Cost-plus-fixed-fee contracts with payment ceilings: impact on commercial markets and indirect cost recoveries

NORTH - ItOI./AND

Cost-plus-Fixed-Fee Contracts with Payment Ceilings: Impact on Commercial Markets and Indirect Cost Recoveries Martin P. Loeb and Krishnamurthy Surysekar

We consider a monopolist selling one product to a commercial market and a related, but distinct, product to the government. In the absence of sales to the government, the usual welfare loss associated with too little quantity being sold at too high a price arises. Using an agency model, the welfare consequences of using a cost-plus contract for procurement are examined. We show that such a contract exacerbates the welfare loss in the commercial market due to cost shifting motivated by cost allocation. We then turn attention to the use of a payment ceiling in connection with a cost-plus contract. Here we show that the payment ceiling reduces the welfare loss in the commercial market by motivating the monopolist both to increase commercial output and to lower the price. Additionally, the use of a payment ceiling is seen to reduce the moral hazard problem associated with cost-plus contracts. © 1997 Elsevier Science Inc.

I. I n t r o d u c t i o n Public policy makers have long been concerned with the welfare loss associated with monopoly. This welfare loss arises because an unregulated monopolist sets price above marginal cost and, thereby, restricts output below the socially optimal level. A different public policy concern centers around the structure of cost-based contracts used in procuring customized products for the government. These two policy issues interact when the government contracts with a producer having monopoly power in the

Address correspondence to: P r o f e s s o r Martin P. Loeb, Department of Accounting, College of Business and Management, University of Maryland, College Park, Maryland 20742, USA. Journal of Accounting and Public Policy, 16, 245 269 (1997) © 1997 Elsevier Science Inc. 655 Avenue of the Americas, New York, NY 10010

0278-4254/97/$17.00 PII S0278-4254(97)00001-X

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commercial market. In this case, the allocation of indirect costs between government and commercial work affects not only the payment by the government and the profits of the contractor, but also the welfare of consumers of the commercial product. In the context of cost-plus contracts, we model the welfare effects of indirect cost allocation on consumers of the commercial product, and analyze the benefits of amending a cost-plus contract with a payment ceiling. Although the issues of procurement contract design, cost allocation, and payment ceilings have received substantial examination in the literature, the model presented here is unique in bringing all three of these issues together. Examples of studies which have used agency models to study the optimal design of procurement contracts include McAfee and McMillan (1986), Cohen et al. (1992), Reichelstein (1992), and Loeb and Surysekar (1994). Cost allocation is one of the recurring issues in the accounting literature, and studies by Rogerson (1992) and Cohen and Loeb (1989) have discussed cost allocation in connection with government contracting. Rogerson (1992, p. 681) has demonstrated that cost allocation provides incentives for the supplier to be inefficient in the use of inputs. In spite of the inefficiencies caused by cost allocations, Cohen and l o e b (1989, pp. 175-177) showed conditions under which the government would be better off using a cost-based contract incorporating cost allocation rather than using a fixed price contract and avoiding the allocation of costs. Sweeney (1982), Braeutigam and Panzar (1989), and Brennan (1990) discussed distortions caused by cost allocations in the context of utility regulation. Issues associated with a payment ceiling--a predetermined maximum payment to a contractor, have received little analysis in the literature. Reichelstein (1992, p. 725) and Laffont and Tirole (1993, pp. 75-76) briefly discussed such ceilings in the context of procurement contracting without cost allocation, l o e b and Surysekar (1996) provide a more complete analysis of payment ceilings, but their model, too, does not incorporate cost allocations. The interaction between cost allocation and payment ceiling is studied here in the context of a cost-plus contract. A cost-plus-fixed-fee contract calls for the purchaser to reimburse the supplier for the actual costs of fulfilling the contract plus a fixed fee. In a report analyzing contract awards, the U.S. Department of Defense (1993, p. 10) indicated that approximately 14.56% of military procurement spending (for contracts above $25,000) for fiscal 1993 used cost-plus contracts. 1 This accounted for

i In addition to the 14.56% of contracts awarded cost-plus fixed fee basis, a n o t h e r 7.76% of all contracts over $25,000 were r e p o r t e d to be awarded as cost-plus-award-fee contracts. With a cost-plus-award-fee contract, the contractor also receives full r e i m b u r s e m e n t for expenses; however the plus may be partly based on the contractor's p e r f o r m a n c e in such areas as quality and timeliness. See section 16.404-2, Title 48 of the Code of Federal Regulations for a complete description of cost-plus-award-fee contracts.

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$16.345 billion. Although the government uses many types of p r o c u r e m e n t contracts, 2 cost-plus contracts are studied here because of both their simplicity and widespread use. However, by limiting our analysis to costplus-fixed-fee contracts, we cannot hope to find the optimal contracting arrangement. Payment ceilings are a c o m m o n feature of many cost-based procurement contracts and affect the welfare properties of these contracts. As evidence of the widespread use of payment ceilings, note that such ceilings are m a n d a t e d for cost-reimbursement contracts with the U.S. government. Section 16.301.1 of Title 48 of the Code of Federal Regulations (1995), known as the Federal Acquisition Regulations, or F A R , states: Cost-reimbursement types of contracts provide for payment of allowable incurred costs, to the extent prescribed in the contract. These contracts establish an estimate of total cost for the purpose of obligating funds and establishing a ceiling that the contractor may not exceed (except at its own risk) without the approval of the contracting officer. (Code of Federal Regulations 1995, Title 48, section 16.301.1)

In our paper, a model is considered in which a risk-neutral purchaser (the government) uses a cost-plus contract to secure a fixed quantity of goods from a sole-source supplier. The supplier faces no uncertainty, and so issues of the supplier's risk aversion do not arise. In addition to the product that is sold to the government, the supplier has a monopoly in the sale of a related product in the commercial market. The direct costs of fulfilling the government contract are assumed equal to the project's intrinsic cost less the supplier's cost-reducing effort. The total cost of producing for the government and the outside market consists of the sum of direct costs plus indirect costs. For expositional ease, we assume all indirect costs are fixed. Any cost-based contract must provide for the allocation of indirect costs between the government output and the output produced for the commercial market. Given any predetermined cost allocation scheme, the supplier must determine both the level of output to sell in the commercial m a r k e t and the level of cost-reducing effort. Although both the government and the supplier can observe the ex post cost of the project, only the supplier knows the project's intrinsic cost and the costreducing level of effort supplied. This effort is costly to the supplier, but as it may not be observed, the purchaser cannot directly reimburse the cost of effort.

2 See Bedingfield and Rosen (1985, pp. 2-4-2-7) for a description of the many p r o c u r e m e n t contracts used by the g o v e r n m e n t .

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With a cost-plus-fixed-fee contract without payment ceiling, we show that the contractor never supplies any productive effort. Because the government picks up some of the firm's fixed indirect costs, the supplier earns positive (incremental) profits on the contract in excess of the contract's fixed fee. The awarding of the government contract also distorts the supplier's output decision in the commercial market. The supplier, having monopoly power in the outside market, produces less for this market in order to shift more indirect costs to the government. In this setting, which does not allow for communication, we show that the cost-plus contract with payment ceiling sometimes increases, and never decreases, the welfare of consumers of the firm's commercial product, while sometimes increasing and never decreasing the supplier's effort aimed at reducing the direct costs of the government contract. As the purchaser will be assumed risk-neutral and the supplier faces no uncertainty, these effects of a payment ceiling are not due to improved risk-sharing. The adoption of a payment ceiling by the government will affect the welfare of the supplier, the consumers of the commercial product, and other consumers who are affected through both the benefits of the project and the taxes collected (or borrowing) to finance the project. However, the use of a payment ceiling will generally affect the welfare of the parties differently. We show that the welfare of consumers (of both the project and commercial output) would be expected to increase by the (optimal) selection of a payment ceiling, while the addition of a payment ceiling would decrease, or at best leave constant, the supplier's welfare(profits). Although we do show how payment ceilings affect the welfare of individual parties, we do not make interpersonal utility comparisons. Thus, we cannot make conclusive statements about the effect on overall social welfare. 3 The remainder of our paper is organized as follows. Section 2 introduces and analyzes the model without payment ceiling. In Section 3, we analyze the effect of payment ceilings. A summary is given in Section 4. In Appendix A, we present a simple numerical example to illustrate our analysis. Formal proofs of the propositions appear in Appendix B.

2. The Cost-Plus Contract without Ceiling Consider a risk-neutral purchaser, the government, who awards a cost-plus contract to a supplier. This is a sole source contract, as, by assumption, the supplier has the unique technological capability to carry out the specified project. The project is for a specified quantity of output, G. Because the quantity of output to be produced for the government is specified exoge-

3 However, in Appendix A, we present an example which shows that when the social welfare function places equal weights on all three sets of interested parties, a p a y m e n t ceiling could lead to an increase in welfare.

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nously, by appropriately defining units, we can, without loss of generality, set G = 1, if the project is awarded, and set G = 0 , if there is no production for the government. The supplier is assumed to have monopoly power in the market for a related product sold in the commercial market. Denote Q to be the number of units of the related product sold in this outside market. The ex post accounting cost of the project is composed of the project's direct cost plus an allocated share of the firm's indirect cost. The direct cost of the project is given by b - e, where b ~ [B 1, B 2] represents the project's intrinsic (direct) cost, and e E [0, E] is the level of cost-reducing effort expended by the supplier. The intrinsic cost is assumed to be known only to the supplier, but the government has a subjective probability distribution over [B1, B2], denoted by F(b) with positive density f(b). The purchaser cannot observe e ~ [0, E], the level of cost-reducing effort expended by the supplier. We assume that this effort itself is costly to the supplier, the dollar cost denoted by ~(e). However, the supplier cannot charge 0(e) to the project. Of course, one could assume that any chargeable effort is embedded in b, the intrinsic cost. We make the standard assumptions about the disutility function, namely 0 ' ( e ) > 0, ~b"(e) > 0, for all e, $(0) = 0, and 0'(0) = 0. Although the government does not know the individual values of b and e, the government is assumed to be able to measure the realized (ex post) direct costs, b - e. This formulation is similar to the Cohen and Loeb (1989, pp. 170-178, 1990, pp. 407-411) models which extend the McAfee and McMillan (1986, pp. 327-330) framework to include indirect costs. A distinguishing feature of our model is that effort affects only the direct cost of the government project. In addition to the direct costs of the government project, the supplier incurs direct costs of producing and selling Q units in the commercial market, denoted C(Q), plus indirect costs. The cost function, C(Q), is assumed to be increasing and convex 4, i.e., C'(Q) > 0 and C"(Q) > 0. In order to define the ex post accounting cost of the project, the government and the supplier must agree on a cost allocation method. Because the government output is fixed, the indirect costs allocated will depend on quantity produced for the commercial market. We let a(Q) denote the fraction of the indirect costs allocated to the government contract. We assume a(Q) is differentiable, 0 < or(Q) < 1 and ot'(Q) < 0, indicating that the more which is produced for the outside market, the more indirect

4 Although the assumption that the direct costs are convex rules out the possibility of economies of scale, this assumption is not strictly needed. We merely require that profits, excluding any cost allocations, are strictly concave. That is, we require R ( Q ) - C ( Q ) to be strictly concave, where R ( Q ) denotes the revenue function.

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costs are allocated to that market. 5 As an example of a cost allocation scheme which is commonly used, consider a ( Q ) = G / G + Q. As the products produced for the government and the outside market will differ, an input measure, such as direct labor hours used, is commonly employed to measure heterogenous outputs. When this is the case, as shown in Rogerson (1992, p. 681), cost based contracts will motivate some input distortions. In order to focus on output distortions rather than input distortions, we assume that G and Q are measured in units which are independent of the inputs. Because we are concerned with the effects of imposing a payment ceiling, and not on the selection of the cost allocation rule, we take a ( Q ) as exogenously given. The common or indirect costs are denoted by J. These costs generally vary with the size of the government project and with the quantity produced for the external market, reflecting variable as well as fixed indirect costs. In order to simplify the exposition, we assume that J is constant, i.e., that the indirect costs are all fixed for the period under consideration. This may be due to excess fixed capacity, e.g., rent for a building which is larger than what is needed for the production of just the commercial output. Rogerson (1994, p. 72) has noted that a 1985 U.S. Department of Defense study (U.S. Department of Defense 1985) showed that divisions dealing with defense contracting derived 17.2% of their business from the commercial market and 82.8% of their business from the government. Thus, much of the contractor's fixed capacity is incurred in the anticipation of business with the government. In order to maintain the long-run viability of the defense industry, the government must cover not only the direct costs, but a share of the fixed capacity costs. 6 For any specified cost allocation rule, a ( Q ) , the accounting cost of the government project, denoted by A, can now be written: A = b - e + a(Q)J.

(1)

With a cost-plus-fixed-fee contract without a payment ceiling, the supplier receives A + K, where K > 0 denotes the fixed fee. In order to analyze

5 O f c o u r s e this a s s u m e s G = 1, i.e., t h e c o n t r a c t is a w a r d e d . I f G = 0, t h e n a ( Q ) =- 0. F u r t h e r m o r e , if a l l o c a t e d cost w e r e a f u n c t i o n o f p r a c t i c a l , i n s t e a d o f actual, capacity, a n d , if p r a c t i c a l c a p a c i t y w e r e a s s u m e d fixed in t h e s h o r t run, t h e n c t ' ( Q ) = 0 for all Q. W i t h s u c h an allocation s c h e m e , t h e a w a r d i n g o f t h e c o n t r a c t (with o r w i t h o u t p a y m e n t ceilin3) w o u l d h a v e no e f f e c t o n s u p p l i e r ' s c h o i c e o f level o f c o m m e r c i a l o u t p u t . T h e s i t u a t i o n is t h u s a n a l o g o u s to o n e in w h i c h t h e r e is n o a l l o c a t i o n o f i n d i r e c t costs. 6 A l t h o u g h t h e a s s u m p t i o n t h a t i n d i r e c t c o s t s a r e all fixed g r e a t l y s i m p l i f i e s t h e analysis, it is n o t crucial. W h a t is r e q u i r e d in t h e a n a l y s i s w h i c h follows is t h a t : (i) [1 - ct(Q)]J(Q, G) is i n c r e a s i n g in Q, i.e., t h e a m o u n t o f i n d i r e c t costs b o r n e by the firm i n c r e a s e s as p r o d u c t i o n for the o u t s i d e m a r k e t i n c r e a s e s , a n d (ii) J(Q, G) - J(Q, O) < ct(Q)J(Q, G), i.e., t h e i n c r e a s e in the i n d i r e c t costs d u e to t h e g o v e r n m e n t p r o j e c t is less t h a n t h e i n d i r e c t cost w h i c h is a l l o c a t e d to t h e project. C o n d i t i o n (i) will h o l d as l o n g as J(Q, G) is n o n d e c r e a s i n g in Q. C o n d i t i o n (ii) holds for m a n y r e a s o n a b l e cost a l l o c a t i o n s a n d J f u n c t i o n s in w h i c h t h e r e a r e e c o n o m i e s o f scope. F o r e x a m p l e , c o n d i t i o n (ii) h o l d s w h e n J ( Q , G ) = v/Q + G a n d a ( Q ) = G / ( G + Q).

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the contractor's problem, we must also consider the commercial market. We assume that the supplier has monopoly power in this market and faces the revenue curve R(Q), where R"(Q) < 0 and, in the relevant portion (where demand is elastic), R ' ( Q ) > 0. In the absence of the government contract, the supplier's problem is written as: m a x R ( Q ) - C ( Q ) - J. Q

(2)

Thus, the optimal quantity, QM, is characterized by the usual marginal revenue equals marginal cost first-order condition, R'(QM) = C'(Q~t). As the monopolist is assumed to face a downward sloping demand curve, price is set above marginal cost, output is restricted, and the dead-weight loss of monopoly arises. For the case of no government contract, the monopolist sets optimal effort, ~ , equal to zero. 7 If the supplier is awarded the cost-plus contract by the government, the supplier's problem becomes: maxR(Q) + K-

C ( Q ) - [1 - a ( Q ) ] J - ~0(e).

(3)

Q,e

Because the realized direct costs of the project are reimbursed by the government, b - e is both a revenue and a cost of the supplier, and thus does not appear in the maximand. From equation (3), we see that there is no gain to the supplier by incurring effort: benefits of effort accrue to the government, while the disutility of effort remains as a non-reimbursed cost. Thus, the supplier's optimal effort, ~1, is still zero. The optimal quantity of commercial output, Ql, is characterized by the first-order condition: R'(0I) = C'(01)-

a'((~,)J.

(4)

As a ' ( Q ) J < 0, the firm's effective marginal cost in the commercial market is shifted upward, resulting in reduced sales to the outside market, i.e, Q1 < QM- Hence, the effect of the government contract is to cause the supplier to reduce output below the already reduced monopoly level, QM, thereby exacerbating any welfare loss in the commercial market. With the cost-plus-fixed-fee contracL the supplier always earns positive rents from obtaining the contract. As Q1 maximizes R(Q) - C(Q) - [1 a(Q)]J, and as a(Q) > 0: R(01)

d- g -

C(01)

-

[1 -

og(0l)]J

> R(OM)

-- C(OM)

-- J,

(5)

where K >_ 0. The fixed fee, K, would be determined by negotiations between the contractor and the government. Because we have a case of 7As the monopolist's effort affects only the direct costs of fulfilling the government contract in our model, the monopolist's profits, expression (2), would be more completely written as R ( Q ) C ( Q ) - J - tp(e).

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bilateral monopoly for this government project, K is not endogenously determined in our model. Note that the strict inequality in (5) holds even if K is set equal to zero. Thus, as Rogerson (1994, p. 72) has stated, "even if prices on sole source defense contracts were set exactly equal to accounting costs, defense firms could still earn economic profits by shifting overhead between products."

3. The Cost-Plus Contract with Payment Ceiling In the remainder of the paper, we consider the case when payment under the contract is subject to an overall ceiling. Let T > K denote the payment ceiling, so that the payment to the supplier is given by Pc = min{A + K, T}, where A, defined by equation (1), represents the accounting cost of the project. 8 Determination of the constants T, K and the allocation rule a ( Q ) fully specify the contract. Thus, the payment to the supplier generally depends on the fixed fee, the payment ceiling, the specified allocation rule, the supplier's cost reducing effort and the supplier's commercial output decision, as well as the project's intrinsic cost and the size of the common cost. In deciding whether or not to accept a given contract, the supplier uses backwards induction. The supplier calculates the optimal commercial output and effort which would be selected if the contract were to be accepted. The supplier then compares the profits when accepting the contract and setting commercial output and effort at these optimal levels with the profits of rejecting the contract. Recall that the supplier faces no uncertainty at the time the decision of whether or not to accept the contract is made. Given that the project is undertaken, the accounting costs are realized and the government l~ays the supplier the lesser of T or b - ec + a(O.c)J + K, where ec and Qc denote, respectively, the optimal levels of effort and commercial output for the contract with ceiling. We saw earlier that, in the absence of a payment ceiling, the supplier will decrease output in the commercial market in order to shift more indirect costs to the government contract. We will now see when a payment ceiling motivates the contractor to reduce the curtailment of commercial output and to increase effort aimed at reducing the direct costs of the government contract. Before turning to the supplier's problem, we make an additional assumption guaranteeing that the relative size of the supplier's indirect costs will be substantial. Letting eF• denote the first-best level of effort, the level of effort which maximizes e - O(e), we assume Ot(OM)J > O(eFB) -K. That is, we assume that the indirect costs which would be allocated to

8 N o t e t h a t if T _< K, t h e n the c o n t r a c t w o u l d be a fixed price contract.

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the government, when the supplier provides the quantity of commercial output which maximizes profits in the absence of the government project, are greater than the disutility costs of supplying the first-best level of effort less the fixed fee. This assumption is made solely for expositional ease. 9 3.1 The Supplier's Problem

We now present our main result: Proposition 1, which characterizes how the level of the payment ceiling, T, influences the supplier's decision for a given fixed fee, K, and allocation rule, a(Q).

Proposition 1. The imposition of a payment ceiling on a cost-plus contract will result in the supplier weakly increasing the level of cost-reducing effort and the level of commercial output for all values of the ceiling. Furthermore, there exist some values of the ceiling for which the levels of cost reducing effort and commercial output are strictly greater than the levels chosen when the contract contains no ceiling. In particular: (a) The supplier accepts the project if and only if T > b - eFB + ~O(eFB), i.e, if and only if the payment ceiling is at least as big as the incremental costs of the project conditioned on taking the first-best level of cost-reducing effort, eFB. For T < b - eFB + ~O(evB), the range where the project is rejected, the supplier puts forth no effort and increases commercial output to the monopoly level, QM. (b) If the payment ceiling is set above the cutoff level, but at a level where the accounting costs of the project plus the fixed fee would be greater than the ceiling if the supplier were to produce the monopoly level for the commercial market and supply the first-best level of effort, i.e., if b - eFB + qt(eFB) < T < b + a(QM)J -- eFR + K, then the supplier will increase commercial output to the monopoly level, O.M, and put forth the first-best level of effort, eFB. (c) Ifthe payment ceiling is set so that b + a(QM)'J -- eVB + K < T < b + a(Q1)J + K, then (i) the supplier will select a commercial output level and effort level so that the ceiling binds exactly, i.e.: T=b

+ o l ( O _ c ) J - ~ c + K;

(6)

(ii) Q, < Qc -< QM; (iii) if b > T, then ~c > O; and (iv) if 0c = O, then

Oc>O.t.

(d) If the payment ceiling is set so that T > b + o~(Q1)J + K, then Q.c = Q_t and ec = O. 9Suppose ~O(eFB) were sufficiently large so that 0 < a ( Q . M ) J ~ qJ(eFB) -- K ; then there exists a level of effort, ~ ~ (0, eFB], such that a ( Q M ) J = qJ(~) -- K . The proofs which follow could be modified by replacing eFB with ~, for the cases where a ( Q M ) J < ~ ( e F B ) -- K .

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The intuition behind this result may best be explained by referring to Figures 1 and 2, where the regions (a), (b), (c) and (d) refer to the levels of the payment ceiling (from the supplier's viewpoint) or the levels of intrinsic cost (from the government's viewpoint) which match up with the four parts of the proposition. The supplier's incremental cost of the project equals the project's intrinsic cost less the cost-reducing effort plus the disutility of providing that effort plus the reduction in profits from the commercial market due to a change in the commercial output level. Thus, the incremental cost of the project is at least as great as b - ec + ~(ec)- Hence, the incremental profits from the project are less than or equal to T - b + e c - ~ b ( e c ) < T - b + e•B -- ~,(eFB). Therefore, in region (a), where T < b - eFB + q , ( e F B ) , the project will be rejected. If the project is rejected, clearly the supplier will not put forth any effort associated with the government project and will set the commercial output at the monopoly level. Given the assumption that a ( f ~ M ) J > ~b(eFB) - - K , the supplier is (at least, weakly) better off accepting a government project when the ceiling payment T > b - eFB + ~ b ( e F B ) . T o see this, note that if the supplier were to accept the contract, produce Qu for the commercial market, and supply the first-best level of effort, the only effect of accepting the contract would

project rejected

project

no effort decision Q=Q.

M

acce

=~ C

C

FB

(~C = (2M

p ted

>0

OC -> (~1

C

c

=0

O-c

>0

or

O. > 01 O

T1 region(a)

T2 region (b)

T3 region (c)

paymentceiling region (d)

Figure 1. Effect of payment ceiling on cost-reducing effort and commercial output level for a given level of intrinsic cost; where T 1 = b - eeB + ~ ( e e B ) ; T 2 = b eFB + o~(QM)J + K ; T 3 = b + ot(01)J + K; eFB > 0; Q~ > Qr

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project accepted

C

project rejected

=0

C

C

Qc ->01 C

=~

->0

FB

0 c = O_M

no effort decision Q = 0M

>0

or

0-c > 0"1

O

b3

region (d)

b2

region (c)

bl

region (b)

intrinsiccost

region (a)

Figure 2. Effect of intrinsic cost on cost-reducing effort and commercial output level for a given payment ceiling; where b 1 = T + e F e - ~O(eFe); b 2 = T + eFB -ot(OM)J K ; b 3 = T - a ( Q 1 ) J - K; eFB > 0; QM > QI. -

-

be to increase revenues by at least b - eFB + q J ( e r n ) , while increasing costs by b - eFB + ~O(erB). Although the supplier's disutility of effort is not directly reimbursed by the government, the government covers this cost through the share of indirect costs and the fixed fee. Next consider region (b) where the ceiling is still low, but not so low as to cause the supplier to reject the contract. In this range, the costs of increasing the commercial quantity and the effort so that the ceiling will not bind are excessive, and the supplier treats the contract as a fixed-price contract. Hence, output for the commercial market will be set at the monopoly level and the effort will be first-best. Now, suppose the ceiling were raised to a level where it would strictly bind if the supplier were to produce at the Q1 level and provide no cost-reducing effort, but would not bind (or at most, weakly bind) if the supplier were to increase commercial output to the no-contract level, QM, and increase effort to the first-best level. For a ceiling in this range, (range (c)), the above proposition shows that the supplier would increase output (from the Q1 level) a n d / o r effort and, therefore, bring down the accounting cost of the government contract so that the ceiling just binds. To better understand the intuition behind case (c), consider the supplier's situation at an output quantity less than the monopoly quantity, QM, and an effort level less than the first-best level, and where the accounting cost

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plus the fixed fee is greater than the ceiling. For this quantity and effort, a small increase in the commercial output increases profits in that market and has no effect on revenues from the government contract. Similarly, a small increase in effort will reduce the supplier's costs by more than the increase in the SUl~plier's disutility of effort without affecting revenues. Thus, the supplier will always choose an effort level and a commercial output level so that the accounting cost plus the fixed fee is less than or equal to the ceiling. However, if the accounting cost plus the fixed fee is less than the ceiling, then the supplier could always do better by a small reduction in effort (as all the benefits accrue solely to the government); thus, effort would be set at zero. By assumption, the ceiling would bind if the commercial output were set at Q1 and effort were set at zero. Thus, the supplier would have to select an output level greater than Q1 in order to shift indirect costs away from the government contract, so that the ceiling would not bind. However, because the supplier's profit function, unconstrained by any payment ceiling, decreases as the commercial output moves away from Q1, any quantity o f commercial output for which the sum of the accounting costs of the government project plus the fixed fee are strictly less than the ceiling, could always be improved upon by increasing the commercial output slightly. Therefore, at the optimal commercial output and effort level, the sum of the accounting costs and the fixed fee cannot be strictly less than the ceiling. Hence, the ceiling exactly binds. As the sum of the accounting costs of the project plus the fixed fee would be greater than the ceiling if effort were set at zero and commercial output were set at ihe no-ceiling maximizing quantity, Q1, it must be the case that effort is positive, output is greater than Q1, or both. A sufficient, though not necessary, condition of effort to be positive for a ceiling in range (c) is that b > T, the intrinsic cost is greater than the payment ceiling. We also note that the commercial quantity selected by the supplier would always lie in the interval [Q1, QM]- If the quantity selected were greater than QM, a reduction in quantity, while leaving the effort level unchanged, would keep the government payment the same (although the accounting cost plus the fixed fee would then be greater than the ceiling) and increase profits in the commercial market. Similarly, when the supplier cuts the commercial output level below Q1, the indirect costs allocated to the government increase, but as the contract exactly binds in region (c), effort increases leading to a decrease in the direct cost of the contract. However, due to the nature of Q1, the supplier suffers a higher loss of commercial market contribution than this level of increased benefit. Hence, the supplier would never set commercial output below Q~. Finally, in region (d), we see that when the payment ceiling is set sufficiently high, the supplier's decisions are unaffected by the ceiling. Hence, the contractor puts forth no cost-reducing effort and restricts commercial output below the monopoly output level.

Fee Contracts with Payment Ceilings

257

The proposition given above shows that there exist some values of the ceiling for which the levels of cost-reducing effort and commercial output are strictly greater than the levels chosen when the contract contains no ceiling. Further, the ceiling never induces levels of cost-reducing effort and commercial output below the no-ceiling levels. Because the government does not know the value of b, the imposition of a payment ceiling would result in higher expected values of the supplier's commercial output level and effort level. Given that the supplier is assumed to face a downward sloping demand curve in the commercial market, the higher expected quantity produced for that market implies a lower expected price. Hence, the addition of a payment ceiling would result in an expected increase in consumers' welfare. Note from Proposition 1, that although the addition of a payment ceiling to a cost-plus contract may result in an increase in the commercial output level from Qi, the quantity selected without the ceiling, the supplierwould never increase the commercial output beyond the monopoly level, QM) °

3.2 The Government's Choice of Optimal Level of Ceiling We now consider the government's problem of selecting the optimal payment ceiling for an exogenously-determined cost allocation rule, c~(Q). In order to analyze the government's design problem, we must make some additional assumptions. We assume that the government, acting as the principal, is the first mover and designs a contract which the supplier must either accept or reject. We also assume that the government's level of expected benefits from the project is exogenously specified, and is denoted by H. The level H is assumed to reflect the benefit, gross of costs, to consumers of the project, exclusive of the welfare effects in the related commercial market. For tractability, we assume that H is independent of Q, the output in the commercial market. The exogenous specification of the expected benefits to the purchaser follows the model of Baron and Besanko (1987, p. 515). 11 The government is assumed to maximize welfare of consumers of the project less expected costs, and this quantity will be referred to as government's welfare. A final assumption is that the government knows o~(Q1).J, the indirect costs which the government would absorb under the cost-plus contract without ceiling. Note that Q1 depends

10/~tS our analysis is limited to cost-plus-fixed-fee contracts, we cannot exclude the very real possibility that some other type of contract (with or without a ceiling) could induce the supplier to set c o m m e r c i a l output at a level exceeding QM. It The exogenous specification of the expected benefits to the purchaser is also employed in footnote 17 of Reichelstein (1992, p. 719).

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M.P. Loeb and K. Surysekar

on R(.), C(-), and a(-), but not on b. Although these assumptions are restrictive, we still believe the following analysis of the choice of payment ceiling level provides additional insight. The government uses backwards induction in calculating the optimal payment ceiling. Using the analysis summarized in the first proposition, the government determines how the supplier will respond to any level of the payment ceiling as a function of the project's intrinsic direct costs, b (known only to the contractor). We saw that the contractor's optimal decision rule was to accept a project whenever b < T + erB - - ~ b ( e F B ) , and that the government will pay exactly the ceiling for accepted contracts for a range of values of b (i.e., whenever T - a ( Q 1 ) J - K < b < T + eF8 -qJ(eFB)). This is because in this range, the supplier, in order to reap the benefits of having the government pick up some of the indirect costs, increases output in the commercial market and effort so that the accounting costs plus the fixed fee just equals the payment ceiling. If the ceiling were set at the expected net benefits, H, there would a range where the government's expected net benefits would be zero. By reducing the ceiling slightly below H, the government increases the range where it earns positive net benefits, even though it increases the probability that the project will not be accepted. The larger the portion of indirect costs which would have been allocated to the government using the contract without ceiling, the larger is the range where the accounting cost of the project plus the fixed fee will be just equal to the payment ceiling. Thus, the larger the allocated indirect cost at the no-ceiling level of commercial output, the smaller will be the optimal payment ceiling. Our second proposition characterizes the government's decision rule for selecting the payment ceiling.

Proposition 2. The optimal level o f the p a y m e n t ceiling is less than the expected gross benefits to the purchaser, i.e., T < H. Furthermore, the larger the indirect costs which would have been allocated to the government under the no-ceiling contract, the smaller will be the optimal p a y m e n t ceiling.

As the government selects the ceiling so as to maximize expected net benefits, and as Proposition 2 shows that the optimal ceiling is finite, the government's welfare (expected net benefits) increases with the implementation of the ceiling. The increase in the government's welfare is due in part to the benefits of avoiding high-cost projects. An additional benefit of the payment ceiling is that, for some values of the intrinsic cost, the ceiling lowers the economic cost of projects which are accepted, by motivating the supplier to put forth more productive effort, a n d / o r lowers accounting costs by motivating the supplier to increase commercial output and shift indirect costs away from the government. Thus, the expected net welfare of

Fee Contracts with Payment Ceilings

259

consumers of the project (government's welfare) and the expected welfare of consumers of the commercial market will increase with the adoption of the optimal ceiling. However, because the supplier's profits may decrease, and can never increase, relative to the no-ceiling case, we cannot make unambiguous statements about total social welfare. The numerical example, given in Appendix A, indicates that when equal weights are put on the welfare of consumers and the supplier, the adoption of a payment ceiling may raise a measure of total welfare.

4. Summary The welfare losses associated with monopoly have been seen to be aggravated when the government awards the monopolist a cost-plus contract for a related product. This comes about because the monopolist cuts back on the commercial output in order to allocate additional indirect costs to the government. We saw that using a payment ceiling in connection with a cost-plus contract helps to alleviate the tendency of cost-plus contracts to motivate reductions in output (below the already diminished monopoly level). Additionally, the imposition of a payment ceiling helps reduce the usual moral hazard problem associated with cost-plus contracts. The monopolist benefits from the government cost-plus contract through the fixed fee and through the allocation of indirect costs which, in the absence of the contract, would have to be charged against commercial product revenue. Because of the payment ceiling, for high cost projects, the monopolist may not be able to benefit from allocating more costs to the government contract by reducing the output level in the commercial market (below the monopoly level). Furthermore, for high cost projects, the payment ceiling converts the cost-plus contract to a near fixed-price contract and, thereby, handles the moral hazard problem. We have seen that the purchaser's expected welfare increases when an optimal payment ceiling is added to a cost-plus contract. The introduction of the ceiling motivates the supplier to reject some projects which would not be beneficial to the purchaser, but would have proceeded in the absence of the ceiling. The ceiling further benefits the purchaser by motivating the supplier to increase productive effort for some cost parameters (and never leads to a decrease). Our simple model enabled us to study the welfare effects of introducing payment ceilings into cost-plus-fixed-fee contracts. However, a number of limitations of our analysis should be highlighted. Although cost-plus-fixedfee contracts are widely used, we did not examine when such contracts are optimal, nor did we search for the optimal contract over the space of all contracts. In a similar vein, we treated cost allocations as exogenously determined, and we did not search for an optimal allocation. In addition, although we assumed that the government faced uncertainty concerning

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M.P. Loeb and K. Surysekar

the direct costs of the project, when we turned to the examination of the optimal ceiling, we made the assumption that the government knew the level of indirect costs which would be allocated to the project in the absence of a payment ceiling. Yet another limitation of our analysis resulted from the fact that the supplier faced no uncertainty in our model and that the government was assumed to be risk-neutral, so that issues of optimal risk-sharing did not arise.

Appendix A: A Numerical Example In this appendix, we provide a simple numerical example which illustrates how the payment ceiling affects various groups. In particular, we will calculate the level of payment ceiling selected by the government, the supplier's equilibrium level of cost-reducing effort, commercial output level, the profits of the supplier, the consumer surplus in the commercial market, and the residual benefits of consumers of the government project. The (inverse) demand curve for the commercial output is assumed to be p ( Q ) = 6 - Q. The direct cost of producing Q units for the commercial market is given by C ( Q ) = Q2/2; the indirect cost of production is given by J = 3, and the exogenously specified fraction of indirect costs which is allocated to the government contract is a ( Q ) = 1/(1 + Q). The government's level of expected benefits from the project is given by H = 5. The government's prior distribution over the intrinsic cost of the project, b, is assumed to be uniformly distributed over [0.5, 6.5]. The supplier's disutility for cost-reducing effort is given by ~O(e) = e2/2, so that the first-best level of effort, eFB , equals 1. Finally, we assume that the reservation utility level of the supplier is 0, and the fixed fee, K, is set equal to 0.

Supplier's Problem Case 1. N o G o v e r n m e n t C o n t r a c t

In this case, the supplier, acting as a monopolist, maximizes profits in the commercial market. Q( = QM) is chosen to maximize the supplier's utility given by: 0 2

R(Q) - C(Q) -J

= Q(6 - Q)

2

3,

QM = 2; the price is 6 - 2 = 4, and the supplier's utility at the optimal QM is 3. The consumer surplus at this level can be computed and is equal to 0.5(2)(6 - 4) = 2.

Fee Contracts with Payment Ceilings

261

2. G o v e r n m e n t C o n t r a c t with N o Ceiling

The supplier, accepting the government contract, maximizes total profits from the commercial market, as well as the government contract. The decision variables are the commercial market output, and the level of cost-reducing effort on the contract. For a given level of commercial output Q, the supplier's payment from the contract is b - e + a ( Q ) J . T h e direct cost of the contract is b - e, and the disutility of effort is qJ(e). The supplier's outside market contribution is R ( Q ) - C ( Q ) . Therefore, the supplier chooses Q( = Q1) and e( = 81) to maximize utility given by: R(Q) - C(Q) - J + b - e + a(Q)J - (b-e)qJ(e) Q2 3 ez =Q(6Q)----3 + (b-e) + - (b-e)-2 I+Q 2 81 = 0 , and Q1 = 1 . 8 7 9 < Q M = 2. The commercial output price is 6 - 1.879 = 4.121, and the supplier's utility at (Ql, el) = 4.020 > 3 (the supplier's profits in the no governmentcontract case (1) above). For a given level of intrinsic cost b, the government's net benefits are: = H - payment to supplier; =H-

[ L b - - 8 1 + a ( 0 )1 J ] = 5 -

[

b-0+

1+11879

=3.958-b.

Therefore, the government's expected net benefits are: 6.5 + 0 . 5 ) = 3.958 - E [ b ] = 3.958 ~ = 0.458. We can now compute the consumer surplus from commercial output as 0.5(6 - 4.121)(1.879) = 1.765 < 2 (the surplus in the no government-contract case (1) above). Thus, the ability to shift indirect costs to the government contract without limits leads to a reduction in commercial market output, and a loss of consumer surplus in that market. Comparing the no government-contract case and the no-ceiling government-contract case, we can observe that: a) the loss of consumer surplus is 2 - 1.765 = 0.235; b) the increase in the supplier's utility is 4.021)- 3 = 1.020, and c) the increase in the government's expected net benefits is 0.458. C a s e 3. G o v e r n m e n t C o n t r a c t with Ceiling A. The Optimal Ceiling

First, we calculate allocated indirect costs a ( Q ) J for Q = Q1 = 1.879 and Q = 0 M = 2. Thus, ot(OM)J 3 / ( 1 + 2) = 1, and a ( Q 1 ) J = 3 / ( 1 + =

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M.P. Loeb and K. Surysekar

1.879) = 1.042. Next, to d e t e r m i n e the optimal p a y m e n t ceiling, 7~, we use (B21) and the a s s u m e d u n i f o r m distribution, to get:

-

-~+1 o5 db ^ 1 iT-1.04e" -6- + (3 -- T ) ~ = 0,

which simplifies to 7~ = 3.458 < H = 5.

B. Supplier's Choice of (O.c, ec) for Different Realizations of b F o r this example, the cutoff values of b c o r r e s p o n d i n g to those given in Figure 2 are b I = 3.958, b 2 = 3.458, and b 3 = 2.416. (i) Let b = 4 (b is in region (a) of Figure 2). F r o m Proposition l(a), the contract is rejected by the supplier, and the supplier's utility = 3 (see (1) above). T h e g o v e r n m e n t ' s net benefits are 0, and c o n s u m e r surplus is 2. N o t e that w h e n b = 4 and the no-ceiling contract is used, the g o v e r n m e n t ' s net benefits are - . 0 4 2 . (ii) Let b = 3.5 (b is in ~gion (b) of Figure 2). Now, f r o m Proposition l(b), supplier chooses Qc = QM = 2, and ~c = CuB 1, and the supplier's utility is given by: =

R(Q.M) - C(O.M) -- J + T -

(b - eF.) - tP(eFn) = 3.458 < 4.02

(the supplier's utility in the no-ceiling case (2) above). Because the ceiling binds, the supplier is paid T = 3.458, and the g o v e r n m e n t ' s net benefits are: H-

7~ = 5 -

3.458 = 1 . 5 4 2 > 0.458

(the g o v e r n m e n t ' s net benefits in the no-ceiling case). Because the m o n o p o l y level o f commercial o u t p u t is provided, the c o n s u m e r surplus in the c o m m e r c i a l m a r k e t is 2. C o m p a r e d to the no-ceiling contract, a) the supplier is worse off by 4.02 - 3.458 = 0.562; b) the c o n s u m e r s o f the c o m m e r c i a l p r o d u c t are better off by 2 - 1.765 = 0.235, and c) the g o v e r n m e n t is better off by 1.542 - 0.458 = 1.084. (iii) Let b = 3 (b is in region (c) of Fig~ure 2). F r o m Proposition l(c), the p a y m e n t ceiling binds. H e n c e , b + a(Qc)J - e c = T, which implies:

a(O_c)J - 3, = 7~ - b = 3.458 - 3 = 0.458. Thus, the supplier maximizes: Q(6-Q)-

Q2 ~--J+

7~ -

(b-e)-

e2 ~-,

subject to a ( Q ) J - e = 0.458.

Fee Contracts with Payment Ceilings

263

S e t t i n g a(Q)l/1 + Q, J = 3 , 7~ = 3 . 4 5 8 , b = 3 , a n d ~ O ( e ) = e 2 / 2 , t h e n u m e r i c a l s o l u t i o n to the a b o v e p r o b l e m is Q¢ = 1.949, ec = 0.559. Thus, t h e s u p p l i e r ' s utility is 3.857, a n d t h e c o n s u m e r s u r p l u s in t h e c o m m e r c i a l m a r k e t is 1.901. T h e g o v e r n m e n t ' s n e t benefits a r e 1.542, w h e r e a s t h e y w o u l d b e .958 using the c o n t r a c t w i t h o u t ceiling. (iv) Let b = 0.5 (b is in region (d) of Figure 2). F r o m P r o p o s i t i o n l(d), the s u p p l i e r sets Qc = Q1 = 1.879, a n d ec = 0. T h e s u p p l i e r ' s utility w o u l d b e t h e s a m e as for t h e n o - c e i l i n g c o n t r a c t , i.e., 4.020 (case 2). H o w e v e r , the g o v e r n m e n t ' s n e t benefits w o u l d be: H-

(b + o t ( Q 1 ) J ) =

5 - 1.542 = 3.458.

T h e s e a r e the s a m e n e t benefits which t h e g o v e r n m e n t w o u l d r e c e i v e with t h e c o n t r a c t w i t h o u t ceiling. F u r t h e r m o r e , the c o n s u m e r surplus in t h e c o m m e r c i a l m a r k e t w o u l d b e 1.765. O u r results for this n u m e r i c a l e x a m p l e a r e s u m m a r i z e d in T a b l e 1 below. F o r t h e v a l u e s o f t h e p a r a m e t e r s chosen, T a b l e 1 shows t h a t t h e g o v e r n m e n t is n e v e r w o r s e off a n d is s o m e t i m e s b e t t e r off with a c o n t r a c t with ceiling; c o m m e r c i a l m a r k e t c o n s u m e r s a r e n e v e r w o r s e off a n d a r e s o m e t i m e s b e t t e r off w h e n t h e p r o d u c e r o f c o m m e r c i a l o u t p u t d o e s n o t p r o d u c e for t h e g o v e r n m e n t p r o j e c t , a n d t h e s u p p l i e r is n e v e r w o r s e off a n d is s o m e t i m e s b e t t e r off with a c o n t r a c t w i t h o u t p a y m e n t ceiling. If t h e social w e l f a r e f u n c t i o n w e r e to w e i g h t t h e utilities o f t h e s e t h r e e g r o u p s

Table 1. Welfare of the Government, Suppliers, and Commercial Market

Consumers under Different Scenarios

Scenario

Government's net benefits

Consumer surplus (commercial)

Supplier's utility

Sum of columns 3, 4, and 5

Intrinsic cost: b = 4.0

No contract No-ceiling contract Contract with ceiling

0 - .042 0

2 1.765 2

3 4.02 3

5 5.743 5

Intrinsic cost: b = 3.5

No contract No-ceiling contract Contract with ceiling

0

2 1.765 2

3 4.02 3.458

5 6.243 7

Intrinsic cost: b = 3.0

No contract No-ceiling contract Contract with ceiling

0 .958 1.542

2 1.765 1.901

3 4.02 3.857

5 6.743 7.3

Intrinsic cost: b = 0.5

No contract No-ceiling contract Contract with ceiling

0 3.458 3.458

2 1.765 1.765

3 4.02 4.02

5 9.243 9.243

.458 1.542

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M.P. Loeb and K. Surysekar

equally, social welfare would increase with the adoption of a payment ceiling for low intrinsic cost environments (b = .5, 3.0, and 3.5).

Appendix B: Proof of Propositions PROOF OF PROPOSITION 1:

We prove the four cases individually.

To show necessity, suppose the supplier accepts the contract setting commercial output at Q c and cost-reducing effort at ~c. If the payment ceiling does not bind, then C a s e a:

R(Oc)

- c(Oc)

> R(Qc)

+ r-

~ + ec - J - ~ )

- C(O_.c) - (1 - o z ( O _ . c ) ) J -

tb(ec) + K.

(B1)

The profits of the supplier would be R(O~c) - C ( O . c ) + T - b + e c - J ~b(~c), if the ceiling binds, and R ( ( ~ c ) - C(O_.c) - (1 - c ~ ( Q c ) ) J - ~P(ec) + K, if the ceiling does not bind. As the supplier accepted the contract, profits with the contract must be at least as great as R(O~M) -- C(O~M) -- J, profits without the contract. Thus, from equation (B1) we see that whether or not the payment ceiling binds, we have: R(O_c) - C(Qc) + T-

b + e c, - J - ~0(e c) > R ( Q . M ) - C ( O . M ) -- J. (B2)

Inequality (B2) may be rewritten as:

r >_ [ R ( 0 ~ )

- c(0~)]

- [R(0c)

- c(0~)]

+ b - ~ec

-

q'(~c)).

(B3) Because (~M maximizes R ( Q ) - C ( Q ) , equation (B3) yields: T >_ b - (~c - ~b(~c)).

(B4)

A s eVB maximizes e -- q,(e), we have: b - ( e c - ~ b ( e c ) ) > b - (eFB -- ~ ( e F B ) ) .

(B5)

Combining equation s(B4) and (B5), we have the desired result: (B6)

T > b - (eFB -- ~ ( e F B ) ) .

Sufficiency follows by noting that if the supplier were to set Qc = QM and ec = eFl~, then T > b - eFB + ~ ( e F B ) implies: R(Q.M)

-- C ( Q . M ) + T -

b + eF8 -- J -

> R ( Q . M ) -- C ( O . M ) -- J ,

qJ(eF8)

(B7)

where RHS of equation (B7) represent the (maximum) profits, given that the project is rejected. As the optimal output and effort levels, given

Fee Contracts with Payment Ceilings

265

acceptance, must yield profits at least as high as LHS of equation (B7), the supplier's profits will be at least as great by accepting the project. ^ Clearly, if the project is rejected, no effort is taken and the supplier sets O c = OM > Ol .

[]

Case b: Suppose b - eFB + ~(eFB) <_ T < b + a ( Q . M ) J -- eFB + K, and suppose the supplier chooses (QM, eFB2" T h e n the profits of the supplier (net of the cost of effort), denoted I I ( Q M, evB) , a r e : II(O-M,eFB) =R(O-M)--C(O-M)

+ T--b

+eFB--J--

O(eFB).

(B8)

Now let Q >_ 0 and e >_ 0 be an arbitrary^choice of commercial output level and cost reducing effort distinct from ( Q u , CFB). If b - e + a ( Q ) J + K < T, then the supplier's profits would be: I I ( O , e ) = R ( Q ) - C ( O ) - (1 - a ( O ) ) J
C(Q) + T-b

+e-J-

- tp(e) + K

$(e).

(B9)

If b - e + o t ( Q ) J + K > T, then the supplier's profits would be: II(Q, e) = R ( Q ) - C ( Q ) + T - b + e - J - O ( e ) .

(B10)

As QM maximizes R ( Q ) - C ( Q ) , for all Q > 0, and as eFB maximizes e -- ~O(e) for all e > 0, RHS of equation (B8) is at least as large as RHS of equations (B9) and (B10). Thus, H ( Q M, eFs) > II(Q, e); that is, (Q.M, eFe) is optimal for payment ceilings in the given range. [] Case c:

(i) We will show that the ceiling binds using^proof by contradiction.

First, we consider the case where b + a ( Q c ) J - ec + K < T and then the case where b + a ( o - c ) J - ec + K > T. Suppose b + a ( o - c ) J - ec + K < T. If ec > 0, there would exist ~ ( 0 , ec), such that b + a ( o - c ) J - ~ + K < T. At ((~c,e) the supplier is better off than at ( Q o e c ) by 0(~ c) - 0(e) > 0. Thus, if b + o t ( o - c ) J - ec + K < T, then ec = 0, so b + c~(o-c)J + K < T. A s b + a ( Q l ) J + K > T, it must be the case that O-c > O-1. But_by continuity of a(Q), there exists L9 ~ (QA, Q c ) such that b + c d Q ) J + K < T. However, as R ( Q ) - C ( Q ) - (1 - a ( Q ) ) J is decreasing in Q >_ Q1, we have: ^

R(~9) - C(Lg) - (1 - o~(Lg))J + K > R ( O - c ) - C ( O - c ) - (1 - a ( o - c ) ) J )

+ K

(Bll)

Thus, (O-c, ec) is not o p t i m a l - - a contradiction. Hence, it must be the case that b + a ( Q c ) J - ec + K > T.

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M.P. Loeb and K. Surysekar

Now suppose b + a ( Q c ) J - ec + K > T. maximizer of R ( Q ) - C(Q), we have:

OM is the unique

As

R(Q.M) - C(Q_.M) + r - b + ec - J - ~O(ec) > n((~c) -C(Qc)

Thus, if b + a ( Q c ) g - e c

+ T-b

+ec-

J-

then Qc O t ( O M ) J - ec + K > T. Because, by assumption, range where b + ot(OM)J - - e F B -[- K < T, it must ec < eFg" But then there exists ~ ~ (~C, gFB), o t ( Q g ) J -- e + K > T. As e - ~O(e) is increasing have: +K>

T,

qs(ec). (B12) = Q M , so b + we are in the be the case that such that b + in e < eFB, we

R(O~M) - C ( ( 2 M ) + T - b + ~ - J - O ( e )

(B13) Thus, (Qc, ec) is not optimal--a contradiction. Hence, T = b + > R(Q_c) - C ( Q c )

+ T-

b + ec - J - ih(~c).

a ( ( ~ c ) J - ec + K.

(ii) We first show that Q1 < Qc. Suppose (Qc, ec) was optimal and suppose Qc < Q1. From (i) we know that the ceiling binds at this point, therefore: (B14) Define el = b + o~(Q1)J + K - T. As Q1 > Q_.c, we have a ( Q l ) J < a(O.c)J, so that e I < ec and el > 0, as T < b + a ( ( ~ l ) J + K. As the payment ceiling binds for both (Qc, ec) and (Q1, el), the profits of the supplier (net of disutility of effort) at these points are: ec = b + a ( Q c ) J

n(0c,~c)

+ K-

r.

= R ( Q c ) - C ( O . c ) - (1 - a(Q_.c))J -

q'(~c)

(B15)

+ K,

and n(0,,e,)

= R(01)

- C(O,)

- (1 -

(B16) As Q1 maximizes R ( Q ) - C ( Q ) - (1 - a ( Q ) ) g and as ~0(~c) > ~b(el), we have H ( Q c , & c) < II(Ql, e 0. This contradicts the assumption that (Qc, e c ) i s optimal, so Qc > 01We now show that Qc_< 0Mx Suppose (Qc, ~c)__wasop timal and suppose Qc > QM. Let Q = ( Q c + Q M ) / 2 . As Q < Q o T < b + a ( Q ) J - ec + K, so that the ceiling remains binding and the revenues from the government remain at T. But at (Q, ~c), the profits from the outside market are greater than at (Qc, ec), as Q is closer than Qc to QM. Because the supplier has the same disutility of effort at both points, (Q, 0c) leaves the supplier better off, thus contradicting the assumption that (Qc, ec) was optimal. Hence, - 0 ( e 1) + K.

QM ~ Qc.

Fee Contracts with Payment Ceilings

267

(iii) A s T = b + a ( Q . c ) J - ec + K , b > T implies ec > 0. (iv) As T = b + f f ( O . c ) J - ec + K and T < b + a ( Q 1 ) J + K , if ec = 0, then Qc > Q1- [] Case d: Since Q1 and ~1 = 0 are optimal for the contract without ceiling, we have for all Q > 0 and e > 0:

R(Q1) - C ( Q , ) - (1 - a ( O l ) ) J

+ K

> R ( Q ) - C ( Q ) - (1 - a ( Q ) ) J

- @(e) + K .

(B17)

Furthermore, if the payment ceiling binds for any Q and e, we have: (B18)

T < b - e + ot(Q)J + K

and, therefore, for such Q and e, we have: R ( Q ) - C ( Q ) - (1 - a ( Q ) ) J > R(Q) - C(Q) + T-

- tp(e) + K

b + e - J - ~0(e).

Hence, Qc = Q1 and ec = el = 0 are optimal.

(B19)

[]

From the first proposition, we know that the supplier rejects the contract for all values of b > T + eFB - - O ( e F B ) , that the supplier is paid b + a ( Q 1 ) J + K when the ceiling does not bind, and that the ceiling binds whenever b > T - o l ( Q 1 ) J - K. Therefore, the purchaser's expected utility with the ceiling is given by: 12 PROOF OF PROPOSITION 2:

Eur(r) = L T - ° ( O ' ) ' K ( , - , - b -

-(0,),

- Kls( )

B1

+ LT+e~R--T--,,(O,)JO(eFB)K ( H - T ) f ( b )

(B20)

dO.

For any exogenous level of fixed fee, we can derive the optimal payment ceiling, denoted T, by the following first-order conditions: 13 ^

= -ff+eF~-q'(er")y(b)db

+ (H-

7~)y('F + e r B -

O(erB) ) =0.

(B21)

aT-a(Q1)J-K

12 F r o m e q u a t i o n (B20), w e s e e t h a t as EUe(T) is d e c r e a s i n g in K f o r all v a l u e s o f T, if the p u r c h a s e r h a d all t h e b a r g a i n i n g p o w e r , t h e p u r c h a s e r w o u l d o p t i m a l l y s e t t h e fixed f e e at 0 ( a s s u m i n g t h a t the fixed f e e m u s t b e n o n - n e g a t i v e ) . 13 W e a s s u m e t h a t t h e p r o b a b i l i t y d i s t r i b u t i o n , t h e disutility o f e f f o r t f u n c t i o n a n d t h e cost allocation method, together, satisfy the corresponding second-order condition: f(7 ~ - a(QI)J

- K ) + ( H - 7~)f'(7~ + e¢ B - 0(e,~B))

2 f ( 7 ~ + eFB

tP(eFB)) < O.

268

M.P. Loeb and K. Surysekar

This implies:

F ( 7 " - a(O_.,)J- K ) -

F ( T + eF8 -- ~O(eF.) )

+ ( H - f ~ ) f ( T + eFB -- ~b(eFB) ) = O.

(B22)

As F ( ' ) is increasing and 7~ - a(Q1)J - K < T + eFB -- ~b(eBB), and as f ( T + eFB -- ~b(eFB)) > 0, e q u a t i o n (B22) yields H > 7~. F u r t h e r m o r e , one can easily s h o w f r o m e q u a t i o n (B22) and the s e c o n d - o r d e r condition that T / O ( a ( Q 1)J) < 0, i.e., as the costs allocated to the g o v e r n m e n t (without a ceiling) increase, the optimal ceiling decreases. []

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