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Baye Morgan Scholten (2006) - Information Search and Price Dispersion (view source)
Revision as of 19:14, 29 September 2020
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{{Article
|Has page=Baye Morgan Scholten (2006) - Information Search and Price Dispersion
|Has bibtex key=
|Has article title=Information Search and Price Dispersion
|Has author=Baye Morgan Scholten
|Has year=2006
|In journal=
|In volume=
|In number=
|Has pages=
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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.
<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>
Each firm lists with probability:
<center><math>\alpha = 1 - \left ( \frac{\frac{n-1}{n-1}\phi}{v-m)S} \right )^{\frac{1}{n-1}}\,</math> with <math>\alpha \in (0,1)\,</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 + \frac{\frac{n-1}{n-1}}{S}\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>.
