1 Moral Hazard: Multiple A gents 1.1 Moral Hazard in a Team • Multiple agents (firm?) Holmstrom(82) Deterministic Q. — Partnership: Q jointly affected ∂Q ∂2QOutput Q(a) ∼ F (q|a), > 0, < 0,∂ai ∂a2 i — Individual qi’s. (tournaments) ∂2Q dqij = ∂ai∂aj ≥ 0, (dq)ij—negative definite. • Common shocks, cooperations, collusion, monitor- ing. Agents: ui(w)=w. Agents: i =1,...,n. Partnership w(Q)={w1(Q),...,wn(Q)}, P Ui(wi,ai)=ui(wi) − ψi(ai) such that for all Q, in =1wi(Q)=Q. Efforts: a =(a1,...,an),with ai ∈ [0, ∞) Problem: free-riding (someone else works hard, I gain) a ∗ Output q =(q1,...,qn) ∼ F (q|a). First-best: ∂Q∂a(i ∗)= ψ0(ai). Principal: R-N.Agents’ choices: FOC: dwi[Q(ai,a ∗ −i)] dQ ∂Q(ai,a ∗ −i) ∂ai = ψ0(ai) ?Nash (a ∗ i ) =FB (a ∗ i )? Locally: dwi[Q(ai,a ∗ −i)] dQ =1,thus wi(Q)= Q + Ci. Budget P n i=1 wi(Q)= Q for all (!) Q. This requires a third party: budget breaker Let zi = −Ci–pa yment from agent i. Thus P n i=1 zi + Q(a ∗) ≥ nQ(a ∗) and zi ≤ Q(a ∗) − ψi(a ∗ i ). At F-B: Q(a ∗) − P n i=1 ψi(a ∗ i ) > 0. Thus ∃z =(z1,...,zn). 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 interp retation (!)). 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) bi if Q = Q(a ∗), penalty k otherwise. (bonuses for certain targets) As long as bi − ψi(a ∗ i ) ≥−k, F-B can be supported, moreover if b’s and k exist so that Q(a ∗) ≥ P n i=1 bi,no b-b needed.Interpretation: Debt financingbythe firm. Firm commits to repay debts of D = Q(a ∗) − P bi,and bi to each i. If cannot, creditors collect Q and each employer pays k. (Hm...) Issues: (1) Multiple equilibria (like in all coordination-type games, and in Mechanism-Design literature). No easy solution unless (2) actions of others are observed by agents, and the prin-cipal can base his compensation on everyone’s reports. Not a problem with Holmstrom though (P ositive effort of one agent increases effort from others). (3) Deterministic Q. 1.2 Special Examples of F-B (approx) via different schemes Legros & Matthews (’93), Legros & Matsushima (’91). • Deterministic Q, finite A’s, detectable deviations. Say, ai ∈ {0, 1}.And Qfb = Q(1, 1, 1). Let Qi = Q(ai =0,a−i =(1, 1)). Suppose Q1 6= Q2 6= Q3. Shirker identified and punished (at the benefit of the oth-ers). Similarly, even if Q1 = Q2 6= Q3.• Approx. efficiency, n =2. Check: Agent 1. Set a2=1, ∙ ¸maxa ((a+1)−1)2 − a2 =0.Idea: use one agent to monitor the other (check with 2 2prob ε). Agent 2. a2 ≥ 1 7→ Q ≥ 1. Implies a2 ∗ =1, U2= ai ∈ [0, ∞) Q = a1+ a2, ψi(ai)=ai 2/2. 1 − ε/2. F-B: ai ∗ =1. a2< 1 guarantees Q< 1 with prob. ε. L & M propose: agent 1 chooses a1=1with pr =1−ε. Obtain a2 ∗ = 21,and U2= 45− εk. When Q ≥ 1, For, k ≥ 21+41 ε, a2∗ =1is optimal. ( w1(Q)=(Q − 1)2/2 w2= q − w1(Q). • Random output. Cremer & McLean works. (condi-tions?) when Q< 1, ( w1(Q)=Q + k w2(Q)=0− k.1.3 Observable individual outputs q1 = a1 + ε1 + αε2, q2 = a2 + ε2 + αε1. ε1,ε2 ∼ iid N(0,σ2). CARA agents: u(w, a)= −e−µ(w−ψ(a)), ψ(a)= 1 2ca2 . Linear incentive schemes: w1 = z1 + v1q1 + u1q2, w2 = z2 + v2q2 + u2q1. No relative performance weights: ui =0, Principal: maxa,z,v,u E(q − w), subject to E h −e−µ(w−ψ(a)) i ≥ u(¯w). Define ˆw(a),as −e−µ ˆw(a) = E h −e−µ(w−ψ(a)) i . Agent’s choice: a ∈ arg max ˆw(a). E(eaε)= ea2σ2/2,for ε ∼ N(0,σ2). (back to General case) Agent i V (w1)= Var(v1(ε1 + αε2)+ u1(ε2 + αε1)) = σ2 h (v1 + αu1)2 +(u1 + αv1)2 i Then, agent’s problem: maxa ⎧ ⎨ ⎩ z1 + v1a + u1a2 − 1 2ca2− −µσ2 2 h (v1 + αu1)2 +(u1 + αv1)2 i ⎫ ⎬ ⎭ . Solution a ∗ 1 = v1 c (as in one A case). ˆw1 = z1+1 2 v2 1 c +u1v2 c −µσ2 2 h (v1 + αu1)2 +(u1 + αv1)2 i . Principal: maxz1,v1,u1 ½ v1 c − µz1 + v2 1 c + u1v2 c ¶¾s.t. ˆw1 ≥ ¯w Principal: maxv1,u1 ½ v1 c − 1 2 v2 1 c µσ2 2 − h (v1 + αu1)2 +(u1 + αv1)2 i¾ . To solve: (1) find u1 to minimize sum of squares (risk) (2) Find v1 (trade-off) risk-sharing, incentives Obtain u1 = − 2α 1+α2 v1. The optimal incentive scheme reduce agents’ exposure to common shock. v1 = 1+α2 1+α2+µcσ2(1−α2)2 . 1.4 Tournaments Lazear & Rosen (’81) Agents: R-N, no common shock. qi = ai + εi. ε ∼ F (·), E =0, Var = σ2 . Cost ψ(ai). F-B: 1= ψ0(a ∗). wi = z + qi. z + E(qi) − ψ(a ∗)= z + a ∗ − ψ(a ∗)= ¯u. Tournament: qi >qj → prize W ,bothagentspaid z. Agent: z + pW − ψ(ai) →ai max.p = Pr(qi >qj)= 1.5 Cooperation and Competition = Pr(ai − aj >εj − εi)= H(ai − aj). • Inducing help vs Specialization EH =0, VarH =2σ2 . • Collusion among agents ∂pFOC: W∂ai = ψ0(ai). Wh(ai − aj)= ψ0(ai). • Principal-auditor-agent 1Symmetric Nash: (+FB): W = h(0). Itoh (’91) z + H(0) − ψ(a ∗)=¯u. 2agents: qi ∈ {0, 1}, (ai,bi) ∈ [0, ∞) × [0, ∞). h(0) √ Result: Same as FB with wages. Ui = ui(w) − ψi(ai,bi), ui(w)= w. Extension: multiple rounds, prizes progressively increas-ψi(ai,bi)= ai 2+ b2 i +2kaibi, k ∈ [0, 1]. ing. Pr(qi =1)= ai(1 + bi). Agents: Risk-averse+Common Shock. i iContract: wi =(wjk), wjk —payment to i when qi = j, Trade-off between (z, q) contracts and tournaments. q−i = k.No Help: bi =0. By itself (igno ring change in a)and if b∗ is small, and since change in w increases risk, it is costly for the prin-w0=0,ai(1 − w1) →w1 max, cipal to provide these incentives. s.t, ai = 12 √ 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 aj,bj,w,w11 > w10, w01 >w00 =0.) Thus, if k is positive, there is a discontinuit y at b =0, a(1 + bj)aj(1 + b) √ w11 +(1 − a(1 + bj))aj(1 + b) √ w01+ thus “a little” of help will not help: for all b<b ∗ principal a(1 + bj)(1 − aj(1 + b)) √ w10 − a 2 − b2 − 2kab → max is worse-off. a,b ³ ´ For k =0, help is always better. FOC+symm: consider ∂ ∂b √ √ √ Two-step a rgument: 1. If ahelp ≥ ab=0 ,