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BEBR 
 
 FACULTY WORKING 
 PAPER NO. 1445 
 
 On the Optimality of Incentive Contracts 
 in the Presence of Joint Costs 
 
 Susan I. Cohen 
 Martin Loch 
 
 WORKING PAPER SERIES ON THE POLITICAL ECONOMY OF 
 INSTITUTIONS NO. 12 
 
 ,or ^ ry of the 
 
 -JUN • 4 1983 
 
 » ILLINOIS 
 
 ■ MOM 
 
 COMMERCE LIBRARY 
 
 College of Commerce and Business Administration 
 
 Bureau of Economic and Business Research 
 
 University of Illinois, Urbana-Champaign MAT ] 1 1988 
 
 UlNivcrtai I Y Or ill-iHUIS 
 URBANA-CHAMPAIGN 
 
BEBR 
 
 FACULTY WORKING PAPER NO. 1445 
 
 College of Commerce and Business Administration 
 
 University of Illinois at Urbana- Champaign 
 
 March 1988 
 
 WORKING PAPER SERIES ON THE POLITICAL ECONOMY OF INSTITUTIONS NO. 12 
 
 On the Optimality of Incentive Contracts 
 in the Presence of Joint Costs 
 
 Susan I. Cohen, Associate Professor 
 Department of Business Administration 
 
 Martin Loeb 
 University of Maryland 
 
ABSTRACT 
 
 Firms that supply goods to the government often produce these goods 
 in conjunction with other goods, incurring joint or common production 
 costs. When the government uses costs as a basis for contracting with 
 such firms, questions of cost allocation naturally arise. This paper 
 presents, in the context of a bidding model, some conditions under which 
 a fixed-price contract is optimal (in the class of linear contracts) for 
 the government. That is we give conditions under which the problem of 
 cost allocation is totally avoidable. 
 
Digitized by the Internet Archive 
 
 in 2011 with funding from 
 
 University of Illinois Urbana-Champaign 
 
 http://www.archive.org/details/onoptimalityofin1445cohe 
 
I. INTRODUCTION 
 
 Firms that supply goods to the government often produce these goods 
 in conjunction with other goods, incurring joint or common production 
 costs. When the government uses cost as a basis for contracting with 
 such firms, questions of cost allocation naturally arise. This paper 
 presents, in the context of a bidding model, some conditions under which 
 a fixed-price contract is optima] (in the class of linear contracts) for 
 the government. That is, we give conditions under which the government 
 is best off when the problem of cost allocation is totally avoidable. 
 
 McAfee and McMillan [1986], hereafter MM, analyze common linear 
 contracts in a principal-agent framework that handles problems of adverse 
 selection (selecting the firm with the lowest expected costs) and moral 
 hazard (motivating the winning firm to expend effort to reduce actual 
 costs). They compare fixed-price contracts, cost-plus contracts, and 
 incentive contracts. With a cost-plus contract the government agrees to 
 pay the contractor full costs plus a fixed fee (or fixed percentage of 
 actual costs) . Using a fixed-price contract the supplier agrees to 
 deliver goods or services for an amount agreed upon before actual costs 
 are realized. Finally, with an incentive contract the government agrees 
 to a certain price (full estimated costs plus a fee) and to share in any 
 deviation between actual and estimated costs. 
 
 MM first show that as long as there are two or more bidders a cost- 
 plus contract is inferior to a fixed-price or an incentive contract. MM 
 further demonstrate that for any finite number of bidders greater than or 
 equal to two, if bidders expect rivals to face different costs, the 
 government will be strictly better off with an incentive contract than 
 
with a fixed-price contract. The advantage of an incentive contract over 
 a fixed-price contract, even when all bidders are risk neutral, is due to 
 what MM refer to as a "bidding-competition effect" , and is associated 
 with the adverse selection problem of finding the low cost producer. 
 Laffont and Tirole [1987] and McAfee and McMillan [1987] show that when 
 all bidders are risk-neutral and when observed costs are as defined by 
 MM, the optimal contract is indeed linear in observed costs. 
 
 Using a generalization of the MM model with risk-neutral bidders, we 
 demonstrate that fixed-price contracts can strictly dominate incentive 
 contracts. The apparent contradiction between our result and that of MM 
 is due to the different interpretations of ex post observable cost. MM 
 assume that the government can write and enforce contracts based on 
 opportunity costs; i.e., they assume that ex post observable costs 
 represent opportunity costs. In coitrast, we assume that the government 
 can only contract on observed costs; these observed costs include an 
 allocated share of joint costs and differ from opportunity costs. This 
 distortion between observable costs and opportunity costs is related to 
 the effect of joint costs. Thus, the distortions caused by joint costs 
 may be sufficient to vitiate the bidding-competition advantage associated 
 with incentive contracts over fixed-price contracts. Note that McAfee 
 and McMillan remark that their results may well depend on accounting 
 costs being "...a close approximation to opportunity cost." [Footnote 16, 
 p. 335]. 
 
 In the next section, we present our model. In Section III, we 
 demonstrate conditions under which a fixed-price contract dominates an 
 incentive contract. We conclude with a brief summary in Section IV. 
 
II. THE MODEL 
 
 Consider the case in which n > 2 risk-neutral agents, indexed 
 i=l,2,..,n have the ability to produce a product or provide a service 
 specified by the government. Also, suppose that each agent currently 
 produces and sells another product or service for the private sector. 
 Let G be a measure of the output produced for the government, and let Q 
 be a measure of (physically distinct) output produced for sale in the 
 private sector. 
 
 Following MM, we employ an independent-private -values model (Milgrom 
 and Weber [1982]) in which agents differ only with respect to a 
 productivity parameter, V^ e [V*,V ]. The ith agent knows the value of 
 the parameter V^. All other agents j^i and the principal (the 
 government) know that V^ represents an independent draw from the 
 distribution of V^. We denote this distribution oy F(V) , and the 
 corresponding density by f (V) , where F(V*) = and F(V*) = 1. Agent i's 
 cost of producing Q and G is made up of separable cost in Q, Cq(Q) , 
 separable cost in G, C G (G) , joint cost, Cj(Q,G) and the agent's dollar 
 cost of supplying unobservable effort, h(e) . The h(e) function is twice 
 continuously differentiable , strictly increasing and strictly convex; 
 Cq(Q) is twice continuously differentiable, non-decreasing, and weakly 
 convex; and Cj(Q,G) is twice continuously differentiable and non- 
 decreasing in Q. The agent's unobservable effort level is assumed to 
 reduce the joint cost of production. Agent i's cost of producing Q and G 
 with effort level e is then: 
 
 Cq(Q) + C G (G) + (Vj + W - e)Cj(Q,G) + h(e) , (1) 
 
where W is a random cost factor that is common to all agents, is 
 independent of V, and is such that E[W] - 0. Since the agents are all 
 assumed to be risk-neutral, we can without loss of generality deal solely 
 with the expected value of (1) : 
 
 Cq(Q) + C G (G) + (Vi - e)Cj(Q,G) + h(e) . (2) 
 
 Purely fixed costs are assumed to be included in the joint cost term, so 
 that Cq(0)=C g (0)=0. 
 
 Suppose that the government uses a first-price, sealed bid auction 
 to award a contract for the production of a fixed quantity, G, of 
 (customized) goods. Let b represent the winning bid, and let a e [0,1] 
 represent the cost share ratio, and suppose that the contract calls for 
 the government to pay the successful bidder: 
 
 b + a(C*(Q,G) - b), (3) 
 
 where C (Q,G) is the bidder's (ex post) observable cost of meeting the 
 contract, defined below. When a = 0, the contract is a fixed-price 
 contract; when a > 0, the contract is an incentive contract. 
 
 In order to implement this incentive contract, the government and 
 bidders must agree on a definition of observable cost. We assume that, 
 without auditing costs, the government can determine ex post separable 
 costs, Cq(G) , and ex post joint costs, (Vj_ - e)Cj(Q,G). The problem of 
 defining observable costs then hinges around an allocation of joint costs 
 between government output and other output. Let k(Q,G) be the allocation 
 
(function) rule, agreed upon by bidders and the government, that 
 represents the share of joint costs assigned to the government contract. 
 In general, a cost allocation k will have the following characteristics: 
 
 (i) < k(Q,G) < 1; 
 
 (ii) k(Q,0) = for all positive Q; 
 
 (iii) k(0,G) = 1 for all positive G; and 
 
 (iv) k(Q,G) is decreasing in Q, for all G > 0. 
 
 One common class of cost allocations may be written as k(Q,G) = 
 G/(Q+G) . With this class of cost allocations the definition of output 
 becomes critical. Thus, the joint costs to be allocated to a contract 
 for 100 jet fighter planes depends on whether output is to be measured in 
 terms of planes, labor hours, labor dollars, or some other measure. 
 
 Given a cost allocation rule, k(Q,G), we may now write observable 
 cost as: 
 
 C*(Q,G) = C G (G) + k(Q,G)(V - e)Cj(Q,G). (4) 
 
 Recall that, by definition, h(e) is not observable and may not be charged 
 to the proj.ect. Note that the existence of joint costs will generally 
 cause observable costs to differ from opportunity cost for two reasons. 
 First, observable costs do not represent incremental production cost of 
 producing G. Second, each bidder would change the production level Q in 
 response to being awarded the contract, so revenues from sale of the 
 outside good, as well as production costs of that good, change. 
 
Let R(Q) be the revenue curve faced by agent i in the outside market 
 where R( • ) is twice continuously differentiable , weakly concave, and 
 R' (0) > 0. We assume that each agent faces the same revenue curve for 
 the sale of its private sector product. Note that we allow for the 
 possibility that all bidders face a competitive outside market. However, 
 the agents may all be selling different products in the outside markets. 
 The assumption of common revenue functions enables us to use an 
 independent-private-values model, and thus focus directly on the effect 
 of joint costs on bidding for the government contract. 
 
 Before presenting the bidding optimization problem of potential 
 agents, we introduce some additional notation. Let P (V^) be the profits 
 of the ith bidder if the agent is not awarded the contract. Then: 
 
 P*(Vi) = (MAX R(Q) - C Q (Q) - (V - e)Cj(Q.O) - h(e) } . (5) 
 
 Q.e 
 
 Given a cost allocation rule k(Q,G) and an incentive share ratio a, the 
 profits of the ith bidder, if successful in winning the contract, are: 
 
 PCbi.Vija) - MAX (R(Q) - Cq(Q) - C G (G) - (V i -e)C J (Q , G) 
 Q,e 
 
 - h(e) + bj + a[C*(Q,G) - bjj }. (6) 
 
 Given our definition of C (Q,G), we can write the winning bidder's 
 maximum profit gross of the net bid payment, (1 - a)b^, as 
 
 P(V i; a) = MAX (R(Q) - Cq(Q) - (1 - a)C G (G) 
 Q,e 
 
 - (1 - ak(Q,G))(V i - e)Cj(Q,G) - h(e) } . (7) 
 
Denote the solution to (7) by (Q(Vi ,G; a) , e(V£ ,G; a) ) . Then using (4) and 
 (7) we can rewrite (6) as: 
 
 P(b i( V i; a) = (1 - a)bi + P(V i; a). (8) 
 
 Finally, we can define the economic cost of fulfilling the contract. 
 
 C(V i; a) - P*(Vi) - P(V i; a). (9) 
 
 Note that C{V^;a) is not required to be positive for all V^. We can i_hus 
 rewrite (8), the incremental profit for the winning bidder, as 
 
 PCbi.Viia) - P*(Vi) - (1 - a)bi - C(V i;a ). (10) 
 
 When there are no joint costs of production, our model reduces to 
 the MM model with risk-neutral bidders. In that case, Cj(Q,0) = and 
 Cj(Q,G) for G > can be taken as identically equal to one (or any 
 constant) . Then the expression in (9) can easily be shown to be equal to 
 (1 - a)[Vi + C G (G) - e(V i( G;a)] - h(e(V i ,G;a). The quantity Vj + C G (G) 
 is what MM refer to as the "opportunity cost" for the ith agent, so that 
 (9) is still the expected cost of the contract to a bidder net of the 
 cost of effort. 
 
 Each agent i chooses his or her bid b^ to maximize the probability 
 of winning times the expression in (10). The winning bidder is chosen 
 via a first-price, sealed bid auction; thus, the probability of agent i 
 inning is just the probability that b^ < bi for all j^i. A symmetric 
 
 w 
 
 8 
 
Nash equilibrium bidding strategy is then a bid function B(V^;a) such 
 that if b-; = B(V-;;a) for all j^i, then b^ = B(V^;a) is the optimal bid 
 for agent i. The probability that agent i is the winning bidder is then 
 [1 - F(B" 1 (b) ) ] ( n "l) since the b-; are all independent draws from the 
 distribution of V. The optimal bid function B(-) is the solution to the 
 following maximization problem, where subscripts have been deleted: 
 
 MAX [1 - F(B _1 (b))]( n_1) [(l-a)b - C(V;a)] 
 
 (11) 
 
 It is straightforward to verify that B(V;a) satisfies the following 
 first-order differential equation: 
 
 s(V)[(l-a)B(V;a) - C(V;a)] + (l-a)B'(V;a) = 0, 
 
 (12) 
 
 where s(V) - - (n-l)f (V)/(1-F(V) ) . Let 
 
 S(V) - 
 
 V 
 s(t)dt 
 
 (13) 
 
 Then we can solve for (1 - a)B(V;a) as^ 
 
 (1 - a)B(V;a) = -exp(-S(V)) 
 
 V 
 
 s(t)exp(S(t))C(t;a)dt 
 
 V 
 
 (14) 
 
 The first thing that we must demonstrate is that B(V;a) is 
 increasing in V , so that the contract is awarded to the most efficient 
 bidder. From (12), and the fact that s(V) < for all V, we see that 
 
B'(V;a) has the same sign as the net benefit of the contract to the 
 winning bidder (equation (10)). Clearly, this -net benefit must be non- 
 negative in equilibrium. If it were equal to zero in equilibrium for 
 some V, then the value of equation (11) would be zero. The bidder could 
 improve his or her position (i.e. make the value of the maximand in (11) 
 strictly positive) by increasing the bid, which makes the profit positive 
 and decreases the probability of winning; however, it could not have been 
 the case that the original bid was an equilibrium bid if this were 
 possible. Therefore, it must be true that the net benefit of winning the 
 contract is strictly positive in equilibrium. This results in B'(V;a) 
 strictly positive, and we have bids increasing in V. 
 
 The observed costs C ( ■ ) can be written in equilibrium as a function 
 of V and the cost share parameter a. For simplicity, we retain the use 
 of C ( • ) and define 
 
 C*(V;a) - C G (G) + k(Q(V,G ; a) , G) (V - e (V,G ; a) ) Cj (Q(V ,G ; a) , G) . (15) 
 
 Given that we have demonstrated that the most efficient (lowest cost) 
 bidder wins the contract, we can now compare fixed price contracts with 
 incentive contracts. In the next section, we give sufficient conditions 
 for the government (principal) to be strictly better off with a fixed- 
 price contract in which the winning bidder bears all of the ex post cost 
 of the contract. 
 
 10 
 
III. OPTIMAL FIXED-PRICE CONTRACTS 
 
 The objective of the government is the ex ante minimization of its 
 expenditures on the contract. If a bidder with cost parameter 
 V e [V*,V ] is the winner, the government's conditional expected 
 expenditure, where expectation is taken with respect to W and conditioned 
 on V, will be 
 
 (1 - a)B(V;a) + aC*(V;a). (16) 
 
 The expected value of the expression in (16) (where expectation is now 
 taken with respect to the distribution of the minimum of n independent 
 draws from the distribution of V) , is the ex ante expected expenditure 
 for the government for a given cost share a. 
 
 We would like to determine conditions under which the expected 
 expenditure is minimized at a = 0. If the expression is (16) is 
 minimized at a = for all V, then we are done. We proceed as follows: 
 we first develop the general conditions under which '(16) will be 
 minimized at a = for all V; we then give conditions on the underlying 
 structure of our model (the revenue and cost functions) that are 
 sufficient for these general conditions. 
 
 THEOREM 1 
 
 If observed costs are decreasing in the cost parameter V in equilibrium, 
 
 then a fixed price contract strictly dominates an incentive contract. 
 
 That is, if C*(V;a) is decreasing in V for all V e [V*,V*], then 
 
 (1 - a)B(V;a) + aC*(V;a) is minimized at a - for all V e [V*,V*]. 
 
 11 
 
Proof: 
 
 To demonstrate that (1 - a)B(V;a) + aC*(V;a) is minimized at a - , 
 we show that 
 
 B(V;0) - (1 - a)B(V;a) - aC*(V;a) < for all < a < 1. (17) 
 
 From (14) we have 
 
 
 (1 - a)B(V;a) = -exp(-S(V)) 
 
 s(t)exp(S(t))C(t;a)dt 
 
 V 
 
 so that the left hand side of (17) can be rewritten as 
 
 exp(-S(V)) 
 
 s(t)exp(S(t) [C(t;0) - C(t;a)]dt - aC*(V;a). (18) 
 
 V 
 
 We first demonstrate that 
 
 C(t;0) - C(t;a) < aC*(t;a) for all t, 
 
 (19) 
 
 Using the definitions of C(t;a) (from (9)), we have 
 
 C(t;0) - C(t;a)"= P*(t) - P(t;0) - P*(t) + P(t;a) 
 
 = P(t;a) - P(t;0) 
 
 (20) 
 
 The last expression in (20) is the difference between the firm's maximum 
 profit when costs are shared and when they are not. From equation (7) 
 (the definition of P(t;a)); the definition of (Q(t ,G ; a) , e( t ,G;a) ) as the 
 
 12 
 
solution to the problem in (7); and equation (15) (the definition of 
 C (t;a)) we have 
 
 P(t;a) - P(t;0) - 
 
 [MAX {R(Q)-C Q (Q)-(l-a)C G (G)-(l-ak(Q,G))(t-e)Cj(Q,G)-h(e)} 
 Q.e 
 
 MAX {R(Q)-C Q (Q)-C G (G)-(t-e)Cj(Q,G)-h(e)} 
 Q,e 
 
 ] < 
 
 (21) 
 
 [MAX {R(Q)-C Q (Q)-C G (G)-(t-e)Cj(Q,G)} 
 Q,e 
 
 + aC G (G) + ak(Q(t,G;a),G)(t - e)Cj(Q(t ,G ;a) , G) 
 
 - MAX {R(Q)-C Q (Q)-C G (G)-(t-e)Cj(Q,G)-h(e)}] 
 Q,e 
 
 aC*(t;a). 
 
 Therefore, we have our desired result that 
 
 * , 
 
 C(t;0) - C(t;a) < aC~(t;a). 
 
 Equations (18), (19), the left hand side of (17), and s(t) < for all 
 t e [V*,V*] yield 
 
 B(V;0) - (1 - a)B(V;a) - aC*(V;a) < 
 
 exp(-S(V)) 
 
 s(t)exp(S(t))aC*(t;a)dt - aC*(V;a) 
 
 (22) 
 
 The major part of our proof demonstrates that the right hand side of 
 
 13 
 
(22) is non-positive. We note that the right hand side of (22) has the 
 same sign as 
 
 s(t)exp(S(t))aC*(t;a)dt - exp(S(V) )aC*(V; a) (23) 
 
 If the expression in (23) is increasing in V and is less than or equal to 
 at V , then (23) is non-positive for all V. In addition, at V = V , 
 the right hand side of (22) can be shown to b° strictly negative. 
 
 To show (23) is increasing in V, we differentiate (23) with respect 
 to V using Liebnitz's Rule: 
 
 s(V)exp(S(V))aC*(V;a) - s (V)exp(S(V) )aC*(V ; a) 
 
 3C*(V;a) aC*(V;a) (24) 
 
 exp(S(V)) = - exp(S(V)) 
 
 3V 3V 
 
 3C* 
 
 since by definition S'(V) = s(V). When < for all V, the 
 
 3V 
 
 right hand side of (24) is positive for all V, and we have (23) 
 
 increasing in V. At V = V*, (23) reduces to 
 
 exp(S(V*))aC*(V*;a) = (25) 
 
 since S(V ) = - » by definition. 
 
 Therefore, (23) is less than or equal to zero. We now must show 
 that the right hand side of (22) is less than zero at V = V*. Taking the 
 limit as V -» V , and applying L'Hopital's rule, we get 
 
 14 
 
lira 
 
 V" 
 
 s ( t ) exp ( S ( t ) ) aC ( t ; a ) dt 
 
 V 
 
 exp(S(V)) 
 
 aC x (V;a) 
 
 lim 
 
 s(v)exp(S(V))aC (V;a) 
 s(v)exp(S(V)) 
 
 aC*(V;a) 
 
 (26) 
 
 lim { aC*(V;a) - aC*(V;a)}- 
 
 V-V* 
 
 We therefore have our desired result that 
 
 B(V;0) - (1 - a)B(V;a) - aC*(V;a) < for all V e [V*,V*]. (27) 
 
 Q.E.D. 
 
 The conditions of Theorem 1 state that even though the cost 
 parameter V is increasing, which increases joint costs for all Q and G, 
 the level of observed costs decrease in equilibrium . This results 
 because the agent has the ability to adjust both the outside output, Q 
 and the level of cost reducing effort, e. In fact, under only our 
 initial assumptions, both Q and e can be shown to be decreasing in V. 
 
 It should be noted that for the MM model, the conditions of 
 Theorem 1 can never be met. In their model, C (V;a) equals 
 (V + C G (G) - e(V,G;a)). Since e(-) maximizes (1 - a)e - h(e) in the MM 
 
 15 
 
model, it is invariant tfith respect to V. Therefore, C (V;a) is strictly 
 increasing with respect to V. 
 
 We can now provide sufficient conditions so that the requirements of 
 Theorem 1 are met. We assume that marginal revenue minus marginal 
 separable cost of the outside output is a positive constant, and that the 
 ratio of allocated joint costs to the marginal joint costs borne by the 
 agent is increasing in Q. 
 
 THEOREM 2 
 
 If (i) R'(Q) - Cq(Q) > for all Q; 
 
 (ii) R"(Q) - Cq(Q) = for all Q; and 
 
 k(Q,G)Cj(Q,G) 
 
 (iii) — — — ^— — is increasing in Q, then 
 
 3(1 - ak(Q,G))Cj(Q,G) 
 
 3Q 
 
 then C (V;a) is decreasing in V for all V and for all < a < 1 
 
 Proof: 5 
 
 For ease of exposition we will (1) denote all first (second) derivatives 
 of all functions with respect to Q, e and V by primes, since functions 
 depend on only one of these arguments, and (2) write functions without 
 explicit reference to their arguments. Differentiating (15) (the 
 definition of C (V;a)) totally with respect to V and collecting terms, 
 gives us the following in equilibrium: 
 
 16 
 
C*'(V;a) = Q'(V - e)[k'Cj + kC j ] + (1 - e')kCj 
 
 (28) 
 
 The first order conditions for the winning agent's maximization problem 
 
 are : 
 
 R' - Cq - (V - e)[-ak'Cj - akCj + C j ] - 
 (1 - ak)Cj - h' = 0. 
 
 Second order conditions for the winning agent's problem are that the 
 principal minors of the following Hessian matrix alternate in sign: 
 
 (29) 
 
 H = 
 
 / it 
 
 R"-CQ-(V-e) [-ak n Cj-2ak'Cj-akCj ] 
 
 -ak'Cj-akCj+Cj 
 
 ■ak'Cj-akCj+Cj 
 
 h" 
 
 (30) 
 
 or equivalently , 
 
 ti ti 
 
 R" - Cq - (V - e)[ak"Cj - 2ak'Cj - akCj + C j ] < 0, 
 - h" < 0, and 
 
 (31) 
 
 | H | > 
 
 Total differentiation of the conditions in (29) with respect to V gives 
 
 us : 
 
 17 
 
H 
 
 Q' 
 
 
 -ak'Cj - akCj + Cj 
 
 e' 
 
 
 
 
 (32) 
 
 Using Cramer's Rule, we can solve for Q' and e' 
 
 Q ' = - h" [Cj - ak'Cj + akCj]/ | H | 
 
 (33) 
 
 [Cj - ak'Cj + akCj] 2 / | H | 
 
 From Cj increasing in Q and k decreasing in Q for all G, we have 
 (1 - ak)Cj increasing in Q; this, along with the second order conditions, 
 give us Q and e strictly decreasing in V. Substituting the explicit 
 expression for H and (33) into (28), and collecting terms gives: 
 
 dC 
 
 av 
 
 - h" 
 
 I H 
 
 i i 
 
 { (V-e) (k'Cj+kCj) (Cj- ak'Cj -akCj) 
 
 (34) 
 
 9 M 
 
 + aCj(R ,, -CQ-(V-e)(Cj-ak"Cj-2ak'C J +akCj) } 
 
 it 
 When R" - Cq(Q) = 0, the expression in (34) reduces to: 
 
 dC 
 
 av 
 
 - h"(V-e) 
 I H | 
 
 i i 
 
 {(k'Cj+kCj)(Cj-ak'Cj-akCj) 
 
 / ii 
 
 aCj(Cj-ak"Cj-2ak'Cj-akCj) } 
 
 (35) 
 
 Since (1 - ak)Cj is increasing in Q, R' - Cq > and the first order 
 conditions give us (V - e) > in equilibrium. To show C* decreasing in 
 
 18 
 
V, we thus need the term in brackets on the right hand side of (35) 
 
 positive. This is immediate from (iii) in the statement of the Theorem. 
 
 Q.E.D. 
 
 A sufficient, but not necessary condition for (i) and (ii) in 
 Theorem 2 to hold is that both the price of Q and the marginal separable 
 cost of Q are constants, with the former larger than the latter. 
 Condition (iii) in Theorem 2 will hold, for example, if 
 
 k(Q,G) = G/(Q+G) and Cj(Q,G) - QG + F, where F is fixed costs and G 2 > F. 
 The conditions in Theorem 2 can thus be met by reasonable revenue, 
 separable cost, joint cost, and cost allocation functions. 
 
 We have thus shown the existence of conditions under which the 
 government is strictly better off not sharing costs with the agents. In 
 that case, the government should solicit bids from the agents, award the 
 contract to the minimum bidder, and let the winning bidder bear all of 
 the ex post cost of the contract. 
 
 19 
 
IV. SUMMARY 
 
 McAfee and McMillan [1986] have shown that in a model in which 
 opportunity costs can be directly observed, risk-sharing in the form of 
 an incentive contract will be optimal even for risk-neutral bidders. We 
 have shown that the presence of joint costs may destroy the optimality of 
 incentive contracts for the case of risk-neutral bidders. In this case, 
 observed costs and opportunity costs are not the same. 
 
 If a cost based incentive contract is to be employed, then observed 
 cost must be clearly defined. When joint costs are present, this 
 requires that joint costs be allocated between che output produced for 
 the government and all other output. In our generalized model, we have 
 seen that the presence of joint costs, and the need to allocate those 
 costs, may cause a significant divergence between opportunity cost and 
 observed cost. Cost sharing has a positive imp<\ct in that it stimulates 
 bidding competition. However, that positive impact can be overwhelmed by 
 cost sharing' s inefficient cross-subsidization of non- government output. 
 When this occurs, the government is strictly better off not sharing costs 
 by using a fixed-price contract. 
 
 As MM observe, in practice most government contracts are fixed- 
 price. They use their results to justify a recommendation that more 
 incentive contracts should be used. However, our results may help to 
 explain the use of fixed-price contracts because of the real world 
 divergence between accounting costs and opportunity costs. 
 
 20 
 
REFERENCES 
 
 Coddington, E.A., An Introduction to Ordinary Differential Equations, 
 Prentice Hall (1961), Englewood Cliffs. 
 
 Laffont, J. -J. and J. Tirole, "Auctioning Incentive Contracts," Journal 
 of Political Economy, vol. 95 (1987), pp. 921-940. 
 
 McAfee, R.P. and J. McMillan, "Competition for Agency Contracts," Rand 
 Journal of Economics, vol. 18 (1987), pp. 296-307. 
 
 McAfee, R.P. and J. McMillan, "Bidding for Contracts: A Principal-Agent 
 Analysis," Rand Journal of Economics, vol. 17, no. 3 (1986), pp. 326- 
 338. 
 
 Milgrom, P.R. and R.J. Weber, "A Theory of Auctions and Competitive 
 Bidding," Econometrica, vol. 50 (1982), pp. 1089-1122. 
 
 21 
 
FOOTNOTES 
 
 1. The product produced for the government does not have exactly the sama 
 characteristics as the product produced for the outside market. If it 
 did, the government could purchase the good on the outside market. For 
 example, the government may want to purchase toilet seats for submarines 
 from manufacturers of toilet seats for the general market. 
 
 2. Note that if auditing costs exist but are known to be the same 
 regardless of which agent wins the contract, then without loss of 
 generality, we can assume that these costs are zero. 
 
 3. This cost can be interpreted as the opportunity cost of the contract; 
 it includes both production cost and effort cost differentials over no 
 government output. 
 
 4. See Coddington, pp. 43-44. 
 
 5. The requirement that {marginal revenue -marginal separable cost} be a 
 positive constant is stronger than we need. As long as the difference is 
 positive and decreasing slowly and condition (iii) is satisfied, the 
 result of the Theorem still holds. 
 
 22 
 
Papers in the Political Economy of Institutions Series 
 
 No. 1 Susan I. Cohen. "Pareto Optimality and Bidding for Contracts." 
 Working Paper # 1411 
 
 No. 2 Jan K. Brueckner and Kangoh Lee. "Spatially-Limited Altruism, 
 Mixed Clubs, and Local Income Redistribution." Working Paper 
 #1406 
 
 No. 3 George E. Monahan and Vijay K. Vemuri . "Mono tonicity of 
 Second-Best Optimal Contracts." Working Paper #1417 
 
 No. 4 Charles D. Kolstad, Gary V. Johnson, and Thomas S. Ulen. " Ex 
 Post Liability for Harm vs. Ex Ante Safety Regulation: 
 Substitutes or Complements?" Working Paper #1419 
 
 No. 5 Lanny Arvan and Hadi S. Esfahani. "A Model of Efficiency Wages 
 as a Signal of Firm Value." Working Paper #1424 
 
 No. 6 Kalyan Chatterjee and Larry Samuelson. "Perfect Equilibria in 
 Simultaneous-Offers Bargaining." Working Paper #1425 
 
 No. 7 Jan K. Brueckner and Kangoh Lee. "Economies of Scope and 
 Multiproduct Clubs." Working Paper #1428 
 
 No. 8 Pablo T. Spiller. "Politicians, Interest Groups, and 
 
 Regulators: A Multiple-Principals Agency Theory of Regulation 
 (or "Let Them Be Bribed." Working Paper #1436 
 
 No. 9 Bhaskar Chakravorti. "Asymmetric Information, 'Interim' 
 Equilibrium and Mechanism Design." Working Paper #1437 
 
 No. 10 Bhaskar Chakravorti. "Mechanisms with No Regret: Welfare 
 
 Economics and Information Reconsidered." Working Paper #1438 
 
 No. 11 Bhaskar Chakravorti. "Communication Requirements and Strategic 
 Mechanisms for Market Organization." Working Paper #1439 
 
 No. 12 Susan I. Cohen and Martin Loeb. "On the Optimality of 
 
 Incentive Contracts in the Presence of Joint Costs." Working 
 Paper #1445 
 
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