W pic ɪ 1 ( n—1 ¢
dP- ι. = μw n v n 7 (33)
dwi lSym 1+ Λw 1 (n—1¢ (33)
μw n n
The market clearing condition Piw - Pid =0(see (13)) has a unique solution
pi = gi (wi ) given that dPiw /dwi > 0 and dPid /dpi < 0 . We have:
dpi(wi) l
dwi lSym=
d 1 Λd 1 ( n—1 ¢
μd 1 + μd ni n J
μw 1 + λW 1 ( n—1 \ .
1 + μw n X n )
(34)
There are two limiting cases of interest. First, without congestion, this ex-
pression reduces to equation (14). This case can also be obtained in the limit
where the product and the labour market diversities are very large compared
to congestion ( μd >> Λd and μw >> Λw ). Second, when congestion costs are
present and very high compared to the product and labour market diversities
(Λd >> μd and Λw >> μw), then:
dpi Λd αd
dwi lSym= - Λw - aw.
In this case the wages and the prices are solely driven by the level of congestion,
since the workers and the shoppers select their destination only as a function of
variable travel times.
6.3 Short run equilibrium
We study first the equilibria in the absence of government interventions: no
congestion pricing, no limit on the number of centers and an exogenous road
capacity.
We know that the marginal cost is ci = c1 + ahti, where ti = t + δρ-, and
road usage ρi is given by (25). Since the travel time ti is variable, the marginal
cost becomes variable and endogenous. We have
ci = c1 + ah (t + δρi´ = c + ΛhPiw,
si
where Λh = αhδNa (using equation (27)), and where we have defined c =
c1 + αh t. This means that the firm bears directly, via the intermediate delivery
cost, part of the congestion costs it creates. Using the market clearing condition,
the profit of Firm i is
ei(wi, w—i,p) = £gi(wi) - wi - c - ΛhPw)] NPw - (F + S). (35)
The first-order condition for optimal wage (and price) setting is: dei/dwi = 0
or
16