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The bargain

Under assumptions A2-A5, the Lagrangian to be maximised by the bargaining parties reads4
$\displaystyle \max_{n,w,\lambda}{\mathcal{L}}$ $\textstyle =$ $\displaystyle \left\{ \frac{n}{m} \left[ u(c^e) - u(c^u) \right] \right\} \cdot \left\{ f(n,x)-n w \right\}
+\lambda (f_{n}-w)$ (2)
    $\displaystyle = G \cdot \pi +\lambda (f_{n}-w),$  

where $G$ denotes the payoff of a union. The outcome of the bargain is given by differentiation of equation (2) with respect to employment, gross wage rate, and $\lambda$, and setting the partial derivatives equal to zero.
$\displaystyle \frac{\partial \mathcal{L}}{\partial n}$ $\textstyle =$ $\displaystyle {G}_{n}\cdot \pi+G {\pi}_{n}+\lambda \cdot f_{nn}=0,$ (3)
$\displaystyle \frac{\partial \mathcal{L}}{\partial w}$ $\textstyle =$ $\displaystyle {G}_{w}\cdot \pi-G\cdot n-\lambda=0,$ (4)
$\displaystyle \frac{\partial \mathcal{L}}{\partial \lambda}$ $\textstyle =$ $\displaystyle f_{n}-w=0.$ (5)

Since firms and unions are equal ex ante, there is no reason, why the outcome of the bargains should differ from each other. Setting $\overline{w}=w$, elimination of $\lambda$ from equations (3) and (4) and rearranging yields the wage equation (Pissarides; 1998, S. 164):
\begin{displaymath}
\frac{{n}_{w}}{n}+\frac{{u^e}_{w}}{u^e-u^u}- \frac{n}{f(n,X)-n w}=0.
\end{displaymath} (6)

The effects of changing the benefit rate or the contribution rate on wages and employment can be derived by means of linear algebra. The bordered Hessian (Jakobian) derived from equations (3-5) reads

$\displaystyle \vert J\vert$ $\textstyle =$ $\displaystyle \left\vert \begin{array}{ccc}
f_{nn}\cdot G +\lambda \cdot f_{nnn...
...2\cdot n \cdot {G}_{w} & -1 \\
\\
f_{nn} & -1 & 0 \\
\end{array} \right\vert$ (7)
       
  $\textstyle =$ $\displaystyle 3\cdot f_{nn}\cdot G -2\cdot f_{nn} \cdot G_{nw} \cdot \pi - f^2_{nn}\cdot G_{ww}\cdot \pi + 2\cdot f^2_{nn}\cdot n\cdot G_{w} -\lambda f_{nnn}$  

and is positive if the second-order conditions for a maximum of the Nash-product are fullfilled, which is assumed in what follows. Employing Cramer's rule, the comparative-static derivatives $\partial w/\partial \beta $, $\partial w/\partial \tau $, $\partial n/\partial \beta $ and $\partial n/\partial \tau $ can be calculated. If the second-order condition holds, the indicated signs of the derivatives are thus5

\begin{eqnarray*}
\frac{\partial w}{\partial \beta}&=&\frac{{G}_{n \beta}\cdot \...
...G_\tau \cdot n \right) + G_{n \tau} \cdot \pi }{\vert J\vert}<0.
\end{eqnarray*}

The indicated signs of the derivatives with respect to $\tau$ only follow if $G_{w \tau}\leq 0$ is sufficiently small. This derivative reads

\begin{displaymath}
G_{w \tau}= \frac{n}{m} \left[ -u_{c^e} - (1-\tau)\cdot w \cdot u_{c^e c^e} \right].
\end{displaymath}

After rearranging, we obtain

\begin{displaymath}
G_{w \tau}= \frac{n}{m} u_{c^e} \left( R -1 \right),
\end{displaymath}

where $R=-(u_{c^ec^e}/u_{c^e})\cdot (1-\tau)\cdot w$ is the Arrow-Pratt measure of relative risk-aversion. If the utility function has the assumed properties, this measure is positive. In the case of the utility function $u=\ln c$, the measure amounts to 1, which yields $G_{w \tau}=0$. In the case of utility functions of the form $u=c^{\frac{1}{\rho}}$, it follows that $R=1-1/\rho$. Risk-aversion ($\rho>1$) then implies $0<R<1$. The more risk-averse workers are, the closer the measure is to unity. This means, that $G_{w \tau}$ is close to zero. In this case, the derivatives with respect to $\tau$ have the indicated signs, which is presumed in what follows.

Thus, a higher benefit rate, or higher contributions to UI bring about higher equilibrium wages and lower equilibrium employment. It should be emphasised, however, that the derivatives with respect to $\tau$ hinge on the degree of risk-aversion. The result that benefits have a positive impact on wages, while the contribution (tax) rate has an ambiguous effect, is parallel to the findings of e.g. Oswald (1985, p. 168) and Vijlbrief and van de Wijngaert (1995, p. 238) for the case of a monopoly union. In comparison, Malcomson and Sator (1987) and Lockwood and Manning (1993), respectively for the cases of a monopoly union and wage bargaining, establish that a higher marginal contribution rate lowers the wage rate. Here, due to the assumption of proportional payroll contributions, marginal and average contribution rate coincide.


next up previous
Next: Endogenous tax rate Up: Assumptions and bargaining setup Previous: Assumptions and bargaining setup
Helge Sanner: Regional Unemployment Insurance, Potsdam 2001