Job search costs and incentives

Abstract

We demonstrate that policies aimed at reducing frictional unemployment may lead to the opposite results. In a labor market with long-term wage contracts and moral hazard, any such policy reduces employees’ opportunity costs of staying on a job. As employees are less worried about losing their job, a smaller share of employees is willing to exert effort, leading to a lower average productivity. Consequently, firms create fewer vacancies, resulting in lower employment and decreased welfare.

Appendix: Proofs

1.1 Proof of Proposition 1

Proposition 1

In the model with no moral hazard, a reduction of the job search cost is welfare improving.

The proof is divided into four subsections.

1.1.1 Labor demand

Denote by x and y the masses of searching individuals and firms, respectively, and by X and Y the masses of active individuals (that are searching or producing) and active firms (that have a filled job or a vacancy), respectively. Let J and V be a firm’s value of a filled job and a vacancy, respectively. Let $\mu _F$ be the probability to fill a vacancy in a given period, $\mu _F= \min \{x/y,1\}$. Then

$$\begin{aligned} V=-s_F+\mu _F J+(1-\mu _F)\delta V. \end{aligned}$$

Assume that firms create vacancies so long as $V>0$ and withdraw them if $V< 0$. If $J< s_F$, then $V<0$ for all $\mu _F$, and hence, there will be no labor demand, $Y=0$. However, if $J\ge s_F$, then in steady state, $V=0$ must hold, and hence, $\mu _F$ must satisfy

$$\begin{aligned} -s_F+\mu _F J=0. \end{aligned}$$

(A.1)

Since $s_F/J=\mu _F\equiv \min \{x/y,1\}$, then either $s_F/J<1$, and hence $x/y< 1$, or $s_F/J=1$, and hence $x/y\ge 1$. Moreover, in the latter case, it must be $x/y= 1$, because we have assumed that firms create vacancies so long as the cost of maintaining one, $s_F$, does not exceed the expected benefit, J.¹⁸ Therefore, if $y<x$, then new vacancies will be created until $y=x$. Consequently, the mass of job searching individuals does not exceed the mass of vacancies, so every individual finds a job immediately with certainty.¹⁹

Let us now derive the mass of active firms, Y. Denote by $\gamma$ the fraction of individuals that are employed at the beginning of each period. Then

$$\begin{aligned} \mu _F = \frac{x}{y}=\frac{X-\gamma X}{Y-\gamma X}. \end{aligned}$$

(A.2)

Combining (A.1) and (A.2) yields:

$$\begin{aligned} Y=X\left( \gamma + (1-\gamma )\frac{J}{s_F}\right) . \end{aligned}$$

(A.3)

1.1.2 Labor supply

In the benchmark model, there is no moral hazard, so all employees exert high effort.²⁰ Hence, firms will never dismiss workers. The fraction of individuals who lose their jobs and return to the job market is given by job destruction rate $1-\alpha$. The total revenue created by a filled job position net of the unemployment income (for short, surplus) is, therefore, equal to:

$$\begin{aligned} S=\pi -u_0+\alpha \delta (\pi -u_0)+\cdots = \frac{\pi -u_0}{1-\alpha \delta }. \end{aligned}$$

Similarly, a firm’s gross value of filling a vacancy is equal to:

$$\begin{aligned} J=\frac{\pi -w}{1-\alpha \delta }. \end{aligned}$$

We assumed that a firm’s share of the surplus is $1-\beta$, so $J=(1-\beta )S$, and hence, the wage is given by²¹:

$$\begin{aligned} w=u_0+\beta (\pi -u_0). \end{aligned}$$

(A.4)

Let us now determine the equilibrium behavior of an individual with cost of effort c.

Lemma 5

An individual of type $c\in {\mathbb {R}}_{+}$ participates in the job market if and only if

$$\begin{aligned} c<\beta (\pi -u_0)-(1-\alpha \delta )s. \end{aligned}$$

Proof

Since we are considering equilibria in steady state only, it is sufficient to compare the lifetime utility of the individual who is always active and the one who is always inactive. The utility of an inactive individual of type c is given by (3) in the main text, restated here:

$$\begin{aligned} U_0 = u_0+\delta U_0= \frac{u_0}{1-\delta }. \end{aligned}$$

(A.5)

Now, consider an active individual of type c. Denote by $U_{\mathrm{H}}(c)$ her lifetime utility starting from a period where she is unemployed (subscript H stands for “high effort” for consistency with notations in further sections). Note that in the period where she is employed, her lifetime utility is simply $U_{\mathrm{H}}(c)+s$, as in a steady state, the only difference between being employed and unemployed is the job search cost (as we have established above that individuals find a job immediately with probability one). This yields (1) in the main text, restated here:

$$\begin{aligned} U_{\mathrm{H}}(c)&=-s+(w-c)+\delta \left[ \alpha (U_{\mathrm{H}}(c)+s)+(1-\alpha )U_{\mathrm{H}}(c)\right] \nonumber \\&= \frac{w-c-(1-\alpha \delta ) s}{1-\delta }. \end{aligned}$$

(A.6)

Consequently, an individual of type c will participate in the labor market if and only if²²$U_{\mathrm{H}}(c)>U_0$, or

$$\begin{aligned} c<w-u_0-(1-\alpha \delta )s, \end{aligned}$$

which, together with (A.4), yields the result. $\square$

Intuitively, an individual will participate in the labor market if her cost of effort is small relative to the expected wage, net of the unemployment income, and expected costs of job search. Denote by ${\bar{c}}(s)$ the critical type who is indifferent between participating or not:

$$\begin{aligned} {\bar{c}}(s)= \beta (\pi -u_0)-(1-\alpha \delta )s. \end{aligned}$$

1.1.3 Steady state

We are now in position to find the masses of active individuals and active firms, X and Y, in steady state.

First, by Lemma 5, only individuals of type $c<{\bar{c}}(s)$ will be active, and hence

$$\begin{aligned} X=F({\bar{c}}(s)). \end{aligned}$$

Next, a firm’s value of a filled job is given by:

$$\begin{aligned} J=(1-\beta )S = \frac{(1-\beta )(\pi -u_0)}{1-\alpha \delta }. \end{aligned}$$

Finally, after every period fraction, $\alpha$ of active individuals remain employed, and thus, $\gamma =\alpha$. Consequently, by (A.3):

$$\begin{aligned} Y=F({\bar{c}}(s))\left( \alpha + (1-\alpha )\frac{(1-\beta )(\pi -u_0)}{s_F(1-\alpha \delta )}\right) . \end{aligned}$$

Note that if the individuals’ search cost is high enough, $s\ge \beta (\pi -u_0)/(1-\alpha \delta )$, then no individuals participate and the labor market collapses (since in that case ${\bar{c}}(s)\le 0$ and $F({\bar{c}}(s))=0$, so we have $X=0$). Similarly, if the firms’ advertising cost is high enough, $s_F> J=(1-\beta )(\pi -u_0)/(1-\alpha \delta )$, no firms are willing to open vacancies, so we have $Y=0$ and the labor market collapses. For the rest of the paper, we assume that $s_F\le (1-\beta )(\pi -u_0)/(1-\alpha \delta )$.

1.1.4 Comparative statics

Let us analyze the relationship between an individual’s job search cost and the welfare. As in this model, search cost s represents a pure waste, it is very intuitive that a reduction of s leads to welfare improvement.

Indeed, consider a reduction of the search cost from s to $s'$. Then, the mass of the individuals who search for jobs weakly increases, as $s>s'$ entails ${\bar{c}}(s)<{\bar{c}}(s')$, and consequently, $F({\bar{c}}(s))\le F({\bar{c}}(s'))$. Any individual who is participating under $s'$ is strictly better off relative to s, as her utility has gone up due to the lower search cost. Any individual who is inactive under $s'$ is indifferent between s and $s'$, as her utility remains unchanged, $u_0/(1-\delta )$. Hence, the consumer surplus strictly increases as s goes down.

Next, firms with filled job positions make profit $(1-\beta )(\pi -u_0)$ in a single period. Under $s'$, the mass of firms engaged in production is larger, and so is the total producers’ surplus. $\square$

1.2 Proof of Theorem 1

Theorem 1

• (A) There exists an equilibrium with positive participation if and only if ${\underline{s}}\le s<{\bar{s}}$.

• (B) The equilibrium with positive participation is unique. If $s\ge \beta (\pi -u_0)$, then it coincides with that in the benchmark model. If $s\le \beta (\pi -u_0)$,²³ then it is characterized by:

  • (i) the equilibrium wage is $w=u_0+s$;

  • (ii) every individual of type $c<\alpha \delta s$ participates with probability one and exerts high effort; every individual of type $c\ge \alpha \delta s$ participates with probability $\lambda ^*$ and exerts low effort,²⁴ where

    $$\begin{aligned} \lambda ^*=\frac{(1-\alpha )F(\alpha \delta s)}{(1-\alpha \delta )(1-F(\alpha \delta s))}\cdot \frac{\beta (\pi -u_0)-s}{\beta u_0+s}. \end{aligned}$$

    (4)

1.2.1 Part (B)

We will prove that if there exists an equilibrium with positive participation, then it is unique and satisfies the conditions stated in Part B.

Recall that we focus on stationary equilibria. In particular, the best reply of a firm to a stationary strategy of the individuals (Lemma 1) and the best reply of individuals to a stationary strategy of firms (Theorem 1) are stationary. In other words, no market participants can benefit by deviating to nonstationary strategies.

Denote by p the probability that a newly hired worker has type $c<c^*(s)$ and thus will exert high effort when employed. Then, surplus S produced by a filled job vacancy is given by:

$$\begin{aligned} S=p\frac{\pi -u_0}{1-\alpha \delta }+(1-p)(-u_0), \end{aligned}$$

(A.7)

where $(\pi -u_0)/(1-\alpha \delta )$ is the surplus from an employee who always exerts high effort (as S in Sect. 3) and $(-u_0)$ is the surplus from an employee who shirks in the first period and loses the job at the end of that period.

Similarly, the value J of a filled job vacancy for a firm is given by:

$$\begin{aligned} J=p\frac{\pi -w}{1-\alpha \delta }+(1-p)(-w)=S-p(\lambda )\frac{w-u_0}{1-\alpha \delta }-(1-p)(w-u_0), \end{aligned}$$

where w is the equilibrium wage. Recall that wage determination rule requires:

$$\begin{aligned} J=(1-\beta )S. \end{aligned}$$

(A.8)

As J is strictly decreasing in w and $J=S$ for $w=u_0$, for any given p, there exists a unique w that solves (A.8).

Suppose that $s>\beta (\pi -u_0)$. By the wage determination rule, the wage cannot exceed its feasible maximum $u_0+\beta (\pi -u_0)$, and hence, $w<u_0+s$. This corresponds to case (a) in Corollary 1, where all individuals who participate exert high effort, $p=1$. Then, we have $S=(\pi -u_0)/(1-\alpha \delta )$ and $J=(\pi -w)/(1-\alpha \delta )$. Solving (A.8) for w yields $w=u_0+\beta (\pi -u_0)$. Note that this is the same equilibrium as in the benchmark model.

Next, suppose that $s<\beta (\pi -u_0)$. Corollary 1 implies the following labor market composition. The total mass of active individuals with type $c<c^*(s)$ is $F(c^*(s))$, of which only fraction $1-\alpha$ lose their jobs and return to the labor market. Hence, the mass of individuals with type $c<c^*(s)$ on the labor market constitutes $(1-\alpha )F(c^*(s))$. Next, the total mass of individuals with type $c\ge c^*(s)$ is $1-F(c^*(s))$, of which some fraction $\lambda \in [0,1]$ are active. Hence, the mass of individuals with type $c\ge c^*(s)$ on the labor market constitutes $\lambda (1-F(c^*(s)))$. Consequently, the probability that a newly hired worker has type $c<c^*(s)$ and thus will exert high effort is given by:

$$\begin{aligned} p=p(\lambda )=\frac{(1-\alpha )F(c^*(s))}{(1-\alpha )F(c^*(s))+\lambda (1-F(c^*(s)))} = \left( 1+\lambda \frac{1-F(c^*(s))}{(1-\alpha )F(c^*(s))}\right) ^{-1}.\nonumber \\ \end{aligned}$$

(A.9)

Consider case (c) in Corollary 1 that stipulates $w=u_0+s$, so condition (i) holds. Then, there is a unique value of $\lambda$ that solves (A.8) and it is equal to $\lambda ^*$ that can be easily verified. Thus, we have proved that the equilibrium must satisfy condition (ii). Yet, we need to verify that $\lambda ^*\in [0,1]$. We have assumed that $s\le \beta (\pi -u_0)$, and thus, $\lambda ^*\ge 0$. Also, we have assumed that F first order stochastically dominates the uniform distribution on $[0,\alpha \delta \beta (\pi -u_0)]$, that is:

$$\begin{aligned} F(c^*(s))\le \frac{c^*(s)}{\alpha \delta \beta (\pi -u_0)}=\frac{\alpha \delta s}{\alpha \delta \beta (\pi -u_0)}=\frac{s}{\beta (\pi -u_0)} \end{aligned}$$

(A.10)

for all $s\in [0,\beta (\pi -u_0]$. Hence, we have $F(c^*(s))\beta (\pi -u_0)\le s$ for all $s\in [0,\beta (\pi -u_0)]$. Consequently

$$\begin{aligned} \lambda ^*=\frac{(1-\alpha )F(\alpha \delta s)}{(1-\alpha \delta )(1-F(\alpha \delta s))}\cdot \frac{\beta (\pi -u_0)-s}{\beta u_0+s}\le \frac{(1-\alpha )s}{\beta (1-\alpha \delta )+s}\le 1. \end{aligned}$$

Next, let us show that cases (a) and (b) in Corollary 1 cannot occur in equilibrium. It was shown above that case (a) entails $s>\beta (\pi -u_0)$, a contradiction. It remains to consider case (b) that stipulates $w>u_0+s$ and $\lambda =1$. Then, we have:

$$\begin{aligned} p=p(1)=\frac{(1-\alpha )F(c^*(s))}{1-\alpha F(c^*(s))}. \end{aligned}$$

Equation (A.8) can be rewritten as:

$$\begin{aligned} p\frac{w-u_0}{1-\alpha \delta }+(1-p)(w-u_0)=\beta \left( p\frac{\pi -u_0}{1-\alpha \delta }+(1-p)(-u_0)\right) . \end{aligned}$$

Solving for $(w-u_0)$, we obtain:

$$\begin{aligned} w-u_0&= \frac{p\beta (\pi -u_0)}{1-\alpha \delta (1-p)}-\frac{(1-\alpha \delta )(1-p)u_0}{1-\alpha \delta (1-p)} \\&\le p\frac{\beta (\pi -u_0)}{1-\alpha \delta (1-p)} = \frac{(1-\alpha )F(c^*(s))}{1-\alpha F(c^*(s))}\cdot \frac{\beta (\pi -u_0)}{1-\alpha \delta (1-p)} \\&\le \frac{(1-\alpha )s}{(1-\alpha F(c^*(s)))(1-\alpha \delta (1-p))} \\&\le s, \end{aligned}$$

where the third line is by (A.10) and the last line is due to:

$$\begin{aligned} 1-\alpha \delta (1-p)= & {} 1-\alpha \delta \frac{1-F(c^*(s))}{1-\alpha F(c^*(s))}=\frac{1-\alpha (\delta +(1-\delta )F(c^*(s)))}{1-\alpha F(c^*(s))}\\\ge & {} \frac{1-\alpha }{1-\alpha F(c^*(s))}. \end{aligned}$$

However, we have assumed $w>u_0+s$, a contradiction.

1.2.2 Part (A)

First, we show that $X=0$ for $s\ge {\bar{s}}$. For $s>\beta (\pi -u_0)$, the equilibrium (if it exists) is the same as in the benchmark model, and by Lemma 5, an individual of type c participates in the labor market if and only if $c<{\bar{c}}(s)$. However, for $s\ge {\bar{s}}$, ${\bar{c}}(s)=\beta (\pi -v_0)-(1-\alpha \delta )s\le 0$, and hence, there is no participation, $X=0$.

Second, we show that $Y=0$ for $s<{\underline{s}}$. Recall that in equilibrium, $Y>0$ only if $J\ge s_F$, where

$$\begin{aligned} J = (1-\beta )S = (1-\beta )\left( p\frac{\pi -u_0}{1-\alpha \delta }-(1-p)u_0\right) . \end{aligned}$$

Substituting the value of $\lambda ^*$ into (A.9), after some manipulations, yields:

$$\begin{aligned} p=p(\lambda ^*)=\frac{(1-\alpha \delta )(\beta u_0+s)}{\beta (\pi -\alpha \delta u_0)-\alpha \delta s}. \end{aligned}$$

Next, substituting $p(\lambda ^*)$ into the expression for J, after some manipulations, yields:

$$\begin{aligned} J=\frac{(1-\beta )\pi s}{\beta (\pi -\alpha \delta u_0)-\alpha \delta s}. \end{aligned}$$

(A.11)

At last, solving inequality $J\ge s_F$ for s yields:

$$\begin{aligned} s\ge \frac{\beta (\pi -\alpha \delta u_0)s_F}{(1-\beta )\pi +\alpha \delta s_F}\equiv {\underline{s}}. \end{aligned}$$

$\square$

1.3 Proof of Lemma 4

Lemma 4

For all $s\in [{\underline{s}},\beta (\pi -u_0)]$, the welfare in the model with moral hazard is equal to:

$$\begin{aligned} W(s) = \int _0^{c^*(s)}\frac{\alpha s-c}{1-\delta }\mathrm{d}F(c)+\frac{\alpha (\pi -u_0-s)}{1-\alpha \delta }F(c^*(s)), \end{aligned}$$

(5)

and it is strictly increasing in s.

For all $s\in (0,{\bar{s}})$, the welfare in the benchmark model is equal to:

$$\begin{aligned} {\overline{W}}(s) =\int _0^{{\bar{c}}(s)}\frac{\left[ \beta (\pi -u_0)-s\right] +\alpha s-c}{1-\delta }\mathrm{d}F(c)+ \frac{\alpha (1-\beta )(\pi -u_0)}{1-\alpha \delta }F({\bar{c}}(s)),\nonumber \\ \end{aligned}$$

(6)

and it is strictly decreasing in s.

Let us calculate the welfare in the benchmark model first. In every period, an individual of type $c<{\bar{c}}(s)\equiv \beta (\pi -v_0)-(1-\alpha \delta )s$ obtains $w-c$. Also, with probability $1-\alpha$, she is unemployed at the beginning of the period, so she pays search cost s. Thus, the lifetime utility of an individual of type $c<{\bar{c}}(s)$ is equal to:

$$\begin{aligned} \frac{w-c-(1-\alpha )s}{1-\delta }=\frac{u_0+\beta (\pi -u_0)-c-(1-\alpha )s}{1-\delta }, \end{aligned}$$

where we used that in equilibrium the wage satisfies $w=u_0+\beta (\pi -u_0)$. The lifetime utility of an individual of type $c\ge {\bar{c}}(s)$ is the unemployment income, $u_0/(1-\delta )$. Hence, the consumer’s surplus (net of the unemployment income) is equal to:

$$\begin{aligned} \overline{CS}(s)&= \int _0^{{\bar{c}}(s)}\frac{\beta (\pi -u_0)-(1-\alpha )s-c}{1-\delta }\mathrm{d}F(c). \end{aligned}$$

Next, at the beginning of the period, the mass of firms who have a filled position is $\alpha F({\bar{c}}(s))$. The lifetime utility of any such firm is $(\pi -w)/(1-\alpha \delta )$. Any firm that has an unfilled vacancy or inactive has zero lifetime utility. Hence, the producers’ surplus is equal to:

$$\begin{aligned} \overline{PS}(s)&= \alpha F({\bar{c}}(s)) \frac{\pi -w}{1-\alpha \delta }=\alpha F({\bar{c}}(s)) \frac{\pi -(u_0+\beta (\pi -u_0))}{1-\alpha \delta } \\&=\alpha F({\bar{c}}(s)) \frac{(1-\beta )(\pi -u_0)}{1-\alpha \delta }. \end{aligned}$$

Summing up $\overline{CS}(s)$ and $\overline{PS}(s)$ gives the expression for welfare ${\overline{W}}(s)$. As ${\bar{c}}(s)$ is strictly decreasing in s, it is easy to verify that $\overline{CS}(s)$ is strictly decreasing and $\overline{PS}(s)$ is decreasing. Hence, ${\overline{W}}(s)$ is strictly decreasing.

Let us now calculate the welfare in the model with moral hazard. In every period, an individual of type $c<c^*(s)\equiv \alpha \delta s$ obtains $w-c$ and also pays search cost s with probability $1-\alpha$ that she has become unemployed at the end of the previous period. Thus, the lifetime utility of an individual of type $c<c^*(s)\equiv \alpha \delta s$ is equal to:

$$\begin{aligned} \frac{w-c-(1-\alpha )s}{1-\delta }=\frac{(u_0+s)-c-(1-\alpha )s}{1-\delta }=\frac{u_0+ \alpha s-c}{1-\delta }, \end{aligned}$$

where we used that in equilibrium the wage satisfies $w=u_0+s$. The lifetime utility of an individual of type $c\ge c^*(s)$ is the unemployment income, $u_0/(1-\delta )$. Hence, the consumer’s surplus (net of the unemployment income) is equal to:

$$\begin{aligned} {CS}(s)&= \int _0^{c^*(s)}\frac{\alpha s-c}{1-\delta }\mathrm{d}F(c). \end{aligned}$$

Next, similarly to in the benchmark model, at the beginning of the period, the mass of firms who have a filled position is $\alpha F(c^*(s))$. The lifetime utility of any such firm is $(\pi -w)/(1-\alpha \delta )$. Any firm that has an unfilled vacancy or inactive has zero lifetime utility. Hence, the producers’ surplus is equal to:

$$\begin{aligned} {PS}(s)&= \alpha F(c^*(s)) \frac{\pi -w}{1-\alpha \delta }=\alpha F(c^*(s)) \frac{\pi -(u_0+s)}{1-\alpha \delta }. \end{aligned}$$

Summing up CS(s) and PS(s) gives the expression for welfare W(s). Taking the derivative of W(s) (where we have used $dc^*(s)/ds=\alpha \delta$) yields:

$$\begin{aligned} \frac{d}{ds}W(s)&=\frac{\alpha }{1-\delta }F(c^*(s))+\alpha \delta f(c^*(s))+\frac{\alpha ^2\delta }{1-\alpha \delta }f(c^*(s))(\pi -u_0-s)\\&\quad -\,\frac{\alpha }{1-\alpha \delta }F(c^*(s)) \\&=\frac{\alpha ^2\delta }{1-\alpha \delta }f(c^*(s))(\pi -u_0-s)+\alpha \delta f(c^*(s)) \\&\quad +\,\alpha F(c^*(s))\left( \frac{1}{1-\delta }-\frac{1}{1-\alpha \delta }\right) >0, \end{aligned}$$

since by assumption $s\le \beta (\pi -u_0)<\pi -u_0$ and $\frac{1}{1-\delta }-\frac{1}{1-\alpha \delta }>0$. $\square$