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The design of supply contracts as a problem of delegation
EUROPEAN JOURNAL OF OPERATIONAL RESEARCH
ELSEVIER
European Journal of Operational Research 86 (1995) 176-187
The design of supply contracts as a problem of delegation Heike Y. Schenk-Mathes FlY Wirtschaftswissenschaften, Universitiit Frankfurt, Mertonstr. 17, D-60325 Frankfurt, Germany
Abstract
We consider the design of a supply contract as a problem of delegation in a buying firm. In general the agent (who acts on behalf of a principal in the firm) cannot be assumed to transform her information in probability estimates like the principal would do possessing the same information. In the investigated case of incomplete information the design of contracts is affected by the beliefs about characteristics of the supplier. Therefore it is important to analyze in which way the expected profit depends on these probabilities in order to find the principal's optimal instruction. The best instruction is determined by a trade-off between the costs due to the time the principal spends on dealing with the problem and formulating the instructions and the costs due to a suboptimal menu of contracts offered by the agent. Keywords: Hierarchical planning; Delegation; Information asymmetry; Self-selection models
I. Introduction
The interaction of individuals is an important characteristic of any organization. Their decisions and actions need to be coordinated in order to reach organizational goals. When a problem is perceived at a certain level of a firm, it is in general not completely solved at the same level. Usually different hierarchical levels within the organization participate in the solution process. As a simplification we consider only two levels of the firm, the first represented by the principal and the second by the agent. There may be different reasons for the principal to charge an agent with at least some parts of the solution process. First, though the principal may be capable of solving the problem in a satisfactory manner all by his own, he lacks the time for a careful decision-making process being en-
gaged in other important tasks. Second, the principal may be less qualified in some aspects of the problem. The principal intends to control the agent by issuing more or less precise instructions. The effect of these instructions on the behaviour of the agent depends on different aspects: the characteristics of the agent (such as qualification, motivation, attitude towards the future) and those of the environment (Laux and Liermann, 1993). The principal can additionally influence this effect, e.g. by verifying the agent's actions a n d / o r by establishing reward systems. He also may consider not only to give one instruction but to wait for reactions (signals) by the agent in order to issue new instructions. The main purpose of the p a p e r is to find an optimal instruction or an optimal (conditional) sequence of instructions. Since this forms a task
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H.Y. Schenk-Mathes/ European Journal of Operational Research 86 (1995) 176-187
of considerable complexity, it appears reasonable to restrict the analysis to some specific (manageable) instances of this problem. We regard the design of supply contracts as a problem of delegation. Three parties are involved: the principal and the agent in the buying firm and a supplier. The agent works on behalf of the principal and has to design the supply contract. First, we specify this principal-agent relationship and describe the principal's set of possible instructions (Section 2). In order to analyze the effect of instructions on the contracting process, we must look at the relationship between the buying firm and the supplier and derive the main characteristics of the optimal solution (Section 3). The consequences for the choice of instructions by the principal are discussed in Section 4.
2. The principal-agent relationship: Assumptions and considered instructions We assume the principal to be risk neutral and sufficiently qualified. Because of lack of time he considers to delegate to an agent the information process and the implementation of the contractual arrangements. The information process comprises information gathering and evaluation for which the agent may not be sufficiently qualified. Her actions may also be guided by objectives which do not comply with those of the principal. So the agent will not always act in the principal's sense. Once a final decision concerning the contract design is taken, the implementation of the contractual arrangements is realized by the agent according to the objectives of the principal. At the initial date 0 the principal makes a factual decision concerning the instructions for the agent and at date 1 the agent makes a final decision concerning the action (i.e. the contract between the buying firm and the supplier is concluded). Here, we have two problems to solve. The first is the organizational problem of the principal. He has to decide in which way the agent should be involved in the decision process referring to the contracting problem. The second
177
problem is the contracting problem itself. The actions concerning the organizational problem are design decisions of the top level (aT). The actions concerning the contracting problem are taken at the base level (aB). In the following we consider three possible actions at the top level (alx, aT, aT). These are design decisions concerning only the information process. The actions of the agent referring to the implementation of the contractual arrangements do not depend on these design decisions and can therefore be ignored. a T is the extreme case in which the principal determines an action for the contracting problem at date 0 and gives the instruction to perform this action at date 1. Information gathering and evaluation that may take place until date 1 are not taken into account, aT implies that the agent gathers information and transmits the signal received until date 1 to the principal. Again but now at date 1 and based on the new information situation, the principal determines the action for the contracting problem to be chosen by the agent. For ala" and a2x we assume it observable whether the agent follows the instructions or not, i.e. the agent chooses the action specified by the principal and the signal is reported truthfully, a3x implies that the agent has to solve the problem without further guidance by or feedback with the principal (for information and delegation analysis see Laux and Liermann, 1993; Demski, 1980). The information situation is the important aspect of the investigated principal-agent relationship since the solution of the contracting procedure depends on beliefs about characteristics of the supplier (like the production technology used by the supplier). Tt, t = 1. . . . . T, is used to denote these possible characteristics of the supplier. The principal has some information about the contracting problem at date 0. This information may change or new information arrive prior to date 1, thus giving rise to a reevaluation of probability estimates. FT(b) describes the principal's rule of mapping an event b into a probability attributed to this event, ,r(b). In the following, FT( •) will be called consolidation function of the top level (the principal). The domain of the consolidation function contains 'simple' events, like the event that
H.Y. Schenk-Mathes/ European Journal of OperationalResearch86 (1995) 176-187
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the supplier has a certain characteristic, and 'conditional' events, like the event that the supplier has a certain characteristic given that a certain signal occurs. For each design decision we will first determine the action at the base level (for the following analysis see Schneeweig, 1995). aT: At date 0, the principal calculates the optimal action based on his a priori beliefs concerning the characteristics of the supplier, F T(T~), and instructs the agent to perform this action. The optimal action a~ ° is determined by Opt
E[Ca~(aB)]FT(T,)].
(1)
a~ ~A
E is the expectation operator and C TB is the criterion applied to the contracting problem by the principal (top level) to evaluate the action concerning the contracting problem and this action is taken at the base level. A is the set of possible actions at the base level. aT: The principal gives the instruction to gather information and to transmit the signal received at the base level. After receiving the signal the principal directs the agent to choose the corresponding optimal action. The principal knows all signals I i, i = 1. . . . . i, the agent (or base level) may receive until date 1 and can form a posteriori beliefs about the characteristics of the supplier conditional on these signals, FT(Tt I//). Given a signal Ii, i = 1. . . . . i, the corresponding optimal action a2U°lI i is determined by Opt E[cTB(a~)IFT(T,[I,)]. a~A
(2)
aT: The principal does not specify an action. Given a signal Ii, the agent may assess other probabilities as the principal. The principal has to anticipate this assessment for every signal. The principal and the agent may also apply different criteria to calculate the optimal solution. Each possible combination of signal received, /~ (i {1,..., I}), consolidation function used, FjB ( f ~ {1. . . . . if}), and applied criterion C ~ (c {1. . . . . C}) by the agent (at the base level) constitutes a scenario
Ss={Ii, F?,C~},
s=l .....
S=i.F.C.
The principal has to anticipate the action a~°l the agent will choose if a certain scenario
Ss={Ii, F?,CB},
s=l
is given. The action aB°l
Ss
. . . . . S,
S s, s = 1. . . . . S, solves
Opt E[C•(aBIIF•(Tt]Ii)]. a~A
(3)
(3) describes the principal's estimation of the agent's optimization problem. The principal has to anticipate the agent's reaction if a certain scenario occurs, i.e. the agent receives a certain signal, derives a certain assessment of probabilities according to one of the possible consolidation functions and chooses a certain criterion. For each design decision we have calculated the corresponding optimal actions of the contracting problem. Only in the case of a T we have determined an unconditional action. If the principal chooses a T we get actions conditional on the signals transmitted by the agent at date 1 and in the case of a T actions conditional on the scenarios at date 1 have to be considered. Since the principal has to decide upon the design decision at date 0 and since he does not know whatever signal or scenario will occur, he has to evaluate the action at the base level given the information situation at date 0, i.e. given the beliefs concerning the signals and scenarios at this date. Now to find the optimal design decision a T* at date 0 the principal has to consider additionally the costs of information and delegation and then to compare the three decisions a T, a T and a T. We will come back to this point in Section 4 where we will specify the action and the criterion of the contracting problem.
3. The buyer-supplier relationship: Assumptions and properties of an optimal menu of contracts
3.1. Assumptions At date 0 the supplier has already been chosen but still there is incomplete information about the type of supplier. E.g. the production technol-
H. Y. Schenk-Mathes/ European Journal of Operational Research 86 (1995) 176-187
ogy used by the supplier may not be known with certainty. Consequently we have precontractual asymmetry of information between buyer (represented by the principal) and supplier. The buying firm considers two types of supplier to be possible ( T = 2). At date 1 the buying firm offers a menu of contracts. A supply contract must be concluded, i.e. for each supplier type there must be at least one contract which is acceptable. The possibility of renegotiation is not considered. If the supplier is indifferent to the contracts offered by the buying firm, then it is assumed that she chooses the contract preferred by the buyer. Buyer and supplier are both risk neutral. Each type of supplier, t ~ {1, 2}, has her own cost function
tion (iii) guarantees the existence of an inner solution for every type of supplier. Consequently we have the following profit G B for the buyer:
(5)
GB(x, P) = V(x) - P , which is the criterion cVa(-) of the principal.
3.2. Optimization problem and properties of an optimal solution If the buyer knew the type of supplier, t = t, then he would face the following optimization problem: Max GB( X, p) = V( x) - e x,P
(6)
subject to
T C ( x l t ) = c , ' x + F t, with x > 0 as the quantity delivered, F t > 0 the fixed costs of supplier type t and c t > O the variable costs per unit. Supplier type 1 has higher variable costs than supplier type 2: (i) Cl > c2 > 0. P denotes the payment the supplier receives from the buyer for the quantity delivered. Then the profit of supplier type t is Gs(x, Pit) =e-TC(xlt)=P-ct'x-F
179
t. (4)
The supplier type t will only cooperate if she gets a profit of at least GMin>_0 which we call minimum profit for type t and which is assumed to be known. E.g. the supplier may have other opportunities and the most attractive of these is providing the minimum profit. The delivered good is a productive input for the buyer. The value of production of the quantity x (without the costs due to the payment P) is denoted by V(x). V ( x ) is twice continuously differentiable with respect to x and fulfills the following conditions: (ii) V~x(X) < 0 and (iii) Vx(O) > c a. The subscripts indicate derivatives. According to (ii) the function V ( x ) is strictly concave. Condi-
P - c i ' x > . _/7"Min , s i -~F i -
Gs~"
An optimal solution to the above problem must strictly fulfill the (cooperation) constraint, because otherwise G s can be increased by reducing P. By substitution of P in the objective function and differentiation we obtain the following necessary (and sufficient) condition for an optimum: Vx(X ) =c-t.
(7)
The optimal solution of (6) we will call the firstbest solution ( X F, pF) for type L The LHS of (7) is decreasing in x. Together with c 1 > c z we have XaF < x F. In the following we investigate the case where the buying firm does not know with certainty the type of supplier. If the buying firm offers only one contract (i.e. only one combination of quantity x and payment P) and if the cooperation constraint must hold for every type of supplier, the optimal contract does not depend on the probability distribution of the type. The buyer can increase the expected profit offering a menu of two contracts {(xl, P1), (x2, Pz)}. The supplier has to choose among these contracts. This implies the idea of a self-selection mechanism. (Recent applications of self-selection mechanisms can be found in Cr6mer and Khalil (1992) and Wagenhofer (1992).)
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H.Y. Schenk-Mathes / European Journal o f Operational Research 86 (1995) 176-187
We get the following optimization problem:
Max E[G(x,, P,, x2, P2)]
Xl,PI,x2,P2
=
(8)
7/'1" (V(Xl) -- P1) + ~'2" ( V ( x 2 ) - P2)
subject to
(2 may be infeasible, i.e. 2 < 0). An optimal menu {(xl*, el* ), (x2*, P2* )} for 7r 1 ~ (0, 1) has the following properties:
Proposition 1. (a) If x F, x F > 2 then
Pl-c~'x 1> P2-Cl.X2,
(xl* < x F, Pl*
P 2 - cz'x2 > P t - c2"xl,
and
(self-selection constraints)
( x ; -- x 2, F p2* >
e t - ct " x t >- t"TMin "-'st "~- Ft = a s t
for every t e {1, 2},
(x; =xL e,*
(cooperation constraints)
and
x,>_0
(x~>xF, p;>PV).
for every t e {1, 2},
where rr t denotes the probability that the supplier is type t. In Harris and Townsend (1981) and Sappington (1983, 1984) one can find examples of this optimization problem, but they do not regard variations of the RHS of the cooperation constraints. The inclusion of such variations leads to a more complex structure of the optimal solution. The self-selection constraint for the supplier type t states that her profit when choosing the 'right' contract (i.e. the contract the buyer intends her to choose) must be at least as large as the profit due to the other contract of the menu. The cooperation constraint must hold for every type. Here, we want to emphasize that the cooperation constraint of a supplier type must be fulfilled even if the profit due to this type is negative. Such a case is given if we consider an additional contract and the realization of this contract is a necessary condition for the fulfillment of all other contracts. Let {(x, P ) : P = G s t + F t + c t - x } be the profit-indifference curve for a given profit Gst of the supplier type t, which we call the minimum-profit-indifference curve (MPI) if Gst + F t = Gst (see Fig. 1). The MPI's have a point of intersection at 2
(b) If x v, x~ <2 then
Gs2 - Gsl C 1 -- C 2
(c) If xF < 2 < X ~ then the optimal menu of contract is {(x F, pF), (x F, p2F)}. An important result is that in the case of x F < 2 < x2v the first-best solutions are attainable. In the two other cases we can describe the optimal menu more precisely.
Proposition 2. (a) If x F, xF2 > ~ then the optimal menu is either
or
((x;,c, (xF,(cI--C2)
" X 1 "4- C 2 " X F "~- G s a ) )
with x I = 0 or 77 1 " ( V x ( X 1 ) - c 1 ) = 77-2 . ( c
I - c2).
(b) If x F, x F <2 then the optimal menu is either
{ ( x ~, P~ ),( 2,c2 "~ + Gs2 ) } or {(xF,Cl'X
F + (c2--el)
"X 2 "Jl- a S 2 ) ,
+ with x;
) -
=
- c,).
s2)}
H. Y Schenk-Mathes/ European Journal of Operational Research 86 (1995) 176-187
~
MPI (t=l)
/
x~ ~
MPI (t=2)
x~
. x
Fig. 1. Indifference curves for XlF < £ < x F.
For the proofs of Propositions 1 and 2 see Appendix A. 3. 3. Interpretation o f the optimal menu o f contracts
Fig. 1 shows the indifference curves of the supplier types and the buyer. The concave indifference curves are those of the buyer. The profit of the buyer increases in southeasterly direction. The linear MPI's of both supplier types are also depicted in Fig. 1. The profits of the supplier types increase in northwesterly direction. In the case considered in Fig. 1 the intercept of the MPI of type 1 on the vertical axis is smaller than the one of type 2. The intercept is the sum of fixed costs and the minimum profit of the type considered. The shape of the curves makes sense assuming that the supplier types have about the same minimum profit which is due to them facing similar opportunities. The different intercepts result from differences in fixed costs. Here the supplier with lower variable costs per unit has higher fixed costs which appears quite reasonable. Each point in the diagram is a possible contractual offer. Point A represents the first-best solution for type 1 and point B the one for type 2. According to (7) the marginal costs c t of a supplier type t equal the marginal value F~(x) of the buyer. Since they correspond to the slopes of the
181
indifference curves, we get points of tangency between the indifference curve of the buyer and the MPI of the supplier type considered. The optimal menu is composed of contracts A and B. By selecting the contract, the supplier will reveal her true type since she attains her minimum profit with only one contract. This case is equivalent to part (c) of Proposition 1. Fig. 2 can be seen as an example of part (a) of Proposition 1, assuming the sum of fixed costs and minimum profit being equal for both types. If the buying firm offers the first-best solutions (contracts A and B), type 2 will mimic type 1, i.e. each supplier type will choose contract A. Contract A is on an indifference curve of type 2 which gives her a profit higher than the minimum profit. The corresponding indifference curve is drawn as a dotted line in Fig. 2. The buyer can improve his position by offering the contracts A and C: Type 1 will choose contract A since C lies on an indifference curve situated below her MPI. The supplier type 2 is indifferent between A and C. According to our assumption she chooses contract C which is the one preferred by the buyer. As the supplier type 2 exists with positive probability the buyer is always better off offering A and C instead of A only. But the buyer can still increase his expected profit. The optimal solution of the optimization problem (8) depends on the probability distribu-
. MPI (t=1 )
x~
x~
Fig. 2. Indifference curves for x F, x ~ > 2.
.x
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H. Y. Schenk-Mathes / European Journal of Operational Research 86 (1995) 176-187
tion. Type 1 always gets exactly her minimum profit and the quantity offered is less than the first-best quantity. For a given probability distribution the contracts of the optimal menu lie on the same indifference curve of type 2. A possible optimal solution is given by E and D in Fig. 2. The contract C for the supplier type 2 leads to a profit higher than her minimum profit. Thus she would be willing to cooperate for even a lower profit. Keeping constant the first-best quantity x F the payment is reduced (contract D). In order to make type 2 choose such a contract, the other offer should not grant her a higher profit. E.g. in Fig. 2 the other contract (for type 1) is E which is given by the intersection of the indifference curve including D with the MPI of type 1 (as it is not optimal to grant type 1 more than her minimum profit). The supplier type 1 now receives a payment P1 < pF and the quantity is smaller than the first-best quantity as well. The higher the probability that the supplier is type 1 the closer are the optimal contracts to A and C and the smaller is the expected profit of the buyer. Part (b) of Proposition 2 can be interpreted analogously.
4. Conclusions for the delegation problem
re(S,) = rc( I i) " "n'(F ~ ) " ~'(CcB): Absolute probability that the scenario S s = {Ii, F~, C~} occurs, s =
1. . . . . ~. re(S, I T t) = ~'( I i I Tt). 7r(F~). ~r(C~): Conditional
probability that the scenario S, = {Ii, F~, Cff} occurs given that the supplier is type T t, s = 1 . . . . . S. "n'(Tt I Ss) = "n'(Ss I T t) "n'(Tt)/'n'(Ss) = "tr(Tt [Ii): Conditional probability that the supplier is type T t given that the scenario S s occurs, t = 1. . . . . T. (This probability equals the probability conditional on the signal because the consolidation function and the criterion of the agent do not depend on the supplier type.) These are all probability estimates of the principal at date 0 when the design decision has to be taken. In the following we refer to a vector if subscripts are omitted, e.g. T/'(T) = (,w(T1), ,w(T2) ). According to (1), (2) and (3) we can now evaluate the different design decisions (see also SchneeweiB, 1995): aT: The principal determines the optimal menu of contracts solving the optimization problem (8) based on his a priori beliefs. This leads to alB°= (x°(Tr(T)), P ° ( ~ - ( T ) ) ) . The principal gives the instruction to use this menu of contracts as a self-selection mechanism. According to (1) we get
e[CT"(a I°)IFT(T,)I Notations for the probabilities: ~-(Tt): Probability (a priori) that the supplier is t y p e Tt, t = 1 , . . . , T. (We now write ~r(T t) instead of TFt in order to avoid confusion.) 7r(IilTt): Conditional probability that signal /~ occurs given that the supplier is type Tt, i = 1. . . . , L ~-(Ii): Absolute probability that signal I i occurs, i = 1 . . . . . i. 7r(T t [ I i) = "rr(Ii I T t)" ~'(Tt)/'n'(Ii): Conditional probability that the supplier is type T t given that the signal I i occurs, t = 1. . . . . T. • -(F~): Probability that the consolidation function Fff is used by the agent, f = 1. . . . . ft. •r(C~): Probability that the criterion C B is applied by the agent, c = 1. . . . . C.
2 = y" 7 r ( T t ) . ( V ( x t ° ( r r ( T ) ) )
-Pt°(~r(T))).
t=l
(9) aT: At date 0 the agent is instructed to gather information on the probability distribution of the supplier types. Until date 1 the agent transmits the information gathered to the principal. Based on this transmission, the principal revises his beliefs about the supplier types according to Bayes rule and determines the optimal menu of contracts with respect to his a posteriori beliefs according to (8). From (2) we get a~°l I , = (x°(zr(T [ I~)), P°(~-(T [ I~))),
H.Y. Schenk-Mathes / European Journal o f Operational Research 86 (1995) 176-187
i = 1, 2 . . . . , i, as the optimal action conditional on the signal Ii, i.e. the optimal menu of contracts with respect to the a posteriori distribution
In order to find the optimal design decision the principal has to consider the costs of delegation and therefore to solve
,n-(T [ I,)= (~(T, II,), "tr(T21I,))
{E [ cTB(AF( aT))l FT( aT)]
Max T
T
T
T
for a given signal I i. Considering the probability distribution of the signal, we obtain the unconditional expected value
a E{al,a2,a 3}
E[cTB(aB°)IFT(Ii)] i =
AF(aix) = a ~ °,
=
2
E~(I,)" E~(T, iIi)'(V(x,°(~(TII,))) i=1
t=l
-Pt°(Tr(T]Ii))).
(10)
aT: The principal does not specify a menu of contracts. Instead the principal describes the procedure to determine the optimal menu of contracts. The menu depends explicitly on the criterion of the agent and on her beliefs about the distribution of the supplier types. It is part of the task of the agent to form the beliefs by gathering and evaluating information. Yet given the same information, the agent may come to a different conclusion as compared to the principal. To evaluate this design decision the principal has to anticipate the menu of contracts offered by the agent given that a certain scenario occurs.
a~° [Ss
=
(xB°(ss), PB°( Ss) )
is the menu of the agent anticipated by the principal according to (3). (The agent will not offer an infeasible menu with respect to the constraint set of (8). The principal is in a position to enforce the feasibility since deviations can be observed by him.) Considering the probability distribution of the scenario we get
E[CTn(a~°)IFT(Ss)] = E r(Ss)'E[CT"(a~°lSs)lFX(T, ISs)] s=l 2 =
E
E
s=l
t=l
ISs)
• (V(xB°(s,) - Pta*(s~)).
-C~(aT)},
(12)
i=1,2,3,
and
IIi)]
i=1 i
183
rFT(Tt)
for ax = aT,
FT(a x) = FT(Ii) FT(S,)
for aX--a -- T 2~ for aX=a~.
crr(a~) is set to zero. C~(a~) denotes the additionally incurred costs if instead of a~x the design decision a2x is taken by the principal. These costs are incurred by the information gathering and transmission but are also attributable to the fact that the principal has to deal with the contracting problem once again, cXT(aT) is the difference of costs between aT and a3x. These costs are essentially due to the information process of the agent. The case x [ _<2 ___x2 F is of particular interest. Here, the principal can prescribe the menu containing the first-best solutions which guarantees both supplier types their minimum profit. The expected profit of the buyer is a linear function of the probability ,r(T 1) in the interval [0, 1]. The menu does not depend on the probability distribution. Improved beliefs consequently do not alter the menu offered. The agent can dispense with gathering and evaluating information and no feedback to the principal is required, alx is always the optimal design decision in that case. If the first-best quantities are both either smaller or greater than 2, a2x or a3x may be the optimal design decision. In the following we will analyze an example of the case x[, x~ > 2. In Fig. 3 the expected profit E[Tr] as a function of the probability that the supplier is type 1 is depicted for Gsl = Gsz and V(x) = a . x - 3 . b1
(11)
"X 2
with a > c I and b > 0 as constant parameters.
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H. Y. Schenk-Mathes / European Journal of Operational Research 86 (1995) 176-187
E[,q
ELI211 .... i-- ,'%.. EtPI] I .... ~ ~--'X."~--: .
.
.
....
.
,",.,.
E[aT] .... i - ' : ........ ~ , ' ~ ~ ~ E[hl
....
i_.i
................
~TIII2)
~(TI)
~TI[I I)
Fig. 3. Comparison of the buyer's expected profit based on different design decisions (a T and aT).
E[rr] is linear for
[o,
c,a-2c2]j
and strictly convex for rr • [~', 1] (see Appendix B). The menu of contracts remains unchanged for rr < 9. In this interval probability variations do not lead to another menu. For zr > ~- the offered menu as well as the expected profit are functions of the probability distribution, i.e. the menu is directly determined by the assumed probabilities according to Proposition 2(a) and the expected profit follows from the derived menu and the probability distribution. If a priori the principal assumes a probability •r(T t) that the supplier is type 1 and applies a T, the resulting expected profit amounts to E[a T] in Fig. 3. This expected value will be used as a point of reference to evaluate the other design decisions. On the assumption that two different signals may be obtained from the information gathering process we can depict the expected profit resulting from aT. Eli i] denotes the conditional maximum expected profit if signal I i (for i = 1, 2) occurs. According to (10) we obtain
preferable to a T if the costs cT~(a~) do not exceed the value of information. Ceteris paribus the higher the difference c I - c 2 and the smaller a, the higher is the probability ~and the larger is the interval in which the expected profit is linear. Signals which lead to probabilities in this interval leave the expected profit unchanged. If the principal expects that the agent knows the type at date 1, he can choose the first-best solution of the corresponding type. This leads to the expected profit E[PI] in Fig. 3 and the difference to E[a T] represents the value of perfect information. If the principal takes the design decision a T, the menu of contracts selected by the agent cannot be easily derived from probabilities. As we have seen, the principal first determines all possible scenarios (comprising not only the signal but also the agent's consolidation function and the agent's criterion) and the corresponding probabilities for the occurrence of these scenarios. For each scenario the principal can anticipate the agent's choice concerning the menu of contracts and according to (3) he is in a position to calculate the conditional expected profit given that a certain scenario occurs. Finally according to (11) he determines the expected profit for this design decision. If the principal expects the agent to use the same criterion and the same consolidation function, it depends only on the costs cTr(a T) and C~(a T) which design decision, a T or a T, he prefers. To estimate the value of delegation in the case of a T is a very complex task, the exact calculation complicated and contrary to the concept of delegation. Therefore one frequently will have to be satisfied with only a rough estimate of the expected profit. Nevertheless the derived formulas provide valuable insight into the determinants of the expected profit and their structure.
5. Extensions
E[a T] = 17"(11)"E[II] + ~'(I2)" ELI2] (see Fig. 3). The value of information is the difference between E[aT] and Elan]. Obviously, a2T is only
The analysis of the buyer-supplier relationship can be extended to transaction specific investments that lead to a closer tie between the par-
H.Y Schenk-Mathes/ European Journal of OperationalResearch 86 (1995) 176-187
ties, e.g. the investments may improve the production technology of the supplier. At date 0 when the investment decision and the delegation decision have to take place, the buyer knows the new cost situation for each supplier type in the case of transaction specific investments. But there is still incomplete information about the type. One could also analyze the case in which contracts can be made with more than one supplier and the cooperation constraint must not always be fulfilled. In all these cases the probability distribution of the supplier type is an important aspect since the optimal solution can depend on these probabilities.
0L(, ) - - 0x ' x tt*
= 0
185
for every t ~ {1, 2},
(A.2)
0L( • ) ¢r 1 + A; - h~ +/x~ = O,
(A.3a)
V z - h ~ +h~ + ~
(A.3b)
~P' 0L(*) = 0,
0P2 OL( * ) 0h~
PI* --Cl "Xl -P2* + c i "x~ >_0,
(A.4a)
OL(,) Oh2
P~ --C2"X~ - P I * +C2"X~ >__0,
Appendix A
(A.4b)
The constraint set of (8) is not empty and convex, the objective function is strictly concave. This implies that the Kuhn-Tucker conditions are both necessary and sufficient for a maximum. The Lagrangian function of problem (8) is given by L ( x , , x2, P I , P2, "~l, *2, /d'l, ]'L2) =
7T1"(V(xI)
--PI)
0L(,) - -Oh"t A
+ 7r2" (V(x2) - - P 2 )
OL( • ) E(x;)
- *;
.c,
+}[~ "C2-- I£ ; "C1__0 ,
3x 2
Vx(X; ) +
(A.la)
t =0
for every t ~ { 1, 2}.
(A.7)
Proof of Propositions 1 and 2. We only have to consider the cases x l > 0 , x ~ > 0 and xl*= 0, x~ > 0: x~* > 0, x2* = 0 cannot be optimal, x ; _
PI* -P2* x~. Since c 1 > c 2 it follows that
¢1
P,* -P2* - - h ~ "C 2 --]£*2 " C 2__ <0,
(A.6b)
as
OL( , ) •
P ; - c 2 - x ; - Gs2 > 0,
O~Lt
(xl*, x2, PI*, P2* ) is an optimal solution of problem (8) if there exist nonnegative values for (h 1, h~,/x~,/x~) such that the following conditions hold: =
(A.6a)
(A.3a) and (A.3b) are fulfilled as equations because the payments P1 and P2 can also take on negative values.
+ 1d'2 " ( P 2 -- C2 "X2 -- a S 2 ) "
-
PI* -- C1 "X; -- Gs1 >_ 0,
0#2 0L( * ) --'/z
+I.zI'(PI-¢I'X1-'GsI)
3x I
(A.5)
O/z,
+ /~2" (P2 - c2 "x2 - P1 + C2 "X1)
-
for every t ~ {1, 2},
0L( • )
OL(*)
+ /~1" ( P , - Cl "x1 - PE + c1 "x2)
t =0
(A.lb)
which contradicts (A.4a).
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H. Y Schenk-Mathes/European Journal of Operational Research 86 (1995) 176-187
xl* = x ~ = 0 cannot be optimal, because from (A.4a) and (A.4b) we get P ( = P ~ * and from (A.6a) and (A.6b),
Substituting the payments in (A.4a) by the optimal payments reveals
PI* = 1°2" > Max{Gsl, Gs2}.
At x = x v the MPI of supplier type 1 does not lie below the MPI of supplier type 2 and we obtain
Since according to (iii), x F, x F > 0 holds, one of the following solutions (consisting of only one contract) leads to a higher profit for the buyer: {(x~, P~)} or {(x~, Pff)}
8Sl + c,
> Gsa + ca
x~ < 2 can be shown similarly with (A.4b). Part (a) o f Proposition 1 and 2: If A2 > 0 and A~ = 0 then x~ = x v and x ( < x v. (A.3a) leads to /zl =~'1 +A2 > 0 .
Case: x 1 > O, x ~ > 0 If x ( > 0, x~ > 0 then (A.2) indicates that (A.la) and (A.lb) must be fulfilled as equations. Substituting /~ and P-a according to (A.3a) and (A.3b) in (A.1) we get
Tl'l" (Vx(xl ) --Cl)-t-1~2 "(C2--C1)=0,
(A.8a)
Tr2"(Vx(Xd)-c2)+A~'(Cl-C2)=O.
(A.8h)
Since the multipliers are nonnegative and c~ > c 2 it follows from (A.Sa) and (A.Sb): If A2 = 0 then V~(xl*) = c , and x ( = x F. IfA~=0then V~(x 2 ) = c a and xd = x F. If A2 > 0 then V~(Xl*) > c 1 and x ( < x F. IfA~ > 0 t h e n V~(xd ) < c 2 and xd > x 2F. In the following we consider all possible cases and it will be shown that in each case the optimal solution leads to a unique condition for the firstbest quantities and the point of intersection between the MPI's. Part (c) o f Proposition 1." If A2 = 0 and A~ = 0 then the optimal quantities are x I = x F and x~ = x 2v. From (A.3a) and (A.3b) we get /x~ =~'1 and p.~ = 7r 2. Since 77"1 ~ (0, 1) the multipliers are positive. (A.7) together with (A.6) then lead to PI* = G s l + C l . x ~ = P ~
and P; = G2 +
Then (A.6a) must be satisfied as equation. Consequently P1* = Gsl + c l "x( < P F from x ( < x v. With respect to A~ > 0, (A.4b) must be satisfied as equation. In addition, substituting the above expression for P1* we arrive at P2* = Gsl + C2"xF + Cl "X( - ° 2 " x ( > P F from x 2 = x F and the cooperation constraint (A.6b) for type 2. Substituting this expression in (A.6b) reveals Gsl + C l ' X 1 >_Gs2+C2"X 1 which corresponds to x ( > 2. At x = x ( the MPI of type 1 does not lie below the MPI of type 2. From x ; < x ~ and x F < x F we get x F , x F > 2 . (A4.a) is satisfied. To complete the proof of part (a) of Proposition 2 we have to consider additionally the two c a s e s J[~,2 ~> 0 a n d /LI,2 = 0. If ~J'a > 0 t h e n
(A.6b) is
satisfied as equation too. Substituting the corresponding payments in (A.6b) yields x ( = 2 . This corresponds to the first possible menu. If p,~ = 0 then from (A.3b) and (A.3a) we get A2 = '/7"2 and / ~ = 1. Substituting "~2 in (A.8a) leads to the condition for x 1 in the second menu. Part (b) o f Propositions 1 and 2." One applies analogous arguments on A~ = 0 and A~ > 0. The case A[ > 0 and A~ > 0 however leads to a contradiction, because from (A.4) we obtain x ( =X 2 =2.
"
= PL
The optimal solutions are the first-best solutions.
Case: x 1 = O, x 2 > 0 For x I = 0 in (A.Sa) we have to regard
_<
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H.Y. Schenk-Mathes / European Journal of Operational Research 86 (1995) 176-187
instead of ' = '. T h e n the only case to consider is A2 > 0 a n d A ~ = 0 . W e g e t P1*=Gsl,
x ~ = x F,
for ~- ~ [~', 1], we get
(c2-c,) 2 E[er]
2"Tr.b a2-c2
and +
P2* = G'si + c2 "x~.
+ 2.q.(e2-a) 2" b
_ - Gsl
for 7r ~ [~', 1]. It can easily be shown that E[~-] is decreasing and convex.
(A.6b) leads to GSl -- Gs2 > 0 and therefore 2 < 0 and x F, x2F > 2 . This corresponds to the second m e n u of part (a) of Proposition 2 with Xl* = 0.
Acknowledgments The author wishes to express her appreciation to Ch. SchneeweiB, the participants of the Summer Institute and anonymous referees for their c o m m e n t s and suggestions.
Appendix B W e consider the case x~, x~ > 2. If 31 " b
V(x) = a-x-
.X 2
with a > c I then according to Proposition 2(a) we have 7r(a - b ' x , * - c , ) = (1 - 7r) " ( c I - c 2 ) which leads to Xl=
~"(a-c2) + c 2 - c l "n-.b
x1* > 0 = 2
and a > c I leads to C 1 -- C 2
0<~-=--<1. a C2 -
Substituting the p a y m e n t s P~*, P2* and the quantity x~ = x~ according to Proposition 2(a) in E [ ~ l = ~r" (a .x 1 - ½ . b - ( x ( ) 2 - P * ) +(1-Tr)'(a'x~-½"b'(x~)2-e;)
References Cr6mer, J., and Khalil, F. (1992), "Gathering information before signing a contract", The American Economic Review 82, 566-578. Demski, J.S. (1980), Information Analysis, Addison-Wesley, Reading, MA. Harris, M., and Townsend, R.M. (1981), "Resource allocation under asymmetric information", Econometrica 49, 33-64. Laux, H. and Liermann, F. (1993), Grundlagen der Organisation, Berlin. Sappington, D. (1983), "Limited liability contracts between principal and agent", Journal of Economic Theory 29,
1-21. Sappington, D. (1984), "Incentive contracting with asymmetric and imperfect precontractual knowledge", Journal of Economic Theory 34, 52-70. Schneeweig, Ch. (1995), "Hierarchical structures in organisations - A conceptual framework", European Journal of Operational Research 86 (1995) 4-31. Wagenhofer, A. (1992), "Verrechnungspreise zur Koordination bei Informationsasymmetrie", in: K. Spremann and E. Zur (eds.), Controlling, Wiesbaden, 637-656.
ELSEVIER
European Journal of Operational Research 86 (1995) 176-187
The design of supply contracts as a problem of delegation Heike Y. Schenk-Mathes FlY Wirtschaftswissenschaften, Universitiit Frankfurt, Mertonstr. 17, D-60325 Frankfurt, Germany
Abstract
We consider the design of a supply contract as a problem of delegation in a buying firm. In general the agent (who acts on behalf of a principal in the firm) cannot be assumed to transform her information in probability estimates like the principal would do possessing the same information. In the investigated case of incomplete information the design of contracts is affected by the beliefs about characteristics of the supplier. Therefore it is important to analyze in which way the expected profit depends on these probabilities in order to find the principal's optimal instruction. The best instruction is determined by a trade-off between the costs due to the time the principal spends on dealing with the problem and formulating the instructions and the costs due to a suboptimal menu of contracts offered by the agent. Keywords: Hierarchical planning; Delegation; Information asymmetry; Self-selection models
I. Introduction
The interaction of individuals is an important characteristic of any organization. Their decisions and actions need to be coordinated in order to reach organizational goals. When a problem is perceived at a certain level of a firm, it is in general not completely solved at the same level. Usually different hierarchical levels within the organization participate in the solution process. As a simplification we consider only two levels of the firm, the first represented by the principal and the second by the agent. There may be different reasons for the principal to charge an agent with at least some parts of the solution process. First, though the principal may be capable of solving the problem in a satisfactory manner all by his own, he lacks the time for a careful decision-making process being en-
gaged in other important tasks. Second, the principal may be less qualified in some aspects of the problem. The principal intends to control the agent by issuing more or less precise instructions. The effect of these instructions on the behaviour of the agent depends on different aspects: the characteristics of the agent (such as qualification, motivation, attitude towards the future) and those of the environment (Laux and Liermann, 1993). The principal can additionally influence this effect, e.g. by verifying the agent's actions a n d / o r by establishing reward systems. He also may consider not only to give one instruction but to wait for reactions (signals) by the agent in order to issue new instructions. The main purpose of the p a p e r is to find an optimal instruction or an optimal (conditional) sequence of instructions. Since this forms a task
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H.Y. Schenk-Mathes/ European Journal of Operational Research 86 (1995) 176-187
of considerable complexity, it appears reasonable to restrict the analysis to some specific (manageable) instances of this problem. We regard the design of supply contracts as a problem of delegation. Three parties are involved: the principal and the agent in the buying firm and a supplier. The agent works on behalf of the principal and has to design the supply contract. First, we specify this principal-agent relationship and describe the principal's set of possible instructions (Section 2). In order to analyze the effect of instructions on the contracting process, we must look at the relationship between the buying firm and the supplier and derive the main characteristics of the optimal solution (Section 3). The consequences for the choice of instructions by the principal are discussed in Section 4.
2. The principal-agent relationship: Assumptions and considered instructions We assume the principal to be risk neutral and sufficiently qualified. Because of lack of time he considers to delegate to an agent the information process and the implementation of the contractual arrangements. The information process comprises information gathering and evaluation for which the agent may not be sufficiently qualified. Her actions may also be guided by objectives which do not comply with those of the principal. So the agent will not always act in the principal's sense. Once a final decision concerning the contract design is taken, the implementation of the contractual arrangements is realized by the agent according to the objectives of the principal. At the initial date 0 the principal makes a factual decision concerning the instructions for the agent and at date 1 the agent makes a final decision concerning the action (i.e. the contract between the buying firm and the supplier is concluded). Here, we have two problems to solve. The first is the organizational problem of the principal. He has to decide in which way the agent should be involved in the decision process referring to the contracting problem. The second
177
problem is the contracting problem itself. The actions concerning the organizational problem are design decisions of the top level (aT). The actions concerning the contracting problem are taken at the base level (aB). In the following we consider three possible actions at the top level (alx, aT, aT). These are design decisions concerning only the information process. The actions of the agent referring to the implementation of the contractual arrangements do not depend on these design decisions and can therefore be ignored. a T is the extreme case in which the principal determines an action for the contracting problem at date 0 and gives the instruction to perform this action at date 1. Information gathering and evaluation that may take place until date 1 are not taken into account, aT implies that the agent gathers information and transmits the signal received until date 1 to the principal. Again but now at date 1 and based on the new information situation, the principal determines the action for the contracting problem to be chosen by the agent. For ala" and a2x we assume it observable whether the agent follows the instructions or not, i.e. the agent chooses the action specified by the principal and the signal is reported truthfully, a3x implies that the agent has to solve the problem without further guidance by or feedback with the principal (for information and delegation analysis see Laux and Liermann, 1993; Demski, 1980). The information situation is the important aspect of the investigated principal-agent relationship since the solution of the contracting procedure depends on beliefs about characteristics of the supplier (like the production technology used by the supplier). Tt, t = 1. . . . . T, is used to denote these possible characteristics of the supplier. The principal has some information about the contracting problem at date 0. This information may change or new information arrive prior to date 1, thus giving rise to a reevaluation of probability estimates. FT(b) describes the principal's rule of mapping an event b into a probability attributed to this event, ,r(b). In the following, FT( •) will be called consolidation function of the top level (the principal). The domain of the consolidation function contains 'simple' events, like the event that
H.Y. Schenk-Mathes/ European Journal of OperationalResearch86 (1995) 176-187
178
the supplier has a certain characteristic, and 'conditional' events, like the event that the supplier has a certain characteristic given that a certain signal occurs. For each design decision we will first determine the action at the base level (for the following analysis see Schneeweig, 1995). aT: At date 0, the principal calculates the optimal action based on his a priori beliefs concerning the characteristics of the supplier, F T(T~), and instructs the agent to perform this action. The optimal action a~ ° is determined by Opt
E[Ca~(aB)]FT(T,)].
(1)
a~ ~A
E is the expectation operator and C TB is the criterion applied to the contracting problem by the principal (top level) to evaluate the action concerning the contracting problem and this action is taken at the base level. A is the set of possible actions at the base level. aT: The principal gives the instruction to gather information and to transmit the signal received at the base level. After receiving the signal the principal directs the agent to choose the corresponding optimal action. The principal knows all signals I i, i = 1. . . . . i, the agent (or base level) may receive until date 1 and can form a posteriori beliefs about the characteristics of the supplier conditional on these signals, FT(Tt I//). Given a signal Ii, i = 1. . . . . i, the corresponding optimal action a2U°lI i is determined by Opt E[cTB(a~)IFT(T,[I,)]. a~A
(2)
aT: The principal does not specify an action. Given a signal Ii, the agent may assess other probabilities as the principal. The principal has to anticipate this assessment for every signal. The principal and the agent may also apply different criteria to calculate the optimal solution. Each possible combination of signal received, /~ (i {1,..., I}), consolidation function used, FjB ( f ~ {1. . . . . if}), and applied criterion C ~ (c {1. . . . . C}) by the agent (at the base level) constitutes a scenario
Ss={Ii, F?,C~},
s=l .....
S=i.F.C.
The principal has to anticipate the action a~°l the agent will choose if a certain scenario
Ss={Ii, F?,CB},
s=l
is given. The action aB°l
Ss
. . . . . S,
S s, s = 1. . . . . S, solves
Opt E[C•(aBIIF•(Tt]Ii)]. a~A
(3)
(3) describes the principal's estimation of the agent's optimization problem. The principal has to anticipate the agent's reaction if a certain scenario occurs, i.e. the agent receives a certain signal, derives a certain assessment of probabilities according to one of the possible consolidation functions and chooses a certain criterion. For each design decision we have calculated the corresponding optimal actions of the contracting problem. Only in the case of a T we have determined an unconditional action. If the principal chooses a T we get actions conditional on the signals transmitted by the agent at date 1 and in the case of a T actions conditional on the scenarios at date 1 have to be considered. Since the principal has to decide upon the design decision at date 0 and since he does not know whatever signal or scenario will occur, he has to evaluate the action at the base level given the information situation at date 0, i.e. given the beliefs concerning the signals and scenarios at this date. Now to find the optimal design decision a T* at date 0 the principal has to consider additionally the costs of information and delegation and then to compare the three decisions a T, a T and a T. We will come back to this point in Section 4 where we will specify the action and the criterion of the contracting problem.
3. The buyer-supplier relationship: Assumptions and properties of an optimal menu of contracts
3.1. Assumptions At date 0 the supplier has already been chosen but still there is incomplete information about the type of supplier. E.g. the production technol-
H. Y. Schenk-Mathes/ European Journal of Operational Research 86 (1995) 176-187
ogy used by the supplier may not be known with certainty. Consequently we have precontractual asymmetry of information between buyer (represented by the principal) and supplier. The buying firm considers two types of supplier to be possible ( T = 2). At date 1 the buying firm offers a menu of contracts. A supply contract must be concluded, i.e. for each supplier type there must be at least one contract which is acceptable. The possibility of renegotiation is not considered. If the supplier is indifferent to the contracts offered by the buying firm, then it is assumed that she chooses the contract preferred by the buyer. Buyer and supplier are both risk neutral. Each type of supplier, t ~ {1, 2}, has her own cost function
tion (iii) guarantees the existence of an inner solution for every type of supplier. Consequently we have the following profit G B for the buyer:
(5)
GB(x, P) = V(x) - P , which is the criterion cVa(-) of the principal.
3.2. Optimization problem and properties of an optimal solution If the buyer knew the type of supplier, t = t, then he would face the following optimization problem: Max GB( X, p) = V( x) - e x,P
(6)
subject to
T C ( x l t ) = c , ' x + F t, with x > 0 as the quantity delivered, F t > 0 the fixed costs of supplier type t and c t > O the variable costs per unit. Supplier type 1 has higher variable costs than supplier type 2: (i) Cl > c2 > 0. P denotes the payment the supplier receives from the buyer for the quantity delivered. Then the profit of supplier type t is Gs(x, Pit) =e-TC(xlt)=P-ct'x-F
179
t. (4)
The supplier type t will only cooperate if she gets a profit of at least GMin>_0 which we call minimum profit for type t and which is assumed to be known. E.g. the supplier may have other opportunities and the most attractive of these is providing the minimum profit. The delivered good is a productive input for the buyer. The value of production of the quantity x (without the costs due to the payment P) is denoted by V(x). V ( x ) is twice continuously differentiable with respect to x and fulfills the following conditions: (ii) V~x(X) < 0 and (iii) Vx(O) > c a. The subscripts indicate derivatives. According to (ii) the function V ( x ) is strictly concave. Condi-
P - c i ' x > . _/7"Min , s i -~F i -
Gs~"
An optimal solution to the above problem must strictly fulfill the (cooperation) constraint, because otherwise G s can be increased by reducing P. By substitution of P in the objective function and differentiation we obtain the following necessary (and sufficient) condition for an optimum: Vx(X ) =c-t.
(7)
The optimal solution of (6) we will call the firstbest solution ( X F, pF) for type L The LHS of (7) is decreasing in x. Together with c 1 > c z we have XaF < x F. In the following we investigate the case where the buying firm does not know with certainty the type of supplier. If the buying firm offers only one contract (i.e. only one combination of quantity x and payment P) and if the cooperation constraint must hold for every type of supplier, the optimal contract does not depend on the probability distribution of the type. The buyer can increase the expected profit offering a menu of two contracts {(xl, P1), (x2, Pz)}. The supplier has to choose among these contracts. This implies the idea of a self-selection mechanism. (Recent applications of self-selection mechanisms can be found in Cr6mer and Khalil (1992) and Wagenhofer (1992).)
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H.Y. Schenk-Mathes / European Journal o f Operational Research 86 (1995) 176-187
We get the following optimization problem:
Max E[G(x,, P,, x2, P2)]
Xl,PI,x2,P2
=
(8)
7/'1" (V(Xl) -- P1) + ~'2" ( V ( x 2 ) - P2)
subject to
(2 may be infeasible, i.e. 2 < 0). An optimal menu {(xl*, el* ), (x2*, P2* )} for 7r 1 ~ (0, 1) has the following properties:
Proposition 1. (a) If x F, x F > 2 then
Pl-c~'x 1> P2-Cl.X2,
(xl* < x F, Pl*
P 2 - cz'x2 > P t - c2"xl,
and
(self-selection constraints)
( x ; -- x 2, F p2* >
e t - ct " x t >- t"TMin "-'st "~- Ft = a s t
for every t e {1, 2},
(x; =xL e,*
(cooperation constraints)
and
x,>_0
(x~>xF, p;>PV).
for every t e {1, 2},
where rr t denotes the probability that the supplier is type t. In Harris and Townsend (1981) and Sappington (1983, 1984) one can find examples of this optimization problem, but they do not regard variations of the RHS of the cooperation constraints. The inclusion of such variations leads to a more complex structure of the optimal solution. The self-selection constraint for the supplier type t states that her profit when choosing the 'right' contract (i.e. the contract the buyer intends her to choose) must be at least as large as the profit due to the other contract of the menu. The cooperation constraint must hold for every type. Here, we want to emphasize that the cooperation constraint of a supplier type must be fulfilled even if the profit due to this type is negative. Such a case is given if we consider an additional contract and the realization of this contract is a necessary condition for the fulfillment of all other contracts. Let {(x, P ) : P = G s t + F t + c t - x } be the profit-indifference curve for a given profit Gst of the supplier type t, which we call the minimum-profit-indifference curve (MPI) if Gst + F t = Gst (see Fig. 1). The MPI's have a point of intersection at 2
(b) If x v, x~ <2 then
Gs2 - Gsl C 1 -- C 2
(c) If xF < 2 < X ~ then the optimal menu of contract is {(x F, pF), (x F, p2F)}. An important result is that in the case of x F < 2 < x2v the first-best solutions are attainable. In the two other cases we can describe the optimal menu more precisely.
Proposition 2. (a) If x F, xF2 > ~ then the optimal menu is either
or
((x;,c, (xF,(cI--C2)
" X 1 "4- C 2 " X F "~- G s a ) )
with x I = 0 or 77 1 " ( V x ( X 1 ) - c 1 ) = 77-2 . ( c
I - c2).
(b) If x F, x F <2 then the optimal menu is either
{ ( x ~, P~ ),( 2,c2 "~ + Gs2 ) } or {(xF,Cl'X
F + (c2--el)
"X 2 "Jl- a S 2 ) ,
+ with x;
) -
=
- c,).
s2)}
H. Y Schenk-Mathes/ European Journal of Operational Research 86 (1995) 176-187
~
MPI (t=l)
/
x~ ~
MPI (t=2)
x~
. x
Fig. 1. Indifference curves for XlF < £ < x F.
For the proofs of Propositions 1 and 2 see Appendix A. 3. 3. Interpretation o f the optimal menu o f contracts
Fig. 1 shows the indifference curves of the supplier types and the buyer. The concave indifference curves are those of the buyer. The profit of the buyer increases in southeasterly direction. The linear MPI's of both supplier types are also depicted in Fig. 1. The profits of the supplier types increase in northwesterly direction. In the case considered in Fig. 1 the intercept of the MPI of type 1 on the vertical axis is smaller than the one of type 2. The intercept is the sum of fixed costs and the minimum profit of the type considered. The shape of the curves makes sense assuming that the supplier types have about the same minimum profit which is due to them facing similar opportunities. The different intercepts result from differences in fixed costs. Here the supplier with lower variable costs per unit has higher fixed costs which appears quite reasonable. Each point in the diagram is a possible contractual offer. Point A represents the first-best solution for type 1 and point B the one for type 2. According to (7) the marginal costs c t of a supplier type t equal the marginal value F~(x) of the buyer. Since they correspond to the slopes of the
181
indifference curves, we get points of tangency between the indifference curve of the buyer and the MPI of the supplier type considered. The optimal menu is composed of contracts A and B. By selecting the contract, the supplier will reveal her true type since she attains her minimum profit with only one contract. This case is equivalent to part (c) of Proposition 1. Fig. 2 can be seen as an example of part (a) of Proposition 1, assuming the sum of fixed costs and minimum profit being equal for both types. If the buying firm offers the first-best solutions (contracts A and B), type 2 will mimic type 1, i.e. each supplier type will choose contract A. Contract A is on an indifference curve of type 2 which gives her a profit higher than the minimum profit. The corresponding indifference curve is drawn as a dotted line in Fig. 2. The buyer can improve his position by offering the contracts A and C: Type 1 will choose contract A since C lies on an indifference curve situated below her MPI. The supplier type 2 is indifferent between A and C. According to our assumption she chooses contract C which is the one preferred by the buyer. As the supplier type 2 exists with positive probability the buyer is always better off offering A and C instead of A only. But the buyer can still increase his expected profit. The optimal solution of the optimization problem (8) depends on the probability distribu-
. MPI (t=1 )
x~
x~
Fig. 2. Indifference curves for x F, x ~ > 2.
.x
182
H. Y. Schenk-Mathes / European Journal of Operational Research 86 (1995) 176-187
tion. Type 1 always gets exactly her minimum profit and the quantity offered is less than the first-best quantity. For a given probability distribution the contracts of the optimal menu lie on the same indifference curve of type 2. A possible optimal solution is given by E and D in Fig. 2. The contract C for the supplier type 2 leads to a profit higher than her minimum profit. Thus she would be willing to cooperate for even a lower profit. Keeping constant the first-best quantity x F the payment is reduced (contract D). In order to make type 2 choose such a contract, the other offer should not grant her a higher profit. E.g. in Fig. 2 the other contract (for type 1) is E which is given by the intersection of the indifference curve including D with the MPI of type 1 (as it is not optimal to grant type 1 more than her minimum profit). The supplier type 1 now receives a payment P1 < pF and the quantity is smaller than the first-best quantity as well. The higher the probability that the supplier is type 1 the closer are the optimal contracts to A and C and the smaller is the expected profit of the buyer. Part (b) of Proposition 2 can be interpreted analogously.
4. Conclusions for the delegation problem
re(S,) = rc( I i) " "n'(F ~ ) " ~'(CcB): Absolute probability that the scenario S s = {Ii, F~, C~} occurs, s =
1. . . . . ~. re(S, I T t) = ~'( I i I Tt). 7r(F~). ~r(C~): Conditional
probability that the scenario S, = {Ii, F~, Cff} occurs given that the supplier is type T t, s = 1 . . . . . S. "n'(Tt I Ss) = "n'(Ss I T t) "n'(Tt)/'n'(Ss) = "tr(Tt [Ii): Conditional probability that the supplier is type T t given that the scenario S s occurs, t = 1. . . . . T. (This probability equals the probability conditional on the signal because the consolidation function and the criterion of the agent do not depend on the supplier type.) These are all probability estimates of the principal at date 0 when the design decision has to be taken. In the following we refer to a vector if subscripts are omitted, e.g. T/'(T) = (,w(T1), ,w(T2) ). According to (1), (2) and (3) we can now evaluate the different design decisions (see also SchneeweiB, 1995): aT: The principal determines the optimal menu of contracts solving the optimization problem (8) based on his a priori beliefs. This leads to alB°= (x°(Tr(T)), P ° ( ~ - ( T ) ) ) . The principal gives the instruction to use this menu of contracts as a self-selection mechanism. According to (1) we get
e[CT"(a I°)IFT(T,)I Notations for the probabilities: ~-(Tt): Probability (a priori) that the supplier is t y p e Tt, t = 1 , . . . , T. (We now write ~r(T t) instead of TFt in order to avoid confusion.) 7r(IilTt): Conditional probability that signal /~ occurs given that the supplier is type Tt, i = 1. . . . , L ~-(Ii): Absolute probability that signal I i occurs, i = 1 . . . . . i. 7r(T t [ I i) = "rr(Ii I T t)" ~'(Tt)/'n'(Ii): Conditional probability that the supplier is type T t given that the signal I i occurs, t = 1. . . . . T. • -(F~): Probability that the consolidation function Fff is used by the agent, f = 1. . . . . ft. •r(C~): Probability that the criterion C B is applied by the agent, c = 1. . . . . C.
2 = y" 7 r ( T t ) . ( V ( x t ° ( r r ( T ) ) )
-Pt°(~r(T))).
t=l
(9) aT: At date 0 the agent is instructed to gather information on the probability distribution of the supplier types. Until date 1 the agent transmits the information gathered to the principal. Based on this transmission, the principal revises his beliefs about the supplier types according to Bayes rule and determines the optimal menu of contracts with respect to his a posteriori beliefs according to (8). From (2) we get a~°l I , = (x°(zr(T [ I~)), P°(~-(T [ I~))),
H.Y. Schenk-Mathes / European Journal o f Operational Research 86 (1995) 176-187
i = 1, 2 . . . . , i, as the optimal action conditional on the signal Ii, i.e. the optimal menu of contracts with respect to the a posteriori distribution
In order to find the optimal design decision the principal has to consider the costs of delegation and therefore to solve
,n-(T [ I,)= (~(T, II,), "tr(T21I,))
{E [ cTB(AF( aT))l FT( aT)]
Max T
T
T
T
for a given signal I i. Considering the probability distribution of the signal, we obtain the unconditional expected value
a E{al,a2,a 3}
E[cTB(aB°)IFT(Ii)] i =
AF(aix) = a ~ °,
=
2
E~(I,)" E~(T, iIi)'(V(x,°(~(TII,))) i=1
t=l
-Pt°(Tr(T]Ii))).
(10)
aT: The principal does not specify a menu of contracts. Instead the principal describes the procedure to determine the optimal menu of contracts. The menu depends explicitly on the criterion of the agent and on her beliefs about the distribution of the supplier types. It is part of the task of the agent to form the beliefs by gathering and evaluating information. Yet given the same information, the agent may come to a different conclusion as compared to the principal. To evaluate this design decision the principal has to anticipate the menu of contracts offered by the agent given that a certain scenario occurs.
a~° [Ss
=
(xB°(ss), PB°( Ss) )
is the menu of the agent anticipated by the principal according to (3). (The agent will not offer an infeasible menu with respect to the constraint set of (8). The principal is in a position to enforce the feasibility since deviations can be observed by him.) Considering the probability distribution of the scenario we get
E[CTn(a~°)IFT(Ss)] = E r(Ss)'E[CT"(a~°lSs)lFX(T, ISs)] s=l 2 =
E
E
s=l
t=l
ISs)
• (V(xB°(s,) - Pta*(s~)).
-C~(aT)},
(12)
i=1,2,3,
and
IIi)]
i=1 i
183
rFT(Tt)
for ax = aT,
FT(a x) = FT(Ii) FT(S,)
for aX--a -- T 2~ for aX=a~.
crr(a~) is set to zero. C~(a~) denotes the additionally incurred costs if instead of a~x the design decision a2x is taken by the principal. These costs are incurred by the information gathering and transmission but are also attributable to the fact that the principal has to deal with the contracting problem once again, cXT(aT) is the difference of costs between aT and a3x. These costs are essentially due to the information process of the agent. The case x [ _<2 ___x2 F is of particular interest. Here, the principal can prescribe the menu containing the first-best solutions which guarantees both supplier types their minimum profit. The expected profit of the buyer is a linear function of the probability ,r(T 1) in the interval [0, 1]. The menu does not depend on the probability distribution. Improved beliefs consequently do not alter the menu offered. The agent can dispense with gathering and evaluating information and no feedback to the principal is required, alx is always the optimal design decision in that case. If the first-best quantities are both either smaller or greater than 2, a2x or a3x may be the optimal design decision. In the following we will analyze an example of the case x[, x~ > 2. In Fig. 3 the expected profit E[Tr] as a function of the probability that the supplier is type 1 is depicted for Gsl = Gsz and V(x) = a . x - 3 . b1
(11)
"X 2
with a > c I and b > 0 as constant parameters.
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H. Y. Schenk-Mathes / European Journal of Operational Research 86 (1995) 176-187
E[,q
ELI211 .... i-- ,'%.. EtPI] I .... ~ ~--'X."~--: .
.
.
....
.
,",.,.
E[aT] .... i - ' : ........ ~ , ' ~ ~ ~ E[hl
....
i_.i
................
~TIII2)
~(TI)
~TI[I I)
Fig. 3. Comparison of the buyer's expected profit based on different design decisions (a T and aT).
E[rr] is linear for
[o,
c,a-2c2]j
and strictly convex for rr • [~', 1] (see Appendix B). The menu of contracts remains unchanged for rr < 9. In this interval probability variations do not lead to another menu. For zr > ~- the offered menu as well as the expected profit are functions of the probability distribution, i.e. the menu is directly determined by the assumed probabilities according to Proposition 2(a) and the expected profit follows from the derived menu and the probability distribution. If a priori the principal assumes a probability •r(T t) that the supplier is type 1 and applies a T, the resulting expected profit amounts to E[a T] in Fig. 3. This expected value will be used as a point of reference to evaluate the other design decisions. On the assumption that two different signals may be obtained from the information gathering process we can depict the expected profit resulting from aT. Eli i] denotes the conditional maximum expected profit if signal I i (for i = 1, 2) occurs. According to (10) we obtain
preferable to a T if the costs cT~(a~) do not exceed the value of information. Ceteris paribus the higher the difference c I - c 2 and the smaller a, the higher is the probability ~and the larger is the interval in which the expected profit is linear. Signals which lead to probabilities in this interval leave the expected profit unchanged. If the principal expects that the agent knows the type at date 1, he can choose the first-best solution of the corresponding type. This leads to the expected profit E[PI] in Fig. 3 and the difference to E[a T] represents the value of perfect information. If the principal takes the design decision a T, the menu of contracts selected by the agent cannot be easily derived from probabilities. As we have seen, the principal first determines all possible scenarios (comprising not only the signal but also the agent's consolidation function and the agent's criterion) and the corresponding probabilities for the occurrence of these scenarios. For each scenario the principal can anticipate the agent's choice concerning the menu of contracts and according to (3) he is in a position to calculate the conditional expected profit given that a certain scenario occurs. Finally according to (11) he determines the expected profit for this design decision. If the principal expects the agent to use the same criterion and the same consolidation function, it depends only on the costs cTr(a T) and C~(a T) which design decision, a T or a T, he prefers. To estimate the value of delegation in the case of a T is a very complex task, the exact calculation complicated and contrary to the concept of delegation. Therefore one frequently will have to be satisfied with only a rough estimate of the expected profit. Nevertheless the derived formulas provide valuable insight into the determinants of the expected profit and their structure.
5. Extensions
E[a T] = 17"(11)"E[II] + ~'(I2)" ELI2] (see Fig. 3). The value of information is the difference between E[aT] and Elan]. Obviously, a2T is only
The analysis of the buyer-supplier relationship can be extended to transaction specific investments that lead to a closer tie between the par-
H.Y Schenk-Mathes/ European Journal of OperationalResearch 86 (1995) 176-187
ties, e.g. the investments may improve the production technology of the supplier. At date 0 when the investment decision and the delegation decision have to take place, the buyer knows the new cost situation for each supplier type in the case of transaction specific investments. But there is still incomplete information about the type. One could also analyze the case in which contracts can be made with more than one supplier and the cooperation constraint must not always be fulfilled. In all these cases the probability distribution of the supplier type is an important aspect since the optimal solution can depend on these probabilities.
0L(, ) - - 0x ' x tt*
= 0
185
for every t ~ {1, 2},
(A.2)
0L( • ) ¢r 1 + A; - h~ +/x~ = O,
(A.3a)
V z - h ~ +h~ + ~
(A.3b)
~P' 0L(*) = 0,
0P2 OL( * ) 0h~
PI* --Cl "Xl -P2* + c i "x~ >_0,
(A.4a)
OL(,) Oh2
P~ --C2"X~ - P I * +C2"X~ >__0,
Appendix A
(A.4b)
The constraint set of (8) is not empty and convex, the objective function is strictly concave. This implies that the Kuhn-Tucker conditions are both necessary and sufficient for a maximum. The Lagrangian function of problem (8) is given by L ( x , , x2, P I , P2, "~l, *2, /d'l, ]'L2) =
7T1"(V(xI)
--PI)
0L(,) - -Oh"t A
+ 7r2" (V(x2) - - P 2 )
OL( • ) E(x;)
- *;
.c,
+}[~ "C2-- I£ ; "C1__0 ,
3x 2
Vx(X; ) +
(A.la)
t =0
for every t ~ { 1, 2}.
(A.7)
Proof of Propositions 1 and 2. We only have to consider the cases x l > 0 , x ~ > 0 and xl*= 0, x~ > 0: x~* > 0, x2* = 0 cannot be optimal, x ; _
PI* -P2*
¢1
P,* -P2* - - h ~ "C 2 --]£*2 " C 2__ <0,
(A.6b)
as
OL( , ) •
P ; - c 2 - x ; - Gs2 > 0,
O~Lt
(xl*, x2, PI*, P2* ) is an optimal solution of problem (8) if there exist nonnegative values for (h 1, h~,/x~,/x~) such that the following conditions hold: =
(A.6a)
(A.3a) and (A.3b) are fulfilled as equations because the payments P1 and P2 can also take on negative values.
+ 1d'2 " ( P 2 -- C2 "X2 -- a S 2 ) "
-
PI* -- C1 "X; -- Gs1 >_ 0,
0#2 0L( * ) --'/z
+I.zI'(PI-¢I'X1-'GsI)
3x I
(A.5)
O/z,
+ /~2" (P2 - c2 "x2 - P1 + C2 "X1)
-
for every t ~ {1, 2},
0L( • )
OL(*)
+ /~1" ( P , - Cl "x1 - PE + c1 "x2)
t =0
(A.lb)
which contradicts (A.4a).
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H. Y Schenk-Mathes/European Journal of Operational Research 86 (1995) 176-187
xl* = x ~ = 0 cannot be optimal, because from (A.4a) and (A.4b) we get P ( = P ~ * and from (A.6a) and (A.6b),
Substituting the payments in (A.4a) by the optimal payments reveals
PI* = 1°2" > Max{Gsl, Gs2}.
At x = x v the MPI of supplier type 1 does not lie below the MPI of supplier type 2 and we obtain
Since according to (iii), x F, x F > 0 holds, one of the following solutions (consisting of only one contract) leads to a higher profit for the buyer: {(x~, P~)} or {(x~, Pff)}
8Sl + c,
> Gsa + ca
x~ < 2 can be shown similarly with (A.4b). Part (a) o f Proposition 1 and 2: If A2 > 0 and A~ = 0 then x~ = x v and x ( < x v. (A.3a) leads to /zl =~'1 +A2 > 0 .
Case: x 1 > O, x ~ > 0 If x ( > 0, x~ > 0 then (A.2) indicates that (A.la) and (A.lb) must be fulfilled as equations. Substituting /~ and P-a according to (A.3a) and (A.3b) in (A.1) we get
Tl'l" (Vx(xl ) --Cl)-t-1~2 "(C2--C1)=0,
(A.8a)
Tr2"(Vx(Xd)-c2)+A~'(Cl-C2)=O.
(A.8h)
Since the multipliers are nonnegative and c~ > c 2 it follows from (A.Sa) and (A.Sb): If A2 = 0 then V~(xl*) = c , and x ( = x F. IfA~=0then V~(x 2 ) = c a and xd = x F. If A2 > 0 then V~(Xl*) > c 1 and x ( < x F. IfA~ > 0 t h e n V~(xd ) < c 2 and xd > x 2F. In the following we consider all possible cases and it will be shown that in each case the optimal solution leads to a unique condition for the firstbest quantities and the point of intersection between the MPI's. Part (c) o f Proposition 1." If A2 = 0 and A~ = 0 then the optimal quantities are x I = x F and x~ = x 2v. From (A.3a) and (A.3b) we get /x~ =~'1 and p.~ = 7r 2. Since 77"1 ~ (0, 1) the multipliers are positive. (A.7) together with (A.6) then lead to PI* = G s l + C l . x ~ = P ~
and P; = G2 +
Then (A.6a) must be satisfied as equation. Consequently P1* = Gsl + c l "x( < P F from x ( < x v. With respect to A~ > 0, (A.4b) must be satisfied as equation. In addition, substituting the above expression for P1* we arrive at P2* = Gsl + C2"xF + Cl "X( - ° 2 " x ( > P F from x 2 = x F and the cooperation constraint (A.6b) for type 2. Substituting this expression in (A.6b) reveals Gsl + C l ' X 1 >_Gs2+C2"X 1 which corresponds to x ( > 2. At x = x ( the MPI of type 1 does not lie below the MPI of type 2. From x ; < x ~ and x F < x F we get x F , x F > 2 . (A4.a) is satisfied. To complete the proof of part (a) of Proposition 2 we have to consider additionally the two c a s e s J[~,2 ~> 0 a n d /LI,2 = 0. If ~J'a > 0 t h e n
(A.6b) is
satisfied as equation too. Substituting the corresponding payments in (A.6b) yields x ( = 2 . This corresponds to the first possible menu. If p,~ = 0 then from (A.3b) and (A.3a) we get A2 = '/7"2 and / ~ = 1. Substituting "~2 in (A.8a) leads to the condition for x 1 in the second menu. Part (b) o f Propositions 1 and 2." One applies analogous arguments on A~ = 0 and A~ > 0. The case A[ > 0 and A~ > 0 however leads to a contradiction, because from (A.4) we obtain x ( =X 2 =2.
"
= PL
The optimal solutions are the first-best solutions.
Case: x 1 = O, x 2 > 0 For x I = 0 in (A.Sa) we have to regard
_<
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H.Y. Schenk-Mathes / European Journal of Operational Research 86 (1995) 176-187
instead of ' = '. T h e n the only case to consider is A2 > 0 a n d A ~ = 0 . W e g e t P1*=Gsl,
x ~ = x F,
for ~- ~ [~', 1], we get
(c2-c,) 2 E[er]
2"Tr.b a2-c2
and +
P2* = G'si + c2 "x~.
+ 2.q.(e2-a) 2" b
_ - Gsl
for 7r ~ [~', 1]. It can easily be shown that E[~-] is decreasing and convex.
(A.6b) leads to GSl -- Gs2 > 0 and therefore 2 < 0 and x F, x2F > 2 . This corresponds to the second m e n u of part (a) of Proposition 2 with Xl* = 0.
Acknowledgments The author wishes to express her appreciation to Ch. SchneeweiB, the participants of the Summer Institute and anonymous referees for their c o m m e n t s and suggestions.
Appendix B W e consider the case x~, x~ > 2. If 31 " b
V(x) = a-x-
.X 2
with a > c I then according to Proposition 2(a) we have 7r(a - b ' x , * - c , ) = (1 - 7r) " ( c I - c 2 ) which leads to Xl=
~"(a-c2) + c 2 - c l "n-.b
x1* > 0 = 2
and a > c I leads to C 1 -- C 2
0<~-=--<1. a C2 -
Substituting the p a y m e n t s P~*, P2* and the quantity x~ = x~ according to Proposition 2(a) in E [ ~ l = ~r" (a .x 1 - ½ . b - ( x ( ) 2 - P * ) +(1-Tr)'(a'x~-½"b'(x~)2-e;)
References Cr6mer, J., and Khalil, F. (1992), "Gathering information before signing a contract", The American Economic Review 82, 566-578. Demski, J.S. (1980), Information Analysis, Addison-Wesley, Reading, MA. Harris, M., and Townsend, R.M. (1981), "Resource allocation under asymmetric information", Econometrica 49, 33-64. Laux, H. and Liermann, F. (1993), Grundlagen der Organisation, Berlin. Sappington, D. (1983), "Limited liability contracts between principal and agent", Journal of Economic Theory 29,
1-21. Sappington, D. (1984), "Incentive contracting with asymmetric and imperfect precontractual knowledge", Journal of Economic Theory 34, 52-70. Schneeweig, Ch. (1995), "Hierarchical structures in organisations - A conceptual framework", European Journal of Operational Research 86 (1995) 4-31. Wagenhofer, A. (1992), "Verrechnungspreise zur Koordination bei Informationsasymmetrie", in: K. Spremann and E. Zur (eds.), Controlling, Wiesbaden, 637-656.