Utility of expected income or expected utility of income?

Question

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 $u(y) = y^a (a > 0)$ and the possibility to study and pay university fees $F$. After studying he may get income $y_1$ with probability $p$ or income $y_2 < y_1$ 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 $$ p(y_1 - y_2) + y_2 > 2F,$$ 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?

Answer 1

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 $y_2>F$ then for all cases you should study. If $y_1<F$ then all cases you should not study. The difficulty happens when $y_2<F$ and $y_1>F$. 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.

Answer 2

Let $w$ denote the existing wealth of agent A. Agent has an option to attain higher education by paying $F$ as fees and in return can earn $y_1$ with probability $p$ and $y_2$ with probability $1-p$, where $y_1>y_2$.

So, the choice is between saying yes or no to higher education.

If he says yes to higher education, his wealth will be $w - F + y_1$ with probability $p$ and $w - F + y_2$ with probability $1-p$. If he says no, then his wealth remains at $w$.

He will say yes if his expected utility from higher education is more i.e.

$p(w - F + y_1)^a + (1-p)(w - F + y_2)^{a} > w^a$

Answer 3

The community bot bumped this to front page, so let's turn my comments into an answer.

The OP makes a mistake in formulating the problem, as follows: For the "not study" case, he says that the agent will have utility from "the saved fees", hence his last-last equation

$$\mathbb{E}[u(\text{wealth if not-study})] = (F+0)^a= F^a \tag{1} $$

It follows that currently the agent has wealth $F$ (otherwise how could he "save" the fees...) This comes for a "state-of-the-workd" approach.

But when considering what will happen to the agent if it studies, the OP adopts a "income-flow" approach. It writes

$$\mathbb{E}[u(\text{wealth if not-study})] = p(y_1 - F)^a + (1-p)(y_2 - F)^a \tag{2}$$

...ignoring the fact that $F$ is already available. If we wanted to describe a different situation, where the agent has no wealth, and it contemplates whether it would be beneficial to study and pay the tuition fees out of his future income, then we should use eq. $(2)$, but together not with eq. $(1)$ but with $\mathbb{E}[u(\text{wealth if not-study})] = 0$.

In Expected Utility Theory we use the "state-of-the world" approach. So the correct equation to go together with eq. $(1)$ (which implies that the agent has already and so certain wealth $F$) is

$$\mathbb{E}[u(\text{wealth if study})] = p(F+y_1 - F)^a + (1-p)(F+y_2 - F)^a \\= py_1^a +(1-p)y_2^a \tag{3}$$

Note what we are doing here. In each of the two possible states of the wrold given that the agent will study, the agent starts with wealth $F$ which he will sacrifice in order to study, and gain income $y_1$ or $y_2$.

Regarding the question

I am a bit puzzled as to which utility to use here? The expected utility or the utility of expected income and why?

the answer is "expected utility of the different states of the world". It is in this way that we internalize the epxeriencing of uncertainty at utility level, and the effects of uncertainty on utility. If we were to compare utility of epxected wealth, it would be the "utility of the average situation", not "average utility from the uncertain situation". But by averaging first, we would not let the uncertainty interact directly with utility.The difference vanishes only if utility is linear in wealth (and we have a "risk-neutral" person).