Consider the following setting:

  1. Profit maximizing firms with production functions $\Pi(w,L)$, where $w$ is the wage and $L$ is employment.

  2. Unions who want to maximize the expected utility of their representative union members. To explicate, let $v(c)$ be the indirect utility function of a union member, where $c$ is consumption. If the union member is employed, he or she gets the wage $c=w$. Otherwise, he or she gets unemployment benefits $c=b$. Then the expected utility of a representative member is $$\nu(w)=lv(w)+(1-l)v(b)$$ where $l=\min(1,L/N)$ and where $N$ is the total amount of union members. (Note: In these problems, one usually assumes $L\leq N$ so that $l=L/N$.)

  3. Firms and unions bargain over the wage $w$; i.e., this is a collective bargaining problem. The collective bargaining problem is modelled as the maximization of the Nash bargaining product w.r.t. $w$ (see below).

Now, consider two outcomes of the bargaining process:

  1. Unions and firms agrees on some wage $w$. In this case, the expected utility of a representative member is $\nu(w)$. The profits to the firm are $\Pi(w,L)$.

  2. Unions and firms do not agree on any wage $w$. In this case, the expected utility to union members is $v(b)$ and profits to the firm is $0$.

In the right-to-manage model the collective bargaining is modelled as a symmetric Nash bargaining solution with $\gamma$ as the relative bargaining strength of the union, given that the firm maximizes its profits with respect to employment. I.e., it is the solution to $$\max_w\Omega(w)$$ such that $$\frac{\partial \Pi(w,L)}{\partial L}=0,$$ where $\Omega(w)=\big(\nu(w)-v(b)\big)^{\gamma}\Pi(w,L)^{1-\gamma}$ is the Nash bargaining product.

Now, when reading about this scenario/optimization problem I see two cases in the academic literature: The first one is called local (or firm-level) wage bargaining and the other is called central (or national) wage bargaining. Even though I have read about them, I do not understand the mathematical difference between them.

So, what is the fundamental, mathematical difference between local (or firm-level) wage bargaining and central (or national) wage bargaining given that we apply the right-to-manage model (i.e., we let firms determine employment unilaterally)? How do I model the two situations?

My guesses and thoughts so far (this will be updated as time goes by):

  • Local wage bargaining is at the firm level. Central wage bargaining is not at the firm level; instead, the firms are organised into an national employers' federation.
  • In central wage bargaining, the firms takes the collective bargaining problem as an exogenous event. This would then mean that when they maximize their profits, they do not take into account the agreed on wage. However, in local wage bargaining, firms take the wage into account, meaning that when they maximize their profit, they take into account that the wage is a function of employment $w=w(L)$. Even though some authors seems to think about it this way, I do not understand why. Maybe it has to do with firms somewhow regarding the wage as exogenous and independent of their own investment decisions since they do not directly engage in the barganing process, but only indirectly through the employers' federation (?).
  • One idea I had was that under central wage bargaining, employment is fixed during the bargaining process, while in local wage bargaining, employment is a function of the wage $w$. This difference would reflect the fact that firms view the wage agreed on as exogenous when wage bargaining is centralised. According to this idea, local wage bargaining would be modelled as $\max_{w}\Omega(w)$ given that $L=L(w)$ is the solution to $\max_w\Pi(w,L)$; and central wage bargaining would be modelled as $\max_w\Omega(w)$ holding $L$ fixed, and the firms chooses $L$ so that it is the solution to $\max_L\Pi(w^*,L)$, where $w^*$ is the centrally determined wage.
  • The timing of events is a bit unclear in the articles I have read about local and central wage bargaining. But it seems to be this: Firstly, the wage is determined through wage bargaining. Secondly, production takes place as firms solve their profit maximization problem. However, since the model is solved by backwards induction, one often begins by solving the profit maximization problem before finding the Nash bargaining solutions.

Examples of articles related to my question:

  1. Hoel, Michael. "Local versus central wage bargaining with endogenous investments." The Scandinavian Journal of Economics (1990): 453-469.

  2. Holden, Steinar. "Local and central wage bargaining." The Scandinavian Journal of Economics 90.1 (1988): 93-99.

  3. Holmlund, Bertil. "Centralized wage setting, wage drift and stabilization policies under trade unionism." Oxford Economic Papers 38.2 (1986): 243-258.

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    $\begingroup$ To me, the beginning of an answer requires to clearly differentiate between employment at the firm-level and at the national level. Yet, you do not seem to do that. What is $L$? Aggregate employment? Also. who is unionised? All workers? Also, can firms coordinate in their bargaining? $\endgroup$ – luchonacho Feb 14 '17 at 8:59
  • $\begingroup$ @luchonacho I agree. I will update the post when I have time. But note that this is the notation used in the articles mentioned; subscript $i$ is often ignored as they view the economy as symmetric. Thus, for firm $i$ we have employment level $L_i$. But since the economy is symmetric we may ingore the subscript. Aggregate employment is $\sum_iL_i\neq L_i$. I guess it does not matter too much who is unionised; the function of the union is to bargain over the wage. W.r.t. the workers, they are employed or unemployed. Firms cannot coordinate. It is unions and firms who wage bargain. $\endgroup$ – AnonymousIGuess Feb 14 '17 at 14:07

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