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Lecture Notes for Econ 702 ProfRios Rull Spring 2003 Prepared by Ahu Gemici and Vivian Zhanwei Yue1 1 Jan 28 Overview and Review of Equilibrium 11 Introduction 0 What is an equilibrium Loosely speaking an equilibrium is a mapping from environments preference technology information market structure to allocations Equilibrium allows us to characterizes What happens in a given environment that is given What people like know have 0 Two requirements of an equilibrium is agents optimize They do as best as possible and ii actions of agents in the economy are compatible to each others Examples of equilibrium concepts Walrasian equilibrium 0 Properties of equilibrium concepts existence and uniqueness We can prove existence of equilibrium by constructing one We need uniqueness because otherwise we do not have sharp prediction on What is likely to happen in a given environment Pareto optimality is not necessary property of equilibrium Neither is tractability 12 ArrowDebreu Competitive Equilibrium o What is Arrow Debreu Competitive Equilibrium ADE All trades happen at time 0 Perfect commitment Everything is tradable and things are traded conditional on date and event 1Makoto Nakajima s Econ 702 Lecture Notes for Spring 2002 were taken as basis when Writing these notes We are thankful to him Agents are price takers 0 Agents Problem 1516a U 0 1 subject to 1750 S 0 2 p S gt R is a continuous linear function S D X may include in nity dimensional subjects Thus p is a linear function but not necessary a vector This problem can be applied to in nite horizon and or stochastic environment In this problem the single objective function is U cc and restriction is St E X and pL S 0 As commitment is perfect in X and trades happen at time 0 people do whatever they choose at time 0 0 Properties of ADE Existence CE 5 ZExistence is relatively easy to get For uniqueness we need more to get su icient condition Pareto optimality CE 5 C 1305 when there is no externalities and non satiation for utility function But we need to be careful here Given agents endowment preferences technology and information structure elements of com petitive equilibrium are price system and ii allocation But CE 5 C PO 5 is only for allocation which means allocation from such competitive equilibrium is Pareto optimal From CE 5 to 1305 three things are required 1 nd price to get CE ii right redistribution transfer t to give agents enough resources for the allocation iii free disposal to ensure the quasi equilibrium is a true equilibrium Second Basic Welfare Theorem Proof uses separation theorem for which the su icient condition is nonempty convex setii interior point For any allocation 5L 6 1305 3p such that ptx is quasi equilibrium with transfers 13 The Road Map 0 In the rst two weeks with Randy we learned how to solve Social planner s problem SPP of neoclassical growth model with representative agent RA NGM using dy namic programming Also we know that solution to SPP is Pareto Optimal P0 in our model Other good things for solution to SPP is that in RA NGM we know that i it exists and ii it s unique Besides we have two welfare theorems FBWT SBWT from Dave s class If we care fully de ne the environment those two theorems guarantee loosely that under certain conditions Arrow Debreu Competitive Equilibrium ADE or Walrasian equi librium or valuation equilibrium is PO and ii also under certain conditions we can construct an ADE from a PO allocation Using those elements we can argue that ADE exists and is unique and we just need to solve SPP to derive the allocation of ADE which is much easier task than solve a monster named ADE Besides we have two welfare theorems FBWT SBWT from Dave s class If we care fully de ne the environment those two theorems guarantee loosely that under certain conditions Arrow Debreu Competitive Equilibrium ADE or Walrasian equi librium or valuation equilibrium is PO and ii also under certain conditions we can construct an ADE from a PO allocation Using those elements we can argue that ADE exists and is unique and we just need to solve SPP to derive the allocation of ADE which is much easier task than solve a monster named ADE But we have another problem The market assumed in ADE is not palatable to us in the sense that it is far from what we see in the world So next we look at an equi librium with sequential markets Sequential Market Equilibrium SME Surprisingly we can show that for our basic RA NGM the allocation in SME and the allocation of ADE turn out to be the same which let us conclude that even the allocation of the equilibrium with sequential markets can be analyzed using the allocation of SPP Lastly we will learn that equilibrium with sequential markets with recursive form Re cursive Competitive Equilibrium RCE gives the same allocation as in SME meaning we can solve the problem using our best friend 2 Dynamic Programming Of course these nice properties are available for limited class of models We need to directly solve the equilibrium instead of solving SPP for large class of interesting models We will see that Dynamic Programming method is also very useful for this purpose We will see some examples later in the course 14 Review of Ingredients of RA NGM Technology 0 Representative agent s problem gt0 5W9 3 max kaz lfio t subject to kt1 Ct fkt 4 Ctykt1 Z 0 5 k0 is given 6 There are many variations of this problem including models with distortion stochastic environment In writing such optimization problem you should always specify control variable initial condition Solutions is a sequence 05 kt10 E loo Existence of a solution We use Maximum theorem to prove existence Su icient condition to use Maximum theorem maximand is continuous function and ii constraint set is compact closedness and boundedness We assume u is continuous f is bounded Uniqueness Su icient condition includes convex constraint set and ii strictly concave function We assume u is strictly concave and f is concave Characterization of the solution If u and f are differentiable and 0 1133 is the unique solution the following condition has to be satis ed u 6f kt u CH1 7 And to rule out corner solution lnada condition is assumed Homework 11 Derive 98 98 can be rewritten as u f kt ktH 5f kt1ulfkt1 kt2 8 This is a second order difference equation We need two initial conditions to pin down the entire sequence We have got one initial capital we have to look for 1 that does not go out of track Therefore to solve the problem as an in nite sequence is di icult Now let s look at another way of solving it as you seen in Randy s class 15 Dynamic Programming De ne V to be the highest utility of the agent by doing right things in life The Bellman equation is W Ewe 6Vk 9 subject to ckl 10 The solution is k gk arg maxuc 6V k 11 Using 9 we can construct the sequence of kt1 k0 9k71 k1 90 92k71 We can show that the sequence constructed this way using Bellman equation satis es the rst order condition 98 o FOC of Bellman equation wk 7 Eggww 7 12 6VW 12 7u f k 7 1mm H 7 o 13 We know V is differentiable vi k 7 n f k 7 gm 6Vgk 7 m k 7 gr am an f k 7 gm 7 69 k v M Ucfk 9k l uc 6 9 011 a 0V8 Uc fk W 1140 0 14 Therefore we got the same FOC as what we have before inc fk W Md 0 15 2 Jan 30 Neoclassical Growth Model 21 Review of growth model 0 The model we studied last class is 5W9 16 max Chkt1oio 7 subject to kt1 0 foty 1 17 0 kt1 Z 0 18 k0 is given 19 We usually assume u and f are strictly concave f is bounded Then there exists a unique solution which can be characterized by rst order condition First Order Condition is a necessary but not su icient condition for the optimal solution Remember the FCC is actually a second order difference equation With only 1 initial condition 1 there can be many sequences indexed by k1 And the sequence may end up to be negative or in nity which is not even feasible for this problem We can also get the optimal solutionc7 1132 from Bellman equation as we see in Randy s class 0 Bellman equation for the growth model Vk c 6V 4 20 subject to 0k fk1 21 The solution is If 9 If 22 The solution is a xed point of an functional operator which is a contraction Using 9 we can construct the sequence of kt1 ko k1 g kg We can show that the sequence constructed this way using Bellman equation satis es the rst order condition 98 o We can go back and forth between these two forms of problem One way is to construct kt1 by g The other direction is to see the sequence ktH satis es kill g kt 22 Social Planner s Problem 0 We can write the growth model as a social planner s problem SPP We assume the economy is populated by a huge number of identical agents The social planner is like the God who tells people what they should do Properties of the solution to SPP l Pareto optimal lt is a oneline proof if the solution to SPP is not optimal there is better allocation to make everyone happier 2 Uniqueness The social planner treats everyone the same So the SPP is symmetric and we get unique solution Since all the Pareto optimal allocation are solution to the SPP PO5 is unique 0 Road map What we want to know is equilibrium price and allocation lf we can apply welfare theorems to the allocation of SPP we can claim that God s will realizes and can analyze allocation of SPP instead of directly looking at an equilibrium allocation ln order to use the argument above we formalize the environment of RA NGM in the way such that we can apply welfare theorems By using existence of solution to SPP ii uniqueness of solution of SPP and iii welfare theorems we can claim that ADE exists ii is unique iii and PO However market arrangement of ADE is not palatable to us in the sense that set of markets that are open in the ADE is NOT close to the markets in our real world ln other words there is notion of time in ADE all the trades are made before the history begins and there is no more choices after the history begins So we would like to proceed to the equilibrium concept that allows continuously open markets which is SME and we will look at it closely next week 23 ArrowDebreu Equilibrium ADE 0 Elements of ADE are commodity space consumption possibility space production possibility space and preference set 0 Commodity space Commodity space is a topological vector space S which is space of bounded real sequences with sup normS includes everything people trade which are sequences Agents have time and rent it to rm labor services owe capital and rent it to rm capital services and buy stuff to consume some and save some for the future Hence S lie 8 517 82 Sgt0 which are goods labor services and capital services respectively 0 Consumption possibility set X XCSand X cc 6 S 0 3ctkt1f0 gt 0 such that kt1 C 1 1 i 5 Vt 23 552 E 07 3 S kt Vt k0 givengt Interpretation is that xltzreceived goods at period t x2tlabor supply at period t 3tcapital service at period t kt 0 1 1 l 7 Diet comes from real accounting Note capital and capital service are not the same thing Think of the difference between a house and to rent a house 0 Preference U X gt R U M 2 3110 24 0 is unique given 5L because each 5L implies a sequence 07kt1390 lf 1 3 kt 0 1 1 5W3 3244 Production possibility set Y Firm s problem is relatively simple as rm do not have intertemporal decision Firms just rent production factors and produce period by period Y 1150 Y 1123032132 glut2793 2 0 1341 S fyatyy2tl 25 lnterpretation is that y1production at period t ygtzlabor input at period t y3tcapital input at period t Note We did not use the convention in general equilibrium that input is negative and output is positive lmplicitly we assume rm is constant return to scale So we do not need to worry about industrial organization Price A price is a continuos and linear function q S gt R q 0 and q E 8 8 is a separating hyperplane in separation hyperplane theorem Continuity for squot gt 8 95quot w 95 8 Linearity q s1 52 q 51 q 52 q may not be represented lnner product representation S lie one candidate space for q is ll 1 is space of sup norm bounded sequences lf 2 6 112 lztl lt 00 Then for s E S Z 6 ll 2 28 lt 00 We use p to denote such price function Not all qc may not be represented in this inner product form But we will see one theorem about inner product representation of price If p 6 13 we can write price as 118 20 push p2t82t p3t83t lt 00 Note here the prices of labor service and capital service are negative as we make input factor to be positive Homework 21 s E S assume sh 5152 5253t 53Vt that is we consider steady 0 is a price vector state for simplicity7 show that for any r gt 0 the discounting 0 4 I 1r t m 13 0 De ne an ADE An Arrow Debreu Competitive Equilibrium is a triad 11 c if such that l 58 solves the consumer s problem x E arg rig Uc 26 subject to q 50 S 0 27 2 if solves the rm s problem 34 E arg 1535mm 3 markets clear ie 58 if 1lt Note that the price system or valuation function p is an element of DnalL and not necessarily represented as a familiar price vector Note there are many implicit assumptions like all the markets are competitive agents are price taker ii absolute commitment economy with a lack of com mitment is also a topic of macroeconomics maybe from your 2nd year on iii all the future events are known with the probability of each events when trade occurs before the history begins 24 Welfare Theorems Theorem 22 FB WT If the preferences of consumers U are locally nonsatiated E X that conueryes to x E X such that Uxn gt Ux then allocation xy of an ADE is PO Homework 23 Show U is locally nonsatiated Theorem 24 SBWT If X is conuex ii preference is conuex for Vx E X if atquot lt x then atquot lt l 7 0c 0x for any 0 6 01 iii Uc is continuous iu Y is conuex WY has an interior point then with any PO allocation xy such that x is not a satiation point there erists a continuous linear functional q such that xyp is a QuasieEquilibrium with transfer 0 We can get rid of transfers in this economy Everyone is the same so given q S t zit 0 t 0 for all i o Quasi equilibrium and true equilibrium Quasi equilibrium is allocation from cost minimization problem That is a for cc 6 X which Uc 2 Uc implies qc 2 qx and b y E Y implies qy S qy lf for 58 y f in the theorem above the budget set has cheaper point than c that is 31 E X such that qc lt qc then xyp is a ADE Homework 25 Show that conditions for SB WT are satis ed in the PO allocation of RA7 NGM Now we established that the ADE of the RA NGM exists is unique and is P0 The next thing we would like to establish is that the price system 1 takes the familiar form of inner product of price vector and allocation vector which we will establish next 25 Inner Product Representations of Prices Theorem 26 based on Prescott and Lucas 1972 If in addition to the conditions to SBWT 6 lt l or some analog stochastic uersion about state and u is bounded then 3p 6 l1 3 such that xyp is a QE That is price system has an inner product representations 10 Remark 27 For OLG overlapping generation model there may be no enough discounting We will see how it works in that case Remark 28 Actually for now the condition 6 lt 1 is what we need to know as you can see on class For bounded utility function remember that most of the familiar period utility functions CRRA including log utility function CARA in macroeconomics do not satisfy the conditions as the utility function is not bounded There is a way to get away with it but we you not need to go into details for those interested see Stohey Lucas and Prescott Section 163 for emample 3 31 0 Now the agent s problem can be written as x E arg rig Uc 28 subject to 2131251 17252 p3t53t 0 29 t0 ln ADE all the trades are made before the history begins and there is no more choices after the history begins However market arrangement of ADE is not palatable to us in the sense that set of markets that are open in the ADE is NOT close to the markets in our real world We will allow people to trade every period and use sequence of budget constraints in agent s problem Next week we will proceed to the equilibrium concept that allows continuously open markets which is sequential market equilibrium Feb 2 Review 0 We established the equivalence between ADE and SPP Unlike the SPP allocation ADE can kind of tell us what happened on the world as people act optimally and compactibly ADE exists And ADE allocation is optimal by FBWT SBWT tells us the any SPP allocation can be got from a QET And there are three key points about QET With identical agents transfer is zero 11 lf there is a cheaper point quasi equilibrium is a true equilibrium Price may have an inner product representation given the condition in Prescott and Lucas 1972 is satis ed 0 ADE de nition 3 11 such that 32 x arg 136a U5L 30 subject to 1350 S 0 31 or Zpi lt JV 1731 p3t3t 0 32 20 Prices in ArrowDebreu Competitive Equilibrium How to get 11 The basic intuition is from college economics Price is equal to marginal rate of sub stitution Remember pit is the price of consumption good at period t in terms of consumption good at time 0 Denote A as Lagrangian multiplier associated with 32 Normalize price of time 0 consumption to be 1 From rst order condition we can get 3U f i 8581 Aplt for t 2 l 33 8U A 34 8x10 Therefore Price of time t consumption good is equal to marginal rate of substitu tion between consumption at time t and consumption at time 0 ADE allocation 58 can be solved from SPP As the functional form U is known we can construct price series pit To get price of labor service p3 related to wage we have to look at the rm s problem under current setting Consumer s problem says nothing about wage because leisure is not valued in utility function although usually the marginal rate of substitution between consumption and leisure is a natural candidate Firm s problem 34 6 Mg HEX phi1t pit2 pityaz 35 subject to 311 f yang2 36 Denote A as Lagrangian multiplier associated with 36First order condition plt A p3 7Asz 343 343 Thus we can construct price pit We have 11 i i 72 fL 343 3421 37 P1 And the wage rate at period t is price of labor service at time t in terms of time t consumption good Capital service price and arbitrage E I P1 kaffynl 38 Homework 31 show 5 8 note there is no 6 in this condition And please relate it to 97 No Arbitrage One freely tradable good can only have one market price Two identical ways of transferring resource have to be priced at same level How people can move one unit of resources from t to t1 There are two ways one is to sell one unit colt at time t and get pit then at time t1 agents can get pi 1712 I I I I I capital hm at time t1 agents can get rental of the additional unit of capital service and also non depreciation part of kn The relative price of doing so is 7 P 21 1 6 T plz1 I unit of consumption good set The other way is to save one more unit of Hint 32 The reason why we can see 1 7 6 is the following In ADE foe all the arguments are x s and y s To relate to hm we make use of consumption possibility spaee de nition ht 0 1 176ht and impose 3 ht Therefore the bene t to saue one more unit of capital is 176 E For 1t1 details see homework solution 33 From no arbitrage argument we know P1 P1t1 17 6 Jr P t1 39 P1t1 In sum we can solve SPP to get allocation of ADE and then construct price using FOC of household and rm s problem Sequential Market Arrangements So far rational expectation does not really apply Agents do not need perfect forecast ing as all the trades are decided at time 0 1n agent s optimization problem there is only one static constraint At time t comes people just execute their decision for this date ADE allocation can be decentralized in different trade arrangement We will look at SME Two things are important here Allocation in equi librium with sequential market arrangements cannot Pareto optimally dominate ADE allocation ii But with various market arrangement the allocation may be worse than ADE allocation Say labor market can be shut down or other arrangement to make people buy and trade as bad as it happens This topic is about endogenous theory of market institution Sequence of Markets With sequential markets people have capital kt and rent it to the rm at rental 1 7 People have time 1 and rent it to rm at wage 111 They also consume 0 and save kt Agents can also borrow and lending one period loan H1 at price qt Then the budget constraint at time t is kt1 7 111 1 C kt1 Qtlt1 40 Principle to choose market structure Enough but not too many There are many ways of arranging markets so that the equilibrium allocation is equivalent to that in ADE as we ll see ENOUGH Note that if the number of markets open is too few we cannot achieve the allocation in the ADE incomplete market Therefore we need enough markets to do as well as possible NOT TOO MANY To the contrary if the number of markets are too many we can close some of the markets and still achieve the ADE allocation in this market arrangement Also it means that there are many ways to achieve ADE allocation because some of the market instruments are redundant and can be substituted by others If the number of markets are not TOO FEW nor TOO MANY we call it JUST RIGHT 14 41 42 With the above structure loans market is redundant as there is only one representative agent in this economy In equilibrium It 0 So we can choose to close loans market We will see that even though there is no trade in certain markets in equilibrium we can solve for prices in those markets because prices are determined even though there is no trade in equilibrium and agents do not care if actually trade occurs or not because they just look at prices in the market having market means agents do not care about the rest of the world but the prices in the market Using this technique we can determine prices of all market instruments even though they are redundant in equilibrium This is the virtue of Lucas Tree Model and this is the fundamental for all nance literature actually we can price any kinds of nancial instruments in this way we will see this soon Feb 6 From ADE to SME We have seen that we need enough markets to get optimal allocation with sequential market arrangement Market structure depends on commodity space There should be markets to trade consumption good capital services and labor services Using consumption good at time t as the numarie and consider relative price we can see that only 2 markets are needed To transfer resources only one intertemporal market is e icient since two ways of trade are equivalent which are to trade 0 with Ct1 and to trade kt and kt The reason why we can normalize price of 0 to 1 at any t is that with in nite horizon future is identical at any time as in the way we write Bellman equation As we know from last class we can write budget constraint with loans lz1 Rt1 kt1nwtlt0tkt1 where 7 rental price of capital and R 2 price of lOUs But we can close the market of loans without changing the resulting allocation This is because we need someone to lend you loans in order that you borrow loans but there is only one agents in the economy Then the budget constraint becomes kt1 T t 111 Ct kt1 41 De ne SME Long de nition rst Consumer s problem is in 42 122IIslza11czlz Z 6 subject to l H1 1 72 wt2t l W 43 Rt1 The producer s problem is for all t 0 l 2 Elaidtylt wit2t 72343 44 11 subject to 341 S Fy3 342 De nition 41 A Sequential Market Equilibrium SME is 3 x3t75cgt 1 ri7 Rw yf ygm ygt0 and there erists cfkf1g390 such that consumer mamimizes giuen rm if fio x37 xii 3 of kg 1 solues optimization problem ii rm maximize giuen muf e yfmygmygt solues the producer problem iii markets clear x yft Vit and l1 0 0 There is a short way to write SME First let s look at the properties of SME 1 As all the solutions are interior and kt or lt1 cannot go to in nity in equilibrium Rt1 1 5 7 t1 2 ln equilibrium so l and so Consumer s Problem in SME can be written as follows Engagiin he 5W9 m 45 subject to ctkt1 wtliortkt Vt012 46 k0 is given 47 De nition 42 A Sequential Market Equilibrium SME is cfhf1 rt w fim such that 1 consumer mamimizes giuen rt w fio c h 1 solues optimization problem 2 0 kill 1 i 5 517 E arg maxi1 7 wig2 rib3 48 subject to 371 F yatyth o FCC to problem 48 1 A wt AZFLh1 rt AtFk h1 then wt FL 1 7 Fk If 1 2 in the above de nition can be substituted With 2 wt FL 7 Fk If 1 0 k 1 1i5 fl FUflyl Homework 43 show 27 2 Homework 44 Emplain the implication of CBS constant return to scale assumption on rms 0 Another way of writing SME is A Sequential Market Equilibrium SME is 07h1 rt wf 30 such that De nition 45 1 consumer maximizes 2 factor prices equal to marginal productivity 5 allocation is feasible 43 Compare ADE and SME 0 Show the equivalence of ADE and SME Theorem 46 If x7y7q 6 AD 5 then there em39sts 07k177 77w7f0 E SME 5 Proof Pick the of 7 If 1 implied by consumer s problem in ADE De ne the following price we 1113 wage at time 0 q 07 1 0 7 0 0 0 7 we wt wage at time t q 07 07 0 7 07 17 0 7 1 HQ 07 ll 7 07 07 0 7 7 3 rental at time 0 q 000001 7 wage at am Thus we have constructed 7 7111 30 Next we need to verify the condition for SME 5 FEBRUARY 11 5 1 Review 0 Last class our purpose was to construct a new market arrangement sequence of mar kets because it is much closer to what we think markets are like in the real world The fact that all trade takes place at time 0 in the Arrow Debreu world is not very realistic so we wanted to allow the agents to trade at each period 0 We used certain properties of equilibrium to write a shorter version of SME that did not bother to distinguish between the choice of the rm and the household for convenience 0 Now we will show that the allocations of the Arrow Debreu equilibrium and the se quence of markets equilibrium are the same Namely we will outline the proof of the following theorem i 1f X 34 11 is an Arrow Debreu equilibrium we can construct the sequence of markets equilibrium with X 34 ii 1f 337215 is an sequence of markets equilibrium we can construct the Arrow Debreu equilibrium with i Proof Outline I Remark 51 Refer to the solution key of Hw Sfor the complete proof First showing ADE SME qxy 3 ck1rwt0 E SME The rst thing we need to do is construct the sequence of markets equilibrium prices from q Remember that q is a function that assigns a value to each commodity bundle in terms of consumption goods AT TlME 0 The prices in the AD world DO NOT correspond to the usual price for consumption goods wage and rent 1n order to get rwt we need to transform these prices in terms of units of consumption at time 0 to prices in terms of units of consumption goods at time t The question is How much does one unit of time 0 consumption exchange for unit of consumption at time t 1 unit of time 0 consumption gt qO O Oh o ohuum ie 1 unit of time 0 consumption can get you units of time 1 con sumption 1000f00 Thus we can write the following ws qlto1oooo rs qltoo1ooo q000010 q000100 w1 Homework In the same way write down the empressions for ri wg Now the following will be our strategy to show that from ADE we can get to SME 0 First construct a candidate g ZO and En th t from of7 L y t0 o For 557 5 pick 07k1 n so that Ct it JV 1 5553 3t1 Vt 71 3 Vt kt 3 Vt o For 77min pick 73 p i Fkltkn v2 P1 a p E Fltkn v2 171 Now verify that these candidates solve the rm s and the consumer s maximization prob lem For rms this is obvious from the condition that marginal productivities equal to the prices of factors of production But for consumer we need to show that mam e argmax ma 1 e n t0 62k1nzo t0 St Ct kt1 at 17 t 6M We know that the objective function is strictly concave The next thing we need is that the constraint set is convex Homework De ne F as FCt7 kt17ntl 0 l Ct kt1 at 1 5W Vt Show that F is conuex Once we know that above ie the strict concavity of the objective function and convexity of the constraint set we can say that the solution to the consumer s problem exists is unique and the First Order Conditions characterize it together with the Transversality Condition Then showing that if 0 hf n satis es the FCC in the AD world given of it also satis es the FCC from the consumer s problem above will be enough to complete the proof Question Can we proue it another way for emample through contradiction Yes but that will not make our life any easier Because euen when you suppose that there is another allocations other than 0lf1 n that solues the consumer s problem in the sequence of markets you will still need the properties that the solution satis es as we deriued to yet the contradiction Now showing SME ADE We need to build the AD objects X s and y s from the SME allocation x1 a miuimt Vt 3 t Vt 3 ht Vt And the candidate for 01 will be 2 1t 215 35 20 q 50 t 1 lt1rgt Note that this is a function on a whole sequence We have to de ne q not just one a point but everywhere The other way ADE to SME was easy because the wage and the rental prices were just numbers 21 Homework Show that this candidate for 11 is indeed a price Hint Show that it is continous and linear Remark 52 What does continous mean in in nite dimensional space Bounded In this content it implies that the ualue of the bundle of commodities has to be nite and for that we need prices to go to zero su iciently fast a 0 A su icient condition for this is that rS gt 0 for some s 12 1r ie that the interest rates are not negatiue too often Thus we can say that a su icient condition for the prices that we contructed to be bounded and thus continous is that the interest rates are positiue BAK Now from market clearing we know that the following has to hold Homework Show that f and if solues the problem of the consumer and the rm And that s the end of the second part of the proof 52 ROAD MAP What have we done so far 0 We know that the social planner s problem can be solved recursively you learned this in Randy s class So with dynamic programming methods we get a good approximation of the optimal policy gk and get 07 hf10 Then we learned that this allocation is Pareto Optimal and that it can be supported as a quasi equilibrium with transfers 0 We also learned that this allocation of 7 hf10 is also the sequence of markets equi librium allocation and it is the ONLY one 53 So now we know that the dynamic programming problem gives us not only what is good but also what wiil happen in the sequence of markets What next The question that we now want to address is What happens if there are heterogenous agents in the economy versus the representative agent model that we have been dealing with so far and if the solution is not Pareto Optimal What can we do when we do not have the luxury of having an economy that does not satisfy the Welfare Theorems or when there are different agents Can we still use dynamic programming to deal with problems like this We will de ne equilibria recursively so that we can write the problem of the households as a dynamic programming problem and we will use the same methods Randy used to nd the optimal policy rule gk But now the objects that the agents are choosing over are not sequences They choose what they will do for today and tomorrow and prices are not a sequence anymore but a function of the states We will do the construction of such equilibria after a short digression on shocks SHOCK AND HISTORY We will now look at the stochastic RA NGM What is a shock Unanticipated change Not really In a stochastic environment we don t know exactly what will happen but we know where 1 it s coming from we know something about the stochastic process ie the process that the shocks are following 531 Markov Chains In this course we will concentrate on Markov productivity shock Markov shock is a sto chastic process with the following properties There are nite number of possible states for each time More intuitively no matter what happened before tomorrow will be represented by one of a nite set 2 The only thing that matters for the realization tomorrow is today s state More in tuitively no matter what kind of history we have the only thing you need to predict realization of shock tomorrow is today s realization 3 Denote Z is the state of today and Z is a set of possible state today ie Z 6 Z 2122 for all t Since the shock follow Markov process the state of tomorrow will only depend on More formally for each period suppose either 21 or 22 happens2 today s state So let s write the probability that Zj will happen tomorrow conditional on today s state being 2i as PM pr0bzt1 Zlet Since PM is a probability we know that 2 PM 1 j for W 49 Notice that 2 state Markov process is summarized by 6 numbers 21 22 F11 F12 F21 F22 The great beauty of using Markov process is we can use the explicit expression of prob ability of future events instead of using weird operator called expectation which very often people don t know what it means when they use 532 Representation of History Let s concentrate on 2 state Markov process In each period state of the economy is 2t E Z 2122 Denote the history of events up to t which of 21 22 happened from period 0 to t respectively by hi 20211 22 2 E H Z0 gtlt Z1 gtlt gtlt Zt In particular H0 ij H1 217 227 H2 217217 21722 Z2721 22722 Note that even if the state today is the same past history might be different By recording history of event we can distinguish the two histories with the same realization today but different realizations in the past think that the current situation might be you do not have a girl friend but we will distinguish the history where you had a girl friend 10 years ago and the one where you didn t tell me if it is not an appropriate example 2In this class superscript denotes the state and subscript denotes the time 3Here we restrict our attention to the 27state Markov process but increasing the number of states to any nite number does not change anything fundamentally 24 0 Let Hht be the unconditional probability that the particular history ht does occur By using the Markov transition probability de ned in the previous subsection it s easy to show that Hh0 1 ii for ht 21 21 Hht F11 iii for ht 21 22 21 32 HUB F12F21F12 0 PI 2H1 Zi l Z Zj72t7172t72 Fji 0 Having nite support of the distribution is very convenient Homework Show that a Markov chain of memory 2 can be represented as a Markov chain of memory 1 533 Social Planner s Problem with Shocks 0 Social Planner s Problem the benevolent God s choice in this world is a state contingent plan ie optimal consumption and saving let s forget about labor leisure choice in this section for simplicity choice for all possible nodes imagine the nodes of a game tree we need to solve optimal consumption and saving for each node in the tree Notice that the number of nodes for which we have to solve for optimal consumption and saving is countable This feature allows us to use the same argument as the de terministic case to deal with the problem The only difference is that for deterministic case the number of nodes is equal to number of periods which is in nite but count able but here the number of nodes is equal to the number of dateevents which is also in nite but countable More mathematically the solution of the problem is the mapping from the set of date events which is speci ed by history to the set of feasible consumption and saving max t H ht a 0 h Czhz kz1hzt6 MEG subject to kt1ht1 6ktht71 ZtFlktht7171l Vt Vb kg 20 given What is the dynamic programming version of this problem When we are writing the dynamic programming version we need to carefully specify what the states are States should be things that matter and change and that are predetermined We will have more on this later Vz k mZIX uc Zl uJz7 lg 2162 subject to 61 ck 176kzFk1 kg 20 given FEB 13 ADE and SME in a stochastic RANGM Review Recall PM Przt1 Zj l 2t l l l ie the probability of going SOMEWHERE given today s state is Zi is 1 Htht l 20 is a function from the set of histories up to t A Markov matrix F is a square matrix such that 1 l 1 2 PM 2 0 Htht l 20 2 Here l denotes possible Markov matrices and Z0 denotes possible initial shocks Why do we have This is because l and 20 are given in the problem They will not be changing while we do the analysis they are like the parameters of the problem 62 ADE We will now go over Arrow Debreu with uncertainty with the inner product representation of prices rather than using a general continous linear function We rst need to de ne the commodity space the consumption possibility set and the production possibility set As in the deterministic environment de ne commodity space as space of bounded real sequences with sup norm L loO But before in the deterministic case we only had 3 commodities for each period Now we have 3 commodities for each dateevent ht De ne the consumption possibility set X as X x e L 1 3cthtkt1ht 50 2 0 such that kt1hczht 1tht176ktht Vcht x2tht 6 01 WVth x3tht Mb Vt Vht kg 20 givengt Notice that the only difference from before is that now all the constraints has to hold for all periods AND all histories De ne the production possibility set Y as Y y E L 1 yltUlt S Fyatht7yztht Vt Vhtl The consumer s problem in ADE is mEa ZBZ Z Hhtu0tht 20 mth subject to gt0 3 Z Z ithtxitht S 0 20 me i1 We know that the solution to this problem is Pareto Optimal Recall the dynamic programming version of the social planner s problem 27 Vz k mix uc Zl u z7 lg 2162 subject to ck 176kzFk1 kg 20 given Remember that the state needs to be changing and predetermined For example l is not a state Solution to the above problem is a policy rule k gk and from this policy rule we can draw the whole path for capital Also Second Welfare Theorem tells us that the solution can be supported as a quasi equilibrium with transfers 63 SME p1h17 Price of one unit of the consumption good in period 17 at history h pl 517 Price of one unit of the consumption good in period 17 at history We want to have sequence of markets that are complete We want the agents to be able to transfer resources not just across time but also across different states of the world For this we need state contingent assets The budget constraint for the representative agent in SME world is QUE f kt1ht Z Qtht73t1lt1ht1 ktht71l17 thtl wht f ltht n1 EZ Here lt1ht1 is the state contingent claim By deciding how much lt1ht1 to get for each possible ht1 the agent decides how much of the good he is buying for each possible realization of tomorrow Homework What should the empresston below be equal to Z qthtz 2162 Note that this is the price of an asset that page one amt of the good to the agent the neat period at each state of the world A sequence of markets equilibrium is ctht ht1ht lt1ht 2H0 111ht7 ht qtht 2H0 such that H Given wht7 htqtht2H0 cththt1ht lt1htzt1 solves the consumer s problem wht ZtF2ktht171 I ht ZtF1ktht7171 lt1ht2t1 0 VhtZt1 E0 03 7 Feb 18 What is it that people buy and sell in the sequence of markets Consider an economy with two periods At t0 the agent s endowment of the good is 2 units At tl two things can happen The good state or the bad state The bad state happens with probability 7T and the bad state with probability l 7T 1n the good state the agent s endowment is 3 units of the good and in the bad state the agent s endowment is 1 unit of the good How many date events are there 3 date events Because in addition to the rst period we also have the two possible events that can take place at tl The consumer s problem in this economy is max Moo 7racb 1 7 7racg 515 00 4717909 pb0b 2 f 3P9 1717 Suppose the solution to the consumer s problem is 2 2 4 What does this mean He signs a contract in period 0 then he consumes c0 regardless of anything After period 0 nature determines whether the good state or the bad state happens NO TRADE happens in period 1 All trade already took place at t0 All that takes place at t1 is the full lment of whatever promises for deliveries were made at t0 For example the given allocation above tells us the following The guy signs a contract at t0 promising that he will give up his endowment of 3 units of the good in the good state for delivery of 2 units AND he will give up his endowment of 1 unit of the good in the bad state for the delivery of 4 units And it also tells us that he will consumer 2 units of the good at period 0 no matter what happens Remember not to think of this concept as just insurance Because insurance is only a subset of possible state contingent claims We are talking about any kind of state contingent claims here not just the ones which are only geared towards insuring you agains the bad state Now let s extend this to three periods We will now have 7 commodities The agent s objective function is We 7 WW 7 1 7 7TUCg 7T2170 7T0 7 7T Mag 7 Wow 7 1 7 70214099 In the Arrow Debreu world in complete markets how many commodities are traded 7 commodities It is 7 commodities because the agent need to decide what he wants for each dateevent For t2 we have four date events for t1 we have two dateevents and for t0 we have one These dateevents are the nodes Recall that in the AD world after period 0 all people do is honour their commitment and deliver promises No trade takes place after period 0 How about in the sequence of markets Trades can occur at more than one node We want to implement the same type of allocation as in AD with a market arrangement that is simpler and recurrent Think of the same world that we described above with two periods and two states And take note of the fact that at each one of those nodes trade CAN take place now unlike in the AD arrangement 1n the sequence of markets how many things are traded at period 0 Only 2 This is because in the sequence of marketts the agent does not trade for two periods ahead Also once we go on to t1 at one of the nodes say the good state the agent again only trades for two commodities he does not do anything about the other state anymore because the bad state has not happened We will characterize what happens in this world through backwards induction We will rst go to the last period t1 in this case and work backwards So at t1 the agent is either at the good state or the bad state Let s rst consider the node associated with the good stateAt this node the agent consumes cg and he chooses what he will consume if tomorrow s period is good again cgg and he chooses what he will consumer if tomorrow s period is bad cgb His objective function consists of the utility that he gets from consuming cg and the expected value of his utility in the next period VMmP max ucg JV 7T cgb JV 1 7 7Tucgg 515 09 091 p cgg x9 1 31 Pg pg pg 1 9 Kg The agent s past choice on what to get at the node associated with the good state at t1 Now consider the node associated with the bad state WWW max 110 401212 1 7TU0bg 515 0b 0171 0179 1 1 3 Pb P p 171 We have basically collapsed what the agent cares for after period 1 to the V functions Now go to time 0 The consumer s problem is 31 max um Wbp1 vrgtvgltxp 51 CO 9119 be 2 3139 171 Constructing ADE from SME and vice versa in this environment This is trivial because this time we don t even need to bother with constructing the prices from one world to the other Notice that in the formulations of the consumer s problem in the sequence of markets we already have been implicitly using the AD prices given that we know the allocations will be the same The p s are the AD prices and the SM prices are for ex ample 173 etc 17 However one thing you should be aware of is that ADE gives us certain prices and allocations whereas in SME we need to determine the prices allocations AND X9 and X From ADE to SME H E0 03 He Construct the SM prices Pb 0 Pg pg Use the same allocation 0070970127099 Construct the missing items X9 and xb Using the budget constraint get Xg from the prices and allocations in state g and get X from the prices and allocations in state b Verify that the following SME conditions are satis ed Markets clear at last period This is trivial from ADE X s add up to 0 across consumers c and X solve the consumer s problem From SME to ADE 1 Get rid of the X s 2 Verify conditions of ADE With two periods In the sequence of markets how many things are traded 9 because we have 3 commodities at each of the 3 nodes In the Arrow Debreu it was 7 Now suppose we have 100 goods instead In Arrow Debreu we will have 700 things to trade On the other hand in SM we will have 102 goods to trade per node and thus we will have only 306 things to trade In the sequence of markets we have minimal number of trades to get the best allocation Arrow Debreu has nice properties but it s messy to deal with 71 Back to the Growth Model 2r 2 minnow max h k h C a 1 2 t0 htth St Gihtl t1ht qht72t1ht72t1 ktht711rtht 6lwhtxhti17Ztht n1 Note The notation Ztht just refers to the 2 that is consistent with history ht In the representative agent model market clearing requires that Xl ltZt1 0 Vht VZt1 Homework Consider an economy with 2 periods There are two states of nature The good state and the bad stateBoth states have equal probabilities There is only one agent in the economy and he has an endowment of Z coconut and 2 scallops In the good state he will have an endowment of5 units of the goods and in the bad state he will have an endowment of 1 unit of each good The agent s utility function takes the following form us c log slog 0 Compute the equilibrium for this economy As before when we write down the equilibrium we do a shortcut and we ignore the X and q This does not mean that markets are not complete If all agents are identical then state contingent claims have to be 0 for all nodes 72 General Overview So far we have shown the following SPP gt AD From the Welfare Theorems SPP gt Dynamic Programming ProblemWhat Randy did ADltgt SME SME lt sc gt RCE RCE lt sc gt Dynamic Programming Problem sc denotes something in common Notice that RCE and DP are not necessarily equivalent Also SME and RCE are not necessarily equivalent Why would it be that SPP b AD Markets may not be complete Externalities Heterogenous Agents So for most equilibria we need to compute the equilibria directly We don t have the luxury of solving the social planner s problem to get the equilibrium allocation Solving the problem from AD and SME it s very messy So we will use the KGB notion to characterize what happens in the economy 8 Feb 20 De ning RCE 8 1 Review Consider the following two period economy The goods A and B at time 0 are denoted by X64 XOB and the goods at time 1 are denoted by X f Xf ln Arrow Debreu the consumer s problem is max ux07 x1 1 m22mso 10 lAB ln SME 9bf q 13w 11Xf750l3 X1 ml 515 x14 1131 bf where bf is what the consumer chose to bring from the past Assume that loans are in the form of good A don t need to transfer resources in the form of all goods Saving in the form of only one good is enough 35 Now go to period 0 A B A Kagggg i uX07O69b1Q bondbA 1 st x64 q0 qOBxOB endowment Homework Take this simple economy and show the equivalence between SME and ADE Homework Given an ADE write two sequence of markets equilibria In one of them take good A as the good used to transfer resources into the futureIn the other take it as good B Show that the two allocations are equivalent 9 Feb 20 9 1 Road map 0 From now on we will look at Recursive Competitive Equilibrium RCE ln Randy s class we learned that a Sequential Problem of SPP can be solved using Dynamic Programming Now we will see that we can use the same Dynamic Programming technique to solve an equilibrium RCE First we know the equivalence between an allocation of SPP and an allocation of ADE using Welfare Theorems And we showed that ADE can be represented as SME where the market arrangements are more palatable From today we will see that SME is equivalent to RCE When Welfare Theorems holds we do not need to directly solve the equilibrium because we know that allocation of SPP can be supported as an equilibrium and it is unique meaning the SPP allocation is the only equilibrium But if assumptions of Welfare Theorems do not hold or ii we have more than one agent thus we have many equilibrium depending on the choice of the Pareto weight in the Social Planner s Problem we can solve the equilibrium directly both in theory and empirically using computer Since solving ADE is almost impossible ii solving SME is very hard but iii solving RCE is possible RCE is important for analyzing this class of economies where Welfare Theorems fail to hold 36 0 ln ADE and SME sequences of allocations and prices characterize the equilibrium but in RCE what characterize the equilibrium are functions from state space to space of controls and values 92 Recursive representation in equilibrium Remember that the consumer s problem in SME is as follows t max u 0 50 sz1Czlzoo 2 6 C kt1 wt f 1 7 tlkt 51 How to translate the problem using recursive formulation First we need to de ne the state variables state variables need to satisfy the following criteria 1 PREDETERMlNED when decisions are made the state variables are taken as given 2 lt must MATTER for decisions of agents there is no sense of adding irrelevant variables as state variable 3 lt VARlES across time and state otherwise we can just take it as a parameter Be careful about the difference between aggregate state and individual state Aggregate state is not affected by individual choice But aggregate state should be consistent with the individual choice we will consider the meaning of consistency more formally later because aggregate state represents the aggregated state of individuals ln particular in our RA NGM as we have only one agent aggregate capital turns out to be the same as individual state in equilibrium but this does not mean that the agent decide the aggregate state or the agent is forced to follow the average behavior but rather the behavior of the agent turns out to be the aggregate behavior in equilibrium Also note that prices wages and rental rates of capital is determined by aggregate capital rather than individual capital and since individual takes aggregate state as given she also takes prices as given because they are determined by aggregate state Again the aggregate capital turns out to coincide with the individual choice but it is not because of the agent s choice rather it is the result of consistency requirement One notational note Victor is going to use a for individual capital and K for aggregate capital in order to avoid the confusion between K and k But the problem with aggregate and individual capital is often called as big K small k problem because the difference of aggregate capital and individual capital is crucial So for our case the counterpart is big K small a problem 37 Having said that we guess that candidates for state variables are K a w 7 But we do not need 73 in Why Because they are redundant K is the su icient statistics to calculate 73111 and K is a state variable we do not need 73111 as state variables Now let s write the representative consumer s problem in the recursive way At this point the time subscript has not be got rid of People care about today s period utility and the continuation utility from tomorrow t1 subject to VXKM 0 IEZXWC Vz1K 7a Gl 52 0a w1r76a 53 w wK 54 r rK 55 K GK 56 Fundamental rules to write a well de ned problem All the variables in the problem above 5VK a have to be either a parameter or an argument of the value function state variable ii a choice variable so appear below max operator 0 and 1 here iii or de ned by a constraint in order for the problem to be well de ned In the case above note c and a is a choice variable ii K is de ned by 56 which we will discuss below iii the variables in 53 especially 7 and w are also de ned by constraints which only contains state variables K thus we know that the problem is well de ned Agents need to make expectations about tomorrow s price to make consumption saving choice Because prices 73111 are given by marginal product of production functions Agents have to make forecast or expectations about the future aggregate state of the world We index the value function with G because the solution of the problem above depends on the choice of G But what is appropriate G This is revealed when we see the de nition of an equilibrium below Homework 91 Show the mapping de ned by 52 is a contraction mapping And prove the emistenee of FF and give the solution s properties 93 Recursive Competitive Equilibrium NOW let s de ne the Recursive Competitive Equilibrium De nition 92 A Representative Agent Recursive Competitive Equilibrium with arbitrary expectation GE is Vg such that Z solves consumer s problem VKa GE mama mm GEN subject to 0a wKlrK76a 57 K GEK 58 Solution is g Ka GE 2 Aggregation of individual choice K GK GE gKK GE 59 Some comments on the second condition The second condition means that if a consumer turns out to be average this period her individual capital stock is K Which is aggregate capital stock the consumer Will choose to be average in the next period she chooses 0K Which is a belief on the aggregate capital stock in the next period if today s aggregate capital stock is You can interpret this condition as consistency condition because this condition guarantees that in an equilibrium individual choice turns out to be consistent With the aggregate laW of motion Agents have rational expectation When G GE To compute this equilibrium we can de ne GE rst then get g and G The Whole sequence of equilibrium choice is obtained by iteration NOW let s de ne A Representative Agent Recursive Competitive Equilibrium With ratio nal expectation De nition 93 A Representative Agent Recursive Competitive Equilibrium with rational expectation is Vg such that Z solues consumer s problem vow 0 mama WK a GE subject to 0a wK1rK76a 60 K GK 61 Solution is g K a G 2 Aggregation of indiuidual ehoiee K GK9K7KG 6 In other words a RA RCE with rational expectation is a RA RCE with expectation GE while with additional condition imposed G K GE GE K 94 Solve SPP and RCE When we look at SPP in recursive form we nd a contraction mapping The SPP is solved as the xed point of contraction mapping ln math we de ne TVoK EE RK7K6V1KI where T maps a continuos concave function to a continuos and concave function And we can show T is a contraction mapping To nd the xed point of this contraction V we can use iteration for any continuos and concave function V0 V lim Tquot V0 such that a 7 I a I V 7EQ RK7K5V But to solve a RE RA RCE we cannot use such xed point theorem because we need nd G g jointly Similarly we can de ne the following mapping T which has three parts corresponding to G V1K7a f1V07G0 Iggch 6V000K7a wK1767 Ka sj c a 40 and the decision rule is a 9K7a00 01 K 71200700 9K7KGo We can see the rst component of j mapping gives V and the second part gives G Fixed point of this mapping T is RE RA RCE But f is not a contraction It is more di icult to nd RE RA RCE in theory but we will see how we can solve the problem on computer later 0 Another comment about RCE If there are multiple equilibria in the economy it is problematic to de ne RCE The reason is that RCE solution is functions Given today s state variable tomorrow s state is unique When we construct SME out of RCE KKj given Ki there is only one unique Kj 10 Feb 25 101 From RCE to SME Homework 101 Proue that a ROE with RE is a SME Hint 102 You can show by construction Suppose we haue a ROE Using a0 yiuen and GK we can deriue a whole sequence of ht7 0532 Using the constructed sequences of allocation we can construct sequence ofprices r w fio Remember that we have necessary and su icient conditions for SME we just need to show that the necessary and su icient conditions are satis ed by the constructed sequences 102 RCE for the Economy with Endogenous LaborLeisure Choice Let s try to write down the problem of consumer The rst try WK a G Ef uwm WK a 0 63 subject to 0a 176rKlawKn 64 K GK 65 41 This is an ill de ned problem Why Something is missing 7 K and 111 K are wrong function of price because now K is not suf cient determinant of w and 7 From rm s problem we know w f2 K7 Now we have two options to add the missing piece Option 1 write V K7 1 G711 7 And the equilibrium condition would be wK f2 KN MK f1K7N Option 2 write V K7 1 G H where H function is agent s expectation about aggregate labor as function of aggregate capital N H K then the price function is WU f2K7H MK f1K7H We will use option 2 to write RCE with RE Homework 103 De ne ROE using option 1 From now on we will only look at RCE with rational expectation Now the consumer s problem is VKa G H nia ucn 6VK a G 66 subject to 00 l1i5f1K7HKlaf2K7HKn 67 K GK 68 And the solutions are 1 9K7a 0 H 69 n hK a G H 70 42 De nition 104 A ROE is a set offunetz39ons V g such that 1 Given H V g solves the consumer s problem 2 CKgKKGH 71 HK hKK G H 72 103 More on solving RCE We have known that we can de ne mapping for V0 K a T V1 maggu a a 6V1K a 2 6 Note that we cannot solve a mapping since that s a mechanic thing Mapping we have here is from a functional space to a functional space We can only solve equation For example the Bellman equation is a functional equation which we can solve VKa ma uaa 6VK a 2 6 When the mapping we de ned above is a contraction mapping suf cient condition is monotonic ity and discounting then there is a unique xed point This xed point can be obtained by iteration For RCE if we x G and H we can construct the contraction mapping and get xed point by iteration The reason why the value function is xed point is that in in nite horizon economy today s View of future is the same as that of tomorrow For nite horizon economy we have to solve problem backward starting from VT1 max 11 BVT To solve RCE there are two steps First given G and H we can solve the problem by some approximation methods you will see this in late May Second we have to verify that G and H are consistent in equilibrium That is agent s expectation is actually correct as what happens in life Since there is no contraction mapping for G K 9 K K G0 H0 H K h KKGOHO it is hard to prove existence directly But we can construct one and verify the equilibrium condition This is the way to solve RCE Although compared with SPP RCE is hard to solve it can be used to characterize more kinds of economies including those environments when welfare theorem does not hold 43 104 RCE for nonPO economies What we did with RCE so far can be claimed to be irrelevant Why Because since the Welfare Theorems hold for these economies equilibrium allocation which we would like to investigate can be solved by just solving SPP allocation But RCE can be useful for analyzing much broader class of economies many of them is not PO where Welfare Theorems do not hold That s what we are going to do from now Let s de ne economies whose equilibria are not PO because of distortions to prices heterogeneity of agents etc 105 Economy with Externality Suppose agents in this economy care about other s leisure We would like to have beer with friends and share time with them So other people s leisure enters my utility function That is the preference is given by uc7 n7 where L1 N is the aggregate leisure One example may be logolog17n17n17N17 With externality in the economy competitive equilibrium cannot be solved from SPP The problem of consumer is as follows lK7 a Eaucn N 6VK 1 73 subject to 0a 176rawn 74 7 FkKN 75 w FNK N 76 K GK 77 N HK 78 And the solutions are a 97 a 44 n MK 1 We can de ne RCE in this economy Homework 105 Please de ne a ROE for this economy Compare the equilibrium with social planner s solution and emplain the di erenee Comments 1 We will not write G and H in value function since this is the way we see in literature But you should feel it 3 What if you only wanna hang out with some friends Write in the utility function This is the way we can work with RA framework We can see how far we can get from RA model To think how to write a problem with unemployment in a RA model for example But if you only wanna hang out with rich guys RA is not enough We will see how to model economy with certain wealth distribution later 106 Economy with tax 1 What is the government It is an economic entity which takes away part of our income and uses it The traditional or right wing way of thinking of the role of the government is to assume that the government is taking away part of our disposable income and throw away into ocean If you are left wing person you might think that the government return tax income to household as transfer or they do something we value Let s rst look at the rst version where income tax is thrown into ocean For now we assume that the government is restricted by period by period budget constraint so the government cannot run de cit nor surplus The consumer s problem is as follows VK a a Mc n 6VG K a 79 subject to 00 afnf2KHKlf1K5lal17 80 Income tax is proportional tax and only levied on income not on wealth Depreciation is exempt from tax too The government period by period constraint is trivial in this case government expenditure TfK H 7 6K 45 Remark 106 Notice that the economy does not achieue Pareto Optimality thus solued by SPP Because in SPP marginal rate of substitution equals to marginal rate of transformation But in this economy income tar a ected equilibrium allocation in the following way the distortion is in fauor of leisure against consumption Why Tax is only on income which is needed to get consumption not on leisure but agent can simply work less to get higher utility ii the distortion is in fauor of today against tomorrow The reason is the return of saying is less due to tax 107 Economy with tax 2 Now let s look at an economy where the tax income is returned to household in the form of lump sum transfers Consumer s problem is VK a a Mc n 6VG a 81 subject to 0a anf2KHKlf1K5la1TT 82 Where T is lump sum transfer From government period by period constraint we know T TfKHK e 6K The equilibrium in this economy is not Pareto optimal The reason is that agents tend to work less in order to pay less tax And they do not realize the lump sum tax they will get from government is affected by their action But we can not blame them because agent only have power to control what she does not other s action Only in a RA world her action happens to be the aggregate state We have to separate agent s problem from equilibrium condition 108 Economy with shocks to production When there is shocks to production should it be included in state variablesYes because shocks matters in two ways 1 it changes rate of return 2 it affects the way that economy evolves Therefore the state variables are ZKa Consumer s problem is ca z VZKa 0412 max MC 5 Z Full2 K a 2 0920 83 46 subject to 0ZqzrzKa z 1767 2Kaw2K 84 7 2K Zf1KHK wzK Zf2KHK K GzK A 00 01 V 86 7 A V 00 0 There is a complete set of markets for all possible contingences So people can sign contract to trade statecontingent goods What we have in the question above is state contingent asset qzz has a fancy name of pricing kernel and it has to induce equilibrium in this economy Since there is only one RA in equilibrium there is no trade The decision rule is alz gz Z7K7aiG7QZ Agent is free to choose any asset holding conditional on any 2 That s Why there are nz decision rules But in equilibrium there is only one K get realized Which cannot depend on 2 First we can get nz market clearing condition for equilibrium 027Kgz39ZK7K 89 But there are nz 1 functions to solve in equilibrium gzr and G So there is one missing condition We Will see in next class that the missing condition is No Arbitrage condition If one is free to store capital rather than trade state contingent claim the result is the same 11 Feb 27 We have talked about stochastic RCE from last class The consumer s problem is Vz K a agragg mc 5 Z run2 K a 23 90 subject to 0Zqzr 2100 1767 2Kaw2K 91 WK zf1KHK 92 wzK zf2KHK 93 K G2K 94 47 The decision rule is alz gz Z7K7ai G7Qz 1n ROE 027Kgz Z7K7K which gives us nz conditions But we need nz 1 conditions The missing condition is NA 111 No Arbitrage condition in stochastic RCE If the agent wanna have one unit of capital good for tomorrow there are two ways to achieve this One is the give up one unit of consumption today and store it for tomorrow s one unit of capital good The cost is 1 The other way is to purchase state contingent asset to get one unit of capital good for tomorrow How to do this Buy one unit of state contingent asset for all the possible 2 That is a 2 1VZ The total cost is ZQZ No Arbitrage condition is Egg27K 1 97 112 Steady State Equilibrimn In a sequential market environment steady state equilibrium is an equilibrium where kt k Vt In a deterministic economy without leisure nor distortion we can rst look at the steady state of SPP To nd steady state we use Euler equation and equate all the 48 Note Euler equation is a second order difference equation so there are k s at three different time involved kt kt 12 In a RCE steady state equilibrium is when K G K When there is shock in economy strictly speaking steady state does not exist in the sense of K G Because now Z is evolving stochastically and K G 2 But we will see the probability measure of K 2 can be found as a stationary once the capital is set at right range And of course the shock has to be stationary somehow itself Comments 48 1 RCE is stationary automatically in the sense that there is no time subscript in value function and decision rule E0 as you may see with Randy 00 For some growing economy we can always transform it into a non growing economy The way that econometricians and macroeconomists look at data are different Econo metricians believe there is a true data generating process underlying the data Macro economists think that real data are generated by people s choice They test models by comparing the properties of data generated by model to the real data We will see how to use model to look at data later in the class 113 FOC in stochastic RCE 1 02 K K 5 2 run 02 mm 1 7 6 7 2 K where from budget constraint 02Ka 1767 2KawzKizqzrzKa 98 Comment in 98 the RA condition a K is used It is allowed because the substitution is done after we derive rst order condition Agent only optimizes with respect to a not K So we get correct FOC rst Then we can apply equilibrium condition that a K To derive FOC envelope condition is used FOC a 15 CZyKyKqu39 27K TBFZ ZI7KI7GCIZ 0 By envelop condition V Zl7Kl7az 17 6 71 27Kllul lcZl7Kl7Kll Homework 111 Derive Envelope condition for this problem Therefore PW 17 6 w axiom ceszK n UICZKK 49 99 100 1f we can get CZK K from SPP 100 is an equation of qzr Z u cz K K Flzz 7 7 Z I 92127K5 WRKKD 1 6 lt 101 101 101 gives the price that induce household to choose the same allocation 0 and a as from SPP And such price ensure that agent s decision gzrz K a does not depend on 2 in equilibrium gz Z7K7K Remark 112 Price q are related to but not the same as probability WW It is also weighted by intertemporal rate of substitution to measure people s eualuation on consumption at some euent One simple ewample in two period economy where good state and bad state happen with equal probability to induce people to choose endowment of 2 and 1 at time 1 price for bad state must be higher since consumption at bad state is more ualuable to people Remark 113 In this uersion of stochastic ROE agent chooses stateicontingent asset a for newt period before shocks are realized When newt period comes 2 realizes and production takes place using the saying a There are other di erent timings Say consumer chooses consumption and saying after shocks for newt period get reuealed an are equi 1 rium con ition or e can a so get in t e o ow1ng 98 d 101 lb 139 I f ROE W l 98 I h fll I way 101 holds for all 2 1f we sum 101 over 2 and use the No Arbitrage condition 97 we can get Z P f ffgfj 1 7 6 w 2210 2111mm 1 Z Therefore u 02 K K 5 Z rug 02 K K 1 7 6 r 2 K Up to this point we know that people will save the same amount regardless of tomorrow s state because the price of statecontingent asset will induce them to do so Therefore an equivalent way to write RCE is to let agent choose tomorrow s capital without trade of state contingent asset And we can de ne RCE without q s Homework 114 Show that if there is a law saying that people have no right to buy statee contingent commodities Then in equilibrium the law is not binding In other word cone sumer s problem is equivalent to V 2 K a malxu c Z Furl2 K a subject to 0a 176rzKawzK K GZK 114 Economy with Two Types of Agents Assume that in the economy there are two types of agents called type A and type B Measure of the agents of type A and type B are the same Without loss of generality we can think of the economy as the one with two agents both of whom are price takers Agents can be different in many ways including in terms of wealth preference ability etc We will rst look at an economy where agents are different in wealth There are 12 population of rich people and 12 population of poor people For simplicity we assume there are no shocks and agents do not value leisure The state variables are aggregate wealth of both types K A and K B Why We know wage and rental only depends on total capital stock K KA KB But K is not su icient as aggregate state variables because agents need know tomorrow s price which depends on tomorrow s aggregate capital Agents preference is the same so the problem for both types are VKAKBa 6VGA KAKB GB KAKB 102 subject to 0a rK176awK 103 Solutions are a 9KA7KB7G ln RCE the equilibrium condition is GAKA7KB 9KA7K37KA GBKA7KB 9KA7K37KB 51 Homework 115 Show that necessary condition for K to be su icient state uariable is that agents decision rules are linear Homework 1 16 Show GAKAKB GB KBKA Homework 117 What does the theory say about the wealth distribution in steady state equilibrium for 27typeeagent economy aboue Compare it with steady state wealth distribution in island economy where markets do not exist 12 Mar 4 121 Review 0 Stochastic RCE With and Without statecontingent asset Consider the economy With shock to production People are allowed to purchase state contingent asset for next period Consumer s problem is VZK7 a G qz agrazn mc 6 Z Furl2 K a 2 0421 104 subject to 0Zqzr 2100 176rzKawzK 105 rzK zf1KHK 106 wzK zf2KHK 107 K G27K 108 Essentially we can get Euler equation 11160 6 Z Fzz ll 7 6 Zf1K71luc cl This condition is what we see more in macro literature But the consumer s problem we have above is a long hand version To derive it we use FOC ucczKa 8 8 Furl3 2 7K 7a 2 qu39 27K az Envelope condition V3 1767 z K uC Thus we have qz Z K uc c BTW17 6 Zf1K1uCrCl 110 Add over 2 and use NA condition 2 QZ Z 7 K 1 and substitute consistency condition a K We will get 109 As we see in the homework the equilibrium of this economy with a complete market can be found in economy without complete market The reason is that state contingent asset price qzr Z K is adjusted in the way such that agents save the same amount independent of 2 o Wealth distribution in economy with heteregenous agents Assume there are I types of agents there are 2 necessary conditions for equilibrium allocation I budget constraint equations ciai wai1r76 111 I FOC conditions u51r75u 112 And there are 2 unknowns 020 in steady state But in steady state the I FOC degenerate to the same one 1617 76 v 1 fk 11gt E i 1 5 113 Therefore the model says nothing about wealth distribution If the economy starts with f1 a l 7 l 7 5 then wealth ranking stays If not asset holding of different types will move parallel toward steady state level 122 Finance We will study Lucas Tree Model Lucas 19784 and look at the things that Finance people talk about Lucas tree model is a simple but powerful model 1221 The Model Suppose there is a tree which produces random amount of fruits every period We can think of these fruits as dividends and use d to denote the stochastic process of fruits production d 6 d1 dquotd Further assume d follows Markov process Formally d N ram d l d dj r 114 Let ht be the history of realization of shocks ie ht d0 d1 dt Probability that certain history ht occurs is Household in the economy consumes the only good which is fruit We assume represen tative agent in the economy and there is no storage technology In an equilibrium the rst optimal allocation is that the representative household eats all the dividends every period We will look at what the price has to be when agents use markets and start to trade First we study the Arrow Debreu world And then we use sequential markets to price all kinds of derivatives where assets are entitlement to consumption upon certain date event 4Lucas R 1978 quotAsset prices in an exchange economy quot Econometrica 46 142971445 54 1222 ArrowDebreu World Consumersgs problem max EH 2 7Thtu0tht 115 Chfio t he subject to Z Z phzczhz 6 Z Z phtdth 116 t ht H t thHg Equilibrium allocation is autarky dtht 117 Now the key thing is to nd the price Which can support such equilibrium allocation Normalize P be 1 Take rst order condition of the above maximization problem and also substitute 117 FOC 6 Wmucdthz Ptltht 118 u do A 119 We get the expression for the price of the state contingent claim in the Arrow Debreu market arrangement t mucdtht piht ucd0 120 Note that the price ptht is in terms of time 0 consumption 1223 Sequences of Markets 1n sequential market we can think of stock market Where the tree is the asset Household can buy and sell the asset Let S be share of asset and qt be the asset price at period t The budget constraint at every time event is then qs c 5q d 121 First we can think of any nancial instruments and use the A D prices ptht to price them 1 The value of the tree in terms of time 0 consumption is indeed Z Z Phtdtht t haEH 2 A contract that gives agent the tree in period 3 and get it back in period 4 This contract is worth the same as price of harvests in period 3 Z Ph3d3h3 hg Hs 3 Price of 3 year bond 3 year bond gives agents 1 unit of good at period 3 with any kinds of history The price is thus 2 PM hg Hs 1224 Market Equilibrium We will write it in a resursive form Then rst can we get rid of hi and write it in a recursive form Or are prices stationary The answer depends on whether the stochastic process is stationary Homework 121 Show prices q are stdtiondy only indewed by Z which is essentially the same as dividend Now the consumer s optimization problem turns out to be V Z s Inaxii c 6 Z Furl2 s 122 subject to 0S QZ Sl 12dzl 123 To solve the problem FOC uc AZ 6 Z F q 2 d 2 Azq z 56 So we get V2 Ms Z 9 Z 5 Z Fzz39lq 2 dZ luC 2 We write out the whole system of equation for all possible Z Us 21 q 21 5 Ea Fzz39lq 2 dZ lUC 2 124 Us 2 q 2 5 El n415 d 2 uc 2 Elements in 124 are marginal utility of consumption in different states and dividends which are numbers and price q s Therefore it is systerm of linear equations in q s And there are n2 linear equations and n2 unknowns We can then solve this system and obtain prices in sequential markets 13 March 18 131 Review 0 Last class we introduced the Lucas 1978 model There is a tree and the tree yields a random number of fruits at each period There is a representative agent 1n the previous class we said that the equilibrium has to be such that ct d since markets clear only if the consumption of the agent equals to her endowment due to the fact that we have a representative agent Then we were able to compute the prices that will support this allocation as the solution to the agent s maximization problem Thus we got the prices that will induce the agent to consume all he has at each period Then we set up a problem where the agent is able to trade at each period and let him maximize by choosing his consumption ct and the amount of share of the tree to buy st From the solution to the agent s problem in the sequence of markets structure we were able to characterize the price of a share of the tree at each node qht We derived these prices by deriving the FCC from consumer s problem and imposing the equilibrium conditions on them Today we see more on the characterization of these prices and we go on to asset pricing using these tools 132 Asset Pricing o What is the state of the economy It s the number of fruits from the tree The dividend is the aggregate state variable in this economy 0 s the share that the consumer has today is the individual state variable 0 Consumer s Problem Vd s max uc ZFderd s C d st 0 sqd sqd d In equilibrium the solution has to be such that cd and s l lmpose these on the FCC and get the prices that induce the agent to choose that particular allocation Homework Show that the prices qi 1 are characterized by the following system of equations Md grijuldj gbg idjl Vi Now let s write the above system of equations in the matrix form so we can write a closed form for q u d u d u d 91 Emmi Hf Hf t q1 d1 rm d2 u d2 g 6F11 6F1J 11 d1 ql d1 5F11 5P122 6F1J Ha z 23 Let q u and d u and A u d2 g l d1 l 112151 BFIJJ 2Zj and let bAd we have qAqb so that q I i A 1b 0 Now consider the same problem but now the agent can also buy bonds for the price of pd which entitles him to get 1 unit of the good the next period The agent s new budget constraint is b pd c s qd sqd d b The equilibrium quantity of bonds is 0 because there is noone to buy those bonds from or sell them to Now using this fact we will nd the pd that induces the agent to choose to buy 0 bonds P di zrijuidj W The price of a bond pd is characterized by the above set of equations You can see the pattern here We can choose any kind of asset and then price it in the same way 1321 Options De nition 131 An option is an asset that gives you the right to buy a share at aprespeei ed price if you choose In general to price any kind of assets and options we only need to know prices of consumption at each node The price of an option is a sum of gain under the option at each node considering the decision of whether to exercise the option or not multiplied by the price of consumption at each node In order to write an expression for the price of an option at a certain node qht we need to rst compute the price of a unit of consumption good at node ht in terms of units of consumption goods at node ht One way to do this is introduce state contingent claims to the problem of the agent y d and compute its price pddr Let s say we want to price options at node ht Assume that the set of the possible aggregate shock contains three elements Start from ht possible nodes in the next period are ht1 1 and hi1 and hf Now denote the price of consumption at node hid in terms of units of consumption at node ht by pill We want to see what pill is as well as pig pita In order to do this we use state contingent claims because the price of a state contingent claim at node ht that entitles the agent to get 1 unit of consumption good at node hi is pal So if we write down the problem of the agent with these state contingent claims and compute the prices of these state contingent claims we ll get exactly what we need The price of consumption at node htl1 or ht1 or hid in terms of units of consumption good at t Once we add the state contingent claims the agent s problem becomes the following vltd say uc 62Pddlvltdz 8 7b 7y d d max cs b y d stZy d pzd b pd c s qd SW d b M d Homework Show that the empression for the price of the state contingent claims pi is as follows y 7 Hwy 12 rm Mm An option that you buy at period t entitles you to max0qht117 gt at node 11max0qht21i a at node hf and max0qh 1 7 a at node hf And therefore all we need to do to compute its price is multiply what it entitles the agent at each node by the price of con sumption at that node in terms of consumption at t which we now know because we already have an expression for and sum over all the nodes 9 mem qj 7 alpi j Homework Price a two period option that can be exercised any time before its mai tarity ie if you buy the option today it can be emercised either tomorrow or the day after 60 ii Price a two period option that can be exercised only at its maturity Homework Come up with an asset and price it Homework Find the formula that relates the price of the bond to the price of the state contingent claims in the aboue problem ie pd to pad Homework Giue a formula for qd in terms of paw 133 RCE with Government When we are considering RCE with government there are several issues that we need to consider before we begin writing down the problem of the agent and the government s budget constraint We need to make some choices about the economy that we are modelling Can the government issue debt If no The government is restricted by his period by period budget constraint He cannot run a budget de cit or a surplus Whatever he gets as revenues from taxes he spends that and no more or no less His budget constraint needs to hold at each period the government cannot borrow from the public Here the government expenditures are exactly equal to the tax revenues If yes The government can issue bonds at each period When it turns out that the government s expenditures are higher than its revenues the tax revenues he can issue debt or when it turns out that the expenditures are less than his revenues he can retire debt that was issued before If we model the economy so that the government is allowed to issue debt then we need to deal with the issue of restricting it to accumulate public debt inde nitely This is where the No Ponzi Scheme comes into play But we ll have more on that later We will do the second case today So the economy is as follows 0 Government issues debt and raises tax revenues to pay for a constant stream of expen ditures 0 Government debt is issued at face value with a strem of interest rate rb t o No shocks 0 No labor leisure choice Let s write down the problem of the consumer Notice that the consumer can transfer resources across time in two ways here He can either save in the form of capital or he can buy bonds In equilibrium the rates of return on both ways of saving should be the same by no arbitrage so that r rk Since the rates of return on both is the same the agent shouldn t care in what form he saves ie the composition of the asset portfolio doesn t 61 matter So let a denote the agent s asset Which consists of physical capital holding k and nancial asset b We don t need to make the distinction between the two And let r denote the rate of return on a Which is in turn the rate of return on capital and rate of return on bonds Aggregate state variables are K and B Where B is the government debt Notice that G the government expenditures is not a state variable here since it s constant across time The individual state variable is a the consumer s asset holdings The consumer s problem VKBa malx uc 6VK Blal subject to K HKB B IKB 739 TKB Solution is a ilJK7 B7 a De nition 132 Giuen TKB a ROE is a set of functions such that Z Household s optimization Giuen solue the household s prob lem 2 Consistency HKB IKB wKBKB 5 No Arbitrage Condition rbKB 1FKHKB717 5 The rate of return on bond is equal to the rate of return on capital notice we already used this fact when we were writing down the problem of the consumer by letting r denote the rate of return on both and not distinguishing between them 4 Government Budget Constraint 1 K73lfK715KfkK15Bl1TKyB GBl1fkK715l So that the government s resources each period are the bonds that it issues IKB plus its reuenues from tax on rental income wage income and income on the interest on bonds Its uses are the youernment erpenditures G and the debt that it pays back 5 No Ponzi Scheme Condition 32 K F such that VKB E K c KyB E 53 Homework Consider an economy with two countries indered by i6 14 B Each country is populated by a continuum of in ntely liued identical agents that is taken to be of measure one Each country has a production function flle Ni di erent technologies across coun tries Assume that output and capital can be transferred between countries at no cost But labor cannot moue across countriesDe ne recursiue equilibria for this economy 14 March 20 Measure Theory 0 We will use measure theory as a tool to describe a society with heterogenous agents Most of the previous models that we dealt with there was a continuum of identical agents so we saw the economy as consisting of only one type of agentBut from now on in the models that we deal with there will be heterogenous agents in the economy The agents will differ in various ways in their preferences in the shocks they get etc Therefore the decisions they make will differ also In order to describe such a society we need to be able to keep track of each type of agent We use measure theory to do that o What is measure Measure is a way to describe society without having to keep track of names But before we de ne measure there are several de nitions we need to learn De nition 141 For a set A A is a set of subsets of A De nition 142 039 7 algebra A is a set of subsets of A such that 1AZJ A 2 B E A BCA E A closed in complementarity where BCA a E A a 3 5 for Bi12m B E A ZBZv E A closed in countable intersections 0 An example for the third property Think of a U algebra de ned on the set of people in a classroom The property of being closed in countable intersections says that if the set of tall people is in the U algebra and the set of women is in the U algebra then the set of tall women should be in the U algebra also Consider the set A1234 Here are some examples of U algebras de ned on the set A A1 93A A2 A1727374ll 43 37 A7 1T7 2 7 27 374l7 17 37 4T7 37 4 17 2 A4 The set of all subsets of A Remark 143 A topology is a set of subsets of a set also just like a oealgebra But the elements of a topology are open intervals and it does not satisfy the property of closedness in complementality since a complement of an element is not an element of the topology Therefore topology is not a oealgebra Remark 144 Topologies and Borel sets are also family of sets but we use them to deal with continuity and oealgebra we use to deal with weight 0 Think of the U algebras de ned on the set A above Which one provides us with the least amount of information It is Al Why Because from A1 we only know whether an element is in the set A or not Think of the example of the classroom From A1all we get to is whether a person is in that classroom or not we learn nothing about the tall people short people males females etc The more sets there are in a U algebra the more information we haveThis is where the Borel sets are useful A Borel set is a U algebra which is generated by a family of open sets Since Borel U algebra contains all the subsets generated by intervals you can recognize any subset of set using Borel U algebra In other words Borel U algebra corresponds to complete information 0 Now we are ready to de ne measure De nition 145 A measure is a function x A a 73 such that 1 scab 0 2 31732 E A and 31 Bg 23 xBl U32 xBl xBg nite addltlulty 5 lfBif1 E A and Bi Bj Z for alll a j xUiBi countable addltlulty De nition 146 Probability measure is a measure such that xA 1 Homework Show that the space of measures ouer the mterual 01 is not a topological uector space Homework Show that the space of sign measures is a topological uector space Homework Show that the countable union of elements of a oialg ebra is also element of the o 7 alg ebra Consider the set A0l where a E A denotes wealth So A is the set of wealth levels normalized to 1 We will de ne X A gt 73 as a probability measure so that the total population is normalized to one Using measure we can represent various statistics in a simple form 4 The total population Adz4l Average wealth adx A We go through each levels of wealth in the economy and multiply the wealth level by the proportion of people that have that wealth level and because the size of the society E0 is normalized to 1 this gives us average wealth 3 Variance of wealth Aai4adx2dcc 4 Coe icient of variation fAla fA WlZd lZ andx 5 Wealth level that seperates the 1 richest and the poorest 99 is a that solves the following equation 099lagt5d A Homework Write the expression for the Gini indem Remark 147 Notation A ligaidxmao mama 141 Introduction to the Economy with Heterogenous Agents Imagine a Archipelago that has a continuum of islands There is a sherman on each island The shermen get an endowment s each period s follows a Markov process with transition my and SE quotquot397 SHE The shermen cannot swim There is a storage technology such that if the shermen store q units of sh today they get 1 unit of sh tomorrow The problem of the sherman is Vsa malx uc Zl ssils stcqa sa ca 0 Homework Can we apply the Contraction Mapping Theorem to this problem 66 15 March 25 Economy with Heterogenous Agents 151 Measure Theory continued 0 Consider the set A1234and the following 039 7 alg ebr as de ned on it A1 WA A2 A127374ll 0 Remember from last class that the more sets there are in the 039 7 alg ebra7 the more we know De nition 151 fA 7 R is measurable with respect to A if Bcb Afb c AVc A Example 1 Consider the following set A the elements of which denote the possible pairs of today s and tomorrow s temperatures A 7273710000 gtlt 727310000l And let A be a 039 7 1 lg ebra de ned on this set A such that A 7273 72735t 72735 7274 7274 72745t 7273 1000mm the subscripts just denote which day the temperatures are for today t or tomorrow t1 Knowing which set in A a certain element lies provides us with no information about tomorrow s temperature The 05 intervals are all for today s temperatures Now imagine we say we will have a party only if tomorrow s temperature is above a certain level Can we have a function de ned on the A above and know whether there will be a party tomorrow as the outcome of this function The answer is no The particular 0397algebra that we de ned above is not appropriate for a function the arguments of which are tomorrow s temperature Any function whose argument is temperatures tomorrow is not measurable with respect to the 0397algebra above For a function to be measurable with respect to a 0397algebra the set of points over which the function changes value should be in that 0397algebra and this is what the above de nition of measurability translates into In other words you should be able to tell apart the arguments over which the function changes value 67 Example 2 Consider the following Uialgebra A de ned on set Al234 W WA 17 2 34 Now think of the function that gives 1 for odd numbers and 0 for even ls this function measurable with respect to A No because the arguments where the functions changes values are not elements of A This function would be measurable with respect to the following 039 7 a lg ebra A 9A1737274 One of the ways we need the notion of measurability in the context of the economies we deal with is the following Every function that affects what people do at time t has to be t measurable in other words cannot depend on the future t1 152 Transition Function De nition 152 A transition function Q A gtlt A a R such that 1 VB 6 A QB A a R is measurable 2 Va 6 A Qa A e R is a probability measure Q function is a probability that a type a agent ends up in the type which belongs to B What the rst condition above says is the following Whatever we need to know for today has to be suf cient to specify what the probability tomorrow is Consider the set A the elements of which are the possiblel states of the world say good and bad Agood bad And let P be the transition matrix associated with A F F 1 PM 99 g f Fbg Pb i The Uialgebra we would want to use is A 7147 good bad Notice that what we need to know to compute the probability of a certain state tomorrow is today s state and the Uialgebra lets us know that 68 Homework Verify that the A 237A7 good bad is a of algebra Homework Take A01 and A Borel sets on A F07 a6 AB 6 A7 QaB de A Verify that Q is a transition function 0 The measure X de ned on a sigma algebra over the set of intervals of wealth is complete description of the society today X B Measure of people who have characteristics in BE A tomorrow The pair XQ will tell us about tomorrow Pick one measure that s de ned over the Uialgebra A on the set of wealth levels A01 553 AQsaBdx So with X and Q we get Lquotie with the measure of people today and the transition function Q we get the measure of people with wealth level in a certain interval Bab tomorrow What people are now X What people do gt 5Lquot Example Consider a society where people are characterized as good guys or bad guys De ne the sigma algebra A as A WA good bad And suppose that today everybody in the society is a good guy so that xgood 1 xbad 0 Let F denote the probability that someone will be a good guy tomorrow given he is a good guy today So the measure of people that are good guys tomorrow is Ilti900dl F99