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

Frank Cowell: Contract Design 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 July

Frank Cowell: Contract Design 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 July

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

Frank Cowell: Contract Design 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 July

Frank Cowell: Contract Design 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) July

Frank Cowell: Contract Design 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 July

Frank Cowell: Contract Design Overview July Design principles Model outline Full information Asymmetric information Contract design A simple owner-and- manager story

Frank Cowell: Contract Design 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 July

Frank Cowell: Contract Design Model basics: manager  A manager’s talent and effort determines output: q =  z q : output produced  : the amount of talent z : the effort put in  Manager’s preferences  =  z  + y  : utility level y : income received  : decreasing, strictly concave, function equivalently:  =  q /  + y  Manager has an outside option  : reservation utility July A closer look at manager’s utility

Frank Cowell: Contract Design The utility function (1) July increasing preference y 1– z  Preferences over leisure and income  Indifference curves   =  (z) + y   z (z) < 0  Reservation utility  ≥  ≥  

Frank Cowell: Contract Design The utility function (2) July increasing preference y q  Preferences over leisure and output  Indifference curves   =  (q/  ) + y   z (q/  ) < 0  Reservation utility  ≥  ≥  

Frank Cowell: Contract Design Model basics: information  There are different talent types j = 1, 2, … type j has talent  j probability of a manger being type j is  j probability distribution is common knowledge owner may or may not know type j of a potential manager  Profits (owner’s payoff) depend on talent: pq j  y j q j =  j z j : the output produced by a type j manager z j : 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 July

Frank Cowell: Contract Design Indifference curves: pattern  Managers of all types have the same preferences  j =  z j  + y j  j =  q j  j  + y j  Function  is common knowledge utility level  j of type j depends on effort z j also depends on payment y j  Take indifference curves in (q, y) space  =  q  j  + y clearly slope of type j indifference curve depends on  j indifference curves of different types cross once only July

Frank Cowell: Contract Design The single-crossing condition July increasing preference y q j = b j = a  Preferences over leisure and output  High talent  q a =  a z a  Low talent  q b =  b z b  Those with different talents have different sloped ICs in this diagram

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

Frank Cowell: Contract Design Full information: setting  Owner may be faced with a manager of any type j  But owner can observe the type (talent)  j therefore can observe effort z j = q j /  j so the contract can be conditioned on effort offer manager of type j the deal ( y j, z j )  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 July

Frank Cowell: Contract Design 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 y j, z j to max  j  j [p  j z j  y j ] subject to y j +  z j  ≥  j  Solve this using standard methods for constrained maximum July

Frank Cowell: Contract Design Full information: solution  Set up standard Lagrangian:  Lagrange multiplier j for participation constraint on type j  choose y j, z j, j to max   j  j [p  j z j  y j ] +  j  j [y j +  z j  −  j ]  First-order conditions: j =  j  z  z* j  = p  j y j +  z* j  =  j  Interpretation “price” of constraint is probability of a type j manager MRS = MRT reservation utility constraint is binding July

Frank Cowell: Contract Design Full-information solution July y q q *a q *b y *b y *a _ bb _ aa p  a type’s reservation utility  b type’s reservation utility  a type’s contract  b type’s contract  Both types get contract where marginal disutility of effort equals marginal product of labour

Frank Cowell: Contract Design 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 July

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

Frank Cowell: Contract Design 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 July

Frank Cowell: Contract Design Another look at the FI solution July y q q *a q *b y *b y *a _ bb _ aa p  a type’s reservation utility  b type’s reservation utility  a type’s contract  b type’s contract  a type’s utility with b type contract  An a type would like to masquerade as a b type!

Frank Cowell: Contract Design 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 q a =  a z a with probability  q b =  b z b with probability  July

Frank Cowell: Contract Design The “second-best” model  Participation constraint for the b type: y b +  z b  ≥  b Have to offer at least as much as available elsewhere  Incentive-compatibility constraint for the a type: y a +  q a /  a  ≥ y b +  q b /  a  must be no worse off than if had taken b contract  Maximise expected profits  [pq a  y a ] + [1  ][pq b  y b ]  Choose q a  q b  y a  y b  to max  [pq a  y a ] + [1  ][pq b  y b  ] + [y b +  q b /  b    b ] +  [y a +  q a /  a   y b   q b /  a  ] July

Frank Cowell: Contract Design Second-best: results  Lagrangian is  [pq a  y a ] + [1  ][pq b  y b ] + [y b +  q b /  b    b ] +  [y a +  qk a /  a   y b   q b /  a  ]  FOC are:  z  q a /  a  = p  a  z  q b /  b  = p  b + k  [1  ] k :=  z  q b /  b   [  b /  a ]  z  q b /  a   Results imply MRS a = MRT a MRS b < MRT b July

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

Frank Cowell: Contract Design 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 July

Frank Cowell: Contract Design 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 July