Changes

'''Case 1)'''&nbsp;&nbsp; <math>h(\overline{p}) < 0\,\,</math> &nbsp; and &nbsp;<math> \int_{\underline{p}}^{\overline{p}} v(p)dF(p) < c\,</math>. In this case it is better not to search.

'''Case 2)'''&nbsp;&nbsp; <math>h(\overline{p}) < 0\,</math> &nbsp; and &nbsp;<math>\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>.

<center><math>h(\underline{p}) = -c < 0\,</math></center>

<center><math>h(\overline(p)) \ge 0\,</math></center>

<center><math>h'(z) = B'(z) > 0\,</math></center>

Using the equation above to find the optimal <math>r \,</math> (i.e. taking the first order condition), and then differentiating with respect to <math>c\,</math>, we can determine an interesting comparative static:

<center><math>\frac{\partial r}{\partial c} = \frac{1}{q(r)F(r)} = \frac{1}{Kr^{\epsilon}F(r)} > 0\,</math></center>

Therefore a the reservation price is increasing in search costs. Note the special case where <math>q(r)=1 \,</math> leads to a magnification effect, but attenuation effects are also possible.