# Expected value of pareto distribution [duplicate]

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?

## marked as duplicate by Glen_b♦ probability StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); May 29 '17 at 0:17

• 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 0:25
• 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 8:27