Expected Utility vs Certainty Equivalent
Expected Utility and Certainty Equivalent are two Microeconomic Theory concepts in AP Economics that students often mix up. Expected utility is the probability-weighted average of the utility of each possible outcome, used to rank risky choices. The certainty equivalent is the guaranteed amount of money that gives a person the same utility as a risky gamble. Here is how they compare side by side.
Expected utility ranks risky options by first converting each outcome into utility and then averaging those utilities using the probabilities. The order matters: you take the utility of each outcome and weight it, rather than taking the utility of the average outcome. For a risk-averse person with a concave utility curve, the expected utility of a gamble is below the utility of its expected value, which is exactly what makes the sure thing more attractive. Expected value is the simpler cousin that averages the dollar payoffs and ignores attitudes toward risk, so it treats a coin flip for $1,000 and a sure $500 as identical. A decision maker who maximizes expected utility picks the option with the highest weighted utility, not the highest average payout.
The certainty equivalent converts a gamble into a single sure dollar figure the decision maker would accept in its place. You find it by computing the gamble's expected utility, then asking what guaranteed wealth produces that same utility level, which means inverting the utility function. For a risk-averse person the certainty equivalent is below the gamble's expected value, and the difference is the risk premium. A risk-neutral person's certainty equivalent equals the expected value exactly, and a risk-loving person's sits above it. The measure is useful because it puts risky and safe options on one scale in dollars, which is how firms compare an uncertain project against a guaranteed contract.
Expected Utility vs Certainty Equivalent: Utils to Rank, Dollars to Compare
| Expected Utility | Certainty Equivalent | |
|---|---|---|
| What it computes | The probability weighted average of utility across outcomes | The safe amount of money worth as much as the gamble |
| Units of the answer | Utils, which have no natural scale | Dollars |
| How it is found | Multiply each outcome's utility by its probability and add | Solve the utility function backwards until it returns the expected utility |
| What it is used for | Ranking two risky options against each other | Putting a price on a single risky option |
| Comparable between people | No, because each person's utility scale is arbitrary | Yes, because the answer is stated in money |
| Role in the risk premium | Not used directly, since the premium is not measured in utils | Subtracted from expected value to give the premium |
Work one gamble all the way through and the two numbers separate cleanly
Suppose someone's utility of wealth is the square root of wealth, a shape that gives diminishing marginal utility and therefore risk aversion. Offer them a coin flip paying 900 dollars on heads and 2,500 dollars on tails. Start with expected value, which is the plain money average: half of 900 plus half of 2,500 is 1,700 dollars. Expected utility is the average of the utilities instead: the square root of 900 is 30, the square root of 2,500 is 50, and half of each gives 40 utils. Now find the certainty equivalent by asking what safe amount delivers 40 utils. Since the square root of 1,600 is 40, the certainty equivalent is 1,600 dollars. Read those three figures together. The gamble is worth 1,700 dollars on average but this person would swap it for a guaranteed 1,600, so the 100 dollar gap is the risk premium, the amount they will pay to be rid of the uncertainty. Notice that 40 is not comparable to 1,700 in any way; one is in utils and the other in dollars. See /calculate/expected-value. The figures are illustrative.
Two gambles with the same certainty equivalent need not have the same expected value
Keep the same square root utility and offer a second coin flip, this time paying 1,296 dollars or 1,936 dollars. The square roots are 36 and 44, so expected utility is again 40 utils and the certainty equivalent is again 1,600 dollars. This person is exactly indifferent between the two gambles. Their expected values, though, are not equal: the second averages 1,616 dollars against the first gamble's 1,700 dollars. The second gamble is worth less money on average and yet is equally attractive, because its outcomes are bunched close together and carry a risk premium of only 16 dollars rather than 100. That is the whole content of risk aversion stated in numbers. It also shows why the two measures on this page do different jobs. Expected utility ranks options and here calls them a tie. The certainty equivalent turns each ranking into a dollar figure you could put in a contract, quote as a selling price, or compare against an insurance premium. See /glossary/risk-aversion for the curvature that produces all of this.
Frequently asked questions
What is the difference between expected utility and the certainty equivalent?
Expected utility is the average satisfaction from a gamble measured in utils, while the certainty equivalent is the guaranteed sum of money that would give exactly that same satisfaction. One is a ranking number and the other is a dollar figure.
How do you find the certainty equivalent?
Compute expected utility first, then invert the utility function to find the wealth level that produces it. Under square root utility that inversion is just squaring, so a gamble worth 30 utils has a certainty equivalent of 900 dollars.
What is the risk premium?
It is the expected value of a gamble minus its certainty equivalent, which measures how much money a person will give up to avoid the uncertainty. A risk averse person has a positive risk premium, and it grows as the outcomes spread further apart.
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