4.2 Short-run equilibrium
Subcenters are competing in wages and prices in a non-cooperative Nash game.
We wish to find the candidate symmetric equilibrium in prices and wages de-
noted by (pe,we). As shown above, the subcenters compete in either wage or
price. We consider here that the strategic variable is the wage, wi .
The best reply of subcenter i to the wages and prices set by the other sub-
centers is:
dei(wi,w-i,p-i) = ½μdf(wɪ - Л + (fi(wi) - Wi - c) (1 -Pw) ¾ NPw = 0.
dwi [ у dwi J μw J
(16)
Note that at the symmetric candidate equilibrium Piw = Pid =1/n and
recall that dfi(wi) /dwi = -μd / μw Therefore (16), set at the symmetric candi-
date equilibrium, leads to:
- μμd + 1) +(pe - we - c)(n-1 =0
∖ μw ) nμw
We prove in Appendix A that the candidate equilibrium is a Nash equilibrium.
Therefore:
Proposition 1 In the absence of congestion, there exists a unique symmetric
Nash equilibrium in prices and wages given by:
pe = c + we + (μd + μw ) —n-. (17)
(n - 1)
The equilibrium markup pe - (c + we) is increasing with product heterogene-
ity and with job heterogeneity. The role of product heterogeneity is well known
(see [8]. The role of job heterogeneity is new: more job heterogeneity means that
workers are also interested in other dimensions than the wage they earn (such
as the proximity of the gym facility or the charms of the boss) so that wage
differences become less important and this increases the profit margin. Interest-
ingly, both types of heterogeneity work in the same direction and are additive.
As in the standard model, the markup decreases when more firms compete in
the market. However, the markup remains bounded away from zero as n →∞,
as in the symmetric monopolistic competition models à la Chamberlin.
4.3 Long-run equilibrium
The equilibrium profit (see (15)) at the symmetric equilibrium is
∏e = [pe - we - c] N - (F + S)
n
or after substitution of the equilibrium price levels (see Proposition 1):
10