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 -
Chapter1: Making Economics Decisions
Section: Chapter Questions
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![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
T 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 T = 1. For e = 0, any T is optimal. Accordingly, let 7(e) be a strategy
for the government such that 7(e) = 1 whenever e > 0. Given this strategy, the unique
optimal 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 owner
chooses e after observing the government's choice. Find all subgame perfect equilibria
of this game.
Solution: Given T, the optimal choice for the owner is e = 1-7. Hence the government
chooses T to maximize T(1-7), which implies that T = 1/2. Therefore, there is a unique
SPE given by (7* — 1/2, e(7) – 1-T).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa7d1c47f-ebeb-4433-a23d-a68418d8e04e%2Fdf68518a-42cd-4f1f-9efe-5751377785ba%2Fbhj4t9_processed.png&w=3840&q=75)
Transcribed Image Text: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
T 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 T = 1. For e = 0, any T is optimal. Accordingly, let 7(e) be a strategy
for the government such that 7(e) = 1 whenever e > 0. Given this strategy, the unique
optimal 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 owner
chooses e after observing the government's choice. Find all subgame perfect equilibria
of this game.
Solution: Given T, the optimal choice for the owner is e = 1-7. Hence the government
chooses T to maximize T(1-7), which implies that T = 1/2. Therefore, there is a unique
SPE given by (7* — 1/2, e(7) – 1-T).
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