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 π₁ = Cį — Si, i = 1, 2, where e; is the effort supplied by a type i worker and 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 = ae², where a > 1. The utility function of a worker of type i is given by: u¡ = Sį — Cį. Find the solution to the firm's problem (assuming that effort is observable and contractible).
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 π₁ = Cį — Si, i = 1, 2, where e; is the effort supplied by a type i worker and 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 = ae², where a > 1. The utility function of a worker of type i is given by: u¡ = Sį — Cį. Find the solution to the firm's problem (assuming that effort is observable and contractible).
Chapter7: Uncertainty
Section: Chapter Questions
Problem 7.7P
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Transcribed Image Text:Question:
=
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 = ei — sį, i = 1, 2, where e; is the
Ti
Si,
effort supplied by a type i worker and and s; is the payment to a type i worker.
The cost function of the more productive worker (type 1) is given by c₁
e and
the cost function of the less productive worker (type 2) is given by c₂ = ae², where
a > 1. The utility function of a worker of type i is given by: ui
Si - Cį. Find the
solution to the firm's problem (assuming that effort is observable and contractible).
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 pį.
● 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 :
• thus, marginal costs for type 2 are also higher:
and 0²
0² 0(e) >
=
. Let s; be the firm's payment to a type i agent.
• The utility function of a type i agent is given by:
0² '(e) > 0¹¹(e) for all e
o² p (ei)
o'(ei) > 0, "(ei) > 0
Oh > 0² = 0¹
0¹6(e) for all e,
● The agent is, therefore, risk-neutral (his utility is linear in s; – 0¹ þ(e;)).
• Assume that the opportunity cost utility for a type i agent is u? (we can take it as u² = 0).
. The Firm:
● Let the firm's profits from a type i agent be given by:
so her utility is:
max{ei
ei
u² = si - 0² (ei)
Si
• The firm is, therefore, also risk-neutral.
4.1
Case I: Complete Information
• If the firm knows the agents' type, for each agent, it solves the problem:
0¹ þ(e;) = 0}
● Or,
Ti = ei Si
• The condition si
oi o(e)
= 0 indicates that all surplus will be extracted (assuming that u = 0).
. We can re-write the problem as:
Si Si
• Assuming an interior solution, the first-order conditions (for the two agents) are:
1 - 0² o' (ei)
o² o² (ei)
max{e; - 0¹ (ei)}
ei
● Let the optimal solution for effort be defined as ež, that is:
0¹¹(e) = 1
● This simply says that the marginal costs of effort (0¹ '(e)) will be set to equal marginal benefits (1).
• This is, what we call, the FIRST BEST, or PARETO OPTIMAL SOLUTION.
Given this solution, the firm will pay an agent of type i:
Si = : 0² (et)
Note that since 0¹ < 0h and from the FOC, we have:
=
=
0³ $(e‡) — 0³ 6(e‡) = 0 = opportunity cost
o'(et)
o' (et)
0, or
1
0¹² o' (et) = 0¹h '(eħ) = 1
See diagram below.
¹This is equivalent to solving: maxe; Σį eį – sį : s₁ - 0²(e) = 0.
-
Ah
= >1
0²
Since the cost functions are convex in effort (ø″ (et) > 0), it follows that:
et > en
(32)
(33)
(34)
(35)
(36)
(37)
(38)
(39)
(40)
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