1 Moral Hazard: Multiple Agents 1.1 Moral Hazard in a Team

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1 1 Moral Hazard: Multiple Agents 1.1 Moral Hazard in a Team Multiple agents (firm?) Partnership: Q jointly affected Individual q i s. (tournaments) Common shocks, cooperations, collusion, monitor- ing. Agents: i = 1,...,n. Holmstrom(8) Deterministic Q. Q Q a i a i Output Q(a) F (q a), > 0, < 0, Q dq ij = ai a j 0, (dq) ij negative definite. Agents: u i (w) =w. Partnership w(q) ={w 1 (Q),...,w n (Q)}, P U i (w i,a i )=u i (w i ) ψ i (a i ) such that for all Q, i n =1 w i(q) =Q. Efforts: a = (a 1,...,a n ),with a i [0, ) Output q = (q 1,...,q n ) F (q a). Problem: free-riding (someone else works hard, I gain) First-best: Q a ( a ) = ψ 0 (a i i ). Principal: R-N.

2 Agents choices: FOC: dw i [Q(a i,a dq i )]?Nash(a i ) = FB (a i )? Locally: Q(a i,a i ) a i = ψ 0 (a i ) dw i [Q(a i,a i )] dq =1,thus w i (Q) = Q + C i. Budget P n i=1 w i(q) = Q for all (!) Q. This requires a third party: budget breaker Let z i = C i payment from agent i. Thus P n i=1 z i + Q(a ) nq(a ) and z i Q(a ) ψ i (a i ). At F-B: Q(a ) P n i=1 ψ i(a i ) > 0. Thus z = (z 1,...,z n ). Note, b-b looses from higher Qs. Comments: b-b is a residual claimant (in fact each agent is a residual claimant in a certain interpretation (!)). Not the same as Alchian & Demsetz (equity for manager s incentives to monitor agents properly).?other ways to support first-best? Mirrlees contract: reward (bonus) b i if Q = Q(a ), penalty k otherwise. (bonuses for certain targets) As long as b i ψ i (a i ) k, F-B can be supported, moreover if b s and k exist so that Q(a ) P n i=1 b i,no b-b needed.

3 Interpretation: Debt financingbythe firm. Firm commits to repay debts of D = Q(a ) P b i,and b i to each i. If cannot, creditors collect Q and each employer pays k. (Hm...) Issues: (1) Multiple equilibria (like in all coordinationtype games, and in Mechanism-Design literature). No easy solution unless () actions of others are observed by agents, and the principal can base his compensation on everyone s reports. Not a problem with Holmstrom though (Positive effort of one agent increases effort from others). (3) Deterministic Q. 1. Special Examples of F-B (approx) via different schemes Legros & Matthews ( 93), Legros & Matsushima ( 91). Deterministic Q, finite A s, detectable deviations. Say, a i {0, 1}. And Q fb = Q(1, 1, 1). Let Q i = Q(a i =0,a i =(1, 1)). Suppose Q 1 6= Q 6= Q 3. Shirker identified and punished (at the benefit of the others). Similarly, even if Q 1 = Q 6= Q 3.

4 Approx. efficiency, n =. Check: Agent 1. Set a =1, ((a+1) 1) max a a =0. Idea: use one agent to monitor the other (check with prob ε). Agent. a 1 7 Q 1. Implies a =1, U = a i [0, ) Q = a 1 + a, ψ i (a i )=a i /. 1 ε/. F-B: a i =1. a < 1 guarantees Q < 1 with prob. ε. L & M propose: agent 1 chooses a 1 =1with pr = 1 ε. Obtain a = 1,and U = 5 4 εk. When Q 1, For, k ε, a =1is optimal. ( w1 (Q) =(Q 1) / when Q < 1, w = q w 1 (Q). ( w1 (Q) =Q + k w (Q) =0 k. Random output. Cremer & McLean works. (conditions?)

5 1.3 Observable individual outputs ε 1,ε iid N(0,σ ). q 1 = a 1 + ε 1 + αε, q = a + ε + αε 1. CARA agents: u(w, a) = e µ(w ψ(a)), ψ(a) = 1 ca. Linear incentive schemes: w 1 = z 1 + v 1 q 1 + u 1 q, w = z + v q + u q 1. No relative performance weights: u i =0, Principal: max a,z,v,u E(q w), subject to E h e µ(w ψ(a)) i u( w). Define ŵ(a), as e µ ŵ(a) = E h e µ(w ψ(a)) i. Agent s choice: a arg max ŵ(a). E(e aε )= e a σ /,for ε N(0,σ ). (back to General case) Agent i V (w 1 )= Var(v 1 (ε 1 + αε )+ u 1 (ε + αε 1 )) = σ h (v 1 + αu 1 ) +(u 1 + αv 1 ) i Then, agent s problem: max a µσ Solution a 1 = v 1 c z 1 + v 1 a + u 1 a 1 ca h (v1 + αu 1 ) +(u 1 + αv 1 ) i. (as in one A case). ŵ 1 = z v h 1 c +u 1v c µσ (v1 + αu 1 ) +(u 1 + αv 1 ) i. ½ µ Principal: max v1 z1,v 1,u 1 c z 1 + v 1 c + u ¾ 1v c

6 s.t. ŵ 1 w 1.4 Tournaments Principal: max v1,u 1 ½ v1 c 1 v1 c µσ h (v 1 + αu 1 ) +(u 1 + αv 1 ) i¾. To solve: (1) find u 1 to minimize sum of squares (risk) () Find v 1 (trade-off) risk-sharing, incentives Obtain u 1 = α 1+α v 1. The optimal incentive scheme reduce agents exposure to common shock. v 1 = 1+α 1+α +µcσ (1 α ). Lazear & Rosen ( 81) Agents: R-N, no common shock. q i = a i + ε i. ε F ( ), E = 0, Var = σ. Cost ψ(a i ). F-B: 1 = ψ 0 (a ). w i = z + q i. z + E(q i ) ψ(a )= z + a ψ(a )= ū. Tournament: q i >q j prize W, bothagentspaid z. Agent: z + pw ψ(a i ) ai max.

7 p = Pr(q i >q j )= 1.5 Cooperation and Competition = Pr(a i a j >ε j ε i )= H(a i a j ). Inducing help vs Specialization E H =0, Var H =σ. p FOC: W ai = ψ 0 (a i ). Wh(a i a j )= ψ 0 (a i ). Collusion among agents Principal-auditor-agent 1 Symmetric Nash: (+FB): W = h(0). Itoh ( 91) z + H(0) ψ(a )=ū. h(0) agents: q i {0, 1}, (a i,b i ) [0, ) [0, ). Result: Same as FB with wages. U i = u i (w) ψ i (a i,b i ), u i (w) = w. Extension: multiple rounds, prizes progressively increas- ψ i (a i,b i )= a i + b i +ka ib i, k [0, 1]. ing. Pr(q i =1)= a i (1 + b i ). Agents: Risk-averse+Common Shock. Contract: w i =(wi i jk ), w jk payment to i when q i = j, Trade-off between (z, q) contracts and tournaments. q i = k.

8 No Help: b i =0. By itself (ignoring change in a) and if b is small, and since change in w increases risk, it is costly for the prin- cipal to provide these incentives. w 0 =0,a i (1 w 1 ) w1 max, s.t, a i = 1 w1 (IC) and IR is met.... Even if a adjusts, since it is different from the first-best for the principal with b = 0, the principal looses for sure. Getting Help: Agent i solves (given a j,b j,w,w 11 > w 10, w 01 >w 00 =0.) Thus, if k is positive, there is a discontinuity at b = 0, a(1 + b j )a j (1 + b) w 11 +(1 a(1 + b j ))a j (1 + b) w 01 + thus a little of help will not help: for all b <b principal a(1 + b j )(1 a j (1 + b)) w 10 a b kab max is worse-off. a,b ³ For k = 0, help is always better. FOC+symm: consider b Two-step argument: 1. If a help a b=0, marginal cost a (1+b)( w 11 w 10 )+a(1 a(1+b)) w 01 =(b+ak) of help is of second order, always good. If (as in No Help) w 11 = w 10,w 01 =0,and k> 0, we. Show that a help a b=0. have RHS = 0,LHS > 0 for any b 0. Therefore, need to change w significantly to get any b close to 0. (Even to get b = 0 with FOC b =0)

9 1.6 Cooperation and collusion. CARA agents: u(w, a) = e µ i(w i ψ i (a)). q i = a i +ε i, (ε 1,ε ) N(0,V), where V = ρ = σ 1 /(σ 1 σ ). Linear incentive schemes: w 1 = z 1 + v 1 q 1 + u 1 q, w = z + v q + u q 1. Ã σ 1 σ 1 σ 1 σ!, s.t (a 1,a ) NE in efforts, and CE i 0. Individual choices: v i = ψ 0 i (a i). u are set to minimize risk-exposure: u i = v i σ i σ j ρ. Total risk exposure: P i=1 µ h i v i σ i (1 ρ) i. Full side-contracting: (?) Enough to consider contracts on (a 1,a ). No side contracts (CE (a 1,a ) analogously): CE 1 (a 1 a ) = z 1 + v 1 a 1 + u 1 a ψ 1 (a 1 ) µ 1 (v 1 σ 1 + u 1 σ +v 1u 1 σ 1 ) Principal (RN): (1 v 1 u )a 1 +(1 u 1 v )a z 1 z max Problem reduces to a single-agent problem with µ 1 = 1 µ µ, with costs ψ(a 1,a )= ψ 1 (a 1 )+ ψ (a ). Full side contracting dominates no s-c iff ρ ρ. (cooperation vs relative-performance evaluation). MD schemes.

10 1.7 Supervision and Collusion Principal: reward Monitor for y with w k. Principal: V > 1, (Punish when there is not y?) Suppose not, that is 1 pk > z. (and thus suppose that Agent: cost c {0, 1}, Pr(c = 0)= 1. w mon =0) Monitor: cost z, Proof y, Pr(y c = 0)=p). Assume: V > (so P =1is optimal without monitor) With monitor (no collusion) 1 ³ pv p (V 1) z ³ Principal: 1 p(v k)+ 1 1 p (V 1). No gain for allowing collusion If k is random, then possible. Compare to V 1. Collision: Agent-Monitor: T agent (kt ) monitor,k 1. max T =1.