Review of Aounting Studies, 4, 5 13 (1999) 1999 Kluwer Aademi Publishers, Boston. Manufatured in The Netherlands. The Impat of Capaity Costs on Bidding Strategies in Prourement Autions JÖRG BUDDE University of Magdeburg, Faulty of Eonomis and Management, PF 4120, 39016 Magdeburg, Germany ROBERT F. GÖX University of Magdeburg, Faulty of Eonomis and Management, PF 4120, 39016 Magdeburg, Germany Abstrat. This paper analyzes the impat of apaity osts on bidding strategies of firms partiipating in prourement autions. More effiient firms will invest in advane due to their high probability of winning the aution while less effiient bidders prefer to wait with their investments until the outome of the aution is known. However, in equilibrium both types of firms inlude a overage for their investment osts in their bids and therefore adopt a full ost priing poliy. The relevane of fixed osts for priing deisions has ontinuously drawn the attention of management aountants at least sine the diret ost ontroversy of the sixties. Despite this long tradition of researh, the issue seems still to be unresolved. While urrent textbooks (see e.g., Horngren, Foster, and Datar (1994), Hansen and Mowen (1994)) and journal artiles (see Burgsthaler and Noreen (1997)) emphasize the importane of marginal ost information for priing deisions, there is also some theoretial evidene that full ost priing might be benefiial for profit maximizing firms. In partiular, Banker and Hughes (1994) have demonstrated the eonomi suffiieny of full ost measures for priing and apaity deisions under unertainty. They onsider a single period model of a multi-produt monopolist who simultaneously ommits to produt pries and initial apaity levels for support ativities before demand is realized. After the demand unertainty is resolved the firm meets the entire demand at the full ost prie by purhases of additional support ativities at a ertain penalty ost if needed. The Banker and Hughes approah has been ritiized for several reasons. Both Balakrishnan and Sivaramakrishnan (1996) and Balahandran, Balakrishnan and Sivaramakrishnan (1997) fous on the apaity planning aspet of the model and demonstrate that apaity planning rules based on full produt osts are neither neessary nor suffiient for optimal apaity planning. Finally, Göx (1997) analyzes the apaity planning problem in a multiperiod setting and demonstrates that a firm may distort its apaity deision when it ommits to full ost priing instead of eonomizing on additional demand information that beomes available within the budgeting yle. Unlike the above mentioned ritiques, the model presented in this paper provides support for the optimality of full ost priing in a ompetitive environment. In partiular, we onsider the impat of apaity osts on the optimal bidding strategies of firms partiipating in a single prourement aution. First, we analyze the apaity planning problem and identify
6 JÖRG BUDDE AND ROBERT F. GÖX two different types of investment strategies: Firms with relatively low osts deide to invest in advane of the bidding ontest beause of their high probability of winning the aution. In ontrast to their low ost ompetitors less effiient firms prefer to wait with their apaity purhases until the outome of the aution ontest beomes known with ertainty although they inur penalty osts if they do not invest ex ante. Despite these different investment strategies, both types of investors inlude a overage for their apaity osts into their equilibrium bids. Bidders of the low ost type, however, are only able to reover their sunk investment ex ante, i.e., in expetation, while high ost type bidders are able to reover their apaity osts ex post, i.e., with ertainty beause these osts are only inurred in ase of winning the aution ontest. Cohen and Loeb (1990) already onsider the alloation of inremental fixed osts to a number of independent autions and find that eah partiipating firm alloates the total amount of its fixed osts ex ante. However, sine the fixed osts in their model are only inurred when the firm wins at least one projet, they are in priniple avoidable and the alloation problem is a matter of indivisibility. Unlike Cohen and Loeb, the fixed osts in our model are aused by a voluntary ommitment to apaity levels before the firms know with ertainty if these apaities will ever be utilized. One the investment is made, the osts are sunk and an not be reovered when the aution ontest is lost. 1 The remainder of this artile proeeds as follows: The first setion introdues the model assumptions. In Setion 2 the optimal solutions for the firm s apaity hoies and bidding strategies are derived under quite general onditions. Setion 3 gives an example for the speial ase of two bidders with uniformly distributed osts, and Setion 4 onludes with a short summary. 1. Model Consider a prourement aution with n risk-neutral firms, indexed by i = 1,...,n, whih are supposed to submit simultaneous sealed bids for a single projet. The projet is awarded to the lowest bidder at the prie offered in the bid. In ase of ommensurate bids, the projet is assigned to eah bidder with the same probability. Thus, the type of aution under onsideration is a first-prie sealed-bid aution, whih is ommon pratie in government prourement ativities and often even required by law (MAfee and MMillan (1987)). If firm i wins the projet, it inurs a diret ost of prodution, denoted by C = [, ] R +. The atual value of is private information of firm i. All other firms and the bid taker share idential beliefs about whih an be haraterized by the probability distribution funtion F i ( ). To keep the analysis tratable, we restrit ourselves to the ase of ex ante symmetri bidders with unorrelated diret ost parameters. Thus, the are i.i.d. random variables with umulative distribution funtion F i ( ) = F( ) for all i. In addition, we assume that f ( ) is absolutely ontinuous with probability density funtion f ( ) = F ( ) over the support C. Up to this point, the setting is strategially equivalent to the well-known symmetrindependent private values model of a first-prie sealed-bid aution (Milgrom and Weber (1982)). The distinguishing feature of our model arises from the additional apaity planning problem faed by the bidders. In partiular, we assume that the winning firm does not only
PROCUREMENT AUCTIONS 7 employ diret inputs like raw materials or diret labor to realize the projet, but also requires inputs whih are usually supplied before the firm knows if it is awarded the ontrat. These inputs will heneforth be referred to as apaity. One the firm has deided on the amount of apaity to be installed in advane of the bidding ontest, it an purhase additional apaity units only if it is willing to pay θr per unit with θ>1, where r denotes the original aquisition ost of the resoure. Thus, ex post provision of apaity implies a penalty (θ 1) > 0. A natural interpretation would be a labor ontrat inluding an agreement about overtime premia where r is the wage per hour and θr the payment per extra work hour. Another example is a long term ontrat for materials where the differene between the ontrated prie and the spot market prie may be due to quantity disounts. These disounts typially arise when a firm is able to pool its input demands over a range of produts or projets by early planning. 2 Let k i [0, 1] be the fration of the required apaity installed in advane of the bidding ontest. In determining k i, the firms partiipating in the aution have to trade off the potential ost savings from early investment in the ase of winning the ontest against the risk of sinking the investment osts when losing. The impat of this trade-off on the firm s bidding strategies is analyzed in the next setion. 2. Equilibrium Bidding Strategies All firms partiipating in the prourement aution fae idential deision problems of submitting a sealed bid given the privately known value of their diret ost parameter. Thus, we entirely fous on a symmetri Bayesian Nash equilibrium where all firms adopt the same bidding strategy b i = B( ), allowing us to restrit the analysis to the deision problem of a representative firm. Sine the ontrat is awarded to the lowest bidder, the players bidding strategies B( ) must be a stritly inreasing and ontinuous funtion of the diret ost parameter (MAfee and MMillan (1987)). 3 Moreover, for reasons that will beome obvious during the analysis, we assume that B( ) is pieewise differentiable. Aordingly, the probability that firm i wins the aution an be written as Prob{i wins} =[1 F(B 1 (b i ))] n 1 H(B 1 (b i )). (1) Aordingly, firm i s expeted profit if it provides a fration k i of the required projet apaity in advane and bids b i for the ontrat is given by E[ i ] = [b i θr(1 k i )]H(B 1 (b i )) rk i. (2) The first term in (2) is the expeted profit ontribution of the projet, whereas the apaity osts are given by the seond term. Consider first the apaity hoie of firm i: Differentiating the expeted profit with respet to k i yields E[ i ] k i = θrh(b 1 (b i )) r, (3) indiating that the firm s expeted profit stritly inreases (dereases) in k i when the expeted marginal penalty osts for additional apaity purhases exeed (fall short of) the marginal
8 JÖRG BUDDE AND ROBERT F. GÖX ost of apaity installed in advane of the bidding ontest. This observation implies the following bang-bang solution: Lemma 1 The optimal apaity hoie of firm i is given by k = { 1 for i < ĉ 0 for ĉ I where ĉ = H 1 (1/θ). (4) Proof: Sine expeted profit is linear in k i, the firm hooses the maximum apaity level k = 1 when (3) is positive and the minimum apaity k i = 0 when (3) is negative. Taking into aount that in equilibrium the ondition B 1 (bi ) = must hold for all i, the ritial value ĉ for the indifferent bidder is obtained by solving (3) for. From Lemma 1 we obtain the intuitively appealing result that the high ost firms prefer to wait until the aeptane of their bid beause the threat of inurring sunk apaity osts outweighs the possible ost savings in the relatively unlikely ase of aeptane. In ontrast, low ost firms prefer to install the entire apaity in advane due to their high probability of winning, and a bidder with ost ĉ is indifferent between installing apaity in advane and waiting until the aeptane of his bid. The hoie between the polar ases of waiting to invest and full ex ante investment may be regarded as a real option (Dixit and Pindyk (1994)). However, the value of waiting to invest is not determined by an exogenous soure of unertainty as in real options models but by the probability of winning the aution whih is in turn determined by the bidder s diret ost parameter in a unique way. Aording to Lemma 1, we have to distinguish two types of apaity supply when analyzing the equilibrium bidding strategy B( ). From the firm s first-order ondition E[ i (b i, ki )] { H( ) + [bi = i ]H ( )(B 1 ) (b i ) = 0 for < ĉ b i H( ) + [b i θr]h ( )(B 1 ) (b i ) for ĉ, (5) where H( ) and H ( ) denote H(B 1 (b i )) and H (B 1 (b i )) respetively, we an derive the following identities: 4 { H( )B ( ) + H ( )B( ) = H ( ) for < ĉ ( + θr)h ( ) for ĉ, (6) requiring that the marginal expeted revenue of the projet on the left hand side of (6) must equal the marginal expeted ost of the projet, given by the right hand side of (6). The first equation in (6) ould also be derived in a setting without additional ost of apaity supply. The distinguishing term θrh ( ) in the seond identity of (6) denotes the ost of ex post installed apaity. However, although the first order ondition does not ontain apaity osts if the firm provides full apaity in advane, apaity osts appear in all ost types equilibrium bids:
PROCUREMENT AUCTIONS 9 Lemma 2 The firm s equilibrium bidding strategy is given by B( ) = + i H(z)dz r + H( ) for < ĉ H( ) θr for ĉ (7) where ĉ is defined in (4) 5. Proof: See Appendix. From (7), the optimal bid onsists of three terms. The first term represents the atual diret projet osts of bidder i, the seond exhibits the firm s rent for its private information, and the third term alloates apaity osts to the bid. Sine the sum of the first two terms in (7), β( ) = + H(z)dz/H( ), would be the optimal bidding strategy in the orresponding standard model without apaity osts (see MAfee and MMillan (1987)), we an derive the following result about the impat of apaity osts on the firm s bidding strategy by omparing both strategies: Proposition 1 Capaity osts are inorporated in the firm s priing deisions as follows: 1. If no apaity is provided in advane, inremental apaity osts θr are fully overed by the bid. 2. If apaity is provided in advane, apaity osts are assigned to the bid suh that they are fully overed ex ante. Proof: 1. The overage for apaity osts in (7) equals the inremental apaity osts θr. 2. The overage for apaity osts in (7) equals r/h( ). Sine in equilibrium firm i s probability of winning the ontest is H( ), the expeted overage of apaity ost is r, the historial apaity aquisition osts. The first result in Proposition 1 is a straightforward extension of the standard model without apaity osts where the diret prodution ost of firm i is substituted by the term = +θr. The seond result, namely the question of why low ost firms alloate apaity osts to their bids, requires some additional onsiderations. Sine high ost firms will wait to invest and inlude their diret ost in their bid funtions, low ost firms an also add a markup for apaity osts beause an optimal bid funtion shedule B( ) must be ontinuous for a ontinuous type spae. This argument is best understood by onsidering the ritial firm of type ĉ. Without alloating apaity ost, its bid would be β(ĉ) =ĉ + ĉ But a ompetitor of type (ĉ + ɛ) bids B(ĉ + ɛ) = θr + (ĉ + ɛ) + (ĉ+ɛ) H(z) dz/h(ĉ). H(z)dz/H(ĉ + ɛ) whih equals β(ĉ) + θr as ɛ approahes zero. Hene, the firm of type ĉ ould inrease its bid to B(ĉ) without dereasing its winning probability. Taking this boundary ondition into aount, we get the bid funtion B(ĉ) inluding apaity osts for all ost types (see proof of Lemma 2).
10 JÖRG BUDDE AND ROBERT F. GÖX Figure 1. Optimal bidding strategies. 3. The Uniform Case The following example illustrates the preeding results. Suppose that there are only n = 2 bidders with uniformly distributed osts on the [0, 1] interval. The umulative distribution funtion beomes F( ) =, and firms i s probability of winning the aution is H( ) = 1. Substituting for H( ) in (7) yields the following optimal bidding strategy: b i = + 1 2 + r 1 for 0 < ĉ θr for ĉ 1. (8) The value of the diret ost parameter for the ritial bidder being indifferent between inurring sunk osts and waiting to invest is ĉ = (θ 1)/θ. Aording to (8), eah bidder demands a reimbursement for his diret osts, an information rent of (1 )/2, and an additional payment for his apaity osts. Obviously, the most effiient type is allowed to extrat the highest possible information rent of 1/2, whereas the information rent of higher ost types stritly dereases in. In ontrast, the apaity ost term stritly inreases in for ost types < ĉ. This effet has a natural interpretation. Sine the most effiient ost type wins the aution with probability one, it an be ertain to reover its investment osts and hene bids B() = 1/2 + r, i.e. it onfines itself with a full ost reimbursement and reeives the full information rent as its surplus. In ontrast, a less effiient firm an only reover its investment outlays with a probability stritly less than one. Therefore, it will adjust its apaity ost term in order to over the apaity osts in expetation. Hene, it demands stritly more than r. Finally, the ost types ĉ are demanding a full ost reimbursement plus their information rent, whih is zero for the least effiient type, i.e., B() = 1 + θr. The results are summarized in Figure 1. Starting from the lowest possible bid B(), the equilibrium bid funtion stritly inreases in with an inreasing rate 6 up to the point where apaity preommitment is no longer optimal. Beyond the ritial value ĉ, the bid funtion inreases linearly in until the maximum possible bid B(), namely the bid of the lease effiient type, is ahieved.
PROCUREMENT AUCTIONS 11 4. Summary and Conlusions This paper has examined the impat of apaity osts on ompetitive bidding strategies of firms partiipating in prourement autions. First, two different types of investment strategies were identified: More effiient firms deide to invest in advane of the bidding ontest due to their high probability of winning the aution. In ontrast, less effiient firms with relatively low probabilities of winning the aution prefer to wait with their apaity purhases until the outome of the aution game beomes known with ertainty. For both types of investors the optimal bidding strategies inlude a overage for their investment osts. While the high ost type s investment osts are avoidable and thus reovered with ertainty if the ontest is won, the low ost types are only able to reover their sunk investment osts in expetation. Nevertheless, full ost priing is identified as a profit maximizing strategy for all ost types in the ontext of our model. Appendix Proof of Lemma 2: To derive the equilibrium bidding strategy we have to solve the pieewise defined differential equation given by (6). The equilibrium bid for ost types ĉ is haraterized by the identity H( )B ( ) + H ( )B( ) = ( + θr)h ( ). Integrating this ordinary differential equation yields B( )H( ) = (z + θr)h (z)dz + K. (9) To determine the onstant K, we look at the maximum ost firm. Substituting by n (9) yields B() H() }{{} =0 = [z + θr]h (z)dz+k. } {{ } =0 Thus, K = 0 and B( ) = zh (z)dz θr H (z)dz H( ) = θr i zh (z)dz H( ) = θr + + i H(z)dz (10) H( ) for ĉ where the first equality follows from H() = 0 and the seond step is obtained from integration by parts.
12 JÖRG BUDDE AND ROBERT F. GÖX From (6), the equilibrium bidding strategy for ost types < ĉ is haraterized by the identity H( )B ( ) + H ( )B( ) = H ( ) as in the standard independent private values model without apaity osts. However, in ontrast to the standard model this part of the bid funtion is only optimal for ost types < ĉ. Thus, the general solution of the differential equation is given by ĉ B( )H( ) = zh (z)dz + K. To determine the onstant K we look at the ritial bidder who is indifferent between investing in advane and waiting. Sine B is ontinuous, his bid must be the same under both investment strategies. Thus, substituting =ĉ yields ĉ B(ĉ)H(ĉ) = zh (z)dz+k. ĉ } {{ } =0 Hene, K = B(ĉ)H(ĉ) and we obtain the partiular solution B( )H( ) = ĉ zh (z)dz + B(ĉ)H(ĉ) or ĉ B( )H( ) = H( ) ĉh(ĉ) + H(z)dz + B(ĉ)H(ĉ) after an integration by parts. Utilizing (10) and substituting ĉ B(ĉ) = θr +ĉ + H(z)dz H(ĉ) yields ( ĉ B( )H( ) = ĉh(ĉ) + H( ) + H(z)dz + θr +ĉ + ĉ H(z)dz ) H(ĉ) H(ĉ) = H( ) + θrh(ĉ) + H(z)dz. Finally, utilizing the fat that H(ĉ) = 1/θ yields the seond part of the equilibrium bid funtion B( ) = + r H( ) + H(z)dz H( ) for ost types < ĉ.
PROCUREMENT AUCTIONS 13 Aknowledgments The authors aknowledge helpful suggestions and omments from Anne Chwolka, Dominique Demougin, Oliver Fabel, Alfred Luhmer, Matthias Raith, James Jordan (the Editor), and session partiipants at the 1998 Annual Conferene of the European Aounting Assoiation in Antwerp, Belgium. Notes 1. To stress this point, onsider the Cohen and Loeb setting with eah firm partiipating only in one aution. Then the indivisibility problem vanishes and the inremental fixed osts beome diret projet osts beause they are entirely avoidable when the firm fails to win the aution. 2. See Atkinson, Banker, Kaplan and Young (1997), Chapter 7 for further examples. 3. This standard result for independent private value autions also holds for the extended setting onsidered here, beause the optimal apaity is also a funtion of the ost type (Equation 4). Denote this funtion by k := h( ), then the optimal bidding strategy of bidder i may equivalently be expressed as b i = ˆB(, h( )) := B( ). 4. This step follows from the ondition for a symmetri equilibrium B 1 (b i ) = and the fat that (B 1 ) (b i ) = 1/B ( ) from the inverse funtion theorem. 5. An impliit assumption for the (soial) optimality of (7) is that the value of the projet to the bid taker exeeds the highest possible bid. Denote this value by v, then v> + θr guarantees optimality beause the least effiient type earns no information rent. 6. Note that bi is stritly onvex in : d 2 B i ( )/d1 2 = 2r/(1 ) 3 > 0. Referenes Atkinson, A. A., Banker, R. D., Kaplan, R. S., and S. M. Young. (1997). Management Aounting. 2nd Edition. Upper Saddle River, NJ: Prentie-Hall International. Balahandran, B. V., Balakrishnan, R., and K. Sivaramakrishnan. (1997). On the Effiieny of Cost-based Deision Rules for Capaity Planning. The Aounting Review 72, 599 619. Balakrishnan, R., and K. Sivaramakrishnan. (1996). Is Assigning Capaity Costs to Individual Produts Neessary for Capaity Planning? Aounting Horizons 10, 1 11. Banker, R. D., and J. S. Hughes. (1994). Produt Costing and Priing. The Aounting Review 69, 479 494. Burgstahler, D., and Noreen, E. (1997). Full-ost Priing and the Illusion of Satisfiing. Journal of Management Aounting Researh 9, 1997, pp. 239 264. Cohen, S. I., and Martin Loeb. (1990). Impliit Cost Alloation and Bidding for Contrats. Management Siene 36, 1133 1138. Dixit, A. K., and Pindyk, R. S. (1994). Investment under Unertainty. Prineton, NJ: Prineton University Press. Göx, R. F. (1997). Short-run Capaity Planning and Priing under Unertainty. Working paper, University of Magdeburg. Hansen, D. R., and Mowen, M. M. (1994). Management Aounting, 3rd Edition. Cininatti, Ohio: South-Western Publishing. Horngren, C. T., Foster, G., and Datar, S. M. (1994). Cost Aounting: A Managerial Emphasis. 8th Edition, Englewood Cliffs, NJ: Prentie Hall. MAfee, P. R., and J. MMillan. (1987). Autions and Bidding. Journal of Eonomi Literature XXV, 699 738. Milgrom, P. R., and Weber, R. J. (1982). A Theory of Autions and Competitive Bidding. Eonometria 50, 1089 1122.