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Amit has von Neumann-Morgenstern utility u(x)=√ over levels of wealth r. He has 4 lollars initially, but faces a 50 percent chance of losing all 4 dollars. Barbara offers to sell nsurance to Amit. If Amit purchases q units of insurance at a price of p dollars per unit, hen Barbara will pay him q dollars if the loss occurs. Barbara is risk neutral and cares only ■bout maximizing her profit from the sale of insurance. (a) Find Amit's optimal demand for insurance for each price p > 0. Assume that Amit cannot sell insurance, so that he must choose q ≥ 0. Solution: The optimal demand is 4(1-P) if p≤ 1 and 0 otherwise. (b) Suppose that trade is determined according to the following sequential game. Barbara first chooses the price p>0 at which she offers insurance to Amit. Amit then chooses the quantity q> 0 of insurance to buy at the offered price. Find a subgame perfect equilibrium of this game. Solution: Using backward induction, the optimal strategy for Amit is to use the demand from part (a). Given this strategy, Barbara's expected profit is marimized by choosing p=1/√2. Hence the SPE is (p=1/√2.q(p) = max{4(1-p)/p,0}). (e) Now suppose instead that first Amit offers a price p>0 and quantity q 20 of insurance to buy. After hearing his offer, Barbara chooses whether to accept it or reject it. If she

Amit has von Neumann-Morgenstern utility u(x)=√ over levels of wealth r. He has 4 lollars initially, but faces a 50 percent chance of losing all 4 dollars. Barbara offers to sell nsurance to Amit. If Amit purchases q units of insurance at a price of p dollars per unit, hen Barbara will pay him q dollars if the loss occurs. Barbara is risk neutral and cares only ■bout maximizing her profit from the sale of insurance. (a) Find Amit's optimal demand for insurance for each price p > 0. Assume that Amit cannot sell insurance, so that he must choose q ≥ 0. Solution: The optimal demand is 4(1-P) if p≤ 1 and 0 otherwise. (b) Suppose that trade is determined according to the following sequential game. Barbara first chooses the price p>0 at which she offers insurance to Amit. Amit then chooses the quantity q> 0 of insurance to buy at the offered price. Find a subgame perfect equilibrium of this game. Solution: Using backward induction, the optimal strategy for Amit is to use the demand from part (a). Given this strategy, Barbara's expected profit is marimized by choosing p=1/√2. Hence the SPE is (p=1/√2.q(p) = max{4(1-p)/p,0}). (e) Now suppose instead that first Amit offers a price p>0 and quantity q 20 of insurance to buy. After hearing his offer, Barbara chooses whether to accept it or reject it. If she

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Amit has von Neumann-Morgenstern utility u(x)=√ over levels of wealth r. He has 4 dollars initially, but faces a 50 percent chance of losing all 4 dollars. Barbara offers to sell insurance to Amit. If Amit purchases q units of insurance at a price of p dollars per unit, then Barbara will pay him q dollars if the loss occurs. Barbara is risk neutral and cares only about maximizing her profit from the sale of insurance. (a) Find Amit's optimal demand for insurance for each price p > 0. Assume that Amit cannot sell insurance, so that he must choose q ≥ 0. Solution: The optimal demand is 4(1-P) if p ≤ 1 and 0 otherwise. P (b) Suppose that trade is determined according to the following sequential game. Barbara first chooses the price p > 0 at which she offers insurance to Amit. Amit then chooses the quantity q ≥ 0 of insurance to buy at the offered price. Find a subgame perfect equilibrium of this game. Solution: Using backward induction, the optimal strategy for Amit is to use the demand from part (a). Given this strategy, Barbara's expected profit is maximized by choosing p=1/√2. Hence the SPE is (p=1/√2,q(p) = max{4(1-p)/p,0}). (c) Now suppose instead that first Amit offers a price p>0 and quantity q≥ 0 of insurance to buy. After hearing his offer, Barbara chooses whether to accept it or reject it. If she accepts the offer, then the proposed trade occurs, and if she rejects it then there is no trade. Find a subgame perfect equilibrium of this game (Note: you only have to find one equilibrium, not every equilibrium). Solution: Barbara is willing to accept any offer with p≥ 0.5. Given this, the optimal offer for Amit is p = 0.5 and q = 4. One SPE is ((p = 0.5,q = 4), accept if p > 0.5). (d) For the equilibria you found in parts (a) and (b), explain in each case hether the outcome is efficient. Solution: The outcome is efficient if and only if q = 4. This is the case for the outcome in (c) but not for the outcome in (b).
Transcribed Image Text:Amit has von Neumann-Morgenstern utility u(x)=√ over levels of wealth r. He has 4 dollars initially, but faces a 50 percent chance of losing all 4 dollars. Barbara offers to sell insurance to Amit. If Amit purchases q units of insurance at a price of p dollars per unit, then Barbara will pay him q dollars if the loss occurs. Barbara is risk neutral and cares only about maximizing her profit from the sale of insurance. (a) Find Amit's optimal demand for insurance for each price p > 0. Assume that Amit cannot sell insurance, so that he must choose q ≥ 0. Solution: The optimal demand is 4(1-P) if p ≤ 1 and 0 otherwise. P (b) Suppose that trade is determined according to the following sequential game. Barbara first chooses the price p > 0 at which she offers insurance to Amit. Amit then chooses the quantity q ≥ 0 of insurance to buy at the offered price. Find a subgame perfect equilibrium of this game. Solution: Using backward induction, the optimal strategy for Amit is to use the demand from part (a). Given this strategy, Barbara's expected profit is maximized by choosing p=1/√2. Hence the SPE is (p=1/√2,q(p) = max{4(1-p)/p,0}). (c) Now suppose instead that first Amit offers a price p>0 and quantity q≥ 0 of insurance to buy. After hearing his offer, Barbara chooses whether to accept it or reject it. If she accepts the offer, then the proposed trade occurs, and if she rejects it then there is no trade. Find a subgame perfect equilibrium of this game (Note: you only have to find one equilibrium, not every equilibrium). Solution: Barbara is willing to accept any offer with p≥ 0.5. Given this, the optimal offer for Amit is p = 0.5 and q = 4. One SPE is ((p = 0.5,q = 4), accept if p > 0.5). (d) For the equilibria you found in parts (a) and (b), explain in each case hether the outcome is efficient. Solution: The outcome is efficient if and only if q = 4. This is the case for the outcome in (c) but not for the outcome in (b).
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