Changes

<center>where <math>\mathbb{E}(p_{min}^{(n)}) = \mathbb{E}(min\{p_1,p_2,\ldots,p_n\}) \,</math> , that is the expected minimum price from n draws</center>

The distribution of the lowest <math>n\,</math> draws is:   <center><math>F_{min}^{(n)}(p) = 1 - (1-F(p))^n\,</math> , where <math>(1-F(p))^n\,</math> is the probability that <math>P\,</math> is less than <math>p\,</math> for all <math>n\,</math> draws. </center>

However, a firm's demand is zero above <math>r\,</math>, so firms will have no sales in the interval <math>\left (r,\frac{\overline{m}\epsilon}{1+\epsilon} \right ]\,</math>, and will set their price at <math>r\,</math> (as the elasticity of demand is constant).

<center><math>\alpha = 1 - \left ( \frac{\frac{n-1}{n-1}\phi}{(v-m)S} \right )^{\frac{1}{n-1}}\,</math></center>  This is obtained by equating the inside and outside options and solving for <math>\alpha\,</math>.  The outside option is: <center><math>(v-m)\left(L-\frac{S}{n}(1-\alpha)^{n-1}\right)\,</math></center> :where <math>(v-m)\,</math> is the mark-up, <math>\frac{S}{n}\,</math> is the traffic if no-one else lists and <math>(1-\alpha)^{n-1}\,</math> is the probability that no-one else lists.  The inside option is: <center><math>(v-m)\left(L - S(1-\alpha)^{n-1}\right)-\phi\,</math></center>  :where <math>S\,</math> is the traffic obtained from listing and <math>\phi\,</math> is the cost of listing.

<center><math>F(p) = \frac{1}{\alpha} \left ( 1 - \left ( \frac{\frac{n-1}{n-1}\phi + (v-p)L}{(p-m)S} \right )^{\frac{1}{n-1}} \right )\,</math> on <math>[p_0,v]\,</math></center>

<center><math>p_0 = m + (v-m)\frac{L}{L+S} + \frac{\frac{n-1}{n-1}}{L+S}\phi\,</math></center>

<center><math>F(p) = \frac{1}{\alpha} \left ( 1 - \left ( \frac{\frac{n-1}{n-1}\phi + (v-p)L}{(p-m)S} \right )^{\frac{1}{n-1}} \right )\,</math> on <math>[p_0,v]\,</math></center>

<center><math>p_0 = m + (v-m)\frac{L}{L+S} + \frac{\frac{n-1}{n-1}}{L+S}\phi\,</math></center>

 Loyal customers expect to pay the average of prices charged:   <center><math>\mathbb{E}(p) = \int_{p_0}^v p dF(p)\,</math></center>  Shoppers expect to pay the lowest of <math>n\,</math> draws from <math>F(p)\,</math>:  <center><math>\mathbb{E} \left [ p_{min}^{(n)} \right ] = \int_{p_0}^{v} p dF_{min}^{(n)}(p)\,</math></center>  As the number of competing firms increases prices increase because it is assumed that more loyals enter the market. The fraction of shoppers in the market is given by:  <center><math>\frac{S}{(S+nL)}\,</math></center>  In addition <math>F\,</math> is stochastically ordered in <math>n\,</math>, so when there is <math>n+1\,</math> firms competing <math>F^{(n+1)}\,</math> [http://en.wikipedia.org/wiki/First-order_stochastic_dominance first-order stochastically dominates] <math>F^(n)\,</math>.   Finally it should be noted that the results in Rosenthal (1980) are essentially identical to those of the fixed-search model of Burdett and Judd (1983). In Burdett and Judd a fixed fraction of consumers sample only one firm and so can be considered "loyal", while the remainder sample two firms are are the "shoppers".  ===The Varian (1980) Model=== The Varian (1980) model gives customers ex-ante different information sets. Shoppers are informed consumers, and Loyals are uniformed consumers. Furthermore, Varian shows that these differences can exist when customers are acting optimally, provided the costs of becoming informed are ordered and surround the price (value) of information. In the Varian (1980) model:*<math>\phi = 0\,</math> (i.e. costless listing)*<math>U > 0\,</math> (i.e. some uniformed customers) such that each firm is visited by <math>L=\frac{U}{n}\,</math> uniformed customers We can use the distribution equations from before substituting in <math>L=\frac{U}{n}\,</math>:  <center><math>F(p) = \left ( 1 - \left ( \frac{(v-p)\frac{U}{n}}{(p-m)S} \right )^{\frac{1}{n-1}} \right )\,</math> on <math>[p_0,v]\,</math></center>  where:  <center><math>p_0 = m + (v-m)\frac{\frac{U}{n}}{\frac{U}{n}+S}\,</math></center>  Suppose that consumers have different costs of accessing the clearinghouse according to their types. The value of information at the clearinghouse can be seen to be: <center><math>VOI^{(n)} = \mathbb{E}(p) - \mathbb{E} \left [ p_{min}^{(n)} \right ]\,</math></center>  If costs are such that <math>K_S\,</math> is the cost for the shoppers and <math>K_L\,</math> is the cost for the loyals, and then costs are such that:  <center><math>K_S \le VOI^{(n)} < K_L\,</math></center>   The shoppers will optimally use the clearinghouse and loyals optimally will not.  It is important to notice that the level of price dispersion is not a monotonic function of the consumer's information costs. When the costs become too high, no shoppers exist (i.e. no-one becomes informed) and all firms charge the monopoly price. Likewise when costs are zero, everyone becomes informed and all firms charge marginal cost (the Bertrand Paradox again). ===Baye and Morgan (2001)=== The Baye and Morgan (2001) model has optimizing firm, optimizing consumers and a monopolist gatekeeper. There is a nice 'story' to match this model that uses geographically distinct local markets that can serve the global market if they list with the gatekeeper. Loyal consumers shop locally, and shoppers are (potentially) global purchasers. The assumptions are as follows:*The gatekeeper optimally sets <math>\phi > 0\,</math>*The gatekeeper optimally sets <math>L=0\,</math> Substituting into the equations we find that:  Each firm lists with probability: <center><math>\alpha = 1 - \left ( \frac{\frac{n}{n-1}\phi}{v-m)S} \right )^{\frac{1}{n-1}}\,</math> with <math>\alpha \in (0,1)\,</math></center>  The price distribution at the clearinghouse is:  <center><math>F(p) = \frac{1}{\alpha} \left ( 1 - \left ( \frac{\frac{n}{n-1}\phi + (v-p)L}{(p-m)S} \right )^{\frac{1}{n-1}} \right )\,</math> on <math>[p_0,v]\,</math></center>  where:  <center><math>p_0 = m + \frac{\frac{n}{n-1}}{S}\phi\,</math></center>  When a firm doesn't list it charges <math>v\,</math> and its equilibrium profits are:  <center><math>\mathbb{E}\pi = \frac{1}{n-1}\phi\,</math></center>  Price dispersion arises from the gatekeeper's incentives to set <math>\phi > 0\,</math>. The expected profits to firms are positive and proportional to <math>\phi\,</math>.