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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