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Incentive contracts when production is subcontracted
European Journal of Operational Research 40 (1989) 169-185 North-Holland
169
Theory and Methodology
Incentive contracts when production is subcontracted K a s h i R. B A L A C H A N D R A N a n d J o s h u a R O N E N New York University, Schools of Business, 424 Tisch Hall, New York 10003, USA
Abstract: The problem of pricing for products under subcontracting is set in an agency theory format. Characteristics of decentralization, moral hazard and adverse selection are inherent in this problem. Optimal incentive compensation function and the transfer price are analyzed. The results are illustrated with an example of logarithm utility functions. Keywords: Agency, pricing, contract, moral hazard, decentralization
1. Introduction Problems related to control and motivation have recently become a focus of the expanding literature on Agency (for a review see Baiman, 1982). From the perspective of this literature new and different insights can be gained into the process of decentralization into profit centers. Just under what circumstances would it be sensible for units of organization to be treated and evaluated as cost centers or alternatively accorded decision making autonomy and judged, rewarded, and penalized on the basis of profits generated? If pertinent information is known by the corporate principal then the decision on quantity to be produced can be made by him and only the implementation delegated to an agent. If under those conditions transfer pricing is observed, the function of such would not reasonably be expected to be the allocation of resources or specifically the decision on what quantity to produce. But, then, transfer prices are merely another form of allocation of overheads which so far has posed equally baffling questions. If allocation serves as a surrogate for the measure on the basis of which allocation is made (or the dimensions of interest regarding the activity under consideration), then allocation of overhead costs is both arbitrary and useless. Why then transfer prices? It would be more plausible, therefore, to assume that transfer prices do genuinely affect resource allocation or quantity production decisions (although empirical evidence on this is shamefully lacking); but then a reasonable necessary condition for transfer prices to usefully serve this function would be information asymmetry: the agent possesses knowledge of input a n d / o r output supply or demand schedules not shared by the principal. Indeed, Ronen and McKinney (1970) devised a procedure for setting transfer prices that eliminated information asymmetry by eliciting truthful revelation of the agent's knowledge by means of the incentives imbedded in the transfer pricing mechanism suggested. But they did not consider the question of uncertainty as to outcomes. Groves and Loeb (1979), while modelling the information asymmetry, did not consider the divergence of preferences among divisions in generality. Harris, Kriebel and Raviv (1982) analyze a multidivisional firm with such assumptions as fixed proporReceived November 1987, revised June 1988 0377-2217/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)
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K.R. Balachandran and J. Ronen / Incentive contracts when production is subcontracted
tions technology for intermediate products, linear production technology for each division's output and linear preferences for divisional managers. Note. Baron (1982) also considered the case of information asymmetry whereby prior to contracting, the agent (banker in his case) has better information (about market demand for securities) than the principal (the issuer of securities). But in this, unlike our case, the principal is one who decides on an offer price for the security (analogous to Q in our case) as a function of a message about the demand state which he receives from the banker. His model, thus, represents a direct revelation mechanism (see Myerson, 1979), where the decision on the relevant variable is centralized, whereas our case is one of decentralization. Further, our scheme does not involve communication for the reasons stated above. Baron and Holmstrom's (1980) modeling of the delegation of the offer price decision in a context similar to Baron (1982) above, on first appearance looks similar to ours in that there is no communication. However, they do not attempt to solve the general problem but rather address limited aspects of it. Specifically, they first suppress the effort decision and for a prespecified compensation function consider the design of incentives for the pricing decision above. Next, for the price given and an assumed linear compensation function they tackle the effort incentive problem. In a limited setting under the linear compensation they finally consider the pricing and incentive problem. Our model differs in terms of the introduction of consumption of nonpecuniary benefits as moral hazard and the resultant usage of a nonseparable utility function for the agent. The purpose of this paper is not to specify all conditions under which transfer pricing would emerge endogenously as an integral part of an optimal contracting scheme. Rather, we intend to explore into a set of conditions, such as when there are bars to communication, wherein it becomes sensible to use a transfer pricing function in the determination of quantities of production. We consider a situation in which the principal pays the agent a total transfer price out of which the agent pays for production cost. This is equivalent to the case where the agent may be viewed as a subcontractor. The difference between the transfer price and the production cost can be considered the equivalent of the traditional compensation where the principal is deemed to pay for the production cost. However, we do not explicitly model this here. Before we address the main focus of the paper, we introduce some essential notations. Q designates quantity of production, ~ the most efficient cost of production (including inputs), # the actual cost of production so that t~ - ~ reflects the inefficiency of the agent including the consumption of nonpecuniary benefits and undue leisure (the opposite of effort). The agent is assumed to derive utility from 6 = / ~ - ~ and disutility from production that increases in Q. ~ is assumed to be a function of a random variable, ~o. No intermediate market for the product manufactured exists, Q is processed additionally by the principal until it is ultimately sold in accordance with some net revenue function R(Q).
The primary organizational designs that one can conceive for these problems are those of a centralized setting and a decentralized setting. Under the first, the principal decides on Q and delegates the productive effort to the agent. Under the second the agent decides himself on Q in addition to undertaking the productive effort. Under the decentralized setting, the principal would compensate the agent in the form of a transfer revenue function O ( Q ) and the agent would pay independently for input costs and decides on how to split the difference between his revenue and input costs into monetary compensation and perquisites or inefficiencies from which he derives utility separately. The consumption, by the agent, of nonpecuniary benefits, 6, affects the output Q, which is observed by both. Thus, the unobservable 6 (for the principal) causes moral hazard in this problem. The introduction of nonpecuniary benefits as an argument in agent's utility is consistent with the observation of organizational slack and its formalization by Jensen and Meckling (1976). It is noteworthy that consideration of decentralization, moral hazard, and adverse selection - - all contained in this problem - - anticipates the observed phenomenon of subcontracting. In other words, transfer pricing coupled with performance evaluation on the basis of profit within decentralization into autonomous
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I
O(Q)
determined by the Principal
I
¢o
observed by A
I
I
dethroned
Q observed produced and by both. Cost paid by A
Q,
I
O(Q)
paid to A
Figure 1
divisions is equivalent to a contractual arrangement whereby the transfer price becomes a 'real' price attached to a product subcontracted by the firm to an independent agent (supplier). The analysis in this paper will show, under these specified conditions which transfer revenue schedules might emerge and whether they make any intuitive sense from the standpoint of their similarity with demand schedules in general and with what one expects to observe in reality in the form of a transfer price that allows for quantity discounts, and so on. In addition, we show in Theorem 2, the precise conditions under which the individual rationality condition is not met with equality when the agent's utility function is not separable, and interpret these conditions. This, we believe, is a new and interesting result. The scenario is depicted on the time-event line. See Figure 1. The agent is viewed as maximizing the expectation of his utility which he derives from three sources: monetary payout, s, the nonpecuniary benefits, 6, which could be viewed as inefficiency, minus the disutility associated with producing Q, Y(Q). We introduce the following notation and assumptions: A1. U(s) is defined as utility in money where U ' > 0, U " < 0. A2. V(8) is the utility in nonpecuniary benefits or inefficiency where V' > 0 and V " < 0. A3. Y(Q) is the disutility in production where Y' > 0. A4. The principal maximizes the expectation of his utility P ( r ) where r is his monetary share of revenue R less his transfer payment 0 to the agent. So, r ( Q ) = R(Q)- O(Q). We assume that P ' > 0, P " < 0. 2. Formulation of the problem and discussion The principal determines O(Q) so that his expected utility is maximized subject to the agent's individual rationality and incentive constraints as follows: Max
Q,O(Q)
fp(Q(,o), O(Q))f(~o) d¢o
(1)
subject to
fq~{~o, O(Q), Q ( ~ ) } f ( ~ ) d ~ 0 > Q(~)~argmax
k,
~b{o~,0(Q), Q(o~)}
(2) vw,
(3)
where qJ[o~, O(Q(o~)), Q(t0)], the composite utility of the agent, is obtained as follows: Given O(Q), the agent's problem is to divide 0 - ~, the excess revenue over the most efficient production cost into s and 8 that will maximize U(s) + V(8). While not a perfect substitute for monetary wealth, 8, which enters separately as an argument into the agent's utility function as non-pecuniary benefits or the inverse of effort, can be measured in pecuniary terms as the present value of cash flows directed away from the principal (see Jensen and Meckling, 1976). The optimafity condition for s* and 8" is given by m a x [ U ( s ) + V(8)] ~ m a x [ U ( s ) + s,8
s,8
V(O- ~ -
~qJ{ o~, O(Q), Q ( ~ ) }
s)]
=
U(s* ) + v(8* ) - Y(Q).
(4)
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s*(0, Q) and 6" (0, Q) refer to the optimal quantities and 0 = s* + 6" + 4. Note that s* and 6 " can be expressed as functions of w and Q. We will refer to ~p as the optimal split utility (OSU) of the agent. We use the symbol 0 to refer to both the price function and the amount as per the context. Few clarifications would be appropriate at this stage. First, consider the role of 6. 6 can have the same interpretation as the inverse of effort in the traditional principal-agent model. Lack of effort on the part of the manager translates immediately into inefficiency which increases cost. Therefore, for a given budget of 0 (the compensation from the principal), the agent can decide on how to sprit it between monetary compensation to himself and inefficiency (which requires less effort and alertness on his part - commodities to which he is averse). Thus, 6 endows the model with features of moral hazard. 6 differs, however, from the traditional concept of effort in that, in our scenarios, part of the compensation 0 can be consumed by A as 6 whereas no part of the compensation in existing agency models can be consumed as leisure. 6 plays an important role in this analysis as it is relevant to the production decision of the agent. The utility the agent derives from inefficiency affects the total cost of production o~Q + 6. Hence, the cost is not independent of 6. So, 6 affects both the transfer pricing function and the quantity of production. Moreover, unlike the traditional analysis whereby a utility function defined over wealth is assumed, the utility function induced in our case need not be separable in wealth and effort, as assumed in most of the existing literature. Note that the optimal s * and 6" are functions of 0 and Q, and ~k is not separable in s * and 6". (When the utility is non-separable, it is not clear whether the individual rationality constraint is satisfied by equality in the optimal solution; see Grossman and Hart, 1983.) The application of this problem is not restricted to the above situation of a principal and a relatively independent subcontractor. It can apply also to transfer pricing attached to internal transfers among divisions of the same corporation in (plausibly numerous) situations, whereby one division would fear that the knowledge of costs it reveals will leak to other divisions that might utilize the information strategically to compete with the communicating division. Thus, the scenario analyzed is one where communication is barred because of the costs it imposes on the agent. In the section below we examine the properties of the optimal split utility function ~k to facilitate further analysis.
3. Properties of ik(o~, O(Q), Q(60)) The proofs are provided in the appendix. The subscripts used below refer to partial derivatives. P1.
~k2 > 0, optimal split utility (OSU) is monotone increasing in the total payment 0 for fixed ~ and
Q. ~3 < 0, that is for a given 0 and co, OSU is monotone decreasing in Q. ~22 < 0. That is, the marginal utility decreases in compensation. lp23 > 0. This reflects the fact that as Q is increased for a given 0 and less money is left to split between s and 6 after paying ~ for the production costs and with utility that is concave in money, the agent moves back along the utility curve into a higher marginal utility point. P5. 1~32 -~" ~23 > 0. This indicates that an initial decrease in ~ caused by an increase in Q (~P3 ~< 0, as in P2) would be mitigated by a marginal increase in 0. P6. q~33 < 0 if Y"(Q) > 0, and ~2~(~, Q)/SQ2 >10.This is caused by the cumulative effect of both the disutility from production and the production costs being convex. P7. d2~/dQ z = 7 < 0 if and only if P2. P3. P4.
O"(O) <
U"(s*) U'(s*) (O'(Q) -~o(w, Q))Zs~(O, ~1 +
U'(s*)~QQ + r"(Q) U'(s*)
Property P7 means that the OSU is concave in Q so that there exists a unique optimal Q = Q* which maximizes the optimal split utility if the specific condition is satisfied. Assuming the condition is satisfied,
K.R. Balachandran and J. Ronen / Incentive contracts when production is subcontracted
173
this now allows us to address the determination of the optimal quantity by the agent. In other words, the property states that if the principal wishes to induce a unique quantity solution to be decided upon by the agent, there must be an upper bound on the slope of the marginal revenue to the agent. To interpret this upper bound, assume first, ~o, the marginal cost to be constant and ¥(Q) = 0 (no disutility in production), This causes the second term to become zero. Given that at optimum (as will be shown below) 0' = ~Q (the familiar marginal revenue = marginal cost condition), we see that P7 reduces to the condition that the marginal revenue should be downward sloping. Allowing ~Q to be nonconstant, we derive the familiar condition, that at optimal production, that the marginal revenue curve crosses the marginal cost curve from above. Further, for all Q, the degree to which 0 can be allowed to be convex increases with (1) absolute risk aversion, - U " / U ' , (2) the absolute value of the decrease in Q's marginal contribution to the agent's monetary payoff, -SQ, for a given marginal cost, ~O (since so = --SQ/~O), (3) the deviation between marginal revenue and marginal cost. Further, a more convex 0 becomes feasible, the greater the convexity of the disutility of production relative to the marginal utility in money for the agent and the higher the slope of the marginal cost curve. These conditions are intuitively plausible as they all contribute to the concavity of ~. We now proceed to state the general problem.
4. Analysis of the problem Condition (3) represents the first order condition of the agent. The agent chooses Q = Q* such that d~=0
dQ dk
V*o,
dO = +2(°~' o ( o * ) , Q*)O'(Q*) + +3(~0, O(Q*), O * ) = 0.
(5)
Hence O'(Q*)= - + 3 / + 2 > 0 since q~z > 0 by P1 and +3 < 0 by P2. Substituting for if2 and q'3, 0'(Q*)---~°(~°'O*)+
Y'(Q*) U'(s*) "
(6)
If Y(Q*) = 0, then Y'(Q*) = 0 and 0' = ~Q(~0, Q*). This restates the standard result that the optimal quantity for the agent is that at which his marginal revenue equals his marginal cost of production. However (6) implies that the optimum is achieved at a point where marginal revenue exceeds the marginal cost by a quantity of Y'(Q* ) / U ' ( s * ). This quantity should be interpreted as the additional dollars that are required to compensate the agent for the disutility of producing one additional unit. This addition is larger, the greater the marginal disutility from production and smaller, the greater is the marginal utility from compensation. We can get a lower bound for 0' by writing
O'(Q*)>~Q(~, Q*)
and
~o(~o, Q * )
s~ (~o, p * ) So* ,
which follows from P5 (see the proof). Then
O'(Q* ) >
s~" "
Note that s~ < 0 from the proof of P4 and sJ' > 0 from the proof of P3. Hence the right hand side is positive. The bound means that the pecuniary benefits the agent obtains from one more unit of Production (O's~') should exceed the pecuniary loss resulting from production of one more Q. For example, if s~' is
K.R. Balachandran and J. Ronen / Incentive contracts when production is subcontracted
174
$0.20 (i.e., for a marginal dollar of revenue the pecuniary benefit to the agent is $0.20) and if s~ is - $0.60 (i.e., marginal cash cost to the agent is $0.60 per unit), the agent needs at least $3 of marginal revenue ( 0 ' ) to break even in the pecuniary sense. The Q* so obtained is optimal and unique provided ~k is strictly concave. The condition for the concavity of ~p is given as P7 and can be used to obtain a condition for optimality. Note. There is a circularity here. For ~b to be concave in Q we need to assume conditions pertaining to the derivatives of the compensation functions, O(Q). Note, however, as the discussion under P7 shows that the observed characteristics of compensation functions do follow the patterns that we need. We admit that we may miss the optimal solution in some unusual cases with our analysis. Q* maximizes ~b (locally) if and only if
U"[Y' as*] U',QQ+Y" O" ( Q* ) <~- -U7 ~U aQ + u' '
(7)
where the right side of (7) is evaluated at Q = Q*. Note that (7) is equivalent to P7 at Q = Q* by substituting ds
Y'
dQ =s~'O'(Q*) + s~ =s~ U " using (6) and the property that ~Qs~' --- --s~. If Y = 0 , then O"(Q*)<~QQ(Q*) where Q = Q*. If, in addition, ~ is linear in Q, we require O"(Q*) < 0 for a local maximum. Note that if ~QQ(Q*) > 0 evaluated at Q = Q*, 0 " need not be negative. For a fixed function 0, Q* is a function of to. We wish to obtain the conditions under which Q*(~0) is monotone in to. Monotonicity implies a one to one correspondence between Q* and to and hence Q* is a global maximum if it satisfies the conditions for local maximum. Q *(to) is monotone increasing if Q *(to) = d Q *(to)/dto > 0 everywhere and monotone decreasing if the derivative is negative everywhere. Equation (6) can be written as follows (where all the terms are evaluated at O = Q*):
U'(s*)O'(O*) = U'(s*)~e(W, Q*) + Y'(Q*). Taking derivatives with respect to to implicitly, . ¢ . ~ O'U"(soO Q~,, +sQQ~ + s * ) + U t 0 Q~, Pt
t!
.
t
,
,
t!
Solving for Q*, -
= v"(s
v"s*
(0' -
O'
v ' o " - v'
Qe- r "
(8)
When (7) is satisfied, the denominator of (8) is negative. Then, Q* is positive everywhere or negative everywhere depending on whether (9) or (10) below is satisfied. Q*<0
**
U t t s ,~o ( 0 ' - ~ O ) <
U
t
~Q~.
(9)
Note that s,o < 0, which can be shown using arguments similar to those in the proof of P4. .
Q~ > 0
•
tr
,
!
U s,o(0'-~Q)>U~Q,o,
(10)
where all functions are evaluated at Q = Q*. Note that if Y - - 0 , 0 ' - ~ Q . = 0 at Q = Q* and (9) will always hold for ~Q,o > 0, at Q = Q*. If to is interpreted as c o s t / u n i t of production, then ~ -- to. Q and ~e~=l >0.
K.R. Balachandran and J. Ronen / Incentive contracts when production is subcontracted
175
We state the following as an assumption: A5. U"(s
*
* * )s*(~o, Q * ){O'(Q*)-~Q(,O, Q*)} 4: U / (s)~Q,o(w, Q ).
Q* satisfying (9) or (10) implies Q* is monotone in ~0 and thus unique for a given w. This means that if Q* satisfies the condition for a local maximum, it is a global maximum and P7 is satisfied for all Q. In addition, we can express ~0 = G(Q* ). The solution to the principal-agent problem is stated as Theorem 1. Theorem 1. Assume A1 to A5.
If the compensation function 0 further satisfies (7); then the optimal solution for 0 and Q to the principal-agent problem as stated in (1), (2) and (3) is characterized by -Po(Q, 0) (a)
~0(~o, 0, Q)
1 dzP a + y . dQ.2
1
dP [ G"(Q*) 1 df + + 7" dQ* [ G'(Q*) f dQ* (b)
f~/(¢o, O, Q)f(co) d~o - K>~ O,
(c)
0' =
1 - d7* y* dQ*
+
U"(s*) U
;r-zr
IS-'-)
S*
]
C(Q )], P
*
(11)
y' + -g
Proof. Provided in the appendix.
To help interpret (11) assume that Y = 0, i.e., the agent suffers no disutility from production. Then we obtain the following:
P'=a+
dP dQ*
d2p ] 0" ;li+ d]l
Of
40 This means that the ratio of marginal utilities which should be a constant under optimal risk-sharing (see Holmstrom, 1979) is a constant (a) plus an amount which varies in Q. The deviation from optimal risk sharing represented by the second term increases in the marginal utility of the principal (dP/dQ*, which could be zero if ~0 were known to the principal) multiplied by fQ/f which represents the degree to which ,0 can be inferred from Q. The deviation from optimal risk sharing also increases in relation to the rate at which the marginal cost to the principal O' increases. The deviation from optimal risk sharing however is inversely related to the sensitivity of the agent's utility to the principal's transfer payment 0, qJ0- The higher the sensitivity, the more capable is the principal to fine-tune 0 so as to induce a more favorable choice (from the principal's standpoint) of Q. Since the agent's utility function is not separable in s *, 3" and Q*, the reservation utility condition need not be binding at optimum. We proceed to find the condition under which the reservation utility restriction is binding at the optimum. Theorem 2.
(1) The reservation utility restriction is satisfied with equality, at optimum, if and only if
f[P2(Q*, 0)
dQ*dP U"(S*)so(O,_,Q.)]f(o~ ) Y
d~0 < 0.
(If Y - O, O' = ~Q. and since P2 < 0, this is always satisfied. If Y> 0, this condition is necessary.)
(12)
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K.R. Balachandran and J. Ronen / Incentiue contracts when production is subcontracted
(2) A sufficient but not necessary condition for (12) to hold is that the derioative of "r(Q) = R ( Q ) - O(Q), with respect to Q, eoaluated at the optimum Q = Q * be non-negatiue. Note. In the above theorem, ad (1): ~k is not separable in 0 and Y since both are functions of Q. Hence 0 cannot be reduced by a constant since that will imply reducing Y and this will change Q. Remark that Q* is a function of to. Proof. Provided in the appendix.
To understand Theorem 2, consider first the impact of perturbations in 0 on Q. As shown in the proof of the theorem in the appendix, Q is positive in e, where e multiplies constant decreases in 0 (as a perturbation). It means, that for a uniform decrease in 6 for every Q, the agent will increase the optimal Q that he wishes to produce. To understand this, consider the optimality condition (6). For a fixed Q, as 0 is decreased, so is s and hence U ' is increased. That is, the right hand side of 0' - ~Q. = Y ' / U ' is decreased relative to the left hand side. To restore equilibrium, Q must increase in order to reduce the left hand side. We shall rewrite (12) as follows: Note that =u"
ds
dQ*
= dU'
dQ* "
(See for the first equality the proofs of P4 and P5.) Also note that P2 = - P , Hence,
f P2f(to) do, = -E~(P). Substituting in (12), the condition becomes E,(P) +
dQ* dQ*/a
f ( t o ) d o > 0.
(13)
For a violation to occur, i.e., for the individual rationality constraint not to be met with equality, we need the following conditions: dr d-"Q < 0
and violation of (13).
Write violation of (13) as
[ d---~-/otJ f(to) dto > 1.
(14)
The term in the first bracket can be interpreted as follows: The numerator is the cost to the principal of increasing Q or equivalently the benefit of reducing Q. The denominator is the cost to the principal of increasing 0 or equivalently the benefit of decreasing 0 - (recall that d ' r / d Q < 0). Hence violation occurs if the benefit of decreasing Q (which entails increasing 6) is greater than the benefit of reducing 9. In other words, the principal has no incentive to reduce 8 until the reservation utility constraint is met with equality. The term in the second brackets can be interpreted as a weight factor which represents the proportion of the impact of Q on the agent's marginal utility in money to the impact of Q on the total marginal utility
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177
(of q~). Intuitively, the larger this proportion, i.e., the larger the relative impact of Q on the agent's utility derived from money, the easier the occurrence of the violation. We now proceed to illustrate these results with an example. Example. Let P ( Q , O(Q)) = d l n ( R ( Q ) - O ( Q ) ) , Y ( Q ) = c In Q,
U(s) = a in s(O, ~),
~(Q, ~0) = wQ,
V(8) = b In 8(0, ~),
f(oa) = 1,
i.e., o0 is uniformly distributed over the range 0 ~< ~0 ~< 1. We assume that J R ( Q ) - 0(Q)] > 0, so that In is well defined. Given O(Q), the agent allocates the compensation optimally between s and 8 to satisfy U ' ( s * ) = V'(8 * ), i.e., s * and 6 * satisfy 0 - - ~ + s * + 8 *, and a/(O - ~ - 6 * ) = b / 8 *. Solving, we obtain b(O-li)
8"(0,4)
a(O-4)
(a+b)'
(a+b)"
The agent's optimal split utility function tp(~o, O, Q) = U(s* ) + V(8* ) - Y ( Q )
1
{--~+ ~
[
+ b In
~-~ + ; ~
1
- c In Q,
(15)
we shall assume that O(Q) - wQ > 0, so that In is well defined. The agent chooses optimal Q = Q* given the compensation function O(Q) and after observing ~0 to satisfy (6). Substituting 4o = o0, Y '(Q *) = c / Q * and U ' ( s * ) = ( a + b )/[ O( Q * ) - oaQ*] into (6) yields 0'(Q*)=~o+
c[O(Q*)-ooQ*] (a+b)Q*
c [O(Q*) =~0+ (a+b-~ Q*
] oa .
(16)
The second term in (14) is (transfer price-cost) multiplied by a weighting factor. Solving for Q*, Q*(oa) =
cO(Q*) (a + b ) O ' ( Q * ) - (a + b - c)oa"
(17)
If Q*(,0) is monotone, then the function can be inverted and ~0 can be expressed as a function of Q*, say, oa = G(Q*). From (17), oa = G ( Q ) = (a + b ) O ' ( Q * ) Q * - cO(Q*) Q*(a+b-c)
(18)
The condition for the concavity of q, from P7 applied to this example gives O"(Q) < (0' - oa)2 (O-~oQ)
c(O - oaQ) QZ(a+b ) "
(19)
To assure local optimum of Q = Q*, the condition to be satisfied is obtained from (7) as c 0"(Q*) < (a+b)Q.
(
0'-
oa)
(O-~oQ*) Q.
(20)
The first term 0 ' - ~ 0 is the marginal profit to the agent and ( 0 - oaQ*)/Q* is the average profit. Formula (19) must hold for all Q in order to assure concavity of ~b and consequently to insure that Q* is the global maximum. Substituting for oa from (18), we see that (20) is equivalent to the following: 0"<
-
c(O-O'Q*) Q*2(a+b)
(21)
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178
In our search for 0(Q), we shall require (21) to hold. We shall now state the following lemma that paves the way for the existence of o: = G(Q* ), the inverse function of Q*(o:). Lemma. Q*(to) is (strictly) monotone in ~, provided a + b -~ c and
c(O-O'Q*) Q*Z(a+b)
8"
ProoL Differentiating Q*(to) given by (17) with respect to o: and substituting for o: from (18), we can obtain the following expression: dO__..~*=
do~
(a + b - c ) Q *z
(22)
Q*ZO" ( a + b ) + c( O - O'Q* ) "
If 0 " < - c ( O - O ' Q * ) / [ Q * 2 ( a + b ) ] , then the denominator in (22) is negative which implies that d Q * / d o : is negative for a + b > c and positive for a + b < c, On the other hand, 0 " > - c ( 0 - 0'Q * ) / [ Q * 2(a + b)] implies that the denominator of (22) is positive and in turn d Q * / d t o is positive for a + b > c and negative for a + b < c. In all cases, d Q * / d t o is strictly either everywhere positive or everywhere negative and the lemma is proved. [] We shall require, for the example, that (21) holds. Then if a + b > c, the derivative of Q* with respect to to is negative and it is positive if a + b < c. Using (17), we can easily show that for 8 - t o Q * > 0, a + b > c implies 0 - O'Q* > 0, and a + b < c implies 0 - O'Q* < 0. For a + b 4: c and (21) satisfied, the Q* obtained locally maximizes ~b and is unique. This implies that Q* is also globally maximizing. The solution to the principal-agent problem is characterized by substituting the following into (11).
(a + b)O" + c ( O - O ' Q * ) (a+b-c)Q*' (a+b-c)
C'(Q*) =
G"(Q*) =
~2
c(O"Q .2 - 2Q*O' + 20) ( a + b - c ) Q .3 '
(a+b)O'" (a+b-c)
(a+b-c) (O-O'Q*)'
(a+b-c) 2 ~22= - ( a + b ) ( O _ O , Q . ) 2 ,
( a + b - c ) [ ( a + b)O'Q* - col 1~32 -~"
P1 ~=
Q.(a+b)(O_O,Q,)2
dR' R(Q*)-O'
'
-d P2= R ( Q * ) - O '
d(O'-R')Q*2(a+b) ( R ( Q ) - O)[Q*2(a+ b)O" + c ( O - O'Q*)] '
T=(R(Q*)-O), d P" R ( Q * ) - O ' dP d dQ* = ( R ' - O ' ) ,
d2~
a+b-c O-O'Q*'
+o
d2p
dQ.2
dO" "r
G'(Q*)(a+b-c)
Y*=dQ2[Q=Q.= (a+b)(O-O'Q*) The solution is formally stated below.
"
d(O'-e') 2 ,r 2
K.R. Balachandran and J. Ronen / Incentive contracts when production is subcontracted
179
For the example with logarithmic utility functions and a + b ~ c, the optimal solution for 0 and Q to the principal-agent problem as stated in (1), (2) and (3) is characterized by the following: P~ +0
--
1 dzP - V* dQ .2
= a +
f[a
c
+
dP (a + b)Q* dQ*
In a(O-~oQ*) +b In b(O-~oQ*) a+b a+b
(23)
' c l n Q * ] d~o >~ K,
O' = o~ + c ( O - o~Q* )
(24) (25)
(a+b)Q* ' 0"<
c(O-O'Q*) Q*2(a+b)
For Theorem 2(2), we note that a sufficient condition for (24) to hold as equality at optimum is that dr(Q)___R,(Q)_0,(Q)>0 dQ
atQ=Q*.
Let R(Q) =pQ so that R'(Q) =p. If a + b > c, we get from (12),
O'(Q*)-o~
c( O- o~Q*) <~O - ~ o Q * (a+b)O* Q*
i.e., 0' < O/Q* < p. The last inequality follows from the assumption pQ * - 0 > 0. This assumption is necessary as otherwise the logarithm in the principal's utility function is not well defined. Thus for the case R(Q) =pQ and a + b > c, we have r'Q = R'(Q*) - O'(Q*) > 0 and the reservation utility constraint (24) is satisfied with equality. The intuition behind the result is that decreasing 0 by a constant decreases the right side of (25). Note that 0" is strictly negative from (22) for a + b > c and consequently, the agent will increase Q* to maintain the equality of (25). This certainly is beneficial to the principal also since r ' is positive. Thus Q* will approach the maximum production capacity of the agent's production facility constrained only by the fact that 0 - ¢0Q* > 0. Note that the disutility parameter c is less than (a + b) and Y"(Q) < 0 for this example. Note also that for any quantity Q, the transfer price is O(Q)/Q. The derivative of the transfer price with respect to Q evaluated at Q = Q* is given by (O'Q* - O)/Q .2 and is negative. That is, the transfer price is a decreasing function of the optimal quantity transferred. This is true for any revenue function R(Q). For a + b < c, the reservation utility constraint may not hold with equality even for the linear revenue function. However, we note that the derivative of the transfer price with respect to Q is positive when evaluated at Q = Q*. Thus for a + b ~< c, the transfer price is an increasing function of the optimal quantity transferred.
5. C o n c l u d i n g
remarks
We considered an agency problem in which the agent chooses both the quantity of output and the level of productive efficiency. There is information asymmetry in the sense that the agent observes the random variable which affects the production cost prior to making the production decision. Any communication of the observation to the principal is barred in the model. The utility of the agent consists of three parts, an increasing function of his monetary net income and the leisure afforded by inefficient production or consumption of perquisites, and a decreasing function of the output quantity. In Theorem 1 we characterized the optimal form of the compensation function.
K.R. Balachandran and J. Ronen / Incentioe contracts when production is subcontracted
180
Decentralization is necessitated when the agent is endowed with superior information and when communication of the cost function is barred for fear of competition. An extreme form of decentralization is implied in the relation with a subcontractor with no intermediate market. A less extreme form is inherent in the case of interdivisional transfer within the same organization when divisions are reluctant to reveal their cost to headquarters and thus possibly to other competing divisions. In these cases, an alternative arrangement to compensating the agent on the basis of profits generated by transfer prices (as in Ronen and McKinney, 1970), would be paying the agent actual prices while the agent assumes the cost of production; i.e., he becomes a pseudosupplier: the two arrangements will produce identical results. Once the transfer price emerges as the only means of compensating because only the quantity transferred is observable by the principal, the question is whether the optimal transfer pricing entails a function that 'makes sense' from the standpoint of these characteristics. We do indeed find that the transfer pricing function is a downward sloping schedule reflecting a typical demand curve, just as might be observed in practice. In other words, the solution indicated is one that is consistent with what one observes in reality in general and that is consistent with the need for attaching compensation to quantity transferred because there is no other observable measure of performance. In principle, the resulting transfer price function can be subject to empirical testing. The degree to which the implications of the model is testable depends on the nature of the data to be gathered. At the very least, it appears that in cases where communication is barred from a business unit transferring goods to a receiving unit, we should observe, in accordance with the implications of the model, a transfer pricing scheme in operation and more specifically a sloping down transfer pricing scheme. Of course, if more data can be gathered such as the nature of uncertainty and preferences, the precise function that is impfied by the model can be tested against the actual scheme. In addition, we are able to specify in Theorem 2, the exact conditions under which the optimal solution will involve providing the agent with an expected utility above his reservation utility. In other words, conditions under which the individual rationality constraint is not met with equality. We also provide a simple intuitive explanation of these conditions. As might be expected, the constraint will not be satisfied with equality if and only if increasing quantity of production decreased the payoff to the principal and the benefit of decreasing quantity to him exceeds the benefit from decreasing the payoff to the agent. An example employing logarithmic utility functions is presented to illustrate the general solution. Appendix
Proofs of properties of Proof of P1.
O~b ~0
~k2 =
dU as* dV ~6" ds* ~ 0 + ~d8 ~0
U'(s*)[-~- +
(since U'(s*) = V ' ( 8 * ) )
= V'(s*)[ = U'(s*)
+ >0.
Proof of P2.
a,
~ 3 = -OQ =
aQ
+ - - ~ - Y'(Q)
= U'(s*)(-a~Q)_y,(Q) <0,
04 since-~>0
and Y ' ( Q ) > 0 .
K.R. Balachandran and J. Ronen / Incentive contracts when production is subcontracted
18l
Proof of P3.
3s* q,= = u " ( s * )
30 "
By assumption U " ( s * ) < 0 . We need to prove that 3s*/30> 0. We know that at optimal split U ' ( s * ) = V' (8) for a given 0. Now perturb 0 to 0 + A0. The corresponding perturbed values in s* and 8 * are s* + -as* ~--k0
and
8" +
A0.
To maintain optimal split, we should have U'[s* + 3s*/30. A0] = V ' [ 8 * + 3 8 * / 3 0 . A0]. Since U " < 0 and V " < 0, we must have that either both ~s*/30 and 3 8 " / 3 0 are positive or both are negative. But 3s*/30 + 3 8 * / 3 0 = 1 from the proof of P1. Hence ~s*/30 > O. Proof of P4. +2, = u " ( s *
) Os* OQ"
By assumption U " ( s * ) ~ O, and we need to show that 3s*/3Q < 0. Following arguments similar to those in the proof of P3, perturb Q to Q +AQ, U'[s* + 3s*/3Q.AQ] = v ' [ 8 * + 38*/~Q.AQ] and we must have the same sign for both 3s*/3Q and 3 8 " / 3 Q . but 3s*/3Q+ 3 8 " / ~ Q = -3~/~Q from the proof of P2. Hence 3s*/3Q < O. Proof of P5.
aO [_ [ ~o~ _ >]0 ~32= U ' ( s * ) as*
s i n c e -~~*- > O ,
-a~ ~>0,
U . <0.
~23 = ~32 follows provided the partial derivatives are continuous and ~ is in c 2. Proof of P6.
OQ [
-~
+ U'(s
- 002]
Q),
since U " < 0, 3s*/3Q < 0 and 3~/3Q > 0, the property follows. Proof of P7. V = ~ 2 2 ( 0 ' ) 2 + 1//230' "{'-+2 0,, -I- +32 O' q-" +33,
T < 0 ,~ 0 " < -~k=(O')2 - ~kz30' - q~320' - ~33
{-st(O')2u"
+ u " *
U2,
0" <
-s~'U"(O'-~Q) 2 ~ 0" <
u '
u"
y,, + -U r + lie°''
upon substituting for the partial derivatives of ~p and using P5.
Y"}
- O'U"
-
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K.R. Balachandran and J. Ronen / Incentive contracts when production is subcontracted
Proof of Theorem 1 For every choice of 0 made by the principal, the agent chooses an optimal Q* after observing o~. Thus any perturbation e?~ in 0 produces a perturbation in Q*. To obtain an expression for the change in Q*, perturb O(Q*) by e?t(Q*) in the agent's first order condition (5). We get qJ2{o~, O(Q* ) + cA(Q*), Q* } { O'(Q* ) + eA'(Q* ) } + ~b3(~o, O(Q* ) + cA(Q*), Q* } = O. Take the derivative with respect to e and set e equal to 0. Let Q = dQ*/de. (The arguments of ~b are suppressed.)
~b22( O'(Q* )0 + ?t(Q* ) ) O'(Q* ) + }23Q0'(Q* ) + ~2 ( O"(Q* )Q + ~t'(Q* )) d-~33Q=O
+tP32{O'(Q*)()+~t(Q*)}
V)k, 02.
Solving for Q and setting 3'
d2~ dQ 2
~p22(0'(Q*)) 2 + q,230'(Q * ) + +20"(Q*) + ~b320'(Q* ) + 1P33,
O= _ ()t(Q*)[q.'z20'(Q*) ++32] + ~t'(Q*)~b2}
(A.1)
7 The principal maximizes the Lagrangian function
L = S P ( Q * ( w ) , O(Q*(w)))f(w)d~o
+a[S+(~o, O(Q*(¢o)), Q*(w)}f(¢o)d~o- K],
where Q* is chosen to satisfy the agent's first order condition (6). To optimize we perturb O(Q*) by eX(Q*). The Lagrangian function is now written as
L~= f p( e.(,o), o(e.(,o)) + ~x(e* (~)))z(~)
d~o
+.{f~.[~, o(o_*(,0))+,X(Q*(~)), e:(~)] f(~)d¢o-
K}.
Taking derivatives of L~ with respect to e and setting e to O, we get the condition for optimality. (Arguments of P are suppressed.)
f[( e,o + p,(o'(o*)o + x(o* )) ) + .{ ,,(o'(e* )o + x(e* )) + ,,o }] f(~) VX.
d~o = 0 (A.2)
Substitute for 0 from (A.1) into (A.2) and express the optimality condition as f X ( Q * ) $ ( , , , 0, 0', 0", Q*) d,o + f X ' ( Q * ) ~ ( , o , 0, 0', 0", Q*) d~o=0 where
= - ( P1 + P20'(Q *)+ a(~b20' + q o ) ) f ( ¢ ° ) - ~ = - (P1 + P20'(Q *)} f(c°)+z, y
V)t,
(A.3)
K.R. BalachandranandJ. Ronen / Incentivecontractswhenproductionis'subcontracted
183
with use of the fact that ~b2O'(Q* ) + @3 = 0 from (5). = (P2 + a@z)f(to) + • (@220'(Q*) + @32)
@2
Assume that (9) or (10) is satisfied so that to can be expressed as a function of Q*. Let to = G(Q*) so that dto = G'(Q*) dQ*. Substituting for to in (A.3), the optimality condition can be written as
fX(Q* )$(G(Q* ), o, o', o", Q* )G'(Q* ) dQ* + fx'(Q*)O(G(Q*), o, o', o", Q*)G'(Q*)dQ*=O
vx.
Integrating the second expression by parts and noting that X(Q*) = 0 evaluated at the boundary values of Q*, the condition becomes,
fx(Q*)l ~ G ' ( Q * ) - - ~ GdO( Q
, )-OG"(Q*) / dQ*=0
VX,
suppressing the arguments of • and ~ for convenience. The optimality condition is given by dO
~G'(Q* ) = ~ - ; a
,.
(Q* ) + OG"(Q* ).
Substituting for ~ in terms of • we get,
G' ( Q* )[ dd~Q* - f( G( Q* ))( P2 + a@2) - O( @220' + @32) ] + OG'' ( Q* ) =
(A.4)
We proceed to simplify this expression further. Note that • = - d P/dQ * f ( G (Q * )) @2/y, where d P/dQ
*
= (P~ + P20'). Also, in all the functions Y, f , @, and their partial derivatives, to has been replaced by allows us to differentiate these functions with respect to Q*. Thus,
dO [ d P , ~ dv d2P OQ* = [ d ~ * 2 J d Q -;- - 7f@2 d - ~ 2
d P d@2 f't d Q * dQ*
G(Q* ). This fact
]±
dP d/ 7d--Q~@2 d~Q-;-, y 2'
where d@2 dQ*
=@220'+@23+@21G'(Q*)
and
,,
.
•
,
.
@21G'(Q*)=U (s )saG (Q ).
Substituting these derivatives into (A.3) and simplifying, the theorem follows, were we have substituted, P, = - P2 and @0 = @2 for • = R - 0, the principal's share after paying 0 to the agent.
Proof of Theorem 2 Consider a compensation function 0(Q) and the corresponding agent's optimal selection Q = Q*. Let X ~< 0; /£ = fq~(to, t~, ~)(e))f(to) dto, where ~) is the agent's optimal Q corresponding to t~. Note that 0 ( Q ) < 0(Q) for e > 0, X < 0 and every Q. We need to find conditions under which [d/£/de]l~= 0 < 0 and [d/3/de]l~=0 > 0; i.e., we wish to obtain conditions under which the change in agent's utility with a reduction in compensation is negative and the corresponding change in the principal's
K, = f@(to, O, Q*)f(to) dto. Consider O(Q) = O(Q(e)) + eX, where e >~ 0 and
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K.R. Balachandran and J. Ronen / Incentive contracts when production is subcontracted
utility is positive. Under these conditions, the reservation utility must be satisfied with equality at optimum.
d/~ de
e=0
= f(,.(o'0
+ x) + ¢ , 0
We obtain 0 = dQ(e) de by substituting h' = 0 in (A.1).
d/~ de e=0
: s{'+.-,I+..,'+ +..)I. +.,'++.)),,.),° = < 0
d~
(since
~k20'+ ~k3 = 0
from (5))
for all X < 0, since @2 ~" 0, by P1.
Hence the agent's utility always decreases when the compensation O(Q) is decreased by a constant. Now consider the principal's utility. dP de e=o
-'~)---'~-So(O'-I~o)}f(~o ) d~o evaluated at (2 = Q*, where we have substituted P1 + P2~' = d P / d Q and for ~22 and q~23 from P3 and P5. Given X < O, the necessary and sufficient condition for [d/3/de] I e=0 > 0 is P2
dQ - -3's o ( O ' - f ~ O )
f(w)do~
This proves (1). Note that P2 < 0 by assumption, U " < 0 by assumption, 3, < 0 by restricting the choice of 0 such that 0 " satisfies P7 at Q = Q*. s o > 0 as shown in the proof of P3, and O' - ~Q > 0 from (6). Hence to prove (2) a sufficient condition for (12) to hold is
dP Q=Q* dQ > O, Note that d P / d ' r > O
dP Q=Q* dP d.c Q=Q*" dQ - d~ d o
by the choice of the utility function. Hence the sufficient condition is
[d'r(O)/dQ]lo=o. >10. [] References Baiman, S. (1983), "Agency research in managerial accounting: A survey", Journal of Accounting Literature 1, 154-213. Baron, D.P. (1982), "A model of the demand for investment banking advising and distribution services for new issues", Journal of Finance 37, 955-976. Baron, D.P., and Holmstron B. (1980), "The incentive contract for new issue under asymmetric information: Delegation and incentive problem", Journal of Finance 35, 1115-1138.
K.R. Balachandran and J. Ronen / Incentive contracts when production is subcontracted
185
Grossman, S, and Hart, O. (1983), "An analysis of the principal-agent problem", Econometrica 51, 7-45. Groves, T., and Loeb, M. (1979), "Incentives in a divisionalized firm", Management Science 25, 221-230. Harris, M., and Raviv, A. (1979), "Optimal incentive contracts with imperfect information", Journal of Economic Theory 20. 231-259. Harris, M., Kriebel, C.H., and Raviv, A. (1982), "Asymmetric information, incentives and intrafirm resource allocation", Management Science 28, 604-620. Holmstrom, B. (1979), "Moral hazard and observability", The Bell Journal of Economics 10, 74-9l. Jensen, M.C., and Meckling, W.H. (1976), "Theory of the firm: Managerial behavior, agency costs and ownership structure", Journal of Financial Economics 4, 304-360. Myerson, R.B. (1979) "Incentive computability of the bargaining problem", Econometrica 47, 61-74. Ronen, J., and McKinney, III, G. (1970), "Transfer pricing for divisional autonomy", Journal of Accounting Research 7.99-112.
169
Theory and Methodology
Incentive contracts when production is subcontracted K a s h i R. B A L A C H A N D R A N a n d J o s h u a R O N E N New York University, Schools of Business, 424 Tisch Hall, New York 10003, USA
Abstract: The problem of pricing for products under subcontracting is set in an agency theory format. Characteristics of decentralization, moral hazard and adverse selection are inherent in this problem. Optimal incentive compensation function and the transfer price are analyzed. The results are illustrated with an example of logarithm utility functions. Keywords: Agency, pricing, contract, moral hazard, decentralization
1. Introduction Problems related to control and motivation have recently become a focus of the expanding literature on Agency (for a review see Baiman, 1982). From the perspective of this literature new and different insights can be gained into the process of decentralization into profit centers. Just under what circumstances would it be sensible for units of organization to be treated and evaluated as cost centers or alternatively accorded decision making autonomy and judged, rewarded, and penalized on the basis of profits generated? If pertinent information is known by the corporate principal then the decision on quantity to be produced can be made by him and only the implementation delegated to an agent. If under those conditions transfer pricing is observed, the function of such would not reasonably be expected to be the allocation of resources or specifically the decision on what quantity to produce. But, then, transfer prices are merely another form of allocation of overheads which so far has posed equally baffling questions. If allocation serves as a surrogate for the measure on the basis of which allocation is made (or the dimensions of interest regarding the activity under consideration), then allocation of overhead costs is both arbitrary and useless. Why then transfer prices? It would be more plausible, therefore, to assume that transfer prices do genuinely affect resource allocation or quantity production decisions (although empirical evidence on this is shamefully lacking); but then a reasonable necessary condition for transfer prices to usefully serve this function would be information asymmetry: the agent possesses knowledge of input a n d / o r output supply or demand schedules not shared by the principal. Indeed, Ronen and McKinney (1970) devised a procedure for setting transfer prices that eliminated information asymmetry by eliciting truthful revelation of the agent's knowledge by means of the incentives imbedded in the transfer pricing mechanism suggested. But they did not consider the question of uncertainty as to outcomes. Groves and Loeb (1979), while modelling the information asymmetry, did not consider the divergence of preferences among divisions in generality. Harris, Kriebel and Raviv (1982) analyze a multidivisional firm with such assumptions as fixed proporReceived November 1987, revised June 1988 0377-2217/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)
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K.R. Balachandran and J. Ronen / Incentive contracts when production is subcontracted
tions technology for intermediate products, linear production technology for each division's output and linear preferences for divisional managers. Note. Baron (1982) also considered the case of information asymmetry whereby prior to contracting, the agent (banker in his case) has better information (about market demand for securities) than the principal (the issuer of securities). But in this, unlike our case, the principal is one who decides on an offer price for the security (analogous to Q in our case) as a function of a message about the demand state which he receives from the banker. His model, thus, represents a direct revelation mechanism (see Myerson, 1979), where the decision on the relevant variable is centralized, whereas our case is one of decentralization. Further, our scheme does not involve communication for the reasons stated above. Baron and Holmstrom's (1980) modeling of the delegation of the offer price decision in a context similar to Baron (1982) above, on first appearance looks similar to ours in that there is no communication. However, they do not attempt to solve the general problem but rather address limited aspects of it. Specifically, they first suppress the effort decision and for a prespecified compensation function consider the design of incentives for the pricing decision above. Next, for the price given and an assumed linear compensation function they tackle the effort incentive problem. In a limited setting under the linear compensation they finally consider the pricing and incentive problem. Our model differs in terms of the introduction of consumption of nonpecuniary benefits as moral hazard and the resultant usage of a nonseparable utility function for the agent. The purpose of this paper is not to specify all conditions under which transfer pricing would emerge endogenously as an integral part of an optimal contracting scheme. Rather, we intend to explore into a set of conditions, such as when there are bars to communication, wherein it becomes sensible to use a transfer pricing function in the determination of quantities of production. We consider a situation in which the principal pays the agent a total transfer price out of which the agent pays for production cost. This is equivalent to the case where the agent may be viewed as a subcontractor. The difference between the transfer price and the production cost can be considered the equivalent of the traditional compensation where the principal is deemed to pay for the production cost. However, we do not explicitly model this here. Before we address the main focus of the paper, we introduce some essential notations. Q designates quantity of production, ~ the most efficient cost of production (including inputs), # the actual cost of production so that t~ - ~ reflects the inefficiency of the agent including the consumption of nonpecuniary benefits and undue leisure (the opposite of effort). The agent is assumed to derive utility from 6 = / ~ - ~ and disutility from production that increases in Q. ~ is assumed to be a function of a random variable, ~o. No intermediate market for the product manufactured exists, Q is processed additionally by the principal until it is ultimately sold in accordance with some net revenue function R(Q).
The primary organizational designs that one can conceive for these problems are those of a centralized setting and a decentralized setting. Under the first, the principal decides on Q and delegates the productive effort to the agent. Under the second the agent decides himself on Q in addition to undertaking the productive effort. Under the decentralized setting, the principal would compensate the agent in the form of a transfer revenue function O ( Q ) and the agent would pay independently for input costs and decides on how to split the difference between his revenue and input costs into monetary compensation and perquisites or inefficiencies from which he derives utility separately. The consumption, by the agent, of nonpecuniary benefits, 6, affects the output Q, which is observed by both. Thus, the unobservable 6 (for the principal) causes moral hazard in this problem. The introduction of nonpecuniary benefits as an argument in agent's utility is consistent with the observation of organizational slack and its formalization by Jensen and Meckling (1976). It is noteworthy that consideration of decentralization, moral hazard, and adverse selection - - all contained in this problem - - anticipates the observed phenomenon of subcontracting. In other words, transfer pricing coupled with performance evaluation on the basis of profit within decentralization into autonomous
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K.R. Balachandran and J. Ronen / Incentive contracts when production is subcontracted
I
O(Q)
determined by the Principal
I
¢o
observed by A
I
I
dethroned
Q observed produced and by both. Cost paid by A
Q,
I
O(Q)
paid to A
Figure 1
divisions is equivalent to a contractual arrangement whereby the transfer price becomes a 'real' price attached to a product subcontracted by the firm to an independent agent (supplier). The analysis in this paper will show, under these specified conditions which transfer revenue schedules might emerge and whether they make any intuitive sense from the standpoint of their similarity with demand schedules in general and with what one expects to observe in reality in the form of a transfer price that allows for quantity discounts, and so on. In addition, we show in Theorem 2, the precise conditions under which the individual rationality condition is not met with equality when the agent's utility function is not separable, and interpret these conditions. This, we believe, is a new and interesting result. The scenario is depicted on the time-event line. See Figure 1. The agent is viewed as maximizing the expectation of his utility which he derives from three sources: monetary payout, s, the nonpecuniary benefits, 6, which could be viewed as inefficiency, minus the disutility associated with producing Q, Y(Q). We introduce the following notation and assumptions: A1. U(s) is defined as utility in money where U ' > 0, U " < 0. A2. V(8) is the utility in nonpecuniary benefits or inefficiency where V' > 0 and V " < 0. A3. Y(Q) is the disutility in production where Y' > 0. A4. The principal maximizes the expectation of his utility P ( r ) where r is his monetary share of revenue R less his transfer payment 0 to the agent. So, r ( Q ) = R(Q)- O(Q). We assume that P ' > 0, P " < 0. 2. Formulation of the problem and discussion The principal determines O(Q) so that his expected utility is maximized subject to the agent's individual rationality and incentive constraints as follows: Max
Q,O(Q)
fp(Q(,o), O(Q))f(~o) d¢o
(1)
subject to
fq~{~o, O(Q), Q ( ~ ) } f ( ~ ) d ~ 0 > Q(~)~argmax
k,
~b{o~,0(Q), Q(o~)}
(2) vw,
(3)
where qJ[o~, O(Q(o~)), Q(t0)], the composite utility of the agent, is obtained as follows: Given O(Q), the agent's problem is to divide 0 - ~, the excess revenue over the most efficient production cost into s and 8 that will maximize U(s) + V(8). While not a perfect substitute for monetary wealth, 8, which enters separately as an argument into the agent's utility function as non-pecuniary benefits or the inverse of effort, can be measured in pecuniary terms as the present value of cash flows directed away from the principal (see Jensen and Meckling, 1976). The optimafity condition for s* and 8" is given by m a x [ U ( s ) + V(8)] ~ m a x [ U ( s ) + s,8
s,8
V(O- ~ -
~qJ{ o~, O(Q), Q ( ~ ) }
s)]
=
U(s* ) + v(8* ) - Y(Q).
(4)
K.R. Balachandran and J. Ronen / Incentive contracts when production is subcontracted
172
s*(0, Q) and 6" (0, Q) refer to the optimal quantities and 0 = s* + 6" + 4. Note that s* and 6 " can be expressed as functions of w and Q. We will refer to ~p as the optimal split utility (OSU) of the agent. We use the symbol 0 to refer to both the price function and the amount as per the context. Few clarifications would be appropriate at this stage. First, consider the role of 6. 6 can have the same interpretation as the inverse of effort in the traditional principal-agent model. Lack of effort on the part of the manager translates immediately into inefficiency which increases cost. Therefore, for a given budget of 0 (the compensation from the principal), the agent can decide on how to sprit it between monetary compensation to himself and inefficiency (which requires less effort and alertness on his part - commodities to which he is averse). Thus, 6 endows the model with features of moral hazard. 6 differs, however, from the traditional concept of effort in that, in our scenarios, part of the compensation 0 can be consumed by A as 6 whereas no part of the compensation in existing agency models can be consumed as leisure. 6 plays an important role in this analysis as it is relevant to the production decision of the agent. The utility the agent derives from inefficiency affects the total cost of production o~Q + 6. Hence, the cost is not independent of 6. So, 6 affects both the transfer pricing function and the quantity of production. Moreover, unlike the traditional analysis whereby a utility function defined over wealth is assumed, the utility function induced in our case need not be separable in wealth and effort, as assumed in most of the existing literature. Note that the optimal s * and 6" are functions of 0 and Q, and ~k is not separable in s * and 6". (When the utility is non-separable, it is not clear whether the individual rationality constraint is satisfied by equality in the optimal solution; see Grossman and Hart, 1983.) The application of this problem is not restricted to the above situation of a principal and a relatively independent subcontractor. It can apply also to transfer pricing attached to internal transfers among divisions of the same corporation in (plausibly numerous) situations, whereby one division would fear that the knowledge of costs it reveals will leak to other divisions that might utilize the information strategically to compete with the communicating division. Thus, the scenario analyzed is one where communication is barred because of the costs it imposes on the agent. In the section below we examine the properties of the optimal split utility function ~k to facilitate further analysis.
3. Properties of ik(o~, O(Q), Q(60)) The proofs are provided in the appendix. The subscripts used below refer to partial derivatives. P1.
~k2 > 0, optimal split utility (OSU) is monotone increasing in the total payment 0 for fixed ~ and
Q. ~3 < 0, that is for a given 0 and co, OSU is monotone decreasing in Q. ~22 < 0. That is, the marginal utility decreases in compensation. lp23 > 0. This reflects the fact that as Q is increased for a given 0 and less money is left to split between s and 6 after paying ~ for the production costs and with utility that is concave in money, the agent moves back along the utility curve into a higher marginal utility point. P5. 1~32 -~" ~23 > 0. This indicates that an initial decrease in ~ caused by an increase in Q (~P3 ~< 0, as in P2) would be mitigated by a marginal increase in 0. P6. q~33 < 0 if Y"(Q) > 0, and ~2~(~, Q)/SQ2 >10.This is caused by the cumulative effect of both the disutility from production and the production costs being convex. P7. d2~/dQ z = 7 < 0 if and only if P2. P3. P4.
O"(O) <
U"(s*) U'(s*) (O'(Q) -~o(w, Q))Zs~(O, ~1 +
U'(s*)~QQ + r"(Q) U'(s*)
Property P7 means that the OSU is concave in Q so that there exists a unique optimal Q = Q* which maximizes the optimal split utility if the specific condition is satisfied. Assuming the condition is satisfied,
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this now allows us to address the determination of the optimal quantity by the agent. In other words, the property states that if the principal wishes to induce a unique quantity solution to be decided upon by the agent, there must be an upper bound on the slope of the marginal revenue to the agent. To interpret this upper bound, assume first, ~o, the marginal cost to be constant and ¥(Q) = 0 (no disutility in production), This causes the second term to become zero. Given that at optimum (as will be shown below) 0' = ~Q (the familiar marginal revenue = marginal cost condition), we see that P7 reduces to the condition that the marginal revenue should be downward sloping. Allowing ~Q to be nonconstant, we derive the familiar condition, that at optimal production, that the marginal revenue curve crosses the marginal cost curve from above. Further, for all Q, the degree to which 0 can be allowed to be convex increases with (1) absolute risk aversion, - U " / U ' , (2) the absolute value of the decrease in Q's marginal contribution to the agent's monetary payoff, -SQ, for a given marginal cost, ~O (since so = --SQ/~O), (3) the deviation between marginal revenue and marginal cost. Further, a more convex 0 becomes feasible, the greater the convexity of the disutility of production relative to the marginal utility in money for the agent and the higher the slope of the marginal cost curve. These conditions are intuitively plausible as they all contribute to the concavity of ~. We now proceed to state the general problem.
4. Analysis of the problem Condition (3) represents the first order condition of the agent. The agent chooses Q = Q* such that d~=0
dQ dk
V*o,
dO = +2(°~' o ( o * ) , Q*)O'(Q*) + +3(~0, O(Q*), O * ) = 0.
(5)
Hence O'(Q*)= - + 3 / + 2 > 0 since q~z > 0 by P1 and +3 < 0 by P2. Substituting for if2 and q'3, 0'(Q*)---~°(~°'O*)+
Y'(Q*) U'(s*) "
(6)
If Y(Q*) = 0, then Y'(Q*) = 0 and 0' = ~Q(~0, Q*). This restates the standard result that the optimal quantity for the agent is that at which his marginal revenue equals his marginal cost of production. However (6) implies that the optimum is achieved at a point where marginal revenue exceeds the marginal cost by a quantity of Y'(Q* ) / U ' ( s * ). This quantity should be interpreted as the additional dollars that are required to compensate the agent for the disutility of producing one additional unit. This addition is larger, the greater the marginal disutility from production and smaller, the greater is the marginal utility from compensation. We can get a lower bound for 0' by writing
O'(Q*)>~Q(~, Q*)
and
~o(~o, Q * )
s~ (~o, p * ) So* ,
which follows from P5 (see the proof). Then
O'(Q* ) >
s~" "
Note that s~ < 0 from the proof of P4 and sJ' > 0 from the proof of P3. Hence the right hand side is positive. The bound means that the pecuniary benefits the agent obtains from one more unit of Production (O's~') should exceed the pecuniary loss resulting from production of one more Q. For example, if s~' is
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$0.20 (i.e., for a marginal dollar of revenue the pecuniary benefit to the agent is $0.20) and if s~ is - $0.60 (i.e., marginal cash cost to the agent is $0.60 per unit), the agent needs at least $3 of marginal revenue ( 0 ' ) to break even in the pecuniary sense. The Q* so obtained is optimal and unique provided ~k is strictly concave. The condition for the concavity of ~p is given as P7 and can be used to obtain a condition for optimality. Note. There is a circularity here. For ~b to be concave in Q we need to assume conditions pertaining to the derivatives of the compensation functions, O(Q). Note, however, as the discussion under P7 shows that the observed characteristics of compensation functions do follow the patterns that we need. We admit that we may miss the optimal solution in some unusual cases with our analysis. Q* maximizes ~b (locally) if and only if
U"[Y' as*] U',QQ+Y" O" ( Q* ) <~- -U7 ~U aQ + u' '
(7)
where the right side of (7) is evaluated at Q = Q*. Note that (7) is equivalent to P7 at Q = Q* by substituting ds
Y'
dQ =s~'O'(Q*) + s~ =s~ U " using (6) and the property that ~Qs~' --- --s~. If Y = 0 , then O"(Q*)<~QQ(Q*) where Q = Q*. If, in addition, ~ is linear in Q, we require O"(Q*) < 0 for a local maximum. Note that if ~QQ(Q*) > 0 evaluated at Q = Q*, 0 " need not be negative. For a fixed function 0, Q* is a function of to. We wish to obtain the conditions under which Q*(~0) is monotone in to. Monotonicity implies a one to one correspondence between Q* and to and hence Q* is a global maximum if it satisfies the conditions for local maximum. Q *(to) is monotone increasing if Q *(to) = d Q *(to)/dto > 0 everywhere and monotone decreasing if the derivative is negative everywhere. Equation (6) can be written as follows (where all the terms are evaluated at O = Q*):
U'(s*)O'(O*) = U'(s*)~e(W, Q*) + Y'(Q*). Taking derivatives with respect to to implicitly, . ¢ . ~ O'U"(soO Q~,, +sQQ~ + s * ) + U t 0 Q~, Pt
t!
.
t
,
,
t!
Solving for Q*, -
= v"(s
v"s*
(0' -
O'
v ' o " - v'
Qe- r "
(8)
When (7) is satisfied, the denominator of (8) is negative. Then, Q* is positive everywhere or negative everywhere depending on whether (9) or (10) below is satisfied. Q*<0
**
U t t s ,~o ( 0 ' - ~ O ) <
U
t
~Q~.
(9)
Note that s,o < 0, which can be shown using arguments similar to those in the proof of P4. .
Q~ > 0
•
tr
,
!
U s,o(0'-~Q)>U~Q,o,
(10)
where all functions are evaluated at Q = Q*. Note that if Y - - 0 , 0 ' - ~ Q . = 0 at Q = Q* and (9) will always hold for ~Q,o > 0, at Q = Q*. If to is interpreted as c o s t / u n i t of production, then ~ -- to. Q and ~e~=l >0.
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We state the following as an assumption: A5. U"(s
*
* * )s*(~o, Q * ){O'(Q*)-~Q(,O, Q*)} 4: U / (s)~Q,o(w, Q ).
Q* satisfying (9) or (10) implies Q* is monotone in ~0 and thus unique for a given w. This means that if Q* satisfies the condition for a local maximum, it is a global maximum and P7 is satisfied for all Q. In addition, we can express ~0 = G(Q* ). The solution to the principal-agent problem is stated as Theorem 1. Theorem 1. Assume A1 to A5.
If the compensation function 0 further satisfies (7); then the optimal solution for 0 and Q to the principal-agent problem as stated in (1), (2) and (3) is characterized by -Po(Q, 0) (a)
~0(~o, 0, Q)
1 dzP a + y . dQ.2
1
dP [ G"(Q*) 1 df + + 7" dQ* [ G'(Q*) f dQ* (b)
f~/(¢o, O, Q)f(co) d~o - K>~ O,
(c)
0' =
1 - d7* y* dQ*
+
U"(s*) U
;r-zr
IS-'-)
S*
]
C(Q )], P
*
(11)
y' + -g
Proof. Provided in the appendix.
To help interpret (11) assume that Y = 0, i.e., the agent suffers no disutility from production. Then we obtain the following:
P'=a+
dP dQ*
d2p ] 0" ;li+ d]l
Of
40 This means that the ratio of marginal utilities which should be a constant under optimal risk-sharing (see Holmstrom, 1979) is a constant (a) plus an amount which varies in Q. The deviation from optimal risk sharing represented by the second term increases in the marginal utility of the principal (dP/dQ*, which could be zero if ~0 were known to the principal) multiplied by fQ/f which represents the degree to which ,0 can be inferred from Q. The deviation from optimal risk sharing also increases in relation to the rate at which the marginal cost to the principal O' increases. The deviation from optimal risk sharing however is inversely related to the sensitivity of the agent's utility to the principal's transfer payment 0, qJ0- The higher the sensitivity, the more capable is the principal to fine-tune 0 so as to induce a more favorable choice (from the principal's standpoint) of Q. Since the agent's utility function is not separable in s *, 3" and Q*, the reservation utility condition need not be binding at optimum. We proceed to find the condition under which the reservation utility restriction is binding at the optimum. Theorem 2.
(1) The reservation utility restriction is satisfied with equality, at optimum, if and only if
f[P2(Q*, 0)
dQ*dP U"(S*)so(O,_,Q.)]f(o~ ) Y
d~0 < 0.
(If Y - O, O' = ~Q. and since P2 < 0, this is always satisfied. If Y> 0, this condition is necessary.)
(12)
176
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(2) A sufficient but not necessary condition for (12) to hold is that the derioative of "r(Q) = R ( Q ) - O(Q), with respect to Q, eoaluated at the optimum Q = Q * be non-negatiue. Note. In the above theorem, ad (1): ~k is not separable in 0 and Y since both are functions of Q. Hence 0 cannot be reduced by a constant since that will imply reducing Y and this will change Q. Remark that Q* is a function of to. Proof. Provided in the appendix.
To understand Theorem 2, consider first the impact of perturbations in 0 on Q. As shown in the proof of the theorem in the appendix, Q is positive in e, where e multiplies constant decreases in 0 (as a perturbation). It means, that for a uniform decrease in 6 for every Q, the agent will increase the optimal Q that he wishes to produce. To understand this, consider the optimality condition (6). For a fixed Q, as 0 is decreased, so is s and hence U ' is increased. That is, the right hand side of 0' - ~Q. = Y ' / U ' is decreased relative to the left hand side. To restore equilibrium, Q must increase in order to reduce the left hand side. We shall rewrite (12) as follows: Note that =u"
ds
dQ*
= dU'
dQ* "
(See for the first equality the proofs of P4 and P5.) Also note that P2 = - P , Hence,
f P2f(to) do, = -E~(P). Substituting in (12), the condition becomes E,(P) +
dQ* dQ*/a
f ( t o ) d o > 0.
(13)
For a violation to occur, i.e., for the individual rationality constraint not to be met with equality, we need the following conditions: dr d-"Q < 0
and violation of (13).
Write violation of (13) as
[ d---~-/otJ f(to) dto > 1.
(14)
The term in the first bracket can be interpreted as follows: The numerator is the cost to the principal of increasing Q or equivalently the benefit of reducing Q. The denominator is the cost to the principal of increasing 0 or equivalently the benefit of decreasing 0 - (recall that d ' r / d Q < 0). Hence violation occurs if the benefit of decreasing Q (which entails increasing 6) is greater than the benefit of reducing 9. In other words, the principal has no incentive to reduce 8 until the reservation utility constraint is met with equality. The term in the second brackets can be interpreted as a weight factor which represents the proportion of the impact of Q on the agent's marginal utility in money to the impact of Q on the total marginal utility
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(of q~). Intuitively, the larger this proportion, i.e., the larger the relative impact of Q on the agent's utility derived from money, the easier the occurrence of the violation. We now proceed to illustrate these results with an example. Example. Let P ( Q , O(Q)) = d l n ( R ( Q ) - O ( Q ) ) , Y ( Q ) = c In Q,
U(s) = a in s(O, ~),
~(Q, ~0) = wQ,
V(8) = b In 8(0, ~),
f(oa) = 1,
i.e., o0 is uniformly distributed over the range 0 ~< ~0 ~< 1. We assume that J R ( Q ) - 0(Q)] > 0, so that In is well defined. Given O(Q), the agent allocates the compensation optimally between s and 8 to satisfy U ' ( s * ) = V'(8 * ), i.e., s * and 6 * satisfy 0 - - ~ + s * + 8 *, and a/(O - ~ - 6 * ) = b / 8 *. Solving, we obtain b(O-li)
8"(0,4)
a(O-4)
(a+b)'
(a+b)"
The agent's optimal split utility function tp(~o, O, Q) = U(s* ) + V(8* ) - Y ( Q )
1
{--~+ ~
[
+ b In
~-~ + ; ~
1
- c In Q,
(15)
we shall assume that O(Q) - wQ > 0, so that In is well defined. The agent chooses optimal Q = Q* given the compensation function O(Q) and after observing ~0 to satisfy (6). Substituting 4o = o0, Y '(Q *) = c / Q * and U ' ( s * ) = ( a + b )/[ O( Q * ) - oaQ*] into (6) yields 0'(Q*)=~o+
c[O(Q*)-ooQ*] (a+b)Q*
c [O(Q*) =~0+ (a+b-~ Q*
] oa .
(16)
The second term in (14) is (transfer price-cost) multiplied by a weighting factor. Solving for Q*, Q*(oa) =
cO(Q*) (a + b ) O ' ( Q * ) - (a + b - c)oa"
(17)
If Q*(,0) is monotone, then the function can be inverted and ~0 can be expressed as a function of Q*, say, oa = G(Q*). From (17), oa = G ( Q ) = (a + b ) O ' ( Q * ) Q * - cO(Q*) Q*(a+b-c)
(18)
The condition for the concavity of q, from P7 applied to this example gives O"(Q) < (0' - oa)2 (O-~oQ)
c(O - oaQ) QZ(a+b ) "
(19)
To assure local optimum of Q = Q*, the condition to be satisfied is obtained from (7) as c 0"(Q*) < (a+b)Q.
(
0'-
oa)
(O-~oQ*) Q.
(20)
The first term 0 ' - ~ 0 is the marginal profit to the agent and ( 0 - oaQ*)/Q* is the average profit. Formula (19) must hold for all Q in order to assure concavity of ~b and consequently to insure that Q* is the global maximum. Substituting for oa from (18), we see that (20) is equivalent to the following: 0"<
-
c(O-O'Q*) Q*2(a+b)
(21)
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In our search for 0(Q), we shall require (21) to hold. We shall now state the following lemma that paves the way for the existence of o: = G(Q* ), the inverse function of Q*(o:). Lemma. Q*(to) is (strictly) monotone in ~, provided a + b -~ c and
c(O-O'Q*) Q*Z(a+b)
8"
ProoL Differentiating Q*(to) given by (17) with respect to o: and substituting for o: from (18), we can obtain the following expression: dO__..~*=
do~
(a + b - c ) Q *z
(22)
Q*ZO" ( a + b ) + c( O - O'Q* ) "
If 0 " < - c ( O - O ' Q * ) / [ Q * 2 ( a + b ) ] , then the denominator in (22) is negative which implies that d Q * / d o : is negative for a + b > c and positive for a + b < c, On the other hand, 0 " > - c ( 0 - 0'Q * ) / [ Q * 2(a + b)] implies that the denominator of (22) is positive and in turn d Q * / d t o is positive for a + b > c and negative for a + b < c. In all cases, d Q * / d t o is strictly either everywhere positive or everywhere negative and the lemma is proved. [] We shall require, for the example, that (21) holds. Then if a + b > c, the derivative of Q* with respect to to is negative and it is positive if a + b < c. Using (17), we can easily show that for 8 - t o Q * > 0, a + b > c implies 0 - O'Q* > 0, and a + b < c implies 0 - O'Q* < 0. For a + b 4: c and (21) satisfied, the Q* obtained locally maximizes ~b and is unique. This implies that Q* is also globally maximizing. The solution to the principal-agent problem is characterized by substituting the following into (11).
(a + b)O" + c ( O - O ' Q * ) (a+b-c)Q*' (a+b-c)
C'(Q*) =
G"(Q*) =
~2
c(O"Q .2 - 2Q*O' + 20) ( a + b - c ) Q .3 '
(a+b)O'" (a+b-c)
(a+b-c) (O-O'Q*)'
(a+b-c) 2 ~22= - ( a + b ) ( O _ O , Q . ) 2 ,
( a + b - c ) [ ( a + b)O'Q* - col 1~32 -~"
P1 ~=
Q.(a+b)(O_O,Q,)2
dR' R(Q*)-O'
'
-d P2= R ( Q * ) - O '
d(O'-R')Q*2(a+b) ( R ( Q ) - O)[Q*2(a+ b)O" + c ( O - O'Q*)] '
T=(R(Q*)-O), d P" R ( Q * ) - O ' dP d dQ* = ( R ' - O ' ) ,
d2~
a+b-c O-O'Q*'
+o
d2p
dQ.2
dO" "r
G'(Q*)(a+b-c)
Y*=dQ2[Q=Q.= (a+b)(O-O'Q*) The solution is formally stated below.
"
d(O'-e') 2 ,r 2
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For the example with logarithmic utility functions and a + b ~ c, the optimal solution for 0 and Q to the principal-agent problem as stated in (1), (2) and (3) is characterized by the following: P~ +0
--
1 dzP - V* dQ .2
= a +
f[a
c
+
dP (a + b)Q* dQ*
In a(O-~oQ*) +b In b(O-~oQ*) a+b a+b
(23)
' c l n Q * ] d~o >~ K,
O' = o~ + c ( O - o~Q* )
(24) (25)
(a+b)Q* ' 0"<
c(O-O'Q*) Q*2(a+b)
For Theorem 2(2), we note that a sufficient condition for (24) to hold as equality at optimum is that dr(Q)___R,(Q)_0,(Q)>0 dQ
atQ=Q*.
Let R(Q) =pQ so that R'(Q) =p. If a + b > c, we get from (12),
O'(Q*)-o~
c( O- o~Q*) <~O - ~ o Q * (a+b)O* Q*
i.e., 0' < O/Q* < p. The last inequality follows from the assumption pQ * - 0 > 0. This assumption is necessary as otherwise the logarithm in the principal's utility function is not well defined. Thus for the case R(Q) =pQ and a + b > c, we have r'Q = R'(Q*) - O'(Q*) > 0 and the reservation utility constraint (24) is satisfied with equality. The intuition behind the result is that decreasing 0 by a constant decreases the right side of (25). Note that 0" is strictly negative from (22) for a + b > c and consequently, the agent will increase Q* to maintain the equality of (25). This certainly is beneficial to the principal also since r ' is positive. Thus Q* will approach the maximum production capacity of the agent's production facility constrained only by the fact that 0 - ¢0Q* > 0. Note that the disutility parameter c is less than (a + b) and Y"(Q) < 0 for this example. Note also that for any quantity Q, the transfer price is O(Q)/Q. The derivative of the transfer price with respect to Q evaluated at Q = Q* is given by (O'Q* - O)/Q .2 and is negative. That is, the transfer price is a decreasing function of the optimal quantity transferred. This is true for any revenue function R(Q). For a + b < c, the reservation utility constraint may not hold with equality even for the linear revenue function. However, we note that the derivative of the transfer price with respect to Q is positive when evaluated at Q = Q*. Thus for a + b ~< c, the transfer price is an increasing function of the optimal quantity transferred.
5. C o n c l u d i n g
remarks
We considered an agency problem in which the agent chooses both the quantity of output and the level of productive efficiency. There is information asymmetry in the sense that the agent observes the random variable which affects the production cost prior to making the production decision. Any communication of the observation to the principal is barred in the model. The utility of the agent consists of three parts, an increasing function of his monetary net income and the leisure afforded by inefficient production or consumption of perquisites, and a decreasing function of the output quantity. In Theorem 1 we characterized the optimal form of the compensation function.
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Decentralization is necessitated when the agent is endowed with superior information and when communication of the cost function is barred for fear of competition. An extreme form of decentralization is implied in the relation with a subcontractor with no intermediate market. A less extreme form is inherent in the case of interdivisional transfer within the same organization when divisions are reluctant to reveal their cost to headquarters and thus possibly to other competing divisions. In these cases, an alternative arrangement to compensating the agent on the basis of profits generated by transfer prices (as in Ronen and McKinney, 1970), would be paying the agent actual prices while the agent assumes the cost of production; i.e., he becomes a pseudosupplier: the two arrangements will produce identical results. Once the transfer price emerges as the only means of compensating because only the quantity transferred is observable by the principal, the question is whether the optimal transfer pricing entails a function that 'makes sense' from the standpoint of these characteristics. We do indeed find that the transfer pricing function is a downward sloping schedule reflecting a typical demand curve, just as might be observed in practice. In other words, the solution indicated is one that is consistent with what one observes in reality in general and that is consistent with the need for attaching compensation to quantity transferred because there is no other observable measure of performance. In principle, the resulting transfer price function can be subject to empirical testing. The degree to which the implications of the model is testable depends on the nature of the data to be gathered. At the very least, it appears that in cases where communication is barred from a business unit transferring goods to a receiving unit, we should observe, in accordance with the implications of the model, a transfer pricing scheme in operation and more specifically a sloping down transfer pricing scheme. Of course, if more data can be gathered such as the nature of uncertainty and preferences, the precise function that is impfied by the model can be tested against the actual scheme. In addition, we are able to specify in Theorem 2, the exact conditions under which the optimal solution will involve providing the agent with an expected utility above his reservation utility. In other words, conditions under which the individual rationality constraint is not met with equality. We also provide a simple intuitive explanation of these conditions. As might be expected, the constraint will not be satisfied with equality if and only if increasing quantity of production decreased the payoff to the principal and the benefit of decreasing quantity to him exceeds the benefit from decreasing the payoff to the agent. An example employing logarithmic utility functions is presented to illustrate the general solution. Appendix
Proofs of properties of Proof of P1.
O~b ~0
~k2 =
dU as* dV ~6" ds* ~ 0 + ~d8 ~0
U'(s*)[-~- +
(since U'(s*) = V ' ( 8 * ) )
= V'(s*)[ = U'(s*)
+ >0.
Proof of P2.
a,
~ 3 = -OQ =
aQ
+ - - ~ - Y'(Q)
= U'(s*)(-a~Q)_y,(Q) <0,
04 since-~>0
and Y ' ( Q ) > 0 .
K.R. Balachandran and J. Ronen / Incentive contracts when production is subcontracted
18l
Proof of P3.
3s* q,= = u " ( s * )
30 "
By assumption U " ( s * ) < 0 . We need to prove that 3s*/30> 0. We know that at optimal split U ' ( s * ) = V' (8) for a given 0. Now perturb 0 to 0 + A0. The corresponding perturbed values in s* and 8 * are s* + -as* ~--k0
and
8" +
A0.
To maintain optimal split, we should have U'[s* + 3s*/30. A0] = V ' [ 8 * + 3 8 * / 3 0 . A0]. Since U " < 0 and V " < 0, we must have that either both ~s*/30 and 3 8 " / 3 0 are positive or both are negative. But 3s*/30 + 3 8 * / 3 0 = 1 from the proof of P1. Hence ~s*/30 > O. Proof of P4. +2, = u " ( s *
) Os* OQ"
By assumption U " ( s * ) ~ O, and we need to show that 3s*/3Q < 0. Following arguments similar to those in the proof of P3, perturb Q to Q +AQ, U'[s* + 3s*/3Q.AQ] = v ' [ 8 * + 38*/~Q.AQ] and we must have the same sign for both 3s*/3Q and 3 8 " / 3 Q . but 3s*/3Q+ 3 8 " / ~ Q = -3~/~Q from the proof of P2. Hence 3s*/3Q < O. Proof of P5.
aO [_ [ ~o~ _ >]0 ~32= U ' ( s * ) as*
s i n c e -~~*- > O ,
-a~ ~>0,
U . <0.
~23 = ~32 follows provided the partial derivatives are continuous and ~ is in c 2. Proof of P6.
OQ [
-~
+ U'(s
- 002]
Q),
since U " < 0, 3s*/3Q < 0 and 3~/3Q > 0, the property follows. Proof of P7. V = ~ 2 2 ( 0 ' ) 2 + 1//230' "{'-+2 0,, -I- +32 O' q-" +33,
T < 0 ,~ 0 " < -~k=(O')2 - ~kz30' - q~320' - ~33
{-st(O')2u"
+ u " *
U2,
0" <
-s~'U"(O'-~Q) 2 ~ 0" <
u '
u"
y,, + -U r + lie°''
upon substituting for the partial derivatives of ~p and using P5.
Y"}
- O'U"
-
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K.R. Balachandran and J. Ronen / Incentive contracts when production is subcontracted
Proof of Theorem 1 For every choice of 0 made by the principal, the agent chooses an optimal Q* after observing o~. Thus any perturbation e?~ in 0 produces a perturbation in Q*. To obtain an expression for the change in Q*, perturb O(Q*) by e?t(Q*) in the agent's first order condition (5). We get qJ2{o~, O(Q* ) + cA(Q*), Q* } { O'(Q* ) + eA'(Q* ) } + ~b3(~o, O(Q* ) + cA(Q*), Q* } = O. Take the derivative with respect to e and set e equal to 0. Let Q = dQ*/de. (The arguments of ~b are suppressed.)
~b22( O'(Q* )0 + ?t(Q* ) ) O'(Q* ) + }23Q0'(Q* ) + ~2 ( O"(Q* )Q + ~t'(Q* )) d-~33Q=O
+tP32{O'(Q*)()+~t(Q*)}
V)k, 02.
Solving for Q and setting 3'
d2~ dQ 2
~p22(0'(Q*)) 2 + q,230'(Q * ) + +20"(Q*) + ~b320'(Q* ) + 1P33,
O= _ ()t(Q*)[q.'z20'(Q*) ++32] + ~t'(Q*)~b2}
(A.1)
7 The principal maximizes the Lagrangian function
L = S P ( Q * ( w ) , O(Q*(w)))f(w)d~o
+a[S+(~o, O(Q*(¢o)), Q*(w)}f(¢o)d~o- K],
where Q* is chosen to satisfy the agent's first order condition (6). To optimize we perturb O(Q*) by eX(Q*). The Lagrangian function is now written as
L~= f p( e.(,o), o(e.(,o)) + ~x(e* (~)))z(~)
d~o
+.{f~.[~, o(o_*(,0))+,X(Q*(~)), e:(~)] f(~)d¢o-
K}.
Taking derivatives of L~ with respect to e and setting e to O, we get the condition for optimality. (Arguments of P are suppressed.)
f[( e,o + p,(o'(o*)o + x(o* )) ) + .{ ,,(o'(e* )o + x(e* )) + ,,o }] f(~) VX.
d~o = 0 (A.2)
Substitute for 0 from (A.1) into (A.2) and express the optimality condition as f X ( Q * ) $ ( , , , 0, 0', 0", Q*) d,o + f X ' ( Q * ) ~ ( , o , 0, 0', 0", Q*) d~o=0 where
= - ( P1 + P20'(Q *)+ a(~b20' + q o ) ) f ( ¢ ° ) - ~ = - (P1 + P20'(Q *)} f(c°)+z, y
V)t,
(A.3)
K.R. BalachandranandJ. Ronen / Incentivecontractswhenproductionis'subcontracted
183
with use of the fact that ~b2O'(Q* ) + @3 = 0 from (5). = (P2 + a@z)f(to) + • (@220'(Q*) + @32)
@2
Assume that (9) or (10) is satisfied so that to can be expressed as a function of Q*. Let to = G(Q*) so that dto = G'(Q*) dQ*. Substituting for to in (A.3), the optimality condition can be written as
fX(Q* )$(G(Q* ), o, o', o", Q* )G'(Q* ) dQ* + fx'(Q*)O(G(Q*), o, o', o", Q*)G'(Q*)dQ*=O
vx.
Integrating the second expression by parts and noting that X(Q*) = 0 evaluated at the boundary values of Q*, the condition becomes,
fx(Q*)l ~ G ' ( Q * ) - - ~ GdO( Q
, )-OG"(Q*) / dQ*=0
VX,
suppressing the arguments of • and ~ for convenience. The optimality condition is given by dO
~G'(Q* ) = ~ - ; a
,.
(Q* ) + OG"(Q* ).
Substituting for ~ in terms of • we get,
G' ( Q* )[ dd~Q* - f( G( Q* ))( P2 + a@2) - O( @220' + @32) ] + OG'' ( Q* ) =
(A.4)
We proceed to simplify this expression further. Note that • = - d P/dQ * f ( G (Q * )) @2/y, where d P/dQ
*
= (P~ + P20'). Also, in all the functions Y, f , @, and their partial derivatives, to has been replaced by allows us to differentiate these functions with respect to Q*. Thus,
dO [ d P , ~ dv d2P OQ* = [ d ~ * 2 J d Q -;- - 7f@2 d - ~ 2
d P d@2 f't d Q * dQ*
G(Q* ). This fact
]±
dP d/ 7d--Q~@2 d~Q-;-, y 2'
where d@2 dQ*
=@220'+@23+@21G'(Q*)
and
,,
.
•
,
.
@21G'(Q*)=U (s )saG (Q ).
Substituting these derivatives into (A.3) and simplifying, the theorem follows, were we have substituted, P, = - P2 and @0 = @2 for • = R - 0, the principal's share after paying 0 to the agent.
Proof of Theorem 2 Consider a compensation function 0(Q) and the corresponding agent's optimal selection Q = Q*. Let X ~< 0; /£ = fq~(to, t~, ~)(e))f(to) dto, where ~) is the agent's optimal Q corresponding to t~. Note that 0 ( Q ) < 0(Q) for e > 0, X < 0 and every Q. We need to find conditions under which [d/£/de]l~= 0 < 0 and [d/3/de]l~=0 > 0; i.e., we wish to obtain conditions under which the change in agent's utility with a reduction in compensation is negative and the corresponding change in the principal's
K, = f@(to, O, Q*)f(to) dto. Consider O(Q) = O(Q(e)) + eX, where e >~ 0 and
184
K.R. Balachandran and J. Ronen / Incentive contracts when production is subcontracted
utility is positive. Under these conditions, the reservation utility must be satisfied with equality at optimum.
d/~ de
e=0
= f(,.(o'0
+ x) + ¢ , 0
We obtain 0 = dQ(e) de by substituting h' = 0 in (A.1).
d/~ de e=0
: s{'+.-,I+..,'+ +..)I. +.,'++.)),,.),° = < 0
d~
(since
~k20'+ ~k3 = 0
from (5))
for all X < 0, since @2 ~" 0, by P1.
Hence the agent's utility always decreases when the compensation O(Q) is decreased by a constant. Now consider the principal's utility. dP de e=o
-'~)---'~-So(O'-I~o)}f(~o ) d~o evaluated at (2 = Q*, where we have substituted P1 + P2~' = d P / d Q and for ~22 and q~23 from P3 and P5. Given X < O, the necessary and sufficient condition for [d/3/de] I e=0 > 0 is P2
dQ - -3's o ( O ' - f ~ O )
f(w)do~
This proves (1). Note that P2 < 0 by assumption, U " < 0 by assumption, 3, < 0 by restricting the choice of 0 such that 0 " satisfies P7 at Q = Q*. s o > 0 as shown in the proof of P3, and O' - ~Q > 0 from (6). Hence to prove (2) a sufficient condition for (12) to hold is
dP Q=Q* dQ > O, Note that d P / d ' r > O
dP Q=Q* dP d.c Q=Q*" dQ - d~ d o
by the choice of the utility function. Hence the sufficient condition is
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K.R. Balachandran and J. Ronen / Incentive contracts when production is subcontracted
185
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