STATIONARY EQUILIBRIA IN ASSET-PRICING MODELS WITH INCOMPLETE MARKETS AND COLLATERAL

Transcription

1 Econometrica, Vol. 71, No. 6 (November, 2003), STATIONARY EQUILIBRIA IN ASSET-PRICING MODELS WITH INCOMPLETE MARKETS AND COLLATERAL BY FELIX KUBLER AND KARL SCHMEDDERS 1 We consider an infinite-horizon exchange economy with incomplete markets and collateral constraints. As in the two-period model of Geanakoplos and Zame (2002), households can default on their liabilities at any time, and financial securities are only traded if the promises associated with these securities are backed by collateral. We examine an economy with a single perishable consumption good, where the only collateral available consists of productive assets. In this model, competitive equilibria always exist and we show that, under the assumption that all exogenous variables follow a Markov chain, there also exist stationary equilibria. These equilibria can be characterized by a mapping from the exogenous shock and the current distribution of financial wealth to prices and portfolio choices. We develop an algorithm to approximate this mapping numerically and discuss ways to implement the algorithm in practice. A computational example demonstrates the performance of the algorithm and shows some quantitative features of equilibria in a model with collateral and default. KEYWORDS: Collateral, default, incomplete markets, Markov equilibrium. 1. INTRODUCTION LUCAS (1978) CONSIDERS AN EXCHANGE ECONOMY with a single perishable consumption good where infinitely lived identical agents trade shares in physical assets ( trees ). We extend the model to allow for heterogeneous agents who not only can trade in trees but can also make promises by selling financial securities. As in the two-period model of Geanakoplos and Zame (2002), agents can default on these promises at any time without any utility penalties or loss of reputation. Financial securities are therefore only traded if the promises associated with these securities are backed by collateral. The only collateral in this economy consists of shares in the physical assets. Under the assumption that all exogenous variables are Markovian, there exist Markov equilibria that can be characterized by a mapping from the exogenous shock and the current distribution of financial wealth to prices and portfolio choices. We develop an algorithm to approximate this mapping numerically 1 We thank seminar participants at the 2001 CEME General Equilibrium Conference at Brown, the SCE conference 2001 at Yale, SAET 2001 in Ischia, the 2001 Workshop on Game Theory in Stony Brook, the Tow conference on Computational Economics at the University of Iowa, the workshop on mathematical economics at IMPA/Rio de Janeiro, the 2002 Latin American meeting of the Econometric Society at Sao Paulo, Arizona State University, UC Davis, HEC Paris, the University of Maastricht, the University of Melbourne, the University of New South Wales, Northwestern University, the University of Rochester, and Stanford University, and in particular Jayasri Dutta, John Geanakoplos, Ken Judd, Dirk Krueger, Mordecai Kurz, Hanno Lustig, George Mailath, Alvaro Sandroni, and Steve Tadelis for helpful comments and conversations. We thank three anonymous referees and the Co-Editor for very useful suggestions on earlier drafts. 1767

2 1768 F. KUBLER AND K. SCHMEDDERS and give details on how to implement the algorithm on a computer. The implementation of the algorithm allows us to compute equilibria for realistically calibrated examples. We consider a simple model with a single tree (representing aggregate equity) and show that significant welfare gains can be obtained from collateralized borrowing. We also illustrate how equilibrium default may be welfare-improving. We include collateral and default in an infinite-horizon economy with incomplete markets because of the necessity to have economically meaningful constraints on agents choices. In models with infinitely lived agents, investors possible trading strategies have to be restricted to avoid Ponzi schemes. Levine and Zame (1996) propose a so-called implicit debt constraint that ensures that, in equilibrium, agents unconstrained Euler equations always hold. Unfortunately, when there are stocks in the economy, which pay dividends over more than one period, and when markets are incomplete, sequential equilibria do not always exist under this constraint. 2 Moreover, in calibrated models, an implicit debt constraint implies unrealistically high levels of individual debt; along the equilibrium path, agents sometimes borrow more than 20 times their yearly income. These problems have led applied researchers to impose much tighter exogenous debt constraints as well as short-sale constraints. Without modeling default, however, these constraints have no economic interpretation. Dubey, Geanakoplos, and Shubik (2000) and Geanakoplos and Zame (2002) incorporate default and collateral into the standard two-period GEI model. In Geanakoplos and Zame (2002), agents have to put up durable goods as collateral when they want to take short positions in financial markets. Economic agents are allowed to default on their promises, but in the case of default, the collateral associated with the promise is seized and distributed among the creditors. Araujo, Pascoa, and Torres-Martinez (2002) extend the model to a dynamic framework with infinitely lived agents and prove equilibrium existence. In this paper we simplify the model in Araujo, Pascoa, and Torres-Martinez (2002) to make it computationally tractable. A key feature of our model is that, just as in Geanakoplos and Zame (2002) and Araujo, Pascoa, and Torres- Martinez (2002), agents default on their promises whenever the market value of the shares they hold as collateral is lower than the face value of their promise. To obtain a better understanding of the impact of collateral constraints and default on the dynamic behavior of economies with incomplete markets, it is necessary to compute equilibria in such models. Computationally, it is only feasible to approximate equilibria that are dynamically simple in the sense that all endogenous variables are functions of some current state that follows a Markov 2 Magill and Quinzii (1996) show existence generically in infinite dividend sequences but their genericity conditions do not translate directly to conditions on an economy with stationary dividends and endowments.

3 ASSET-PRICING MODELS 1769 process. It is standard in modern macroeconomics to study the equilibrium dynamics indirectly by using recursive methods. According to Ljungqvist and Sargent (2000), recursive methods characterize a pair of functions: a transition function mapping the state of the model today into the state tomorrow and a policy function mapping the state into the other endogenous variables of the model. Previous computational work on models with incomplete markets and heterogeneous agents (e.g., Heaton and Lucas (1996), Judd, Kubler, and Schmedders (2003)) focuses on equilibria that can be described recursively, using as a state the current shock combined with last period s portfolio holdings. Unfortunately, for interesting specifications of the model, there are no known conditions that guarantee the existence of such equilibria (even without collateral constraints and default; see Kubler and Schmedders (2002)). When the state is taken to include all current endogenous variables, a Markov equilibrium does exist (see Duffie et al. (1994)), but the transition function that maps the state today into the state tomorrow can be arbitrarily complicated, and it is often impossible to determine even the set over which it is defined. In this paper we develop an alternative approach by merging a recursive method with the ideas of Duffie et al. (1994). The endogenous state is the collection of all current endogenous variables. Our notion of a stationary equilibrium is similar to a recursive equilibrium in that it also consists of a transition map and a policy map. However, in our approach, the policy map is a correspondence that maps beginning-of-period financial wealth (cash at hand) into possible current-period equilibrium prices and portfolios. The transition maps the current state into next periods endogenous states under the restriction that these have to lie in the graph of the policy map. Adapting an argument from Duffie et al. (1994) we construct a nonempty policy correspondence. Since it is computationally impossible to compute exact equilibria, the objective of our algorithm is to search for approximate equilibria that can be described via a policy function. We define a notion of ɛ-equilibrium and discuss its relation to exact equilibria. While the described algorithm is not globally convergent (i.e., we cannot prove that for arbitrary economies and arbitrary ɛ>0 it always computes a Markov ɛ-equilibrium), we find it to be very robust for a wide variety of calibrated economies. We use the algorithm to determine the impact of different collateral requirements on equilibrium welfares. In the model, the collateral requirement is exogenously given, and it is a natural question whether there are margin requirements that are socially optimal. In our examples, the households disagree on optimal margin requirements. Agents who own most of the financial wealth in the economy prefer low collateral requirements although they lead to frequent default in equilibrium while poor households prefer high requirements that lead to no default. The remainder of this paper is organized as follows. We introduce the model in Section 2. In Section 3 we give an existence proof of a Markov equilibrium that provides the basis for the development of an algorithm. Section 4 defines

4 1770 F. KUBLER AND K. SCHMEDDERS and discusses ɛ-equilibria and describes the implementation of the algorithm. A computational example in Section 5 demonstrates the performance of our algorithm and shows some quantitative features of equilibria in a model with collateral and default. Section 6 concludes. 2. THE ECONOMIC MODEL We examine a model of an exchange economy that extends over an infinite time-horizon and is populated by infinitely-lived heterogeneous agents. The Physical Economy At each period t 0oneofY possible exogenous shocks y Y ={1 Y} occurs. We represent the resolution of uncertainty by an event tree Σ. The root of the tree σ 0 is given by a fixed state y 0 Y in which the economy starts at time 0. Each node of the tree is characterized by a history of shocks σ = (y 0 y t ).Eachnodeσ has Y immediate successors, (σy) y = 1 Y and a unique predecessor σ. To simplify notation we sometimes include the root node s predecessor σ 0 in the event tree Σ. The exogenous shocks follow a Markov process with transition matrix π At each node σ Σ there is a single perishable consumption good. There are H agents, which we collect in a set H. Atnodeσ Σ agent h s individual endowment in the consumption good, e h (σ), depends on the current shock alone, i.e. e h (σ y) = e h (y) where e h : Y R ++ is a time-invariant function. In addition, the agent owns shares in physical assets (i.e., Lucas trees these assets can be interpreted as firms, machines, land, or houses). There are A different such assets, which we collect in A. At period 0 each agent h owns a share θ h(σ ) 0oftreea and we normalize a 0 h H θh(σ a 0 ) = 1foralla A. At a node σ = (σ y) let θ h a (σ) denote agent h s (end-of-period) share in tree a. We assume that trees are traded cum dividends, that is, buying a tree allows the agent to harvest the fruit in the same period: If agent h holds a share θ h(σ) of tree a, then he obtains a dividend payment a θh a (σ) dh a (y) The dividend that asset a produces when it is held by agent h, d h : Y R a + depends solely on the current state y Y.Ifd h a is identical across all agents h H then we have the standard Lucas model. We allow the payoff of some collateralizable assets to depend on their current owner. This generalization of the model makes no sense when the trees represent equity as in Lucas model. However, there is at least one alternative interpretation of trees for which the generalization is useful: In Shleifer and Vishny (1992) and Kiyotaki and Moore (1997), the collateralizable asset can only be used productively if owned by agents who can operate it. Harvesting the trees requires human capital that cannot be traded: the tree may be full of fruit, but if the agent who owns it has to pick the fruit, he may lose some of it in the process.

5 ASSET-PRICING MODELS 1771 With this construction of dividends, aggregate consumption depends on the distribution of assets and is therefore endogenous. We define an upper bound on aggregate consumption for each shock y by ē(y) = h H e h (y) + max d h (y) a h H a A Agent h H maximizes a time-separable expected utility function { } U h (c) = E β t u h h(c t y t ) t=0 We assume that the possibly state dependent Bernoulli function u h ( y) : R ++ R is strictly monotone, C 2, strictly concave, bounded above, and unbounded below, i.e., u h (x y) as x 0, for all y Y. In order to rule out that agents are almost satiated at aggregate endowments, we assume that there exists a consumption level ĉ such that for all h H, min u h (ĉ y) > 1 max u h (ē(y) y) y Y 1 β h y Y We also assume that each agent s discount factor lies between zero and one, β h (0 1), and that all agents agree on the transition matrix for the shock process. In the following discussion we will suppress the dependence of u h on y whenever there is no possibility of confusion. Markets We consider an economy with security trading in every time period. Agent h can buy θ h(σ) shares of tree a at node σ for a price q a a(σ). Aslongasθ h 0 a there is no possibility of default since no promises are made when shares of the physical assets change hands. In addition to the physical assets, there are J financial assets, which we collect in a set J. These assets are one-period securities; asset j traded at node σ promises a payoff b j (σy) = b j (y) > 0 at the successor nodes (σy), y = 1 Y.Wedenoteagenth s portfolio in financial assets by φ h and write p j (σ) for the price of asset j. Agents can only sell (short) a financial security if they hold shares of trees as collateral. At each node σ, weassociatean A-dimensional vector k j (σ) > 0 of collateral requirements with each financial security j J. If an agent sells 1 unit of security j she is required to hold k j a (σ) dollars worth of each tree a = 1 A as collateral. If an asset a can be used as collateral for different financial securities, the agent is required to invest k j a (σ) for each j = 1 J. In the next period the agent can default on her promise b j In this case the buyer of the financial security gets the collateral associated with the promise.

6 1772 F. KUBLER AND K. SCHMEDDERS We allow the margin requirement k j a to vary across nodes, but to retain stationarity it must be a function of the current shock and current period endogenous variables. We write k j (σ) = k ( j a a p(σ) q(σ) (θ h (σ) c h (σ)) h H y ) for some continuous functions k j a ( y), y Y. In order to rule out trivial equilibria we need to assume that for all j J, y Y, q c, andθ, (1) inf k j (p q c θ y)>0 a p 0 for at least one a A For notational simplicity we sometimes suppress the dependence of k on σ. Since there are no penalties for default, a seller of an asset defaults at a node σ = (σ y) whenever b j (y) > a A kj (σ a )(q a (σ)/q a (σ )). By individual rationality, the actual payoff of security j at node σ = (σ y) is therefore always given by { f j (σ) = min b j (y) k j (σ a ) q } a(σ) q a A a (σ ) Note that an agent can default on individual promises without declaring personal bankruptcy and giving up all the assets he owns. Since this implies that the decision to default on a promise is independent of the debtor, we do not need to consider pooling of contracts as in Dubey, Geanakoplos, and Shubik (2000), even though there may be default in equilibrium. This treatment of default is somewhat unconvincing since default does not affect a household s ability to borrow in the future and it does not lead to any direct reduction in consumption at the time of default. Moreover, declaring personal bankruptcy typically results in a loss of all assets, and it is rarely possible to default on some loans while keeping the collateral for others. However, there do exist laws for collateralizable borrowing where default is possible without declaring bankruptcy. Examples include pawn shops and the housing market in some U.S. states, in which households are allowed to default on their mortgages without defaulting on other debt. Conversely, in some U.S. states such as Texas and Florida, property can be seized in a personal bankruptcy, but houses are exempt from seizure. Our model therefore captures some aspect of default even though it is certainly not a compelling model of bankruptcy. Our model is analytically tractable and default provides a rationale for collateral constraints. Following Geanakoplos and Zame (2002), we can endogenize the margin requirements by introducing a menu of financial securities. All securities promise the same payoff, but they distinguish themselves by the margin requirement. In equilibrium only some of them will be traded, thereby determining an endogenous margin requirement.

7 ASSET-PRICING MODELS 1773 We can also determine the existence of socially optimal margin requirements. In Section 5 we consider a simple example, which shows that there is often disagreement between agents on welfare-maximizing requirements. Financial Markets Equilibrium with Collateral An economy E is a collection of Bernoulli functions, impatience parameters, state-dependent endowments and asset promises, transition probabilities and collateral requirements, E = ( (u h β h e h θ h (σ 0 ) dh ) h H b π k ) We define a financial markets equilibrium as follows: DEFINITION 1: A financial markets equilibrium for an economy E with initial tree holdings (θ h (σ 0 )) h H and initial shock y 0 is a collection ( ( θ 1 (σ) φ 1 (σ) c 1 (σ)) ( θ H (σ) φ H (σ) c H (σ)) q 1 (σ) q A (σ) p 1 (σ) p J (σ) ) σ Σ satisfying the following conditions: (i) Markets clear: θ h (σ) = 1 and h H h H (ii) For each agent h: φ h (σ) = 0 forallσ Σ ( θ h (σ) φ h (σ) c h (σ)) arg max U h (c) θ 0 φ c 0 such that for all σ = (σ y) Σ c h (σ) = e h (y) + φ h (σ ) f(σ)+ θ h (σ ) q(σ) + θ h (σ) (d h (y) q(σ)) φ h (σ) p(σ) q a θ h (σ) + a k j a φh(σ) 0 j j J :φ h j (σ)<0 (a = 1 A) The following observation from Duffie et al. (1994) is important throughout the analysis of this paper: In each financial markets equilibrium, agents optimality implies that there exists a positive lower bound on an agent s optimal consumption. For each agent h and shock y we can define a lower bound on consumption, c h (y) > 0 by u h (c h (y) y) + β h max 1 β h y Y u h (ē(y ) y ) 1 min u h (e h (y ) y ) 1 β h y Y

8 1774 F. KUBLER AND K. SCHMEDDERS Let c h = min y c h (y). This positive lower bound for possible equilibrium consumption always exists, since all utility functions u h are unbounded from below. The term on the right-hand side is a lower bound on the agent s lifetime utility if the agent has sold all her shares in trees, because she can still afford to consume her labor endowments e h (y). The second term on the left-hand side is an upper bound on the agent s utility after the first period because consumption is bounded above by aggregate endowments. Financial Wealth In any financial markets equilibrium, tree prices are positive, q>0 We define ω h (σ) as household h s fraction of financial wealth (cash at hand) in the economy at the beginning of node σ = (σ y),i.e. ω h (σ) := θh (σ ) q(σ) + φ h (σ ) f(σ) A q a=1 a(σ) Let Ω(σ) = (ω 1 (σ) ω H (σ)) Note that by the definition of f, in equilibrium Ω always lies in the (H 1)-dimensional simplex H 1 i.e. ω h 0 h H and h H ωh = 1. If all initial tree holdings are proportional (i.e. if for all h H, thereisa θ h such that θ h a (σ 0 ) = θ h for all a A), we must have that ω h (σ 0 ) = θ h. 3. EXISTENCE OF EQUILIBRIUM Araujo et al. (2002) consider a model that is related to ours, but with durable and storable goods instead of long-lived assets, and prove that equilibria always exist. However, since we are interested in approximating equilibria numerically, an adaptation of their existence proof to our model would not help us. We want to describe equilibria as a mapping from the beginning-of-period wealth distribution to current period equilibrium prices and portfolio holdings. From such a mapping we are able to infer a transition mapping the current state of the economy into next period s state. We call equilibria that can be described in such a way Markov Markov Equilibrium In order to define our notion of Markov equilibrium, we need to adapt some concepts from Duffie et al. (1994) to our model The State Space We let the state space S consist of all exogenous and endogenous variables that occur in the economy at some node σ,i.e.s = Y Z,whereY is the finite

9 ASSET-PRICING MODELS 1775 set of exogenous shocks and Z is the set of all possible endogenous variables. Let z(σ) = (Ω(σ) (c h (σ) θ h (σ) φ h (σ)) h H q(σ) p(σ)) be the endogenous state at node σ. We define the endogenous state space as Z = { z H 1 R H + RAH + RHJ R A + RJ + : h H θh a = 1foralla A h H φh j = 0forallj J } By definition, in any financial markets equilibrium, all endogenous variables lie in Z We will make frequent use of the set of endogenous variables without the wealth variables, and we denote this set by Z ˆ Note that Z = H 1 Z ˆ Expectations Correspondence For a set X,letX Y be the Cartesian product of Y copies of X. Given a state (y z) S, the expectations correspondence g : S Z Y describes all next period states that are consistent with market clearing and agents first-order conditions. A vector of endogenous variables (z + 1 z+ Y ) g(s) if for all households h H the following conditions hold. (a) For all y = 1 Y, ω h+ y = θh q + + y j J φh min{b j j(y) a A kj q ay + a q a } A a=1 q+ ay and c h+ y = e h (y) + ω h+ y A a=1 q + ay + θh+ y (d h (y) q + y ) φh+ y p + y ch (b) There exist multipliers λ h R A and + µh R A + such that for all a A the following equations and inequalities hold: (2) (3) (4) (5) µ h a q a + λ h a + (dh a q a)u h (ch ) + β h E s{ q + a u h (ch+ ) } = 0 λ h a θh = 0 a ( q a θ h + a µ h a q a θ h a + j J :φ h j <0 k j a φh j {j J :φ h j <0} k j a φh j 0 ) = 0

10 1776 F. KUBLER AND K. SCHMEDDERS (c) In addition, if we define φ h( ) = max(0 j φh) and j φh j (+) = max(0 φ h), there exist multipliers j νh (+) ν h ( ) R J +,suchthatforallj J the following conditions are satisfied: (6) µ h a kj p a ju h (ch ) + β h E s{ f j u h (ch+ ) } ν h j ( ) = 0 a A (7) p j u h (ch ) + β h E s{ f j u h (ch+ ) } + ν h j (+) = 0 ν h (+) φ h (+) = 0 (8) (9) ν h ( ) φ h ( ) = 0 Conditions (a) follow from the definition of ω and the budget constraint. The conditions (b) are part of the standard first-order conditions with respect to θ h a Finally, conditions (c) consist of the Kuhn Tucker conditions with respect to φ h j, where we treat the choices of long and short positions in the asset separately (i.e. the agent chooses both φ h( ) and j φh j (+) and therefore has two first-order conditions for each financial asset). There is a redundancy in the Kuhn Tucker conditions in (b): if the collateral constraint for an asset a is satisfied, the short-sale constraint on this collateralizable asset holds automatically. We keep the redundant conditions for illustrative purposes. We need the following lemma to show existence. All proofs are collected in the Appendix. LEMMA 1: The graph of g is a closed subset of S Z Y Definition of Markov Equilibrium A Markov equilibrium consists of a (nonempty valued) policy correspondence, P, and a transition function, F, P : Y H 1 ˆ Z and F : graph(p) Z Y such that for all s graph(p) and all y Y F(s) g(s) and (y F y (s)) graph(p) We say a Markov equilibrium is simple if the associated policy correspondence is single valued. The following lemma verifies that Markov equilibria are in fact competitive equilibria in the usual sense. LEMMA 2: A Markov equilibrium is a financial markets equilibrium according to Definition 1.

11 ASSET-PRICING MODELS Constructing Markov Equilibria In constructing a Markov equilibrium (P F), we closely follow the existence proof in Duffie et al. (1994). The basic idea of our approach is very similar to backward induction. We start with a compact set, T Z ˆ that is large enough to ensure that for all finitely truncated economies the endogenous variables take equilibrium values in T. The fact that such a set exists is established in Lemma 3 below. Assuming that next period s equilibrium variables can be described by some correspondence, V n : Y H 1 T we define for each exogenous shock y and wealth distribution Ω a new correspondence, V n+1 where V n+1 (y Ω) is the set of all endogenous variables today that are consistent (in the sense that all agents first-order conditions and market clearing hold) with some ((Ω 1 ẑ 1 ) (Ω Y ẑ Y )) tomorrow for which ẑ y V n (y Ω).Withthis construction, V n n=1 (y Ω) is nonempty for all y Y, Ω H 1. Formally, given a compact set T Z ˆ and a correspondence V : Y H 1 T define an operator, G T that maps the correspondence V : Y H 1 T to a new correspondence W : Y H 1 T as follows: W = G T (V ) if for all Ω H 1 and all y Y, W(y Ω)= { (c θ φ q p) T : (z 1 z Y ) g(y Ω c θ φ q p) such that for all y Y (c y θ y φ y q y p y ) V(y Ω y ) where z y = (Ω y c y θ y φ y q y p y ) } In order to construct a policy correspondence we first need a suitable set T that is large enough to contain all endogenous variables. We define a truncated economy with T + 1 periods to be a finite-horizon economy with an event tree, where all nodes of the form σ = (y 0 y 1 y T ) are terminal nodes. Endowments and asset payoffs at nodes of the truncated tree, as well as agents preferences over consumption at these nodes, are the same as in the infinite-horizon economy. LEMMA 3: For all T 1 there exists a financial markets equilibrium for the truncated economy in which values of all prices, asset holdings, and consumption allocations lie in a compact set T Z ˆ Define V 0 by V 0 (y Ω) = T for all Ω H 1 and all y Y. Given a correspondence V n n= 0 1 define recursively V n+1 = G T (V n ) Finally let (10) V (y Ω) := V n (y Ω) n=1 We can now state our main theorem. for all y Y Ω H 1 THEOREM 1: The correspondence V is nonempty valued and there exists a Markov equilibrium with policy correspondence V.

12 1778 F. KUBLER AND K. SCHMEDDERS Equilibrium Correspondence Remarkably, the correspondence V as constructed above contains all possible values of first period endogenous variables for all equilibria with values in the bounded set T Let the equilibrium correspondence, B y (Ω), contain all ((c h θ h φ h ) h H q p)for which there exists a financial markets equilibrium for initial shock y and initial tree-holdings θ h(0 a ) = ω h for all h H, a A with ( (c h θ h φ h ) h H q p ) = ( ( c h (σ 0 ) θ h (σ 0 ) φ h (σ 0 )) h H q(σ 0 ) p(σ 0 ) ) and with all endogenous variables lying in T. For any policy correspondence P,weobviouslyhaveP(y Ω) B y (Ω) for all Ω H 1 y Y. More importantly, we have V (y Ω) = B y (Ω).Forcomputational purposes, it would be important to know under which conditions there exists a (single-valued) selection of B that is also a policy correspondence. We know of no nontrivial conditions. 4. COMPUTATION OF MARKOV EQUILIBRIA To examine quantitative features of equilibria, we develop an algorithm to approximate Markov equilibria numerically. The algorithm searches for a continuous policy function that describes prices and allocations, such that markets clear and agents make small optimization errors. Since computed equilibria suffer from rounding and truncation errors, they are (almost) never exact. We therefore define an ɛ-equilibrium. The purpose of the described algorithm is then to find ɛ-equilibria for small ɛ>0. We will interpret ɛ-equilibria purely as a computational necessity. To assess the quality of a solution from a numerical procedure, it is now common in computational work to examine the relative errors in agents first-order conditions (see den Haan and Marcet (1994), Judd (1998), and Santos (2000)) Epsilon-Equilibria We define an ɛ-equilibrium by requiring that in the expectations correspondence relative errors in Euler equations (i.e. the relative error in equations (2), (6), and (7)) are smaller than ɛ for all (Ω y) H 1 Y. In the definition, we assume that market clearing and budget constraints hold exactly. Since in numerical work all equations hold at most with machine precision, this is an idealization. In Section 4.3 below, we argue how this can be justified.

13 ASSET-PRICING MODELS 1779 To define an ɛ-markov equilibrium, we define an approximate expectations correspondence g ɛ as in Section above, with the exception that we replace equation (2) by 1 + µhq a a + λ h + β a he s {q + a u h (ch+ )} (2 ) (q a d q )u h (ch ) ɛ equation (6) by 1 + a A µh a kj + β a he s {f j u h (ch+ )} ν h( ) j (6 ) p j u h (ch ) ɛ and (7) by 1 + β he s {f j u h (ch+ )}+ν h(+) j (7 ) p j u h (ch ) ɛ An ɛ-markov equilibrium is then a pair (P ɛ F ɛ ) such that for all s graph(p ɛ ) and all y Y, (y F ɛ y (s)) graph(p ɛ ) and F ɛ (s) g ɛ (s) An ɛ-markov equilibrium is simple if P ɛ is a function. While simple exact equilibria may fail to exist (see Kubler and Schmedders (2002) 3 ), simple ɛ-equilibria always exist. The following argument from Kubler and Polemarchakis (2002) proves how this follows from the existence of an exact Markov equilibrium. By construction, the equilibrium correspondence V has a compact graph. Therefore, for each ɛ 1 > 0 there exists a finite set F ɛ 1 graph(v ) such that [ ] sup inf s s <ɛ 1 s graph(v ) s F ɛ 1 which can be written as the graph of a function v ɛ 1 : Y Z, with the finite sets Z Z ˆ and H 1 ensuring that if ( c θ φ q p) Z then for all y Y there exist Ω and (c θ φ q p)= v ɛ 1 (y Ω) such that for all h, ω h = θ h q + φ h f A a=1 q a 3 Krebs (2000) also discusses existence in these models, but his analysis rests crucially on the absence of trading constraints and has therefore no implications for our model.

14 1780 F. KUBLER AND K. SCHMEDDERS Given an ɛ>0, by continuity of the utility functions, for sufficiently small ɛ 1 the function v ɛ 1 describes a simple Markov ɛ-equilibrium. Evidently, by construction, this function can generally not be extended to a continuous function over H 1. Before we describe our algorithm for computing ɛ-equilibria in Section 4.3 we discuss possible interpretations of ɛ-equilibria Interpretation of ɛ-equilibria As mentioned above, we view our focus on ɛ-equilibria as a computational necessity. A different motivation would be to argue that economic agents have bounded computational capacity and can only find approximately optimal choices. Any improvement over their choices results in an extra gain of at most ɛ (see, e.g., Akerlof and Yellen (1985)). However, in our model there is a tension between assuming that agents have rational expectations about future prices and assuming that they make errors in choosing their consumptions given the correct forecasts for prices. Furthermore, since markets clear exactly, future prices must already reflect the agents optimization errors. In order to relate computed ɛ-equilibria to exact equilibria we suggest estimation of perturbations that need to be imposed on the exogenous variables in order to obtain the results actually computed as approximate solutions. 4 In the context of dynamic games, Mailath, Postlewaite, and Samuelson (2002) show that a pure ɛ-nash equilibrium is an exact Nash equilibrium of a game with nearby discount factors and reward functions. Similarly, we want to interpret acomputedɛ-equilibrium as an approximate equilibrium of some other economy with endowments and preferences that are close to those in the original economy. As an example, consider a simple economy without financial assets and with a single tree. In this case, perturbing the individual discount factors at every node is sufficient for obtaining an economy for which a computed ɛ-equilibrium is exact: 5 We can assume without loss of generality that in equation (2 ) λ h = 0. If for some agent h, λ h > 0 and the left-hand side of equation (2 ) is nonzero, the error in the equation can be reduced by varying λ h.in a the optimum, either the error or λ h must be equal to zero. In order for a (2 )to hold exactly, it suffices that the true impatience parameter β h is replaced by some β h satisfying β 1 + ɛ β h β 1 ɛ 4 In numerical analysis this method is called backward error analysis (see, e.g., Judd (1998, p. 48)). 5 If there are several trees or financial securities in the economy, the perturbations need to involve individual endowments as well. Deriving exact bounds for this case is subject to further research.

15 ASSET-PRICING MODELS 1781 Unfortunately, for a given ɛ>0 it is generally impossible to infer how close computed allocations and prices are to exact equilibrium allocations and prices for the same economy. In simple finite-dimensional fixed-point problems, it is obvious that f(x) <ɛdoes not always imply (reasonable) bounds on x x for an x with f(x ) = 0. General conditions that ensure that approximate fixed points are close to true fixed points have to involve higher derivatives of f at x and are often difficult to obtain (see, for example, Blum et al. (1998, Chapter 8)). In our infinite-horizon models, matters are even worse. As ɛ 0 there is no guarantee that the ɛ-equilibrium allocation converges to an exact equilibrium allocation. It is easy to construct an ɛ-equilibrium for the above simple example that is not close to an exact equilibrium of the original economy. The intuition is that small errors in the Euler equations may propagate over time and lead to large errors in the long run behavior of the model. This poses a challenge for computational methods that claim to approximate an exact equilibrium by an ɛ-equilibrium and it is the reason why we include perturbations of exogenous parameters. In applications of dynamic general equilibrium models most exogenous variables such as discount factors, endowments, or asset dividends are often not known exactly. The approximation of equilibria for a close-by economy might then be almost as interesting as the approximation of equilibria for the given economy The Algorithm We compute an approximate 6 policy function via an iterative algorithm. Denote this function by ρ : Y H 1 T As a starting point, we choose a continuous function ρ 0 : Y H 1 T In each iteration of the algorithm, given functions ρ n we construct ρ n+1 y (Ω) for all y Y Ω H 1 by taking a selection of G T (ρ n ) Intuitively, we assume that the values of the endogenous variables in the next period, as a function of agents wealth, is determined by the function ρ n We then move backwards one period and determine ρ n+1 by solving for an equilibrium in the current period. The algorithm terminates if for some prespecified δ>0 (11) sup ρ n+1 ρ n <δ y Ω We define a candidate policy function, ρ = ρ n+1 and examine the quality of the solution by computing the maximum relative error in Euler equations. If this error lies above some prespecified ɛ we restart the algorithm with a lower value for δ. 6 For the remainder of the paper, we drop the adjective approximate. It is understood that all equilibria we compute are only ɛ-equilibria.

16 1782 F. KUBLER AND K. SCHMEDDERS Our algorithm will not converge if there does not exist a continuous policy function. In our practical computational experience using realistically calibrated models, this problem has not occurred. The implemented algorithm has always produced a policy function. Outline of Main Steps The actual implementation of the algorithm involves several tedious details, many of which are common in the literature on finding equilibrium functions for infinite-horizon problems. We do not discuss them in detail here and instead refer to the survey by Judd, Kubler, and Schmedders (2003). We focus on the aspects of the algorithm that are new to the literature. We restrict attention to economies with H = 2 agents. In this case, the wealth distribution is represented by a single number in the unit interval. Therefore the equilibrium policy function ρ is a function of only the discrete exogenous state variable y and the single continuous wealth variable ω 1 [0 1]. We approximate the equilibrium functions by cubic splines (i.e. twice differentiable piecewise cubic functions; see deboor (1978)). In order to handle nondifferentiabilities in the policy functions, we use several hundred knots (i.e. pieces in the spline-approximation). We solve for the spline coefficients using a time-iteration collocation algorithm similar to the one described in detail in Judd, Kubler, and Schmedders (2003). As the example in Section 5 demonstrates, in economies with collateral constraints, policy functions may not be smooth on the entire domain, but instead exhibit some kinks. At these kinks cubic splines are not well behaved approximating functions. Therefore we need to use many knots in order to obtain a good approximation. On the other hand, we cannot use lower order piecewise polynomials, such as piecewise linear functions, since we want to approximate the smooth portions of the policy and price functions with differentiable functions. Our choices regarding the approximation of the equilibrium functions represent a trade-off between the goal of achieving approximations with small errors and the need for keeping running times at acceptable levels. More knots would increase the precision and the running times; fewer knots would reduce the precision and running times. Given a wealth distribution Ω = (ω 1 1 ω 1 ), the current shock y today, and the spline coefficients of an approximate map from tomorrow s wealth distribution into tomorrow s prices and portfolio holdings ξ, we need to solve for prices and optimal choices today that lie in G(ρ n ) In order to do so, we simultaneously solve two systems of nonlinear equations. Given a value for wealth shares next period, ω 1+, (p q (θ h φ h ) h=1 2 ) lie in G(ρ n ) at ω 1 if and only if they solve all first order conditions and markets today clear. The Kuhn Tucker inequalities in these first order conditions can easily

17 ASSET-PRICING MODELS 1783 be replaced by equalities via a change of variable (see Garcia and Zangwill (1981)) and the system can be written as (12) F(q p (θ h φ h ) h=1 2 ; ω 1 y ξ)= 0 In order to infer tomorrow s wealth share, ω 1+ (which is implicit in equation (12)), from today s choices, we solve a second nonlinear system of equations. Given choices (θ h φ h ) h=1 2 today, as well as approximate equilibrium price functions ρ n q (y Ω; ξ), tomorrow s wealth distribution Ω+ must solve (13) ω 1+ (y) := θ1 ρ n(y q ω1+ ; ξ) + φ 1 f(y ω 1+ ) A a=1 ρn q a (y ω 1+ ; ξ) We can solve this single equation in the one unknown ω 1+ using a standard Newton solver. Via equation (13), we have then tomorrow s optimal choices implicitly defined as functions of today s portfolio choices, and so equation (12) is well defined. The system can be solved with standard nonlinear equation solvers. At the beginning of iteration n the spline coefficients ξ n for the policy function iterate ρ n are given. For each shock y = 1 Y and on agridofω 1 [0 1] the algorithm solves the nonlinear system of equations (12) for prices and choices (q p (θ h φ h ) h H ) The resulting functions (q p (θ h φ h ) h H )(y ω 1 ) are then approximated by splines with new coefficients, ξ n+1 and the next iteration begins. Since we use interpolating splines, we need to solve (12) at as many points as there are spline-knots for each shock y. Our algorithm can be summarized as follows: Time Iteration Spline Collocation Algorithm Step 0: Select an error tolerance δ for the stopping criterion (11), a finite grid W [0 1] of collocation points, and the spline coefficients ξ 0 for a starting point ρ 0 Step 1: Given spline coefficients ξ n and thus the function ρ n, solve system (12) ω 1 W y Y finding a solution (q p (θ h φ h ) h H )(y ω 1 ) Step 2: The implicit wealth share ω 1+ in (12) is found by solving (13). Compute the new approximations ρ n+1, that is, the new coefficients ξ n+1 by interpolation of (q p (θ h φ h ) h H )(y ω 1 ) ω 1 W y Y Step 3: Check stopping criterion (11). If sup y Ω ρ n+1 ρ n <δ,thengotostep4. Otherwise increase n by 1 and go to Step 1. Step 4: The algorithm terminates. Set ρ = ρ n+1.

18 1784 F. KUBLER AND K. SCHMEDDERS Epsilon-Equilibria and Computed Equilibria In order to assess the accuracy of a candidate solution we compute the maximal relative error in Euler equations on a grid of 10,000 points. Evidently this error only provides a lower bound on the true maximal error in Euler equations. However, since our approximated functions are continuous and (except at a small number of kinks) smooth, the true maximal error is not likely to be much higher. As mentioned above, market-clearing and all budget constraints will also not hold exactly but only with machine precision. The machine precision lies around (on most PC s). Therefore, errors in budget constraints and market clearing are always several orders of magnitude below the errors in Euler equations and can thus be suppressed in our analysis. We revisit the issue of numerical errors (and running times) in Section 5.1 in the context of a concrete example Recursive Equilibria and Simple Markov Equilibria Existing algorithms to approximate equilibria in infinite-horizon models with incomplete markets focus on recursive equilibria. In these equilibria, the endogenous state space consists of agents previous period s portfolio choices (see, for example, Heaton and Lucas (1996) and Judd, Kubler, and Schmedders (2003)). In order to approximate such a recursive equilibrium, the set of possible portfolios must be compact and known ex ante. Exogenous portfolio bounds are used to ensure this compactness requirement. Obviously, we cannot use this equilibrium concept in a model with default and collateral constraints where portfolios are only bounded endogenously. In addition, it appears unlikely that the portfolio choices from the previous period are sufficient state variables, since the payoffs of financial assets depend on previous-period prices. If there are no financial securities in our model (and the tree s dividends are independent of its owner), there are no collateral constraints and thus our model essentially reduces to a version of Lucas (1978) model with heterogeneous agents. In this case, we can use our algorithm to compute recursive equilibria. The set of admissible portfolios is known and compact, and a recursive equilibrium is described by a function mapping portfolios and the current shock, (Θ y) ( H 1 ) A Y to current-period prices and portfolios. Although we cannot show that the existence of simple Markov equilibria is equivalent to the existence of recursive equilibrium, we can construct a recursive equilibrium (if it exists 7 ) from a simple Markov equilibrium with a continuous policy corre- 7 Recursive equilibria do not always exists (see Kubler and Schmedders (2002)), but it is not clear whether there are examples where simple Markov equilibria exist but recursive equilibria do not.

19 ASSET-PRICING MODELS 1785 spondence. The construction is as follows: For all Θ = (θ h ) h H ( H 1 ) A the function Γ Θ : H 1 H 1 defined by Γ Θ (Ω) = θh q(ω) A a=1 q a(ω) is continuous and therefore always has a fixed-point. Numerically, we solve for this fixed point by solving equation (13). The subproblem of solving equation (13) is an innovation, which is made necessary by the use of the state variable wealth. Contrary to the literature of computing recursive equilibria, the agents choice variables (portfolios) are not state variables. Agents portfolio decisions today do not immediately yield tomorrow s wealth distribution. Since the dimension of the domain of the equilibrium map is independent of the number of securities traded, this setup is a significant practical improvement over existing algorithms, which, due to a curse of dimensionality, cannot consider models with more than two assets (see Judd, Kubler, and Schmedders (2003) for an overview). In particular, our algorithm can be used to compute equilibria in a model in which all assets are several Lucas trees (i.e. Lucas (1978) model with heterogeneous agents). 5. AN EXAMPLE We compute equilibria for a stylized example that illustrates the impact of collateral requirements on equilibrium welfares. In the model setup, the asset structure and the collateral requirements are exogenously fixed. By maintaining the assumption of exogenously given asset payoffs, we show that it is very well possible that agents disagree on the optimal (exogenous) margin requirements. Model Parameterization There are two agents with identical constant relative risk aversion utility and a coefficient of risk aversion of 2. They have identical impatience parameters β 1 = β 2 = There are 4 exogenous shocks. Aggregate endowments are given by ē = ( ). We assume that the payoffs of the tree are independent of its owner and given by d(y) = 0 3 ē(y) for all y. Individual endowments are e 1 = ( ) e 2 = 0 7ē e 1 The transition probabilities are given by { π(y y ) = 0 4 fory y = 1 2ory y = 3 4, 0 1 otherwise.

20 1786 F. KUBLER AND K. SCHMEDDERS FIGURE 1. Agent 1. We first examine the welfare gains from introducing collateralized borrowing. We compare an economy where the only asset available is the tree to an economy, where this tree can be used as collateral for trading in a risk-free bond. We assume that the collateral requirement in the latter economy is fixed across states at k = 1 3 Figures 1 and 2 depict the welfare gains from the additional asset in the economy with collateral as a function of the wealth distribution for an economy with initial shock y 0 = 1 Welfare changes are in terms of percentage consumption equivalent variation. FIGURE 2. Agent 2.

21 ASSET-PRICING MODELS 1787 Agent 1, who starts off in his bad idiosyncratic shock, gains significantly. When he owns only 10 percent of the stock, his consumption in the one-asset economy would have to increase by 7.4% in each state in order to make him as well off as he is with collateralized borrowing. Agent 2 s welfare losses are generally much smaller, except in situations where agent 1 owns most of the tree. For a very extreme initial wealth distribution, where agent 2 owns the entire tree, he loses when a new asset is introduced because of the price effect on the tree. Margin Requirements We expect equilibrium welfares to depend on the margin requirement k since different values for k result in different probabilities of default and different consumption allocations. Quantitatively, these welfare effects turn out to be relatively small. Table I reports default percentages as a function of k In addition, it depicts welfare gains and losses (in %) from reducing the collateral requirement from 1.3 to a smaller value. In the computed equilibria, default never occurs if collateral requirements are larger than k = 1 2 Collateral requirements exceeding 1.2 are not optimal from a welfare point of view; higher requirements only decrease agents ability to borrow and therefore lead to welfare losses compared to an economy with k = 1 2 Remarkably, when agents initial wealth levels are unequal, the agents disagree on the optimal requirement. The rich agents tend to favor low collateral requirements that lead to high default rates, while poor agents tend to favor requirements that lead to low default rates. Although the above example is very stylized, it matches some key features of U.S. data. The variation in total endowments, as well as the capital share, seem reasonable. In order to examine the robustness of the qualitative result, we also computed equilibria for an eight-state model using the calibration of Heaton and Lucas (1996). In their calibration, idiosyncratic uncertainty is much smaller, but aggregate shocks are about the same. In this model, the resulting welfare gains from introducing an asset in addition to a single tree are generally very small. For k = 1 2 welfare gains lie between 0.1 and 0.6 percent of consumption equivalent variation. While different collateral requirements TABLE I MARGIN REQUIREMENTS, DEFAULT, AND WELFARE k Default rate Ω = ( ) ( ) ( ) ( ) ( ) ( ) Ω = ( ) ( ) ( ) ( ) ( ) ( ) Ω = ( ) ( ) ( ) ( ) ( ) ( )

22 1788 F. KUBLER AND K. SCHMEDDERS do lead to slightly different equilibrium welfares that exhibit the same qualitative features as those in our stylized model, the differences always lie below 0.01 percent. (Since we compute ɛ-equilibria with ɛ lying around 10 5,itis unclear whether these welfare differences are meaningful.) 5.1. Some Remarks on the Computation We implemented the algorithm in Fortran 77 on a 500 MHz Pentium PC. In the examples in this section, maximal errors lie between and 10 5 (evaluated on a grid of points). In the case where there is only a tree and no financial security, this implies that the computed equilibrium is exact for economies where the β h vary stochastically between and For the example with 4 states, running times lie around 1 5 hours. For the example with 8 shocks and the calibration from Heaton and Lucas (1996), running times lie around 5 6 hours. As mentioned above, collateral constraints provide additional computational challenges because policy functions fail to be smooth when a constraint becomes binding. Figures 3 and 4 depict the portfolio policy functions for y = 1 and k = 1 2. It is apparent from the figures that cubic splines perform well, although the portfolio policy functions exhibit two nondifferentiabilities. In the example, we used 300 knots for the spline approximation. Although there are only two kinks in each policy function, we do not know where they occur and therefore use equi-spaced points. FIGURE 3. Stock-holding.