Imperfect competition and congestion in the City

π^e = (μ^d + μ^w )    N - (F + S).

(n - 1)

The profit is a decreasing function of the number of subcenters: further entry
drives profits to zero.

In the long run, we assume that there is free entry or exit of subcenters. The
long run equilibrium is such that the profit of each subcenter is zero (we neglect
integer problems). The long-run n^f number of subcenters is:

n^f = 1+ (μ^d + μ^w ) N 1.
F+S

At the free entry equilibrium, the consumption g^f of the homogeneous good
is:

g^f = g (1) - (μ^d + μ^w) (-F+-K,
(F + )

where g (1) is given by equation (2). Less homogeneous good is consumed when
the product differentation and/or the job heterogeneity increases since both
factors increase profit margins and with free entry, also the number of firms.
In this case, a larger number of firms increases the resource cost needed to
produce the differentiated good, n (F +K), and therefore decreases the amount
of residual consumption of the homogeneous good.

5 Optimum without congestion

5.1 The welfare function

We now compute the first-best optimum. The welfare function can be derived
from the indirect utility function (5) in the symmetric case (all subcenters are
of equal size). At symmetry, the utility of an individual who works at i and
purchases from subcenter k is (see equation (5)): Uik = [Ω^o + (w — p)o]+ μ^wεi +
μ^dεk.Note that, at the symmetric optimum, (p — w)o = c so that:

Ω^o = (—awt — β) + θ (1 — β) + h — α^dt — n (F + S) — T

Recall the government budget equation is nS +NT = nK. Then:
n

Ω^o + (w — p)o = Ψ — NN (F + K),

where:

Ψ = —β + θ (1 — β) + h — c¹ — (α^h + α^d + α^w) t.

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