Changes
Jump to navigation
Jump to search
Baye Morgan Scholten (2006) - Information Search and Price Dispersion (view source)
Revision as of 22:11, 25 January 2010
, 22:11, 25 January 2010no edit summary
*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 (i.e. additively-seperable in income): <center><math>u(q) + y\,</math> where <math>y\,</math> is a numeraire good</center>*The indirect utility of consumers is: <center><math>V(p,M) = v(p) + M\,</math> where <math>v(\cdot)\,</math> in nonincreasing in <math>p\,</math>, and <math>M\,</math> is income.</center>
*By [http://en.wikipedia.org/wiki/Roy%27s_identity Roy's identity]:
<center><math>q(p) \equiv -v'(p)\,</math>.</center>
*There is a search cost <math>c\,</math> per price quote
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>:
<center>
<math>\frac{d(v(M,p))}{dp} = \frac{\partial v(M,p)}{\partial m} \cdot \frac{\partial M}{\partial p} + \frac{\partial v(M,p)}{\partial p} = 0,\,</math> where
<math>\frac{\partial M}{\partial p} = \frac{\partial e(p,u)}{\partial p}\,</math>.
<math>\therefore q(m,p) = \frac{\partial e(p,u)}{\partial p} = -\frac{\frac{\partial v(M,p)}{\partial p}}{\frac{\partial v(M,p)}{\partial m}}\,</math>
<math>\therefore q(m,p) = -\frac{d}{dp(v(p))}\,</math> in our case.
The consumer seeks to minimize the expected cost (purchase + search) given by:
<center><math>\mathbb{E}(C) = K \mathbb{E}(p_{min}^{(n)}) + cn\,</math> </center>
<center>where <math>\mathbb{E}(p_{min}^{(n)}) = \mathbb{E}(min\{p_1,p_2,\ldots,p_n\}) \,</math> </center>
The distribution of the lowest <math>n\,</math> draws is: <center><math>F_{min}^{(n)}(p) = 1 - (1-F(p))^n\,</math> </center>
<center><math>\therefore \mathbb{E}(C) = K \int_{\underline{p}}^{\overline{p}} p \; dF_{min}^{(n)}(p) + cn\,</math></center>
<center><math>\therefore \mathbb{E}(C) = K \left [ \underline{p} + \int_{\underline{p}}^{\overline{p}} (1-F(p))^n \; dp \right ] + cn\,</math></center>
To see this, first recall that <center><math>\mathbb{E}(X) = \int_{\underline{x}}^{\overline{x}} x f(x) dx \,</math></center>
Then use the [http://en.wikipedia.org/wiki/Integration_by_parts integration by parts] formula <center><math>\int u\, \frac{dv}{dx}\; dx=uv-\int v\, \frac{du}{dx} \; dx\!</math></center>
