Search and matching model: effect of hiring cost on wage curve

Question

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?

Answer 1

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.