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The Role of Effort for Self-Insurance and Its Consequences for the Wealth Distribution

This paper is based on the second chapter of the author's dissertation at UCLA. He would like to acknowledge the comments of Andy Atkeson, Francisco Buera, Roger Farmer, Christian Hellwig, Gonzalo Llosa, Bentley McLeod, Andy Neumeyer, Lee Ohanian, Venky Venkateswaran, and Pierre-Olivier Weill; as well as participants in the World Bank ABCDE Conference, Macro Lunch Proseminar at UCLA, LACEA, Universidad de los Andes, the Midwest Macro Conference, the Central Bank of Colombia, the University of Leipzig and The Guanajuato Workshop for Young Economists. The author is also very grateful for the financial support given by the Central Bank of Colombia. The valuable research assistance of Felipe Acero is greatly acknowledged. The usual disclaimer applies.

Author Notes

Abstract

I explore the effect of effort as a mechanism to alleviate the idiosyncratic risk faced by individuals in the presence of incomplete markets. I construct a DSGE model where costly effort determines the probability of being employed the next period and a riskless asset can be used to smooth consumption. I first show how effort and assets are inverse related, and that a unique stationary equilibrium exists. Then, in a calibrated version of the model to the US economy, I show that in the stationary equilibrium a positively skewed wealth distribution arises, which is closer to the observed data and has not been obtained by models without ex-ante heterogeneity. I then use the model to evaluate the effect of unemployment insurance on the wealth distribution.

Introduction

Heterogenous agents models with incomplete markets allow one to study the distribution of key economic variables in an economy such as earnings, consumption, and wealth. The first models in this literature, where agents have only access to a single riskless asset to smooth consumption, generate left-skewed wealth distributions since most agents accumulate assets for precautionary motives (Huggett 1993; Aiyagari 1994). However, real wealth distributions are skewed to the right: few individuals hold most of the wealth, whereas most agents have some degree of debt or have little savings (see Fig. 1). The purpose of this paper is to study the role of effort as a mechanism for self-insurance and evaluate its consequences for the wealth distribution.

Empirical Distribution of Wealth Survey of Consumer Finances, 2010

Figure 1.

Empirical Distribution of Wealth Survey of Consumer Finances, 2010

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The model builds on the framework proposed by Huggett (1993) but includes effort as a variable determining the transition dynamics between states. In our two-state model, it can be interpreted as search effort when the individual is unemployed or effort in the job when the agent is employed. Our first result suggests a negative relationship between effort and asset holdings. The smoothing role of the assets loses importance when they are close to the debt limit, there effort plays a major role by increasing the likelihood of being employed next period. Thus, effort partially completes the financial markets.

We then calibrate the model to the US economy and obtain a unique right skewed stationary distribution. The intuition behind this result is that diversification between effort and the riskless asset eliminates the need of accumulating precautionary savings. Finally, we perform an experiment by doubling the income an unemployed agent would get. Remarkably, the wealth distribution does not change importantly, thus suggesting unemployment insurance benefits do not have an important effect on wealth inequality.

Several models have been recently developed to obtain a right-skewed wealth distribution. Their strategy consists of allowing for a source of ex ante heterogeneity and calibrates it to match the observed wealth distribution. Krusell and Smith (1998) propose ex ante heterogeneity in discount rates, Quadrini (2000) in rates of return, Cagetti and De Nardi (2006) in ability, and Huggett (1996) and Castaneda, Díaz-Giménez, and Ríos-Rull (2003) in the persistence of shocks. Our paper is the first one to our knowledge to obtain such result without assuming ex ante heterogeneity.

The organization of the paper is as follows. The next section describes the environment faced by the individuals. The third section defines the equilibrium in this scenario. We then describe the calibration used in the model, and discuss the results and its implications. The last section concludes.

Environment

Consider an exchange economy with a continuum of agents with total mass equal to one who face idiosyncratic risk. There are two commodities: one perishable consumption good c and asset holdings a. Each period, each agent receives an stochastic endowment of the consumption good w_t. Assume the endowment can take two possible values w_L < w_H, which are usually associated with unemployed/employed status, respectively.

Effort e is made in order to increase the probability of having a good endowment next period. This probability is defined as $Pr (w_{t + 1} = w_{H} | w_{t}) = P (e_{t}; w_{t})$ ⁠, which is assumed to be increasing concave and satisfying Inada conditions with respect to e. We let $P (e_{t}; w_{H}) > P (e_{t}; w_{L})$ for all e_t, which implies that effort to remain employed is more effective than the effort to become employed when previously unemployed.¹

Each agent is able to smooth her consumption by holding a riskless asset a. This asset entitles the individual to receive one unit of future consumption at a price q > 0. The amount of claims held must remain above the limit $a_{min} \leq 0$ ⁠, thus preventing perpetual debt. The budget constraint faced by an individual who holds a claims, has a current endowment w, and chooses consumption c and future claims a′, is given by $c + q a^{'} \leq w + a$ ⁠.

Each agent maximizes her expected additive separable concave utility function over her infinite lifetime,² and future is discounted at a rate

β \in (0, 1)

⁠. The agent's problem can be represented in recursive formulation as

v (a, w; q) = max_{c, e, a^{'}} {u (c) - e + β [P (e; w) v (a^{'}, w_{H}) + (1 - P (e; w)) v (a^{'}, w_{L})]}

(1)

subject to the budget constraint.

Lemma 1

For q > 0 and $a_{min} + e_{L} - a_{min} q > 0$ , there exist a unique solution to v (a, w; q). Such value function v (a, w; q) is strictly increasing, strictly concave and continuous differentiable in a, and strictly increasing in w. Moreover, the optimal decision rules c (a, w; q), e (a, w; q), and $a^{'} (a, w; q)$ are continuous in a.

The previous lemma states that the problem has standard properties. Its proof relies on a straightforward extension of the corresponding proof of theorem 1 in Huggett (1993). Just note that the problem remains bounded and that P(·) is independent of a. Therefore the first order conditions are necessary and sufficient, and the optimal decision rules c (a, w; q), e (a, w; q), and

a^{'} (a, w; q)

are given by

1 = β P_{e}^{'} (e; w) [v (a^{'}, w_{H}; q) - v (a^{'}, w_{L}; q)],

(2)

\begin{aligned} u_{c}^{'} (c) \geq \frac{β}{q} [P (e; w) \frac{\partial v (a^{'}, w_{H}; q)}{\partial a^{'}} + (1 - P (e; w)) \frac{\partial v (a^{'}, w_{L}; q)}{\partial a^{'}}], \\ w i t h e q u a l i t y i f a^{'} > a_{min} \end{aligned}

(3)

c + q a^{'} \leq w + a

(4)

Lemma 2

Optimal effort e (a, w; q) is decreasing in a and decreasing (increasing) in w_L (w_H).

When the agent has more assets, it is more able to smooth consumption and thus the difference of the value function when employed or unemployed becomes smaller; hence, there are less incentives to exert effort. However, when the agent is close to the maximum level of debt, she must rely heavily on effort to increase the chances of being employed next period since the assets cannot be longer used to smooth consumption. In other words, effort is used to partially complete the markets. The lemma also states that effort will be lower (greater), the greater are the benefits when unemployed (employed). The latter is a standard result in the unemployment insurance literature (see, e.g., Hopenhayn and Nicolini 1997).

Condition (3) shapes the assets' behavior: when the individual is employed she accumulates assets, while she decreases her holdings when unemployed. Using the theory of supermartingales, it can be shown that to have an equilibrium, where consumption and asset holdings remain finite, it must be the case that $β < q$ (see Williams 1991). The price q will be determined in equilibrium according to a market clearing condition that we describe in the next section. Moreover, it can be shown, extending Huggett's (1993) proof of his theorem 2, that there exists an upper bound for the assets a_max, a fixed point that will be achieved if an agent remains employed forever. The boundedness of this optimal rule will be required for the existence of the stationary equilibrium defined in the next section.

Equilibrium

The aggregate state at time t in this economy is described by the joint distribution of wealth and endowment

λ_{t} (a, w; q) = Pr (a_{t} = a, w_{t} = w; q)

⁠. Since this distribution is always evolving, the price q that clears the market must also change. To avoid this difficulty we focus on a stationary equilibrium where the distribution remains invariant, that is,

λ_{t + 1} (a, w; q) = λ_{t} (a, w; q) = λ (a, w; q)

⁠, which law of motion is described by

λ_{t + 1} (a^{'}, w^{'}; q) = \sum_{i \in L, H} \int_{{a : a^{'} (a, w; q)}} λ_{t} (a, w_{i}; q) \cdot Pr (w_{t + 1} = w^{'} | e (a, w_{i}), w_{i}) d a

Lemma 3

There exists a unique stationary distribution $λ (a, w; q)$ ⁠.

Proof.

To establish the existence one needs to adapt slightly the argument in Huggett (1993). First note that there exists a compact set $W = [a_{min}, a_{max}] \times {w_{L}, w_{H}}$ such that if an agent starts in any point in W, the next period will remain in W. Then define the natural order over W, where (a_min, w_L) is the minimum and (a_max, w_H) is the maximum. Finally, note that the associated transition is increasing since the optimal decisions are monotone and $P (e; w_{H}) > P (e; w_{L})$ ⁠. For uniqueness, it remains to prove that the monotone mixing condition in Hopenhayn and Prescott (1992) is satisfied. Define a sequence $x_{1} = a_{(min)}, x_{2} = a^{'} (x_{1}, w_{H}), x_{3} =$ $a^{'} (x_{2}, w_{H}), \dots$ and a sequence $y_{1} = a_{(max)}, y_{2} = a^{'} (y_{1}, w_{L}), y_{3} = a^{'} (y_{2}, w_{L}), \dots$ ⁠. Both sequences are feasible because of the Inada conditions satisfied by P(·). Moreover $lim_{n \to \infty} x_{n} = a_{max}$ and $lim_{n \to \infty} y_{n} = a_{min}$ ⁠. Therefore, there exists $w^{*} \in W$ and N such that, after N periods, the probability that someone in a_min will be in some $w > w^{*}$ is positive, and that someone in a_max will be in some $w < w^{*}$ is positive.▪

The lemma suggests that starting from any initial distribution, a sufficient number of iterations will converge to the invariant one. Moreover, since $a^{'} (a, w; q)$ is bounded, the sequence of averaged assets will also converge. Therefore, it implies the existence of a unique stationary equilibrium.

Definition 4

A stationary equilibrium is defined by policy rules c (a, w; q), e (a, w; q), and $a^{'} (a, w; q)$ ; a value function v (a, w; q); a price q; and a stationary distribution $λ (a, w; q)$ , such that

The policy and value functions solve the agent's problem (1)
Markets clear:
1. $\int_{a} \sum_{i \in L, H} a^{'} (a, w_{i}; q) λ (a, w_{i}; q) d a = 0$
2. $\int_{a} \sum_{i \in L, H} c (a, w_{i}; q) λ (a, w_{i}; q) d a = \int_{a} \sum_{i \in L, H} w_{i} λ (a, w_{i}; q) d a$
The stationary distribution $λ (a, w; q)$ is induced by the policy functions and the endogenous Markov chains generated by P(e(a, w; q); w).

Numerical Exercise

Since we are interested on the effects of effort on the wealth distribution, we shut down any ex ante heterogeneity, including income. Thus we normalize the endowment to w_H = 1 and w_L = 0.1. We will later perform simulations in the hypothetical case where unemployment benefits were to double. We assume the utility function takes the form $u (c) = c^{1 - σ} / 1 - σ$ ⁠, which is typically used in the literature. According to Mehra and Prescott (1985), estimates of the risk aversion coefficient σ are around 1.5. We model the probability of having a high state tomorrow as a cdf of an exponential distribution with parameter $μ = 10 w_{i}$ ⁠, that is $P (e; w_{i}) = 1 - \exp^{- 10 w_{i} e}$ ⁠.

The rest of the parameters are calculated according to periods of 8.5 weeks approximately, that is 6 periods per year.³ The discount rate is calibrated to $β = 0.99322$ to match an annual discount rate of 0.96. The lower bound a_min is set to −5, which is close to the annual average endowment of this economy and close to the natural borrowing limit of $- s_{L} / r$ described by Aiyagari (1994). This parameterization satisfies our initial assumptions of first order stochastic dominance and the ones described by Hopenhayn and Nicolini (1997) to characterize the optimal unemployment insurance. Moreover, the optimal probabilities in equilibrium will wander around Huggett's calibration. This calibration replicates a coefficient of variation for the annual earnings of 20%, which is close enough to the actual data.

The computation follows the standard procedure of value function iteration for a guessed price. Then we calculate the average assets, and the price is adjusted accordingly until we reached a market clearing price. The price of assets that clears the market in the benchmark case is 0.994, which is equivalent to an annual interest rate of 3.86%; the unemployment rate is 6.39%.

The distribution of wealth in the stationary distribution differs from the one found by Huggett (1993) and similar parsimonious models. Figure 2 shows the stationary distribution is skewed to the right, just as the normalized wealth distribution we computed using the Survey of Consumer Finances (2010) in figure 1.⁴ Wealth is concentrated in fewer agents, while most of agents hold a slightly negative amount of assets. In the long run, individuals do not need to accumulate assets for precautionary motives since they have another nonmarket mechanism to smooth consumption. In other words, the incomplete markets are complemented by effort. The model also replicates the fact that consumption inequality is lower than the wealth one as a consequence of the smoothing process.

Stationary Distribution of Assets

Figure 2.

Stationary Distribution of Assets

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Although the model replicates a positively skewed distribution, it is not able to obtain the long right tail. Part of the explanation is that the grid is finite and the computed upper bound for assets becomes smaller. This is why there is a concentration of agents at the end of the distribution that otherwise will become part of a longer right tail. However, this would not account for most of the right tail. This is also a shortcoming of models with ex ante heterogeneity in the persistence of shocks, as Huggett (1996) recognizes. Castaneda et al. (2003) estimate that to obtain such concentration of wealth, agents must receive a shock about 1,060 times the median income level with a small probability. Krusell and Smith (1998) are also able to achieve such dispersion using ex ante heterogeneity in discount rates.

We also perform an experiment by doubling the unemployment income w_L to 0.2 to evaluate its consequences for the wealth distribution. The interest rate increased to 4.24%, while the unemployment rate increased to 7.15%. The result arises because agents have less incentives to exert effort, as it was shown in lemma 2. This increases unemployment and incentivizes agents to rely more on assets for self-insurance; therefore its demand increases and the interest rate must increase to achieve an equilibrium. However, and remarkably, the stationary wealth distribution does not change importantly, it only seems a bit less dispersed in figure 2. This result suggests that unemployment insurance benefits decrease wealth inequality, although its effect is very modest.

Concluding Remarks

We have studied a model of heterogenous agents who face idiosyncratic risk and smooth their consumption using a riskless asset and effort, which determines the transition distribution to the next state. We have found that there is a negative relationship between assets and effort, which could be interpreted as a role of effort to partially complete the financial incomplete markets. We also examine the effect of effort on the wealth distribution. It is shown that our parsimonious model is able to replicate a wealth distribution skewed to the right, which have not been achieved by similar models. Moreover, it is shown that such distribution is robust to changes in the relative income perceived by unemployed agents, suggesting that unemployment insurance is not an important determinant of wealth inequality.

1

This assumption follows empirical data that has been studied in search models and emphasize the role of the depreciation of human capital during unemployment (Addison and Portugal 1989; Neal 1995).

2

Separability is obtained if we assume the existence of lotteries (Hansen (1985)). The linearity in e is just an innocuous normalization.

3

Huggett (1993) chose this length to match the average duration of unemployment spells of 17 weeks (Bureau of Labor Statistics), which is a underestimation of the current average duration of 21.6, but it fits the 5-year trend.

4

Wealth is calculated according to Wolff (2010). We used both total wealth and non-housing wealth, obtaining similar results.

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Author notes

Andres Zambrano is an assistant professor in the Department of Economics at Universidad de los Andes;

This paper is based on the second chapter of the author's dissertation at UCLA. He would like to acknowledge the comments of Andy Atkeson, Francisco Buera, Roger Farmer, Christian Hellwig, Gonzalo Llosa, Bentley McLeod, Andy Neumeyer, Lee Ohanian, Venky Venkateswaran, and Pierre-Olivier Weill; as well as participants in the World Bank ABCDE Conference, Macro Lunch Proseminar at UCLA, LACEA, Universidad de los Andes, the Midwest Macro Conference, the Central Bank of Colombia, the University of Leipzig and The Guanajuato Workshop for Young Economists. The author is also very grateful for the financial support given by the Central Bank of Colombia. The valuable research assistance of Felipe Acero is greatly acknowledged. The usual disclaimer applies.

© The Author 2015. Published by Oxford University Press on behalf of the International Bank for Reconstruction and Development / the world bank. All rights reserved. For permissions, please e-mail: [email protected].

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