Teoria das organizações e contratos

Transcription

1 Teoria das organizações e contratos Chapter 6: Adverse Selection with two types Mestrado Profissional em Economia 3 o trimestre 2015 EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

2 Outline 1 Introduction 2 A Model of Adverse Selection 3 Principals Competing for Agents EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

3 Outline 1 Introduction 2 A Model of Adverse Selection 3 Principals Competing for Agents EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

4 Informational structure Very often, parties to a contract do not have all the relevant information about each other Consider a person who hires a carpenter The task to be done may be well defined However, the worker s ability, cleanliness, and manners are not Adverse selection problem Before the signing of the contract, the agent has more information than the principal concerning certain aspects of his personal characteristics EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

5 Other examples A driver knows more than his insurance company about his driving habits: if he uses motorways or local roads, or the number of daily hours spent behind the wheel A firm has more information than the government about the costs of carrying out a certain project EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

6 Adverse selection in general Adverse selection refers not only when the agent s informational advantage concerns his own personal characteristics But also when there is asymmetric information regarding any variable relevant to the contractual relationship Consider the example of the homeowner who hires a carpenter He may not be aware of the job s difficulty Or the cost of the required materials A lawyer has more information than his client regarding the legal history concerning similar cases, which relates to the probability of winning the case A regulated firm may know more than the government about the market in which it operates EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

7 Informational asymmetries and efficiency The agent will only reveal his information to the principal if it is in his interest The agent may try to profit from information by keeping it private The principal s problem is to find a way to reduce her informational disadvantage The presence of asymmetric information can result in modifications with respect to the first-best agreement (when information is symmetric) EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

8 Lemon s problem: Akerlof (1970) Consider the market for second-had cars The owner knows the quality of the car he is selling, but the buyer does not Some cars are placed on the market because their owners simply want a bigger or better one Others because they have been involved in a major accident or after having been used has driving-school vehicles EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

9 Lemon s problem: Akerlof (1970) Represent the quality of a second-hand car by a real number Quality k [0, 1] is uniformly distributed (for simplicity) Quality 0 is the worst and 1 the best We assume that both the seller and buyer are risk-neutral A seller is willing to sell a car of quality k for a price p s k A buyer values a car of quality k at p b k There are gains to trade if p b > p s Let s assume that p b = (3/2)p s EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

10 Lemon s problem: symmetric case Assume information is symmetric Any car s type would be sold A car of quality k would sell in the market at a price P (k) satisfying p s k P (k) p b k The value of P (k) depends on the trade protocol and the relative bargaining powers of the buyer and the seller If the buyer had all the bargaining power, the selling price of the car would be P (k) = p s k EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

11 Lemon s problem Assume now that the buyer does not know the quality of the car sold Denote by P the price at which a car is being offered The buyer needs to form expectations about the quality of the car sold at price P What is the available information? The buyer knows that the owner of a car with quality k is willing to sell at price P only if P p s k EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

12 Bayesian expectations Denote by µ the prior belief of the buyer on the cars quality µ is a probability measure on the set [0, 1] of a car s quality We assume (for simplicity) it is uniform µ([a, b]) = b a When observing the price P being offered, the buyer knows that the quality of the car belongs to K(P ) = {k [0, 1] : p s k P } = [0, P/p s ] We assume that buyer updates his beliefs using the Bayes rule µ( K(P )) : A [0, 1] µ(a K(p)) µ(k(p)) EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

13 Bayesian expectations Since the agent is risk-neutral, he is willing to buy the car only if P p b [0,1] kµ(dk K(p)) = p b p s P P/ps 0 kdk = p b P 2p s Since p b = 3/2p s, no transactions will take place if P > 0 There will only be markets for lemons, i.e., cars with the lowest quality The informational problem is large enough to cause the disappearance of the market for second-hand cars EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

14 Outline 1 Introduction 2 A Model of Adverse Selection 3 Principals Competing for Agents EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

15 Production technology Consider a risk-neutral principal who contracts an agent to carry out some effort The relationship allows a certain result to be obtained Its monetary value is denoted by x X R represents the set of possible values for x The final result obtained depends on the effort that the agent dedicates to the task, denoted by e the realization of a random variable E R + represents the set of possible efforts The probability of outcome x i when effort e is exerted is denoted by p(x i e) or p i (e) EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

16 Assumptions Effort e is assumed to be verifiable Denote by Π(e) the expected production Π(e) = n p i (e)x i = p(x e)e = E[x e] x X i=1 We assume that Π is twice differentiable with Π > 0 and Π < 0 EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

17 Asymmetric information The agent could be either of two types {g, b} Types differ only with respect to disutility of effort U g (w, e) = u(w) v(e) and U b (w, e) = u(w) kv(e) where k > 1 Type-g agents are also called more efficient How much he requires to exert some effort level is lower Type-b agents are called less efficient EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

18 Assumption Assumption The functions u and v are twice differentiable with u > 0, u 0, v > 0 and v > 0 We have v : E R + R with v(0) = 0 EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

19 Timing of the game 1 Nature chooses the type of the agent 2 The principal designs the contract 3 The agent accepts or rejects 4 The agent supplies effort (required by the accepted contract) 5 Nature chooses the random shock on the technology 6 Outcomes are realized and payoff are made EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

20 Symmetric information: type-g agent Assume there is no adverse selection: the principal observes the type of the agent With a type-g agent, the principal should solve subject to max e,w Π(e) w u(w) v(e) U There is no reason to allow for stochastic payments since the principal is risk-neutral and effort is observable (PC) EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

21 Symmetric information: type-g agent The optimal contract (e g, w g ) is characterized by The binding participation constraint The first-order condition u(w g ) v(e g ) = U Π (e g ) = v (e g ) u (w g ) EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

22 Symmetric information: type-b agent The optimal contract (e b, w b ) is characterized by The binding participation constraint The first-order condition u(w b ) kv(e b ) = U Π (e b ) = kv (e b ) u (w b ) EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

23 The symmetric case Given a type i {g, b} we let Iso i (U) := {(w, e) : U i (w, e) = U} The curve Iso b (U) is beneath Iso g (U) because effort is more costly for type-b agents Let and FO b := {(w, e) : Π (e) = kv (e)/u (w)} FO g := {(w, e) : Π (e) = v (e)/u (w)} The curve FO b is beneath the curve FO g EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

24 The symmetric case EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

25 The symmetric case Proposition It is optimal for the principal to demand more effort from the agent to whom effort if less costly, i.e., e g > e b We cannot be sure about wages There are two effects of opposite sign For given a particular effort, type-b agent requires a greater wage in order to participate However, the principal demands less effort from B than from G EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

26 The symmetric case EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

27 Asymmetric information Assume now that the principal does not observe the agent s type He can offer a menu of contracts {(e 1, w 1 ), (e 2, w 2 ), (e 3, w 3 )} and let the agent choose A possible menu is {(e b, w b ), (e g, w g )} Proposition Type-B agent chooses (e b, w b ) Type-G agent also chooses (e b, w b ) EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

28 Asymmetric information Type-G agents choose the contract (e b, w b ) EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

29 Asymmetric information Symmetric information contracts are not optimal under asymmetric information EESP (FGV) Teoria das organizac o es e contratos 3o trimestre / 104

30 Principal s expected profits Assume that the principal has a prior about the distribution of types He believes that the fraction of agents being of type-g is q where 0 < q < 1 Alternatively, the principal believes that the probability of an agent being of type-g is q If type-g agent accepts contract (e g, w g ) and type-b agent accepts contract (e b, w b ) then the principal expected profit is q [Π(e g ) w g ] + (1 q) [Π(e b ) w b ] EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

31 Menus of contracts We assume that the principal offers a menu of contracts {(e b, w b ), (e g, w g )} The contract (e i, w i ) is directed to type-i agent It is as if the principal ask the agent what is his type and offers the corresponding contract Since the type is not observable, it is equivalent to let the agent choose among the two contracts EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

32 Truth-telling constraints The principal understands that each agent chooses the contract that suits him Therefore, the menu of contracts must be self-selective Type-B agent must prefer contract (e b, w b ) to (e g, w g ), i.e., u(w b ) kv(e b ) u(w g ) kv(e g ) (IC b ) Type-G agent must prefer contract (e g, w g ) to (e b, w b ), i.e., u(w g ) v(e g ) u(w b ) v(e b ) (IC g ) EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

33 More general contracts The principal could offer a menu with more than two contracts However, only two (or one) of them will be selected Therefore, without any loss of generality, we can restrict attention to menus with two contracts EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

34 The principal s problem max q (e g,w g ),(e b,w b ) [Π(eg ) w g ] + (1 q) [Π(e b ) w b ] subject to participation constraint of type-g participation constraint of type-b incentive compatibility of type-g incentive compatibility of type-b u(w g ) v(e g ) U (PC g ) u(w b ) kv(e b ) U (PC b ) u(w g ) v(e g ) u(w b ) v(e b ) (IC g ) u(w b ) kv(e b ) u(w g ) kv(e g ) (IC b ) EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

35 Contracts with adverse selection Proposition The only participation constraint that the principal need to be concerned with is that corresponding to the least efficient agent Consider a menu of contracts ((e g, w g ), (e b, w b )) satisfying (PC b ) and (IC g ) Then (PC g ) is automatically satisfied If e b > 0 then (PC g ) is not binding EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

36 The principal s problem: reduced form subject to max q (e g,w g ),(e b,w b ) [Π(eg ) w g ] + (1 q) [Π(e b ) w b ] participation constraint of type-b incentive compatibility of type-g incentive compatibility of type-b u(w b ) kv(e b ) U (PC b ) u(w g ) v(e g ) u(w b ) v(e b ) (IC g ) u(w b ) kv(e b ) u(w g ) kv(e g ) (IC b ) EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

37 Contracts with adverse selection Proposition If a menu of contracts satisfies the incentive-compatibility constraints then greater effort is demanded of the most efficient agent Consider a menu of contracts ((e g, w g ), (e b, w b )) satisfying (IC g ) and (IC b ) Then we must have e g e b Proof. Show that v(e g ) v(e b ) u(w g ) u(w b ) k[v(e g ) v(e b )] (IC) EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

38 Optimal contracts with adverse selection Assume that ((e g, w g ), (e b, w b )) solves the principal s problem Denote by λ, µ and δ the Lagrange multipliers associated to (PC b ), (IC g ) and (IC b ) respectively µ δ = q u (w g ) λ µ + δ = 1 q u (w b ) µ δk = qπ (e g ) v (e g ) λk µ + δk = (1 q)π (e b ) v (e b ) (FO w g) (FO w b) (FO e g) (FO e b) EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

39 Optimal contracts with adverse selection Proposition The participation constraint of the less efficient agent (type-b) binds, i.e., u(w b ) kv(e b ) = U This is because λ > 0 The principal extracts all the surplus of the type-b agent EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

40 Optimal contracts with adverse selection Proposition The self-selection constraint of the more efficient agent (type-g) binds, i.e., u(w g ) v(e g ) = u(w b ) v(e b ) This is because µ > 0 In particular we have u(w g ) v(e g ) = U + (k 1)v(e b ) The principal cannot extract all the surplus of the agent with the lowest cost The most efficient agent benefits from the adverse selection problem EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

41 Optimal contracts with adverse selection Proposition The principal does not demand the same effort from both agent s types, i.e., e g > e b Assume, by way of contradiction, that e g = e b Show that we necessarily have w g = w b Induce a contradiction EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

42 Optimal contracts with adverse selection Proposition The incentive condition for low-efficient agents does not bind This follows from (IC) and the fact that e g > e b EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

43 Optimal contracts with adverse selection Proposition The first-best first order condition is satisfied for the type-g agents, i.e., 1 u (w g ) = Π (e g ) v (e g ) There is no distortion at the top : this is the same FO condition as in the symmetric case 1 u (w g ) = Π (e g ) v (e g ) The presence of adverse selection does not affect the efficient allocation of effort and wage for the agent with the best characteristics The contracts (e g, w g ) and (e g, w g ) differ because the expected utility is different EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

44 Optimal contracts with adverse selection Proposition The first order condition of the type-b agents is distorted Π (e b ) = kv (e b ) u (w b ) q(k 1) v (e b ) + (1 q) u (w g ) }{{} distortion The principal needs to make the contract (e b, w b ) less attractive to type-g agents By distorting, the principal looses efficiency with respect to type-b agents But she pays less informational rent to type-g agents EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

45 Optimal contracts with adverse selection: summary Participation constraints The participation constraint only binds for the agent with the highest cost (type-b) The most efficient agent (type-g) receives an informational rent of (k 1)v(e b ) The type-g agent receives utility greater than his reservation level due to his private information EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

46 Optimal contracts with adverse selection: summary Incentive compatibility constraints The IC condition binds for the high-efficiency agent The IC condition of the low-efficiency agent does not bind First best contracts The efficiency condition of the first best contract is satisfied for the good agent A distortion is introduced into the efficiency condition of the bad agent EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

47 Optimal contracts with adverse selection: distortion Why does the principal choose to distort the efficiency condition of the contract offered to the type-b agent? The intuition is to make the contract (e b, w b ) less attractive to type-g agents By distorting, the principal loses efficiency with respect to type-b agents However, she pays less informational rent to type-g agents The trade-off between these two effects is favourable to the distortion EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

48 Optimal contracts with adverse selection: distortion EESP (FGV) Teoria das organizac o es e contratos 3o trimestre / 104

49 Optimal contracts with adverse selection: distortion Why the trade-off is favourable to the distortion? Start from agent B s efficiency effort level (e b, w b ) satisfying Π (e b ) = kv (e b ) u (w b ) Make a marginal change (de b, dw b ) but keeping the (PC b ) binding u (w b )dw b = kv (e b )de b We have to evaluate the impact on the principal s profit The first order term is zero Π(e b + de b ) Π(e b ) dw b EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

50 Optimal contracts with adverse selection: distortion A marginal change in the agent B s contract... induces a second order negative effect on the welfare the principal extracts from agent B while it causes a first order negative effect on agent G s information rent which implies a first order positive effect on the principal s profit EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

51 Optimal contracts with adverse selection: Size of the distortion The distortion in the efficiency of agent B is C = q(k 1) (1 q) u (e b ) u (w b ) When q 0, the probability of type-g agents vanishes and the distortion is not anymore beneficial When q 1, the distortion is maximal to reduce the informational rent to the type-g agents EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

52 Asymmetric vs symmetric: agent G The efficiency condition is the same in both situations The participation constraint binds in the symmetric information case But it does not under asymmetric information u(w g ) v(e g ) = U + A > U = u(w g ) v(e g ) where A = (k 1)v(e b ) EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

53 Asymmetric vs symmetric: agent G EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

54 Asymmetric vs symmetric: agent B EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

55 Asymmetric vs symmetric Due to the distortion, for the type-b agent, we have e b < e b and w b < w b We also have e g > e b and w g > w b EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

56 Role of risk-aversion The problem of adverse selection is present independently of the agent s risk-aversion Even if both types of agents are risk-neutral, the problem has the same basic characteristics The reason is that there is no insurance incentive problem Rather, the principal is unsure to whom she is offering a contract EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

57 Selecting only the good ones We have analyzed the principal s problem when she wants to contract the agent independently of his type There is an alternative option The principal could simply offer the contract (e g, w g ) Only the type-g agents will accept the contract: with probability 1 q, the transaction will not take place Since there is only one contract in the menu, there is no informational rent for the type-g agents The principal will prefer to offer the optimal menu {(e g, w g ), (e b, w b )} only if q[π(e g ) w g ] + (1 q)[π(e b ) w b ] q[π(e g ) w g ] EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

58 Outline 1 Introduction 2 A Model of Adverse Selection 3 Principals Competing for Agents EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

59 Agents differences We continue to consider the case with two types of agents: G and B Now, there are not different with respect to their effort disutility Rather, agent G is more productive than agent B We assume that agent G is more careful than B He commits fewer errors and his average result is better EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

60 Production technology The productive process is not deterministic The final result depends on a random variable We assume that effort is not a choice variable: it is unique and observable The principal cannot separate the two agent types by demanding greater effort from one of them EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

61 Production technology: success or failure When the agent exerts effort, the result could be either a success (S) or a failure (F) The probability of success if p i for type i {g, b} with p g > p b The result in case of success is x s, in case of failure it is x f x s > x f The result is verifiable: the principal can pay the agent according to the result Denote by w s the payment in case of success and w f in case of failure EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

62 Competing principals There are several competitive principals trying to contract with the agents Principals are identical and risk-neutral If the probability of success is p, and the wages offered to agents are (w s, w f ) Then a principal s expected profit is E[x w p] := p(x s w s ) + (1 p)(x f w f ) We also use the notation Π b (w) = E[x w p b ] and Π g (w) = E[x w p g ] EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

63 Risk-averse agents Agents are risk averse with Bernoulli function u We assume that u > 0 and u < 0 Fix a wage contract (w s, w f ) The expected utility of type G agents is U g (w) = E[u(w) p g ] := p g u(w s ) + (1 p g )u(w f ) The expected utility of type B agents is U b (w) = E[u(w) p b ] := p b u(w s ) + (1 p b )u(w f ) Since only one effort is possible and effort implies the same disutility for each agent type, we can transfer effort into the reservation utility EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

64 Competition between principals Principals are competing to attract agents We are then interested in the set of equilibrium contracts in the game played by principals Equilibrium contracts are Nash equilibria We will focus on symmetric Nash equilibria EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

65 Symmetric equilibria A principal can choose to offer a menu of contracts {(w g s, w g f ), (w b s, w b f)} Given a type t {g, b}, the contract (w t s, w t f) is intended to type-t agents This means that, either information is symmetric and this contract is restricted to type t Or, information is asymmetric and this contract is self-selecting EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

66 Symmetric equilibria Consider two principals, the first one is offering the menu {(w g s, w g f ), (w b s, w b f)} The second one is offering the menu {(ŵ g s, ŵ g f ), (ŵ b s, ŵ b f)} Both principals must make the same expected profit Otherwise, one will switch to the other s principal menu This motivates our interest to symmetric equilibria where all principals offer the same menu If two principals offer the same menu, half of agents choose one principal and the other half choose the other principal EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

67 Symmetric information Assume that principals can perfectly distinguish agent types Then we can analyze two separate markets: one for type-g contracts and another for type-b contracts We look for the characteristics of an equilibrium contract for type-t agents Proposition (ws t, wf t ) At equilibrium, the expected profits of the principals must be zero EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

68 Proof Let (ws t, wf t ) be an equilibrium contract Assume that the expected profit for each agent hired is positive, i.e., Π := E[x w t p t ] = p t [x s ws t ] + (1 p t )[x f wf t ] > 0 Remember that all principal have the same expected profit Agents will split among principals, each principal expecting to get a fraction µ (0, 1) of the agents, say µ = 1 2 Or equivalently, each principal s probability to hire a type-t agent is µ EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

69 Proof Consider a principal deviating and offering the contract (w t s + επ, w t f + επ) where 0 < ε < 1 This contract is strictly preferred by type-t agent Therefore the principal will hire all agents, and not only the fraction µ The expected profit for each agent hired is (1 ε)π The principal makes less profits from each agent but hires a larger fraction of agent If ε is close enough to 0, we get (1 ε)π > µπ This contradicts the requirement that (ws t, wf t ) is a Nash equilibrium EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

70 Efficiency Proposition Any equilibrium contract (ws t, wf t ) must be Pareto optimal No other contract exists that would be preferred by both principal and agent If such a contract did exist, some principal would have sufficient incentives to deviate and offer the new contract The new contract gives positive expected profits And the agent has sufficient incentives to accept it since it gives him greater expected utility EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

71 Efficiency Fix an equilibrium contract (ws t, wf t ) Denote by U the type-t agent s expected utility U := p t u(ws t ) + (1 p t )u(ws t ) Since the equilibrium contract is Pareto optimal It must solve the following constrained maximization problem subject to max (w pt [x s w s ] + (1 p t )[x f w f ] s,w f) p t u(w s ) + (1 p t )u(w f ) U EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

72 Symmetric information: characterizing equilibrium contracts If follows from the Kuhn-Tucker conditions that w t s = w t f This was predictable: the optimal risk-sharing requires the risk-neutral principal to fully insure the agent Principals make zero profits w t s = w t f = E[x p t ] EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

73 Symmetric information EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

74 Indifference curves Denote by Iso t (U) the type-t agent indifference curve with expected utility U Iso t (U) := {(w s, w f ) : E[u(w) p t ] = U} We can construct an implicit function where (w s, w f ) Iso t (U) w f = h t (w s ) ( 1 h t (x) = u 1 [ U p t 1 p t u(x) ] ) The slope of the indifference curve at the contract w = (w s, w f ) is defined by [h t ] dw f (w s ) and denoted by dw s U t =cst EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

75 Single crossing For any given contract w = (w s, w f ), the indifference curves of G is steeper than that of B dw f = pg u (w s ) dw s (1 p g )u (w f ) < pb u (w s ) (1 p b )u (w f ) = dw f dw s U g =cst Indifference curves cross at most once U b =cst EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

76 Asymmetric information The menu {C g, C b } = {(ws g, wf g ), (ws b, wf b )} is not information revealing in the sense that each type-t agent prefers his corresponding contract C t Indeed, all type-b agents have incentives to pass themselves off as type-g Since all principals anticipate this behaviour, the menu {C g, C b } will not be offered Indeed, the actual expected profits with type-b agents would be negative p b [x s ws g ]+(1 p b )[x f wf g ] < p b [x s ws b ]+(1 p b )[x f wf b ] = 0 and the expected profits with type-g agents remains zero EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

77 Asymmetric information Assume that the principal cannot observe the type of agent When a principal offers a contract intended for agents She must make sure that, given the rest of contracts offered, type-g agents are effectively interested in signing the contract, and type-b agents are not The same reasoning is applied to a principal who wants to offer a contract intended for both agent types (a contract that pools all agents) EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

78 Equilibrium with asymmetric information We look for an equilibrium where all principals offer a menu {C g, C b } = {(ws g, wf g ), (ws b, wf)} b This menu of contracts should be such that U g (C g ) U g (C b ) and U b (C b ) U b (C g ) In that case, the principal expected profits for each agent hired is qπ g (C g ) + (1 q)π b (C b ) EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

79 Equilibrium with asymmetric information Proposition At equilibrium, contracts give zero expected profits to principals, i.e., qπ g (C g ) + (1 q)π b (C b ) = 0 Proof. If not, one of the principals would be prepared to increase wages (reducing her expected profits per agent), but take on all the agents in the market Be careful with the self-selecting conditions EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

80 Necessary conditions Fix an equilibrium menu of contracts {C g, C b } Proposition There does not exists a contract C = (w s, w f ) such that it is preferred to C g by good agents it is not preferred to C b by bad agents gives strictly positive expected profits to the principal offering it, given that only type-g agents will agree to sign it, i.e., Π G (C) = p g [x s w s ] + (1 p g )[x f w f ] > 0 EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

81 Necessary conditions Fix an equilibrium menu of contracts {C g, C b } Proposition There does not exists a contract C = (w s, w f ) such that it is preferred to C b by bad agents it is not preferred to C g by good agents gives strictly positive expected profits to the principal offering it, given that only type-b agents will agree to sign it, i.e., Π b (C) = p b [x s w s ] + (1 p b )[x f w f ] > 0 EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

82 Necessary conditions Fix an equilibrium menu of contracts {C g, C b } Proposition There does not exists a contract C = (w s, w f ) such that it is preferred to C b by bad agents it is preferred to C g by good agents gives strictly positive expected profits to the principal offering it, given that both types of agent will agree to sign it, i.e., qπ G (C) + (1 q)π b (C) > 0 EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

83 Necessary conditions Proposition At an equilibrium {C g, C b } principals make zero profits with each agent s types, i.e., Π g (C g ) = 0 and Π b (C b ) = 0 Split the proof in two cases Case 1: Π g (C g ) > 0 and Π b (C b ) < 0 Case 2: Π g (C g ) < 0 and Π b (C b ) > 0 Use the property that indifference curves have different slopes when they cross EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

84 Pooling and separating equilibria Definition An equilibrium menu of contracts {C g, C b } is said to be pooling if C g = C b separating if C g C b EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

85 Non-existence of pooling equilibria Proposition Pooling equilibria do not exist If an equilibrium does exist, it must be such that each type of agent is offered a different contract EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

86 Non-existence of pooling equilibria: proof Assume C i is a pooling equilibrium then we must have 0 = Π i (C i ) := p i [x s w i s] + (1 p i )[x f w i f] where p i = qp g + (1 q)p b is the probability that the result will be successful when the principal does not know which type accepts the contract We first analyze the case where Π g (C i ) > 0 EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

87 Non-existence of pooling equilibria EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

88 Non-existence of pooling equilibria: proof We can follow a similar argument when Π b (C i ) > 0 We still have to analyze the case Π g (C i ) = Π b (C i ) = 0 This is only possible when C i = x = (x s, x f ) EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

89 Non-existence of pooling equilibria: proof EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

90 Finding the separating equilibrium Assume there exists a separating equilibrium {C g, C b } Proposition We must have C b = C b EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

91 Finding the separating equilibrium EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

92 Finding the separating equilibrium Consider a principal offering a contract C in the shaded area, on the 45 o line below C b The type-b agent will be better off and the principal makes strictly profits with this type of agents The type-g agent may want to sign this contract too The principal will also make positive profits with type-g agents EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

93 Finding the separating equilibrium Proposition The contract C g belongs to the zero profit line Π g ( ) = 0 and satisfies U b (C g ) = U b (C b ) EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

94 Finding the separating equilibrium EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

95 Finding the separating equilibrium We must have U b (C b ) U b (C g ) This conditions locates C g on the zero profit line Π g ( ) = 0 and below the intersection with the indifference curve U b ( ) = U b (C b ) It cannot be strictly lower, otherwise a principal can deviate and make strictly positive profits Observe that we have U g (C g ) U g (C b ) EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

96 Separating equilibrium: necessary conditions Assume there exists a separating equilibrium {C b, C g } Then we must have C b = C b The contract C g is characterized by the condition that U b (C b ) = U b (C g ), i.e., u(w b ) = p b u(ws g ) + (1 p b )u(wf g ) the condition of zero profits p g x s + (1 p g )x f = p g w g s + (1 p g )w g f EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

97 Separating equilibrium: sufficient conditions It is not possible to find C b that is strictly preferred by type-b agents, not preferred by type-g agents and with strictly positive profits Π b (C b ) > 0 It is not possible to find C g that is strictly preferred by type-g agents, not preferred by type-b agents and with strictly positive profits Π g (C g ) > 0 EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

98 Separating equilibrium: sufficient conditions Depending on the value of parameters, a principal can deviate offering a unique contract C that both types of agents strictly prefer and that gives strictly expected profits, i.e., Π i (C ) > 0 There exists q 0 (0, 1) such that for each q > q 0, the above situation occurs, implying that there is no equilibrium When q is large enough, the incentive to attract all the agents increases, even though there is a risk that the agent will turn out to be bad (this is a low-probability event) In that case, separating the agent types is too difficult EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

99 Separating equilibrium: no equilibrium when q > q 0 EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

100 What happens when q q 0 EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

101 What happens when q q 0 Macho-Stadler and Pérez-Castrillo claim that the candidate {C b, C g } is a separating equilibrium How can we prove that a principal cannot deviate by offering a menu {C g, C b } where contracts are self-selecting, strictly preferred by each corresponding type and with positive expected profits? EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

102 Possible deviations? EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

103 Possible deviations? EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104

104 Summary Adverse selection may provoke the absence of any equilibrium If an equilibrium does exist, it will be a separating equilibrium Contracts with contingent pay-offs permit the more efficient agents to be separated from the less efficient ones The existence of contracts that include contingent pay-offs need not be attributed to the existence of a moral hazard problem, this type of contract may also be the consequence of adverse selection In the equilibrium, the least efficient agent is offered the same contract as under symmetric information The more efficient agent loses expected utility, while the expected pay-off is the same The more efficient agent needs to sacrifice efficiency so that type-b agents do not prefer the type-g contract to their own contract EESP (FGV) Teoria das organizações e contratos 3 o trimestre / 104