Let $q(p) = \frac{1}{p}$ denote the demand function and $p^*$ some equilibrium price. Consumer surplus is defined as \begin{align} CS = \int_{p^*}^\infty\frac{1}{p} dp= \ln(\infty) - \ln(p^*) = \infty \end{align} which is quite unsatisfactory. What's the convention here?
Edit: Since I wanna compare welfare under price and quantity competition I found a workaround. Let $n$ the number of firms in the market, $\bar \pi^j$, equilibrium profits of the firm, $\bar p^j$ equilibrium prices and $j \in \{b,c\}$ is the index for the compeition type. Now we may calculate the welfare difference as \begin{align} \Delta W := W^c - W^b &= n(\bar \pi^c - \bar \pi^b) + \int_{\bar p^c}^\infty\frac{1}{p}dp - \int_{\bar p^b}^\infty\frac{1}{p}dp\\ & = n(\bar \pi^c - \bar \pi^b) + \ln(\bar p^b) - \ln(\bar p^c) \end{align}
Suppose we get the following relationship \begin{align} \Delta W = \frac{(n-1)(\sigma-1)^2}{\sigma n^2(\sigma(n-1)+1)} + \ln\left(\frac{\sigma(n-1)+1}{n \sigma}\right) \end{align} with $n \geq 2$ and $\sigma \in [0,\infty)$ is a parameter. Clearly $\Delta W = 0$ for $\sigma = 1$ and for all $n$. I wanna further show that $\Delta W > 0$ for $\sigma < 1$ and $\Delta W < 0$ for $\sigma > 1$. Any ideas how to proceed?
