THE ECONOMY

In this section, I introduce a simple endowment economy populated by two types of Epstein-Zin agents who differ only in their degree of risk aversion. I consider two alternative structures for the asset market. In the first one, the asset market is complete, whereas in the second one a borrowing constraint limits the scope for risk sharing between agents. In the former case, the long-run distribution of wealth in the economy is always degenerate. According to the parameters of the model either the more- or the less-risk-averse type of agent dominates the distribution of wealth. In the latter case, instead, the long-run distribution of wealth in the economy may be stationary. Even in this circumstance though, the distribution of wealth can be concentrated in the hands of more-risk-averse agents.

Preferences

The economy is populated by two types of agents, individually denoted by i=1,2. The size of each type is normalized to 1. Denoting by U_it agent i's utility from time t onward, the preferences of each agent are represented by the following recursive utility function⁴

The expression for the utility function when ρ=0 (unit intertemporal elasticity of substitution) is easily obtained using de l'Hospital's rule to compute the limit of

for ρ→0. Tallarini (2000) considers the case of unit intertemporal elasticity using Epstein-Zin preferences in a real-business-cycle representative-agent model.

:

where β is the discount factor, E_t[·] denotes the expectation conditional on information available at the beginning of time t, and C_it denotes consumption by an agent of type i at time t. These preferences have been introduced by Epstein and Zin (1989) to allow for the independent parameterization of an agent's attitudes toward intertemporal substitution and his degree of risk aversion. In this specification, (1-ρ)⁻¹ is the elasticity of intertemporal substitution and γ_i<1 parameterizes risk aversion, with higher values of γ_i denoting lower risk aversion.⁵

These preferences have been extensively used in the asset pricing literature. See, for example, Epstein (1988), Weil (1989), Epstein and Zin (1991), and Campbell (1993). Epstein and Zin (1989, pp. 949–950) discuss the interpretation of the parameters ρ and γ_i.

To evaluate the effects of heterogeneity in risk aversion, I assume that the agents in this economy share the same intertemporal elasticity parameter ρ and have different degrees of risk aversion. By convention, agents of type 1 are more risk averse than agents of type 2: γ₁<γ₂.

In the following, I focus on the case of unit intertemporal elasticity of substitution (ρ=0). This assumption simplifies the interpretation of the results because it implies that agents with different risk aversion have the same propensity to save out of current wealth. Therefore, the dynamics of the wealth distribution are entirely driven by their different portfolio choices. The main results of the paper easily generalize to the case of elasticity of substitution different from one.⁶

The results for the case ρ≠0 are available from the author upon request.

Endowment

The aggregate endowment, in each period, consists of a perishable dividend D_t. This evolves according to the process

The growth rate λ_t+1 follows an i.i.d. two-state Markov process

with

and Pr

Budget and Borrowing Constraint

Both types of agents maximize their time-zero utility U_i0 subject to the sequence of budget constraints

where c_it denotes the ratio

. Without loss of generality, I assume that at time zero,

and

.

In addition to this frictionless complete-markets economy, I also evaluate the effects of borrowing constraints on the long-run distribution of wealth. In particular, to keep the analysis simple, I consider situations where an agent has to finance a fraction κ<1 of his stocks' purchase out of his own savings⁷

This constraint can be interpreted as a margin requirement. It simplifies the analysis with respect to a constraint of the type

for some constant

because it implies, with the homotheticity of preferences, that decision rules are linear in agents' wealth. See the agent's optimization problem and Coen-Pirani (in press) for details.

:

Recursive Formulation and Equilibrium

In this economy, aggregate demand for assets and the consumption good, as well as the bond price, depend, in general, on the way aggregate wealth is distributed across agents. Let

denote the fraction of aggregate wealth, 1+_pt, held by the more-risk-averse type of agent at time t:

The law of motion for Γ_t, which summarizes the state of the economy at the beginning of period t, is denoted by

This function relates the distribution of wealth at time t+1 to the current state and the realization of the endowment shock λ_t+1.

Letting w represent individual wealth, appropriately standardized by D, homotheticity of preferences implies that the agents' problem can be written recursively as follows:

The definition of recursive equilibrium for this economy is standard, requiring individual optimization and market clearing. The only feature of this definition that is worth noticing is the fact that in equilibrium the perceived law of motion for the distribution of wealth H(Γ,λ′) must be consistent with agents′ behavior:

Notice that the first argument of the stock and bond demand functions s₁′(.,.) and b₁′(.,.) in equation (4) is type-1 agents' wealth in equilibrium, while the second argument is the aggregate state of the economy Γ.

Agents' Optimization Problem

Homotheticity of preferences and the linearity of the budget and borrowing constraints imply that the value function for both types of agents and the decision rules for consumption and asset holdings can be written as the product of individual wealth w and a term involving only the distribution of wealth Γ. For example, the value function and the decision rule for consumption take, respectively, the form

and

for some positive functions

and

.

Moreover, the optimization problem in (3) can be recast in a consumption-savings problem and a portfolio decision problem. To do this, denote the fraction of an agent's savings invested in stock by

and the rate of return on his portfolio by

where the rate of return on the risky and riskless assets are, respectively,

An agent's next-period wealth can then be rewritten as

while, using the definition of x, the borrowing constraint (2) amounts to

.

Taking into account the linearity of the value function and of the decision rules in w, the recursive problem (3) becomes

As it is clear from equation (6), the consumption-savings decision can be analyzed independently from the portfolio choice problem. The assumption of unit elasticity of substitution implies that in every state Γ, an agent of type i chooses to consume a fraction

of his wealth, independently of his degree of risk aversion. Thus, in this case the dynamics of the wealth distribution are driven exclusively by the different portfolio decisions made by the agents. Moreover, their common propensity to save out of wealth implies that the stock price dividend ratio p(Γ) is a constant equal to

⁸

This can be seen by imposing the market-clearing condition for consumption,

and solving for p(Γ). The rate of return on stocks is therefore

which simplifies to λ′β since p(Γ)=β/(1–β).

DYNAMICS OF THE DISTRIBUTION OF WEALTH

In this section, I study the dynamics of the wealth distribution in this economy. I am particularly interested in exploring the robustness of the standard result, derived within the expected-utility framework, according to which less-risk-averse agents always tend to dominate the long-run distribution of wealth if the economy is growing.

The focus of the analysis is represented by the equilibrium law of motion for the distribution of wealth

Using the notation introduced in the preceding section, it is easy to see that

Notice that the mapping

has only two fixed points:

and

If

type-2 agents own all the wealth in the economy, whereas the opposite occurs at

. Taking logarithms, the law of motion for the wealth distribution (7) can be rewritten as

where the innovation

is defined as

and by construction has a conditional mean equal to zero. Equation (8) suggests that the share of wealth of type-2 agents will tend to grow, on average, as long as the conditional expectation term is positive or, equivalently, if

Equation (9) contains the key to understanding why in the long run less-risk-averse agents may tend to become poorer and poorer. The term on the left-hand side of this equation is the expected value of the logarithmic return on type-2 agents' portfolio. The term on the right-hand side is the logarithm of the rate of return on a portfolio that contains only stock. The fact that less-risk-averse agents are levered and therefore choose portfolios where stocks receive a weight larger than 1

implies that, on average, the rate of return on their portfolio exceeds the rate of return on stock:

However, levered portfolios are also riskier. The concave transformation of returns operated by the logarithm implies that equation (9) may not be satisfied, even if equation (10) is. In other words, the share of aggregate wealth owned by type-2 agents may shrink over time, even if their portfolio pays, on average, a higher return than the one on stock. This is the basic intuition behind the fact that less-risk-averse agents may lose all their wealth in the long run in this kind of model.

The main difference with respect to the standard expected-utility framework is that in the latter, less-risk-averse agents also have a higher propensity to save out of current wealth, due to the higher intertemporal elasticity of substitution. In turn, a higher propensity to save increases the growth rate of their wealth with respect to more-risk-averse agents, leading them to become richer and richer over time. Epstein-Zin preferences, instead, allow attitudes toward risk to be parameterized separately from attitudes toward intertemporal substitution. As a result, in this economy the dynamics of the wealth distribution are exclusively driven by the different portfolio choices made by the agents.

Of course, even if type-2 agents' portfolios are riskier, I have not shown yet that this effect can be strong enough to lead them to lose all their wealth in the long run. Since the model does not admit an analytical solution, I proceed in two complementary ways. First, I evaluate the conditional expectation on the left-hand side of equation (9) at the steady state Γ=1. Second, I solve the model numerically and verify that the dynamics of the wealth distribution around Γ=1 are in fact consistent with the intuition obtained at Γ=1.

Consider the first approach. The inequality (9) evaluated at the steady state Γ=1 determines the stability of the mapping

at Γ=1. That is, it determines whether the share of wealth owned by type-2 agents tends to decline or increase over time in a neighborhood of Γ=1. The conditional expectation on the left-hand side of equation (9) at Γ=1 depends on type-2 agents' portfolio decision x₂(1) and the bond return R_b(1). Notice that at the steady state, Γ=1 type-2 agents own zero wealth and therefore save zero and buy zero stock.⁹

Both R_b(1) and x₂(1) can be computed analytically. The return R_b(1) is simply the rate of return on one-period bonds in a representative-agent economy populated only by type-1 agents. The portfolio decision x₂(1) is found by solving the optimization problem (6) at Γ=1.¹⁰

In particular, it can be easily shown that

Notice that, as remarked in note 8,

for all values of Γ ∈[ 0,1]. Weil (1989) analyzes asset prices in a representative-agent economy with Epstein-Zin preferences. The analytical expression for x₂(1) is obtained from equation (A.2) in the Appendix by setting Γ=1 and noticing that the terms involving the value function υ_i(·) cancel out since

where

The term in z on the left-hand side of this equation represents the difference between the expected logarithmic return on type-2 agents' portfolio and the expected logarithmic return on stock. The direction of the inequality depends only on whether this term is positive or negative. After some simplifications, one obtains that it is positive if and only if

Thus, these computations suggest that it is in fact possible that inequality (9) is not satisfied so that a riskier portfolio strategy may lead less-risk-averse agents to become poorer and poorer in a neighborhood of Γ=1. This circumstance is more likely to occur [i.e., condition (11) is less likely to be satisfied] the less risk averse agents of type-2 are. Lower risk aversion, in fact, leads to the choice of riskier portfolios, decreasing the growth rate of their wealth. Condition (11) also depends on the degree of risk aversion of type-1 agents because the latter determine the rate of return of the riskless asset. In particular, the less risk averse type-1 agents are (the higher γ₁ is), the higher the riskless rate must be to reduce their incentive to sell riskless debt. In turn, a high riskless return reduces the rate of return on type-2 agents' portfolio. Therefore, the higher γ₁, the more likely it is that type-2 agents tend to become poorer and poorer when Γ is close to 1.

Repeating the same analysis at Γ=0, one obtains that condition (11) is also necessary and sufficient to guarantee that the share of wealth held by type-1 agents tends to decrease over time in a neighborhood of Γ=0. Therefore, this approach suggests that the long-run distribution of wealth in this economy under complete markets is always degenerate. If condition (11) is satisfied, type-2 agents dominate the long-run distribution of wealth. Conversely, if condition (11) is not satisfied, the long-run distribution of wealth is dominated by type-1 agents.

To confirm these intuitions and to consider the dynamics of the wealth distribution away from the rest points Γ=1 and Γ=0, it is necessary to use numerical methods and assign numerical values to the parameters of the model.¹¹

I first consider the complete-markets version of the model, without the borrowing constraint (2). Figure 1 shows the dynamics of Γ_t when the two types have risk aversion parameters γ₁=−0.1 and γ₂=0.3, so that condition (11) is not verified. The figure shows how the process Γ_t eventually converges to 1; that is, the more-risk-averse type dominates the long-run distribution of wealth. It is important to stress the fact that {Γ_t} converges to 1 only asymptotically.¹³

0.01, and satisfy condition (11).

More-risk-averse agents dominate the long-run distribution of wealth.

Less-risk-averse agents dominate the long-run distribution of wealth.

As discussed above, the intuition that explains why type-2 agents can lose all their wealth in the long run has to do with the fact that their portfolio choices are relatively risky. This suggests that introducing an upper bound on leverage and therefore reducing the riskiness of their portfolios may prevent this extreme result from occurring, leading to a stationary long-run distribution of wealth.

Figure 3 represents the dynamics of the wealth distribution when the parameters are the same as in Figure 1, with the exception of κ, which is set equal to 0.84, so that at Γ=1, the borrowing constraint is binding for agents of type-2: x₂(1)=1κ. The introduction of the borrowing constraint is enough to make the growth rate of type-2 agents' wealth positive in a neighborhood of Γ=1. The figure shows how the distribution of wealth does not converge to Γ=1. The process Γ_t gets extremely close to 1 but always bounces back.

Dynamics of the wealth distribution in the economy with binding borrowing constraints.

The introduction of this borrowing constraint, while giving rise to a stationary distribution of wealth, does not qualitatively affect the main result of the paper. Figure 4 is a histogram of the wealth distribution corresponding to the simulation of Figure 3. The long-run distribution of wealth is still very much concentrated in the hands of the more-risk-averse agent, because risky portfolio strategies induce less-risk-averse agents to remain poor over time.

Histogram of the stationary distribution of wealth in theeconomy with binding borrowing constraints

SUMMARY

This paper has explored the role played by differences in risk aversion in affecting the long-run distribution of wealth across agents. To disentangle risk aversion from intertemporal substitution I have adopted the recursive preferences introduced by Epstein and Zin (1989). According to the parameters of the model, the long-run distribution of wealth is dominated by either less- or more-risk-averse agents. By choosing riskier portfolio strategies, less-risk-averse agents enjoy returns on their investments characterized by a higher mean and a higher variance than the ones enjoyed by more-risk-averse agents. The former effect tends to make less-risk-averse agents wealthier over time, whereas the latter tends to make them poorer. The paper shows how, differently from a model based on expected utility, the latter effect prevails for some parameter values. Therefore, it is possible that agents characterized by higher risk aversion dominate the long-run distribution of wealth.