Wage and employment determination in a dynamic insider–outsider model

Abstract

In this paper, I develop a differential insider–outsider game in which a union of corporative incumbents chooses the wage of its members by taking into account the optimal employment policy of a firm that, in turn, is assumed to decide the number of outsiders to hire in a spot labour market. Under the assumption that incumbents cannot be fired and commit themselves to a given path of wages, I demonstrate that such a game displays an open-loop Stackelberg equilibrium in which the initial stock of insiders pins down the trajectories of incumbents, entrants and insider wages. Moreover, resorting to numerical simulations, I show that adjustments towards the steady-state equilibrium occur through asymmetric oscillations that mimic the decline of union membership and union wage premia observed in the US all over the last 20 years. In addition, I show that the model provides a positive relationship between the labour market power of the insider union and the impatience of the firm.

Appendix: Controllability of the firm problem

Consider the first two differential equations of the dynamic system in (14). They read as

$$\begin{aligned} \overset{\cdot }{L}_{\text {I}}\left( t\right)= & {} m\frac{A\phi -w_{\text {E}}+m\Lambda \left( t\right) }{\alpha +h}-bL_{\text {I}}\left( t\right) \nonumber \\ \overset{\cdot }{\Lambda }\left( t\right)= & {} \Lambda \left( t\right) \left( b+\rho _{{\text {F}}}\right) -A+\alpha L_{\text {I}}\left( t\right) -\frac{1-\beta }{\Psi \left( t\right) }. \end{aligned}$$

(A.1)

Taking into account Eq. (10), the system of autonomous differential equations in (A.1) can be written as the following linear time-invariant (LTI) system that describes the solution of the firm problem:

$$\begin{aligned} \left( \begin{array}{c} \overset{\cdot }{L}_{\text {I}}\left( t\right) \\ \overset{\cdot }{\Lambda }\left( t\right) \end{array} \right) =\left( \begin{array}{c} \frac{m\left( A\phi -w_{\text {E}}\right) }{\alpha +h}\\ -A \end{array} \right) +\left[ \begin{array}{cc} -b &{} \frac{m^{2}}{\alpha +h}\\ \alpha &{} b+\rho _{{\text {F}}} \end{array} \right] \left( \begin{array}{c} L_{\text {I}}\left( t\right) \\ \Lambda \left( t\right) \end{array} \right) +\left( \begin{array}{c} 0\\ -1 \end{array} \right) w_{\text {I}}\left( t\right) , \end{aligned}$$

(A.2)

where $w_{\text {I}}\left( t\right)$ is the control of the union.

The controllability matrix of the LTI system in (A.2) is given by

$$\begin{aligned} H_{2\times 2}=\left[ \begin{array}{cc} 0 &{} -\frac{m^{2}}{\alpha +h}\\ -1 &{} -\left( b+\rho _{{\text {F}}}\right) \end{array} \right] . \end{aligned}$$

(A.3)

Since H has full rank, i.e. $rank\left( H\right) =2$, the problem of the firm is controllable by the union.