EC476 Contracts and Organizations, Part III: Lecture 2

Transcription

1 EC476 Contracts and Organizations, Part III: Lecture 2 Leonardo Felli 32L.G January 2015

2 Moral Hazard: Consider the contractual relationship between two agents (a principal and an agent) The principal hires the agent to perform a task. The agent chooses his effort intensity, a, which affects the outcome of the task, q. The principal only cares about the outcome, but effort is costly for the agent, hence the principal has to compensate the agent for incurring the cost of effort. Effort is observable only to the agent, (it is the agent s private information). Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

3 General Setup Assume that the outcome of the task can take only two values: q {0, 1}. We assume that when q = 1 the task is successful and when q = 0 the task is a failure. The probability of success is: P{q = 1 a} = p(a), p ( ) > 0, p ( ) < 0. where p(0) = 0, lim a p(a) = 1, and p (0) > 1. The principal s preferences are represented by: V (q w), V ( ) > 0, V ( ) 0 where w is the transfer to the agent. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

4 General Setup (2) The agent s preferences are represented by the utility function separable in income and effort: U(w) φ(a), U ( ) > 0, U ( ) 0 where φ ( ) > 0, φ ( ) 0. For convenience we take φ(a) = a and we normalize the agent s outside option: U = 0. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

5 General Setup (3) Assume that: a is chosen by the agent before uncertainty is realized; a is only observed by the agent. It is his private information. q is verifiable information (observable to all agents involved in the contract Court included). the transfer w can only be contingent on the verifiable information q. q is not in a one-to-one relation with the effort a. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

6 First Best Benchmark The contract theory literature defines the first best world as a world where there are no frictions. In the current setting this implies that the contract offered by the principal can be contingent on the effort a. In other words, the effort a is verifiable (observable to all agents involved in the contract Court included). Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

7 First Best Contract The first best contract is obtained as the solution to the problem: max a,w i p(a) V (1 w 1 ) + (1 p(a)) V ( w 0 ) s.t. p(a) U(w 1 ) + (1 p(a)) U(w 0 ) a The optimal pair of transfers w1 and w 0 are such that the following FOC (Borch optimal risk-sharing rule) are satisfied: V (1 w1 ) U (w1 ) = V ( w0 ) U (w0 ) Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

8 First Best Contract (2) These transfers are paid only if the effort level coincides with a that satisfies the following FOC: [ V (1 w p (a ) 1 ) V ( w0 ) V (1 w1 ) + U(w 1 ) U(w 0 ) ] U (w1 ) = 1 U (w1 ) Finally the agent s expected utility coincides with the outside option: p(a ) U(w 1 ) + (1 p(a )) U(w 0 ) = a Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

9 First Best Contract Risk Neutrality If the principal is risk neutral: V (x) = x Then the conditions above become: and w 1 = w 0 = w U(w ) = a, p (a ) = If, instead, the agent is risk neutral: Then the optimum entails: U(x) = x 1 U (w ) w 1 w 0 = 1, p (a ) = 1. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

10 Second Best Contract If a is not verifiable then, for every w 1 and w 0, a is determined so that: max a p(a) U(w 1 ) + (1 p(a)) U(w 0 ) a (1) The latter is the agent s incentive problem. Only the agent controls a and hence incentives for the agent to choose the principal s desired level of a have to be induced through the contingent trasfer w(q). In other words, the second best contract can be contingent only on q. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

11 Second Best Contract (2) The second best contract can be obtained as the solution to the problem: max â,w i p(â) V (1 w 1 ) + (1 p(â)) V ( w 0 ) s.t. p(â) U(w 1 ) + (1 p(â)) U(w 0 ) â â arg max a p(a) U(w 1 ) + (1 p(a)) U(w 0 ) a The first constraint is known as the agent s individual rationality constraint, The second constraint is known as the agent s incentive compatibility constraint. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

12 Second Best Contract (3) The FOC of the incentive compatibility constraint are: p (â) [U(w 1 ) U(w 0 )] = 1 (2) A first observation: from this condition it is clear that full insurance: w 1 = w 0 leads to no incentives: p(0) = 0 Assumptions on p( ) imply that the solution to this condition is unique for any pair (w 0, w 1 ). We can replace the agent s (IC) by the set of FOC in (2). In general replacing the (IC) constraint with the FOC of the agent s effort choice problem is not a valid approach. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

13 Risk Neutral Agent Consider now the case in which the agent is risk neutral: U(x) = x we have seen that first best optimality requires p (a ) = 1 In this case the FOC of the (IC) constraint becomes: Therefore setting p (â)(w 1 w 0 ) = 1 w 1 w 0 = 1 leads to the first best allocation: optimal risk sharing and optimal incentives. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

14 Risk Neutral Agent (2) The reason is that: optimal risk sharing requires that the agent bears all the risk in the environment, optimal incentives requires that the agent is residual claimant. This is achieved by selling the activity to the agent at a fix price w 0 > 0 so that the risk averse principal receives full insurance. Notice that in this case we need the agent to have deep enough pockets: when the outcome is q = 0 the agent s payoff is w 0 < 0. The agent must be willing to incur a loss with a strictly positive probability. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

15 Resource Constrained Agent It is often natural to assume that the agent has no resources to put in the activity. This implies a resource constraint: w i 0. In this case the problem becomes: max â,w i p(â) V (1 w 1 ) + (1 p(â)) V ( w 0 ) s.t. p(â) w 1 + (1 p(â)) w 0 â p (â)(w 1 w 0 ) = 1 w i 0 i {0, 1} Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

16 Resource Constrained Agent (2) In the situation in which the agent is resource constrained not all the risk can be transferred to the agent: the constraint w i 0 will be binding for the transfer w 0 : w 0 = 0 It is still possible to create first best incentives but for this purpose the agent s needs to be rewarded. If w 1 w 0 = 1 then the agent s payoff is: p(a ) a > 0 since p (0) > 1 and p (a ) = 1. In other words the (IR) constraint is not binding. This is not necessarily optimal for the principal. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

17 Resource Constrained Agent (3) In particular, if we assume that the principal is risk neutral as well: V (x) = x then the principal s problem is: max â,w i p(â) (1 w 1 ) (1 p(â)) w 0 s.t. p(â) w 1 + (1 p(â)) w 0 â The solution implies that p (â)(w 1 w 0 ) = 1 w i 0 i {0, 1} w 0 = 0, w 1 = 1 p (â) Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

18 Resource Constrained Agent (4) Moreover, â solves the constrained problem: max â p(â) (1 w 1 ) s.t. p (â) w 1 = 1 or p (â) = 1 p(â) p (â) (p (â)) 2 Given that p ( ) < 0 then we conclude: â < a. The resource constraint implies a second best level of effort. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

19 Resource Constrained Agent (5) The principal trades off the lower effort choice by the agent against the higher compensation that the agents needs to provide for the first best level of effort. However the agent still gets a strictly positive payoff: p(â) p (â) â > 0 Indeed, by Taylor expansion we can show that there exists ξ (0, â) such that p(â) p (â) â = p (ξ) â2 2 > 0 Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

20 Risk Averse Principal and Agent: First Best Contract Recall now that the first best contract with risk averse principal and agent led to the Borch optimal risk-sharing rule: V (1 w1 ) U (w1 ) = V ( w0 ) U (w0 ) and the optimal effort level a that satisfies: p (a ) {V (1 w 1 ) V ( w 0 ) + λ[u(w 1 ) U(w 0 )]} = λ or [ V (1 w p (a ) 1 ) V ( w0 ) V (1 w1 ) + U(w 1 ) U(w 0 ) U (w1 ) ] = 1 U (w 1 ) Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

21 Risk Averse Principal and Agent: Second Best Contract Consider now the second best contract, this is the solution to the following problem max â,w i p(â) V (1 w 1 ) + (1 p(â)) V ( w 0 ) s.t. p(â) U(w 1 ) + (1 p(â)) U(w 0 ) â p (â) [U(w 1 ) U(w 0 )] = 1 Let λ and µ be the lagrange multipliers of the (IR) and (IC) constraints, respectively. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

22 Risk Averse Principal and Agent: Second Best Contract (2) The FOC with respect to w 1 and w 0 imply: V (1 w 1 ) U (w 1 ) = λ + µ p (â) p(â) and V ( w 0 ) U (w 0 ) = λ µ p (â) 1 p(â) Clearly for µ = 0 we get back Borch rule, however in general µ > 0: optimal insurance is distorted. Since V ( ) < 0 and U ( ) < 0 the agent faces in equilibrium more risk than he would face in the absence of moral hazard: w 1 > w 1, w 0 < w 0 Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

23 Risk Averse Principal and Agent: Second Best Contract (3) The second best level of effort ã is such that: p (ã) {V (1 w 1 ) V ( w 0 ) + λ[u(w 1 ) U(w 0 )]} = λ µ p (ã) [U(w 1 ) U(w 0 )] Recall that the first best level of effort a is such that: p (a ) {V (1 w 1 ) V ( w 0 ) + λ[u(w 1 ) U(w 0 )]} = λ We therefore can conclude that if the (IC) constraint is binding µ > 0 then in equilibrium the agent under-invests: ã < a Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

24 Adverse Selection Static Adverse Selection problem: one principal facing one agent who has private information on his type (preferences, intrinsic productivity). We consider the simple monopolist pricing model: a transaction between a buyer (the agent) and a seller (the principal). The seller sets the terms of the contract (tioli from the principal to the agent). The seller does not know how much the buyer is willing to pay for the commodity. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

25 Static Adverse Selection: Setup The buyer s preferences are represented by: U(q, T, θ i ) = q 0 p(x, θ i ) dx T T total transfer from the buyer to the seller, θ i preference characteristics of the buyer, p(x, θ i ) inverse (Marshallian) demand curve of the buyer. A special and convenient case is: U(q, T, θ i ) = θ i u(q) T where u ( ) > 0, u ( ) < 0, u(0) = 0, lim q 0 u (q) = +. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

26 Static Adverse Selection: Setup (2) The seller is risk neutral, the unit s cost of production is c > 0 and her profit for selling q units in exchange for T is: Π = T c q Question: what is the profit maximizing set of pairs (T, q) the seller will be able to induce the buyer to choose (price discriminating monopolist)? Assume that: θ i {θ L, θ H } and λ = Pr{θ i = θ L } Let U be the buyer s outside option, normalized: U = 0. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

27 Static Adverse Selection: First best Assume that the seller is perfectly informed on each buyer s type θ i. The contract is then (Ti, qi ), for i {L, H} The seller s problem is: max T i,q i T i c q i s.t. θ i u(q i ) T i 0 The constraint is known as the individual rationality (IR) constraint of the agent. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

28 Static Adverse Selection: First best (2) The solution is such that: θ i u (q i ) = c, i {L, H} and T i = θ i u(q i ), i {L, H} The seller chooses a quantity qi so that marginal utility equals marginal cost (efficiency), extracts the consumer s total willingness to pay by means of the transfer T i. The seller s total expected profit: λ (T L c q L ) + (1 λ) (T H c q H ) Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

29 Static Adverse Selection: Second best: If the seller cannot observe the buyer s type then she has to offer the same contract to both types. In other words the seller may offer to the agent (whatever his type) a set of choices {(T L, q L ), (T H, q H )} The problem is that the contract space is potentially very large: the set of functions T (q), of all shapes and features. Fortunately, the Revelation Principle simplifies the search for the best contract from the principal s perspective. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

30 Revelation Principle: General procedure: we transform a problem we cannot solve (contract space not well defined) in a problem in which all possible contracts are well defined and simple to manage. Each agent i observes its own preference characteristic: θ i. If the principal has all the bargaining power he chooses the mechanism (from the set of all possible games) which has the best equilibrium from her view point (mechanism design). The principal is a (Stackelberg) leader, she selects the game the agents will play so that the equilibrium of the agent s subgame is the best one from her viewpoint. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

31 Centralize the Mechanism: The first step consists in moving from the problem where the agent chooses the price (transfer) and quantity to a communication mechanism. A communication mechanism is a mechanism where the agent reports his private information ˆθ (the agent of course can lie) and the principal associates this report with a transfer T (ˆθ i ) and quantity q(ˆθ i ). Clearly there is no loss in generality in this first step. The revelation principle (Green and Laffont 1977, Myerson 1979, Harris and Townsend 1981, Dasgupta, Hammond and Maskin 1979) identifies the set of mechanisms among which the principal selects. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

32 Using Revelation Principle The second step consists in focussing on an equilibrium of the direct revelation mechanism (communication game) where the agent reports his true type. Revelation Mechanism guarantees that there exists no loss in generality in focussing on this type of equilibrium: direct revelation mechanism with truth-telling. The equilibrium is selected so as to maximize the principal s (mechanism designer s) payoff. Result (Revelation Mechanism) Every equilibrium of the communication game (indirect revelation game) corresponds to an equilibrium of the direct relegation game where it is optimal for each participating party to tell the truth about his type. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

33 Timing of revelation game: The principal selects the contract: the communication game offered to the agent. The agent decides whether to participate (accept the contract or not). The agent sends his message ˆθ i to the principal. The principal implements the allocation associated with the message received (T (ˆθ i ), q(ˆθ i )). We focus on the communication games where it is optimal for the agent to tell the truth: ˆθ i = θ i. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

34 The Static Adverse Selection Problem: Step 1: By revelation principle the principal s problem identifies the direct revelation mechanism (T i, q i ) = (T (q(ˆθ i )), q(ˆθ i )), i {L, H} that solves the problem: max λ (T L c q L ) + (1 λ)(t H c q H ) T i,q i s.t. θ H u(q H ) T H θ H u(q L ) T L θ L u(q L ) T L θ L u(q H ) T H θ H u(q H ) T H 0 θ L u(q L ) T L 0 Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

35 The Static Adverse Selection Problem (2): Two constraints guarantee individual rationality (IR), in other words they guarantee that both type of agents are willing to accept the contract offered by the principal: θ H u(q H ) T H 0 θ L u(q L ) T L 0 Two constraints guarantee incentive compatibility (IC), in other words they guarantee that the contract offered by the principal is such that in equilibrium both types of agent report the truth about their type: θ H u(q H ) T H θ H u(q L ) T L θ L u(q L ) T L θ L u(q H ) T H Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

36 The Static Adverse Selection Problem (3): Step 2: The individual rationality constraint of the type H will not bind at the optimum. Indeed since θ H > θ L : θ H u(q H ) T H θ H u(q L ) T L > θ L u(q L ) T L 0 Step 3: Solve the relaxed problem that ignores the (IC L ) constraint. To select which constraint to omit consider the two (IC) constraints at the first best optimum: θ H u(q H ) T H = 0, θ H u(q L ) T L = (θ H θ L ) u(q L ) > 0 clearly θ H u(q H ) T H < θ H u(q L ) T L Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

37 The Static Adverse Selection Problem (4): While θ L u(q L ) T L = 0, θ L u(q H ) T H = (θ L θ H ) u(q H ) < 0 clearly θ L u(q L ) T L > θ L u(q H ) T H Therefore the key (IC) constraint is the one of the H-type. The reason why the (IC) constraint of only one type of agent binds is Spence-Mirrlees Single Crossing Condition: [ U/ q ] = u (q) > 0 θ U/ T Marginal utility of consumption (relative to the marginal utility of money) rises with θ. This is key to be able to separate the two types. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

38 The Static Adverse Selection Problem (5): Step 4: Notice that the relaxed problem is such that both constraints bind at the optimum: max λ (T L c q L ) + (1 λ)(t H c q H ) T i,q i s.t. θ H u(q H ) T H θ H u(q L ) T L θ L u(q L ) T L 0 Proof: If (IC H ) does not bind then the principal can raise T H without affecting (IR L ), while improving the maximand. In other words the (IC H ) is binding. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

39 The Static Adverse Selection Problem (6): Solve the binding (IC H ) for T H : T H = θ H (u(q H ) u(q L )) + T L Substituting in the maximand, the relaxed problem becomes: max λ (T L c q L ) + (1 λ)[(θ H (u(q H ) u(q L )) + T L c q H ] T i,q i s.t. θ L u(q L ) T L 0 The maximand is monotonic increasing in T L while (IR L ) is monotonic decreasing in T L. Hence, at the optimum (IR L ) must be binding. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

40 The Static Adverse Selection Problem (7): Step 5: Solve the binding (IR L ) for T L and substitute them into the maximand. We get: or max q i λ [θ L u(q L ) c q L ] + + (1 λ) [θ H u(q H ) (θ H θ L ) u(q L ) c q H ] max q i [λ θ L (1 λ) (θ H θ L )] u(q L ) λ c q L + + (1 λ) [θ H u(q H ) c q H ] The second best contract (qi, Ti ) is then the solution to the unconstraint maximization problem above. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

41 The Static Adverse Selection Problem (8): To characterize the solution we distinguish two cases. Case 1: [λ θ L (1 λ) (θ H θ L )] 0 In this case the slope of the maximand with respect to q L is strictly negative for every q L 0: [λ θ L (1 λ) (θ H θ L )] u (q L ) λ c < 0 Therefore the principal chooses q L at a corner: q L = 0, T L = 0 Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

42 The Static Adverse Selection Problem (9): In other words the principal decides not to serve the type θ L of the agent. The principal then serves only the type θ H of the agent: qh = q H, T H = T H Recall that in this case the (IC L ) constraint we omitted is satisfied since: θ L u(q H ) T H < 0 In other words, the type θ L agent is strictly better off by announcing the truth about his type. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

43 The Static Adverse Selection Problem (10): Case 2: [λ θ L (1 λ) (θ H θ L )] > 0 In this case the optimal contract (q i, Ti ) is such that: 1) it satisfies efficiency at the top: q H θ H u (q H ) = c = q H In other words, according to the optimal contract the θ H agent receives the efficient quantity q H. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

44 The Static Adverse Selection Problem (11): 2) it safisfies inefficiency at the bottom: ql < q L [ ] θ L u (ql ) = c λ θ L > c λ θ L (1 λ) (θ H θ L ) In other words, according to the optimal contract the θ L agent receives an inefficiently low quantity q L. 3) inefficient premium to the top type: T H T H < T H = θ H u(q H ) (θ H θ L ) u(q L ) < θ H u(q H ) = T H In other words, according to the optimal contract not all the consumer surplus θ H u(q H ) is extracted from the θ H agent. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

45 The Static Adverse Selection Problem (12): 4) it prescribes an efficient transfer for the bottom type: TL < T L TL = θ L u(ql ) In other words, according to the optimal contract all the consumer surplus θ L u(ql ) is extracted from the θ L agent: perfect price discrimination. Notice that from the first two conditions we conclude q L < q H Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

46 The Static Adverse Selection Problem (13): Step 6: We still need to check that the omitted constraint (IC L ) holds. This is indeed the case: θ L u(ql ) T L θ L u(qh ) T H Since we do know that θ H > θ L and θ H [u(qh ) u(q L )] = TH T L > 0 Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

47 Taxation Principle: The result obtained can be re-interpreted in terms of the taxation principle. The principal offers a menu of (two) two-part tariff contracts: {(qh, T H ), (q L, T L )} These contracts are such that the L-type agent self-selects in choosing the contract (ql, T L ), While the H-type agent self-selects in choosing the contract (qh, T H ). Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48

48 Quantity Discounts This re-interpretation corresponds to a realistic indirect mechanism. An alternative indirect mechanism that is quite frequently observed in real life is the following: The good is offered at the price T L, If the consumer is willing to buy any quantity in excess of q L then he is offered a discount in the amount of (TH T L ), the balk quantities offered are either q L or q H. Leonardo Felli (LSE) EC476 Contracts and Organizations, Part III 19 January / 48