Introduction to Principal-Agent Theory, Personnel Economics,
and Efficiency Wage Models
1. The Principal-Agent Problem
In the basic Principal-Agent problem, the
Agent
chooses
e
(effort) to maximize:
W
=
R
(
q
(
e
)) –
V
(
e
)
(1)
Where
V
(
e
) is disutility of effort,
q
is the agent’s output, and
R
is the agent’s monetary
reward.
Note two important things:
First, the utility function is linear in income,
R
.
This
means there are no income effects on effort supply.
The usual justification for this is that
the rewards we are thinking about are small relative to lifetime income.
Second, for
simplicity (1) ignores uncertainty in the relation between effort and results but the vast
majority of actual treatments include it, for example in the form
q
=
Q
(
e
,
ε
), where
ε
is a
random variable outside the agent’s and principal’s control.
In that case, risk aversion
matters.
When risk aversion is considered in these models, it is pretty much always
modelled by letting the the utility function be
U
[
R
(
q
(
e
)) ] –
V
(
e
), where
U
′ >0,
U
″<0.
This is still considerably simpler and more restrictive than the general case
U
(
R, e
),
because
U
CL
equals zero.
The
Principal
, in turn, chooses
R
(
q
) (note this is a
function
) to maximize expected
profits:
q
–
R
(
q
)
(2)
subject to two contraints:
1.
The agent’s maximization as in (1) (“incentive compatibility”).
Sometimes this is
characterized by a first-order condition, but care must be taken if you do that….
.
2. The agent’s participation constraint,
W
W
≥
, where
W
is the agent’s alternative (or
target) utility level.
Thus, note that this is a “Stackelberg” type problem:
The principal moves first, but
chooses her actions
anticipating
how the agent will respond to
R
(
q
).
To solve these
problems, the economist proceeds backwards in time, solving (1) first, then using that
information to solve (2).
Also, note that the participation constraint ensures that the solution is (constrained)
Pareto-Optimal, i.e. it maximizes one party’s utility (the principal’s) subject to a
minimum level of the agent’s.
Thus, by varying
W
, we can find not just the best