Imperfect competition and congestion in the City

dP^w

pw + gi(wi) - Wi - c - 2Λ^hPW]     = 0.

dwi

Substituting the expressions (34) and (33), the first-order condition, at the
symmetric candidate equilibrium reduces to:

Therefore, the candidate equilibrium price is given by the solution of (36):

pe = c + —+ we + (μd + μ^w)
n

where b = yMah+ad+âw] ⁶and where Λ^h = α^hδ N a.

Proposition 4 With congestion but without congestion pricing, there exists a
unique symmetric Nash equilibrium in prices and wages given by:

e   I ^Л I e∣∕d∣ wʌ ⁿ I ^{δ N} "2

p — c +    + w + μ + μ J i + b          (37)

Proof. See Appendix C. ■

Note that without congestion, the equilibrium price reduces to equation (17).
With congestion, there are two additional positive terms in the RHS. First the
marginal production cost is now c + Λ^h /n + w^e and contains a congestion term
translating the increased cost of intermediate deliveries. The second term is
related to the congestion created by shopping, commuting and intermediate
delivery traffic and represents the increased market power effect.

The markup (p^e - c - Λ^h /n - w^e) now has two components. The first one
is the product/wage heterogeneity term (proportional to (μ^d + μ^w)), as in the
non-congested case. The second term, represents the externality due to conges-
tion⁷ As discussed, congestion reduces the incentive to cut prices, and therefore,
increases equilibrium prices. This may explain why shops often lobby against
policy measures which aim to improve traffic conditions in general although a
firm individually will be in favour of local improvements of traffic, i.e. measures
which improve the accessability to.workers and consumers.

The short run equilibrium profit (see 35 and 37 is:

N δ αN ²

π^e(s) = (μ^d + μ^w) (n-₁) + s J - (F + S)         (38)

which is an increasing function of the congestion level.

⁶ Note than when κ =1, αb = α.

⁷The total cost is TC = αNt^e = αN Çt + δα-N^ . Therefore the externality, which is the
difference between the marginal cost and the average cost is equal to: δα² -П = α (te — t) .

17

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