Corneo (1993, 1995) uses a social custom approach in order to discuss the effects of centralisation in wage bargaining on membership and wages. He considers workers with the following utility function:
Ui ={w+ (1−di)δ}L+w0(1−L) +di(ri−c) (6.1) wherewdenotes the bargained wage,Lis employment,di is a dummy, taking on value 1 for union members and 0 otherwise, cis the membership fee and ri denotes utility from reputation. We have normalised the mass of workers (denoted byN in Corneo’s model) to unity without loss of generality.
3Note that the term ‘reputation’ does not have the special meaning (together with its implications) it represents in information-theoretic models.
Heterogeneity of workers is represented by the distribution of ri. For sake of simplicity, Corneo assumesri to follow a uniform distribution with support [0,Θ].
For convenience and readability we ‘expand’ the compact representation (using a variable) of the utility function above into separate formulas for members and free riders
Uiu = w L−w0(1−L) +ri−c Uif = (w+δ)L−w0(1−L)
and give a verbal summary: union members obtain wage w if employed and w0 as outside option. They receive reputation ri and have to pay the fee c irrespective of employment status. Free riders obtain the union wage wplus a free rider bonus δ if employed and w0 otherwise. They save c, but have to do without reputation utility.
The model is a simple three stage game, solved by backward induction.
In the first stage, the firm determines a bonusδ as reward for non-members.
Then workers decide on membership and finally the bargain is struck.
The profit function of the firm is
π =R(L)−w M−(w+δ) (L−M) (6.2) whereM denotes union membership. In order to simplify things considerably, Corneo assumes efficient bargaining which implies full employment (L= 1).4 The wage is determined in a generalised Nash bargaining solution, i.e. by maximisation of
(U −U0)α(π−π0)1−α (6.3) where U and π denote utility of the median member and profit of the firm in case of an agreement, and U0 and π0 denote the respective thread points.
The components of this expression are U = w+rU −c U0 = w0+rU −c
π = R(1)−w M −(w+δ) (1−M) π0 = R(1−M)−(w0+δ) (1−M)
4The plausibility of efficient bargaining deteriorates considerably with the transition to centralised wage setting, since the level of employment determined in efficient bargaining is too high ex post from the firm’s point of view. And the employers’ association is faced with the difficulty to distribute the high level of employment over firms. Here the assumption is made, however, only in order to isolate the relevant effects, to retain the applicability of the ceteris paribus clause, and to keep the model as simple as possible.
To the best of my understanding,π0 is specified incorrectly. Corneoassumes that the firm pays the bonusδ (as markup on the outside wage) to free riders also during a strike. This is not optimal for the firm, sincew0is – by definition – the wage workers are willing to work for if they are not unionised. If the rules of the game allowed free riders to reverse their membership decision,π0 had to be determined (recursively) in a bargaining process. Consequently, we see that the firm will not pay a bonus to strike breakers for two reasons (a direct and an indirect one) if we take the rules of the game seriously.
The bonus lowers profits during a strike and weakens the bargaining position against the union by lowering the firm’s thread point.
Another minor flaw of the specification concerns the constraint δ ≥ 0 in the firm’s profit maximisation problem. δ ≥ 0 is not a natural exogenous constraint.5 It is simple to show that −c < δ <0 is an optimising choice for certain values of the exogenous parameters. In the conclusion we will discuss several other reasons for this constraint. We will correct these points here, but note that Corneo’s minor lapse does not change the qualitative results of his model.
The solution of the model (by backward induction) is straightforward.
Maximisation of the Nash product gives the wage6
w=w0(1−αM) +α{R(1)−R(1−M) +δ(1−M)} (6.4) Straightforward comparison of the member and nonmember utility positions shows that reputation must be higher than the sum of the lost bonus and the membership fee to make membership attractive. From this Corneo derives
M =
1−(δ+c)/Θ ifδ <Θ−c
0 otherwise (6.5)
if r is distributed uniformly in [0,Θ]. Again, we have some reservations against this computation. We think that is valid only when a union is already present. If no union exists, the worker has not to comparew+r−cwith w, butw+r−cwithw0, the relevant wage for the union-free firm. Consequently a worker enters the union if r > w0 −w+c. At a glance, this detail seems
5In most countries wage differentiation between members and nonmembers is not banned. ‘Allgemeinverbindlichkeitserkl¨arungen’ in Germany are an exception here. How-ever, they are not general in scope, but relate only to certain industries or regions, and the are applied seldomness now.
6The corresponding solution in Corneo’s paper (for bonus payments during a strike) is w=w0(1−M) +α{R(1)−R(1−M)}
to be not important for the central results of the model. Therefore we will discuss the issue in the conclusion.
After Insertion of M and w into the profit function we can perform the maximisation procedure of the firm in the first stage of the game. To show the existence of an outcome with δ∗ >0, we have to consider three cases. In the ‘trivial’ case c≥Θ, i.e. even maximum reputation does not compensate the membership fee and no worker joins the union. Thus δ∗ = 0. If contrary c <Θ, we have to check whetherδ <Θ−cfor the optimum choice ofδ, since M = 0 ifδ ≥Θ−c. Ifδ∗ meets this condition, an interior solution exists if.7
∂π
∂δ δ=0
>0 ⇔ R0(c/Θ) > w0+ 1−α
α c (6.6)
The second order conditions are fulfilled. After an extensive comparative statics investigation of the model properties (which is of minor interest for us here), Corneo applies this model to the centralisation debate and finds that it predicts gross membership and wages to be larger in a centralised bargaining environment, since an individual firm’s union density has a smaller effect on wages than for local bargaining. Consequently, the gain to the firm associated with the bonus payment shrinks with the number of firms in the economy.
If a central employers’ association cannot enforce the payment of a bonus, each single firm will save this cost, management opposition shrinks and union membership and wages rise. It should be emphasised that Corneo’s argument rests heavily on the fact that bonus payment violates equal treatment laws.
Otherwise bonus payment could be fixed in collective labor agreements and free riders could enforce payment by law. This implies that bonus payments have to be masked in most cases, e.g. the have to take place in biased promotion or firing procedures.