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Baye Morgan Scholten (2006) - Information Search and Price Dispersion (view source)
Revision as of 03:34, 26 January 2010
, 03:34, 26 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 intuition for this, that can be demonstrated graphically. 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:
===The Rothschild Critique===
Rothschild (1973) pointed out that the search procedure used in Stigler (1961) may not be optimal. Specifically the customer's commitment to a fixed number of searches may not be 'credible'. The strategy fails to incorporate new information as it becomes available: Once a sufficiently low price quote has been obtained, the benefit of additional searches may drop below the marginal cost.
<center>
Consumers know the distribution of prices, which is exogenously specified.
The reservation price is endogenously determined.
Recall is free.
</center>
Then a firm with constant marginal cost (<math>m\,</math>) has expected profits:
<center><math>\pi(p) = (p-m)Q(p)\,</math></center>
All firms have the same marginal cost and so the same expected profit function. So then why would profit-maximizing firms choose the same distribution of prices? <center> '''The Rothschild Critique''' The Stigler (1961) model has only optimizing consumers, and not optimizing firms This is therefore a 'partial-partial equilibrium' approach. Why would profit-maximizing firms choose the same profit-maximizing price?</center>
===Diamond's Paradox===
Diamond (1971) provides conditions under which the unique equilibrium in undominated strategies has firms all charging the monopoly price, with costly search by the consumers.
To see Diamond's result, suppose:
*Consumers have identical downward sloping demand:
<center><math>-v''(p)=q'(p)<0\,</math></center>
*Consumers engage in optimal sequential search
*Firms all charge the unique monopoly price <math>p^*\,</math>
*The consumer with the highest marginal cost <math>\overline{m}\,</math> still earns sufficient surplus to cover the cost of one price quote:
<center><math>v \left ( \frac{\epsilon}{1+\epsilon}\overline{m} \right ) > c\,</math></center>
First we must determine the optimum reservation price in a sequential search. Suppose that following n searches the consumer has found a best price (to date) of <math>z\,</math>. Then the benefit of an additional search is:
<center><math>B(z) = \int_{\underline{p}}^{z} (v(p) - v(z))dF(p) = \int_{\underline{p}}^{z} v'(p)dF(p)\,</math></center>
By Liebnitz' rule:
<center><math>B'(z) = -v'(z)dF(z) = Kz^{\epsilon}F(z) > 0\,</math></center>
This is smaller when <math>z\,</math> is small, that is the benefits are lower when the best price already identified is lower. Furthermore, search is costly, so consumers must make a trade-off. The expected net benefit of an additional search is:
<center><math>h(z) \equiv B(z) - c\,</math></center>
The optimal consumer strategy is:
Case 1) <math>h(\overline{p}) < 0 and \int_{\underline{p}}^{\overline{p}} v(p)dF(p) < c\,</math>. In this case it is better not to search.
Case 2) <math>h(\overline{p}) < 0 and \int_{\underline{p}}^{\overline{p}} v(p)dF(p) \ge c\,</math>. In this case the net benefit at the current price is negative, but the consumer is best off by searching until they get a price quote at (or below) <math>\underline{p}\,</math>.
Case 3) <math>h(\overline{p}) > 0\,</math>. This is the interior solution. The customer should search until they get a reservation price <math>r\,</math> given by:
<center><math>h(r) = \int_{\underline{p}}^{r} (v(p) - v(r))dF(p) - c = 0\,</math></center>
