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1. Consider a setting with two workers į 1, 2 who choose how much effort to exert on their individual tasks. Suppose that the effort choice is binary with ej = {0, 1}. The probability of task completion is 1 if the worker exerts ei 1 and p G (0, 1) if ei 0. The cost of exerting effort ei 1 is c> 0, while ei — 0 is costless. Both tasks need to be completed in order for the entire project to be realized. Completing only one of the tasks is useless to the hiring firm. The objective of the firm is to ensure the project completion at the lowest possible cost. Suppose that the firm cannot observe the completion of the individual tasks, but only whether the entire project is realized. The firm commits to public bonus payments to the workers if the project is successfully completed. That is, worker į receives a bonus bi if the project is a success and 0 otherwise. Such a bonus contract induces a simultaneous move game between the two workers who independently decide whether to exert effort on their individual tasks. The

1. Consider a setting with two workers į 1, 2 who choose how much effort to exert on their individual tasks. Suppose that the effort choice is binary with ej = {0, 1}. The probability of task completion is 1 if the worker exerts ei 1 and p G (0, 1) if ei 0. The cost of exerting effort ei 1 is c> 0, while ei — 0 is costless. Both tasks need to be completed in order for the entire project to be realized. Completing only one of the tasks is useless to the hiring firm. The objective of the firm is to ensure the project completion at the lowest possible cost. Suppose that the firm cannot observe the completion of the individual tasks, but only whether the entire project is realized. The firm commits to public bonus payments to the workers if the project is successfully completed. That is, worker į receives a bonus bi if the project is a success and 0 otherwise. Such a bonus contract induces a simultaneous move game between the two workers who independently decide whether to exert effort on their individual tasks. The

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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1. Consider a setting with two workers į 1, 2 who choose how much effort to exert on their individual tasks. Suppose that the effort choice is binary with ej = {0, 1}. The probability of task completion is 1 if the worker exerts ei 1 and p G (0, 1) if ei 0. The cost of exerting effort ei 1 is c > 0, while ei – 0 is costless. Both tasks need to be completed in order for the entire project to be realized. Completing only one of the tasks is useless to the hiring firm. The objective of the firm is to ensure the project completion at the lowest possible cost. Suppose that the firm cannot observe the completion of the individual tasks, but only whether the entire project is realized. The firm commits to public bonus payments to the workers if the project is successfully completed. That is, worker į receives a bonus bi if the project is a success and 0 otherwise. Such a bonus contract induces a simultaneous move game between the two workers who independently decide whether to exert effort on their individual tasks. The workers are risk-neutral, expected payoff maximizers. (a) Suppose that worker į believes that j exerts effort % 1. What is the minimum bonus bli that would induce į to exert effort ei 1? (b) Suppose that worker į believes that j exerts effort ei 0. What s the minimum bonus b} that would induce į to exert effort ei 1? (c) What is the least costly bonus contract (bl* b'2) that would induce both workers to exert effort (i.e., el* e2 =1) as a Nash equilibrium of this simultaneous move game? (d) What is the least costly bonus contract b3) that would induce both workers to exert effort (i.e., el* = e3 1) as the unique Nash equilibrium of this game? Consider now the possibility of the firm offering a stochastic contract. In particular, suppose that worker 1 is offered a bonus bl that takes a value of bl + (from part b)) with probability a and bl (from part a)) with probability (1 a) where € > 0 is arbitrary close to 0. For our purposes, we can treat e as being zero. Worker 1 is privately informed of her bonus before deciding whether or not to exert effort, but worker 2 only knows the bonus distribution.
Transcribed Image Text:1. Consider a setting with two workers į 1, 2 who choose how much effort to exert on their individual tasks. Suppose that the effort choice is binary with ej = {0, 1}. The probability of task completion is 1 if the worker exerts ei 1 and p G (0, 1) if ei 0. The cost of exerting effort ei 1 is c > 0, while ei – 0 is costless. Both tasks need to be completed in order for the entire project to be realized. Completing only one of the tasks is useless to the hiring firm. The objective of the firm is to ensure the project completion at the lowest possible cost. Suppose that the firm cannot observe the completion of the individual tasks, but only whether the entire project is realized. The firm commits to public bonus payments to the workers if the project is successfully completed. That is, worker į receives a bonus bi if the project is a success and 0 otherwise. Such a bonus contract induces a simultaneous move game between the two workers who independently decide whether to exert effort on their individual tasks. The workers are risk-neutral, expected payoff maximizers. (a) Suppose that worker į believes that j exerts effort % 1. What is the minimum bonus bli that would induce į to exert effort ei 1? (b) Suppose that worker į believes that j exerts effort ei 0. What s the minimum bonus b} that would induce į to exert effort ei 1? (c) What is the least costly bonus contract (bl* b'2) that would induce both workers to exert effort (i.e., el* e2 =1) as a Nash equilibrium of this simultaneous move game? (d) What is the least costly bonus contract b3) that would induce both workers to exert effort (i.e., el* = e3 1) as the unique Nash equilibrium of this game? Consider now the possibility of the firm offering a stochastic contract. In particular, suppose that worker 1 is offered a bonus bl that takes a value of bl + (from part b)) with probability a and bl (from part a)) with probability (1 a) where € > 0 is arbitrary close to 0. For our purposes, we can treat e as being zero. Worker 1 is privately informed of her bonus before deciding whether or not to exert effort, but worker 2 only knows the bonus distribution.
(e) Consider a pessimistic belief by worker 2 that worker 1 exerts effort (i.e., el 1) only if offered bl and does not exert effort (i.e., el O) if exert effort e2 | 9 (f) Does the bonus contract (in, b!}_) implement (el* 1 e (g) |What is the optimal value of that minimizes the expected bonus payments for the firm? (h) (5) Suppose that p = 21 and a = a* (5. How does the stochastic contract you characterized in part f) compare to the contract characterized in part d)? In particular, which one results in lower expected bonus costs for the firm? Given this belief, what is the minimum bonus that would induce 2 to 1) as the unique NE? Why/why not?
Transcribed Image Text:(e) Consider a pessimistic belief by worker 2 that worker 1 exerts effort (i.e., el 1) only if offered bl and does not exert effort (i.e., el O) if exert effort e2 | 9 (f) Does the bonus contract (in, b!}_) implement (el* 1 e (g) |What is the optimal value of that minimizes the expected bonus payments for the firm? (h) (5) Suppose that p = 21 and a = a* (5. How does the stochastic contract you characterized in part f) compare to the contract characterized in part d)? In particular, which one results in lower expected bonus costs for the firm? Given this belief, what is the minimum bonus that would induce 2 to 1) as the unique NE? Why/why not?
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