Homework Help is Here – Start Your Trial Now!
Business
Economics
Consider the following principal agent problem with adverse selection. A firm faces a worker who may be one of two types, with equal probabilities. The firm's profits from a type i worker are given by i = ei Si, i = 1, 2, where e; is the effort supplied by a type i worker and s; is the payment to a type i worker. The cost function of the more productive worker (type 1) is given by c₁ = el and the cost function of the less productive worker (type 2) is given by c₂ = 2e2. The utility function of a worker of type i is given by: u; = c; and his opportunity cost utility is u= 0. Find the solution to the firm's problem (assuming that effort is observable and contractible). = s1/2 - -

Consider the following principal agent problem with adverse selection. A firm faces a worker who may be one of two types, with equal probabilities. The firm's profits from a type i worker are given by i = ei Si, i = 1, 2, where e; is the effort supplied by a type i worker and s; is the payment to a type i worker. The cost function of the more productive worker (type 1) is given by c₁ = el and the cost function of the less productive worker (type 2) is given by c₂ = 2e2. The utility function of a worker of type i is given by: u; = c; and his opportunity cost utility is u= 0. Find the solution to the firm's problem (assuming that effort is observable and contractible). = s1/2 - -

ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
14th Edition
ISBN:9780190931919
Author:NEWNAN
Publisher:NEWNAN
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
See similar textbooks
icon
Related questions
Question

advanced microeconomics, principal agent

Question: Consider the following principal agent problem with adverse selection. A firm faces a worker who may be one of two types, with equal probabilities. The firm's profits from a type i worker are given by πį = ¤¿ li Si, i 1, 2, where e; is the effort supplied by a type i worker and s; is the payment to a type i worker. The cost function of the more productive worker (type 1) is given by c₁ = e² and the cost function of the less productive worker (type 2) is given by c₂ = 2e2. The utility function of a worker of type i is given by: uį = si Si C₂ and his opportunity cost utility is u 0. Find the solution to the firm's problem (assuming that effort is observable and contractible). 1/2 = Use below Methods to solve above question: 4 Principal-Agent - Hidden Information . Consider the following principal-agent problem. • Effort (e) is observable, so we can contract on it. ● But, the agent has some private information. Agent: ● • For example, suppose the agent can be either one of two "types": type 1 or 2. ● Each agent knows her type, but the principal does not. • The principal's probability that the agent is type i is pi. . An alternative interpretation is as follows. ● There are many agents in the population, some of type 1, others of type 2, where the proportion of type i agents is pi. • Now, assume that the two types differ in terms of their "productivity." • Specifically, their effort cost functions are different. . Let the cost functions of the two types be given by: c² where and 0² = 0² 6(e) > ● thus, marginal costs for type 2 are also higher: Let s; be the firm's payment to a type i agent. • The utility function of a type i agent is given by: = . The firm is, therefore, also risk-neutral. u² : = si − 0¹ þ(ei) oi p(ei)). • Assume that the opportunity cost utility for a type i agent is u? (we can take it as u = 0). The Firm: ● Let the firm's profits from a type i agent be given by: The agent is, therefore, risk-neutral (his utility is linear in s; – ● 0² (ei) oʻ(ei) > 0, ø″(ei) > 0 Oh > 0² = 0¹ ⇒ 0¹ (e) for all e, 0² þ′(e) > 0¹ þ′(e) for all e $1 Ti = ei Si - 4.3 Case II: Hidden Information ● Again, remember that effort is observable, but whereas agents know their type, the firm does not. • The firm's problem is to choose "two packages," (e₁, 8₁) and (e2, 82), each designed to be attractive only to one (the correct one) type, such that expected profits are maximized: - max (e₁ - 81)p1 + (e₂ - $2)p2 : ei, si participation constraints ● This can be re-written as: $₁0¹0(e₁) $2 - 0¹0(e₂) ≥ 52-0² (е₂) ≥ 8₁- 0² 0(e₁) S2 $1 8₁ - 0¹6(e₁) ≥ 0 pc1 = pcl S20² (₂) ≥ 0 pc2 = pch incentive compatibility constraints S2 = 0² 0(e₂) ICC1= ICCI ICC2 = ICCh . Note that this implies that type 1 will also participate. • Why? ● Since type 1 can always "pretend" to be type 2 and then receive: $2 − 0¹ 0(e₂) > s2 – 0² (€₂) = 0 - ● The constraint ICC1 ensures that a type 1 agent is better off accepting the package designed for her (rather than pretending to be a type 2 agent). ● The constraint ICC2 ensures that a type 2 agent is better off accepting the package designed for her (rather than pretending to be a type 1 agent). Consider and pc2: pc1 • For a type 2 to accept, we need s2 ≥ 0² 0(e₂). • But, since the firm wants to make the payment as low as possible, it will set: = (32) $₁0¹0(e₁) ≥ s₂ - 0¹¹ (е₂): S2 0² 0(e2) — 0¹ 0(e₂) (0² - 0¹) (e₂) > 0 since 0² > 01 (33) ● Again, note that from the above, we see that type 1 extracts a surplus: $1 $₁ - 0¹¹ (е₁) > 0 • But, to prevent type 1 from acting as type 2, it is sufficient to set: $₁ - 0¹6(e₁) = (02² - 0¹) (e₂) (43) (44) i.e., if she acts like a type 2, she will get a strictly positive surplus. • This says that if type 2 participates, type 1 will also participate. Next, consider the two incentive compatibility constraints ICC1 and ICC2: (45) (46) (47) = (48) (49) • Primarily, we are interested in revealing type 1 (low-cost type). • We will, therefore, concentrate on Icci rather than IcC2. Later, we will show that if Icc₁ holds, so does ICC2. • Hence, plugging pc2 (50) into ICC1(48), we get that in order to prevent type 1 from acting as type 2, we have to satisfy the ICC1: (50) (51) (52) (53) (54) (55) (56) $₁ = 0¹ 6(e₁) + (0² – 0¹) (e2) ● Condition (57) will make type 1 (just) indifferent between acting her type or pretending to be a type 2. ● In other words, it does not leave any extra surplus over and above what is necessary to prevent type 1 from pretending to be type 2. (57)
Transcribed Image Text:Question: Consider the following principal agent problem with adverse selection. A firm faces a worker who may be one of two types, with equal probabilities. The firm's profits from a type i worker are given by πį = ¤¿ li Si, i 1, 2, where e; is the effort supplied by a type i worker and s; is the payment to a type i worker. The cost function of the more productive worker (type 1) is given by c₁ = e² and the cost function of the less productive worker (type 2) is given by c₂ = 2e2. The utility function of a worker of type i is given by: uį = si Si C₂ and his opportunity cost utility is u 0. Find the solution to the firm's problem (assuming that effort is observable and contractible). 1/2 = Use below Methods to solve above question: 4 Principal-Agent - Hidden Information . Consider the following principal-agent problem. • Effort (e) is observable, so we can contract on it. ● But, the agent has some private information. Agent: ● • For example, suppose the agent can be either one of two "types": type 1 or 2. ● Each agent knows her type, but the principal does not. • The principal's probability that the agent is type i is pi. . An alternative interpretation is as follows. ● There are many agents in the population, some of type 1, others of type 2, where the proportion of type i agents is pi. • Now, assume that the two types differ in terms of their "productivity." • Specifically, their effort cost functions are different. . Let the cost functions of the two types be given by: c² where and 0² = 0² 6(e) > ● thus, marginal costs for type 2 are also higher: Let s; be the firm's payment to a type i agent. • The utility function of a type i agent is given by: = . The firm is, therefore, also risk-neutral. u² : = si − 0¹ þ(ei) oi p(ei)). • Assume that the opportunity cost utility for a type i agent is u? (we can take it as u = 0). The Firm: ● Let the firm's profits from a type i agent be given by: The agent is, therefore, risk-neutral (his utility is linear in s; – ● 0² (ei) oʻ(ei) > 0, ø″(ei) > 0 Oh > 0² = 0¹ ⇒ 0¹ (e) for all e, 0² þ′(e) > 0¹ þ′(e) for all e $1 Ti = ei Si - 4.3 Case II: Hidden Information ● Again, remember that effort is observable, but whereas agents know their type, the firm does not. • The firm's problem is to choose "two packages," (e₁, 8₁) and (e2, 82), each designed to be attractive only to one (the correct one) type, such that expected profits are maximized: - max (e₁ - 81)p1 + (e₂ - $2)p2 : ei, si participation constraints ● This can be re-written as: $₁0¹0(e₁) $2 - 0¹0(e₂) ≥ 52-0² (е₂) ≥ 8₁- 0² 0(e₁) S2 $1 8₁ - 0¹6(e₁) ≥ 0 pc1 = pcl S20² (₂) ≥ 0 pc2 = pch incentive compatibility constraints S2 = 0² 0(e₂) ICC1= ICCI ICC2 = ICCh . Note that this implies that type 1 will also participate. • Why? ● Since type 1 can always "pretend" to be type 2 and then receive: $2 − 0¹ 0(e₂) > s2 – 0² (€₂) = 0 - ● The constraint ICC1 ensures that a type 1 agent is better off accepting the package designed for her (rather than pretending to be a type 2 agent). ● The constraint ICC2 ensures that a type 2 agent is better off accepting the package designed for her (rather than pretending to be a type 1 agent). Consider and pc2: pc1 • For a type 2 to accept, we need s2 ≥ 0² 0(e₂). • But, since the firm wants to make the payment as low as possible, it will set: = (32) $₁0¹0(e₁) ≥ s₂ - 0¹¹ (е₂): S2 0² 0(e2) — 0¹ 0(e₂) (0² - 0¹) (e₂) > 0 since 0² > 01 (33) ● Again, note that from the above, we see that type 1 extracts a surplus: $1 $₁ - 0¹¹ (е₁) > 0 • But, to prevent type 1 from acting as type 2, it is sufficient to set: $₁ - 0¹6(e₁) = (02² - 0¹) (e₂) (43) (44) i.e., if she acts like a type 2, she will get a strictly positive surplus. • This says that if type 2 participates, type 1 will also participate. Next, consider the two incentive compatibility constraints ICC1 and ICC2: (45) (46) (47) = (48) (49) • Primarily, we are interested in revealing type 1 (low-cost type). • We will, therefore, concentrate on Icci rather than IcC2. Later, we will show that if Icc₁ holds, so does ICC2. • Hence, plugging pc2 (50) into ICC1(48), we get that in order to prevent type 1 from acting as type 2, we have to satisfy the ICC1: (50) (51) (52) (53) (54) (55) (56) $₁ = 0¹ 6(e₁) + (0² – 0¹) (e2) ● Condition (57) will make type 1 (just) indifferent between acting her type or pretending to be a type 2. ● In other words, it does not leave any extra surplus over and above what is necessary to prevent type 1 from pretending to be type 2. (57)
Expert Solution
steps

Step by step

Solved in 3 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Knowledge Booster
Property Damage
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
ENGR.ECONOMIC ANALYSIS
ENGR.ECONOMIC ANALYSIS
Economics
ISBN:
9780190931919
Author:
NEWNAN
Publisher:
Oxford University Press
Principles of Economics (12th Edition)
Principles of Economics (12th Edition)
Economics
ISBN:
9780134078779
Author:
Karl E. Case, Ray C. Fair, Sharon E. Oster
Publisher:
PEARSON
Engineering Economy (17th Edition)
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)
Principles of Economics (MindTap Course List)
Economics
ISBN:
9781305585126
Author:
N. Gregory Mankiw
Publisher:
Cengage Learning
Managerial Economics: A Problem Solving Approach
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-…
Managerial Economics & Business Strategy (Mcgraw-…
Economics
ISBN:
9781259290619
Author:
Michael Baye, Jeff Prince
Publisher:
McGraw-Hill Education
SEE MORE TEXTBOOKS