Employment Contracting in the Labor Market
Employment Contracting in the Labor Market
Joseph Mullins
October 29, 2013
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Employment Contracting in the Labor Market
Introduction
Today:
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Begin to consider alternative pay structures in the labor
market.
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Employment Contracting in the Labor Market
Introduction
Today:
I
Begin to consider alternative pay structures in the labor
market.
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Examine and solve a “Principal-Agent” problem.
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Employment Contracting in the Labor Market
Introduction
Today:
I
Begin to consider alternative pay structures in the labor
market.
I
Examine and solve a “Principal-Agent” problem.
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Consider extensions, empirical plausibility.
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Employment Contracting in the Labor Market
Introductory Concepts
Why Contracts?
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So far, we have seen the simplest possible employment
contract: a flat wage rate.
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Employment Contracting in the Labor Market
Introductory Concepts
Why Contracts?
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So far, we have seen the simplest possible employment
contract: a flat wage rate.
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Labor is analogous to any other product: simply bought and
sold.
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Employment Contracting in the Labor Market
Introductory Concepts
Why Contracts?
I
So far, we have seen the simplest possible employment
contract: a flat wage rate.
I
Labor is analogous to any other product: simply bought and
sold.
I
This is useful abstraction in many cases, but sometimes we
would like to think in more detail about what determines
wages in the labor market.
.
.
.
.
.
.
Employment Contracting in the Labor Market
Introductory Concepts
Why Contracts?
I
So far, we have seen the simplest possible employment
contract: a flat wage rate.
I
Labor is analogous to any other product: simply bought and
sold.
I
This is useful abstraction in many cases, but sometimes we
would like to think in more detail about what determines
wages in the labor market.
I
Certain patterns that this simple wage structure cannot
replicate (positive).
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.
.
.
.
.
Employment Contracting in the Labor Market
Introductory Concepts
Why Contracts?
I
So far, we have seen the simplest possible employment
contract: a flat wage rate.
I
Labor is analogous to any other product: simply bought and
sold.
I
This is useful abstraction in many cases, but sometimes we
would like to think in more detail about what determines
wages in the labor market.
I
Certain patterns that this simple wage structure cannot
replicate (positive).
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Particular questions about how to elicit performance
(normative).
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Employment Contracting in the Labor Market
Introductory Concepts
Extensions
We’re going to extend the standard model with three intuitive
concepts:
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Employment Contracting in the Labor Market
Introductory Concepts
Extensions
We’re going to extend the standard model with three intuitive
concepts:
1. Workers can choose the effort they provide on their jobs.
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Employment Contracting in the Labor Market
Introductory Concepts
Extensions
We’re going to extend the standard model with three intuitive
concepts:
1. Workers can choose the effort they provide on their jobs.
2. Effort is not costless to provide.
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Employment Contracting in the Labor Market
Introductory Concepts
Extensions
We’re going to extend the standard model with three intuitive
concepts:
1. Workers can choose the effort they provide on their jobs.
2. Effort is not costless to provide.
3. This effort may not be perfectly observed by the firm.
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Employment Contracting in the Labor Market
A Principal-Agent Model
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Let’s focus on points (1) and (2) from the previous slide.
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Employment Contracting in the Labor Market
A Principal-Agent Model
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Let’s focus on points (1) and (2) from the previous slide.
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Make “effort”, e, costly. Utility cost to worker c(e).
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Employment Contracting in the Labor Market
A Principal-Agent Model
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Let’s focus on points (1) and (2) from the previous slide.
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Make “effort”, e, costly. Utility cost to worker c(e).
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Choice of e is connected to output, q. Simple case:
q(e) = e
(1)
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Employment Contracting in the Labor Market
A Principal-Agent Model
Efficiency
If we were deciding how much output to produce here (suppose we
want to maximise output subject to utility cost of effort), the
“efficient solution” would be to solve:
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Employment Contracting in the Labor Market
A Principal-Agent Model
Efficiency
If we were deciding how much output to produce here (suppose we
want to maximise output subject to utility cost of effort), the
“efficient solution” would be to solve:
max {e − c(e)}
(2)
e
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Employment Contracting in the Labor Market
A Principal-Agent Model
Efficiency
If we were deciding how much output to produce here (suppose we
want to maximise output subject to utility cost of effort), the
“efficient solution” would be to solve:
max {e − c(e)}
(2)
1 − c 0 (e) = 0 ⇔ c 0 (e) = 1
(3)
e
FOC:
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Employment Contracting in the Labor Market
A Principal-Agent Model
Moral Hazard
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Now suppose that someone, the Principal, owns the
production technology, but must hire a worker, the Agent, to
supply the labor (effort) in production.
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Employment Contracting in the Labor Market
A Principal-Agent Model
Moral Hazard
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Now suppose that someone, the Principal, owns the
production technology, but must hire a worker, the Agent, to
supply the labor (effort) in production.
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This introduces Moral Hazard. Why?
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Employment Contracting in the Labor Market
A Principal-Agent Model
Moral Hazard
I
Now suppose that someone, the Principal, owns the
production technology, but must hire a worker, the Agent, to
supply the labor (effort) in production.
I
This introduces Moral Hazard. Why?
I
Want to consider a pay structure that provides incentives for
effort. Let’s try:
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Employment Contracting in the Labor Market
A Principal-Agent Model
Moral Hazard
I
Now suppose that someone, the Principal, owns the
production technology, but must hire a worker, the Agent, to
supply the labor (effort) in production.
I
This introduces Moral Hazard. Why?
I
Want to consider a pay structure that provides incentives for
effort. Let’s try:
w = α + βq = α + βe
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(4)
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Employment Contracting in the Labor Market
A Principal-Agent Model
Solving for the Agent
The agent will solve:
max {α + βe − c(e)}
(5)
e
⇒
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Employment Contracting in the Labor Market
A Principal-Agent Model
Solving for the Agent
The agent will solve:
max {α + βe − c(e)}
(5)
c 0 (e) = β
(6)
e
⇒
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Employment Contracting in the Labor Market
A Principal-Agent Model
Solving for the Agent
The agent will solve:
max {α + βe − c(e)}
(5)
c 0 (e) = β
(6)
e
⇒
Efficiency only if β = 1.
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Employment Contracting in the Labor Market
A Principal-Agent Model
Stating the Principal’s Problem
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Now the Principal knows how the agent will choose their
effort given choices of (α, β).
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Employment Contracting in the Labor Market
A Principal-Agent Model
Stating the Principal’s Problem
I
Now the Principal knows how the agent will choose their
effort given choices of (α, β).
I
We need to formalise the Principal’s problem:
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.
.
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Employment Contracting in the Labor Market
A Principal-Agent Model
Stating the Principal’s Problem
I
Now the Principal knows how the agent will choose their
effort given choices of (α, β).
I
We need to formalise the Principal’s problem:
max {q(e) − α − βe}
(7)
α,β
Subject to:
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.
.
.
.
.
Employment Contracting in the Labor Market
A Principal-Agent Model
Stating the Principal’s Problem
I
Now the Principal knows how the agent will choose their
effort given choices of (α, β).
I
We need to formalise the Principal’s problem:
max {q(e) − α − βe}
(7)
c 0 (e) = β
(8)
α,β
Subject to:
.
.
.
.
.
.
Employment Contracting in the Labor Market
A Principal-Agent Model
Stating the Principal’s Problem
I
Now the Principal knows how the agent will choose their
effort given choices of (α, β).
I
We need to formalise the Principal’s problem:
max {q(e) − α − βe}
(7)
c 0 (e) = β
(8)
α,β
Subject to:
What else? We need the supply of effort to, ultimately, be
worth it for the agent:
α + βe ≥ c(e)
(9)
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Employment Contracting in the Labor Market
A Principal-Agent Model
Solving the Principal’s Problem
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First, is it optimal for the principal to choose α, β such that
α + βe > c(e)?
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Employment Contracting in the Labor Market
A Principal-Agent Model
Solving the Principal’s Problem
I
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First, is it optimal for the principal to choose α, β such that
α + βe > c(e)?
No! Require that: α + βe = c(e).
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Employment Contracting in the Labor Market
A Principal-Agent Model
Solving the Principal’s Problem
I
I
I
First, is it optimal for the principal to choose α, β such that
α + βe > c(e)?
No! Require that: α + βe = c(e).
Restate the problem:
max{e − c(e)}
(10)
α,β
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Employment Contracting in the Labor Market
A Principal-Agent Model
Solving the Principal’s Problem
I
I
I
First, is it optimal for the principal to choose α, β such that
α + βe > c(e)?
No! Require that: α + βe = c(e).
Restate the problem:
max{e − c(e)}
(10)
α,β
I
First, effort does not depend on α, so
∂e
∂α
.
= 0.
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Employment Contracting in the Labor Market
A Principal-Agent Model
Solving the Principal’s Problem
I
I
I
First, is it optimal for the principal to choose α, β such that
α + βe > c(e)?
No! Require that: α + βe = c(e).
Restate the problem:
max{e − c(e)}
(10)
α,β
I
I
First, effort does not depend on α, so
Then, for β we get the FOC:
∂e
∂α
= 0.
∂e
∂e
− c 0 (e)
=0
∂β
∂β
.
(11)
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Employment Contracting in the Labor Market
A Principal-Agent Model
Solving the Principal’s Problem
I
I
I
First, is it optimal for the principal to choose α, β such that
α + βe > c(e)?
No! Require that: α + βe = c(e).
Restate the problem:
max{e − c(e)}
(10)
α,β
I
I
First, effort does not depend on α, so
Then, for β we get the FOC:
∂e
∂α
= 0.
∂e
∂e
− c 0 (e)
=0
∂β
∂β
(11)
Which rearranges to the condition:
c 0 (e) = 1
(12)
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Employment Contracting in the Labor Market
A Principal-Agent Model
Solving the Principal’s Problem
I
Ok, so for the optimal solution, the Principal needs to choose
β such that the c 0 (e) = 1. How can we do that?
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Employment Contracting in the Labor Market
A Principal-Agent Model
Solving the Principal’s Problem
I
Ok, so for the optimal solution, the Principal needs to choose
β such that the c 0 (e) = 1. How can we do that?
I
Recall that c 0 (e) = β, this is the optimal choice for the Agent.
So we set β = 1.
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.
.
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Employment Contracting in the Labor Market
A Principal-Agent Model
Solving the Principal’s Problem
I
Ok, so for the optimal solution, the Principal needs to choose
β such that the c 0 (e) = 1. How can we do that?
I
Recall that c 0 (e) = β, this is the optimal choice for the Agent.
So we set β = 1. Efficiency!
.
.
.
.
.
.
Employment Contracting in the Labor Market
A Principal-Agent Model
Solving the Principal’s Problem
I
Ok, so for the optimal solution, the Principal needs to choose
β such that the c 0 (e) = 1. How can we do that?
I
Recall that c 0 (e) = β, this is the optimal choice for the Agent.
So we set β = 1. Efficiency!
I
To pin down α, we choose it make the worker indifferent:
α + βe = c(e)
(13)
.
.
.
.
.
.
Employment Contracting in the Labor Market
A Principal-Agent Model
Solving the Principal’s Problem
I
Ok, so for the optimal solution, the Principal needs to choose
β such that the c 0 (e) = 1. How can we do that?
I
Recall that c 0 (e) = β, this is the optimal choice for the Agent.
So we set β = 1. Efficiency!
I
To pin down α, we choose it make the worker indifferent:
α + βe = c(e)
(13)
Substituting β = 1 and rearranging:
α = c(e) − e.
(14)
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Employment Contracting in the Labor Market
A Principal-Agent Model
Interpreting the solution
I
By setting β = 1, we make the Agent a full claimant on
profits from production.
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Employment Contracting in the Labor Market
A Principal-Agent Model
Interpreting the solution
I
By setting β = 1, we make the Agent a full claimant on
profits from production.
I
This ensures that they make the right choice at the margin
when it comes to the supply of effort (think about β in the
Agent’s first-order condition).
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.
.
.
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Employment Contracting in the Labor Market
A Principal-Agent Model
Interpreting the solution
I
By setting β = 1, we make the Agent a full claimant on
profits from production.
I
This ensures that they make the right choice at the margin
when it comes to the supply of effort (think about β in the
Agent’s first-order condition).
I
For the Principal to make profit, they essentially rent the
production technology at a price −α.
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Employment Contracting in the Labor Market
A Principal-Agent Model
A Two Person Extension
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See blackboard notes.
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Employment Contracting in the Labor Market
Back to Reality
I
Some strong assumptions going on:
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Employment Contracting in the Labor Market
Back to Reality
I
Some strong assumptions going on:
1. Fully observable effort
2. The Principal and the Agent have same preferences.
3. Simple behavioural model (no notions of “fairness”,
“teamwork”, etc).
I
Still, we do see similar pay schedules that reflect the role of
incentives:
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Employment Contracting in the Labor Market
Back to Reality
I
Some strong assumptions going on:
1. Fully observable effort
2. The Principal and the Agent have same preferences.
3. Simple behavioural model (no notions of “fairness”,
“teamwork”, etc).
I
Still, we do see similar pay schedules that reflect the role of
incentives:
1. Taxi cabs (almost exactly how we described above).
2. Commissions, bonuses.
3. Firm hierarchies (see: Tournament Model, next lecture).
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