# Consumer surplus with isoelastic demand

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?

• Possible duplicate of Consumer surplus in case of perfectly inelastic demand – Giskard Oct 4 '17 at 18:32
• I actually saw that question. But here demand is not perfectly inelastic (approximatively maybe). Problem is that there is no upper price bound here and the consumer will never be budgeted out. In fact, since quantities are continuous demand will be approximatively zero for high prices. That's why I was thinking that the question might be of interest. – clueless Oct 4 '17 at 18:52
• Note that the consumer always pays $p q(p) = 1$, so the problem of not having enough budget does not apply here. – wythagoras Oct 4 '17 at 18:59
• I find that similar practical considerations would apply. Namely that you cannot possibly consume an infinite amount of something. This is a physical constraint on your consumption, and not unlike the budget constraint it renders infinity meaningless. – Giskard Oct 4 '17 at 19:21

This simply tells us that the mathematical object $f(x) = 1/x$ is not friendly to all the economic concepts we want to associate/obtain from its use as a demand function.