both agents strictly prefer trading to continuing to search When a consumable good is received it is immediately con sumed and the agent goes back to the unemployment pool and begins searching for a production opportunity again The basic idea here is that the decision about what production opportunities to accept from an individual s point of view is affected by how easy it is to trade once something is produced This in turn is determined by how many would be trading partners there are which is determined by what production opportunities are accepted by other agents Thus there is a strategic complementarity sometimes referred to as an externality though this is bad language which introduces the possibility of multiple equilibria A story which helps in visualizing what is going on in this economy is related by Diamond 1982 in his paper though this need not be much help is thinking about how the model captures what is going on in real economies The story is as follows There are some people wandering on a beach who happen on coconut trees at random Each tree has one coconut and the trees are of varying heights On encountering a tree a person must decide whether she should climb the tree and pick the coconut or continue to wander the beach and hopefully nd a smaller tree If she picks a coconut unfortunately she cannot eat it herself and she wanders the beach until she nds another person with a coconut they trade eat their coconuts and then go back to searching for trees In terms of capturing real world phenomena the agents searching for production opportunities are like the unemployed agents in the search model we examined in the previous section in that they are searching for employment a production opportunity and their de cision as to what production opportunities to accept will be like the reservation wage strategy discussed previously There are no rms in the model but we can view the employment activities here as produc tion and sales and once the goods are produced and sold employment 100 CHAPTER 7 SEARCH AND UNEMPLOYMENT ends and the agent can be viewed as laid off or discharged from a rm which has gone out of business 722 Determining the Steady State First let V and Vu denote the values of being employed and unem ployed respectively as of the end of the period Then we can summa rize the agent s dynamic optimization problem in the steady state by the following two Bellman equations Ve 5W My lul 1 MWe Vu goomaX e 7aVufczdcz170Vu As in the one sided search model it is convenient to substitute in the above two Bellman equations and manipulate to get 7 177 My Vu 7 Val 77 W 0 mame 7 a 7 nofada 78 Now note that from 78 since V and Vu are constants then assuming that V237Vu gt g there exists some C such that lg7cz7Vu 2 0 for CM S Cz and V 7 Cl 7 Vu lt 0 for C1 gt C That is the agent will accept any production opportunities with Cl S Cz and decline any with C1 gt C so that C is a reservation cost note that we suppose that if the agent is indi erent she accepts We can then rewrite 78 as TV 6Fczcz 7 Oa szCzdCz 79 where cf v i Vu 710 Then subtracting 79 from 77 and substituting using 710 we obtain raw mum 7 a1 7 6Fltcrgtcr a leWWCZ on Q 72 DIAMOND S COORDINATION FAILURE MODEL 101 which can be simpli ed by using integration by parts to get w 1m uy 7 Cr 7 a mama 711 Further using 710 to substitute in 77 we have rmmwmwwy am Given the reservation cost strategy the ow of agents into employ ment in the steady state each period is 0FCz1 7 y and the ow out of employment is yb y 1n the steady state these ows must be equal or 0Fcz 1 7 7 Now equations 711 and 713 determine Cz and y and then 79 and 712 solve for Vu and 123 respectively In 713 one solution is C g and y 0 and since the right hand side of 713 is strictly increasing in y and the left hand side is strictly increasing in Cf 713 713 determines an upward sloping locus AA as depicted in Figure 72 Further in 711 one solution is C g and y 1 where lgt 0 and solves g b11uy Q17 and totally di erentiating 711 and solving we get dwbwmwim d7 7 bcz 0FCz Therefore 711 describes an upward sloping locus BB as depicted in Figure 72 so that there may be multiple steady states A B and D as shown Again multiple steady states can occur due to a strategic complementarity That is if all agents believe that other agents will choose a low high Cf then they also believe that there will be a small large number of agents in the trading sector which implies that they will choose a low high C as well Steady states are Pareto ranked according to the value of C To show this rst note that again using integration by parts we can rewrite 79 as 7 00 Fczdcz 102 CHAPTER 7 SEARCH AND UNEMPLOYMENT pha gamma Fgumvz 72 DIAMOND S COORDINATION FAILURE MODEL 103 so that Vu increases as steady state C increases Second 710 tells us that V 7 Vu increases with Cf so that Vu increases with steady state C as well Thus both the unemployed and the employed are better 03 in a steady state with a higher Cz and by implication with a higher W 723 Example This example is quite simple but it is somewhat di erent from the model laid out above as in the steady state agents will wish in general to engage in mixed strategies Here suppose that all production op portunities have the same cost given by 51 gt 0 and b y b y where b gt 0 Here in the steady state we want to determine V23 Vu y and 7T which is the probability that an unemployed agent accepts a produc tion opportunity We can write the modi ed versions of the Bellman equations as rMMw7W7 HQ TV 6 max 7rl3 7 Vu 7 cl 715 7r 01 In the steady state the ow into employment is 1 7 y07r and the ow out of employment is 721 so in the steady state where the ows are equal we have by2 91 i 7 Now we want to consider the three possible cases here in turn ie 7T 0 7T 6 01 and 7T 1 The rst and third involve pure strategies and in the second unemployed agents follow a mixed strategy The procedure will be to conjecture that an equilibrium of a particular type exists and then to use 714 and 715 to characterize that steady state equilibrium First consider the case where 7T 1 Then from 715 we have 716 7139 TVu9Ve7Vu51v 717 and then using 714 and 717 we get i b39yuy 0a Veilu rb76 104 CHAPTER 7 SEARCH AND UNEMPLOYMENT Now for this to be an equilibrium ie for 7T l to be optimal we must have V 7 Vu 2 51 or bluw 7 5 Second consider the case where 0 lt 7T lt 1 Here we must have V 7 Vu 51 so that 715 gives Vu 0 which implies given 714 that v 2 718 VB 7 Vu WM 7 7 and V 7 Vu 51 implies that 7462 A 719 7 may 7 a lt In a similar manner it is straightforward to show that 7T 0 implies that we must have lt rcz Y 7 A 17M Gil Thus from 716 and 718 720 there are three steady states as de picted in Figure 73 These are 7T 7 00 n71 l QEM W 720 where WM 7 03f As in the general model above these steady states are Pareto ranked V 724 Discussion The spatial separation in a search model provides a convenient rationale for the lack of ability of the agents in the model to get together to co ordinate on better outcomes than might be achieved in a decentralized equilibrium It is possible though that if agents could communicate freely ie if information could ow among agents though goods can not that bad outcomes like the Pareto dominated steady states studied here would not result Of course in practice it is dif cult for economic agents to get to gether and they certainly can not communicate perfectly so there is the 72 DIAMOND S COORDINATION FAILURE MODEL 105 Fxgure73 gamma Figure 7 3 106 CHAPTER 7 SEARCH AND UNEMPLOYMENT possibility that Diamond s coordination failure model tells us something about how real economies work However in practice there are many coordinating types of institutions which do not exist in the model such as nancial intermediaries which could serve to eliminate or mitigate the effects studied here Coordination failure models have been extended to dynamic envi ronments to study whether they can explain the statistical features of business cycles Some success has been achieved in this area by Farmer and Guo 1994 for example but it is dif cult to distinguish the im plications of these dynamic models with multiple equilibria from those of real business cycle models which typically have unique equilibria A further problem with multiple equilibrium models in general is that they may not have testable implications That is the implications for what we should see in economic data may differ across equilibria 73 References Bryant J 1983 A Simple Rational Expectations Keynes Type Model Quarterly Journal of Economics 98 525 528 Cooper R and John A 1988 Coordinating Coordination Failures in Keynesian Models Quarterly Journal of Economics 103 441 463 Diamond P 1982 Aggregate Demand in Search Equilibrium Jour nal of Political Economy 90 881 894 Farmer R and Guo J 1994 Real Business Cycles and the Animal Spirits Hypothesis Journal of Economic Theory 63 42 72 Keynes J 1936 The General Theory of Employment Interest and Money Macmillan London McCall J 1970 Economics of lnformation and Job Search Quar terly Journal of Economics 84 113 126 Phelps E et al 1970 l39u ic 4 F 39 4 of r39 1 39 and In ation Theory Norton New York


Buy Material

Are you sure you want to buy this material for

25 Karma

Buy Material

BOOM! Enjoy Your Free Notes!

We've added these Notes to your profile, click here to view them now.


You're already Subscribed!

Looks like you've already subscribed to StudySoup, you won't need to purchase another subscription to get this material. To access this material simply click 'View Full Document'

Why people love StudySoup

Steve Martinelli UC Los Angeles

"There's no way I would have passed my Organic Chemistry class this semester without the notes and study guides I got from StudySoup."

Allison Fischer University of Alabama

"I signed up to be an Elite Notetaker with 2 of my sorority sisters this semester. We just posted our notes weekly and were each making over $600 per month. I LOVE StudySoup!"

Bentley McCaw University of Florida

"I was shooting for a perfect 4.0 GPA this semester. Having StudySoup as a study aid was critical to helping me achieve my goal...and I nailed it!"

Parker Thompson 500 Startups

"It's a great way for students to improve their educational experience and it seemed like a product that everybody wants, so all the people participating are winning."

Become an Elite Notetaker and start selling your notes online!

Refund Policy


All subscriptions to StudySoup are paid in full at the time of subscribing. To change your credit card information or to cancel your subscription, go to "Edit Settings". All credit card information will be available there. If you should decide to cancel your subscription, it will continue to be valid until the next payment period, as all payments for the current period were made in advance. For special circumstances, please email


StudySoup has more than 1 million course-specific study resources to help students study smarter. If you’re having trouble finding what you’re looking for, our customer support team can help you find what you need! Feel free to contact them here:

Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.