Imperfect competition and congestion in the City

W pic         ɪ 1 ( n—¹ ¢

^dP- ι. = μw n ^v n ⁷                        (33)

dwi ^lSym 1+ Λw 1 (n—1^{¢                    (33)}

μ^w n   n

The market clearing condition P_i^w - P_i^d =0(see (13)) has a unique solution
pi = gi (wi ) given that dP_i^w /dwi > 0 and dP_i^d /dpi < 0 . We have:

_{d 1}    Λ^d 1 ( n—1 ^¢

μ^{d 1 +} μ^d ni n J
_μw 1 + ^λW 1 ( n^—1 \ .
^{1   +} μ^w n X n )

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 ( μ^d >> Λ^d and μ^w >> Λ^w ). Second, when congestion costs are
present and very high compared to the product and labour market diversities
(Λ^d >> μ^d and Λ^w >> μ^w), then:

dpi          Λ^d    α^d

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 + a^ht_i, where ti = t + δ^ρ-, and
road usage ρ_i is given by (25). Since the travel time ti is variable, the marginal
cost becomes variable and endogenous. We have

ci = c¹ + a^h (t + δ^ρi´ = c + Λ^hPi^w,
sⁱ

where Λ^h = α^hδNa (using equation (27)), and where we have defined c =
c¹ + α^h 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 - Λ^hPw)] NPw - (F + S).     (35)

The first-order condition for optimal wage (and price) setting is: dei/dwi = 0
or

16

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