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UBC Theses and Dissertations

Incentive compatible compensation mechanism for centrally planned industry with multiple agents and communiction Zhang, Guochang 1986

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I N C E N T I V E C O M P A T I B L E C O M P E N S A T I O N M E C H A N I S M FOR C E N T R A L L Y P L A N N E D INDUSTRY W I T H M U L T I P L E A G E N T S A N D C O M M U N I C A T I O N by G U O C H A N G Z H A N G B . A . S c , Shanghai Jiao Tong University, 1983 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF M A S T E R OF S C I E N C E IN BUSINESS A D M I N I S T R A T I O N in T H E F A C U L T Y OF G R A D U A T E STUDIES The Faculty of Commerce and Business Administration We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y OF BRITISH C O L U M B I A December 1986 © G u o c h a n g Zhang In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Q>nm0ft^ QMJ. BuSf/tfSZ, Ad^W&rCttlSYX The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date Det.n, (ivb Abstract This thesis applies the existing agency theory into the problem of production planning in a centrally planned industry. The planner's objective is to maximize social welfare contributed by the industry, while the firms individually want to maximize utility over money compensation minus disutility over effort. The problem contains both moral hazard and adverse selection because each agent privately observes a predecision information about the production process. A model is built for determining the optimal incentive compatible scheme. The analysis starts with the problem of fixed proportions production. An optimal incentive compatible scheme is first derived in single agent settings. It is then extended to multiple agent settings. Under the optimal incentive scheme, the principal is able to derive all the rent. The solution is the first-best when the agents are all risk neutral, and strictly second-best otherwise. The subgaming issues amongst the agents are investigated. When the agents are not cooperative, a sufficient condition is given for the incentive scheme to be effective, i.e., the equilibrium induced by the scheme is implementable. It is also concluded that, if the agents are able to cooperate, there always exist some state realizations under which the scheme is not effective. Finally, a different type of production problem, namely, production with substitutable inputs, are studied. And an incentive compatible compensation scheme is again proposed. n T A B L E OF C O N T E N T S Abstract ii Table of Contents iii List of Figures v Acknowledgement vi Chapter 1 I N T R O D U C T I O N 1 Chapter 2 T H E G E N E R A L M O D E L 3 2.1 Basic Information 3 2.2 The Principal's Objective Function 5 2.3 Incentive Compatibility Condition 5 2.4 Reservation Utility Constraint 7 Chapter 3 P R O D U C T I O N W I T H F I X E D P R O P O R T I O N INPUTS 8 3.1 The Principal Has Perfect Information 9 3.2 Single Agent Problem With Information Asymmetry 11 3.2.1 The State Set Is Finite 12 3.2.2 The Agent Is Risk Neutral And The State Set Is Infinite 13 3.2.3 The Agent Is Risk Averse And The State Set Is Infinite 15 3.3 Multiple Agent Problem With Information Asymmetry 19 3.4 The Subgame Of The Agents 22 3.4.1 The Agents Do Not Cooperate 23 iii 3.4.2 The Agents May Cooperate 24 Chapter 4 P R O D U C T I O N W I T H S U B S T I T U T A B L E I N P U T S 26 4.1 Single Agent Problem 27 4.2 Multiple Agent Problem 31 Chapter 5 C O N C L U D I N G R E M A R K S 33 Bibliography 35 Appendix A 38 Appendix B 43 Appendix C 45 Appendix D 49 Appendix E 51 Appendix F 53 iv List of Figures Figure 1 Economic Structure 36 Figure 2 Collusion With Side Payments 37 v Acknowledgement I would like to express my sincere thanks to Professor Peter Cheng, Professor John Hughes and Professor A s h a Sadanand for their enlightening advice and helpful suggestions. The time and effort that they have individual ly devoted are truly invaluable to every part of this thesis. M y principal advisor, Professor Peter Cheng, has guided me to the domain of agency theory and stimulated my init ial interest in this area. I would also like to thank many others for their help and contributions relating to this thesis and to my master's program. v i Chapter 1. I N T R O D U C T I O N An agency exists when some people (agents) perform a task on behalf of other people (princi-pals). This is a typical form of modern economic organizations. A problem of incentive conflicts between these groups is widely seen in these organizations, and is perhaps one of the most impor-tant issues to resolve. Incentive conflicts and information asymmetry together make this problem nontrival. Theoretically, if the principal has perfect information, i.e., the principal knows the action choice of the agent and the underlying condition, he or she can force the agent to take the most desirable action by imposing a potentially severe penalty. However, the existence of information asymmetry makes this forcing contract a practical impossibility. Accordingly, the principal seeks to design an appropriate incentive compensation scheme in order to induce the agent, at his or her own interest, to take actions that the principal prefers. This paper studies the incentive problem in the context of a centrally. planned industry. There are a number of previous studies on planning and regulating industrial production. Domar (1974) has presented an incentive compensation scheme for a single firm industry. The scheme bases the compensation on a weighted average of sales and profit. Baron and Myerson (1982) have derived the optimal policy of prices and subsidies to regulate a monopolistic firm whose cost coefficient is unknown to the regulator. In their setting, the authors assumed that the cost function is linear. More recently, Donaldson and Neary (1984) have proposed an incentive scheme, in which the bonus or penalty a firm gets is based on its profit relative to the industrial average. This scheme is effective when each firm's cost function is identical to that of at least one of the other firms in the industry. These studies assumed that firms are reward-maximizers. Therefore, they only considered the problem of adverse selection, not moral hazard. In this paper, we assume that there are disutilities associated with effort levels. So, we consider the joint problems of moral hazard and 1 adverse selection. The analysis is conducted in both single firm settings and multiple firm settings in which firms' cost functions are unrelated. Chapter 2 presents a general model of the problem. The model is a special case of the gen-eralized principal-agent model in Myerson (1982). Chapter 3 and Chapter 4 present the analysis. We study two types of production process in Chapter 3 and Chapter 4 respectively. In Chapter 3, the production is of fixed proportions, that is, different types of input are not substitutable. An incentive compatible compensation scheme is first derived for a single firm industry. The scheme is then extended to a multiple firm industry. The subgaming issues amongst the agents are discussed, particularity on the issue of whether the incentive scheme is collusion immune. In Chapter 4, we assume that the inputs of the production are substitutable. Using a similar approach, we derive an incentive compatible scheme. Chapter 5 provides concluding remarks. 2 Chapter 2 THE GENERAL MODEL 2.1 Basic Information The economic structure of the industry is shown in Figure 1. There is one central planner (the principal) and n firms (agents). The principal is delegated the responsibility of planning and controlling the industrial production for the benefit of the society. The principal makes policy on production and compensation, requests local information, and finally decides the actual compen-sation for each agent based on the agent's performance. Each agent, on the other hand, works in his or her own interest. He or she acquires material from the society and provides effort in order to operate the production process. It should be pointed out that it is the society, not the agents, who bears the cost of material input. All firms produce a homogeneous product. We denote I = {l,... ,n} as the set of all agents, and agent i, i £ 7, as any individual agent. For agent i, the production output, g,- £ Qi, is a function of material input m,, TO, £ M,-, labor input e,-, e, £ Ej, and the state of production efficiency s,-, s, £ 5,-, fi : Mi X Ei x Si -> R, i e I. For the industry as a whole, the production function is the following mapping, / : M x E x S -> Rn, 3 where M = Yli-i % is the set of material input combinations, m — {m\,..., mn); E = Yli=i &i is the set of labor (or effort) input combinations, e = [e\,..., en); S = njLi $ i s fcne s e* °f state combinations, s = (si,... , sn); and / = liiLi /»• The total output of the industry is q = ft; q G Q C R. The total material input of the industry is m = zDi=i m i ; m £ M. The following information is common knowledge: — Product demand function, P = P(<?), with P'(q) < 0. — Material cost function, c = C(m), and marginal material cost function, MC{m) = C'(m), with MC'(m) > 0. — Agents' utility functions over money z and effort e, uf = Ui{z) - v,(e), with U\{z) > 0, U"(z) < 0, v[{e) > 0, and t/'(e) > 0, Vi€ /. — Material set, M,, effort set, state set, 5,-, and its probability distribution, p,(s,), Vi£ 7. — Realization of material input, TO,-, and production output, <fr, Vi£ /. These are known ex post. The sequence of the events is 1. The principal offers a contract, which specifies the rules to determine the production level, 1 — (9i,--->?n)> material usage, TO = (TO1;...,TO„), and compensation, z = [z\,... , zn). These rules are functions of the reported state, s = (si,...,sn), and other commonly known variables. Once an agent accepts the contract, he or she can not quit. 2. Agent i observes the local state of production , s,- € 5,-, Vi £ I. 3. Agent i reports a message to the principal, s,-, Vie I. Consequently, agent i is required to produce g,(s) and allowed to use material w,(s),Vi£ /. 4. Agent i chooses an input combination (m,-;e,),Vi£ /. 5. The actual output, q = (qi,..., qn), and the actual material input, TO = (mi,... ; mn), are realized and observed by all the parties. 6. Agent i gets monetary compensation 2,-(s, TO,-, qi, TO,); Vi £ I. 4 2.2 The Principal's Objective Function The principal is responsible for increasing social welfare through industrial production. We assume that the principal is committed to fulfill this task to the maximum extent. If output q is produced, material m is used, and compensation z is made, the realized social welfare is measured by w= / P(q0)dq0~ / MC(m0) dm0 - 2. Zi. (1) The principal seeks to design an incentive scheme, 7r = (q, m,z), to maximize the expected social welfare, / P(q0)dq0- I MC(m0) dm0 - > Zi Jo Jo Where Es means the expectation taken over the set S. (2) 2.3 Incentive Compatibility Condition For the production function qi = /t(m„ e,v, s,), if we substitute u,- (the disutility) for e,, it becomes Ii — fi(mi, v^ivi); Si) = fi(mi,v;, Si). (Here, function is slightly redefined.) Assume it is invertible with respect to v,-, then agent «'s disutility of effort, if qi is produced, m,- is used, and s,- is realized, is « i = >T1 (?i; m<> Si) = V{( qit mi, s^. 5 If agent i observes s,, a n d r e p o r t s s,, w i t h a l l the other agents observing s' and r e p o r t i n g s', t h e n agent i's u t i l i t y is u?{zi(h, s', •), si, s%) = Ui[zi{s, qi(s), rhi(s), q{, m,)] - V,-(ft, m,-, s,), (3) where s' = ... ; s,_ 1 ; s,+i, • • • , s n ) , a n d s' = (si,... , s , _ i ; s 1 + 1 ; . . . ; sn). If an in c e n t i v e scheme TT — (q,m,z) is i n c e n t i v e c ompatible, i.e., i t induces a l l the agents to repor t the true state and to take an a c t i o n preferred by the p r i n c i p a l , i t must meet the f o l l o w i n g N a s h c o n d i t i o n , (s{, rhi(si), qi{si)) G Argmax^.m.^ u^ (z,-(s,-, s%, •), s{, s'), Vi € /. (4) where (s, ; m „ qi) represents agent i's possible strategies. L e t 2,(3) = Zi(s, qi(s), mi(s), qi(s), rhi(s)) be the comp e n s a t i o n to agent i i f agent i is obedient, i.e., m, = m,(s) a n d g, = qi(s). T h e p r i n c i p a l c a n force agents to be obedient by using the f o l l o w i n g f o r m of compensation, M „ , * ,<„ , , , „,) = { W J ^ = j f ^ - = ( 5 ) In t h i s paper, we only focus on the t r u t h - i n d u c i n g aspect of incentive c o m p a t i b l e schemes. It is i m p l i c i t t h a t obedience aspect is e l i m i n a t e d by using compensation of the f o r m given by (5). F r o m (3), we see t h a t agent i's u t i l i t y is affected by other agents' message, s 1, b u t not by other agents' state r e a l i z a t i o n , s\ i.e., uiCzt{si, sh «') = s>; s') • (6) 6 This implies that a Nash equilibrium in our problem is automatically a dominant strategy equilibrium. Therefore, (4) is also the necessary and sufficient condition of a dominant strategy equilibrium. 2.4 Reservation Utility Constraint At the time of contracting, every agent has to be offered no less than his or her minimum utility level, which may represent either the utility from alternative employment or some welfare standard specified by the goverment. If the principal restricts the incentive scheme to be incentive compatible, then (7) becomes The complete model of the problem consists of (2), (4) and (8), if the incentive scheme is restricted to be incentive compatible. According to Myerson (1982), this restriction on the incentive scheme does not affect the optimality of the solution with communication. Proposition 1 (Myerson (1982), Proposition 2). The optimal incentive schemes which are incentive compatible are also optimal in the class of all incentive schemes with communication. In his paper, Myerson has also shown that when the state set and decision sets are all finite, a linear program is sufficient to solve this type of problem. We do not show how this works in our problem since it merely requires some mathematical transformations. In this paper, we mainly focus on the type of problem with continuous variables. E,[Ui{zi) - Vi{qi, mi: s,)] > Vie I. (7) E.[Ui(zi(8i)) - Vi(§(«), &,•(«),*,•)] > u,-,V*€ / . (8) 7 Chapter 3 PRODUCTION WITH FIXED PROPORTION INPUTS In a fixed proportions production process, one type of input usage will not be reduced by using more of other types of input. In other words, there is no substitution among different types of input. Later, in Chapter 4, we shall discuss the problem in which there are substitutions among inputs. To be more general, let the state of production be two dimensional, i.e., let s,- = (#,-, Si), Vi G I, where 0, is the state of material efficiency, and is the state of labor efficiency. Assume these states are independent. Since the production is of a fixed proportions type, the production function can be decomposed into two parts, m, = Mi(qi, 9i) and i>,- = V,-(ft, Si), Vi G I. Since both the material realization, m,-, and the output realization, qi, are observed ex post by the principal, can be inferred. Therefore, the principal is able to force agents to report the true state of material efficiency by imposing a penalty if they do not do so. Without loss of generality, we use a deterministic function m, = M,(<7,); Vi G I, for material usage. Hence, the state degenerates to one dimension, s, = We make additional assumptions as follows: Al. Material input is strictly increasing in production output, M'i(qi)>0, M,(0)=0; V i e / . 8 Moreover, the incremental material cost is increasing in production output, n diMCC^MiiqMMfaMdqi > 0;Vi6 I. A2. The state of production takes positive values on the real line, Si e Si c [s,-, oo) Vi e I. Finite state sets are also used for illustrations. A3. Distility function is strictly increasing and weakly convex in output, d Vi{qt, Si)/dqi > 0, d2 Vi{qi, Si)/dq? > 0, Vi(0,Si)=0, Vs,- G Si,Vie I. The larger the value of s,, the less efficient the production, which reflects that the greater the incremental disutility. In other words, d2Vi(qi,Si) — ^ — ^ > 0, V ? 1 £ Qi,Vi e I, oqiOSi with strict inequality for some s,-. It is not within our interest to consider the problem with d2 V,-/9g,-3s,- = 0 for all s,, since no information asymmetry exists in this case. 9 3.1 The Principal Has Perfect Information When the principal has perfect information, he or she observes the state realization and the agents' action choice. The principal can make an optimal central planning decision and force all agents to take the specified action. Therefore, the first-best solution is achieved. Let q* be the optimal production and z' be the optimal compensation. We then have the following model: Pl: Max ft* jZ") / / p{lo)dq0- MC{m0)dm0-) zi Js I Jo Jo p(s) ds. (9) St. I [Ui(zi) - V,-(§,•(«),*,-)]p(s) ds > Hi. (9A) Compared with the general model in Chapter 2, this model does not have the incentive com-patibility constraint. This is because the principal observes state realization, so, he or she is able to force the agents to produce what is desired. By using the Lagrange formula, we characterize the sufficient conditions to determine (q*,z*). P(£ * ) - MC(J2 Mt(qt)mq,) ~ J = ° ' ( 1 0 ) ,= 1 ,= i Ui\Z0 °ti zi^u;1 [«, + I vt(Us), 8i)p{a) ds], (ii) Js Vs £ S and Vi £ /. When all the agents are risk neutral, we set U\{zi) = 1, V i e I. Then q* = ( ? i , • • •; ?n) a n d z" — {z\,... , z„) can be easily solved from (10) and (11). However, when some or all the agents are 10 risk averse, i t is d i f f i c u l t t o determine q" and z*. Nevertheless, f o l l o w i n g p r o p e r t i e s can be derived f r o m (10) and (11). Proposition 2. If the principal has perfect information, given the assumptions on common knowl-edge, then (l) the optimal production level for agent i increases as his or her state becomes more efficient, dq'/dsi < 0 ; and decreases as any other agent's state becomes more efficient, d'q^Jdsj > 0; 1 ^ JihJ ^ Ii (%) the compensation functions of all agents are constants, regardless of the state realization. Proof: T h e p r o p o s i t i o n follows d i r e c t l y f r o m (10) a n d (11), given c ommon knowledge and assumptions A l , A 2 , and A 3 . Q.E.D. 3.2 Single Agent Problem With Information Asymmetry Here, we assume t h a t the p r i n c i p a l does not observe e i t h e r state r e a l i z a t i o n s or agent effort. A ssume t h a t there is b u t one f i r m (agent) i n the i n d u s t r y . T h i s s i t u a t i o n may be uncommon, however the analysis on single agent p r o b l e m p r o v i d e s an essential guide towards the so l u t i o n of mu l t i p l e agent problem. P2: r rq(s) rM(~q(s)) Afaz(5* | S*j E3 / P{lo) dq0 - / MC{m0) dm0 - z(s) Uo Jo M(q(s)) St. U.(z(s)) - V(q(s),s) > U(z(s)) - V(q(s),s) (12) (12A) V s G S . V s G S, E,[U{z{s)) - V{q(s),s) > u. (UB) 11 It should be pointed out that both q* and "z" are functions of the reported state. 3.2.1 The State Set Is Finite In Chapter 2, we used Myerson (1982)'s result and stated that when the sets Q, Z, M, E, and 5 are all finite, a linear programming technique can be used to solve the problem. In fact, when just S is finite, constraint (12A) contains a finite number of equations, and the problem becomes solvable without requiring the finiteness of Q,Z,M, and E. A simple example will serve as an illustration. Example 1. Suppose there is the following information, P(q) = 10 - 2q,MC(m) — 1 + 0.2m, M(q) = 2q, U(z) - z,V(q,s) = sq,u = 1,5 = {1,2}, with p ( l ) = 0.5, and p(2) = 0.5. Then / (l0-2q)dq- (1 + 0.2m) dm - 2(1) -'o Jo f'q(2) f2'q(2) / (10 - 2q) dq- (1 + 0.2m) dm - z{2)} Jo Jo + 0.5 St. *(1) - q(l) > z{2) - q(2), z{2) - 2q{2) > z(l) - 2q(l), 0.5[z(l) - q{l)} +0.5[2(2) - 2q{2)} > 1. Applying the Lagrange formula, we get the optimal solution, q*(l) = 2.5, q*(2) = 2.143, z'(l) 4.661, z" (2) = 4.125, and EW = 6.589, where EW stands for the expected social welfare. 12 3.2.2 The Agent Is Risk Neutral And The State Set Is Infinite In the usual one principal-one agent setting described in Holmstrom (1979), the principal's objective is to maximize the monetary return (or profit) from the operation. When the agent is risk neutral, the first-best solution is achieved by renting the production facility to the agent, regardless of the existence of an information asymmetry. This is because there is no risk-sharing aspect when the agent is risk neutral. Moreover, the agent automatically chooses the production which maximizes the net utility—profit minus disutility of effort since the agent keeps the residual. This is exactly the same production level as the principal would choose. In our problem, because the principal wants to maximize social welfare, not profit, the rental scheme is not directly applicable. If the principal rents the production process to the agent, the agent again chooses the producton to maximize net utility, but this is not the production level the principal wants. Nonetheless, we can design an incentive scheme which is equivalent to the rental scheme and achieve the first-best solution of the problem. Proposition 3. For the problem P2, if the agent is risk neutral, then the following compensation scheme induces optimal production, and achieves the first-best solution, where q is the actual output, and q is some constant predetermined by the following condition (13) (denote g*(s) as the optimal production rule), r r rf\>) r / / P(q0)dq0- MC(m0) dm0 p(s) ds — / V(q*(s), s)p(s) ds = u (14) M(q) s Proof. If s € S is observed and q is produced, then the agent's utility is 13 rq rM(q) uA = / P(q0) dq0 - / MC(m0) dm0 - V(q, s). Jq J M{q) The agent chooses the production by satisfying the following condit ion, ^ = P{q) - MC{M{q))M'{q) - = 0. This is the same as (10), the condition to determine the opt imal product ion when the pr incipal has perfect information. Therefore, the production induced by (13) is opt imal. Since q is chosen such that the minimun util ity constraint is binding, the solution is the first-best. Q . E . D . It is immediate that by applying (13) to Example 1, we get the same result. F r o m = 10 - 2q - (2 + O.Bq) -3 = 0, dq we have q*(s = 1) = 2.5, q*(s = 2) = 2.134. From (14), we get q = 0.9979 given u = 1. So EW = 6.589. Proposi t ion 3 says that for the single agent problem, if the agent is risk neutral , then the first-best solution can always be achieved. 14 3.2.3 The Agent Is Risk Averse And The State Set Is Infinite When the agent is risk averse, the first-best solution is generally not achievable if there exists information asymmetry. First, the forcing contract can no longer be used because the principal does not observe either state realization or agent's action choice. Second, there are both risk-sharing aspects and motivational aspects to consider. It is usually not optimal to let the agent absorb all the risk. The following numerical example is given to illustrate that only a second-best solution is achieved when the agent is risk averse with an information asymmetry. For simplicity, we use a finite state set. Example 2. The information is the same as Example 1 except that the agent is risk averse, U(Z) = yfz. (1). The principal has perfect information. Mazfa*^-) 0.5 + 0.5 rq[l) r2q[\) I (10 - 2q) dq- (1 + 0.2m) dr, Jo Jo • r'q(2) r2q{2) I (10 - 2q) dq- j (1 + 0.2m) Jo Jo dm -0.5[e(l) + 2(2)] St. 0.5[v/i(l) - ?(!)] + 0.5[v^ (2y - 2q{2)\ > 1. The solution is j*(l) = 1.502,?*(2) = 0.1456,^(1) = 2*(2) = 3.60, and EW = 1.40. This is the first-best solution. (2). The principal does not observe the state realization. Maz(3*;j*)0.5 + 0.5 rq(l) rzq(l) / (10 - 2q) dq- I (1 + 0.2m) Jo Jo - r'q(2) r2q{2) / (10-2q)dq- / (1 + 0.2m) St. ^/^X) - q{l) > v^) - q{2), V/I(2T-2?(2) > v /KlT-2g(l) ; 0.5[ v^ OT - ?(1)] + 0.5[\fW) - 2?(2)] > 1. dm dm -0.5[2(1) + z(2)\ 15 The solution is f (1) = 1.1745, §* (2) = 0.3624, z*(l) = 5.549,i*(2) = 2.3827, and EW = 1.2404. We see that, from the principal's point of view, this solution is strictly inferior to that in Now, we start to analyse the general problem of P2. We assume that the set S is infinite, with common prior probability density function p(s). Since the constraint (12A) contains an infinite number of equations, it is impossible to solve the numerical values of "z and q under each state. Rather, we have to look for functions z(s) and q(s). When all externally determined functions are continuous and differentiable, and S is infinite, then we are able to concentrate on those z(s) and q(s) which are continuous and differentiable. The necessary and sufficient condition for a scheme , (z(s), q(s)), to be incentive compatible, i.e., it satisfies (12A), is, duA(z(s),s) _ dU(z('s)) dz dV(g,s) dq ds dz ds dq ds v s e s. if s < s; if s = s; (15) if s > s: The next step is to search for the optimal scheme (q*(s), z*(s)) which satisfies the necessary and sufficient condition (15). The result is presented in proposition 4. Proposition 4. For the single agent problem P2, if assumptions Al, A2, and AS hold, let z{s) = U~l f dV(q,s) dq(s) dq ds ds+ h (16) 16 where s is the reported state, and h is some constant determined by the minimum utility constraint (12B), and q(s) is chosen such that P(q) - MC(M(q)) • M'(q) -U'(z(s)) 1 dV(q,s) dq = 0, (17) V s € S, then, 1. the incentive scheme is the optimal incentive compatible scheme; 2. the optimal incentive compatible scheme is unique; S. the optimal production q" increases as the state becomes more efficient; 4- the principal derives all the rent. Proof: see Append ix A . Corollary 4.1 For the single agent problem P2, if assumptions Al to AS hold, and if the incremental disutility function, dV [q, s)/dq is strictly increasing in s, then the optimal production q* is strictly decreasing in s. Proof: This follows directly f rom the proof of proposition 4. Proposit ion 4 says that there is no other incentive compatible scheme which is superior to the one determined by (16) and (17). Moreover, by Myerson's revelation principle, there is no other incentive scheme of any type which is superior to the one determined by (16) and (17). When the agent is risk neutral , ( l e t U'(z) = 1 ), then it is straight forward to solve q{s) from (17), and then solve z(s) from (16). Since the opt imal incentive compatible scheme is unique, the scheme from (16) and (17) should be the same as (13) if the agent is risk neutral. We show this in the following. 17 From (16), we have d'z(s) __ dV(q, s) dq{s) ds dq ds Since (q,z) is incentive compatible, then, s — s, imply ing that dz(s) _ dV(q, s) dq(s) ds dq ds ' and, hence Therefore, (17) becomes dz _ dz 1 _ d V (q, s) dq~ ds £ ~ dq dz, - = P(q) - MC(M(q)) • M'(q), which yields rq fM{q) z = / P(q0) dq0 - I MC(m0) dm0 + h. (18) Jq JM(q) This compensation is the same as the one derived in Proposit ion 3. When the agent is risk averse, determining the function form of q(s) and z(s) is very complex. We may even be required to take a numerical solution approach. Part (3) of Porposit ion 4 is quite intuit ive. When the state is more efficient, the incremental cost of production is smaller. Since the incremental value stays the same, then we should increase the production to maximize the welfare. Since the min imum uti l i ty constraint (12A) is binding, it is the pr inc ipal who derives all the rent from the private information of the agent. In order to have this result, the assumption that the agent can not quit once he or she accepts the contract is crucial . 18 3.3 Multiple Agent Problem With Information Asymmetry In this section, we extend our analysis to the multiple agent problem. We sti l l assume that the set S is infinite, where S = Y\i=i Let p(s) be the joint probabil ity density function over S. We repeat the model for convenience. P3 / / P{q0) dq0 - / MC{m0) dm0 - > z,-(s) st. ufcisi, a1') - v;-(§,•(«,-, v), *,-) > #,•(*,•(*,•, a1')) - v,-( j,-(s,-, a1'), *,•) V(«,-, 3') € 5 , Va,- G 5,- and V i € / , "Efci (^5,-(')) 1=1 V i G 7, p(s) ds > u,-p(s) rfs (19) (194) (19B) where ?(3) = (? i (3 ) , . . . , ? n(3)) and z(s) = {h{s), ••• , zn{s)). When there is more than one agent, the opt imal production has to be determined by the overall reported state. Hence, g,- is a funct ion of both s,- and s' = ( s j , . . . , S,-_i, s,+ i ; . . . , sn). B y extending the opt imal incentive compatible scheme of the single agent problem, we have the following result. Proposition 5. For the multiple agent problem PS, if assumptions Al to AS hold, if we let the compensation be i) dqi{si,s') z,{si,si)= f/-1 L; 9 ft 3 s,-dsj + h{ (20) VS' G 5* and V i G I, 19 where hi is some constant determined by the minimum utility constraint, and if we choose qi such that P(£ ft) - MC(J2 Mm • M>(qi)) - ^ • = o, (21) Vs,- G Si, and V i G /. 1. equations (20) and (21) give the optimalincentive compatible scheme, implying that truth-telling is a dominant strategy for every agent; 2. the optimal production for agent i increases as agent i's state becomes more efficient, and decreases as any other agent's state becomes more efficient; S. the principal derives all the rent from the private information of the agents; 4- the solution is the first-best when all the agents are risk neutral, and strictly second-best other-wise. Proof: see Append ix B . The economic intuit ion behind Proposit ion 5 is the same as Proposi t ion 4. In (21), the first term is the incremental value of product ion, and the sum of the second and the third term is the total incremental cost of the product ion. A t the opt imum, the total incremental cost of production for each agent is identical, and the total incremental cost for any agent equals the incremental value of production. However, another issue which arises in the mult iple agent case is that the principal's incentive scheme creates a subgame among the agents. Th is increases the dimensions of the problem. The following numerical example is given to show the calculat ion of q and z. 20 E x a m p l e 3. Suppose for a one princip a l - t w o agent problem, there is the f o l l o w i n g i n f o r m a t i o n , P(q) = 10 - 2q, MC(m) = 2m, Mi{qi) = qi, fli) = (1 + s i ) ? ? , M2{q2) = 2q2, V 2 ( g 2 ; s 2) = (1 + s2)q\, s i E [1, 2], p x ( a i ) = 1, s 2 G [1, 2], p 2 ( s 2 ) = 1. L e t ( s i , s 2 ) be the r e p o r t e d state, the o p t i m a l p r o d u c t i o n is d e t e r m i n e d by 10 - 2{qi + q2) - 2{qi + 2§j) - 2(1 + h)qi = 0, 10 - 2{qi + q2) - 4(ft + 2q2) - 2(1 + s2)q2 = 0. Therefore, 5 ( S 2 + 3) , , 5«i 1l = — — — ~ . — — r : — — , and q2 — sis2 + 6 s i + 33 2 + 9 sis2 + 6 s i + 3 s 2 + 9 Since s x = 1 an d s 2 = 1, t h e n * . x _ / 5 l ( - 5 0 ) ( g 2 + 3 ) 2 ( S 2 + 6 ) ( S l ) ( s i s 2 + 6 s ! + 3 s 2 + 9 ) 3 2 5 ( s 2 + 3 ) 2 ( s i + 1) 2 5 ( s 2 + 3 ) 2 (S1S2 + 6S1 + 3 s 2 + 9 ) 2 + (S1S2 + 6s! + 3 s 2 + 9 ) ( S 2 + 6) 5 0 ( s 2 + 3 ) 2 2 5 ( s 2 + 3 ) 2 ( 4 s 2 + 1 5 ) 2 ( 4 s 2 + 1 5 ) ( s 2 + 6) A n d u s n = 2 5 § ' ( ' 2 ) I ^ n l' 2 > ( s 1 s 2 + 6 s 1 + 3 3 2 + 9 ) 2 ^ ( s i S 2 + 6 s i + 3 s 2 + 9 ) ( S l + 3) 50S? 25s 2 1 - - ^ 1 +h2. (7h + 1 2 ) 2 (7ai + 12)(3i + 3) It is easy t o ver i f y t h a t uA(si,s2) is ma x i m i z e d at s i = s l 5 a n d M 2 ( S I , S 2 ) is m a x i m i z e d at h = s2. 21 3.4 The Subgame Of The Agents We have proposed an incentive compatible compensation scheme which induces a dominant strategy equilibrium. However, as has been pointed out by Amershi and Cheng (1986), a dominant strategy equilibrium is not always implementable. It is possible that the agents can be made better off, with at least one agent being strictly better off and all others weakly better off, through implicit or explicit collusion. Therefore, we need to study the subgame of the agents. Specifically, we would like to see under what conditions the equilibrium preferred by the principal can be implemented. The following terms are defined to facilitate our discussion. Definition 1. A non-cooperative subgame is the one in which the agents do not play non-Nash strategies and do not use side payments. Definition 2. A subgame induced by an incentive scheme is a prisoners' dilemma if there exists at least one non-Nash strategy combination which, without side payments, strictly Pareto-dominates the equilibrium specificed by the scheme. Definition 3. An equilibrium is non-cooperatively implementable if, in the subgame, the equilibrium at least weekly Pareto-dominates all the other Nash strategies in the non-cooperative subgame. Definition 4. An equilibrium is implementable if, in the subgame, there exists no strategy combina-tion, either cooperative or non-cooperative, which strictly Pareto-dominates this equilibrium. Definition 5. An incentive scheme is collusion immune if the equilibrium induced by the scheme is implementable under every state realization. 22 3.4.1 The Agents Do Not Cooperate When the agents do not cooperate, i.e., they neither use side payments nor play non-Nash strategies, then the following result is obtained. Proposition 6. If the agents do not cooperate, and if every agent's incremental disutility is strictly increasing in his or her state realization, 32 V,(g,; s,)/3g,-3s, > 0, Vg,- £ Q, and Vi £ I, then the dominant strategy equilibrium induced by the incentive scheme (20) and (21) is non-cooperatively implementable under every state realization. Proof: If 3 V,/3g,- is strictly increasing in s,-, then from (21), we know that g, is strictly decreasing in Si, i.e., 3g,(s,-, s')/3S,- < 0,Vs £ S, and Vi£ I. Since duf = dVi{qi,Sj) _ dVj{qi, s.) 3 g, (§,•,§') dsi dqi dqi 3 s,- ' then uf is strictly increasing when s, < s,, and strictly decreasing when s,- > s,. Therefore, agent i is strictly worse off by not reporting the true state. This, together with the fact that telling the truth is a dominant strategy for every agent, implies that the equilibrium induced by the scheme is the unique Nash equilibrium. Since the agents do not cooperate, this implies that the equilibrium is non-cooperatively implementable under every state realization. Q.E.D. When we allow 3V,/3g, to be weakly increasing in s,, then Proposition 6 may not be true. Suppose there are two agents, agent 1 and agent 2. If 3^/3^1 is weakly increasing in s\, then it is possible that reporting s\, an untruthful state, instead of s\, the true state, does not change qi, nor ui, but increases agent 2's utility, U2- Moreover, it is possible that (si,S2) is also a Nash equilibrium. (Amershi and Cheng (June 1986) have shown this situation.) Therefore, (Si,S2) is implemented rather than (si,Si). 23 3.4.2 The Agents May Cooperate Without loss of generality, we assume there are two agents in 3.4.2. If the agents coooper-ate, there appear to be many obstacles to implementing the equilibrium induced by the scheme. The following example shows that a collective strategy of not reporting truthfully may be strictly preferred by the agents. Example 3 Continued. The utilities of the agents, if (si,s2) is realized and (3i,S2) is reported are A , . ^ x 25(s2 + 3)2(s1 - sx) + (S152 + 6 S 1 + 3 S 2 + 9)2 25(32 + 3)2 50(s2 + 3)2 (s1s2 + 6s1 + 3s2 + 9)(s2 + 6) (4s2 + 15)2 25(§2 + 3)2 (4s2 + 15)(s2 + 6) 1 and Al- l- - \ ^ 25s?(32 - s2) (hh + 6s! + 3s2 +9)2 25s? (sis2 + 6si + 3S2 + 9) (Si + 3) 503? 25s? 1 1 +A 2. (73i + 12)2 (7si + 12)(«i + 3) Suppose that the state realization is (2;2). For simplicity, we consider four reporting strategies, (h,h) — (2, 2), (2,1), (I, 2), and (1,1). The result is shown in Figure 2. We see that reporting (2,1) with any side payment within the range of (0.024,0.214) from agent 1 to agent 2 makes both agents strictly better off. There are two possible types of collusion when the agents cooperate. One, as is shown in the numerical example, is that a side payment is necessary to make no one worse off. The other is the prisoners' dilemma situation. The following proposition states that the incentive scheme we have derived is not immune against these types of collusion when considered simultaneously. 24 Proposition 7. If the agents cooperate, then the incentive scheme determined by (20) and (21) is not collusion immune. Proof: see appendix C. Proposition 7 says there exists at least one state realization under which the equilibrium induced by (20) and (21) is not implementable, because the agents are better off through collusion. Even though the incentive scheme is not collusion immune, we are able to say something about the agents' possible dishonest behavior. Proposition 8. Under the incentive scheme determined by (20) and (21), if the incremental disutility dVi/dqi is strictly increasing in s,, then the agents' possible collusion is either (1) s,- < s,-, W £ / with at least one strict inequality, or (2) s,- > s,; Vi£ /. In other words, every agent lies in the same direction. Proof: see Appendix D. Intuitivly, we think that agents tend only to report a state worse than the actual one so they can reduce the amount of effort. This is true when the problem is solely moral hazard. In the problem we are studying, there are both moral hazard and adverse selection present. We try to solve both by a scheme which induces the agents to report the true state. The reason for the result in Proposition 8 is that the principal pays very dearly to induce the truthful reporting in good states. Nevertheless, by getting the true state, the principal gets the best overall. Finally, we want to point out one more issue. When the agents cooperate, they can collude with a side payment. However, in our single period setting, this side payment is not credible, because there is nothing to enforce its payment. This, of course, reduces the possibility of collusion. 25 Chapter 4 PRODUCTION WITH SUBSTITUTABLE INPUTS In Chapter 3, we assumed that the production is of the fixed proportions type. Now, we are going to study the problem with substitutable inputs. The following additional assumptions are made for this chapter. These assumptions hold for every agent. A4. The production function is strictly increasing and weakly concave in both material input and labor input, df{m,v,s) df{m,v,s) m dv d2f(m,v,s) < Q d2f{m,v,s) < Q dm2 ~ ' dv2 ~ ' 3 d2f d2f . d2f dm* o r d mo v A5. The marginal productivity of either input is increasing in the level of the other input, d2f dmdv > 0 \/sE S. A6. The state of production, s, is some positive value on real line, s 6 S C R, s > 0, with s being the smallest member of the set. Also, marginal production is decreasing in s, Uf)<0, and Ud/)<0, ds dm os dv with at least one strict inequality for every (m, v, s) combination. 26 A 7 . The output is zero when either input is zero, / (0 , v, s) = f{m, 0, s) = 0 , V m G M, Vv G V, Vs G S. Th is , together with A 5 , implies that < 0. A 8 . The marginal substitution between inputs is decreasing, i.e., —dv/dm increases in v and de-creases in m, given (q, s). In addit ion, all the common knowledge stated in Chapter 2 applies here. Aga in , it is implicit that the agents are all obedient under the type of contract given in (5). Agent i's disuti l i ty, when s = (si,..., s„) is realized and s = (s\,... , sn) is reported, is The analysis in this section is parallel to that in Chapter 3. We focus on the t ruth revealing aspect of the incentive compatible scheme. When the principal restricts the incentive scheme to be incentive compatible, the problem is then described by the following model: Vi{h, s\ s{, s') = / 1{qi(s), m,(s) ; s,). (22) So agent i's uti l i ty is ufiziisi/s')^,^1) = Ui{zi(s)) - Vi(si,s\si,s'). (23) 4.1 Single A g e n t P r o b l e m P4: (24) 27 St. U(z{s)) - v(s, s) > U{z(s)) - v(s, s) (2AA) Vs£S and Vs e S, / [U(z(s)) - v{s, s)]p{s) ds > u, (245) J s where q(s) = f[m(s), v(s, s), s], Vs € S. Lemma 1. The optimal incentive compatible scheme (q*,hY,z~) satisfies the following conditions, P(q) • ~ MC{m) = 0, (25) am p ( ' > - f { - c 4 = o ' (26) Vs€ S. Proof: The result follows directly by taking the first order conditions of the Lagrange formula. It is easy to see from (25) and (26) that at the optimum, the incremental costs of production in material input and labor input are identical. Q.E.D. Lemma 2. For the single agent problem P4, if assumptions A4 to A8 hold, then the agent's actual disutility, v{s,s), has the following properties, , .„ . dv , dv , n d2v os os osos Proof: see Appendix E. Lemma 2 says that the agent can reduce the disutility of effort by reporting a less efficient state, given the state realization; and the more efficient the state realization, the less the disutility 28 of effort is, given the reported state. The agent has incentive to mispresent the state of production. Therefore, it is important for the principal to design an incentive scheme which induces the agent to behave properly. Proposition 9. For the problem P4, if assumptions A4 to A8 hold, then the following compensation scheme is truth inducing, K's) = U~l [ f9 v[{s,s) ds+h], (27) where h is some constant chosen to make the minimum utility constraint binding. Proof: The agent's utility , under the compensation (27), is uA('z(s),s) = U(z(s)) - v(s,s) = I v[(s} s) ds - f - h — v{$, s) duA , , ds From Lemma 2, i/[2 < 0, therefore, %->0, if a < 3; ^ = 0; if ^ < 0, if S > a. So the agent's utility is maximized at S = s. Q.E.D. Following is a numerical example to show the calculation of the incentive scheme. Example 4. Suppose P(q) = 10, MC(m) = 0.1m, U(z) = z, v(e) = e, q = -yjme = -y/mv. Then the optimal incentive scheme is determined by (25), (26) and (27). 29 Therefore, 10 - 2s2 ql = 0, 10 - 23 2 ql (0.1m) = 0, q = i V n w . 1250 ,„ A , 6250 1 s ) = ~^r~, = - 7 4 - , , . * r . v 250 and m (s) = ^ - r -s J . The actual disuti l i ty, if s is observed and s is reported, is , A , 6250s 2 From (27), we get the compensation 2(3) = 9 3 7 5 ( ^ - i ) + A. So the actual ut i l i ty the agent gets is Ai~t~\ \ n o , r , 1 1 , . 6250s 2 uA{z{s),s) = 9 3 7 5 ( - - - ) + h-s ' s* s" Since 3 u A 37500, s 2 3s s 5 s 2 therefore « A is maximized at s = s, i.e., the agent is truth tell ing. Suppose that u = 10, s E S = [1,2], w i th p(s) = 1, then f rom the m in imum uti l i ty constraint, we get h = 8473.5. Therefore, the compensation function is 9375 r(3) = _ _ - 9 0 1 . 5 . 30 In the above example, we see that the agent's disuti l i ty, v(s, s), decreases as the reported state, s, increases (less efficient). A lso , compensation z('s) decreases as s increases. Hence, dz/dv > O.'If this condit ion holds, then we have the following proposit ion. Proposition 10. Under the incentive compensation scheme determined by (25),(26), and (27), if dz/dv> 0, then, the optimal production, q", increases as the state becomes more efficient. Proof: see Append ix F 4.2 Multiple Agent Problem Analogous to the analysis in Chapter 3, the incentive compatible scheme for the single agent problem can be extended to mult iple agent problem. It is given below. V) = UT1 [ r dV'{'Sf'Si) \ ^ 3 . dst + hi], (28) J ^  OSi P ( E * ) - ^ - M C ( E ™ < ) = 0 , (29) i=l ' »=1 «=1 I V ' VS e S, Vi e /. Aga in , under the incentive compatible scheme determined by (28), (29) and (30), t ruth telling is a dominant strategy equi l ibr ium. (30) 31 The subgame of the agents is essentially of the same nature as the one described in section 3.4. We do not expect much to change in the result. The procedure for analysing the subgame is not repeated here. 32 Chapter 5. C O N C L U D I N G R E M A R K S We have developed an incentive compatible compensation scheme for a social planning problem. The restriction of the scheme to be incentive compatible does not change the optimality of the solution. In the problem we haved studied, both moral hazard and adverse selection are present. It is interesting to see that inducing truthful reporting of the state is able to deal with both aspects simultaneously. It is assumed in the analysis that the agents can not quit once they accept the contract. Re-laxing this assumption does not significantly change the anslysis, since the basic incentive problem does not change—making the agents worse off if they do not report the true state. However, the principal is no longer able to derive the rent from the private information of the agents. Even though the incentive scheme is derived in the context of a planned economy, it may be applicable to other types of problems, such as the planning of a multi-divisional company, as long as they contain the same incentive problem. Technically, this means that the form of the objective function, whether maximizing social welfare or profit, is not crucial in the analysis. At this point, we would like to make a comparison of the incentive scheme derived in this study with the well-known Groves' scheme. The schemes are derived in different types of social planning problems. In this study, the principal wants to maximize the welfare of the society which is a separate entity from the agents; Groves' scheme applies to the situation where the principal wants to maxmize the sum of all the agents' welfare. The similarities in these two schemes are (1) the compensation to any agent is not only based on this agent's reporting, but on all the other agents' reporting as well; (2) telling the truth is a dominant strategy equilibrium under both schemes; (3) neither scheme is collusion immune. This study is preliminary, and much further research is necessary. First, the incentive scheme does not provide an obvious economic interpretation. Second, solving for the optimal production 33 and compensation scheme is very complicated when the agents are not all risk neutral. Th i r d , a more thorough exploration of the agents' subgame is possible. For example, it would be interesting to see under what conditions the prisoners' di lemma does not occur. 34 Bibliography Amershi, A. and P. Cheng (June 1986), "Implementable Equilibria In Accounting Contexts: An Exploratory study", working paper, The University of British Columbia. Aumann, R. (1976), "Agreeing To Disagree", The Annual of Statistics, Vol.4 No 6. Baron, D. and R. Myerson (July 1982), "Regulating A Monopolist With Unknown Cost", Econometrica, Vol. 50, No 4. Domar, E (Feburary 1974), "On The Optimal Compensation Of A socialist Manager", The Quarterly Journal Of Economics, Vol. lxxxviii. Donaldson, D. and H. Neary (1984), "Decentralized Control Of A Socialist Industry", Canadian Journal Of Econocmics, xvii, Nol. Groves, T. (July 1973), "Incentives In Teams", Econometrica, Vol.41, No4. Holmstrom, B. (1979), "Moral Hazard And Observability", The Bell Journal Of Economics 10. Myerson, R. (1982), "Optimal Coordination Mechanisms In Generalized Principal-Agent Prob-lems", Journal Of Mathematical Economics 10. 35 PLANNER C1 C i FIRM 1 FIRM i FIRM n Cn LABOR 1 M1 S1 PRODUCTION PROCESS 1 M l S i LABOR i PRODUCTION PROCESS i Mn Sn LABOR n PRODUCTION PROCESS n M: MATERIAL S: STATE OUTPUT 1 OUTPUT l OUTPUT n F i g u r e 1 Economic S t r u c t u r e 36 Agent 2 2 1 -3.240 -3.026 -0.420 -0.444 -3.544 -3.324 -0.196 -0.208 * Constants hi and h.2 are ignored. The true s t a t e i s (2,2), but the agents are b e t t e r o f f by r e p o r t i n g (2,1). F i g u r e 2 C o l l u s i o n With Side Payment 37 Appendix A . Proof Of Proposition 4: (1) The scheme is incentive compatible Suppose part (3) of the theorem is true for the moment. (We will verify this later on.) The agent's utility, if s is observed and s is reported, is uA(z(s),s)= U(z(s))- V(q(s),s) = /£ ~ d q — - T s d s + h - v { q { s ] ' s ) -duA =dV{q,s) dV(q,s) dq ds dq dq ds Since dq/ds < 0, and <92 Vjdqds > 0, then ^ > 0 ; ifS< S; # = 0, if S = a ; ^ < 0 , if S > s. Therefore, uA is maximized at s = s,VsG S. The compensation z(s) is truth-inducing. By applying the form of compensation specified in (5), the agent is obedient. So we have a incentive compatible compensation. The production rule q(s) is optimal given z(s) Given any incentive compatible compensation z(s), we can write the model as 38 / / P(q0) dq0 - j MC(m0) dm0 - z(s) Js I Jo Jo St. I [U(z(s)) - V(q(s)!8)}p(s) ds > u. Js p{s) ds (A3) (A4) So L(q(s),z(s),X) =(A3) + \(A4). The first order conditions are and | | = P(q) ~ MC(M(q)) • M'(q) - = 0, - 1 + A{7 ' ( z ) = 0 . We see that A is positive, so the solution is opt imal. Therefore, the opt imal product ion rule, q(s), given z(s), is determined by P(q) - MC(M(q)) • M'(q) - J 9 V ^ s ) = 0. (A5) Since z(s) is incentive compatible, so s = s, therefore, (A5) is the same as (17). (2) Suppose there exists another incentive compatible scheme, say (q0(s), z0(s)), such that (17) holds. Since both scheme are incentive compatible, the state reported is the same as the state real ized. Since z0(s) is different from z(s), there must exist at least one point, s\ G S, where Zo(si) 7 ^ z(si)- Suppose z0(si) > z(si), then from (A5), the corresponding product ion outputs must satisfy q0(si) < q(s\), and the corresponding util it ies satisfy 39 uA{Sl) = U{z0) - V{q0;Sl) > uA{Sl) = U{z) - V{q,Sl). A t the opt imum, both schemes should make the min imum uti l i ty constraint (A4) binding, i.e., / uA(s)p{s) ds = u = / uA(s)p(s) ds. (AQ) Js Js Since uA(si) > uA(si), for (A6) to hold, there must exist another point S2 € 5 , where uA(s2) < (S2). F rom (A5) , this implies that ^o («2) < z{s2), and qQ(s2) > q(s2). Without loss of generality, we assume that s i < S2- Let Az(s) = z0(s) - z(s), Aq(s) - q0(s) - q(s). Because all the functions are continuous and differentiable, there must exist one point, SQ,SI < < s2, where AZ(SQ) = zo(s0) - z(s0) = 0, [Al) dAz(s) . , . . ., . , — 7 ^ |s=,0= z'0(so) - z'{so) < 0, {A&) A ? ( s 0 ) = q0{so) - q(s0) = 0, (A9) 40 dAq(s) . , „, ds %(so) ~ q (so) > 0. (A10) Since both schemes are incentive compatible, they must meet the necessary condit ion at every state realization. So U'(z0(so))z'0(so) - 3 V <>»«>) . q ' o [ s o ) = o, (All) oq0 U'(z(s0))z'(s0) - d V { ^ S ° ] • ? ' ( S 0 ) = 0. (A12) B u t clearly, if (A7) to (A10) hold, ( A l l ) and (A12) can not be hold simultaneously. Therefore, we conclude that the optimal incentive compatible scheme determined by (16) and (17) is unique. (3) Ve r i f y tha t q'(s) < 0 Suppose this is not true in some small area s £ [o, 6], that is q'(s) > 0. Then for any two points, si, S2, w i th a < Si < S2 < b, there is' q(s2) > q(si). (A13) From the common knowledge, this implies that P(q(s2)) - MC(M(q(s2))) • M'{q(s2)) < P(?(«i)) - MC(M(q(sj))) • M'(q(Sl)). (AU) From (16), (A13) also implies that z(s2) > z(Sl). (Alb) 41 When s2 > si, and q(s2) > <Ksi)> we have dV(q(s2),s2) . c 9 V(g( g 2); S l) ^ ay(g( S l) ; S l) <9g <9(7 3g However, equations (A14), (A15) and (A16) upset the optimal condition (17) either at Si or at Therefore, it must be q'(s) < 0 for all s E S. (4) Since the minimum utility constraint is binding, the principal derives all the rent. Q.E.D. 42 Appendix B Proof Of Proposition 5 (1) The same procedure can be used, as in part (1) of Appendix A, to prove that agent f's utility is maximized at s,- = s,-,Vsv G St-,Vz G I, given s' G S', Vs* G 5'. So, the scheme is incentive compatible. Since this is true for every agent, then truth-telling becomes a dominant strategy equilibrium. From the proof of Proposition 4 in Appendix A, we know that, the optimal compensation to agent i must be z> = U{ / — dsi + hi], Ju dqi dsi where A, is some constant chosen to make the expected utility of agent i over the set S to be The optimal production condition (21) is again derived by taking the first order conditions of the Lagrange formula. (2) First, we assume there are two agents. Then (21) yields P(h + ft) ~ MCiMtfa) + M2(q2))M[(qi) - 1 . d V l £ 1 , h ) = 0, (53) dqi P(qi + ft) - MC(M1(g1) + M2{q2))M'2{q2) - 1 dV^'h) = 0. (54) u2(z2) Oq2 Suppose that Si increases while s2 stays the same. Then dVi/dqi increases, so equation (B3) is unbalanced. We look at all the possible effects on q\ and q2 to restore the equality of both (B3) and (B4). Notice that P — MC • M\ decreases in output, and the incremental disutilities increase in corresponding outputs. If q\ increases as "s increases, then in all cases, (B3) is unbalanced. If 43 qi and q2 both decrease as Si increases, then (B4) is unbalanced. The only possible way to restore the equalities of both (B3) and (B4) simultaneously is to decrease q\ and increase q2. Therefore, we have dqi/dh < 0, and dq2/ds\ > 0. In the same way, we can prove dqi/ds2 > 0, and dq2/ds2 '< 0. This conclusion can be extended to the problems with more than two agents. This is because a change of s, should have a symmetrical effect on all the other agents. (3) The principal derives all the rent because every agent gets the minimum utility ex ante. (4) When all the agents are risk neutral, the optimal production is the same as when the principal has perfect information. Moreover, the principal does not have to pay any risk premium. Therefore, the expected social welfare is the same as when the principal has perfect information. The solution is then the first-best. When at least one agent is risk averse, the principal has to pay some risk premium since the amounts of compensation are not constant. Therefore, only a second-best solution is achieved. Q.E.D. 44 Appendix C Proof Of Proposition 7 Assume there are two agents, agent 1 and agent 2. U n d e r the i n c e n t i v e scheme de t e r m i n e d by (20) and (21), th e i r u t i l i t i e s are u f f o f r , * ) , * , * ) = / h a V ^ ^ ) . g g l ^ ^ + ^ - V ^ ^ ) ^ ! ) , (Cl) «£(h{h,h),si,S2) = T dVf*>S2) • ds2 + A 2 - V2(q2(h,h),s2). (C2) i> 2 oq2 ds2 where qi = g i ( 3 i , s 2 ) a n d q2 = q2{h,h)-L e t ti(si, s2) be the amount of money has to be s u b t r a c t e d f r o m or ad d e d t o uA(zi(si, s2), s\, s2) i n order to make agent I's u t i l i t y the same as uA(zi(si, s 2 ) , si, s 2 ) . L e t t2(si,s2) be the same t e r m w i t h respect to agent 2. A n d let r . ( s 1 ; s 2 ) = h + t2 be the amount of e x t r a money t h a t the agents get by r e p o r t i n g ( S i , s2) instead of the true state ( s i , s 2 ) . L e t uf (£v(si, s 2 ) , s\, s2) be agent i's u t i l i t y i f ( s i , s 2) is report e d a n d i , is transferred given ( s i , s 2 ) . T h e n , «1 ( ^ l ( s i , S 2 ) , S l , S 2 ) = / «1 3 1 5 v x ag!( S l,s 2) dqi U, - l c9si * i a V ^ _ c J g i ( s i , 3 2 ) a <3<h asi rfsi + A i - V l ( ? l ( s i , S 2 ) , S l ) dsi + A i ) - i i ( s i , s 2 ) - V r 1 ( ? i ( s i , s 2 ) , s 1 ) = uA(zi{s1,h))h,si,s2). 45 (C3) «2 (h(si,S2),Si,S2) - I J | 2 s2dy2 a g 2 ( s 1 ; s 2 ) 3ft ds2 rfs2 + A 2 - V 2 ( g 2 ( s 1 ; s 2 ) , s 2 ) u: °2dV2 dq2(h,S2) 2 V4 dh - V2{q2{hh),S2) = uA(z2(si,h),k,si,S2). ds2 ds2 + h2 ~t2{h,h) From ( C 3 ) and (C4), we have t(h,h) = Ut 1 + U21 '^dvx dqi{Sl,h) dqi dsi 1 f'1 dVi . dqi{Sl,S2) Jix dqi dsi °ldV2 dq2{si,s2) - U. -1 / Jsn da ds2 s 2 d V 2 dq2{si,s2) i 2 q2 ds2 U-\A) - U~\B) + U;\C) - U-\D), dsi + hi dsi + hi - ^(^(ai, s2), si) + Vi(qi(si, s2), si) ds2 + h2 ds2 + h2 - V2(q2{si, s2), s2) + V2{q2{si, s2), s 2) where A --B = C D L J i d Vi 3 ? i ( s i , s 2 ) 1 dqi dsi s i dVi dqi{si, s2) ! dqi dsi 32 d v2 dq2{h,s2) ds2 '2 d V2 dq2{si,s2) , 3 f t ds2 dsi -t- hi, dsi + hi- V 1 ( ? i ( s i , s 2 ) , s i ) + Vi(qi(si,s2),si), ds2 + h2, ds2 + h2 - V2(q2{si,s2),s2) + V2(q2(si, s2), s2). 46 So dt = 1 dVi dqi(h,h) 1 dVi dqi(si,s2) + U[(ur\B)) dh 1 d s\ u'2(uz\c)) I U'2{U2\D)) s2 ra 2y 2 dq2(h,s2) dq2(Sl,s2) + dV^ _ d2q2{si, s2)-j £ 2 <9?| 3s x 3s 2 3y 2(92(si/s2); s2) dg 2(si,s 2) 3g 2 3s x 3s 2 3ft 3 Si (C76) The necessary and sufficient condition for the incentive scheme to be collusion immune is t{si,s2) > i(si,s 2) ; (CT) V(si, s2) G Si X S2, and V(si, s2) G 5i X 5 2 . Since all the functions are continuous and differentiable with respect to si and s2, then (C7) requires, at (s 1 ; S2) = (si ; s 2 ), ^ ' ^ = 0 and MfllM =0. 3 Si 3s 2 (C8) Using (C6), (C8) becomes s2 S3 d2V2 3 ? 2 (s i , s 2 ) dq2(slrs2) + d V 2 d2q2(si,s2) dq2 dsi ds2 _ 3 V2{q2,s2) _ dq2(si, s2) dq2 dsi dq2 ds\ds2 ds (C9) In order for the scheme to be collusion immune, condition (C9) has to hold for every (si,s 2) G Si X S2. Differentiating both sides of (C9) with respect to s2, we get 47 d2V2 dq2{Sl,s2) = 0, V ( s i , s 2 ) € S i X 62. (CIO) dq2ds2 dsx Tha t means either 3 l ? 2 a 2 = 0 or g,* ' ' 2 = 0, V ( s 1 ; s 2 ) G S i x 5 2 . In fact, the latter condit ion implies the former one. However, this means that the incremental disuti l i ty dV2/ds2 is not affected by the state of product ion s 2 , which contradicts assumption A 3 . Therefore, for the scheme to be collusion immune, it requires that there is no uncertainty on product ion function. So the scheme can not be col lusion immune. Q . E . D . 48 Appendix D Proof Of Proposition 8 Agent I's uti l i ty is ds2 dqi ds2 dqi d~s2 dsidfc ds2 1 dqi ds2 If dVi/dqi is strictly decreasing in S\, then dqi/dsi is str ict ly negative. Therefore, according to (D2), we can conclude that U l < 0 when S l < S l . (D3) d"s2 For the same reason, we have - r ^ 2 - < 0 when s 2 < s 2 . [D4) osi 49 Now, look at all the possible ways of colluding. A. If S2 | while Si stays the same, so uA ] (according to (D3)) and uA [ (truth-telling is a strictly dominant strategy for agent 2). Therefore, A is a possible means of collusion (but not necessary). B. If s\ i, and S2 stays the same, then uA [ and uA J • So, B is a possible means of collusion. C. If si I and 3 2 j, then the outcome can not determined, C is also a possible means of collusion. D. If Si | while s2 [, then the outcome is strictly dominated by A. So, D is not likely to be a possible means of collusion. E. If Si I while S2 |, then the outcome is dominated by B. So, E is not likely to be a possible means of collusion. F. If either Sx or J 2 increases, and the other stays the same, then the solution is strictly dominated by truth-telling. So F is not a possible means of collusion. G. If both Si and s2 increase, then the solution can not be determined or ordered. So, G may be a possible means of collusion. Therefore, we have the conclusion in Proposition 8. Q.E.D. 50 Appendix E Proof Of Lemma 2 Suppose s realized and s is reported, then the principal makes optimal decision on output and material input according to s, and the agent makes the production as is specified by the principal, but with the real state s. That is q(s) = f{m, v, s) = f(rh, v, s). (El) where v= v(s, s) and v= v(s,s). Let F = f(m, v, s) - /(TO, v, s) = 0. (E2) then p. d_F_ __ p J _ a v _ _ a, _ _ J, > 0 , E 4 ) ds $£L -f ^ = -/.-t4-/. = 0, (E5) 51 (07) - fv • Vl2 - fs'rh • m'(s) - C • v[ = 0. F r o m (E6) and (E7), we have C-A-Vi+fv-vli+C-v'^Q. (08) In (E8), the first t e r m and the t h i r d t e r m are b o t h p o s i t i v e , a n d also fv is p o s i t i v e , therefore have V» = dWs< °" Q.E.D. 52 Appendix F Proof Of Proposition 10 Under the conditions given in the proposition, v decreases as s increases. Suppose that q" increases as s increases. Then m must increase. From P'{q) < 0, and MC'(m) > 0, df/dm must increases in order to maintain the optimal condition (21), Which means = fL • m'(S) + fL • v'(s) + fL > 0. (7^ 1) The only way to hold (Fl) is v'(s) > 0, which contradicts the condition of the proposition. Therefore, q{s)/ds < 0. Q.E.D. 53 


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