Economics 401 Problem Set 9 Answer Key Prof. David A. Miller

Economics 401 Problem Set 9 Answer Key Prof. David A. Miller Tis version: November 6, 2021 1 Choice under uncertainty 1.1 Insurance Emily faces an unavoidable lottery of having income mG = M 1 in state G and income mB = 1 in state B, where πG is the probability of state G and πB = 1 − πG is the probability of state B. She has an expected utility preference with a state utility function of the following form in each state: v(c) = ln(c) Let cG be her consumption in state G and cB be her consumption in state B. We will not prove it here, but Emily is risk averse, since her state utility function v is strictly concave. a) What is Emily’s expected utility function? Answer: u(cG, cB) = πG ln(cG) + πB ln(cB) Suppose Emily is ofered insurance. Tis means an insurance company ofers to pay her a coverage amount I in state B, in return for a premium payment of γI that she must pay in both states. Assume that 0 γ 1. Ten Emily will consume cG = M − γI in state G and cB = 1 + (1 − γ)I in state B. Emily’s budget constraint across the two states is (1 − γ)cG + γcB = (1 − γ)M + γ 1 b) Suppose the insurance is actuarially fair, meaning γ = πB. Prove that Emily purchases full insurance, i.e., that 1 + I = M. Answer: Emily chooses I to set the marginal rate of substitution across states equal to the efective price ratio across states. MRS = ∂u/∂cG ∂u/∂cB = πGcB πBcG = 1