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Notes on Macroeconomic Theory Steve Williamson Dept of Economics University of Iowa Iowa City 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 pricetakers 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 K where c is consumption and Z is leisure Here u is strictly increasing in each argument strictly concave and 2 CHAPTER 1 SIMPLE REPRESENTATIVE AGENT MODELS twice differentiable Also assume that limc0 u1c Z 2 oo 8 gt 0 and limg0 M 05 2 oo 0 gt 0 Here uzc Z is the partial derivative with respect to argument 73 of uc 8 Each consumer is endowed with one unit of time which can be allocated between work and leisure Each consumer also owns 1 units of capital which can be rented to firms There are M firms which each have a technology for producing consumption goods according to y 210067 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 differentiable and homogeneous of degree one That is produc tion is constant returns to scale so that Ag 2 zfkn 11 for A gt 0 Also assume that limk0 f1k n 2 oo limkOO f1k n 0 limn0 f2k n 2 00 and limOO f2k n 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 1 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 39r Consumer s Problem Each consumer treats to as being xed and maximizes utility subject to his her constraints That is each solves a 11 A STATIC MODEL 3 subject to c g w1 Z 7498 12 160 lt ks lt 1 0 N 3 0 S K S 1 14 c 2 0 1 5 Here ks 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 ks 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 8 1 as then nothing would be produced so we can safely ignore this case The optimization problem for the con sumer is therefore much simplified and we can write down the following Lagrangian for the problem uc uwr w c where u is a Lagrange multiplier Our restrictions on the utility func tion assure that there is a unique optimum which is characterized by the following firstorder conditions 8L Ezul Mzo a L 0 85 Hw g zwr w c0 Here u is the partial derivative of u with respect to argument i The above firstorder conditions can be used to solve out for u and c to obtain 16 k wu1w TWO 108 Z u2w TWO 1055 2 0 16 4 CHAPTER 1 SIMPLE REPRESENTATIVE AGENT MODELS which solves for the desired quantity of leisure Z in terms of w 39r and 1 Equation 16 can be rewritten as 2 w U1 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 he she maximizes utility at E where the budget constraint which has slope w is tangent to the highest indifference curve where an indifference curve has slope Firm s Problem Each firm chooses inputs of labor and capital to maximize pro ts treat ing 10 and 39r as being xed That is a firm solves maxzfk n 39rk um and the firstorder conditions for an optimum are the marginal product conditions Zfl 7quot Zf2 U where denotes the partial derivative of f with respect to argu ment 73 Now given that the function f is homogeneous of degree one Euler s law holds That is differentiating 11 with respect to A and setting A 1 we get zfk n zflk zfgn 19 Equations 17 18 and 19 then imply that maximized profits equal zero This has two important consequences The first is that we do not need to be concerned with how the firm s profits are distributed through shares owned by consumers for example Secondly suppose 6 and 71 are optimal choices for the factor inputs then we must have zfkn rk wn0 110 11 A STATIC MODEL Figure 11 rEII ti Figure 11 6 CHAPTER 1 SIMPLE REPRESENTATIVE AGENT MODELS for k 6 and n 71 But since 110 also holds for k Ak and n An 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 firm as the number of firms will be irrelevant for determining the competitive equilibrium 113 Competitive Equilibrium A competitive equilibrium is a set of quantities c K n k and prices 10 and 7 which satisfy the following properties 1 Each consumer chooses c and Z optimally given 10 and 39r 2 The representative firm chooses n and k optimally given 10 and 39r 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 N1 2n 111 y N0 112 kg k 113 That is supply equals demand in each market given prices Now the total value of excess demand across markets is Nc ywln N1 lTk k07 but from the consumer s budget constraint and the fact that profit maximization implies zero profits we have Nc ywn N1 rk k00 114 Note that 114 would hold even if profits were not zero and were dis tributed lumpsum to consumers But now if any 2 of 111 112 11 A STATIC MODEL 7 and 113 hold then 114 implies that the third marketclearing 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 1 of those markets are in equilibrium then the additional mar ket is also in equilibrium We can therefore drop one marketclearing 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 five equations in the five unknowns Z n k w and 39r 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 different if there were many firms and many consumers We can substitute in equation 16 to obtain an equation which solves for equilibrium Z zf2k0 1 u1zfk0 1 U2Zfk0 1 0 115 Given the solution for Z we then substitute in the following equations to obtain solutions for 7 w n k and c zf km1 zr 116 z hh1 zu Lin n1 z km ca 1amp 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 1 4 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 rm 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 5 06 subject to c zfk0 1 z 119 Given the restrictions on the utility function we can simply substitute using the constraint in the objective function and differentiate with respect to Z to obtain the following rstorder condition for an optimum zf2k0 1 u1zfk0 1 08 Ugzfk0 1 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 f is strictly quasiconcave there is a unique Pareto optimum and the competitive equilibrium is also unique Note that we can rewrite 120 as 2 Zf2 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 STATIC MODEL 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 w is equal to Q 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 nontechnical assumptions required for 1 and 2 to go through include the absence of externalities completeness of markets and ab sence of distorting taxes e g 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 first 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 first solving for c and K from the social planner s problem and then finding 10 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 Figure 12 rHIII F 1 leisure Figure 12 11 A STATIC MODEL 11 1 1 5 Example Consider the following speci c functional forms For the utility func tion we use 01quoty 1 uc K 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 Cl ry 1 ie1 Y10gc 1 lim lim 7 d 39y gt1 1 y 39y gt1 31 y log 0 using L Hospital s Rule For the production technology use fk n kan1 where 0 lt 04 lt 1 That is the production function is CobbDouglas The social planner s problem here is then a 1 a 1 max zk01 6 7 1 g e 1 7 7 and the solution to this problem is z 1 1 ozzk8 17 a1lagtv 121 As in the general case above this is also the competitive equilibrium solution Solving for c from 119 we get c lt1 a1 zk6 alt1lagtm 122 and from 117 we have w 1 oz1 zk8 l WM 123 From 122 and 123 clearly c and w are increasing in z and k0 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 effect goes depends on whether 7 lt 1 or y gt 1 With 7 lt 1 an increase in z or in kg 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 fluctuations 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 simplified technology 92 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 meaxuz1 Z Z and the firstorder condition for a maximum is zu1z1 Z Z U2Z1 Z Z 0 124 Here in contrast to the example we cannot solve explicitly for K but note that the equilibrium real wage is av an so that an increase in productivity 2 corresponds to an increase in the real wage faced by the consumer To determine the effect of an increase 10 z 11 A STATIC MODEL 13 in z on 5 apply the implicit function theorem and totally differentiate 124 to get 711 U11 U210 22u11 222112 Ug2d 0 We then have dg U1 Ml11 U210 dz 227111 227112 22 125 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 3 gt 0 and where 3 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 2 211 Z 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 K but higher 0 Effectively the repre sentative consumer faces a higher real wage and his her 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 Z h 126 where h is a constant and zu1c Z u2c Z 0 127 Totally differentiating 126 and 127 with respect to c and Z and us ing 127 to simplify we can solve for the substitution effect 3 sub8t as follows 1 lt 0 subst dZ 227111 227112 U22 From 125 then the income effect 3 7ch is just the remainder 7111 U210 dz 2 gt 0 Z U11 227112 U22 14 CHAPTER 1 SIMPLE REPRESENTATIVE AGENT MODELS 1 leiaure Figure 13 12 GOVERNMENT 15 provided 8 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 1 2 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 lumpsum taxes on the representative consumer Let g be the quantity of government purchases which is treated as being exogenous and let 739 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 simplifies 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 1007579 16075 19 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 1amp9 0 As in the previous section labor is the only factor of production ie assume a technology of the form y zn Here the consumer s optimization problem is max uc 5 06 16 CHAPTER 1 SIMPLE REPRESENTATIVE AGENT MODELS subject to c w1 Z 739 and the firstorder condition for an optimum is wu1 log 0 The representative firm s pro t maximization problem is m7axz wn Therefore the firm s demand for labor is in nitely elastic at w z A competitive equilibrium consists of quantities c K n and 739 and a price 10 which satisfy the following conditions 1 The representative consumer chooses c and Z to maximize utility given 10 and 739 2 The representative firm chooses n to maximize pro ts given 10 3 Markets for consumption goods and labor clear 4 The government budget constraint 128 is satisfied 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 maxuc subject to c g 21 5 Substituting for c in the objective function and maximizing With re spect to Z the firstorder condition for this problem yields an equation Which solves for Z zu1z1 Z 9 Z U2Z1 Z 95 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 z w 12 GOVERNMENT 17 1 gfz leisure Figure 14 18 CHAPTER 1 SIMPLE REPRESENTATIVE AGENT MODELS We can now ask what the effect of a change in government eXpen ditures would be on consumption and employment In Figure 15 g increases from g1 to g2 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 differentiate 129 and the equation 0 21 Z g and solve to obtain d5 ZU11 U12 lt 0 d9 Z2U11 2212 U22 d c 12 22 lt 0 130 dg 227111 227112 U22 Here the inequalities hold provided that zu11 U12 gt 0 and ZU12 7122 gt 0 ie if leisure and consumption are respectively normal goods Note that 130 also implies that lt 1 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 financing 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 effects of changes in the timing of taxes Here we deal with an infinite horizon economy where the represen tative consumer maximizes timeseparable utility Z gilleta 07 t0 where is the discount factor 0 lt lt 1 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 firm 13 A DYNAMIC ECONOMY 19 1 g2fz 1g1fz leisure Figure 15 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 lumpsum taxation and by issuing oneperiod gov ernment bonds The government budget constraint is 975 l 1 l Ttbt 7397 bt17 131 t 0 1 2 where 1 is the number of oneperiod bonds issued by the government in period t 1 A bond issued in period t is a claim to 1rt1 units of consumption in period t l 1 where n1 is the oneperiod 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 130 0 The optimization problem solved by the representative consumer is max imam st1 Cta et10 t0 subject to Ct wt1 gt Tt St1 Tt8t t 0 1 2 so 0 where St1 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 Sn Rhino 1111 Ti 0 133 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 00 wt1 10 Tt C w 1 Z 739 0 7 1121 l Ti 0 O 0 1131 H 134 13 A DYNAMIC ECONOMY 21 Now maximizing utility subject to the above intertemporal budget constraint we obtain the following rstorder conditions A ta 5 0t 123 A tu2ct t wt 2 015 1 2 3 11111 l Ti u100750 A 0 260750 wO 0 Here A is the Lagrange multiplier associated with the consumer s in tertemporal budget constraint We then obtain u2ct7 gt were t ie the marginal rate of substitution of leisure for consumption in any period equals the wage rate and u1ct17 t1 1 136 1Ct7gt 1 n1 ie the intertemporal marginal rate of substitution of consumption equals the inverse of one plus the interest rate The representative firm simply maximizes profits in each period ie it solves mzaxzt want and labor demand m is perfectly elastic at wt 2 21 A competitive equilibrium consists of quantities 075 75 nt St1 bt1 Tt 0 and prices 1075 rt1 0 satisfying the following conditions 1 Consumers choose 075 ft St1 EEO optimally given 73 and 1075 rt1f0 2 Firms choose n f o optimally given wt 0 3 Given gt 0 bt1 720 satisfies 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 marketclearing conditions 5t1 bt17 t 071727quot397 and 1 gt ntt 012 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 210 n 210 m 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 00 Ct 00 wt1 10 gt 0 w 1 K 139 0 7 H 11 l Ti 0 0 90 752231 1121 l Ti Now suppose that wtrt1 0 are competitive equilibrium prices Then 139 implies that the optimizing choices given those prices re main optimal given any sequence Tt 0 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 Ricardz39an Equivalence Theorem For example holding the path of government purchases constant if the representative consumer receives a taX cut today he she knows that the government will have to make this up with higher future taxes The government issues more debt today to finance an increase in the government deficit 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 gt 913813 ethic 130 This breaks down into a series of static problems and the firstorder conditions for an optimum are ZtU1Zt1 gt 913813 U2Zt1 gt 913813 I 0 t 0 1 2 Here 140 solves for 87515 2 012 and we can solve for ct from ct zt1 Kt 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 infinitelived 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 effects of taxation That is the present discounted value of each individual s tax burden is unaffected 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 semiserious 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 effects 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 nitelived 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 1 2 00 Each period Lt twoperiodlived 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 L01 nt 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 0 1 2 are given by ctya c17 where 0 denotes the consumption of a young consumer in period t and c is the consumption of an old consumer Assume that u is strictly increasing in both arguments strictly concave and de ning au vcyc E E 80 assume that limcyo Ucy co 00 for cquot gt 0 and limcoovcy co 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 oneforone into capital and viceversa Capital con structed in period 15 does not become productive until period t 1 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 FKt Lt 22 OPTIMAL ALLOCATIONS 27 where Y is output and Kt and L 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 first 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 confine 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 FKt Lt Kt Kt1 Clzgth cht1 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 percapita quantities converge to constants Thus it proves to be convenient to express everything here in percapita terms using lower case letters Define kt E g the capital labor ratio or percapita capital stock and Lt fkt E Fact 1 We can then use 21 to rewrite 22 as O C De nition 1 A Pareto optimal allocation is a sequence cg kt1 0 satisfying and the property that there exists no other allocation 835623 kt1 0 which satis es and quotO 0 c1 2 c1 aif1gt 2 6137 cg1 for all t 0 1 2 3 with strict inequality in at least one instance 2SCHAPTER 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 cf 2 Cy and c 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 kg 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 noncomparable 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 0quot subject to Co 1 n Substituting for cquot in the objective function using 24 we then solve the following ma nk C 24 maxucy 1 n 71k Cyl Cy The firstorder conditions for an optimum are then ul 1 nu2 0 01 u 1 1 n 25 U2 intertemporal marginal rate of substitution equal to 1 n and was n 26 marginal product of capital equal to Note that the planner s prob lem splits into two separate components First the planner finds the 23 COMPETITIVE EQUILIBRIUM 29 capitallabor 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 y o yIQaX ct 7 Ct1gt ct ct1 st subject to 0 2 wt st 27 C1 5t1 Ttlgt 28 Here 10 is the wage rate 7quot 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 30 CHAPTER 2 GROWTH WITH OVERLAPPIN G GENERATIONS cans elf culd l ikJ k Figure 21 Figure 21 111ka mans 23 COMPETITIVE EQUILIBRIUM 31 Substituting for c and c 1 in the above objective function using 27 and 28 to obtain a maximization problem with one choice vari able st the rstorder condition for an optimum is then u1ut st st1 rt1 u2ut st st1 rt11 n1 0 29 which determines st ie we can determine optimal savings as a function of prices st 2 sut n1 210 Note that 29 can also be rewritten as 3 1 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 goods we have 8878 gt 0 however the sign of 888 is t 739t1 indeterminate due to opposing income and substitution effects 232 Representative Firm s Problem The firm solves a static profit maximization problem maXFKt Lt tht Tth KtLt The rstorder conditions for a maXimum are the usual marginal con ditions F1Kt7 Lt T1 0 F2ltKt7 Lt wt Since F is homogeneous of degree 1 we can rewrite these marginal conditions as flfkt Tt 0 211 fkt ktf39kt wt 2 0 212 233 Competitive Equilibrium De nition 2 A competitive equilibrium is a sequence of quantities kt1 5in and a sequence of prices 1075 rtf0 which satisfy con sumer optimization rm optimization market clearing in each period t 0 1 2 given the initial capital labor 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 15 savings is equal to the capital that will be rented in period t I 1 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 n 8wt7 Tt1a 213 and we can substitute in 213 for 10 and n1 from 211 and 212 to get M1 n swat kf kt mm 214 Here 214 is a nonlinear firstorder difference equation which given k0 solves for kt 1 Once we have the equilibrium sequence of capital labor ratios we can solve for prices from 211 and 212 We can then solve for 5in from 210 and in turn for consumption allocations 2 4 An Example Let ucyco ln Cy lnco and FK L VKO L1O where gt 0 y gt 0 and 0 lt 04 lt 1 Here a young agent solves msaxlnwt St anl Tt18tl7 and solving this problem we obtain the optimal savings function 10 1 1539 215 575 Given the CobbDouglass production function we have f 7k and yozko 1 Therefore from 211 and 212 the firstorder conditions from the firm s optimization problem give rt 2 70463 1 216 24 AN EXAMPLE 33 wt 2 71 ozk 217 Then using 214 215 and 217 we get m unw f gu aw3 2 Now equation 218 determines a unique sequence k fil given k0 see Figure 2m which converges in the limit to 6 the unique steady state capitallabor ratio which we can determine from 218 by setting kt1 kt 6 and solving to get k lm 0 219 1 n1 3 Now given the steady state capitallabor ratio from 219 we can solve for steady state prices from 216 and 217 that is Za m 51 04 7 wl1al1ngt1 gtl 39 We can then solve for steady state consumption allocations In the long run this economy converges to a steady state where the capitallabor ratio consumption allocations the wage rate and the rental rate on capital are constant Since the capitallabor ratio is constant in the steady state and the labor input is growing at the rate 71 the growth rate of the aggregate capital stock is also n in the steady state In turn aggregate output also grows at the rate 71 Now note that the socially optimal steady state capital stock is is determined by 26 that is Aa l 704k n 34CHAPTER 2 GROWTH WITH OVERLAPPIN G GENERATIONS 01 1 1 1 220 n Note that in general from 219 and 220 16 7E 16 ie the competi tive equilibrium steady state is in general not socially optimal so this economy suffers from a dynamic inefficiency There may be too little or too much capital in the steady state depending on parameter values That is suppose 1 and n 3 Then if 04 lt 103 16 gt 16 and if 04 gt 103 then 16 lt 25 Discussion The above example illustrates the dynamic inefficiency that can result in this economy in a competitive equilibrium There are essentially two problems here The first 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 inefficiency is a form of market incomplete ness in that agents currently alive cannot trade with the unborn To correct this inefficiency 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 financial 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 Bt1 denote the quantity of oneperiod bonds issued by the government in period 15 Each of these bonds is a promise to pay 1 rt1 26 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 Bt1 bLt where b is a constant That is the quantity of government debt is xed in percapita terms The government s budget constraint is Bt1 CF15 T39tBt ie the revenues from new bond issues and taxes in period 15 T15 equals the payments of interest and principal on government bonds issued in period t 1 Taxes are levied lumpsum on young agents and we will let 73975 denote the taX per young agent We then have I TtLt A young agent solves HEXth 8t 73975 1 l Tt18t7 where 8 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 st 8wt Tt n1 224 As before profit maximization by the firm implies 211 and 212 A competitive equilibrium is defined as above adding to the defini tion that there be a sequence of taxes 73i0 satisfying the government budget constraint From 221 222 and 223 we get T 2 ft n b 225 1 l n The asset market equilibrium condition is now kt11 n b 8wt Tt n1 226 36 CHAPTER 2 GROWTH WITH OVERLAPPIN G GENERATIONS that is per capita asset supplies equals savings per capita Substituting in 226 for wt 73975 and n1 from 211 we get mimrwwbsfaa mfmw i 3atho222 We can then determine the steady state capitallabor ratio kb by setting kb kt kt1 in 227 to get f kb n 1 n kwwumbs ww kwwfcww Achw 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 IA such that Now given that n from 228 we can solve for IA as follows 13 41 n 8 mg 2n n 229 In 229 note that IA may be positive or negative If 13 lt 0 then debt is negative ie the government makes loans to young agents which are nanced by taxation Note that from 225 73975 2 0 in the steady state with b 13 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 71 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 26 1 Example Consider the same example as above but adding government debt That is ucyco lncy lnco and FK L VKO L1O where 5 gt 0 y gt 0 and 0 lt 04 lt 1 Optimal savings for a young agent is s w 7 230 15 2 7 REFERENCES 37 Then from 216 217 227 and 230 the equilibrium sequence k f o is determined by M1 n b lt1 04W and the steady state capitallabor ratio kb is the solution to kb1n b 1 ozy Mama 7k1b quotb Then from 229 the optimal quantity of percapita debt is IA 2 i 1 a7 1n 1 n n 047 51 04 oz 7 n 1 n 39 Here note that given 7 n and A lt 0 for 04 suf ciently large and b gt 0 for 04 suf ciently small ozyktohl 77 1 n 7 262 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 off the currentlyissued 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 27 References Diamond P 1965 National Debt in a Neoclassical Growth Model American Economic Review 55 11261150 3SCHAPTER 2 GROWTH WITH OVERLAPPIN G GENERATIONS Blanchard O and Fischer S 1989 Lectures on Macroeconomics Chapter 3 Kareken J and Wallace 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 rst 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 7 will be something of a misnomer in this case as the model will not exhibit longrun growth One objective of this chapter will be to introduce and illustrate the use of discretetime dynamic programming methods which are useful in solving many dynamic models 39 40 CHAPTER 3 NEOCLASSI CAL GROWTH AND DYNAMIC PROGRAMMING 31 Preferences Endowments and Tech nology There is a representative infinitelylived consumer with preferences given by Z WCt t0 where 0 lt 5 lt 1 and C is consumption The period utility function is continuously differentiable strictly increasing strictly concave and bounded Assume that limc0 u39c 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 yt 2 F097 m 31 where yt is output kt is the capital input and 71 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 limk0 F1k1 00 and limknoo F106 The capital stock obeys the law of motion kt1 it where it is investment and 6 is the depreciation rate with 0 S 6 S 1 and k0 is the initial capital stock which is given The resource constraints for the economy are Ct it E yt 33 and 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 perperiod profits Given this it is a standard result that the competitive equilibrium is unique and that the first 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 00 max in c Ctnt itkt10 1220 t subject to Ct it E Fkt nt kt1 it 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 satisfied with equality As there is no disutility from labor if 37 does not hold with equality then 71 and ct 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 define E Fk 1 as nt 2 1 for all t Then the problem can be reformulated as 00 max t39U C ctkt10t 0 0 subject to Ct kt1 ffkt l 1 31975 15 0 1 2 k0 given This problem appears formidable particularly as the choice set is infinitedimensional 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 NEOCLASSI CAL GROWTH AND DYNAMIC PROGRAMMING variable for the problem Within the period the choice variables or control variables are ct and kt1 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 first period That is in period 0 the social planner solves gagluko MUM subject to CO k1 This is a simple constrained optimization problem which in principle can be solved for decision rules k1 gk0 where is some function and co f k0 1 3160 gk0 Since the maximization problem is identical for the social planner in every period we can write vase ggglmco WM subject to Co k1 ko 1 51907 or more generally vast We vkt1l 38 subject to Ct kt1 l 1 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 kt1 gkt and 0 1 31975 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 find it In most of the cases we will deal with the Bellman equation satisfies a contraction mapping theorem which implies that 1 There is a unique function which satisfies the Bellman equa tion 32 SOCIAL PLANNER S PROBLEM 43 2 If we begin with any initial function 00 and de ne vi1k by vi1k Iggxluk vzk subject to ck M 1 6k for i 0 1 2 then limOO vi1k 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 vkt1 into the right side of 38 and then verify that 220675 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 find an approximation to the value function in the following manner First allow the capital stock to take on only a finite number of values ie form a grid for the capital stock k 6 161162 S where m is finite and k lt CZ1 Next guess an initial value function that is m values of v0ki 1 2 m Then iterate on these values determining the value function at the jth iteration from the Bellman equation that is L e 39Uj Iggxlufc l vj ll subject to c k2 Na 1 6 Iteration occurs until the value function converges Here the accu racy of the approximation depends on how fine the grid is That is if k ki1 y i 2 m 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 m grid points This problem of computational burden as n gets large is sometimes referred to as the curse of dimensionality 44 CHAPTER 3 NEOCLASSI CAL GROWTH AND DYNAMIC PROGRAMMING 321 Example of Guess and Verify Suppose that Fktnt 2 19377320 lt 04 lt 1 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 22kt maXlnkt kt1 vkt1 310 kt1 Now guess that the value function takes the form Ukt A l Bll lkt where A and B are undetermined constants Next substitute using 311 on the left and right sides of 310 to get A B ln kt maXlnkt kt1 A B ln kt1 312 kt1 Now solve the optimization problem on the right side of 312 which gives k 53k H1 1 B and substituting for the optimal kt1 in 312 using 313 and col lecting terms yields 313 A I Blnkt z Bln B 1 Bln1l B A 1 Bozlnkt 314 We can now equate coefficients on either side of 314 to get two equa tions determining A and B Az Bln B 1 Bln1 B A 315 B 1 Boz 316 Here we can solve 316 for B to get oz 1 oz 39 317 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 kt1 km a kf 318 Since 0 16 16t1 we have Ct 1 0451191 That is consumption and investment which is equal to kt1 given 100 depreciation are each constant fractions of output Equation 318 gives a law of motion for the capital stock ie a firstorder nonlinear difference equation in 167 shown in Figure 31 The steady state for the capital stock 16 is determined by substituting 16 kt1 16 in 318 and solving for 16 to get k am Given 318 we can show algebraically and in Figure 1 that 16 con verges monotonically to 16 with 1675 increasing if 160 lt 16 and 165 decreas ing if 160 gt 16 Figure 31 shows a dynamic path for kt where the initial capital stock is lower than the steady state This economy does not eXhibit longrun growth but settles down to a steady state where the capital stock consumption and output are constant Steady state con sumption is c 1 oz 16 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 firstorder conditions Substituting in the objective function for 0 using in the constraint we have Ukt 133qu kt 1 5161 l 11l vkt1 319 46 CHAPTER 3 NEOCLASSI CAL GROWTH AND DYNAMIC PROGRAMMING Figure 31 katIlj k1 H H if quotx km Figure 31 32 SOCIAL PLANNER S PROBLEM 47 Then the rstorder condition for the optimization problem on the right side of 38 after substituting using the constraint in the objective function is wyuwrr am kng wwngo 92m The problem here is that without knowing v we do not know 0 However from 319 we can differentiate on both sides of the Bellman equation with respect to kt and apply the envelope theorem to obtain 0th ull kt 1 5197 kt1llflkt 1 Sly or updating one period Ulkt1 ullfkt1 1 6kt1 kt2llflfkt1l 1 6l 321 Now substitute in 320 for v kt1 using 321 to get ulfkt 1 3191 kt1l ullfkt1 1 6kt1 kt2f kt1 1 6 07 322 or u0t ulct1lflkt1 1 6l 07 The first term is the benefit 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 benefit obtained in period t 1 discounted to period 15 from investing the foregone consumption in capital At the optimum the net benefit must be zero We can use 322 to solve for the steady state capital stock by setting kt kt1 kt2 6 to get fwa ia 3 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 NEOCLASSI CAL GROWTH AND DYNAMIC PROGRAMMING 323 Competitive Equilibrium Here I 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 rm s optimization problems Consumer s Problem Consumers store capital and invest ie 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 00 max Ema ct aktl 10 subject to Ct kt1 2 10 T39tk t 66 15 t 0 1 2 k0 given where wt is the wage rate and rt is the rental rate on capital If we simply substitute in the objective function using 324 then we can reformulate the consumer s problem as 00 max 257516007 Ttkt l 1 5197 kt1 kt l 11gt0 subject to kt 2 0 for all t and k0 given Ignoring the nonnegativity constraints on capital in equilibrium prices will be such that the con sumer will always choose kt1 gt 0 the rstorder conditions for an optimum are t39Ulwt Ttkt kt1 t1ulwt1 Irtlkt1 6kt1 kt2gtltTt1 1 0 325 32 SOCIAL PLANNER S PROBLEM 49 Using 324 to substitute in 325 and simplifying we get ulct1 1 u ct 1 n1 67 326 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 profits each period ie it solves rknaXFkt n75 1075711 Ttktla and the rstorder conditions for a maximum are F1kt 7713 T39t F2kt nt 39LUt Competitive Equilibrium Prices The optimal decision rule kt1 gkt which is determined from the dynamic programming problem 38 allows a solution for the compet itive equilibrium sequence of capital stocks k fil given k0 We can then solve for ct 0 using 39 Now it is straightforward to solve for competitive equilibrium prices from the rstorder conditions for the rm s and consumer s optimization problems The prices we need to solve for are 1075 rt0 the sequence of factor prices To solve for the real wage plug equilibrium quantities into 328 to get F2ltkt 7015 To obtain the capital rental rate either 326 or 327 can be used Note that rt 6 f kt 6 is the real interest rate and that in the steady state from 326 or 323 we have 1 39r 6 or if we let 5 where 77 is the rate of time preference then 39r 6 77 ie the real interest rate is equal to the rate of time preference 50 CHAPTER 3 NEOCLASSI CAL GROWTH AND DYNAMIC PROGRAMMING Note that when the consumer solves her optimization problem she knows the whole sequence of prices 1075 r fio 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 firms 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 727732 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 233240 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 discretetime 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 growthenhancing policies Before we can hope to evaluate the efficacy 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 00 quoty Z mi en 730 7 where 0 lt 5 lt 1 y lt 1 0 is per capita consumption and N75 is population where N 1 ntN0 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 Y2 KtaNtAt1 a7 43 where Y is aggregate output Kt is the aggregate capital stock and At is a laboraugmenting technology factor where A 1 am 44 41 A NEOCLASSICAL GROWTH MODEL EXOGENOUS GROWTH53 with a constant and A0 given We have 0 lt 04 lt 1 and the initial capital stock K0 is given The resource constraint for this economy is NtCt Kt1 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 4245 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 define quantities normalized by efficiency units of labor for example yt E Ag Also let mt E 2 With substitution in 41 and 45 using 4244 the social planner s problem is then 00 max 2W1 n1 amt mt akt39l l Rio tO 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 61 and kt1 That is given the value function vkt the Bellman equation is mm u mm agtivltkt1gtj subject to 46 Note here that we require the discount factor for the problem to be less than one that is 61 n1 a y lt 1 Substituting in the objective function for 61 using 46 we have 191 l 111 n1 0W 7 tact maX kt1 M1 mm agtivltkt1gtj 47 54 CHAPTER 4 ENDOGENOUS GROWTH The rstorder condition for the optimization problem on the right side of 47 is lt1 M1 maul 1 131 mm Mam o 48 and we have the following envelope condition v39kt 614163 15132 1 49 Using 49 in 48 and simplifying we get 1 a1 YaY 1 akg xgrf 0 410 Now we will characterize balanced growth paths that is steady states where 51375 513 and kt 15 where 513 and 15 are constants Since 410 must hold on a balanced growth path we can use this to solve for 15 that is 19 411 Then 46 can be used to solve for 513 to get a Z 12 1n1a 412 Also since yt 2 kg then on the balanced growth path the level of output per ef ciency unit of labor is m W 413 1quot 06 In addition the saVings rate is Kt1 kt11 quotMl a Y k 7 573 so that on the balanced growth path the saVings rate is 8 151 1 n1 a 41 A NEOCLASSICAL GROWTH MODEL EXOGENOUS GROWTH55 Therefore using 411 we get 8 oz1 n1 a 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 56 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 a Thus longrun 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 04 or 7 have no effect on longrun growth Note in particular that an increase in any one of 04 or 7 results in an increase in the longrun 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 04 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 aquoty lt 1 implies that an increase in steady state 16 will result in an increase in steady state 513 Therefore an increase in leads to an increase in 513 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 04 or 7 But if all countries have access 56 CHAPTER 4 ENDOGENOUS GROWTH to the same technology then 04 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 longrun 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 Y ahtutNta where 04 gt 0 Y1 is output ht is the human capital possessed by each agent at time t and at is time devoted by each agent to production That is the production function is linear in qualityadjusted labor in put Human capital is produced using the technology ht1 Ut where 6 gt 0 1 at is the time devoted by each agent to human capital accumulation ie education and acquisition of skills and ho is given Here we will use lower case letters to denote variables in per capita terms for example yt 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 ht1 and at That is the Bellman equation for the social planner s problem is mm max LawnWm Ct utht1 7 subject to Ct Odht Ut and 415 Then the Lagrangian for the optimization problem on the right side of the Bellman equation is Y L 2 51 nvht1 Atmhtut c ut6ht1 at hm where At and at are Lagrange multipliers Two rstorder conditions for an optimum are then 3 cgl A 0 417 8L ahtH 51 n ht1 Mt 07 418 415 and 416 In addition the rst derivative of the Lagrangian with respect to at is 8 L AtOdht Ltt ht a Ut Now if g C gt 0 then at 1 But then from 415 and 416 we have hs 2 cs Ut0 for s t 1 t 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 3 S 0 If 3 lt 0 then at 0 and ct 0 from 416 Again this could not be optimal so we must have 8L a Ut AtOdht Ltt ht 0 at the optimum 58 CHAPTER 4 ENDOGENOUS GROWTH We have the following envelope condition Ulht Od UtAt 7113 or using 417 201 0462 1 420 From 417420 we then get 51 M601 02 1 0 421 Therefore we can rewrite 421 as an equation determining the equi librium growth rate of consumption Ct1 cmunwp 23 ct Then using 415 416 and 422 we obtain 51 n6 61 utut1 Ht 7 or 1 51 l mml m 1 Ut 1 Ut1 2 Now 423 is a firstorder difference equation in at depicted in Figure 41 for the case where 1 77 l397639y lt 1 a condition we will assume holds Any path 7in satisfying 423 which is not stationary a stationary path is at u a constant for all t has the property that limt00 at 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 mu1 munww for all 15 Therefore substituting in 415 we get ht1 h t W1 nl5l 7 43 ENDOGENOUS GROWTH WITH PHYSICAL CAPITAL AND HUMAN CAPITAL59 and human capital grows at the same rate as consumption per capita If gt 1 which will hold for 6 sufficiently 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 Y2 KtaNthtut1 a7 where Kt is physical capital and 0 lt 04 lt 1 and the economy s resource constraint is NtCt l Kt1 K3Nthtutl a 60 mij CHAPTER 4 ENDOGENOUS GROWTH Figure 41 45 deg line Figure 41 110 uEstati na 3311 um 43 ENDOGENOUS GROWTH WITH PHYSICAL 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 ht and four choice variables at ct ht1 and kt1 The Bellman equation for this dynamic program is Y c Ukt ht rianh t 51 nvkt1 ht1 Cum t1 t1 7 subject to Ct nkt1 kht39Ut1 a ht1 Ut The Lagrangian for the constrained optimization problem on the right side of the Bellman equation is then C L 7t51n 0kt17ht1tlkhtut1a Ct 1nkt1l tl5ht1 Ut ht1l The rstorder conditions for an optimum are then 8L 1 A 42 act ct t 07 8L At1 ozkfhl uta ut ht 0 427 a Ut 8L 51 nv2kt17 ht1 Mt 07 428 3ht1 8L 2 t1 n nv1kt17 ht1 07 3kt1 424 and 425 We also have the following envelope conditions 39U1kt ht Atalkta 1ht Ut1 a U2kt ht OdkhaU a Ut Next use 430 and 431 to substitute in 429 and 428 respec tively then use 426 and 427 to substitute for At and at in 428 and 429 After simplifying we obtain the following two equations Cy1 ciyFllo kfljllhtlutll a 07 432 62 CHAPTER 4 ENDOGENOUS GROWTH cty 1ktaht aut a l 651 l ncty11k1hflufl 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 uk ML and MC denote the growth rates of physical capital human capital and consumption respectively on the balanced growth path From 425 we then have 1 uh 61 ut which implies that 1 uh 6 7 a constant along the balanced growth path Therefore substituting for at ut1 and growth rates in 433 and simplifying we get 7113 1 Mal 41 Mia 1 m 651 M 434 Next diViding 424 through by kt we have k 1 n k31hu1a 435 Then rearranging 432 and backdating by one period we get 1 l Hal W og kf 1htut1 436 Equations 435 and 436 then imply that ct uc1 fy 1 1 kt lt ngtlt w 0 But then is a constant on the balanced growth path which implies kt that pa 2 uk Also from 436 since at is a constant it must be the case that m 2 ML 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 1uc1uk1uh1upawnW 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 S Kt1 kt11 77 t Y1 ktkta 1htut1a Using 436 and 437 on the balanced growth path we then get 5 a 6W1 mm 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 22 342 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 rst 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 73 gives the consumer 01 units of consumption with probability pi and 02 units of consumption with probability 1 p where 0 lt p lt 1 i 1 2 65 66 CHAPTER 5 CHOICE UNDER UN CERTAIN TY Then the expected utility the consumer receives from lottery 73 is lame 1 1901403 and the consumer would strictly prefer lottery 1 to lottery 2 if p1ucl 1 p1uci gt 192140 1 1991403 would strictly prefer lottery 2 to lottery 1 if p1ucl 1 WW lt 192140 1 MINCE and would be indifferent if p1uci 1 P1uci 192140 1 MINCE 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 EMCH S 16 EM 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 Ec uE see Figure 1 This tangent is described by the function 90 04 50 52 where 04 and are constants and we have 04 Ec 53 Now since uc is strictly concave we have as in Figure 1 04 0 2 uc 54 51 EXPECTED UTILITY THEORY 67 for c 2 0 with strict inequality if 0 7E E Since the expectation operator is a linear operator we can take expectations through 54 and given that c is random we have 04 ElCl gt Elucl or using 53 ElCD gt EMM As an example consider a consumption lottery which yields cl units of consumption with probability p and 02 units with probability 1 p where 0 lt p lt 1 and 02 gt cl In this case 51 takes the form PU01 1 PUC2 lt U 1901 1 PC2 In Figure 2 the difference U 1901 1 W2 Pu01 1 PUC2l is given by DE The line AB is given by the function 02ucl c1uc2 u02 ucl c f0 C2 01 C2 01 A point on the line AB denotes the expected utility the agent receives for a particular value of p for example p 0 yields expected utility ucl or point A and B implies p 1 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 111 gt 1115 89111 01110 OI 11111 gt 1115 01110 55 Similarly a preference for lottery 4 over lottery 3 gives 11111 89110 lt 1115 9110 OI 11111 lt 1115 9110 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 affine transformations of the utility func tion That is suppose a utility function maaama 51 EXPECTED UTILITY THEORY 69 where 04 and are constants with gt 0 Then we have El UCl 04 EMdl since the expectation operator is a linear operator Thus lotteries are ranked in the same manner with vc or uc as the utility function Any measure of risk aversion should clearly involve u 0 since risk aversion increases as curvature in the utility function increases However note that for the function 220 that we have v c u c ie the second derivative is not invariant to affine transformations which have no effect on behavior A measure of risk aversion which is invariant to affine transformations is the measure of absolute risk aversion u c We 39 A utility function which has the property that ARAC is constant for ARAC all c is uc e c 04 gt 0 For this function we have O2 occ ARAC 04 Ole ac 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 u c We 39 A utility function for which RRAC is constant for all c is RRAC c where y 2 0 Here 70 CHAPTER 5 CHOICE UNDER UN CERTAIN TY Note that the utility function uc lnc has RRAC 1 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 Elu0l ElCla so that the consumer cares only about the expected value of consump tion Since u c 0 and u c we have ARAC RRAC 0 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 00 E0 2 tuct t0 where 0 lt 5 lt 1 ct 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 ct 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 yt ZtFkt7 7715 where F is strictly quasiconcave homogeneous of degree one and increasing in both argument Here kt is the capital input nt is the labor input and 25 is a random technology disturbance That is z fio is a sequence of independent and identically distributed iid random variables each period 275 is an independent draw from a fixed probability distribution C In each period the current realization 25 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 2713 66 13 where it is investment and 6 is the depreciation rate with 0 lt 6 lt 1 The resource constraint for this economy is Ctityt 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 first 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 he she 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 15 0 At t 0 the representative firm and the representative consumer get together and trade contingent claims on competitive markets A contingent claim is a promise to deliver a specified 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 20 21 22 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 statecontingent 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 rt 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 15 for every 72 CHAPTER 5 CHOICE UNDER UN CERTAIN TY possible realization of the random shocks 20 21 22 25 In equilib rium expectations are rational in the sense that agents 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 275 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 maX E0 203 tO ctakt110 subject to Ct kt1 thkt 1 51975 where f E F k 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 21 where kt is determined by past decisions and 21 is given by nature and known when decisions are made concerning the choice variables ct and kt1 The Bellman equation is written as ktaztl max lufct Etvkt1azt1l Ctkt1 subject to Ct kt1 thkt 52 STOCHASTIC DYNAMIC PROGRAMMING 73 Here 2 is the value function and E is the expectation operator conditional on information in period 15 Note that in period 15 ct is known but 0H 2 1 23 is unknown That is the value of the problem at the beginning of period t 1 the expected utility of the representative agent at the beginning of period t 1 is uncertain as of the beginning of period 15 What we wish to determine in the above problem are the value function v and optimal decision rules for the choice variables ie kt1 gkt 2t and 0 ztfkt 1 661 90917215 5 23 Example Let Fktnt kf n with 0 lt 04 lt 1 lnct 6 1 and Eln 25 u Guess that the value function takes the form vktzt A I Blnkt i Dlnzt The Bellman equation for the social planner s problem after substi tuting for the resource constraint and given that nt 2 1 for all t is then A I B ln kt I D ln 2 max lnztkt kt1l EtM B 111 kt1 D 111 Zt17 t1 OI A I B ln kt l D ln 2 max lnztkt kt1l A 3 ln kt1 Di kt1 57 Solving the optimization problem on the righthand side of the above equation gives 53 a kt1 mask Then substituting for the optimal kt1 in 57 we get Ztk ta BZtk ta A Bl k D1 1 A Bl D 1115 1127 nlt1 B 5 nlt1 B 5 M 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 coef cients on either side of equation 59 gives 1 BB A l lt1 B A Bl lt1 B DL 510 B oz I oz B 511 D 1 B 512 Then solving 510512 for A B and D gives 04 le oz 1 DZl oz 1 Az ln1 oz 1i i lnoz 1 We have now shown that our conjecture concerning the value function is correct Substituting for B in 58 gives the optimal decision rule kt1 oz ztkta 513 and since 0 ztkf kt1 the optimal decision rule for 0 is ct 1 awash 514 Here 513 and 514 determine the behavior of time series for 0 and kt Where kt1 is investment in period 15 Note that the economy Will not converge to a steady state here as technology disturbances Will cause persistent uctuations 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 k0 determine a sequence 2ttT0 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 postwar detrended US time series though there will be obvious ways in which this model does not fit 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 ln kt1 ln oz ln yt and lnct ln1 oz lnyt We therefore have uarln kt1 uarln ct uarln gt 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 RBC analysis is essentially an exercise in mod ifying this basic stochastic growth model to fit the postwar 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 longrun 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 fit 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 53 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 chrea 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 in 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 consumptionsavings 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 6 1 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 5 lt 1 ct is consumption and is increasing strictly concave and twice differentiable The consumer s budget constraint is At1 1 7 At 0 wt 62 for t 0 1 2 where 39r is the oneperiod interest rate assumed con stant over time and 10 is income in period t where income is exoge nous We also assume the noPonZischeme condition At lim t gtoo TV This condition and 62 gives the intertemporal budget constraint for the consumer 7520 1 I t39rV A0 7 1 t7gtt 63 The consumer s problem is to choose ctAt1E0 to maximize 61 subject to 62 Formulating this problem as a dynamic program with the value function 12At assumed to be concave and differentiable the Bellman equation is Am 1 39r 22At max u 10 At At1 vAt1 The rstorder condition for the optimization problem on the right hand side of the Bellman equation is 1 17 and the envelope theorem gives I A I U 10 At 1 1 At1 0 UAt u39 10 At Alt H 65 61 CONSUMPTION 79 Therefore substituting in 64 using 65 and 62 gives Wet u39fctH That is the intertemporal marginal rate of substitution is equal to one plus the interest rate at the optimum Now consider some special cases If 1 39r Ir ie if the interest RIP rate is equal to the discount rate then 66 gives Ct Ct1 C for all t where from 63 we get 0 11 67 Here consumption in each period is just a constant fraction of dis counted lifetime wealth or permanent income The income stream given by 1155in 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 51 6 6398 80 CHAPTER 6 CONSUMPTION AND ASSET PRICING so that consumption grows at a constant rate for all 15 Again the consumption path is smooth From 68 we have Ql CtCOW1Tgtl 7 and solving for co using 63 we obtain c0 1 51 1771 A0 1tr 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 25757400 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 15 Given a value function UAt wt for the consumer s problem the Bellman equation associated with the consumer s problem is A vAt wt 2 max uAt wt 1 EtvAt1 w151 7 At1 1 7 and the rstorder condition for the maximization problem on the right hand side of the Bellman equation is Am 1 39r UAt 10 gt Et Ul At1 107511 1r We also have the following envelope condition At1 1 39r v1At wt 2 u39At wt 610 61 CONSUMPTION 81 Therefore from 62 69 and 610 we obtain 61 i r 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 is a martingale with drift How ever without knowing the utility function this does not tell us much about the path for consumption If we suppose that is quadratic Etu39ct1 611 ie E ct2 where E gt 0 is a constant 611 gives EtCt 1 r 1 Ct 51 r 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 fit 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 fit the data The first 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 fit 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 ICAPM intertemporal capital asset pricing model or the consumptionbased capital asset pricing model This is a representative agent economy where the representative consumer has preferences given by 00 E0 Z tuct 612 where 0 lt 5 lt 1 and is strictly increasing strictly concave and twice differentiable Output is produced on n productive units where git is the quantity of output produced on productive unit 73 in period 15 Here gas 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 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 73 in terms of the consumption good and zit the 62 ASSET PRICING 83 quantity of shares in productive unit 73 held at the beginning of period t the representative consumer s budget constraint is given by Epitziat l Ct Z Zit Pit yit7 614 i1 i1 for t 0 1 2 The consumer s problem is to maximize 612 subject to 614 Letting pt 21 and yt denote the price vector the vector of share holdings and the output vector for example yt ylt ygt ynt we can specify a value function for the consumer UZt pt yt and write the Bellman equation associated with the consumer s problem as 27571973790 2 max l EtvZt1iPt1iyt1l Ctzt1 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 I L UZt7Pt7 yt U ZitPit yit PitZit1l EtvZt17pt17yt1 i1 Now the firstorder conditions for the optimization problem on the righthand side of the above Bellman equation are n 81 PitU39 Zit Pit yit PitZit1l Eta 0 615 i1 Zit1 for 73 1 2 n We have the following envelope conditions 8 I n a Pit yitu zit Pit yit pitzm j 616 it i1 Substituting in 615 using 613 614 and 616 then gives PitU39 Pit Et Pit1 yit1ul yit1 07 617 i1 i1 for 73 12 n or Pit Et Pit1 yit131 0 618 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 mt denote the gross rate of return on share 73 between period t and period t 1 ie pit1 y t1 7113 7 pit and let mt denote the intertemporal marginal rate of substitution 5 CH1 mt 1639 Ct Then we can rewrite equation 618 as Et Tlitmt 1 or using the fact that for any two random variables X and Y 001 X Y EXY EXEY CO Ut Tlit mt Et Tlit Et mt 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 5615 E Etszcts EtzctS 8 2 s 2 0 to get 00 s tucs pi E 619 t t 32131 ILct 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 yit1ul yit1 Ai 620 i1 for 73 1 2 n where Az gt 0 is a constant That is the expression inside the expectation operator in 620 is a function of pi 1 and yit1 73 1 2 71 each of which is unpredictable given information in period 15 therefore the function is unpredictable given information in period 15 Given 617 and 620 we get 9 3 m Zt I 2211 yit 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 Et pit1 yit17 ie the current price is the discount value of the expected price plus the dividend for next period or z39 z39 z39 1 Pt1yt1 Pt 1 621 pit 5 Equation 621 states that the rate of return on each asset is unpre dictable given current information which is sometimes taken in the E7 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 00 pit Et 2 styisa st1 or the current price is the expected present discounted value of divi dends A third example considers the case where uc 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 y1 7r 0 lt 7r lt 1 Let p denote the price of a share when yt y for 73 1 2 Then from 617 we obtain two equations which solve for p1 and p2 y1 y1 P1 W Pl 91 1 7Tl P2 92 y1 M 92 M P2 W Pl 91 1 7Tl P2 92 91 y2 Since the above two equations are linear in p1 and p2 it is straightfor ward to solve obtaining 591 P1 1 592 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 riskfree asset which trades on 62 ASSET PRICING 87 a competitive market at each date This is a oneperiod riskfree bond which is a promise to pay one unit of consumption in the following period Let bt1 denote the quantity of riskfree bonds acquired in period t by the representative agent note that bt1 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 Epitzi 1 Ct tht1 Z Ztt pit ytt bt i1 i1 In equilibrium we will have 1 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 resolving the consumer s problem but it is more straightforward to simply use equa tion 617 setting pmH 0 since these are oneperiod bonds they have no value at the end of period t 1 and yi 1 1 to get 39CtH E 622 Qt 5 t uct gt The oneperiod riskfree interest rate is then 1 rt 2 1 623 It If the representative agent is risk neutral then qt 2 and rt 2 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 1 and there are two assets an equity share which is a claim to aggregate output and a oneperiod riskfree asset as discussed above They consider preferences of the form 1 1 6 c 1 7 In the data set which Mehra and Prescott examine which includes annual data on riskfree 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 riskfree 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 twostate Markov process that is Prlyt1 9939 l yt yil 7 We will assume that 7n 0 for 73 1 2 where 0 lt 0 lt 1 Here we want to solve for the asset prices 1 pi 73 12 where qt 2 q and pt 2 p when yt yi for 73 1 2 From 617 we have plyy 5 p691 my 1 692 y2y239yl 624 P29y 0692 92M 1 0P1 y1y1 fy 625 Also 622 implies that Q1 0 1 0 626 12 0 1 0 627 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 riskfree asset is certain and given by rt in 623 Let rt 2 7 when yt y for 73 1 2 For the equity share the eXpected return denoted R75 is given by Rt 2 Et ltPt1 yt1 Pt Pt 62 ASSET PRICING 89 Therefore letting R denote the expected rate of return on the equity share when yt y we get R12pltp1y1 1p 192y2 1 P1 P1 R2 ZpltP2y2 1plt191yi 1 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 longrun probability of being in either state is here Therefore the average equity premium is 1 R1 1 5R2 quot 27 DIH 6 7 77 107 yla y2 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 ypy1y2 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 riskfree rate of inter est rt 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 riskfree 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 CyclePermanent lncome Hypothesis Journal of Political Economy 86 971987 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 14261445 Mehra R and Prescott E 1985 The Equity Premium A Puzzle Journal of Monetary Economics 15 145162 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 If 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 efficacy 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 onesided 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 twosided search models for a summary see Pissarides 1990 in which workers and firms 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 first 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 onesided 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 00 where 0 lt 5 lt 1 and is concave and strictly increasing Let 1 where 39r 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 10 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 10 which has associated with it a probability density func tion Assume that w 6 0122 ie the set 0122 is the support of the distribution If an agent is employed receiving wage to assume that each job requires the input of one unit of labor each period then her consumption is also 10 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 benefit I from the government at the beginning of the period and then receives a wage offer that she may accept or decline Let V and Vw 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 f maX lw Vu fwdw 71 Vw 6V 1 6lw 72 71 A ONESIDED 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 F The wage offer is accepted if 110 2 V 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 110 2 V otherwise she would not have accepted the job in the first place In search models a useful simplification of the Bellman equations is obtained as follows For 71 divide both sides by substitute 2 TL and subtract Vu from both sides to obtain TV 2 111 f maX l10 V 0 f10d10 73 On the righthand side of 73 is the flow return when unemployed plus the eXpected net increase in eXpected utility from the unemployed state Similarly 72 can be simplified to obtain mm ultwgt 6M mm 74 We now want to determine what wage offers an agent will accept when unemployed From 74 we obtain 1110 614 V7110 39 Therefore since u10 is strictly increasing in 10 so is 110 Thus there is some 10 such that 110 2 V for 10 2 10 and 110 S V for 10 g 10 see Figure 71 The value 10 is denoted the reservation wage That is an unemployed agent will accept any wage offer of 10 or more and decline anything else Now let 7 denote the fraction of agents who are employed in period t The flow 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 7t 1 F Further the 94 CHAPTER 7 SEARCH AND UNEMPLOYMENT Figure 7quot1 VI w Figure 71 71 A ONESIDED SEARCH MODEL 95 flow of agents out of employment to unemployment is the number of separations 7156 Therefore the law of motion for 715 is 7t1 71 1 7t 1 Fwl 715 75 1 Fwawwa Since 1 F 10 6 llt 1 7t converges to a constant 7 which is deter mined by setting 7t1 75 y in 75 and solving to get 1 Fw 761 Fw39 76 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 1 7 so that if the separation rate increases increasing the flow 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 flow 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 110 Vu decreases for each 10 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 benefits acts to make unemployed agents more choosy concerning the jobs they will accept decreasing the flow 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 12 with probability 7r and an offer of zero with probability 1 7r where 0 lt 7r lt 1 Suppose first that 0 lt b lt 17 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 offers are always accepted Low wage offers 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 TV 2 ub 7rV Vu TV 2 u1I 6Vu V and we can solve these two equations in the unknowns V and V to obtain 7 7ru1I 6ub 39r39r 6 7r V19 7 7ru1D 39r 6ub 39r39r 6 7r Vu 2 Note that V Vu depends critically on the difference between 12 and b and on the discount rate 39r The number of employed agents in the steady state is given by 7r 67r7 7 so that the number employed unemployed increases decreases as 7r increases and falls rises as 6 increases Now for any I gt 12 clearly we will have 7 0 as no offers of employment will be accepted due to the fact that collecting unemploy ment insurance dominates all alternatives However if b if then an unemployed agent will be indifferent between accepting and declining a high wage offer 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 777T 6 777r7 7 which is increasing in 77 This is a rather stark example where changes in the UI benefit have no effect before some threshold level but increasing benefits above this level causes everyone to turn down all job offers 72 DIAMOND S COORDINATION FAILURE MODEL 97 7 12 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 Twosided 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 lumpsum taxes on employed agents Note in the example above that it would be infeasible to have b gt if 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 Keynesiantype lowemployment steady state where there is a possibility of welfareimproving 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 Paretoranked 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 mostpreferred 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 justifiable 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 flexible money policy pp 267268 721 The Model There is a continuum of agents with unit mass each with preferences given by E0 Met at where 0 lt 5 lt 1 is strictly increasing with u0 0 ct 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 9 one production opportunity arrives during the period and with prob ability 1 9 no production opportunities arrive A given production opportunity is defined by its production cost 04 Given that a produc tion opportunity arrives it is a draw from a probability distribution F 04 with corresponding probability density function f 04 Assume that 04 E 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 y 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 37 where is a strictly increasing function with 130 2 0 and 131 3 1 and y is the fraction of employed agents in the popula tion Thus the probability of finding a trading partner increases with the number of wouldbe 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 wouldbe 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 find a smaller tree If she picks a coconut unfortunately she cannot eat it herself and she wanders the beach until she finds another person with a coconut they trade eat their coconuts and then go back to searching for trees In terms of capturing realworld 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 firms 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 V 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 V9 hOW VFWlH 6WH V 9 00 maXV 04 Vuf04d04 1 9Vu As in the onesided search model it is convenient to substitute 1 in the above two Bellman equations and manipulate to get W be mm v Va 77 TV 2 6 maXV 04 Vu 0f04d04 78 Now note that from 78 since V and V are constants then assuming that V Vu gt g there eXists some 04 such that V 04 Vu 2 0 for 04 g 04 and V 04 Vu lt 0 for 04 gt 04 That is the agent will accept any production opportunities with 04 g 04 and decline any with 04 gt 04 so that 04 is a reservation cost note that we suppose that if the agent is indifferent she accepts We can then rewrite 78 as TV 2 9F0404 9 04f04d04 79 where M mm Then subtracting 79 from 77 and substituting using 710 we obtain 404 bvgtuy or 6Fltagta e aroma 72 DIAMOND S COORDINATION FAILURE MODEL 101 which can be simpli ed by using integration by parts to get 404 97 uy 04 9 F04d04 711 Further using 710 to substitute in 77 we have W be My 04 712 Given the reservation cost strategy the ow of agents into employ ment in the steady state each period is 9F041 y and the ow out of employment is 707 In the steady state these ows must be equal or 9F04 713 1 7 Now equations 711 and 713 determine 04 and y and then 79 and 712 solve for V and lIe respectively In 713 one solution is 04 g and y 0 and since the righthand side of 713 is strictly increasing in y and the lefthand side is strictly increasing in 04 713 determines an upwardsloping locus AA as depicted in Figure 72 Further in 711 one solution is 04 g and 7 1 where 1gt 0 and solves m 131 My gl and totally differentiating 711 and solving we get dor Home or d7 7 b04 6FO4 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 04 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 04 as well Steady states are Pareto ranked according to the value of 04 To show this rst note that again using integration by parts we can rewrite 79 as TV 2 9a F04d04 2 102 CHAPTER 7 SEARCH AND UNEMPLOYMENT pha gamma Figure 72 72 DIAMOND S COORDINATION FAILURE MODEL 103 so that V increases as steady state 04 increases Second 710 tells us that V Vu increases with 04 so that V increases with steady state 04 as well Thus both the unemployed and the employed are better off in a steady state with a higher 04 and by implication with a higher 7 7 23 Example This example is quite simple but it is somewhat different 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 34 gt 0 and 137 2 by where b gt 0 Here in the steady state we want to determine V13 V y and 7r 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 rnmwwn m cm TV 2 9 max 7rlI3 Vu 34 715 7139601 In the steady state the ow into employment is 1 y97r and the ow out of employment is 721 so in the steady state where the ows are equal we have 2 197 7r 91 7 716 Now we want to consider the three possible cases here in turn ie 7r 2 0 7r 6 0 1 and 7r 2 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 7r 2 1 Then from 715 we have TV 60 V a 717 and then using 714 and 717 we get 37743 904 Vg Vu 39rIb ylQ 104 CHAPTER 7 SEARCH AND UNEMPLOYMENT Now for this to be an equilibrium ie for 7r 2 1 to be optimal we must have V V 2 34 or A bluty 34 Second consider the case where 0 lt 7r lt 1 Here we must have V V 34 so that 715 gives V 0 which implies given 714 that 7 2 718 b n n by 39r and V V 34 implies that To A 719 1 My a gt In a similar manner it is straightforward to show that 7r 2 0 implies that we must have A lt TO 7 A 51169 Oil Thus from 716 and 718720 there are three steady states as de pi ted in Figure 73 These are 7r 7 0 07 0123gt2777 17 249101 w ere 720 A m As in the general model above these steady states are Paretoranked 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 Paretodominated 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 In Figure T3 gamma Figure 73 106 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 financial 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 difficult 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 KeynesType Model Quarterly Journal of Economics 98 525528 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 881894 Farmer R and Guo J 1994 Real Business Cycles and the Animal Spirits Hypothesis Journal of Economic Theory 63 4272 Keynes J 1936 The General Theory of Employment Interest and Money Macmillan London McCall J 1970 Economics of Information and Job Search Quar terly Journal of Economics 84 113126 Phelps E et al 1970 Microeconomic Foundations of Employment and Inflation Theory Norton New York 7 3 REFERENCES 107 Pissarides C 1990 Equilibrium Unemployment Theory Basil Black well Cambridge MA 108 CHAPTER 7 SEARCH AND UNEMPLOYMENT Chapter 8 Search and Money Traditionally money has been viewed as having three functions it is a medium of exchange a store of value and a unit of account Money is a medium of exchange in that it is an object which has a high velocity of circulation its value is not derived solely from its intrinsic worth but from its wide acceptability in transactions It is hard to conceive of money serving its role as a medium of exchange without being a store of value ie money is an asset Finally money is a unit of account in that virtually all contracts are denominated in terms of it J evons 1875 provided an early account of a friction which gives rise to the mediumof exchange role of money The key elements of J evons s story are that economic agents are specialized in terms of what they produce and what they consume and that it is costly to seek out would be trading partners For example suppose a world in which there is a finite number of different goods and each person produces only one good and wishes to consume some other good Also suppose that all trade in this economy involves barter ie trades of goods for goods In order to directly obtain the good she wishes it is necessary for a particular agent to nd someone else who has what she wants which is a single coincidence of wants A trade can only take place if that other person also wants what she has ie there is a double coincidence of wants In the worst possible scenario there is an absence of double coincidence of wants and no trades of this type can take place At best trading will be a random and timeconsuming process and agents will search on average a long time for trading partners 109 110 CHAPTER 8 SEARCH AND MONEY Suppose now that we introduce money into this economy This money could be a commodity money which is valued as a consumption good or it could be fiat money which is intrinsically useless but difficult or impossible for private agents to produce lf money is accepted by everyone then trade can be speeded up considerably Rather than having to satisfy the double coincidence of wants an agent now only needs to find someone who wants what she has selling her production for money and then find an agent who has what she wants purchasing their consumption good with money When there is a large number of goods in the economy two single coincidences on average occur much sooner than one double coincidence The above story has elements of search in it so it is not surprising that the search structure used by labor economists and others would be applied in monetary economics One of the first models of money and search is that of Jones 1976 but the more recent monetary search lit erature begins with Kiyotaki and Wright 1989 Kiyotaki and Wright s model involves three types of agents and three types of goods the sim plest possible kind of absence of double coincidence model and is use ful for studying commodity monies but is not a very tractable model of fiat money The model we study in this chapter is a simplification of Kiyotaki and Wright 1993 where symmetry is eXploited to ob tain a framework where it is convenient to study the welfare effects of introducing fiat money 81 The Model There is a continuum of agents with unit mass each having preferences given by aims t0 where 0 lt 5 lt 1 ct denotes consumption and is an increasing function There is a continuum of goods and a given agent can produce only one of the goods in the continuum An agent gets zero utility from consuming her production good Each period agents meet pairwise and at random For a given agent the probability that her wouldbe trading partner produces a good that she likes to consume is c and 82 ANALYSIS 111 the probability that she produces what her wouldbe trading partner wants is also c There is a good called money and a fraction M of the population is endowed with one unit each of this stuff in period 0 All goods are indivisible being produced and stored in oneunit quantities An agent can store at most one unit of any good including money and all goods are stored at zero cost Free disposal is assumed so it is possible to throw money or anything else away For convenience let u 2 711 denote the utility from consuming a good that the agent likes and assume that the utility from consuming a good one does not like is zero Any intertemporal trades or giftgiving equilibria are ruled out by virtue of the fact that no two agents meet more than once and because agents have no knowledge of each others trading histories 82 Analysis We con ne the analysis here to stationary equilibria ie equilibria where agents trading strategies and the distribution of goods across the population are constant for all t In a steady state all agents are holding one unit of some good ignoring uninteresting cases where some agents hold nothing Given symmetry it is as if there are were only two goods and we let V denote the value of holding a commodity and Vm the value of holding money at the end of the period The fraction of agents holding money is u and the fraction holding commodities is 1 u If two agents with commodities meet they will trade only if there is a double coincidence of wants which occurs with probability 5132 If two agents with money meet they may trade or not since both are indifferent but in either case they each end the period holding money If two agents meet and one has money while the other has a commodity the agent with money will want to trade if the other agent has a good she consumes but the agent with the commodity may or may not want to accept money From an individual agent s point of view let 7r denote the proba bility that other agents accept money where 0 3 7r 3 1 and let 7r denote the probability with which the agent accepts money Then we 112 CHAPTER 8 SEARCH AND MONEY can write the Bellman equations as Wmaer39eo11 W Vm 1 7 qu M1 SEW V9 l 1 M 5620 V9 1 2M 8391 Vm uVm 1 u 67ru Vg 1 67TVm 82 In 81 an agent with a commodity at the end of the current pe riod meets an agent with money next period with probability p The moneyholder will want to trade with probability cc and if the money holder wishes to trade the agent chooses the trading probability 7r to maximize endofperiod value With probability 1 u the agent meets another commodityholder and trade takes with probability 132 If the agent trades she consumes and then immediately produces again Similarly in 82 an agent holding money meets another agent holding money with probability p and meets a commodityholder with probability 1 a Trade with a commodityholder occurs with proba bility cmr as there is a single coincidence with probability cc and the commodityholder accepts money with probability 7r It is convenient to simplify the Bellman equations as we did in the previous chapter defining the discount rate 39r by 2 TLquot and manipulating 81 and 82 to get TV M26 3 7 Vm V9 1 103321 83 39er 1 uc7ru V Vm 84 Now we will ignore equilibria where agents accept commodities in exchange that are not their consumption goods ie commodity equilibria In these equilibria if two commodityholders meet and there is a single coincidence they trade even though one agent is indifferent between trading and not trading It is easy to rule these equilibria out as in Kiyotaki and Wright 1993 by assuming that there is a small trading cost 5 gt 0 and thus a commodity holder would strictly prefer not to trade for a commodity she does not consume Provided a is very small the analysis does not change Now there are potentially three types stationary equilibria One type has 7r 2 0 one has 0 lt 7r lt 1 and one has 7r 2 1 The first we 82 ANALYSIS 113 can think of as a nonmonetary equilibrium money is not accepted by anyone and the latter two are monetary equilibria Suppose rst that 7r 2 0 Then an agent holding money would never get to consume and anyone holding money at the rst date would throw it away and produce a commodity so we have u 0 Then from 83 the expected utility of each agent in equilibrium is V 85 Next consider the mixed strategy equilibrium where 0 lt 7r lt 1 In equilibrium we must have 7r 2 7r so for the mixed strategy to be optimal from 83 we must have Vm K From 83 and 84 we then must have 7r 2 c This then gives expected utility for all agents in the stationary equilibrium 1 Mm T Vm 2 V9 2 86 Now note that all moneyholders are indifferent between throwing money away and producing and holding their money endowment Thus there is a continuum of equilibria of this type indexed by u E 0 M Further note from 85 and 86 that all agents are worse off in the mixed strategy monetary equilibrium than in the nonmonetary equilibrium and that expected utility is decreasing in p This is due to the fact that in the mixed strategy monetary equilibrium money is no more acceptable in exchange than are commodities 7r 2 13 so introducing money in this case does nothing to improve trade In ad dition the fact that some agents are holding money in conjunction with the assumptions about the inventory technology implies that less consumption takes place in the aggregate when money is introduced Next consider the equilibrium where 7r 2 1 Here it must be opti mal for the commodityholder to choose 7r 2 7r 2 1 so we must have Vm 2 V5 Conjecturing that this is so we solve 83 and 84 for Vm and V2 to get 1 mm Vq 39r39r c u1 c 39r 13 87 114 CHAPTER 8 SEARCH AND MONEY M 41 mu1 mwxb 8 and we have 1 cc 1 cc u m M 39r cc for u lt 1 Thus our conjecture that 7r39 2 1 is a best response to 7r 2 1 is correct and we will have u M as all agents with a money endowment will strictly prefer holding money to throwing it away and producing Now it is useful to consider what welfare is in the monetary equi librium with 7r 2 1 relative to the other equilibria Here we will use as a welfare measure gt0 ie the expected utilities of the agents at the first date weighted by the population fractions If money is allocated to agents at random at t 0 this is the expected utility of each agent before the money allocations occur Setting u M in 87 and 88 and calculating IV we get 1 M ccu 7 Note that for M 0 W 273 which is identical to welfare in the nonmonetary equilibrium as should be the case Suppose that we imagine a policy experiment where the monetary authority can consider setting M at t 0 This does not correspond to any realworld policy experiment as money is not indiVisible in any essential way in practice but is useful for purposes of examining the welfare effects of money in the model Differentiating W with respect to M we obtain W 2 cc M1 a 89 dW ccu 1 2 2M 1 d2W dM2 2 1 cc lt 0 Thus if cc 2 then introducing any quantity of money reduces welfare ie the optimal quantity of money is M 0 That is if the absence of double coincidence of wants problem is not too severe then introducing 83 DISCUSSION 115 money reduces welfare more by crowding out consumption than it in creases welfare by improving trade lf 1 lt then welfare is maximized for M Thus we need a sufficiently severe absence of double coincidence problem before welfare improves due to the introduction of money Note that from 86 and 89 for a given u welfare is higher in the pure strategy monetary equilibrium than in the mixed strategy monetary equilibrium 83 Discussion This basic search model of money provides a nice formalization of the absenceof doublecoincidence friction discussed by J evons The model has been extended to allow for divisible commodities Trejos and Wright 1995 Shi 1995 and a role for money arising from informational fric tions Williamson and Wright 1994 Further it has been used to address historical questions Wallace and Zhou 1997 Velde Weber and Wright 1998 A remaining problem is that it is difficult to al low for divisible money though this has been done in computational work Molico 1997 If money is divisible we need to track the whole distribution of money balances across the population which is ana lytically messy However if money is not divisible it is impossible to consider standard monetary experiments such as changes in the money growth rate which would affect in ation In indivisible money search models a change in M is essentially a meaningless experiment While credit is ruled out in the above model it is possible to have creditlike arrangements even if no two agents meet more than once if there is some knowledge of a wouldbe trading partner s history Kocherlakota and Wallace 1998 and Aiyagari and Williamson 1998 are two examples of search models with credit arrangements and mem ory Shi 1996 also studies a monetary search model with credit arrangements of a different sort 84 References Aiyagari S and Williamson S 1998 Credit in a Random Matching 116 CHAPTER 8 SEARCH AND MONEY Model with Private Information forthcoming Review of Eco nomic Dynamics Jevons W 1875 Money and the Mechanism of Emchange Appleton London Jones R 1976 The Origin and Development of Media of Exchange Journal of Political Economy 84 757775 Kiyotaki N and Wright R 1989 On Money as a Medium of EX change Journal of Political Economy 97 927954 Kiyotaki N and Wright R 1993 A Search Theoretic Approach to Monetary Economics 83 63 77 Kocherlakota N and Wallace N 1998 Optimal Allocations With Incomplete RecordKeeping and No Commitment forthcoming Journal of Economic Theory Molico M 1997 The Distributions of Money and Prices in Search Equilibrium working paper University of Pennsylvania Shi S 1995 Money and Prices A Model of Search and Bargaining Journal of Economic Theory 67 467496 Shi S 1996 Credit and Money in a Search Model With Divisible Commodities Review of Economic Studies 63 627652 Trejos A and Wright R 1995 Search Bargaining Money and Prices Journal of Political Economy 103 118141 Velde F Weber W and Wright R 1998 A Model of Commodity Money With Applications to Gresham s Law and the Debasement Puzzle Review of Economic Dynamics forthcoming Wallace N and Zhou R 1997 A Model of a Currency Shortage Journal of Monetary Economics 40 555572 Williamson S and Wright R 1994 Barter and Monetary Exchange Under Private Information American Economic Review 84 104 123 Chapter 9 Overlapping Generations Models of Money The overlapping generations model of money was rst introduced by Samuelson 1956 who did not take it very seriously Earlier in Chap ter 2 we studied Peter Diamond s overlapping generations model of growth which was an adaptation of Samuelson s model used to exam ine issues in capital accumulation Samuelson s monetary model was not rehabilitated until Lucas 1972 used it in a business cycle context and it was then used extensively by Neil Wallace his coauthors and students in the late 1970s and early 1980s see Kareken and Wallace 1980 for example As in the search model of money money is used in the overlapping generations environment because it overcomes a particular friction that is described explicitly in the model In this case the friction is that agents are nitelived and a particular agent can not trade with agents who are unborn or dead In the simplest overlapping generations mod els agents hold money in order to consume in their old age In terms of how money works in real economies this may seem silly if taken lit erally since the holding period of money is typically much shorter than thirty years However the overlapping generations friction should be interpreted as a convenient parable which stands in for the spatial and informational frictions which actually make money useful in practice In fact as we will see the overlapping generations model of money in cludes an explicit representation of the absence of double coincidence 117 118OHAPTER 9 OVERLAPPIN G GENERATIONS MODELS OF MONEY problem 91 The Model In each period t 123 Nt agents are born who are each two periodlived An agent born in period t has preferences given by uc 0 1 where cf denotes consumption in period t by a member of generation 8 Assume that u is strictly increasing in both arguments strictly concave and twice continuously differentiable and that 8ucl 02 801 lim OO cl0 811161622 7 802 for 02 gt 0 and 8ucl02 801 321310 m 0 802 for cl gt 0 which will guarantee that agents want to consume positive amounts in both periods of life We will call agents young when they are in the first period of life and old when they are in the second period In period 1 there are NO initial old agents who live for only one period and whose utility is increasing in period 1 consumption These agents are collectively endowed with M0 units of fiat money which is perfectly divisible intrinsically useless and can not be privately produced Each young agent receives y units of the perishable consumption good when young and each old agent receives nothing except for the initial old who are endowed with money Assume that the population evolves according to Nt TLNt1 for t 1 23 where n gt 0 Money can be injected or withdrawn through lumpsum transfers to old agents in each period Letting 73975 denote the lump sum transfer that each old agent receives in period t in terms of the period t consumption good which will be the numeraire throughout and letting Mt denote the money supply in period t the government budget constraint is PtMt Mt l Nt tha 92 92 PARETO OPTIMAL ALLOCATIONS 119 for t 1 2 3 where pt denotes the price of money in terms of the period t consumption good ie the inverse of the price level1 Further we will assume that the money supply grows at a constant rate ie Mt ZMt1 for t 1 2 3 with z gt 0 so 92 and 93 imply that 1 ptMt1 Nt1739t 92 Pareto Optimal Allocations Before studying competitive equilibrium allocations in this model we wish to determine what allocations are optimal To that end suppose that there is a social planner that can confiscate agents endowments and then distribute them as she chooses across the population This planner faces the resource constraint Ntcg NHcg 1 g Nty 95 for t 123 Equation 95 states that total consumption of the young plus total consumption of the old can not exceed the total endowment in each period Now further suppose that the planner is restricted to choosing among stationary allocations ie allocations that have the property that each generation born in periods 15 1 2 3 receives the same allocation or cf of 1 2 0102 for all t where cl and 02 are nonnegative constants Note that the initial old alive in period 1 will then each consume 02 We can then rewrite equation 95 using 91 to get QSy pm We will say that stationary allocations 0102 satisfying 96 are fea sible 1It is convenient to use the consumption good as the numeraire as we will want to consider equilibria where money is not valued ie pt 2 0 120 CHAPTER 9 OVERLAPPIN G GENERATIONS MODELS OF MONEY De nition 3 A Pareto optimal allocation chosen from the class of stationary allocations c1 c2 is feasible and satis es the property that there exists no other feasible stationary allocation 6162 such that ucl 62 2 uc1 c2 and 62 2 c2 with at least one of the previous two inequalities a strong inequality Thus the de nition states that an allocation is Pareto optimal within the class of stationary allocations if it is feasible and there is no other feasible stationary allocation for which all agents are at least as well off and some agent is better off Here note that we take account of the welfare of the initial old agents To determine what allocations are Pareto optimal note rst that any Pareto optimal allocation must satisfy 96 with equality Further let cf c3 denote the stationary allocation that maximizes the welfare of agents born in generations t 1 2 3 ie cf c3 is the solution to subject to 96 Then the Pareto optimal allocations satisfy 96 with equality and c1 3 c 97 To see this note that for any allocation satisfying 96 where 97 is not satis ed there is some alternative allocation which satis es 96 and makes all agents better off 93 Competitive Equilibrium A young agent will wish to smooth consumption over her lifetime by ac quiring money balances when young and selling them when old Thus letting mt denote the nominal quantity of money acquired by an agent born in period t the agent chooses mt 2 0 cf 2 0 and c 1 2 0 to solve maX uc cal 98 subject to Cl l Ptmt 2 ya 99 Cl1 Pt1mt 739t1 910 93 COMPETITIVE EQUILIBRIUM 121 De nition 4 A competitive equilibrium is a sequence of prices pt 1 a sequence of consumption allocations c c 1 1 a sequence of money supplies Mt 1 a sequence of individual money demands mt 1 and a sequence of tapes Ttf1 giuen M0 which satis es cg cal and mt are chosen to solve 98 subject to 99 and 910 giuen pt pt1 and n1 for all t 123 and 94 for t 123 ptMt thtmt for all t 1 2 3 In the de nition condition says that all agents optimize treating prices and lumpsum taxes as given all agents are pricetakers condi tion ii states that the sequence of money supplies and lump sum taxes satis es the constant money growth rule and the government budget constraint and iii is the marketclearing condition Note that there are two markets the market for consumption goods and the market for money but Walras7 Law permits us to drop the marketclearing condition for consumption goods 931 Nonmonetary Equilibrium In this model there always exists a nonmonetary equilibrium ie a competitive equilibrium where money is not valued and pt 2 0 for all t In the nonmonetary equilibrium cg cg 1 2 y 0 and 73975 2 0 for all t It is straightforward to verify that conditions iiii in the de nition of a competitive equilibrium are satis ed It is also straightforward to show that the nonmonetary equilibrium is not Pareto optimal since it is Pareto dominated by the feasible stationary allocation c c 1 2 ex ea Thus in the absence of money no trade can take place in this model due to a type of absenceofdoublecoincidence friction That is an agent born in period t has period t consumption goods and wishes to trade some of these for period t 1 consumption goods However there is no other agent who wishes to trade period t 1 consumption goods for period t consumption goods 122 CHAPTER 9 OVERLAPPIN G GENERATIONS MODELS OF MONEY 932 Monetary Equilibria We will now study equilibria where pt gt 0 for all 15 Here given our assumptions on preferences agents will choose an interior solution with strictly positive consumption in each period of life We will also suppose as has to be the case in equilibrium that taxes and prices are such that an agent born in period t chooses mt gt 0 To simplify the problem 98 subject to 99 and 910 substitute for the constraints in the objective function to obtain 111718 y ptmta Pt1mt n1 and then assuming an interior solution the firstorder condition for an optimum is PtU1y Ptmt7Pt1mt 739t1 Pt1u2 9 Ptmt7Pt1mt 739t1 0 911 where artcl 02 denotes the first partial derivative of the utility function with respect to the ith argument Now it proves to be convenient in this version of the model though not always to look for an equilibrium in terms of the sequence qt 1 where qt E 1 is the real per capita quantity of money We can then use this definition of qt and conditions ii and iii in the definition of competitive equilibrium to substitute in the firstorder condition 911 to arrive after some manipulation at n JtU1y It Qt1n l Qt1gu2fy It Qt1n 0 912 Equation 912 is a firstorder difference equation in qt which can in principle be solved for the sequence qt1 Once we solve for qt 1 we can then work backward to solve for pt 1 given that pt 2 The sequence of taxes can be determined from 73975 2 nqt1 i and the sequence of consumptions is given by 0 of 1 2 y qt qt1n One monetary equilibrium of particular interest and in general this will be the one we will study most closely is the stationary monetary equilibrium This competitive equilibrium has the property that qt 2 q a constant for all 15 To solve for q simply set qt1 2 qt 2 q in equation 94 EXAMPLES 123 912 to obtain u1y 1 1n gum 1 1n 0 913 Now note that ifz 1 then y q 2 cf and qn 03 by virtue of the fact that each agent is essentially solving the same problem as a social planner would solve in maximizing the utility of agents born in periods 15 1 2 3 Thus 2 1 implies that the stationary monetary equilibrium is Pareto optimal ie a xed money supply is Pareto optimal independent of the population growth rate Note that the rate of in ation in the stationary monetary equilibrium is 1 so optimality here has nothing to do with what the in ation rate is Further note that any 2 g 1 implies that the stationary monetary equilibrium is Pareto optimal since the stationary monetary equilibrium must satisfy 96 due to market clearing and 2 g 1 implies that 97 holds in the stationary monetary equilibrium lf 2 gt 1 this implies that intertemporal prices are distorted ie the agent faces intertemporal prices which are different from the terms on which the social planner can exchange period t consumption for period t 1 consumption 94 Examples Suppose rst that ucl 02 ln cl ln 02 Then equation 912 gives It l y It 2 and solving for qt we get qt 2 3 so the stationary monetary equi librium is the unique monetary equilibrium in this case though note that qt 2 0 is still an equilibrium The consumption allocations are cf of 1 2 1 112 lz so that consumption of the young increases with the money growth rate and the in ation rate and consumption of the old decreases 1 1 Alternatively suppose that UC1C2 2 0111 025 Here equation 912 gives after rearranging 1322 qt1 9 6115 124OHAPTER 9 OVERLAPPIN G GENERATIONS MODELS OF MONEY Now 914 has multiple solutions one of which is the stationary mon etary equilibrium where qt 2 There also exists a continuum of equilibria indexed by ql E 0 Each of these equilibria has the property that limtOO qt 2 0 ie these are nonstationary monetary equilibria where there is convergence to the nonmonetary equilibrium in the limit see Figure 91 Note that the stationary monetary equi librium satis es the quantity theory of money in that the in ation rate i g 1 so that increases in the money growth rate are essentially re ected oneforone in increases in the in ation rate the velocity of money is xed at one However the nonstationary monetary equilibria do not have this property 95 Discussion The overlapping generations model s virtues are that it captures a role for money without resorting to adhoc devices and it is very tractable since the agents in the model need only solve twoperiod optimization problems or threeperiod problems in some versions of the model Further it is easy to integrate other features into the model such as credit and alternative assets government bonds for example by al lowing for suf cient withingeneration heterogeneity see Sargent and Wallace 1982 Bryant and Wallace 1984 and Sargent 1987 The model has been criticized for being too stylized ie for do ing empirical work the interpretation of period length is problematic Also some see the existence of multiple equilibria as being undesirable though some in the profession appear to think that the more equi libria a model possesses the better There are many other types of multiple equilibria that the overlapping generations model can exhibit including sunspot equilibria Azariadis 1981 and chaotic equilibria Boldrin and Woodford 19901 96 References Azariadis C 1981 SelfFul lling Prophecies Journal of Economic Theory 25 380396 9 6 REFERENCES QEFF1 125 Figure 9 1 45 degree line I a a f 11erf nf2 Catt Figure 91 126 CHAPTER 9 OVERLAPPIN G GENERATIONS MODELS OF MONEY Boldrin M and Woodford M 1990 Equilibrium Models Display ing Endogenous Fluctuations and Chaos A Survey Journal of Monetary Economics 25 189222 Bryant J and Wallace N 1984 A Price Discrimination Analysis of Monetary Policy Review of Economic Studies 51 279288 Kareken J and Wallace N 1980 Models of Monetary Economies Federal Reserve Bank of Minneapolis Minneapolis MN Lucas R 1972 Expectations and the Neutrality of Money Journal of Economic Theory 4 103124 Samuelson P 1956 An Exact Consumption Loan Model of Interest With or Without the Social Contrivance of Money Journal of Political Economy 66 467482 Sargent T 1987 Dynamic Macroeconomic Theory Harvard Univer sity Press Cambridge MA Sargent T and Wallace N 1982 The Real Bills Doctrine vs the Quantity Theory A Reconsideration Journal of Political Econ omy 90 12121236 Chapter 10 A CashInAdvance Model Many macroeconomic approaches to modeling monetary economies pro ceed at a higher level than in monetary models with search or overlap ping generations one might disparagingly refer this higher level of mon etary model as implicit theorizing One approach is to simply assume that money directly enters preferences moneyin theutilityfunction models or the technology transactions cost models Another ap proach which we will study in this chapter is to simply assume that money accumulated in the previous period is necessary to finance cur rent period transactions This cash inadvance approach was pioneered by Lucas 1980 1982 and has been widelyused particularly in quan titative work eg Cooley and Hansen 1989 101 A Simple CashinAdvance Model With Production In its basic structure this is a static representative agent model with an added cashinadvance constraint which can potentially generate dynamics The representative consumer has preferences given by E0 we 22mm 101 where 0 lt 5 lt 1 ct is consumption and nf is labor supply Assume that is strictly increasing strictly concave and twice differentiable 127 128 CHAPTER 10 A CASHIN ADVANCE MODEL and that v is increasing strictly convex and twice differentiable with v 0 0 and v39h 2 00 where h is the endowment of time the consumer receives each period The representative firm has a constantreturnstoscale technology yt 7mg 102 where yt is output n is labor input and 71 is a random technology shock Money enters the economy through lumpsum transfers made to the representative agent by the government The government budget constraint takes the form Mt1 2 Mt P137 where Mt is the money supply in period 15 Pt is the price level the price of the consumption good in terms of money and T75 is the lumpsum transfer that the representative agent receives in terms of consumption goods Assume that Mt1 QtMt 104 where 975 is a random variable In cashinadvance models the timing of transactions can be critical to the results Here the timing of events within a period is as follows 1 The consumer enters the period with Mt units of currency 3 oneperiod nominal bonds and 2t oneperiod real bonds Each nominal bond issued in period t is a promise to pay one unit of currency when the asset market opens in period t 1 Similarly a real bond issued in period t is a promise to pay one unit of the consumption good when the asset market opens in period t 1 2 The consumer learns 9t and 75 the current period shocks and receives a cash transfer from the government 3 The asset market opens on which the consumer can exchange money nominal bonds and real bonds 4 The asset market closes and the consumer supplies labor to the firm 101 A SIMPLE CASHIN ADVANCE MODEL WITH PRODUCTI ON 129 5 The goods market opens where consumers purchase consumption goods with cash 6 The goods market closes and consumers receive their labor earn ings from the firm in cash The consumer s problem is to maximize 101 subject to two con straints The first is a cashin advance constraint ie the constraint that the consumer must finance consumption purchases and purchases of bonds from the asset stocks that she starts the period with StBt1 PtCItZt1 35 3 Mt Bt 3 PtTt lo5 Here 5 is the price in units of currency of newlyissued nominal bond and qt is the price in units of the consumption good of a newlyissued real bond The second constraint is the consumer s budget constraint StBt1 Pttht1 Mt1 PtCt S MtIBtIHztlPtTtlthtnf where 10 is the real wage To make the consumer s dynamic opti mization stationary it is useful to divide through constraints 105 and 106 by Mt the nominal money supply and to change variables defining lowercase variables except for previouslydefined real vari ables to be nominal variables scaled by the nominal money supply for example pt E Constraints 105 and 106 can then be rewritten as Stbt19t l thtZt1 19150 3 mt bt l P1521 l P1573 107 and Stbt19t PtCItZt1 mt19t Pt0t 3 mt bt PtZt Pt739t thtn lo8 Note that we have used 104 to simplify 107 and 108 The con straint 108 will be binding at the optimum but 107 may not bind However we will assume throughout that 107 binds and later estab lish conditions that will guarantee this The consumer s optimization problem can be formulated as a dy namic programming problem with the value function vmt bt 2t 9t 75 The Bellman equation is then vmt7bt7zt76t77t maXCtnfmt1bt1zt1 luct Et39Umt17 bt17 Zt17 6t17 7141 130 CHAPTER 10 A CASHIN ADVANCE MODEL subject to 107 and 108 The Lagrangian for the optimization prob lem on the righthand side of the Bellman equation is L Ct 71 Etvmt17 bt17 Zt17 9t17 7t1 tmt bt 1 1975215 1 197573 Stbt19t PtQtZt1 PtCt Mtmt bt 1 191521 1 291573 1 19751075 Stbt19t thtZt1 mt19t Ptct Where At and at are Lagrange multipliers Assuming that the value function is differentiable and concave the unique solution to this op timization problem is characterized by the following rstorder condi tions 8L a ct ulct tpt Mtpt 0 lo9 8L an 2 v39n utptwt 0 1010 8L 81 E 9 0 1011 amtH tamt1 Mt t 7 8L 8v abt1 Et abt1 AtStQt 1113813613 0 8L 8v 53 ttht MtPtQt 0 1013 aZis1 aZis1 We have the following envelope conditions 8390 81 amt a ljt At 1 Mt 8 a At 1 MtPt 1015 A binding cashinadvance constraint implies that At gt 0 From 1011 1012 and 1014 we have At Therefore the cashinadvance constraint binds if and only if the price of the nominal bond St is less than one This implies that the nominal interest rate Sit 1 gt 0 101 A SIMPLE CASHIN ADVANCE MODEL WITH PRODUCTI ON 131 Now use 1014 and 1015 to substitute for the partial derivatives of the value function in 10111013 and then use 109 and 1010 to substitute for the Lagrange multipliers to obtain 3E aw 0 10 16 Pt1 tht 7 39 5E7 Stef 0 1017 pt1 Pt Etu39ct1 qtu39ct 0 1018 Given the de nition of pt we can write 1016 and 1017 more infor matively as 3E lb0 207 1019 and I I Now equation 1018 is a familiar pricing equation for a risk free real bond In equation 1019 the righthand side is the marginal disutility of labor and the lefthand side is the discounted expected marginal utility of labor earnings ie this period s labor earnings cannot be spent until the following period Equation 1020 is a pricing equation for the nominal bond The righthand side is the marginal cost in terms of foregone consumption from purchasing a nominal bond in period t and the lefthand side is the expected utility of the payoff on the bond in period t 1 Note that the asset pricing relationships 1018 and 1020 play no role in determining the equilibrium Pro t maximization by the representative firm implies that 10 775 1021 in equilibrium Also in equilibrium the labor market clears 71 nil nt 1022 the money market clears ie Mt 2 Mt or 132 CHAPTER 10 A CASHIN ADVANCE MODEL and the bond markets clear I 25 0 1024 Given the equilibrium conditions 10211024 103 104 and 108 with equality we also have ct ytnt 1025 Also 10211024 103 104 and 107 with equality give 191507 2 6t or using 1025 19757th 9t Now substituting for 0 and pt in 1016 using 1025 and 1026 we get I 3E lt1nt1gm1nt1gt ntv39nt0 1027 t1 Here 1027 is the stochastic law of motion for employment in equi librium This equation can be used to solve for 71 as a function of the state 7t 975 Once nt is determined we can then work backward to obtain the price level from 1026 QtMt Pt 77 nt 1028 and consumption from 1025 Note that 1028 implies that the in come velocity of money de ned by v 2 55 3115 tMt is equal to 1 Empirically the velocity of money is a measure of the intensity with which the stock of money is used in exchange and there are regularities in the behavior of velocity over the business cycle which we would like our models to explain In this and other cashinadvance models the velocity of money is xed if the cashinadvance constraint 102 EXAMPLES 133 binds as the stock of money turns over once per period This can be viewed as a defect of this model Substituting for pt and ct in the asset pricing relationships 1017 and 1018 using 1025 and 1026 gives I Et Vt1nt1g 7t1nt1 St ytntu ytnt 0 11 Etu t1nt1 qtu tnt 0 1030 From 1028 and 1029 we can also obtain a simple expression for the price of the nominal bond 0 n St l t 71 715 Note that for our maintained assumption of a binding cashinadvance constraint to be correct we require that 5 lt 1 or that the equilibrium solution satisfy 1031 v39nt lt ytu39tnt 1032 102 Examples 1021 Certainty Suppose that 775 y and 95 9 for all t where y and 9 are positive constants ie there are no technology shocks and the money supply grows at a constant rate Then 71 n for all t where from 1027 n is the solution to W Un 0 1033 Now note that for the cashinadvance constraint to bind from 1032 we must have 6gt that is the money growth factor must be greater than the discount factor From 1028 and 104 the price level is given by 6t1M P 0 1034 yn 134 CHAPTER 10 A CASHIN ADVANCE MODEL and the in ation rate is 16 1 mew Here money is neutral in the sense that changing the level of the money supply ie changing M0 has no effect on any real variables but only increases all prices in proportion see 1034 Note that M0 does not enter into the determination of n which determines output and con sumption in 1033 However if the monetary authority changes the rate of growth of the money supply ie if 9 increases then this does have real effects money is not super neutral in this model Comparative statics in equation 1033 gives dn vulvn 6572164772 62vquotngt lt 039 Note also that from 1035 an increase in the money growth rate im plies a oneforone increase in the in ation rate From 1034 there is a level effect on the price level of a change in 9 due to the change in n and a direct growth rate effect through the change in 9 Em ployment output and consumption decrease with the increase in the money growth rate through a labor supply effect That is an increase in the money growth rate causes an increase in the in ation rate which effectively acts like a taX on labor earnings Labor earnings are paid in cash which cannot be spent until the following period and in the intervening time purchasing power is eroded With a higher in ation rate the representative agent s real wage falls and he she substitutes leisure for labor With regard to asset prices from 1029 and 1030 we get It 2 and St E The real interest rate is given by 1 1 rt 2 1 It 5 102 EXAMPLES 135 ie the real interest rate is equal to the discount rate and the nominal interest rate is 1 9 R 1 1 t 51 5 Therefore we have 9 1 Rt T39t T g 1 7113 which is a good approximation if 0 is close to 1 Here 1036 is a Fisher relationship that is the difference between the nominal interest rate and the real interest rate is approximately equal to the in ation rate Increases in the in ation rate caused by increases in money growth are reflected in an approximately oneforone increase in the nominal interest rate with no effect on the real rate 1022 Uncertainty Now suppose that 9 and 75 are each iid random variables Then there exists a competitive equilibrium where nt is also iid and 1029 gives W ntv nt 0 1037 where w is a constant Then 1037 implies that nt 2 n where n is a constant From 1029 and 1030 we obtain 5 St YtnU tn 0 1038 and u qtu39ytn 0 1039 where w is a constant Note in 10371039 that 9 has no effect on output employment consumption or real and nominal interest rates In this model monetary policy has no effect except to the extent that it is anticipated Here given that 91 is iid the current money growth rate provides no information about future money growth and so there are no real effects Note however that the probability distribution for 91 is important in determining the equilibrium as this well in general affect in and w 136 CHAPTER 10 A CASHIN ADVANCE MODEL The technology shock 71 will have real effects here Since yt 7m high 7 implies high output and consumption From 1039 the in crease in output results in a decrease in the marginal utility of con sumption and qt rises the real interest rate falls as the representa tive consumer attempts to smooth consumption into the future From 1028 the increase in output causes a decrease in the price level Pt so that consumers eXpect higher inflation The effect on the nominal interest rate from 1038 is ambiguous Comparative statics gives L57 1 1 61 7t 39th Therefore if the coefficient of relative risk aversion is greater than one 5 rises the nominal interest rate falls otherwise the nominal interest rate rises There are two effects on the nominal interest rate First the nominal interest rate will tend to fall due to the same forces that cause the real interest rate to fall That is consumers buy nominal bonds in order to consume more in the future as well as today and this pushes up the price of nominal bonds reducing the nominal interest rate Second there is a positive anticipated in ation effect on the nominal interest rate as in ation is eXpected to be higher Which effect dominates depends on the strength of the consumptionsmoothing effect which increases as curvature in the utility function increases 103 Optimality In this section we let 7 y a constant for all t and allow 9 to be determined at the discretion of the monetary authority Suppose that the monetary authority chooses an optimal money growth policy 92 so as to maximize the welfare of the representative consumer We want to determine the properties of this optimal growth rule To do so first consider the social planner s problem in the absence of monetary arrangements The social planner solves maX i t U nt ntll 7 Int 1gt0 104 PROBLEMS WITH THE CASHIN ADVANCE MODEL 137 but this breaks down into a series of static problems Letting n denote the optimal choice for mg the optimum is characterized by the rst order condition wens v n2 0 1040 and this then implies that n n a constant for all 15 Now we want to determine the 92 which will imply that nt 2 n is a competitive equi librium outcome for this economy From 1027 we therefore require that I 8Et nvln 07 t1 and from 1040 this requires that 62 5 1041 ie the money supply decreases at the discount rate The optimal money growth rule in 1041 is referred to as a Friedman rule see Friedman 1969 or a Chicago rule 7 Note that this optimal money rule implies from 1029 that St 1 for all t ie the nominal interest rate is zero and the cashinadvance constraint does not bind In this model a binding cashinadvance constraint represents an inef ciency as does a positive nominal interest rate If alternative assets bear a higher real return than money then the representative consumer economizes too much on money balances relative to the optimum Producing a de ation at the optimal rate the rate of time preference eliminates the distortion of the labor supply decision and brings about an optimal allocation of resources 104 Problems With the Cashin Advance Model While this model gives some insight into the relationship between money interest rates and real activity in the long run and over the business cycle the model has some problems in its ability to t the facts The rst problem is that the velocity of money is xed in this model but is highly variable in the data There are at least two straightforward 138 CHAPTER 10 A CASHIN ADVANCE MODEL means for curing this problem at least in theory The first is to de ne preferences over cash goods 7 and credit goods as in Lucas and Stokey 1987 Here cash goods are goods that are subject to the cashinadvance constraint In this context variability in in ation causes substitution between cash goods and credit goods which in turn leads to variability in velocity A second approach is to change some of the timing assumptions concerning transactions in the model For example Svennson 1985 assumes that the asset market opens before the current money shock is known Thus the cashinadvance con straint binds in some states of the world but not in others and velocity is variable However neither of these approaches works empirically Hodrick Kocherlakota and Lucas 1991 show that these models do not produce enough variability in velocity to match the data Another problem is that in versions of this type of model where money growth is serially correlated as in practice counterfactual re sponses to surprise increases in money growth are predicted Empiri cally money growth rates are positively serially correlated Given this if there is high money growth today high money growth is expected tomorrow But this will imply in this model that labor supply falls output falls and given anticipated in ation the nominal interest rate rises Empirically surprise increases in money growth appear to gen erate short run increases in output and employment and a short run decrease in the nominal interest rate Work by Lucas 1990 and Fuerst 1992 on a class of liquidity effect models which are versions of the cashinadvance approach can obtain the correct qualitative responses of interest rates and output to money injections A third problem has to do with the lack of explicitness in the basic approach to modeling monetary arrangements here The model is silent on what the objects are which enter the cashinadvance constraint Implicit in the model is the assumption that private agents cannot pro duce whatever it is that satisfies cashinadvance If they could then there could not be an equilibrium with a positive nominal interest rate as a positive nominal interest rate represents a profit opportunity for private issuers of money substitutes Because the model is not explicit about the underlying restrictions which support cashinadvance and because it requires the modeler to define at the outset what money is the cashinadvance approach is virtually useless for studying sub 1 05 REFERENCES 139 stitution among money substitutes and the operation of the banking system There are approaches which model monetary arrangements at a deeper level such as in the overlapping generations model Wallace 1980 or in search environments Kiyotaki and Wright 1989 but these approaches are not easily amenable to empirical application A last problem has to do With the appropriateness of using a cash inadvance model for studying quarterly or even monthly uctuations in output prices and interest rates Clearly it is very dif cult to argue that consumption expenditures during the current quarter or month are constrained by cash acquired in the previous quarter or month given the low cost of visiting a cash machine or using a credit card 105 References Cooley T and Hansen G 1989 The In ation TaX in a Real Business Cycle Model American Economic Review 79 733748 Friedman M 1969 The Optimum Quantity of Money and Other Essays Aldine Publishing Chicago Fuerst T 1992 Liquidity Loanable Funds and Real Activity Journal of Monetary Economics 29 324 Hodrick R Kocherlakota N and Lucas D 1991 The Variabil ity of Velocity in CashinAdvance Models Journal of Political Economy 99 358384 Kiyotaki N and Wright R 1989 On Money as a Medium of EX change Journal of Political Economy 97 927954 Lucas R 1980 Equilibrium in a Pure Currency Economy in Mod els of Monetary Economies Kareken and Wallace eds Federal Reserve Bank of Minneapolis Minneapolis MN Lucas R 1982 Interest rates and Currency Prices in a TwoCountry World Journal of Monetary Economics 10 335359 Lucas R 1990 Liquidity and Interest Rates Journal of Economic Theory 50 237264 140 CHAPTER 10 A CASHIN ADVANCE MODEL Lucas R and Stokey N 1987 Money and Interest in a CashIn Advance Economy Econometrica 55 491514 Wallace N 1980 The Overlapping Generations Model of Fiat Money 7 in Models of Monetary Economics Kareken and Wallace eds Federal Reserve Bank of Minneapolis Minneapolis MN
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