Why can equal expected wealth deliver different utility under risk?
Average utility across states because curvature captures risk attitudes.
Follow the cause and effect
Calculate u(Ws) in every state.
Weight by ps to obtain EU rather than substituting u(EW).
Solve u(CE)=EU and compute EW−CE.
The model you are using
KEY MODEL
EU=Σpₛu(Wₛ)
u(CE)=EU
risk premium=E(W)−CE
Conditions for this model
u(W)=√W, W>0, with known probabilities summing to one.
The exam trap
Do not average wealth before applying utility; generally E[u(W)]≠u(E[W]).
Connect the reasoning to the graph
What changes
- Concavity of u(W) places expected utility below utility at expected wealth by Jensen’s inequality.
What stays fixed
u(W)=√W, W>0, with known probabilities summing to one.
Keep these conditions throughout the comparison; change only what the case above specifies.
What to inspect
Solve u(CE)=EU and compute EW−CE.
Compare before and after, and locate the conclusion on the graph.
5 MINUTES · TRANSFER THE IDEA
Can you explain it without the lesson?
Compare certainty equivalents for risk-neutral and risk-averse people facing the same lottery.