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Baye Morgan Scholten (2006) - Information Search and Price Dispersion (view source)
Revision as of 23:36, 25 January 2010
, 23:36, 25 January 2010no edit summary
[[Image:Two_Normals.png|thumb|right|400px|A mean preserving spread on a Normal distribution]]The Stigler model also implies that expected transaction prices will be lower when prices have the same mean but are more dispersed. There is a simple graphical proof of this. Suppose that the customer is drawing from one of the two distributions pictured - a draw from the green distribution (that has the higher variance) would be more likely to yield a lower price and have lower total costs.
These results are proved formally (as propositions 1 and 2), and the essence of the proofs are as follows: Let <center><math> \Delta = \mathbb{E}_F \left [ p_{min}^{(n)} \right ] - \mathbb{E}_G \left [ p_{min}^{(n)} \right \,</math></center> So that <math>\Delta\,</math> represents the difference in expected transaction prices. Then show that for <math>n>1\,</math> that <math>\Delta >0\,</math>. Then suppose that the expected number of searches under <math>F\,</math> is <math>n^*\,</math>. The consumer's expected total cost under <math>F\,</math> is: <center><math>\mathbb{E}(C_F) = \mathbb{E}_F \left [ p_{min}^{(n^*)} \right ] \times K - c n^* \,</math></center> <center><math> > \mathbb{E}_G \left [ p_{min}^{(n^*)} \right ]\,</math></center> <center><math> \ge \mathbb{E}(C_G)\,</math></center> Note that the strick inequality follows from the proof that <math>\Delta > 0\,</math>, and the weak inequality follows as <math>n^*\,</math> may not be optimal under <math>G\,</math>.
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