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Baye Morgan Scholten (2006) - Information Search and Price Dispersion (view source)
Revision as of 17:46, 25 April 2010
, 17:46, 25 April 2010→Information Clearinghouses
To solve this note that the expected profits must be the same across the entire support (for it to be a mixed strategy) and are equal to the profit from the outside option. The inside option (above) is made up of the following components:
*<math>(p-m) \,</math> is difference between the price and consumer's willingness to pay*This is gained for sure for the <math>L \,</math> loyal consumers*This is gained for the <math>S \,</math> shoppers on the basis of:*<math>\sum_{i=0}^{n-1} \,</math> is the sum over the number of people on the site*<math>\binom{n-1}{i} \,</math> is the <math>n-1 \,</math> choose <math>i \,</math> ways that this could occur*<math>\alpha^i \,</math> is the probability that <math>i \,</math> firms list*<math>(1-\alpha)^{n-1-i} \,</math> is the probability that the other firms <math>(n-1-i) \,</math> didn't list*<math>(1-F(p))^i \,</math> is the probability that everyone who did list prices above <math>p\,</math>
Using the [http://en.wikipedia.org/wiki/Binomial_theorem binomial theorem]:
:<math>\sum_{k=0}^{n} \binom{n}{k} \gamma^k \cdot \beta^{n-k} = (\gamma + \beta)^n\,</math>
Let <math>\gamma = \alpha(1-F(p))\,</math> and <math>\beta = (1-\alpha)\,</math> and solve to get:
<center><math>\mathbb{E}\pi(p) = (p-m) \left ( L + \left ((1-\alpha F(p))^n-1 \right ) S \right) - \phi\,</math></center>
Then use the outside option to solve for <math>F(p)\,</math> to get:
<center><math>\mathbb{E}\pi(p) = (v-m)L + \frac{\phi}{n-1}\,</math></center>
