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Belief Heterogeneity, Collateral Constraint, and Asset Prices with a Quantitative Assessment Dan Caoy Department of Economics, Georgetown University Abstract The recent economic crisis highlights the role of …nancial markets in allowing economic agents, including prominent banks, to speculate on the future returns of di¤erent …nancial assets, such as mortgage-backed securities. This paper introduces a dynamic general equilibrium model with aggregate shocks, endogeneously incomplete markets and heterogeneous agents to investigate this role of …nancial markets. In addition to their risk aversion and endowments, agents di¤er in their beliefs about the future exogenous states (aggregate and idiosyncratic) of the economy. This di¤erence in beliefs induces them to take large bets under frictionless complete …nancial markets, which enable agents to leverage their future wealth. Consequently, as hypothesized by Friedman (1953), under complete markets, agents with incorrect beliefs will eventually be driven out of the markets. In this case, they also have no in‡uence on asset prices in the long run. In contrast, I show that under incomplete markets generated by collateral constraints, agents with heterogeneous (potentially incorrect) beliefs survive in the long run and the movement in the …nancial wealth distribution between agents with di¤erent beliefs permanently drive up asset price volatility. The movement in the wealth distribution also generates various patterns of booms and busts in asset prices observed in Burnside, Eichenbaum, and Rebelo (2011). Laslty, I use this framework to study the e¤ects of …nancial regulation and of the …nancial wealth distribution on leverage and asset price volatility. 1 Introduction The events leading to the …nancial crisis of 2007-2008 have highlighted the importance of belief heterogeneity and how …nancial markets create opportunities for agents with di¤erent I am grateful to Daron Acemoglu and Ivan Werning for their in…nite support and guidance during my time at MIT. I wish to thank Guido Lorenzoni and Robert Townsend for their advice since the beginning of this project, to thank Ricardo Caballero and MIT Macro seminar, Macro Lunch, Theory Lunch, International breakfast participants for helpful comments and discussions. I also thank Markus Brunnermeier, John Geanakoplos, Felix Kubler, and other participants at conferences and seminars at UCLA, Princeton University, UCL, LSE, University of Wisconsin-Madison, Cowles Foundation, SED meeting in Montreal, Stanford Institute for Theoretical Economics, NYU-Columbia NBER mathematical economics meeting for comments and discussions on the later versions of the paper. y Email: [email protected] 1 beliefs to leverage up and speculate. Several investment and commercial banks invested heavily in mortgage-backed securities, which subsequently su¤ered large declines in value. At the same time, some hedge funds pro…ted from the securities by short-selling them. One reason for why there has been relatively little attention, in economic theory, paid to heterogeneity of beliefs and how these interact with …nancial markets is the market selection hypothesis. The hypothesis, originally formulated by Friedman (1953), claims that in the long run, there should be limited di¤erences in beliefs because agents with incorrect beliefs will be taken advantage of and eventually be driven out of the markets by those with the correct belief. Therefore, agents with incorrect beliefs will have no in‡uence on economic activity in the long run. This hypothesis has been formalized and extended in recent work by Blume and Easley (2006) and Sandroni (2000). However these papers assume …nancial markets are complete and this assumption plays a central role in allowing agents to pledge all their wealth. In this paper, I present a dynamic general equilibrium framework in which agents di¤er in their beliefs but markets are endogenously incomplete because of collateral constraints. Collateral constraints limit the extent to which agents can pledge their future wealth and ensure that agents with incorrect beliefs never lose so much as to be driven out of the market. Consequently, all agents, regardless of their beliefs, survive in the long run and continue to trade on the basis of their heterogeneous beliefs. This leads to additional asset price volatility (relative to a model with homogeneous beliefs or relative to the limit of the complete markets economy). The dynamic general equilibrium approach adopted here is central for the investigation of survival and disappearance of agents as well as their e¤ect on asset prices. Since the approach permits the use of well-speci…ed collateral constraints, it enables me to look at whether agents with incorrect beliefs will be eventually driven out of the market. It also allows for a comprehensive study of leverage and a characterization of the e¤ects of …nancial regulation on economic ‡uctuations.1 More speci…cally, I study an economy in dynamic general equilibrium with both aggregate shocks and idiosyncratic shocks and heterogeneous, in…nitely-lived agents.2 The shocks follow a Markov process. Consumers di¤er in terms of their beliefs on the transition matrix of the Markov process (for simplicity, these belief di¤erences are never updated as there is no learning; in other words agents in this economy agree to disagree).3 There is a unique …nal 1 In Cao (2010), I show that the dynamic stochastic general equilibrium model with endogenously incomplete markets presented here is not only useful for the analysis of the e¤ects of heterogeneity in the survival of agents with di¤erent beliefs, but also includes well-known models as special cases, including recent models, such as those in Fostel and Geanakoplos (2008) and Geanakoplos (2009), as well as more classic models including those in Kiyotaki and Moore (1997) and Krusell and Smith (1998). For instance, a direct generalization of the current model allows for capital accumulation with adjustment costs in the same model in Krusell and Smith (1998) and shows the existence of a recursive equilibrium. The generality is useful in making this framework eventually applicable to a range of questions on the interaction between …nancial markets, heterogeneity, aggregate capital accumulation and aggregate activity. 2 In…nite horizon and in…nitely-lived agents allow the use of sationary equilibria and the analysis of short run versus long run. 3 Alternatively, one could assume that even though agents di¤er with respect to their initial beliefs, they partially update them. In this case, similar results would apply provided that the learning process is su¢ ciently slow (which will be the case when individuals start with relatively …rm priors). In the paper, I 2 consumption good, and several real and …nancial assets. The real assets, which I sometimes refer to as Lucas trees as in Lucas (1978), are in …xed supply. I assume that agents cannot short sell these real assets. Endogenously incomplete (…nancial) markets are introduced by assuming that all loans have to use …nancial assets as collateralized promises as in Geanakoplos and Zame (2002). Selling a …nancial asset is equivalent to borrowing and in this case, agents need to put up some real assets as collateral. Loans are non-recourse and there is no penalty for defaulting. Consequently, whenever the face value of the security is higher than the value of its collateral, the seller of the security can choose to default without further consequences. In this case, the security buyer seizes the collateral instead of receiving the face value of the security. I refer to equilibria of the economy with these …nancial assets as collateral constrained equilibria 4,5 . Several key results involve the comparison of collateral constrained equilibria to the standard competitive equilibrium with complete markets. Households (consumers) can di¤er in many aspects, such as risk-aversion and endowments. Most importantly, they di¤er in their beliefs concerning the transition matrix governing transitions across the exogenous states of the economy. Given the consumers’ subjective expectations, they choose their consumption and real and …nancial asset holdings to maximize their intertemporal expected utility. The framework delivers several results. The …rst set of results, already mentioned above, is related to the survival of agents with incorrect beliefs. As in Blume and Easley (2006) and Sandroni (2000), with perfect, complete markets, in the long run, only agents with correct beliefs survive. Their consumption is bounded from below by a strictly positive number. Agents with incorrect beliefs see their consumption go to zero, as uncertainties realize overtime. However, in any collateral constrained equilibrium, every agent survives because of the constraints. When agents lose their bets, they can simply walk away from their collateral while keeping their current and future endowments to come back and trade in the …nancial markets in the same period.6 They cannot do so under complete markets because they can commit to delivering all their future endowments.7 More importantly, the survival or disappearance of agents with incorrect beliefs a¤ects asset price volatility. Under complete markets, agents with incorrect beliefs will eventually be driven out of the markets in the long run. The economies converge to economies with also show how to incorporate learning into the framework. 4 I avoid using the term incomplete markets equilibria to avoid confusion with economies with missing markets. Markets can be complete in the sense of having a complete spanning set of …nancial assets. But the presence of collateral constraints introduces endogenously incomplete markets because not all positions in these …nancial assets can be taken. 5 Collateral constrained equilibria are closer to liquidity constrained equilibria than to debt-constrained equilibria in Kehoe and Levine (2001), in which the authors show that the dynamics of the former is much more complex than the one of the latter. Liquidity constrained economies are special cases of collateral constrained economies when the set of …nancial assets is chosen to be empty. 6 The collateral constraints are a special case of limited commitment because there will be no need for collateral if agents can fully commit to their promises. Even though the survival mechanism due to limited commitment here is relatively simple (but also realistic), characterizing equilibrium variables such as asset prices and leverage in this environment is not an easy exercise. 7 Even though the survival mechanism here is relatively simple (but also realistic), characterizing equilibrium variable such as asset prices and leverage in this environment is not an easy exercise. 3 homogeneous beliefs, i.e., the correct belief. Market completeness then implies that asset prices in these economies are independent of the past realizations of aggregate shocks. In addition, asset prices are the net present discounted values of the dividend processes with appropriate discount factors. As a result, asset price volatility is proportional to the volatility of dividends if the aggregate endowment, or equivalently the equilibrium stochastic discount factor, only varies by a limited amount over time and across states. These properties no longer hold in collateral constrained economies. Given that agents with incorrect beliefs survive in the long run, they exert permanent in‡uence on asset prices. Asset prices are not only determined by the aggregate shocks as in the complete markets case, but also by the evolution of the wealth distribution across agents. This also implies that asset prices are history-dependent as the realizations of past aggregate shocks a¤ect the current wealth distribution. The additional dependence on the wealth distribution raises asset price volatility under collateral constraints above the volatility level under complete markets.8 I establish this result more formally using a special case in which the aggregate endowment is constant and the dividend processes are I.I.D. Under complete markets, asset prices are asymptotically constant. Asset price volatility, therefore, goes to zero in the long run. In contrast, asset price volatility stays well above zero under collateral constraints as the wealth distribution changes constantly, and asset price depends on the wealth distribution. Although this example is extreme, numerical simulations show that its insight carries over to less special cases. In general, long-run asset price volatility is higher under collateral constraints than under complete markets. The volatility comparison is di¤erent in the short run, however. Depending on the distribution of endowments, short run asset price volatility can be greater or smaller under complete markets or under collateral constraints. This result happens because the wealth distribution matters for asset prices under both complete markets and under collateral constraints in the short run. This formulation also helps clarify the long-run volatility comparison. In the long run, under complete markets, the wealth distribution becomes degenerate as it concentrates only on agents with correct belief. In contrast, under collateral constraints, the wealth distribution remains non-degenerate in the long run and a¤ects asset price volatility permanently. However, the wealth of agents with incorrect beliefs may remain low as they tend to lose their bets. Strikingly, under collateral constraints and when the set of actively traded …nancial assets is endogenous, the poorer the agents with incorrect beliefs are, the more they leverage to buy assets. High leverage generates large ‡uctuations in their wealth, and as a consequence, large ‡uctuations in asset prices.9 It is also useful to highlight the role of dynamic general equilibrium for some results mentioned above. In particular, the in…nite horizon nature of the framework allows a com8 I establish this result more formally using a special case in which the aggregate endowment is constant and the dividend processes are I.I.D. Under complete markets, asset prices are asymptotically constant. Asset price volatility, therefore goes to zero in the long run. In contrast, asset price volatility stays well above zero under collateral constraints as the wealth distribution changes constantly, and asset prices depend on the wealth distribution. Although this example is extreme, numerical simulations show that its insight carries over to less special cases. 9 Similarly, in Cao (2010) I show that the results concerning volatility of asset prices also translate into volatility of physical investment, i.e., capital accumulation. Physical investment under collateral constraints and hetereogenous beliefs exhibits higher volatility than under complete markets. 4 prehensive analysis of short-run and long-run behavior of asset price volatility. Such an analysis is not possible in …nite horizon economies, including Geanakoplos’s important study on the e¤ects of heterogeneous beliefs on leverage and crises. For example, in page 35 of Geanakoplos (2009), he observes similar volatility as the economy moves from collateral constrained economies to complete markets economies. In my model, the …rst set of results described above shows that the similarity holds only in the short run. The long run dynamics of asset price volatility totally di¤ers from complete markets to collateral constrained economies. In my model, the results are also based on insights in Blume and Easley (2006) and Sandroni (2000) regarding the disappearance of agents with incorrect beliefs. However, these authors do not focus on the e¤ect of their disappearance on asset price or asset price volatility. Proposition 4 in this paper also strengthens the result on asset prices in Sandroni (2000). The dynamic general equilibrium of the economy also captures the "debt-de‡ation" channel as in Mendoza (2010), which models a small open economy. The economy in my paper also follows two di¤erent dynamics in di¤erent times, "normal business cycles" and "debtde‡ation cycles," depending on whether the collateral constraints are binding for any of the agents. In a debt-de‡ation cycle, the collateral constraint binds. Then, when a bad shock hits the economy, the constrained agents are forced to liquidate their physical asset holdings. This …re sale of the physical assets reduces the price of these assets and tightens the constraints further and starting a vicious circle of falling asset prices. This paper shows that the debt-de‡ation channel still operates when we are in a closed-economy with endogenous interest rate, as opposed to exogenous interest rates as in Mendoza (2010). Moreover, due to this mechanism, asset price volatility also tends to be higher at low levels of asset price near the debt-de‡ation region. This pattern has been documented in several empirical studies, including Heathcote and Perri (2011). The movement in wealth distribution also generates the patterns of booms and busts observed in Burnside, Eichenbaum, and Rebelo (2011). The second set of results that follows from this framework concerns collateral shortages. I show that collateral constraints will eventually be binding for every agent in any collateral constrained equilibrium provided that the face values of the …nancial assets with collateral span the complete set of state-contingent Arrow-Debreu securities, i.e., markets are complete in the spanning sense but endogenously incomplete due to collateral constraints. Intuitively, if this was not the case, as proved for complete markets, the unconstrained asset holdings would imply arbitrarily low levels of consumption at some state of the world for every agent, contradicting the result that consumption is bounded from below. In other words, there are always shortages of collateral. This result sharply contrasts with those obtained when agents have homogenous beliefs but still have reasons to trade due to di¤erences in endowments or utility functions. In these cases, if the economy has enough collateral, then collateral constraints may not bind and the complete markets allocation is achieved. Heterogeneous beliefs, therefore, guarantee collateral shortages, but not other dimensions of heterogeneity, such as heterogeneity in risk-aversion or endowment. The above mentioned results are derived under the presumption that collateral constrained equilibria exist. However, establishing existence of collateral constrained equilibria is generally a challenging task. The third set of results establishes the existence of collateral constrained equilibria with a stationary structure. I look for Markov equilibria, i.e., in which equilibrium prices and quantities depend only on the distribution of normalized …nancial 5 wealth. I show the existence of the equilibria under standard assumptions. I also develop an algorithm, to compute these equilibria. A similar algorithm can be used to compute the complete markets equilibrium benchmark. The fourth set of results attempts to answer some normative questions in this framework. Simple and extreme forms of …nancial regulations such as shutting down …nancial markets are not bene…cial. Using the algorithm described above, I provide numerical results illustrating that these regulations fail to reduce asset price volatility. The intuition is for the greater volatility under such regulations is similar to the intuition for why long run asset price volatility is higher under collateral constrained economies than under complete markets economies explained earlier. Financial regulations act as further constraints protecting the agents with incorrect beliefs. Thus, in the long run these agents hold most of the assets which they believe, incorrectly, to have high rates of return. The shocks to the rates of return on these assets then create large movements in the marginal utilities of the agents, hence large volatility of the prices of the assets. This paper is related to the growing literature studying collateral constraints, started with a series of papers by John Geanakoplos. The dynamic analysis of collateral constrained equilibria is related to Kubler and Schmedders (2003). They pioneer the introduction of …nancial markets with collateral constraints into a dynamic general equilibrium model with aggregate shocks and heterogeneous agents. The technical contribution of this paper relative to Kubler and Schmedders (2003) is to introduce heterogeneous beliefs using Radner (1972) rational expectations equilibrium concept: even though agents assign di¤erent probabilities to both aggregate and idiosyncratic shocks, they agree on the equilibrium outcomes, including prices and quantities, once a shock is realized. This rational expectations concept di¤ers from the standard rational expectation concept, such as the one used in Lucas and Prescott (1971), in which subjective probabilities should coincide with the true conditional probabilities given all the available information.10 Related to the survival of agents with incorrect beliefs, Coury and Sciubba (2005) and Beker and Chattopadhyay (2009) suggest a mechanism for agents’survival based on explicit debt constraints as in Magill and Quinzii (1994). These authors do not consider the e¤ects of the agents’survival on asset prices. My framework is tractable enough for a simultaneous analysis of survival and its e¤ects on asset prices. As mentioned in footnote 6, collateral constraints are a special case of limited commitment. However, this special case of limited commitment is in contrast to the usual limited commitment literature where agents are assumed to be banned from trading in …nancial markets after their defaults such as in Kehoe and Levine (1993) and Alvarez and Jermann (2001). In this paper, agents can always come back to the …nancial markets and trade starting with their endowment after defaulting and loosing all their …nancial wealth. Given this better outside option, the …nancial constraints are more stringent then they are in the other papers. Beker and Espino (2010) has a similar survival mechanism to mine based on the limited commitment framework in these papers. However, my approach to asset pricing is di¤erent because asset prices are computed explic10 Another technical contribution in Cao (2010), is to introduce capital accumulation and production in a tractable way. Capital accumulation or physical investment is modelled through intermediate asset producers with convex adjustment costs that convert old units of assets into new units of assets using …nal good. Lorenzoni and Walentin (2009) models capital accumulation with adjustment cost using used capital markets. Through asset producers, I assume markets for both used and new capital. 6 itly as a function of wealth distribution. Moreover, my approach also allows a comprehensive study of asset-speci…c leverage. Kogan, Ross, Wang, and Wester…eld (2006) and Borovicka (2010) explore yet another survival mechanism based on the preferences of agents but use complete markets instead. My paper is also related to the literature on the e¤ect of heterogeneous beliefs on asset prices studied in Xiong and Yan (2009) and Cogley and Sargent (2008). These authors, however, consider only complete markets. The survival of irrational traders is studied Long, Shleifer, Summers, and Waldmann (1990) and Long, Shleifer, Summers, and Waldmann (1991) but they do not have a fully dynamic framework to study the long run survival of the traders. Simsek (2009b) also studies the e¤ects of belief heterogeneity on asset prices, but in a static setting. He assumes exogenous wealth distributions to investigate the question whether heterogeneous beliefs a¤ect asset prices. In contrast, I study the e¤ects of the endogenous wealth distribution on asset prices as well as asset price volatility. Simsek (2009a) focuses on consumption volatility. He shows that as markets become more complete, consumption becomes more volatile as agents can speculate more. My …rst set of results suggests that this comparative statics only holds in the short run. In the long run, the reverse statement holds due to market selection. The channel through which asset prices deviate from their fundamental values is di¤erent from the limited arbitrage mechanism in Shleifer and Vishny (1997). In their paper, the deviation arises because agents with correct beliefs hit their …nancial constraints before being able to arbitrage away the price anomalies. In this paper, agents with incorrect beliefs hit their …nancial constraint more often and are protected by the constraint. Moreover, in the equilibria computed in Section 5.2, agents with the correct belief (the pessimists) never hit their borrowing constraint. When capital accumulation is introduced, in Cao (2010), the model presented here is a generalization of Krusell and Smith (1998) with …nancial markets and adjustment costs.11 In particular, the existence theorem 2 shows that a recursive equilibrium in Krusell and Smith (1998) exists. Krusell and Smith (1998) derives numerically such an equilibrium, but they do not formally show its existence. My paper is also related to Kiyotaki and Moore (1997), although I provide a microfoundation for the …nancial constraint (3) in their paper using the endogeneity of the set of actively traded …nancial assets. At the time of the …rst draft of this paper in 2009, I was not aware of the recent papers that discuss some issues related to the ones I consider in this paper. Brumm, Grill, Kubler, and Schmedders (2011) show the importance of collateral requirements on asset price volatility in a similar model but with two trees and Epstein-Zin recursive preferences. Kubler and Schmedders (2011) show the importance of beliefs heterogeneity and wealth distribution on asset prices in a model with overlapping-generations. The rest of the paper proceeds as follow. In Section 2, I present the general model of an endowment economy and preliminary analysis of survival, asset price volatility under the complete markets benchmark as well as under collateral constraints. In Section 3, I de…ne and show the existence of collateral constrained equilibria under the form of Markov equilibria. In this section, I also prove important properties of Markov equilibria in this model. In Section 11 See Feng, Miao, Peralta-Alva, and Santos (2009) for an existence proof in an environment without …nancial markets. 7 4, I derive a general numerical algorithm to compute Markov and competitive equilibria. Section 5 focuses on assets in …xed supply with an example with only one asset to illustrate the ideas in Sections 2 and 3. Section 6 presents preliminary assessment of the quantitative signi…cant of belief heterogeneity, collateral constraint and wealth distribution on asset prices using the parameters used in Heaton and Lucas (1995). In Cao (2010), I develop the most general model with capital accumulation, labor supply and production. Section 7 concludes with potential applications of the framework in this paper. Lengthy proofs and constructions are in Appendices A and B and in Cao (2010). 2 General model In this general model, there are heterogeneous agents who di¤er in their beliefs about the future streams of dividends. There are also di¤erent types of assets (for examples trees, land, housing and machines) that di¤er in their dividend process and their collateral value. For example, some of the assets can be used as collateral to borrow and others cannot. These assets are in …xed supplies as in Lucas (1978) in order to study the e¤ects of belief heterogeneity on asset prices. In Cao (2010), I show that the model can also allow for assets in ‡exible supply and production in order to study the e¤ects of belief heterogeneity on aggregate physical investment and aggregate economy activity. Assets in …xed supply presented in this paper are special cases of assets in ‡exible supply with adjustment costs approaching in…nity. 2.1 The endowment economy Consider an endowment, a single consumption (…nal) good economy in in…nite horizon with in…nitely-lived agents (consumers). Time runs from t = 0 to 1. There are H types of consumers h 2 H = f1; 2; : : : ; Hg in the economy with a continuum of measure 1 of identical consumers in each type. These consumers might di¤er in many dimensions including per period utility function Uh (c) (i.e., risk-aversion), discount rates h , and endowments of good eh . The consumers might also di¤er in their initial endowment of real assets, Lucas’trees,12 that pay o¤ real dividends in terms of the consumption good. However, the most important dimension of heterogeneity is the heterogeneity in beliefs over the evolution of the exogenous state of the economy. There are S possible exogenous states (or equivalently shocks) s 2 S = f1; 2; : : : ; Sg : The states capture both idiosyncratic uncertainties, i.e., individual endowments, and aggregate uncertainties, i.e., the dividends from the physical assets.13 12 See Lucas (1978) A state s can be a vector s = (A; shocks. 13 1 ; :::; H ) where A consists of aggregate shocks and 8 h are idiosyncratic The evolution of the economy is captured by the past and current realizations of the shocks over time: st = (s0 ; s1 ; : : : ; st ) is the series of realizations of shocks up to time t. Notice that the space S can be chosen large enough to encompass both aggregate shocks, such as shocks to the aggregate dividends, and idiosyncratic shocks, such as individual endowment shocks. I assume that the shocks follow a Markov process with the transition probabilities (s; s0 ). In order to rule out transient states, I make the following assumption. Assumption 1 S is ergodic. Now, in contrast to the standard rational expectation literature, I assume that the agents do not have a perfect estimate of the transition matrix . Each of them has their own estimate of the matrix, h :14 However, these estimates are not very far from the truth, .i.e., there exist u and U strictly positive such that h u< (s; s0 ) < U 8s; s0 2 S and h 2 H (s; s0 ) h (1) 0 (s;s ) where (s; s0 ) = 0 if and only if h (s; s0 ) = 0 in which case let (s;s 0 ) = 1:This formulation allows for time varying belief heterogeneity as in He and Xiong (2011). In particular, agents 0 might share the same beliefs in good states, h (s; :) = h (s; :) ; but their beliefs can start 0 diverging in bad states, h (s; :) 6= h (s; :).15 Notice that (1) implies that every agent believes that S is ergodic. Real Assets: As mentioned above, there are A real assets a 2 A = f1; 2; : : : ; Ag. These assets pay o¤ state-dependent dividends da (s) in …nal goods. These assets can both be purchased and be used as collateral to borrow. This gives rise to the notion of leverage on each asset. The ex-dividend price of each unit of asset a in history st is denoted by qa (st ). I assume that agents cannot short-sell these real assets.16 The total supply Ka of asset a is given at the beginning of the economy, under the form of asset endowments to the consumers. Financial Assets: In each history st , there are also (collateralized) …nancial assets, j 2 J . Each …nancial asset j (or …nancial security) is characterized by a pair of vectors, (bj ; kj ), of promised payo¤s and collateral requirements using di¤erent assets. Promises are a standard feature of …nancial asset similarly to Arrow’s securities, i.e., asset j traded in history st promises next-period pay-o¤ bj (st+1 ) = bj (st+1 ) > 0 in terms of …nal good at the successor nodes st+1 = (st ; st+1 ). The non-standard feature is the collateral requirement. Agents can only sell the …nancial asset j if they hold shares of real assets as collateral. We associate j with an A dimensional vector k j = (kaj )a2A of collateral requirements. If an 14 Learning can be easily incorporated into this framework as into this framework by allowing additional state variables which are the current beliefs of agents in the economy. As in Blume and Easley (2006) and Sandroni (2000), agents who learn slower will dissappear under complete markets. However they all survive under collateral constraints. The dynamics of asset prices describe here will corresponds to the short-run behavior of asset prices in the economy with learning. 15 Simsek (2009b) shows in a static model that only the divergence in beliefs about bad states matters for asset prices. 16 I can relax this assumption by allowing limited short-shelling. 9 agent sells one unit of security j, she is required to to hold a portfolio kaj 0 units of asset for each a 2 A, as collateral.17 Since there are no penalties for default, a seller of the …nancial asset defaults at a node st+1 whenever the total value of collateral assets falls below the promise at that state. By individual rationality, the actual pay-o¤ of security j at node st is therefore always given by ) ( X : (2) kaj qa st+1 + da st+1 fj;t+1 st+1 = min bj (st+1 ) ; a2A Let pj;t (st ) denote price of security j at node st . Assumption 2 Each …nancial asset requires at least a strictly positive collateral min max kaj > 0 j2J a2A If a …nancial asset j requires no collateral then its e¤ective pay-o¤, determined by (2) will be zero, it will be easy to show that in equilibrium its price, pj ; will be zero as well. We can thus ignore these …nancial assets. Remark 1 The …nancial markets incomplete endogenously even if J is complete in the usual sense of complete spanning, i.e., J S. Because agents are constrained in the positions they can take due to the collateral requirement and the fact that the total supply of collateral assets is …nite. The collateral requirement is a special case of limited commitment, as if borrowers (sellers of the …nancial assets) have full commitment ability, they will not be required to put up any collateral to borrow.18 Remark 2 Consider the case in which a …nancial asset j requires only kaj units of asset a as collateral. Selling one unit of …nancial asset j is equivalent to purchasing kaj units of asset a and at the same time pledging these units as collateral to borrow pj;t . It is shown in Cao (2010) that kaj qa;t pj;t > 0; that is the seller of the …nancial asset always has to pay some margin. So the decision to sell the …nancial asset j using the real asset a as collateral corresponds to the desire to invest into asset a at margin rather than the simple desire to borrow. Remark 3 We can then de…ne the leverage ratio on asset a associated with the transaction as k j qa;t qa;t Lj;t = j a = (3) pj;t : qa;t ka qa;t pj;t kaj Even though there are many …nancial assets available, in equilibrium only some …nancial asset will be actively traded, which in turn determines which leverage levels prevail in the economy. In this sense, both asset price and leverage are simultaneously determined in equilibrium, as emphasized in Geanakoplos (2009). 17 Notice that, there are only one-period ahead …nancial assets. See He and Xiong (2011) for a motivation why longer term collateralized …nancial assets are not used in equilibrium. 18 Alvarez and Jermann (2000) is another example of asset pricing under limited commitment. 10 Remark 4 It is shown in Cao (2010) that the set J of …nancial securities can be assumed dependent on the history of the economy st , provided that there exists a k > 0 such that inf max kaj > k j2Jt a2A For example, Jt may contain a …nancial security j that requires only an asset a as collateral with the collateral requirement depending on the history st and j ka;t = max t+1 t s js bj (st+1 ) t+1 qa (s ) + da (st+1 ) : (4) j So in this example ka;t is the minimum collateral level that ensures no default, that is fj;t+1 st+1 = bj (st+1 ) 8st+1 2 S: This constraint captures the situation in Kiyotaki and Moore (1997) in which agents can borrow only up to the minimum across future states of the future value of their land. With S = 2, and state non-contingent debts, i.e., bj (st+1 ) = bj ; Geanakoplos (2009) argues that even if we allow for a wide range of collateral level, that is the unique collateral level that prevails in equilibrium (thus there is a unique level of leverage in each instance, according to the remark above). This statement for two future states still holds in this context of in…nitelylived agents as proved later in Section 5. However, this might not be true if we have more than two future states. Consumers: Consumers are the most important actors in this economy. They can be hedge fund managers or banks’traders in …nancial markets. In each state st , each consumer is endowed with a potentially state dependent endowment eht = eh (st ) units of the consumption good. I suppose there is a strictly positive lower bound on these endowments. This lower bound guarantees a lower bound on consumption if a consumer decides to default on all her debt. Assumption 3 minh;s eh (s) > e > 0. An example of this assumption is that commercial banks receive deposits from their retail branches while these banks also have trading desks that trade independently in the …nancial markets. Consumers maximize their intertemporal expected utility with the per period utility functions Uh (:) : R+ ! R that satisfy Assumption 4 Uh is concave and strictly increasing. Notice that I do not require Uh to be strictly concave. This assumption allows for linear utility functions in Geanakoplos (2009) and Harrison and Kreps (1978). Consumer h takes the sequences of prices fqa;t ; pj;t g as given and solves19 "1 # X t h max Eh0 (5) h Uh ct h h h fct ;ka;t ; j;t g t=0 19 I also introduce the disutility of labor in the general existence proof in Cao (2010) in order to study employment in this environment. The existence of equilibria for …nite horizon allows for labor choice decision. When we have strictly positive labor endowments, lh , we can relax Assumption 3 on …nal-good endowments, eh . 11 and in each history st , she is subject to the budget constraint X X X X h h (qa;t + da;t ) ka;t fj;t hj;t 1 + cht + qa;t ka;t + pj;t hj;t eht + a2A the collateral constraints h + ka;t j: X (6) a2A j2J j2J 1 h j j;t ka 0 8a 2 A j;t <0 (7) One implicit condition from the assumption on utility functions is that consumptions are positive, i.e., cht 0. In the constraint (7), if the consumer does not use asset a as collateral to sell any …nancial security, then the constraint becomes the no-short sale constraint h ka;t (8) 0: The most important feature of the objective function is the superscript h in the expectation operator, Eh [:] that represents the subjective beliefs when agents their future P estimate t expected utility. The expectation can also be re-written explicitly as t;st Ph (s ) t Uh cht (st ) ; where Ph (st ) is the probability of history st under agent h’s belief. Entering period t, agent h h h holds ka;t 1 old units of real asset a and j;t 1 units of …nancial asset j. She can trade old h units of real asset a at price qa;t and buy new units of asset ka;t for time t + 1 at the same h price. She can also buy and sell …nancial securities j;t at price pj;t . If she sells …nancial securities she is subject to collateral requirement (7). At …rst sight, the collateral constraint (7) does not have the usual property of …nancial constraints in the sense that higher asset prices do not seem to enable more borrowing. However, using the de…nition of the e¤ective pay-o¤, fj;t ; in (2), we can see that this e¤ective pay-o¤ is weakly increasing in the prices of physical assets, qa;t+1 . As a result, …nancial asset prices, pj;t , are also weakly increasing in physical asset prices. So borrowers can borrow more if qa;t+1 ’s increase. Given that the borrowing constraints is e¤ective through future asset prices, when we embed this channel in a production economy in Cao (2010), this constraint creates a feed-back mechanism from the …nancial sector to the real sector similar to Kiyotaki and Moore (1997).20 Assumption 5 Agents have the same discount factor h = 8h 2 H. Equilibrium: In this environment, I de…ne an equilibrium as follows De…nition 1 An collateral constrained equilibrium for an economy with initial asset holdings h ka;0 h2f1;2;:::;Hg and initial shock s0 is a collection of consumption, real and …nancial asset holdings and prices in each history st , h st ; ( cht st ; ka;t qa;t s 20 t a2A h j;t ; pj;t s st h2f1;2;:::;Hg t j2Jt (st ) ) This channel is di¤erent from the one in Brunnermeier and Sannikov (2010). 12 satisfying the following conditions: i) Asset markets for each real asset a and for each …nancial asset j in each period clear: X h ka;t st = Ka h2H X h j;t st = 0: h2H h (st ) ; hj;t (st ) solves the individual maximization probii) For each consumer h, cht (st ) ; ka;t lem subject to the budget constraint, (6), and the collateral constraint, (7). Notice that by setting the set of …nancial securities J empty, we obtain a model with no …nancial markets in which agents are only allowed to trade in real assets, but they cannot short-sell these assets. This case corresponds to Lucas (1978)’s model and the liquidity constrained economy in Kehoe and Levine (1993) with several trees and heterogeneous agents. As a benchmark, I also study equilibria under complete …nancial markets. Consumers can borrow and lend freely by buying and selling Arrow-Debreu state contingent securities, only subject to the no-Ponzi condition.21 In each node st ; there are S …nancial securities. Financial security s delivers one unit of …nal good if state s happens at time t + 1 and zero units otherwise. Let ps;t denote time t price and let hs;t denote consumer h’s holding of this security. The budget constraint (6) of consumer h becomes X X X h h cht + qa;t ka;t + ps;t hs;t eht + hst ;t 1 + (qa;t + da;t ) ka;t (9) 1 a2A s2S a2A De…nition 2 A complete markets equilibrium is de…ned similarly to an incomplete markets equilibrium except that each consumer solves her individual maximization problem subject to the budget constraint (9) and the no-Ponzi condition, instead of the collateral constraints (7). In the next subsection, I establish some properties of collateral constrained markets equilibrium. I compare each of these properties to the one in the complete markets equilibrium. 2.2 General properties of collateral constrained and complete markets equilibria Even though the formulation and solution method presented below allow for heterogeneity in the discount rates, to focus on beliefs heterogeneity, I assume from now on that agents have the same discount factor. 21 No-Ponzi condition lim t !1 X tY1 psr+1 (sr ) s2S r=0 13 h s;t 0: Given the endowment economy, we can easily show that total supply of …nal good in each period is bounded by a constant e: Indeed in each period, total supply of …nal good is bounded by ! X X e = max eh (s) + da (s) Ka (10) s2S a2A h2H The …rst term on the right hand side is the total endowment of each individual. The second term is total dividends from the real assets. In collateral constrained or complete markets equilibria, the market clearing condition for the …nal good implies that total consumption is bounded from above by e. Given that consumption of every agent is always positive, consumption of each agent is bounded from above by e, i.e., e 8t; st : ch;t st (11) Under the …xed supply of real assets, we can show that in any collateral constrained equilibrium, the consumption of each consumer is bounded from below by a strictly positive constant c. Two assumptions are important for this result. First, no-default-penalty allows consumers, at any moment in time, to walk away from their past debts and only lose their collateral assets. After defaulting, they can always keep their non-…nancial wealth (inequality (13) below). Second, increasingly large speculation by postponing current consumption is not an equilibrium strategy, because in equilibrium, consumption is bounded by e, in inequality (14). This assumption prevents agents from constantly postponing their consumption to speculating in the real assets and is main di¤erence with the survival channel in Alvarez and Jermann (2000) used by Beker and Espino (2010) for heterogeneous beliefs which will be explained in detail below. Formally, we have the following theorem Theorem 1 Suppose that there exists a c such that Uh (c) < 1 1 | Uh (e) {z } endowment 1 | Uh (e); 8h 2 H; {z } (12) speculation where e is de…ned in (10).Then in a collateral constrained equilibrium, consumption of each consumer in each history always exceeds c. Proof. This result is shown in an environment with homogenous beliefs (Lemma 3.1 in Du¢ e, Geanakoplos, Mas-Colell, and McLennan (1994)). It can be done in the same way under heterogenous beliefs. However, I replicate the proof in order to provide the economic intuition in this environment. As in (11), we can …nd an upper bound for consumption of each consumer in each future period. Also, in each period, one of the feasible strategies of consumer h is to default on all her past debts at the only cost of losing all the collateral assets, but she can still at least consume her endowment from the current period on, therefore "1 # X 1 r Uh (ch;t ) + Eht Uh (ch;t+r ) Uh (e) : (13) 1 r=1 14 Notice that in equilibrium, P e therefore ch;t+r h ch;t+r Uh (ch ) + 1 Uh (e) 1 1 e. So Uh (e) (14) This implies Uh (ch ) 1 1 Uh (e) 1 Uh (e) > Uh (c) : Thus, ch c. Condition (12) is satis…ed immediately if limc !0 Uh (c) = 1, for example, with log utility or CRRA utility with CRRA constant exceeding 1. The survival mechanism here is similar to the one in Alvarez and Jermann (2000) and Beker and Espino (2010). In particular, the …rst term on the right hand side of (12) captures the fact that the agents always have the option to default and go to autarky in which they only consume their endowment which exceeds e in each period, which is the lower bound for consumption in Alvarez and Jermann (2000) and Beker and Espino (2010). However, the two survival mechanisms also di¤er because, in this paper, agents can always default on their promises and lose all their physical asset holdings, but they can always go back to …nancial markets to trade right after defaulting. The second term in the right hand side of (12) captures the fact that, this possibility might hurt the agents if they have incorrect beliefs. The prospect of higher reward for speculation, i.e. high e, will induce these agents to constantly postpone consumption to speculate. As a result, their consumption level might fall well below e. Indeed, the lower bound of consumption, c; is decreasing in e: the more there is of the total available …nal good, the more pro…table speculative activities are and the more incentives consumers have to defer current consumption to engage in these activities. One immediate corollary of Proposition 1 is that every consumer survives in equilibrium. Therefore, collateral constrained equilibrium di¤ers from complete markets equilibrium when consumers di¤er in their beliefs. The proposition below shows that in a complete markets equilibrium, with strict di¤erence in beliefs, consumption of certain consumers will come arbitrarily close to 0 at some history. The intuition for this result is that if an agent believes that the likelihood of a state is much smaller than what other agents believe, the agent will want to exchange his consumption in that state for consumption in other states. Complete markets allow her to do so but collateral constraints limit the amount of consumption that she can sell in each state. Proposition 1 Suppose there are consumers with the correct belief and some consumers with incorrect beliefs. Moreover, the utility functions satisfy the Inada-condition lim Uh0 (c) = +1 8h 2 H: c !0 (15) Then, in a complete markets equilibrium, almost surely lim cht = 0 t !1 for each h such that h di¤ers from , that is h has incorrect belief. 15 (16) Proof. In Appendix A, I show that the conditions in Proposition 5 in Sandroni (2000) are satis…ed. Thus the proposition implies that almost surely, for agents h with incorrect beliefs, we have Ph (st ) = 0: lim t !1 P (st ) (16) is a direct consequence given the …rst-order conditions on the holding of Arrow-Debreu securities: Uh0 (ch (st )) Ph0 (st ) Uh0 (ch (s0 )) = 8h; h0 2 H, (17) Uh0 0 (ch0 (st )) Ph (st ) Uh0 (ch0 (s0 )) together with the facts that there are agents with correct beliefs, i.e. Ph0 = P , the Inada condition (15), and bounded consumption (11) applied for ch0 . Corollary 1 In a complete markets equilibrium, under the Inada condition (15), if agents strictly di¤er in their beliefs, then consumption of some agent approaches zero at some history of the world. Formally, inft ch st = 0: (18) h;s Proof. If agents strictly di¤er in their beliefs, there exists a pair of agents h; h0 such that 0 h 6= h . Applying Proposition 1 for P = Ph0 we have, under Ph0 , (16) happens almost surely. This directly implies (18). Notice that this result is rather surprising because even if agents are strictly risk-averse, they can also disappear over time if they have incorrect beliefs.22 Because they can perfectly commit to pay their creditor using their future income. The de…nition of complete markets equilibrium 2 shows that they can commit by using short-term debts and by rolling over their debts while using their present income to pay interests, which grows over time as their indebtedness grows. This result also sheds light on the survival mechanism in Theorem 1: agents have limited ability to pledge their future income, for example their labor income, to their creditors. As a result, they can always default and keep their future income. This limited commitment is even stronger in my setting than in Alvarez and Jermann (2000) and Beker and Espino (2010) because after defaulting, agents can always come back and trade in the …nancial markets by buying new physical assets and then use these physical assets to sell …nancial assets.23,24 This mechanism is also di¤erent from the limited arbitrage mechanism in 22 If the consumers with incorrect beliefs are risk-neutral, their consumption will go to zero immediately after a certain date. 23 In Chien and Lustig (2009), in equilibrium, borrowers are indi¤erent between defaulting and not defaulting due to the complete spanning properties of …nancial assets. In my model, when there is not complete spanning, there will be strict defaults in equilibrium. The numerical method in Section 4 can be used to solve for equilibria in Chien and Lustig (2009) as well. See Hopenhayn and Werning (2008) for a model in which equilibrium defaults happen due to stochastic outside options. 24 The following story of the founder of Long Term Capital Management shows that traders in the …nancial markets often have limited commitment: John Meriwether worked as a bond trader at Salomon Brothers. At Salomon, Meriwether rose to become the head of the domestic …xed income arbitrage group in the early 1980s and vice-chairman of the company in 1988. In 1991, after Salomon was caught in a Treasury securities trading scandal, Meriwether decided to leave the company. Meriwether founded the Long-Term Capital Management hedge fund in Greenwich, Connecticut in 1994. Long-Term Capital Management spectacularly collapsed in 1998. A year after LTCM’s collapse, in 1999, Meriwether founded JWM Partners LLC. The 16 Shleifer and Vishny (1997), where asset prices di¤er from their fundamental values because agents with correct beliefs hit their …nancial constraints before being able to arbitrage away the price di¤erence. Here, agents with incorrect beliefs hit their …nancial constraint more often and are protected by the constraint. Moreover, in the equilibria computed in Section 5.2, agents with the correct belief (the pessimists) never hit their borrowing constraint. Due to di¤erent conclusions about agents’survival, the following corollary asserts that complete markets and collateral constrained allocations strictly di¤er when some agents strictly di¤er in their beliefs. Corollary 2 Suppose that conditions in Theorem 1 and Proposition 1 are satis…ed. Then, a collateral constrained equilibrium never yields an allocation that can be supported by a complete markets equilibrium. By the Second Welfare Theorem, collateral constrained equilibrium allocations are Pareto-ine¢ cient. Proof. In a collateral constrained equilibrium, consumptions are bounded away from 0, but in a complete markets equilibrium, consumptions of some agents will approach 0. Therefore, the two sets of allocations never intersect. Using this corollary, we can formalize and show the shortages of collateral assets. Proposition 2 (Collateral Shortages) If …nancial markets are complete in terms of spanning, i.e., the set of the vectors of promises bj has full rank. Then, for any given time t, with positive probability, the collateral constraints must be binding for some agent after time t. Proof. We prove this corollary by contradiction. Suppose none of the collateral constraints are binding after a certain date. Then we can take the …rst-order condition with respect to the state-contingent securities. This leads to consumption of some agents to approach zero at in…nity, as shown in the proof of Proposition 1. This contradicts the conclusion of Theorem 1 that consumption of each agent is bounded away from zero. Notice that, as in Lucas (1978), agents can hold the real assets for the risk-return and consumption-investment trade-o¤s. However, when their collateral constraints are binding, the agents only purchase these assets at the lowest margin allowed, i.e, pledging these assets as collateral to borrow at the maximum possible. I interpret the binding collateral constraints as collateral shortages. Araujo, Kubler, and Schommer (2009) argue that when there is enough collateral we might reach the Pareto optimal allocation. However, in the complete markets case, there will never be enough collateral.25 I also emphasize here the di¤erence between belief heterogeneity and other forms of heterogeneity such as heterogeneity in endowments or in risk-aversion. The following proposition, in the same form as Theorem 5 in Geanakoplos and Zame (2007), shows that if consumers share the same belief and discount rate, there exist endowment pro…les with which collateral equilibria attain the …rst-best allocations. Greenwich, Connecticut hedge fund opened with $250 million under management in 1999 and by 2007 had approximately $3 billion. The Financial crisis of 2007-2009 badly battered Meriwether’s …rm. From September 2007 to February 2009, his main fund lost 44 percent. On July 8, 2009, Meriwether closed the fund. 25 I show in Cao (2010) that even if the supply of these collateral assets is endogeneous, there will still be collateral shortages. 17 Proposition 3 If consumers share the same belief and discount factor, there is an open set of endowment pro…les with the properties that the competitive equilibrium can be supported by an collateral constrained equilibrium. Proof. We start with an allocation such that there is no trade in the complete markets equilibrium, then as we move to a neighborhood of that allocation, all trade can be collateralized. Even though other dimensions of heterogeneity such as risk-aversion and endowments also create trading in …nancial markets, this proposition shows the importance of belief heterogeneity in driving up trading volume and resulting in binding collateral constraints. Before moving to show the existence and study the properties of collateral constrained equilibria, we go back to the complete markets benchmark to study the behavior of asset price volatility. We will compare this volatility with volatility under collateral constraints and show that, in general, in the long run, asset price is more volatile in a collateral constrained equilibrium than it is in a complete markets equilibrium. Proposition 4 Suppose that there are some agents with the correct belief, in the complete markets equilibrium, almost surely asset prices converge to the prices prevailing in an economy in which there are only agents with the correct belief. In particular, the prices are independent of the past realizations of the aggregate shocks, as they are functions of only the current aggregate shock. Formally, there exists a set of asset prices q a (s) as functions of the aggregate state of the economy such that, almost surely, for any sequence of history fst g: lim t !1 sup qa st+r r 0 q a (st+r ) = 0: (19) Proof. The detailed proof is in Appendix A. Proposition 1 shows that in the long run, only agents with the correct belief survive. Therefore, in the long run, the economy converges to the economy with homogeneous belief (rational expectation). In such an economy, given market completeness, there exists a representative agent with an instantaneous utility function URep , and her marginal utility evaluated at the total endowment determines asset prices 0 q a st URep (e (st )) = 0 Et q a st+1 + da (st+1 ) URep (e (st+1 )) (1 ) X 0 = Et da (st+r ) r URep (e (st+r )) (20) r=1 in which e (s) is the aggregate endowment in the aggregate state s. We can see easily from this expression that q a (st ) is history-independent. Under complete markets, asset price does depend on the endogenous wealth distribution and the exogenous state s, as shown in the example below. However, in the long run, as the economy converges to an homogenous beliefs economy, the wealth distribution converges to a limiting distribution. So in the long run and under complete markets, asset prices only depend on the aggregate state. In the short run, as the wealth distribution also moves over time, asset price volatility might be very large. 18 Remark 5 Proposition (4) is stronger than Proposition 6 in Sandroni (2000) in two ways. First, the functions q a (s) are independent of history path. Second, the convergence in (19) is strong in the sense that the sup is taken over r 0 instead of any …nite number as in Sandroni (2000). To illustrate Proposition 4, in Appendix A, I derive the following closed form solution of asset price under complete markets Example 1 In the case of log utility, the equilibrium price of assets is a weighted sum of the prices 8 9 1 X <X X e (st ) = r Ph st+r jst d (st ) qa st = ! b h st ; (21) : r=0 t+r t e (st+r ) ; h2H s js where ! b h st = P and wh st = 1 X X r=0 st+r jst wh (st ) ; t h0 2H wh0 (s ) (22) p st+r jst ch st+r : In the long run, as agents h with incorrect beliefs disappear in the limit, i.e., limt !1 wh (st ) = 0, or equivalently limt !1 ! b h (st ) P = 0, all the wealth in the economy concentrates on agents with correct beliefs, i.e., limt !1 h2H;Ph =P ! b h (st ) = 1 so qa (st ) approaches q a st = 1 X X r=0 st+r jst r Ph st+r jst d (st ) e (st ) ; e (st+r ) which depends only on st due to the Markov property of the evolution of the exogenous state. A direct corollary of Proposition 4 is the following: Corollary 3 When the dividend process of an asset is I.I.D. and the aggregate endowment is constant across states, with probability one, the price of the asset converges to a constant in the long run. 0 Proof. By assumption e (st+r ) = e for all t and r. So URep (e) cancels out in both sides of (20). As a result (1 ) X X q a st = Et da (st+r ) r = (s) da (s) : 1 r=1 s2S The second equality comes from the fact that shocks are I.I.D. In contrast to complete markets equilibrium, in the next section I show that, in a collateral constrained equilibrium, asset prices can be history-dependent, as past realizations of aggregate shocks always a¤ect the wealth distribution, which in turn a¤ects asset prices. 19 One issue that might arise when one tries to interpret Proposition 4 is that, in some economy, there might not be any consumer whose belief coincides with the truth. For example, in Scheinkman and Xiong (2003), all agents can be wrong all the time, except they constantly switch from over-optimistic to over-pessimistic.26 To avoid this issue, I again use the language in Blume and Easley (2006) and Sandroni (2000). I reformulate the results above using the subjective belief of each consumer. Proposition 5 Suppose that the Inada condition (15) is satis…ed. Then each agent believes that: 1) In complete markets equilibrium, only her and consumers sharing her belief survive in the long run. However, in collateral constrained equilibrium, everyone survives in the long run. 2) In complete markets equilibrium, asset prices are history-independent. However, in collateral constrained equilibrium, asset price can be history-dependent. The properties in this section are established under the presumption that collateral constrained equilibria exist. The next section is devoted to show the existence of these equilibria with a stationary structure. Then Section 4 presents an algorithm to compute the equilibria. 3 Markov Equilibrium In this section, I show that collateral constrained equilibrium exists with a stationary structure. The equilibrium prices and allocation depend on the exogenous state of the economy and a measure of the wealth distribution. 3.1 The state space and de…nition I de…ne the normalized …nancial wealth of each agent by P P h a (qa;t + da;t ) ka;t 1 + j h P !t = (q + d ) K a;t a;t a a h j;t fj;t 1 : (23) Let ! (st ) = ! 1 (st ) ; :::; ! H (st ) denote the normalized …nancial wealth distribution. Then in P h equilibrium ! (st ) always lies in the (H-1)-dimensional simplex , i.e., ! h 0 and H h=1 ! = 1. ! h ’s are non-negative because of the collateral constraint (7) that requires the value of each agent’s asset holdings to exceed the liabilities from their past …nancial assets holdings. And the sum of ! h equals 1 because of the asset market clearing and …nancial market clearing conditions. In Cao (2010), I show that, under conditions detailed in Subsection 3.2 below, there exists a Markov equilibrium over the compact state space S , i.e., an equilibrium in which equilibrium prices and allocation depend only on the state (st ; ! t ) 2 S . In particular, for each (st ; ! t ) 2 S , we need to …nd a vector of prices and allocation t 26 b = RH 2V + RAH + RJH RA + RJ+ (24) Sandroni (2000) shows that if none of the agents has the correct beliefs, then only agents with beliefs closest to the truth survive, where distance is measured using entropy. 20 that consists of the consumers’decisions: consumption of each consumer ch ( ) h2H 2 RH +, h RJH ; the prices of physical real and …nancial asset holdings kah ( ) ; j ( ) h2H 2 RAH + J must satisfy assets (qa ( ))a2A 2 RA + and the prices of the …nancial assets (pj ( ))j2J 2 R+ . the market clearing conditions and the budget constraint of consumers bind. Moreover, for each future state s+ t+1 2 S succeeding st , we need to …nd a corresponding wealth distribution + b ! t+1 and equilibrium allocation and prices + t+1 2 V such that for each household h 2 H the following conditions holds a) For each s+ 2 S succeeding s27 P h j + h + (q + + d+ )g ) + + d k (q s s j2J j min fbj (s) ; k h+ P + !s = : (25) + a (q + d ) K h a b) There exist multipliers 0= h a 0= h a 0 qa Uh0 ch + @kah + j2J : c) De…ne hj ( ) = max 0; and hj ( ) 2 R+ such that X h j 0 = a ka hE X j2J : X kah + 0 2 R+ corresponding to collateral constraints such that kaj h j <0 kaj h j: h 1 0 h+ qa+ + d+ a Uh c (26) hA j h j <0 h j h j and (+) = max 0; pj Uh0 ch + h h j , there exist multipliers E h fj+ Uh0 ch+ h j h j (+) (27) ( ) a2A 0 = pj Uh0 ch + 0 = h j h j 0 = h E h fj+ Uh0 ch+ + h j (+) h j (+) h j ( ): (+) ( ) Condition a) guarantees that the future normalized wealth distributions are consistent with the current equilibrium decision of the consumers. Conditions b) and c) are the …rstorder conditions from the maximization problem (5) of the consumers. Given that the maximization problem is convex, the …rst-order conditions are su¢ cient for a maximum. As a result, as in Du¢ e, Geanakoplos, Mas-Colell, and McLennan (1994), a Markov equilibrium is a collateral constrained equilibrium. Before continue, let me brie‡y discuss asset prices in a Markov equilibrium. We can rewrite that …rst-order condition with respect to asset holding (26) as qa Uh0 ch = h a + hE h h+ 0 qa+ + d+ a Uh c By re-iterating this inequality we obtain (1 X qa;t Eth r=1 27 We use the vector product notation a b = P i hE Uh0 cht+r r d h t+r Uh0 cht a i bi . 21 h ) 0 h+ qa+ + d+ a Uh c : : We have a strict inequality if there is a strict inequality ha;t+r > 0 in the future. So the asset price is higher than the discounted value of the stream of its dividend because in future it can be sold to other agents, as in Harrison and Kreps (1978) or it can be used as collateral to borrow as in Fostel and Geanakoplos (2008). Proposition 2 shows some conditions under which collateral constraints will eventually be binding for every agent when they strictly di¤er in their belief. As a results, the prices of the real assets are strictly higher than the discounted value of their dividends.28 Equation (26) also shows that asset a will have collateral value when some ha > 0, in addition to the asset’s traditional pay-o¤ value weighted at the appropriate discount factors. Unlike in Alvarez and Jermann (2000), attempts to …nd a pricing kernel which prices assets using their pay-o¤ value might prove fruitless because assets with the same payo¤s but di¤erent collateral values will have di¤erent prices. This point is also emphasized in Geanakoplos’s’papers. Equation (27) implies that " # P 0 h+ h j a2A a ka + Uh c h + h E fj : pj = Uh0 (ch ) Uh0 (ch ) As in Garleanu and Pedersen (2010), the price of …nancial asset j does not only depend on U 0 (ch+ ) its promised payo¤s in future states h E h fj+ Uh0 ch but also on its collateral requirements h( ) P h j a2A a ka Uh0 ch ( ) 3.2 : Existence and Properties of Markov equilibrium The existence proof is similar to the ones in Kubler and Schmedders (2003) and Magill and Quinzii (1994). We approximate the Markov equilibrium by a sequence of equilibria in …nite horizon. There are three steps in the proof. First, using Kakutani …xed point theorem to prove the existence proof of the truncated T-period economy. Second, show that all endogenous variables are bounded. And lastly, show that the limit as T goes to in…nity is the equilibrium of the in…nite horizon economy. To prove that the Markov equilibrium exists, we need to …rst show that there exists a compact set in which …nite horizon equilibria lie. We need the following assumption Assumption 6 There exist c; c > 0 such that29 Uh (c) + max min 1 1 h 1 Uh (e) ; 0 h min Uh (e) ; min Uh (e) h s2S s2S 28 8h 2 H: (28) We can also derive a formula for the equity premium that depends on the multilipliers similar to the equity premium formula in Mendoza (2010) 29 This assumption is slightly di¤erent from the one in Kubler and Schmedders (2003) as Uh might be negative, for example with log utility. 22 and Uh (c) + min max 1 1 h 1 Uh (e) ; 0 h Uh (e) ; Uh (e) h 8h 2 H: (29) The intuition for (28) is detailed in the proof of Theorem 1; it ensures a lower bound for consumption. (29) ensures that prices of real assets are bounded from above. Both inequalities are obviously satis…ed by log utility. Lemma 1 Suppose Assumption 6 is satis…ed then there is a compact set that contains the equilibrium endogenous variables constructed in Cao (2010) for every T and every initial condition lying inside the set. Proof. Appendix. Theorem 2 Under the same conditions, a Markov equilibrium exists. Proof. In Cao (2010), I show the existence of Markov equilibria for a general model also with capital accumulation. As in Kubler and Schmedders (2003), we extract a limit from the T-…nite horizon equilibria. Lemma 1 guarantees that equilibrium prices and quantities are bounded as T goes to in…nity. The proof use an alternative de…nition of attainable sets and also corrects several errors in the Appendix of Kubler and Schmedders (2003). As argued in the last subsection, any Markov equilibrium is a collateral constrained equilibrium. So Markov equilibria inherit all properties of collateral constrained equilibria. In particular, in a Markov equilibrium, every consumer survives, Theorem 1, and Markov equilibrium allocations are Pareto-ine¢ cient if agents strictly di¤er in their beliefs, Corollary 18. Regarding asset prices, the construction of Markov equilibria provides us with the following Proposition Proposition 6 In contrast to the complete markets benchmark, in these Markov equilibria, asset prices can be history-dependent in the long run. Proof. Proposition 4 shows that under complete markets, asset prices do depend on the wealth distribution but the wealth distribution converges in the long run, so asset prices only depend on the current exogenous state st . However, in a Markov equilibrium, the normalized …nancial wealth distribution constructed in (23) constantly moves over time even in the long run. For example, if some agent h with incorrect belief lose all her real asset holding due to leverage. Next period, she can always use her endowment to speculate in the real assets again. In this case, ! b ht will jump from 0 to a strictly positive number. So asset prices depend on the past realizations of the exogenous shocks, which determine the evolution of the normalized wealth distribution ! bt. Proposition 7 When the aggregate endowment is constant across states s 2 S, and shocks are I.I.D., long run asset price volatility is higher in Markov equilibria than it is in complete markets equilibria. 23 Proof. Corollary 3 shows that, in the long run, under complete markets and the assumptions above, the economy converges to the one with homogenous beliefs because agents with incorrect beliefs will eventually be driven out of the markets and asset price qa (st ) converges to prices independent of time and state. Hence, under complete markets, asset price volatility converges to zero in the long run. In Markov equilibrium, asset price volatility remains well above zero as the exogenous shocks constantly change the wealth distribution, which, in turn, changes asset prices. There are two components of asset price volatility. The …rst and standard one comes from the volatility in the dividend process and the aggregate endowment. The second one comes from wealth distribution, when agents strictly di¤er in their beliefs. In general, it depends on the correlation of the two components, that we might have asset price volatility higher or lower under collateral constrained versus under complete markets. However, the second component disappears under complete markets because only agents with correct beliefs survive in the long run. Whereas, under collateral constraints, this component persists. As a result, when we shut down the …rst component, asset price is more volatile under collateral constraints than it is under complete markets in the long run. In general, the same comparison holds or not depending on the long-run correlation between the …rst and the second volatility components under collateral constrained markets. 4 Numerical Method The construction of Markov equilibria in the last section also suggests an algorithm to compute them. The following algorithm is based on Kubler and Schmedders (2003). There is one important di¤erence between the algorithm here and the original algorithm. The future wealth distributions are included in the current mapping instead of solving for them using sub-…xed-point loops. This innovation reduces signi…cantly the computing time, given that solving for a …xed-point is time consuming in MATLAB. In section 5, as we seek to …nd the set of actively traded …nancial assets and the equilibrium leverage level in the economy, we need to know the future prices of the physical asset. As we know future wealth distributions, these future prices can be computed easily. As in the existence proof, we look for the following correspondence :S b !V S (s; !) 7 ! vb; ! + s; ; L (30) b is the set of endogenous variables excluding the wealth distribution as de…ned in (24). vb 2 V + (! s )s2S are the wealth distributions in the S future states and ( ; ) 2 L are Lagrange multipliers as de…ned in subsection 3.1. From a given continuous initial mapping 0 = ( 01 ; 02 ; : : : ; 0S ), we construct the sequence n of mappings f n = ( n1 ; n2 ; : : : ; nS )g1 , for each n=0 by induction. Suppose we have obtained state variable (s; !), we look for n+1 s (!) = vbn+1 ; ! + s;n+1 ; n+1 ; n+1 (31) that solves the …rst-order conditions (26), (27), market clearing conditions, and the consistency of the future wealth distribution (25). 24 We construct the sequence f n g1 . So from n to n=0 on a …nite discretization of S n+1 , we will have to extrapolate the values of n to outside the grid using extrapolation n methods in MATLAB. Fixing a precision , the algorithm stops when k n+1 k< . There are two important details in implementing this algorithm: First, in order to calculate the (n + 1)-th mapping n+1 from the n th mapping, we need to only keep track of the consumption decisions ch and asset prices qa and pj . Even though other asset holding decisions and Lagrange multipliers might not be di¤erentiable functions of the normalized …nancial wealth distribution, the consumption decisions and asset prices normally are.30 Relatedly, when there are redundant assets, there might be multiple asset holdings that implement the same consumption policies and asset prices.31 Second, if we choose the initial mapping 0 as an equilibrium of the 1 period economy as in Subsection 3.2, then n corresponds to an equilibrium of the (n + 1) period economy. I follow this choice in computing an equilibrium of the two agent economy presented below. The algorithm to compute complete markets equilibria and is presented in Appendix A. 5 Asset price volatility and leverage This section uses the algorithm just described to compute collateral constrained and complete markets equilibria and study asset price and leverage. To make the analysis as well as the numerical procedure simple, I allow for only one real asset and two types of agents: optimists and pessimists. The general framework in Section 2 allows for a wide range of …nancial assets with di¤erent promises and collateral requirements. However, given that the total quantity of collateral is exogenously bounded, in equilibrium, only certain …nancial assets are actively traded. I choose a speci…c setting based on Geanakoplos (2009), in which I can …nd exactly which assets are traded. The setting requires that promises are state-incontingent and in each exogenous state there are only two possible future exogenous states. The only …nancial assets that are traded are the assets that allow maximum borrowing while keeping the payo¤ to lenders riskless. As noticed in Remark 3, the leverage level corresponding to this unique …nancial asset can be called the leverage level in the economy. Endogenous …nancial assets interestingly generate the most volatility in the …nancial wealth distribution as agents borrow to the maximum and lose most of their …nancial wealth as they lose their bets but their wealth increases largely when they win. This volatility in the wealth distribution in turn feeds into asset price volatility. To answer questions related to collateral requirements asked in the introduction, in Subsection 5.2.5, I allow regulators to control the sets of …nancial assets that can be traded. Given the restricted set, the endogenous active assets can still be determined. One special case is the extreme regulation that shuts down …nancial markets. There are surprising consequences of these regulations on the welfare of agents, on the equilibrium wealth distribution and on asset prices. An endogenous set of traded assets also implies endogenous leverage which has been the object of interest during the current …nancial crisis. In order to match the observed 30 See Brumm and Grill (2010) for an algorithm with adapted grid points that deals directly with nondi¤erentiabilities in the policy functions. 31 See Cao, Chen, and Scott (2011) for such an example. 25 pattern of leverage, i.e., high in good states and low in bad states, I introduce the possibility for changing volatility from one aggregate state to others. This feature is introduced in Subsection 5.2.6. 5.1 One asset economy Consider a special case of the general model presented in Section 2. There are two exogenous states S = fG; Bg and one single asset of which the dividend depends on the exogeneous state: d (G) > d (B) : The state follows an I.I.D. process, with the probability of high dividends, , unknown to agents in this economy. The supply of the asset is exogenous and normalized to 1. Let q (st ) denote the ex-dividend price of the asset at each history st = (s0 ; s1 ; : : : ; st ). To study the standard debt contracts, I consider the set J of …nancial assets which promise state-independent payo¤s next period. I normalize these promises to bj = 1. Asset j also requires kj units of the real asset as collateral. The e¤ective pay-o¤ is therefore fj;t+1 st+1 = min 1; kj q st+1 + d st+1 : Due to the …nite supply of the real asset, in equilibrium only a subset of the …nancial assets in J are traded. Determining which asset are traded allows us to understand the evolution of leverage in the economy (Remark 3). This is also important for computing collateral constrained equilibria in Subsection 4. It turns out that in some special cases, we can determine exactly which …nancial assets are traded. For example, Fostel and Geanakoplos (2008) and Geanakoplos (2009) argue that if we allow for the set J to be dense enough then in equilibrium the only …nancial asset traded in equilibrium is the one with the minimum collateral level to avoid default. This statement also applies for my general set up under the condition that in a history node, there are only two future exogeneous states. Proposition 8 below makes it clear. The proposition uses the following de…nition De…nition 3 Two collateral constrained equilibria are equivalent if they have the same allocation of consumption to the consumers and the same prices of real and …nancial assets. They might di¤er in the consumers’portfolios of real and …nancial assets. Proposition 8 Consider a collateral constrained equilibrium and suppose in a history st , there are only two possible future exogenous states st+1 . Let u = d = and max q st+1 + d (st+1 ) st+1 jst min q st+1 + d (st+1 ) st+1 jst 1 k = : d 26 We can …nd another collateral constrained equilibrium equivalent to the initial one such that only the …nancial assets with the collateral requirements max j2J ;kj k kj and min j2J ;kj k kj are actively traded. In particular, when k 2 J , we can always …nd an equivalent equilibrium in which only …nancial assets with the collateral requirement k are traded. Proof. Intuitively, this proposition is true because with only two future states, two assets, a …nancial asset and the real asset, can e¤ectively replicate the pay-o¤ all other …nancial assets. But we need to make sure that we do not have to use short-selling in any replication. The detail of the proof is in Appendix B. Imagine that the set J includes all collateral requirements kj 2 R+ ; kj > 0.32 . Proposition 8 says, for any collateral constrained equilibrium, we can …nd an equivalent collateral constrained equilibrium in which the only …nancial asset with the collateral requirement exactly equals to k (st ) are traded in equilibrium. Therefore in such an equilibrium the only actively traded …nancial asset is riskless to its buyers. Let p (st ) denote the price of this …nancial asset. The endogenous interest rate is therefore r (st ) = p(s1t ) 1: Proposition 8 is also useful to study …nancial regulations which correspond to choosing the set J in Subsection 5.2.5.33 Coming back to consumers, there are only two types of agents in this economy, optimists, O, and pessimists, P, each in measure one of identical agents. They di¤er in their belief: h suppose agent h 2 fO; P g estimates the probability of high dividends as hG = 1 B :We P O suppose G > G , i.e. optimists always think that good states are more likely than the pessimists think they are. Each agent maximizes the inter-temporal utility (5) given their belief of the evolution of the aggregate state, and is subject to the budget constraint: ct + q t k t + p t t et + (qt + dt ) kt 1 + ft t 1; (32) no short-sale constraint (33) 0; t and collateral constraint kt + t kt 0; (34) for each h 2 fO; P g. At time t, each agent choose to buy t units of real asset at price qt and t units of …nancial asset at price pt . Moreover, Proposition 8 allows us to focus on only one level of collateral requirement kt . As a result, the set of collateral constraints (7) can be replaced by the constraints (33) and (34). Given prices q and p, this program yields solution cht (st ) ; kth (st ) ; ht (st ) : In equilibrium prices fqt (st )g and fpt (st )g are such that asset and …nancial markets clear, i.e., ktO + ktP = 1 O P = 0 t + t 32 To apply the existence theorem 2 I need J to be …nite. But we can think of J as a …ne enough grid. The uniqueness of actively traded …nancial assets established in Geanakoplos (2009) and He and Xiong (2011) is in the "equivalent" sense in Proposition 8. 33 27 for each history st . With only one physical asset and one …nancial asset, the general formula for normalized …nancial wealth (23) translates into ! ht = (qt + dt ) kth 1 + ft qt + dt h t 1 : Again, due to the collateral constraint, in equilibrium, ! ht must always be positive and P !O t + ! t = 1: The pay-o¤ relevant state space O !O t ; st : ! t 2 [0; 1] and st 2 fG; Bg is compact.34 Section 2 showed the existence of collateral constrained equilibria under the form of Markov equilibria in which prices and allocations depend solely on that state de…ned above. Section 4 provides an algorithm to compute such equilibria. As explained in Proposition 6, the equilibrium asset prices depend not only on the exogenous state but also on the normalized …nancial wealth. 5.2 Numerical Results In this subsection, I will chose plausible parameters to quantify the magnitude of the properties presented in Sections 2 and 3.2. These parameters are chosen such that the size of the …nancial sector is about 5% (eO )of the US economy, and the size of the housing markets (the real asset) is about 20% of the US annual GDP. In particular, let = 0:95 d (G) = 1 > d (B) = 0:2 U (c) = log (c) ; and the beliefs are O = 0:9 > P = 0:5. Given the high discount factor, the algorithm takes about 2 hours to converge. To study the issue of survival and its e¤ect on asset prices, I assume that the pessimists have the correct belief, i.e., = P = 0:5. Thus the optimists are over-optimistics. I …x the endowments of the pessimists and the optimists at eP = O e = 100 100:8 5 5 : The aggregate endowment is kept constant by choosing the pessimists’ endowment to be state dependent. 34 Given that the optimists prefer holding the physical asset, i.e., ktO > 0, ! O t corresponds to the fraction of the asset owned by the optimists. 28 5.2.1 Asset Prices The numerical algorithms presented in Section 4 for collateral constrained and Appendix A for complete markets allow us to examine and compare the properties of asset prices under the two market arrangements. The numerical example here illustrates Propositions 4 and 6 on asset prices. Collateral Constrained Equilibrium: I rewrite the budget constraint of the optimists (32) using normalized …nancial wealth, ! O t ; O cO t + qt kt + pt t O eO t + (qt + dt ) ! t : O Therefore, their total wealth eO t + (qt + dt ) ! t a¤ects their demand for the asset. If non…nancial endowment eO t of the optimists is small relative to price of the asset, their demand for the asset is more elastic with respect to their …nancial wealth (qt + dt ) ! O t . In Cao (2010), I investigate this relationship by varying the endowment of the optimists. Figure 1 plots the price of the physical asset as a function of the optimists’normalized …nancial wealth ! O for the good state st = G on the left axis, the solid line. The function for the bad state st = B is similar. Interest rate r is also endogenously determined in this economy, however most of the time it hovers around the common discount factors of the two agents, i.e. r (st ) 1 1: The mechanism for the high slopes at the right and at the left are not the same. On the left side of the …gure, when ! O is close to zero, the optimists are highly leveraged to buy the physical asset. If a bad shock hits in the next period, they have to sell o¤ their asset holdings to pay o¤ their debts. Their next period …nancial wealth plummets and contributes to the fall in asset price. Consequently, we can see in Figure 1, the slope of the price function is steeper for ! O < 0:2: This also corresponds to the region in which the borrowing constraint binds, or is going to bind in the near future. Then when a bad shock hits the economy, that is st+1 = B, the optimists are forced to liquidate their physical asset holdings. This …re sale of the physical asset reduces its price and tightens the constraints further, thus setting o¤ the vicious cycle of falling asset price. Notice that, this channel also explains a ‡atter slope at the lowest level of …nancial wealth of the optimists as they hold less of the physical asset, the asset …resale has less bite. This dynamics of asset price under borrowing constraint corresponds to the "debt-de‡ation" channel in a small-open economy in Mendoza (2010). This example shows that the channel still operates when we are in a closed-economy with endogenous interest rate, r (st ) as opposed to exogenous interest rates in Mendoza (2010) or in Kocherlakota (2000). On the right side of the …gure, the "debt-de‡ation" channel is not present as the collateral constraint is not binding or going to be binding in the near future. High asset price elasticity with respect to the normalized …nancial wealth of the optimists is due to their high exposure to the asset. As the optimists own most of the real asset, shocks to the dividend of the asset directly impacts the wealth, or cash in hand of the optimists. These shocks feed into large changes in the optimists’marginal utilities, thus large changes in the price of the real asset because the optimists are the marginal buyers of the asset. Due to the same reason, long run asset price volatility is higher when the optimists cannot lever using the real asset, analyzed in Subsection 5.2.5. 29 17.8 0.2 0.15 Asset Price Asset Price Volatility 17.4 0.1 17.2 0.05 17 0 0.1 0.2 0.3 0.4 0.5 0.6 Normalized Financial Wealth of the Optimists 0.7 0.8 0.9 1 Asset P rice Volatility Asset Price 17.6 0 Figure 1: Asset Price and Asset Price Volatility Under Collateral Constraints Figure 1 also plots asset price volatility as function of the normalized …nancial wealth of the optimists, the dashed line on the right axe. This function shows that at low and medium levels of …nancial wealth of the optimists, asset price is invertedly related to asset price volatility. This negative relationship between price level and price volatility has been observed in several empirical studies, for example, Heathcote and Perri (2011). In order to study the dynamics of asset price, we need to combine the fact that asset price as a function of the normalized …nancial wealth shown in Figure 1 with the evolution of the exogenous state and the evolution of the "normalized …nancial wealth" distribution, ! O t . . The left panel corresponds to the current good state s Figure 2 shows the evolution of ! O t = t G, and the right panel corresponds to the current bad state st = B: The solid lines represent next period normalized wealth of the optimists as a function of the current normalized wealth, if good shock realizes next period. The dashed lines represent the same function when the bad shock realize next period. I also plot the 45 degree lines for comparison. This …gure shows that, in general, good shocks tend to increase and bad shocks tend to decrease the normalized wealth of the optimists. This is because the optimists bet more for the good state to happen (buy borrowing collateralized and investing into the real asset). A similar evolution of the wealth distribution holds for complete markets. We can also think of normalized wealth as the fraction of the trees that the optimists owns. When the current state is good, and the fraction is high, the optimists will get a lot of dividends from their tree holdings, due to consumption smoothing they will not consume all the dividends, but will use some part of the dividends to buy new trees, so we see that on the left panel, the tree holding of the optimists normally increases at high ! O . Similarly when the current state is bad, and the fraction is high, the optimists will sell o¤ some of their tree holdings to smooth consumption. As a result, we see on the right panel that the tree holding of the optimists normally decreases at high ! O . Complete Markets Equilibrium: In a complete markets equilibrium, as shown in Appendix A, the state variable is the consumption of the optimists. However, there is a one-to-one 30 Next period Norm alized F inancial Wealth Good State, s = G Bad State, s = B 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 sn = G sn = B 0.3 0.3 0 45 line 0.2 0.2 0.1 0.1 0 0 0.2 0.4 0.6 Normalized Financial Wealth of the Optimitsts 0.8 0 1 0 0.2 0.4 0.6 0.8 1 Figure 2: Dynamics of Wealth Distribution under Collateral Constraints mapping from this state variable to a more meaningful state variable which is the relative wealth of the optimists, ! b O , de…ned in (22). Similar to the collateral constrained equilibrium, this variable determines asset price and constantly changes as aggregate shocks hit the economy. Due to log utility and constant aggregate endowment, apply the general formula (21), we obtain the relationship between asset price and relative wealth q (b !) = X h2fO;P g ! bh h 1 d (G) + 1 h d (B) : (35) This expression is the counterpart of Figure 1 for complete markets.35 Notice that at two extreme ! bO t = 0 or 1, we go back to the representative agent economy in which there are either only the optimists or the pessimists.36 In this special case where the aggregate endowments are constant across states and shocks are I.I.D., applying Corollary 3, we have in the long run, with probability 1, the price of the 35 Unlike under collateral constraints, under complete markets the asset price function does not depend on the exogenous state st due to the I.I.D. assumption and constant aggregate endowment. 36 When ! bO t = 0, asset price is the discounted value of average dividends evaluated at the pessimists’belief qP = P 1 P d (G) + 1 d (B) ; which is smaller than when ! bO t = 1, where asset price is the discounted value of average dividends evaluated at the optimists’belief qO = O 1 d (G) + 1 31 O d (B) > q P : 1 0 0.9 45 degree line sn = G sn = B Normalized Wealth Next Period 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 Normalized Wealth of the Optimists 0.8 1 Figure 3: Dynamics of Wealth Distribution under Complete Markets real asset, q (st ) converges to a constant q= 1 (P (G) d (G) + P (B) d (B)) ; (36) i.e., asset price volatility decreases to zero in the long run. Another way to see this convergence, is to notice that the wealth distribution (b !O ; ! b P ) converges to (0; 1); thus according to (35), qt converges to q (0) given by (36). In the short-run, however, the wealth distribution constantly changes as shocks hit the economy. Figure 3 depicts the evolution of the relative wealth distribution that determines the evolution of asset price under complete markets. This …gure is the counterpart of Figure 2 under complete markets. Given that the aggregate endowment is constant, the transition of the wealth distribution does not depend on current aggregate state, unlike under collateral constraints. The optimists buy more Arrow-Debreu assets that deliver in the good future states and buy less Arrow-Debreu assets that deliver in bad future states. Therefore, when a good shock hits, the relative wealth of the optimists increases (solid line) and vice versa when a bad shock hits (dashed line). Notice that, as opposed to the Figure 2, ! b O = 0 and ! b O = 1 are two absorbing states. So the optimists disappear under complete markets but not under collateral constraints. 5.2.2 Survival under collateral constraints This subsection illustrates the survival and disappearance results in Theorem 1 and Proposition 1. Figure 4 shows a realization of the …nancial wealth of the optimist starting at ! O = 0:1: The optimists always lever up to buy the real asset and often they will lose all their asset holdings (selling o¤ their asset holdings to pay o¤ their debt), in which case, their …nancial wealth reverts to zero. However, they can always use their non-…nancial endowment 32 0.12 N ormalized Financial Wealth of the Optimists 0.1 0.08 0.06 0.04 0.02 0 0 10 20 30 40 50 Time 60 70 80 90 100 Figure 4: A Sample Path of the Normalized Financial Wealth Distribution 0.25 0.2 D ensity 0.15 0.1 0.05 0 0 0.02 0.04 0.06 0.08 0.1 0.12 N ormalized Financial W ealth of the Optimists 0.14 0.16 0.18 0.2 Figure 5: Stationary Normalized Financial Wealth Distribution to come back to the …nancial markets by investing in the physical asset again (leveraged). Sometime they are lucky, that is, when the asset pays high dividends and its price appreciates, their …nancial wealth can increase rapidly. Given this dynamics, there exists a non-degenerate stationary …nancial wealth distribution of the optimists, shown in Figure 5. The spikes of the distribution (including the one at 0) shows that the …nancial wealth of the optimists often reverts to 0 and after that the optimists come back to the …nancial markets with certain levels of …nancial wealth, see Figure 2. This is in contrast with the results from the same simulation exercise for complete markets, where with probability 1 the wealth of the optimists will go to zero in the long run, thus the stationary distribution of the wealth of the optimists will be a degenerate mass at 0. 33 0.45 0.4 0.35 As s et Price Volatility 0.3 Collateral Cons traied Ec onomy Complete Mark ets Ec onomy 0.25 0.2 0.15 0.1 0.05 0 0 5 10 15 20 Time 25 30 35 40 Figure 6: Asset Price Volatility Over Time 5.2.3 Asset Price Volatility In this subsection, I compare asset price volatility under collateral constraints against the complete markets benchmark. I measure price volatility as one-period ahead standard deviation of price. This measure is the discrete time equivalence of the continuous instant volatility, see for example Xiong and Yan (2009). Figure 6 shows the evolution of asset price volatility under the two cases, the optimists with high or low non …nancial wealth. The …gure shows that, in the short run, asset price is more volatile under complete markets than under collateral constraints. However, in the long run, as the optimists are driven out in the complete markets equilibrium; that makes asset price volatility converge to zero. This property does not hold in collateral constrained equilibria, the overly optimistic agents constantly speculate on asset price using the same asset as collateral. Asset price becomes more volatile than in the complete markets equilibrium, given that the wealth of the optimists constantly change as they win or loose their bets. This result is an illustration of Proposition 7 because this economy has constant aggregate endowment.37 5.2.4 The …nancial crisis 2007-2008 Geanakoplos (2009) argues that the introduction of credit default swap (CDS) triggered the …nancial crisis 2007-2008. The reason is that the introduction of CDS moves the markets close to being complete. CDS allow pessimists to leverage their pessimism about the assets. I do the same exercise here by simulating a collateral constrained equilibrium in its stationary state from time t = 0 until time t = 49. At t = 50 markets suddenly become complete. In 37 Strikingly, the smaller the non-…nancial wealth of the optimist is, the higher the short-run asset price volatility in the collateral constrained equilibrium but the lower the short-run asset price volatility in the complete markets equilibrium. This is because, under complete markets, it takes less time to drive out the optimists if they have lower non-…nancial wealth. As we increase the non-…nancial wealth of the optimists, we increase the short-run volatility of asset price with complete markets and decrease the short-run volatility of asset price under collateral constraints. Above some certain level of non-…nancial wealth of the optimists, in the short-run asset price is more volatile under complete markets. But in the long run, the reverse inequality holds, see Cao (2010). 34 18 0.45 0.4 17 0.35 16 Asset Price Volatility 0.3 Asset Price 15 14 0.25 0.2 0.15 13 0.1 12 0.05 11 0 20 40 60 80 0 100 0 20 Time 40 60 80 100 Time Figure 7: Financial Crisis 2007-2008 Figure 7, the left panel plots asset price level and the right panel plots asset price volatility over time. The simulation shows that asset price decreases but asset price volatility increases in the short run after the introduction of CDS. The reason for the fall in asset price is that the pessimists can now "leverage their view" with a complete set of …nancial assets. The reason for increasing asset price volatility in the short run is the movement in the wealth distribution toward the long-run wealth distribution, which concentrates on pessimists. Nevertheless in the long run asset price volatility goes to zero when wealth distribution fully concentrates on the pessimists. 5.2.5 Regulating Leverage Proposition 7 suggests that the variations in the wealth distribution drive up asset price volatility relative to the long run complete markets benchmark. It is then tempting to conclude that by restricting leverage, we can reduce the variation of wealth of the optimists, therefore reduce asset price volatility. However, this simple intuition is not always true. The reason is that, similar to collateral constraints, …nancial regulation may act as another device to protect the agents with incorrect beliefs from making wrong bets and from disappearing from the economy. The higher wealth increases their impact on asset prices, thus make asset prices more volatile. To show this result, I consider an extreme form of …nancial regulation, that is, strictly forbidding leverage. This corresponds to setting the set of …nancial assets J empty, or equivalently require in…nite collateral k kr where kr is very high. Figure 8 plots the volatility of asset price as functions of the …nancial wealth of the optimists in two cases, with …nancial markets and without …nancial markets38 . We can see that at the low levels of …nancial wealth of the optimists, asset price volatility is higher with …nancial markets 38 Without …nancial market, "…nancial wealth" is asset holding itself. 35 Good S tate, s = G 1.4 1.2 A sset P rice V olatility 1 0.8 Unregulated Collateral Constrained E conomy Regulated Collateral Constrained E conomy 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 Normalized Financial W ealth of the Optimists 0.7 0.8 0.9 1 Figure 8: Asset Price Volatility in Unregulated and Regulated Economies than without …nancial markets. This is due to the debt-de‡ation, analyzed in Subsection 5.2.1, present under unregulated …nancial markets, but absent without …nancial markets. However, the opposite holds at higher level of …nancial wealth of the optimists. When the optimists hold a large fraction of the real asset, a drop in its dividends leads to a large drop in consumption and a large increase in the marginal utility of the optimists. As they are the marginal buyers, the increase in their marginal utility decreases the price of the real asset. This mechanism leads to high asset price volatility at the right side of Figure 8. But …nancial markets, allow the optimists to borrow to reduce the drop in consumption, thus mitigate the drop in the price of the real asset. It then makes asset price less volatile than when the …nancial markets are complete shut down. The numerical solution also shows that, without …nancial markets, the optimists always accumulate assets to move up to the high …nancial wealth (asset holding) region.39 This dynamics makes asset price more volatile without …nancial markets then it is with …nancial markets. Figure 9 shows the Monte-Carlo simulation for an economy starting in good state and ! O = 0. The …gure plots the evolution of the average of the normalized …nancial wealth of the optimists, left panel, and asset price volatility, right panel, over time (the solid lines represent the unregulated economy and the dashed lines represent the regulated economy). As discussed above, the wealth of the optimists remains low on average in the unregulated economy but increases to a permanently high level under regulation. Thus, initially asset price volatility is higher in the unregulated economy than in the regulated economy. The reverse inequality holds, however, as over time, the wealth of the optimists increase more in the regulated economy than in the unregulated economy. I conclude this part with two additional remarks. First, intermediate regulations can be computed using Proposition 8. If the regulator requires collateral k kr , then the proposition shows that in equilibrium, only the leverage level max (kt ; kr ) prevails. The 39 With leverage, the optimists will want to hold more of the real assets using leverage, but given that they have incorrect beliefs, they will tend to lose all their shares, and remain …nancially poor. 36 1.2 1.6 1.4 1 1 Asset Price Volatility N ormalized Financial Wealth of the Optimists 1.2 U nregulated C ollateral C onstraied E conomy R egulated C ollateral C onstrained E conomy 0.8 0.6 0.4 0.8 0.6 0.2 0.4 0 -0.2 0.2 0 5 10 15 20 Time 25 30 35 0 40 0 5 10 15 20 Time 25 30 35 40 Figure 9: Wealth Distribution and Asset Price Volatility over Time numerical solutions for intermediate regulations con…rms the conclusion in the paragraphs above. Second, regulation not only fails to reduce asset price volatility, it also reduces welfare of both types of agents as it reduces trading possibilities. 5.2.6 Dynamic leverage cycles Even though the example in Subsection 5.2.3 generates high asset price volatility, leverage is not consistent with what we observe in …nancial markets: high leverage in good times and low leverage in bad times, as documented in Geanakoplos (2009). In order to generate the procyclicality of leverage, I use the insight from Geanakoplos (2009) regarding aggregate uncertainty: bad news must generate more uncertainty and more disagreement in order to reduce equilibrium leverage signi…cantly. The economy also constantly moves between low uncertainty and high uncertainty regimes. To formalize this idea, I assume that in the good state, s = G, next period dividend has low variance. However, when a bad shocks hits the economy, s = GB or BB, the variance of next period dividend increases. In this dynamic setting, the formulation translates to a dividend process that depends not only on current exogenous shock but also on the last period exogenous shock. Therefore we need to use three exogenous shocks, instead of the two exogenous shocks in the last subsections: s 2 fG; GB; BBg : Figure 10, left panel, shows that the good state, the variance of next period dividend is low, d = 1 or 0:8. However in bad states, the variance of next period dividend is higher, d = 1 or 0:2. The right panel of the …gure shows the evolution through time of the exogenous states using Markov chain representation. Even though we have three exogenous states in this set-up, each state has only two immediate successors. So we can still use Proposition 8 to show that in any history there is only one leverage level in the economy. The uncertainty structure generates high leverage at the good states G and low leverage in bad states GB and BB: Figure 11 shows this pattern of leverage. The dashed line represents 37 Figure 10: Evolution of the Aggregate States Leverage = q/(q-p/k) 100 90 80 Leverage 70 60 50 40 s =G s = GB s = BB 30 20 0 0.1 0.2 0.3 0.4 0.5 0.6 Normalized Financial W ealth of the Optimists 0.7 0.8 0.9 1 Figure 11: Leverage and Wealth Distribution leverage level in good states s = G as a function of the normalized wealth distribution. The two solid lines represent leverage level in bad states s = GB or BB. In addition to the fact that uncertainty a¤ects leverage emphasized in Geanakoplos (2009), we also learn from Figure 11 that …nancial wealth distribution is another important determinant of leverage. For example, we learn from the …gure that leverage decreases dramatically from good states to bad states. However, in contrast to the static version in Geanakoplos (2009), changes in the wealth distribution do not amplify the decline in leverage from good states to bad states as leverage is relatively insensitive to the wealth distribution in bad states. Moreover, this version of dynamic leverage cycles generates a pattern of leverage build-up in good times. Good shocks increase leverage as they increase the wealth of the optimists relative to the wealth of the pessimists and leverage is increasing the wealth of the optimists. Figure 12 shows the evolution of the wealth distribution and leverage over time. The economy starts at good state and ! O = 0. It experiences 9 consecutive good shocks from t = 1 to 9 and two bad shocks at t = 10; 11 then another 9 good shocks from t = 12 to 19. This …gure shows that, in good states, both the wealth of the optimists and leverage increase. However 38 O Leverage over time 80 0.11 70 0.1 60 Leverage Normalized F inancial Wealth of the Optimists ω over time 0.12 0.09 50 0.08 40 0.07 30 0.06 0 5 10 Time 15 20 20 0 5 10 Time 15 20 Figure 12: Leverage Cycles their wealth and leverage plunge when bad shocks hit the economy. Notice, however, that even though leverage decreases signi…cantly from 80 to 25 when a bad shock hits the economy, that leverage level is still too high compared to what observed during the last …nancial crisis. Gorton and Metrick (2010) document that leverage on some class of assets declined to almost 1 on some classes of assets. The reason that leverage is always very high in this model is due to high discount factor and pj in formula (3). Of course, there are some other channels absent in this paper that might have caused the rapid decline in leverage. 6 Quantitative assessment In this section, I apply the numerical solution method to a more seriously calibrated setup used in Heaton and Lucas (1995). In order to do so, I need to modify the economy in Section 2 to allow for the possibility that aggregate endowment grows overtime. As in Heaton and Lucas (1995), the aggregate endowment e (st ) evolves according to the process e (st+1 ) = 1 + g st : e (st ) There is only one Lucas tree that pays o¤ the aggregate dividend income at time, Dt and s t D (st ) = ; e (st ) the remaining endownment belong to the individual incomes under the form of labor income X D st : eh st = e st h2H 39 Individual h’s labor income as a fraction of aggregate labor income is given by h eh (st ) : h t h2H e (s ) st = P To use the numerical method used in Section 2, I use the following normalized variables b cht = and cht h eht b dt ; ebt = ; dt = ; et et et qbt = qt : et 1 Assuming CRRA for the agents, Uh (c) = c1 hh , the expected utility can also be re-written using the normalized variables "1 # "1 # 1 h X X e t+r 1 r r Eht Uh cht+r = (et ) h Eht Uh b cht+r et r=0 # " r=0 r 1 Y X 1 h 1 h h r t+r0 h 1+g s Uh b ct+r Et = (et ) r=0 r0 =1 I focus on debt contracts by assuming that, in each node st , only …nancial contracts, i.e., bj (st+1 ) = 1 for all st+1 jst , are allowed to be traded. I consider the following collateral requirements 1 1 kj st = max ; t+1 1 m st+1 jst q (s ) + d (st+1 ) where 0 m < 1. These collateral constraints correspond to the borrowing constraints h t (1 m) kth min q st+1 + d st+1 t+1 t s js : When m = 1, no borrowing is possible, the agents are allowed to only trade on the real asset.40 In these environment, I also use the normalized variables for the choice of debt h h holding bt = ett . 40 Alternatively, we can consider the collateral requirements kj st = 1 1 max ; t+1 t m s js q (st+1 ) which correspond to the borrowing constraints h t mkth min q st+1 : st+1 jst This constraint is similary to the one used in Kiyotaki and Moore (1997), when borrowers can only commit to deliver the part of the collateral asset without the current dividend. The quantiative implications of such requirements are very similar to the one used here. 40 We rewrite the optimization of the consumer as "1 t Y X t h h ct (1 + g (sr ))1 max E0 h Uh b h h h fct ;ka;t ; j;t g r=1 t=0 h # (37) and in each history st , she is subject to the budget constraint h b cht + qbt kth + bt the collateral constraints bh + mk h min t t t+1 t s js n ebht + 1 bh t 1 + g (s ) t 1 h + qbt + dbt ka;t 1; qb st+1 + db st+1 1 + g st+1 and …nally the no short-sale constraint in the real asset, kth 6.1 o (38) 0 (39) 0, as before. Homogeneous beliefs The authors further assume the variables depend only on the current state of the economy, that is g st st = g (st ) = (st ) h = st h (st ) : They use the annual aggregate labor income and dividend data from NIPA and individual income from PSID to calibrate g (:) ; (:) ; (:) and the transition matrix (sjs0 ) : The estimates for a two-agent economy are taken from Table 1 in Heaton and Lucas (1995). As comparision, the economy in Heaton and Lucas (1995) is the same as in my economy except for the fact that the collateral constraint (39) are replaced by the exogenous borrowing constraint h Bh: (40) t Table 1 shows that collateral constraints do not alter sign…cantly the quantitative result in Heaton and Lucas (1995). The standard deviations of consumption and stock return are slightly lower in my economy. 6.2 Belief heterogeneity Now, I introduce belief heterogeneity by assuming the …rst agent in Heaton and Lucas (1995) are more optimistic about the growth rate of the economy. Table 2 shows that the standard deviation of stock returns does not change signi…cantly. 41 Moments Consumption Growth Average Standard deviation Bond return Average Standard deviation Stock return Average Standard deviation Data CM HL m = 0 m = 0:9 m = 1 0.020 0.018 0.018 0.019 0.030 0.028 0.044 0.041 0.019 0.045 0.008 0.080 0.077 0.078 0.026 0.009 0.012 0.012 0.077 0.013 0.089 0.082 0.079 0.079 0.173 0.029 0.032 0.030 0.080 0.035 0.019 0.045 0.080 0.036 Table 1: Summary Statistics for Baseline Case Moments Consumption Growth Average Standard deviation Bond return Average Standard deviation Stock return Average Standard deviation Data CM HL m = 0 m = 0; 1 m = 0; 0.020 0.018 0.018 0.019 0.030 0.028 0.044 0.041 0.027 0.048 0.020 0.057 0.008 0.080 0.077 0.078 0.026 0.009 0.012 0.012 0.084 0.011 0.080 0.018 0.089 0.082 0.079 0.079 0.173 0.029 0.032 0.030 0.086 0.029 0.076 0.033 Table 2: Summary Statistics with Belief Heterogeneity 42 2 7 Conclusion In this paper I develop a dynamic general equilibrium model to examine the e¤ects of belief heterogeneity on the survival of agents and on asset price volatility under di¤erent …nancial markets structures. I show that, when …nancial markets are collateral constrained (endogenously incomplete), agents with incorrect beliefs survive in the long run. The survival of these agents leads to higher asset price volatility. This result contrasts with the frictionless complete markets case, in which agents holding incorrect beliefs are eventually driven out and as a result, asset prices and investment exhibit lower volatility. In addition, I show the existence of stationary Markov equilibria in this framework with collateral constrained …nancial markets and with general production and capital accumulation technology. I also develop an algorithm for computing the equilibria. As a result, the framework can be readily used to investigate questions about the interaction between …nancial markets and the macroeconomy. For instance, it would be interesting in future work to apply these methods in calibration exercises using more rigorous quantitative asset pricing techniques, such as in Alvarez and Jermann (2001). This could be done by allowing for uncertainty in the growth rate of dividends rather than uncertainty in the levels, as modeled in this paper, in order to match the rate of return on stock markets and the growth rate of aggregate consumption. Such a model would provide a set of moment conditions that could be used to estimate relevant parameters using GMM as in Chien and Lustig (2009). A challenge in such work, however, is that …nding the Markov equilibria is computationally demanding. I started this exercise in Section 6 and further follow that path in Cao, Chen, and Scott (2011). A second avenue for further research is to examine more normative questions in the framework developed in this paper. My results suggest, for example, that …nancial regulation aimed at reducing asset price volatility should be state-depedent, as conjectured by Geanakoplos (2009). It would also be interesting to consider the e¤ects of other intervention policies, such as bail-out or monetary policies. 43 8 Appendices Appendix A: The analysis of survival, disappearance, and asset pricing under complete markets. Proof of Proposition 1. This is an application of Proposition 5 in Sandroni (2000). We need to show that, for any agent h with incorrect belief, there exists l and st > 0 such that for any path dl Psht ; Pst > : (41) Indeed, given that h 6= , there exists an s and s such that h (s ; s ) 6= (s ; s ). Now Assumption 1 implies that for each s 2 S there exists n such that (n) (s; s ) > 0:41 (1) then (n) implies that h (s; s ) > 0. Let l be the maximum of these n’s over S. Given the …niteness of S, we can …nd that satis…es (41). Proof of Propostion 4. Let I denote the set of agents with the correct belief. The asset price is the presented discounted value of dividends weighted by the stochastic discount factor: qa st = 1 X r=1 We know that or ch st+r da (st+r ) : Uh0 (ch (st )) 0 r Uh Ph st+r jst Ph st+r Uh0 ch st+r Uh0 (ch (s0 )) = (42) Ph0 st+r Uh0 0 ch0 st+r Uh0 0 (ch0 (s0 )) Ph0 st+r Uh0 ch s0 Uh0 ch st+r = : Uh0 0 (ch0 (st+r )) Ph (st+r ) Uh0 0 (ch0 (s0 )) For h and h0 2 I; we have, Ph0 st+r = Ph st+r , so Uh0 ch st+r Uh0 ch s0 = : Uh0 0 (ch0 (st+r )) Uh0 0 (ch0 (s0 )) Let CI st+r = X ch st+r (43) : h2I Using the equations (43), we can solve for ch as functions of CI . ch st+r = Ch CI st+r for each h 2 H. Applying Proposition 1, we have, almost surely, lim CI st e (st ) = 0 t !1 or lim ch st t !1 41 Ch (e (st )) = 0: See Stokey and Lucas (1989) for standard notations with transition matrices. 44 (44) Let q a (st ) = 1 X r=1 0 r Uh (Ch (e (st+r ))) da (st+r ) Uh0 (Ch (e (st ))) P st+r jst for any h 2 H. (45) Finally, (42), (45), and (44) implies lim sup qa st+r q a (st+r ) = 0: t !1 r 0 This is asset price in an complete markets economy.42 Closed form solution in Example 1. The natural state variable is the wealth distribution wh st = 1 X X r=0 st+r jst p st+r jst ch st+r : In case of log utility, ch st = (1 ) wh st The good market clearing condition implies X e (st ) = ch st = (1 ) h2H We have t qa s = 1 X X r=1 st+r jst X wh st h2H p st+r jst da (st ) where, also given log utility, p st+r jst = = = ch st u0 st+r Ph st+r jst h 0 t = r Ph st+r jst uh (s ) ch (st+r ) P P r r t+r jst c t t+r jst c t h s h s h2H Ph s h2H Ph s P = t+r ) e (st+r ) h2H ch (s X X 1 1 = (1 ) wh st r Ph st+r jst : ch st r Ph st+r jst e (st+r ) e (st+r ) r h2H So …nally h2H 8 1 X <X X qa st = ! ht st : e (st ) t+r t r=1 s h2H where ! ht st = (1 js r Ph st+r jst 9 d (st+r ) = ; e (st+r ) ; ) wh st wh st =P t e (st ) h0 2H wh0 (s ) Numerical procedure for complete markets. The state space should be s; fch gh2H 42 See Ljungqvist and Sargent (2004) for a standard treatment of asset pricing under complete markets with similar notations but with homogeneous beliefs. 45 such that X ch = e (s) h2H We …nd the mapping from that n state space o into the set of current prices and consumption levels + , and fps gsS the Arrow-Debreu state prices. There fqa ga2A ; future consumptions ch h2H s2S are therefore A + (S + 1) H + S unknowns. For each a 2 A we have A equations qa = X ps qa+ + d+ a s Regarding ps , the inter-temporal Euler equation implies ps = that give SH equations and …nally X h h = c+ h h + s; s X = e Uh0 c+ h Uh0 (ch ) e+ h + X e+ a Ka a2A h + which give other S equations. With these A + (S + 1) H + S equations, we can solve for the A + (S + 1) H + S unknowns. That solution determines the mapping T : In order to …nd an equilibrium corresponding to an initial asset holdings ( h;a )h2H;a2A we …nd the value of stream of consumption and endowment of each consumers X wch = ch + ps wch+ (s) s2S and weh = eh + X ps weh+ (s) s2S Then we solve for H unknowns (ch )h2H using H equations X wch = weh + h;a qa : a2A Appendix B: One asset economy Proof of Proposition 8. Let kd = maxj2J ;kj k kj . Let pd denote the price of the …nancial assets with collateral requirement kd . Suppose that there is another …nancial asset in J that is actively traded and have collateral requirement k k . Then by de…nition k < kd . Let pk denote the price of the …nancial asset. We have two cases: Case 1) d1 kd > k > u1 : Consider the optimal portfolio choice of a seller of …nancial asset k. The pay-o¤ from selling the asset is (ku 1; 0) and she has to pay kq pk in cash: she buys k units of the real asset but she get pk from selling the …nancial asset. So the return on the …nancial asset is kd u 1 ku 1 kq pk (when good state happens). Similarly, the return on selling …nancial asset kd is kd q pd . If there are sellers for …nancial asset k, it then implies that ku 1 kq pk kd u 1 : kd q pd 46 or equivalently pk ku kd u 1 kd k pd + q; 1 kd u 1 (46) otherwise, sellers will strictly prefer selling …nancial asset kd to …nancial asset k. Now from the perspective of the buyers of the …nancial assets, the pay-o¤ of …nancial asset k is (1; kd). We can write this pay-o¤ as a portfolio of …nancial asset kd and the real asset: ku 1 = kd kd u 1 kd k u 1 + : 1 kd d kd u 1 d (47) As a result, if there are buyers for …nancial asset k, we must have pk ku kd u 1 kd k pd + q 1 kd u 1 (48) otherwise the buyers will buy the portfolio (47) of …nancial asset kd and the real asset instead. Thus we have both (46) and (48) happen with equality if …nancial asset k is actively traded. Armed with the equality, we can now prove the proposition. Consider each pair of seller and buyer of a unit of …nancial asset k: the buyer buy one unit and the seller sells one unit of …nancial asset k at the same time is required to buy k units of the real asset. We alter the their portfolios in the following 1 way: instead of buying one unit of …nancial asset k, the buyer buys kku units of …nancial asset du 1 kd from the seller and kkddu k1 of the real asset. Given (47) and the equality (48), this changes in portfolio let the consumption and future wealth of the buyer unchanged. Now the seller instead of 1 1 selling one unit of …nancial asset k, sells kku units of …nancial asset kd and holds kku kd units du 1 du 1 of the real asset as collateral. Similarly, due to the equality (46), this transaction costs the same and yields the same returns to the seller compared to selling one unit of …nancial asset k. So the consumption and the future wealth of the seller remains unchanged. Now we just need to verify that the total quantity of the real asset used remain unchanged. Indeed this is true because kd k ku + kd u 1 kd u 1 kd = k: 1 Case 2) u1 k: Financial asset k’s pay-o¤ to the buyers is k (u; d) and to the seller is 0. So the …nancial asset is essential the real asset. This implies, pk = kq. The proposition follows immediately. Now let ku = minj2J , kj k kj . Let pu denote the price of the …nancial asset ku . The proof of the proposition is similar. First, we show that the price of any actively traded …nancial asset k with k k is pk = pu and we can alter the portfolio of the buyers and sellers of …nancial asset k to transfer all the trade in the asset to …nancial asset ku . 47 References Alvarez, F. and U. J. Jermann (2000, July). E¢ ciency, equilibrium, and asset pricing with risk of default. Econometrica 68 (4), 775–798. Alvarez, F. and U. J. Jermann (2001). Quantitative asset pricing implications of endogenous solvency constraints. 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