Imperfect competition and congestion in the City

The welfare function is defined as the expected maximum utility: W (n)=

This has the property of a welfare function and satisfies Roy’s
identity (see Anderson, et al. 1992). The first—best optimum in the short run
(exogeneous number of subcenters) and in the long run are characterized by:

Proposition 2 In the absence of congestion, the short-run first-best optimum
welfare function is given by:

W (n) = Ψ - Nn (F + K)+ (μ^d + μ^w ) log (n),           (21)

where Ψ is given by 20. The long-run first-best optimum number of subcenters
is

no=<μ^d+μ^w ) (FNK).                   (²²)

Proof. Using equation (19) and the i.i.d. property, we get

.

Recall that with the double exponential distribution: E maxεi = ln (n), Then:

W (n) = Ψ - NN (F + K) + (μ^d + μ^w ) log (n) .

This function is concave in n. The optimal number of subcenters, n^o is obtained
by differentiation of W (n) where n is treated as a real number. ■

At the optimum, the consumption of the homogenous good is g^o = g(n^o) or

go = θ - c¹ - (α^w + α^d + α^h) t - (μ^d + μ^w) = g(0) - (μ^d + μ^w) .

Where g(n) is given by 2. Note that this expression is independent of the fixed
costs F and K . The comparative statics on the first-best number of subcenters
and on the consumption of the goods are left to the reader.

5.2 Equilibrium versus optimum number of subcenters

We can now compare the equilibrium and the optimum numbers of subcenters.
Note that:

(p - w)^e = (p - w)o + (μ^d + μ^w) -----,

n-1

that is to say firms charge a price (net of wage) above the socially optimal
level (c). However the excessive price level will not induce distortions in the

12

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