Financial Economics Lecture notes. Alberto Bisin Dept. of Economics NYU

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1 Financial Economics Lecture notes Alberto Bisin Dept. of Economics NYU September 25, 2010

2 Contents Preface ix 1 Introduction 1 2 Two-period economies Arrow-Debreu economies Financial market economies The stochastic discount factor Arrow theorem Aggregation Asset pricing Pareto optimality Corporate nance economies Asymmetric information economies Moral hazard insurance economies Corporate agency economies In nite-horizon economies Arrow-Debreu economies Financial markets economies Asset pricing Bubbles Pareto optimality Bewley economies Earning risk Investment risk Limit incomplete market economies Asymmetric information economies v

3 vi CONTENTS Lack of Commitment Economies

4 Preface These notes constitute the material for the graduate course on Financial Economics I at NYU. ix

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6 Chapter 1 Introduction The aim of these lecture notes is to provide an introduction to Asset pricing and Corporate nance in the context of general equilibrium models. While not standard, this approach is consistent and related to the practice of macroeconomics and it has the advantage of facilitating a coherent understanding of nance in both its asset pricing and of corporate nance manifestations. Besides providing an introduction to Financial economics, these notes have therefore also the ambition of suggesting a useful approach to theoretical and empirical research in the eld. We will rst (very quickly) review two-period economies, an environment in which concepts can be de ned and results proved with minimal notation. We will then study in nite horizon economies, the typical workhorse in - nance and macroeconomics. In the context of in nite horizon economies we will study exchange economies without commitment and with asymmetric information: these economies are rapidly becoming the frontier in macroeconomics. Finally, we shall also study production economies without commitment and with asymmetric information. While not much studied yet, these are the natural environments to integrate asset pricing and corporate nance. 1

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8 Chapter 2 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. As this chapter is intended as a review of known material, we shall not provide any proof, for which we shall refer the reader to Bisin (2010). 2.1 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 +; 3

9 4 CHAPTER 2 TWO-PERIOD ECONOMIES for any s = 0; :::S. Let u i : X will assume:! R denote agent i s utility function. We 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 (2.1) De nition An Arrow-Debreu equilibrium is a (x ; ) such that s=0 2: IX i=1 1: x i 2 arg max u i (x i ) s.t. 0(x i 0! i 0) + x i s! i s = 0, for any s = 0; 1; :::; S SX s(x i s s=0! i s) = 0; and 1 We write the budget constraint with equality. This is without loss of generality under monotonicity of preferences, an assumption we shall maintain.

10 2.1 ARROW-DEBREU ECONOMIES 5 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 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. 2 Theorem Any Arrow-Debreu equilibrium allocation x is Pareto Optimal. Recall that the proof exploits strict monotonicity of preferences and little else. Arrow-Debreu economies are easily extended to account for production. Suppose h = 1; ::::H (types of) rms produce at date 1 using as only input the amount k h of the numeraire good invested as capital at time 0: The output depends on k h 0 according to the function ys h = f h (k h ; s) 2 R L +; for any h = 1; ::::H. We assume that f h (k h ; s) is continuously di erentiable, increasing and concave in k h ; and it satis es f h (0; s) = 0; for any h = 1; ::::H and any s = 1; :::; S: De nition An Arrow-Debreu equilibrium of the economy with production is a (x ; k ; y ; ) such that SX 1: x i 2 arg max u i (x i ) s.t. 0(x i 0! i 0) + s(x i s! i s) = 0; 3: 2: (k h ; y h ) 2 arg max 0k h + IX i=1 x i 0! i 0 = HX h=1 ks h ; and IX i=1 s=0 SX sy h, s.t. ys h = f h (k h ; s); for any s = 1; :::; S s=0 x i s! i s = HX h=1 ys h, for any s = 1; :::; S The First and Second Welfare Theorems are straightforwardly extended. 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 successive notes.

11 6 CHAPTER 2 TWO-PERIOD ECONOMIES 2.2 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 ): To summarize the payo s of all the available assets, de ne the S J asset payo matrix 0 1 a 1 1 ::: a J 1 A = B ::: ::: A : a 1 S ::: aj 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 : Note the di erence. 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 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) and the asset structure A, any agent i picks a consumption vector x i 2 X and a portfolio z i 2 R J to s.t. max u i (x i ) p 0 (x i 0! i 0) = qz i p s (x i s! i s) = A s z i ; for s = 1; :::S: 3 The non-negativity restriction on asset payo s is just for notational simplicity. 4 Quantities will be row vectors and prices will be column vectors, to avoid the annoying use of transposes.

12 2.2 FINANCIAL MARKET ECONOMIES 7 De nition A Financial markets equilibrium is a (x ; z ; p ; q ) such that 2: 1: x i 2 arg max u i (x i ) s.t. p 0(x i 0! i 0) = q z i ; and p s(x i s! i s) = a s z i ; for s = 1; :::S; and furthermore IX IX x i! i s = 0, for any s = 0; 1; :::; S; and z i 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 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, zw; i that is, letting! i 1 = (! i 11; :::;! i 1S ), we can write! i 1 = w i 1 + Az i w and proceed straightforwardly by constructing the budget constraints and the equilibrium notion. 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 = 2 4 q A 3 5 : i=1

13 8 CHAPTER 2 TWO-PERIOD ECONOMIES De nition W satis es the No-arbitrage condition if there does not exist a 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 (No-Arbitrage theorem) < W > \ R S+1 + = f0g () 9^ 2 R S+1 ++ such that ^W = 0: First, observe that there is no uniqueness claim on the ^, just existence. Next, notice how ^W = 0 implies ^ = 0 for all 2< W > : It then provides a pricing formula for assets: ::: ::: ^W = B 0 q j + ^ 1 a j 1 + ::: + ^ S a j C S A = B 0 A ::: ::: and, rearranging, we obtain for each asset j, q j = 1 a j 1 + ::: + S a j S ; for Jx1 s = ^ s ^ 0 (2.2) Note how the positivity of all components of ^ is necessary to obtain (2.2). A few nal remarks to this section. 5 W z > 0 requires that all components of W z are 0 and at least one of them > 0:

14 2.2 FINANCIAL MARKET ECONOMIES 9 Remark An asset which pays one unit of numeraire in state s and nothing in all other states (Arrow security), has price s according to (2.2). Such asset is called Arrow security. Remark Is the vector ^ obtained by the No-arbitrage theorem unique? Notice how (??) 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 dimensional set. If we had S linearly independent assets, the solution set has dimension zero, and there is a unique vector that solves (??). The case of S linearly independent assets is referred to as Complete markets. Remark Let preferences be Von Neumann-Morgernstern: u i (x i ) = u i (x i 0) + X prob s u i (x i s) where X s=1;:::;s prob s = 1: Let then m s = s=1;:::;s s prob s. Then q j = E (ma j ) In this representation of asset prices the vector m 2 R S ++ is called Stochastic discount factor 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 MRS i s(x i ) denote agent i s marginal rate of substitution between consumption of the

15 10 CHAPTER 2 TWO-PERIOD ECONOMIES numeraire good 1 in state s and consumption of the numeraire good 1 at date 0: MRS i s(x i ) i (x i s i i (x i 0 i 10 Let MRS i (x i ) = (: : : MRS i s(x i ) : : :) denote the vector of marginal rates of substitution for agent i, an S dimentional vector. Note that, under the assumption of strong monotonicity of preferences, MRS i (x i ) 2 R S ++: By taking the rst order conditions (necessary and su cient for a maximum under the assumption of strict quasi-concavity of preferences) with respect to z i j of the individual problem for an arbitrary price vector q, we obtain that q j = SX prob s MRSs(x i i )a j s = E MRS i (x i ) a j ; (2.3) 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 smrss(x i i )a j s. Compare equation (2.3) to the previous equation (2.2). Clearly, at any equilibrium, condition (2.3) 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 MRSs(x i i ) : : :) belongs to R(q) and is hence a viable for the asset pricing equation (2.2). But recall that 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 (sloppily said, the larger nancial markets) the more aligned are agents marginal rates of substitution at equilibrium (sloppily 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), the set R(q) is in fact a singleton and hence the MRS i (x i ) are equalized across agents i at equilibrium: MRS i (x i ) = MRS; for any i = 1; :::; I:

16 2.2 FINANCIAL MARKET ECONOMIES 11 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 Arrow theorem The Arrow theorem is the fondamental 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, on the span of A, < A >= 2 R S : = Az; z 2 R J in particular. Such a reformulation is important not only in itself but as a lemma for welfare analysis in Financial market economies. Proposition Let (x ; ) represent an Arrow-Debreu equilibrium. Suppose rank(a) = S ( nancial markets are Complete). Then (x ; z ; p ; q ) is a Financial market equilibrium, where s = s p s; for any s = 1; :::; S: and SX q = prob s MRSs(x i i )A s s=1 Futhermore, the converse also holds: if (x ; z ; p ; q ) is a Financial market equilibrium of an economy with rank(a) = S, (x ; ) represents an Arrow- Debreu equilibrium, where s = s p s; for any s = 1; :::; S: A sketch of the argument of the proof is important to understand several results in this chapter. Financial market equilibrium prices of assets q satisfy No-arbitrage. 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! 0 i + q z i = 0 p s x i s! s i = As z i ; for s = 1; :::S:

17 12 CHAPTER 2 TWO-PERIOD ECONOMIES Substituting q = A; expanding the rst equation, and writing the constraints at time 1 in vector form, we obtain: p 0 SX x i 0! i 0 + s p s s=1 x i s! i s = 0 (2.4) 2 3 : : p s (x i s! i s) 6 : < A > (2.5) : 2 3 : : If rank(a) = S; it follows that < A >= R S ; and the constraint p s (x i s! i s) 2< 6 : : A > is never binding. Each agent i s problem is then subject only to p 0 SX x i 0! i 0 + s p s s=1 x i s! i s = 0; the budget constraint in the Arrow-Debreu economy with Aggregation s = s p s; for any s = 1; :::; S: 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

18 2.2 FINANCIAL MARKET ECONOMIES 13 for goods and assets are all determined simultaneously from the system of equations formed by (??) and (??). 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 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 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 u i (x i ) = u(x i 0) + SX 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 This result is a consequence of the fact that 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

19 14 CHAPTER 2 TWO-PERIOD ECONOMIES nancial income y i following: = (y i 0; y i 1; :::; y i S ) 2 RS+1 ++ in units of numeraire, is the max x i 2R L(S+1) + s:t: u i (x i ) 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, and 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, is the following: max z i 2R J ;y i 2R S+1 ++ s:t: p 0! i 0 qz i = y i 0 v i (y i ; p) p s! i s + a s z i = y i s; for s = 1; :::S: 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 ; y i s) for each state. Because of identical and homothetic preferences, then, spot prices p s for each state s; are determined independently of asset prices q and of the distribution of endowments f! i g i2i. Remark 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 ++ s:t: y i 0 = p 0! i 0 qz i y i s v i (y i ; p ) = p s! i s + a s z i ; for s = 1; :::S

20 2.2 FINANCIAL MARKET ECONOMIES 15 If preferences separable across states, identical, homothetic within states, and von Neumann-Morgenstern, it is straightforward to show that v i (y i ; p ) is identical across agents i and, seen as a function of y i, it satis es the assumptions we have imposed on u i as a function of x i, in Assumption A.2. Let w 0 = p 0! i 0; w s = p s! i s; for any s = 1; :::; S; and disregard for notational simplicity the dependence of v i (y i ; p ) on p : The income allocation problem becomes: max y i 2R S+1 ++ s:t: y i 0 w 0 = qz i v(y i ) y i s w s = a s z i ; for s = 1; :::S 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 s is interpreted as agent i s income endowment in state s: The representative agent theorem A representative agent is the following theoretical construct. De nition 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 Representative agent. Suppose there exists an homothetic u : R L! R such that u i (x i ) = u(x i 0) + SX prob s u(x i s); for all i = 1; ::; I: s=1

21 16 CHAPTER 2 TWO-PERIOD ECONOMIES Let p denote equilibrium spot prices. If p s! i s 2< A >; then there exist a map u R : R S+1! R such that: IX! R =! i s; i=1 U R (x) = u R (y 0 ) + SX prob s u R (y s ) where y s = p s=1 IX x i s; s = 0; 1; :::; S i=1 constitutes a Representative agent. Since the Representative agent is the only agent in the economy, her consumption allocation and portfolio at equilibrium, x R ; z R ; are: x R =! R = z R = 0 IX i=1! i 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 ): 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, q = SX MRS s (w)a s ; for MRS s (w) = R (w R (w 0 0 That is, asset prices can be computed from agents preferences u R : R! R and from the aggregate endowment w: This is called the Lucas trick for pricing assets. Another interesting but misleading result is the "weak" representative agent theorem, due to Constantinides (1982).

22 2.2 FINANCIAL MARKET ECONOMIES 17 Theorem 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, IX! R =! i ; i=1 U R (x) = max (x i ) I i=1 IX i u i (x i ) s.t. i=1 constitutes a Representative agent. Clearly, then, q = IX i=1 x i = x; where i = ( i ) 1 and i (x i i 10 SX MRSs R (! R s )a s ; s=1 where MRSs R (x) R R 0 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. 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? q = E(mA)

23 18 CHAPTER 2 TWO-PERIOD ECONOMIES Single-factor beta representation 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 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 (2.6) 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 (2.6) 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.

24 2.2 FINANCIAL MARKET ECONOMIES 19 Multi-factor beta representations A multi-factor beta representation for asset returns has the following form: ER j = R f + FX jf mf (2.7) 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 A single factor beta representation ER j = R f + j m 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. 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.

25 20 CHAPTER 2 TWO-PERIOD ECONOMIES 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 Then, s=1 x s = R w s (w 0 x 0 ) m s = x s x # x 0 x # = (w 0 x 0 ) (x 0 x # ) Rw s x # x 0 x # i=1 xi s = 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 (m; Rj) 1 denotes the correlation coe cient and (:), 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

26 2.2 FINANCIAL MARKET ECONOMIES 21 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. Corollary Let (A; q) satisfy No-arbitrage. Then, there exists a unique 2< A > such that q = A : 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.

27 22 CHAPTER 2 TWO-PERIOD ECONOMIES Pareto optimality Under Complete markets, the First Welfare Theorem holds for a Financial market equilibrium. This is a direct implication of Arrow theorem. Proposition Let (x ; z ; p ; q ) be a Financial market equilibrium of an economy with Complete markets (with rank(a) = S): Then x is a Pareto optimal allocation. However, under Incomplete markets 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. Theorem At a Financial Market Equilibrium (x ; z ; p ; q ) of an incomplete nancial market economy, that is, of an economy with rank(a) < S, the allocation x is generically 6 not Pareto Optimal. 7 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 6 We say that a statement holds generically when it holds for a full Lebesgue-measure subset of the parameter set which characterizes the economy. In these notes we shall assume that the an economy is parametrized by the endowments for each agent, the asset payo matrix, and a two-parameter parametrization of utility functions for each agent; see Magill-Shafer, ch. 30 in W. Hildenbrand and H. Sonnenschein (eds.), Handbook of Mathematical Economics, Vol. IV, Elsevier, The proof can be found in Magill-Shafer, ch. 30 in W. Hildenbrand and H. Sonnenschein (eds.), Handbook of Mathematical Economics, Vol. IV, Elsevier, It requires mathematical tecniques from di erential topology which are not appropriate to be introduced in this course.

28 2.2 FINANCIAL MARKET ECONOMIES 23 of an e ciency notion capturing this idea follows. Let x i t=1 = (x i s) S s=1 2 RSL + ; and similarly p t=1 = (p s ) S s=1 2 RSL + De nition (Diamond, 1968; Geanakoplos-Polemarchakis, 1986) Let (x ; z ; p ; q ) represent a Financial market equilibrium of an economy whose consumption set at time t = 1 is restricted by x i t=1; 2 B(p t=1 ); for any i = 1; :::; I In this economy, the allocation x is Constrained Pareto optimal if there does not exist a (y; ) 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(!; )); for any i = 1; :::; I where g t=1(!; ) is a vector of equilibrium prices for spot markets at t = 1 opened after each agent i = 1; :::; I has received income transfer A 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(!; ), 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(!; ); that is, to internalize the e ects of di erent transfers on spot prices at equilibrium. Proposition Let (x ; z ; p ; q ) represent a Financial market equilibrium of an economy with complete markets (rank(a) = S) and 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.

29 24 CHAPTER 2 TWO-PERIOD ECONOMIES 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. Constraint Pareto optimality of Financial market equilibrium allocations is guaranteed as long as the constraint set B is exogenous. Proposition Let (x ; z ; p ; q ) represent a Financial market equilibrium of an economy with Incomplete markets (rank(a) < S). In this economy, the allocation x is not Constrained Pareto optimal. 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 gs(! s ; )(x i s! i s) = A s i ; for all i and s; while any 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. 8 Several examples of such form of externality/ine ciency have been developed recently in macroeconomics. Some examples are: Thomas (1995), Krishnamurthy (2003), Caballero and Krishnamurthy (2003), Lorenzoni (2008). Kocherlakota (2009), Davila, Hong, Krusell, and Rios Rull (2005). Remark 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. 8 The name is due to Joe Stiglitz (or is it Greenwald-Stiglitz?).

30 2.3 CORPORATE FINANCE ECONOMIES 25 Corollary 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. In fact, 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 ; and it is hence ndependent of prices, of the form x i t=1 2 B. Remark Consider an alternative de nition of Constrained Pareto optimality, due to Grossman (1970), in which constraints 3 are substituted by 2 3 : 6 : : 6 4 p s (x i s! i s) : : = Az i ; for any i = 1; :::; I 7 5 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): Remark Consider a 1-good (L = 1) Incomplete market economy (rank(a) < S) which lasts 3 periods. Note that Financial market equilibrium allocations of such an economy are not Constrained Pareto optimal. 2.3 Corporate nance economies 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

31 26 CHAPTER 2 TWO-PERIOD ECONOMIES k of the commodity invested in capital at time 0: 9 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. 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. 10 The rm chooses its production plan k so as to maximize its value. The rm s problem is then: max k k + Q(k) (2.8) 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. 9 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.

32 2.3 CORPORATE FINANCE ECONOMIES 27 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 (2.9) max ui x i x i 0 ;xi ;z i ; i 0 + Eu i x i (2.10) subject to (2.9) and x i 0 =! i 0 + [ k + Q] i 0 Q i q z i (2.11) x i (s) =! i (s) + f(k; s) i + A(s)z i ; 8s 2 S (2.12) 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: 11 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 or, equivalently, for the assets: X z i = 0 (2.13) i X i = 1 (2.14) i 11 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 as, for instance, in the example of Section??.

33 28 CHAPTER 2 TWO-PERIOD ECONOMIES 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 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

34 2.3 CORPORATE FINANCE ECONOMIES 29 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 k - clear at zero trade. 12 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 ) (2.15) 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. Symmetric equilibria are then not guaranteed to exist, and asymmetric equilibria might obtain, in which di erent rms choose di erent production plans. Starting with the initial contributions of Diamond (1967), Dreze (1974), Grossman-Hart (1979), and Du e-shafer (1986), a large literature has dealt 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, (2.9), 12 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.

35 30 CHAPTER 2 TWO-PERIOD ECONOMIES 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 (2010). 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. 13 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 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. 14 Consider for example the consistency condition proposed by Dreze (1974): Q D (k) = E " X i i MRS i f(k) # ; 8k (2.16) 13 Independence of the kernel is guarantee by the fact that MRS i (s); for any i, is evaluated at equilibrium. 14 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.

36 2.3 CORPORATE FINANCE ECONOMIES 31 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. Pareto optimality 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 (2.17) i i X x i (s) X! i (s) + f(k; s); 8s 2 S (2.18) i i 15 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.

37 32 CHAPTER 2 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 (2.19) Next we present the notion of e ciency restricted by the admissibility constraints: De nition 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. Theorem (First Welfare) Competitive equilibria are constrained Pareto e cient. 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 (2.8). 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 ) :

38 2.3 CORPORATE FINANCE ECONOMIES 33 Proposition 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. Modigliani-Miller theorem 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 (2.20) 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: Proceeding as in the previous section, at equilibrium we shall require that Q(k) = max E MRS i [f(k) B] ; 8k; i p = max E MRS i i 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;

39 34 CHAPTER 2 TWO-PERIOD ECONOMIES 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. 2.4 Asymmetric information economies 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. 16 The fundamental contribution on competitive markets for insurance contracts is Prescott and Townsend (1984a); see also Prescott and Townsend (1984b). 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. 17 In these notes we concentrate on the sim- 16 In the context of these economies M. Harris and R. Townsend (1981) prove a version of the Revelation principle. 17 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 equi-

40 2.4 ASYMMETRIC INFORMATION ECONOMIES 35 pler case of moral hazard. We refer to Bisin-Gottardi (2006) for the case of adverse selection Moral hazard insurance economies 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. 19 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 s (e) be the probability of the realization! s given e. Obviously P s2s s(e) = 1; for any e 2 E; a compact, convex set: By the Law of Large Numbers, 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 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) librium 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 (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)). 18 Other references include Bennardo and Chiappori (2003), Bisin and Gottardi (1999, 2006), Dubey, Geanakoplos and Shubik (2004), Gale (1992, 1996). 19 Measurability issues arise in probability spaces with a continuum of indipendent random variables. We adopt the usual abuse of the Law of Large Numbers.

41 36 CHAPTER 2 TWO-PERIOD ECONOMIES is strictly increasing, strictly convex, twice continuously di erentiable, and sup e2e v 0 (e) = 1: 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 e, for any s = 2; :::; S: The symmetric information benchmark s(e) s 1 (e) is strictly increasing in 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 s (e)u(x s ) v(e); (2.21) s2s X 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; not just at the equilibrium 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 each agent solves s.t. and markets clear max x;e X s (e)u(x s ) v(e); (2.22) s2s X q s (e)(x s! s ) = 0 s2s X s (e)(x s! s ) = 0 (2.23) s2s

42 2.4 ASYMMETRIC INFORMATION ECONOMIES 37 We impose the following consistency condition on the price conjecture: q s (e) = s (e): Under the consistency condition it is now straightforward to prove the First and Second Welfare theorems for this economy under symmetric information. Moral hazard 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: s.t. and max x;e X s (e)u(x s ) v(e); (2.24) s2s X s (e)(x s! s ) = 0 s2s e 2 e(x) = arg max X s2s 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. At a Prescott-Townsend competitive equilibrium prices cannot have the form q s (e), as the agent s choice for e is not observed. What is observed, however 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 s (e)u(x s ) v(e); (2.25) max x;e s2s

43 38 CHAPTER 2 TWO-PERIOD ECONOMIES s.t. and markets clear X q s (e(x))(x s! s ) = 0 s2s X s (e)(x s! s ) = 0 (2.26) s2s At an equilibrium (x ; e ), we impose the following consistency condition on the price conjecture: q s (e(x)) = s (e(x)) The First and Second Welfare theorems, in its incentive constrained versions, hold straightforwardly for the moral hazard economy. Remark An equivalent notion of equilibrium is possible, which is equivalent to the one just proposed. Suppose at a competitive equilibrium each agent solves X s (e)u(x s ) v(e); (2.27) s.t. X q s (e)(x s s2s X s (e)u(x s ) s2s and markets clear max x;e s2s! s ) = 0; and v(e) X s2s s (e 0 )u(x s ) v(e 0 ); for any e 0 2 E X s (e)(x s! s ) = 0 (2.28) s2s Prices depend on e ort e; q(e): As e ort is not observed, this is to be interpreted that prices depend on the (implicit) declaration of e ort on the part of the agent; for instance di erent markets could be present with di erent prices - and by choosing one of these where to trade each agent implicitly declares his e ort choice. The condition X s (e)u(x s ) v(e) X s (e 0 )u(x s ) v(e 0 ); for any e 0 2 E s2s s2s ther requires that in the market with prices q(e) only incentive compatible allocations are o ered, that is, only allocations which induce the agent to choose

44 2.4 ASYMMETRIC INFORMATION ECONOMIES 39 e ort e: 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. As we noted, the exclusivity assumption guarantees that x is observable. Suppose it is not (this is the non-exclusivity case). The next problem deals with this case in a simple but instructive example Corporate agency economies [...Bisin, Gottardi, Ruta...]

45

46 Chapter 3 In nite-horizon economies Note that, when economies have an in nite horizon, their commodity space is in nite dimensional. Good discussion of the spaces we typically study is in Lucas-Stokey with Prescott (1989); see also Zame (.). 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. 3.1 Arrow-Debreu economies 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 41

47 42 CHAPTER 3 INFINITE-HORIZON ECONOMIES i. given p ; x i 2 arg maxfu(x 0 ) + E 0 [ P 1 t=1 t u(x t )]g s:t: P 1 t=0 p t (x t! t ) = 0 ii. and P i xi! i = 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 economies 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 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 s:t: 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 De nition ii. P i xi! i = 0 and P i zi = 0:

48 3.2 FINANCIAL MARKETS ECONOMIES 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 ) (3.1) (x t ) u 0 (x t+1 ) 1 = E t R u 0 t+1 : (3.2) (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 (3.1), 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 De ne now A t = 0; t < T; and A T = maxfq T 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) ; 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 (3.2) applied to a 1-period bond yields 1 u 0 (x t+1 ) = E t : u 0 (x t ) R f t+1 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]

49 44 CHAPTER 3 INFINITE-HORIZON ECONOMIES Conditional asset pricing Conditional versions of the beta representation hold in this economy: E t (R t+1 ) R f t+1 = Cov t(m t+1 ; R t+1 ) = (3.3) 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 : Recall that our basic pricing equation is a conditional expectation: q t = E t (m t+1 A t+1 ); (3.4) 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 (3.4): E(q t ) = E(m t+1 A t+1 ); (3.5) where we have invoked the law of iterated expectations. It should be clear that equation (3.4) implies but it is not implied by (3.5). 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 ) 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. 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/ rengle/ = 13.

50 3.2 FINANCIAL MARKETS ECONOMIES 45 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 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+1 q t ; the theorem yields the desired result. Predictability of returns Recall the asset-pricing equation for stocks: 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. Frictions [...He-Modest... Luttmer...]

51 46 CHAPTER 3 INFINITE-HORIZON ECONOMIES Bubbles For assets whose payo is made of a dividend and a capital gain, the foc s dictate q t = E t (m t+1 (q t+1 + d t+1 )) ; where 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 : j=1 A more general nancial market economy is useful to study the conditions for the existence bubbles; we follow Santos and Woodford (1997). Let N = Xt=0S 1 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. Denote by s t 1 the single (immediate) predecessor node to s t. Use s T js t to indicate that s T is some successor of s t, for T > t: At each node, there are J securities traded. 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. Assumption 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 :

52 3.2 FINANCIAL MARKETS ECONOMIES 47 Assumption 2. The economy is connected across time and states. This is achieved when at any state there is some agent alive and non-terminal. Formally, 8s t ; 9i : s t 2 N i nn i : 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 that pay at node s t. We shall assume that a security can pay in dividends (units of consumption) and in units of assets. Mapping b : N! R J 2 de nes at each node s t a J J matrix, whose j-th column denotes the vector of assets that are paid at node s t by asset j: Assumption 3. Assets payo s, both in terms of dividends and of other assets, are non-negative: d(s t ) 0 and b(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!(s i 0 ): Therefore, the initial net supply of assets is given by z! (s 0 ) = X z!(s i 0 ). i2i(s 0 ) De ne the net supply of securities at any node s t, z! (s t ), recursively by z! (s t ) = b(s t )z! (s t 1): We shall assume that z! (s 1 ) 0: This ensures that z! (s t ) 0, at all nodes s t js 0 ; t > 0. Observe that a portfolio held at the end of s t, say z(s t ), will generally pay dividends (and assets) at several future states s t+1 ; s t+2, etc. We shall now construct a mapping that will assign to each node the dividends paid by portfolio z(s t ). As an intermediate step, let us construct the dividends mapping for a portfolio made of one unit of each asset that can be traded at s t. To determine the stream of dividends generated by this canonical portfolio at each successor node, de ne for all s r js t with r t, the J J matrix e(s r js t ) by e(s t js t ) = I JJ e(s r js t ) = b(s r )e(s r 1js t ); for all s r js t and r > t: Matrix e(s r js t ) is just a counter of the asset payo s at s r of the canonical portfolio. Now, for all s r js t with r > t, de ne

53 48 CHAPTER 3 INFINITE-HORIZON ECONOMIES x(s r js t ) = d(s r )e(s r 1js t ): This is a J-dimensional vector x(s r js t ) and its j-th component is the dividends that the j-th asset of the canonical portfolio produces at node s r, for j = 1; :::; J. Finally, the dividends mapping for the vector z(s t ) is simply x(s r js t )z(s t ); for each node s r js t with r > t. De ne an asset j to have nite maturity if there exists a T such that e(s r js t ) = 0; for all s r js t and r T > 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. 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! (s t 1) 0: The utility function of any agent i is written 1X U(x) = X t st u i (x(s t )): s t t=0 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 )b(s t ): Agent i chooses, at each node s t 2 N i a level of consumption x i (s t ) and a J vector of securities z i (s t ) to hold at the end of trading, subject to the budget constraints: x i (s 0 ) + q(s 0 )z i (s 0 )! i (s 0 ) + q(s 0 )z i!(s 0 ); and at each node s t 6= s 0, 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 );

54 3.2 FINANCIAL MARKETS ECONOMIES 49 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. Market Clearing conditions. At each s t ; X x i (s t ) = ew(s t ) i2i(s t ) X i2i(s t ) z i (s t ) = z! (s t ): 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. Lemma When q satis es the no-arbitrage condition at s t ; there exists a set of state prices (a SDF) fm(s t+1 )g with m(s t+1 ) > 0 for all s t+1 js t, such that the vector of asset prices at s t can be written as q(s t ) = X s t+1 js t m(s t+1 )A(s t+1 ): (3.6) Proof. As usual, proof follows from applying the separation result. Note that if for a given price process q, there are no arbitrage opportunities at any s t individually, then we can apply the lemma repeatedly and de ne some state-price process m for which the pricing equation holds. Let M(s t ) denote the set of such processes for the subtree with root s t. Only under complete markets is the set M(s t ) a singleton. As a remark, note that completeness is an endogenous property since one-period payo s A contain asset prices. Therefore, completeness cannot be assessed ex ante but only at each given equilibrium. De nition For any state-price process m 2 M(s t ); de ne the J vector of fundamental values for the securities traded at node s t by 1X X f(s t ; m) = m(s T )x(s T js t ): (3.7) T =t+1 s T js t

55 50 CHAPTER 3 INFINITE-HORIZON ECONOMIES 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 At each s t 2 N, f(s t ; m) is well-de ned for any m 2 M(s t ) and satis es 0 f(s t ; m) q(s t ): Proof. First of all, 0 f(s t ; m) follows directly from non-negativity of m, the dividend, the asset payo, and the price processes. We therefore turn to f(s t ; m) q(s t ): From equation (3.6), we have q(s t ) = X X m(s t+1 )x(s t+1 js t ) + m(s t+1 )q(s t+1 )e(s t+1 js t ) s t+1 js t s t+1 js t and, iterating on this equation we obtain q(s t ) = btx T =t+1 X X m(s T )x(s T js t ) + m(s bt )q(s bt )e(s bt js t ) s T js t s bt js t for any b T > t: Since e(s bt js t ) 0 by construction, q(s bt ) is non-negative by de nition of equilibrium (in the paper) and m 2 M(s t ) is a positive stateprice vector, the second term on the right-hand-side is non-negative. So, q(s t ) btx T =t+1 X s t+1 js t m(s T )x(s T js t ): Note that the right-hand-side is a nondecreasing series in b T. It is bounded above and, therefore, must converge. So, q(s t ) m(s t )f(s t ; m) We can correspondingly de ne the vector of asset pricing bubbles as (s t ; m) = q(s t ) f(s t ; m); (3.8) for any m 2 M(s t ) for the J securities traded at s t. proposition that 0 (s t ; m) q(s t ); It follows from the

56 3.2 FINANCIAL MARKETS ECONOMIES 51 for any m 2 M(s t ): This corollary is known as the impossibility of negative bubbles result. Substituting (3.8) and (3.7) into (3.6) yields (s t ) = X s t+1 js t m(s t+1 )(s t+1 )e(s t+1 js t ): 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 some securities 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. The proposition also o ers a corollary that deals with nite-maturity assets. Even with incomplete markets, we have that f j (s t ; m) = f j (s t ) for all m 2 M(s t ) and q j (s t ) = f j (s t ); if asset j has nite maturiry: there are no pricing bubbles for securities with nite maturity. Notice that we reached this conclusion just by no arbitrage. In the case of securities of in nite maturity in an economy with incomplete markets, the fundamental value need not be the same for all state-price processes consistent with the available securities returns. But even in this case, we can de ne the range of variation in the fundamental value, given the restrictions imposed by no-arbitrage. De nition (Present value of a stream of dividends) Let x : N! R + denote a stream of non-negative dividends. For any s t, pick any m 2 M(s t ). Then we de ne the present value at s t of x with respect to m by V x (s t ; m) = 1X T =t+1 X s T js t m(s T )x(s T ). Since this present value depends on the stochastic discount factor m picked, let us now de ne the bounds for the present value at s t of dividends x. De nition For any s t, suppose x(s r ) 0; 8s r js t ; r > t: De ne x (s t ) = x (s t ) = inf fv x(s t ; m)g m2m(s t ) sup fv x (s t ; m)g: m2m(s t )

57 52 CHAPTER 3 INFINITE-HORIZON ECONOMIES A few remarks follow from these de nitions. First note that these de nitions are conditional on a given price process q since the set of no-arbitrage stochastic discount factors are de ned with respect to q. Next, observe that for any security with payo process x j, x j(s t ) f j (s t ; m) x j(s t ), for all m 2 M(s t ). Finally, note that x j(s t ) < q j implies that there is a pricing bubble for security j. 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. De nition We shall say that 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. B i (s t ) = ~! i(s t ); (3.9) It can be shown that these borrowing limits never bind at any nite date (see Magill-Quinzii, Econometrica, 94). They are just a constraint on the asymptotic behavior of a household s debt. 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. An important consequence of this speci cation is the following. Proposition Suppose that household i has borrowing limits of the form (3.9). 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 there is a nite borrowing limit at each node. Note that if more stringent borrowing limits were imposed, i.e. B i (s t ) < ~! i(s t ), we can have equilibria where ~! i(s t ) = 1. This is a crucial point to understand bubbles (recall this comment in the Bewley s example): if an agent s endowment (that is, the whole - or part of the whole - stochastic process) is 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. But note also that even if ~! i(s t ) < 1, it is still possible that ~! (s t ) = 1 (and, a fortiori, that ~! (s t ) = 1) because the economy allows for a countable in nity of agents, though we require the number of agents at each node to be nite (recall this comment in Samuelson s overlapping generations example).

58 3.2 FINANCIAL MARKETS ECONOMIES 53 Proposition Consider an equilibrium fq; x i ; z i g. Suppose that the (maximum) value of aggregate wealth is nite, i.e. ~! (s t ) < 1. 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. Then, 8K > 0; there exists a time T and s T js t such that (s T )z(s T ) > K X s T js t m(s T )e!(s T ): Proof. The martingale property of pricing bubbles implies, (s t )z(s t ) = X s T js t m(s T )(s T )z(s T ) while P s T js t m(s T )e!(s T ) must converge to 0 in T! 1 to guarantee that ~! (s t ) < 1: 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. The proof exploits crucially the martingale property of bubbles. It follows from this result that some agent must accumulate vast wealth. We already learned that no bubbles can arise in securities with nite maturity. The next theorem extends the result to securities in positive net supply as long as we are at equilibria with nite aggregate wealth. The proof uses the nonoptimality of the behavior implied by the previous proposition. Theorem Let fq; x i ; z i g be an equilibrium. Suppose that at each node s t 2 N; there exists m 2 M(s t ) such that V ~! (s t ; m) < 1: Then q j (s T ) = f j (s T ; m); for all s T js t and m 2 M(s t ), for each security j traded at s T maturity or positive net supply. that has nite Note that if we have that at equilibrium ~! (s t ) < 1; the condition of the theorem is satis ed. 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.

59 54 CHAPTER 3 INFINITE-HORIZON ECONOMIES Corollary Suppose that there exists a portfolio bzr J + such that x(s t js 0 )bz ~!(s t ); 8s t 2 N: Then the theorem holds at any equilibrium. Intuitively, if the existing securities allow such a portfolio 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 Suppose that there exists an (in nitely lived) agent 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. The intuition for this result is as follows. The borrowing limits imposed allow all agents (and in particular the in nitely lived one) to issue an IOU against his or her stream of endowments. Again, in equilibrium, the asset issued by the in nitely lived agent must have a nite price. This implies that a fraction of aggregate wealth has a nite value in equilibrium, which obviously implies that aggregate wealth is nite. As a remark, note that these two corollaries share the same spirit. They both state that by adding some assets to the economy, we can rule out bubbles in equilibrium (for securities in positive net supply). Therefore, the conclusion of the paper is that bubbles are a non-robust phenomenon (in securities in positive net supply). Examples 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 net supply and has a positive price in equilibrium, that is a bubble. The following two models have equilibria with such a property. Theorem 3.3. implies that in those equilibria aggregate wealth must have an in nite value. See the paper for the primitives of each model. 1. Samuelson (1958) s OLG model: Observe how all the assumptions of corollary 3.5 hold except the existence of an agent that owns a fraction of aggregate wealth.

60 3.2 FINANCIAL MARKETS ECONOMIES Bewley (1980): Observe that now the only assumption of corollary 3.5 that fails is that the borrowing constraints are more stringent than those required in the corollary. 3. More recent examples: Jovanovic (2007), Hong, Scheinkman, Xiong (2006), Abreu and Brunnermeier (2003), Allen, Morris, and Shin (2003) Pareto optimality 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 sequences endowed with the sup norm. 2 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 2 See Lucas-Stokey with Prescott (1989), Recersive methods in economic dynamics, Harvard University Press, for de nitions. t=1

61 56 CHAPTER 3 INFINITE-HORIZON ECONOMIES (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 (x ; p ) is an Arrow-Debreu Equilibrium if i. given p ; x i 2 arg maxfu(x 0 ) + E 0 [ P 1 t=1 t u(x t )]g s:t: P 1 t=0 p t (x t! i t) = 0 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.

62 3.2 FINANCIAL MARKETS ECONOMIES 57 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 i! i t 1X p t (yt i! i t) 0; strictly for one i 0: t=0 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;

63 58 CHAPTER 3 INFINITE-HORIZON ECONOMIES - there exist economies which have Arrow-Debreu equilibria whose allocations are NOT Pareto optimal. Remark 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 1: Overlapping generation 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. 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; 3 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)

64 3.2 FINANCIAL MARKETS ECONOMIES 59 The restriction p 0 = 1 is the standard normalization due to the homogeneity of the Arrow-Debreu budget constraint. Proof. First of all, x0 1 =! 0 1 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. It is convenient to specialize this economy to a simple stationary example where! 1 0 = ;! 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 :

65 60 CHAPTER 3 INFINITE-HORIZON ECONOMIES 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 i t=0 t=0 The value of aggregate wealth is nite when < a 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 in OLG economies 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

66 3.2 FINANCIAL MARKETS ECONOMIES 61 money, in units of the consumption good at time t = 0; at time t; be denoted q m t : 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 + qt m s t =! t t p t+1 (x t t+1! t t+1) = qt+1s m t : The budget constraint for agent i = 1 is: x 1 0 =! q m 0 m: Turning back to the stationary economy example where! 1 0 = ;! 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:

67 62 CHAPTER 3 INFINITE-HORIZON ECONOMIES 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 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 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 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), Bewley economies In this section we consider two economies with incomplete markets and idiosynchratic shocks which have been studied in macroeconomics. The rst, economies with earning risk, are economies characterized by: agents face stationary idiosynchratic endowment shocks (with/out an aggregate component), but can only trade a riskless bond, typically with a no-short-sales constraint.

68 3.3 BEWLEY ECONOMIES 63 The second class of economies, referred to as economies with investment risk, are characterized by: agents face stationary idiosynchratic shocks (with/out an aggregate component) on the rate of return of savings, but can only trade a riskless bond, typically with a no-short-sales constraint. Both economies can be somewhat extended to allow for production. We discuss these extension in a future chapter Earning risk The prototypical earning risk economy is an economy with idiosynchratic shocks in which asset trading is restricted to a bond, which trades at time t for a price q t normalized 1 for any t 0; and pays pays 1 + r t+1 at time t + 1: This economy, originally studied by Huggett (1993); see Ljungvist-Sargent (2004), ch. 17. This economy is straightforwardly modi ed to the case in which agents face a no-borrowing constraint, z t 0; and the rate of return on savings is an exogenous sequence r t ; see Aiyagari (1994) and Ljungvist and Sargent (2004), ch. 17. An equilibrium is still characterized by a policy function of the form z t+1 = g t (z t ; s t ); 4 and a distribution t (z t ; s t ); de ned recursively from an initial given distribution 0 (z 0 ; s 0 ): The distribution of wealth at time t is: t (z t ) = X s t2s t (z t ; s t ): The limit distribution of wealth, if it exists, satis es: (z) = lim t!1 t (z t ): A straightforward modi cation/reinterpretation/extension of this economy, has production to endogeneize the rate of return (on capital; re-interpret 4 We drop the dependence of the policy function from the whole sequence r t ; for notational simplicity.

69 64 CHAPTER 3 INFINITE-HORIZON ECONOMIES wealth as capital) and a wage rate. Let F (K t ; N t ) the aggregate production function at time t, in terms of aggregate capital K t and labor N t : Assume e.g., that labor supply is xed and! t (s) = w t s, where w t is the wage rate at time t, and the shock s is interpreted as labor productivity. Then, de ning Z K t+1 = t+1 (z t+1 )dz t+1 ; N t = N = X (s)s s2s we have r t (K t; N) ; w t (K t; N) Aggregate shocks to the production function can be easily added. Typically we assume write it as a t F (K t ; N t ): Investment risk Suppose instead each agent faces an idiosynchratic exogenous rate of return on savings, a mapping r : f1; 2; :::; Sg! R +. We maintain the assumption that each agent can only save (not borrow) at the rate r: z t 0 for any t 0. Given the process r; 5 each agent solves: v t (z; s) = max u ((1 + r(s)) z +!(s) z 0 0 z0 ) + X s 0 2s prob (s 0 js)v t (z 0 ; s 0 ) The solution of this problem is a policy function of the form: Let t+1 (z t+1 ; s t+1 ) = X Z prob (s 0 js) s t2s z t+1 = g(z t ; s t ): z t:z t+1 =g(z;s) (z t ; s t ) t (z t ; s t )dz t ; for any t 0: 5 Formally, the realization for s 1 is also given. Assume also agents are endowed with no portfolio positions on bonds at time 0: z i (s 1 ) = 0; for all i 2 I:

70 3.3 BEWLEY ECONOMIES 65 Thenn the distribution of wealth at time t + 1 in this economy is de ned recursively, from an initial given distribution 0 (z 0 ; s 0 ): t+1 (z t+1 ) = X t+1 (z t+1 ; s t+1 ) s t+1 2S The limit distribution of wealth, if it exists, satis es: (z) = lim t!1 t (z): Limit incomplete market economies Let s go back to economies with no trading restrictions and in nitely lived agents. Of course, we still need to impose some borrowing limit to rule out Ponzi schemes but we know how to do that in a non-binding way, as we learned from Santos and Woodford (1997). In a series of papers, Telmer (1993), Aiyagari (1994) and Krusell and Smith (1998) among others, di erent authors have found support for a puzzling result. Even though theoretically the e ects of complete or incomplete markets on equilibrium allocations and prices are crucial, empirically, they it does not seem to matter signi cantly. These are all in nite-horizon economies with agents facing idiosyncratic risk due to stochastic endowments and a seriously incomplete asset structure. The next two papers illuminate the question from the theoretical viewpoint. The current state of the literature is that the stationarity or not of the individual endowment process is key to the e ects of incomplete markets. This is the result from Levine and Zame (2002). A second contribution is Constantinides and Du e (1996), who show that with incomplete markets and nonstationary endowment processes (i.e. permanent shocks), anything is possible. For short horizons, we know that market incompleteness generally matters because of agents inability to insure against bad shocks. However, for long horizons, market incompleteness may not matter if traders can self insure i.e. if they can borrow in bad times and save in good times. Consider the same economy we studied for bubbles, but let s specialize it to our purposes. Let uncertainty de ne a Markov process on the tree N = Xt=0S 1 t as follows: each node s t has S successors, that is, s t+1 js t = f1; 2; :::; Sg ;

71 66 CHAPTER 3 INFINITE-HORIZON ECONOMIES prob( s t+1 js t = s : s t = s 0 ) = prob (sjs 0 ), for any s; s0 2 f1; 2; :::; Sg; obviously prob (sjs 0 ) de nes a transition probability of a Markov chain with state space f1; 2; :::; Sg. We assume the Markov chain is recurrent (e.g., it is su cient that prob (sjs 0 ) > 0, for any s; s0 2 f1; 2; :::; Sg) At each node s t, J short-lived, actually one-period, securities are traded. Let the dividend process be the map d : f1; 2; :::; Sg! R + ; a J- dimensional Markov process in our probability space. Assume also that at each node s t, a riskless bond, yielding one unit of consumption at each s t+1 js t, is available for trade. All agents are alive and trade at all periods t > 0. Furthermore, each agent s i endowment process is a mapping! i : f1; 2; :::; Sg! R +, once again a Markov process in our probability space. De nition of equilibrium in this economy are just specialized versions of the de nitions in the economy we studied for bubbles (in particular, we impose the same No Ponzi scheme condition). Let s assume the following: the economy has no aggregate risk, that is,! = X i! i (s) is independent of s; preferences satisfy precautionary savings, that is, Du i (x) is (weakly) convex: Let! i denote the long-run average endowment (permanent income) for agent i: It is well de ned because under our assumptions, the Markov process for endowment has a stationary distribution. In fact, let such stationary distribution associate probability prob (s) to any s 2 f1; 2; :::; Sg ; then! i = X s prob (s)! i (s) It is easy to show that, the Pareto e cient allocations of this economy are given by the I-tuples of xed shares of the constant long-run average endowment! = P i!i. In particular, the complete market equilibrium allocation (hence Pareto e cient) for this economy is characterized by each agent i consuming his/her long-run average endowment (permanent income) at each node s t :

72 3.4 ASYMMETRIC INFORMATION ECONOMIES 67 Theorem For this economy there exist a discount factor su ciently close to 1, such that every equilibrium allocation process x i is "close" to perfect risk sharing, x i (s t )! i (in the sense that the time-discounted probability that equilibrium consumptions deviate from the perfect risk sharing allocation by more than a given amount is small). The proof provides a lower bound on equilibrium utility by constructing a budget feasible plan whose utility is almost that of constant average consumption. A crucial step in the argument is establishing that the riskless interest rate is bounded above, with a bound close to zero. This is important because the budget feasible plan they construct is nanced by borrowing, and a low interest rate makes borrowing easy. In general, in the presence of aggregate risk, market incompleteness matters even if endowment processes are stationary (i.e. shocks are transitory). The reason is the following. When there is aggregate risk, the upper bound on the interest rate need not obtain; when the aggregate endowment is low, many traders will want to borrow, and this demand for loans may drive up the riskless interest rate. A high interest rate interferes with risk sharing because it makes borrowing di cult. Summing up, aggregate risk matters because it a ects asset prices. When there is more than one consumption good, market incompleteness matters again, even without aggregate risk. The reason is that commodity prices provide another source of untraded risk. We conclude that in a one-good economy populated by in nitely-lived, patient agents, market incompleteness will not matter if shocks are transitory and risk is purely idiosyncratic. When there is aggregate uncertainty or more than one consumption good, market incompleteness matters, in general. The next paper adds to the question by showing that if shocks are permanent, market incompleteness matters too. 3.4 Asymmetric information economies [...survey...]

73 68 CHAPTER 3 INFINITE-HORIZON ECONOMIES Lack of Commitment Economies Consider the following economy, populated by a nite set of in nitely-lived agents, i 2 I: Let N = Xt=0S 1 t be the set of nodes of the tree, s 0 the root of 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

74 3.4 ASYMMETRIC INFORMATION ECONOMIES 69 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 U i (x i ; s 0 ) x i 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 ) (x i ) i2i X i2i s.t. 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:

75 70 CHAPTER 3 INFINITE-HORIZON ECONOMIES It follows easily then that Any Arrow-Debreu Equilibrium allocation of this economy is constraint Pareto optimal. 6 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 ) s:t: 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 ): 6 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.

76 3.4 ASYMMETRIC INFORMATION ECONOMIES 71 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; 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

77 72 CHAPTER 3 INFINITE-HORIZON ECONOMIES 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. 7 Remark 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 )) : 7 Bloise and Reichlin (2009) prove 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.

78 Bibliography [1] D. Abreu and M. Brunnermeier (2003): Bubbles and Crashes, Econometrica, 71(1), [2] V. Acharya and A. Bisin (2008): Managerial Hedging, Equity Ownership and Firm Value, Rand Journal of Economics. [3] Y. Algan, O. Allais, and E. Carceles-Poveda (2009): Macroeconomic implications of nancial policy, Review of Economic Dynamics, 12(4), [4] F. Allen and D. Gale (1996): Optimal Security Design, MIT Press. [5] F. Allen and D. Gale (2000): Bubbles and Crises, The Economic Journal, 110(460), [6] F. Allen, S. Morris, and H.S. Shin (2003), Beauty Contests, Bubbles and Iterated Expectations in Asset Markets, Cowles Foundation Discussion Paper [7] F. Alvarez and U. Jermann (2000): E ciency, Equilibrium, and Asset Pricing with Risk of Default, Econometrica, 2000, 68(4), [8] J. Amaro de Matos (2001): Theoretical Foundations of Corporate Finance, Princeton University Press. [9] M.G. Angeletos (2005), Uninsured Idiosyncratic Investment Risk and Aggregate Saving, MIT, mimeo. [10] M.G. Angeletos and J. La O (2009): Noisy business cycles, mimeo, MIT. 73

79 74 BIBLIOGRAPHY [11] A. Attar, T. Mariotti, and F. Salanie (2009): Non-exclusive competition in the market for lemons, mimeo, Toulouse School of Economics. [12] Bennardo, A. and P.A. Chiappori (2003): Bertrand and Walras Equilibria under Moral Hazard, Journal of Political Economy, 111(4), [13] A. Bisin (2010): Lecture Notes on General Equilibrium Theory, mimeo, NYU. [14] A. Bisin and P. Gottardi (1999): Competitive Equilibria with Asymmetric Information, Journal of Economic Theory, 87, [15] A. Bisin and P. Gottardi (2006): E cient Competitive Equilibria with Adverse Selection, Journal of Political Economy, 114(3), [16] A. Bisin, P. Gottardi, and A. Rampini (2007): Managerial Hedging and Portfolio Monitoring, Journal of the European Economic Association, 6, [17] A. Bisin, P. Gottardi, and G. Ruta (2008): Equilibrium Corporate Finance and Macroeconomics, NYU, mimeo. [18] P. Bolton and M. Dewatripont (2005): Contract Theory, MIT Press. [19] M. Brunnermeier (2001): Asset Pricing Under Asymmetric Information: Bubbles, Crashes, Technical Analysis, and Herding, Oxford University Press. [20] R. Caballero and A. Krishnamurthy (2003): Excessive Dollar Debt: Financial Development and Underinsurance, Journal of Finance, 58(2), [21] C. Campanale, R. Castro, and G.L. Clementi (2010): Asset Pricing in a Production Economy with Chew-Dekel Preferences, Review of Economic Dynamics, 13(2), [22] E. Carceles-Poveda and D. Coen-Pirani (2005): Capital Ownership under Incomplete Markets: Does it Matter? Stony Brooks, mimeo. [23] R. Castro, G.L. Clementi and G. MacDonald (2004): Investor Protection, Optimal Incentives, and Economic Growth, Quarterly Journal of Economics, 119(3),

80 BIBLIOGRAPHY 75 [24] A. Citanna and P. Siconol (2010): Recursive equilibrium in stochastic OLG economies: Incomplete Markets, forthcoming, Econometrica. [25] J. Cochrane (1995): Time-consistent health insurance, Journal of Political Economy, 103(3), [26] J. Cochrane (2001): Asset Pricing, Princeton University Press. [27] H. Cole and N. Kocherlakota (2001): E cient allocations with hidden storage, Review of Economic Studies, 68, [28] G. Constantinides and D. Du e (1996): Asset pricing with heterogeneous consumers, Journal of Political Economy, 104(2), [29] T. Cooley, R. Marimon, and V. Quadrini (2004): Aggregate Consequences of Limited Contract Enforcement, Journal of Political Economy, 112, [30] J. Davila, J. Hong, P. Krusell, and V. Rios Rull (2005): "Constrained E ciency in the Neoclassical Growth Model with Uninsurable Idiosyncractic Shocks," mimeo, University of Pennsylvania. [31] J. Dreze, E. Minelli and M. Tirelli (2007): Production and Financial Policies under Asymmetric Information, Economic Theory. [32] Dubey, P., J. Geanakoplos (2004): Competitive Pooling: Rothschild- Stiglitz Re-considered, Quarterly Journal of Economics. [33] D. Gale (1992): A Walrasian Theory of Markets with Adverse Selection, Review of Economic Studies, 59, [34] D. Gale (1996): Equilibria and Pareto Optima of Markets with Adverse Selection, Economic Theory, 7, [35] K. Gerardi, H. Rosen and P. Willen (2010): The Impact of Deregulation and Financial Innovation on Consumers: The Case of the Mortgage Market, Forthcoming in the Journal of Finance. [36] J.F. Gomes, A. Yaron, and L. Zhang (2006): Asset Pricing Implications of Firms Financing Constraints, Review of Financial Studies.

81 76 BIBLIOGRAPHY [37] S. Grossman and O. Hart (1983): An Analysis of the Principal-Agent Problem, Econometrica, 51(1). [38] R.S. Gurkaynak (2005): Econometric testing of asset price bubbles: taking stock, Finance and Economics Discussion Series, Federal Reserve Board. [39] J. Heathcote, K. Storesletten, and G.L. Violante (2009): Quantitative macroeconomics with heterogeneous households, Annual Review of Economics, 1, forthcoming. [40] J. Heaton and D. Lucas (1996): Evaluating the E ects of Incomplete Markets on Risk Sharing and Asset Pricing, Journal of Political Economy 104(3), pp [41] M. Hellwig (1987): Some Recent Developments in the Theory of Competition in Markets with Adverse Selection, European Economic Review, 31, [42] H. Hong, J. Scheinkman, and W. Xiong (2006): Asset Float and Speculative Bubbles, Journal of Finance, 61(3), [43] U. Jermann (2007): The Equity Premium Implied by Production, Wharton, mimeo. [44] B. Jovanovic (2007), Bubbles in Prices of Exhaustible Resources, NBER Working Paper No , [45] G. Kaplan and G. Violante (2009): How much insurance in Bewley models?, mimeo, NYU. [46] K. Kaufmann and L. Pistaferri (2009): Disentangling insurance and information in intertemporal consumption choices," American Economic Review P&P. [47] T. Kehoe, D. Levine and E.C. Prescott (2002): Lotteries, Sunspots, and Incentive Constraints, Journal of Economic Theory, 107, [48] T. Kehoe and D. Levine (1993): Debt Constrained Asset Markets, Review of Economic Studies, 60(4),

82 BIBLIOGRAPHY 77 [49] N. Kocherlakota (2009): Bursting Bubbles: Consequences and Causes, [50] N. Kocherlakota and L. Pistaferri (2009): Asset pricing implications of pareto optimality with private information, Journal of Political Economy, 117(3), [51] A. Krishnamurthy (2003): Collateral Constraints and the Ampli cation Mechanism, Journal of Economic Theory, 111(2), [52] P. Krusell and A. Smith (1998): Income and Wealth Heterogeneity in the Macroeconomy, Journal of Political Economy, 106(5), [53] F. Kubler and K. Schmedders (2003): Stationary Equilibria in Asset- Pricing Models with Incomplete Markets and Collateral, Econometrica, 71(6), [54] D. Levine and W. Zame (2001): Does Market Incompleteness Matter? Econometrica, 70, [55] L. Ljungvist and T.J. Sargent (2004): Recursive macroeconomic theory, second edition, MIT Press. [56] G. Lorenzoni (2008): Ine cient Credit Booms, Review of Economic Studies, 75 (3), [57] R.E. Lucas, N. Stokey, with E.C. Prescott (1989), Recursive economic dynamics, Harvard Univ. Press. [58] S. Ludvigson and M. Lettau (2001): Resurrecting the (C)CAPM: A Cross-Sectional Test When Risk Premia are Time-Varying, Journal of Political Economy, 109(6): [59] H. Lustig and D. Krueger (2007): When is Market Incompleteness Irrelevant for the Price of Aggregate Risk, UPenn, mimeo. [60] M. Magill and M. Quinzii (1996): Theory of Incomplete Markets, Vol. 1, MIT Press. [61] M. Magill and M. Quinzii (2005): An Equilibrium Model of Managerial Compensation, IEPR Working Papers

83 78 BIBLIOGRAPHY [62] E. Mendoza, V. Quadrini and V. Rios-Rull (2008): Financial Integration, Financial Development and Global Imbalances, mimeo. [63] L. Pastor and P. Veronesi (2006): Was there a Nasdaq bubble in the late 1990s? Journal of Financial Economics, 81, [64] E.C. Prescott and R. Townsend (1984a): Pareto Optima and Competitive Equilibria with Adverse Selection and Moral Hazard, Econometrica, 52, 21-45; and extended working paper version dated [65] E.C. Prescott and R. Townsend (1984): General Competitive Analysis in an Economy with Private Information, International Economic Review, 25, [66] E.S. Prescott and R. Townsend (2006): Firms as Clubs in Walrasian Markets with Private Information, Journal of Political Economy, 114, [67] V. Quadrini and U. Jermann (2006): Financial Innovation and Macroeconomic Volatility, Wharton, mimeo. [68] V. Quadrini and U. Jermann (2010): Macroeconomic e ects of nancial shocks, mimeo. [69] M. Rothschild and J. Stiglitz (1976): Equilibrium in Competitive Insurance Markets: An Essay in the Economics of Imperfect Information, Quarterly Journal of Economics, 80, [70] A. Rustichini and P. Siconol (2008): General Equilibrium in Economies with Adverse Selection, Economic Theory, 37, [71] M. Santos and M. Woodford (1997): Rational Asset Pricing Bubbles, Econometrica, 65(1), [72] C. Thomas (1995): The role of scal policy in an incomplete markets framework, Review of Economic Studies, 62, [73] J. Tirole, (2006): Corporate Finance, MIT Press. [74] C. Wilson (1977): A Model of Insurance Markets with Incomplete Information, Journal of Economic Theory, 16,

84 BIBLIOGRAPHY 79 [75] W. Zame (1987): Competitive equilibria in production economies with an in nite dimensional commodity space, Econometrica, 55(5), [76] W.R. Zame (2007): Incentives, Contracts, and Markets: A General Equilibrium Theory of Firms, Econometrica, 75(5),