Imperfect competition and congestion in the City

We assume that the random variable γ are i.i.d., with cdf (strictly increasing
and absolutely continuous) denoted by G (.) and density denote by g defined on
R. We get:

• Consumer choices

The probability that a household located in the center patronizes subcenter
k is

Pd = ©-Pk + μ^dη_fc ≥ -pi + μ^dηι, ι = ι...n}.

We assumed random variable η are i.i.d., with cdf (strictly increasing and abso-
lutely continuous) denoted by H (.) and density denote by h defined on R.We
get:

• Market clearing conditions

We still require that the market clearing condition holds: P_i^w = P_i^d , where
P_i^w is given by (50) and P_i^d is given by (51). We get:

g (x) Yg μ ⁱ w ^j + -'ɔ dχ = ʃ h (χ) Yh μ~—dpi + x^ dx

As before, the demand for the differentiated product sold in subcenter i is
Di = NP_i^d = NP_i^w .

• The profit function

Consider the price adjustment for subcenter i. The LHS of (52) is strictly
increasing in wi and the RHS is strictly decreasing in pi since F and G are
strictly increasing and absolutely continuous. We denote by r the one to one
relation between w_i and p_i: p_i = r_i (w_i ), and let r_i (w_i )=r(w_i ,w_-i,p_-i ). Note
that:

dP_i^{w             ∞}       wi - wj           wi - wk

-w- = (n^- 1)X /  g^(x)g( —— ^{+ x} HGl _nW ^{+ x dχ},i = 1...n.

^dwi       j=i ^-∞       ^{μ μ}       к k=i,j ^{μ μ}        к

(53)

Therefore:

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