Prerequisites Almost essential Adverse selection CONTRACT DESIGN MICROECONOMICS

Prerequisites Almost essential: Adverse selection CONTRACT DESIGN MICROECONOMICS Principles and Analysis Frank Cowell April 2018 Frank Cowell: Contract Design 1

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 April 2018 Frank Cowell: Contract Design 2

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 April 2018 Frank Cowell: Contract Design 3

Overview Contract design Design principles Roots in social choice and asymmetric information Model outline Full information Asymmetric information April 2018 Frank Cowell: Contract Design 4

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 April 2018 Frank Cowell: Contract Design 5

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) April 2018 Frank Cowell: Contract Design 6

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 April 2018 Frank Cowell: Contract Design 7

Overview Contract design Design principles A simple owner-andmanager story Model outline Full information Asymmetric information April 2018 Frank Cowell: Contract Design 8

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 April 2018 Frank Cowell: Contract Design 9

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 April 2018 A closer look at manager’s utility Frank Cowell: Contract Design 10

The utility function (1) y § Preferences over leisure and income § Indifference curves ing s a re e inc erenc f pre § Reservation utility § u = y(z) + y § yz(z) < 0 § u≥u u 1– z April 2018 Frank Cowell: Contract Design 11

The utility function (2) y § Preferences over leisure and output inc r pre easing fere nce § Indifference curves § Reservation utility § u = y(q/t) + y § yz(q/t) < 0 § u≥u u q April 2018 Frank Cowell: Contract Design 12

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 same preferences, independent of type April 2018 Frank Cowell: Contract Design 13

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 April 2018 Frank Cowell: Contract Design 14

The single-crossing condition §Preferences over leisure and output § a: High talent § b: Low talent y inc r pre easing fere nce § Those with different talents have different sloped ICs in this diagram § qa = taza j=b § qb = tbzb j=a q April 2018 Frank Cowell: Contract Design 15

Overview Contract design Design principles Where talent is known to all Model outline Full information Asymmetric information April 2018 Frank Cowell: Contract Design 16

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 April 2018 Frank Cowell: Contract Design 17

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 April 2018 Frank Cowell: Contract Design 18

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 April 2018 Frank Cowell: Contract Design 19

Full-information solution y §a type’s reservation utility §b type’s reservation utility §a type’s contract §b type’s contract _ub _ua p y*a y*b § Both types get contract where marginal disutility of effort equals marginal product of labour q q*b April 2018 q*a Frank Cowell: Contract Design 20

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 April 2018 Frank Cowell: Contract Design 21

Overview Contract design Design principles Where talent is private information Model outline Full information Asymmetric information April 2018 Frank Cowell: Contract Design 22

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 April 2018 Frank Cowell: Contract Design 23

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 April 2018 q*a Frank Cowell: Contract Design 24

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 April 2018 Frank Cowell: Contract Design 25

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)] April 2018 Frank Cowell: Contract Design 26

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 April 2018 Frank Cowell: Contract Design 27

Two types of Agent: contract design §a-type’s reservation utility §b-type’s contract §incentive-compatibility constraint §a-type’s contract §a contract schedule y ~a y ~b y q ~b q April 2018 ~a q Frank Cowell: Contract Design 28

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 April 2018 Frank Cowell: Contract Design 29

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 April 2018 Frank Cowell: Contract Design 30