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:

g (χ) YGfwiTwwj
∞      j=i   × μ


+x


dx, i = 1...n.


(50)


Consumer choices

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

Pd = ©-Pk + μdηfc ≥ -pi + μdηι, 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:

Pd = £ h (x) Y ' p


+ x dx, k = 1...n.


(51)


Market clearing conditions

We still require that the market clearing condition holds: Piw = Pid , where
Piw is given by (50) and Pid is given by (51). We get:

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

(52)


As before, the demand for the differentiated product sold in subcenter i is
Di = NPid = NPiw .

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
wi and pi: pi = ri (wi ), and let ri (wi )=r(wi ,w-i,p-i ). Note
that:

dPiw                    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:

29



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