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Baye Morgan Scholten (2006) - Information Search and Price Dispersion (view source)
Revision as of 19:14, 29 September 2020
, 19:14, 29 September 2020no edit summary
{{Article
|Has page=Baye Morgan Scholten (2006) - Information Search and Price Dispersion
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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.
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>, 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}\,</mathcenter> is the probability that no-one else lists.
: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>
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>.
