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
Problem 1QTC
Related questions
Question
PLEASE CHECK THIS HOW TO SOLVE
![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).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, economics and related others by exploring similar questions and additional content below.Recommended textbooks for you
Principles of Economics (12th Edition)
Economics
ISBN:
9780134078779
Author:
Karl E. Case, Ray C. Fair, Sharon E. Oster
Publisher:
PEARSON
Engineering Economy (17th Edition)
Economics
ISBN:
9780134870069
Author:
William G. Sullivan, Elin M. Wicks, C. Patrick Koelling
Publisher:
PEARSON
Principles of Economics (12th Edition)
Economics
ISBN:
9780134078779
Author:
Karl E. Case, Ray C. Fair, Sharon E. Oster
Publisher:
PEARSON
Engineering Economy (17th Edition)
Economics
ISBN:
9780134870069
Author:
William G. Sullivan, Elin M. Wicks, C. Patrick Koelling
Publisher:
PEARSON
Principles of Economics (MindTap Course List)
Economics
ISBN:
9781305585126
Author:
N. Gregory Mankiw
Publisher:
Cengage Learning
Managerial Economics: A Problem Solving Approach
Economics
ISBN:
9781337106665
Author:
Luke M. Froeb, Brian T. McCann, Michael R. Ward, Mike Shor
Publisher:
Cengage Learning
Managerial Economics & Business Strategy (Mcgraw-…
Economics
ISBN:
9781259290619
Author:
Michael Baye, Jeff Prince
Publisher:
McGraw-Hill Education