A consumer has log utility, so u′(C) = 1/C. Next period's consumption will be 100 for sure, or else 50 or 150 with equal probability. Compare expected marginal utility of next-period consumption in the two cases.
AIt is 0.01 in both cases
BIt is 0.01 when certain and about 0.0133 when risky
CIt is higher when certain
DIt cannot be computed without knowing the interest rate
Certain case: 1/100 = 0.01. Risky case: 0.5 × (1/50) + 0.5 × (1/150) = 0.5 × (0.02 + 0.00667) ≈ 0.0133. Higher expected future marginal utility makes saving more valuable, so the consumer saves more today.
Question 2 True / False
With quadratic utility, an increase in future income uncertainty that leaves expected income unchanged has no effect on current consumption.
TTrue
FFalse
Answer: True
Quadratic utility has linear marginal utility, so expected marginal utility depends only on expected consumption. This is called certainty equivalence. Without convex marginal utility there is no precautionary motive.
Question 3 Multiple Choice
Which condition on the utility function produces a precautionary saving motive?
Au′ > 0
Bu″ < 0
Cu‴ > 0
Du″ = 0
A positive third derivative makes marginal utility convex. By Jensen's inequality, spreading out future consumption then raises expected marginal utility, which raises desired saving. Concavity alone (u″ < 0) gives risk aversion but not prudence.
Question 4 Short Answer
Why might precautionary saving help explain why young households with uncertain earnings hold liquid savings even when they expect higher income later?
Think about your answer, then reveal below.
Model answer: The permanent income hypothesis with certainty would have them borrow against future income. But with prudent preferences, uncertainty about earnings raises the value of a cash buffer against bad outcomes, especially if borrowing is limited. They therefore hold a small stock of savings and consume less than a certainty model predicts.
Buffer-stock models by Angus Deaton and Christopher Carroll combine impatience, income risk, and borrowing limits to produce this pattern.