Difference between revisions of "Baye Morgan Scholten (2006) - Information Search and Price Dispersion"
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'''A note on the derivation of demand''' | '''A note on the derivation of demand''' | ||
− | Recall that | + | Recall that <math>M=e(p,u)\,</math>, |
− | |||
− | <math>M=e(p,u)\,</math>, | ||
so that <math>v(e(p,u),p)=u\,</math> when the expenditure function is evaluated at <math>p\,</math> and | so that <math>v(e(p,u),p)=u\,</math> when the expenditure function is evaluated at <math>p\,</math> and | ||
− | <math>u\,</math>. | + | <math>u\,</math>. Taking the derivitive with respect to <math>p\,</math>: |
− | <math>\frac{d | + | <center> |
+ | <math>\frac{d(v(M,p))}{dp} = \frac{dv(M,p)}{dm} \cdot \frac{dM}{dp} + \frac{dv}{dp} = 0,\,</math> where | ||
<math>\frac{dM}{dp} = \frac{de(p,u)}{dp}\,</math>. | <math>\frac{dM}{dp} = \frac{de(p,u)}{dp}\,</math>. | ||
Revision as of 20:39, 25 January 2010
- This page is part of a series under PHDBA279B
Key Reference(s)
Introduction
Baye et al. (2006) provides a survey of models of search and clearing-house that exhibit price dispersion. The survey is undertaken through two specializable frameworks, one for search and one for cleaning-houses, which are then adapted to show the key results from the literature. There are a number of equivalent results across the two frameworks.
Search Theoretic Models of Price Dispersion
The general framework used through-out is as follows:
- A continuum of price-setting firms with unit measure compete selling an homogenous product
- A mass [math]\mu[/math] is interested in purchasing the product
- Consumers have quasi-linear utility:
- The indirect utility of consumers is:
- By Roy's identity:
- There is a search cost [math]c\,[/math] per price quote
- The customer purchases after [math]n\,[/math] price quotes
- The final indirect utility of the customer is
A note on the derivation of demand
Recall that [math]M=e(p,u)\,[/math], so that [math]v(e(p,u),p)=u\,[/math] when the expenditure function is evaluated at [math]p\,[/math] and [math]u\,[/math]. Taking the derivitive with respect to [math]p\,[/math]:
[math]\frac{d(v(M,p))}{dp} = \frac{dv(M,p)}{dm} \cdot \frac{dM}{dp} + \frac{dv}{dp} = 0,\,[/math] where [math]\frac{dM}{dp} = \frac{de(p,u)}{dp}\,[/math].
[math]\therefore q(m,p) = \frac{de(p,u)}{dp} = -\frac{\frac{dv}{dp}}{\frac{dv(M,p)}{dm}}\,[/math]
[math]\therefore q(m,p) = -\frac{d}{dp(v(p))}\,[/math]\\
[math][/math] [math][/math] [math][/math] [math][/math]