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First we must determine ===Sequential Search Models=== The first step to solving sequential search models is determining the optimum reservation price in a sequential search. Suppose that following n searches the consumer has found a best price (to date) of <math>z\,</math>. Then the benefit of an additional search is:
Baye Morgan Scholten (2006) - Information Search and Price Dispersion (view source)
Revision as of 03:57, 26 January 2010
, 03:57, 26 January 2010→Diamond's Paradox
<center><math>v \left ( \frac{\epsilon}{1+\epsilon}\overline{m} \right ) > c\,</math></center>
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'''Diamond's Paradox'''
Even though there is a continuum of competing firms (i.e. perfect competition)
in the presence of any search frictions whatsoever the monopoly price is the equilibrium.
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The optimal consumer strategy is:
'''Case 1)''' <math>h(\overline{p}) < 0 and \int_{\underline{p}}^{\overline{p}} v(p)dF(p) < c\,</math>. In this case it is better not to search.
'''Case 2)''' <math>h(\overline{p}) < 0 and \int_{\underline{p}}^{\overline{p}} v(p)dF(p) \ge c\,</math>. In this case the net benefit at the current price is negative, but the consumer is best off by searching until they get a price quote at (or below) <math>\underline{p}\,</math>.
'''Case 3)''' <math>h(\overline{p}) > 0\,</math>. This is the interior solutionand the interesting case. The customer should search until they get a reservation price <math>r\,</math> (or below) which makes them exactly indifferent between buying now and making another search. This is given by:
<center><math>h(r) = \int_{\underline{p}}^{r} (v(p) - v(r))dF(p) - c = 0\,</math></center>
Note that this is uniquely defined because:
h(\underline{p}) = -c < 0
h(\overline(p)) \ge 0
h'(z) = B'(z) > 0
Using the equation above to find the optimal r (i.e. taking the first order condition), and then differentiating with respect to c, we can determine an interesting comparative static:
\frac{\partial r}{\partial c} = \frac{1}{q(r)F(r)} = \frac{1}{Kr^{\epsilon}F(r)} > 0
Therefore a the reservation price is increasing in search costs. Note the special case where q(r)=1 leads to a magnification effect, but attenuation effects are also possible.
