Imperfect competition and congestion in the City

the wages and the prices, Thus P_i^w = P_i^d, where P_i^w is given by (11) and P_i^d 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 = NP_i^d = NP_i^w.

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 w_i 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 i^th component missing.
We shall use the following result:

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
μ^d is larger because then the consumers are more loyal to their ideal product.
Conversely, the price reduction is smaller when μ^w 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] NP_i^w - (F + S) ,          (15)

where we use the identity Di = NP_i^w , and where c = c¹ + α^ht.

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