General equilibrium theory Lecture notes

Transcription

1 General equilibrium theory Lecture notes Alberto Bisin Dept. of Economics NYU 1 November 16, These notes constitute the material for the second section of the rst year graduate Micro course at NYU. The rst section on decision theory, is taught by Ariel Rubinstein. The notes owe much to the brilliant TA s Ariel and I had, Sevgi Yuksel and Bernard Herskovic.

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3 Contents 1 Introduction Preliminaries Abstract exchange economies Allocation mechanisms Competitive market Jungle Pareto e ciency Characterization of Pareto e cient allocation Competitive market equilibrium Welfare Existence Uniqueness Local uniqueness Characterization of the structure of equilibria Competitive equilibrium in production economies Strategic foundations Two-period economies Arrow-Debreu economies Financial market economies The stochastic discount factor Arrow theorem Existence Constrained Pareto optimality Aggregation Asset pricing Some classic representation of asset pricing iii

4 iv CONTENTS Production Matching and search equilibrium 67 5 Monopolistic competitive equilibrium 69 6 Asymmetric information A simple insurance economy The Symmetric information benchmark The moral hazard economy The adverse selection economy Information revealed by prices References In nite-horizon economies Asset pricing Arrow-Debreu economy Financial markets economy Conditional asset pricing Predictability or returns Fundamentals-driven asset prices Bubbles (Famous) Theoretical Examples of Bubbles Double In nity Overlapping generations economies Default To add Appendix 1 A quick review of consumer and producer theory Consumer theory Duality Aggregate demand Producer theory Appendix 2 Some useful math Separating hyperplane theorems Fixed point theorems

5 CONTENTS v 10.3 Di erential topology References

6 vi CONTENTS

7 Chapter 1 Introduction The standard approach to graduate teaching of general equilibrium theory involves introducing a series of theorems on existence, characterization, and welfare properties of competitive equilibria under weaker and weaker assumptions in larger and larger commodity spaces. Such an approach introduces the students to precise rigorous mathematical analysis and invariably impresses them with the elegance of the theory. Various textbooks take this approach, in some form or another A. Mas-Colell, M. Whinston, and J. Green (1995) Microeconomic Theory, Oxford University Press, Part 4 - the main modern reference; it also contains a short introduction to two-period economies. L. McKenzie (2002), Classical General Equilibrium Theory, MIT Press - a beautiful modern treatment of the classical theory. K. Arrow and F. Hahn (1971) General Competitive Analysis, North Holland - the classical treatment of the classical theory. 1 B. Ellickson (1994) Competitive Equilibrium Theory and Applications, Cambridge University Press - somewhat heterodox in the choice of the main themes; it contains a useful chapter on non-convex economies. 1 And so is G. Debreu (1972), Theory of Value An Axiomatic Analysis of Economic Equilibrium, Cowles Foundation Monographs Series, Yale University Press, an invaluable little book for several generations of theorists. 1

8 2 CHAPTER 1. INTRODUCTION M. Magill and M. Quinzii (1996) Theory of Incomplete Markets, MIT Press - a classical theory approach to nancial market equilibrium in twoperiod economies. The approach adopted in these notes aims instead at introducing general equilibrium theory as the canonical theoretical structure of economics in its application, e.g., as the main microfoundation for macroeconomics and - nance. To this end, the standard theory of general equilibrium is introduced in its rigour and elegance, but only under restrictive assumptions, allowing some shortcuts in analysis and proofs. On the other hand, we shall be able to introduce nancial market equilibria in two-period economies rather quickly, exposing students to fundamental conceptual notions like complete and incomplete markets, no-arbitrage pricing, constained e ciency, equilibria in moral hazard and adverse selection economies, and many more. The course ends with a treatment of dynamic economies and recursive competitive equilibria. Pedagogically, from two-period to fully dynamic economies the step is rather short, so that we can concentrate on purely dynamic concepts, like bubbles. 1.1 Preliminaries For any x; y 2 R N +; we say x > y if x n y n ; for any n = 1; 2; ; N; and x n > y n for at least one n For any map f R N! R we let the gradient vector be de ned as i i2i

9 Chapter 2 Abstract exchange economies Consider an economy populated by agents with exogenously given preferences over and endowments of commodities. There is no production. Nonetheless agents do not necessarily consume their own endowments but rather participate in an allocation mechanism. We now formalize this structure. The economy is populated by an in nity of agents. Agents are categorized in a nite set I = f1; ; Ig of types, with generic element i. 1 We also assume an in nite number of agents is of type i; for any i 2 IThe consumption set X is the set of admissible levels of consumption of L existing commodities. We shall assume X = R L +; with generic element x X is then a convex set, bounded below. Each agent of type i 2 I has a utility function U i R L +! R which represents his preferences. 2 Each agent of type i 2 I also has an 1 We are conscious of the notational abuse. 2 In these notes we shall adopt the utility function as a primitive. That is, we shall assume that any agent s underlying preference ordering % on X (is complete, transitive, and continuous; so that it) can be represented by a utility function; see Rubinstein (2009). We shall in fact typically also assume struct convexity. A preference ordering % on X which (is not continuous and hence it) cannot be represented by a utility function is the Lexicographic ordering x % y if x 1 y 1 or x 1 = y 1 and x l y l ; for l = 2; ; L 3

10 4 CHAPTER 2. ABSTRACT EXCHANGE ECONOMIES endowment! i 2 X De nition 1 An allocation for the economy is an array x =(x 1 ; ; x I ) 2 R LI +. An allocation is feasible if it satis es IX x i i=1 IX! i In the special case in which L = 2, I = 2, feasible allocations can be graphically represented using the Edgeworth Box. i=1 2.1 Allocation mechanisms An allocation mechanism is a rule which maps the preferences and endowments of each of the agents in the economy into an allocation. The allocation mechanism standing at the core of most of economics is that of competitive markets. But this is not the only possible mechanism. In this section we shall then de ne and compare two di erent mechanism the competitive market and the jungle Competitive market Each of the commodities is traded on a market characterized by a linear (non-negative) price and each agent is a price taker in the market. Prices are such that markets clear the allocation is feasible. Formally, let p 2 R L + denote the vector of commodity prices; and let B(p;! i ) denote the budget set of agent i 2 I B(p;! i ) = x 2 X p x p! i A competitive market allocation is an allocation x 2 R LI + such that i) x i 2 X maximizes the utility of each agent i 2 I subject to his budget set, x i 2 arg for given prices for given p 2 R L +; max x2b(p;! i ) U i (x)

11 2.2. PARETO EFFICIENCY 5 ii) prices p 2 R L + are such that IX x i i=1 IX! i i= Jungle Allocations are determined by strength, whereby stronger agents can obtain the endowments of weaker agents. Formally, let S denote a binary relationship on the set I isj; for i; j 2 I is to be interpreted as "agent i is stronger than agent j"a jungle allocation is an allocation x 2 R LI + such that i) the allocation, jointly with a vector of commodities x 0 2 R L + not allocated to anyone, is feasible x 0 + IX x i i=1 IX! i ; i=1 ii) stronger agents can obtain the endowments of weaker j 2 I such that isj and U i (x i +x j ) > U i (x i ) or U i (x i +x 0 ) > U i (x i ). 2.2 Pareto e ciency An interesting possible property of an allocation is Pareto e ciency. De nition 2 An allocation x 2 R LI + is Pareto e cient if it is feasible and there is no other feasible allocation ^x 2 R LI + which Pareto dominates it; that is, if there is no ^x 2 R LI + such that U i (^x i ) U i (x i ) for all i 2 I, > for at least one i. Pareto dominance de nes a (social) preference relation over the set of allocations R LI +. It is however incomplete, in the sense that given two allocations x; y 2 R LI + it might very well that that neither x Pareto dominates y, nor viceversa. It is useful to impose the following strong (but not outrageous) assumptions on the economic environment.

12 6 CHAPTER 2. ABSTRACT EXCHANGE ECONOMIES Assumption 1 U i R L +! R is C 2 in any open subset of R L +; strictly monotonic in its arguments and strictly quasi-concave. Furthermore,! 2 R LI ++ Pareto E cient allocations are solutions of the following problem (convince yourself this is just a formal translation of the de nition) max U i (x i ) x2x s.t. X x i i X! i i (PE pb) U j (x j ) U j for all j 6= i for some given vector U j 2 j6=i RI 1 Varying the values of U j for j 6= i we obtain the set of Pareto e cient allocations. Let the Utility possibility set be de ned as ( U = U 2 R I U U i (x i ) I, for some x 2 RLI i=1 + such that X i Note that U is possibly larger than U 0 = x i X i n U U 2 R I = (U i (x i )) I i=1, for some x 2 RLI + such tha the image of the feasible allocations in the space of utility levels; in particular U is unbounded below, which is not necessarily the case for U 0. Problem 3 Show that U is bounded above and closed. Under which conditions is U also convex? Do these properties translate to U 0 under the same conditions? The Pareto utility frontier is de ned as UP = fu 2 U j@u 0 2 U such that U 0 > U g It contains the image of all Pareto optimal allocations in the space of utility levels. Problem 4 Show that UP ( bdry(u). Show that instead UP =bdry(u 0 )! i )

13 2.2. PARETO EFFICIENCY 7 There exist another formal characterization of Pareto e ciency. Theorem 5 (Negishi) (If) Suppose U is convex. Let x 2 R LI + be a Pareto e cient allocation. Then there exist a 2 R I +, 6= 0, such that x = arg max X i2i i U i (x i ) (Negishi pb) s.t. X i2i x i X ii! i (Only if) Furthermore, any solution of the Negishi pb, for some 2 R I +, 6= 0; is a Pareto e cient allocation. Proof. Consider a Pareto e cient allocation x 2 R LI + ; so that U(x) = fu i (x i )g i2i 2 UP. Furthermore, U(x) 2 bdry(u), and U is closed, convex, bounded above. The Supporting hyperplane theorem (see Math Appendix) then implies that there exists a 2 R I, 6= 0, such that that is, U(x) U; for any U 2 U; U(x) = arg max X i2i i U i s.t. U 2 U Trivially, then, by the de nition of U s.t. x = arg max X i2i X x i X! i i i i U i (x i ) Finally, U is unbounded below, which implies that 2 R I + In fact, suppose by contradiction i < 0 for some i 2 I Then there would exist a ^U 2 U with ^U i < 0 and small enough that U(x) < ^U. This proves the if part of the theorem. The only if part is straightforward and hence left as a problem.

14 8 CHAPTER 2. ABSTRACT EXCHANGE ECONOMIES Characterization of Pareto e cient allocation Both the PE pb and the Negishi pb are well-behaved convex maximization problems and hence rst order conditions are necessary and su cient (see Math Appendix). Recall also we assumed utility functions are strictly monotonic increasing (convince yourself that this implies that all constraints in either problem hold with equality). The rst order conditions (for an interior solution) of the PE pb are ru i = i0 ru i0 = for all i 0 6= i X x i = X! i i i where 2 R L + and ( j ) j6=i 2 R I 1 + and the Lagrange multipliers of, respectively, the feasibility constraint and the minimal utility constraints. Thus ru i = j ru j for all j 6= i and utility gradients are co-linear for all agents. As a consequence, marginal rates of substitution are equalized across agents. Problem 6 Compare the Negishi and the PE pb. Show that an allocation x 2R LI + is a solution of both problems i i j = j ; for any j 6= i 2.3 Competitive market equilibrium We now study in detail one of the two mechanisms we introduced, competitive markets. The jungle is (introduced formally) and studied in detail by Piccione and Rubinstein (2007). Agents trade in perfectly competitive markets, where prices are linear the unit price p l of each commodity l is xed independently of level of individual trades and is the same for all agents;

15 2.3. COMPETITIVE MARKET EQUILIBRIUM 9 prices are non-negative this is typically justi ed under free disposal, that is, by the assumption that agents can freely dispose of any amount of any commodity; 3 markets are complete for each commodity l in X there is a market where the commodity can be traded. De nition 7 A competitive equilibrium is an allocation x 2 R LI + and a price p 2 R L ++ such that i) each agent i 2 I solves max x i 2R L + U i (x i ) s.t. px i p! for given price p 2 R L ++; and (Consumer pb) ii) markets clear (the allocation is feasible) X x i X i i Trade is voluntary, hence equilibrium allocations will satisfy individual rationality U i (x i ) U i (! i ) for all i = 1; ; I It is convenient to represent the solution to the Consumer pb as a demand function x i p;! i ; x i R L ++ R L ++! R L + Let z i (p;! i ) denote agent i 2 I s excess demand z i (p;! i ) = x i (p;! i )! i Finally, the aggregate excess demand is de ned as z (p;!) = X z i p;! i ; i2i! i where! = (! i ) i2i ; z R L ++ R LI ++! R L +; It is convenient to identify and parametrize an economy with its endowment vector! 2 R LI ++; keeping utility functions given. 3 We assumed utilities are strictly monotonic increasing and hence, as we shall later see, prices will be strictly positive at any equilibrium.

16 10 CHAPTER 2. ABSTRACT EXCHANGE ECONOMIES Proposition 8 For any economy! 2 R LI ++ the aggregate excess demand z (p;!) satis es the following properties smoothness z (p;!) is C 1 ; homogeneity of degree 0 z (p;!) = z (p;!) ;for any > 0; Walras Law pz (p;!) = 0; 8p >> 0; lower boundedness 9s such that z l (p;!) > s, 8l 2 L; boundary property p n! p 6= 0; with p l = 0 for some l; ) max fz 1 (p n ;!); ; z L (p n ;!)g! 1 The simple proof of this Proposition is left to the reader. The somewhat counterintuitive aspect of the boundary property is due to the fact that if the prices of more than one commodity converge to 0, it is possible that only a subset of the correspondig excess demands explodes Welfare Let s rst study the welfare properties of competitive equilibrium allocations. Theorem 9 (First welfare) Consider an economy! 2 R LI ++ All competitive equilibrium allocations are Pareto e cient. Proof. Suppose x 2 R LI + is a competitive equilibrium allocation for some price p and it is not Pareto e cient. Then there must be another allocation ^x 2 R LI + which is feasible and Pareto dominates x. But since x i is the optimal choice of consumer i at prices p and preferences are strongly monotone, U i (^x i ) U i (x i ) implies p ^x i p x i for all i 2 I and also p ^x i p! i, with all the previous inequalities being strict for at least some i. Summing the latter inequality over i yields p P i ^xi > p P i!i which contradicts the feasibility of ^x Note that while monotonicity is crucial in the proof of the First welfare theorem, to go from U i (^x i ) U i (x i ) to p ^x i p x i, convexity of the Consumer pb is never used.

17 2.3. COMPETITIVE MARKET EQUILIBRIUM 11 Theorem 10 (Second welfare) Consider an economy! 2 R LI ++ For any Pareto e cient allocation x 2 R LI ++, there exist transfers t 2 R LI, P i ti = 0; such that x 2 R LI ++ is a competitive equilibrium allocation (for some prices p 2 R L +) of the economy! + t 2 R LI +. Proof. First of all choose t 2 R LI + so that t i = x i! i for all i. We then just need to nd prices p 2 R L ++ such that at these prices any agent i 2 I; with x i 2 R L ++ as endowment will not trade. Recall that x = (x i ) i2i is Pareto e cient by assumption. The theorem is then an implication of the separating hyperplane theorem (see Math Appendix). Consider the Better than set for agent i B i (x i ) = y i 2 R L + ju i (y i ) > U i (x i ) It is a convex set and x i =2 B i (x i ) The separating hyperplane theorem implies that there exist a p 2 R L such that py i px i ; for any y i 2 B i (x i ) It remains to show that in fact p 2 R L + and py i > px i ; for any y i 2 B i (x i ) We show p 2 R L + rst This is easily shown by constradiction if p l < 0 for some l = 1; ; L; we can construct y i as yl i = 0 xi l 0; for any l0 6= l; and yl i > x i l so that y i 2 B i (x i ) and py i < px i, the desidered contradiction with the implication of the separating hyperplane theorem just obtained. Then, since x i is strictly positive, px i > 0 As consequence, there exist a cheaper bundle ^x i, such that p^x i < px i Consider now any y i 2 B i (x i ) and assume by contradiction py i px i. Construct the bundle y i +(1 )^x i ; for some 2 [0; 1) Then p (y i + (1 )^x i ) < px i and, for some is close enough to 1; U i (y i + (1 )^x i ) > U i (x i ) by continuity; a contradiction with the implication of the separating hyperplane theorem obtained above Note that, di erently from the case of the First welfare theorem, convexity is crucial for the Second welfare theorem, to be able to apply the separating hyperplane theorem Existence Competitive equilibrium prices are solutions of z(p;!) X X x i (p; p! i )! i = 0; i i a system of L equations in L unknowns, the prices p. By Walras law p X x i (p; p! i ) i! X! i = 0; for all p 2 R L + i

18 12 CHAPTER 2. ABSTRACT EXCHANGE ECONOMIES and hence at most L 1 equations are independent (the market clearing equation for one market can be o,itted without loss of generality). By homogeneity of degree 0 in p of x i (p; p! i ); for any i 2 I; prices can always be normalized, e.g., restricted without loss of generality to ( ) p 2 L 1 p 2 R L + X l p l = 1 ; the L-simplex, a compact ad convex set. The equilibrium equations can thus be always reduced to L 1 equations in L 1 unknowns. Since equations are typically nonlinear, having number of unknowns less or equal than number of independent equations does not ensure a solution exists. Existence proof 1 Trimmed simplex The rst existence proof is based on the following Lemma. Lemma 11 Let z L 1! R L be a continuous function, such that pz(p) = 0 for all p. Then there exists p such that z(p ) 0 Proof. Let ' l (p) = p l + max f0; z l (p)g P L j=1 [p, l = 1; L j + max f0; z j (p)g] Note that L 1 is convex and compact, and ' L 1! L 1. Hence by the Brouwer Fixed Point theorem there is a xed point p p l = p l + max f0; z l (p)g P L j=1 [p, l = 1; L. j + max f0; z j (p)g] Then z l (p)p l L X j=1 and summing over l yields [p j + max f0; z j (p)g] = z l (p)p l + z l (p) max f0; z l (p)g 0 = X l z l (p) max f0; z l (p)g ) z l (p) 0 for all l = 1; ; L

19 2.3. COMPETITIVE MARKET EQUILIBRIUM 13 This is not quite an existence proof because aggregate excess demand functions, di erently from the the map z L 1! R L in the Lemma, are not de ned for prices on the boundary of L 1 ; that is, when the price of some commodity is zero. We need then to use a limit argument. Let L " 1 p 2 R L + P l p l = 1, p l " for all l de ned a "trimmed" simplex. The aggregate excess demand is indeed well-de ned on it z L " 1! R L. Consider now a sequence of "trimmed" simplexes, as "! 0. A boring and also a little intricate argument can be used to show that, as "! 0; xed points of the map ' " L 1! L 1 constructed on the the aggregate excess demand z i) exist (we need Kakutani s xed point rather than Brouwer, but this is straightforward) and ii) must be in the interior of the trimmed simplex (this involves exploiting the boundary condition of excess demand). But then restricting the analysis to the trimmed simplex L " 1 is without loss of generality for some " > 0 small enough. We hence have that the xed point p satis es z(p) 0 and Walras law, pz(p) = 0. As a consequence, z l (p) < 0 i p l = 0; which is impossible under strong monotonicity and hence Existence proof 2 Debreu map z(p) = 0 [see Mas Colell et al. (1995), Proposition 17.C.1, p ] Problem 12 Which steps of the proof of a) Negishi s theorem, b) First and Second Welfare theorem, c) Existence, relies crucially on i) strict monotonicity of preferences, ii) strict quasi-concavity of preferences, iii) strictly positive endowments? Uniqueness Existence of a competitive equilibrium can be proved under quite general conditions. 4 Equilibria are however unique only under very strong restrictions. 4 We shall leave this statement essentially unsubstantiated. General equilibrium theory, for more than half a century, has considered this as one of its main objectives.

20 14 CHAPTER 2. ABSTRACT EXCHANGE ECONOMIES Several examples of such restrictions are listed in the following, without any detail. Pareto e ciency. If endowments are Pareto e cient, there exists a unique equilibrium which is autarchic x i =! i for all i 2 I Aggregation. If preferences are identical and homothetic, then an aggregation result implies that the economy is equivalent to one with a single representative agent and hence the exists a unique equilibrium which is e ectively autarchic. Gross substitution. If the aggregate demand satis es the gross substitution property, p 0 l > p l and p 0 j = p j for all j 6= l =) z j (p 0 ) > z j (p); the law of demand holds at any equilibrium price and there exists a unique equilibrium. Gross substitution holds for instance for Cobb Douglas and CES utility functions under restrictions on the elasticity of substitutions across goods. Problem 13 Prove that indeed uniqueness holds for Pareto e ciency and Gross substitution Local uniqueness Let an economy be parametrized by the endowment vector! 2 R LI ++ keeping preferences (u i ) i2i xed. Furthermore, normalize p L = 1 and eliminate the L-th component of the excess demand. Then z R L 1 ++ R LI ++! R L 1 represents an aggregate excess demand for an exchange economy! = (! i ) i2i 2 R LI ++. De nition 14 A p 2 R L ++ such that z (p;!) = 0 is regular if D p z (p;!) has rank L 1

21 2.3. COMPETITIVE MARKET EQUILIBRIUM 15 It is convenient to rely on standard notions from Linear algebra to better understand the concept of regularity. For given (p;!) ; D p z (p;!) is an L 1 L 1 matrix. Its rank being L 1 implies that the matrix spans the whole R L 1 space; that is, for any z 0 2 R L 1 ; there exists a p 0 2 R L 1 such that D p z (p;!) p 0 = z 0 Since we deal with non-linear maps z R L 1 ++ R LI ++! R L 1, these kind of arguments only hold locally, and z 0 needs to be restricted to an open ball around z (p;!) and p 0 to an open ball around p De nition 15 An economy! 2 R LI ++ is regular if D p z (p;!) has rank L 1 for any p 2 R++ L 1 such that z (p;!) = 0. De nition 16 An equilibrium price p 2 R L 1 ++ is locally unique if 9 an open set P such that p 2 P and for any p 0 6= p 2 P, z (p 0 ;!) 6= 0 Proposition 17 A regular equilibrium price p 2 R L 1 ++ is locally unique. Proof. Fix an arbitrary! 2 R LI ++ Since D p z (p;!) has rank L 1; by regularity of p, the Inverse function theorem - Local (see Math Appendix) applied to the map z R L ++ 1! R L 1, directly implies local uniqueness of p 2 R L Proposition 18 Any economy! in a full measure Lebesgue subset of R LI ++ is regular. We say then that regularity is a generic property in R LI ++ (or equivalently that it holds generically) if it holds in a full measure Lebesgue subset of R LI ++. Proof. The statement follows by the Transversality theorem (see Math Appendix), if z t 0 We now show that z t 0 Pick an arbitrary agent i 2 I It will be su cient to show that, for any (p;!) 2 R L ++ 1 R LI ++ such that z(p;!) = 0; we can nd a perturbation d! i 2 R L such that dz = D! iz(p;!)d! i, for any dz 2 R L 1. Consider any perturbation d! i such that d! i L + pd!i L = 0; for d!i L = (!i l )L 1 l=1 Any such perturbation, leaves each agent i 2 I demand unchanged and hence it implies D! iz(p;!)d! i = d! i L, for any arbitrary d! i L 2 RL Proposition 19 The set of equilibrium prices of an economy! 2 R LI ++ is a smooth manifold (see Math Appendix) of dimension LI

22 16 CHAPTER 2. ABSTRACT EXCHANGE ECONOMIES Characterization of the structure of equilibria The di erential techniques exploited to study generic local uniqueness can be expanded to provide a general characterization of competitive equilibria as a manifold parametrized by endowments. This characterization implies an existence result. We sketch some of the analysis, just to provide the reader with the avor of the arguments. De nition 20 The index i(p;!) of a price p 2 R L 1 ++ such that z (p;!) = 0 is de ned as i(p;!) = ( 1) L 1 sign jd p z (p;!)j The index i(!) of an economy (! i ) i2i is de ned as i(!) = X i(p;!) pz(p;!)=0 Theorem 21 (Index) For any regular economy! 2 R LI ++, i(!) = 1 Proof. The theorem is a deep mathematical result whose proof is clearly beyond the scope of this class. Let it su ce to say that the proof relies crucially on the boundary property of excess demand. Adventurous reader might want to look at Mas Colell (1985), section 5,6, p Corollary 22 Any regular economy! 2 R LI ++ has an odd number of equilibria. Existence proof 3 Index theory and homothopy theory Corollary 23 Any economy! 2 R LI ++ has at least one equilibrium price p 2 R L ++ 1 Proof. By contradiction. Suppose there exist an economy! 2 R LI ++ with no equilibrium. Then,! 2 R LI ++ is regular, by de nition of regularity - a contradiction with previous corollary. The existence result is then a corollary of the Index theorem. It is useful to study the simple case in which L = 2 In this case, then, z R ++! R. The boundary properties of the excess demand z(p;!) imply that, for any! 2 R 2I ++; z(p;!)! +1 as p! 0 z(p;!)! L as p! 1

23 2.3. COMPETITIVE MARKET EQUILIBRIUM 17 As a consequence, an equilibrium exists by continuity of z(p;!). Furthermore, suppose! is regular, and let the prices p j such that z(p;!) = 0 be ordered, so that p j < p j+1 ; j = 1; 2; p=p1 < 0 Actually, < 0 for j p=pj As a consequence, i(!) = 1 > 0for j even We can also try and give more intuitive sense of the arguments, o of the proof of the Index theorem, required for this approach to the existence question. To this end we need to use some construction used in homothopy theory (see Milnor (1965). Let! 2 R LI ++ be an arbitrary regular economy. Pick an economy! 0 2 R LI ++ such that there exist a unique price p 2 R L ++ 1 such that z(p;!) = 0; and D p z(p) has rank L 1 One such economy can always be constructed by choosing! 0 2 R LI ++ to be a Pareto optimal allocation. In fact, [we can show that] generic regularity holds in the subset of economies with Pareto optimal endowments. Let t! + (1 t)! 0 ; for 0 t 1; represent a 1- dimensional subset of economies. Let Z(p; t) be the map Z R L ++ 1 [0; 1]! R L 1 induced by Z(p; t) = z(p; t! + (1 t)! 0 ) for given (!;! 0 ) We say that Z(p; t) is an homotopy, or that z(p;!) and z(p;! 0 ) are homotopic to each other. [We can show that] DZ(p; t) has rank L 1 in its domain. It follows from the Corollary of the Inverse function theorem - Global (see Math Appendix) that the set (p; t) 2 Z 1 (0); is a smooth manifold of dimension 1 [We can show that] prices p can, without loss of generality, be restricted to a compact set P such that Z 1 (0) never intersects the boundary of P Z 1 (0) \ [bdry(p ) [0; 1]] =? 5 As a consequence Z 1 (0) is a compact smooth manifold of dimension 1 By the Classi cation theorem (see Math Appendix), Z 1 (0) is then homeomorphic to a countable set of segments in R and of circles S. 6 Regularity of Z 1 (0) at the boundary, t = 0 and t = 1 and the property that Z 1 (0)\[bdry(P ) [0; 1]] =? imply that at least one component of Z 1 (0) is homeomorphic to a line with boundary at t = 0 and t = 1 It looks confusing, but it s easier with a few gures. The only possible representation of Z 1 (0) is then as in the following gure. Therefore an equilibrium price p 2 P exists, for any t 2 [0; 1]; and furthermore, for a full-measure Lebesgue subset of [0; 1] the number of equilibria is odd. 5 This is a consequence of the boundary conditions of excess demand systems. In other words, we could adopt the alternative normalization, restricting prices in the simplex ;

24 18 CHAPTER 2. ABSTRACT EXCHANGE ECONOMIES Figure 2.1 Characterization of Z 1 (0) - impossible

25 2.3. COMPETITIVE MARKET EQUILIBRIUM 19 Figure 2.2 Characterization of Z 1 (0) - impossible

26 20 CHAPTER 2. ABSTRACT EXCHANGE ECONOMIES Figure 2.3 Characterization of Z 1 (0) - impossible

27 2.3. COMPETITIVE MARKET EQUILIBRIUM 21 Figure 2.4 Characterization of Z 1 (0)

28 22 CHAPTER 2. ABSTRACT EXCHANGE ECONOMIES Competitive equilibrium in production economies [...] Strategic foundations [...] a compact set, and show that equilibrium prices are never 6 Along a component of Z 1 (0) (a line or a circle), a change in index occurs when the manifold folds.

29 Chapter 3 Two-period economies In a two-period pure exchange economy we study nancial market equilibria. In particular, we study the welfare properties of equilibria and their implications in terms of asset pricing. In this context, as a foundation for macroeconomics and nancial economics, we study su cient conditions for aggregation, so that the standard analysis of one-good economies is without loss of generality, su cient conditions for the representative agent theorem, so that the standard analysis of single agent economies is without loss of generality. The No-arbitrage theorem and the Arrow theorem on the decentralization of equilibria of state and time contingent good economies via nancial markets are introduced as useful means to characterize nancial market equilibria. 3.1 Arrow-Debreu economies Consider an economy extending for 2 periods, t = 0; 1. Let i 2 f1; ; Ig denote agents and l 2 f1; ; Lg physical goods of the economy. In addition, the state of the world at time t = 1 is uncertain. Let f1; ; Sg denote the state space of the economy at t = 1. For notational convenience we typically identify t = 0 with s = 0, so that the index s runs from 0 to S De ne n = L(S +1) The consumption space is denoted then by X R n +. Each agent is endowed with a vector! i = (! i 0;! i 1; ;! i S ), where!i s 2 R L +; for any s = 0; S. Let u i X! R denote agent i s utility function. We will assume 23

30 24 CHAPTER 3. TWO-PERIOD ECONOMIES Assumption 1! i 2 R n ++ for all i Assumption 2 u i is continuous, strongly monotonic, strictly quasiconcave and smooth, for all i (see Magill-Quinzii, p.50 for de nitions and details). Furthermore, u i has a Von Neumann-Morgernstern representation u i (x i ) = u i (x i 0) + SX s=1 prob s u i (x i s) Suppose now that at time 0, agents can buy contingent commodities. That is, contracts for the delivery of goods at time 1 contingently to the realization of uncertainty. Denote by x i = (x i 0; x i 1; ; x i S ) the vector of all such contingent commodities purchased by agent i at time 0, where x i s 2 R L +; for any s = 0; ; S Also, let x = (x 1 ; ; x I ) Let = ( 0 ; 1 ; ; S ); where s 2 R L + for each s; denote the price of state contingent commodities; that is, for a price ls agents trade at time 0 the delivery in state s of one unit of good l Under the assumption that the markets for all contingent commodities are open at time 0, agent i s budget constraint can be written as 1 0 (x i 0! i 0) + SX s (x i s! i s) = 0 (3.1) s=0 De nition 24 An Arrow-Debreu equilibrium is a (x; ) 2 R ni ++ R n ++ such that 2 SX 1 x i 2 arg max u i (x) s.t. 0 (x 0! i 0) + s (x s IX x i! i s = 0, for any s = 0; 1; ; S i=1 s=0! i s) = 0; and Observe that the dynamic and uncertain nature of the economy (consumption occurs at di erent times t = 0; 1 and states s 2 S) does not manifests itself in the analysis a consumption good l at a time t and state s is treated simply as a di erent commodity than the same consumption good 1 We write the budget constraint with equality. This is without loss of generality under monotonicity of preferences, an assumption we shall maintain.

31 3.2. FINANCIAL MARKET ECONOMIES 25 l at a di erent time t 0 or at the same time t but di erent state s 0. This is the simple trick introduced in Debreu s last chapter of the Theory of Value. It has the fundamental implication that the standard theory and results of static equilibrium economies can be applied without change to our dynamic) environment. In particular, then, under the standard set of assumptions on preferences and endowments, an equilibrium exists and the First and Second Welfare Theorems hold Financial market economies Consider the 2-period economy just introduced. Suppose now contingent commodities are not traded. Instead, agents can trade in spot markets and in j 2 f1; ; Jg assets. An asset j is a promise to pay a j s 0 units of good l = 1 in state s = 1; ; S. 3 Let a j = (a j 1; ; a j S ) 2 RS + To summarize the payo s of all the available assets, de ne the S J asset payo matrix 0 A a 1 1 a J 1 a 1 S a J S It will be convenient to de ne a s to be the s-th row of the matrix. Note that it contains the payo of each of the assets in state s. Let p = (p 0 ; p 1 ; ; p S ), where p s 2 R L ++ for each s, denote the spot price vector for goods. That is, for a price p ls agents trade one unit of good l in state s Recall the de nition of prices for state contingent commodities in Arrow-Debreu economies, denoted and note the di erence (di erent commodity spaces are everything in the world of general equilibrium theory)! Let good l = 1 at each date and state represent the numeraire; that is, p 1s = 1, for all s = 0; ; S. Let x i sl denote the amount of good l that agent i consumes in good s. Let 2 Having set de nitions for 2-periods Arrow-Debreu economies, it should be apparent how a generalization to any nite T -periods economies is in fact e ectively straightforward. In nite horizon will be dealt with in the next chapter. 3 The non-negativity restriction on asset payo s is just for notational simplicity, essentially without loss of generality. Note also that each asset pays in units on good 1. We call such assets numeraire assets, for obvious reasons. This assumption instead not without loss of generality. We ll see this afterwards. 1 A

32 26 CHAPTER 3. TWO-PERIOD ECONOMIES q = (q 1 ; ; q J ) 2 R J +, denote the prices for the assets. 4 Note that the prices of assets are non-negative, as we normalized asset payo to be non-negative. Given prices (p; q) 2 R n ++R j + and the asset structure A 2 R SJ +, any agent i picks a consumption vector x i 2 X and a portfolio z i 2 R J to maximize presend discounted utility. s.t. p 0 (x i 0! i 0) = qz i p s (x i s! i s) = A s z i ; for s = 1; S De nition 25 A Financial markets equilibrium is a (x; z; p; q) 2 R ni ++R j R n ++ R j + such that 2 1 x i 2 arg max u i (x) s.t. IX x i! i s = 0, for any s = 0; 1; ; S; and i=1 p 0 (x i 0! i 0) = qz i p s (x i s! i s) = a s z i ; for s = 1; S ; IX z i = 0 i=1 Financial markets equilibrium is the equilibrium concept we shall care about. This is because i) Arrow-Debreu markets are perhaps too demanding a requirement, and especially because ii) we are interested in nancial markets and asset prices q in particular. Arrow-Debreu equilibrium will be a useful concept insofar as it represents a benchmark (about which we have a wealth of available results) against which to measure Financial markets equilibrium. Remark 26 The economy just introduced is characterized by asset markets in zero net supply, that is, no endowments of assets are allowed for. It is straightforward to extend the analysis to assets in positive net supply, e.g., stocks. In fact, part of each agent i s endowment (to be speci c the projection of his/her endowment on the asset span, < A >= f 2 R S = Az; z 2 R J g) can be represented as the outcome of an asset endowment, z i w! i s1 = w i s1 + a s z i w; for any s 2 S and proceed straightforwardly by constructing the budget constraints and the equilibrium notion. 4 Quantities will be row vectors and prices will be column vectors, to avoid the annoying use of transposes.

33 3.2. FINANCIAL MARKET ECONOMIES 27 No Arbitrage Before deriving the properties of asset prices in equilibrium, we shall invest some time in understanding the implications that can be derived from the milder condition of no-arbitrage. This is because the characterization of noarbitrage prices will also be useful to characterize nancial markets equilbria. For notational convenience, de ne the (S + 1) J matrix W = q A De nition 27 W satis es the No-arbitrage condition if there does not exist a z 2 R J such that W z > 0 5 The No-Arbitrage condition can be equivalently formulated in the following way. De ne the span of W to be < W >= f 2 R S+1 = W z; z 2 R J g This set contains all the feasible wealth transfers, given asset structure A. Now, we can say that W satis es the No-arbitrage condition if < W > \ R S+1 + = f0g Clearly, requiring that W = ( q; A) satis es the No-arbitrage condition is weaker than requiring that q is an equilibrium price of the economy (with asset structure A). By strong monotonicity of preferences, No-arbitrage is equivalent to requiring the agent s problem to be well de ned. The next result is remarkable since it provides a foundation for asset pricing based only on No-arbitrage. Theorem 28 (No-Arbitrage theorem) < W > \ R S+1 + = f0g () 9^ 2 R S+1 ++ such that ^W = 0 5 Recall that W z > 0 requires that all components of W z are 0 and at least one of them > 0

34 28 CHAPTER 3. TWO-PERIOD ECONOMIES Note we are using the following de nition ^W = 0 implies ^ = 0 for all 2< W > Observe that there is no uniqueness claim on the ^, just existence. Importantly, ^W = 0 provides a pricing formula for assets 0 ^W ^ 0 q j + ^ 1 a j ^ S a j S 1 0 A z a condition which must hold for any z 2 R J, hence implying, after rearranging q j = 1 a j S a j S ; for 0 1 A Jx1 s = ^ s ^ 0 and any asset j 2 J (3.2) Note how the positivity of all components of ^ was necessary to obtain (3.2). Proof. =) De ne the simplex in R S+1 + as = f 2 R S+1 + P S s=0 s = 1g. Note that by the No-arbitrage condition, < W > T is empty. The proof hinges crucially on the following separating result which we shall take without proof. Lemma 29 Let X be a nite dimensional vector space. Let K be a nonempty, compact and convex subset of X. Let M be a non-empty, closed and convex subset of X. Furthermore, let K and M be disjoint. Then, there exists ^ 2 Xnf0g such that sup ^ < inf ^ 2M 2K Let X = R S+1 +, K = and M =< W >. Observe that all the required properties hold and so the Lemma applies. As a result, there exists ^ 2 Xnf0g such that sup ^ < inf ^ (3.3) 2<W > 2 We rst show that ^ 2 R S+1 ++ Suppose, on the contrary, that there is some s for which ^ s 0. Then note that in (3.3 ), the RHS 0 By (3.3), then, LHS < 0 But this contradicts the fact that 0 2< W >. We still have to show that ^W = 0, or in other words, that ^ = 0 for all 2< W >. Suppose, on the contrary that there exists 2< W > such that ^ 6= 0 Since < W > is a subspace, there exists 2 R such that ;

35 3.2. FINANCIAL MARKET ECONOMIES 29 2< W > and ^ is as large as we want. However, RHS is bounded above, which implies a contradiction. T (= The existence of ^ 2 R S+1 ++ such that ^W = 0 implies < W > R S+1 + = f0g By contradiction, suppose 9 2< W > and such that 2 R+ S+1 nf0g Since ^ is strictly positive, ^ > 0; the desired contradiction. A few nal remarks to this section. Remark 30 An asset which pays one unit of numeraire in state s and nothing in all other states (Arrow security), has price s ; this is an immediate consequence of (3.2). Such asset is called Arrow security. Remark 31 Is the vector ^ obtained by the No-arbitrage theorem unique? Notice how (3.2) de nes a system of J equations and S unknowns, represented by. De ne the set of solutions to that system as R(q) = f 2 R S ++ q = Ag Suppose, the matrix A has rank J 0 J (that it, A has J 0 linearly independent column vectors and J 0 is the e ective dimension of the asset space). In general, then R(q) will have dimension S J 0. It follows then that, in this case, the No-arbitrage theorem restricts to lie in a S J 0 dimensional set (or, equivalently, ^ in a S J dimensional set). If we had S linearly independent assets, the solution set has dimension zero, and there is a unique vector that solves (3.2). The case of S linearly independent assets is referred to as "complete markets." Remark 32 Recall we assumed preferences are Von Neumann-Morgernstern u i (x i ) = u i (x i 0)+ X s=1;;s prob s u i (x i s); with prob s > 0, for any s 2 S; and X s=1;;s We never used this assumption until now. But in this case, let then m s = s prob s. Then q j = E (ma j ) (3.4) In this representation of asset prices the vector m 2 R S ++ is called Stochastic discount factor. prob s = 1

36 30 CHAPTER 3. TWO-PERIOD ECONOMIES The stochastic discount factor In the previous section we showed the existence of a vector that provides the basis for pricing assets in a way that is compatible with equilibrium, albeit milder than that. In this section, we will strengthen our assumptions and study asset prices in a full- edged economy. Among other things, this will allow us to provide some economic content to the vector Recall the de nition of Financial market equilibrium. Let IMRS i s(x i ) denote agent i s marginal rate of substitution between consumption of the numeraire good 1 in state s and consumption of the numeraire good 1 at date 0 IMRS i s(x i ) i (x i s i i (x i 0 i 10 Let IMRS i (x i ) = ( IMRSs(x i i ) ) 2 R S + denote the vector of intertemporal marginal rates of substitution for agent i, an S dimentional vector. Note that, under the assumption of strong monotonicity of preferences, IMRS i (x i ) 2 R S ++ By taking the First Order Conditions (necessary and su cient for a maximum under the assumption of strict quasi-concavity of preferences) with respect to zj i of the individual problem for an arbitrary price vector q, we obtain that SX q j = prob s IMRSs(x i i )a j s = E IMRS i (x i ) a j ; (3.5) s=1 for all j = 1; ; J and all i = 1; ; I; where of course the allocation x i is the equilibrium allocation. At equilibrium, therefore, the marginal cost of one more unit of asset j, q j, is equalized to the marginal valuation of that agent for the asset s payo, P S s=1 prob simrs i s(x i )a j s. Compare equation (3.5) to the previous equation (3.4). Clearly, at any equilibrium, condition (3.5) has to hold for each agent i. Therefore, in equilibrium, the vector of marginal rates of substitution of any arbitrary agent i can be used to price assets; that is any of the agents vector of marginal rates of substitution (normalized by probabilities) is a viable stochastic discount factor m In other words, any vector ( prob s IMRSs(x i i ) ) belongs to R(q) and is hence a viable for the asset pricing equation (??). But recall that

37 3.2. FINANCIAL MARKET ECONOMIES 31 R(q) is of dimension S J 0 ; where J 0 is the e ective dimension of the asset space. The higher the the e ective dimension of the asset space (intuitively said, the larger the set of nancial markets) the more aligned are agents marginal rates of substitution at equilibrium (intuitively said, the smaller are unexploited gains from trade at equilibrium). In the extreme case, when markets are complete (that is, when the rank of A is S) and the set R(q) is a singleton, IMRS i (x i ) are equalized across agents i at equilibrium IMRS i (x i ) = IMRS; for any i = 1; ; I Let MRSls i (xi ) denote agent i s marginal rate of substitution between consumption the good l and consumption of the numeraire good 1 in state s = 0; 1; ; S MRS i ls(x i ) i (x i i i (x i i 1s let also MRS i s(x i ) = ( MRS i ls (xi ) ) 2 R L + and MRS i (x i ) = ( MRS i s(x i ) ) 2 R LS + Problem 33 Write the Pareto problem for the economy and show that, at any Pareto optimal allocation, x; it is the case that IMRS i (x i ) = IMRS MRS i (x i ) = MRS for any i = 1; ; I Furthermore, show that an allocation x which satis es the feasibility conditions (market clearing) for goods and is such that IMRS i (x i ) = IMRS MRS i (x i ) = MRS for any i = 1; ; I, is a Pareto optimal allocation. We conclude that, when markets are Complete, equilibrium allocations are Pareto optimal. That is, the First Welfare theorem holds for Financial market equilibria when markets are Complete. Problem 34 (Economies with bid-ask spreads) Extend our basic two-period incomplete market economy by assuming that, given an exogenous vector 2 R J ++ the buying price of asset j is q j + j ;

38 32 CHAPTER 3. TWO-PERIOD ECONOMIES while the selling price of asset j is q j for any j = 1; ; J. Write the budget constraint and the First Order Conditions for an agent i s problem. Derive an asset pricing equation for q j in terms of intertemporal marginal rates of substitution at equilibrium Arrow theorem The Arrow theorem is the fundamental decentralization result in nancial economics. It states su cient conditions for a form of equivalence between the Arrow-Debreu and the Financial market equilibrium concepts. It was essentially introduced by Arrow (1952). The proof of the theorem introduces a reformulation of the budget constraints of the Financial market economy which focuses on feasible wealth transfers across states directly, that is, on the span of A < A >= 2 R S = Az; z 2 R J Such a reformulation is important not only in itself but as a lemma for welfare analysis in Financial market economies. Proposition 35 Let (x; z; p; q) 2 R ni ++R j R n ++R j + represent a Financial market equilibrium of an economy with rank(a) = S. Then (x; ) 2 R ni ++ R n ++ represents an Arrow-Debreu equilibrium if s = s p s ; for any s = 1; ; S and some 2 R++. S The converse also holds. Let (x; ) 2 R ni ++ R n ++ represent an Arrow-Debreu equilibrium. Then (x; z; p; q) 2 R ni ++R j R n ++ R j + represents a Financial market equilibrium of a complete market economy (that is, whose asset structure satis es rank(a) = S) if s = s p s ; for any s = 1; ; S; and some 2 R S ++ SX q = prob s IMRSs(x i i )a s s=1 Proof. =) Financial market equilibrium prices of assets q satisfy Noarbitrage. There exists then a vector ^ 2 R S+1 ++ such that ^W = 0; or q = A. The budget constraints in the nancial market economy are p 0 x i 0! i 0 + qz i = 0 p s x i s! s i = as z i ; for s = 1; S

39 3.2. FINANCIAL MARKET ECONOMIES 33 Substituting q = A; expanding the rst equation, and writing the constraints at time 1 in vector form, we obtain SX SX p 0 x i 0! i 0 + s a s z i = p 0 x i 0! i 0 + s p s x i s! s i = 0(3.6) s=1 p s (x i s! i s) 3 s= < A > (3.7) But if rank(a) = S; it follows that < A >= R S ; and the constraint 6 4 A > is never binding. Each agent i s problem is then subject only to SX p 0 x i 0! i 0 + s p s x i s! s i = 0; the budget constraint in the Arrow-Debreu economy with s=1 s = s p s ; for any s = 1; ; S (= The converse is straightforward. By No-arbitrage 2 p s (x i s! i s) < q = SX prob s IMRSs(x i i )a s s=1 and using s = prob s IMRSs(x i i ), for any s = 1; ; S; proves the result. (Recall that, with Complete markets IMRS i (x i ) = IMRS; for any i = 1; ; I.) Existence We do not discuss here in detail the issue of existence of a nancial market equilibrium when markets are incomplete (when they are complete, existence

40 34 CHAPTER 3. TWO-PERIOD ECONOMIES follows from the equivalence with Arrow-Debreu equilibrium provided by Arrow theorem). A sketch of the proof however follows. The proof is a modi cation of the existence proof for Arrow-Debreu equilibrium. By Arrow theorem, fact, we can reduce the equilibrium system to an excess demand system for consumption goods; that is, we can solve out for the asset portfolios z i s. An equilibrium will now be a zero of the excess demand function z F M R S ++ R (L 1)(S+1) ++! R L(S+1) 1 ++ X x i (; p)! i = z F M (; p) = 0 i2i Note that equations and unknowns match in Financial Market economies (after Arrow Theorem is applied to them), the prices (; p) are S + L(S + 1), but the normalizations (the budget constraints) are S + 1 and hence we get to L(S + 1) 1 unknowns for the same number of equations. Note also that the count applies to Arrow-Debreu economies, where the prices are L(S + 1) and they become L(S + 1) 1 after the normalization (1 single budget constraint). We can then apply to z F M (; p) = 0 the techniques used to prove existence for Arrow-Debreu economies. The only conceptual problem with the proof is then that the boundary condition on the excess demand system 2 might not be 3 guaranteed as each agent s excess demand is restricted by 6 p s (x i s! i s) 7 2< A > This is where the Cass trick comes 4 5 in handy. It is in fact an important Lemma. Cass trick. For any Financial 2market economy, 3 consider a modi ed economy where the constraint 6 p s (x i s! i s) 7 2< A > is imposed on all 4 5 agents i = 2; ; I but not on agent i = 1 Any equilibrium of the Financial Market economy is an equilibrium of the modi ed economy, and any equilibrium of the modi ed economy is a Financial market equilibrium.

41 3.2. FINANCIAL MARKET ECONOMIES 35 Proof. Consider an equilibrium of the modi ed economy in the statement. At equilibrium, P I i=1 p s (x i s! i s) = 0 Therefore, P I 2 3 i=2 p s (x i s! i s) = p s (x 1 s! 1 s) But 6 p s (x i s! i s) 7 2< A >; for any i = 2; ; I; and 4 5 hence P I i=2 p s (x i s! i s) 2< A > Since P I i=2 p s (x i s! i s) = p s (x 1 s! 1 s) ; it follows that p s (x 1 s! 1 s) 2< A >; and hence that p s (x 1 s! 1 s) 2< A > Therefore, the constraint p s (x 1 s! 1 s) 2< A > must necessarily hold at an equilibrium of the modi ed economy. In other words, the constraint p s (x 1 s! 1 s) 2< A > is not binding at a Financial market equilibrium. The equivalence between the modi ed economy and the Financial Market economy is now straightforward. In the modi ed economy, now, agent 1 faces complete markets without loss of generality. His excess demand, therefore, will satisfy the boundary conditions; these properties will transfer than to the aggregate excess demand and the existence proof will proceed exactly as in the standard Arrow-Debreu economy Constrained Pareto optimality Under Complete markets, the First Welfare Theorem holds for Financial market equilibrium. This is a direct implication of Arrow theorem. Proposition 36 Let (x; z; p; q) 2 R ni ++R j R n ++R j + be a Financial market equilibrium of an economy with Complete markets (with rank(a) = S) Then x 2 R ni ++ is a Pareto optimal allocation. However, under Incomplete markets (with rank(a) < S); Financial market equilibria are generically ine cient in a Pareto sense. That is, a planner could nd an allocation that improves some agents without making any other agent worse o. Note that of course a Pareto optimal allocation is a Financial Market equilibrium (with no trade), independently of the asset matrix A in the economy. As a consequence, it is immediate that, even with Incomplete markets, equilibria are a most generically (not always) Pareto ine cient. Theorem 37 Let (x; z; p; q) 2 R ni ++ R j R n ++ R j + be a Financial market

42 36 CHAPTER 3. TWO-PERIOD ECONOMIES equilibrium of an economy with Incomplete markets (with rank(a) < S) Then x 2 R ni ++ is generically not a Pareto optimal allocation. Proof. From the proof of Arrow theorem, we can write the budget constraints of the Financial market equilibrium as SX p 0 x i 0! i 0 + s p s x i s! s i = 0 (3.8) s= p s (x i s! i s) < A > (3.9) for some 2 R S ++ Pareto optimality of xrequires that there does not exist an allocation y such that 2 1 u(y i ) u(x i ) for any i = 1; ; I (strictly for at least one i), and IX y i! i s = 0, for any s = 0; 1; ; S i=1 Reproducing the proof of the 2 First Welfare 3theorem, it is clear that, if such a y exists, it must be that 6 p s (ys i! i s) 7 =2< A >; for some i = 1; ; I; 4 5 otherwise the allocation y would be budget feasible for all agent i at the equilibrium prices. Generic Pareto sub-optimality of x follows then directly from the following Lemma. Lemma 38 Let (x; z; p; q) 2 R ni ++ R j R n ++ R j + be a Financial market equilibrium of an2 economy with3rank(a) < S For a generic set of economies, the constraints 6 p s (x i s! i s) 7 2< A > are binding for some i = 1; ; I. 4 5

43 3.2. FINANCIAL MARKET ECONOMIES 37 Proof. We shall only sketch the proof here. Consider Financial market equilibria as the zeroes of the excess demand system for this economy, as de ned earlier in this section (but making explicit the dependence on endowments! 2 R ni ++) z F M (; p;!) = 0. Take any two distinct agents i and j and note that Pareto optimality requires that IMRS i (x i ;!) = IMRS j (x j ;!); where once again we make explicit the dependence on endowments! 2 R ni ++ Consider now the system h(; p;!) = z F M (; p;!) IMRS i (x i ;!) IMRS j (x j ;!) = 0 Because of the normalizations, the system maps R++ n 1 R ni ++ into R n ++ (recall that n = L(S + 1)). Suppose we could show that, at any (; p;!) 2 R n ++ 1 R ni ++ such that h(; p;!) = 0, D! h(; p;!) has rank n Then, the Transversality Theorem would immediately imply that h(; p;!) = 0 has generically no solutions in! 2 R ni ++ The proof that D! h(; p;!) has rank n at equilibrium can be found in Magill-Shafer, ch. 30 in W. Hildenbrand and H. Sonnenschein (eds.), Handbook of Mathematical Economics, Vol. IV, Elsevier, Pareto optimality might however represent too strict a de nition of social welfare of an economy with frictions which restrict the consumption set, as in the case of incomplete markets. In this case, markets are assumed incomplete exogenously. There is no reason in the fundamentals of the model why they should be, but they are. Under Pareto optimality, however, the social welfare notion does not face the same contraints. For this reason, we typically de ne a weaker notion of social welfare, Constrained Pareto optimality, by restricting the set of feasible allocations to satisfy the same set of constraints on the consumption set imposed on agents at equilibrium. In the case of incomplete markets, for instance, the feasible wealth vectors across states are restricted to lie in the span of the payo matrix. That can be interpreted as the economy s nancial technology and it seems reasonable to impose the same technological restrictions on the planner s reallocations. The formalization of an e ciency notion capturing this idea follows. Let x i t=1 = (x i s) S s=1 2 RSL ++; and similarly! i t=1 = (! i s) S s=1 2 RSL ++, p t=1 = (p s ) S s=1 2 RSL ++ Let g t=1 (! t=1 ; ); mapping R SL ++ R J into R SL ++, denote the equilibrium map for t = 1 spot markets at when each agent i = 1; ; I has endowment (! i s1+ a s i ;! i s2; ;! i sl ); for any s 2 S

44 38 CHAPTER 3. TWO-PERIOD ECONOMIES De nition 39 (Diamond, 1968; Geanakoplos-Polemarchakis, 1986) Let (x; z; p; q) 2 R ni ++ R j R n ++ R j + be an Arrow-Debreu equilibrium of an economy whose consumption set at time t = 1 is restricted by x i t=1 2 B(p t=1 ) R SL ++; for some set B(p t=1 ) and any i = 1; ; I In this economy, the allocation x is Constrained Pareto optimal if there does not exist a (y; ) 2 R ni ++ R j such that 2 and 1 u(y i ) u(x i ) for any i = 1; ; I, strictly for at least one i IX ys i! i s = 0, for any s = 0; 1; ; S i=1 3 y i t=1 2 B(g t=1 (! t=1 ; )); for any i = 1; ; I The constraint on the consumption set restricts only time 1 consumption allocations. More general constraints are possible but these formulation is consistent with the typical frictions we encounter in economics, e.g., on nancial markets. It is important that the constraint on the consumption set depends in general on g t=1 (! t=1 ; ), that is on equilibrium prices for spot markets opened at t = 1 after income transfers to agents. It implicit identi- es income transfers (besides consumption allocations at time t = 0) as the instrument available for Constrained Pareto optimality; that is, it implicitly constrains the planner implementing Constraint Pareto optimal allocations to interact with markets, speci cally to open spot markets after transfers. On the other hand, the planner is able to anticipate the spot price equilibrium map, g t=1 (! t=1 ; ); that is, to internalize the e ects of di erent transfers on spot prices at equilibrium. Consider rst a degenerate case Proposition 40 Let (x; z; p; q) 2 R ni ++ R j R n ++ R j + be a Financial market equilibrium of an Arrow-Debreu economy whose consumption set at time t = 1 is restricted by x i t=1 2 B R SL ++; for any i = 1; ; I In this economy, the allocation x is Constrained Pareto optimal.

45 3.2. FINANCIAL MARKET ECONOMIES 39 Crucially, markets are complete and B is independent of prices. The proof is then a straightforward extension of the First Welfare theorem combined with Arrow theorem. 6 Of course, using Arrow s theorem, this result implies the Constraint Pareto optimality of Financial market equilibrium allocations of economies with Complete markets as long as the constraint set B is exogenous. But note that we can apply the de nition of Constraint Pareto optimality also to Financial market equilibria with Incomplete markets. By Arrow s theorem Financial market economies with Incomplete markets are indeed Arrow-Debreu economies whose consumption set at time t = 1 is restricted by x i t=1 2 B(p t=1 ) R SL ++; for any i = 1; ; I; where B(p t=1 ) = x i t=1 2 R SL g t=1 (! t=1 ; ) x i t=1! t=1 i 2< A > ++ Consider a weaker parametrization of the economy rather than simply xing utility functions fu i g i2i and having economies parametrized by endowments! 2 R ni ++; we also parametrize utility functions by 2 R 2I, letting u i (x) = v i (x) + i 1x + i 2x 2 for some well-behaved v i (x) Proposition 41 Let (x; z; p; q) 2 R ni ++R j R n ++R j + be a Financial market equilibrium of an economy with Incomplete markets (with rank(a) < S) In this economy, the allocation x is, generically in (!; ) 2 R ni ++ R 2I, not Constrained Pareto optimal. 7 Proof. Note rst of all that, by construction, p s 2 g s (! s ; z) Following the proof of Pareto sub-otimality of Financial market equilibrium allocations, it then follows that if a Pareto-improving y exists, it must be that 6 To be careful, we need to guarantee that monotonicity of preferences on R SL ++ results in monotonicity on B R SL ++ This is the case e.g., if is a subspace in R SL ++ or in any case if it is an open set 7 The genericity result is then weaker in this theorem than we are used to in the previous sections. We ll get back to this later, but we anticipate here that the parametrization of the utility functions is necessary to produce perturbatiopns away from homethtic utility functions, which have the property that spot prices are independent of the distribution of income across states.

46 40 CHAPTER 3. TWO-PERIOD ECONOMIES p s (ys i! i s) 3 7 =2< A >; for some i = 1; ; I; while g s (! s ; ) (ys i! i s) A i, for all i = 1; ; I Generic Constrained Pareto sub-optimality of x follows then directly from the following Lemma, which we leave without proof = Lemma 42 Let (x; z; p; q) 2 R ni ++ R j R n ++ R j + be a Financial market equilibrium of an economy with Incomplete markets (with rank(a) < 2 S) For a generic set of 3 economies (!; ) 2 R ni ++ R 2I, the constraints 6 g s (! s ; z + dz) (ys i! i s) = A (zi + dz i ), for some dz 2 R JI n f0g such that X i2i one. 9 dz i = 0; are weakly relaxed for all i = 1; ; I, strictly for at least There is a fundamental di erence between incomplete market economies, which have typically not Constrained Optimal equilibrium allocations, and economies with constraints on the consumption set, which have, on the contrary, Constrained Optimal equilibrium allocations. It stands out by comparing the respective trading constraints g s (! s ; )(x i s! i s) = A s i ; for all i and s, vs. x i t=1 2 B, for all i The trading constraint of the Incomplete market economy is determined at equilibrium, while the constraint on the consumption set is exogenous. Another way to re-phrase the same point is the following. A planner choosing (y; ) will take into account that at each (y; ) is typically associated a different trading constraint g s (! s ; )(x i s! i s) = A s i ; for all i and s; while any 8 The proof is due to Geanakoplos-Polemarchakis (1986). It also requires di erential topology techniques. 9 The Lemma implies that a Pareto improving allocation can be found locally around the equilibrium, as a perturbation of the equilibrium.

47 3.2. FINANCIAL MARKET ECONOMIES 41 agent i will choose (x i ; z i ) to satisfy p s (x i s! i s) = A s z i ; for all s, taking as given the equilibrium prices p s The constrained ine ciency due the dependence of constraints on equilibrium prices is sometimes called a pecuniary externality. 10 Several examples of such form of externality/ine ciency have been developed recently in macroeconomics. Some examples are - Thomas, Charles (1995) "The role of scal policy in an incomplete markets framework," Review of Economic Studies, 62, Krishnamurthy, Arvind (2003) "Collateral Constraints and the Ampli - cation Mechanism," Journal of Economic Theory, 111(2), Caballero, Ricardo J. and Arvind Krishnamurthy (2003) "Excessive Dollar Debt Financial Development and Underinsurance," Journal of Finance, 58(2), Lorenzoni, Guido (2008) "Ine cient Credit Booms," Review of Economic Studies, 75 (3), Kocherlakota, Narayana (2009) "Bursting Bubbles Consequences and Causes," http// - Davila, Julio, Jay Hong, Per Krusell, and Victor Rios Rull (2005) "Constrained E ciency in the Neoclassical Growth Model with Uninsurable Idiosyncractic Shocks," mimeo, University of Pennsylvania. Remark 43 Consider an economy whose constraints on the consumption set depend on the equilibrium allocation x i t=1 2 B(x t=1 ; z ); for any i = 1; ; I This is essentially an externality in the consumption set. It is not hard to extend the analysis of this section to show that this formulation introduces ine ciencies and equilibrium allocations are Constraint Pareto sub-optimal. 10 The name is due to Joe Stiglitz (or is it Greenwald-Stiglitz?).

48 42 CHAPTER 3. TWO-PERIOD ECONOMIES Corollary 44 Let (x; z; p; q) represent a Financial market equilibrium of a 1-good economy (L = 1) with Incomplete markets (rank(a) < S). In this economy, the allocation x is Constrained Pareto optimal. Proof. The constraint on the consumption set implied by incomplete markets, if L = 1, can be written (x i s! i s) = A s z i It is independent of prices, of the form x i t=1 2 B. Remark 45 Consider an alternative de nition of Constrained Pareto optimality, due to Grossman (1970), in which constraints 3 are substituted by p s (x i s! i s) = Azi ; for any i = 1; ; I where p is the spot market Financial market equilibrium vector of prices. That is, the planner takes the equilibrium prices as given. It is immediate to prove that, with this de nition of Constrained Pareto optimality, any Financial market equilibrium allocation x of an economy with Incomplete markets is in fact Constrained Pareto optimal, independently of the nancial markets available (rank(a) S) Problem 46 Consider a Complete market economy (rank(a) = S) whose feasible set of asset portfolios is restricted by A typical example is borrowing limits z i 2 Z ( R J ; for any i = 1; ; I z i b; for any i = 1; ; I Are equilibrium allocations of such an economy Constrained Pareto optimal (also if L > 1)? Problem 47 Consider a 1-good (L = 1) Incomplete market economy (rank(a) < S) which lasts 3 periods. De ne an Financial market equilibrium for this economy as well as Constrained Pareto optimality. Are Financial market equilibrium allocations of such an economy Constrained Pareto optimal?

49 3.2. FINANCIAL MARKET ECONOMIES 43 Problem 48 Extend our basic two-period complete market economy by assuming that, given an exogenous vector 2 R J ++ while the buying price of asset j is q j + j the selling price of asset j is q j for any j = 1; ; J. 1) Suppose the asset payo matrix is full rank. Do you expect nancial market equilibria to be Pareto e cient? Carefully justify your answer. (I am not asking for a formal proof, though you could actually prove this.) How would you de ne Constrained Pareto e ciency in this economy? Do you expect nancial market equilibria to be Constrained Pareto e cient? Carefully justify your answer. (I am really not asking for a formal proof.) Aggregation Agent i s optimization problem in the de nition of Financial market equilibrium requires two types of simultaneous decisions. On the one hand, the agent has to deal with the usual consumption decisions i.e., she has to decide how many units of each good to consume in each state. But she also has to make nancial decisions aimed at transferring wealth from one state to the other. In general, both individual decisions are interrelated the consumption and portfolio allocations of all agents i and the equilibrium prices for goods and assets are all determined simultaneously. The nancial and the real sectors of the economy cannot be isolated. Under some special conditions, however, the consumption and portfolio decisions of agents can be separated. This is typically very useful when the analysis is centered on nancial issue. In order to concentrate on asset pricing issues, most nance models deal in fact with 1-good economies, implicitly assuming that the individual nancial decisions and the market clearing conditions in the assets markets determine the nancial equilibrium, independently of the individual consumption decisions and market clearing in the goods markets; that is independently of the real equilibrium prices and allocations. In this section we shall identify the conditions under which this can be done without loss of generality. This is sometimes called "the problem of aggregation." The idea is the following. If we want equilibrium prices on the spot markets to be independent of equilibrium on the nancial markets, then the aggregate spot market demand for the L goods in each state s should

50 44 CHAPTER 3. TWO-PERIOD ECONOMIES must depend only on the incomes of the agents in this state (and not in other states) and should be independent of the distribution of income among agents in this state. Theorem 49 Budget Separation. Suppose that each agent i s preferences are separable across states, identical, homothetic within states, and von Neumann-Morgenstern; i.e. suppose that there exists an homothetic u R L! R such that SX u i (x i ) = u(x i 0) + prob s u(x i s); for all i = 1; ; I s=1 Then equilibrium spot prices p are independent n of asset prices q and of the income distribution; that is, constant in! i 2 R L(S+1) ++ P o I i=1!i given Proof. Normalize all spot prices of good 1 p 10 = p 1s = 1; for any s 2 S The consumer s maximization problem in the de nition of Financial market equilibrium can be decomposed into a sequence of spot commodity allocation problems and an income allocation problem as follows. The spot commodity allocation problems. Given the current and anticipated spot prices p = (p 0 ; p 1 ; ; p S ) and an exogenously given stream of nancial income y i = (y0; i y1; i ; ys i ) 2 RS+1 ++ in units of numeraire, agent i has to pick a consumption vector x i 2 R L(S+1) + to max u i (x i ) s.t. p 0 x i 0 = y i 0 p s x i s = y i s; for s = 1; S Let the L(S + 1) demand functions be given by x i ls (p; yi ), for l = 1; ; L; s = 0; 1; S. De ne now the indirect utility function for income by v i (y i ; p) = u i (x i (p; y i )) The Income allocation problem. Given prices (p; q); endowments! i, and the asset structure A, agent i has to pick a portfolio z i 2 R J and an income stream y i 2 R S+1 ++ to max v i (y i ; p) st p 0! i 0 qz i = y i 0 p s! i s + a s z i = y i s; for s = 1; S

51 3.2. FINANCIAL MARKET ECONOMIES 45 By additive separability across states of the utility, we can break the consumption allocation problem into S +1 spot market problems, each of which yields the demands x i s(p s ; ys) i for each state. By homotheticity, for each s = 0; 1; S; and by identical preferences across all agents, x i s(p s ; y i s) = y i sx i s(p s ; 1); and since preferences are identical across agents, y i sx i s(p s ; 1) = y i sx s (p s ; 1) Adding over all agents and using the market clearing condition in spot markets s, we obtain, at spot markets equilibrium, x s (p s ; 1) Again by homothetic utility, IX i=1 y i s IX x s (p s ; ys) i i=1 IX! i s = 0 i=1 IX! i s = 0 (3.10) i=1 Recall from the consumption allocation problem that p s x i s = y i s; for s = 0; 1; S By adding over all agents, and using market clearing in the spot markets in state s, IX IX ys i = p s x i s; for s = 0; 1; S (3.11) i=1 i=1 IX = p s! i s; for s = 0; 1; S i=1 By combining (3.10) and (3.11), we obtain x s (p s ; p s IX! i s) = i=1 IX! i s (3.12) Note how we have passed from the aggregate demand of all agents in the economy to the demand of an agent owning the aggregate endowments. Observe also how equation (3.12) is a system of L equations with L unknowns i=1

52 46 CHAPTER 3. TWO-PERIOD ECONOMIES that determines spot prices p s for each state s independently of asset prices q Note also that equilibrium spot prices p s de ned by (3.12) only depend! i through P I i=1!i s The Budget separation theorem can be interpreted as identifying conditions under which studying a single good economy is without loss of generality. To this end, consider the income allocation problem of agent i, given equilibrium spot prices p max y i 2R S+1 ++ ;zi 2R J v i (y i ; p) s.t. y i 0 = p 0! i 0 qz i y i s = p s! i s + a s z i ; for s = 1; S If preferences u i (x i ) are identical, homothetic within states, and von Neumann- Morgenstern, that is, if they satisfy u i (x i ) = u(x i 0) + SX prob s u(x i s); with u(x) homothetic, for all i = 1; ; I s=1 it is straightforward to show that indirect preferences v i (y i ; p ) are also identical, and von Neumann-Morgenstern v i (y i ; p) = v(y i 0; p) + SX prob s v(ys; i p) Note that homotheticity in (y0; i y1; i ; ys i ) is guaranteed by the von Neumann- Morgenstern property. Let w0 i = p 0! i 0; ws i = p s! i s; for any s = 1; ; S; and disregard for notational simplicity the dependence of v(y; p) on p The income allocation problem can be written as s.t. y i 0 w i 0 = qz i s=1 max y i 2R S+1 ++ ;zi 2R J v(y i 0) + y i s w i s = A s z i ; for s = 1; S SX prob s v(ys) i which is homeomorphic to any agent i s optimization problem in the de - nition of Financial market equilibrium with l = 1. Note that y i s gains the interpretation of agent i s consumption expenditure in state s, while w i s is interpreted as agent i s income endowment in state s s=1

53 3.2. FINANCIAL MARKET ECONOMIES 47 The representative agent theorem A representative agent is the following theoretical construct. De nition 50 Consider a Financial market equilibrium (x; z; p; q) of an economy populated by i = 1; ; I agents with preferences u i X! R and endowments! i A Representative agent for this economy is an agent with preferences U R X! R and endowment! R such that the Financial market equilibrium of an associated economy with the Representative agent as the only agent has prices (p; q). In this section we shall identify assumptions which guarantee that the Representative agent construct can be invoked without loss of generality. This assumptions are behind much of the empirical macro/ nance literature. Theorem 51 Representative agent. Suppose preferences satisfy u i (x i ) = u(x i 0) + SX prob s u(x i s); with homothetic u(x); for all i = 1; ; I s=1 Let p denote equilibrium spot prices. If a map u R R S+1 +! R such that! R = IX! i s; i=1 U R (x) = u R (y 0 ) p s! i s 3 SX prob s u R (y s ); where y s = p s=1 constitutes a Representative agent. 7 2< A >; then there exist 5 IX x i s; s = 0; 1; ; S Since the Representative agent is the only agent in the economy, her consumption allocation and portfolio at equilibrium, x R ; z R ; are i=1 x R =! R = z R = 0 IX i=1! i

54 48 CHAPTER 3. TWO-PERIOD ECONOMIES If the Representative agent s preferences can be constructed independently of the equilibrium of the original economy with I agents, then equilibrium prices can be read out of the Representative agent s marginal rates of substitution evaluated at P I i=1!i. Since P I i=1!i is exogenously given, equilibrium prices are obtained without computing the consumption allocation and portfolio for all agents at equilibrium, (x ; z ) Proof. The proof is constructive. Under the assumptions on preferences in the statement, we need to shown that, for all agents i = 1; ; I, equilibrium asset prices q are constant in! i 2 R L(S+1) ++ P o I i=1!i given.if preferences satisfy u i (x i ) = u(x i 0) + P S s=1 prob su(x i s); for all i = 1; ; I, with an homothetic u(x); then by the Budget separation n theorem, equilibrium spot prices p are independent of q and constant in! i 2 R L(S+1) ++ P o I i=1!i given There fore, 6 4 p s! i s 7 2< A > can be written as an assumption on fundamentals; 5 in particular on! i Furthermore, we can restrict our analysis to the single good economy, whose agent i s optimization problem is s.t. y i 0 w 0 = qz i max y i 2R S+1 ++ ;zi 2R J v(y i 0) + y i s w s = A s z i ; for s = 1; S SX prob s v(ys) i where v(y) is homothetic. We show next that u R (y) = v(y) and! R = P I i=1!i s constitute a Representative agent. By Arrow theorem we can write budget constraints as y i 0 w i 0 + s=1 SX s ys i ws i = 0 s= y i s w i s < A >

55 3.2. FINANCIAL MARKET ECONOMIES 49 2 But, 6 4 w i s < A > implies that there exist a zi w such that w i s = Azi w Therefore, 6 ws i < A > implies that yi s = A s (z i + zw), i for any s 2 S We can then write each agent i s optimization problem in terms of (y0; i z i ); and the value of agent i 0 s endowment is w0 i + P S s=1 sws i = w0 i + P S s=1 sa s zw i = w0 i + qzw i The consumer i s problem becomes y i 0 + max y i 2R S+1 + st SX s ys i = W i s= y i s < A > v(y i 0) + SX prob s v(ys); i s=1 where W i = w i 0 + qz i w Note that the solution is homogeneous of degree 1 y i (q; W i ) = y i (q; W i ) Hence By the fact that preferences are identical across agents and by homotheticity of v(y); then we can write y i 0 q; W i = w i 0 + qz i w y0 (q; 1) y i s q; W i = w i 0 + qz i w ys (q; 1) ; for any s 2 S

56 50 CHAPTER 3. TWO-PERIOD ECONOMIES At equilibrium then y 0 (q; 1) X i2i w0 i + qzw i = y0 q; X i2i! w0 i + qzw i = X i2i w i 0 y s (q; 1) X i2i w0 i + qzw i = ys q; X i2i and prices q only depend on P I i=1 wi 0 and P I i=1 zi w 2 6 4! X w0 i + qzw i = A s zw; i for any s 2 S Make sure you understand where we used the assumption w i s 3 i2i p! i s = 7 2< A >. Convince yourself that the assumption is necessary in the 5 proof. The Representative agent theorem, as noted, allows us to obtain equilibrium prices without computing the consumption allocation and portfolio for all agents at equilibrium, (x; z)let w = P I i=1 wi Under the assumptions of the Representative agent theorem, let w 0 = P i2i wi 0; and w s = P i2i wi s; for any s 2 S Then q = SX prob s MRS s (w)a s ; for MRS s (w) = 0 0 That is, asset prices can be computed from agents preferences u R = v R! R and from the aggregate endowment (w 0 ; ; w s ; ) This is called the Lucas trick for pricing assets. Problem 52 Note that, under 2 the3 Complete markets assumption, the span restriction on endowments, 6 p! i s 7 2< A >; for all agents i; is trivially satis ed. Does this assumption imply Pareto optimal allocations in 4 5 equilibrium?

57 3.2. FINANCIAL MARKET ECONOMIES 51 Problem 53 Assume all agents have identical quadratic 2 3preferences. Derive individual demands for assets (without assuming 6 p! i s 7 2< A >) and show 4 5 that the Representative agent theorem is obtained. Another interesting but misleading result is the "weak" representative agent theorem, due to Constantinides (1982). Theorem 54 Suppose markets are complete (rank(a) = S) and preferences u i (x i ) are von Neumann-Morgernstern (but not necessarily identical nor homothetic). Let (x; z; p; q) be a Financial markets equilibrium. Then,! R = IX! i ; i=1 U R (x) = max (x i ) I i=1 IX i u i (x i ) s.t. i=1 where i = ( i ) 1 and i (x i i 10 constitutes a Representative agent. Clearly, then, q = IX x i = x; i=1 SX prob s MRS s (w)a s ; for MRS s (w) = R (w R (w 0 0 Proof. Consider a Financial market equilibrium (x; z; p; q). By complete markets, the First welfare theorem holds and x is a Pareto optimal allocation. Therefore, there exist some weights that make x the solution to the planner s problem. It turns out that the required weights are given i i (x i 1 ) i 10 This is left to the reader to check; it s part of the celebrated Negishi theorem.

58 52 CHAPTER 3. TWO-PERIOD ECONOMIES This result is certainly very general, as it does not impose identical homothetic preferences, however, it is not as useful as the real Representative agent theorem to nd equilibrium asset prices. The reason is that to de ne the speci c weights for the planner s objective function, ( i ) I i=1; we need to know what the equilibrium allocation, x; which in turn depends on the whole distribution of endowments over the agents in the economy Asset pricing Relying on the aggregation theorem in the previous section, in this section we will abstract from the consumption allocation problems and concentrate on one-good economies. This allows us to simplify the equilibrium de nition as follows Some classic representation of asset pricing Often in nance, especially in empirical nance, we study asset pricing representation which express asset returns in terms of risk factors. Factors are to be interpreted as those component of the risks that agents do require a higher return to hold. How do we go from our basic asset pricing equation to factors? Single factor beta representation q = E(mA) Consider the basic asset pricing equation for asset j; q j = E(ma j ) Let the return on asset j, R j, be de ned as R j = A j q j. Then the asset pricing equation becomes 1 = E(mR j ) This equation applied to the risk free rate, R f, becomes R f = 1. Using the Em fact that for two random variables x and y, E(xy) = ExEy + cov(x; y), we can rewrite the asset pricing equation as ER j = 1 Em cov(m; R j ) Em = R f cov(m; R j ) Em

59 3.2. FINANCIAL MARKET ECONOMIES 53 or, expressed in terms of excess return Finally, letting and ER j R f = cov(m; R j) Em j = cov(m; R j) var(m) = var(m) Em we have the beta representation of asset prices ER j = R f + j m (3.13) We interpret j as the "quantity" of risk in asset j and m (which is the same for all assets j) as the "price" of risk. Then the expected return of an asset j is equal to the risk free rate plus the correction for risk, j m. Furthermore, we can read (3.13) as a single factor representation for asset prices, where the factor is m, that is, if the representative agent theorem holds, her intertemporal marginal rate of substitution. Multi-factor beta representations A multi-factor beta representation for asset returns has the following form ER j = R f + FX jf mf (3.14) where (m f ) F f=1 are orthogonal random variables which take the interpretation of risk factors and jf = cov(m f; R j ) var(m f ) is the beta of factor f, the loading of the return on the factor f. f=1 Proposition 55 A single factor beta representation ER j = R f + j m

60 54 CHAPTER 3. TWO-PERIOD ECONOMIES is equivalent to a multi-factor beta representation FX ER j = R f + jf mf with m = f=1 FX f=1 b f m f In other words, a multi-factor beta representation for asset returns is consistent with our basic asset pricing equation when associated to a linear statistical model for the stochastic discount factor m, in the form of m = P F f=1 b fm f. Proof. Write 1 = E(mR j ) as R j = R f cov(m;r j ) Em m = P F f=1 b fm f and the de nitions of jf, to have and then to substitute mf = var(m f)b f Em f The CAPM The CAPM is nothing else than a single factor beta representation of the following form where ER j = R f + jf mf m f = a + br w the return on the market portfolio, the aggregate portfolio held by the investors in the economy. It can be easily derived from an equilibrium model under special assumptions. For example, assume preferences are quadratic u(x i o; x i 1) = 1 2 (xi x # ) 2 1 SX 2 prob s (x i s x # ) 2 Moreover, assume agents have no endowments at time t = 1. Let P I x s ; s = 0; 1; ; S; and P I i=1 wi 0 = w 0. Then budget constraints include s=1 x s = R w s (w 0 x 0 ) i=1 xi s =

61 3.2. FINANCIAL MARKET ECONOMIES 55 Then, m s = x s x # x 0 x # = (w 0 x 0 ) (x 0 x # ) Rw s x # x 0 x # x which is the CAPM for a = # x 0 and b = (w 0 x 0 ) x # (x 0 x # ) Note however that a = x# x 0 and b = (w 0 x 0 ) x # (x 0 are not constant, as they x # ) do depend on equilibrium allocations. This will be important when we study conditional asset market representations, as it implies that the CAPM is intrinsically a conditional model of asset prices. Bounds on stochastic discount factors Write the beta representation of asset returns as ERj R f = cov(m; Rj) Em = (m; Rj)(m)(Rj) Em where 0 j(m; Rj)j 1 denotes the correlation coe cient and (m), the standard deviation. Then j ERj (Rj) Rf j (m) Em The left-hand-side is the Sharpe-ratio of asset j. The relationship implies a lower bound on the standard deviation of any stochastic discount factor m which prices asset j. Hansen-Jagannathan are responsible for having derived bounds like these and shown that, when the stochastic discount factor is assumed to be the intertemporal marginal rate of substitution of the representative agent (with CES preferences), the data does not display enough variation in m to satisfy the relationship. A related bound is derived by noticing that no-arbitrage implies the existence of a unique stochastic discount factor in the space of asset payo s, denoted m p, with the property that any other stochastic discount factor m satis es m = m p + where is orthogonal to m p. The following corollary of the No-arbitrage theorem leads us to this result.

62 56 CHAPTER 3. TWO-PERIOD ECONOMIES Corollary 56 Let (A; q) satisfy No-arbitrage. Then, there exists a unique 2< A > such that q = A Proof. By the No-arbitrage theorem, there exists 2 R S ++ such that q = A We need to distinguish notationally a matrix M from its transpose, M T We write then the asset prices equation as q T = A T T. Consider p T p = A(A T A) 1 q Clearly, q T = A T T p ; that is, T p satis es the asset pricing equation. Furthermore, such T p belongs to < A >, since T p = Az p for z p = (A T A) 1 q Prove uniqueness. We can now exploit this uniqueness result to yield a characterization of the multiplicity of stochastic discount factors when markets are incomplete, and consequently a bound on (m). In particular, we show that, for a given (q; A) pair a vector m is a stochastic discount factor if and only if it can be decomposed as a projection on < A > and a vector-speci c component orthogonal to < A >. Moreover, the previous corollary states that such a projection is unique. Let m 2 R++ S be any stochastic discount factor, that is, for any s = 1; ; S, m s = s prob s and q j = E(mA j ); for j = 1; ; J Consider the orthogonal projection of m onto < A >, and denote it by m p. We can then write any stochastic discount factors m as m = m p + ", where " is orthogonal to any vector in < A >; in particular to any A j. Observe in fact that m p + " is also a stochastic discount factors since q j = E((m p +")a j ) = E(m p a j )+E("a j ) = E(m p a j ), by de nition of ". Now, observe that q j = E(m p a j ) and that we just proved the uniqueness of the stochastic discount factors lying in < A > In words, even though there is a multiplicity of stochastic discount factors, they all share the same projection on < A >. Moreover, if we make the economic interpretation that the components of the stochastic discount factors vector are marginal rates of substitution of agents in the economy, we can interpret m p to be the economy s aggregate risk and each agents " to be the individual s unhedgeable risk. It is clear then that (m) (m p ) the bound on (m) we set out to nd.

63 3.2. FINANCIAL MARKET ECONOMIES Production Assume for simplicity that L = 1, and that there is a single type of rm in the economy which produces the good at date 1 using as only input the amount k of the commodity invested in capital at time 0 11 The output depends on k according to the function f(k; s), de ned for k 2 K, where s is the state realized at t = 1. We assume that - f(k; s) is continuously di erentiable, increasing and concave in k; - ; K are closed, compact subsets of R + and 0 2 K. In addition to rms, there are I types of consumers. The demand side of the economy is as in the previous section, except that each agent i 2 I is also endowed with i 0 units of stock of the representative rm. Consumer i has von Neumann-Morgernstern preferences over consumption in the two dates, represented by u i (x i 0) + Eu i (x i ), where u i () is continuously di erentiable, strictly increasing and strictly concave. Competitive equilibrium Let the outstanding amount of equity be normalized to 1 the initial distribution of equity among consumers satis es P i i 0 = 1. The problem of the rm consists in the choice of its production plan k. Firms are perfectly competitive and hence take prices as given. The rm s cash ow, f(k; s); varies with k. Thus equity is a di erent product for di erent choices of the rm. What should be its price when all this continuum of di erent products are not actually traded in the market? In this case the price is only a conjecture. It can be described by a map Q(k) specifying the market valuation of the rm s cash ow for any possible value of its choice k. 12 The rm chooses its production plan k so as to maximize its value. The rm s problem is then max k k + Q(k) (3.15) 11 It should be clear from the analysis which follows that our results hold unaltered if the rms technology were described, more generally, by a production possibility set Y R S These price maps are also called price perceptions.

64 58 CHAPTER 3. TWO-PERIOD ECONOMIES When nancial markets are complete, the present discounted valuation of any future payo is uniquely determined by the price of the existing assets. This is no longer true when markets are incomplete, in which case the prices of the existing assets do not allow to determine unambiguously the value of any future cash ow. The speci cation of the price conjecture is thus more problematic in such case. Let k denote the solution to this problem. At t = 0, each consumer i chooses his portfolio of nancial assets and of equity, z i and i respectively, so as to maximize his utility, taking as given the price of assets, q and the price of equity Q. In the present environment a consumer s long position in equity identi es a rm s equity holder, who may have a voice in the rm s decisions. It should then be treated as conceptually di erent from a short position in equity, which is not simply a negative holding of equity. To begin with, we rule out altogether the possibility of short sales and assume that agents can not short-sell the rm equity The problem of agent i is then i 0; 8i (3.16) max ui x i x i 0 ;xi ;z i ; i 0 + Eu i x i (3.17) subject to (3.16) and x i 0 =! i 0 + [ k + Q] i 0 Q i q z i (3.18) x i (s) =! i (s) + f(k; s) i + A(s)z i ; 8s 2 S (3.19) Let x i 0; x i ; z i ; i denote the solution to this problem. In equilibrium, the following market clearing conditions must hold, for the consumption good 13 X x i 0 + k X! i 0 i i X x i (s) X! i (s) + f(k; s); 8s 2 S i i 13 We state here the conditions for the case of symmetric equilibria, where all rms take the same production and nancing decision, so that only one type of equity is available for trade to consumers. They can however be easily extended to the case of asymmetric equilibria.

65 3.2. FINANCIAL MARKET ECONOMIES 59 or, equivalently, for the assets X z i = 0 (3.20) i X i = 1 (3.21) i In addition, the equity price map faced by rms must satisfy the following consistency condition i) Q(k ) = Q; This condition requires that, at equilibrium, the price of equity conjectured by rms coincides with the price of equity, faced by consumers in the market rms conjectures are correct in equilibrium. We also restrict out of equilibrium conjectures by rms, requiring that they satisfy ii) Q(k) = max i E [MRS i f(k)], 8k, where MRS i denotes the marginal rate of substitution between consumption at date 0 and at date 1 in state s for consumer i; evaluated at his equilibrium consumption allocation (x i 0; x i ). Condition ii) says that for any k (not just at equilibrium!) the value of the equity price map Q(k) equals the highest marginal valuation - across all consumers in the economy - of the cash ow associated to k. The consumers marginal rates of substitutions MRS i (s) used to determine the market valuation of the future cash ow of a rm are taken as given, una ected by the rm s choice of k. This is the sense in which, in our economy, rms are competitive each rm is small relative to the mass of consumers and each consumers holds a negligible amount of shares of the rm. To better understand the meaning of condition ii), note that the consumers with the highest marginal valuation for the rm s cash ow when the rm chooses k are those willing to pay the most for the rm s equity in that case and the only ones willing to buy equity - at the margin - when its price satis es ii). Given i) such property is clearly satis ed for the rms equilibrium choice k. Condition ii) requires that the same is true for any other possible choice k the value attributed to equity equals the maximum any consumer is willing to pay for it. Note that this would be the equilibrium

66 60 CHAPTER 3. TWO-PERIOD ECONOMIES price of equity of a rm who were to deviate from the equilibrium choice and choose k instead the supply of equity with cash ow corresponding to k is negligible and, at such price, so is its demand. In this sense, we can say that condition ii) imposes a consistency condition on the out of equilibrium values of the equity price map; that is, it corresponds to a "re nement" of the equilibrium map, somewhat analogous to bacward induction. Equivalently, when price conjectures satisfy this condition, the model is equivalent to one where markets for all the possible types of equity (that is, equity of rms with all possible values of k) are open, available for trade to consumers and, in equilibrium all such markets - except the one corresponding to the equilibrium k - clear at zero trade. 14 It readily follows from the consumers rst order conditions that in equilibrium the price of equity and of the nancial assets satisfy Q = max i E MRS i f(k) (3.22) q = E MRS i A The de nition of competitive equilibrium is stated for simplicity for the case of symmetric equilibria, where all rms choose the same production plan. When the equity price map satis es the consistency conditions i) and ii) the rms choice problem is not convex. Asymmetric equilibria might therefore exist, in which di erent rms choose di erent production plans. The proof of existence of equilibria indeed requires that we allow for such asymmetric equilibria, so as to exploit the presence of a continuum of rms of the same type to convexify rms choice problem. A standard argument allows then to show that rms aggregate supply is convex valued and hence that the existence of (possibly asymmetric) competitive equilibria holds. Proposition 57 A competitive equilibrium always exist. Objective function of the rm Starting with the initial contributions of Diamond (1967), Dreze (1974), Grossman-Hart (1979), and Du e-shafer (1986), a large literature has dealt 14 An analogous speci cation of the price conjecture has been earlier considered by Makowski (1980) and Makowski-Ostroy (1987) in a competitive equilibrium model with di erentiated products, and by Allen-Gale (1991) and Pesendorfer (1995) in models of nancial innovation.

67 3.2. FINANCIAL MARKET ECONOMIES 61 with the question of what is the appropriate objective function of the rm when markets are incomplete.the issue arises because, as mentioned above, rms production decisions may a ect the set of insurance possibilities available to consumers by trading in the asset markets. If agents are allowed in nite short sales of the equity of rms, as in the standard incomplete market model, a small rm will possibly have a large e ect on the economy by choosing a production plan with cash ows which, when traded as equity, change the asset span. It is clear that the price taking assumption appears hard to justify in this context, since changes in the rm s production plan have non-negligible e ects on allocations and hence equilibrium prices. The incomplete market literature has struggled with this issue, trying to maintain a competitive equilibrium notion in an economic environment in which rms are potentially large. In the environment considered in these notes, this problem is avoided by assuming that consumers face a constraint preventing short sales, (3.16), which guarantees that each rm s production plan has instead a negligible (in nitesimal) e ect on the set of admissible trades and allocations available to consumers. Evidently, for price taking behavior to be justi ed a no short sale constraint is more restrictive than necessary and a bound on short sales of equity would su ce; see Bisin-Gottardi-Ruta (2009). When short sales are not allowed, the decisions of a rm have a negligible e ect on equilibrium allocations and market prices. However, each rm s decision has a non-negligible impact on its present and future cash ows. Price taking can not therefore mean that the price of its equity is taken as given by a rm, independently of its decisions. However, as argued in the previous section, the level of the equity price associated to out-of-equilibrium values of k is not observed in the market. It is rather conjectured by the rm. In a competitive environment we require such conjecture to be consistent, as required by condition ii) in the previous section. This notion of consistency of conjectures implicitly requires that they be competitive, that is, determined by a given pricing kernel, independent of the rm s decisions. 15 But which pricing kernel? Here lies the core of the problem with the de nition of the objective function of the rm when markets are incomplete. When markets are incomplete, in fact, the marginal valuation of out-of-equilibrium production plans di ers across di erent agents at equilibrium. In other words, equity 15 Independence of the kernel is guarantee by the fact that MRS i (s); for any i, is evaluated at equilibrium.

68 62 CHAPTER 3. TWO-PERIOD ECONOMIES holders are not unanimous with respect to their preferred production plan for the rm. The problem with the de nition of the objective function of the rm when markets are incomplete is therefore the problem of aggregating equity holders marginal valuations for out-of-equilibrium production plans. The di erent equilibrium notions we nd in the literature di er primarily in the speci cation of a consistency condition on Q (k), the price map which the rms adopts to aggregate across agents marginal valuations. 16 Consider for example the consistency condition proposed by Dreze (1974) " # X Q D (k) = E i MRS i f(k) ; 8k (3.23) i Such condition requires the price conjecture for any plan k to equal the pro rata marginal valuation of the agents who at equilibrium are the rm s equity holders (that is, the agents who value the most the plan chosen by rms in equilibrium). It does not however require that the rm s equity holders are those who value the most any possible plan of the rm, without contemplating the possibility of selling the rm in the market, to allow the new equity buyers to operate the production plan they prefer. Equivalently, the value of equity for out of equilibrium production plans is determined using the - possibly incorrect - conjecture that the rms equilibrium shareholders will still own the rm out of equilibrium. Grossman-Hart (1979) propose another consistency condition and hence a di erent equilibrium notion. In their case " # X Q GH (k) = E i 0MRS i f(k) ; 8k i We can interpret such notion as describing a situation where the rm s plan is chosen by the initial equity holders (i.e., those with some predetermined stock holdings at time 0) so as to maximize their welfare, again without contemplating the possibility of selling the equity to other consumers who value it more. Equivalently, the value of equity for out of equilibrium production plans is again derived using the conjecture belief that rms initial shareholders stay in control of the rm out of equilibrium. 16 A minimal consistency condition on Q (k) is clearly given by i) in the previous section, which only requires the conjecture to be correct in correspondence to the rm s equilibrium choice. Du e-shafer (1986) indeed only impose such condition and nd a rather large indeterminacy of the set of competitive equilibria.

69 3.2. FINANCIAL MARKET ECONOMIES 63 Unanimity Under the de nition of equilibrium proposed in these notes, equity holders unanimously support the rm s choice of the production and nancial decisions which maximize its value (or pro ts), as in (3.15). This follows from the fact that, when the equity price map satis es the consistency conditions i) and ii), the model is equivalent to one where a continuum of types of equity is available for trade to consumers, corresponding to any possible choice of k the representative rm can make, at the price Q(k). Thus, for any possible value of k a market is open where equity with a payo f(k; s) can be traded, and in equilibrium such market clears with a zero level of trades for the values of k not chosen by the rms. For any possible choice k of a rm, the (marginal) valuation of the rm by an agent i is E MRS i f(k) ; and it is always weakly to the market value of the rm, given by max i E MRS i f(k) Proposition 58 At a competitive equilibrium, equity holders unanimously support the production k; that is, every agent i holding a positive initial amount i 0 of equity of the representative rm will be made - weakly - worse o by any other choice k 0 of the rm. E ciency A consumption allocation (x i 0; x i ) I i=1 is admissible if it is feasible there exists a production plan k such that X x i 0 + k X! i 0 (3.24) i i X x i (s) X! i (s) + f(k; s); 8s 2 S (3.25) i i 17 To keep the notation simple, we state both the de nition of competitive equilibria and admissible allocations for the case of symmetric allocations. The analysis, including the e ciency result,extends however to the case where asymmetric allocations are allowed are admissible; see also the next section.

70 64 CHAPTER 3. TWO-PERIOD ECONOMIES 2. it is attainable with the existing asset structure for each consumer i, there exists a pair z i ; i such that x i (s) =! i (s) + f(k; s) i + A(s)z i ; 8s 2 S (3.26) Next we present the notion of e ciency restricted by the admissibility constraints Constrained e ciency. A competitive equilibrium allocation is constrained Pareto e cient if we can not nd another admissible allocation which is Pareto improving. The validity of the First Welfare Theorem with respect to such notion can then be established by an argument essentially analogous to the one used to establish the Pareto e ciency of competitive equilibria in Arrow-Debreu economies. First welfare theorem. Competitive equilibria are constrained Pareto ef- cient. Modigliani-Miller We examine now the case where rms take both production and nancial decisions, and equity and debt are the only assets they can nance their production with. The choice of a rm s capital structure is given by the decision concerning the amount B of bonds issued. The problem of the rm consists in the choice of its production plan k and its nancial structure B. To begin with, we assume without loss of generality that all rms debt is risk free. The rm s cash ow in this context is then [f(k; s) B] and varies with the rm s production and nancing choices, k; B. Equity price conjectures have the form Q(k; B), while the price of the (risk free) bond is independent of (k; B); we denote it p. The rm s problem is then max k;b k + Q(k; B) + p B (3.27) The consumption side of the economy is the same as in the previous section, except that now agents can also trade the bond. Let b i denote the bond portfolio of agent i; and let continue to impose no-short sales contraints i 0 b i 0; 8i

71 3.2. FINANCIAL MARKET ECONOMIES 65 Proceeding as in the previous section, at equilibrium we shall require that p = i max E MRS i i Q(k) = max E MRS i [f(k) B] ; 8k; where MRS i denotes the marginal rate of substitution between consumption at date 0 and at date 1 in state s for consumer i; evaluated at his equilibrium consumption allocation (x i 0; x i ). Suppose now that nancial markets are complete, that is rank(a) = S At equilibrium then MRS i = MRS, 8i. Therefore, in this case Q(k; B) + p B = E [MRS f(k)] ; 8k; and the value of the rm, Q(k; B) + p B; is independent of B This proves the celebrated Modigliani-Miller theorem. If nancial markets are complete the nancing decision of the rm, B; is indeterminate. It should be clear that when nancial markets are not complete and agents are restricted by no-short sales constraints, the Modigliani-Miller theorem does not quite necessarily hold.

72 66 CHAPTER 3. TWO-PERIOD ECONOMIES

73 Chapter 4 Matching and search equilibrium [...Assignment problem. Search with complete information (Moen JPE ) and incomplete information (Guerrieri and Shimer); is there also a static/dynamic distinction? look for literature...] 67

74 68 CHAPTER 4. MATCHING AND SEARCH EQUILIBRIUM

75 Chapter 5 Monopolistic competitive equilibrium [...Use also Eaton-Kortum (in Alvarez-Lucas form) as example...] 69

76 70 CHAPTER 5. MONOPOLISTIC COMPETITIVE EQUILIBRIUM

77 Chapter 6 Asymmetric information Do competitive insurance markets function orderly in the presence of moral hazard and adverse selection? What are the properties of allocations attainable as competitive equilibria of such economies? And in particular, are competitive equilibria incentive e cient? For such economies the interaction between the private information dimension (e.g., the unobservable action in the moral hazard case, the unobservable type in the adverse selection case) and the observability of agents trades plays a crucial role, since trades have typically informational content over the agents private information. In particular, to decentralize incentive e cient Pareto optimal allocations the availability of fully exclusive contracts, i.e., of contracts whose terms (price and payo ) depend on the transactions in all other markets of the agent trading the contract, is generally required. The implementation of these contracts imposes typically the very strong informational requirement that all trades of an agent need to be observed. The fundamental contribution on competitive markets for insurance contracts is Prescott and Townsend (1984). They analyze Walrasian equilibria of economies with moral hazard and with adverse selection when exclusive contracts are enforceable, that is, when trades are fully observable. 1 In these 1 The standard strategic analysis of competition in insurance economies, due to Rothschild-Stiglitz (1976), considers the Nash equilibria of a game in which insurance companies simultaneously choose the contracts they issue, and the competitive aspect of the market is captured by allowing the free entry of insurance companies. Such equilibrium concept does not perform too well equilibria in pure strategies do not exist for robust examples (Rothschild-Stiglitz (1976)), while equilibria in mixed strategies exist 71

78 72 CHAPTER 6. ASYMMETRIC INFORMATION notes we concentrate on the simpler case of moral hazard. We refer to Bisin- Gottardi (2006) for the case of adverse selection. 6.1 A simple insurance economy Agents live two periods, t = 0; 1; and consume a single consumption good only in period 1. Uncertainty is purely idiosyncratic and all agents are exante identical. In particular, each agent faces a (date 1) endowment which is an identically and independently distributed random variable! on a nite support S. 2 Moral hazard (hidden action) is captured by the assumption that the probability distribution of the period 1 endowment that each agent faces depends from the value taken by a variable e 2 E, an unobservable level of e ort which is chosen by the agent. Let prob s (e) be the probability of the realization! s given e. Obviously P s2s prob s(e) = 1; for any e 2 E; a compact, convex set By the Law of Large Numbers, prob s (e) is also the fraction of agents who have chosen e ort e for which state s is realized. Agents preferences are represented by a von Neumann-Morgenstern utility function of the following form X prob s (e)u(x s ) s2s v(e) where v(e) denotes the disutility of e ort e. We assume the following regularity conditions The utility function u(x) is strictly increasing, strictly concave, twice continuously di erentiable, and lim x!0 u 0 (x) = 1 The cost function v(e) is strictly increasing, strictly convex, twice continuously di erentiable, and sup e2e v 0 (e) = 1 (Dasgupta-Maskin (1986)) but, in this set-up, are of di cult interpretation. Even when equilibria in pure strategies do exist, it is not clear that the way the game is modelled is appropriate for such markets, since it does not allow for dynamic reactions to new contract o ers (Wilson (1977) and Riley (1979); see also Maskin-Tirole (1992)). Moreover, once sequences of moves are allowed, equilibria are not robust to minor perturbations of the extensive form of the game (Hellwig (1987)). 2 Measurability issues arise in probability spaces with a continuum of indipendent random variables. We adopt the usual abuse of the Law of Large Numbers.

79 6.1. A SIMPLE INSURANCE ECONOMY 73 Essentially without loss of generality, let the state space S be ordered so that! s >! s 1 ; for all s = 2; ; S We then impose the following standard restriction. Single-crossing property. The odds ratio in e, for any s = 2; ; S probss(e) probs s 1 (e) is strictly increasing The Symmetric information benchmark Consider now the benchmark case of symmetric information, in which e is commonly observed. An allocation (x ; e ) 2 R S + E of consumption and e ort is optimal under symmetric information if it solves s.t. max x;e X prob s (e)u(x s ) v(e); (6.1) s2s X prob s (e)(x s! s ) = 0 s2s Let q s (e) denote the (linear) price of consumption in state s for agents who chose e ort e. By allowing the prices of the securities whose payo is contingent on the idiosyncratic uncertainty to depend on e, we e ectively are introducing price conjectures we read q s (e) as the price of consumption contingent to state s if the agent chooses e ort e, for any e 2 E. In other words, we cannot just specify the prices at the equilibrium choice not just at the equilibrium e ort choice e ; we need specify prices at all possible e ort choices e 2 E. This is the same problem we found in production economies with incomplete markets, where the price faced by the rm was a whole map Q(k); interpreted as a price conjecture. At a competitive equilibrium, s.t. (x ; e ) 2 arg max X s2s prob s (e)u(x s ) v(e); (6.2) X q s (e)(x s! s ) = 0; s2s furthermore, at an equilibrium, markets clear X prob s (e )(x s! s ) = 0; (6.3) s2s

80 74 CHAPTER 6. ASYMMETRIC INFORMATION and the following consistency condition is satis ed q s (e) = prob s (e); for any e 2 E This is a bona de consistency condition normalizing prices so that P s2s q s(e ) = 1; optimization and market clearing only imply q s (e ) = prob s (e ) It is now straightforward to prove the First and Second Welfare theorems for this economy under symmetric information. 6.2 The moral hazard economy Consider now the case of asymmetric information, in which his choice of e ort e is private information of each agent. In this context, an allocation (x; e) 2 R S + E of consumption and e ort is incentive constrained optimal if it solves X s.t. and max x;e s2s prob s (e)u(x s ) v(e); (6.4) X prob s (e)(x s! s ) = 0 s2s e 2 e(x) = arg max X s2s prob s (e)u(x s ) v(e); given x The last constraint, called incentive constraint, requires that the allocation (x; e) must be such that the agent prefers (x; e) to any other allocation (x; e 0 ), for any e; e 0 2 E. A Prescott-Townsend competitive equilibrium is an allocation (x ; e ) 2 R S + E and prices q s (e ) such that (x ; e ) 2 arg max X s2s prob s (e)u(x s ) v(e); (6.5) s.t. X q s (e)(x s! s ) = 0 s2s

81 6.2. THE MORAL HAZARD ECONOMY 75 and e 2 e(x ) = arg max X s2s prob s (e)u(x s) v(e); given x ; furthermore, at an equilibrium, markets clear X prob s (e )(x s! s ) = 0; (6.6) s2s and the following consistency condition is satis ed q s (e) = prob s (e); for any e 2 E It is immediate to check that at a Prescott and Townsend competitive equilibrium, prices satisfy q s (e ) = prob s (e ) and allocations are incentive constrained optimal. It is di cult however to justify conceptually a formulation like this, in which agents self-impose the incentive-compatibility constraint, that is, in which non-incentive compatible allocations are not traded. A possible alternative - but equivalent - formulation can be motivated as an extension of the equilibrium concept we adopted in the symmetric information economy. In the moral hazard economy, however, prices cannot have the form q s (e), as the agent s choice for e is not observed. What is observed is the consumption allocation x demanded by the agent in the market. (Note that the exclusivity assumption guarantees that this is the case, that x is observable). Price conjectures can then be de ned as a function of x as q s (x) = q s (e(x)) At a competitive equilibrium then each agent solves X prob s (e)u(x s ) v(e); (6.7) max x;e s.t. X q s (x)(x s! s ) = 0 s2s s2s where q s (x) = q s (e (x)) and e(x) 2 arg max X s2s prob s (e)u(x s ) v(e), given x;

82 76 CHAPTER 6. ASYMMETRIC INFORMATION furthermore, at equilibrium markets clear X prob s (e)(x s! s ) = 0 (6.8) s2s and the following consistency condition is satis ed q s (e(x)) = prob s (e(x)) It is straightforward to show that this equilibrium concept is equivalent to Prescott and Townsend s. Yet another equivalent notion of equilibrium is possible, which is equivalent to the one just proposed. It is a (sort of) mechanism design formulation of the competitive equilibrium. Since prices cannot depend on e ort e, which is unobservable, they depend on agents message/declaration about e ort, which we denote m 2 E and which his optimally chosen by the agents themselves. An agent declaring m will face prices q s (m) but also a restriction on the space of allocations B(m) R S ++ designed to guarantee that his declaration coincides with his e ort choice, that is, designed to guarantee truth-telling m = e Speci cally, we write the agent s problem as follows X max prob s (e)u(x s ) v(e); (6.9) x2r S ++ ;(e;m)2e2 s2s s.t. X q s (m)(x s! s ) = 0 and x 2 B(m); s2s where B(m) = ( x 2 R S ++ m 2 arg max "2E X prob s (")u(x s ) s2s v(") ) At equilibrium then markets clear X prob s (e)(x s! s ) = 0; (6.10) s2s and the following consistency condition is satis ed q s (m) = prob s (m)

83 6.2. THE MORAL HAZARD ECONOMY 77 In other words, the truth-telling constraint, x 2 B(m), requires that in the market with prices q(m) only incentive compatible allocations are o ered, that is, only allocations which induce the agent to choose e ort m; for any m 2 E 3 It is straightforward to see that this equilibrium notion is equivalent to the one with rational price conjectures we have proposed. The interpretation is di erent however with rational conjectures it is the conjectures on prices of non incentive compatible allocations which are restricted, while with the equilibrium notion in this remark it is tradable allocations which are restricted to exclude non incentive compatible ones. It is important to note that the allocation (x s ) s2s is assumed observable. This corresponds to the exclusivity case, as it is called in the literature. The next problem deals with the non-exclusivity case in a simple but instructive example. Problem 59 In the context of these moral hazard economies M. Harris and R. Townsend (1981) prove a version of the Revelation principle. Formulate the statement and sketch the proof. Problem 60 Characterize Incentive constrained optima and competitive equilibria (with the consistency condition on price maps) of the moral hazard economy specialized so that S = 1; 2; e = h; l; v(h) > v(l); 1 (h) > 1 (l);! 1 >! 2 Set-up and characterize equilibria of this economy under the constraint that prices are linear (independent of x) q s (e(x)) = q s Are equilibrium allocations in this case Incentive constraint e cient? 3 Typically, the declaration of e ort on the part of the agent is implicit and the problem of the agent is written s.t. X X max prob s (e)u(x s ) v(e); (6.11) x2r S ++ ;e2e s2s s2s q s (e)(x s! s ) = 0; x 2 B(e) = ( x 2 R S ++ e 2 arg max "2E X prob s (")u(x s ) s2s v(") )

84 78 CHAPTER 6. ASYMMETRIC INFORMATION 6.3 The adverse selection economy Consider now an asymmetric information economy in which e is private information of each agent, but it is not chosen by the agent it is rather an exogenous type. Assume that a fraction e of agents has type e 2 E (E is nite). In this context, an allocation is x = (x e ) e2e 2 R SE ++ An allocation x 2 R SE ++ incentive constrained optimal if, for some ( e ) e2e 2 E ; it solves s.t. and max x2r SE ++ X X e prob s (e)u(x e s); (6.12) e2e s2s X X e prob s (e)(x e s! s ) = 0 e2e e 2 arg max "2E s2s X prob s (e)u(x " s); for any e 2 E s2s The last constraint, called incentive constraint or truth-telling, requires that, for any type e 2 E; the allocation x e 2 R S ++ must be such that the agent prefers x e to any other allocation x ", for any " 2 E. Consider now a Prescott-Townsend competitive equilibrium (in the two alternative formulations discussed in the moral hazard section). At a competitive equilibrium the allocation of each agent of type e 2 E; x e 2 R S ++, solves s.t. and max x2r S ++ X prob s (e)u(x s ) s2s X q s (e)(x s! s ) = 0; s2s X prob s (")u(x " s) X prob s (")u(x s ); for any " 2 Ene; s2s s2s at a competitive equilibrium, also, markets clear X X e prob s (e)(x e s! s ) = 0 e2e s2s

85 6.3. THE ADVERSE SELECTION ECONOMY 79 A more explicit formulation of the same equilibrium concept would have an explicit message choice on the part of the agent. In this case, at a competitive equilibrium the allocation of each agent of type e 2 E; x e 2 R S ++; solves s.t. with X max prob s (e)u(x s ) (x;m)2r S ++ E s2s X q s (m)(x s! s ) = 0; x 2 B(m; [x " s] "2Enm ) s2s B(m; [x " s] "2Enm ) = ( x 2 R S ++ X s2s prob s (")u(x " s) X s2s prob s (")u(x s ); for any " 2 Enm; ) ; at a competitive equilibrium, also, markets clear X X e prob s (e)(x e s! s ) = 0 e2e s2s Yet another equiavalent formulation of a competitive equilibrium concept with adverse selection is motivated by the fact that prices cannot have the form q s (e), as the agent s type e is not observed. But once again price conjectures can be de ned as a function of the observables. In this case, the budget set following message m 2 E is unrestricted, but prices are as follows, q s (m; x; [x " s] "2Enm ) = X e2e(x) { e prob s (e); with { e = e P e2e(x) e and ( e(x) = e 2 E X prob s (e)u(x s ) X ) prob s (e)u(x e s) s2s s2s Note that, in all these formulations, the problem of each agent of type e 2 E depends on the whole allocation x 2 R SE ++; that is on the agent s allocation x e as well on all the other types allocations x " ; " 2 E In other words, each type s consumption problem contains an externality. As a consequence, the First and Second Welfare theorems, in its incentive constrained versions, do

86 80 CHAPTER 6. ASYMMETRIC INFORMATION not hold for the adverse selection economy; see Bisin-Gottardi (2006) for details (many more than you might ever want to). In the adverse selection case, as well in the moral hazard case, it is important to note that the allocation (x s ) s2s is assumed observable. The next problem deals with the non-exclusivity case in a simple but instructive example. Problem 61 Characterize Incentive constrained optima and competitive equilibria (with the consistency condition on price maps) of the adverse selection economy specialized so that S = 1; 2; e = h; l; 1 (h) > 1 (l);! 1 >! 2 Set-up and characterize equilibria of this economy under the constraint that prices are linear (independent of x). Are equilibrium allocations in this case Incentive constraint e cient? 6.4 Information revealed by prices In the previous sections, moral hazand and adverse selection regarded asymmetric information about idiosyncratic shocks which they acquired insurance against. In this section we study instead economies in which agents are endowed with asymmetric information regarding aggregate shocks, e.g., because di erent agent types observe di erent signals about the aggregate endowment process. In this case, under rational expectations agents choices contain implicitly information about the signal they have received, which in turn implies that prices might contain some of this information, and hence that agents at equilibrium condition on the information contained in prices. If equilibrium prices aggregate in a signi cant manner the information which is asymmetrically distributed across agents in the economy, then competitive equilibrium allocations might in fact Pareto dominate the allocations which a benevolent planner could choose without observing equilibrium prices. This is the fundamental insight due to Friederick August Hayek in The Road to Serfdom, University of Chicago, On the other hand, economies in which prices do not reveal completely the information asymmetrically distributed across agents in the economy, are economies in agents have di erent probability distributions across aggregate endowments, and hence trade in nancial markets on account of risk sharing

87 6.4. INFORMATION REVEALED BY PRICES 81 as well as information, a positive property which seem to characterize real nancial markets. Consider a nancial market economy with one good, L = 1; and 2 periods, t = 0; 1. Let i 2 f1; ; Ig denote agents. Let s 2 f1; ; Sg denote the realization of the state of the world of the economy at t = 1. The consumption space is denoted then by X R S+1 +. Each agent is endowed with a vector! i 2 R++ S+1. Let u i X! R denote agent i s utility function. Financial markets are characterized by J assets with payo matrix A 2 R SJ + ; J S; with rank J Each agent of type i 2 f1; ; Ig observes, before markets are open at t = 0; the realization of a signal i 2 = f1; ; g; which is possibly correlated with the state of the world s 2 S. All agents have identical prior over the distribution of states of the world and signals in S I ; denoted prob s; Let prob i s; denote the posterior after conditioning on i 2 Furthermore, let = ( i ) i2i 2 I ; and let prob s denote the posterior distribution over s 2 f1; ; Sg after conditioning on 2 I Finally, given a price map I! R J +; such that q = (); let prob i ; 1 (q ) s denote the posterior distribution over s 2 f1; ; Sg after conditioning on i 2 and 2 1 (q ) Each agent type i 2 I s choice problem is the following s.t. max u i (x i 0) + X prob i ; 1 (q ) (x i ;z i )2R S+1 s u i (x i s) + RJ + s2s x i 0 + q z i! i 0 x i s A s z i +! i s Note that agent i s expected utility is calculated after conditioning on the private signal i and the information on contained in prices q Let z i (q ; ) denote the solution to this problem in terms of portfolios z i Let then Z(q ; ) = X i2i z i (q ; ) denote the excess demand system for this economy, J equations, one for each asset. Note that the condition Z(q ; ) = 0 is su cient for all markets to clear, as in this case x i s = A s z i +! i s implies P i2i xi s! i s = 0, for any s 2 S; and P i2i xi 0! i 0 = 0 by Walras Law.

88 82 CHAPTER 6. ASYMMETRIC INFORMATION We are now ready for the de nition of equilibrium, Rational Expectation Equilibrium (REE). De nition 62 A REE price is a map I! R J + such that Z( () ; ) = 0; for any 2 I This notion of equilibrium is due to Robert E. Lucas (1972) and has been taken up by a series of important papers by S. Grossman and by S. Grossman and J. Stiglitz in the late 70 s. Several examples in the literature show that REE prices might not exist for some economies (the rst example is due to D. Kreps). Furthermore, even when they exist, characterizing REE prices is typically a daunting task. On the other hand, it is not (that) hard to characterize the information contained in the prices, for generic economies. This result is due to Radner (1979). Theorem 63 For a generic subset of economies parametrized by endowments! 2 R S+1 +, a REE price I! R J + is fully revealing, that is, () 6= ( 0 ) ; for any ; 0 2 I Proof. Let each agent type i 2 I s choice problem be the following s.t. max u i (x i 0) + X prob (x i ;z i )2R S+1 s u i (x i s) + RJ + s2s x i 0 + q z i! i 0 x i s A s z i +! i s Let z i (q ; prob ) denote the solution to this problem in terms of portfolios z i when the signals pro le is 2 I Let then Z(q ; prob ) = X i2i z i (q ; prob ) denote the excess demand system for this economy, J equations, one for each asset. The statement of the theorem is a consequence of the application of the Transversality theorem to the following system, Z(q ; prob ) = 0 Z(q 0; prob 0 ) = 0 q = q 0

89 6.4. INFORMATION REVEALED BY PRICES 83 for ; 0 2 I While we avoid the perturbation argument, we note that the system has 3J equations in 2J unknowns, prices q ; q 0 Notice that a fully revealing price map is invertible, by de nition, and hence prob i ; 1 (q ) s = prob s At a fully revealing REE, therefore, agents posterior probability distributions over s 2 S are identical. As a consequence, it follows that existence is generic and that at these equilibria agents do not trade on account of information. Consider however the following information structure. Each agent of type i 2 f1; ; Ig observes, before markets are open at t = 0; the realization of a signal i 2 R ; = f1; ; g; which is possibly correlated with the state of the world s 2 S. All agents have identical prior over the distribution of states of the world and signals in S R I ; denoted prob s; Let prob s denote the posterior distribution over s 2 f1; ; Sg after conditioning on 2 R I In this economy, a REE price is a map R I! R J + such that Z( () ; ) = 0; for any 2 R I Theorem 64 Suppose I > J. For a generic subset of economies parametrized by endowments! 2 R S+1 +, a REE price exists and it is not is fully revealing, that is, () is not invertible. While we shall not prove the theorem (see Allen, 1981, for a proof), we notice that maps a I dimensional set into a J dimensional one; if I > J; then, the result is at least not surprising References Bennardo, A. and P.A. Chiappori (2003) Bertrand and Walras Equilibria under Moral Hazard, Journal of Political Economy, 111(4), Bisin, A. and P. Gottardi (1999) Competitive Equilibria with Asymmetric Information, Journal of Economic Theory, 87, Bisin, A. and P. Gottardi (2006) E cient Competitive Equilibria with Adverse Selection, Journal of Political Economy, 114(3),

90 84 CHAPTER 6. ASYMMETRIC INFORMATION Dubey, P., J. Geanakoplos (2004) Competitive Pooling Rothschild-Stiglitz Re-considered, Quarterly Journal of Economics. Gale, D. (1992) A Walrasian Theory of Markets with Adverse Selection, Review of Economic Studies, 59, Gale, D. (1996) Equilibria and Pareto Optima of Markets with Adverse Selection, Economic Theory, 7, Grossman, S. and O. Hart (1983) An Analysis of the Principal-Agent Problem, Econometrica, Vol. 51, No. 1. Hellwig, M. (1987) Some Recent Developments in the Theory of Competition in Markets with Adverse Selection, European Economic Review, 31, Prescott, E.C. and R. Townsend (1984) Pareto Optima and Competitive Equilibria with Adverse Selection and Moral Hazard, Econometrica, 52, 21-45; and extended working paper version dated Prescott, E.C. and R. Townsend (1984b) General Competitive Analysis in an Economy with Private Information, International Economic Review, 25, Rothschild, M. and J. Stiglitz (1976) Equilibrium in Competitive Insurance Markets An Essay in the Economics of Imperfect Information, Quarterly Journal of Economics, 80, Wilson, C. (1977) A Model of Insurance Markets with Incomplete Information, Journal of Economic Theory, 16,

91 Chapter 7 In nite-horizon economies Note that the commodity space is in nite dimensional. Good discussion of the spaces we typically study is in Lucas-Stokey with Prescott (1989), Recursive economic dynamics, Harvard Univ. Press. Existence and rst welfare theorems are straightforwardly extended as long as the number of agents is nite. See Zame, "Competitive equilibria in production economies with an in nite dimensional commodity space," Econometrica, 55(5), Asset pricing Assume a representative-agent economy with one good. Let time be indexed by t = 0; 1; 2; Uncertainty is captured by a probability space represented by a tree. Suppose that there is no uncertainty at time 0 and call s 0 the root of the tree. Without much loos of generality, we assume that each node has a constant number of successors, S. At generic node at time t is called s t 2 S t. Note that the dimensionality of S t increases exponentially with time t (abusing notation it is in fact S t ). When a careful speci cation of the underlying state space process is not needed, we will revert to the usual notation in terms of stochastic processes. Let x = fx t g 1 t=0 denote a stochastic process for an agent s consumption, where x t S t! R + is a random variable on the underlying probability space, for each t. Similarly, let! = f! t g 1 t=0 be a stochastic processes describing an agent s endowments. Let 0 < < 1 denote the discount factor. 85

92 86 CHAPTER 7. INFINITE-HORIZON ECONOMIES Arrow-Debreu economy Suppose that at time zero, the agent can trade in contingent commodities. Let p = fp t g 1 t=0 denote the stochastic process for prices, where p t S t! R +, for each t. Then ((x i ) i ; p ) is an Arrow-Debreu Equilibrium if i. given p ; ii. and P i xi! i = 0 x i 2 arg maxfu(x 0 ) + E 0 [ P 1 t=1 t u(x t )]g st P 1 t=0 p t (x t! t ) = 0 The notation does not make explicit that the agent chooses at time 0 a whole sequence of time and state contingent consumption allocations, that is, the whole sequence of x(s t ) for any s t 2 S t and any t Financial markets economy Suppose that throughout the uncertainty tree, there are J assets. We shall allow assets to be long-lived. In fact we shall assume they are and let the reader take care of the straightforward extension in which some of the assets pay o only in a nite set of future times. Let z = fz t g 1 t=1 denote the sequence of portfolios of the representative agent, where z t S t! R J. Assets payo s are captured at each time t by the S J matrix A t. Furthermore, capital gains are q t q t 1, and returns are R t = At+qt q t 1. In a nancial market economy agents do not trade at time 0 only. They in fact, at each node s t receive endowments and payo s from the portfolios they carry from the previous node, they re-balance their portfolios and choose state contingent consumption allocations for any of the successor nodes of s t, which we denote s t+1 j s t. De nition 65 f(x i ; z i ) i ; q g is a Financial Markets Equilibrium if i. given q ; at each time t 0 (x i ; z i ) 2 arg maxfu(x t ) + E t [ P 1 =1 j u(x t+js t)]g st x t+ + q t+z t+ =! t+ + A t+ z t+ 1 ; for = 0; 1; 2; ; with z 1 = 0 some no-ponzi scheme condition

93 7.1. ASSET PRICING 87 De nition 66 ii. P i xi! i = 0 and P i zi = Conditional asset pricing From the FOC of the agent s problem, we obtain or, u 0 (x t+1 ) q t = E t A u 0 t+1 = E t (m t+1 A t+1 ) (7.1) (x t ) u 0 (x t+1 ) 1 = E t R u 0 t+1 (7.2) (x t ) Example 1. Consider a stock. Its payo at any node can be seen as the dividend plus the capital gain, that is, R t+1 = q t+1 + d t+1 q t ; for some exogenously given dividend stream d. By plugging this payo into equation (7.1), we obtain the price of the stock at t. Example 2. For a call option on the stock, with strike price k at some future period T > t, we can de ne A t = 0; t < T; and A T = maxfq T k; 0g De ne now m t;t = T t u 0 (x T ) u 0 (x t ) and observe that the price of the option is given by q t = E t (m t;t maxfq T k; 0g) ; Note how the conditioning information drives the price of the option the price changes with time, as information is revealed by approaching the execution period T. Example 3. The risk-free rate is know at time t and therefore, equation (7.2) applied to a 1-period bond yields 1 u 0 (x t+1 ) = E t u 0 (x t ) R f t+1

94 88 CHAPTER 7. INFINITE-HORIZON ECONOMIES Once again, note that the formula involves the conditional expectation at time t. Therefore, while the return of a risk free 1-period bond paying at t+1 is known at time t, the return of a risk free 1-period bond paying at t+2 is not known at time t. [... relationship between 1 and period bonds... from the red Sargent book] Conditional versions of the beta representation hold in this economy E t (R t+1 ) R f Cov t (m t+1 ; R t+1 ) t+1 = = (7.3) E t (m t+1 ) = Cov t(m t+1 ; R t+1 ) V art (m t+1 ) = V ar t (m t+1 ) E t (m t+1 ) t t Unconditional moment restrictions Recall that our basic pricing equation is a conditional expectation q t = E t (m t+1 A t+1 ); (7.4) In empirical work, it is convenient to test for unconditional moment restrictions. 1 However, taking unconditional expectations of the previous equation implies in principle a much weaker statement about asset prices than equation (7.4) E(q t ) = E(m t+1 A t+1 ); (7.5) where we have invoked the law of iterated expectations. It should be clear that equation (7.4) implies but it is not implied by (7.5). The theorem in this section will tell us that actually there is a theoretical way to test for our conditional moment condition by making a series of tests of unconditional moment conditions. De ne a stochastic process fi t g 1 t=0 to be conformable if for each t, i t belongs to the time-t information set of the agent. It then follows that for any such process, we can write i t q t = E t (m t+1 i t A t+1 ) 1 Otherwise, speci c parametric assumptions need be imposed on the stochastic process of the economy underlying the asset pricing equation. For instance this is the route taken by the literature on autoregressive conditionally heteroschedastic (that is, ARCH - and then ARCH-M, GARCH,...) models. See for instance the work by??rengle/http//pages.stern.nyu.edu/ rengle/ = 13.

95 7.1. ASSET PRICING 89 and, by taking unconditional expectations, E(i t q t ) = E(m t+1 i t A t+1 ) This fact is important because for each conformable process, we obtain an additional testable implication that only involves unconditional moments. Obviously, all these implications are necessary conditions for our basic pricing equation to hold. The following result states that if we could test these unconditional restrictions for all possible conformable processes then it would also be su cient. We state it without proof. Theorem 67 If E(x t+1 i t ) = 0 for all i t conformable then E t (x t+1 ) = 0 By de ning x t+1 = m t+1 A t Predictability or returns Recall the asset-pricing equation for stocks q t ; the theorem yields the desired result. q t = E t (m t+1 (q t+1 + d t+1 )) It is sometimes argued that returns are predictable unless stock prices to follow a random walk. (Where in turn predictability is interpreted as a property of e cient market hypothesis, a fancy name for the asset pricing theory exposed in these notes). Is it so? No, unless strong extra assumptions are imposed Assume that no dividends are paid and agents are risk neutral; then, for values of close to 1 (realistic for short time periods), we have q t = E t (q t+1 ) That is, the stochastic process for stock prices is in fact a martingale. Next, for any f" t g such that E t (" t+1 ) = 0 at all t, we can rewrite the previous equation as q t+1 = q t+1 + " t+1 This process is a random walk when var t (" t+1 ) = is constant over time. A more important observation is the fact that marginal utilities times asset prices (a risk adjusted measure of asset prices) follow approximately a martingale (a weaker notion of lack of predictability). Again under no dividends, u 0 (c t )q t = E t (u 0 (c t+1 )(q t+1 + d t+1 )); which is a supermartingale and approximately a martingale for close to 1.

96 90 CHAPTER 7. INFINITE-HORIZON ECONOMIES Fundamentals-driven asset prices For assets whose payo is made of a dividend and a capital gain, FOC dictate where q t = E t (m t+1 (q t+1 + d t+1 )) ; m t+1 = u0 (c t+1 ) u 0 (c t ) By iterating forward and making use of the Law of Iterated Expectations,!! TX TX q t = lim E t m t;t+j d t+j + lim E t m t;t+j q t+j ; T!1 T!1 j=1 As we shall see, in nite horizon models (with in nitely lived agents) usually satisfy the no-bubbles condition, or lim E t T!1 j=1! TX m t;t+j q t+j = 0 j=1 In that case, we say that asset prices are fully pinned down by fundamentals since 1! X q t = E t m t;t+j d t+j Conditional factor models and the conditional CAPM [...from Cochrane...] Frictions He-Modest Luttmer 7.2 Bubbles j=1 Let 2 N = X 1 t=0s t be the set of nodes of the tree. Recall we denoted with s 0 denote the root of the tree and with s t an arbitrary node of the tree at time t. Use s t+ js t to indicate that s t+ is some successor of s t, for > 0 2 In this section we follow closely Santos-Woodford (1997), Econometrica.

97 7.2. BUBBLES 91 At each node, there are J securities traded. Also, it is important that the notation include Overlapping Generation economies. We need therefore to account for nitely lived agents. Let I(s t ) be the set of agents which are active at node s t. Let N i be the subset of nodes of the tree at which agent i is allowed to trade. Also, denote by N i the terminal nodes for agent i. The following assumptions will not be relaxed. 1. If an agent i is alive at some non-terminal node s t, she is also alive at all the immediate successor nodes. That is, s t N i nn i =) fs t+1 N s t+1 js t g N i 2. The economy is connected across time and states at any state there is some agent alive and non-terminal. Formally, 8s t ; 9i s t 2 N i nn i Assets are long lived. Let q N! R J be the mapping de ning the vector of security prices at each node s t. Similarly, let d N! R J denote the vector-valued mapping that de nes the dividends (in units of numeraire) that are paid by the assets at node s t. We assume that d(s t ) 0 for any s t Each of the households alive at s 0 enters the markets with an initial endowment of securities z! i 0 Therefore, the initial net supply of assets is given by z! = X z!. i i2i(s 0 ) As assets are long lived, a supply z! of assets is available at any s t At each node s t, each households in I(s t ) has an endowment of numeraire good of! i (s t ) 0. We shall assume that the economy has a well-de ned aggregate endowment!(s t ) = X! i (s t ) 0 i2i(s t ) at each node s t. This is the case, e.g., if I(s t ) is nite, for any s t Taking into account the dividends paid by securities in units of good, the aggregate good supply in the economy is given by e!(s t ) =!(s t ) + d(s t )z! 0

98 92 CHAPTER 7. INFINITE-HORIZON ECONOMIES The utility function of any agent i is written 1X U(x) = u i (x(s 0 ) + X t prob(s t ) u i (x(s t )) s t js 0 t=1 De ne the 1-period payo vector (in units of numeraire) at node s t by A(s t ) = d(s t ) + q(s t ) Any agent i faces a sequence of budget constraints, one for every node s t 2 N i Let x i (s t ) denote the consumption of agent i at node s t 2 N i and z i (s t ) his a J-dimensional portfolio vector at s t 2 N i The agent budget constraint at any s t js 0 2 N i is with x i (s t ) + q(s t )z i (s t )! i (s t ) + A(s t )z i (s t 1); x i (s t ) 0 q(s t )z i (s t ) B i (s t ); where B i N! R + indicates an exogenous and non-negative household speci c borrowing limit at each node. We assume households take the borrowing limits as given, just as they take security prices as given. At s 0 ; if s 0 2 N i, the budget constraint is x i (s 0 ) + q(s 0 )z i (s 0 )! i (s 0 ) + q(s 0 )z i!; At equilibrium, markets clear that is, at each s t 2 N; X x i (s t ) = e!(s t ) i2i(s t ) X i2i(s t ) z i (s t ) = z! Given the price process q, we say that no arbitrage opportunities exist at s t if there is no z 2 R J such that A(s t+1 )z 0; for all s t+1 js t ; q(s t )z 0; with at least one strict inequality.

99 7.2. BUBBLES 93 Lemma 68 When q satis es the no-arbitrage condition at s t 2 N; there exists a set of state prices f(s t+1 js t )g with (s t+1 js t ) > 0 for all s t+1 js t, such that the vector of asset prices at s t 2 N can be written as q(s t ) = X (s t+1 js t )A(s t+1 js t ) (7.6) s t+1 js t Proof. As usual, proof follows from applying an appropriate separation theorem. Applying the Lemma at any s t 2 N we can construct a stochastic process N! R++ S recursively as follows (s t+2 js t ) = (s t+2 js t+1 )(s t+1 js t ); for t = 0; 1; Let (s t ) denote the set of such processes for the subtree with root s t. Only under complete markets is the set (s t ) a singleton. As a remark, note that nancial market completeness is an endogenous property in this economy since one-period payo s A contain asset prices. Therefore, the rank property which de nes completeness can only be assessed at each given equilibrium. For any state-price process 2 (s t ); de ne the J vector of fundamental values for the securities traded at node s t by f(s t ; ) = 1X T =t+1 X s T js t (s T js t )d(s T js t ) (7.7) Observe that the fundamental value of a security is de ned with reference to a particular state-price process, however the following properties it displays are true regardless of the state prices chosen. Proposition 69 At each s t 2 N, f(s t ; ) is well-de ned for any 2 (s t ) and satis es 0 f(s t ; ) q(s t ) Proof. First of all, 0 f(s t ; ) follows directly from non-negativity of, the dividend process d, and the price process q. We therefore turn to f(s t ; ) q(s t ) From equation (7.6), we have q(s t ) = X X (s t+1 js t )d(s t+1 js t ) + (s t+1 js t )q(s t+1 ) s t+1 js t s t+1 js t

100 94 CHAPTER 7. INFINITE-HORIZON ECONOMIES and, iterating on this equation we obtain q(s t ) = btx T =t+1 X X (s T js t )d(s T js t ) + (s bt js t )q(s bt ) s T js t s bt js t for any b T > t Since by construction, q(s bt ) is non-negative and 2 (s t ) is a positive state-price vector, the second term on the right-hand-side is non-negative. So, q(s t ) btx T =t+1 X s T js t (s T )d(s T js t ); for any b T > t and 1X X q(s t ) (s T js t )d(s T js t ) T =t+1 s T js t We can correspondingly de ne the vector of asset pricing bubbles as (s t ; ) = q(s t ) f(s t ; ); (7.8) for any 2 (s t ) for the J securities. It follows from the proposition that 0 (s t ; ) q(s t ); for any 2 (s t ) This corollary is known as the impossibility of negative bubbles result. Substituting (7.8) and (7.7) into (7.6) yields (s t ; ) = X s t+1 js t (s t+1 js t )(s t+1 ; ) This is known as the martingale property of bubbles if there exists a (nonzero) price bubble on any security at date t, there must exist a bubble as well on the security at date T, with positive probability, at every date T > t. Furthermore, if there exists a bubble on any security at node s t, then there must have existed a bubble as well on some security at every predecessor of the node s t.

101 7.2. BUBBLES 95 Remark 70 Suppose our economy is deterministic. Some of the earliest analyses on bubbles dealt with this case, e.g., Samuelson s and Bewley s models of money. In this case, let t denote the bubble at time t The martingale property of bubbles implies that t = t t+1 We shall get back to this example when we ll study the Overlapping Generation economy at the end of this section. What does this imply for securities with nite maturity? Your answer must depend on how you de ne securities with nite maturity in this context. In particular, are the ownership rights to an asset maintained (and tradable) after the end of its maturity (even if the assets ceases to pay dividends) or not? [Discuss!] In an economy with incomplete markets, the fundamental value need not be the same for all state-price processes consistent with no arbitrage. But even in this case, we can de ne the range of variation in the fundamental value, given the restrictions imposed by no-arbitrage. Let x N! R + denote a non-negative stream of consumption goods. For any s t, pick any 2 (s t ) and de ne the present value at s t of x with respect to by 1X X V x (s t ; ) = (s T js t )x(s T ). T =t+1 s T js t Since this present value depends on the stochastic discount factor, let us now de ne the bounds for the present value at s t of dividends x. For any s t, de ne x (s t ) = inf fv x (s t ; )g 2(s t ) x (s t ) = sup fv x (s t ; )g 2(s t ) We can now prove the following fundamental lemma. Lemma 71 Suppose that the (supremum of the) value of aggregate wealth is nite at equilibrium, i.e., ~! (s t ) < 1; for each node s t 2 N. Suppose also that there exists a bubble on some security in positive net supply at s t so that (s t )z(s t ) > 0; for some z(s t ) > 0. Then, 8K > 0; there exists a time T and s T js t such that (s T )z(s T ) > Ke!(s T )

102 96 CHAPTER 7. INFINITE-HORIZON ECONOMIES Proof. The martingale property of pricing bubbles implies, (s t )z(s t ) = X s T js t (s T js t )(s T )z(s T ) and hence P s T js (s T js t )(s T )z(s T ) is constant for any T > t On the other t hand, P s T js (s T js t )e!(s T ) must converge to 0 in T! 1 to guarantee that t ~! (s t ) = P P T s T js (s T js t )e!(s T ) < 1 t That is, there is a positive probability that the total size of the bubble on the securities becomes an arbitrarily large multiple of the value of the aggregate supply of goods in the economy. A contradiction. The proof exploits crucially the martingale property of bubbles. It follows from this result that some agent must accumulate, at some node s T, a wealth whose value is larger than the value of the aggregate endowment. The next theorem follows directly from the Lemma and shows that no bubbles can arise to securities in positive net supply as long as we are at equilibria with nite aggregate wealth. Theorem 72 Suppose that the (supremum of the) value of aggregate wealth is nite at equilibrium, i.e., ~! (s t ) < 1, at each node s t 2 N Then q j (s T ) = f j (s T ; ); for all s T js t and 2 (s t ), for each security j traded at s T that has positive net supply, z j! > 0. To evaluate under which conditions we can guarantee that the value of aggregate wealth is nite at equilibrium, i.e., ~! (s t ) < 1, it is useful to consider no Ponzi scheme conditions in some detail. Recall that to rule out Ponzi schemes when agents are in nitely lived, a lower bound on individual wealth is needed. Let us de ne a particular type of borrowing limit. An agent s borrowing ability is only limited by her ability to repay out of her own future endowment if for each s t N i nn i. 3 B i (s t ) = ~! i(s t ); (7.9) 3 For any a non-negative stream of consumption goods x N! R + it can be shown

103 7.2. BUBBLES 97 It can be shown that these borrowing limits never bind at any nite date (see Magill-Quinzii, Econometrica, 94), but rather only constrain the asymptotic behavior of a household s debt. 4 This constraint is to the e ect that each agent can e ectively trade his own future endowment stream any node s t An important consequence of this speci cation is the following. Proposition 73 Suppose that agent i has borrowing limits of the form (7.9). Then the existence of a solution to the agent s problem for given prices q implies that ~! i(s t ) < 1; at each s t 2 N i ; so that the borrowing limit is nite at each node. This is because, if agent i can borrow o of the value of ~! i ; this value must be nite at equilibrium prices for the agent s problem to be well-de ned. If it is nite at equilibrium prices, it must be nite at all implied no-arbitrage state prices. This is a crucial if an agent s endowment can be traded, then its value is on the right hand side of the present value budget constraint of the agent, and hence it must be nite. The next two corollaries to the theorem provide conditions on the primitives of the model that guarantee that the value of aggregate wealth is nite at any equilibrium. Corollary 74 Suppose that there exists a portfolio bzr J + such that d(s t js 0 )bz ~!(s t ); 8s t 2 N Then the theorem holds at any equilibrium. that x j (s t ) = sup such that q (s t ) z (s t ) where the sup is taken over all plans fz (s r js t ) ; r tg and x s r js r 1 + d s r js r 1 z s r 1 q (s r ) z (s r ) ; for any s r js t and any r t 9T such that q (s r ) z (s r ) 0; for any s r js t with r T In words, x j (s t ) is the least upper bound for the amount that an agent can borrow at node s t if the endowment of the agent is x s r js r 1 for any s r js t and any r t; under the constraint that the agent has to maintain a non-negative wealth after some time T 4 They are equivalent to requiring that the consumption process lies in the space of measurable bounded sequences. In the case of nitely lived agents, these borrowing limits are equivalent to imposing no-borrowing at all nonterminal nodes.

104 98 CHAPTER 7. INFINITE-HORIZON ECONOMIES Intuitively, if the existing securities allow such a portfolio bz to be formed, it must have a nite price at any equilibrium. But since the dividends paid by this portfolio are higher at every state than the aggregate endowment, the equilibrium value of the aggregate endowment is bounded by a nite number. Corollary 75 Suppose that there exists an (in nitely lived) agent i and an " > 0 such that i)! i (s t ) "!(s t ); 8s t 2 N and ii) B i (s t ) =! i(s t ); 8s t 2 N and for all i. Then the theorem holds at any equilibrium. Again, the result follows because in equilibrium, "!(s t ) must have a nite value as it appears on the right hand side of agent i s budget set. If a positive fraction of aggregate wealth has a nite value in equilibrium, then aggregate wealth has nite value. As a remark, note that these two corollaries share the same spirit and somewhat imply that bubbles are not a robust equilibrium phenomenon (for securities in positive net supply) (Famous) Theoretical Examples of Bubbles Recall that at money is a security that pays no dividends. Its only return comes from paying one unit of itself in the next period. Therefore if at money is in positive net supply and has a positive price in equilibrium, that is a bubble. The following two models have equilibria with such a property. Samuelson (1958) s OLG model Consider an economy in which [ I(s t ) is (countably) in nite, even though s t 2N I(s t ) is nite for any s t 2 N In this case even if ~! i(s t ) < 1, it is still possible that ~! (s t ) = 1 (and hence that ~! (s t ) = 1). The theorem does not apply and bubbles are possible. Bewley (1980) s turnpike model Consider the case in which stringent borrowing limits are imposed on trading, i.e., B i (s t ) < ~! i(s t ). In this case, nothing will exclude the possibility that ~! i(s t ) = 1. The theorem does not apply and bubbles are possible. For a detailed but simple treatment, see L. Ljungqvist and T. Sargent, Recursive macroeconomic theory, MIT Press, 2004; Ch. 25.

105 7.3. DOUBLE INFINITY 99 More recent examples of bubbles Many papers have studied bubbles recently. This is so even though the previous analysis leaves relatively little space for bubbles in the classes of models we are studying. Nonetheless, see, e.g., - Boyan Jovanovic (2007), Bubbles in Prices of Exhaustible Resources, NBER Working Paper No , http// Others have studied models with either behavioral agents (e.g., overcon- dent). See, e.g., - Harrison Hong, Jose Scheinkman, Wei Xiong (2006), Asset Float and Speculative Bubbles, Journal of Finance, American Finance Association, vol. 61(3), pages Dilip Abreu and Markus Brunnermeier (2003), Bubbles and Crashes, Econometrica, 71(1), Yet others have studied economies where the rational expectations assumption is relaxed (more precisely, the common knowledge assumption of rational expectation models). See, e.g., - Franklin Allen, Stephen Morris, and Hyun Song Shin (2003), Beauty Contests, Bubbles and Iterated Expectations in Asset Markets, Cowles Foundation Discussion Paper 1406, http//cowles.econ.yale.edu/p/cd/d14a/d1406.pdf 7.3 Double In nity In this section we consider in nite horizon economies with an in nite number of agents (hence double in nity). The classic example is an overlapping generation economy, where at each time t = 0; 1; 2; 1 a nitely-lived generation is born. We consider rst the case of an Arrow-Debreu economy with a complete set of time and state contingent markets. Assume a representative-agent economy with one good. Let time be indexed by t = 0; 1; 2; Uncertainty is captured by a probability space represented by a tree with root s 0 and generic node, at time t, s t 2 S t. Let `1+ denote the space of non-negative bounded

106 100 CHAPTER 7. INFINITE-HORIZON ECONOMIES sequences endowed with the sup norm. 5 Let x i = fx i tg 1 t=0 2 `1+ denote a stochastic process for an agent i s consumption, where x i t S t! R + is a random variable on the underlying probability space, for each t. Similarly, let! i = f! i tg 1 t=0 2 `1+ be a stochastic processes describing an agent i s endowments. Each agent i 0 s preferences, U i `1+! R; satisfy 1X U i (x) = u i 0(x 0 ) + E 0 [ t u i t(x t )] where u i t R +! R satisfy the standard di erentiability, monotonicity, and concavity properties, for any i > 0 and any t > 0. Obviously, 0 < < 1 denotes the discount factor. We say that an agent i is alive at s t 2 N if! i (s t ) > 0 and u i t(x t ) > 0 for some x t 2 R + We assume that! i (s t ) > 0 implies u i t(x t ) > 0 for some x t 2 R + and conversely, u i t(x t ) > 0 for some x t 2 R + implies! i (s t ) > 0 for all s t 2 S t We maintain the previous section assumptions that if an agent i is alive at some non-terminal node s t, she is also alive at all the immediate successor nodes, and that the economy is connected across time and states (at any state there is some agent alive and non-terminal). We nally assume that, i) each agent is nitely-lived! i > 0 for nitely many times t 0; ii) at each node s t 2 S t ; the set of agents alive (with positive endowments and preferences for consumption), I( s t ) is nite. Suppose that at time zero, the agent can trade in contingent commodities. Let p = fp t g 1 t=0 2 `1+ denote the stochastic process for prices, where p t S t! R +, for each t. The de nition of Arrow-Debreu equilibrium in this economy is exactly as for the one with nite agents i 2 I Let x = (x i ) i0 De nition 76 (x ; p ) is an Arrow-Debreu Equilibrium if t=1 i. given p ; x i 2 arg maxfu(x 0 ) + E 0 [ P 1 t=1 t u(x t )]g st P 1 t=0 p t (x t! i t) = 0 5 See Lucas-Stokey with Prescott (1989), Recersive methods in economic dynamics, Harvard University Press, for de nitions.

107 7.3. DOUBLE INFINITY 101 ii. and P i xi! i = 0 Note that, under our assumptions, 1X p t! i t < 1; for any i 0 t=0 X! i 1 i While the de nition of Arrow-Debreu equilibrium is unchanged once an in nite number of agents is allowed for, its welfare properties change substantially. As always, we say that x is a Pareto optimal allocation if there does not exist an allocation y 2 `1+ such that U i (y i ) U i (x i ) for any i 0 (strictly for at least one i), and IX y i! i = 0, i=1 Does the First welfare theorem hold in this economy? Is any Arrow-Debreu equilibrium allocation x Pareto Optimal? Well, a quali cation is needed. First welfare theorem for double in nity economies. Let (x ; p ) be an Arrow-Debreu equilibrium. If at equilibrium aggregate wealth is nite,! 1X X < 1; t=0 then the equilibrium allocation x Pareto Optimal. p t The important result here is that the converse does not hold. An equilibrium with in nite P 1 t=0 p t ( P i!i t) might not display Pareto optimal allocation. In other words, with double in nity of agents the proof of the First welfare theorem breaks unless P 1 t=0 p t ( P i!i t) < 1 Let s see this. Proof. By contradiction. Suppose there exist a y 2 `1+ which Pareto dominate x Then it must be that 1X p t (yt i! i t) 0; strictly for one i 0 t=0 i! i t

108 102 CHAPTER 7. INFINITE-HORIZON ECONOMIES Summing over i 0, even though X y i ; X! i are nite, P 1 t=0 p t ( P i!i t) i0 i0 might not be. In this case, we cannot conclude that!! 1X X 1X X > t=0 p t i y i t t=0 p t The quali cation P 1 t=0 p t ( P i!i t) < 1 is however su cient to obtain P 1 t=0 p t ( P P i yi t) > 1 t=0 p t ( P i!i t) and hence the contradiction i! i t X yt i > X i i! i t, for some t 0 Importantly, the proof has no implication for the converse. In other words, the proof is silent on P 1 t=0 p t ( P i!i t) < 1 being necessary for Pareto optimality. We shall show by example that - P 1 t=0 p t ( P i!i t) < 1 is not necessary for Pareto optimality; that is, there exist economies which have Arrow-Debreu equilibria whose allocations are Pareto optimal and nonetheless P 1 t=0 p t ( P i!i t) is in nite; - there exist economies which have Arrow-Debreu equilibria whose allocations are NOT Pareto optimal. Remark 77 The double in nity is at the root of the possibility of ine cient equilibria in these economies. The First welfare theorem in fact holds with nite agents i 2 I even if the economy has an in nite horizon, t 0. This we know already. But it follows trivially (check this!) from the proof above that the First welfare theorem holds for nite horizon economies, t = 0; 1; ; T even if populated by an in nite number of agents i 0; provided of course P i!i Overlapping generations economies We will construct in this section a simple overlapping generations economy which displays i) Arrow-Debreu equilibria with Pareto ine cient allocations (aggregate wealth is necessarily in nite in this case, ii) Arrow-Debreu equilibria with in nite aggregate wealth whose allocations are Pareto e cient.

109 7.3. DOUBLE INFINITY 103 The economy is deterministic and is populated by two-period lived agents. An agent s type i 0 indicates his birth date (all agent born at a time t ae identical, for simplicity). Therefore, the stochastic process for the endowment of an agent i 0;! i, satis es! i t > 0 for t = i; i + 1 and = 0 otherwise. We assume there is also an agent i = 1 with! 1 0 > 0. The utility functions are as follows U i (x i ) = u(x i i) + (1 )u(x i i+1); for any i 0; 6 U 1 (x 1 ) = u(x 1 0 ) Arrow-Debreu equilibria are easily characterized for this economy. Autarchy The economy has a unique Arrow-Debreu equilibrium (x ; p ) which satis es and x t t =! t t; x t t+1 =! t t+1; for any t 0; x0 1 =! 0 1 p 0 = 1; and p t+1 = (1 )u0 (! t t+1) ; for t 0 p t u 0 (! t t) The restriction p 0 = 1 is the standard normalization due to the homogeneity of the Arrow-Debreu budget constraint. Proof. First of all, x 1 0 =! 1 0 follows directly from agent i = 1 s budget constraint. Then, market clearing at time t = 0 requires x0 1 + x 0 0 =! 0 1 +! 0 0 Substituting x0 1 =! 0 1 ; we obtain x 0 0 =! 0 0 We can now proceed by induction to show that x t t =! t t implies x t t+1 =! t t+1 (using the budget constraint of agent t), which in turn implies x t+1 t+1 =! t+1 t+1 (using the market clearing condition at time t + 1 The characterization of equilibrium prices then follows trivially from the rst order conditions of each agent s maximization problems.

110 104 CHAPTER 7. INFINITE-HORIZON ECONOMIES It is convenient to specialize this economy to a simple stationary example where! 0 1 = ;! t t = 1 ;! t t+1 = ; for any t 0; u(x) = ln x Then, at the Arrow-Debreu equilibrium, x 1 0 = ; x t t = 1 ; x t t+1 = ; for any t 0; t 1 p 1 t = 1 A symmetric Pareto optimal allocation is as follows (it is straightforward to derive, once symmetry is imposed) at the Arrow-Debreu equilibrium, It follows then that, x 1 0 = ; x t t = 1 ; x t t+1 = ; for any t 0 The Arrow-Debreu equilibrium (autarchy) is Pareto e cient if and only if t=0 i t=0 Proof. The case = is obvious. If < all agents i 0 prefer the allocation x i i = 1 ; x i i+1 = to autarchy, but agent i = 1 prefers x0 1 = to x0 1 =. Autarchy is then Pareto e cient. If otherwise > all agents i 0 prefer the allocation x i i = 1 ; x i i+1 = to autarchy, and agent i = 1 as well prefers x0 1 = to x0 1 =. Autarchy is then NOT Pareto e cient. Furthermore, note that in this economy, the aggregate endowment P i0!i t = 1; for any t 0; and hence the value of aggregate wealth is! 1X X 1X 1X t 1 p t! i t = p 1 t = 1 It follows that t=0 The value of aggregate wealth is nite when < a

111 7.3. DOUBLE INFINITY 105 and autarchy is e cient. On the other hand, the value of aggregate wealth is in nite when a In this case, autarchy is e cient if = a and ine cient when > a Note that we have in fact shown what we were set to from the beginning of this section i) Arrow-Debreu equilibria with Pareto ine cient allocations (aggregate wealth is necessarily in nite in this case, ii) Arrow-Debreu equilibria with in nite aggregate wealth whose allocations are Pareto e cient. Bubbles We have seen previously that, when the value of the aggregate endowment is in nite, bubbles in in nitely-lived positive net supply assets might arise. We shall now show that this is the case in the overlapping generation economy we have just studied. Interestingly, it will turn out that bubbles, in this economy might restore Pareto e ciency. (This is special to overlapping generations economies, by no means a general result). Suppose agent i = 1 is endowed with a in nitely-lived asset, in amount m The asset pays no dividend ever it is then interpreted to be at money.any positive price for money, therefore, must be due to a bubble. Let the price of money, in units of the consumption good at time t = 0; at time t; be denoted qt m We continue to normalize p 0 = 1 The Arrow-Debreu budget constraints in this economy for agent i = t 0 can be written as follows 1X p t (x t! i t) = p t (x t t! t t) + p t+1 (x t t+1! t t+1) = 0; t=0 or, denoting s t the demand for money of agent i = t 0 p t x t t + q m t s t =! t t p t+1 (x t t+1! t t+1) = q m t+1s t The budget constraint for agent i = 1 is x 1 0 =! q m 0 m

112 106 CHAPTER 7. INFINITE-HORIZON ECONOMIES Turning back to the stationary economy example where! 0 1 = ;! t t = 1 ;! t t+1 = ; for any t 0; u(x) = ln x; we can easily characterize equilibria. Suppose in particular that > and autarchy is ine cient. We restrict the analysis to following stationarity restriction qt m = q m for any t 0 The autarchy allocation is obtained as an Arrow-Debreu equilibrium of this economy, for q m = 0 and p t+1 = 1 p t 1 The Pareto optimal allocation x 1 0 = ; x t t = 1 ; x t t+1 = ; for any t 0 is also obtained an Arrow-Debreu equilibrium of this economy, for q m = m and p t+1 p t = 1 It is straightforward to check that these are actually Arrow-Debreu equilibria of the economy. Note that Pareto optimality is obtained at equilibrium for q m > 0; that is, when money has positive value and hence a bubble exist. At this equilibrium, prices p t are constant over time and hence the value of the aggregate wealth is in nite. Remark 78 The stationary overlapping generation example introduced in this section has been studied by Samuelson (1958). The fundamental intuition for the role of money in this economy is straightforward money allows any agent i 0 to save by acquiring money (in exchange for goods) at time t = i from agent i 1 and then transfering the same amount of money (in exchange for the same amount of goods) to agent i + 1 at time t = i + 1 Money, in other words, serves the purpose of a pay-as-you-go social security system; and in fact such a system, implemented by an in nitely lived agent (like a benevolent government), could substitute for money in this economy. (Try and set it up formally!) Furthermore, note that, for money to have value in

113 7.4. DEFAULT 107 this economy, we need >. Consistently with the interpretation of money as a social security mechanism, the condition > is interpreted to require that i) the endowment of any agent i 0 at time t = i + 1, ; be relatively small and that ii) any agent i 0 discounts relatively little the future, that is, = 1 is high. Remark 79 Large-square economies, with a continuum of agents and commodities, are studied by Kehoe-Levine-Mas Colell-Woodford (1991), "Grosssubstitutability in large-square economies," Journal of Economic Theory, 54(1), Problem 80 Consider our basic Overlapping Generation economy with no money, when specialized to specialize this economy to a simple stationary example where! 1 0 = ; and! t t = 1 ;! t t+1 = ; for any t 0; U 1 (x 1 ) = ln x 1 0 ; and U t (x t ) = ln x t t + ln x t t+1; for any t 0. Assume however that population grows at rate n > 0 at time t = 0 the economy is populated by a mass 1 of agents of generation i = 1 and a mass 1 + n of agents of generation i = 0; and so on. Suppose also that a social security system can be imposed on the economy. It works as follows any generation i 0 pays to the social security system 0 1 units of the consumption good at time t = i and receive b = (1 + n) units of the consumption good at time t = i For given 0 1 solve for a competitive equilibrium (quantities and prices). 2. Derive a condition on which guarantees that the equilibrium allocation Pareto dominates autarchy (with no social security system). 3. When this conditions is satis ed, characterize the subset of s which induce equilibrium allocations which are Pareto optimal. Does aggregate wealth have nite value at such Pareto optimal allocations. 7.4 Default Consider the following economy, populated by a nite set of in nitely-lived agents, i 2 I Let N = X 1 t=0s t be the set of nodes of the tree, s 0 the root of

114 108 CHAPTER 7. INFINITE-HORIZON ECONOMIES the tree, s t an arbitrary node of the tree at time t. Use s t+ js t to indicate that s t+ is some successor of s t, for > 0 At each node, there are S securities in zero-net supply traded with linearly independent payo s nancial markets are complete. Without loss of generality we let the nancial assets be a full set of Arrow securities A = I S (the S dimensional identity matrix). Let q N! R S be the mapping de ning the vector of asset prices at each node s t. At each node s t, each households has an endowment of numeraire good of! i (s t ) > 0; aggregate endowment is then!(s t ) = X i2i! i (s t ) 0 at each node s t. At the root of the tree the expected utility of agent i 2 I is U i (x; s 0 ) = u i (x(s 0 ) + 1X X t prob(s t s 0 ) u i (x(s t s 0 )) s t js 0 t=1 Any agent i 2 I; at any node s t 2 N can default (more precisely cannot commit not to default). If he does default at a node s t, he is forever kept out of nancial markets and is therefore limited to consume his own endowment! i (s t+ js t ) at any successor node s t+ js t 2 N An agent i 2 I; therefore, will default at a node s t on allocation x i if U i (x i ; s t ) < U i (! i ; s t ) The notion of equilibrium we adopt for this economy will be the one used by Prescott and Townsend for economies with asymmetric information where a no-default constraint U i (x i ; s t ) U i (! i ; s t ); for any s t 2 N; takes the place of the incentive compatibility constraint. In particular we shall choose the notion of equilibrium introduced in the remark, where these constraints are interpreted as constraints on the set of tradable allocation (rather than rationality restrictions on price conjectures). Let ((x i ) i ; p ) be an Arrow-Debreu Equilibrium if x i ; for any agent i 2 I; solves max x i U i (x i ; s 0 )

115 7.4. DEFAULT 109 s.t. and markets clear p (x i! i ) = 0; and U i (x i ; s t ) U i (! i ; s t ); for any s t 2 N X x i! i = 0 i This problem is formulated and studied by T. Kehoe and D. Levine, Review of Economic Studies, Note that i) the value of aggregate wealth for each agent i 2 I is necessarily nite at equilibrium p! i < 1 (and I is nite by assumption) ii) the set of constraints which guarantee no-default at equilibrium do not depend on equilibrium prices U i (x i ; s t ) U i (! i ; s t ); for any s t 2 N Let now incentive constrained Pareto optimal allocations be those which solve max i U i (x i ; s 0 ) s.t. (x i ) i2i X i2i X x i! i = 0; and i2i U i (x i ; s t ) U i (! i ; s t ); for any s t 2 N; for some 2 R I + such that P i2i i = 1 It follows easily then that Any Arrow-Debreu Equilibrium allocation of this economy is constraint Pareto optimal. 7 7 In fact, in the Kehoe and Levine paper, L > 1 commodities are traded at each node and the conquence of default is that trade is restricted to spot markets at any future node. In this economy the non-default constraints depend on spot prices and Arrow-Debreu equilibrium allocations are not incentive constrained optimal.

116 110 CHAPTER 7. INFINITE-HORIZON ECONOMIES Consider now a nancial market equilibrium for this economy. The objective is to capture the no-default constraints by means of appropriate borrowing constraints. In other words we want to set borrowing constraints as loose as possible provided no agent would ever default. To this end we need to write exploit the recursive structure of the economy. Let i (s t+1 js t ) denote the portfolio of Arrow security paying o in node s t+1 acquired at the predecessor node s t The borrowing constraints an agent i 2 I will face at node s t will then take the form i (s t+1 s t ) B i (s t+1 s t ); for any s t+1 s t ; and B i (s t+1 js t ) must be chosen to be as loose as possible provided it induces agent i 2 I not to default at any node s t+1 js t Formally, the value function of the problem of agent i 2 I at any node s t 2 N when i) the agent enters state s t 2 N with i units of the Arrow security which pays at s t 2 N; and the agent faces borrowing limits B i (s t+1 js t ), can be constructed as follows V i ( i ; s t ) = max x i ; i (s t+1 js t ) u i (x i ) + P s t+1 js t prob(s t+1 js t )V i ( i (s t+1 js t ); s t ) st x i + P s t+1 js t i (s t+1 js t )q(s t+1 js t ) = i +! i (s t+1 js t ); and i (s t+1 js t ) B i (s t+1 js t ) The condition that the borrowing limits B i (s t+1 js t ) be as loose as possible is then endogenously determined as V i (B i (s t+1 s t ; s t ) = U i (! i ; s t+1 ). It can be shown that V i (B i (s t+1 js t ; s t ) is monotonic (decreasing) in its rst argument, for any s t 2 N Borrowing limits B i (s t+1 js t ) are then uniquely determined at any node. Note that they are determined at equilibrium, however, as they depend on the value function V i (; s t ) Market clearing brings no surprises We can now show the following. The autarchy allocation X x i! i = 0 i x i (s t ) =! i (s t ); for any s t 2 N;

117 7.4. DEFAULT 111 can always be supported as a nancial market equilibrium allocation with borrowing constraints B i (s t+1 s t ) = 0; for any s t 2 N Note in particular that V i (B i (s t+1 js t ; s t ) = U i (! i ; s t+1 ) is satis ed at this equilibrium. Let q(s t+2 js t ) = q(s t+2 js t+1 )q(s t+1 js t ); for t = 0; 1; We can then show the following. Any nancial market equilibrium allocation (x i ) i2i whose supporting prices q (s t+1 js t ) satisfy! i s 0 + X X q (s t+1 s 0 )! i (s t+1 s 0 ) < 1 t0 s t+1 js 0 is an Arrow-Debreu equilibrium allocation supported by prices p (s 0 ) = 1; p (s t ) = q (s t s 0 ) This can be proved by repeatedly solving forward the nancial market equilibrium budget constraints, using the relation between Arrow-Debreu and nancail market equilibrium prices in the statement. The solution is necessarily the Arrow-Debreu budget constraint only if lim T!1 q (s T js 0 ) = 0, which is the case if the value of aggregate wealth is nite at the nancial market equilibrium. In this case, then the nancial market equilibrium allocation (x i ) i2i is constrained Pareto optimal. When (x i ) i2i is not autarchic (easy to show by example that robustly, such nancial market equilibrium allocations exist) it Pareto dominates autarchy (as autarchy is always budget feasible and satis es the no-default constraint). In this case the autarchy allocation is not constrained Pareto optimal and the value of some agent s wealth is in nite at the autarchy equilibrium. 8 8 A?? by Gaetano Bloise and Pietro Reichlin (2009) proves that in this economy nancial market equilibria with in nite value of aggregate wealth other than autrachy exist. Some of these equilibria are e cient and some are not. The machinery used to prove these results is related to the classical analysis of Overlapping Generations and Bewley models. Not surprisingly, bubbles also arise.

118 112 CHAPTER 7. INFINITE-HORIZON ECONOMIES Problem 81 Show that an the autarchy allocation asset prices must satisfy q (s t+1 s t ) max i2i u i0 (! i (s t+1 )) u i0 (! i (s t )) Problem 82 Write down the nancial market equilibrium notion in which rational price conjecture substitute borrowing constraints. How do prices look like? Problem 83 Agents live two periods, t = 0; 1; and consume a single consumption good only in period 1. Uncertainty is purely idiosyncratic. Each agent faces a (date 1) endowment which is an identically and independently distributed random variable! = (! H ;! L ) ; with! H >! L Agents come in 2 types, e = h; l; which are di erentiated in terms of the probability distribution of their endowments e s is the probability of state s for agents of type e Let h H > l H (Notation is set so that H is the "high state" (the state where endowment is high) and h is the "high type" (the type with a higher probability of the high state). The fraction of h types in the economy is 1 Types are only 2 privately observable. (Yes, this is an adverse selection economy). Agents preferences are represented by a von Neumann-Morgenstern utility function of the following form X e s ln x s s2s 1. Write the incentive constrained optimal planning problem for the planner (with equal weight across types). 2. Write the de nition of competitive equilibrium with rational conjectures (where prices are function of the allocations). Anything "strange" in these prices?

119 Chapter 8 To add 1. Constantinedes and Du e on Asset Pricing with incomplete markets 2. aggregation in Angeletos (two periods and in nite horizon) 3. Citanna-Siconol on decentralization of adverse selection 4. papers by Townsend with Weerachart Kilenthong <wkilenthong@gmail.com>; http//cier.uchicago.edu/papers/working/moral%20hazard_submitted.pdf 5. Papers by Meier-Minelli-Polemarchakis, Correia-da-Silva on two-sided asymmetric information; write them in the context of a nancial market economy (see also Tirelli et al. on ET); see also Acharya-Bisin 6. Paper by Aguiar and Amador - On small open economy with debt and taxes nice example of ine ciency in kehoe Levine with 2 goods (labor and consumption) 7. Boldrin and Levine on activity analysis; and Long and Plosser on multisector models of real business cycles 8. MacKenzie s constant return to scales is withput loss of generality - managerial rents - result. Linked to Makowski-Ostroy eq m concept. 113

120 114 CHAPTER 8. TO ADD

121 Chapter 9 Appendix 1 A quick review of consumer and producer theory The consumption set X is the set of admissible levels of consumption of L existing commodities. In this section we shall assume X = R L +; with generic element x X is then a convex set, bounded below. For any x; y 2 X; we say x > y if x l y l ; for any l = 1; 2; ; L; and x l > y l for at least one l = 1; 2; ; L Assumption 2 The consumer has a utility function 1 U R L +! R which satis es the following properties, Strong Monotonicity y > x ) U(y) > U(x); for any x; y 2 R L +; 1 In these notes we shall adopt the utility function as a primitive. That is, we shall assume that any agent s underlying preference ordering % on X (is complete, transitive, and continuous; so that it) can be represented by a utility function; see Rubinstein (2009). A preference ordering % on X which (is not continuous and hence it) cannot be represented by a utility function is the Lexicographic ordering x % y if x 1 y 1 or x 1 = y 1 and x l y l ; for l = 2; ; L 115

122 116CHAPTER 9. APPENDIX 1 A QUICK REVIEW OF CONSUMER AND PRODUCER T Strict Convexity U R L +! R is strictly quasi-concave, that is, U (x + (1 ) y) U(x) + (1 ) U(y); for any x; y 2 R L + and 2 [0; 1]; and U (x + (1 ) y) > U(x) + (1 ) U(y); if x 6= y and 2 (0; 1) Di erentiability U R L +! R is C 2 on the interior of its domain. Let ru denote the gradient of the map U R L +! R Exploiting di erentiability strong monotonicity can be equivalently written as while strict quasi-concavity as ru(x) 2 R L ++; for any x 2 R L ++; vr 2 U(x)v < 0 for any v 6= 0 2 R L such that ru(x)v = 0; for any x 2 R L ++. Common properties of utility functions at times studied in applications include Quasi-linearity U(x) = x 1 + (x 2 ; ; x L ); for some R L + 1! R satisfying strong monotonicity, strict convexity, and di erentiability. Homotheticity U(x) = U(x); for any 2 R ++ Examples of homothetic utility functions include Cobb Douglas U(x) = x 1 x 1 2 ;CES U(x) = x 1 + x 2. Di erentiability is violated, for instance, by Leontief preferences U(x) = min f; l x l; g ; for some 2 R L Consumer theory Markets are competitive, that is, the consumer takes market prices as given, independent of his decisions (each agent is a price taker). In addition we consider the case where prices are linear unit price p l of each commodity l is xed, independent of level of individual trades (and the same for all agents);

123 9.1. CONSUMER THEORY 117 prices are non-negative this is justi ed under free disposal, that is, when agents can freely dispose of any amount of any commodity; markets are complete for each commodity l in X there is a market where the commodity can be traded. Given wealth level m, the budget set is ( B(p; m) = x 2 X p x = ) LX p l x l m l=1 The budget set is convex, compact, and non-empty for p 2 R L ++; m 0Furthermore, the budget set is homogeneous of degree 0 B(p; m) = B(p; m); for all > 0 Consider the consumer s utility maximization problem, max U(x) x2b(p;m) For any p 2 R L ++; m 0; under our assumptions on U(x), by the Maximum theorem, a solution of the consumer s problem exists. Furthermore, the solution of the consumer s problem for every p; m induces the consumer s demand correspondence x R L ++ R +! R L +; x(p; m) Strict quasi-concavity of U(x); again by the Maximum theorem, implies that x(p; m) is a in fact a continuous function. Proposition 84 The individual consumer s demand x(p; m) satis es the following properties Homogeneity of degree zero in p; m x(p; m) = x(p; m) for all p 2 R L ++; > 0; Continuity x(p; m) is a continuous function in p; m; Walras Law px(p; m) = m; WARP for any (p; m); (p 0 ; m 0 ) 2 R L ++ R + such that x = x(p; m) 6= x 0 = x(p 0 ; m 0 ) if px 0 m; then p 0 x > m 0

124 118CHAPTER 9. APPENDIX 1 A QUICK REVIEW OF CONSUMER AND PRODUCER T Proof. These properties are straightforward consequences of the assumptions. Homogeneity of degree zero is a consequence of homogeneity of degree zero of the budget set. Continuity follows from the Maximum theorem under strict quasi-concavity of U(x) (weak quasi-concavity would induce an upperhemi-continuous, convex valued correspondence x R L ++ R +! R L + WARP and Walras law follow easily from strict monotonicity. It is important to notice that WARP does not imply the (uncompensated) Law of demand, that is, (p p 0 ) (x(p; m) x(p 0 ; m 0 )) 0; for m = m 0 Note also that the (uncompensated) Law of demand is equivalently written, exploiting di erentiability, as dpd p x(p; m)dp 0 Therefore, WARP does not imply that D p x(p; m) is negative semi-de nite. In general in fact, D p x(p; m) is NOT negative semi-de nite. If preferences are homothetic, however, the individual consumer s demand x(p; m) does satisfy the (uncompensated) Law of demand. Furthermore, in this case, the individual consumer s demand x(p; m) is homogeneous of degree 1 in m x(p; m) = x(p; m) for all p 2 R L ++; m; > 0 2 WARP on the other hand does imply the (compensated) Law of demand (p p 0 ) (x(p; m) x(p 0 ; m 0 )) 0; for m m 0 = (p p 0 )x(p 0 ; m 0 ) Once again, exploiting di erentiability we can express the (compensated) Law of demand through the properties of the Slutsky matrix of compensated price e ects. Consider the following compensated price change The induced demand change is dp; dm dm = x dp dx = D p x(p; m)dp + D m x(p; m)dm = = D p x(p; m) dp + D m x(p; m)(xdp) 2 This is a consequence of the fact that homotheticity implies that marginal rates of substitution, @x l 0 ; for any l; l 0 ; are invariant with respect to expansions along rays from

125 9.1. CONSUMER THEORY 119 De ne then the Slutsky matrix of compensated price e ects as WARP then implies the following. S(p; m) = D p x(p; m) + D m xx Proposition 85 S(p; m) is negative semi de nite; that is, dps(p; m)dp 0; for any dp 2 R L Finally, any solution of the consumer s problem satis es the following system of Kuhn Tucker necessary and su cient conditions. In the case of interior solutions, these are ru p = 0 m p x = 0; for some > 0, the Lagrange multiplier associated to the budget constraint. Local properties of x(p; m) can then be obtained using the Implicit Function theorem on the previous rst order conditions (foc s) r 2 U p p T 0 dx d = x T dp Note that the second order conditions (soc s) guarantee that r 2 U p p T is invertible 0 dm when evaluated at an (x; ; p) satisfying foc s. A simple proof of this statement follows for completeness. Proof. We show z) 2 R L+1, (y; z) 6= 0 such that r 2 U p p T 0 y z r 2 Uy pz = = 0 py 3 r 3 For any strict quasi concave and C 2 function f R L 2 f rf +! R; the matrix rf T 0 r 2 U p is called bordered Hessian and has a non-zero determinant. Note that p T is 0 the bordered Hessian of the Lagrangian.

126 120CHAPTER 9. APPENDIX 1 A QUICK REVIEW OF CONSUMER AND PRODUCER T By contradiction, suppose such a (y; z) exists. Then pre-multiplying r 2 Uy pz by y, yields yr 2 Uy = ypz But ypz = 0; and hence yr 2 Uy = 0 But the second order conditions (soc s; in turn guaranteed by strict quasi-concavity of the Lagrangian, U(x) (px m)), require that vr 2 Uv < 0 on all v 6= 0 such that (ru p) v = 0; a contradiction Duality Let V R L ++ R + ; V (p; m), be de ned by V (p; m) = U(x(p; m)) V (p; m) is the indirect utility function. Proposition 86 The indirect utility function V (p; m) satis es the following (p; m)=@m = > 0 the consumer s marginal utility of wealth equals the shadow value of relaxing the budget constraint; V (p; m) is homogenous of degree zero in p; (p; m)=@p l 0 for all l; p >> 0; V (p; m) is quasi-convex in p the lower contour set p V (p; m) V convex. is Proof. The properties are straighforward consequences of the assumptions on U(x) and the properties of x(p; m) We leave them to the reader, except quasi-convex in p; which is proved as follows. Take any pair p 0 ; p 00 such that V (p 0 ; m); V (p 00 ; m) V and consider ^p = p 0 + (1 )p 00 for 2 [0; 1]. Note that for all x such that ^p x m we must have either p 0 x m and/or p 00 x m; thus U(x) v. Consider the consumer s cost minimization problem, min x2x s.tu(x) u

127 9.1. CONSUMER THEORY 121 A solution exists for all u U(0) and p 2 R L ++ The solution h(p; u); with h R L ++ R! R L +; is typically referred to as compensated - or Hicksian - demand. By the soc s, D p h(p; u) is symmetric, negative semi-de nite. We say that Hicksian demands h(p; u) satisfy the law of demand. Let e R L ++R! R + ; e(p; u) = ph(p; u) de ne the expenditure function. Proposition 87 The expenditure function e(p; u) has the following u)=@u > u)=@p l 0 for all l = 1; ; L; e(p; u) is homogeneous of degree one in p; e(p; u) is concave in p Proof. The properties are straighforward consequences of the Maximum theorem and the assumptions on U(x). We leave them to the reader, except quasi-concavity in p; which is proved as follows. For any pair p 0 ; p 00, consider ^p = p 0 + (1 )p 00 for 2 [0; 1]. Then, for any u, e(^p; u) = ^p h(^p; u) = p 0 h(^p; u) + (1 )p 00 h(^p; u); which is in turn e(p 0 ; u) + (1 )e(p 00 ; u) For all p 2 R L ++, m > 0; u > U(0), the following identities hold x(p; m) = h(p; u) for u = V (p; m) and e(p; u) = m (9.1) By the envelope theorem, the compensated demand can be obtained from the expenditure function h(p; u) = D p e(p; u) Hence the properties of h(p; u) can also be obtained from those of e(p; u). Similarly, di erentiating the equation de ning the indirect utility, V (p; m) = U(x(p; m) (px m) with respect to p, we obtain Roy s identity r p V (p; m) = x(p; m) = r m V (p; m) x(p; m) A brief summary of the duality relationships can be helpful

128 122CHAPTER 9. APPENDIX 1 A QUICK REVIEW OF CONSUMER AND PRODUCER T Figure 9.1 Brief summary of duality relations

129 9.1. CONSUMER THEORY 123 x(p; m)! h(p; u) Slutsky D p x(p; u) D m x x T = D p h(p; u) V (p; m)! x(p; m) Roy s identity x(p; m) = 1 r m V (p; m) r pv (p; m) e(p; u)! h(p; u) Expenditure function properties h(p; u) = r p e(p; u) V (p; m) $ e(p; u) Utility-expenditure duality, for given p 2 R L ++ V (; m) = e 1 (; m); e(; u) = V 1 (; u) x(p; m) $ h(p; u) Marshallian-Hicks demand duality x(p; m) = h(p; V (p; m)) h(p; u) = x(p; e(p; u)) Aggregate demand It is useful to study in detail the properties of aggregate demand, as a function of wealth. With some abuse of notation, let introduce the notation to index agents, i = 1; ; I. Let the wealth of agent i be denoted m i Fix the distribution of wealth as follows, m i = i m for some given i 0, for all i, X i i = 1 The Marshallian demand of any agent i is then denoted x i (p; m i ) and x(p; m) = X i2i x i (p; m i ) is the aggregate demand.

130 124CHAPTER 9. APPENDIX 1 A QUICK REVIEW OF CONSUMER AND PRODUCER T Aggregate demand trivially inherits several properties from individual demand, i.e., di erentiability, homeogeneity of degree 0, Walras Law. But does x(p; m) satisfy WARP? Not, typically. Take (p; m) and (p 0 ; m 0 ) such that x (p 0 ; m 0 ) 6= x (p; m) and p x (p 0 ; m 0 ) m It is immediate to see that the following inequality can still hold p 0 x (p; m) m 0 (because px (p 0 ; m 0 ) m does not imply px i (p 0 ; m i0 ) m i for all i; similarly for p 0 x (p; m) m 0.) A su cient condition for aggregate demand x (p; m) to satisfy WARP is that individual Marshallian demands satisfy the (uncompensated) law of demand, that is, substitution e ects prevail over income e ects. Proposition 88 Suppose x i (p; m i ) satis es the (uncompensated) law of demand, that is, D p x i (p; m i ) is negative semi-de nite for all i Then x (p; m) also satis es the (uncompensated) law of demand and hence WARP. 9.2 Producer theory We shall study the production activity of rms operating in competitive markets. A production plan is a y 2 R L, interpreted as the net output of the L goods y i < 0 input y i > 0 output The production set is the set of (technologically feasible) production plans Y R L Whenever the commodities which are outputs in the production set are xed, O f1; ; Lg (and hence also the complementary set of those which are inputs), the outer boundary of Y can typically be represented by a (continuous)

131 9.2. PRODUCER THEORY 125 production function, describing the maximal output level attainable for any level of inputs. In the case where O = f1g ; e.g., y 1 = f(z) i (y 1 ; z) > y 1 (y 0 1; z) 2 Y We assume that Y and and f R L 1 +! R + satisfy the following assumptions Regularity Y is nonempty, closed, Y \ R L + = f0g (no free lunch and possibility of inaction), and it satis es free disposal y 2 Y and y 0 y ) y 0 2 Y Correspondingly, f R L 1 +! R + is monotonically increasing. Convexity Y is strictly convex y; y 0 2 Y ) y + (1 )y 0 2 inty Correspondingly, f R L 1 +! R + is strictly concave (has decreasing returns to scale). (Y is convex, y 2 Y ) y 2 Y for all 2 [0; 1] corresponds to f R L 1 +! R + is concave - has non-increasing returns to scale.) Any rm chooses a production plan so as to maximize pro ts s.t. y 2 Y max p y Existence of a solution requires conditions ensuring that Y is bounded above. For every p 0, a solution induces a net supply correspondence y(p) Proposition 89 The net supply correspondence y(p) satis es the following properties

132 126CHAPTER 9. APPENDIX 1 A QUICK REVIEW OF CONSUMER AND PRODUCER T y(p) is homogeneous of degree 0 in p; y(p) is a convex-valued correspondence (a single valued function if f R L 1 +! R + is strictly concave). The value of the solution is a pro t function (p) = p y(p) Proposition 90 The pro t function (p) satis es the following properties (p) is homogeneous of degree 1 in p; (p) is a convex function. Proof. The properties are straighforward consequences of the properties of y(p) We leave them to the reader, except convexity; which is proved as follows. Take any pair p 0 ; p 00 and consider ^p = p 0 + (1 )p 00 for 2 (0; 1). Note that (^p) = y(^p) (p 0 + (1 )p 00 ) y(p 0 ) p 0 + (1 )y(p 00 ) p 00 = (p 0 ) + (1 )(p 00 ) Suppose Y is a convex cone, y 2 Y ) y 2 Y for all > 0; that is, the technology has constant returns to scale Correspondingly, the production function f R L + 1! R + is homogeneous of degree 1 in its arguments and, - y 2 y(p) ) y 2 y(p) for all > 0; - (p) = 0 for all p Whenever f is di erentiable, any solution of the rm s problem satisfy the following system of foc s (stated here for the case of interior solutions) p 1 Df = w Focs are also su cient if f is concave. Assume f is continuously di erentiable and strictly concave. Applying the Implicit Function theorem to foc s, we obtain D w z = 1 p 1 D 2 f 1 is symmetric, negative de nite is Furthermore, by the envelope theorem, y(p) = D p, so that D p y = D 2

133 9.2. PRODUCER THEORY symmetric, - positive - recall z = (y 2 ; ; y L )! - semide nite (by the convexity of ) and - such that D p y p = 0 (by the homogeneity of y(p)). p1 y1 Let p = and y = The input level z which solves the rm s w z choice problem also solves the following problem min z2r L 1 + C = wz st f(z) y 1 This is perfectly analogous to expenditure minimization problem of consumer. Hence we know that - C(w; y 1 ) is concave in w and such 1 > 0 l 0; l = 1; ; L 1; - z(w; y 1 ) = D w C exhibits the properties of a compensated demand function.

134 128CHAPTER 9. APPENDIX 1 A QUICK REVIEW OF CONSUMER AND PRODUCER T

135 Chapter 10 Appendix 2 Some useful math 10.1 Separating hyperplane theorems Theorem 91 (Separating hyperplane) Suppose A; B R N are convex disjoint sets. Then there exist a p 2 R N ; p 6= 0; and a c 2 R such that px c; for any x 2 A; and py c; for any y 2 B Theorem 92 (Supporting hyperplane) Suppose B R N is a convex set and suppose x =2 int(b). Then there exist a p 2 R N ; p 6= 0; such that px py, for any y 2 B 10.2 Fixed point theorems 10.3 Di erential topology Theorem 93 (Inverse function theorem - Local). Let f R n! R n be C 1 If Df has rank n; at some x 2 R n, there exist an open set V R n and a function f 1 V! R n such that f(x) 2 V and f 1 (f(z)) = z in a neighborhood of x 129

136 130 CHAPTER 10. APPENDIX 2 SOME USEFUL MATH Figure 10.1 Parametrization of a 1-manifold X

137 10.3. DIFFERENTIAL TOPOLOGY 131 De nition 94 A subset X R m is a smooth manifold of dimension n if for any x 2 X there exist a neighborhood U X and a C 1 function f U! R m such that Df has rank n in the whole domain. Let f(u) = V A smooth manifold of dimension n is then locally parametrized by a restriction of the function f 1 on the open set V \ R n f0g m n, in the sense that f 1 maps V \ R n f0g m n onto U; a neighborhood of x on X. Example 95 An example of a 1-manifold of R 2 is S = x 2 R 2 (x1 ) 2 + (x 2 ) 2 = 1, the circle. An explicit parametrization for S can be constructed as follows. Seeing a restriction of f 1 on the open set V \ R n f0g m n as a map i R n! R m ; the following four maps are su cient to parametrize S 1 (x 1 ) = 2 (x 1 ) = 3 (x 2 ) = 4 (x 2 ) = x 1 ; q1 (x 1 ) 2 if x 2 > 0 x 1 ; q1 (x 1 ) 2 if x 2 < 0 q1 (x 2 ) 2 ; x 2 if x 1 > 0 q1 (x 2 ) 2 ; x 2 if x 1 < 0 De nition 96 Let f R m! R n ; m n; be C 1 f is transversal to 0, denoted f t 0; if Df has rank n for any x 2 R m such that f(x) = 0 Theorem 97 (Transversality). Let f R m ++! R n ++; m n; be C 1 and transversal to 0, f t 0 Decompose any vector x 2 R m ++ as x = x 1 x 2 ; with x 1 2 R m ++ n ; x 2 2 R n ++ Then D x2 f(x) has rank n for all x 1 in a Lebesgue measure-1 subset of R m n ++ De nition 98 A subset X R m is a smooth manifold with boundary of dimension n if for any x 2 X there exist a neighborhood U X and a C 1 function f U! R m 1 R + such that Df has rank n in the whole domain. The of X is de ned = f 1 (f0g m 1 R + )\X It can be shown that, if X R m is a smooth manifold with boundary of dimension n; is a smooth manifold (without boundary) of dimension n 1

138 132 CHAPTER 10. APPENDIX 2 SOME USEFUL MATH Figure 10.2 Parametrization of a 2-manifold with boundary