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#The distribution of firms' prices is given exogenously by the non-degenerate CDF <math>F(p)\,</math> on <math>[\underline{p}, \overline{p}]\,</math>.

'''Fixed sample search'''

In a fixed sample search the consumer commits to conducting <math>n\,</math> searches

and then buys from the firm offering the lowest price

The consumer seeks to minimize the expected cost (purchase + search) given by:

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.

'''Sequential search'''

In sequential search there is an optimal stopping rule.

The reservation price is endogenously determined.

Recall is free.

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?

'''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?

===Diamond's Paradox===

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: