Uncertainty and Risk


Expected Utility and Risk Aversion

Recall from probability theory that if you have a random variable that takes on different possible values, the expected value of that variable is the weighted average of those values, where the weights are the probability of each value occurring.

For example, if $x = 16$ with probability $\frac{3}{4}$ and $x = 64$ with probability $\frac{1}{4}$, the expected value of $x$ is

$$\mathbb{E}[x] = \frac{3}{4} \times 16 + \frac{1}{4} \times 64 = 28$$

More generally, if we think about a lottery in which an agent has $c_1$ dollars with probability $\pi$ and $c_2$ dollars with probability $1 - \pi$, their expected consumption is

$$\mathbb{E}[c] = \pi c_1 + (1-\pi) c_2$$

The same logic may apply to their utility: that is, if we assume that they are perfectly rational people whose utility is the expected value of the within-state utility function $u(c)$ over all states of the world, then their expected utility is

$$\mathbb{E}[u(c)] = \pi u(c_1) + (1 - \pi) u(c_2)$$

Visually, if we plot the points $(c_1, u(c_1))$ and $(c_2, u(c_2))$, the point $(\mathbb{E}[c], \mathbb{E}[u(c)])$ lies fraction $\pi$ of the way along a line connecting those two points:

Notice that when $r < 1$, the line connecting $(c_1, u(c_1))$ and $(c_2, u(c_2))$ lies below the utility curve. In other words, the utility of consuming one’s expected consumption, $u(\mathbb{E}[c])$, is greater than the expected utility $\mathbb{E}[u(c)]$. The opposite is true when $r > 1$; and when $r = 1$, the consumer is indifferent between the lottery and the expected result of the lottery.

This leads to our formal definition of risk aversion: given a choice between facing a lottery (e.g., consume $c_1$ with probability $\pi$ and $c_2$ with probability $1-\pi$) and having the expected consumption from the lottery for sure (e.g., consume $\pi c_1 + (1-\pi) c_2$ with certainty):

Visually, you can see this in the following diagram. Notice that the height of the purple dot is the utility from consuming the expected value of the lottery for sure – that is, $u(\mathbb{E}[c])$. The height of the orange dot is the expected utility of the lottery, $\mathbb{E}[u(c)]$. When the purple dot is higher, the consumer is risk averse; when the orange dot is higher, the consumer is risk loving. Change $r$ to see how the curvature of the utility function affects the risk aversion of the consumer:

Another way we can think about these kinds of preferences is to relate this graph to our usual “good 1 - good 2” space. The lottery $(c_1,c_2)$ is a point in this space. We can also plot the point $(\mathbb{E}[c],\mathbb{E}[c])$ – i.e., a point that would represent consuming the expected value of the lottery in both states of the world. If a consumer is risk averse, they prefer to consume $\mathbb{E}[u(c)]$ for sure than to face the lottery, so the point $(\mathbb{E}[u(c)],\mathbb{E}[u(c)])$ lies on a higher indifference curve:

Now that we have a good sense of what we mean by preferences over risk, let’s look at some of the ways consumers might try to improve their lot by paying to reduce their risk: that is, to move closer to consuming $c_1 = c_2$.

Next: Certainty Equivalence and Risk Premia

Copyright (c) Christopher Makler / econgraphs.org