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It is instrumental to temporarily ignore that a firm's demand is zero above <math>r\,</math>. The profit maximizing price from above is (from the first order condition):
Baye Morgan Scholten (2006) - Information Search and Price Dispersion (view source)
Revision as of 05:25, 26 January 2010
, 05:25, 26 January 2010no edit summary
</math></center>
<center><math>p_j=\left ( \frac{\epsilon}{1+\epsilon} \right ) m_j\,</math></center>
If firm's firms were to do this then consumers would face a distribution of prices:
<center><math>\hat{F}(p)=G \left (p \left ( \frac{\epsilon}{1+\epsilon} \right ) \right)\,</math> on the interval <math>\left [ \frac{\underline{m}\epsilon}{1+\epsilon} , \frac{\overline{m}\epsilon}{1+\epsilon} \right ]\,</math></center>
Given this distribution of prices consumers will set their reservation price using:
<center><math>h(r) = \int_{\underline{p}}^r (v(p) -v(r)d\hat{F}(p)-c=0\,</math></center>
However, a firm's demand is zero above <math>r\,</math>, so firms will have no sales in the interval <math\left (r,\frac{\overline{m}\epsilon}{1+\epsilon} \right ]\,</math>, and will set their price at <math>r\,</math> (as the elasticity of demand is constant).
Therefore:
<center><math>
F(p) =
\begin{cases}
\hat{F}(p) & \mbox{if}\; p_j < r \\
1 & \mbox{if}\; p_j = r
\end{cases}
</math></center>
To verify that this is an equilibrium, we must check that it is a best response for consumers to set their reservation price as before. The reservation price is:
<center><math>h(r) = \int_{\underline{p}}^r (v(p) -v(r)d{F}(p)-c\,</math></center>
<center><math>h(r) = \int_{\underline{p}}^r (v(p) -v(r)d\hat{F}(p) + \left [1-\hat{F}(r) \right ] \left[ v(r)-v(r) \right ] -c\,</math></center>
<center><math>h(r) = \int_{\underline{p}}^r (v(p) -v(r)d\hat{F}(p)-c\,</math></center>
Therefore we can see that downward sloping demand and cost heterogeneity together give rise to price dispersion with optimizing consumers and firms. In fact both conditions are required. This is shown in detail in the paper. However, briefly, if consumers only want a single unit of the product then it is easy to show that <math>r=v\,</math>, and if firms all had the same marginal cost then the Diamond Paradox would re-assert itself.
A final result in the Reinganum (1979) model is that a reduction in search costs leads to a reduction in equilibrium prices. However, this is NOT a general result.
<math>
