Moral Hazard in Teams

Transcription

1 Moral Hazard in Teams Ram Singh Department of Economics September 23, 2009 Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

2 Outline 1 Moral Hazard in Teams: Model 2 Unobservable Individual Output 3 First Best without Budget Breaker 4 Risk-Averse Teams Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

3 Model I Moral Hazard in Teams: Model Many agents; at least two agents Effort on the part of each agent affects the output; Effort is not observable or contractible; Cost of effort by an agent is private Risk-neutral parties Example Firm as Team and Profit as Output; Cooperative (farm) as Team and Produce or profit as Output; Sales-persons as Team and sales as Outputs; Advocate as Team and Judicial judgement as Output Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

4 Model II Moral Hazard in Teams: Model General Model: Holmstrom (1982, BJE) n Agents; n 2 e = (e 1,..., e n ) Output Q = (e 1,..., e n ), { (q1,..., q Q = n ) R n, or; Q R,. Agents are weakly risk-averse. Team/partnership Contract: w(q) = (w 1 (Q),..., w n (Q)) where w i (Q) = s i (Q) is the output sharing rule such that s i 0. Typically, we have wi (Q) = s i (Q) = Q. Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

5 Unobservable Individual Output Unobservable Individual Output I Simple Model: Q = Q(e 1,..., e n ) R is scalar deterministic output Q is increasing and concave; for all i, j, Q e i > 0, 2 Q e 2 i < 0, 2 Q e i e j 0, Matrix of second derivatives Q ij is Negative Definite Agent is risk neutral in wealth; u i (w i, e i ) = u i (w i ) ψ(e i ) = w i ψ(e i ) and ψ(e i ) is increasing and convex. w i (Q) = s i (Q) is continuously differentiable and ( Q)[ w i (Q) = s i (Q) = Q] Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

6 Unobservable Individual Output Unobservable Individual Output II The first best is solution to s.t. max e 1,...,e n {Q(e 1,..., e n ) ψ i (e i )} ( Q)[ w i (Q) = Q] (1) Let e i = (e 1,..., e i 1, e i+1,..., e n ). Therefore, the first best effort ei solves the following foc Q(e i,e i ) e i = ψ (e i ), for every i = 1,..., n. That is, for every i = 1,..., n Q(e i, e i ) e i = ψ (e i ) (2) Let e = (e 1,..., e i,..., e n) solve system 2. Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

7 Unobservable Individual Output Is First Best Achievable? I In SB, e is not contractible but Q is Given e i = (e 1,..., e i 1, e i+1,..., e n ), agent i solves max e i {w i (Q(e i, e i ) ψ(e i )}. Therefore, a (Nash) equilibrium is characterized by the following n equations for every i = 1,..., n. dw i (Q(e i, e i )) Q(e i, e i ) = ψ (e i ), (3) dq e i Now e = (e 1,..., e i,..., e n) can solve (2) iff for every, we have ( i {1,..., n})[ dw i(q(e i, e i )) dq = 1] (4) Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

8 Unobservable Individual Output Is First Best Achievable? II But from 1, we have dw i (Q(ei, e i )) = 1 (5) dq 4 and 5 give us a contradiction. Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

9 Unobservable Individual Output First Best with Budget Breaker I Consider the following contract: BB demands an upfront payment of z i and appropriate the output; and pays ( Q)[w i (Q) = Q] to each agent Under this contract it is easy to see that BB and each agent is a residual claimant on the entire output; and e = (e 1,..., e i,..., e n) is a N.E. Is such a contract feasible? Yes, if for all i Q(e 1,..., e n) ψ i (e i ) z i, i.e., nq(e 1,..., e n) ψ i (e i ) z i (6) and zi + Q(e 1,..., e n) nq(e 1,..., e n) (7) Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

10 Unobservable Individual Output First Best with Budget Breaker II That is, if which is clearly true. Q(e 1,..., e n) ψ i (e i ) > 0, Is e = (e 1,..., e i,..., e n) a unique N.E.? Suppose e i = (0,..., 0). Agent i solves max e i {Q(0,.., e i,..., 0) ψ(e i )}. > ψ(0) e i, the agent i will choose a positive effort. Now 0 implies that other agents will also increase their effort. If Assuming Q(0,..,0,...,0) e i 2 Q e i e j e = (e 1,..., e i,..., e n) a unique optimizer, iteration will continue till they reach e. Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

11 Unobservable Individual Output First Best without BB I Consider{ the following Mirrlees Contract: bi 0, if Q = Q w i (Q) = ; k i, if Q Q where b. i 0 and k i < 0 BB pays b i if output Q = Q, where b i ψ i (e i ); and imposes penalty of k i if Q Q Can choose b i = Q Do not need external intervention in equilibrium Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

12 Unobservable Individual Output First Best without BB II Under this contract it is easy to see that e = (e 1,..., e i,..., e n) is a N.E. Multiple equilibria: Let ê i solve Now if holds (0,..., 0) is a N.E. If for some i, Q(0,..., ê i,..., 0) = Q(e 1,..., e i,..., e n) b i ψ i (ê i ) k i (8) b i ψ i (ê i ) > k i (9) there exist N.E. (ẽ 1,..., ẽ i,..., ẽ n ) such that for some j, ẽ j < e j Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

13 Unobservable Individual Output Problematic Features I Remark Under Holmstrom scheme, the payoff of the BB is w BB = z i + Q(e) Q(e) = z i (n 1)Q(e), i.e., dw BB dq = (n 1) < 0 Remark Note the results do not depend on output being stochastic or Risk aversion of agents BB want the scheme to fail BB may collude with one of the agents Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

14 Unobservable Individual Output Problematic Features II A side contract between BB and an agent gives back original problem Agents may collude to borrow Q and game with BB Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

15 First Best without Budget Breaker Deterministic Output and Finite Effort Space I Legros and Matthews (1993) Let Three agents, i = 1, 2, 3 Q = Q(e 1, e 2, e 3 ) e i {0, 1}, i = 1, 2, 3 ψ i (e i ) = ψ i (1) > ψ i (0) > 0, i = 1, 2, 3 The FB solves max{q(e 1, e 2, e 3 ) ψ i (e i )} Let (e1, e 2, e 3 ) = (1, 1, 1) Q i = Q(0, e i ), where e i = (1, 1) Q 1 Q 2 Q 3, a generic feature Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

16 First Best without Budget Breaker Deterministic Output and Finite Effort Space II Consider the following contract wi if Q = Q ; Q w i (Q) = 2 + δ if Q Q & Q Q i ; k i, if Q = Q i. δ 0. where w i = w i (Q ) ψ i > 0 and This contract implements the FB. However, if Q 1 = Q 2 = Q 3 the FB cannot be implemented. Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

17 First Best without Budget Breaker Approximating FB with Deterministic Output I Legros and Matthews (1993) Let Two agents, i = 1, 2 Q = Q(e 1, e 2 ) = e 1 + e 2 e i [0, + ), i = 1, 2 ψ i = ψ i (e i ) = e2 i 2, i = 1, 2 The FB solves max{q(e 1, e 2 ) e1,e 2 ψ i (e i )} = max e 1,e 2 {e 1 + e 2 e2 1 2 e2 2 2 } Clearly (e1, e 2 ) = (1, 1) Consider the following contract Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

18 First Best without Budget Breaker Approximating FB with Deterministic Output II If Q 1 { w 1 (Q) = w 2 (Q) = (Q 1)2 2 and ; Q w 1 (Q). { w1 (Q) = Q + k and ; If Q < 1 w 2 (Q) = k. Proposition Under the above contract if agent acts as principal, then ((ɛ, 1 ɛ), (0, 1)) is a N.E. in which the first agent plays e = 0 and e = 1 with probability ɛ and 1 ɛ, respectively; and agent two plays e = 1 with probability one. Proof: Given e 2 = 1 opted by 2, agent 1 solves, max{w 1 (e 1 + 1) e2 1 e 1 2 } = max { e2 1 e 1 2 e2 1 2 } = 0 Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

19 First Best without Budget Breaker Approximating FB with Deterministic Output III i.e., all effort levels are equally good. So, (ɛ, 1 ɛ) is a best response for agent 1. Note agent 2 will never opt for e 2 > 1. Given that agent 1 opts for (ɛ, 1 ɛ), a choice of e 2 = 1 gives agent 2, (1 ɛ)[2 1 2 ] + ɛ[1 0] 1 2 = 1 ɛ 2. In contrast, when e 2 < 1 agent 2 s payoff is, (1 ɛ)[1 + e 2 e2 2 2 ] ɛk e e 2 e 2 2 ɛk, which is uniquely maximized at e 2 = 1 2. At e 2 1 2, agent 2 s payoff is e 2 = 1 is the best response for 2, if 5 4 ɛk k ɛ Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

20 Risk-Averse Team I Risk-Averse Teams Q = Q(e 1,..., e n ) R is scalar deterministic output Q is increasing and concave; for all i, j, Q e i > 0, 2 Q e 2 i < 0, 2 Q e i e j 0, Matrix of second derivatives Q ij is Negative Definite Agents are risk-averse in wealth; ũ i (w i, s i (Q), e i ) = u i (w i, s i (Q)) ψ i (e i ) = e r i s i (Q) ψ i (e i ) and ψ i (e i ) is increasing and convex. ( Q)[ s i (Q) = Q] Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

21 Risk-Averse Team II Risk-Averse Teams The First Best: max { ũ i (s i (Q), e i )}, i.e., e 1,...,e i,...,e n;s i s.t. max { [u i (s i (Q)) ψ i (e i )]} e 1,...,e i,...,e n;s i ( Q)[ s i (Q) = Q] Let e = (e 1,..., e i,..., e n) along with a sharing scheme s (Q) be the unique F.B. profile in this context. Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

22 Risk-Averse Teams Risk-Averse Team III Remark For a sharing scheme s i (Q) and a profile of efforts (e 1,..., e i,..., e n ) (e 1,..., e i,..., e n), the following holds: There exists a sharing scheme s (Q) such that ( i)[e(s i, e i ) E(s i, e i )] (10) ( j)[e(s j, e j ) > E(s j, e j )] (11) If a sharing scheme ŝ i (Q) induces e = (e 1,..., e i,..., e n) as a N.E., then for any sharing scheme s i (Q) that induces (e 1,..., e i,..., e n ), the following cannot hold ( i)[e(s i, e i ) E(ŝ i, e i )] (12) ( j)[e(s j, e j ) > E(ŝ j, e j )] (13) Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

23 Risk-Averse Team IV Risk-Averse Teams If a sharing contract does not induce e = (e 1,..., e i,..., e n) as a N.E., it cannot be F.B. Therefore, a P.O. sharing scheme will necessarily induce e = (e 1,..., e i,..., e n) as a N.E. We know that if agents are risk neutral, i.e., if u(x) = x, then no BB sharing scheme can induce e = (e 1,..., e i,..., e n) as a N.E. Can a BB sharing scheme can induce e = (e 1,..., e i,..., e n) as a N.E. if agents are risk-averse? Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

24 Risk-Averse Team V Risk-Averse Teams Consider the following BB Scapegoat sharing contract: If Q = Q(e ), then s i (Q) = b i, where b i s are such that b i = Q(e ); If Q > Q(e ), then s i (Q) = b i + Q Q(e ) n If Q < Q(e ), choose one agent j randomly and fix shares such that s j (Q) = w j ( i j) s i (Q) = b i + b j + w j + Q Q(e ) n 1 Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

25 Risk-Averse Team VI Risk-Averse Teams Remark Note when Q < Q(e ), n s i (Q) = s j (Q) + i=1 n s i (Q) = w j + i j n [bi + b j + w j + Q Q(e ) ] = Q. n 1 i j Therefore, the above contract meets the BB constraint. Suppose, e i = e i, i.e., all agents apart from i have opted for FB effort. If i opts for ei, his payoff is u i (bi ) ψ i (ei ). If he opts for some e i > ei, his payoff is u i (bi + Q Q(e ) ) ψ i (e i ). n Since e is P.O. profile, u i (bi + Q Q(e ) ) ψ i (e i ) > u i (bi ) ψ i (ei ) n Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

26 Risk-Averse Team VII Risk-Averse Teams cannot hold. Now, if i opts for some e i < ei, his share { wi with probability 1 s i (Q) = n ; bi + z i with probability 1 n n, where z i is a random variable. For each j i, probability of z i = b i + b j +w j +Q Q(e ) for some e i < ei, his payoff is n 1 is 1 n 1. Therefore, if i opts n 1 n Eu i(b i + z i ) + 1 n u( w i) ψ i (e i ) (14) n 1 n n [ 1 n 1 u i(bi + z i )] + 1 n u( w i) ψ i (e i ) (15) i j Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

27 Risk-Averse Teams Risk-Averse Team VIII For e i < e i, agent i s payoff function is concave. Let ê i uniquely solve in region e i < e i. Now let Y i = u i (b i ) ψ i (e i ) [ n 1 n Eu i(b i + z i ) + 1 n u( w i)] ψ i (ê i ) (16) Clearly, if Y i > 0, e i is a unique best response for agent i. Now, using envelop theorem Moreover, concavity of u i implies dy i dw i = 1 n u i > 0 (17) d 2 Y i dw 2 i = 1 n u i > 0 (18) Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

28 Risk-Averse Team IX Risk-Averse Teams That is Y i is increasing in w i at an increasing rate. So, there exits w i such that for all w i w i, Y i > 0. That is, for all w i w i, ei is a unique best response to e i. Therefore, Proposition If w i is sufficiently large for all i, then e = (e1,..., e i,..., en) is a N.E. Proposition If r i is sufficiently large for all i, then e = (e 1,..., e i,..., e n) is a N.E. Proof: Rewriting as Y i = u i (b i ) ψ i (e i ) [ n 1 n Eu i(b i + z i ) + 1 n u( w i) ψ i (ê i )] Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

29 Risk-Averse Team X Risk-Averse Teams Y i = u i (bi ) ψ i (ei ) [ n 1 n n [ 1 n 1 u i(bi + z i )] + 1 n u( w i) ψ i (ê i )] i j i.e., as Y i = e r i b i ψ i (e i ) + 1 n ( i j e r i {b i + 1 n 1 [b j +w j Q(e )+Q(ê i,e i )]} ) (19) + 1 n er i w i + ψ i (ê i ) Note as r i goes up, the first and the third terms approach zero. The second term is unaffected and the fifth one is bounded by ψ i (0) and ψ i (e i ). But, the fourth term exploded towards infinity. Therefore, for sufficiently large r i, Y i > 0 holds. Again, e = (e 1,..., e i,..., e n) is a N.E. Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30

30 Risk-Averse Teams Scapegoats Versus Massacres When agents are identical, the scapegoat contract is: Q n if Q Q(e ); Q+w n 1 s i (Q) = n 1 with probability n if Q < Q(e ); w with probability 1 n if Q < Q(e ). When agents are identical, the massacre contract is: Q n if Q Q(e ); s i (Q) = Q + (n 1)w with probability 1 n if Q < Q(e ); w with probability n 1 n if Q < Q(e ). Reference: Rasmusen (1984, RJE) Ram Singh (Delhi School of Economics) Moral Hazard September 23, / 30