ECONOMICS OF INFORMATION:
INCENTIVES AND CONTRACTS
INTRODUCTION
David Pérez-Castrillo (UAB and BGSE)
THE ELEMENTS OF THE PROBLEM
A bilateral relationship in which one party contracts another to
carry out some action or to take some decision.
The contractor: principal
The contractee: agent
Both principal and agent could be individuals, institutions,
organizations, or decision centres
Examples: (1) shareholders of the firm and the firm's manager
(2) contract between a laboratory and a researcher
The principal designs the contract, and then offers it to the agent,
who, after having studied the terms of the contract, must decide
whether or not to sign it
THE ELEMENTS OF THE PROBLEM
The agent will accept the contract if the utility obtained is greater
than that he would get from not signing (the reservation utility)
The agent cannot make a counter offer to the principal (no
bilateral bargaining), i.e., the principal has all the bargaining
power. The agent decides whether or not to accept
If the agent does not sign the contract, the relationship does not
take place. If the agent does accept the offer, he must carry out
the actions for which he has been contracted
In the example of shareholders and manager, the latter must
decide the firm's strategy, which implies a certain effort dedicated
to different tasks and projects
THE TIMING
rejects
END
A
P
contract
offered to
A
accepts
A
N
effort
Result and
payments
THE TIMING
P offers the
contract to A
N (random)
A accepts (or
rejects)
A exerts effort
Result and
payment
accrued
THE ELEMENTS OF THE PROBLEM
A contract is a reliable promise by both parties, in which the
obligations of each, for all possible contingencies, are
specified
It includes the payment mechanism under which the agent will
be compensated for his effort. It is a very important point that
a contract can only be based on verifiable variables
Example: the shareholders in a firm and the manager
can contract on bonus or milestones, shares of the firm,
relative performance evaluation, …
Information is related to the set of variables that are
verifiable in a contractual relationship
6
SYMMETRIC INFORMATION
If principal and agent share common information as to all
relevant characteristics and the agent's effort is verifiable
The principal offers a different form of contract to each
possible agent type (depending on abilities, knowledge,
behaviour, etc.)
Each agent type will accept different contract formats,
depending on which principal offers the contract, and the
tasks for which he is being contracted
It is possible to introduce the agent's effort and the final result
of the relationship explicitly into the terms of the contract
7
SYMMETRIC INFORMATION
P designs the
contract
A accepts A supplies N determines Outcome
the state of
and
effort
(or rejects)
the world
payoffs
8
MORAL HAZARD
A moral hazard problem exists when the agent's action is not
verifiable, or when the agent receives private information after
the relationship has been initiated
Participants have the same information when the relationship
is established
The informational asymmetry arises once the contract has
been signed (the principal cannot observe or cannot verify the
action or the effort of the agent)
9
MORAL HAZARD
A supplies N determines Outcome
P designs the A accepts
contract
(or rejects) non verifiable the state of
and
effort
the world payoffs
10
ADVERSE SELECTION
Adverse Selection appears when the agent holds private
information before the relationship is begun
The principal can verify the agent's behaviour, but the optimal
decision, or the cost of this decision, depends on the agent's
type (of which the agent is the only informed party)
The principal knows that the agent could be any one of several
different types between which she cannot distinguish
It is a game with asymmetric information previous to the signing
of the contract
11
ADVERSE SELECTION
N chooses the type
of A which is only
observed by A
P designs the A accepts A supplies
contract
(or rejects)
effort
N chooses Outcome
the state of
and
the world payoffs
12
SIGNALING
This situation is similar
(asymmetric information)
to
adverse
selection
But, after learning his type, and before signing the
contract, the agent can send a signal that is observed
by the principal. Before the principal offers the contract,
the agent takes some sort of decision that may
influence the principal's beliefs about the agent's
identity
13
SIGNALING
N chooses the
type of A
which is only
observed by A
A sends
a signal
A accepts
N chooses
the state
(or rejects) A supplies
of the
P designs the
effort
world
contract
Outcome
and
payoffs
14
ECONOMICS OF INFORMATION:
INCENTIVES AND CONTRACTS
CONTRACTS UNDER SYMMETRIC INFORMATION
David Pérez-Castrillo (UAB and BGSE)
THE MODEL WITH SYMMETRIC INFORMATION
The set of possible results is finite
X = {x1, . . . , xn}
Probability of result xi conditional on effort e is:
Prob [x = xi | e] = pi (e) , for i ∈ {1, 2, . . . , n}.
with Σi=1,…,n pi (e) = 1
We assume pi (e) > 0 for all e, i
16
THE PRINCIPAL
The principal receives the production (result), and must pay the
agent for his part in the relationship
B (.) is the P’s utility function. Her objective is to obtain the greatest
profit:
B (x – w)
w is the payoff to the agent
We assume B' > 0, B" ≤ 0. The concavity of the function B(.)
indicates that the principal is either risk-neutral or risk averse
17
THE AGENT
The agent receives a monetary pay-off for his participation in the
relationship, and he supplies an effort which implies some cost
to him
The agent's utility function is:
U (w, e) = u (w) – v (e)
We assume
u' (w) > 0, u" (w) ≤ 0,
v' (e) > 0, v" (e) ≥ 0.
The expected utility that external opportunities offer the agent,
his reservation utility, is denoted by U
So long as the contract allows the agent to earn expected utility
not less than his reservation utility, he will accept it
18
SYMMETRIC INFORMATION CONTRACTS
The principal decides the effort e that she demands of the
agent, and the wages {w (xi)}i = 1, . . . , n that will be paid according
to the result
She must find out what contracts are acceptable by the agent,
given the effort demanded, and chooses from the contracts that
achieve this effort, the cheapest one
19
SYMMETRIC INFORMATION CONTRACTS
Max[e, {w (x)}] Σi pi (e) B(xi – w (xi) )
s.t.
Σi pi (e) u (w (xi) ) – v (e) ≥ U
(backward induction)
20
SYMMETRIC INFORMATION CONTRACTS
Let e° denote the efficient effort level and let {w° (xi)i = 1, . . . , n}
denote the associated payoffs
The FOC with respect to w° (xi) is
- pi (e°) B' (xi – w° (xi) ) + λ° pi (e°) u' (w° (xi) ) = 0
21
SYMMETRIC INFORMATION CONTRACTS
B' (x – w° (x ) )
λ° = u' (w° (x ) ) , for all i ∈ {1, 2, . . . , n}.
i
i
i
Then:
The multiplier λ°is strictly positive; hence PC binds
The optimal risk sharing leads to
B' (x – w° (x ) )
u' (w° (x ) ) = constant
i
i
i
22
SYMMETRIC INFORMATION CONTRACTS
From the FOC:
– B' (xi – w° (xi) ) + λ u' (w° (xi) ) = 0.
“Differentiating” with respect to xi and substituting λ
(using again the FOC) we have
rp
dw°
dxi = rp + ra .
23
ECONOMICS OF INFORMATION:
INCENTIVES AND CONTRACTS
CONTRACTS UNDER MORAL HAZARD
David Pérez-Castrillo (UAB and BGSE)
CONTRACTS UNDER MORAL HAZARD
n
Max
[e, {w (x )}
i
i = 1, . . . , n
]
∑ pi (e) B (xi – w (xi) )
i =1
n
s.t.
∑ pi (e) u (w (xi) ) – v (e) ≥ U
i =1
n
e ∈ arg Max { ∑ pi (ê) u (w (xi) ) – v (ê)}
ê
i =1
25
CONTRACTS UNDER MORAL HAZARD
The incentive compatibility constraint takes the form of a
maximum, difficult to treat
Two solutions:
Finite set of effort levels (the ICC takes the form of a finite
set of inequalities)
Substitute the program by its FOC and assume conditions
that make the “FIRST ORDER APPROACH” (F.O.A.) correct
26
CONTRACTS UNDER MORAL HAZARD
P is risk-neutral (this simplifies the analysis)
Effort can only take two values: e ∈ {eH, eL}.
Disutility of effort: v (eH) > v (eL).
The set of results X is ordered :
x 1 < x 2 < . . < x n.
piH = pi (eH) probability of xi when A offers high effort
Similarly, piL = pi (eL)
P prefers high effort to low
27
CONTRACTS UNDER MORAL HAZARD
n
Max
∑ piH [xi – w (xi)]
{w (xi)}i = 1, . . . , n i = 1
n
s.t.
∑ piH u (w (xi) ) – v (eH) ≥ U
i=1
n
∑ [piH – piL] u (w (xi) ) ≥ v (eH) – v (eL) .
i=1
28
CONTRACTS UNDER MORAL HAZARD
p iH
u' (w( x i )) = λ piH + µ [piH – piL],
for all i = 1, . . . , n .
Summing from i = 1 to i = n, we get λ > 0.
29
CONTRACTS UNDER MORAL HAZARD
1
u '( w ( xi )) = λ + µ

p iL 
1 − H  ,
pi 

for all i = 1, . . . , n .
µ ≠ 0; because for µ = 0, this equation indicates that w (xi) would
be constant
Since µ > 0, the agent's wage varies with the result
The wage will be greater the smaller is the ratio piL/piH
30
CONTRACTS UNDER MORAL HAZARD
The ratio piL/piH is called the likelihood ratio. It indicates the
precision with which the result xi signals that the effort level
was eH. The smaller is the likelihood quotient, the greater is
piH with respect to piL, and so the signal that the effort used
was eH is stronger
A reduction in the likelihood ratio is an increase in the
probability that the effort was eH when the result xi is
observed. Therefore, the wage must be greater if we want the
agent to exert high effort
Is it always optimal to set the wage increasing in the result?
31
A FIRM WANTS ITS SUBCONTRACTOR TO DO R&D
R&D
effort
High
(success)
0.40
100
Profits
Medium
20
Low
(failure)
0
0.40
No
effort
In this case:
0.10
0.10 / 0.40 <
< 0.10 / 0.20 <
< 0.80 / 0.40
0.80
Hence:
W(100) > w(0) > w(20)
0.20
0.10
32
PRINCIPALPRINCIPAL-AGENT MODEL
WITH MORAL HAZARD
AND LIMITED LIABILITY
David Pérez-Castrillo (UAB and BGSE)
PRINCIPAL, AGENT AND PROJECT
One risk-neutral shareholder (the Principal)
she owns a project
she lacks the skills to develop the project
One risk-neutral manager (the Agent)
he has the ability to conduct the project
he enjoys limited liability: his wage can not be negative
his reservation utility is U
The project
it yields R > 0 for the shareholder if successful and 0 if failure
probability of success depends on manager's effort e: p(e) = e
manager's effort is his own private information
manager's effort has a cost c(e) = c e²/2, with c > 0
THE CONTRACT AND THE CONSTRAINTS
The contract: (wR, w₀)
Incentive compatibility constraint ICC
e = argmaxê {w₀₀ + ê(wR – w₀)
₀ – c ê²/2}, or
e = (wR – w₀) /c
Participation constraint PC
w₀ + e (wR – w₀) – c e²/2 ≥ U
Limited Liability constraints LLC
wR ≥ 0
w₀ ≥ 0
THE PROGRAM
The shareholder solves:
max (wR, w₀) { e (R – wR) – (1 – e) w₀₀ }
s.t.
w₀ + e (wR – w₀) – c e²/2 ≥ U
e = (wR – w₀)
₀ /c
wR ≥ 0
w₀ ≥ 0
THE OPTIMAL CONTRACT
(a) If U > R² / (2c), the relationship is not profitable
(b) If U ∈ [R² / (8c), R² / (2c)], then (wR = (2cU)1/2, w₀₀ = 0)
manager's effort: e = (2U/c)1/2
manager's expected utility: U
shareholder's expected profits: R (2U/c)1/2 – 2U
(c) If U < R² / (8c), then (wR = R/2, w₀ = 0)
manager's effort: e = R / (2c)
manager's expected utility: R² / (8c)
shareholder's expected profits: R² / (4c)
THE OPTIMAL CONTRACT
SOME REFERENCES
Macho-Stadler, I. and D. Pérez-Castrillo (1997): “An Introduction
to the Economics of Information”, Oxford University Press.