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Dynamic collective bargaining and labor adjustment costs


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.

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  1. The notion of a union’s “influence” nowadays seems to be more important than a union’s “presence” (see Boeri and Ours 2013, p. 64). The concept of “excess coverage” (i.e. collective bargaining coverage rate minus union density rate) appears to be central to understanding industrial and labor relations currently. A proof of this statement is that coverage is usually higher (specially relevant in cases like France, Austria or Spain) than union density in most of the countries (Boeri and Ours 2013, p. 65).

  2. For instance, Cahuc et al. (2014, p. 404) point out: “(...) in France and Spain collective agreements do not have the right to discriminate between union members and non-unionized workers”.

  3. We analyze the steady-state equilibria, although considering the capital stock as an exogenous constant. Thus, only the employment level adjusts, and not the capital stock and therefore we do not characterize a long-run equilibrium.

  4. Despite the fact that one of the first representations was not symmetric (Holt et al. 1960), symmetry has been a standard assumption ever since Eisner et al. (1963).

  5. Cahuc et al. (2014, p. 120) stated that: “(...) in countries where strong legal measures are in place to enhance job security, the costs of separation outstrip recruitment costs”. A well-documented example of this empirical regularity is France (Abowd and Kramarz 2003; Goux et al. 2001). This is also true for other Southern European countries. Boeri and van Ours (2013, p. 278) shows how France (3.0), Greece (3.0), Portugal (3.2), Italy (2.6), and Spain (3.1) exhibit extremely high levels in the overall employment protection legislation index. By contrast, English-speaking countries like Australia (1.4), Canada (1.0), Ireland (1.4), New Zealand (1.2), the United Kingdom (1.1), and the United States (0.9) tend to show the lowest levels for this index.

  6. Alternatively, these wage savings for the firm can be explained based on the existence of a payroll tax subsidy for newly-hired employees (which is a common economic policy in European countries).

  7. As Booth (2014) points out, although labor economists’ interest in trade unions has declined in recent years, trade unions are still important agents in many OECD countries. One of the reasons argued by some authors is the negligible role of trade unions in the US labor market. Notwithstanding, our theoretical framework is thought to model some features more connected to European labor markets.

  8. Both the right-to-manage model and the monopoly union model are studied in Koba (2003), who analyzes the effect of deregulation on employment and wages.

  9. Some interesting works in this strand of research are Chang and Lai (1997) and, more recently, Dittrich and Schirwitz (2011a), Dittrich and Schirwitz (2011b), or Kazanas and Miaouli (2014).

  10. We follow the idea that the union’s objective is based on the utility of insider employees as in the seminal papers by Lindbeck and Snower (1988), Blanchard and Summers (1986) and Carruth and Oswald (1987). However, we do not consider “an insider dominated union” (see Creedy and McDonald 1991 and McDonald 1991).

  11. To clarify the asymmetric effect of hiring and firing, assume that the firm fires some employees and hires the exact same amount. Then the welfare improvement associated with the arrival of new employees is more than offset by the strong decrease in the union’s welfare from their dismissal.

  12. A level of employment above a is not feasible as it would imply lower output and higher labor costs.

  13. This standard assumption is not exempt from criticism. Nickell (1987) states that for low levels of hiring it is hard to think of good reasons why hiring costs should increase at the margin.

  14. Here and henceforth we omit the time argument when no confusion arises.

  15. This assumption has been made for tractability. However, it is consistent with the positive empirical correlation between firing costs and wage inequality observed by Dias da Silva and Turrini (2015).

  16. Superscript s refers to the static scenario, or equivalently, the scenario without adjustment costs for the firm and the union. Similarly, a hat over a decision variable refers to the best response function of the player who controls this variable.

  17. Superscript AU refers to the scenario with adjustment costs only for the union.

  18. A bar over a variable refers to its steady-state equilibrium.

  19. In general, one might presume that the firm welcomes new entrants no matter how large their number. By assuming \(H/2>1\) we could drop the possibility that the marginal benefit for the union from additional hirings can become negative. Then, the adjustment costs for the union would always lead to lower wages and unemployment rates.

  20. Superscript AF refers to the scenario with adjustment costs only for the firm.

  21. We will show that the former is always true and thus \(w^{{\tiny \text{ AF }}}(L)=w_{\max }=a\) for any \(t\ge 0\). See the “Appendix” for more details.

  22. The highly cumbersome expressions for \(c_{{\tiny \hbox {{F}}}}^{{\tiny \text{ AF }}}(\bar{w})\) and \(c_{{\tiny \hbox {{U}}}}^{{\tiny \text{ AF }}}(\bar{w})\) are not relevant and, hence, are not presented here. They are available from the authors upon request.

  23. A higher wage has a twofold effect on hirings. A positive effect as it represents higher wage savings for the firm. And a negative effect as the marginal valuation of employment by the employer decreases. The partial derivative of \(\hat{h}^{{\tiny \text{ AF }}}(w,L)\) in (9) w.r.t. w is negative under Condition 1 and hence the latter indirect effect is stronger, and the net effect is negative.

  24. No superscript is used in this general case with adjustment costs for the firm and the union.

  25. The solutions are obtained with the help of Mathematica. Since the analytical expressions for these coefficient are highly cumbersome and hence not relevant, we do not present them here.

  26. We have computed the coefficients of the value functions for parameters’ values: \(c=0.1,\,\rho =0.05,\,\delta =0.15,\, d=1,\,B=0.1,\, H=0.1,\, a=0.77588,\,L0=0, \beta = 0.3\). For these parameters’ values, the value functions read: \(V_{{\tiny \hbox {{F}}}}(L)=0.09L^2/2 -0.17L+ 0.39\), \(V_{{\tiny \hbox {{U}}}}(L)=-0.89L^2/2+0.58L+5.92\). The numerical results in this section are robust to changes in parameters’ values. We analyze a 10% increase/decrease in each parameter’s value (\(L_0\) moving from 0 to 0.1), keeping all other parameters constant.

  27. However, firings do not necessarily occur in this case. The firm would initially fire workers if \(L_0>>\bar{L}\) but, after some time, it would end up hiring those who voluntary quit at the steady-state.

  28. This result can be obtained considering, for example, the same parameters values as in footnote 26, except for a lower productivity of labor \(a=0.6\) (to reduce employment at the steady-state equilibrium \(\bar{L}= 0.66\)) and a higher initial level of employment \(L_0=1>\bar{L}\).

  29. A tilde denotes functions and parameters related to the adjustment costs of firing as opposed to the adjustment costs of hiring. Superscript ± labels the solution in this section in which hiring and firing are separately determined.


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The authors acknowledge financial support from the Spanish Government (Projects ECO2014-52343-P and ECO2017-82227-P), as well as financial aid from Junta de Castilla y León VA024P17, co-financed by FEDER Funds. We would like to thank Guiomar Martín-Herrán for her useful comments and suggestions. We are also grateful to the two anonymous reviewers. The final version of this article has greatly benefited from their critics, comments and remarks.

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Correspondence to Francisco Cabo.



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:

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

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

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

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.

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Cabo, F., Martín-Román, A. Dynamic collective bargaining and labor adjustment costs. J Econ 126, 103–133 (2019).

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  • Dynamic labor demand
  • Collective wage bargaining
  • Monopoly union model
  • Adjustment costs
  • Stackelberg differential game

JEL Classification

  • J5
  • J23
  • C73
  • C61