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Due to an upcoming exam I came across the following past exam question which at first seemed easy. The question is as follows.

Let $f(y_i) = (\frac{k}{y_i})^2$ be the density function of the (random) income of agent A where $k$ denotes the minimum amount. As such the income follows a pareto distribution with support $[k, \infty)$. Determine the expected income.

At first, my thoughts were pretty routine: Apply the definition $\mathbb{E}[y_i] = \int\limits_{k}^{\infty} f(y_i)y_i dy_i$ and integrate. The result is the expected income. But after doing that I got the integral $\int\limits_{k}^{\infty} \frac{1}{y_i} dy_i$ which of course does not converge. Hence, the expected income would be infinite which is not sensible - not economically nor in the exam situation. But what do I miss here?

Taufi
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  • The fact that you conclude that it's not economically sensible tells you it's a poor choice if you want to model real incomes, not that you made a mistake in algebra. Not all densities have finite expectation. – Glen_b May 29 '17 at 00:11
  • See https://stats.stackexchange.com/questions/232967/what-makes-the-mean-of-some-distributions-undefined ... and ...

    https://stats.stackexchange.com/questions/70088/when-does-a-distribution-not-have-a-mean-or-a-variance ... and perhaps ... https://stats.stackexchange.com/questions/114511/do-mean-variance-and-median-exist-for-a-continuous-random-variable-with-continu

     – Glen_b May 29 '17 at 00:16

1 Answers1

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You are missing the fact that not all distributions have an expectation. The Pareto can either have one or not have one. In this case, it does not have one. In this case, there is no such thing as an expected income. It isn't that it is infinite, it is that the expectation does not exist. There is no population mean even under infinite repetition. The most you can say is that you anticipate income to be in the support region and finite given the form of the question. Alternatively, you could describe the quantile generating function. You could discuss likely regions as well, to some degree of confidence.

Dave Harris
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  • Thanks, Dave Harris. To me this seems odd since 1) we never had this case in any form neither in the lectures nor in the classes and 2) I have to calculate the expected income's utility under the utility function $u(y) = y^a$ for some $a > 0$. If the expected income is not defined then so is its utility. It seems odd to me to ask this in an exam. – Taufi May 28 '17 at 23:23
  • @Taufi The expected utility of income can exist when no expected income exists. Consider the case of $U(x)=log(x)$ then the expected utility of income is $(log(k)+1)/k$. An undefined return expectation is not the same thing as an undefined utility expectation. – Dave Harris May 29 '17 at 00:25
  • Yes, I am aware of that. But I wrote the utility of the expected income, i.e. $u(\mathbb{E}[y_i])$ and not $\mathbb{E}[u(y_i)]$. Am I still right to suppose that the former does not exist/is not defined when the expected income is infinite or undefined? – Taufi May 29 '17 at 08:27
  • Yes u(E(x)) is undefined. – Dave Harris May 29 '17 at 15:03