Choice under Uncertainty

The situations we have discussed up to this point have all lacked any elements of uncertainty. Individuals and firms have made their choices knowing what the outcomes would be. That is, of course, very unrealistic. Most of the time, we cannot be certain about which consequences our actions will have, even though we can perhaps know which consequences they will probably have.
A few examples of important decisions under uncertainty are:

• Buying a house or an apartment. You know what you pay for it, but what will it be worth when you sell it? What happens if the house burns down?
• Investments in an education. It is often easy to get statistics on today’s salaries but in the future, they might change substantially.
• A firm invests in a new factory. Will the goods produced in the factory still be in demand in the future? Think, for instance, about the computer market where development is very rapid.

It is common in economics to view uncertainty as a sort of lottery. In a lottery one often knows which outcomes are possible: There might, for instance be a list showing how much you can win. It is also possible to calculate the probabilities for the different outcomes.

Expected Value

In statistics, “expected value” is a technical term. Suppose we toss a coin. If “heads” comes up, we win 5; if “tails” comes up we lose 5. The expected value of that lottery is then

Here, Pr(.) is the probability that the event within the parentheses will occur. If there are more than two possible outcomes, the expected value is the probability of each outcome multiplied with the value of that outcome, and then summed together. Note that the expected value need not to be something you outcome to be 0. We expect it to be either +5 or -5. can be seen as a sort of average over the outcomes, where an outcome with high probability has a higher weight than one with lower probability. A lottery with an expected value of zero is called a fair lottery. Note that most real-world lotteries are not fair.

Expected Utility

Consider a case where an agent has to choose between several alternatives, all of which will lead to an uncertain outcome. A naïve method could then be to analyze them as if they were lotteries, and then choose the one with the highest expected value. There are several reasons why such a method would not be a good one.

To begin, we need to define a utility function (compare to Section 3.8) over wealth. Usually, more wealth is better for an individual but as she becomes wealthier additional wealth matters less and less. The utility an individual has of wealth is often written as

U stands for utility and W for wealth. The expression can then be read as “the utility level is a function of wealth.” What form the utility function takes varies between individuals, but a function that is often used for illustrations is

In Figure 6.1, we have drawn this function (sqrt = “the square root”). Note that the slope of the function becomes less and less steep. That means that the individual, as we just noted, receives less utility of extra wealth as she becomes wealthier. Note that how much extra utility the individual gets from a small increase in wealth corresponds to the slope of the utility function. The slope is called the marginal utility, MU.

We differentiate between three different kinds of utility functions

• Diminishing marginal utility. The slope decreases with increased wealth, such as the utility function in Figure 6.1.
• Constant marginal utility. The utility function is straight line, i.e. the slope is constant.
• Increasing marginal utility. The slope increases with increased wealth.

If we use the utility function together with expected value, we can calculate the expected utility. Suppose our wealth is 5. If we participate in the lottery above, we will either win 5 or lose 5, both with a 50% probability. The result will then be that we have either 0 or 10. The utility we would have of these outcomes, and the expected utility is then

E(U) stands for expected utility. Earlier, we calculated the expected value of the lottery to 0. Our expected wealth, as opposed to expected utility, is then 5 + 0 = 5 (the sum of what we have plus the expected value of the lottery). Note now that, the utility we have from a certain wealth of 5 is

The utility from getting the expected value with certainty (the utility of 5) is usually higher than the expected utility of participating in a fair lottery (the utility of either 0 or 10, both with a 50% probability).

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