Prerequisites
Almost essential
Adverse selection
Frank Cowell: Microeconomics
August 2006
Contract Design
MICROECONOMICS
Principles and Analysis
Frank Cowell
Purpose of contract design
Frank Cowell: Microeconomics
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A step in moving from how we would like to
organise the economy…
…to what we can actually implement
Plenty of examples of this issue:
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Purpose and nature of the design problem
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hiring a lawyer
employing a manager
construct a menu of alternatives…
…to induce appropriate choice of action
Key: takes account of incomplete information
Informational issues
Frank Cowell: Microeconomics
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Two key types of informational problem:
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Hidden action:
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the adverse selection problem
concerned with unseen attributes…
…and unseen effort
Here focus on the hidden information problem
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The moral hazard problem
concerned with unseen/unverifiable events…
…and unseen effort
Hidden information:
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each is relevant to design question
each can be interpreted as a version of “Principal and Agent”
How to design a payment system ex ante…
…when the quality of the service/good cannot be verified ex ante
Attack this in stages:
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outline a model
examine full-information case
then contrast this with asymmetric information
Overview...
Contract design
Frank Cowell: Microeconomics
Design
principles
Roots in social
choice and
asymmetric
information
Model outline
Full
information
Asymmetric
information
The essence of the model
Frank Cowell: Microeconomics
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The Principal employs the Agent to produce some output
But Agent may be of unknown type
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The Principal designs a payment scheme
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takes into account that type is unknown…
…and that one type of Agent might try to masquerade as another
Provides an illustration of second best problem
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type here describes Agent’s innate productivity
…how much output per unit of effort
because of delegation under imperfect information may have to
forgo some output
… “Agency cost”
Use a parable to explain how it works
A parable: paying a manager
Frank Cowell: Microeconomics

An owner hires a manager
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A problem of hidden information
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It makes sense to pay the manager according to talent
But how talented is the manager?
Similar to adverse selection problem
But here with a monopolist – the owner
The nature of the design problem
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Owner acts as designer
Wants to maximise expected profits
Wants to ensure that manager acts in accordance with
this aim
“Mechanism” here is the design of contract (s)
Frank Cowell: Microeconomics
The employment contract:
information
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Perhaps talent shows
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Perhaps it doesn’t
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Ability can be observed or …
…costlessly verified
Full-information solution
Ability cannot be observed in advance of the contract
Will low ability applicants misrepresent themselves?
Will high ability applicants misrepresent themselves?
The approach
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Examine full-information solution
Get rules for contract design in this case
Remodel the problem for the second-best case
Modify contract rules
Overview...
Contract design
Frank Cowell: Microeconomics
Design
principles
A simple ownerand-manager
story
Model outline
Full
information
Asymmetric
information
Model basics: owner
Frank Cowell: Microeconomics
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Owner makes first move
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Has market power
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designs payment schedule for the manager
makes a take-it-or-leave-it offer
Can act as a monopolist
Appropriates the gains from trade
Gets profit after payment to manager:
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utility (payoff) to owner is just the profit pq - y
p: price of output
q: amount of output
y : payment to manager
Model basics: manager
Frank Cowell: Microeconomics
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A manager’s talent and effort determines output:
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Manager’s preferences
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q = tz.
q : output produced
t : the amount of talent
z : the effort put in
u = y(z) + y
u : utility level
y : income received
y() : decreasing, strictly concave, function
equivalently: u = y(q / t) + y.
Manager has an outside option
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u : reservation utility
A closer look at
manager’s utility
The utility function (1)
Frank Cowell: Microeconomics
Preferences over leisure and
income
y
 Indifference curves
 Reservation utility
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u = y(z) + y
yz(z) < 0
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u≥u
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u
1– z
The utility function (2)
Frank Cowell: Microeconomics
Preferences over leisure and
output
y
 Indifference curves
 Reservation utility

u = y(q/t) + y
yz(q/t) < 0

u≥u

u
q
Model basics: information
Frank Cowell: Microeconomics
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There are different talent types j = 1,2,…
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Profits (owner’s payoff) depend on talent:
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Type j has talent tj
Probability of a manger being type j is pj
Probability distribution is common knowledge
Owner may or may not know type j of a potential manager
pqj - yj.
qj = tjzj: the output produced by a type j manager
zj : effort put in by a type j manager
Managers’ preferences are common knowledge
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Utility function is known
Also known that all managers have the same preferences,
independent of type
Indifference curves: pattern
Frank Cowell: Microeconomics
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Managers of all types have the same preferences
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Function y() is common knowledge
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uj = y(zj) + yj.
uj = y(qj/tj) + yj.
But utility level uj of type j depends on effort zj and
payment yj.
Take indifference curves in (q, y) space
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u = y(q/tj) + y.
Clearly slope of type j indifference curve depends on tj.
Indifference curves of different types cross once only
The single-crossing condition
Frank Cowell: Microeconomics
Preferences over leisure and
output
y
 High talent
 Low talent
 Those with different
talent will have
different sloped ICs in
this diagram
j=b
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j=a
q

qa = taza
qb = tbzb
Overview...
Contract design
Frank Cowell: Microeconomics
Design
principles
Where talent is
known to all…
Model outline
Full
information
Asymmetric
information
Full information: setting
Frank Cowell: Microeconomics
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Owner may be faced with a manager of any type j
But owner can observe the type (talent) tj
j
j j
 Therefore can observe effort z = q /t
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Owner prepares menu of such contracts in
advance
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So the contract can be conditioned on effort
Offer manager of type j the deal (yj, zj)
Aims to maximise expected profits
Manager then chooses effort in response
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Aims to maximise utility
This choice is correctly foreseen by the owner
designing the contract
Full information: problem
Frank Cowell: Microeconomics
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Owner aims to maximise expected profits
Expectation is over distribution of types.
 Maximisation subject to (known) manager
behaviour
 Participation constraint of type j.
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Choose yj, zj to
max Sj pj [ptjzj - yj]
 subject to yj + y(zj) ≥ uj.
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Solve this using standard methods for
constrained maximum
Full information: solution
Frank Cowell: Microeconomics
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Set up standard Lagrangean:
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First-order conditions:
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Lagrange multiplier lj for participation constraint on
type j.
Choose yj, zj, lj to max
Sj pj [ptjzj - yj] +Sj lj [yj + y(zj) − uj]
lj = pj
- yz(z*j) = ptj
yj + y(z*j) = uj
Interpretation
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“Price” of constraint is probability of a type j manager
MRS = MRT
Reservation utility constraint is binding
Full-information solution
Frank Cowell: Microeconomics
y
a type’s reservation utility
ub
_
b type’s reservation utility
a type’s contract
b type’s contract
_ua
p
y*a
 Both types get
contract where
marginal disutility of
effort equals marginal
product of labour
y*b
q
q*b
q*a
Full information: conclusions
Frank Cowell: Microeconomics
“Price” of constraint is probability of
getting a type-j manager
 The outcome is efficient:
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MRS = MRT
 …for each type of manager
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Owner drives manager down to reservation
utility
complete exploitation
 owner gets all the surplus
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Overview...
Contract design
Frank Cowell: Microeconomics
Design
principles
Where talent is
private
information
Model outline
Full
information
Asymmetric
information
Asymmetric information: approach
Frank Cowell: Microeconomics
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Full-information contract is simple and efficient
However, this version is not very interesting.
Problem arises when contract has to be drawn up
before talent is known
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Agent may have an incentive to misrepresent his talents
this will impose a constraint on the design of the
contract
Re-examine the Full-information solution
Another look at the FI solution
Frank Cowell: Microeconomics
y
a type’s reservation utility
ub
_
b type’s reservation utility
a type’s contract
b type’s contract
a type’s utility with b type contract
_ua
p
y*a
 An a type would
like to masquerade as
a b type!
y*b
q
q*b
q*a
Asymmetric information again
Frank Cowell: Microeconomics
As we have seen a type would want to
mimic a b type
 We can exploit a standard approach to the
problem.
 Assume that the distribution of talent is
known.
 For simplicity take two talent levels
 qa = taza with probability p
b
b b
 q = t z with probability 1-p
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The “second-best” model
Frank Cowell: Microeconomics
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Participation constraint for the b type:
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Incentive-compatibility constraint for the a type:
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yb + y(zb) ≥ ub.
Have to offer at least as much as available elsewhere
ya + y(qa/ta) ≥ yb + y(qb/ta).
Must be no worse off than if took b contract
Maximise expected profits
a
a
b
b
 p[pq - y ] + [1-p][pq - y ].
Choose qa, qb, ya, yb to max
p[pqa - ya] + [1-p][pqb - yb ]
+ l [yb + y(qb/tb) - ub]
+ m [ya + y(qa/ta) - yb - y(qb/ta)]
Second-best: results
Frank Cowell: Microeconomics
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Lagrangean is
p[pqa - ya] + [1-p][pqb - yb]
+ l [yb + y(qb/tb) - ub]
+ m [ya + y(qka/ta) - yb - y(qb/ta)]
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FOC are:
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- yz(qa/ta) = pta
- yz(qb/tb) = ptb + kp/[1-p]
k := yz(qb/tb) - [tb/ta] yz(qb/ta) < 0
Results imply
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MRSa = MRTa
MRSb < MRTb
Two types of Agent: contract design
Frank Cowell: Microeconomics
a type’s reservation utility
y
b type’s reservation utility
b type’s contract
incentive-compatibility constraint
b type’s contract
a contract schedule
y~ a
y~ b
q
q~ b
q~ a
Second-best: lessons
Frank Cowell: Microeconomics
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a-types
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b-types
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for high-talent people…
…marginal rate of substitution equals marginal rate of
transformation
no distortion at the top
for low-talent people…
…MRS is strictly less than MRT
Principal
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will make lower profits than in full-information case
this is the Agency cost
Summary
Frank Cowell: Microeconomics
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Contract design fundamental to economic relations
Asymmetric information raises deep issues:
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Second-best approach builds these issues into the problem
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Principal cannot know the productivity of the agent beforehand
Agent may have incentive to misrepresent information
important not to have a manipulable contract
known distribution of types
incentive-compatibility constraint
Solution
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satisfies “no-distortion-at-the-top” principle
gives no surplus to the lowest productivity type