Hotelling模型

Suppose that we have two firms located on a line of length 1. The unit costs of the good for each store is c. Consumers incur a transportation cost of tx2 for a length of x. Consumers have unit demands, and are uniformly distributed along the line. Firm 1 is
located at point a >= 0 and firm 2 at point 1 − b, where b >= 0 and without loss of generality, 1−a−b >= 0 (firms 1 is to the left of firms 2; a = b = 0 corresponds to maximal differentiation and a + b = 1 corresponds to minimal differentiation, i.e. perfect substitutes). Assumes that the market is covered and firms sell positive quantities.
(a) Show that given a and b, firm 1 will charge p1 = c + t(1 − a − b)(1 +(a − b)/3), whereas firm 2 will charge p2 = c + t(1 − a − b)(1 +(b − a)/3)
(b) Find firms’ market shares of the market.
(c) Now consider a first stage to this game where the two firms choose their locations, knowing that prices will be chosen in the second stage as in (a). Where do they locate?
(d) What would be socially optimal locations of the two firms? Compare with the market outcome.

这是我们这学期高微的一道作业题。但是不知道该从什么地方入手，所以想请高手给予指点。。。Thanks~~~

这个问题要从一个ab之间的x点的人来考虑，对于这个人来说，去a和b买东西是无差异的P1+t(x-a)^2=P2+t(b-x)^2。这样的话，x左边的人选择去firm1更优，而x右边的人去firm2更优，因而x就是firm1的mkt share 而1-x就是firm2 的mkt share。每个firm的目标是利润最大化，继而解出定价策略，and so on~ 明白了？

siline 发表于 2012-3-11 05:44
这个问题要从一个ab之间的x点的人来考虑，对于这个人来说，去a和b买东西是无差异的P1+t(x-a)^2=P2+t(b-x)^2 ...

了解。。。thx

谢谢

P1+t(x-a)^2=P2+t(1-b-x)^2