Utility of expected income or expected utility of income? Currently I am reviewing microeconomic material related to utility maximization due to an upcoming examination. One old exam question asks me the following for which I am not sure whether to use the utility of expected income or the expected utility of the income. 

Suppose agent A has the utility function $u(y) = y^a (a &gt; 0)$ and the possibility to study and pay university fees $F$. After studying he may get income $y_1$ with probability $p$ or income $y_2 &lt; y_1$ with probability $1 - p$. If he does not study he will have income $y_3 = 0$. When will the agent start studying and what does it depend on? 

My thoughts on this question are as follows. Intuitively, it will highly depend on the fees $F$. On the one hand, the expected income of the agent if he studies is 

$$\mathbb{E}[y_i] = p(y_1 - F) + (1 - p)(y_2 - F) = p(y_1 - y_2) + y_2 - F$$

If he does not study the expected income is just $F$, i.e. the saved fees. 

So the utility of the expected income is just $U(\mathbb{E}[y_i]) = (p(y_1 - y_2) + y_2 - F)^a$ in case he studies or $U(y_3) = F^a$ if he does not. I could just set up an inequality now and say that if 
$$ p(y_1 - y_2) + y_2 &gt; 2F,$$ 
that is, if the expected net income is greater than twice the fees, the agent decides to study. But then I remembered that I might be wrong in looking at the utility of the expected income and that I should rather look into the expected utility. Then, if the agent studies he will have expected utility 
$$ \mathbb{E}[u(y_i)] = p(y_1 - F)^a + (1-p)(y_2 - F)^a = p((y_1 - F)^a - (y_2 - F)^a) + (y_2 - F)^a$$ 

and, if he does not study, 

$$\mathbb{E}[y_3] = F^a $$

Similar to above I now could set up an inequality which is different from the one before. Therefore, I am a bit puzzled as to which utility to use here? The expected utility or the utility of expected income and why?
Your problem has problems with it.  There is no disutility of studying and there is no interest rate or it is zero percent.  However, let's consider the three possible cases.

If $y_2&gt;F$ then for all cases you should study.  If $y_1&lt;F$ then all cases you should not study.  The difficulty happens when $y_2&lt;F$ and $y_1&gt;F$.  Then you should study if $E(U(s))\gg{U}(F)$ and not if $E(U(S))\ll{U}(F)$.  The difficulty with the phrasing of the problem arises when $E(U(s))\approx{U}(F)$.

From the form of your problem, indifference should happen at: $$(y_1^\alpha-y_2^\alpha)p+y_2^\alpha=F^\alpha.$$  It should have been written as: $$(y_1^\alpha-y_2^\alpha)p+y_2^\alpha-U(s=study)=F^\alpha(1+\bar{r})^\alpha.$$  You cannot differentiate over $s$ because it is a binary choice.  In fact, $y_1,y_2$ are really $y_1(s=study)$ and $y_2(s=study)$ and $F$ is really $F(s=\text{don't study})$.

For the form of your problem, you should be indifferent if: $$p=\frac{F^\alpha-y_2^\alpha}{y_1^\alpha-y_2^\alpha}.$$  This permits an objective solution that does not depend upon the person.  For the more expansive problem at the indifference point, the solution depends upon the market rate of interest and the subjective cost of studying.

This implies two important things.  The first is that the value of an education depends on market rates of interest.  Because of this, there should be a decrease in the rate of education when rates are high.  The second is that the indifference point for the probability depends on the specific person.  This implies that probability has a subjective component, even when the probability is external to the person.  Indeed, if the probability is a function of all the choices of all the people, then the probability actually depends upon the marginal rate of disutility experienced in studying.