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) - ².

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Chapter1: Making Economics Decisions
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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
7 after observing the owner's choice. Find all subgame perfect equilibria of this game.
(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.
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 7 after observing the owner's choice. Find all subgame perfect equilibria of this game. (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.
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