EXISTENCE OF STATIONARY EQUILIBRIUM IN AN INCOMPLETE‐MARKET MODEL WITH ENDOGENOUS LABOR SUPPLY

• Author: Shenghao Zhu
• DOI: http://doi.org/10.1111/iere.12451
• Date: 01 August 2020
• Published date: 01 August 2020

INTERNATIONAL ECONOMIC REVIEW

Vol. 61, No. 3, August 2020 DOI: 10.1111/iere.12451

EXISTENCE OF STATIONARY EQUILIBRIUM IN AN INCOMPLETE-MARKET

MODEL WITH ENDOGENOUS LABOR SUPPLY∗

BYSHENGHAO ZHU1

University of International Business and Economics, Beijing, China

In this article, I first study an income fluctuation problem with endogenous labor supply. Let βbe the agent’s

time discount factor and R>0 be the constant gross rate of return on assets. For βR=1, I show that the agent’s

wealth either approaches infinity almost surely or converges to a finite level almost surely. For βR<1, I prove

the existence, uniqueness, and stability of the stationary distribution of state variables. I then show the existence

of the stationary general equilibrium in an incomplete-market model with endogenous labor supply.

1. INTRODUCTION

The aim of this article is to show the existence of the stationary general equilibrium in an

incomplete-market model with endogenous labor supply. There is a continuum of households

with measure 1 in the economy. Households have uninsurable idiosyncratic labor efficiency

shocks. Each household faces an income fluctuation problem with endogenous labor supply.

The labor efficiency shock follows a Markov chain along time and is independent and identically

distributed (i.i.d.) across households.

Aiyagari and McGrattan (1998) use an incomplete-market heterogeneous agents model with

endogenous labor supply to study the optimum quantity of government debt. Marcet et al.

(2007) show that incomplete insurance to idiosyncratic employment shocks introduce an ex

post wealth effect, which reduces labor supply. The ex post wealth effect on labor supply runs

counter to the precautionary savings motive.2These articles did not show the existence of the

stationary general equilibrium. This article fills this gap.

I first study an income fluctuation problem with endogenous labor supply. Let βbe the agent’s

time discount factor and R>0 be the constant gross rate of return on assets. The net rate of

return is r=R−1. For βR=1, I show that the agent’s wealth either approaches infinity almost

surely or converges to a finite level almost surely as t→∞. If wealth converges to a finite level

almost surely, then the agent’s labor supply approaches zero almost surely as t→∞. As long as

the agent does not stop working, income shocks always exist. Precautionary savings cause wealth

accumulation to eventually reach infinity. If the agent stops working due to the income effect,

then wealth accumulation stops. Moreover, there exists upper bounds of wealth accumulation.

This general result holds as long as both consumption and leisure are normal goods. Therefore, I

extend the well-known result of Chamberlain and Wilson (2000) to situations with endogenous

labor supply.

∗Manuscript received January 2017; revised October 2019.

1I would like to thank Jushan Bai, Jess Benhabib, Alberto Bisin, Greg Kaplan, Sydney Ludvigson, Yulei Luo,

Albert Marcet, Efe Ok, Thomas Sargent, Yeneng Sun, Rangarajan Sundaram, John Stachurski, Gianluca Violante,

Philippe Weil, Bin Wu, and Charles Wilson for their comments and suggestions. I acknowledge the financial support

from National Natural Science Foundation of China (71873034) and the Fundamental Research Funds for the Central

Universities in UIBE (CXTD10-01). Please address correspondence to: Shenghao Zhu, School of International Trade,

and Economics, University of International Business and Economics, Beijing, China. E-mail: zhushenghao@yahoo.com.

2In their numerical examples, the wealth effect often dominates the precautionary saving effects. Thus, the output

and savings in incomplete markets are lower than those in complete markets.

1115

C

(2020) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

1116 ZHU

I find sufficient conditions guaranteeing that wealth accumulation has upper bounds for

cases of βR=1 and βR<1. I also find that the ratio between marginal utility functions of

consumption in different shock states plays an important role in determining precautionary

savings. To confine this ratio, we can always obtain a lower bound of the consumption policy

function for βR<1 in a model with endogenous labor supply. Recent works by Acikg ¨

oz (2018)

and Stachurski and Toda (2019) find the lower bound of the consumption policy function for

βR<1 in models with exogenous labor supply. I provide a general framework to investigate

income fluctuation problems with exogenous labor supply and with endogenous labor supply.

The unified framework brings us more insight into research on incomplete-market models.

I prove the existence, uniqueness, and stability of the stationary distribution of state variables

for βR<1.3Aiyagari (1993), Huggett (1993), and Marcet et al. (2007) employ the monotone-

Markov-process method of Hopenhayn and Prescott (1992) to show this. Kamihigashi and

Stachurski (2014) extend this method to the unbounded state space. In contrast, I use a new

method to show the existence, uniqueness, and stability of the stationary distribution, and I do

not need the monotonicity assumption of the Markov chain. The crucial observation is that the

lower bound of the state space for βR<1 is an accessible atom. Starting from any asset level,

the state variables have positive probability to hit the lower bound in finitely many periods. That

the borrowing constraint is binding infinitely often in an income fluctuation problem implies that

the lower bound of the state space is an accessible atom. The new result highlights the impact

of borrowing constraints and precautionary savings on the stationary wealth distribution.

To show the existence of the stationary equilibrium, I find the intersection of the “supply”

and “demand” curves for the capital–labor ratio in the economy. The aggregate capital supply

is the total wealth of households in the stationary distribution of state variables. The aggregate

labor supply is the total labor supply in the stationary distribution. The “supply” curve for the

capital–labor ratio is the ratio of the aggregate capital supply to the aggregate labor supply. I

show that the “supply” curve is a continuous function of the interest rate rand tends to infinity

as rapproaches ¯r=1

β−1 from below. However, the infinity limit could be due to infinite wealth

accumulation or zero labor supply. From the firm’s profit-maximization problem the “demand”

curve for the ratio is derived, which approaches infinity as rtends to −δ. Following Aiyagari

(1994), I show the existence of the stationary equilibrium by finding the intersection of these

two continuous curves. Simply replacing capital by the capital–labor ratio, I extend the idea of

Aiyagari (1994) to models with endogenous labor supply. Thus I provide a general framework to

show the existence of the stationary equilibrium in incomplete-market models with exogenous

labor supply and with endogenous labor supply.

My existence proof of the stationary equilibrium also provides new insight into income

fluctuation problems. If the agent’s wealth approaches infinity almost surely as t→∞for the

case of βR=1, then the aggregate capital supply converges to infinity as r↑¯r. If the agent’s

wealth converges to a finite level almost surely as t→∞for the case of βR=1, then aggregate

labor supply approaches zero as r↑¯r. These limit results are due to the continuity of optimal

policy functions with respect to parameters, including interest rate r.

After I weaken the monotonicity of the Markov chain shocks, these results are more appli-

cable in simulation exercises. My existence proof of the stationary equilibrium also shows that

a bisection algorithm can find a stationary general equilibrium. Therefore, my article offers

guidance to simulation works on incomplete-market models with endogenous labor supply.

Using the concept of “tightness” of a collection of probability measures, I provide a new

probability-limit-theory tool, which extends the frequently used Theorem 12.13 by Stokey and

Lucas (1989), to investigate the parametric continuity of stationary distributions. Specifically,

I use tightness to replace compactness in the previous famous theorem. Therefore, I relax

the assumption and extend the application scope of the theorem. Equipped with this new

tool, I investigate how stationary distributions move when model parameters change and find

3The stability here means that, starting from any initial distribution of state variables, the stochastic process converges

to the unique stationary distribution.