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Baye Morgan Scholten (2006) - Information Search and Price Dispersion (view source)
Revision as of 15:18, 26 January 2010
, 15:18, 26 January 2010no edit summary
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>
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'''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
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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-1}{n}\phi}{v-m)S} \right )^{\frac{1}{n-1}}\,</math></center>
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}\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}}{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>
Using the [http://en.wikipedia.org/wiki/Binomial_theorem binomial theorem]:
<center><math>\mathbb{E}\pi(p) = (p-m) \left ( L + \left ((1-\alpha F(p))^n-1 \right ) S \right) - \phi\,</math></center>
<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".
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'''Ed's observation'''
In clearinghouse models, the use of mixed strategies by firms who are indifferent between listing and not,
drives many of the price-dispersion results.
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