Dynamic collective bargaining and labor adjustment costs

Abstract

Collective bargaining between a trade union and a firm is analyzed within the framework of a monopoly union model as a dynamic Stackelberg game. Adjustment costs for the firm are comprised of the standard symmetric convex costs plus a wage-dependent element. Indeed, hiring costs can turn into benefits assuming wage discrimination against new entrants. The union also bears increasing marginal costs in the number of layoff workers and decreasing marginal benefits in the number of new entrants. Starting from a baseline scenario with instantaneous adjustment, we characterize the conditions under which the adjustment costs for the firm, or for the union, lead to higher employment and lower wages or vice versa. More generally, these adjustment costs, when they affect both the union and the firm, are generally detrimental to employment. However, the standard symmetric element of the adjustment costs for the firm positively affects employment, even with lower wages. Finally, if hiring and firing costs are defined separately, then hiring and firing could take place simultaneously if the wage discrimination towards new entrants is strong, because the firm would agree to pay the costs of firing incumbent employees, in order to enjoy wage savings from new entrants.

Appendices

Appendix:

Solutions with adjustment costs only for the firm

• Feedback solution under the assumption$w=\bar{w}$, for all$t\ge 0$. Under the assumption of $w=\bar{w}$, and taking into account from (9) that

  $$\begin{aligned} \hat{h}^{{\tiny \text{ AF }}}(\bar{w},L)=\frac{\bar{w}\beta +(V_{{\tiny \hbox {{F}}}}^{{\tiny \text{ AF }}})'(L)}{c}, \end{aligned}$$

  the feedback solution to problems (7)–(8) and (10)–(11) is obtained by solving the Hamilton–Jacobi–Bellman equations:

  $$\begin{aligned} \rho V^{{\tiny \text{ AF }}}_ {{\tiny \hbox {{U}}}}(L)= & {} \bar{w}L+(1-L)B+(V^{{\tiny \text{ AF }}}_ {{\tiny \hbox {{U}}}})'(L)(\hat{h}^{{\tiny \text{ AF }}}(\bar{w},L)-\delta L),\\ \rho V^{{\tiny \text{ AF }}}_ {{\tiny \hbox {{F}}}}(L)= & {} aL-\frac{L^2}{2}-\bar{w}L-c\frac{\hat{h}^2(\bar{w},L)}{2}+\beta \bar{w} \hat{h}(\bar{w},L)\\&+\,\left( V^{{\tiny \text{ AF }}}_ {{\tiny \hbox {{F}}}}\right) '(L)\left( \hat{h}(\bar{w},L)-\delta L\right) . \end{aligned}$$

  Or equivalently,

  $$\begin{aligned} \rho \left( b^{{\tiny \text{ AF }}}_ {{\tiny \hbox {{U}}}}L+c^{{\tiny \text{ AF }}}_ {{\tiny \hbox {{U}}}}\right)= & {} \bar{w}L+(1-L)B+b^{{\tiny \text{ AF }}}_ {{\tiny \hbox {{U}}}}\left( \frac{\bar{w}\beta +(a^{{\tiny \text{ AF }}}_ {{\tiny \hbox {{F}}}}L+b^{{\tiny \text{ AF }}}_ {{\tiny \hbox {{F}}}})}{c}-\delta L\right) ,\\ \rho \left( a^{{\tiny \text{ AF }}}_ {{\tiny \hbox {{F}}}}\frac{L^2}{2}+b^{{\tiny \text{ AF }}}_ {{\tiny \hbox {{F}}}}L+c^{{\tiny \text{ AF }}}_ {{\tiny \hbox {{F}}}}\right)= & {} aL-\frac{L^2}{2}-\bar{w}L-\frac{c}{2}\left( \frac{\bar{w}\beta +(a^{{\tiny \text{ AF }}}_ {{\tiny \hbox {{F}}}}L+b^{{\tiny \text{ AF }}}_ {{\tiny \hbox {{F}}}})}{c}\right) \\&+\,\beta \bar{w} \frac{\bar{w}\beta +\left( a^{{\tiny \text{ AF }}}_ {{\tiny \hbox {{F}}}}L+b^{{\tiny \text{ AF }}}_ {{\tiny \hbox {{F}}}}\right) }{c}\\&+\,\left( a^{{\tiny \text{ AF }}}_ {{\tiny \hbox {{F}}}}L+b^{{\tiny \text{ AF }}}_ {{\tiny \hbox {{F}}}}\right) \left( \frac{\bar{w}\beta +\left( a^{{\tiny \text{ AF }}}_ {{\tiny \hbox {{F}}}}L+b^{{\tiny \text{ AF }}}_ {{\tiny \hbox {{F}}}}\right) }{c}-\delta L\right) . \end{aligned}$$

  Identifying quadratic coefficients, linear coefficients and constant terms, in the LHS and the RHS one gets a system of 2+3 algebraic Ricatti equations. The solution to this system of equations is obtained with the help of Mathematica and it is presented in (13) and (14) for the quadratic and the linear coefficients.

• Alternative solutions The maximization problem for the union in (10)–(11) with

  $$\begin{aligned} \hat{h}^{{\tiny \text{ AF }}}(w,L)=\frac{w\beta +a_{{\tiny \hbox {{F}}}}^{{\tiny \text{ AF }}}L+b_{{\tiny \hbox {{F}}}}^{{\tiny \text{ AF }}}}{c}, \end{aligned}$$

  is a linear state optimization problem. Therefore, the optimal strategy settled on by the union is assumed constant, $w=\bar{w}$ for all $t\ge 0$. Consequently, for this specific structure of the game, a solution that switches from a to B or vice versa is not feasible.

In Sect. 4.3 it is proven that the solution $\bar{w}=a$ for any $t\ge 0$ satisfies the necessary conditions for optimality. Conversely, it can be shown that a solution with $\bar{w}=B$ for any $t\ge 0$ is not optimal. For this solution, expression (14) would imply $b_{{\tiny \hbox {{U}}}}^{{\tiny \text{ AF }}}(B)=0$, and hence from (12), $w_{{\tiny \text{ AF }}}(L)$ could not be given by $w_{min}=B$, unless L(t) remained equal to 0 forever. This solution cannot be optimal because it would imply zero production forever.

A constant wage $\bar{w}\in [B,a]$ could also appear in a singular path. This type of solution should satisfy:

$$\begin{aligned} L(t)=-\frac{\beta }{c}b_{{\tiny \hbox {{U}}}}^{{\tiny \text{ AF }}}(\bar{w}). \end{aligned}$$

(33)

From the expressions of $a_{{\tiny \hbox {{F}}}}^{{\tiny \text{ AF }}}$ and $b_{{\tiny \hbox {{U}}}}^{{\tiny \text{ AF }}}(\bar{w})$ in (13) and (14), it immediately follows that:

$$\begin{aligned} b_{{\tiny \hbox {{U}}}}^{{\tiny \text{ AF }}}=2\frac{\bar{w}-B}{\rho +\sqrt{(\rho +2\delta )^2+4/c}}\ge 0\quad \forall \bar{w}\in [B,a]. \end{aligned}$$

Thus, from (33), employment should remain constant and either negative or zero along the whole time path, $[0,\infty )$. A solution with negative employment is not feasible and a solution with no employment cannot be optimal. Hence, a singular path is not feasible.

Proof of Proposition 1

From (5) and (15) it follows that $\bar{L}^{{\tiny \text{ AF }}}(w^s)<L^s$ if and only if $\beta (a+B)<\delta c(a-B)$, or equivalently, if and only if:

$$\begin{aligned} h^s\equiv \delta L^s>\beta \frac{w^s}{c}. \end{aligned}$$

Moreover, from (15)

$$\begin{aligned} \left( \bar{L}^{{\tiny \text{ AF }}}\right) '(w)=-\frac{1-\beta \left( \rho +\delta \right) }{1+c\delta \left( \rho +\delta \right) }<0. \end{aligned}$$

Since $w^{{\tiny \text{ AF }}}=a>w^s$ then under condition above $L^s<\bar{L}^{{\tiny \text{ AF }}}(w^s)<\bar{L}^{{\tiny \text{ AF }}}(a)$.

Proof of Proposition 2

The expression for $a_{{\tiny \hbox {{U}}}}$ reads

$$\begin{aligned} a_{{\tiny \hbox {{U}}}}=\frac{-\left[ 4c^2-d\beta \varPhi \varTheta \right] -\sqrt{\varDelta }}{2\beta \varTheta }, \end{aligned}$$

with $\varPhi =\rho +2\delta$, $\varTheta =2(c+d)-d\beta \varPhi$ and

$$\begin{aligned} \varDelta =\left[ 4c^2-d\beta \varPhi \varTheta \right] ^2-4\varTheta \left[ c^2\left( 2c-2d-d\beta \varPhi \right) -2d^2\beta ^2\right] . \end{aligned}$$

Under Condition 1, $\varTheta >0$, and hence a sufficient condition for $a_{{\tiny \hbox {{U}}}}<0$ would be $\varDelta >[4c^2-d\beta \varPhi \varTheta ]^2$, which can be guaranteed under sufficient condition $c\le d$.$\square$

Proof of Proposition 3

From the optimal feedback strategies in (21) and (22), one gets

$$\begin{aligned} \phi _w^1=\frac{c(\beta a_{{\tiny \hbox {{U}}}}+c)-d\beta a_{{\tiny \hbox {{F}}}}}{d\beta ^2},\quad \phi _h^1=\frac{\beta a_{{\tiny \hbox {{U}}}}+c}{d\beta }. \end{aligned}$$

We have numerically seen that $a_{{\tiny \hbox {{F}}}}>0$. Therefore, a necessary condition for $\phi _w^1<0$, and a necessary and sufficient condition for $\phi _h^1<0$, is $\beta a_{{\tiny \hbox {{U}}}}+c<0$. This expression can be written as

$$\begin{aligned} \beta a_{{\tiny \hbox {{U}}}}+c=\frac{-4c^2+d\beta \varPhi \varTheta +2c\varTheta -\sqrt{\varDelta }}{2\beta \varTheta }<0. \end{aligned}$$

Since $\varTheta >0$, a sufficient condition for a negative sign of this expression is $-4c^2+d\beta \varPhi \varTheta +2c\varTheta <0$, or equivalently, after some rearrangements

$$\begin{aligned} -(d\beta \varPhi )^2+2d(d\beta \varPhi )+4cd<0. \end{aligned}$$

(34)

The LHS of this inequality can be interpreted as a second order polynomial in $d\beta \varPhi$, with roots: $d\pm \sqrt{d^2+cd}$. The in Eq. (34) holds true if $d\beta \varPhi <d+\sqrt{d^2+cd}$. And this condition immediately holds under Condition 1.$\square$

Numerical analysis in Section 6

The optimization problem in (26)–(28) is a dynamic problem, subject to the dynamic evolution of the state variable in (28). Furthermore, it is also subject to (algebraic) non-negativity control constraints in (27). To fully characterize the solution one should define a Lagrangian appending the non-negativity constraints to the objective function with their corresponding multipliers, and then derive the necessary conditions including Kuhn–Tucker conditions. Our approach has been to solve the problem (with the help of Mathematica) for the parameters’ values specified, ignoring the non-negativity constraints, and once the solution is found check whether these conditions are indeed satisfied.

The Hamilton–Jacobi–Bellman equation associated with problem (26)–(28) is:

$$\begin{aligned} \rho V^{\pm }_{{\tiny \hbox {{F}}}}(L)= & {} \max _{h^+,h^-}\Bigg \{aL-\frac{L^2}{2}-wL-c\frac{(h^+)^2}{2}+\beta w h^+-\tilde{c}\frac{(h^-)^2}{2}-\tilde{\beta }wh^-\\&+\,\left( V^{\pm }_{{\tiny \hbox {{F}}}}\right) '(L)\left( h^+-h^--\delta L\right) \Bigg \}. \end{aligned}$$

From this equation, the reaction functions in (29) immediately follow. Plugging these policies into the union’s maximization problem, the dynamic problem (31)–(32) is obtained. The Hamilton–Jacobi–Bellman equation for this problem is:

$$\begin{aligned} \rho V^{\pm }_{{\tiny \hbox {{U}}}}(L)= & {} \max _{w}\Bigg \{wL+(1-L)B-d\frac{\hat{h}^+(w,L)(\hat{h}^+(w,L)-H)}{2}\\&-\,\tilde{d}\frac{\hat{h}^-(w,L)(\hat{h}^-(w,L)+\tilde{H})}{2}+(V^{\pm }_{{\tiny \hbox {{U}}}})'(L)(\hat{h}^+(w,L)-\hat{h}^-(w,L)-\delta L)\Bigg \}. \end{aligned}$$

From this equation the optimal wage is obtained, and plugging it into the reaction functions in (29) the optimal hiring and firing decisions follow. The three optimal controls depend on the parameters, the stock of employment, L, and the value functions of the firm and the union (we do not present the expressions for brevity).

Plugging these optimal controls in the two equations above, assuming linear-quadratic value functions, $V^{\pm }_{{\tiny \hbox {{F}}}}(L)=a^{\pm }_{{\tiny \hbox {{F}}}}L^2/2+b^{\pm }_{{\tiny \hbox {{F}}}}L+c^{\pm }_{{\tiny \hbox {{F}}}}$ and $V^{\pm }_{{\tiny \hbox {{U}}}}(L)=a^{\pm }_{{\tiny \hbox {{U}}}}L^2/2+b^{\pm }_{{\tiny \hbox {{U}}}}L+c^{\pm }_{{\tiny \hbox {{U}}}}$, and identifying quadratic coefficients, linear coefficients and constant terms, one gets a system of six algebraic Ricatti equations. At this point we numerically obtain four different solutions for this system of equations. Only two of them satisfy convergence to the steady-state equilibrium. From these two stable solutions, we chose the one that brings higher welfare to the firm and the union: $V^{\pm }_{{\tiny \hbox {{F}}}}(L)=5.48 + 0.12 L - 0.09 L^2$ and $V^{\pm }_{{\tiny \hbox {{U}}}}(L)=0.8 + 0.29 L - 0.23 L^2$. For the chosen solution, the differential Eq. (28) can be solved. Therefore, the time path and the steady-state value of employment are computed. From these the time paths and the steady-state values of hiring, firing and wage rates follow.