Prerequisites
Almost essential:
Adverse selection
CONTRACT DESIGN
MICROECONOMICS
Principles and Analysis
Frank Cowell
July 2015
Frank Cowell: Contract Design
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Purpose of contract design
 A step in moving the argument:
• from how we would like to organise the economy
• to what we can actually implement
 Plenty of examples of this issue:
• hiring a lawyer
• employing a manager
 Purpose and nature of the design problem
• construct a menu of alternatives
• to induce appropriate choice of action
 Key: takes account of incomplete information
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Informational issues
 Two key types of informational problem:
• each is relevant to design question
• each can be interpreted as a version of “Principal and Agent”
 Hidden action:
• The moral hazard problem
• concerned with unseen/unverifiable events
• and unseen effort
 Hidden information:
• the adverse selection problem
• concerned with unseen attributes
• and unseen effort
 Here focus on the hidden information problem
• How to design a payment system ex ante
• when the quality of the service/good cannot be verified ex ante
 Attack this in stages:
• outline a model
• examine full-information case
• then contrast this with asymmetric information
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Overview
Contract design
Design
principles
Roots in social choice and
asymmetric information
Model outline
Full
information
Asymmetric
information
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The essence of the model
 The Principal employs the Agent to produce some output
 But Agent may be of unknown type
• type here describes Agent’s innate productivity
• how much output per unit of effort
 The Principal designs a payment scheme
• 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
• because of delegation under imperfect information may have to forgo
some output
• “Agency cost”
 Use a parable to explain how it works
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A parable: paying a manager
 An owner hires a manager
• it makes sense to pay the manager according to talent
• but how talented is the manager?
 A problem of hidden information
• similar to adverse selection problem
• but here with a monopolist – the owner
 The nature of the design problem
• 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)
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The employment contract: information
 Perhaps talent shows
• ability can be observed
• or costlessly verified
• get a full-information solution
 Perhaps it doesn’t
• ability cannot be observed in advance of the contract
• will low ability applicants misrepresent themselves?
• will high ability applicants misrepresent themselves?
 The approach
• examine full-information solution
• get rules for contract design in this case
• remodel the problem for the second-best case
• modify contract rules
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Overview
Contract design
Design
principles
A simple owner-andmanager story
Model outline
Full
information
Asymmetric
information
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Model basics: owner
 Owner makes first move
• designs payment schedule for the manager
• makes a take-it-or-leave-it offer
 Has market power
• can act as a monopolist
• appropriates the gains from trade
 Gets profit after payment to manager:
• utility (payoff) to owner is just the profit pq – y
• p: price of output
• q: amount of output
• y: payment to manager
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Model basics: manager
 A manager’s talent and effort determines output:
•
•
•
•
q = tz
q : output produced
t : the amount of talent
z : the effort put in
 Manager’s preferences
•
•
•
•
•
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
• u : reservation utility
July 2015
A closer look at
manager’s utility
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The utility function (1)
 Preferences over leisure and income
y
 Indifference curves
 Reservation utility
 u = y(z) + y
 yz(z) < 0
 u≥u
u
1– z
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The utility function (2)
 Preferences over leisure and output
y
 Indifference curves
 Reservation utility
 u = y(q/t) + y
 yz(q/t) < 0
 u≥u
u
q
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Model basics: information
 There are different talent types j = 1, 2, …
• 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
 Profits (owner’s payoff) depend on talent:
• 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
• utility function is known
• also known that all managers have the same preferences, independent of
type
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Indifference curves: pattern
 Managers of all types have the same preferences
• uj = y(zj) + yj
• uj = y(qj/tj) + yj
 Function y() is common knowledge
• utility level uj of type j depends on effort zj
• also depends on payment yj
 Take indifference curves in (q, y) space
• u = y(q/tj) + y
• clearly slope of type j indifference curve depends on tj
• indifference curves of different types cross once only
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The single-crossing condition
Preferences over leisure and output
 High talent
 Low talent
y
 Those with different talents have
different sloped ICs in this diagram
 qa = taza
j=b
 qb = tbzb
j=a
q
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Overview
Contract design
Design
principles
Where talent is
known to all
Model outline
Full
information
Asymmetric
information
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Full information: setting
 Owner may be faced with a manager of any type j
 But owner can observe the type (talent) tj
• therefore can observe effort zj = qj/tj
• so the contract can be conditioned on effort
• offer manager of type j the deal (yj, zj)
 Owner prepares menu of such contracts in advance
• aims to maximise expected profits
 Manager then chooses effort in response
• aims to maximise utility
• this choice is correctly foreseen by the owner designing the
contract
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Full information: problem
 Owner aims to maximise expected profits
• expectation is over distribution of types
• maximisation subject to (known) manager behaviour
• participation constraint of type j
 Choose yj, zj to
• max Sj pj [ptjzj - yj]
• subject to yj + y(zj) ≥ uj
 Solve this using standard methods for constrained
maximum
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Full information: solution
 Set up standard Lagrangian:
 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]
 First-order conditions:
• lj = pj
• - yz(z*j) = ptj
• yj + y(z*j) = uj
 Interpretation
• “price” of constraint is probability of a type j manager
• MRS = MRT
• reservation utility constraint is binding
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Full-information solution
a type’s reservation utility
y
b type’s reservation utility
_ub
a type’s contract
b type’s contract
_ua
p
y*a
y*b
 Both types get contract where
marginal disutility of effort equals
marginal product of labour
q
q*b
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q*a
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Full information: conclusions
 “Price” of constraint is probability of getting a type-j
manager
 The outcome is efficient:
• MRS = MRT
• for each type of manager
 Owner drives manager down to reservation utility
• complete exploitation
• owner gets all the surplus
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Overview
Contract design
Design
principles
Where talent is
private
information
Model outline
Full
information
Asymmetric
information
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Asymmetric information: approach
 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
• 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
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Another look at the FI solution
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
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q*a
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Asymmetric information again
 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
• qb = tbzb with probability 1 - p
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The “second-best” model
 Participation constraint for the b type:
• yb + y(zb) ≥ ub
• Have to offer at least as much as available elsewhere
 Incentive-compatibility constraint for the a type:
• ya + y(qa/ta) ≥ yb + y(qb/ta)
• must be no worse off than if had taken b contract
 Maximise expected profits
• p[pqa - ya] + [1-p][pqb - yb]
 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)]
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Second-best: results
 Lagrangian is
p[pqa - ya] + [1-p][pqb - yb]
+ l [yb + y(qb/tb) - ub]
+ m [ya + y(qka/ta) - yb - y(qb/ta)]
 FOC are:
• - yz(qa/ta) = pta
• - yz(qb/tb) = ptb + kp/[1-p]
• k := yz(qb/tb) - [tb/ta] yz(qb/ta) < 0
 Results imply
• MRSa = MRTa
• MRSb < MRTb
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Two types of Agent: contract design
a-type’s reservation utility
b-type’s reservation utility
y
b-type’s contract
incentive-compatibility constraint
a-type’s contract
a contract schedule
y~ a
y~ b
q
q~ b
July 2015
q~ a
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Second-best: lessons
 a-types
• for high-talent people
• marginal rate of substitution equals marginal rate of
transformation
• no distortion at the top
 b-types
• for low-talent people
• MRS is strictly less than MRT
 Principal
• will make lower profits than in full-information case
• this is the Agency cost
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Summary
 Contract design fundamental to economic relations
 Asymmetric information raises deep issues:
• Principal cannot know the productivity of the agent beforehand
• Agent may have incentive to misrepresent information
• important not to have a manipulable contract
 Second-best approach builds these issues into the problem
• known distribution of types
• incentive-compatibility constraint
 Solution
• satisfies “no-distortion-at-the-top” principle
• gives no surplus to the lowest productivity type
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