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Prerequisites. Almost essential Adverse selection. Contract Design. MICROECONOMICS Principles and Analysis Frank Cowell . August 2006 . Purpose of contract design. A step in moving from how we would like to organise the economy… …to what we can actually implement
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Prerequisites Almost essential Adverse selection Contract Design MICROECONOMICS Principles and Analysis Frank Cowell August 2006
Purpose of contract design • 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: • 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
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
Overview... Contract design Design principles Roots in social choice and asymmetric information Model outline Full information Asymmetric information
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
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)
The employment contract: information • Perhaps talent shows • Ability can be observed or … • …costlessly verified • 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
Overview... Contract design Design principles A simple owner-and-manager story Model outline Full information Asymmetric information
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
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 A closer look at manager’s utility
The utility function (1) • Preferences over leisure and income y • Indifference curves increasing preference • Reservation utility • u = y(z) + y • yz(z) < 0 • u≥u u 1– z
The utility function (2) • Preferences over leisure and output y • Indifference curves increasing preference • Reservation utility • u = y(q/t) + y • yz(q/t) < 0 • u≥u u q
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
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 • But utility level ujof type j depends on effort zj and 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
The single-crossing condition • Preferences over leisure and output y • High talent increasing preference • Low talent • Those with different talent will have different sloped ICs in this diagram j=b • qa = taza j=a • qb = tbzb q
Overview... Contract design Design principles Where talent is known to all… Model outline Full information Asymmetric information
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
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
Full information: solution • Set up standard Lagrangean: • 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
ub ua _ _ Full-information solution • a type’s reservation utility y • b type’s reservation utility • a type’s contract • b type’s contract 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 • “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
Overview... Contract design Design principles Where talent is private information Model outline Full information Asymmetric information
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
ub ua _ _ Another look at the FI solution • a type’s reservation utility y • b type’s reservation utility • a type’s contract • b type’s contract • a type’s utility with b type contract p y*a • An a type would like to masquerade as a b type! y*b q q*b q*a
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 = tbzbwith probability 1-p
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 took 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)]
Second-best: results • Lagrangean 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
~ a y ~ b y ~ b ~ a q q Two types of Agent: contract design • 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 q
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
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