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Baye Morgan Scholten (2006) - Information Search and Price Dispersion (view source)
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|Has article title=Information Search and Price Dispersion
|Has author=Baye Morgan Scholten
|Has year=2006
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*This page is part of a series under [[PHDBA279B]]
*This page is referenced in [[BPP Field Exam Papers]]
Baye, Michael R., John Morgan, and Patrick Scholten (2006), "Information, Search, and Price Dispersion," Handbook of Economics and Information Systems (T. Hendershott, ed.), Elsevier Press, Amsterdam.
==Key Reference(s)==
*Baye, M.R. and J. Morgan (2001), "Information Gatekeepers on the Internet and the Competitiveness of Homogeneous Product Markets", American Economic Review, 91 (3), 454-474.
*Diamond, P. (1971), "A Model of Price Adjustment", Journal of Economic Theory, 3, 156-168.
*MacMinn, R.D. (1980), "Search and Market Equilibrium", Journal of Political Economy, 88 (2), 308-327.
*Reinganum, J.F. (1979), "A Simple Model of Equilibrium Price Dispersion", Journal of Political Economy, 87, 851-858.
*Rosenthal, R.W. (1980), "A Model in Which an Increase in the Number of Sellers Leads to a Higher Price", Econometrica, 48(6), 1575-1580.
*Rothschild, M. (1974), "Searching for the Lowest Price When the Distribution of Prices is Unknown", Journal of Political Economy, 82(4), 689-711
*Stigler, G. (1961), "The Economics of Information", Journal of Political Economy, 69 (3), 213-225.
*Varian, H.R. (1980), "A Model of Sales", American Economic Review, 70, 651-659.
==Introduction==
<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>
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>.
===The Rothschild Critique===
<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:
<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.
If a firm lists then its price is drawn from:
<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>
where:
<center><math>p_0 = m + (v-m)\frac{L}{L+S} + \frac{\frac{n-1}{n-1}}{L+S}\phi\,</math></center>
<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>
<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>
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>.
