Contract Design

1. Prerequisites Almost essential Adverse selection Contract Design MICROECONOMICS Principles and Analysis Frank Cowell August 2006

2. 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

3. 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

4. Overview... Contract design Design principles Roots in social choice and asymmetric information Model outline Full information Asymmetric information

5. 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

6. 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)

7. 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

8. Overview... Contract design Design principles A simple owner-and-manager story Model outline Full information Asymmetric information

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. Overview... Contract design Design principles Where talent is known to all… Model outline Full information Asymmetric information

17. 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

18. 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

19. 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

20. 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

21. 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

22. Overview... Contract design Design principles Where talent is private information Model outline Full information Asymmetric information

23. 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

24. 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

25. 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

26. 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)]

27. 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

28. ~ 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

29. 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

30. 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