The Economics of an Uncertain World – Expected Utility Theory

My Microeconomics class this year deals with information economics. To be more exact, it looks at decision-making under imperfect and incomplete information in order to be able to explain individual behaviour in our economy as well as aggregate market outcomes.

The first block of the class is devoted to the analysis of decision-making under uncertainty. This is important because the agents in an economy face substantial uncertainty about the future. For example, firms’ future profits depend on prospective GDP/ GDP growth, the prospective interest rate and the prospective rate of inflation. Economic theory does recognise that we live in an uncertain world: Firms, as well as investors and households are generally confronted with ‘nature’ or ‘chance’ when making decisions.

The ‘workhorse’ theory for decision making under uncertainty in Economics is ‘Expected Utility Theory’. Its name already suggests that under uncertainty the rational individual does not maximise her utility but their ‘expected utility’. Expected utility, or EU in short, sounds like an abstract concept but it is essentially the weighted sum of the utilities of the payoffs of the possible outcomes, where the weights correspond to the probabilities of the payoffs (Campbell, 2006).

lottery-a-2

An example should make the concept of EU clearer. Suppose that you have a lottery A with a 50% probability of winning 10,000 and a 50% probability of winning nothing, as shown in the figure above. The expected value (EV) of this lottery is be the average of the payoffs weighted by their corresponding probabilities. Hence the EV of A is 5,000. The EU is actually very similar to this, the only difference being that one weighs the utility of the payoffs and not the payoffs themselves. Therefore, one needs to ‘plug in’ the payoffs into the utility function of an individual first. I have done this for a fictitious individual, say Berta, with the utility function U(w) = w^1/2. So Berta derives 100 ‘utils’ from the payoff of 10,000, and 0 ‘utils’ from the payoff of 0. Having calculated the utility of each payoff, we can now weigh them (that is 100 and 0) by their probabilities (that is 0.5 and 0.5) to get the expected utility of the lottery. It follows that EU(A) = 50, meaning that Berta’s predicted utility value of the lottery is 50 ‘utils’.

Now suppose that Berta could receive a “sure thing” of 5,000 instead of participating in lottery A. This “sure thing” which we offer Berta is equivalent to the average payout, i.e. the ‘expected value’, of the lottery. But what would be Berta’s utility from a “sure thing” paying her 5,000 with certainty? For this we need to ‘plug in’ 5,000 into her utility function, which yields U(5,000) = 50√2. The result should be surprising: lottery A and the “sure thing” have the same expected value of 5,000. Yet Berta derives a higher utility from the “sure thing” than from the lottery.

This apparent puzzle leads us to the concept of risk aversion. Expected utility theory does not only allow us to compute the predicted utility value of lotteries and gambles (as examples for economic decisions with uncertain outcomes); it accounts for the risk preferences of individuals and categorises them as either risk-averse, risk-neutral or risk-seeking. This is important, because people tend to be risk-averse in decisions involving gambling and the like. This feature can also explain why Berta derived a higher utility from the “sure thing” than from the lottery. She preferred the “sure thing” because she has risk-averse preferences. What is more, one can immediately see that Berta is risk-averse when examining her utility function in more detail. In particular, the parameter ‘a’ in the utility function defines an individual’s risk preferences. Because I have chosen a to be 0.5 in Berta’s case, her utility function exhibits diminishing marginal utility of wealth, which is equivalent to saying that she is risk-averse. The decision rule for determining an individual’s risk preferences from her utility function is as follows:

lottery-a-4
Risk Preferences and the Utility Function

But what does diminishing marginal utility of wealth mean exactly? It means is that a one-unit increase in an individual’s wealth yields a higher marginal utility, i.e. a higher increase in utility, at low levels of wealth compared to an equal-sized wealth increase at high levels of wealth. One might think of it like this: a one-unit increase in wealth matters a lot if an individual has no wealth at all but the same one-unit increase is negligible for individuals with a wealth of 1,000,000.

Lastly, I want to look at how risk preferences regarding our hypothetical lottery A can be analysed graphically.

lottery-a-5
Graphical Representation of Risk Preferences in Expected Utility Theory

A risk-averse individual like Berta has a convex utility function. She prefers the sure thing over the gamble. A risk-neutral individual does not care about risk. The utility she derives from the gamble and the sure thing are the same and her utility function is a straight line. A risk-seeking individual has a concave utility function. She prefers the gamble over the sure thing.

In sum, my post today looked the economic theory underlying an uncertain world. The ‘workhorse’ theory for decision making under uncertainty is Expected Utility Theory in which the rational individual maximises her ‘expected utility’. An important feature of this theory is that is allows for both risk aversion and risk loving, depending on the individual’s utility function. However, I would like to stress that there are other approaches for decision making under uncertainty which highlight the drawbacks of expected utility theory; for example, that the ‘utility-function’ is defined over absolute levels wealth rather than gains and losses. The most prominent alternative is prospect theory, formulated by Kahneman and Tversky in 1979.

Thanks for reading!

Jasse

 


References

Campbell, D.E. (2006). Incentives: Motivation and the Economics of Information (2nd ed.). Cambridge: The Cambridge University Press.
Kahneman, D. and Tversky, A. (1979). Prospect Theory: An Analysis of Decision under Risk. Econometrica, 47(2), pp. 263-291.

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