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:

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>

===The Reiganum Reinganum (1979) Model===

Reiganum Reinganum (1979) showed that price dispersion could exist with sequential search as well as with optimizing consumers and optimizing firms.

Reiganum Reinganum (1979) shows that under these assumptions there is an equilibrium in which firms optimally set prices, consumers engage in optimal sequential search, and yet there is still price dispersion. We return to the Reiganum (1979) model after a discussion of sequential search.

By [http://en.wikipedia.org/wiki/Leibniz's_rule Liebnitz' rule]:

Recall that Reinganum (1979) has firms with marginal costs drawn from a distribution <math>G(m)\,</math>. Suppose that an individual firm's cost is <math>m_j\,</math>, and that a fraction <math>\lambda\,</math> (, where <math>\lambda \in \left [0,1 \right )\,</math>) , of firms price above <math>r\,</math>. Then, with a mass <math>\mu \,</math> of consumers as before: 

\mathbb{E}\pi_j =

(p_j-m_j)q(p_j)\left ( \frac{\mu}{1-\lambda} \right ) & \mbox{if}\; p_j \le r \\0 & \mbox{if}\; p_j > r /\end{cases}</math></center> The profit maximizing price from above is (from the first order condition):  <center><math>\left [ (p_j - m_j) q'(p_j) + q(p_j) \right ] \left ( \frac{\mu}{1 - \lambda} \right ) = 0\,</math></center> Which implies (given the consumer's demand function above):  <center><math>p_j=\left ( \frac{\epsilon}{1+\epsilon} \right ) m_j\,</math></center> If firms were to do this then consumers would face a distribution of prices:  <center><math>\hat{F}(p)=G \left (p \left ( \frac{\epsilon}{1+\epsilon} \right ) \right)\,</math> on the interval <math>\left [ \frac{\underline{m}\epsilon}{1+\epsilon} , \frac{\overline{m}\epsilon}{1+\epsilon} \right ]\,</math></center> Given this distribution of prices consumers will set their reservation price using: <center><math>h(r) = \int_{\underline{p}}^r (v(p) -v(r)d\hat{F}(p)-c=0\,</math></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). Therefore:

F(p) = \begin{cases}\hat{F}(p) & \mbox{if}\; p_j < r \\1 & \mbox{if}\; p_j = r \end{cases}</math></center>  To verify that this is an equilibrium, we must check that it is a best response for consumers to set their reservation price as before. The reservation price is:  <center><math>h(r) = \int_{\underline{p}}^r (v(p) -v(r)d{F}(p)-c\,</math></center>  <center><math>h(r) = \int_{\underline{p}}^r (v(p) -v(r)d\hat{F}(p) + \left [1-\hat{F}(r) \right ] \left[ v(r)-v(r) \right ] -c\,</math></center>  <center><math>h(r) = \int_{\underline{p}}^r (v(p) -v(r)d\hat{F}(p)-c\,</math></center>  Therefore we can see that downward sloping demand and cost heterogeneity together give rise to price dispersion with optimizing consumers and firms. In fact both conditions are required. This is shown in detail in the paper. However, briefly, if consumers only want a single unit of the product then it is easy to show that <math>r=v\,</math>, and if firms all had the same marginal cost then the Diamond Paradox would re-assert itself. A final result in the Reinganum (1979) model is that a reduction in search costs leads to a reduction in equilibrium prices. However, this is '''not''' a general result. ===The MacMinn (1980) Model=== The MacMinn (1980) Model provides an example in which fixed search is optimal. The constraints in MacMinn (1980) are:*Customers have unit valuation <math>v\,</math>*Customers engage in optimal fixed sample search*Firms have privately observed marginal costs drawn from <math>G(m)\,</math> on <math>[\underline{m},\overline{m}]\,</math>, where <math>\overline{m} < v\,</math>. The MacMinn model can be solved using the [http://en.wikipedia.org/wiki/Revenue_equivalence_theorem#Revenue_equivalence Revenue Equivalence Theorem]. Each <math>n^*\,</math> firms competes with <math>n^*-1\,</math> firms. The firm offering the lowest price 'wins' the 'auction'. The revenue to any auction where firms have a marginal cost <math>m\,</math>, the lowests price firm cost wins and the the firms with highest marginal cost earns zero surplus is:  <center><math>R(m) = m( 1-G(m))^{n^*-1} + \int_{m}^{\overline{m}} (1-G(t))^{n^*-1}dt\,</math></center> In the MacMinn model, the firm's expected revenues are:   <center><math>p(m) \times ( 1-G(m))^{n^*-1}.\,</math></center> This allows use to solve (using integration by parts) to find:  <center><math>p(m) = \mathbb{E} \left [ m_{min}^{n^*-1} | m_{min}^{n^*-1} \ge m \right ]\,</math></center>  where <math>m_{min}^{n^*-1}\,</math> is the lowest <math>n^*-1\,</math> draws from <math>G\,</math>.   This gives rise to a distribution of posted <math>F(p) = G(p(m))\,</math>.  For this to be optimal it must be optimal for consumers to sample <math>n^*\,</math> firms, so that:  <center><math>\mathbb{E}(B^{n^*+1}) < c \le \mathbb{E}(B^{n^*})\,</math></center>  So, if the search costs are low enough, a dispersed price equilibrium can exist! In this model there are ex-post differences in consumers information sets (they observe different prices). Furthermore, the greater the variance in marginal costs, the greater the variance in prices. Somewhat counter-intuitively the dispersion in prices also increases as the sample size increases. ==Information Clearinghouses== In information clearinghouse models a subset of the consumers consults a clearinghouse that displays prices for a sub-set of firms. In these models, for simplicity, <math>n\,</math> will denote the number of firms in the market. The general environment is:*a finite <math>n>1\,</math> number of price-setting firms competes in a homogeneous product market.*Firms have unlimited capacity and marginal cost <math>m\,</math>*It costs firms <math>\phi \ge 0\,</math> to list their prices in the clearinghouse*Consumers have unit demand with a willingness to pay of <math>v > m\,</math>*<math>S > 0\,</math> are shoppers that consult the clearinghouse (if no prices are listed below <math>v\,</math>, the shopper chooses at random and buys if <math>p \le v\,</math>)*<math>L \ge 0\,</math> are loyal customers that buy if <math>p \le v\,</math> <center><big> '''Information Clearinghouse General Result''' If <math>L > 0\,</math> '''or''' if <math>\phi > 0\,</math> then equilibrium price dispersion exists providing <math>\phi\,</math> is not so large that all firms refuse to list their products</big></center> Assume that:   <center><math>0 \le \phi < \frac{n-1}{n}(v-m)S\,</math></center> Then 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></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.  If a firm lists then its price is drawn from:  <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 + (v-m)\frac{L}{L+S} + \frac{\frac{n}{n-1}}{L+S}\phi\,</math></center>  (so <math>p_0 > 0\,</math> if <math>L > 0\,</math> or <math>\phi > 0\,</math>, and <math>p_0 = m\,</math> otherwise fulfilling the [http://en.wikipedia.org/wiki/Bertrand_paradox_(economics) Bertrand Paradox]) If a firm does not list it charges <math>v</math> (this is a dominant strategy) and each firm's expected profits are:  <center><math>\mathbb{E}\phi = (v-m)L + \frac{1}{n-1}\phi\,</math></center> The condition on <math>\phi\,</math> implies that <math>\alpha \in \left ( 0,1 \right]\,</math>. We must show that a firm can do no better than pricing accoring to <math>F\,</math>. Pricing outside the support of <math>F\,</math> is dominanted and it will transpire that pricing in the support of <math>F\,</math> leads to constant profits through-out the support.   The expected profits of the firm pricing in <math>F\,</math> are:  <center><math>\mathbb{E}\pi(p) = (p-m) \left ( L + \left ( \sum_{i=0}^{n-1} \binom{n-1}{i} \alpha^i (1-\alpha)^{n-1-i}(1-F(p))^i \right ) S \right ) - \phi\,</math></center>  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>  Therefore the firm's profits are constant on the support and so must be a best-response. When <math>\phi = 0\,</math> it is weakly dominant to list. When <math>\phi > 0\,</math> and <math>\alpha \in (0,1)\,</math> a firm's expected profits when it doesn't list are:  <center><math>\mathbb{E}\pi(p) = (v-m) \left ( L + \frac{S}{n}(1-\alpha)^n-1 \right )\,</math></center>  <center><math>\mathbb{E}\pi(p) = (v-m)L + \frac{\phi}{n-1}\,</math></center>  Therefore the firm earns the same expected profit whether it lists or not. This leads to "Ed's observation". <center><big> '''Ed's observation''' In clearinghouse models, the use of mixed strategies by firms drives the price-dispersion results.</big></center> ===The Rosenthal (1980) Model=== In the Rosenthal (1980) model we suppose:*<math>\phi = 0\,</math> (i.e. costless listing)*<math>L > 0\,</math> (i.e. some loyal customers) Since <math>\phi=0\,</math>,&nbsp; <math>\alpha=1\,</math> and all firms list at the clearinghouse. The equilibrium distribution of prices is therefore:  <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>  <center><math>F(p) = \left ( 1 - \left ( \frac{(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 + (v-m)\frac{L}{L+S} + \frac{\frac{n}{n-1}}{L+S}\phi\,</math></center>  <center><math>p_0 = m + (v-m)\frac{L}{L+S}\,</math></center>  Price dispersion results from exogenous differences in preferences of consumers. Firm's wish to price at <math>v\,</math> for their loyal customers but if they did so, they could be undercut and loose their shoppers... however, this does not lead to the Bertrand outcome as at some point firms are better off pricing <math>v\,</math> and giving up on serving the shoppers. The equilibrium is therefore in mixed strategies - sometimes firms price low to attract shoppers and sometimes price high to maintain margins.  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>.