Recursive Risk Sharing: Microfoundations for RepresentativeAgent Asset Pricing∗

David K. Backus,† Bryan R. Routledge,‡ Stanley E. Zin§

First draft: October 12, 2006 This draft: September 14, 2007

Abstract We explore the properties of Pareto optimal allocations when agents have heterogeneous recursive preferences. The dynamics of individual consumption growth reflect not just standard mean-variance tradeoffs as in the expected-utility model, but also tradeoffs involving the timing of the resolution of uncertainty. We also explore the implications of these optimal allocations for the aggregate asset-pricing kernel. In our specific Gaussian log-linear environment, the representative agent will appear to have recursive preferences of the same form as the individual agents. However, the representative agent’s preference parameters will reflect both the heterogenous preference parameters of the individuals and the parameters governing the stochastic process for income growth. Empirical findings of unusual values for the representative agent’s preference parameters, therefore, can be a reflection of this co-mingling of individual preferences and the dynamics of the opportunity set. Moreover, evidence of parameter instability of the representative agent’s preferences may simply reflect changes in the dynamics of income growth. JEL Classification Codes: D81, D91, E1, G12. Keywords: aggregation,time preference; risk preference; risk sharing; asset pricing

∗

We thank Tom Sargent for helpful comments and encouragement. Thanks also to seminar participants at Carnegie Mellon University and at the 2007 SED meetings in Prague, CZ. This paper is preliminary – more so than most working papers – so comments are welcome and apologies for the rough edges. † Stern School of Business, New York University, and NBER; [email protected]. ‡ Tepper School of Business, Carnegie Mellon University; [email protected]. § Tepper School of Business, Carnegie Mellon University, and NBER; [email protected].

1

Introduction

Recursive preferences in representative agent settings are helpful (even necessary) for understanding macro-economic facts like dynamic properties of the equity premium (e.g., Routledge and Zin (2004), Bansal and Yaron (2005)). However, these models effectively must assume that all agents in the economy have the same preference parameters. This assumption is more prominent since much of the debate in representative-agent asset pricing hinges on functional forms and parameter values. For example, much of the equity premium puzzle is that the risk-aversion parameter implied by a representative agent model is “too high.” (Mehra and Prescott (1985)). More recently, Bansal and Yaron (2005) fit asset prices only with an elasticity of intertemporal substitution larger than one. In this paper we explore the aggregation properties of recursive preferences. We consider multiple agents with recursive preferences in the CES class of EpsteinZin and solve the optimal risk sharing or Pareto allocations. From this we can determine the properties of a representative agent. In particular, we show that, in general, a representative agent with recursive preferences in the same CES class exists. In addition we show that the representative agent preference “parameters” depend on, naturally, the preferences of the agents in the economy. However, the representative agent parameters also depend on the assumption about the income endowment process. Therefore, caution is warranted when assessing the parameters of a representative agent implied from asset prices. Their unusual characteristics may simply be the result of aggregation of heterogeneous, but “reasonable,” agents. To explore aggregation we need to solve for the Pareto optimal consumption allocations. Allocation problems or risk-sharing solutions are of fundamental interests to many problems in macroeconomics and finance. In particular, through the Welfare Theorems, they offer insight into competitive equilibrium and aggregation in asset pricing. However, such problems are also at the core of international trade puzzles, dynamic contracting problems, and models of incomplete markets. Unfortunately, even in an endowment economy, where the resource constraint is exogenous, this allocation problem is difficult to solve, particularly in closed form. Even in a static expected-utility setting,closed-form expressions to the Pareto problem are typically not available. As is well-known, the first-order conditions are linear in logs and the

1

resource constraint is linear in levels. In dynamic problems with recursive preferences the difficulty is exacerbated since optimality involves considering innovations to future utility. The value function typically is not available in closed form in recursive problems. We solve the Pareto problem with a log-linear approximation. The paper is organized by starting with well-known static example and dynamic example with time-adaptive expected utility examples. Next we turn to characterizing the risk-sharing problem in a recursive utility setting using a log-linearizion. The solution to this problem lets us explore the pricing kernel and the implied representative agent preferences.

2

Recursive Utility Pareto Problem

To consider a dynamic Pareto sharing problem, let yt be a stochastic process for income transition probabilities p(yt |yt−1 , ...). We use the usual tree setup with states s and histories st . The short-hand notation, xt means x(st ) for some specific history st . In addition we use Et and µt to indicate expected value and certainty equivalents calculated based on date-t information in st . When convenient, we use x′ for to indicate next-period, xt+1 . We focus on the the Pareto problem for agents with recursive preferences (as in Kreps and Porteus (1978) and Epstein and Zin (1989)). This allows us to explore how differences in risk aversion and intertemporal substitution influence consumption and aggregation. Recursive preferences, in the usual Epstein-Zin setting, are characterized by a risk-aversion parameter α < 1 (coefficient of relative risk aver ion of 1 − α), an intertemporal substitution parameter, ρ < 1 (intertemporal elasticity of substitution of 1/(1 − ρ)), and a discount factor β. Throughout the paper, we consider allocating income, yt , for consumption, ci,t , of agents i = 1, 2. We focus on two agent examples. However, it is straightforward to consider additional agents. Preferences for the agent 1 are Wt =

h

(1 −

β)cρ1,t1

ρ1

+ βµ1,t (Wt+1 )  1 α1 α1 µ1,t (Wt+1 ) = Et Wt+1 2

i

1 ρ1

and for agent 2 are Vt =

h

(1 − β)cρ2,t2 + βµ2,t (Vt+1 )ρ2 1  α2  α2 µ2,t (Vt+1 ) = Et Vt+1

i

1 ρ2

The form of the two agents utility is identical. However, we use W for agent one and V for agent two. Calculating the inter-temporal marginal rate of substitution for recursive preferences is an exercise in the chain rule. Define, m1,t+1 = β1



c1,t+1 c1,t

ρ1 −1 

Wt+1 µ1,t (Wt+1 )

α1 −ρ1

(1)

For agent 1, the M RS1,t+1 = p(st+1 |st )m1,t+1 . Since agents agree on probabilities throughout the paper, we work with the m-process. In particular, m is a pricing kernel. The specification of m2,t+1 for agent 2 is similar. In the log-linear model in the next section, we equate marginal rates of substitution to characterize optimality. Equation (1) illustrates an important feature of recursive preferences. The MRS includes the growth rate in consumption. This is familiar from time-additive expected utility. In addition, recursive preferences include a term that compares the innovation in next-period’s utility with its certainty equivalent. Note that this second term goes away when α1 = ρ1 and preferences are the familiar time-additive expected utility. A natural way to state the Pareto problem is to maximize the weighted average of date-0 utility. max λW0 + (1 − λ)V0

{c1,t ,c2,t }

s.t. c1,t + c2,t = yt

for all st

However, this problem is not recursive and therefore does not lead to time-consistent planning. To impose time-consistent planning, the problem must be written as a constrained optimization. Following, Lucas and Stokey (1984), and Kan (1995), the

3

two-agent Pareto problem is J(V, s) =

max

c1 ,c2 ,V

′ (s′ )

1

[(1 − β)cρ11 + βµ1,t (J(V ′ , s′ ))ρ1 ] ρ1 1

[(1 − β)cρ22 + βµ2,t (V ′ )ρ2 ] ρ2 ≥ V

s.t.

(2)

c1 + c2 = y(s) where s is the current information set, µi,t is the certainty equivalent based on information s, and s′ is next periods information set. The optimal policy involves choosing agent two’s consumption, c2 , and a future utility V ′ (one for each future state, s′ ) to maximize agent one’s utility given an initial promise to agent 2 of utility, V . The promise serves as a law of motion for the state variable, V .

3

Recursive Utility Pareto Problem - Log-linear approximations

As is the case with time additive expected utility, the Pareto problem does not typically have a closed-form solutions. To attack the problem, we use a log-linear approximation. To see how it works, we start with a static setting and then time additive expected utility. We start with expected utility to highlight what part of the log-linear approximation is common to the Pareto problem and what part is specific to recursive utility.

3.1

Static Deterministic Example

To start, consider a simple Pareto or allocation problem that is static and deterministic. Allocate income, y, to consumption, ci , to agents i = 1, 2 with Pareto weight λ. max

{c1 ,c2 }

s.t.

cρ11 cρ2 + (1 − λ) 2 ρ1 ρ2 c1 + c2 = y

λ

(3)

This is the familiar statement of the Pareto problem. We could, equivalently, specify the problem in constraint form as with equation (2). Optimality is implied by the 4

first-order condition (ρ1 − 1) log c1 − (ρ2 − 1) log c2 = log



1−λ λ



and the resource constraint c1 + c2 = y. The first-order condition, of course, equates the marginal utilities of the two agents (weighted by the Pareto weight). This is illustrated in Figure 1. Despite the extreme simplicity in this example (no risk), the figure illustrates the challenges to the general problem. First note that the first-order-condition is linear in logs and the resource constraint is linear in levels. A closed-form solution for is not available outside the case of identical agents (ρ1 = ρ2 ). Next, note that in the case ρ1 6= ρ2 , one of the agents will receive less of the aggregate resource, y. In this case, for larger y (compare (a) to (b) in Figure 1)., it is optimal for only one of the agents to receive a small fraction of y.This points out that in an economy with a growing income, consumption shares are not likely to be stationary. To construct an analytical solution to the optimality condition, there are two choices for an approximation. We can approximate the first-order condition with a linear function. Alternatively, we can approximate the linear resource constraint with a log function. Although it seems counter intuitive, it turns out that this approach is most helpful. A first-order Taylor approximation of the resource constraint around c¯1 and y¯ is k log(c1 ) + (1 − k) log(c2 ) ≈ k¯ + log(y)

(4)

where k = c¯1 /¯ y and k¯ = k log(k) + (1 − k) log(1 − k). Appling this approximation to the first-order condition gives the approximate solution log c1 log c2



 1 − ρ2 ≈ log y + constant 1 − ρˆ   1 − ρ1 log y + constant ≈ 1 − ρˆ

where ρˆ = kρ2 + (1 − k)ρ1 . This solution is linear in logs. Log-consumption shares depend on the curvature parameter, ρ, relative to the weighted-average ρ. For 5

the approximation to be accurate, k, needs to be chosen carefully. In particular, we would like c1 = c¯. This is, of course, common in any Taylor approximation approach. Solving this numerical fixed-point calculation is straightforward. (As we will see below, this is a bit more complicated in a dynamic setting.)

3.2

Static Expected Utility Example

Extending the previous example to the case of random income is straighforward. Denote the random variable y(s) for s = 1, ..., S with probabilities, p(s). The Pareto problem is max

c1 (s),c2 (s)

s.t.

λ

ρ s p(s)c1 (s)1

P

+ (1 − λ)

P

ρ s p(s)c2 (s)2

(5)

c1 (s) + c2 (s) = y(s), s = 1, ..., S

Since the individuals agree on probabilities, p(s), the solution is state-by-state the same as the deterministic problem in equation (3). That is, we can approximate the solution as 

 1 − ρ2 log c1 (s) ≈ log y(s) + constant 1 − ρˆ   1 − ρ1 log c2 (s) ≈ log y(s) + constant 1 − ρˆ As you might expect, the log-approximation in the static expected utility case is quite good. The approximation becomes a bit more challenging in the dynamic setting. [todo: Inset Figure 1]

3.3

Dynamic Time-Additive Expected Utility Example

Now consider a dynamic setting, with income yt with probabilities p(yt |yt−1 , ...). For two agents with time-aditive expected utility, the Pareto problem is max

{c1,t },{c2,t }

s.t.

cρ1 t 1,t β1 λE0 ρ1 t=0 ∞ X

+ (1 −

cρ2 t 2,t β2 λ)E0 ρ2 t=0 t

c1,t + c2,t = yt , for all s 6

∞ X

(6) (7)

Since time-additive expected utility is linear in both time and states, the Pareto problem in (6) is period-by-period and state-by-state equivalent to the deterministic problem in equation (3). Optimality is implied by the first-order condition (ρ1 − 1) log c1,t − (ρ2 − 1) log c2,t = log



1−λ λ



+ t log



β2 β1



and the resoruce constraint, c1,t + c2,t = yt . Using the same Taylor approximation of the resource constraint as in the static example, equation (4), log-consumption -growth is linear in log-income. 

         c1,t+1 1−k ρ2 − 1 β2 yt+1 ≈ − + log log c1,t ρˆ − 1 β1 ρˆ − 1 yt           c2,t+1 k ρ1 − 1 β2 yt+1 log ≈ + log log c2,t ρˆ − 1 β1 ρˆ − 1 yt log

where, recall, ρˆ = kρ2 + (1 − k)ρ1 and k is an approximation coefficient. We come back to the role of k in a moment. To introduce the example we work with extensively below, consider a situation where income growth is an infinite order moving average: log(yt+1 /yt ) = χ ¯+

∞ X

χj ǫt+1−j

(8)

j=0

with {ǫt } ∼ NID(0, 1) and

P

j

χ2j < ∞ (“square summable”). This is general enough

to allow a wide variety of dynamics. Given this dynaimc, we “conjecture” (it is actually quite obvious) that log(c1,t+1 /c1,t ) = f¯ + log(c2,t+1 /c2,t ) = g¯ +

∞ X

j=0 ∞ X

fj ǫt+1−j

(9)

gj ǫt+1−j

(10)

j=0

The log-linear approximation of the resource constraint in equation (4) can be restated to growth rates as k log(c1,t+1 /c1,t ) + (1 − k) log(c2,t+1 /c2,t ) = log(yt+1 /yt ) 7

(11)

The resource constraint implies kfj + (1 − k)gj

= χj

j≥0

kf¯ + (1 − k)¯ g = χ ¯

We can solve for the coefficients in equations (9) and (10) that solve the optimality condition. They are: fj gj



 ρ2 − 1 = χj ρˆ − 1   ρ1 − 1 = χj ρˆ − 1

and    β2 ρ − 1 2 χ ¯ − (1 − k) log f¯ = ρˆ − 1 β1     ρ1 − 1 β2 g¯ = χ ¯ + k log ρˆ − 1 β1 

where, recall, ρˆ = kρ2 + (1 − k)ρ1 . [Check f¯ and g¯] With time additive expected utility, the dynanmic prperites in income growth are passed one-for-one to each agent. There is nothing interesting in the consumption allocation dynamics. For example, sicne consumption is a constant share of income, the agent with the larger conditional mean consumption will also have a larger conditional variance in consumption growth. The development here is useful since we use the same approach for the recursive utility Pareto problem below.

3.3.1

Aggregation

As in the linear-Leonteif example in the previous section, having solved for the income allocations, we can now describe asset prices. In particular, optimality in 8

the Pareto problem implies m1,t+1 = m2,t+1 = mt+1 . Using equation (1) implies: log mt+1 = log β1 + (ρ1 − 1)f¯ + (ρ1 − 1)

∞ X

fj ǫt+1−j

j=0

∞

(ρ1 − 1)(ρ2 − 1) (ρ1 − 1)(ρ2 − 1) X = ... + χj ǫt+1−j χ ¯+ ρˆ − 1 ρˆ − 1 j=0

By the usual logic,a pricing kernel for a representative agent with time additive preferences with preference parameters β˜ and ρ˜ is log m′ = log β˜ + (˜ ρ − 1) log(y ′ /y). Therefore, our economy with two agents has a representative agent with preferece paramters (ρ1 − 1)(ρ2 − 1) ρˆ − 1 β˜ = ...

ρ˜ − 1 =

This is the standard result with time additive expected utility. The representative agents parameters are simply a weighted average of the agents parameters. The weighting depends on the cross-section of income (see the discussion of the approximation parametr, k, below), but does not depend on the income process itself. In particular, note that the χj do not matter for aggegation. It turns out, as we will explore below, that this is particular to the time additive expected utility case and does not hold for general recursive preferences.

3.3.2

Income Distribution

From the solution above, the differences in the drift of the two agnet’s consumption is

−1 f¯ − g¯ = ρˆ − 1



(ρ1 − ρ2 )χ ¯ + log



β1 β2



(12)

If β1 = β2 , the solution to the Pareto problem in (6) is a sequence of static problems. In this case, if income growth is mean-zero, χ ¯ = 0, there is no drift in consumption of either agent. if income has positive drift, χ ¯ > 0, then the cross-sectional distribution degenerates asymptotically and the agent with the smallest coefficient of relative risk aversion (largets ρ) gets everything in the limit. Note the same thing happens even in the static and deterministic case. As y increases, the agent with the larger ρ gets 9

a larger share of income. Lastly, in the case where β1 6= β2 , then the combination of the two preference parameters determines the relative drifts in consumption growth as in equation (12). The fact that consumption shares, in general, drift at different rates (f¯ 6= g¯), means that the approximation for the resource constraint must adjust the level of k. For example, in equation (11), if k = 0.5, the equations imply that if income grows by 5%, and person one’s consumption grows by 5%, then person two’s consumption growth is pinned at 5%. As an approximation of the resource constraint, this is sensible in the region where the consumption of the two agents is similar. [We return to this issue in our numerical implimentation. ]

3.4

Recursive Utility

We now return to the general recursive utility Pareto problem introduced in Section 2 in equation (2). The first-order conditions to the recursive Pareto problem in (??), among other things, imply that the marginal rates of substitution for the two agents should be state-by-state equal. That is β1



c′1 c1

ρ1 −1 

W′ µ1 (W ′ )

α1 −ρ1

= β2



c′2 c2

ρ2 −1 

V′ µ2 (V ′ )

α2 −ρ2

(13)

(using ′ to denote next period where the certainty equivalent functions is understood to reflect conditional expectations). To begin, re-scale preferences by the consumption consumption. Define x′1 = c′1 /c1 , x′2 = c′2 /c2 , w = W/c1 , v = V /c2 . Therefore, 1/ρ1 (1 − β1 ) + β1 µ1 (w′ x′1 )ρ1  1/ρ2 v = (1 − β2 ) + β2 µ2 (v ′ x′2 )ρ2

w =



The optimality condition in equation (13) is
ρ −1 β1 x′1 1



w′ x′1 µ1 (w′ x′1 )

α1 −ρ1

=

10


ρ −1 β2 x′2 2



v ′ x′2 µ2 (v ′ x′2 )

α2 −ρ2

(14)

We proceed in the same fashion as in section 3.3 above. Assume endowment growth is an infinite order moving average process as in equation (8). Conjecture a similar process for the consumption growth for the two agents as in equations (9) and (10). We use the same log-linear approximation of the resource constraint as in equation (11). The innovation over time-additive expected utility is that the marginal rate of substitution includes the utility terms. A log-linear approximation of v around µ2 = v¯ is: log v ≈

ρ−1 2 log [(1

− β2 ) + β2 exp(ρ2 v¯)] +

= κ2,0 + κ2 log µ2



β2 exp(ρ2 v¯) 1 − β2 + β2 exp(ρ2 v¯)



(µ2 − v¯)(15) (16)

Note this is an exact equality when ρ2 = 0 (i.e., a log time-aggregator). In this case (or when v¯ = 0), κ2 = β2 . We focus on this example below. Since preferences are recursive, we guess that log vt = ν¯ +

∞ X

νj ǫt−j

j=0

and verify the conjecture with the calculation log v = κ2,0 + κ2 log µ(v ′ c′2 /c2 ) = κ2,0 + κ2

ν¯ + 21 α2 (ν0 + g0 )2 +

∞ P

!

(νj+1 + gj+1 )ǫt−j + g¯

j=0

(17)

The conjecture is verified and we can solve for the νj terms by recursive substitution. It turns out below, that ν0 is important (the others are not). Matching up terms and iteratively substituting, gives ν0 =

∞ X

κj2 gj

j=1

Since κ2 < 1, the sum is well-defined. Repeat these steps for agent 1; log w = ω ¯+

∞ X j=0

11

ωj ǫt−j

Again, the important coefficient is ω0 and is ω0 =

∞ X

κj1 fj

j=1

We can now solve for the new term in the MRS condition in (13)  w′ x′1 α1 log = − (ω0 + f0 )2 + (ω0 + f0 )ε′ ′ ′ µ1 (w x1 ) 2   ′ ′ α2 v x2 = − (ν0 + g0 )2 + (ν0 + g0 )ε′ log µ2 (v ′ x′2 ) 2 

Take logs of (13) log β1 + (ρ1 − 1) f¯ +

∞ P

fj ǫt+1−j

j=0

!


2 +(α1 − ρ1 ) (ω0 + f0 )ǫt+1 − 12 α1 (ω! 0 + f0 ) ∞ P = log β2 + (ρ2 − 1) g¯ + gj ǫt+1−j j=0
+(α2 − ρ2 ) (ν0 + g0 )ǫt+1 − 12 α2 (ν0 + g0 )2

(18)

solve by mathcing terms and imposing a imposing the the resource constraint, kgj + (1 − k)fj = χj . Consumption allocations for the two agents are given by parameters f0 = g0 =

(α2 − 1)χ0 (1 − k) ((α1 − ρ1 )ω0 − (α2 − ρ2 )ν0 ) − α ˆ−1 α ˆ−1 (α1 − 1)χ0 k ((α1 − ρ1 )ω0 − (α2 − ρ2 )ν0 ) + α ˆ−1 α ˆ−1

where α ˆ = kα2 + (1 − k)α1 and k is our approximation coefficient on the resource constraint. Next, for j ≥ 1 fj = gj

=

ρ2 − 1 χj ρˆ − 1 ρ1 − 1 χj ρˆ − 1

where, as before, ρˆ = kρ2 + (1 − k)ρ1 . We can now calculate the utility terms ω0

12

and ν0 ∞

ω0 = ν0 =

ρ2 − 1 X j κ1 χj ρˆ − 1 ρ1 − 1 ρˆ − 1

j=1 ∞ X

κj2 χj

j=1

Lastly, we solve for the drift in consumption allocations, f¯ = g¯ =

1−k ρ2 − 1 χ ¯− D ρˆ − 1 ρˆ − 1 k ρ1 − 1 χ ¯+ D ρˆ − 1 ρˆ − 1

where D =



1 log β1 − α1 (α1 − ρ1 )(ω0 + f0 )2 2





1 − log β2 − α2 (α2 − ρ2 )(ν0 + g0 )2 2



So, that happened. With the characterization of the optimal allocations we can explore how the preference parameters influence asset prices and the evolution of the consumption shares. To get an idea for the properties of the model, we consider a few parametric examples.

3.4.1

Example: Focus on risk-aversion: ρ1 = ρ2 = 0, β1 = β2

First, we focus on risk-aversion, holding constant intertemporal elasticity and discount factor. We can see how differences in risk aversion affect asset prices. Let ρ1 = ρ2 = 0 and β1 = β2 = β. Considering the case of a log-time aggregation, ρ1 = ρ2 = 0, is a good place to begin since the utility representation in (16) is exact. Moreover, assuming β1 = β2 = β also implies κ1 = κ2 = β. This implies the utility P j terms ω0 = ν0 = ∞ j=1 β χj . Therefore, the consumption allocations are governed by:

f0 =

(1 − α2 )χ0 (1 − k) (α2 − α1 ) − χ(β) 1−α ˆ 1−α ˆ 13

(1 − α1 )χ0 k (α2 − α1 ) + χ(β) 1−α ˆ 1−α ˆ = gj = χj , for j ≥ 1

g0 = fj

where, as before, α ˆ = kα2 + (1 − k)α1 . Lastly, to solve for the drift in consumption P j allocations, define the operator, χ(β) = ∞ j=0 β χj and note ω0 + f0 ν0 + g0



 1 − α2 = χ(β) 1−α ˆ   1 − α1 = χ(β) 1−α ˆ

The drifts are given by f¯ = χ ¯ + (1 − k)D g¯ = χ ¯ − kD where D =

1 2



α22 (1 − α1 )2 − α21 (1 − α2 )2 (1 − α ˆ )2



χ(β)2

Note f¯− g¯ = D. Hence, in the case where α1 < 0 and α2 < 0, α2 < α1 implies D > 0 and agent 1’s consumption grows faster than agent 2. In this case, the cross-section distribution degenerates to the point where agent 1 gets everything (in the limit). It is not surprising that the consumption of the two agents have the same sensitivity to past shocks, fj = gj = χj j ≥ 1, since they have an identical elasticity of intertemporal substitution. The sensitivity to current shock, f0 and g0 , reflect the different attitudes to risk. Interestingly, the attitudes to risk do not easily translate into a simple mean-variance interpretation. Based on the optimal allocation, calculate the difference in conditional means and conditional variances Et log(

c1t+1 c2t+1 ) − E log( ) t c1t c2t =

  1 α22 (1 − α1 )2 − α21 (1 − α2 )2 χ(β)2 2 (1 − α ˆ )2 14

and Vt log(

=



c2t+1 c1t+1 ) − V log( ) t c1t c2t

χ0 + (1 − k)



α1 − α2 1−α ˆ



 2   2 α1 − α2 χ(β) − χ0 − k χ(β) 1−α ˆ

Note that the difference in conditional mean growth rates is simplified but the fact that both agent’s consumptions weight past shocks identically (fj = gj for j ≥ 1). For the case of α1 < 0 and α2 < 0, α2 < α1 implies that agent one has a larger expected growth rate of consumption. However, this does not imply that agent one also has a higher conditional variance in log-consumption growth. Recall that for time additive expected utility preferences, the familiar mean-variance tradeoff applies. aggregatin solve for pricing kenrel pictures 2. consumption dynamics

3.4.2

Example: Focus on intertemporal differences: α1 = α2 = 0, β1 = β2

Next, we focus on differences in intertemporal elasticity of substitution (along deterministic paths) while holding constant the risk aversion levels. Focusing on log risk aversions makes the analysis somewhat tractable. However, the approx coed kappa matters. LOOK AT TERMS...

3.5

put this somewhere

Since equation (18) equates the marginal rates of substitution for the two agents, this equation also gives the pricing kernel. 15

log mt+1 = log βa + (ρa − 1) g¯ + g0 ǫt+1 +

∞ P

gj ǫt+1−j
+(αa − ρa ) (η0 + g0 )ǫt+1 − 12 αa (η0 + g0 )2 j=1

!

= log βa + (ρa − 1)¯ g − 0.5(αa − ρa )αa (η0 + g0 )2 ∞ P +(ρa − 1) gj ǫt+1−j

(19)

j=1

+ [(αa − 1)g0 + (αa − ρa )η0 ] ǫt+1

(By similar logic, we can calculate the pricing kernel from a representative agent ˜ ρ˜, and α economy where the preference parameters are β, ˜ log mt+1 = log β˜ + (˜ ρ − 1)¯ x − 0.5(˜ α − ρ˜)˜ α(χ0 + X0 )2 ∞ P +(˜ ρ − 1) gj ǫt+1−j j=1

(20)

+ [(˜ α − 1)χ0 + (˜ α − ρ˜)X0 ] ǫt+1

where X0 =

∞ P

κ ˜j gj . Note that this term depend on the approximation of the

j=1

utility function since κ ˜ is defined by the approximation of i1/˜ρ h ˜ + βµ ˜ t (˜ v˜ = (1 − β) v ′ x′ )ρ˜ A log-linear approximation [around???] is: log v˜ = κ ˜0 + κ ˜ log µ

3.6

Numerical analysis

Make a table drift means variances pricing kernel all for different par mater values. implement the dynamic-k model Check results: An advantage to approximating resource constraint (rather than foc) is that it is easy to check our approximation. Just simulate paths and see how big (variable) 16


the error is. a simple dynamic approximation with kt = kt−1 exp f¯t−1 − χ ¯ should work well to ensure exp(log(c1,t )) + exp(log(c2,t )) ≈ exp(log(yt )) over many sample paths. Preference Aggregation and the Pricing Kernel...

17

References Bansal, R., and A. Yaron (2005): “Risks For the Long Run: A Potential Resolution of Asset Pricing Puzzles,” Journal of Finance (forthcoming), Wahrton Working Paper. Epstein, L. G., and S. E. Zin (1989): “Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework,” Econometrica, 57(4), 937–969. Kan, R. (1995): “Structure of Pareto Optima When Agents Have Stochastic Recursive Preferences,” Journal of Economic Theory, 66(2), 626–631. Kreps, D. M., and E. L. Porteus (1978): “Temporal Resolution of Uncertainty,” Econometrica, 46, 185–200. Lucas, R. E., and N. L. Stokey (1984): “Optimal Growth with Many Consumers,” Journal of Economic Theory, 32(1), 139–171. Mehra, R., and E. Prescott (1985): “The Equity Premium: A Puzzle,” Journal of Monetary Economics, 15, 145–161. Routledge, B. R., and S. E. Zin (2004): “Generalized Disappointment Aversion and Asset Prices,” NBER Working Paper No. w10107.

18

Figure 1: Risk sharing in time-aditive expected utility. The optimal allocations are seperable in time and states. Optimality implies marginal utilities are equated. This depends on Pareto weight, aggregate income, and the degree of heterogeneity. Shown is the case where ρ1 = 0.5 and ρ2 is 0.5, 0 (log), or -1. Aggregate income is y = 10 (a) or y = 20 (b) (a) y=10

0.5

Marginal Utilities

ρ1=0.5

λ(y−c)ρ1−1

ρ −1

(1−λ) c

2

0.4

0.3

0.2

ρ =0.5 2

0.1

ρ =0.0 2

0

ρ =−1.0 1

2

3

4

5

6

7

8

9

2

c = person 2 consumption

(b) y=20

0.5

Marginal Utilities

λ(y−c)ρ1−1

ρ −1

(1−λ) c

2

ρ1=0.5

0.4

0.3

0.2

ρ2=0.5

0.1

0

ρ2=0.0 ρ =−1.0 2

4

6

8

10

12

14

c = person 2 consumption

19

16

18

2

Figure 2: Representive agent alpha “risk” parameter implied by opitimal risk sharing. Shown is the solution to equation (XX) for various income processes

INCOME: x~ARMA(1,1) φ=0.9 PREFS: β =β =0.98 , ρ =ρ =0, α =0 1

2

1

2

1

1

Representative Agent α

0

−1

−2 θ=−1.0 θ=−0.9 IID θ=−0.8 θ=−0.7 θ=−0.6 θ=−0.5 θ=−0.4 θ=−0.3 θ=−0.2 θ=−0.1 θ=0.0 AR(1)

−3

−4

−5 −4

−3.5

−3

−2.5

−2

−1.5

α2

20

−1

−0.5

0

0.5

1

Figure 3: Representive agent beta “discount factor” parameter implied by opitimal risk sharing. Shown is the solution to equation (XX) for various income processes

INCOME: x~ARMA(1,1) φ=0.9 PREFS: β =β =0.98 , ρ =ρ =0, α =0 1

2

1

2

1

1

0.9

Representative Agent β

0.8

0.7

0.6

0.5 θ=−1.0 θ=−0.9 IID θ=−0.8 θ=−0.7 θ=−0.6 θ=−0.5 θ=−0.4 θ=−0.3 θ=−0.2 θ=−0.1 θ=0.0 AR(1)

0.4

0.3

0.2

0.1

0 −4

−3.5

−3

−2.5

−2

−1.5

α

2

21

−1

−0.5

0

0.5

1

Who Holds Risky Assets?∗

David K. Backus,† Bryan R. Routledge,‡ Stanley E. Zin§

First draft: June, 2007 This draft: September 12, 2008

Abstract [Re-do/shorten // slip this for now] Preference heterogeneity is a natural explanation for portfolio heterogeneity. In a dynamic environment in which preference heterogeneity is as extreme as possible, we show that intuition about the relationship between risk aversion and holdings of risky assets derived from static choice problems can be very misleading. In equilibrium, an agent with recursive utility who is infinitely risk averse over static gambles will hold a portfolio composed almost entirely of risky assets. Conversely, an agent with recursive utility who is risk neutral over static gambles will hold a portfolio composed almost entirely of risk-free assets. Moreover, there is no added compensation for holding the risky asset since equilibrium asset prices will appear to be generated by a risk-neutral representative agent. This counter-intuitive result highlights the relative roles of static risk preferences and deterministic substitution preferences in recursive utility. Since portfolio choice is fundamentally a decision about intertemporal consumption lotteries, both characteristics are important. Specifically, we show that the preference for the timing of the resolution of uncertainty plays a major role in allocation of investments between risky and risk-free assets. A strong preference for the late resolution of uncertainty translates into a strong preference for smooth consumption paths and, hence, a portfolio choice heavily skewed toward risk-free assets. This strong preference can exist even when the agent is risk neutral with respect to static gambles. Conversely, a strong preference for the early resolution of uncertainty translates into a strong preference for smooth utility which can be achieved even when portfolio choice heavily skewed ∗

We thank Tom Sargent for helpful comments and encouragement. Thanks also to seminar participants at Carnegie Mellon University, Baruch University and at Society for Economic Dynamics (Prague, CZ), the North American Summer Meetings of the Econometric Society (Pittsurgh, PA), and the Northern Finance Association (Kananaskis, AB). This paper is preliminary – more so than most working papers – so comments are welcome and apologies for the rough edges. If you are interested in the related paper “Recursive Risk Sharing: Microfoundations for Representative-Agent Asset Pricing” see: http://beeks.tepper.cmu.edu/rrs.pdf † Stern School of Business, New York University, and NBER; [email protected]. ‡ Tepper School of Business, Carnegie Mellon University; [email protected]. § Tepper School of Business, Carnegie Mellon University, and NBER; [email protected].

toward risky assets. JEL Classification Codes: D81, D91, E1, G12. Keywords: aggregation,time preference; risk preference; risk sharing; asset pricing

2

1

Introduction

Preference heterogeneity is a natural explanation for portfolio heterogeneity. The natural assumption, for example, is that more risk aversion dictates a portfolio more heavily weighted towards bonds. However, in a dynamic setting this intuition can be misleading. In particular, in a dynamic setting with recursive preferences (Epstein and Zin (1989), Kreps and Porteus (1978)), preferences involve both a static risk preferences and deterministic substitution preferences. Since portfolio choice is fundamentally a decision about intertemporal consumption lotteries, both characteristics are important. That is, a portfolio affects the variability of your consumption and the variability of future utility. Models of recursive preferences are difficult to make tractable. This is especially so when considering preference heterogeneity (see the related paper of Backus, Routledge, and Zin (2007)). In this paper, we solve a model explicitly where preference heterogeneity is as extreme as possible. The result underscores how our intuition about the relationship between risk aversion and holdings of risky assets derived from static choice problems can be very misleading. In equilibrium, an agent with recursive utility who is infinitely risk averse over static gambles will hold a portfolio composed almost entirely of risky assets. Conversely, an agent with recursive utility who is risk neutral over static gambles will hold a portfolio composed almost entirely of risk-free assets. Moreover, there is no added compensation for holding the risky asset since equilibrium asset prices will appear to be generated by a risk-neutral representative agent. This counter-intuitive result highlights the relative roles of static risk preferences and deterministic substitution preferences in recursive utility. Since portfolio choice affects intertemporal consumption lotteries, both characteristics are important. Specifically, we show that the preference for the timing of the resolution of uncertainty plays a major role in allocation of investments between risky and risk-free assets. A strong preference for the late resolution of uncertainty translates into a strong preference for smooth consumption paths and, hence, a portfolio choice heavily skewed toward risk-free assets. This strong preference can exist even when the agent is risk neutral with respect to static gambles. Conversely, a strong preference for the early resolution of uncertainty translates into a strong preference for smooth utility which can be achieved even when portfolio choice heavily skewed toward risky assets. 3

The paper structure is straightforward. We consider the Pareto sharing problem in a setting with maximal preference heterogeneity. We can then address aggregation and asset prices and consider decentralization. We focus primarily on a setting with an aggregate income process that is iid. We also consider a income process that is persistent yielding similar results. [Sections to consider adding: (1) General ARMA income process (infinite order MA); (2) Risk Sensitive agent 1 (Stan’s notes)]

1.1

Related literature

To be added... [ Epstein, Anderson, Chapman and Pol..., Coen-Pirrani]

2

Recursive Utility Pareto Problem

To consider a dynamic Pareto sharing problem, let yt be a Markov stochastic process for aggregate income. We use the usual tree setup with states s and histories st . The short-hand notation, xt means x(st ) for some specific history st . In addition we use Et and µt to indicate expected value and certainty equivalents calculated based on date-t information in st . When convenient, we use x0 for to indicate nextperiod, xt+1 . Transition probabilities are, p(st+1 |st ) (or, alternatively p(yt |yt−1 , ...)) are common across all agents. We consider a social planner allocates aggregate income yt across two agents. Preferences for the agent 1 are Wt =

h

(1 −

Vt =

h

(1 − β2 )cρ2,t2 + β2 µ2,t (Vt+1 )ρ2

β1 )cρ1,t1

+ β1 µ1,t (Wt+1 ) 1  α 1  α1 µ1,t (Wt+1 ) = Et Wt+1

ρ1

i

1 ρ1

and for agent 2 are

4

i

1 ρ2

1  α 2  α2 µ2,t (Vt+1 ) = Et Vt+1

The form of the two agents utility is identical. However, we use W for agent one and V for agent two. Preference parameters are ρi ≤ 1, αi ≤ 1 and 0 < βi < 1. The preferences aggregate consumption over time with a CES aggregator. The parameter ρi determines the inter-temporal rate of substitution over deterministic consumption at consecutive dates. The inter-temporal rate of substitution is 1/(1−ρi ) and a small (negative) value of ρi indicates a strong preference to smooth consumption across time. Risk preferences are captured in the risk aggregator or certainty equivalence function µi,t that is conditional on information at date t. Over static gambles, the parameter αi determines the risk aversion (the coefficient of relative risk aversion is 1 − αi ). The preferences are homothetic. In addition, for convenience, utility is scaled to be in units of consumption. That is, if consumption is risk-free and constant c1,t = c, Wt = c (See Backus, Routledge, and Zin (2005) for more on recursive preferences.) Calculating the inter-temporal marginal rate of substitution for recursive preferences is an exercise in the chain rule. For agent 1, define,  m1,t+1 = β1

c1,t+1 c1,t

ρ1 −1 

Wt+1 µ1,t (Wt+1 )

α1 −ρ1 (1)

For agent 1, the marginal rate of substitution is M RS1,t+1 = p(st+1 |st )m1,t+1 . Since agents agree on probabilities throughout the paper, we can focus on the m-process. In particular, m is a pricing kernel. The specification of m2,t+1 for agent 2 is similar. The marginal rate illustrates an important feature of recursive preferences. The MRS includes the growth rate in consumption. This is familiar from time-additive expected utility. In addition, recursive preferences include a term that compares the innovation in next-period’s utility with its certainty equivalent. Note that this second term goes away when α1 = ρ1 and preferences are the familiar time-additive expected utility. The two-agent Pareto problem is a sequence of consumption allocations for each agent {c1,t , c2,t } that maximizes the weighted average of date-0 utilities subject to the aggregate resource constraint which binds at each date and state: max λW0 + (1 − λ)V0

{c1,t ,c2,t }

5

s.t. c1,t + c2,t = yt

for all st

Note that even though each agent has recursive utility, the objective function of the social planner is not recursive (except in the case of time-additive expected utility). However, we can rewrite this as a recursive optimization problem following, Lucas and Stokey (1984), and Kan (1995): 1

J(V, s) =

max

c1 ,c2 ,V 0 (s0 )

[(1 − β)cρ11 + βµ1,t (J(V 0 , s0 ))ρ1 ] ρ1 1

s.t.

[(1 − β)cρ22 + βµ2,t (V 0 )ρ2 ] ρ2 ≥ V

(2)

c1 + c2 = y(s) where s is the current information set, µi,t is the certainty equivalent based on information s, and s0 is next periods information set. The optimal policy involves choosing agent two’s consumption, c2 , and a future utility V 0 (one for each future state, s0 ) to maximize agent one’s utility given an initial promise to agent 2 of utility, V . The promise serves as a law of motion for the state variable, V . The solution to this problem is “perfect” or optimal risk sharing. Since we consider complete and frictionless markets, there is no need to specify the individual endowment process. This is in contrast to many papers that measure financial market fricitons by comparing, for example, the variance of an individual’s consumption to the variance of an individual’s income. In our setting, such a measure is not needed since allocations are, by construction, Pareto efficient.

3

Linear-Leontief Preferences

Recursive preferences allow the flexibility specifying risk aversion, α, and intertemporal substitution, ρ separately. The difference in these two parameters, α − ρ, determines the preference for the timing of the resolution of uncertainty. It turns out that an example with two agents who differ wildly along this dimension is explicitly solvable. To develop a better understanding of how the Pareto problem in equation (2) works, consider preferences W

= min{(c1 , E[W 0 ]}

V

= (1 − β)c2 + β min{V 0 } 6

Agent 1 is risk neutral (α1 = 1) with perfectly inelastic deterministic substitution (ρ1 = −∞). In contrast, agent 2 has infinite risk aversion (α2 = −∞) and perfectly elastic deterministic intertemporal substitution (ρ2 = 1). Formally, we assume that β1 = β2 = β and the two agents have common beliefs. However, note that the limit as the curvature parameter (ρ1 ) goes to infinity eliminates the discount factor for agent 1. Similarly, as agent 2’s curvature parameter (α2 ) goes to infinity, the probabilities are eliminated. This leads to the “min” in agent 2’s preferences. The agents, by construction, are very different. Agent 1 dislikes long-run risk, but tolerates short-run risk. Agent 2 is the opposite. In this extreme setting we can explicitly solve the Pareto problem to see how the consumption allocations vary over time. We can also, interestingly, solve for aggregate asset prices. Surprisingly (perhaps), this setting leads to a simple representative agent.

3.1

IID income endowment

To begin, we consider an aggregate income process that is IID. This simplifies the analysis and facilitates exposition. In the subsequent section we consider a persistent process. For now, let income be IID as yt = y¯ +t with Et = 0. The structure of the problem implies that no other features of the income process matter (the variance, for example).¶ J(V, y) = max0 min{y − c, EJ(V 0 , y 0 )} c,V

s.t.

(1 − β)c + β min{V 0 } ≥ V

That is choose agent two’s consumption, c, and future utility, V 0 (one for each future state) to maximize agent one’s utility from current consumption y − c (applying the resource constraint) and future utility, EW 0 = EJ(V 0 , y 0 ). This is given an initial utility promise, V , to agent two. The utility promised to agent two, V 0 , defines the law of motion for the state variable. We solve this problem with a guess and verification of the value function. Conjecture that the value function is linear (the obvious guess after you work out a few ¶

For tractability we do not explicitly impose the requirement that consumption be positive. [Mention more? less?]

7

recursions). J(V, y) = p0 + py y − pv V for parameters (p0 , py , pv ) to be determined. We solve for these parameters and the allocations in three steps. (i) Since agent 2 has infinite risk aversion, utility promises are constant across states, V 0 = V¯ 0 . (ii) The level of current promised utility, V implies a current consumption allocation of V β V¯ 0 − 1−β 1−β

c =

(iii) The optimization, after substituting in the conjecture for J(V, y), is n  V p0 + py y − pv V = max min y − 1−β − V¯ 0

β V¯ 0 1+β



, p0 + py y¯ − pv V¯ 0

o

Given the Leontief Optimality occurs where the two quantities inside the min are equal (where current consumption equals future utility). This implies  0 ¯ V pv +

β 1−β

 = p0 + py y¯ − y +

V 1−β

Substitutes this relation into the Bellman equation and equate terms, p0 = β y¯,

py = 1 − β,

pv = 1

This solution defines the Pareto frontier as W = J(V, y) = (1 − β)¯ y + βy − V Substituting these coefficients into for consumption (along with a bit of algebra) solves for the the consumption of the two agents. y − c = y¯ − V + (1 − β)(y − y¯) c = V + β(y − y¯) In addition, we solve for the controlled law of motion for the promised utility of

8

agent 2 as V¯ 0 = V − (1 − β)(y − y¯) Lastly, we have the law of motion for the utility of agent 1 W 0 = y¯ − V + (1 − β)(y − y¯) + (1 − β)0 = W + (1 − β)0 The exact solutions offer some insight into the optimal allocation rules. If there is an increase in income y, agent 2 consumes a fraction β of it and agent 1 consumes fraction (1 − β). In agent 1’s case, optimality requires current consumption and expected future expected utility go up one-for-one; hence, the identical increase in W 0 . In agent 2’s case, an increase in y leads to an increase in current consumption and a fall in future utility. This exploits the fact that agent 2 has infinite intertemporal substitution. The feature that perhaps appears counter intuitive is that agent 2, with the infinite risk averion bears most of the consumption risk (for β close to one) . This emphasizes an important feature of recursive utility. With recursive utility, “risk aversion” (ie, α) is a statement about the risk aversion over one-step-ahead utility lotteries. “Resolution of uncertainty” (ie, α − ρ) is a statement about the preference over consumption lotteries. In our example with extreme heterogeneity, person 2 (utility denoted V ) wants his utility next period to be perfectly predictable. In addition, he doesn’t care how much consumption fluctuates over time as long as utility is smooth. Person 1 (utility denoted W ), in contrast, is unconcerned about utility fluctuations. However, their preference dictates that future consumption be perfectly predictable. Here, with iid income, it is easy to see how Pareto optimal allocation satisfies these preferences. To see this, set β be close to 1 (of course β < 1 is required for things to be well defined). In the iid-income case, person 2, infinitely risk averse (over utility lotteries), absorbs all consumption risk: ct = V0 + εt . By doing this, utility is always perfectly predictable: Vt = V0 . The future utility from a stream of iid shocks is constant since β close to one and perfect intertemporal elasticty, a law of large numbers applies. On the other hand, the person 1 who has an infinite

9

preference for late resolution of uncertainty (again, over consumption lotteries) has perfectly predictable consumption every period: yt − ct = y¯ − V0 , which (in the case of β = 1) is constant. In the the iid case, the mean y¯, is the only predictable component of future income. (It turns out that his utility is constant too, but that’s an artifact he doesn’t care about). If the horizon is shorter, less of the risk is borne by the risk-averse agent, agent 2. For example, consider the case where beta is close to zero. Here agent 2 will bear almost none of the endowment risk. The iid endowment stream, in this case, is very risky since the utility value comes mostly from the first realization and not the mean of a long series of draws. [More?]

3.2

Income Distribution

An important characteristics in models with heterogeneous agents is the dynamic properties of the cross-section distribution of consumption (or wealth). If the two agents have consumption that grow at different rates, then the cross-section distribution degenerates to the point where one agent gets everything (in the limit). In the Linear-Leontief example does infinite risk aversion or inelastic substitution dominate the consumption cross-section? The answer is neither. The controlled laws of motion above both imply the consumption of both agents is a drift-less random walk. In addition, note that utility levels are also a random walk. [Another spot to discuss the fact we do not impose consumption is positive. Mention, discuss?]

3.3

Aggregation

Having solved for the income allocations, we can now describe asset prices in these economies. Given the extreme nature of the Linear-Leontief preferences (α1 = 1, ρ1 = −∞, α2 = −∞, and, ρ2 = 1), it is, perhaps, surprising that there exists a representative agent and that the representative agent has preferences in the class of recursive preferences. In fact, the representative agent has linear preferences; that is recursive utility with parameters α ˜ = ρ˜ = 1. 10

To start, consider the MRS of agent 2 m02

 0 ρ2 −1  α2 −ρ2 c V0 =β c µ2 (V 0 )

The first term in brackets is one since ρ2 = 1. Optimality implies agent 2’s next period’s utility is a constant, V 0 = V¯ 0 . Hence, the second term in brackets is one. This implies, the pricing kernel for agent 2 is β. For Agent 1, α1 = 1 implies m01

y 0 − c0 y−c



(y 0 − c0 )EW (y − c)W 0

= β = β

ρ1 −1 



W0 µ1 (W 0 )  0 ρ−1

α1 −ρ1

Recall, that for agent 1, optimality requires that we equate current and future utility, hence y − c = EW 0 . Second, the Bellman equation implies y 0 − c0 = W 0 (a bit of algebra using the controlled laws of motion). This implies that the pricing kernel for agent 1 is, again, β. It is, of course, not surprising that optimal allocations imply m01 = m02 . Despite the extreme preferences in this example, asset prices are equivalent to an economy with a representative agent with linear preferences, α ˜ = ρ˜ = 1, implying a pricing kernel of m0 = β.

3.4

Decentralized Equilibrium

Given the pricing kernel implied by the Pareto sharing of the aggregate income, we can calculate an equity and bond price process. Note that the pricing kernel is mt+1 = β. The one-period risk-free bond price is Bond price: Ptb = Et mt+1 = β which is constant for all t for any income process. Defining the equity price as the price of a claim to the aggregate income process implies Pts = Et

" j ∞ Y X j=1

11

i=1

# mt+i yt+j

=

∞ X

β j Et yt+j

j=1

Since aggregate income is iid, Et yt+j = y¯ so the equity price is Pts = [β/(1 − β)]¯ y, constant for all t. As noted above, the bond price is also constant, Ptb = β. We can verify these equilibrium prices by finding a feasible portfolio strategy for each agent that achieves the Pareto optimal allocations. Consider agent 2’s portfolio choice in a stock-bond economy. The date-t budget constraint is s b θt−1 (Pts + yt ) + θt−1 = ct + θts Pts + θtb Ptb ,

where θts and θtb investments in equity and one-period bonds respectively. At the proposed equilibrium prices, the budget constraint becomes s b θt−1 (P s + y¯ + εt ) + θt−1 = ct + θts Ps + θtb β.

Conjecture a solution for equity holdings of θts = β constant for all t. This implies b β y¯ + βεt + θt−1 = ct + θtb β.

Recall that optimal consumption in this case satisfies ct = Vt + βεt = Vt−1 + (β − 1)εt−1 + βεt t X = V0 + (β − 1) εt−j + βεt , j=1

Therefore any sequence of bond holdings such that β y¯ +

b θt−1

−

βθtb

= V0 + (β − 1)

t X

εt−j ,

j=1

will support the optimal consumption allocation. In other words, the optimal bond holdings for Agent 2 solves the stochastic difference equation b θtb = κ0 + κ1 θt−1 + ηt ,

12

where κ0 = y¯ −V0 /β, κ1 = 1/β and ηt −ηt−1 = [(1−β)/β]εt−1 . Initial bond holdings follow from the initial condition for agent two’s utility, V0 (i.e., the initial Pareto weight for agent 2) Note that the asset holdings of Agent 1 can be found from the aggregate resource constraint. The bonds are in zero net supply and there is one unit of the stock that pays a per-period dividend of yt . This implies that agent 1’s equity holding is constant at θt = (1 − β) which, by construction, achieves agent 1’s optimal consumption. The interesting feature of this competitive equilibrium in our example, in the case where β is close to one, is that the “infinitely risk averse” agent, Agent 2, holds almost all of the risky asset, θts = β. The “risk neutral” agent, Agent 1, holds almost none, 1 − β, of the risky asset. This example highlights the importance of the preference for the early resolution of uncertainty as a dimension of risk that can be captured with recursive utility. With recursive utility, “risk aversion” (ie, α) is a statement about the risk aversion over one-step-ahead utility lotteries. However, portfolio choice determines consumption lotteries, thus the “Resolution of uncertainty” (ie, α − ρ). Agent 2’s portfolio produces variable consumption. But the value of the portfolio remains constant since it is a claim to a stream of iid dividends.

4

Persistent income endowment

The previous example shows how differences in attitude towards intertemproal risk effects the optimal income allocations. Therefore, it is interesting to see how the allocations behave if the endowment process follows a persistent process. Assume income yt is AR(1) process y 0 = (1 − ϕ)¯ y + ϕy + 0 where y¯ is the unconditional mean, ϕ is the persistence parameter, and E0 = 0. Again, the structure of the problem implies that no other features of the process matter (the conditional variance, for example).

13

We solve the Pareto problem in the same manner as above. We conjecture that the value function is linear, J(V, y) = p0 + py y − pv V , and solve for the coefficents following the same steps as above. In particular, the key steps are the same. First, the infinite risk aversion implies agent 2’s future utility is risk free at V¯ . Second, the Leonteif intertemporal preferences of agent 1 means it is optimal for current consumption and future utility to move together. With a bit of algebra, we get the value function parameters p0 =

β(1 − ϕ) y¯, 1 − βϕ

py =

(1 − β) , 1 − βϕ

pv = 1

Plugging these values, we can solve for consumption 1−ϕ 1−β y¯ − V + (y − y¯) 1 − βϕ 1 − βϕ β(1 − ϕ) c = V + (y − y¯) 1 − βϕ

y−c =

Again, we can also solve for the controlled law of motion of promised utility to agent 2, V 0 (1 − β)(1 − ϕ) V¯ 0 = V − (y − y¯) 1 − βϕ

W 0 = y¯ − V +

(1 − β) (1 − β) 0 (y − y¯) +  1 − βϕ 1 − βϕ

Comparing these results to the IID case above, shows the role of a persistent endowment process. In particular, contrast this case with the iid case when the parameter β is close to one. Persistent income, ϕ > 0, increases the sensitivity of agent 1’consumption, y − c, to current income. The persistence of the shock to income makes it easier to increase agent 1’s future utility, EW 0 to move lockstep with current consumption as required by optimality and person 1’s inelastic intertemporal preferences. For example, if consumption has close to a unit root, ϕ ≈ 1, then agent 1 bears most of the variability in aggregate income. Unlike in the iid case, variability in current income is not tolerated by the infinitely risk averse agent 2 since the persistence implies variability in future utility as well.

14

4.1

Income Distribution

In this case, with a stationary prcess An important characteristics in models with heterogeneous agents is the dynamic properties of the cross-section distribution of consumption (or wealth). If the two agents have consumption that grow at different rates, then the cross-section distribution degenerates to the point where one agent gets everything (in the limit). In the Linear-Leontief example does infinite risk aversion or inelastic substitution dominate the consumption cross-section? The answer is neither. The controlled laws of motion imply the consumption of both agents is a drift-less random walk. In addition, note that utility levels are also a random walk.

4.2

Aggregation

Again, having solved for the income allocations, we can now describe asset prices in these economies. In this setting, exactly as in the setting with iid aggregate income, the representative agent has linear preferences; that is recursive utility with parameters α ˜ = ρ˜ = 1. The construction of the representative agent relies on the two key optimality conditions: (i) Agent 2 has constant continuation utility, (ii) Agent 1’s consumption and future utility move one-for-one. These two conditions are independent of the assumption for the income process, so the exact same steps can be used to construct the representative agent and verify that m0 = β.

4.3

Decentralization

In the case of perisitant income, Et yt+j = y¯(1 − ϕj ) + ϕj yt . This implies an equity price of Pts = yt

∞ X

  β j y¯(1 − ϕj ) + ϕj

j=1

 =

 β βϕ βϕ − + yt . 1 − β 1 − βϕ 1 − βϕ

15

The bond price is unchanged. We can construct portfolio holdings of the two agents to verify these equilibrium pricess in an analogous way. [...Portfolio holdings in the AR(1) algebra to be added...]

5

Conclusion

With recursive utility, “risk aversion” (ie, α) is a statement about the risk aversion over one-step-ahead utility lotteries. “Resolution of uncertainty” (ie, α − ρ) is a statement about the preference over consumption lotteries. In our example with extreme heterogeneity, person 2 (utility denoted V ) wants his utility next period to be perfectly predictable. In addition, he doesn’t care how much consumption fluctuates over time as long as utility is smooth. Person 1 (utility denoted W ), in contrast, is unconcerned about utility fluctuations. However, their preference dictates that future consumption is perfectly predictable. In the iid case, it is easy to see how Pareto optimal allocation satisfies these preferences. To see this, set β be close to 1 (of course β < 0 is required for things to be wll defined). In the iid-income case, person 2, infinitely risk averse (over utility lotteries), absorbs all consumption risk: ct = V0 +εt . The result is that by doing this, utility is always perfectly predictable: Vt = V0 . On the other hand, the person 1 who has an infinite preference for early resolution of uncertainty (again, over consumption lotteries) has perfectly predictable consumption every period: yt −ct = y¯ −V0 , which is constant. In the the iid case, the mean y¯, is the only predictable component of future income. (It turns out that his utility is constant too, but that’s an artifact he doesn’t care about). A similar explanation holds for the persistent-income case (and when β is less not close to one). The open question, of course, is does this intuition carry over to the case where preferences are not this extreme. For example, what do the Pareto allocations look like when one agent has a stronger preference for predictable utility and the other has a stronger preference for predictable consumption. In a related paper, Backus, Routledge, and Zin (2007) we explore these issues in a Gaussian/log-approximation setting.

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References [1] David K. Backus, Bryan R. Routledge, and Stanley E. Zin, Exotic preferences for macroeconomists, Nber macroeconomics annual 2004, 2005, pp. 319–391. [2] , Recursive risk sharing: Microfoundations for representative-agent asset pricing, 2007. Carnegie Mellon University Working Paper. [3] Larry G. Epstein and Stanley E. Zin, Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework, Econometrica 57 (1989July), no. 4, 937–969. [4] Rui Kan, Structure of pareto optima when agents have stochastic recursive preferences, Journal of Economic Theory 66 (1995August), no. 2, 626–631. [5] David M. Kreps and Evan L. Porteus, Temporal resolution of uncertainty, Econometrica 46 (1978January), 185–200. [6] Robert E. Lucas and Nancy L. Stokey, Optimal growth with many consumers, Journal of Economic Theory 32 (1984), no. 1, 139–171.

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