The Cyclicality of On-the-Job Search Effort

1.1 Overview

Consider a two-period consumption model of a worker who already has a job in the first period.^[29] The worker is risk averse, and derives utility from consumption. However, she derives disutility from the effort she spends on searching for a new job. Assume that workers do not have access to complete insurance market. Suppose, for simplicity, that saving technology does not exist in this economy.^[30] The worker self-insures against possible income loss in the future by raising job-search effort in the first period, when she expects the labor market conditions to deteriorate in the second period or feels uncertain about the labor market situation in the future. By searching hard for a new job, it becomes easier for her to find a job in case she loses her current job, or to switch to a better-paying employer when she experiences a wage cut at her current job. The model is a partial equilibrium model where labor market variables such as the job-arrival rate, job-destruction rate, employment volatility, and wages are determined exogenously.^[31]

1.2 Model Timing

1. First period: In the first period, the worker earns wage, w₁, and decides how much of her income she will spend on consumption, c₁, and job search, s₁.^[32]

2. Second period: In the second period, separation takes place. The worker becomes unemployed with probability δ₂. When the worker loses her job, she remains unemployed and receives the unemployment insurance benefit b if the worker did not search for a job in the first period. Or she can find a new job with probability p₂ and earn wage w₂, if the worker did job-search activities in the first period. When the worker does not lose her job, she can either stay with her current employer, and gets wage w₂ if she did not search for a new job. Or she can switch to a new employer with probability p₂ and earn wage w₂ + μ with μ > 0 if she looked for a new job in the first period.^[33][34]

The worker has imperfect knowledge about the labor market condition in the second period. Therefore, her second-period consumption is a random variable that depends on her second-period income.

Let f₂ be the aggregate job-finding probability. Then, an employed individual’s probability of finding a new job, p₂, is a function of s₁ and f₂ as follows:

We assume that ∂p(s1,f2)∂s1>0 and ∂2p(s1,f2)∂s12<0.

1.3 Model

In this set-up, the employed worker’s problem is written into the following.

If we retain the assumption β = 1, for simplicity, the first-order necessary condition for optimal consumption and search effort is obtained from the following Euler equation

Equation (1) says that the worker should be indifferent between, on the one hand, consuming one more unit in period one and, on the other hand, spending one unit for job search in period one and consuming the units equivalent to the expected outcome of job search in period two. Note that the term in brackets is the rate of return from job search. The key difference from the usual consumption Euler equation is that the rate of return from one unit of job search depends on the expected values of stochastic parameters, δ₂, w₂, and f₂, while the rate of return from saving is a fixed parameter in a usual model of consumption and saving.

Suppose that there is a sufficient statistic of labor market condition in period two, denoted as ξ2, that determines the labor market variables in the second period. Let V(ξ2) be the realized utility from consumption in the second period, then

Assume that ∂V(ξ2)∂ξ2,∂V2(ξ2)(∂ξ2)2, and ∂V3(ξ2)(∂ξ2)3 exist. If we take derivatives of both sides of Equation (2) with respect to s₁, then we have

Let ξ¯2 be the value of ξ2 expected in period 1. Then, the optimal level of job search in the first period is also determined by ξ¯2. Suppose that s1=g(ξ¯2) where ξ¯2=h(s1), and ∂ξ¯2∂s1=∂h(s1)∂s1=h. We assume that h = 1 to simplify our analysis, as suppressing the notation h does not influence the key feature of the model. Suppose in addition that ξ2=ξ¯2+e2 where e2∼IID(0,σ2).

Approximating Equation (3) using Taylor expansion, it follows that

Let V′″(ξ¯2)=∂V3(ξ2)(∂ξ2)3|ξ2=ξ¯2 for notational convenience. If we assume that the utility function is a log function, Equation (1) is rearranged into

where E(δ2)=δ¯2, E(w2)=w¯2, and E(f2)=f¯2, and

If we are willing to assume that δ, w, and f are persistent processes, then the expected values of δ₂, w₂, and f₂ are positively correlated with δ₁, w₁, and f₁(∂δ¯2∂δ1>0,∂w¯2∂w1>0,∂f¯2∂f1>0).^[35] In addition, note that 1−12V′″(ξ¯2)σ2(w1−s1) is not zero, as Equation (4) implies that consumption in the first period should be zero, otherwise.

In addition, the rate of return from one unit of job search should be higher when there is uncertainty to compensate for the risk of job search. The following lemma provides the necessary condition for the risk premium associated with job search to be positive.

If 1−12V′″(ξ¯2)σ2(w1−s1) is between 0 and 1, the rate of return from job search is higher when there is uncertainty.

The Euler Equation (4) suggests that the rate of return from job search when there is uncertainty is

For the rate of return from job search in the presence of uncertainty to be higher than that without uncertainty, 0<1−12V′″(ξ¯2)σ2(w1−s1)<1.

Note that 1−12V′″(ξ¯2)σ2(w1−s1) is the parameter determining the size of risk premium associated with the rate of return from OJS.

Based on the Euler equation (1), now we can analyze how w¯2, δ¯2, f¯2 and σ₂ affect the job search effort in the first period.

Using the implicit function theorem, the derivative is written into the following,

where ∂F∂σ2 and ∂F∂s1 are the following:

If V′″(ξ¯2)>0 and ∂F∂σ2<0, then ∂s1∂σ2>0.

Therefore, ∂s1∂δ¯2>0, and thus ∂s1∂δ1>0.

The corollary implies the following. When workers expect the wage from current job to fall in the future, then they will search harder for a new job, if they can get a much higher wage at a new job. Otherwise, when wages fall, it is more beneficial for a worker to reduce job-search effort and to stay unemployed receiving the unemployment insurance benefit when losing the current job.