<p>It depends on what you define as wage curve. I did not check in the textbooks, but you suggest that Pissarides and Cahuc et al. do not use the same mathemaical expression for the "wage curve".</p>

<p>1) In Cahuc et al., the expression combines the surplus sharing equation with the Bellman equations without using the free entry condition of firms. The wage <span class="math-container">$w$</span> is increasing in <span class="math-container">$\theta$</span> because a tighter market tightness improves the worker's bargaining position. There is no labor demand effect in this wage equation. Therefore, the vacancy cost does not appear.</p>

<p>2) The Pissarides' wage curve is less straightforward to interpret. Given the positive relationship between <span class="math-container">$w$</span> and <span class="math-container">$c$</span>, one could erroneously think that the vacancy cost increases the wage. The interpretation is the following.
At <span class="math-container">$\theta$</span> fixed, a higher vacancy cost means that firms spend more to recruit. More spendings capture the fact that firms make larger job profits (free-entry condition). When firms make larger profits, workers also obtain a larger surplus due to the Nash bargaining. Thus, the wage is higher.</p>

<p>This definition of the wage curve does not depend on the matching technology (<span class="math-container">$m$</span>). To understand this result, we can write the model with an abstract matching market.
With your notations, we define the value of unemployment <span class="math-container">$U$</span>, the value of employment <span class="math-container">$W$</span> and the value of a job for a firm <span class="math-container">$J$</span>.
We can write the three Bellman equations, and the Nash bargaining condition with the wage:
<span class="math-container">\begin{align}
    &amp; rU = z + R\\
    &amp; rW = w+q(U-W)\\
    &amp; rJ = y-w-qJ\\
    &amp; w=\gamma y + (1-\gamma)rU
\end{align}</span>
If a worker is employed, she produces <span class="math-container">$y$</span> and receive <span class="math-container">$w$</span>, with a risk to lose the job <span class="math-container">$q$</span>.
If the worker is not employed, she receives <span class="math-container">$z$</span> plus the returns from participating to a matching market <span class="math-container">$R$</span>.</p>

<p>In the full model <span class="math-container">$R=\theta m(\theta) (W-U)$</span>. Here, we do not use this structure.</p>

<p>We assume that each unemployed worker participating to the "matching market" produces <span class="math-container">$G$</span> and obtain <span class="math-container">$R$</span>. To participate to the market and obtain part of the gains, firms have to pay a cost <span class="math-container">$c$</span>. If we assume that <span class="math-container">$G$</span> is Nash-bargained, each worker obtains <span class="math-container">$R=\gamma G$</span>, and recruiting firms share <span class="math-container">$(1-\gamma)Gu$</span>. The average returns for a firm is <span class="math-container">$\frac{(1-\gamma)G u}{v}=(1-\gamma)\frac{G}{\theta}$</span>.
The free entry condition implies that firms make no profits from participating to the market. Therefore, <span class="math-container">$(1-\gamma)\frac{G}{\theta}=c$</span>.
Using the wage bargaining solution, we obtain Pissarides' wage curve using the wage bargaining solution and eliminating <span class="math-container">$rU$</span>, <span class="math-container">$R$</span> and <span class="math-container">$G$</span>.</p>

<p>The wage curve holds without assuming how the "matching market" works, we only assume that the outcome is Nash-bargained. To come back to a change in <span class="math-container">$c$</span>. An increase in the vacancy cost <span class="math-container">$c$</span> necessarily implies that the gain <span class="math-container">$G$</span> is higher, ans so the wage. </p>

<p>3) This is partial equilibrium analysis. If you solve for the equilibrium level of wages, you will find that <span class="math-container">$c$</span> actually reduces the equilibrium level of wages.</p>

CC BY-SA 4.0
2019-05-05T12:39:41.497Z
Thanks! Could you please go into more detail about the interpretation of Pissarides' wage curve? If I understood you correctly, job profits (presumably through $y$), drive both higher $c$ and wage in the wage curve? But what happens to labor demand and the wage curve after an exogenous increase in $c$ (with $y$ fixed, i.e. with job profit a priori fixed)?
Mark
CC BY-SA 4.0
2019-04-30T19:40:29.270Z
2019-04-30T19:40:29.270Z
GuiWil
<p>What effect does the hiring cost have on the wage bargaining outcome (wage curve)?</p>

<p>In Cahuc, Carcillo, and Zylberberg: <em>Labor Economics</em>, they derive the following wage curve:
<span class="math-container">$$w = z + (y-z) \frac{\gamma[r+q+\theta m(\theta)]}{r+q+\gamma \theta m(\theta)}$$</span></p>

<p>where <span class="math-container">$z$</span> = unemployment benefit, <span class="math-container">$y$</span> = marginal product of labor, <span class="math-container">$\gamma$</span> = bargaining power of the worker, <span class="math-container">$r$</span> = interest rate, <span class="math-container">$q$</span> = rate of exogenous job destruction, <span class="math-container">$\theta$</span> = labor market tightness, <span class="math-container">$m(\theta)$</span> rate of job filling.</p>

<p>On the other hand, following Pissarides (2000) we get the following wage curve:
<span class="math-container">$$w = (1-\gamma)z + \theta \gamma c + \gamma y$$</span>
where <span class="math-container">$c$</span> = hiring cost.</p>

<p>The first wage curve (from Cahuc et al.) would suggest that <span class="math-container">$c$</span> does not affect the wage curve while the second one (from Pissarides) would suggest that it does.</p>

<p>My thoughts so far are:
1. The wage curve tells us the outcome of bargaining but <span class="math-container">$c$</span> does not affect bargaining because <span class="math-container">$c$</span> is only directly relevant for <span class="math-container">$\Pi_V$</span> (the employer's outside option) but <span class="math-container">$\Pi_V$</span> is always zero in equilibrium. Therefore, the employer's outside option is not any worse if <span class="math-container">$c$</span> is higher. 2. In deriving the wage curve, Pissarides substitutes labor demand into the wage curve. So, <span class="math-container">$c$</span> enters the Pissarides' wage curve only through labor demand and not in any other way, suggesting, again, that bargaining itself is unaffected by <span class="math-container">$c$</span>.</p>

<p>So, what is the effect of a change in <span class="math-container">$c$</span> on the wage curve? Does an increase in <span class="math-container">$c$</span> shift the wage curve?</p>

<macroeconomics><labor-economics><unemployment><search-and-matching>
Search and matching model: effect of hiring cost on wage curve
Mark
# Search and matching model: effect of hiring cost on wage curve

(Asked by: username_0 on 2019-04-28)

What effect does the hiring cost have on the wage bargaining outcome (wage curve)?

In Cahuc, Carcillo, and Zylberberg: *Labor Economics*, they derive the following wage curve:
$$w = z + (y-z) \frac{\gamma[r+q+\theta m(\theta)]}{r+q+\gamma \theta m(\theta)}$$

where $z$ = unemployment benefit, $y$ = marginal product of labor, $\gamma$ = bargaining power of the worker, $r$ = interest rate, $q$ = rate of exogenous job destruction, $\theta$ = labor market tightness, $m(\theta)$ rate of job filling.

On the other hand, following Pissarides (2000) we get the following wage curve:
$$w = (1-\gamma)z + \theta \gamma c + \gamma y$$
where $c$ = hiring cost.

The first wage curve (from Cahuc et al.) would suggest that $c$ does not affect the wage curve while the second one (from Pissarides) would suggest that it does.

My thoughts so far are:
1. The wage curve tells us the outcome of bargaining but $c$ does not affect bargaining because $c$ is only directly relevant for $\Pi\_V$ (the employer's outside option) but $\Pi\_V$ is always zero in equilibrium. Therefore, the employer's outside option is not any worse if $c$ is higher. 2. In deriving the wage curve, Pissarides substitutes labor demand into the wage curve. So, $c$ enters the Pissarides' wage curve only through labor demand and not in any other way, suggesting, again, that bargaining itself is unaffected by $c$.

So, what is the effect of a change in $c$ on the wage curve? Does an increase in $c$ shift the wage curve?

**Tags:** macroeconomics, labor-economics, unemployment, search-and-matching

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## 1 answers

### Answer by username_1 on 2019-04-30. Score: 1 | Accepted answer

It depends on what you define as wage curve. I did not check in the textbooks, but you suggest that Pissarides and Cahuc et al. do not use the same mathemaical expression for the "wage curve".

1) In Cahuc et al., the expression combines the surplus sharing equation with the Bellman equations without using the free entry condition of firms. The wage $w$ is increasing in $\theta$ because a tighter market tightness improves the worker's bargaining position. There is no labor demand effect in this wage equation. Therefore, the vacancy cost does not appear.

2) The Pissarides' wage curve is less straightforward to interpret. Given the positive relationship between $w$ and $c$, one could erroneously think that the vacancy cost increases the wage. The interpretation is the following.
At $\theta$ fixed, a higher vacancy cost means that firms spend more to recruit. More spendings capture the fact that firms make larger job profits (free-entry condition). When firms make larger profits, workers also obtain a larger surplus due to the Nash bargaining. Thus, the wage is higher.

This definition of the wage curve does not depend on the matching technology ($m$). To understand this result, we can write the model with an abstract matching market.
With your notations, we define the value of unemployment $U$, the value of employment $W$ and the value of a job for a firm $J$.
We can write the three Bellman equations, and the Nash bargaining condition with the wage:
\begin{align}
& rU = z + R\\
& rW = w+q(U-W)\\
& rJ = y-w-qJ\\
& w=\gamma y + (1-\gamma)rU
\end{align}
If a worker is employed, she produces $y$ and receive $w$, with a risk to lose the job $q$.
If the worker is not employed, she receives $z$ plus the returns from participating to a matching market $R$.

In the full model $R=\theta m(\theta) (W-U)$. Here, we do not use this structure.

We assume that each unemployed worker participating to the "matching market" produces $G$ and obtain $R$. To participate to the market and obtain part of the gains, firms have to pay a cost $c$. If we assume that $G$ is Nash-bargained, each worker obtains $R=\gamma G$, and recruiting firms share $(1-\gamma)Gu$. The average returns for a firm is $\frac{(1-\gamma)G u}{v}=(1-\gamma)\frac{G}{\theta}$.
The free entry condition implies that firms make no profits from participating to the market. Therefore, $(1-\gamma)\frac{G}{\theta}=c$.
Using the wage bargaining solution, we obtain Pissarides' wage curve using the wage bargaining solution and eliminating $rU$, $R$ and $G$.

The wage curve holds without assuming how the "matching market" works, we only assume that the outcome is Nash-bargained. To come back to a change in $c$. An increase in the vacancy cost $c$ necessarily implies that the gain $G$ is higher, ans so the wage.

3) This is partial equilibrium analysis. If you solve for the equilibrium level of wages, you will find that $c$ actually reduces the equilibrium level of wages.

> **Comments:**
> **username_0**: Thanks! Could you please go into more detail about the interpretation of Pissarides' wage curve? If I understood you correctly, job profits (presumably through $y$), drive both higher $c$ and wage in the wage curve? But what happens to labor demand and the wage curve after an exogenous increase in $c$ (with $y$ fixed, i.e. with job profit a priori fixed)?  

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