Recursive Utility and Parameter Uncertainty

Recursive Utility and Parameter Uncertainty Nabil I. Al-Najjar∗ Eran Shmaya† First draft: June 2016 This version: March 22, 2018

Abstract We explore how the Epstein-Zin utility captures an agent’s sensitivity to parameter uncertainty. Our main result is a closed-form representation of the Epstein-Zin utility for an i.i.d. consumption process with unknown parameter, as the discount factor approaches 1. Using this representation, and under the usual assumption about the relationship between risk aversion and the attitude towards time smoothing, we show that the agent is averse to parameter uncertainty.

∗ Department of Managerial Economics and Decision Sciences, Kellogg School of Management, Northwestern University, Evanston IL 60208. † Department of Managerial Economics and Decision Sciences, Kellogg School of Management, Northwestern University, Evanston IL 60208.

Contents 1 Introduction 2 Model and Main Result 2.1 Epstein-Zin Utility . . . . . 2.2 Main Theorems . . . . . . . 2.3 Epstein-Zin Preferences and 2.4 Epstein-Zin Preferences and

1

. . . .

4 4 5 6 7

3 Proofs 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . .

8 8 11 13

. . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameter Uncertainty Expected Utility . . . .

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1

Introduction

In their seminal paper, Epstein and Zin (1989) proposed a functional form for intertemporal preferences that has become a building block in many asset pricing and macroeconomic models. The key advantage of the Epstein-Zin model relative to the standard additive utility is that risk aversion and the elasticity of intertemporal substitution can be separated. To motivate the Epstein-Zin utility and illustrate our contribution, consider the following example. In each period t = 1, 2, . . ., consumption is random with realization xt ∈ {h, l}, with h > l > 0. Consider two stochastic consumption processes: Process P1 :

Consumption is i.i.d. where the probability of h in each period is 0.50.

Process P2 :

Consumption is determined in two stages: first a parameter θ is drawn from the uniform distribution on [0, 1], then consumption is i.i.d. where the probability of h in each period is θ. The parameter θ itself is never directly observed.

The standard additive utility model cannot distinguish between P1 and P2 . The reason is that an additive utility depends only on the marginal distributions on period consumptions, and thus ignores any correlation between periods. Since P1 (xt = h) = P2 (xt = h) = 0.50 for each t, the consumption processes P1 and P2 lead to identical additive utilities. This feature of additive utility is counter-intuitive for two reasons. First, additive utility ties down the preference for smoothing consumption over time to the attitude towards risk. This restriction is both empirically questionable and lacks compelling theoretical justification. Second, additive utility is insensitive to uncertainty about the long-run fundamentals of the process—or parameter uncertainty, for short. In the example above, the agent is indifferent between process P1 , where long-run average consumption is known with certainty, and P2 , where that average is unknown. Epstein-Zin utility captures, in a very subtle way, these phenomena in a recursive power utility form. Their model evaluates a stochastic consumption process using a discount factor, an elasticity of intertemporal substitution, and a coefficient of relative risk aversion. While this makes it clear how the Epstein-Zin utility separates risk aversion from attitude towards 1

time smoothing, it is less obvious whether, and how, this model captures the sensitivity to parameter uncertainty. Part of the difficulty is that the Epstein-Zin utility is a fixed point of an operator that often lacks a closed form solution, making for a non-trivial dependence on the structure of the consumption process. This paper considers Epstein-Zin utility for conditionally i.i.d. consumption processes and establishes a simple closed-form representation of utility as the discount factors approaches 1. Using this closed form representation, it is easy to see how the intertemporal substitution and risk aversion parameters characterize the impact of parameter uncertainty. To make this precise, assume that the set of possible consumption levels C ⊂ R is finite, and that consumption is i.i.d. with an unknown distribution θ over C. Let µ be the distribution over θ representing uncertainty about this parameter. Our main result states that, as β → 1, the Epstein-Zin utility approaches: 1

!α

 Z 

α

ρ

X

Θ

ρ

θ(x) x

dµ(θ) ,

(1)

x∈C

1 where 1−α is the coefficient of relative risk aversion, and 1−ρ is the elasticity of intertemporal substitution. Intuitively, a patient agent with Epstein-Zin utility evaluates a consumption process in two steps. First, once θ is known, the agent will almost surely face a sequence where each consumption level x occurs with a known frequency θ(x). Although the time periods in which these consumption levels occur remains uncertain, this uncertainty is irrelevant to a patient agent, 1 P who will therefore evaluate such sequence according to ( x θ(x) xρ ) ρ . Note that risk aversion plays no role in this expression since the agent is concerned only with smoothing consumption. Risk aversion is relevant only with respect to the uncertainty about how the utility from long-run consumption 1 P ( x θ(x) xρ ) ρ varies with θ. To illustrate this, consider the two processes introduced earlier. Since consumption takes only two values, 0 or 1, we have θ ∈ [0, 1] and the above expression simplifies to:

Z

α ρ

(θ hρ + (1 − θ) lρ ) dµ(θ)

[0,1]

2

!1

α

.

(2)

1

For process P1 , θ = 0.50 and this expression reduces to (0.5 hρ + 0.5 lρ ) ρ . For Process P2 , by contrast, long-run consumption is uncertain, and the limiting utility is given by (2) with µ being the uniform distribution. For example, if ρ = 1, so the agent cares only about the long-run consumption, the Epstein-Zin utility of a patient agent from P1 is just the long-run consumption 0.5 h + 0.5 l, while the utility from P2 for a risk averse agent
1 R (α < 1) is (θ h + (1 − θ) l)α dµ(θ) α < 0.5 h + 0.5 l. Uncertainty about long-run consumption is “penalized” by α, reducing overall utility. Since the Epstein-Zin utility is defined as a fixed point of an operator, the closed form we obtain is useful and non-trivial. Finding the utility for Process P1 is not too difficult, as it can be obtained from solving one equation in one unknown. However, even for a simple process like P2 , a closed form for a fixed β is not known and utility must be computed via simulations. The Epstein-Zin utility of a process like P2 is computed recursively as a function of the future utilities, which are indexed by the posterior beliefs about the parameter that can arise from Bayesian updating. An interesting feature of our result is that the limiting Epstein-Zin utility does not depend on the timing of resolution of uncertainty. This is in contrast with the case of a fixed discount factor. Consider, again, Process P2 with a fixed discount factor. The Epstein-Zin utility in this case will be different under: (1) the standard information structure, where only past consumptions are observed, and (2) the finest information structure, where the entire future path of consumptions is revealed in period 1. This feature of the Epstein-Zin utility was highlighted by Epstein et al. (2014), who considered a number of calibrated macro-finance models. Under the usual assumption that ρ ≥ α, the agent will display a preference for early resolution of uncertainty. They find that, although information has no instrumental value, agents would be willing to give up a substantial portion of their future income to have the uncertainty resolved early. Our result implies that this does not occur with a patient agent and a conditionally i.i.d. process. The Epstein-Zin utility belongs to the class of models developed by Kreps and Porteus (1978). Although this utility is extensively used in empirical work, the literature on its theoretical properties is relatively small. Within this literature, we make use of the results of Marinacci and Montrucchio (2010) on the existence, uniqueness, and continuity of the Epstein-Zin utility.

3

Bommier et al. (2017), who characterize ordinal monotonicity for recursive utilities, show that the Epstein-Zin utility is not ordinally monotone. We further discuss the connection with their work below. Finally, the present paper is motivated by our earlier work, Al-Najjar and Shmaya (2015), where we characterize the implications of disagreement and parameter uncertainty in dynamic stochastic models. In the language of that paper, the parameter α in (1) above represents the attitude to structural uncertainty.

2

Model and Main Result

The time horizon is infinite with periods indexed by t = 1, 2, . . .. In each period t, consumption is a random variable Xt defined on some probability space and taking values in a strictly positive interval [c, c¯].1 We use upper case letters to denote random variables and lower case letters for their realizations (so x is a realization of X). Consider an agent with power utility xα , where 1−α ≥ 0 is the coefficient of relative risk aversion. The certainty equivalent of a random consumption X given a σ-algebra F ⊂ A is defined as: CEα (X|F) = (E (X α |F))1/α . We will simply write CE when α is clear from the context.

2.1

Epstein-Zin Utility

Intertemporal utility is defined for a stochastic process of consumptions X1 , X2 , . . . adapted to a filtration F = (F0 , F1 , . . .), where F0 is trivial.2 That is, Ft is the information available at time t. Define the Epstein-Zin utility to be:   1/ρ ρ ρ ˆ ˆ Zt−1 = CE (1 − β) Xt + β Zt (3) Ft−1 . 1 The definition and properties of Epstein-Zin utility can be (and usually are) extended to the case of real-valued consumption, provided additional technical conditions are satisfied. See Epstein and Zin (1989) original paper, or Marinacci and Montrucchio (2010). In this paper it is more convenient to restrict attention to the case where the set of consumption levels is a strictly positive compact interval. 2 That is, Xt is Ft -measurable for every t.

4

The random variable Zˆt−1 represents the certainty equivalent of the stream of future payoffs Xt , Xt+1 , . . . given the information at time t − 1.3 We shall (β) write Zˆt−1 when we want to highlight the dependence on β. To keep things simple we assume throughout the paper that α ≤ ρ and 0 ≤ ρ ≤ 1 even though these assumptions are not required for the main theorem.4 The parameter ρ determines the elasticity of intertemporal 1 substitution as 1−ρ . We note for future use the special case ρ = 1 where the elasticity of intertemporal substitution is infinite. In this case the agent has no preference for smoothing consumption and the limiting utility simplifies to:   (4) Zˆt−1 = CE (1 − β)Xt + β Zˆt Ft−1 .

2.2

Main Theorems

We present two theorems. The first theorem considers the case where consumption is i.i.d., like the process P1 in the introduction, while the second considers correlation, as in process P2 . Theorem 1. Let X1 , X2 , . . . be a sequence of i.i.d. random variables with distribution θ. Then Z 1/ρ (β) ρ ˆ lim Z0 = x dθ(x) . (5) β→1

To intuitively understand the conclusion of the theorem, assume that θ has finite support C. Then (5) becomes !1

ρ

X

θ(x) x

ρ

.

(6)

x∈C

By the strong law of large numbers, in any realized consumption stream x1 , x2 , . . . , consumption level x would occur with frequency θ(x). Since a patient agent cares about the frequency in which a consumption level occurs, and not the timing of its occurrence, this agent computes the utility of the 3

We depart from the convention of assuming that the first period consumption is deterministic. In our setting of a conditionally i.i.d. process, it is more natural to highlight that the first period consumption is governed by the same probability law as other periods. 4 Without this assumptions some monotonicity assertions such as Proposition 3.1.4 should be with opposite directions.

5

stream as the weighted average of the utilities along that stream, xρ . This gives the “certainty equivalent”-like expression (6), where the parameter of risk aversion is ρ instead of α. Of course, no probabilities are involved in (6), where the risk aversion parameter α does not play a role. Next, we turn to the case of correlated consumption–as in process P2 in the introduction. As before, we consider an i.i.d. consumption process X1 , X2 , . . . with parameter θ, except that now this parameter is unknown to the agent. Our proof requires that consumption takes values in a finite set C ⊂ [c, c¯], and that the agent’s belief about the parameter has a strictly positive density over ∆(C). We discuss the need for these assumptions in our proof. Theorem 2. Suppose that consumption is a conditionally i.i.d. process whose parameter θ has distribution µ0 with a strictly positive density over ∆(C). Then: !α

 (β) lim Zˆ0 = 

β→1

Z

ρ

X

ρ

θ(x) x

1

α

dµ0 (θ) .

(7)

x∈C

Since the filtration does not appear in the expressions of the limiting Epstein-Zin utility above, Theorems 1 and 2 imply that this utility is independent of the filtration. This is in contrast with the fixed discount factor case, where the filtration plays an important role, as discussed in the introduction. In particular, the sensitivity to the filtration diminishes as the discount factor approaches 1. To understand the intuition for Theorem 2, consider (6) above, which reflects the agent’s Epstein-Zin utility if he knew the parameter θ. When the parameter is unknown, this utility is itself random, so the agent computes its expected utility the usual way—using the risk aversion parameter α and distribution µ0 . The Epstein-Zin utility of the entire process is the corresponding certainty equivalent given by (7).

2.3

Epstein-Zin Preferences and Parameter Uncertainty

Our theorems explain how the Epstein-Zin utility distinguishes between parameter uncertainty and ordinary risk. As noted in the introduction, an agent with the standard additive utility will be indifferent to correlation in 6

consumption across periods and, therefore, to parameter uncertainty, which is often a major source of such correlation. To express the sensitivity of the Epstein-Zin utility to parameter uncertainty, we begin with some definitions. Given a distribution µ on parameters, let its barycenter be: Z ¯ θ dµ0 . θµ = 0

Θ

The barycenter θ¯µ0 is the “average parameter” in the sense that, in a single period, θ¯µ0 and µ0 induce the same distribution on consumption levels. For a process P , we write Zˆ (β) (P ) to emphasize the dependence of the Epstein-Zin utility on P . The next corollary considers the usual case where the agent displays a strict preference for early resolution of uncertainty: Corollary 3. When α < ρ, we have: lim Zˆ (β) (Pθ¯µ ) < lim Zˆ (β) (Pµ0 ). 0

β→1

β→1

Proof: Theorem 1 implies that: !1

ρ

ˆ (β)

lim Z

β→1

X

(Pθ¯µ ) = 0

θ¯µ0 (x) x

ρ

x∈C

while Theorem 2 implies: !α

 lim Zˆ (β) (Pµ0 ) = 

β→1

Z

ρ

X

θ(x) x

ρ

1

α

dµ0 (θ) .

x∈C

When α < ρ, the concavity of the function y 7→ y α/ρ and Jensen’s inequality establish the result.

2.4

Epstein-Zin Preferences and Expected Utility

Our main theorem implies that as the discount factor approaches 1, the Epstein-Zin utility approaches an expected utility functional. This is despite the double integral and the use of two probability measures, θ and µ0 . To see this, we first need some notation. 7

Define the empirical frequency of a consumption level x by #{i : xi = x} . n→∞ n

fx (ω) = lim

By the strong law of large numbers, fx (ω) is well-defined and equals θ(x), P -almost surely. With this notation, Theorem 2 implies:  lim

β→1

(β) Zˆ0

α

Z = Ω

!α ρ

X

fx (ω) x

ρ

dP (ω).

x∈C

The RHS is an expected utility expression: each deterministic sequence
α P ρ ρ that does not involve probabilities. f (ω) x ω is assigned a utility x∈C x Expected utility is then taken by integrating with respect to P . This observation formalizes the intuition that θ(x) in (7) is an empirical frequency of consumption levels; the only uncertain term in that expression is the parameter θ. An interesting consequence of this result is that, as in the case for expected utility preferences, the limiting Epstein-Zin utility is ordinally monotone while, as shown by Bommier et al. (2017), this property generally fails in the case of fixed discount factor. Thus, like the sensitivity to the filtration discussed earlier, the failure of ordinal monotonicity in the conditionally i.i.d. case diminishes as the discount factor approaches 1.

3

Proofs

3.1 3.1.1

Preliminaries Reduction to the Special Case ρ = 1

We first show that it is sufficient to prove the theorems for the special case ρ = 1. To prove this, it will be useful to index the certainty equivalent operators by the risk aversion parameter. Consider a consumption process X1 , X2 , . . . as in Section 2.1. Define the process Y1 , Y2 , . . . by Yt = Xtρ and ˆ t = Zˆ ρ . Then Y1 , Y2 , . . . is adapted to the same filtration, so from (3) let W t

8

we get: ˆ t−1 W

  1/ρ = CEα (1 − β) Xtρ + β Zˆtρ

ρ Ft−1

ˆ t ). = CEα/ρ ((1 − β)Yt + β W

ˆ t−1 is the Epstein-Zin utility of the Y1 , Y2 , . . . when the risk Thus, W aversion parameter is α/ρ ≤ 1 (since we assumed that α ≤ ρ and 0 < ρ ≤ 1.) and no preference for smoothing consumption. Given Theorem 1 for the case ρ = 1, if X1 , X2 , . . . are i.i.d. then so are Y1 , Y2 , . . . and it would follow that ˆ (β) = EY1 = EX ρ , lim W 1 0

β→1

which implies (β) 1/ρ lim Zˆ0 = (EX1ρ ) ,

β→1

establishing the claim for general values of ρ. By a similar argument it is sufficient to prove Theorem 2 for the case ρ = 1. 3.1.2

Properties of the Certainty Equivalent Operator

In the proof, we use the following elementary properties of the certainty equivalent operator: 1. Iterated certainty equivalence (or reduction of compound lotteries): For every random variable X and every sigma algebra F it holds that CE(CE(X|F)) = CE(X). 2. Homogenuity: For every b > 0 and every random variable X it holds that CE(b X) = b CE(X). 3. Monotonicity: If X ≤ X 0 a.s., then CE(X) ≤ CE(X 0 ). 4. Risk aversion: CE(X) ≤ EX for every X. 5. Decreasing relative risk aversion: CE(b + X) ≥ b + CE(X) for every random variable X and b > 0. 9

6. Continuity: Let 0 < c < c¯ < ∞. Then the map X 7→ CE(X) defined over random variables X which assume values in [c, c¯] is continuous in the topology of weak convergence. 7. Deterministic evaluation: If X = r a.s. for some r ∈ R (i.e., X is constant) then CE(X) = r. 3.1.3

Fixed Point Formulation of Epstein-Zin Utility

Let X be the metric space of all adapted stochastic processes Z = (Z0 , Z1 , . . .) with values in [c, c¯] equipped with the metric d(Z, Z 0 ) = supt≥0 kZt − Zt0 k∞ , where k · k∞ denotes the essential supremum of a random variable. Define the Epstein-Zin operator T : X → X by T (Z)t−1 = CE ((1 − β)Xt + βZt |Ft−1 ) for every t = 1, 2, . . .. Then the Epstein-Zin utility Zˆ introduced in (4) can ˆ be equivalently defined as a fixed point of this operator: Zˆ = T (Z). Marinacci and Montrucchio (2010) proved that T is continuous and has a unique fixed point and that it has the global attraction property, i.e. Zˆ = lim T n Z n→∞

for every Z ∈ X . Note that our theorems identify the limit of the fixed point of the Epstein-Zin operator T when the discount factor β goes to 1. A similar issue arises in the study asymptotic value of zero-sum stochastic games, viewed as a fixed point of the so-called Shapley operator, when the discount factor goes to 1. See Neyman (2003) for a survey and Ziliotto (2016) for the most recent breakthrough. The results in these papers cannot be used in our framework, as they are formulated for finite dimensional Banach spaces, and our underlying Banach space X is infinite-dimensional. 3.1.4

Preference for more information

Let X1 , X2 , . . . be as in Section 2.1. We consider two filtrations F = (F0 ⊆ F1 ⊆ F2 ⊆ . . .) and F 0 = (F00 ⊆ F10 ⊆ F20 ⊆ . . .) with F0 and F00 trivial. We let Zˆ0 , Zˆ1 , . . . and Zˆ00 , Zˆ10 , . . . be the corresponding Epstein-Zin utilities defined in Section 2.1. Say that F 0 is more informative than F if Ft ⊆ Ft0 10

for every t. The following proposition is implicit in several papers but we have not found it stated anywhere in the generality we need. Proposition 4. Assume ρ ≥ α. If F and F 0 are two filtrations such that F 0 is more informative than F then Zˆt ≤ CE(Zˆt0 |Ft ) for every t = 0, 1, 2, . . .. In particular, the proposition implies that Zˆ0 ≤ Zˆ00 , i.e., the agent prefers more information. Proof: Assume, without loss of generality, that ρ = 1. Following the notations of Section 3.1.3, consider the spaces XF and XF 0 of stochastic processes with values in [c, c¯] adapted to the filtrations F and F 0 , and the Epstein-Zin operators T and T 0 over these spaces. Let G ⊂ XF ×XF 0 be the closed set of all pair of all (Z, Z 0 ) such that Zt ≤ ˆ Zˆ 0 ) ∈ G where CE(Zt0 |Ft ), for every t = 0, 1, . . . . We have to prove that (Z, 0 0 Zˆ and Zˆ are the fixed points of T and T . Because of the global attraction property, it is sufficient to prove that (Z, Z 0 ) ∈ G implies (T (Z), T 0 (Z 0 )) ∈ G. Indeed, if (Z, Z 0 ) ∈ G then: T (Z)t−1 = CE ((1 − β)Xt + βZt |Ft−1 )
≤ CE (1 − β)Xt + βCE(Zt0 |Ft ) |Ft−1

≤ CE CE (1 − β)Xt + βZt0 |Ft |Ft−1 = CE((1 − β)Xt + βZt0 |Ft−1 ) 0

|Ft−1 = CE CE (1 − β)Xt + βZt0 Ft−1 = CE(T 0 (Z 0 )t−1 |Ft−1 ), where the first inequality follows from the assumption that (Z, Z 0 ) ∈ G and the monotonicity of the risk aversion operator; the second inequality follows from monotonicity, and from homogeneity and decreasing absolute risk aversion of the certainty equivalent given Ft ; and the next two equalities from the law of iterated certainty equivalent.

3.2

Proof of Theorem 1

We begin with a sketch of the proof. Following the argument in Section 3.1.1 we assume that ρ = 1. We need to prove that (β) lim Zˆ0 = EX1 .

β→1

11

Because of the monotonicity of the Epstein Zin utility with respect to information, it is sufficient to prove the result for the two extreme cases of information flow: the case when all uncertainty is resolved in period 1, F1 = σ(X1 , X2 , . . . ), and the case where at each period only past consumptions are observed, Ft = σ(X1 , X2 , . . . , Xt ) for every t. Section 3.2.1 considers the first case. Since all uncertainty is resolved in the first period, and since the agent does not care about smoothing consumption, it follows that the EZ utility of the future at day 1 is given by P t β ∞ t=1 (1 − β) Xt . By the law of large numbers (from the perspective of day 0), this value is close to the deterministic value EX1 . Therefore, regardless of risk aversion, the EZ utility at day 0 is also approximately equal to EX1 . Section 3.2.2 considers the second case. There we prove the theorem using the fixed point property of the Epstein-Zin utility. 3.2.1

Upper bound

We first prove that (β) lim sup Zˆ0 ≤ EX1 . β→1

From Proposition 4 it is sufficient to prove the assertion for the case that F1 = σ(X1 , X2 , . . . ), i.e., all uncertainty is realized after the first period. P (β) In this case Zˆ1 = (1 − β) n≥0 β n X2+n since the future consumption stream is known at period 1 and therefore the Epstein Zin utility is just the discounted sum of future consumptions. Therefore     X (β) (β) Zˆ0 = CE (1 − β)X1 + β Zˆ1 = CE (1 − β) β n X1+n  . n≥0

P From weak law of large numbers it follows that (1−β) n≥0 β n X1+n → EX1 in distribution when β → 1. By the continuity of the certainty equivalent and by the fact that the certainty equivalent of a constant is a constant it (β) therefore follows that Zˆ0 → EX1 , as desired. 3.2.2

Lower bound

Next, we prove that: (β) lim inf Zˆ0 ≥ EX1 . β→1

12

From Proposition 4 it is sufficient to prove the assertion for the case that Ft = σ(X1 , X2 , . . . , Xt ) for t ≥ 1, i.e. that the only information available at period t is the realization of the process up to that period. The idea is to use the recursive structure of the Epstein-Zin utility, and the fact that when the sequence is i.i.d. the situation from day t onwards is always the same as the situation from day 0 onwards. Thus there exists some number z (β) such that Zˆt = z (β) for every t. Moreover z (β) must satisfy the following recursive equation z (β) = CE((1 − β)X1 + βz (β) ). Thus z (β) is a fixed point of the function f : z 7→ CE ((1 − β)X1 + βz). We claim that for every  > 0 the fixed point must satisfy z (β) ≥ EX1 −  for sufficiently large β. Because of the global attraction property of the function f , it is sufficient to prove that f (EX1 − ) ≥ EX1 − . Indeed, let κ = 1 − β. From the Arrow-Pratt approximation we get that for sufficiently large β f (EX1 − ) = CE((1 − β)X1 + β(EX1 − )) = CE(EX1 −  + κ(X1 − EX1 + )) 1 = EX1 −  + κ − κ2 Var(X1 )AEX1 − + o(κ2 ) 2 ≥ EX1 − , where AEX1 − is the absolute coefficient of risk aversion at consumption level EX1 − .

3.3

Proof of Theorem 2

We prove the main theorem using a similar method as the one used in the proof of the i.i.d. case by relying on the recursive structure of the EpsteinZin utility. The upper bound is again easy to prove using the intuitive argument described after the statement of Theorem 2. The lower bound is more involved since, in contrast to the i.i.d. case where the problem facing the agent was the same at every period, the problem now is defined by the current belief about the parameter. We use the fixed point property of the Epstein-Zin utility to prove the theorem simultaneously for every current belief µ. First, in Section 3.3.1, we replace the Arrow-Pratt approximation used in the proof for the i.i.d. case by a similar argument for the case in which the initial wealth is not constant. Then, in Section 3.3.2, we use 13

a well-known result on Bayesian learning to show that the belief on the parameter will not change by much in any period. This, in turn, implies that the change in the EZ utility is always small. As we have done before, assume that ρ = 1. We need to prove that lim

β→1

(β) Zˆ0

Z =

α

1

α

e(θ) dµ0 (θ)

.

P where for every θ ∈ ∆(C), e(θ) = x θ(x) x is the expected payoff from a roll of a lottery which results in the value x with probability θ(x). It will be convenient to introduce a ∆(C)-valued random variable Θ with distribution µ0 that represents the unknown parameter of the dice. Thus the variables X1 , X2 , . . . are i.i.d. conditional on Θ and their conditional distribution given Θ is Θ. The right-hand side of 3.3 is CE(e(Θ)) 3.3.1

Small Lotteries

Recall the Arrow-Pratt Approximation: for every initial wealth c and every lottery Y we have, as κ → 0, CE(c + κY ) = CE(c) + κ EY −

1 2 κ Ac Var(C) + o(κ2 ), 2

where Ac is the absolute coefficient of risk aversion at consumption level c. It follows from the Arrow-Pratt Approximation that a risk averse agent with some wealth c will always be willing to invest some small amount in any lottery Y such that E(Y ) > 0. Lemma 3.1 below does two things: it extends this observation for the case in which the initial wealth is random with small variance (instead of constant) and it provides an explicit bound instead of the asymptotic term o(κ2 ). The proof is similar to that of the Arrow-Pratt Approximation. Lemma 3.1. For every  > 0 there exist u, η, δ > 0 such that for every random variable C that assumes values in [c, c¯] and such that V ar(C) < δ, every random variable Y with values in [−1, 1] and such that EY > , and every κ < η it holds that CE(C + κY ) ≥ CE(C) + uκ.

14

Proof: Let φ(c) = cα , 0 < v < c, and L be a Lipschitz constant of φ0 on [c − v, c¯ + v]. Then it follows from Taylor’s formula that φ(c + y) ≥ φ(c) + φ0 (c)y − Ly 2 /2 whenever c, c + y ∈ [c − v, c¯ + v]. Let C, Y be random variables as in the lemma. Then it follows from the last inequality that, for κ < v,
1 Eφ(C + κY ) ≥ Eφ(C) + κE φ0 (C)Y − Lκ2 EY 2 . 2 Since φ−1 is convex and increasing it follows that φ−1 (Eφ(C)+h) ≥ φ−1 (Eφ(C))+ th for every h ∈ R, where t = 1/φ0 (φ−1 (Eφ(C))) = 1/φ0 (CE(C)). Therefore CE(C + κY ) = φ−1 (Eφ(C + Y ))
1 ≥ CE(C) + κ tE φ0 (C)Y − Lκ2 t EY 2 . 2

(8)

The Cauchy-Schwartz inequality implies: p Eφ0 (C)Y ≥ Eφ0 (C) EY − Var(φ0 (C))Var(Y ). Combining this inequality with (8), we have:

  p CE(C + κY ) ≥ CE(C) + κ t Eφ0 (C) EY − Var(φ0 (C))Var(Y ) −

1 Lκ2 t EY 2 2 

≥ CE(C) + κt

 p 1 φ (¯ c)EY − L Var(C) − Lκ , (9) 2 0

where the last inequality used the facts that Var(Y ) ≤ EY 2 ≤ 1 from the assumptions on Y ; that Eφ0 (C) ≥ φ0 (¯ c) from the assumption that C assumes values in [c, c¯] and the concavity of φ; and that Var(φ0 (C)) ≤ L2 Var(C), which follows from the fact that L is the Lipschitz constant of φ0 on [c, c¯] and the assumption on C. √ √ c) − L δ − Choosing δ, η, u such that φ0 (¯ c) − L δ − Lη/2 > 0 and (φ0 (¯ Lη/2)/φ0 (c) > u > 0, we get from (9) and the fact that t = 1/φ0 (CE(C))) ≥ 1/φ0 (c) that CE(C + κY ) ≥ CE(C) + uκ when Var(C) < δ, EY >  and κ < η, as desired. 15

3.3.2

Uniform consistency of the Bayesian Estimator

Let ft be the empirical distribution of consumption levels at time t, so that ft is the Ft -measurable ∆(C)-valued random variable given by ft (c) =

#{1 ≤ s ≤ t : Xs = c} . t

(10)

The following theorem, due to Diaconis and Freedman (1990), captures the intuitive idea that in the long run the data swamps the prior and the posterior belief about the parameter is concentrated around the empirical distribution. In what follows, we use | · | to denote the max norm for finite dimensional vectors. Proposition 5. Assume that Ft = σ(X1 , . . . , Xt ) for every t, and that the prior distribution µ0 of Θ has a density that is bounded away from zero. Then for every h > 0 there exists t¯ such that
P |Θ − ft | < h Ft > 1 − h a.s. For every t ≥ t¯. The result is not obvious and requires regularity assumptions about the initial belief µ0 .5 It is of course intuitive that the empirical frequency ft should be close to the true parameter Θ when t is large. The strength of the proposition lies in requiring that this occurs after any sequence of observed outcomes, including sequences that are unlikely to occur under µ0 . Contrast this with the weaker statement: P (|Θ − ft | < h) > 1 − h for t > t¯, which follows from a martingale argument and requires no assumption on µ0 or on the filtration. For our purposes, we will need the following immediate corollary of Proposition 5: Corollary 6. Assume that Ft = σ(X1 , . . . , Xt ) and that the prior distribution µ0 of Θ has a density that is bounded away from zero. Then for every h > 0 there exists t¯ such that P (|e(Θ) − e(ft )| < h |Ft ) > 1 − h a.s. for every t ≥ t¯. 5

We use the stronger assumption that µ0 has a density that is bounded away from zero for clarity. It clearly implies the weaker assumption used in the proof in Diaconis and Freedman (1990).

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3.3.3

Upper bound

We first prove that: (β) lim sup Zˆ0 ≤ CE(e(Θ)),

(11)

β→1

From Proposition 4 it is sufficient to prove the assertion for the case that F1 = F2 = . . . = σ(X1 , X2 , . . .), i.e., when all uncertainty is resolved at period 1. In this case ! ∞ X (β) t−1 Zˆ = CE β (1 − β) Xt . 0

t=1

P t−1 X By the weak law of large numbers the random variable β ∞ t t=1 (1 − β) converges to e(Θ) in distribution. Therefore (11) follows from the continuity of the Epstein-Zin operator 3.3.4

Lower bound

In this section we prove that (β)

lim inf Zˆ0 β→1

≥ CE(e(Θ)).

(12)

From Proposition 4 it is sufficient to prove the assertion for the case that Ft = σ(X1 , . . . , Xt ) for t ≥ 1, i.e. that the only information available at period t is the realization of the process up to that period. Next, we prove the following claim: Given Ft = σ(X1 , . . . , Xt ), then for every 0 < λ < 1 there exists t¯ and η > 0 such that (β) Zˆt ≥ λ CE(e(Θ)|Ft ),

(13)

for every t ≥ t¯ and every β > 1 − η. To prove the claim, write ζt = CE(e(Θ)|Ft ) and let  < c (1/λ−1) and let u, η, δ > 0 as in Lemma 3.1. Fix h > 0 with the following property: If X is a random variables which assumes values √ in [c, c¯] and P (|X − e| < h) > 1 − h for some e ∈ [c, c¯] then |CE(X) − e| < δ/2. The existence of such h follows from the continuity of the certainty equivalent and the certainty equivalence of the constant. Let ft be the empirical distribution of consumption levels at time t, so that ft is the Ft -measurable ∆(C)-valued random variable given by (10). Let t¯ be large enough so that P (|e(Θ) − e(ft )| < h |Ft ) > 1 − h a.s. 17

for every t ≥ t¯. The existence of such T follows from Corollary 6. From the choice of h it follows that √ |ζt − e(ft )| < δ/2, a.s. √ ¯ to also satisfy t¯ > 2/ δ then for every t ≥ 1. Moreover, if we choose t √ |e(ft ) − e(ft−1 )| < δ/2 (that is, the empirical distribution ft at time t can’t be too far √ from the empirical distribution ft−1 ), and so it follows that |ζt − e(ft−1 )| < δ a.s., which implies that Var(ζt |Ft−1 ) < δ

(14)

since ft−1 is Ft−1 -measurable. We are now in a position to use the global attraction property to prove (13). Fix β > 1 − η so that (1 − β) < η. Following the notations of Section 3.1.3, let G be the subset of X of all processes (Z0 , Z1 , . . .) such that Zt ≥ λζt for every t ≥ t¯. We need to prove that Zˆ (β) ∈ G. From the global attraction property, it is sufficient to prove that T (Z) ∈ G whenever Z ∈ G. Indeed, if Z = (Z0 , Z1 , . . .) ∈ G then T (Z)t−1 = CE((1 − β)Xt + βZt |Ft−1 ) ≥ CE((1 − β)Xt + βλζt |Ft−1 ) = λ CE(ζt + (1 − β)Y |Ft−1 ).

(15)

where Y = Xt /λ − ζt . The first equality follows from the definition of the Epstein-Zin operator T , the inequality follows from the assumption that Z ∈ G, and the second equality follows from homegenuity of the certainty equivalent. Now E(ζt |Ft−1 ) = E(CE(e(Θ)|Ft )|Ft−1 ) ≤ E(E(e(Θ)|Ft )|Ft−1 ) = E(e(Θ)|Ft−1 ) = E(Xt |Ft−1 ), where the first equality is the definition of ζt , the inequality follows from risk aversion, the second equality is the law of iterated expectation, and the last equality follows from the assumption that Ft−1 = σ(X0 , . . . , xt−1 ). Therefore E(Xt − ζt |Ft−1 ) ≥ 0 and E(Y |Ft−1 ) = E(Xt /λ − ζt |Ft−1 ) ≥ c(1/λ − 1) + E(Xt − ζt |Ft−1 ) >  (16) 18

since Xt ≥ c and from the choice of . Therefore, from (14), (15), (16), and Lemma 3.1, we get T (Z)t−1 ≥ λ CE(ζt ) = λζt−1 , where the equality follows from the law of iterated certainty equivalent. This completes the proof of the claim. Returning to the proof of (12), let λ < 1 and let t¯ be as in the claim above. Then for every β   ¯ ¯ (β) (β) Zˆ0 ≥ β t CE Zˆt¯ ≥ β t λ CE(e(Θ)) −−−→ λ CE(e(Θ)). β→1

Since this holds for every λ < 1, (12) follows.

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