ECON90034 Chap.3 Expected Utility and Risk Aversion
Expected Utility and Risk Aversion
Choice under uncertainty begins with lotteries: outcomes with probabilities. Expected value and variance describe them, but decisions depend on a Bernoulli utility function over wealth and on expected utility. The shape of that function sets the attitude to risk: concave for risk aversion, linear for neutrality and convex for risk loving.
The subject measures risk aversion in four ways, the two Arrow-Pratt coefficients, absolute and relative, the certainty equivalent and the risk premium, and expects you to remember the formulas and compute them for simple power functions. It closes with the empirical evidence, including the equity premium puzzle, where observed share returns imply far more risk aversion than experiments find.
Every formula here is worked on a fresh lottery, and the chapter ends with the two framing traps the subject itself warns about.
What this chapter covers
- 01
Expected value and variance of a lottery
- 02
Expected utility with a Bernoulli utility function
- 03
Risk-averse, risk-neutral and risk-loving behaviour and the shape of utility
- 04
Absolute and relative risk aversion: formulas and calculation
- 05
Certainty equivalent and risk premium, with and without initial wealth
- 06
Insurance decisions and the most a person would pay
- 07
Comparing two people's risk aversion
- 08
Empirical evidence and the equity premium puzzle
Worked example · free
Risk aversion with logarithmic utility
- 1u' = 1/w and u'' = −1/w2, so rA = 1/w, which falls as wealth rises.
- 1rR = w × rA = 1, constant relative risk aversion.
- 1EU = 0.5 ln 100 + 0.5 ln 400 = ln √(100 × 400) = ln 200, so CE = 200.
- 1Expected wealth is 250, so the risk premium is 250 − 200 = 50.
Key terms
- Bernoulli utility function
- A utility function defined over wealth whose expected value ranks lotteries; its curvature captures the person's attitude to risk.
- Absolute risk aversion
- The coefficient minus u'' over u', a measure of how curved utility is at a given wealth; a larger value means a stronger dislike of a fixed-size gamble.
- Relative risk aversion
- Wealth times the coefficient of absolute risk aversion; it measures dislike of gambles that are a fixed share of wealth.
- Risk premium
- Expected value minus certainty equivalent: the most a risk-averse person would pay to swap a lottery for its expected value for sure.
- Equity premium puzzle
- The finding that the average return of shares over the risk-free rate is too large to be explained by the degree of risk aversion measured in experiments.
- Fair gamble
- A gamble whose expected gain is zero; a risk-averse person rejects it, a risk-neutral person is indifferent, and a risk lover accepts it.
Expected Utility and Risk Aversion FAQ
How do you calculate the coefficient of absolute risk aversion?
Differentiate the utility function twice, then take minus the second derivative divided by the first. For u(w) = w to the power one half, this gives 1/(2w), which is positive and falls as wealth grows.
What is the difference between the certainty equivalent and the risk premium?
The certainty equivalent is the sure amount worth the same as the lottery; the risk premium is the expected value minus that amount. A more risk-averse person has a lower certainty equivalent and a larger premium.
When should I solve u(w0 + CE) = EU instead of u(CE) = EU?
Use the first when the question gives initial wealth and the lottery is a gain or loss added to it; the certainty equivalent is then a change in wealth. Use the second when the lottery is defined over final wealth.
What is the equity premium puzzle?
Shares have earned far more than the risk-free rate, and explaining that gap with a standard model requires relative risk aversion around 50, while experiments suggest values between about 3 and 10.
Why should absolute risk aversion fall as wealth rises?
A wealthier person can absorb a fixed loss of, say, a thousand dollars more easily, so they should be more willing to take the same gamble. Utility functions whose coefficient rises with wealth, such as quadratic utility, are therefore regarded as implausible even though they are concave.
Exam move
Memorise the two coefficients and the two certainty-equivalent equations, and practise them on square-root, cube-root, logarithmic and quadratic utility until each takes under two minutes. For every lottery, compute expected value, expected utility, certainty equivalent and premium in that order, keeping expected utility unrounded.
Draw the concave curve with the chord for at least one problem a week, labelling expected wealth, the certainty equivalent and the premium, because the exam rewards a labelled graph. Practise the comparison questions too: given two people's certainty equivalents or coefficients, state who is more risk averse and whose premium is larger.
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