Heterogeneous Agent Economies With Knightian. Uncertainty. Wen-Fang Liu y. October Abstract

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1 Heterogeneous Agent Economies With Knightian Uncertainty Wen-Fang Liu y October 1998 Abstract This paper studies decentralized economies in which decision-makers do not have full knowledge of the probabilities of the states of nature. This ambiguity isre- ferred to as Knightian uncertainty. Using a general equilibrium heterogeneous-agent model, we show how prices and consumption allocations are altered by the presence of Knightian uncertainty. Itisknown from previous work that models with Knightian uncertainty can have indeterminant security prices. In a multi-agent setup, we show the range of the indeterminacy is determined by the agents with the least amount of Knightian uncertainty. In contrast to models with `full risk-sharing', in a twoperiod setup we show consumption allocations have a threshold feature in which the consumptions of some individuals can be constant over a range of variations in the aggregate endowment. In an innite-horizon setting, the threshold feature shows up in the expected utility values. Furthermore, consumption allocations in the random aggregate endowment economy are history-dependent. JEL: D50, D80, E21, G12 Address: Department of Economics, University ofwashington, Box , Seattle, WA liuwf@u.washington.edu ythis paper is a revision of my PhD dissertation at the University of Chicago. I am grateful to Tom Sargent, Robert Lucas, Grace Tsiang, Nancy Stokey, Jose Scheinkman, Gerald Cubbin, Evan Anderson, Andreas Lehnert, Georey Shuetrim, Tim Conley and Rose-Anne Dana for valuable comments and criticism on earlier drafts, and especially to Lars Hansen for guidance, encouragement and constant support. Financial support from the University of Chicago and Chiang Ching-Kuo Foundation is gratefully acknowledged. 1

2 1 Introduction General equilibrium model-builders typically assume that the decision-makers in their models use von Neumann Morgenstern expected utility functions with no ambiguity in the probability assessments. By contrast, this paper follows Epstein and Wang (1994) by studying decentralized economies in which agents do not have full knowledge of the underlying probabilities. 1 This ambiguity is referred to as Knightian uncertainty. We extend Epstein and Wang's representative agent model by considering models in which there is heterogeneity across agents. Heterogeneous-agent versions of rational expectations models with a complete array of security markets are known to behave invery much the same way as their single-agent counterparts. As we will demonstrate, this similarity does not extend to models with Knightian uncertainty. Our interest in models with decision-makers that have imprecise probability assessments stems in part from previous demonstrations of how observable implications are altered by the presence of Knightian uncertainty. For example, Dow and Werlang (1992) show that in the presence of Knightian uncertainty aninvestor will neither buy nor sell a risky security for a range of prices. 2 As emphasized by Liu (1997), this result is useful in understanding why only a quarter of American families own stocks in spite of the historically high equity premium (see Mankiw and eldes (1991), Haliassos and Bertaut (1995), and Heaton and Lucas (1996)). In related work, Melino and Epstein (1995) use a model featuring Knightian uncertainty to account for the equity-premium puzzle. Finally, Liu (1997) shows that the presence of Knightian uncertainty in labor income prompts an economic agent tosave more as a form of insurance for future contingencies. This paper explores how the heterogeneity in degrees of Knightian uncertainty alters the consumption allocations and asset pricing implications in a general equilibrium model. For 1 Most of the studies on Knightian uncertainty are from the perspective of individual decision making. See, for example, the literature review by Camerer and Weber(1992). 2 See also Bewley (1987). 2

3 simplicity we adopt a one-parameter formulation of Knightian uncertainty borrowed from the literature on robust statistics. We use the so-called -contamination model in which with probability the known distribution is replaced by an `arbitrary distribution'. We assume agents are uncertainty averse, i.e., they do not like the presence of Knightian uncertainty. Following Gilboa and Schmeidler (1989), this leads us to model decision-makers as guarding against the `worst case' among the family of -contaminated probability distributions. We use this -contamination formulation to show how equilibrium consumption allocations dier from those in a benchmark rational expectations model. By introducing heterogeneity in the contamination parameters, we are able to understand better the nature of the price indeterminacy known to exist in models with Knightian uncertainty (Epstein and Wang, 1994). In independentwork, Chateauneuf, Dana, and Tallon (1997) investigate comonotonicity properties of consumption allocations under the non-additive expected utility theory in a two-period setting. This paper complements their work. By adopting a more specic formulation of non-additive expected utility, this paper is able to go farther and get more implications from the model. This paper also adds to their work in exploring dynamics of the economy over time. 1.1 Overview of this Paper Knightian uncertainty introduces a form of rst order risk aversion in the sense of Segal and Spivak (1990), which means that, roughly speaking, the risk premium of an asset is proportional to the standard deviation of the asset (See Liu (1997)). As shown by Segal and Spivak (1990), a property of preferences with rst order risk aversion is that when we draw indierence curves in a two-state world, there are kinks along the certainty line (forty-ve degree line). See Figure 1.1. This is in contrast to preferences of a smooth utility function under the expected utility theory, in which indierence curves are smooth everywhere and the risk premium is pro- 3

4 consumption at state degree consumption at state 1 Figure 1: Indierence Curve for Preferences With First Order Risk Aversion portional to the variance of the asset (second order risk aversion). This paper investigates how this rst order risk aversion property from Knightian uncertainty is transmitted to prices and consumption allocations in general equilibrium. The economies that we study in this paper are populated with agents who have heterogeneous degrees of Knightian uncertainty about individual endowments and possibly about the aggregate endowment. We will distinguish economies with constant aggregate endowments and with random aggregate endowments and examine them separately Equilibrium Prices Preferences incorporating Knightian uncertainty can result in indeterminacy in decisionmaking because of agents' inability to distinguish the probability ofeachevent. As shown by Epstein and Wang (1994) and Melino and Epstein (1995), price indeterminacy is a likely consequence in models with Knightian uncertainty due to the kinked indierence curves that are induced. When price indeterminacy occurs in these models, there is a continuum of equilibrium prices. We show that the degree of equilibrium price indeterminacy is determined by the type of agents who have the least Knightian uncertainty. 4

5 The indeterminacy in the equilibrium prices implies that the volatility of the observed prices may partly come from dierent realizations of the indeterminacy range. This volatility implication can potentially explain the excess volatility of stock prices, as proposed by Epstein and Wang (1994). This paper shows that with complete markets, the price of a security whose payo is a function of the aggregate endowment will be uniquely determined. On the other hand, the equilibrium price of a security whose payo is not a function of the aggregate endowment is generally indeterminant. In particular, the price of a statecontingent claim for consumption goods contingent on idiosyncratic shocks that are not a function of the aggregate resource will not be uniquely determined. 3 We also show that in a constant aggregate endowment economy, the price indeterminacy of contingent claims on individual endowments leads to indeterminacy in the wealth of each agent, and hence the equilibrium consumption allocation is not unique Equilibrium Consumption Allocations In an economy where the aggregate endowment is constant but there is Knightian uncertainty about individual endowments, equilibrium consumption allocations share the same feature as an equilibrium allocation in a rational expectations economy { individual consumption is completely smoothed despite the stochastic nature of the individual endowment. On the other hand, in a two-period economy where the aggregate endowment is random, individual consumption will only weakly comove with the aggregate endowment. For instance, gure 2 shows the equilibrium consumption allocation among the two types of agents as functions of the aggregate endowment in an economy with continuum aggregate endowment states. Agents with a higher degree of Knightian uncertainty (solid line) have smooth consumption when the aggregate endowment islow. Agents with a lower degree of Knightian uncertainty (dash line) therefore absorb all the uctuations when the aggregate 3 See also the discussion by Epstein and Wang (1994). 5

6 endowment is low and are compensated by higher consumption on the rest of aggregate endowment states. When we extend to an innite horizon economy, the threshold feature 20 INDIVIDUAL CONSUMPTION Type B agent Type A agent y AGGREGATE ENDOWMENT y+h Figure 2: Optimal Consumption Allocation for a two-period economy with two types of agents: type A and type B. Both types of agents have the same log utility function but dierent degrees of ambiguity in their probability assessments. Type A agents (solid line) have a higher degree of Knightian uncertainty than type B agents (dash line). shows up in the expected utility values. In contrast to the equilibrium of the benchmark rational expectations economy, the ecient consumption allocations are history-dependent. The rest of this paper is organized as follows. Section 2 lays down the formulation of Knightian uncertainty that we adopt in this paper. Section 3 is devoted to economies with a constant aggregate endowment; and Section 4 to economies with a random aggregate endowment. Section 5 concludes. 2 Modeling Knightian Uncertainty In a model without Knightian uncertainty, an economic agent's beliefs about the probability law of the stochastic economic environment are specied by a unique prior. In contrast, to model Knightian uncertainty we allow an agent's beliefs to be a set of probability distributions. We formulate an agent's set of priors by the -contamination parameterization. 6

7 2.1 -contamination Beliefs Assume the agent's beliefs P have an-contamination form: P = f(1, ) + m : m 2Mg (1) where is a constant in the unit interval [0, 1], is a reference probability distribution, and M is the collection of all possible probability measures on the support of. The interpretation for the -contamination beliefs is the following. An economic agent uses his information and knowledge about the stochastic environment and forms an estimate for the true probability law. We call this estimate the reference probability distribution. The agent is not absolutely sure the reference probability distribution is the true one. He thinks that with (1-) probabilities, the true distribution is, and with probabilities it is drawn randomly at an unknown distribution from the set M. 4 Roughly speaking, we can view P as a collection of perturbations of the reference measure. Note that when is greater than zero, the belief set P is non-singleton. This nonsingleton property reects the agent's ambiguity about the probability law. For a given, the larger the value, the larger the set of priors. When equals 1, P equals M. In this case, the agent is completely ignorant about the probability distribution. When approaches zero, the belief set P reduces to a singleton that contains only the reference probability measure, i.e., P =0 = fg. In most part of the paper, we assume all agents have the same reference probability distribution. Heterogeneity in agents' beliefs is modeled through dierent values. We can interpret this homogeneity of the reference probability measure as agents having the same information such as a common data set. This homogeneous reference measure assumption also makes the model an immediate generalization of the rational expectations model. 4 See Huber (1981). 7

8 When the values of 's approach zeros, the model serves as a robustness check of the benchmark rational expectations model. An agent is said to have a higher degree of Knightian uncertainty than the other agent if the agent's set of priors is a superset of the other agent's set of priors. Consider two agents A and B with -contamination beliefs and with the same reference probabilities but dierent values. Since P B P A if and only if A B, A has more Knightian uncertainty than B if and only if A B. Hence we use the value to index an agent's degree of uncertainty about the probability distribution when agents have an identical reference measure. An agent with a higher value is less condent about the the estimated probability measure. 2.2 Generalized Expected Utility There are several approaches to formulating decision-making under Knightian uncertainty (see, for example, Bewley(1986), and Schmeidler(1987)). The approach this paper adopts follows the maxmin expected utility theory of Gilboa and Schmeidler (1989), Epstein and Wang (1994), and Melino and Epstein (1995). To compute the \expected utility" under non-unique priors, rst compute the expected utility for each probability distribution in the set of priors, and then take the minimum over these values: udp min 2P ud : (2) This minimization operator reects the agent's uncertainty-averse preferences: the agent is guarding against the worst possible cases. Loosely speaking, the larger the set P, the more uncertain the agent and, thus, the smaller the value of \expected utility". Note that when P is singleton, i.e., the agent has a unique prior, equation (2) reduces to the usual expected utility denition. Thus this maxmin expected utility framework is a generalization of the expected utility theory. Aninteresting feature of Knightian uncertainty is that preferences with uncertainty-aversion are observationally equivalent to preferences with a pessimistic 8

9 character. This pessimistic characteristic is also found in the risk-sensitive control theory (see, for example, Hansen, Sargent and Tallarini (1997)), and the construction of robust decision rules (see Huber (1981)). Under the assumption of -contamination beliefs, the calculation of the generalized expected utility becomes : udp =(1, ) ud + minu: (3) where is the set of states. As it will be clear later, this simple form facilitates our analysis. This equation shows that the value controls the eect of ambiguity in the utility function. When is very small, for a given the impact of the minimization can be quite limited. 3 Heterogeneous Agent Economies With Constant Aggregate Endowments This section studies heterogeneous agent economies in which the aggregate endowment is constant but there is Knightian uncertainty about individual endowments. The degrees of Knightian uncertainty among agents can dier. We show that in this economy, equilibrium consumption allocations share the same full insurance feature as those in a rational expectations economy: individual consumption is completely smoothed across states. However, in the presence of Knightian uncertainty, equilibrium indeterminacy occurs in the constant aggregate-endowment economy. Specically, there will be price indeterminacy for the state contingent claims on individual endowments. Individual wealth and consumption shares will be indeterminant. Section 3.1 investigates a two-period model with two types of agents. The simple setup enables us to gain insights prior to our analysis for a more general setting. Section 3.2 then extends the results to an innite-horizon economy with n 2types of 9

10 agents. 3.1 A Two-Period Model Consider a two-period exchange economy with heterogeneous agents. There are two types of agents in the economy who dier in their endowments and in their degrees of Knightian uncertainty. We denote them as type A and type B. There is a continuum of each type of agent with unit mass. At period 0, each agent is endowed with one unit of consumption good. At period 1, a type A agent will be endowed with y=2 units of consumption good if state! 1 occurs, and y=2 + units ( > 0) if state! 2 occurs. Atype B agent will be endowed with y=2 + units of consumption good at! 1, and y=2 units at! 2. Notice that the aggregate endowment at period 1 is constant across states. Assume both types of t =0 t =1! 1! 2 A 1 y=2 y=2+ B 1 y=2+ y=2 aggregate 2 y + y + Table 1: Endowment Process agents have -contamination beliefs. Let =(p; 1, p) denote the reference probabilities used by both types of agents for states! 1 and! 2. Types A and B agents have dierent degrees of ambiguity on the probabilities of next period's endowment shocks, denoted by A and B. The belief mappings are P i = f(1, i ) + i m : m 2Mg for i = A; B (4) where M is the collection of all possible probability distributions for states! 1 and! 2. Let q i and q i be the maximal and minimal probabilities in P i that can be assigned to state! 1. Then q i =(1, i )p + i, and q i =(1, i )p. We can express the set of uncertainty adjusted 10

11 probability measures in the following way. P i = f(q;1, q) : q i q q i g for i = A; B: (5) Assume that both types of agents have log utility functions: U i (c i 0 ; ~ci 1 )=lnci 0 + ln ~c i dp i 1 ; i = A; B: (6) where c i 0 and ~c i 1 are consumptions of a type i agent at period 0 and 1 respectively. We set = 1 for simplicity. Assume there are complete markets in the sense that agents can trade all the state-contingent claims in the economy. To see why trade occurs, we rst compute each agent's valuations of state-contingent claims under autarky Autarky We will obtain agents' valuations of state-contingent claims from the rst order conditions of agents" maximization problems. Dene the set of uncertainty adjusted probability measures foratype i agent, Q i, to be the collection of probability distributions in P i that minimize the agent's expected utility level for a choice of a consumption plan: Q i f 2P i j u(c i )d = min 2P i u(c i )dg: Epstein and Wang (1994) show that rst order conditions from a maxmin expected utility have the property that they are evaluated using the uncertainty adjusted measures. Under autarky, each agent consumes his own endowment. Because a type A agent has alow endowment on! 1, the probability distribution in P A that minimizes A's expected utility is the one that maximizes the probability of state! 1. Similar analysis can be applied 11

12 for a type B agent. Thus, we have that Q A = f(q A ; 1, q A )g and Q B = f(q B ; 1, q B )g; where q A =(1, A )p + A ; q B =(1, B )p: Notice that q A q B and (1,q A ) (1,q B ) for any A and B in the interval [0,1]. We now compute agents' valuations of state contingent claims on state! j 's consumption goods. The formula is the usual one, i.e. marginal rates of substitution weighted by probabilities, except that it is now weighted using the uncertainty adjusted measures. Denote by p i 1 and p i 2 atype i agent's valuations of contingent claims on states! 1 and! 2 respectively. Given the uncertainty adjusted measures and the log utility function, we obtain p i and 1 pi for 2 i = A; B under autarky. p A 1 = 2 y qa ; p B 1 = 2 y +2 qb ; p A 2 = 2 y+2 (1, qa ); p B 2 = 2 y (1, qb ): Because A puts higher probability weight on state! 1 on which A has a lower consumption, the state-contingent claim on! 1 has a relative higher value for A. Since p A 1 >pb 1 and p A 2 <p B 2,type A agents and type B agents can benet from trade. Type A agents will sell to type B agents state contingent claims on! 2 consumption goods and buy state contingent claims on! 1 consumption goods from type B agents. Observation : Type A and type B agents will trade state contingent claims. With this observation, we are ready to study the equilibrium of this economy. 12

13 3.1.2 Equilibrium Consumption Allocations and Prices As long as a type A agent has a higher consumption level in state! 2 than in state! 1, his uncertainty adjusted probability measure on state! 1, i.e., q A, will remain (1, A )p + A. Similarly, as long as a type B agent has a higher level of consumption in state! 1 than in state! 2, q B remains (1, B )p. Therefore we have (p A 1 =pa 2 ) > (pb 1 =pb 2 ) as long as ca 1 <ca 2 and c B 1 >c B 2. Hence type A and B will continue to trade until both types have constant consumption across states, i.e., c i 1 = c i 2 for i = A; B. Table 2 parameterizes the equilibrium consumption allocation with a parameter a, which can be pinned down by agents' budget constraints. t =0 t =1! 1! 2 y+ A a a y+ a 2 2 y+ B 2, a (2, a) y+ (2, a) 2 2 Table 2: Equilibrium Consumption Allocation We nowcharacterize the equilibrium prices, and then solve a. To obtain the equilibrium prices from agents' rst order conditions, we rst need to nd out the sets of uncertainty adjusted measures. Since an agent's consumption is constant across states, the agent is indierent among the occurrence of dierent states. In this case, the set of uncertainty adjusted measures is no longer a singleton; it is equal to the set of priors, i.e., Q i = P i. Denote by p j the equilibrium price of a state contingent claim at! j for j =1; 2. The rst order conditions under a maxmin expected utility with a non-singleton set of uncertainty adjusted measure have the following form 5 min 2Q i u 0 (c i t=1 (! j))(! j ) u 0 (c i t=0 ) p j max 2Q i u 0 (c i t=1 (! j))(! j ) u 0 (c i t=0 ) i = A; B; j =1; 2 (7) 5 For the derivation of the rst order condition under maxmin expected utility, see Epstein and Wang (1994). 13

14 p 1 + p 2 = 2 y+ : (8) There are two features in the Euler equations for the maxmin expected utility. One is the inequalities. The other is the maximization and minimization on the left and right hand sides of the inequalities, and they are taken over the set of uncertainty adjusted measures Q i. If the set of uncertainty adjusted measures were singleton, then the max and min would be equal, and the euler equations would hold as equalities, just like the usual formula we got for the expected utility, except they would be evaluated under the uncertainty adjusted measure. With Q i being non-singleton, the maximum and minimum are not equal, and the Euler equations hold as inequalities. The inequalities demonstrate that Knightian uncertain induces kinks in the indierence curves, and these kinks occur along the certainty line, as shown in Figure 1.1. It shows that agent's strong dislike of consumption variation across states, as suggested by the rst order risk aversion property introduced by Knightian uncertainty. Using equations (5), (6), (7) and (8), and table 2, we get 2q i y+ p 1 2qi y+ 2(1,q i ) p y+ 2 2(1,qi ) y+ ; i = A; B ; (9) ; i = A; B ; (10) and p 1 + p 2 = 2 y+ : (11) These are the necessary and sucient conditions for the equilibrium prices. From equations (9) and (10), because i > 0 if and only if q i < q i for i = A; B, price indeterminacy occurs if both A and B are positive. The price indeterminacy implies asset prices can uctuate even when there is no uncertainty or uctuations in aggregate resources. This implication on the volatility of prices is consistent with the observation that high frequency stock price changes occur even though there are no changes in the economic fundamentals, which are 14

15 expected to occur mostly in low frequencies. With a smooth consumption allocation, type A agents are in an inaction position for any prices computed using the set of uncertainty adjusted probability distributions Q A, and the same analysis applies to type B agents, so when we take anintersection of these two sets of prices, we get the equilibrium prices. But taking the intersection of the two sets of prices is equivalent to rst taking the intersection of Q A and Q B, and then computing the price range using the intersection. That is, the range of indeterminacy of equilibrium state contingent claim prices will be determined by the intersection of the Q A and Q B. Without loss of generality, we assume A B. This assumption implies Q B Q A, and the range of the equilibrium price indeterminacy hinges only on the size of Q B. Equations (9) and (10) can be simplied into 2q B y + p 1 2(1, q B ) y + p 2 2(1, qb ) y + 2qB y + ; (12) (13) The following proposition establishes a relationship between agents' dierent degrees of uncertainty and price indeterminacy. Proposition 3.1 Given the agents' endowments specied by table 1, preferences and uncertainty dened inequations (4) and (6), and complete markets, if both types of agents have Knightian uncertainty, there will be acontinuum of equilibria. Furthermore, the degree of indeterminacy in the equilibrium prices is determined by agents with the least Knightian uncertainty. In the case where B is zero, i.e. type B agents have no uncertainty, Q B is equal to the probability distribution (p; 1, p), and the equilibrium prices and consumption ratio are uniquely determined regardless of how much Knightian uncertainty type A agents have. In fact, in this case the equilibrium coincides with a rational expectations equilibrium in 15

16 which is the true probability distribution. The result that if there is one type of agents with no Knightian uncertainty, then there is no price indeterminacy is analogous to the result that in an economy without Knightian uncertainty if there is one type of agents who are risk neutral, then there will be no risk premium in equilibrium. In table 2, a is a parameter to be determined. The parameter a is related to the ratio of each type of agent's value of endowments. To solve a, we equate each agent's value of endowments with the value of consumption. After some calculations, we obtain a relation between a and p 1, p 2. a = 2k 1+k ; where k = 1+p 1y=2+p 2 (y=2+) 1+p 2 y=2+p 1 (y=2+) : (14) When there is price indeterminacy, i.e., p 1 and p 2 are not uniquely pinned down by the equilibrium condition, indeterminacy occurs for the value of a also. Using equations (9), (10), (11) and (14), we can show the indeterminacy range of a is an interval. To see how the share of individual endowment processes aects price indeterminacy and the value of a, we can change type A and type B agent's endowment att = 0 in such a way so that the aggregate endowment at period 0 remains the same as before. It can be easily veried that this change has no eect on the equilibrium prices. It only aects the center point of the indeterminacy range for a. The following theorem summarizes the equilibria of this two-period economy. Theorem 3.2 Given the agents' endowment specied by table 1, preferences and uncertainty by equations (4) and (6), and complete markets, the equilibria of this economy have the following properties: 1. The necessary and sucient conditions for the equilibrium prices are equations (9), (10), and (11). Price indeterminacy occurs if all agents have Knightian uncertainty. 2. For a given equilibrium price, consumption allocation is given by table 3, where the 16

17 value of the consumption share, a, can be computed from (14). All agents have constant consumption across two states at period The values of the endowment, and thus the consumption ratios between the types of agents depend on the prices in the particular equilibrium. Indeterminacy in prices results in indeterminacy in the wealth distribution and consumption ratios. 3.2 An Innite-Horizon Model This section generalizes the previous two-period economy in several aspects while maintaining the assumption of a constant aggregate endowment. In particular, we allow there to be more than two types of agents in the economy,we relax the logarithmic utility assumption, and we extend the two-period setup into an innite horizon setting. As we will see, all the results we derived from the two-period model hold in this general setting. Consider an exchange economy with a single perishable consumption good. The economy is populated with heterogeneous agents who have Knightian uncertainty about the underlying economic environment. There are N types of agents and there are a continuum of each type of agents with unit mass. Each type of agent has a dierent degree of Knightian uncertainty. Aggregate endowment is constant, while each agent's endowments are subjected to idiosyncratic shocks. Let f! t g be a rst order Markov process that governs the distribution of aggregate endowments among individuals, and be the state space of the Markov process. Denote by e i t atype i agent's period t endowment, which is a function of! t : e i t(! t ). Thus, we have that NX i=1 e i t(! t )=y 8! t 2 ; 8 t: (15) Each period, an agent' non-unique priors about next period can be characterized by an \-contamination" correspondence. We allow agents' reference transition probability 17

18 matrices to dier. Denote by i the reference transition probability matrix of agent i, and by i! t the reference transition probability distribution conditional on state! t. The belief mappings of a type i agent at period t conditional on state! t are P i ;! t = f(1, i ) i! t + i m : m 2Mg (16) where M is the collection of all transition probability measures on. We arrange agents' types in such away that agents of type 1 are the ones with the highest value, and agents of type N have the lowest value : 1 > 1 2 ::: N 0: Assume the intersection of all agents' priors is non-empty for all periods and states. \ fi=1;2;::;ng P i ;! t 6= ; for all t and all states (17) Let c i t denote a type i agent's consumption at period t, and c t;i denote a type i agent's consumption starting at t, c t;i = fc i t ;ci t+1 ;ci t+2 ; g. Let ct = fc t;1 ;c t;2 ; :::; c t;n g. The preferences of a type i agent are described by a recursive utility function U(c t;i ;! t ) = u(c i t(! t )) + U(c t+1;i ;! t+1 )dp i ;! t ; (18) where u is strictly increasing, strictly concave and twice dierentiable, is a discount factor, and the integration over a set of measures P i ;! t should be interpreted as the operation described in section 2, i.e., taking the minimum of the expectation values computed under the measures in the set P i ;! t. There are complete state-contingent-claim markets. 18

19 We can use the same argument as in the two-period model to verify the following properties of equilibrium consumption allocations for this economy. Proposition 3.3 Individual consumption is constant for all states and dates, i.e., For all i;! t 2 ; and t; c i t(! t )=c i : Proof : See the appendix. This proposition generalizes the results from the two-period model in section 3.1 to a setting in which we allow for N types of agents who live innitely long, and no special assumption about the form of utility function is needed other than the recursive property. A similar result also shows up in Chateauneuf, Dana and Tallon (1997) in a general but static setup. Equilibrium consumption for each agent is completely smoothed across time and states. All idiosyncratic risk is diversied. This result is in contrast to the results from models with private or incomplete information(see, for example, Atkeson and Lucas(1992), Thomas and Worrall(1990), and Phelan and Townsend(1991)), and from models without commitment technology (see, for example, Kocherlakota(1996)). Incomplete information models and models without commitment technology generally can have constrained optimal allocations in the sense that idiosyncratic risk in individual consumption is not completely diversied. This model shows that the presence of Knightian uncertainty does not prevent agents from risk diversication. Knightian uncertainty does, however, result in multiple equilibria, as we see next Equilibrium Prices and Consumption Shares Let p t;!t (!) be the price, in units of period-t consumption goods, of a state-contingent claim for a unit of consumption good if state! occurs at t+1 given the state at period t is! t, and let p t;!t = fp t;!t (!)g!2, and p t = fp t;!t g!t2. Given individuals' endowment sequence, we want to nd a price sequence fp t g 1 t=0 that supports the consumption allocation in a competitive equilibrium. Because equilibrium consumption is constant across states for 19

20 every agent, the sets of uncertainty adjusted measures are equal to the sets of priors. From agents' Euler equations and the fact that individuals' consumption is constant across both states and time, we have min q i (!) p t;!t (!) max q i (!); 8 i; 8! 2 ; (19) q i 2P i ;!t q i 2P i ;!t X!2 p t;!t (!) =: (20) Equations (19) and (20) are necessary and sucient conditions for the determination of equilibrium prices. The assumption that the intersections of sets of priors are non-empty (equation (17)) guarantees the solution for equation (19) exists. Equations (19) and (20) imply that price indeterminacy occurs if the intersection of sets of priors is non-singleton. In the case where agents have common reference transition matrices, price indeterminacy occurs if everyone in the economy has Knightian uncertainty, i.e., i > 0 for all i, or equivalently P i ;! t is non-singleton for all i. Notice that the upper bound and lower bound of the one period state contingent claim prices in general can vary over time, which means the ranges of indeterminacy in the state contingent claim prices can vary over time. Epstein and Wang(1994, pp ) use the aggregation results from a heterogeneous agent economy to suggest that the origin of the price indeterminacy in an Knightian uncertainty economy may be the analyst's incomplete information about individual endowments. This model shows that even if an analyst has complete information about individual endowments, price indeterminacy can still exist. On the other hand, if there are agents in this economy who have no Knightian uncertainty, then equilibrium prices will be uniquely determined. This is because for those agents without Knightian uncertainty, their rst order conditions hold as equalities, and that will pin down all prices. Equation (20) also implies that one period risk-free rates are constant and equal 1= for all t. 20

21 Given an equilibrium price sequence, the corresponding consumption shares are determined from agents' present value budget constraints. As in the two period model, dierent equilibrium price sequences can result in dierent valuations of individual's endowments, and consequently dierent equilibrium consumption shares among agents. On the other hand, for a given consumption allocation, there may exist more than one price sequence that supports the allocation in a competitive equilibrium. The following theorem summarizes the equilibria of this economy. Theorem 3.4 Given agents' preferences and uncertainty specied by equations (16), (17) and (18), agents' endowment sequence as in equation (15), and complete markets, the equilibria of this economy has the following characteristics: 1. Equilibrium consumption for each agent is constant across time and states. 2. Price indeterminacy occurs if all agents in the economy have Knightian uncertainty. 3. For a given equilibrium price sequence fp t g 1 t=0, the equilibrium consumption allocation is uniquely determined. 4. Given a consumption allocation that can be supported in a competitive equilibrium, there may be more than one price sequence that can support it as an equilibrium allocation. 5. When there is price indeterminacy, if the true probability distribution is in all agents' belief sets, then there exists a price selection rule such that the resulting equilibrium coincides with a rational expectations equilibrium. The following results apply to the case in which agents in the economy have common reference measures. Corollary 3.5 If all agents have the same reference transition probability matrix, then 21

22 1. The degree of price indeterminacy is determined by the type of agents who have the least Knightian uncertainty. 2. If there is one type of agent which has rational expectations(i.e. do not have Knightian uncertainty, and the reference probability is the true probability), the equilibrium is unique and coincides with a rational expectations equilibrium. 4 Heterogeneous Agent Economies With Random Aggregate Endowment In this section, we study economies in which the aggregate endowment is random, but there is no idiosyncratic shock. Assume agents have identical reference measures but dierent degrees of Knightian uncertainty. We start from a two-period economy with a continuum of aggregate endowment states. We then look at an innite horizon economy. In the two period setup we show that consumption allocations have a threshold feature and the consumptions of some individuals can be constant over a range of variations in the aggregate endowment. When we extend the model to an innite horizon setting, the threshold feature shows up in the expected utility values. In contrast to the equilibrium of the benchmark rational expectations economy, consumption allocations in the random aggregate endowment economy are history-dependent. We will also show that there is no indeterminacy in the contingent claim prices for the aggregate endowment. 4.1 A Two-Period and Continuum-State Model This section studies a two-period economy with aggregate endowment uncertainty. There are two types of agents in the economy. To highlight the consequences of aggregate endowment uncertainty, we let the two types of agents have identical endowment processes. They 22

23 also have the same log utility functions given by equation (6), but they may dier in their degrees of uncertainty about the probability distributions of the endowment process. The aggregate endowment is normalized to be 1 at period 0. At period 1, there are continuum of aggregate endowment states ranging from a lower bound y to an upper bound y +h with h>0. Agents' endowment streams are shown in table 3. t =0 t =1 Continuous States A 1 y=2 (y + h)=2 B 1 y=2 (y + h)=2 Aggregate 2 y y + h Table 3: Endowment Process : Continuous Endowment States at period t=1 Both types of agents have -contamination beliefs about endowments. The common reference probability measure for the period-one aggregate endowment is assumed to be uniform on the support [y;y + h]. The belief mappings for each type of agents are P i = f(1, i ) + i m : m 2Mg; 0 i 1; i = A; B; (21) where M is the collection of all probability measures on the support. Without loss of generality, we assume type A agents have a higher degree of Knightian uncertainty than type B agents, A > B. To study the equilibrium of this economy, rst notice that since both types of agents have the same endowment stream, they have the same amount ofwealth. The equal wealth and log utility functions imply they will spend the same fraction of wealth on the rst period(t = 0) consumption. This pins down the equilibrium consumption allocation at t = 0 : each agent consumes one unit of consumption good in the rst period (t = 0). We therefore only need to nd the equilibrium consumption allocation at t = 1. Toachieve this, we construct a corresponding planner's problem, and characterize its solution. It can 23

24 be shown that equality inwealth results in equality in the Pareto weights for the two types of agents. Denote by c(s) and s, c(s) respectively the consumption of a type A agent and atype B agent att = 1 when the aggregate endowment iss. The following proposition characterizes the equilibrium consumption allocations. Proposition 4.1 For any ecient allocations (c A ;c B ), there exists, >0, k, 0 <k<1, and c, 0 < c <ysuch that fc A (s) =c(s);c B (s) =s, c(s)g and either 8 >< >: c(s) =a (y + kh) for s 2 [ y;y+ kh ]; c(s) =a s for s 2 [ y + kh; y + h] where a = or c(s) =c for s 2 [ y; y+ h ]: (1, A ) (1, A )+(1, B ) Furthermore, the equilibrium consumption allocation for the economy can be obtained for =1. Proof : See the Appendix. This proposition says that in this economy there will be a threshold in the aggregate endowment states such that agents with more Knightian uncertainty (type A agents) will have constant consumption if the aggregate endowment islower than the threshold. In this case, any aggregate endowment uctuations in the region [y;y + kh] are absorbed by the consumption of agents with a low degree of Knightian uncertainty (type B agents). Figure 2 of section 2 shows the individual consumptions as functions of aggregate endowment. Since a<(1, a), the consumption of a type B agent is higher than that of a type A agent when the aggregate endowment is on the range [y + kh;y + h]. Because of the threshold feature, individual consumptions are only weakly increasing in aggregate endowment. This equilibrium consumption allocation rule contrasts the \full risk-sharing" rule in a corresponding rational expectations economy in which individual consumption is strictly increasing in the aggregate consumption. Notice also how Knightian uncertainty enhances 24

25 agents' incentive to smooth consumption. An agent with more Knightian uncertainty(type A) has relatively smoother consumption than agents with less uncertainty (type B). Because a type A agent has a at consumption prole on the interval [y;y + kh], his uncertainty adjustment measure is not unique. On the other hand, the uncertainty adjustment measure for a type B agent is unique, for his consumption is strictly increasing in the aggregate endowment. Therefore, we can infer the equilibrium price of a security from the type B agent's shadow value for the asset. It can be easily veried that B's uncertainty adjusted measure is uniform on the interval (y;y + h] with hight (1, B )=h and has probability B at the point y. Notice that this uncertainty adjusted measure is stochastically dominated by the reference probability measure. This implies that the resulting stochastic discount factor for asset pricing should be constructed with a stochastically dominated distribution (i.e., the uncertainty adjusted distribution), and not the estimated distribution ( i.e, the reference distribution). In this economy, there is only one shock that governs both individual and aggregate endowments. Therefore, once the prices of state contingent claims on aggregate endowment states are uniquely determined, so do the prices of state contingent claims on individual endowment states. There is no multiple equilibria. Alternatively, we can introduce idiosyncratic shocks to individual endowments. By combining the results from section 3 and from this section, we can infer the properties of the equilibrium in this new economy. For example, individual consumption will not vary with idiosyncratic shocks. The prices of state contingent claims on aggregate endowment states are still uniquely determined once the values of individual endowments are determined, but there will be indeterminacy in the prices of state contingent claims on idiosyncratic shocks, as in the economy studied in section 3, which results in indeterminacy in the distribution of wealth. 25

26 4.1.1 Heterogeneous Reference Probability Measures In the constant aggregate endowment economy, most results extend to the setting in which agents have heterogeneous reference measures, provided the intersection of agents' sets of priors is non-empty. This section addresses the same issue in the context of random aggregate endowments. It is known that in a random endowment economy populated by agents with heterogeneous beliefs but without Knightian uncertainty, the equilibrium allocation rule in general does not imply that individual consumption is monotonic in aggregate consumption. Therefore, when we extend the current random endowment economyby allowing agents to have heterogeneous reference measures, we do not expect monotonicity of individual consumption to hold, even with the condition that the intersection of agents' sets of priors is non-empty. To illustrate, note that both A's and B's uncertainty adjusted measures have the property that in the region [y + kh;y + h], they are equal to the reference measures scaled by (1, i ). Therefore we can construct an example that results in the non-monotonicity in the consumption allocation by changing A's and B's reference measures on the [y + kh;y + h] region such that one agent's consumption is decreasing for some region. As long as each agent's resulting equilibrium consumption on the region [y + kh;y + h] is higher than the agent's lowest consumption levels on the worst aggregate endowment state y, the uncertainty adjusted measures on the region [y + kh;y + h] remains equal to the reference measures scaled by (1, i ). This can be achieved without violating the condition on the non-emptiness of the intersection of agents' sets of priors. 4.2 An Innite-Horizon Model This section extends the previous two-period economy to an innite horizon setting. The aim is to investigate how the threshold feature in the consumption allocation obtained in the two-period model generalizes to the longer horizon setting. There are two types (A and B) of agents in the economy. Each period, both types of 26

27 agents' beliefs about the next period endowments follow an -contamination form with a common reference probability measure which is iid and has support equal to Y =[y;y+h]. Agents may have dierent degrees of -contamination. P i = f(1, i ) + i m : m 2Mg i = A; B (22) where M is the collection of all probability measures on Y. Agents have recursive utility functions as specied in equation (18). We will use a social planner's maximization problem to solve for the Pareto optimal allocations, and then nd prices that support the ecient allocations in a competitive equilibrium A Dynamic Program Denote by i the Pareto weight for a type i agent, and let = ( A ; B ). Normalize the Pareto weights such that they sum up to one. Denote by 2 the two dimensional unit simplex: 2 = f( A ; B ) 2 R 2 + : i > 0 for i = A; B; A + B =1g: Then 2 2. The planner's maximization problem is to nd a sequence of state-contingent consumption allocations, subject to the resource constraint, such that the sum of the values of future discount utility,weighted by a given set of Pareto weights, are maximized. Denote by V (;y) the value function for the planner's maximization problem. Given a set of Pareto weights, we have V (;y) = max c t X i=a;b i fu(c i t (y)) + s:t: c A t (y)+c B t (y) y U(c t+1;i ;y t+1 )dp ig c A t+l + c B t+l y t+l 8 l>0: 27

28 In light of the recursive structure of the utility function in equation (18), we adopt the method developed by Lucas and Stokey (1984) and Kan (1995) to solve Pareto-optimal allocations for this heterogeneous agent economy. The basic idea of this approach is that instead of directly solving a sequence of consumption, which is a complex task in our context, we solve the program recursively via the device of adding Pareto weights as one of the state variables for the optimization problem. The Pareto weights in this economy are time-varying and state-dependent. Though out the paper, variables with a tilde denote random variables that will realize next period. For example, ~ A (~y) denotes a type A agent's next period Pareto weight which is a function of next period's states. Following Lucas and Stokey (1984), the planner's problem can be formulated by the following recursive program : V (;y) = max fc i ;fz i (~y)g ~y2y g i=a;b X i=a;b u(c i i )+ z i (~y)dp i s:t: c A + c B y (23) min [V ( ; ~ ~y), X ~(~y)2 2 i=a;b ~ i z i (~y)] 0; 8 ~y 2 Y (24) Given a set of Pareto weights, instead of solving a sequence of state-contingent consumption allocations, the program solves a one period optimal consumption allocation fc i g i=a;b, and a set of state-contingent next period utilityentitlement fz i (~y)g i=a;b, which is the statedepend discounted expected-utility-from-next-period-on. Equation (23) says current period consumption allocation is constrained by the aggregate resource available this period, and equation (24) puts a restriction on the feasible set of state-contingent entitlement of utilityfrom-next-period-on. The solutions ~ to the minimization problems in equation (24) are functions of next period's state ~y and this period's Pareto weights. As shown by Lucas and Stokey (1984), ~(; ~y) will be the set of state-contingent Pareto weights for the nextperiod's recursive program if state ~y is realized next period. Since ~ depends on ~y and, 28

29 it implies the Pareto weights vary with time and states. The following lemma establishes a property of the function V which will be useful in solving the maximization problem. Lemma 4.2 V (;y) is strictly convex in and strictly increasing in y. Proof : See the Appendix. Lemma 4.2 implies that for each and y, the program has a unique solution for the current consumption allocation fc i g and state-contingent entitlement of utility-from-next-periodon fz i (~y)g ~y2y, and that for each ~y 2 Y and each value of z i (~y), the minimization problem described in equation (24) has a unique solution for (~y), ~ as a function of. To solve the maximization problem, we rst form a Lagrangian. L = X i=a;b u(c i i )+ z i (~y)dp i + fy, c A, c B g X + (~y)f V ( ; ~ ~y), Y i=a;b ~z i (~y)+(~y)[1, X i=a;b ~ i (~y)] gd~y where, (~y) and (~y) are Lagrangian multipliers. Denote by Q i the set of uncertainty adjusted measures of agent i, i.e., a subset of P i that realizes the minimum of the expectation of a type i agent's entitlement of the sum-of-discounted-utility-from-next-period-on, Q i = 2P i j z i (~y)d = z i (~y)dp i (25) where fz i (~y)g ~y2y are from the solutions to the maximization problem. Since the solutions for agent i's next-period state-contingent utility entitlement fz i (~y)g ~y2y depend on this period's Pareto weights, the set Q i Euler Equations in general will be a function of also. 29