Abstract This paper studies decentralized economies populated with households who have heterogeneous degrees of Knightian uncertainty about endowments

Transcription

1 Heterogeneous Agent Economies With Knightian Uncertainty Wen-Fang Liu 1 Department of Economics University of Washington Box , Seattle, WA liuwf@u.washington.edu Phone: ; Fax: This paper is a revision of my PhD dissertation at the University of Chicago. I am grateful to Robert Lucas, Rose-Anne Dana, Andreas Lehnert, Tim Conley, Tom Sargent, and Bill Dupor for valuable comments and criticism on earlier drafts, and especially to Lars Hansen for guidance, encouragement and constant support. 1

2 Abstract This paper studies decentralized economies populated with households who have heterogeneous degrees of Knightian uncertainty about endowments. We investigate how equilibrium prices and consumption allocations are altered by the presence of heterogeneous Knightian uncertainty. It is known from previous work that models with Knightian uncertainty can have indeterminate security prices. In a multi-agent setup, we show the range of the indeterminacy is determined by the households with the least amount of Knightian uncertainty. In contrast to models with `full risksharing', in a two-period 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 infinite-horizon setting, the threshold feature appears in the expected utilityvalues. Furthermore, equilibrium consumption allocations in the random aggregate endowment economy are history-dependent. JEL: D50, D80, E21, G12 2

3 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 their probability assessments. By contrast, this paper studies decentralized economies in which households do not have full knowledge of the underlying probabilities. 2 This ambiguity is referred to as Knightian uncertainty. We extend Epstein and Wang's (1994) representative agent model by considering household heterogeneity. Unlike rational expectations models, in which representative agent and heterogeneous agent models with complete markets are quite similar, with Knightian uncertainty, agent heterogeneity directly affects equilibrium outcomes. 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 an investor will neither buy nor sell a risky security for a range of prices. 3 As emphasized by Liu (2002), 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 Zeldes (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 (20002) shows that the presence of Knightian uncertainty in labor income prompts households to save more as a form of insurance for future contingencies. We consider economies in which some households have a greater degree of uncertainty than other. We characterize the effects of this heterogeneity on equilibrium consumption allocations and asset prices. For simplicity we adopt a one-parameter formulation of 2 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). 3 See also Bewley (1987). 3

4 Knightian uncertainty borrowed from the literature on robust statistics. We use the socalled ffl-contamination model in which the known distribution is replaced by an `arbitrary distribution' with probability ffl. We assume households are uncertainty averse, i.e., they do not like thepresence of Knightian uncertainty. This leads us to model decision-makers as guarding against the `worst case' among the family of ffl-contaminated probability distributions. We use this ffl-contamination formulation to show how equilibrium consumption allocations differ from those in a benchmark rational expectations model. By introducing heterogeneity in the contamination parameters, we are able to better understand the nature of the price indeterminacy known to exist in models with Knightian uncertainty (Epstein and Wang, 1994). As shown by Epstein and Wang (1994) and Melino and Epstein (1995), price indeterminacy is a likely consequence in models with Knightian uncertainty. When price indeterminacy occurs in these models, there is a continuum of equilibrium prices. This paper shows that the degree of equilibrium price indeterminacy is determined by thetype of consumers who have the least Knightian uncertainty. In contrast to models with `full risk-sharing', in a two-period setup we derive that 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 infinite-horizon setting, the threshold feature appears in the expected utility values. Furthermore, consumption allocations in the random aggregate endowment economy are history-dependent. The rest of this paper is organized as follows. Section 2 lays down the formulation of Knightian uncertainty. Section 3 studies two-period economies. Section 4 extends the analysis to an infinite horizon setting. Section 5 concludes. 4

5 2 Modeling Knightian Uncertainty In a model without Knightian uncertainty, a household's beliefs about the probability lawof the stochastic economic environment are specified by a unique prior. In contrast, to model Knightian uncertainty and uncertainty aversion we allow a household's beliefs to be a set of probability distributions. We formulate a household's set of priors by the ffl-contamination parameterization. ffl-contamination Beliefs Assume the household's beliefs P ffl have an ffl-contamination form: P ffl = f(1 ffl)ß + fflm : m 2Mg (1) where ffl is a constant in the unit interval [0, 1], ß is a reference probability distribution, and M is the collection of all possible probability measures on the support of ß. The interpretation for the ffl-contamination beliefs is the following. A household uses its information and knowledge about the stochastic environment and forms an estimate for the true probability law. This estimate is called the reference probability distribution. Unsure about the reference probability distribution being the true one, the household postulates that with probability (1-ffl), the true distribution is ß, and with probability ffl it is drawn randomly at an unknown distribution from the set M. 4 Roughly speaking, we can view P ffl as a collection of ffl perturbations of the reference measure ß. Note that when ffl is greater than zero, the belief set P ffl is non-singleton if Mis not the empty set. For a given ß, the larger the ffl value, the larger the set of priors. When ffl equals 1, P ffl equals M. In this case, the household is completely ignorant about the probability distribution. When ffl approaches zero, the belief set P ffl reduces to a singleton that contains only the reference probability measure ß, i.e., P ffl=0 = fßg. We assume households have the same reference probability distribution. Heterogeneity 4 See Huber (1981). 5

6 in households' beliefs is modeled through different ffl values. We can interpret this homogeneity of the reference probability measure as households 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. When the values of ffl's approach zero, the model serves as a robustness check of the benchmark rational expectations model. Throughout the paper, we assume households are uncertainty averse and have the same amount of uncertainty aversion. However, we allow heterogeneity in households' degrees of Knightian uncertainty. A household is said to have a higher degree of Knightian uncertainty than the other household if the household's set of priors is a superset of the other household's set of priors. 5 Consider two households A and B with ffl-contamination beliefs and with the same reference probabilities but different ffl values. Since P B ffl P A ffl if and only if ffl A ffl B, A has more Knightian uncertainty than B if and only if ffl A ffl B. Hence we use the ffl value to index a household's degree of uncertainty about the probability distribution when households have an identical reference measure. Generalized Expected Utility with Knightian Uncertainty There are several approaches to formulating decision-making under Knightian uncertainty (see, for example, Bewley (1986), and Schmeidler (1987)). This paper adopts the maxmin expected utility theory of Gilboa and Schmeidler (1989), and Epstein and Wang (1994). To compute the expected utility" under non-unique priors, first compute the expected utility for each probability distribution in the set of priors, and then take the minimum over these values: E Pffl (u) min μ2p ffl fe μ (u)g : (2) 5 Strictly speaking, both the degree of uncertainty aversion and the amount ofknightian uncertainty affect the size of the set of priors. With the assumption of homogeneity in degrees of uncertainty aversion among households, we can use the size of the set of priors as a measure of the degree of Knightian uncertainty. 6

7 This minimization operator reflects the household's uncertainty-averse preferences: the household is guarding against the worst possible cases. Loosely speaking, the larger the set P ffl, the more uncertain the household and, thus, the smaller the value of expected utility". Note that when P ffl is singleton, i.e., the household has a unique prior, equation (2) reduces to the usual expected utility definition. 6 : With ffl-contamination beliefs, the calculation of the generalized expected utility becomes E Pffl (u) =(1 ffl)e ß (u)+fflmin u: (3) Ω where Ω is the collection of states. As it will be clear later, this simple form facilitates our analysis. Throughout the paper, we consider economies populated with N types of households, indexed by i =1; 2;:::;N. These N types of households may differ in their endowments and the degrees of Knightian uncertainty ffl i. Assume there is a continuum of each type with unit mass. Denote by ffl i the degree of Knightian uncertainty ofatype i household. Let y t be the aggregate endowment at period t. Let s i t be the endowment share of a type i household at t so that the type i household's period t endowment is equal to s i ty t. Individual endowment shares are strictly positive and sum up to one. We will study two-period economies in section 3, and then extend the model to an infinite horizon environment in section 4. 3 Two-period Economy Section 3.1 investigates a constant aggregate endowment economy where y t is degenerated while s i t is random. Section 3.2 turns to a random endowment economy where s i t is degenerated while y t is random. 6 An interesting feature of Knightian uncertainty is that preferences with uncertainty-aversion are observationally equivalent to preferences with a pessimistic character. This pessimistic characteristic is also found in the risk-sensitive control theory (see, for example, Hansen, Sargent and Tallarini (1999)), and the construction of robust decision rules (see Huber (1981)). 7

8 3.1 Constant Aggregate Endowment Economy We start with the simplest case where there are two types of households (N = 2): type A and type B. These two types differ both in their endowments and in the degrees of Knightian uncertainty. There is no aggregate endowment uncertainty, i.e. y t is constant. The source of uncertainty comes from households' period 2 endowment shares. At period 1, each type is endowed with half of the aggregate endowment. We normalize the period 1 aggregate endowment to be 1. At period 2, the aggregate endowment is a constant y. There are two states for individual endowment share:! 1 and! 2. Since there are only two types of households, we simplify the notation by denoting type A household's endowment share at period 2 by s and type B household's by (1 s). For a type A household, state! 1 is the low endowment state with endowment s L y, while state! 2 is the high endowment state with endowment s H y, where 0 <s L <s H < 1. A type B household is endowed with (1 s L )y at! 1, and (1 s H )y at! 2. Table 1 summaries the endowment processes. t =1 t =2! 1! 2 Type A 1/2 s L y s H y Type B 1/2 (1 s L )y (1 s H )y Aggregate 1 y y Table 1: Endowment Process Both types of households have ffl-contamination beliefs with common reference probabilities ß s =(p; 1 p) for states! 1 and! 2. A household's beliefs are: P i ffl = f(1 ffl i )ß s + ffl i m : m 2Mg; 0» ffl i» 1; i = A; B; (4) where M is the collection of all probability distributions on these two states. Let q i and q i be the maximal and minimal probabilities in Pffl i for state! 1. Then q i =(1 ffl i )p + ffl i (5) q i =(1 ffl i )p (6) 8

9 Given (5) and (6), equation (4) can be written as P i ffl = f(q; 1 q) : q i» q» q i g for i = A; B: (7) Assume that both types of households have log utility functions: U i (c i 1;c i 2)=lnc i 1 + fie P i ffl (ln c i 2); i = A; B: (8) where c i t is the consumption of a type i household at period t for t = 1; 2. We set fi = 1 for simplicity. There are complete markets in the sense that households can trade all the state-contingent claims. To see how the presence of Knightian uncertainty motivates trade between households, we first compute each household's valuations of state-contingent claims under autarky. Autarky Define a type i household's set of uncertainty adjusted probability distributions, Q i, to be the collection of probability distributions in P i ffl that minimizes the household's expected utility level for a given consumption plan: Q i (c i ) fν 2PffljE i ν (u(c i )) = min E μ (u(c i ))g: μ2pffl i For household i, Q i represents the set of probability measures associated with the worst case scenario given that the household has chosen a particular consumption plan. These probability measures, as in Epstein and Wang (1994), will enter into the household's first order conditions in the utility maximization problem. Under autarky, each household consumes its own endowment. Because a type A household has a low consumption level at! 1, and a type B household has a low consumption level at! 2, the sets of uncertainty adjusted probability distributions for A and for B are each singleton and equal to the distributions that maximizes the probabilities of the low 9

10 consumption states. Q A = f(q A ; 1 q A )g; and Q B = f(q B ; 1 q B )g: Here q A and q B are defined in equations (5) and (6). Notice that q A q B and (1 q A )» (1 q B ) for any ffl A and ffl B in the interval [0,1]; Knightian uncertainty causes an uncertainty averse household to weight the low consumption state more. The households' valuation of contingent claims in terms of period 1 consumption is equal to the marginal rates of substitution weighted by the uncertainty adjusted probabilities. Denote by p i 1 and p i 2 a type i household's valuation of contingent claims on states! 1 and! 2. With the uncertainty adjusted measures and the log utility function, we obtain p i 1 and p i 2 under autarky: p A 1 = 1 2s L y qa ; p B 1 = p A 2 = 1 2s H y (1 qa ); p B 2 = 1 2(1 s L )y qb ; and 1 2(1 s H )y (1 qb ): Because of the lower consumption at state! 1 compared at state! 2, type A has a higher marginal utility and applies a higher probabilityweight μq A at! 1, which results in A's higher valuation for the contingent claim on! 1. The opposite is true for type B. It is straightforward to show that p A 1 =p A 2 > p B 1 =p B 2, which implies both types can benefit from trade: type A will buy consumption at state! 1 from type B and sell to type B consumption at state! 2. This simple example illustrates that in response to the presence of Knightian uncertainty households apply higher probability weights on bad states, and thus generate a stronger incentive to smooth consumption across states. Equilibrium Consumption Allocations and Prices As long as a type A household has a higher consumption at state! 2 than state! 1,type A's 10

11 uncertainty adjusted probabilities remain (μq A ; 1 μq A ). Similarly, as long as a type B household has a higher consumption at state! 2 than state! 1, type B's uncertainty adjusted probabilities remain (q B ; 1 q B ) Both together imply the inequality (p A 1 =p A 2 ) > (p B 1 =p B 2 ) holds. As a result, types A and B will continue to trade until both types have smooth consumption across states. Table 2 characterizes the resulting equilibrium consumption allocation with a parameter a, which can be pinned down by households' budget constraints. 7 t =0 t =1! 1! 2 type A a a y a y type B (1 a) (1 a)y (1 a)y aggregate 1 y y Table 2: Equilibrium Consumption Allocation We can obtain the equilibrium prices from households' first order conditions. Denote by p j the equilibrium price of aunitof consumption contingent on state! j. The household's first order conditions under a maxmin expected utility have the following form. 8 min μ2q i u0 (c i t=2 (! j ))μ(! j ) u 0 (c i t=1 )» p j» max μ2q i u0 (c i t=2 (! j ))μ(! j ) u 0 (c i t=1 ) i = A; B; j =1; 2 (10) p 1 + p 2 = 1 y : (11) The Euler equations for the maxmin expected utility exhibits two features: one is the inequality form; the other is the minimization and maximization on the two sides of the inequalities, taken over the set of uncertainty adjusted probability distributions Q i. If Q i 7 The parameter a in table 2, which determines the consumption ratio between households, is solved by equating each household's total expenditure with the value of endowment. a = 1 2» 1 2 +(p 1s L + p 2 s H )y : (9) 8 For the derivation of the first order conditions under the maxmin expected utility, see Epstein and Wang (1994). 11

12 is singleton, then the max and the min are equal, and the first order conditions hold as equalities, just as in the formula from the standard expected utility maximization. With Q i being non-singleton, which occurs especially when consumption (or the utility) is constant across states, the maximum and minimum are not equal, and the first order conditions hold as inequalities. The inequality on the first order condition demonstrates how Knightian uncertainty introduces a mechanism on the preferences that imitates first 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 return (See Liu (2002)). As shown by Segal and Spivak (1990), a property of preferences with first order risk aversion is that there are kinks on the indifference curves along the certainty equivalence line (fortyfive degree line). See Figure 1. This is in contrast to preferences of a smooth utility function consumption at state degree consumption at state 1 Figure 1: Indifference Curve for Preferences With First Order Risk Aversion under the expected utility theory, in which indifference curves are smooth everywhere and the risk premium is proportional to the variance of the asset return (second order risk aversion). The inequalities on the first order conditions produce kinks in the indifference curves along the certainty equivalence line. It shows household's enhanced dislike (first 12

13 order aversion) of consumption variations across states induced by Knightian uncertainty. Given that the equilibrium consumption is constant across states, the set of uncertainty adjusted distributions is equal to the set of belief: Q i = P i ffl. Using the log utility function and table 2, the first order conditions (10) and (11) become: q i y» p 1» qi ; i = A; B ; (12) y (1 q i ) y» p 2» (1 qi ) y ; i = A; B (13) p 1 + p 2 = 1 y : (14) These are the necessary and sufficient conditions for the prices to constitute an equilibrium. Because ffl i > 0 if and only if q i < q i, equations (12) and (13) imply that price indeterminacy occurs if both ffl A and ffl B are positive. The price indeterminacy implies asset prices can fluctuate even when there are no changes 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. When p 1 and p 2 are not uniquely pinned down by the equilibrium conditions, indeterminacy occurs for the value of a also. Using equations (12), (13), (14) and (9), we can show the indeterminacy range of a is an interval. 9 When we take intersections of each pricing equation from types A and B, we get the range for the equilibrium prices. But taking the intersection of the pricing equations from types A and B is equivalent to computing the price ranges using the intersection of Q A and Q B. Without loss of generality, we assume ffl A ffl B, which implies Q B ρ Q A. Thus, the range of the equilibrium price indeterminacy hinges only on the size of Q B,andequations 9 To see how the share of individual endowment processes affects price indeterminacy and the value of a, we canchange type A and type B household's endowment at period 1 in such away so that the aggregate endowment at period 1 remains the same as before. It can be easily verified that this change has no effect on the equilibrium prices. It only affects the center point of the indeterminacy range for a. 13

14 (12) and (13) can be simplified into q B y» p 1» qb y ; (15) (1 q B ) y» p 2» (1 qb ) y (16) This shows that the degree of price indeterminacy is determined by households with the least Knightian uncertainty. In the case where ffl B is zero, i.e. type B households have no uncertainty, Q B is equal to the reference probability distribution ß s = (p; 1 p), and the equilibrium prices and consumption ratio are uniquely determined regardless of how much Knightian uncertainty type A households have. In fact, in this case the equilibrium coincides with a rational expectations equilibrium if ß s happens to be the true probability distribution. The result that if there are households with no Knightian uncertainty, then there is no price indeterminacy" is analogous to the result that in an economy without Knightian uncertainty if there are risk neutral households, then there is no risk premium". In the following theorem, we summarize the indeterminacy property of this two-period economy. Theorem 3.1 Price indeterminacy occurs if all households have Knightian uncertainty. Furthermore, the degree of indeterminacy in the equilibrium prices is determined by households with the least Knightian uncertainty. 3.2 Random Aggregate Endowment Economy In this section, we study an economy with random aggregate endowment. In particular, we investigate how variations in aggregate endowment affect the equilibrium consumption allocation when households have different degrees of Knightian uncertainty. Consider an economy which is similar to the previous economy except now the period 2 aggregate endowment is random. To highlight the consequences of the aggregate endowment uncertainty, we assume the two types of households have identical endowment 14

15 streams: each type has half of the aggregate endowment: s i t =1=2 for i = A; B and for all t. Aggregate endowment at period one, y 1, is normalized to be 1. To simplify the notation, we will drop the time index for the period 2 aggregate endowment sothaty y 2, and now y is a random variable. Assume there are continuous aggregate endowment states, ranging from a lower bound y L to an upper bound y H. The endowment processes are shown in table 3. t =1 t =2 Continuous States type A 1/2 y L =2 ο y H =2 type B 1/2 y L =2 ο y H =2 Aggregate 1 y L ο y H Table 3: Endowment Process : Continuous Endowment States at period 2 A type i household's beliefs about the period 2 endowment process is: P i ffl = f(1 ffl i )ß y + ffl i m : m 2Mg; 0» ffl i» 1; i = A; B; (17) where M is the collection of all probability distributions, and ß y is the common reference probability distribution assumed to be uniform on [y L ;y H ]. Without loss of generality, we assume type A households have more Knightian uncertainty than type B households: ffl A >ffl B. To study the equilibrium of this economy, first note that since both types of households have the same endowment stream, they have the same amount of wealth. The equal wealth property together with log utility functions means that they will spend the same amount onperiod 1 consumption. This pins down the equilibrium consumption allocation at t = 1 : each household consumes 1=2 units of consumption. To find the equilibrium consumption allocation at t = 2. we first characterize the efficient allocations by looking at the social planner's maximization problem. We then find the particular Pareto weights in the planner's problem that correspond to the competitive equilibrium for the economy. 15

16 In the economy, equal wealth among the two types of households results in equal Pareto weights. The following proposition summarizes the equilibrium consumption allocation for the economy. The proof is in the appendix. Proposition 3.2 There exists 0 < k» 1, and 0 < μc < y L such that the equilibrium consumption allocation for the economy in the second period satisfies fc A + c B = yg and either ρ ca (y) =a (y L + kh) for y 2 [ y L ; y L + kh ] c A (y) =a s for y 2 [ y L + kh; y H ] or c A (y) =μc for all y where a = (1 ffl A ) (1 ffl A )+(1 ffl B ) ; and h =(y H y L ): This proposition says that there is a threshold in the aggregate endowment states such that households with more Knightian uncertainty (type A households) have constant consumption if the aggregate endowment is below the threshold. Households with a lower degree of Knightian uncertainty therefore absorb all the fluctuations in the low aggregate endowment states. Note that a<0:5 < (1 a) for ffl A >ffl B. This implies type B households are compensated by higher consumption on the rest of aggregate endowment states. Figure 2shows the individual consumptions as functions of aggregate endowment. Having more uncertainty about the endowment process, type A households use higher probability weights on low endowment states, and thus value more the consumption on bad states. Hence they are willing to trade their consumption on good states for consumption on those bad states to achieve a smoother consumption plan. The higher amount of Knightian uncertainty enhances economic households' consumption smoothing motive. 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 benchmark rational expectations economy in which individual consumption is strictly increasing in the aggregate consumption. Notice also how Knightian 16

17 20 18 INDIVIDUAL CONSUMPTION Type B agent Type A agent Figure 2: Optimal consumption allocation for a two-period economy with two types of consumers: A and B. Both types have the same log utility function and equal wealth, but type A (solid line) have a higher degree of Knightian uncertainty thantype B (dash line) y AGGREGATE ENDOWMENT y+h uncertainty enhances households' incentive to smooth consumption: households with more Knightian uncertainty (type A) choose relatively smoother consumption than households with less uncertainty (type B). Because a type A household has constant consumption when the aggregate endowment is below the threshold, his uncertainty adjusted probability distribution is not unique. On the other hand, the uncertainty adjusted probability distribution for a type B household is unique, for his consumption is strictly increasing in the aggregate endowment. Therefore, we caninfer the equilibrium price of a security from type B households' shadow values for the asset. It can be easily verified that B's uncertainty adjusted probability distribution is uniform on the interval (y L ;y H ] with density (1 ffl B )=(y H y L ) and has probability ffl B at the point y L. Notice that this uncertainty adjusted distribution is stochastically dominated by the reference probability distribution. This implies that the resulting stochastic discount factor used for consumption based asset pricing should be constructed not with the estimated distribution (or the reference distribution), but with the uncertainty adjusted distribution which is stochastically dominated by 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 17

18 states are uniquely determined, so are the prices of state contingent claims on individual endowment states. Equilibrium asset prices and allocations are not indeterminate. However, we can introduce idiosyncratic shocks to individual endowments. By combining the results from the constant aggregate endowment economy and the random aggregate endowment economy, we can infer the properties of the equilibrium in a new economy with both aggregate and idiosyncratic shocks. Individual consumption will be weakly increasing in aggregate endowment but will not vary with idiosyncratic shocks, and there will be indeterminacy in the prices of state contingent claims on idiosyncratic shocks. As before, this causes wealth to be indeterminate. The prices of state contingent claims on aggregate endowment states are still uniquely determined once the values of individual endowments are determined. 4 Infinite Horizon We now generalize the previous two-period economy in several aspects. In particular, we relax the logarithmic utility assumption, and extend the two-period setup into an infinite horizon setting. 4.1 Constant Aggregate Endowment Consider an exchange economy that is similar to the ones in the previous section except that now households live forever. There are N types of households. We first look at the economy where aggregate endowment is constant (y t = y for all t), while individual endowment share s i t is subjected to idiosyncratic shocks. Let s t = s 1 t ;s 2 t ; ::::; s N t. Assume fs t g is a first order Markov process, and let S be the state space of the Markov process. Denote by e i t(s t )=s i ty t atype i household's period t endowment. Thus, we have that NX i=1 e i t(s t )=y; 8 s t 2 S; 8 t: (18) 18

19 We allow households to have different reference probability distributions. Denote by ß i the reference transition matrix of household i, and by ß i s t the reference probability distribution conditional on state s t. The belief mappings of a type i household at t conditional on s t are : P ffl i ;s t = f(1 ffl i )ß i s t + ffl i m : m 2Mg; (19) where M is the collection of all probability distributions on s. We arrange households' types such that households of type 1 have the highest ffl value, and households of type N have the lowest ffl value : 1 >ffl 1 ffl 2 ::: ffl N 0: Assume the intersection of all households' priors is non-empty for all periods and states. fi=1;2;::;ng P ffl i ;s t 6= ; for all s t 2 S and for all t: (20) Let c i t denote a type i household's consumption at t, let c t;i fc i t;c i t+1;c i t+2; g denote a type i household's consumption sequence starting at t, and let c t fc t;1 ;c t;2 ; :::; c t;n g. The preferences of a type i household are described by the recursive utility function U(c t;i ;s t ) = u(c i t(s t )) + fie Pffl i ;st [U(c t+1;i ;s t+1 )]; (21) where u is strictly increasing, strictly concave and twice differentiable, and fi is the time discount factor. Markets are complete. We can apply the same argument used in the two-period model to show that equilibrium consumption for each household is completely smoothed across time and states. All idiosyncratic shocks are diversified. This result is similar to the equilibrium allocation of a rational expectations economy. A similar result also shows up in Chateauneuf, Dana, and Tallon (1997) in a static setup. Equilibrium Prices Let p t (s t ;s t+1 ) denote the period t price, conditional on s t, of a contingent claim for a unit of 19

20 period (t + 1) consumption on s t+1. Define p t;st fp t (s t ;s t+1 )g st+1 2S, andp t fp t;st g st2s. Given the individuals' endowment sequence, we want to find a price sequence fp t g 1 t=0 that supports the consumption allocation in a competitive equilibrium. Because in equilibrium each household's consumption is constant across states, a household's set of uncertainty adjusted measures is equal to his set of beliefs. From households' Euler equations and the fact that individuals' consumption is constant across both states and time, we have : fi min q i (s)» p t (s t ;s t+1 = s)» fi q i 2P ffl i ;st X s2s max q i (s); 8 i; 8s 2 S; (22) q i 2P ffl i ;st p t (s t ;s t+1 = s) =fi: (23) Equations (22) and (23) are necessary and sufficient conditions for the determination of equilibrium prices. The assumption that the intersections of sets of priors are non-empty, equation (20), guarantees that the solution for equation (22) exists. Equations (22) and (23) imply that price indeterminacy occurs if the intersection of sets of priors is non-singleton. In the case where households have common reference transition matrices, price indeterminacy occurs if everyone in the economy has Knightian uncertainty, i.e., ffl i > 0 for all i. Notice that the upper bound and lower bound of the one-period state contingent claim prices can vary over time, which means the ranges of indeterminacy in the state contingent claim prices can vary over time and states. Epstein and Wang (1994, pp ) suggest that prices may be indeterminate in an economy with Knightian uncertainty because analysts in the economy have incomplete information about idiosyncratic endowment. Here, using an economy with completely specified heterogeneity, we have shown that even if analysts have complete information, asset prices are still indeterminate. On the other hand, if there are households in this economy who have no Knightian uncertainty, then equilibrium prices will be uniquely determined. This is because for those households without Knightian uncertainty, their first order conditions hold as equalities, and that will pin down all prices. Equation (23) also 20

21 implies that one period risk-free rates are constant and equal 1=fi. Given an equilibrium price sequence, the corresponding consumption shares are determined from households' present value budget constraints. As in the two period model, different equilibrium price sequences can result in different valuations of individual's endowments, and consequently different equilibrium consumption shares among households. 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 properties of equilibrium consumption and prices for this economy. Theorem 4.1 Given households' preferences and uncertainty specified by equations (19), (20) and (21), households' endowment sequence as in equation (18), and complete markets, the equilibria of the constant aggregate endowment economy have the following properties: 1. Equilibrium consumption for each household is constant across time and states. 2. Price indeterminacy occurs if fi=1;2;::;ng P ffl i ;s t is non-singleton for some s t 2 S. For a given equilibrium price sequence fp t g 1 t=0, the equilibrium consumption allocation is uniquely determined. 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. When there is price indeterminacy, if the true probability distribution is in all households' belief sets, then there exists a price selection rule such that the resulting equilibrium coincides with a rational expectations equilibrium. Given the economy described in theorem 4.1, if all households have the same reference transition probability matrix, then the degree of price indeterminacy is determined by the household type with the least Knightian uncertainty. If there is one type of households which has rational expectations (i.e. the households do not have Knightian uncertainty and 21

22 the reference probability is the true probability), the equilibrium is unique and coincides with a rational expectations equilibrium. 4.2 Random Aggregate Endowment We now extend the previous two-period random aggregate endowment economy to an infinite horizon setting. The aim is to investigate how the threshold feature in the consumption allocation for the two-period model generalizes to the longer horizon setting. We maintain the assumption that there are twotypes (A and B) of households who have identical endowment streams. Each period, both types of households' beliefs about the next period endowments follow an ffl-contamination form with a common reference probability distribution ß y which is i:i:d: with support Y [y L ;y H ]: P ffl i = f(1 ffl i )ß y + ffl i m : m 2Mg i = A; B; (24) where M is the collection of all probability distributions on Y and ffl A ffl B. Households have recursive utility functions as specified in equation (21). As a way to find the equilibrium consumption allocations, we set up a social planner's maximization problem and solve for the Pareto optimal allocations. The planner's maximization problem is to find a sequence of state-contingent consumption allocations, subject to the resource constraint, such that the sum of the values of households' future discount utilities, weighted by a given set of Pareto weights, are maximized. Normalize the Pareto weights such that they sum to one. Let 2 be the two dimensional unit simplex: 2 = f(x 1 ;x 2 ) 2 R 2 + : x 1 ;x 2 > 0 x 1 + x 2 =1g: Define = ( A ; B ) 2 2; where i is the Pareto weight foratype i household. Denote by V ( ;y)the value function for the planner's maximization problem: V ( ;y) = max c t X i=a;b i fu(c i t)+fie Pffl i [U(c t+1;i ;y t+1 )]g 22

23 s:t: c A t + c B t» y; and c A t+l + c B t+l» y t+l for all states, 8 l>0: In light of the recursive structure of the utility function in equation (21), we adopt the method developed by Lucas and Stokey (1984) and Kan (1995). The basic idea of this approach is adding Pareto weights as one of the state variables, and solving the program recursively. We use variables with a tilde to denote random variables that will be realized next period. For example, ~ A (~y) denotes a type A household's next-period Pareto weight as a function of next period's endowment state ~y. The planner's problem can be reformulated with the following recursive program : V ( ;y) = max fc i ;fz i (~y)g ~y2y g i=a;b X i=a;b i Φ u(c i )+fie Pffl i [z i (~y)] Ψ s:t: c A + c B» y (25) min [V ( ; ~ ~y) X ~ i z i (~y)] 0; 8 ~y 2 Y (26) ~ (~y)2 2 i=a;b 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 utility entitlements fz i (~y)g i=a;b, which are the discounted expected utilities from next period on. Equation (25) says current period consumption allocation is constrained by the aggregate resources available this period. Equation (26) puts a restriction on the feasible set of state-contingent entitlements of utilityfrom next period on. The solution ~ to the minimization problems in (26), denoted by ~ ( ; ~y), is a function of next period's state ~y and current period's Pareto weights. As shown by Lucas and Stokey (1984), ~ ( ; ~y) will be the set of Pareto weights for the next-period's recursive program contingent on next period's state being ~y. Because of the dependence of ~ on ~y and, the Pareto weights vary with states and over time. This implies the optimal consumption allocation rule can be state-dependent. 23

24 It can be shown that u being strictly increasing and strictly concave implies that V ( ;y) is strictly convex in and strictly increasing in y. This means that (i) for each and y, the program has a unique solution for the current consumption allocation fc i g and statecontingent entitlement of utility from next period on, fz i (~y)g ~y2y, and (ii) for each ~y 2 Y and each value of z i (~y), the minimization problem described in (26) has a unique solution for (~y). ~ To solve the maximization problem, we first form a Lagrangian. L = X Φ Ψ i u(c i )+fie Pffl i [z i (~y)] i=a;b + fy c A c B g Z + μ(~y)f V ( ~ ; ~y) X ~ z i (~y)+ (~y)[1 X Y i=a;b i=a;b ~ i (~y)] gd~y where, μ(~y) and (~y) are Lagrange multipliers. Denote by Q ffl i the set of a household i's uncertainty adjusted measures: Q ffl i = Φ ν 2P ffl i j E ν [z i (~y)] = E Pffl i [z i (~y)] Ψ ; (27) where fz i (~y)g ~y2y are from the solutions to the maximization problem. Since the solutions for household i's next-period utility entitlement fz i (~y)g ~y2y depend on current period's Pareto weights, the set Q ffl i in general will be a function of also. The first order conditions from the Lagrangian, together with the constraints (25) and (26), imply [c i ] : i u 0 (c i )= ; 8i (28) [z i (~y)] : min q i (~y)» μ(~y)~ i (~y) q i 2Q ffl i fi i» max q i (~y); 8i 8~y 2 Y (29) q i 2Q ffl i X i fi = μ(~y) ~ i (~y); 8i (30) X ~y2y [ ] : y = c i ; (31) i=a;b 24

25 X [μ(~y)] : V ( (~y); ~ ~y) = ~ i (~y)z i (~y) (~y)f1 NX i=a;b i=1 ~ i (~y)g; 8~y 2 Y (32) [ ~ i (~y)] ( ~ (~y); ~y)=@ ~ i (~y) =z i (~y)+ (~y); 8i 8~y 2 Y (33) X i=a;b ~ i (~y) =1; 8~y 2 Y (34) Equation (28) equates across all households the marginal utilities of current consumption weighted by Pareto weights. Equations (29) and (30) are the conditions for the current and future Pareto weights such that valuations of next period state-contingent consumption are equated across households. Equation (31) is the current period resource constraint. Equations (32), (33), and (34) are the conditions derived from the minimization from the constraint (27). From equations (28) to (34), we can solve fc i g i=a;b, fz i (~y)g i=a;b; ~y2y, f ~ i (~y)g i=a;b; ~y2y, and the multipliers, f (~y)g ~y2y and fμ(~y)g ~y2y. Instead of directly equating all households' valuations of state-contingent consumption for all future dates and states, this program uses the device of possible time and state varying Pareto weights to achieve the same goal. The reason for doing this is that when households have heterogeneous beliefs, direct computation of the households' valuations of state-contingent consumption becomes more involved and less tractable as we look at more distant future. To circumvent this complication, the recursive Pareto weight approach takes advantage of the recursive structure of the utility function. Again, the special features of the Euler equations under Knightian uncertainty show up, in equation (29), in the inequality form and the appearance of maximization and minimization, taken over the set of uncertainty adjusted probability measures Q ffl i. If the set Q ffl i is singleton, then the maximum and minimum in equation (29) are equal, and equation (29) takes a simple equality form: i fiq i (~y) =μ(~y) ~ i (~y); (35) where q i are the uncertainty adjusted probabilities. Using equation (28) to write i in terms 25

26 of c i and, substituting it into equation (35), dividing the resulting equation for type A households by that for type B households, and rearranging the terms, we get: fiu 0 (~c A (~y)) u 0 (c A ) q A (~y) = fiu0 (~c B (~y)) u 0 q B (~y) for all ~y 2 Y: (36) (c B ) Here c i isatype i household's current period consumption and ~c i (~y) isatype i household's next period consumption at state ~y for i = A; B. Equation (36) amounts to equate the marginal rates of substitution across households, which is similar to the usual optimality condition except that here it is the uncertainty adjusted probabilities that is used in evaluating the state contingent consumption. In the case where the set of uncertainty adjusted probability distributions Q ffl i is not a singleton on Y, the maximum and minimum in equation (29) are not equal for some (or all) states, and the first order conditions have inequality forms for those states. Recall that Q ffl i is the set of probability distributions that give the household the same expectation on the utility function. Because the Euler equations are evaluated under Q ffl i, the non-uniqueness of Q ffl i causes the non-uniqueness in the values of the Euler equations. 10 Denote by W i ( ;y) a type i household's current-period utility entitlement, i.e., the sum of discounted single-period utility from this period on, as a function of current-period Pareto weights and the aggregate endowment state. Then W i ( ;y)=u(c i ( ;y)) + fie Pffl i [z Λi ( ; ~y)] (37) and W i ( ~ ( ; ~y); ~y) =z Λi ( ; ~y) (38) where c Λi ( ;y), z Λi ( ; ~y) and ~ ( ; ~y) are from the solution to the planner's maximization problem. Notice that with i.i.d. assumption, the set of priors P ffl i is independent of the current state y, so the expected value of promised utility in equation (37) is independent of 10 However, the strict concavity ofu and the strict convexity of the function V guarantee the uniqueness of the solution to the planner's maximization problem. 26

27 the current state once the Pareto weights are given. Using equation (37), the planner's value function can be expressed as X V ( ;y)= i W i ( ;y): We now explore the function W i. i=a;b Lemma y k <y l ; 8 2 2; and 8 i 1. W i ( ;y k ) <W i ( ;y l ), and 2. W i ( ~ ( ; ~y k ); ~y k )» W i ( ~ ( ; ~y l ); ~y l ): Proof : See the appendix. The part 1 of lemma 4.2 states that optimality implies that holding fixed the Pareto weights, everyone in the economy strictly prefers good states (high aggregate endowment) to bad states (low aggregate endowment). The part 2 of the lemma is similar to the part 1 but is a stronger statement about households' welfare at different states. When comparing next period's utility entitlement across different states, there are two forces in action. On the one hand, if a good state occurs, there will be more aggregate resources to be allocated among households. On the other hand, because the Pareto weights are state-dependent, some households may get lower Pareto weights on a high aggregate endowment state. The lemma states that optimality requires individual's future utilityentitlement to be positively related to the amount of future aggregate resources. Notice that we only have a weak inequality in the second part of the lemma. It is possible that some households may be indifferent among the occurrence of some states. Indeed, a household with more Knightian uncertainty has a stronger incentive to guard against the worst outcomes and thus maywant to smooth future utilityentitlementover the low endowment states. As we will see next, efficient allocation implies that the household 27

28 with more Knightian uncertainty has constant utility for a range of aggregate endowment states. The following theorem characterizes the properties of Pareto optimal allocations for the economy. The proofisintheappendix. Theorem 4.3 The efficient allocations of the economy have the following properties. a. Consumption: c i ( ~ ( ; ~y l ); ~y l ) <c i ( ~ ( ; ~y k ); ~y k ) 8 ~y l ; ~y k 2 Y, ~y l < ~y k ; 8 i. b. Pareto weight: 1. ~ i ( ; ~y k ) is strictly increasing in i. 2. ~ ( ; ~y k ) is not constant in ~y k. c. Welfare: There exists a constant ^k in (0; 1], such that W A ( ~ ( ; ~y); ~y) is constant in ~y for ~y 2 [y L ;y L + ^k(y H y L )], and strictly increasing for ~y 2 [y L + ^k(y H y L );y H ]. W B ( ~ ( ; ~y); ~y) is strictly increasing for all ~y Similar to the benchmark rational expectations economy, part (a) of the theorem says that individual household's consumption in the economy with Knightian uncertainty is strictly increasing in aggregate endowment. From part (b) of the theorem, we can infer that the Pareto weights at period t depend on all the past realizations of aggregate endowment states. This time-varying Pareto weight result, together with the planner's optimality condition that weighted marginal utilities being equated, implies that consumption allocation rules are history-dependent. 11 This is in contrast to the rational expectation economy where consumption allocation rule is independent of the history of the shock. Also, here equilibrium consumption allocations depend on the history of aggregate endowments in a special way: households' current marginal utilities are a sufficient statistic to summarize the history of aggregate endowments for the evolution of equilibrium consumption allocation. This sufficient statistic property 11 Note that it is the history of the aggregate endowment that matters. The history of idiosyncratic shocks does not affect allocations because of the complete markets. 28

29 of marginal utilities also shows up in Kocherlakota's (1996) model without commitment technology. Part (c) of the theorem is a threshold result on the expected utility entitlement for the household with more Knightian uncertainty. It extends the threshold result on consumption from the two period model. Note that constant consumption in the second period for the two period model is equivalent to constant expected utility. It clarifies that it is the low expected utility states that households with Knightian uncertainty want to protect themselves from, not the low consumption states, although the two are closely related. For an household with a higher degree of Knightian uncertainty (a type A household), although he is indifferent among the realizations of states next period that are below a threshold, his consumption next period is strictly increasing in aggregate endowments. This is because for any two next-period states that are lower than the threshold, the one with a lower aggregate endowment will imply not only a lower next-period consumption but also a higher (stochastically dominant) consumption stream starting from the period after next. The effects from the two factors exactly balance each other, and thus leave the expected utility constant. 5 Conclusion This paper studies how heterogeneity in Knightian uncertainty affects equilibrium allocations and prices. Knightian uncertainty is modeled using the ffl-contamination formulation from robust statistics, and households are allowed to have different ffl's. We use constant aggregate endowment economies to highlight the occurrence of price indeterminacy. The range of the price indeterminacy is governed by the consumers with the most confidence in their beliefs (the smallest ffl). In our heterogeneous household economies the resulting price indeterminacy carries over to indeterminacy in the distribution of wealth and consumption shares. Second, in a two-period economy with random aggregate endowments and a contin- 29