Uncertainty and Risk


Lotteries

Generally speaking, we talk about preferences over certain quantities of goods, or amounts of money. But the world isn’t a certain place: chance determines a lot of outcomes.

Some chance we bring upon ourselves: we play the lottery, or we bet on the outcome of sports games, or we invest in a stock that could go up or down. Some chance occurrences are called “acts of God” - whether you get into a car accident, or your house burns down due to a freak accident.

A great deal of economic activity is centered around transferring risk from one person to another. Insurance contracts pay you money if something bad happens to you; hedge funds invest in securities that are negatively correlated with one another. Understanding how preferences over risk drive these markets is one of the critical tasks of modern economic theory.

To analyze situations like this, let’s define a “lottery” as a set of possible outcomes, each occurring with a certain probability. For example, suppose we bet $\$150$ on a coin toss; heads I win, tails you win. There are two possible outcomes: the coin could come up heads, and you would lose $\$150$, or it could come up tails, and you could win $\$150$. Each occurs with probability $\frac{1}{2}$.

Suppose, as you consider whether to take this bet, you have $\$250$ in your pocket. Therefore, from your perspective, this lottery would give you an outcome of $c_1 = \$100$ if the coin comes up heads, and $c_2 = \$400$ if it comes up tails. Of course, you could reject the bet, and have $c_1 = c_2 = \$250$ regardless of whether the coin comes up heads or tails.

We can picture this lottery in “good 1 - good 2 space,” where “good 1” (or “state 1”) is consumption in the state of the world in which the coin comes up heads (written $c_1$), and “good 2” (or “state 2”) is consumption in the state of the world in which the coin comes up tails (written $c_2$).

Should you take the bet? It depends on your preferences. We can draw indifference curves through the point $(100,400)$. If the “don’t bet” point is preferred, then you shouldn’t take the be; on the other hand, if the “bet” point is preferred, you should take the bet:

Let’s think about this another way. Let’s assume that the way you feel about money doesn’t depend on whether you win the bet or not: that is, you have some utility function $u(c)$ which says how much utility you get from having $c$ dollars, and that this function is independent of the state of the world.

If you do not take the bet, therefore, your utility is just $u(250)$. If you win the bet, your utility would be $u(400)$; if you lose, your utility will be $u(100)$.

Given this framework, your utility gain from winning the bet is $\textcolor{#31a354}{u(400) - u(250)}$, and your utility loss from losing the bet would be $\textcolor{#d62728}{u(250) - u(100)}$. Since each of the two outcomes is equally likely, you should therefore take the bet if

$$\textcolor{#31a354}{u(400) - u(250)} > \textcolor{#d62728}{u(250) - u(100)}$$

Let’s see when this is the case. The following diagram shows a particular kind of utility function where $u(c) = c^r$. The horizontal axis shows consumption, in dollars; the vertical axis shows utility, in “utils.” Initially it shows the case where $r = 0.5$, but you can use the slider to change $r$ to be anything from 0.25 to 2.

As you can see, for this particular case, it’s better to take the bet if $r > 1$, and better not to take the bet if $r < 1$. In order to think about this more generally, though, we need to introduce the notion of expected utility.

Next: Expected Utility

Copyright (c) Christopher Makler / econgraphs.org