the wages and the prices, Thus Piw = Pid, where Piw is given by (11) and Pid is
given by (12). We get a relation between the price pi and the wage wi set by
Firm i:
exp fe) exp (-di)
---------` =---------( ∖. (13)
P exp (wj) P exP -j)
j=1...n j=1...n
Therefore, the demand for the differentiated product sold in subcenter i is
Di = NPid = NPiw.
4 Equilibrium without congestion
4.1 The profit function
We look for a symmetric Nash equilibrium in prices and wages between firms
(or subcenters). The strategic variables of subcenter i are wi and pi . Given the
market clearing condition (13), the choice of wi determines the choice of pi and
vice-versa.
Consider subcenter i which takes all other wages and prices as given. Since
the LHS of (13) is strictly increasing in wi and the RHS of this equation is strictly
decreasing in pi , there is a one to one relation between wi and pi ,the other prices
and wages being fixed. Let pi = fi (wi).Notethatfi (wi)=f (wi ,w-i ,p-i)
where w-i and p-i are the vectors w and p with the ith component missing.
We shall use the following result:
dfi(wi)
dwi
Piw (1-Piw)
μw
Pd(i-Pd
μd
μd
— < 0.
μw
(14)
This expression is negative since when a firm raises its wage, it increases the
number of workers hired. In order to be able to sell the additional production,
a firm needs to reduce its prices. The price reduction needs to be larger when
μd is larger because then the consumers are more loyal to their ideal product.
Conversely, the price reduction is smaller when μw is larger, since in this case
the workers are more loyal to their preferred workplace and less amenable to
changing jobs for a wage increase.
Given the relation between price and wage of Firm i, the profit of subcenter
i only depends on a single strategic variable (we select the wage as the strategic
variable). In this case πi(wi, w-i, fi (wi) ,p-i)=πei(wi,w-i,p-i) with:
πei(wi, w-i, p-i) = [fi(wi) - wi - c] NPiw - (F + S) , (15)
where we use the identity Di = NPiw , and where c = c1 + αht.