PLEASE CHECK THIS HOW TO SOLVE A small business owner can invest any amount of effort e ≥ 0 to produce output valued at2e. The cost to the owner of effort level e is e². The government, which seeks to maximizeits own revenue, may expropriate some fraction 7 € [0, 1] of the owner's output. The owner'spayoff to choosing e if the government expropriates fraction 7 is 2e(1-7) – e².(a) Consider a sequential game in which the owner first chooses e and the government choosesT after observing the owner's choice. Find all subgame perfect equilibria of this game.Solution: Use backward induction. For each e > 0, the unique optimal choice for thegovernment is T = 1. For e = 0, any T is optimal. Accordingly, let 7(e) be a strategyfor the government such that 7(e) = 1 whenever e > 0. Given this strategy, the uniqueoptimal choice for the owner is e = 0. Therefore, the SPE are the strategy profiles(e* = 0, 7(e)) where 7(e) = 1 for all e > 0.(b) Now consider a sequential game in which the government first chooses 7 and the ownerchooses e after observing the government's choice. Find all subgame perfect equilibriaof this game.Solution: Given T, the optimal choice for the owner is e = 1-7. Hence the governmentchooses T to maximize T(1-7), which implies that T = 1/2. Therefore, there is a uniqueSPE given by (7* — 1/2, e(7) – 1-T).

Value of product = 2e Cost of production = e2 Government expropriates a fraction = 2e (1-t ) - e2 In…

A small business owner can invest any amount of effort e ≥ 0 to produce output valued at 2e. The cost to the owner of effort level e is e². The government, which seeks to maximize its own revenue, may expropriate some fraction 7 € [0, 1] of the owner's output. The owner's payoff to choosing e if the government expropriates fraction 7 is 2e(1-7) - e². (a) Consider a sequential game in which the owner first chooses e and the government chooses 7 after observing the owner's choice. Find all subgame perfect equilibria of this game. Solution: Use backward induction. For each e > 0, the unique optimal choice for the government is I = 1. For e= = 0 any I is antimal Accordingly let T(e) be a strateau -

A small business owner can invest any amount of effort e ≥ 0 to produce output valued at 2e. The cost to the owner of effort level e is e². The government, which seeks to maximize its own revenue, may expropriate some fraction 7 € [0, 1] of the owner's output. The owner's payoff to choosing e if the government expropriates fraction 7 is 2e(1-7) - e². (a) Consider a sequential game in which the owner first chooses e and the government chooses 7 after observing the owner's choice. Find all subgame perfect equilibria of this game. Solution: Use backward induction. For each e > 0, the unique optimal choice for the government is I = 1. For e= = 0 any I is antimal Accordingly let T(e) be a strateau -

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter7: Uncertainty

Section: Chapter Questions

Problem 7.7P

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