ECOS2001 · Intermediate Microeconomics
Choices under Uncertainty & Risk
This is the first topic on the 50% final. Choices over risky outcomes are ranked by expected utility, EU = Σ πᵢ·U(cᵢ), not by expected money value. The curvature of U reveals the attitude to risk: a concave U (U″ < 0) means risk-averse, with the certainty equivalent below the expected value and a positive risk premium; a linear U is risk-neutral (CE = EV); a convex U is risk-loving. The certainty equivalent solves U(CE) = EU, and the risk premium is EV − CE. Insurance applies the same ideas: at an actuarially fair premium (= probability × loss) a risk-averse person fully insures, ending with certain wealth.
What this chapter covers
- 01Expected value EV vs expected utility EU = Σ πᵢU(cᵢ)
- 02Risk attitudes from curvature: averse U″ < 0, neutral U″ = 0, loving U″ > 0
- 03Certainty equivalent U(CE) = EU and risk premium = EV − CE
- 04Actuarially fair premium = probability × loss
- 05A risk-averse agent fully insures at a fair premium (certain wealth)
- 06State-contingent consumption and the insurance optimum
Expected utility, certainty equivalent and risk premium
- 1 markU(W) = √W is concave (U″ < 0), so the investor is risk-averse.
- 2 marksExpected value: EV = 0.4·1600 + 0.6·400 = 640 + 240 = 880.
- 2 marksExpected utility: EU = 0.4·√1600 + 0.6·√400 = 0.4·40 + 0.6·20 = 16 + 12 = 28.
- 2 marksCertainty equivalent solves U(CE) = EU, i.e. √CE = 28, so CE = 28² = 784.
- 1 markRisk premium = EV − CE = 880 − 784 = 96.
Key terms
- Expected utility (EU)
- The probability-weighted average of utility across outcomes, EU = Σ πᵢU(cᵢ); risky prospects are ranked by EU, not by expected money value.
- Certainty equivalent (CE)
- The sure amount that gives the same utility as a gamble, defined by U(CE) = EU; for a risk-averse person CE < EV.
- Risk premium
- The gap EV − CE: how much expected value a person will sacrifice to swap the gamble for a sure thing. Positive for risk-averse agents.
- Actuarially fair premium
- A premium equal to the expected loss (probability × loss), at which the insurer expects zero profit; a risk-averse consumer fully insures when offered it.
Choices under Uncertainty & Risk FAQ
Why does a risk-averse person fully insure at a fair premium?
Because a fair premium lets them trade risky wealth for certain wealth at no expected cost, and a concave utility always prefers the sure equivalent of a gamble. Equalising wealth across states (full insurance) maximises expected utility when the odds are fair.
How do I read risk attitude from the utility function?
Look at the second derivative (curvature). U″ < 0 (concave, like √W or ln W) is risk-averse with CE < EV; U″ = 0 (linear) is risk-neutral with CE = EV; U″ > 0 (convex) is risk-loving with CE > EV.
Exam move
Drill the four-number routine — EV, EU, CE, risk premium — for both √W and ln W utilities, and always invert the utility to get the CE in dollars. Be ready to explain in one line why a concave utility implies full insurance at fair odds.
