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Notes on The Principal-Agent Model
Brian Jenkins
University of California, Irvine
September 29, 2015
Contents
1 Symmetric Information with One Agent
2
1.1
The Principal and the Agent . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
The Socially Optimal (Efficient) Allocation . . . . . . . . . . . . . . . . . . .
4
1.4
The Optimal Contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.4.1
The Indifference Curves of the Agent . . . . . . . . . . . . . . . . . .
6
1.4.2
The Indifference Curves of the Principal . . . . . . . . . . . . . . . .
7
1.4.3
Graphical Solution to the Contracting Problem . . . . . . . . . . . .
8
1.4.4
Formal Solution to the Contracting Problem . . . . . . . . . . . . . .
9
2 Symmetric Information with Two Agent Types
10
2.1
The Socially Optimal (Efficient) Allocation . . . . . . . . . . . . . . . . . . .
10
2.2
The Optimal Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
3 Two Agent Types with Hidden Characteristics
14
3.1
The Socially Optimal (Efficient) Allocation . . . . . . . . . . . . . . . . . . .
15
3.2
The Optimal Menu of Contracts . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.2.1
Participation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
3.2.2
Incentive Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . .
16
3.2.3
Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
1
A Figures
21
2
1
Symmetric Information with One Agent
1.1
The Principal and the Agent
- A principal delegates the production of q units of a good to an agent.
- The agent is risk-neutral.
- The principal receives value from the good that the agent produces denoted by V (q).
The function V (·) satisfies:
V (0) = 0
(1)
V 0 (·) > 0
(2)
V 00 (·) < 0,
(3)
which means that the principal receives no value when the agent doesn’t produce
anything and that the marginal value to the principal from having the agent produce
the good is positive, but decreasing.
- Graphically, the assumptions that we make about V (·) imply a curve that looks like
that which is presented in Figure 1.
- We will work with two explicit functional forms for V (·).
1. The first is a logarithmic function:
V (q) = a log(q + 1)
(4)
where a is a positive constant that reflects the degree to which the principal values
the agent’s output and log(·) is the natural logarithm (base e). The derivative of
V (·) is then given by:
V 0 (q) =
a
q+1
(5)
2. We will also consider a power function:
V (q) = aq b
(6)
3
where a is a positive constant that reflects the degree to which the principal values
the agent’s output and 0 < b < 1 is a constant that reflects how the marginal
value of q changes as q increases. In this case, the derivative of V (·) is then given
by:
V 0 (q) = (ab) · q b−1
(7)
- The agent incurs a cost that depends on the quantity of the good produced.
- The cost function for the agent is given by:
C(q, k) = F + k · q
(8)
where F and k are both nonnegative constants.
- In the cost function, F represents the fixed cost to the agent of producing any quantity.
F includes two things:
1. The fixed real resource cost of producing any quantity of the good. For example,
an new employee – an agent of the employer – might have to go through training
before she can begin working.
2. The forgone utility from the next-best alternative. For example, a new employee
at a job gives up the opportunity to work at a different job or gives up leisure
time.
- We will set the fixed cost F equal to zero because it will make our analysis easier and
will have no real effect on our results. Therefore, we will write:
C(q, k) = k · q
(9)
- Refer to Figure 2 for a sample cost function.
- The constant k is the agent’s marginal cost of production.
- For now, we assume:
1. k is the same for all agents that the principal might come into contact with.
4
2. The principal and the agents know the value of k with certainty.
We will changes each of these assumptions later.
1.2
Contracts
- The principal and the agent enter into a contractual obligation.
- A contract is a binding agreement enforceable in a court of law that specifies action(s)
to be taken by the agent and payments to be made by the principal under all possible
observable contingencies.
- We assume that the principal makes take it or leave it contracts that the agent can
either accept or reject: if the agent does not like the contract, there is no opportunity
for the agent to make a counter-offer.
- Based on the problem that we are considering, it makes sense to restrict our attention
to contracts that specify:
1. a quantity q that the agent is to produce.
2. an amount t that the principal promises to pay – or transfer – to the agent upon
delivery of the good.
We can represent a contract succinctly as an ordered pair: (q, t).
1.3
The Socially Optimal (Efficient) Allocation
- A benevolent social planner 1 chooses q to maximize the sum of the utility received by
the principal and the agent:
max UP + UA ,
(10)
q
where, the utility to the principal is:
UP = V (q) − t
(11)
1
The benevolent social planner is a hypothetical person who chooses how much of each good is to be
produced in order to maximize the utility of the people in the economy.
5
and the utility to the agent is:
UA = t − k · q
(12)
- Substituting the definitions of UP and UA into the social planner’s problem, we obtain:
max V (q) − k · q,
(13)
q
- We will use q̃ to denote the socially optimal or efficient quantity of the good.
- To solve the social planner’s problem, we find a first-order condition by taking the
derivative of the expression to be maximized and setting it equal to zero:
V 0 (q̃) − k = 0
(14)
which implies:
V 0 (q̃) = k
(15)
- This means that it is optimal from a social perspective to have enough units of the
good produced so that the marginal value to the principal is equal to the marginal cost
of production to the agent.
- Example:
◦ Suppose that the function V is logarithmic:
V (q) = a log(q + 1)
(16)
◦ Then, the efficient quantity of the good is found by using the first-order condition
to solve for q̃:
a
= k
1 + q̃
(17)
6
which implies:
q̃ =
a
−1
k
(18)
◦ The answer appears totally reasonable. Larger values of a mean that the principal
values the good more and the solution reveals that the efficient level of production
would also increase. Similarly, a larger value of k means that the agent incurs
a greater marginal cost to produce the good and the solution implies that the
efficient level of production would therefore decrease.
◦ Plugging in some numbers – like a = 20 and k = 4 – we find:
q̃ =
100
− 1 = 24
4
(19)
- Of course in reality there is no social planner – certainly no benevolent one – but we
will use the result as a benchmark for interpreting the allocation induced when the
principal and agent contract with each other.
1.4
The Optimal Contract
- Now we examine the optimal contract between the principal and the agent.
- We will characterize the optimal contract by considering the indifference curves of the
principal and the agent over combinations of q and t.
1.4.1
The Indifference Curves of the Agent
- The utility that the agent receives from a contract (q, t) is:
UA = t − k · q
(20)
- Solve the previous equation for t and obtain an expression for an indifference curve:
t = k · q + ŪA
(21)
where ŪA simply denotes a given level of utility.
7
- Figure 3 depicts some examples of indifference curves for the agent.
- Notice that the agent’s utility is increasing to the northwest. This is because the agent
enjoys having more transfer t and enjoys producing less units of the good q.
- Also notice that the agent will not accept any contract that provides him with less
utility than his best outside alternative. Since we have assumed F = 0, we have been
implicitly assuming that the most utility that the agent would receive elsewhere is 0.
Therefore, the agent will never accept a contract that falls below the line:
t=k·q .
(22)
- Equation (22) is called the agent’s participation constraint. It describes the minimum
transfer t that the principal has to promise the agent for any desired quantity q.
- Example: Suppose that k = 5.
◦ If the principal desires q = 2, then the participation constraint means that the
principal must pay the agent at least $10 for the agent to be to accept the contract.
◦ If the principal desires q = 12, then the participation constraint means that the
principal must pay the agent at least $60.
1.4.2
The Indifference Curves of the Principal
- The utility that the principal receives from the contract is:
UP = V (q) − t
(23)
- Solve the previous equation for t and obtain an expression for an indifference curve for
the principal:
t = V (q) − ŪP
(24)
where ŪP simply denotes a given level of utility.
8
- Figure 4 depicts some examples of indifference curves for the principal.
- Notice that the principal’s utility is increasing to the southeast because the principal
enjoys paying less transfer t and enjoys having more units of the good q.
1.4.3
Graphical Solution to the Contracting Problem
- The principal writes a contract to present to the agent that maximizes the principal’s
utility subject to the constraint that the agent is just willing to accept the contract.
- This is another way of saying that the principal chooses a contract that lies along the
participation constraint for the agent that places the principal on the indifference curve
with the highest level of utility.
- The optimal contract is found by finding the quantity q ∗ such that the an indifference
curve for the principal is tangent to the participation constraint. See figure 5.
- Since the slope of the indifference curves of the principal and agent are equal at q ∗ , it
follows that at q ∗ :
V 0 (q ∗ ) = k
(25)
- Note that this expression is identical to equation (15). This means that the optimal
contract under symmetric information and with a single agent is efficient or socially
optimal. There is no other production level that would lead to a greater cumulative
level of utility between the principal and agent.
- The optimal transfer t∗ is found using the participation constraint:
t∗ = k · q ∗
(26)
- Example:
◦ Again, suppose that V (q) = 100 · log(q + 1) and that C(q) = 4 · q.
9
◦ Then:
a
−1
k
100
−1
=
4
= 24
q∗ =
(27)
(28)
(29)
◦ And:
1.4.4
t∗ = k · q ∗
(30)
= 4 · 24
(31)
= 96
(32)
Formal Solution to the Contracting Problem
- That is, the principal chooses q and t to maximize equation (22) subject to equation
(22).
- Formally:
max
q,t
V (q) − t
(33)
s.t. t = k · q
(34)
- We can eliminate t from the problem by substituting participation constraint into the
principal’s utility function and writing the maximization problem with respect to q
only:
max V (q) − k · q
(35)
q
- Obtain the first-order condition:
V 0 (q ∗ ) − k = 0
(36)
10
which implies:
V 0 (q ∗ ) = k
(37)
- Then use the participation constraint to find the optimal transfer:
t∗ = k · q ∗
2
(38)
Symmetric Information with Two Agent Types
- In this section, we allow for the possibility that the are two types of agents
◦ Type 1: low-cost (efficient) producers with marginal cost: k = k1 .
◦ Type 2: high-cost (inefficient) producers with marginal cost: k = k2 .
where k1 < k2 .
- The principal has no control over which type of agent he confronts.
- But since we are still supposing that there is symmetric information, the principal can
prepare a separate contract to offer agents from each type.
2.1
The Socially Optimal (Efficient) Allocation
- Since the principal writes a contract specifically for each type of agent, the socially
optimal level of production for each agent will mirror what we found previously.
- The benevolent social planner chooses q1 to maximize the sum of the utility received
by the principal and the agent:
max V (q1 ) − k1 · q1 ,
(39)
q1
implying:
V 0 (q̃1 ) = k1
(40)
11
- And the benevolent social planner chooses q2 to maximize the sum of the utility received
by the principal and the agent:
max V (q2 ) − k2 · q2 ,
(41)
q2
implying:
V 0 (q̃2 ) = k2
(42)
- That is, it is socially optimal for each agent to produce enough so that their marginal
costs of production equal the marginal value to the principal.
- Note that since the function V is increasing at a decreasing rate, q˜2 < q˜1 .2 . This means
that it is socially optimal for the low-cost agent to produce more than the high-cost
agent.
- Example:
◦ Suppose that V (q) = 100 · log(q + 1) and that k1 = 4 and k2 = 5.
◦ Then, from before, we know that:
a
= k
q̃ + 1
(43)
implies:
q̃ =
a
−1
k
(44)
◦ Then the socially optimal production for a type 1 agent is:
a
−1
k1
100
=
−1
4
= 24
(45)
q̃1 =
2
0
(46)
(47)
00
Recall: V (·) > 0, V (·) < 0.
12
◦ And the socially optimal production for a type 2 agent is:
a
−1
k2
100
−1
=
5
= 19
q̃2 =
2.2
(48)
(49)
(50)
The Optimal Contracts
- Symmetric information means that the principal can offer agents contracts that are
designed specifically for their type.
- Therefore, the optimal contract to each type of agent will have the same form as the
contract to a single type of agent under symmetric information.
- That is:
1. The optimal contract to a type 1 agent will satisfy:
V 0 (q1∗ ) = k1
(51)
t∗1 = k1 · q1∗
(52)
and:
2. The optimal contract to a type 2 agent will satisfy:
V 0 (q2∗ ) = k2
(53)
t∗2 = k2 · q2∗
(54)
and:
- Note that with symmetric information, the optimal contract produces the socially
optimal allocation:
q1∗ = q̃1
(55)
q2∗ = q̃2
(56)
13
- Figure 6 reflects the determination of the optimal contracts to type 1 and type 2 agents.
Notice that each contract is independent of the details of the other.
- Since the principal knows the agent’s type with certainty, the principal will compel each
agent to produce such that the marginal benefit to the principal equal the marginal
cost to the agent. The principal offers each type a transfer that is just enough so that
the agents are indifferent between accepting or rejecting their contracts.
- Example:
◦ Suppose that V (q) = 100 · log(q + 1) and that k1 = 4 and k2 = 5 and that the
principal can perfectly observe k1 and k2 .
◦ Since q1∗ = q̃1 and q2∗ = q̃2 , we know from previous calculations that:
q1∗ = 24
(57)
q2∗ = 19
(58)
◦ Now, use the participation constraints for each type of agent to find the optimal
transfers:
t∗1 = k1 · q1∗
(59)
= 4 · 24
(60)
= 96
(61)
t∗2 = k2 · q2∗
(62)
= 8 · 19
(63)
= 95
(64)
and:
- Of course, the principal is not indifferent to which type of agent shows up. Using the
numbers from the previous example, find that the utility to the principle from a type
14
1 contract is:
UP = V (q1∗ ) − t∗1
(65)
= 100 · log (24 + 1) − 96
(66)
= 225.89
(67)
and the utility to the principal from a type 2 contract is:
UP = V (q2∗ ) − t∗2
(68)
= 100 · log (19 + 1) − 95
(69)
= 204.57
(70)
So the principal would definitely prefer to only work with type 1 agents, but if the
principal did not have a choice, the he would be willing to work with either.
3
Two Agent Types with Hidden Characteristics
- Finally, we are at a point where we can examine the how asymmetric information
affects the contracting equilibrium.
- We continue to work with the assumption that an agent can have one of two types:
type 1 agents have marginal costs of production k1 and type 2 agents have marginal
costs of production k2 , where k1 < k2 .
- However, now we suppose that the principal does not observe the type of an agent.
We say the the agent’s type is a hidden characteristic.
- We suppose that the principal meets agents that are randomly from one type or the
other.
◦ Denote by p the share of all agents that are type 1.
◦ Then 1 − p is the share of all agents that are type 2.
15
◦ For example, if p = 0.6, then 60% of all agents are type 1 and so the principal
believes that there is a 60% chance that any given agent is a type 1 (or efficient)
agent.
- Of course, the principal could just ask agents to reveal their types before the contract is
offered, but there is no mechanism in place that would force the agents to be truthful.
- So the principal will create a menu of contracts with the goal to have a contract that
type 1 agents will accept and a contract that type 2 agents will accept. That is, the
principal wants to create contracts that cause the agents to reveal their types.
3.1
The Socially Optimal (Efficient) Allocation
- The presence of asymmetric information does not change what would be socially optimal.
- Therefore, the efficient quantity of the good to be produced by a type 1 and a type 2
agent is:
V 0 (q̃1 ) = k1
(71)
V 0 (q̃2 ) = k2
(72)
and:
3.2
The Optimal Menu of Contracts
- The principal prepares a menu of contracts (q1∗ , t∗1 ) and (q2∗ , t∗2 ) that will maximize the
principal’s expected utility subject to the conditions that:
1. Each agent is willing to accept at least one contract (participation).
2. Each agent selects the contract that the principal intends for them to select (incentive compatibility).
16
3.2.1
Participation
- Recall that in the symmetric information case, the type 1 (low cost) agent and the
type 2 (high cost) agent preferred the contract that was written for the type 2 agent.
- This means that if a menu consisted of the symmetric information contracts, then
both agents would choose the type 2 contract. The type 2 agent would earn no utility
and the type 1 agent would receive positive utility: the type 2 agent would be right on
the edge between accepting and rejecting.
- With this in mind, we will take the following statement as given:
If a menu of contracts has at least one contract that the type 2 (high cost)
agent will accept, then it also has at least one contract that the type 1 (low
cost) agent will accept.3
- Put more succinctly, the principal only has to make sure that the type 2 agent is willing
to participate.
- Therefore, the participation constraint for the principal is:
t2 = k2 · q2
3.2.2
(73)
Incentive Compatibility
- Next, the principal needs to be sure that each agent wants that the contract that was
designed for her.
- Again, we appeal to the intuition from the symmetric information case. Recall that
the type 1 agent had an incentive to pose as a type 2 agent.
- With this in mind, we will take the following statement as given:
For a menu of contracts, if the type 1 agent prefers the contract intended
for type 1 agents, then the type two agent will prefer the contract for type 2
agents.
3
This can be proven formally, but we are not concerned with that now.
17
- The principal needs to be sure that the contract written for the type 1 agent is at least
as appealing to the type 1 agent as the other contract.
- The relative constraint here is:
t1 − k1 · q1 = t2 − k1 · q2
(74)
- The lefthand side of equation (74) is the utility that aa type 1 agent would receive
under a type 1 contract. The righthand side is the utility that a type 1 agent would
receive under a type 2 contract.
◦ If the righthand side were greater, then type 1 agents would always take the
contract for type 2 agents.
◦ If the lefthand side were greater, then the principal would be paying the type 1
agents more than necessary.
- Imposing (74) means that the menu will incentive compatible.
- Incentive compatibility means that each agent voluntarily chooses the contract that the
principal wants them to choose: the incentives of the principal and the agents align.
3.2.3
Solution
- The principal chooses a menu of contracts to maximize his expected utility subject to
the participation constraint (73) and the incentive compatibility constraint (74).
- Formally:
max
q1 ,t1 ,t2 ,q2
p [V (q1 ) − t1 ] + (1 − p) [V (q2 ) − t2 ]
s.t.
(75)
t2 = k2 · q2
(76)
t1 − k1 · q1 = t2 − k1 · q2
(77)
- To solve the problem, we will eliminate the transfers t1 and t2 from the problem by
substituting the constraints into the principal’s utility function.
18
1. Eliminate t2 by substituting the participation constraint into the utility function
and the incentive compatibility constraint:
max
q1 ,t1 ,q2
p [V (q1 ) − t1 ] + (1 − p) [V (q2 ) − k2 · q2 ]
(78)
t1 − k1 · q1 = k2 · q2 − k1 · q2
(79)
s.t.
2. Eliminate t1 by substituting the incentive compatibility constraint into the utility
function:
max
q1 ,q2
p [V (q1 ) − k1 · q1 − k2 · q2 + k1 · q2 ] + (1 − p) [V (q2 ) − k2 · q2 ]
(80)
3. Find the first-order or optimality conditions by taking derivatives with respect to
q1 and q2 and setting the results equal to zero:
p [V 0 (q1∗ ) − k1 ] = 0
(81)
−p [k2 − k1 ] + (1 − p) [V 0 (q2∗ ) − k2 ] = 0
(82)
and:
- After a little bit of algebra, find:
V 0 (q1∗ ) = k1
(83)
and:
V 0 (q2∗ ) = k2 +
p
(k2 − k1 )
1−p
(84)
- The optimal transfers are found by using the participation and incentive compatibility
constraints:
t∗2 = k2 · q2∗
(85)
19
and:
t∗1 = k1 · q1∗ + t∗2 − k1 · q2∗
(86)
- Immediately, the following facts are apparent about the equilibrium induced by the
optimal menu:
1. The type 1 agent produces the efficient amount for type 1 agents.
2. The type 2 agent produces less than the efficient amount for type 2 agents because
V 0 (q̃2 ) < V 0 (q2∗ ).
3. The type 2 agent receives zero utility because the participation constraint holds
with equality.
4. The type 1 agent receives positive utility because the principal has to pay the
type 1 agent enough to keep incentives compatible.
- Example:
◦ Let’s consider the same example example that we’ve been carrying along this
entire time.
◦ Suppose that V (q) = 100 · log(q + 1) and that k1 = 4 and k2 = 5.
◦ Also, suppose that p = 5/6 so that the principal expects that about 5 out of every
6 agents that he encounters will be type 1, low cost agents.
◦ Then, find q1∗ using:
V 0 (q1∗ ) = k1
100
= 4
∗
q1 + 1
(87)
(88)
which means:
q1∗ = 24
(89)
20
◦ Next, find q2∗ :
p
(k2 − k1 )
1−p
5 6
100
= 5+ · ·1
∗
q2 + 1
6 1
= 10
V 0 (q2∗ ) = k2 +
(90)
(91)
(92)
which means:
q2∗ = 9
(93)
◦ Then, find t∗2 :
t∗2 = k2 · q2∗
(94)
= 5·9
(95)
so:
t∗2 = 45
(96)
◦ Finally, find t∗1 :
t∗1 = k1 · q1∗ + t∗2 − k1 · q2∗
(97)
= 4 · 24 + 45 − 4 · 9
(98)
so:
t∗1 = 105
(99)
◦ Notice that the type 1 agent produces the socially optimal quantity of 24 units.
The principal is willing to pay the type 1 agent more than the agent would accept
and to allow the type 2 agent to produce less than the efficient amount.
◦ The amount by which the type 1 agent’s transfer payment exceeds the agent’s cost
is called (economic) rent. The cost to the type 1 agent of producing 24 units is 96.
So the principal pays the type 1 agent an additional 9 units to prevent him from
taking the type 2 contract. The type 1 agent has valuable private information
and is able extract rent from the principal because of the hidden information.
21
A
Figures
Figure 1: The value to the principal of having the good. The function V is increasing
with respect to q which means that the principal is always better off from having an additional
unit of the good. Also, the function V becomes less steep as q increases which means that
the marginal value to the principal of having the good declines as more is produced.
V
V (q)
0
q
22
Figure 2: The cost to the agent of producing the good. The cost function C is
increasing with respect to q which means that the total cost to the agent increases with the
amount produced. The cost function C has a constant slope which means that the marginal
cost to the agent is constant.
C
k·q
0
q
Figure 3: Agent’s indifference curves. The agent prefers contracts that feature more t
and less q. Each indifference curve has the equation: t = k · q − U , where U is a given level
of utility.
t
+
U2
U1
U =0
Reject
0
q
23
Figure 4: Principle’s indifference curves. The principle prefers contracts that feature
more q and less t. Each indifference curve has the equation: t = V (q) − U , where U is a
given level of utility.
t
U1
U2
U3
U4
+
0
q
Figure 5: Optimal contract: symmetric information and one agent. The optimal
contract to a single agent under symmetric information is a quantity q ∗ and transfer t∗
such that the principal’s indifference curve that passes through (q ∗ , t∗ ) is just tangent to the
agent’s participation constraint.
t
PC
UP∗
t∗
0
q∗
24
q
Figure 6: Optimal contracts: symmetric information and two agents.
t
∗
UP,2
P C2
∗
UP,1
t∗1
t∗2
0
P C1
q2∗
q1∗
25
q