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Entry and R & D in procurement contracting
JOURNAL OF ECONOMIC THEORY
58, 41-60 (1992)
Entry and R&D In Procurement Contracting GUOFU TAN*
Department of Economics, The UniveYSlty of Brillsh ColumbIa, Vancouver, B. C. Canada, V6T 1Z1
Received November 14, 1989, revised October I, 1991 A model of competltlve procurement and contractmg is presented in thiS paper. The key features of the model include precontract R&D, an endogenous number of homogeneous firms, and sealed-bid auctIOn rules. Both diminishing and constant returns to scale (DRS and CRS) in R&D expenditure technologies are considered. I find that the sealed-bid first-price auction (FPA) is eqUivalent to the second-price auction (SPA) under DRS technology. The Revenue EqUivalence Theorems breaks down under CRS because multiple equilibria arise m the R&D stage when SPA IS used However, SPA yields a unique perfect equilibrium given suffiCient heterogeneity among potentlal firms. The free-entry perfect eqUilibrium IS also characterized. The buyer prefers free entry through an appropnate selectIOn of the reservatIOn pnce Journal of Economic Lllerature Classification Numbers' D44,
D82. H57, 032. ,(,
1992 AcademIC Press. Inc
1. INTRODUCTION
The existing literature on auctions and procurement 1 typically assumes that the number of bidders is exogenous and constant 2 and that each bidder has certain private information. Auction mechanisms are designed to elicit the bidders' information and to maximize the auctioneer's expected
* Fmancial support from the Sloan FoundatIOn m the form of an Alfred P. Sloan Doctorial DissertatIOn Fellowship and the Jet Propulsion Laboratory of Caltech IS gratefully acknowledged. I thank my adviser, John Ledyard, for his encouragement and many suggestions, and an associate editor and a referee for helpful comments. I also benefited from discussions with Charles Blackorby, Richard Boylan, DaVid Green, Ken Hendricks, Preston McAfee, Thomas Palfrey, Michele Piccione, and Daniel Spulber. 1 See McAfee and McMillan [7] for a survey on auctions and bidding and Besen and Terasawa [1] for a survey on procurement contracting. 2 The exceptIOns are French and McConmck [3] and McAfee and McMillan [8] which consider fixed entry costs and entry equilibria. McAfee and McMillan [9] allow the number of actual bidders to be stochastic, but the probability of any subset of potential bidders becommg the set of actual bidders is assumed to be exogenous and Independent of their types. They have shown that the optimal auction IS the same whether or not the risk-neutral bidders know who their competitors are. Considering stochastic number of risk-averse bidders, Matthews [10] has offered a comparison study of auctIOn mechamsms from a bidder's point of view. 41 0022-0531/92 $5.00 Copynght
CO
1992 by AcademIC Press, Inc
All nghts of reproductIOn
III
any form reserved
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GUOFU TAN
payoff. As the number of bidders increases, the auction mechanisms become more attractive to the auctioneer. 3 The determination of the number of bidders is therefore of central importance. Clearly, that determination is related to decisions by potential bidders on whether to acquire information before competitive bidding. This paper examines how such decisions are made under different auction rules. There are many procurement situations in which only one, or a small number, of potential suppliers are plausible candidates to be the prime contractor. In other words, only a small number of firms find the procurement contract profitable given the current technology and other information. The current attempt by the U.S. Air Force to procure a new jet fighter is a case in point. Congress required that the contract be awarded through competitive bidding, but only two groups of military contractors found it worthwhile to compete. Each of those two groups invested over $600 million of its own funds in research and development (R & D) prior to bidding. 4 A further example is found in Hendricks, Porter, and Boudreau [4]. In their study of the federal auctions for leases on the outer continental shelf, they observed that potential firms do decide how much information to collect before participating in competitive bidding and that only a fraction of the set of potential bidders typically choose to submit bids in any auction. Standard analysis suggests certain relationships between precontract R&D and contract bidding. In general, higher costs of R&D reduce the number of competing bidders. The decisions to acquire information and to enter the bidding competition depend on the nature of R&D processes of the firms and the type of bidding procedure in place. Further, if fewer firms participate in the competitive bidding, the contract is more profitable to the winning firm and each firm invests more in R&D. But if the expected profit of the winning firm is high, more firms enter the R&D process and compete for the contract. These arguments suggest that the buyer, being interested in the number of bidders, may want to affect the firms' precontract R&D behavior. One potential mechanism for this is the choice of contract auction rules. In this paper, I examine the question of whether different auction rules yield the same equilibrium outcome given precontract R&D behavior. In addition, I answer a set of related questions. What is the equilibrium number of
3 See RIOrdan and SappIngton [13] and Dasgupta and Spulber [2] for diSCUSSIOns on procurement contracts In the case of a downward-sloping demand. 4 Each also received $691 million from the AIr Force for deSignIng the prototypes for a new fighter jet See the New York Times (December 27, 1989) for the report in detail. For other examples about private R&D Investment response to pubhc defence procurement, see Lichtenberg [6].
ENTRY IN CONTRACT AUCTIONS
43
bidders under free entry and how does each potential firm make precontract R&D decisions? Is free entry of firms an optimal policy for the buyer? Moreover, is it socially optimal? The relationship between R&D and procurement has also been studied in Rob [14]. In that paper, Rob formally models R&D activity in procurement as searching behavior. He emphasizes the second sourcing problem, using a model in which the government decides whether to bring in a new round of bidding after the initial contract has been awarded. In contrast, I concentrate on an environment without second sourcing where a number of firms make R&D-related decisions both on whether to enter a competition and on how much to bid. I model the R&D behavior of a firm as an one-shot stochastic process with certain costs of R&D. 5 Given certain auction rules, each firm that decides to enter the R&D process invests in R&D and privately observes its production costs. Based on the R&D information, firms submit bids and compete for the production contract. A model of competitive procurement with precontract R&D is presented in Section 2. The number of firms is viewed as an endogenous variable in the model and each firm is assumed to be risk neutral. Both constant and diminishing returns to scale (CRS and DRS) in R&D expenditure technologies are considered. In Section 3, I compare the perfect equilibrium outcomes in the two-stage game when the sealed-bid first-price auction (FPA) and the sealed-bid second-price auction (SPA) with the same reservation price are used by the buyer. I find that the two auction rules are equivalent when R&D technology is subject to DRS. The Revenue Equivalence Theorem breaks down under CRS R&D technology. In this latter case, if SPA is used, multiple equilibria of R&D decisions arise because the R&D outcomes of n firms together are indentical to the R&D outcomes of one firm that invests the same amount as all n firms. In general, SPA yields a unique perfect equilibrium outcome given sufficient heterogeneity among potential firms. Thus, SPA works better than FPA when potential bidders are heterogeneous in the R&D stage. Section 4 examines free-entry perfect equilibrium for a particular reservation price of the buyer. The equilibrium number of firms and the R&D expenditure are simultaneously determined and depend on the R&D process of the firm and the buyer's reservation price. Section 5 analyzes the buyer's optimal strategy. The buyer always prefers free entry of firms and 5 Tan [16] has also incorporated this type of R&D activity into competitive procurements for a given number of firms and has characterized the equilibrium investment level on R&D and the optimal procurement contract. There are other ways to model R&D activity, e.g., see Li, McKelvey, and Page [5] and Vives [18] for the discussions on mformation acquisition m the oligopoly theory.
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GUOFU TAN
selects a particular free-entry perfect equilibrium by choosing a relatively low reservation price. As a result, each active firm invests less, which causes underinvestment in R&D in the industry relative to the social optimum.
2. THE MODEL
The model I consider contains a single buyer (e.g., the government) who seeks to procure one unit of a certain novel good or service. The buyer is assumed to minimize the expected total costs of this procurement. The buyer is also assumed to face many potential suppliers; each of which is risk neutral and can produce a unit of the good at a potential unknown cost y. Each firm can invest in R&D for information about cost reduction and will privately observe a potential cost y which is drawn from the same random distribution
H( Y Ix) = 1 - [1 - F( y) ] x
(1)
with the fixed support [y,yJcR~, y>y, where F(y) is a continuously differentiable cumulative distribution function over [y, yJ and density function f( y), and x E R~ is the level of investment in R&D. 6 For convenience I define the function G( y) == 1 - F( y) for all y. The R&D cost function is assumed to be the same for all firms and is represented by C=C(x)+K,
(2)
°
where K> is the fixed cost of R&D and C(x) is the variable cost which is assumed to be twice differentiable in x. I also assume C(o) = 0, C' > 0, and Crt ~o. When C( x) = ex and e > 0, the R&D activity can be viewed as an independent experimental search process. For example, if the firm invests one unit (or one experiment), i.e., x = 1, a cost level y 1 is observed from the distribution F(yd at the cost e + K. Ifthe firm repeats this experiment x times (x is an integer now), each additional experiment is independent of the others and costs e. Then x numbers of production costs (Y1, ... , Yx) are observed at the cost ex + K. The minimum cost level y of (y 1, ... , Y x) is subject to the distribution of the lowest order statistic, which takes the form of (1). This independent experimental process exhibits constant returns to R&D scale CRS. When crt > 0, the cost function of R&D is strictly 6 We can also think of x as the number of R&D projects or different levels of R&D activities.
ENTRY IN CONTRACT AUCTIONS
45
convex. Increasing R&D scale becomes more and more expensive. In this case, R&D activity exhibits DRS.7 Competitive procurement is modeled as a three-stage process in this paper. In the first stage, the buyer announces and commits to the general rules of procurements. I consider both a sealed-bid FPA and a sealed-bid SPA with announced reservation price. In the second stage R&D is conducted, with each firm investing in R&D and acquiring information about the production cost. In the final stage, a competitive bidding procedure is conducted in which the buyer procures the good via the sealed-bid auction announced at the beginning. 8 The process can be described more specifically as follows. First, the buyer announces the rules of the sealed-bid auction (either FPA or SPA) including a reservation price r that is no higher than the highest possible cost level y. The lowest bid will be accepted unless it is below r. Second, each firm will calculate its expected profit from bidding and decide to invest in R&D if this profit is no less than its R&D costs. The active firms playa noncooperative Nash game in the R&D stage given that a Bayesian-Nash game is played in the bidding stage. The winner is chosen via the sealed-bid rule as the contractor for production. The contractor produces the good and gets paid according to the contract. Cost reduction during the production is not considered. The buyer is able to procure the good elsewhere at the cost Yo if the lowest bid is higher than the reservation price r. One can consider an extreme case in which Yo is very high, meaning that there are no substitutes available for the buyer and Yo does not play any role in the analysis. I consider the general case where Yo E Cv, y], whic means that the procurement is more competitive. Finally, the structure of the game is common knowledge.
3. A COMPARISON BETWEEN FPA AND SPA Given the rules of the sealed-bid auction (FPA or SPA) with a reservation price r, suppose that a firm believes that n firms including itself might invest in R&D and compete for the procurement contract in the auction. 7 One can always rescale the Illvestment varIable by lettlllg .x = K + C(x). then x = li(x), li' > O. and li" ,::; O. The R&D technology can then be represented by a random dlstnbution
H(yl.'i')=I-G(yl"t'I. In the speCIal case when G(Yl=e-> for YER~, one gets H(ylx)= l-e-hl'l" that IS an exponential distrIbution and IS widely used in the literature
on stochastIC R&D races (see Reinganum [12] for a survey). 8 In an early versIOn of thIS paper, I also allow each firm to have a bid-preparation cost SImIlar to that III Samuelson [IS]. Both the costs of R&D and costs of bId-preparatIOn affect the number of actIve firms and the buyer's optimal reservation price. The analysis IS similar.
642/58/1-4
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GUOFU TAN
If the firms' expectations over n are Bayesian consistent then on the equilibrium path the expectations among these symmetric firms are identical and the expected n is the same as the equilibrium n that is determined by free-entry conditions. In this section, I compare the equilibrium outcomes under the two sealed-bid auction rules for a given number of active firms. I will show how the number of firms is determined in the later sections. I first consider the sealed-bid FPA with a reservation price r. Given the rules of FPA, each firm invests in the R&D stage. Firm i invests Xl on R&D at a cost of C(xJ + K and privately learns the new production cost Yl (of supplying the good being procured) that is independently drawn from the distribution HCVllxJ represented by (1). A higher investment level Xl gives firm i a higher probability of observing a low production cost Yl and hence a greater chance of winning in the contract auction. In the bidding stage, each firm submits a bid that is equal to or lower than the buyer's reservation price r. The firm with the lowest bid wins the contract and gets paid according to its bid. Suppose that firm i uses a strategy BI=BI(yJ, iEN= {1, ... ,n}. I consider the Bayes-Nash equilibrium (BI(yd, ... , Bn(Yn)). Given the bidding equilibrium, each firm calculates its ex ante profit in the R&D stage, which depends on all the firms' investment levels (x I, ... , xn). A perfect eqauilibrium of this two-stage game consists of an investment strategy profile (Xl' ... , xn) and a bidding strategy profile (BI(yd, ... , Bn(Yn)) such that (BI(yd, ... , Bn(Yn)) is a Bayes-Nash equilibrium and (Xl' ... , x n ), taking into account the bidding equilibrium, is a noncooperative Nash equilibrium. A perfect equilibrium is symmetric if BI(y)=B(y) and x,=x for all iEN. Using a standard derivation, I find a symmetric perfect equilibrium (x, B( y)) that satisfies f~ G(t)(n-l)x dt B(y)= y+ . G(y)(n-l)x
for y
E
[y, r) and B(y) = r for y
r
E
[r, jiJ, and
G(tt In G(t) dt + C(x) x
(3)
= 0,
(4)
y
where X is the equilibrium investment level for each firm. On the equilibrium path, B( y) actually depends on the firms' beliefs about the equilibrium investment strategy x. PROPOSITION 1. Assume that the buyer uses the sealed-bid FP A with a reservation price r and that there are n active firms.
(a) There is a unique symmetric perfect equilibrium (x, B(y)) in the two-stage game, which is determined by (3) and (4).
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ENTRY IN CONTRACT AUCTIONS
(b) If G(y)/f(y) is weakly decreasing in y, no asymmetric perfect equilibrium exists. 9
When R&D technology is represented by a general distribution H( y I x) with H~ > 0 and Hu < 0 Vx E R ~ and y E (,Y, ),), similar results to those in Proposition 1 hold if the hazard rate [1 - H( Y Ix) J/h( y I x) is decreasing in both x and y. Now consider the sealed-bid SPA with a reservation price r. Under SPA, it is dominant strategy for a firm with a production cost lower than r to bid its true production cost. If the firm has production cost higher than r, not bidding is a dominant strategy. The lowest cost firm wins and receives a payment that equals the second lowest bid if that bid is also below rand receives r otherwise. This result does not require any assumptions about the expectations that firms have about the number of firms or their production costs. Given the dominant strategy equilibrium in the bidding stage, each firm invests in the R&D stage. Suppose that each firm has a consistent Bayesian belief that n firms will invest in R&D. Firm i is able to calculate its ex ante expected profit En,(x j , ... , xn). The buyer also calculates his expected costs EBC(x j , ••• , xn). Let .\'=L7~j x, and .\'_,=.\'-x,. LEMMA 1. Under the sealed-bid SPA with a reservation price r and a given number of firms, n,
En,(x j , ... , xn) =
f; (1- G(ty') G(t)L, dt -
C(x,) - K,
EBC(xj"",xn)='y+(Yo-r)G(r)~-(n-1)
r
G(t)~dt+
}'
i: l=
r
G(t)L'dt.
1 l
A noncooperative Nash game is played among n firms in the R&D stage given the dominant strategy equilibrium in the bidding stage. In this game, the marginal revenues of the investment on R&D are the same for all firms and depend on the total investment level x only. At the equilibrium, the marginal revenue should equal the marginal cost of the investment for each firm. Under DRS R&D technology, the marginal cost is increasing; equating marginal costs among all firms yields the symmetric equilibrium. PROPOSITION 2. Given DRS, SPA, and n active firms, there exists a unique Nash equilibrium at the R&D stage that is symmetric and is determined by (4). 9 More precisely, If we restrict ourselves to differentiable bidding strategies. then there is no asymmetric perfect eqUlhbrium in this two-stage game. In general, the sealed-bid FPA with asymmetnc bidders might have a non-dlfferenliable biddmg strategy eqUlhbnum.
48
GUOFU TAN
Given the payoff function for each firm in Lemma 1, the proof of Proposition 2 is straightforward and is omitted here (see the proof for a general result in Proposition 4). The results in Proposition 2 hold when R&D technology is represented by a general distribution function H( y Ix) with Hx > 0 and Hu < 0 'Ix E R ~ and y E (y, y). In other words, there exists a unique Nash equilibrium that is symmetric when C(x) is convex and P(ylx)= -In[l-H(ylx)] is weakly concave in x (DRS), where P( y Ix) is a production measure of the R&D process. When P( y I x) is linear in x, it can be shown that H( y 1.Xl) takes the form given by (1), which corresponds to CRS R&D technology if C(x) is also linear in x. From Lemma 1, under CRS, both the marginal revenues and the marginal costs of investment are identical for all firms and hence the equilibria are not uniquely determined. But for any (pure or mixed) strategies used by other firms, each firm has a unique best response. Thus, no mixed strategy equilibrium exists. PROPOSITION
3.
Given C RS, SPA, and n active firms, the following hold:
(a) There is a continuum of pure strategy equilibria, among which the total investment level i is uniquely determined by
r
G(t)~lnG(t)dt+c=O,
(5)
and no mixed strategy equilibrium exists. (b) Among the multiple equilibria, Enl(x b ... , x n ) is maximized at the equilibrium Xl = i, Xl = 0 Vj # i, where i is determined by (5), and EBC(x I, ... , x n ) is minimized at the symmetric equilibrium Xl = i/n Vj EN.
Among the multiple equilibria under CRS, each firm prefers a particular equilibrium at which it invests i and all other firms invest O. This is because each firm's expected profit and marginal revenue are decreasing in the other firms' investment levels. The best equilibrium for one firm is the worst for all the other. 10 The R&D game is thus very competitive since 10 When n = 2, by reordering the strategy variable for one player, e.g., letting X2 = -x 2 , the R&D game has the strategIc complementarity property (i.e., such player's strategy set is partially ordered and the marginal returns to increasing one's strategy rise with increases in the competitors' strategies). As Milgrom and Roberts [11] have shown, in this type of game the sets of pure strategy Nash equilibria, correlated equilibria, and rationalizable strategies have identical bounds. Also, the largest equilibrium is the most preferred equilibrium for player 1 and the least preferred equilibrium for player 2, while the smallest equilibrium is least preferred by player 1 and most preferred by player 2. When n > 2, the equilibrium outcomes of this R&D game have similar properties; but the game cannot be simply transformed into the strategic complementarity game.
49
ENTRY IN CONTRACT AUCTIONS
individual firms do not like the symmetric equilibrium even if they collude. More precisely, if the firms choose (Xl' ... , X n ) to maximize joint expected profits, they will choose x, = x for some i and x] = 0 for all}"# i. Then the firms probably divide the profit of firm i equally among themselves. On the other hand, the buyer prefers the symmetric Nash equilibrium since the competition in the R&D stage helps him to obtain a lower bid in the bidding stage. Society is indifferent about the equilibrium the firms play. The expected social costs depend only on the total investment level x. Comparing the equilibrium outcomes between FPA and SPA, one can see that, under DRS, both auction rules give the same equilibrium outcome in the R&D stage, the same expected profits for each firm, and the same expected cost for the buyer. Thus, FPA and SPA are equivalent under DRS. But, under CRS, there are many equilibria in the R&D stage when SPA is used and only one equilibrium (symmetric) when FPA is used. From Lemma 1, the buyer has different expected costs at different equilibria. Since the firms do not prefer the symmetric equilibrium, there is no reason to believe that the firms will play this equilibrium. Thus, the buyer has different expected costs when he uses FPA and SPA. The Revenue Equivalence Theorem in the auction literature breaks down when there is pre-auction R&D that is subject to constant returns to. scale on expenditure. SPA, given homogeneity, is very sensitive to pre-auction R&D activities. However, SPA works well when firms are heterogeneous. Assume that firm i has R&D technology that is represented by H, (y Ix), Y E (y, y), x E R~ and that all firms have the same cost function C(x) + K. Assume, further, H x, > 0 and H"x, < 0 for all y E (y, y) and x E R~. Let P,(ylx)= -In[l-H,(Ylx)]. If, for some i, ((oP,(ylx»/ox» ((oP](ylx»/ox) for all YE(Y,y), xER~, and }"#i, and C(O) < ((oH, (t IO»/ox) dt, then firm i is called a dominant firm.
g,
PROPOSITION
4.
the R&D stage
Under SPA, there exists a unique Nash equilibrium in conditions are satisfied:
if the following
(a) P,(Y I x) is weakly concave with respect to x and iEN, and C(x) is convex in xER~; or
(b)
E R~
for all y
E
(y, y) -
(( oP 1(y Ix) )/ox), ... , (( oP n( y Ix) )/ox) are linearly independent; or
(c) P,(ylx) and C(x) are all linear in xER~ for all YE(Y, y) and i EN, and there exists a dominant firm. Moreover, if firm i is a dominant firm then x I> Xl ~ 0 V}"# i at the equilibrium. Under condition (c), in particular, Xl = 0 V}"# i.
Since PI (y I x) represents an output measure of firm i's R&D process, condition (a) means that the R&D process for each firm exhibits
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GUOFU TAN
diminishing returns to scale. Under DRS, even though the firms are heterogeneous, there is only one Nash equilibrium. Alternatively, if the productivity measures (oP,/ox) among all firms are linearly independent even if (a) is not satisfied, the Nash equilibrium is also unique. Given sufficient heterogeneity among firms, the multiplicity of Nash equilibria disappears even if R&D technology is subject to CRS. When ((oP,(ylx))/ox»((oPj(ylx))/ox for all y and x, firm i has a higher marginal product of the investment in the R&D process than has firm j. This means that firm i has a better R&D technology. For instance, this condition implies H, (y Ix) stochastically dominates Hj (y I x) when both P, and Pj are linear in x. When SPA is used, in equilibrium, the dominant firm always invests more on R&D than other firms. In fact, under CRS the dominant firm invests and the rest of the firms do not. In summary, firms with better technology invest more and hence have a higher probability of winning the contract, and the firm with the lowest production cost always wins the contract. This is an efficient outcome. In contrast, if the firms are heterogeneous, the noncooperative equilibria of FPA are usually not symmetric and are difficult to analyze. This implies that the firm with a lower bid is not the same firm with lower production costs. Thus, efficient allocations need not be achieved under FPA and SPA is preferable when there are private pre-auction R&D activities among heterogeneous firms.
4.
FREE-ENTRY PERFECT EQUILIBRIUM
I turn now to the question of how many firms will invest in the R&D stage and compete for the contract. Let n(r) == Maxx{f~ [1- G(W] dtC(x) - K} be the maximum profits when there is only- one firm participating in R&D activity and bidding for the contract with the reservation price r. I start with the assumption n(r) > for a given reservation price (i.e., at least one firm is interested in R&D and procurement contract). With free entry, more firms will enter the R&D and bidding process. When R&D technology is subject to CRS, multiple perfect equilibria exist under SPA. But the total investment level x is uniquely determined by (5) and is independent of the number of firms. Among the equilibria, from Lemma 1, firm i has the profit:
°
En,(xj,x)=
r
G(t)~-X'dt-r G(tYdt-cx,-K.
!'
!'
Since En,(O, x) < 0, En,(x, x) = n(r) > 0, and En,(x" x) is increasing in x" there exists Xo E (0, x) such that En,(x o , x) = 0, where Xo is also
ENTRY IN CONTRACT AUCTIONS
51
independent of the number of firms. Each active firm invests X,? Xo such that L7~ 1 X, =.x and the equilibrium number of firms, n e , is less than or equal to x/x o. When R&D technology is subject to DRS, under either FPA or SPA, the perfect equilibrium is unique and symmetric and each firm has the same expected profits:
Enn(x,r)=
t
(l-G(ty)G(t)(n-llxdt-C(x)-K.
(6)
For brevity, I refer to FPA in the following discussions, but the arguments apply equally to SPA. With free entry, each potential firm enters the R&D process if its expected profits are nonnegative. That is, equilibrium entry gives (7)
and any additional entrant n + 1 earns negative profits,
Enn+1(x',r)<0,
(8)
where x' is the individual R&D expenditure that is determined by (4) when n + 1 firms simultaneously enter the R&D process. Entry decisions are simultaneous, not sequential. For any given reservation price, r, Eqs. (4), (7), and (8) determine the free-entry pure-strategy symmetric perfect equilibrium (ne' x e) with the bidding function in (3). In order to show the existence and uniqueness of a free-entry purestrategy symmetric perfect equilibrium for a given reservation price, I first allow the number of firms n to be a continuous variable. Each firm enters the R&D process until its expected profit is zero and hence the equilibrium-entry conditions (7) and (8) can be represented by the equation,
f [1- G(tYJ G(t)(n-l)x dt - C(x) - K = 0. r
(9)
l'
Then (n e, xe) is determined by (4) and (9). PROPOSITION 5. Suppose n(r)? 0, then there exists a unique solution (ne' xe) with ne E [1,00) and Xe E (0, 00). Furthermore, let xe == neXe' then
~
(a)
(oxe/oK)
(b)
(OXe/or) >0, (OXe/or) <0, and (one/or) >0. 11
0, (oxe/oK) > 0, and (one/oK) < 0;
11 If the solutIOn ne above is an integer, each informed firm earns exactly zero expected profit at the equilibrium. If ne is not an integer, the equilibrium can be easily adjusted. See
Tan [17J for the discussion.
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GUOFU TAN
If at least one firm enters the R&D and bidding process then free entry results in more active firms. The expected profits for each firm decrease as the number of active firms increases and, thus, competition makes the procurement contract less profitable to each firm. As the number of active firms goes to infinity, each firm invests zero on R&D and expects to earn a profit - K. Therefore, if there is a positive fixed cost of R&D then only a finite number of firms decide to enter the R 84 D process. The fixed cost of R&D, K, is the key determinant of the free-entry equilibrium number of firms, although the latter is also affected directly or indirectly by the variable cost of R&D, C(x), the reservation price, r, and the distribution of production costs, H(y I x). For a given number of active firms, the distribution of the production costs y of the winning firm is 1- G(ytx. From (3) and (6), the buyer has the following ex ante expected costs in the competitive procurement,
t
B(y)d(1-G(y)nx)=
f
yd(1-G(ytX)+nC(x)+nK+nEnn(x,r),
(10)
where Enn(x, r) is a firm's expected profits under the perfect equilibrium defined by (6). The buyer's expected costs include the expected minimum production cost, the total R&D costs among all firms, and the total expected profits among all firms. With free entry, each firm enters the R&D and bidding processes until its expected profit Enn(x, r) equals zero. The buyer has to pay not just the expected minimum production cost, but also the total R&D cost neC(x e ) + neK of all informed firms. The latter result may seem counterintuitive: one expect that the buyer has only to pay the R&D costs of the winner. However, since firms are symmetric and adopt the same investment and bidding strategies, each has an equal probability to be the winner. Therefore, the buyer actually ex ante expects to pay all of the costs of R&D among active firms. The winning firm's expected profits g,(B(y)-y)d(l-G(yt X ) from the bidding equal the total costs on R&D among all of the firms. In other words, if free entry is allowed, the rents for the firms from contracting are dissipated by precontract R&D competition. One wonders whether these R&D activities are good for the buyer and society. I turn now to address this question.
5.
THE BUYER'S OPTIMAL STRATEGY
In this section, I go back to the first stage of the three-stage game and look at the buyer's optimal decision problem. ,One wants to know whether there exists a reservation price under which the free-entry perfect
53
ENTRY IN CONTRACT AUCTIONS
equilibrium characterized in Section 4 is optimal for the buyer. In other words, if the buyer can select the number of firms, can his optimal selection be supported by free entry? Remember that, under FPA, the buyer actually procures the good at a cost, Yo, elsewhere with probability G(rtx. Thus, the buyer's total expected costs, EBC, consist of (10) and yoG(rYx. More explicitly, EBC(r,n,x)=}'+CYo-r)G(ryx+ n
I;
G(t)(n-1)Xdt-(n-1)
t
G(ttxdt. (11 )
The buyer wants to minimize EBC(r, n, x) by selecting r, n, and possibly x. In general, the buyer may require each active firm to invest more than it wants to. This implies that there exists a moral hazard problem if the buyer is unable to observe or verify the R&D decision, x. He has to take each firm's R&D decision as a constraint and should then minimize his costs (11) subject to the R&D constraint (4) and the participation constraint (7). Suppose (rb' n b, Xb) is the solution to the buyer's optimization problem and Xs is the investment level that minimizes the total expected social costs. PROPOSITION
6.
(a)Ennb(xb,rb)=O, and (b)
if n(yo»O then rb
and Xb < Xs'
Proposition 6 implies that, at the buyer's optimum, the firm's nonnegative profits constraint (7) is binding. Therefore, under free entry the buyer is able to control the number of active firms and the R&D investments by offering an appropriate reservation price tb' Given the selection of rb , the number of active firms and the R&D investments are simultaneously determined as the free-entry perfect equilibrium characterized in Section 4. The buyer will select a reservation price, r b , lower than the opportunity cost, Yo, and extract surplus from firms whenever the surplus exists. As a result, each firm invests less aggressively, which causes underinvestment in the industry relative to the social optimum. 12 As Tan [17] has shown, the social planner who minimizes the total expected social costs of procurement prefers only one firm to conduct all of the R&D and production when R&D technology is subject to constant returns to scale on expenditure. Since CRS R&D activity is an independent search process, the R&D outcomes of n firms are equivalent to the outcome of one firm that invests the same amount as all n firms. However, because of fixed costs of investing, more firms participating in 12 See Tan [17] for diSCUSSIOns on a comparison between the buyer's optimal strategy and the social optimum.
54
GUOFU TAN
R&D activities imply higher total costs of R&D. For society, one firm conducting all of the R&D and production is more efficient than several firms and competitive bidding is not necessarily socially efficient. This result contrasts with the conventional view that it is always more efficient to award procurement contracts through a very competitive procedure (see Dasgupta and Spulber [2], Riordan and Sappington [13], among others for more discussions on the advantage of competitive procedures). The fact that it is costly for bidders to acquire similar information before bidding reduces the attractiveness of competitive mechanisms. In many cases of defense procurements, Congress requires the DOD to award the contract in very competitive ways. The above analysis implies that the optimal procurement and R&D policies from the point of view of the buyer and society can be very different under two types of R&D technology. When deciding to choose auction rules and to promote more bidders, both Congress and the DOD should consider the objectives they pursue, the types of projects they procure, and the types of R&D technologies potential firms face.
6.
CONCLUSION
Most results in the theory of auctions and procurement contracting depend on the assumption that there is an exogenously fixed number of well-informed bidders. But potential bidders are often not well informed initially. Different auction mechanisms will in general provide potential bidders different incentives to acquire private information and to enter the competitive bidding. This fact will in turn affect the design of the auction mechanisms. In this paper, I present a model of procurement contracting with entry and private R&D investments. I find that the nature of the private R&D processes plays an important role in determining both the performance of the different auction rules and the optimal procurement policies for the buyer. When the R &D process is subject to diminishing returns to scale in expenditure, FPA and SPA are equivalent as in earlier work. But the Revenue Equivalence Theorem breaks down when the R&D process is subject to constant returns to scale. This is true because multiple equilibria arise when SPA is used; the firms do not select the equilibrium that the buyer prefers. The buyer as a profit maximizer prefers free entry to the industry in general. However, underinvestments in the industry relative to the social optimum usually arise due to the monoposony power of the buyer. While FPA has an advantage over SPA when potential bidders are ex ante homogeneous, SPA works very well when the bidders are ex ante heterogeneous. The bidder with a better R&D technology invests more
ENTRY IN CONTRACT AUCTIONS
55
and is more likely to observe a better R&D outcome (low production costs). Each firm also bids its true production cost in SPA and, in this sense, SPA works efficiently. The interesting issue is then the optimal mechanism design when potential bidders are not well informed ex ante and each may decide to acquire more information before the bidding. The number of actual bidders should be endogeneously determined with the optimal mechanism.
ApPENDIX
Proof of Proposition 1. I prove (b) in the case n = 2. Consider a differentiable equilibrium bidding strategy B, (y,) of firm i, i = 1, 2. It is easy to establish that, B,(y,) is monotonically increasing in y" BI(r) = B 2(r) = r, and B I (1)=B 2(1). Let l!=B,(l) and ¢J,(b)=B,-I(b) VbE[l!,r). Then ¢Ji(b) is also increasing and differentiable, ¢JI(r) = ¢J2(r) = r, and ¢JI(l!) = ¢J2(l!) = }'- The profit for firm 1 of type y I to bid b is
The first order condition yields (12) where I(y) = G(y)/f(y}. Similarly, firm 2 bids b such that (13 )
Since ¢J:(b) >0, (12) and (13) imply that, VYE(l,r), ¢J,(b)
(14 )
Claim 1. Suppose XI ¢JI (b) Vb E [l!, bolo This contradicts ¢J2(l!) = ¢JI(l!), i.e., ~boE (l!, r) such that ¢JI(b o) = ¢J2(b o)· Second, since ¢J2(l!) = ¢JI(l!) < l!, (14) again implies ¢J;(l!) < ¢J~(l!). Then ¢J I (b) > ¢J2( b) for b > l! and b close l! enough. The continuity of ¢J i (b) then implies ¢Jl(b»¢J2(b) VbE(l!, r) and Claim 1 follows.
56
GUOFU TAN
The same argument shows: Suppose x, > x 2, then B,(y) > B 2(y) Vy E (y, r). Now, using the Envelope Theorem, I obtain that, VYE (y, r),
Claim 2.
(15) Given the bidding equilibrium (B,(yd, B 2 (Y2)), the ex ante expected profit for firm 1 is En,(x"x 2)=
=
r
n,(B,(y,), yddH(y,lx,)-C(xd-K
r
[1- G(tyl] G(¢J2(B,(t))Y2 dt - c(xd - K,
}'
where the second equality holds because of (15), n,(r, r) = 0, and integration by parts. The first order condition with respect to x, yields
-r
G(tYIG(¢J2(B 1 (t))yZin G(t) dt- C(xd =0.
(16)
y
Similarly, firm 2 chooses X2 such that
If x,
< -G(t)XIG(¢JAB,(t))Y2In G(t), VtE (y, r). (18)
Integrating both sides of (18) violates the equilibrium conditions (16) and (17). Similarly, x, >x 2 also leads to a contradiction. Therefore, x, =x 2 , which also implies B,(y) = B 2 (y) Vy E (y, r]. I Proof of Lemma 1. The expression of En;(x" ... , x n ) is easy to establish and I only derive EBC(x I, ... , x n ) here. Let yl and y2 be the lowest and the second lowest order statistics of (YI, ... , Yn). It can be shown that the joint distribution of yl and y2 has a density function
if yl ~ y2, otherwise,
57
ENTRY IN CONTRACT AUCTIONS
EBC(xj, ... , xn) =
I; t
f'
y2p(y\ y2) di dy2 + r (
p(yl, y2) di dy2 + yoG(r)X,
where the first term can be written as
I; ,t!
(rlJf,(r)- yllJf,(yl)-
= =
rI rf
yllJf;(i)H(ilx,)di
!
,~I
y
1=
=y-r
(1 1Jf,(y2) dy2 ) h(i Ix,) di
i·x_,G(yl),,-,-lf(i)di-(n-l)x
r
iG(yly,-lf(yl)di
r
1
I
G(r)~-'+
r f G(i)~-'di+(n-l)rG(r)X }'
,~I
,~I
and the second term is r
n
n
,~l
,~l
L H(rlx,)G(r)L'=r L G(r)~-'-nrG(r)~.
Combine these three terms of EBC(x 1 ,
... ,
x n ) and Lemma 1 follows.
I
Proof of Proposition 3. It is easy to establish (a) and the first part of (b). I prove the second part of (b) here. From Lemma 1, given X, minimizing EBC(x I ' ... , xn) subject to L7~ I x, = x is equivalent to minimizing
Ln fr G(tt-x'dt ,~I
subject to
L7~
I
(19)
!
x, = x. The first order conditions yield - ( G(tY-x, in G(t) dt - JI. = 0,
"liEN,
(20)
where JI. is the Lagrange multiplier. Equation (20) implies x, = Xl Vi,} EN. Since (19) is a convex function of (XI' ... , Xn), EBC(x 1 , ... , x n) is minimized at the symmetric equilibrium Xl = x/n V} EN. I
58
GUOFU TAN
Proof of Proposition 4.
En,(x I ,
••• ,
xn) =
r
Similar to Lemma 1, firm i has a payoff function
H,(t I·')
}'
TI
[1- H;(t 1x)] dt - C(xJ - K.
}#l
Since H~,\,
(21 )
where ¢(tlxl, ... ,xn)=Il;~1 [1-Hj (tlx)J. The Jacobian of the system (21) is symmetric, and negative definite Vx, ~ 0, i EO N under the assumption (a) or (b), i.e., Vv EO R n and v # 0
"
L.
',jEN
I -(L, O v, OP,(t XJ)2) dt - LV,C"(X,) X, , r
(12En, (" Jo2p,(tlxJ t',v]-;;--;-= ,¢(tlxl, ... ,xn) L. v; 0. 2 uX, uX j
X,
'EN
}
1
<0. By Rosen (1965) Theorem, there exists a unique Nash equilibrium of the R&D game. Now, if firm i is a dominant firm, at the equilibrium (x I, ... , x n ), it cannot be xj=O Vi because of the assumption C(O)< -g, ((oH,(tIO))/ox) dt. If x] > 0 for some i, then (21) holds for i and hence -
I
r oEn, (OP,(tIXJ -",-=/fi(tIXl""'X n ) 0 ux, } x,
OPj(tIX)) i3 dt-C(xJ+C(x) Xj I
I
is positive at x, = x)' Thus, at the equilibrium, x, > Xj ~ 0 Vi # i. Under condition (c), let i be the dominant firm and (XI' ... , xn) be an equilibrium, then x, > x; ~ 0 Vi # i. Suppose Xj > 0 for some i, then (21) holds for i and hence
r oEn'=f d.( 1 ) (OP,(tIXJ 'f' t Xl' ... , Xn '" ox; }' uX,
(1Pj (t IX)) d '" UX;
0 t> ,
i.e., x, cannot be an equilibrium strategy for firm i. This contradiction shows that x; = 0 Vi # i, x, is the unique response and is determined by
Irop,(tlxJdt_C(xJ=O. ox, y
I
59
ENTRY IN CONTRACT AUCTIONS
Proof of Proposition 6. I treat n E [1, + 00) as a continuous variable. Since E7r n(r, x) is continuously differentiable with respect to r E [y, y], nE [1, +00), and xER~, the constraints (4) (and (7) form a compact set S of (r, n, x). EBC(r, n, x) is continuous with respect to (r, n, x) E Sand hence there exists a solution (rb' n b, Xb) to the buyer's optimization problem. The constraint (7) and K> 0 imply X h > O. Let L== -EBC(r,n,X)+AE7r n(r,X)+,u( where Then
A~O
-L
G(ttxlnG(t)dt-C'(X)),
and,u are the Lagrange multipliers for constraints (7) and (4).
oL = nxf(r) G(r) G(rt x - 1 or
lyO _(r + (1 _~) H(r Ix) _ t: HAr IX))]. n h(rlx) n h(rlx)
Since Yo>}', limr~y+.(oL/or»O for small B>O. Then rb>Y' The first order conditions with respect to n and x yield oL on = -xCvo - r) G(rt' In G(r)
-r
G(t)(n-l)x[l_ G(ty + xG(tY In G(t)] dt
-'
+ X(A - n) - x,u
r
r
G(t)(1l-1)XH(t I x) In G(t) dt
y
G(tr X In 2 G(t) dt
}'
:( O.
(22)
oL ox = -n(yo- r) G(r)nx In G(r) + (n -l)(A -n)
x
r
G(t)(n-l)x H(t Ix)ln G(t) dt - n,u
!
r
G(tr x In 2 G(t) dt - ,uC"(x)
!
= O. If rb=YO'
(23)
then
(22)
and
(23)
imply
,u~O,
A
hence
< 0 for small B > 0 because of yo:(.v. Thus rb < y. The first order condition with respect to r becomes limr~v_e(oL/or)
yo=r+(l-~) H(rlx) _t:H,(r1x). n h(rlx)
n h(rlx)
(24)
60
GUOFU TAN
Suppose A= 0, then (23) and (24) imply J.1 >
-aL = an
fr G(t)(n-l)x[l -
°
and J.1X
G(tY + x in G(t)] dt + - C(x) > 0,
n
!
which contradicts (22). Therefore A> 0, which implies ETCnb(r b, Xb) = 0. Now suppose rb~Yo, then (23) and (24) imply A~nb and J1~0. Then (22) holds with inequality, which implies nb = 1. But n(yo) = ETCt(Yo, Xb) = that violates the assumption. Therefore, rb < Yo. I
°
REFERENCES
1. S. M. BESEN AND K. L. TERESAWA, "An Introduction to the Economics of Procurement," Working draft-3555-PA & E, The Rand CorporatIOn, 1987. 2. S. DASGUPTA AND D. F. SPULBER, Managing procurement auctions, In! Econ. Policy 4 (1990), 5-29. 3. K. R. FRENCH AND R. E. MCCORMICK, Sealed bids, sunk costs, and the process of competition, J. Bus. 57 (1984), 417--442. 4. K. HENDRICKS, R. H. PORTER, AND B. BOUDREAU, Information, returns, and bidding behavior in OCS auctions: 1954-1969. J. Ind. Econ. 35 (1987), 517-542. 5. L. LI, R. D. MCKELVEY, AND T. PAGE, Optimal research for Cournot oligopolists, J. Econ. Theory 42 (1987), 140--166. 6. F. LICHTENBERG, The pnvate R&D investment response to federal design and technical competitions, Amer. Econ. Rev. 78 (1988), 550--559. 7. R. P. MCAFEE AND J. McMILLAN, Auctions and bidding, J. Econ. Lzt. 25 (1987), 699-738. 8. R. P. McAFEE AND J. McMILLAN, Auctions with entry, Econ. Lett. 23 (1987), 343-347. 9. R. P. MCAFEE AND J. McMILLAN, Auctions with a stochastic number of bidders, J. Econ. Theory 43 (1987), 1-19. 10. S. A. MATTHEWS, Comparing auctions for risk averse buyers: A buyer's point of view, Econometrica 55 (1987), 633-646. 11. M. MILGROM AND J. ROBERTS, Rationalizabihty, learning and equilibrium in games with strategic complementarities, Econometrica 58 (1990), 1255-1277. 12. J. F. REINGANUM, The timing of innovation: Research, development and diffusion, in "Handbook of Industrial Organization" (R. Schmalense and R. D. Willig, Eds.), NorthHolland, Amsterdam, 1989. 13. M. H. RIORDAN AND D. E. SAPPINGTON, Awarding monopoly franchises, Amer. Econ. Rev. 77 (1987), 375-387. 14. R. ROB, The design of procurement contracts, Amer. Econ. Rev. 76 (1986), 378-389. 15. W. F. SAMUELSON, Competitive biddings with entry costs, Econ. Lett. 17 (1985), 53-57. 16. G. TAN, "Incentive Procurement Contracts with Costly R&D," Caltech Social Sciences Working paper 702, 1989. 17. G. TAN, "Entry and R&D Costs in Procurement Contracting," Working paper 14-91, University of British Columbia, 1991. 18. X. VIVES, Duopoly informational equilibrium: Cournot and Bertrand, J. Econ. Theory 34 (1984), 71-94.
58, 41-60 (1992)
Entry and R&D In Procurement Contracting GUOFU TAN*
Department of Economics, The UniveYSlty of Brillsh ColumbIa, Vancouver, B. C. Canada, V6T 1Z1
Received November 14, 1989, revised October I, 1991 A model of competltlve procurement and contractmg is presented in thiS paper. The key features of the model include precontract R&D, an endogenous number of homogeneous firms, and sealed-bid auctIOn rules. Both diminishing and constant returns to scale (DRS and CRS) in R&D expenditure technologies are considered. I find that the sealed-bid first-price auction (FPA) is eqUivalent to the second-price auction (SPA) under DRS technology. The Revenue EqUivalence Theorems breaks down under CRS because multiple equilibria arise m the R&D stage when SPA IS used However, SPA yields a unique perfect equilibrium given suffiCient heterogeneity among potentlal firms. The free-entry perfect eqUilibrium IS also characterized. The buyer prefers free entry through an appropnate selectIOn of the reservatIOn pnce Journal of Economic Lllerature Classification Numbers' D44,
D82. H57, 032. ,(,
1992 AcademIC Press. Inc
1. INTRODUCTION
The existing literature on auctions and procurement 1 typically assumes that the number of bidders is exogenous and constant 2 and that each bidder has certain private information. Auction mechanisms are designed to elicit the bidders' information and to maximize the auctioneer's expected
* Fmancial support from the Sloan FoundatIOn m the form of an Alfred P. Sloan Doctorial DissertatIOn Fellowship and the Jet Propulsion Laboratory of Caltech IS gratefully acknowledged. I thank my adviser, John Ledyard, for his encouragement and many suggestions, and an associate editor and a referee for helpful comments. I also benefited from discussions with Charles Blackorby, Richard Boylan, DaVid Green, Ken Hendricks, Preston McAfee, Thomas Palfrey, Michele Piccione, and Daniel Spulber. 1 See McAfee and McMillan [7] for a survey on auctions and bidding and Besen and Terasawa [1] for a survey on procurement contracting. 2 The exceptIOns are French and McConmck [3] and McAfee and McMillan [8] which consider fixed entry costs and entry equilibria. McAfee and McMillan [9] allow the number of actual bidders to be stochastic, but the probability of any subset of potential bidders becommg the set of actual bidders is assumed to be exogenous and Independent of their types. They have shown that the optimal auction IS the same whether or not the risk-neutral bidders know who their competitors are. Considering stochastic number of risk-averse bidders, Matthews [10] has offered a comparison study of auctIOn mechamsms from a bidder's point of view. 41 0022-0531/92 $5.00 Copynght
CO
1992 by AcademIC Press, Inc
All nghts of reproductIOn
III
any form reserved
42
GUOFU TAN
payoff. As the number of bidders increases, the auction mechanisms become more attractive to the auctioneer. 3 The determination of the number of bidders is therefore of central importance. Clearly, that determination is related to decisions by potential bidders on whether to acquire information before competitive bidding. This paper examines how such decisions are made under different auction rules. There are many procurement situations in which only one, or a small number, of potential suppliers are plausible candidates to be the prime contractor. In other words, only a small number of firms find the procurement contract profitable given the current technology and other information. The current attempt by the U.S. Air Force to procure a new jet fighter is a case in point. Congress required that the contract be awarded through competitive bidding, but only two groups of military contractors found it worthwhile to compete. Each of those two groups invested over $600 million of its own funds in research and development (R & D) prior to bidding. 4 A further example is found in Hendricks, Porter, and Boudreau [4]. In their study of the federal auctions for leases on the outer continental shelf, they observed that potential firms do decide how much information to collect before participating in competitive bidding and that only a fraction of the set of potential bidders typically choose to submit bids in any auction. Standard analysis suggests certain relationships between precontract R&D and contract bidding. In general, higher costs of R&D reduce the number of competing bidders. The decisions to acquire information and to enter the bidding competition depend on the nature of R&D processes of the firms and the type of bidding procedure in place. Further, if fewer firms participate in the competitive bidding, the contract is more profitable to the winning firm and each firm invests more in R&D. But if the expected profit of the winning firm is high, more firms enter the R&D process and compete for the contract. These arguments suggest that the buyer, being interested in the number of bidders, may want to affect the firms' precontract R&D behavior. One potential mechanism for this is the choice of contract auction rules. In this paper, I examine the question of whether different auction rules yield the same equilibrium outcome given precontract R&D behavior. In addition, I answer a set of related questions. What is the equilibrium number of
3 See RIOrdan and SappIngton [13] and Dasgupta and Spulber [2] for diSCUSSIOns on procurement contracts In the case of a downward-sloping demand. 4 Each also received $691 million from the AIr Force for deSignIng the prototypes for a new fighter jet See the New York Times (December 27, 1989) for the report in detail. For other examples about private R&D Investment response to pubhc defence procurement, see Lichtenberg [6].
ENTRY IN CONTRACT AUCTIONS
43
bidders under free entry and how does each potential firm make precontract R&D decisions? Is free entry of firms an optimal policy for the buyer? Moreover, is it socially optimal? The relationship between R&D and procurement has also been studied in Rob [14]. In that paper, Rob formally models R&D activity in procurement as searching behavior. He emphasizes the second sourcing problem, using a model in which the government decides whether to bring in a new round of bidding after the initial contract has been awarded. In contrast, I concentrate on an environment without second sourcing where a number of firms make R&D-related decisions both on whether to enter a competition and on how much to bid. I model the R&D behavior of a firm as an one-shot stochastic process with certain costs of R&D. 5 Given certain auction rules, each firm that decides to enter the R&D process invests in R&D and privately observes its production costs. Based on the R&D information, firms submit bids and compete for the production contract. A model of competitive procurement with precontract R&D is presented in Section 2. The number of firms is viewed as an endogenous variable in the model and each firm is assumed to be risk neutral. Both constant and diminishing returns to scale (CRS and DRS) in R&D expenditure technologies are considered. In Section 3, I compare the perfect equilibrium outcomes in the two-stage game when the sealed-bid first-price auction (FPA) and the sealed-bid second-price auction (SPA) with the same reservation price are used by the buyer. I find that the two auction rules are equivalent when R&D technology is subject to DRS. The Revenue Equivalence Theorem breaks down under CRS R&D technology. In this latter case, if SPA is used, multiple equilibria of R&D decisions arise because the R&D outcomes of n firms together are indentical to the R&D outcomes of one firm that invests the same amount as all n firms. In general, SPA yields a unique perfect equilibrium outcome given sufficient heterogeneity among potential firms. Thus, SPA works better than FPA when potential bidders are heterogeneous in the R&D stage. Section 4 examines free-entry perfect equilibrium for a particular reservation price of the buyer. The equilibrium number of firms and the R&D expenditure are simultaneously determined and depend on the R&D process of the firm and the buyer's reservation price. Section 5 analyzes the buyer's optimal strategy. The buyer always prefers free entry of firms and 5 Tan [16] has also incorporated this type of R&D activity into competitive procurements for a given number of firms and has characterized the equilibrium investment level on R&D and the optimal procurement contract. There are other ways to model R&D activity, e.g., see Li, McKelvey, and Page [5] and Vives [18] for the discussions on mformation acquisition m the oligopoly theory.
44
GUOFU TAN
selects a particular free-entry perfect equilibrium by choosing a relatively low reservation price. As a result, each active firm invests less, which causes underinvestment in R&D in the industry relative to the social optimum.
2. THE MODEL
The model I consider contains a single buyer (e.g., the government) who seeks to procure one unit of a certain novel good or service. The buyer is assumed to minimize the expected total costs of this procurement. The buyer is also assumed to face many potential suppliers; each of which is risk neutral and can produce a unit of the good at a potential unknown cost y. Each firm can invest in R&D for information about cost reduction and will privately observe a potential cost y which is drawn from the same random distribution
H( Y Ix) = 1 - [1 - F( y) ] x
(1)
with the fixed support [y,yJcR~, y>y, where F(y) is a continuously differentiable cumulative distribution function over [y, yJ and density function f( y), and x E R~ is the level of investment in R&D. 6 For convenience I define the function G( y) == 1 - F( y) for all y. The R&D cost function is assumed to be the same for all firms and is represented by C=C(x)+K,
(2)
°
where K> is the fixed cost of R&D and C(x) is the variable cost which is assumed to be twice differentiable in x. I also assume C(o) = 0, C' > 0, and Crt ~o. When C( x) = ex and e > 0, the R&D activity can be viewed as an independent experimental search process. For example, if the firm invests one unit (or one experiment), i.e., x = 1, a cost level y 1 is observed from the distribution F(yd at the cost e + K. Ifthe firm repeats this experiment x times (x is an integer now), each additional experiment is independent of the others and costs e. Then x numbers of production costs (Y1, ... , Yx) are observed at the cost ex + K. The minimum cost level y of (y 1, ... , Y x) is subject to the distribution of the lowest order statistic, which takes the form of (1). This independent experimental process exhibits constant returns to R&D scale CRS. When crt > 0, the cost function of R&D is strictly 6 We can also think of x as the number of R&D projects or different levels of R&D activities.
ENTRY IN CONTRACT AUCTIONS
45
convex. Increasing R&D scale becomes more and more expensive. In this case, R&D activity exhibits DRS.7 Competitive procurement is modeled as a three-stage process in this paper. In the first stage, the buyer announces and commits to the general rules of procurements. I consider both a sealed-bid FPA and a sealed-bid SPA with announced reservation price. In the second stage R&D is conducted, with each firm investing in R&D and acquiring information about the production cost. In the final stage, a competitive bidding procedure is conducted in which the buyer procures the good via the sealed-bid auction announced at the beginning. 8 The process can be described more specifically as follows. First, the buyer announces the rules of the sealed-bid auction (either FPA or SPA) including a reservation price r that is no higher than the highest possible cost level y. The lowest bid will be accepted unless it is below r. Second, each firm will calculate its expected profit from bidding and decide to invest in R&D if this profit is no less than its R&D costs. The active firms playa noncooperative Nash game in the R&D stage given that a Bayesian-Nash game is played in the bidding stage. The winner is chosen via the sealed-bid rule as the contractor for production. The contractor produces the good and gets paid according to the contract. Cost reduction during the production is not considered. The buyer is able to procure the good elsewhere at the cost Yo if the lowest bid is higher than the reservation price r. One can consider an extreme case in which Yo is very high, meaning that there are no substitutes available for the buyer and Yo does not play any role in the analysis. I consider the general case where Yo E Cv, y], whic means that the procurement is more competitive. Finally, the structure of the game is common knowledge.
3. A COMPARISON BETWEEN FPA AND SPA Given the rules of the sealed-bid auction (FPA or SPA) with a reservation price r, suppose that a firm believes that n firms including itself might invest in R&D and compete for the procurement contract in the auction. 7 One can always rescale the Illvestment varIable by lettlllg .x = K + C(x). then x = li(x), li' > O. and li" ,::; O. The R&D technology can then be represented by a random dlstnbution
H(yl.'i')=I-G(yl"t'I. In the speCIal case when G(Yl=e-> for YER~, one gets H(ylx)= l-e-hl'l" that IS an exponential distrIbution and IS widely used in the literature
on stochastIC R&D races (see Reinganum [12] for a survey). 8 In an early versIOn of thIS paper, I also allow each firm to have a bid-preparation cost SImIlar to that III Samuelson [IS]. Both the costs of R&D and costs of bId-preparatIOn affect the number of actIve firms and the buyer's optimal reservation price. The analysis IS similar.
642/58/1-4
46
GUOFU TAN
If the firms' expectations over n are Bayesian consistent then on the equilibrium path the expectations among these symmetric firms are identical and the expected n is the same as the equilibrium n that is determined by free-entry conditions. In this section, I compare the equilibrium outcomes under the two sealed-bid auction rules for a given number of active firms. I will show how the number of firms is determined in the later sections. I first consider the sealed-bid FPA with a reservation price r. Given the rules of FPA, each firm invests in the R&D stage. Firm i invests Xl on R&D at a cost of C(xJ + K and privately learns the new production cost Yl (of supplying the good being procured) that is independently drawn from the distribution HCVllxJ represented by (1). A higher investment level Xl gives firm i a higher probability of observing a low production cost Yl and hence a greater chance of winning in the contract auction. In the bidding stage, each firm submits a bid that is equal to or lower than the buyer's reservation price r. The firm with the lowest bid wins the contract and gets paid according to its bid. Suppose that firm i uses a strategy BI=BI(yJ, iEN= {1, ... ,n}. I consider the Bayes-Nash equilibrium (BI(yd, ... , Bn(Yn)). Given the bidding equilibrium, each firm calculates its ex ante profit in the R&D stage, which depends on all the firms' investment levels (x I, ... , xn). A perfect eqauilibrium of this two-stage game consists of an investment strategy profile (Xl' ... , xn) and a bidding strategy profile (BI(yd, ... , Bn(Yn)) such that (BI(yd, ... , Bn(Yn)) is a Bayes-Nash equilibrium and (Xl' ... , x n ), taking into account the bidding equilibrium, is a noncooperative Nash equilibrium. A perfect equilibrium is symmetric if BI(y)=B(y) and x,=x for all iEN. Using a standard derivation, I find a symmetric perfect equilibrium (x, B( y)) that satisfies f~ G(t)(n-l)x dt B(y)= y+ . G(y)(n-l)x
for y
E
[y, r) and B(y) = r for y
r
E
[r, jiJ, and
G(tt In G(t) dt + C(x) x
(3)
= 0,
(4)
y
where X is the equilibrium investment level for each firm. On the equilibrium path, B( y) actually depends on the firms' beliefs about the equilibrium investment strategy x. PROPOSITION 1. Assume that the buyer uses the sealed-bid FP A with a reservation price r and that there are n active firms.
(a) There is a unique symmetric perfect equilibrium (x, B(y)) in the two-stage game, which is determined by (3) and (4).
47
ENTRY IN CONTRACT AUCTIONS
(b) If G(y)/f(y) is weakly decreasing in y, no asymmetric perfect equilibrium exists. 9
When R&D technology is represented by a general distribution H( y I x) with H~ > 0 and Hu < 0 Vx E R ~ and y E (,Y, ),), similar results to those in Proposition 1 hold if the hazard rate [1 - H( Y Ix) J/h( y I x) is decreasing in both x and y. Now consider the sealed-bid SPA with a reservation price r. Under SPA, it is dominant strategy for a firm with a production cost lower than r to bid its true production cost. If the firm has production cost higher than r, not bidding is a dominant strategy. The lowest cost firm wins and receives a payment that equals the second lowest bid if that bid is also below rand receives r otherwise. This result does not require any assumptions about the expectations that firms have about the number of firms or their production costs. Given the dominant strategy equilibrium in the bidding stage, each firm invests in the R&D stage. Suppose that each firm has a consistent Bayesian belief that n firms will invest in R&D. Firm i is able to calculate its ex ante expected profit En,(x j , ... , xn). The buyer also calculates his expected costs EBC(x j , ••• , xn). Let .\'=L7~j x, and .\'_,=.\'-x,. LEMMA 1. Under the sealed-bid SPA with a reservation price r and a given number of firms, n,
En,(x j , ... , xn) =
f; (1- G(ty') G(t)L, dt -
C(x,) - K,
EBC(xj"",xn)='y+(Yo-r)G(r)~-(n-1)
r
G(t)~dt+
}'
i: l=
r
G(t)L'dt.
1 l
A noncooperative Nash game is played among n firms in the R&D stage given the dominant strategy equilibrium in the bidding stage. In this game, the marginal revenues of the investment on R&D are the same for all firms and depend on the total investment level x only. At the equilibrium, the marginal revenue should equal the marginal cost of the investment for each firm. Under DRS R&D technology, the marginal cost is increasing; equating marginal costs among all firms yields the symmetric equilibrium. PROPOSITION 2. Given DRS, SPA, and n active firms, there exists a unique Nash equilibrium at the R&D stage that is symmetric and is determined by (4). 9 More precisely, If we restrict ourselves to differentiable bidding strategies. then there is no asymmetric perfect eqUlhbrium in this two-stage game. In general, the sealed-bid FPA with asymmetnc bidders might have a non-dlfferenliable biddmg strategy eqUlhbnum.
48
GUOFU TAN
Given the payoff function for each firm in Lemma 1, the proof of Proposition 2 is straightforward and is omitted here (see the proof for a general result in Proposition 4). The results in Proposition 2 hold when R&D technology is represented by a general distribution function H( y Ix) with Hx > 0 and Hu < 0 'Ix E R ~ and y E (y, y). In other words, there exists a unique Nash equilibrium that is symmetric when C(x) is convex and P(ylx)= -In[l-H(ylx)] is weakly concave in x (DRS), where P( y Ix) is a production measure of the R&D process. When P( y I x) is linear in x, it can be shown that H( y 1.Xl) takes the form given by (1), which corresponds to CRS R&D technology if C(x) is also linear in x. From Lemma 1, under CRS, both the marginal revenues and the marginal costs of investment are identical for all firms and hence the equilibria are not uniquely determined. But for any (pure or mixed) strategies used by other firms, each firm has a unique best response. Thus, no mixed strategy equilibrium exists. PROPOSITION
3.
Given C RS, SPA, and n active firms, the following hold:
(a) There is a continuum of pure strategy equilibria, among which the total investment level i is uniquely determined by
r
G(t)~lnG(t)dt+c=O,
(5)
and no mixed strategy equilibrium exists. (b) Among the multiple equilibria, Enl(x b ... , x n ) is maximized at the equilibrium Xl = i, Xl = 0 Vj # i, where i is determined by (5), and EBC(x I, ... , x n ) is minimized at the symmetric equilibrium Xl = i/n Vj EN.
Among the multiple equilibria under CRS, each firm prefers a particular equilibrium at which it invests i and all other firms invest O. This is because each firm's expected profit and marginal revenue are decreasing in the other firms' investment levels. The best equilibrium for one firm is the worst for all the other. 10 The R&D game is thus very competitive since 10 When n = 2, by reordering the strategy variable for one player, e.g., letting X2 = -x 2 , the R&D game has the strategIc complementarity property (i.e., such player's strategy set is partially ordered and the marginal returns to increasing one's strategy rise with increases in the competitors' strategies). As Milgrom and Roberts [11] have shown, in this type of game the sets of pure strategy Nash equilibria, correlated equilibria, and rationalizable strategies have identical bounds. Also, the largest equilibrium is the most preferred equilibrium for player 1 and the least preferred equilibrium for player 2, while the smallest equilibrium is least preferred by player 1 and most preferred by player 2. When n > 2, the equilibrium outcomes of this R&D game have similar properties; but the game cannot be simply transformed into the strategic complementarity game.
49
ENTRY IN CONTRACT AUCTIONS
individual firms do not like the symmetric equilibrium even if they collude. More precisely, if the firms choose (Xl' ... , X n ) to maximize joint expected profits, they will choose x, = x for some i and x] = 0 for all}"# i. Then the firms probably divide the profit of firm i equally among themselves. On the other hand, the buyer prefers the symmetric Nash equilibrium since the competition in the R&D stage helps him to obtain a lower bid in the bidding stage. Society is indifferent about the equilibrium the firms play. The expected social costs depend only on the total investment level x. Comparing the equilibrium outcomes between FPA and SPA, one can see that, under DRS, both auction rules give the same equilibrium outcome in the R&D stage, the same expected profits for each firm, and the same expected cost for the buyer. Thus, FPA and SPA are equivalent under DRS. But, under CRS, there are many equilibria in the R&D stage when SPA is used and only one equilibrium (symmetric) when FPA is used. From Lemma 1, the buyer has different expected costs at different equilibria. Since the firms do not prefer the symmetric equilibrium, there is no reason to believe that the firms will play this equilibrium. Thus, the buyer has different expected costs when he uses FPA and SPA. The Revenue Equivalence Theorem in the auction literature breaks down when there is pre-auction R&D that is subject to constant returns to. scale on expenditure. SPA, given homogeneity, is very sensitive to pre-auction R&D activities. However, SPA works well when firms are heterogeneous. Assume that firm i has R&D technology that is represented by H, (y Ix), Y E (y, y), x E R~ and that all firms have the same cost function C(x) + K. Assume, further, H x, > 0 and H"x, < 0 for all y E (y, y) and x E R~. Let P,(ylx)= -In[l-H,(Ylx)]. If, for some i, ((oP,(ylx»/ox» ((oP](ylx»/ox) for all YE(Y,y), xER~, and }"#i, and C(O) < ((oH, (t IO»/ox) dt, then firm i is called a dominant firm.
g,
PROPOSITION
4.
the R&D stage
Under SPA, there exists a unique Nash equilibrium in conditions are satisfied:
if the following
(a) P,(Y I x) is weakly concave with respect to x and iEN, and C(x) is convex in xER~; or
(b)
E R~
for all y
E
(y, y) -
(( oP 1(y Ix) )/ox), ... , (( oP n( y Ix) )/ox) are linearly independent; or
(c) P,(ylx) and C(x) are all linear in xER~ for all YE(Y, y) and i EN, and there exists a dominant firm. Moreover, if firm i is a dominant firm then x I> Xl ~ 0 V}"# i at the equilibrium. Under condition (c), in particular, Xl = 0 V}"# i.
Since PI (y I x) represents an output measure of firm i's R&D process, condition (a) means that the R&D process for each firm exhibits
50
GUOFU TAN
diminishing returns to scale. Under DRS, even though the firms are heterogeneous, there is only one Nash equilibrium. Alternatively, if the productivity measures (oP,/ox) among all firms are linearly independent even if (a) is not satisfied, the Nash equilibrium is also unique. Given sufficient heterogeneity among firms, the multiplicity of Nash equilibria disappears even if R&D technology is subject to CRS. When ((oP,(ylx))/ox»((oPj(ylx))/ox for all y and x, firm i has a higher marginal product of the investment in the R&D process than has firm j. This means that firm i has a better R&D technology. For instance, this condition implies H, (y Ix) stochastically dominates Hj (y I x) when both P, and Pj are linear in x. When SPA is used, in equilibrium, the dominant firm always invests more on R&D than other firms. In fact, under CRS the dominant firm invests and the rest of the firms do not. In summary, firms with better technology invest more and hence have a higher probability of winning the contract, and the firm with the lowest production cost always wins the contract. This is an efficient outcome. In contrast, if the firms are heterogeneous, the noncooperative equilibria of FPA are usually not symmetric and are difficult to analyze. This implies that the firm with a lower bid is not the same firm with lower production costs. Thus, efficient allocations need not be achieved under FPA and SPA is preferable when there are private pre-auction R&D activities among heterogeneous firms.
4.
FREE-ENTRY PERFECT EQUILIBRIUM
I turn now to the question of how many firms will invest in the R&D stage and compete for the contract. Let n(r) == Maxx{f~ [1- G(W] dtC(x) - K} be the maximum profits when there is only- one firm participating in R&D activity and bidding for the contract with the reservation price r. I start with the assumption n(r) > for a given reservation price (i.e., at least one firm is interested in R&D and procurement contract). With free entry, more firms will enter the R&D and bidding process. When R&D technology is subject to CRS, multiple perfect equilibria exist under SPA. But the total investment level x is uniquely determined by (5) and is independent of the number of firms. Among the equilibria, from Lemma 1, firm i has the profit:
°
En,(xj,x)=
r
G(t)~-X'dt-r G(tYdt-cx,-K.
!'
!'
Since En,(O, x) < 0, En,(x, x) = n(r) > 0, and En,(x" x) is increasing in x" there exists Xo E (0, x) such that En,(x o , x) = 0, where Xo is also
ENTRY IN CONTRACT AUCTIONS
51
independent of the number of firms. Each active firm invests X,? Xo such that L7~ 1 X, =.x and the equilibrium number of firms, n e , is less than or equal to x/x o. When R&D technology is subject to DRS, under either FPA or SPA, the perfect equilibrium is unique and symmetric and each firm has the same expected profits:
Enn(x,r)=
t
(l-G(ty)G(t)(n-llxdt-C(x)-K.
(6)
For brevity, I refer to FPA in the following discussions, but the arguments apply equally to SPA. With free entry, each potential firm enters the R&D process if its expected profits are nonnegative. That is, equilibrium entry gives (7)
and any additional entrant n + 1 earns negative profits,
Enn+1(x',r)<0,
(8)
where x' is the individual R&D expenditure that is determined by (4) when n + 1 firms simultaneously enter the R&D process. Entry decisions are simultaneous, not sequential. For any given reservation price, r, Eqs. (4), (7), and (8) determine the free-entry pure-strategy symmetric perfect equilibrium (ne' x e) with the bidding function in (3). In order to show the existence and uniqueness of a free-entry purestrategy symmetric perfect equilibrium for a given reservation price, I first allow the number of firms n to be a continuous variable. Each firm enters the R&D process until its expected profit is zero and hence the equilibrium-entry conditions (7) and (8) can be represented by the equation,
f [1- G(tYJ G(t)(n-l)x dt - C(x) - K = 0. r
(9)
l'
Then (n e, xe) is determined by (4) and (9). PROPOSITION 5. Suppose n(r)? 0, then there exists a unique solution (ne' xe) with ne E [1,00) and Xe E (0, 00). Furthermore, let xe == neXe' then
~
(a)
(oxe/oK)
(b)
(OXe/or) >0, (OXe/or) <0, and (one/or) >0. 11
0, (oxe/oK) > 0, and (one/oK) < 0;
11 If the solutIOn ne above is an integer, each informed firm earns exactly zero expected profit at the equilibrium. If ne is not an integer, the equilibrium can be easily adjusted. See
Tan [17J for the discussion.
52
GUOFU TAN
If at least one firm enters the R&D and bidding process then free entry results in more active firms. The expected profits for each firm decrease as the number of active firms increases and, thus, competition makes the procurement contract less profitable to each firm. As the number of active firms goes to infinity, each firm invests zero on R&D and expects to earn a profit - K. Therefore, if there is a positive fixed cost of R&D then only a finite number of firms decide to enter the R 84 D process. The fixed cost of R&D, K, is the key determinant of the free-entry equilibrium number of firms, although the latter is also affected directly or indirectly by the variable cost of R&D, C(x), the reservation price, r, and the distribution of production costs, H(y I x). For a given number of active firms, the distribution of the production costs y of the winning firm is 1- G(ytx. From (3) and (6), the buyer has the following ex ante expected costs in the competitive procurement,
t
B(y)d(1-G(y)nx)=
f
yd(1-G(ytX)+nC(x)+nK+nEnn(x,r),
(10)
where Enn(x, r) is a firm's expected profits under the perfect equilibrium defined by (6). The buyer's expected costs include the expected minimum production cost, the total R&D costs among all firms, and the total expected profits among all firms. With free entry, each firm enters the R&D and bidding processes until its expected profit Enn(x, r) equals zero. The buyer has to pay not just the expected minimum production cost, but also the total R&D cost neC(x e ) + neK of all informed firms. The latter result may seem counterintuitive: one expect that the buyer has only to pay the R&D costs of the winner. However, since firms are symmetric and adopt the same investment and bidding strategies, each has an equal probability to be the winner. Therefore, the buyer actually ex ante expects to pay all of the costs of R&D among active firms. The winning firm's expected profits g,(B(y)-y)d(l-G(yt X ) from the bidding equal the total costs on R&D among all of the firms. In other words, if free entry is allowed, the rents for the firms from contracting are dissipated by precontract R&D competition. One wonders whether these R&D activities are good for the buyer and society. I turn now to address this question.
5.
THE BUYER'S OPTIMAL STRATEGY
In this section, I go back to the first stage of the three-stage game and look at the buyer's optimal decision problem. ,One wants to know whether there exists a reservation price under which the free-entry perfect
53
ENTRY IN CONTRACT AUCTIONS
equilibrium characterized in Section 4 is optimal for the buyer. In other words, if the buyer can select the number of firms, can his optimal selection be supported by free entry? Remember that, under FPA, the buyer actually procures the good at a cost, Yo, elsewhere with probability G(rtx. Thus, the buyer's total expected costs, EBC, consist of (10) and yoG(rYx. More explicitly, EBC(r,n,x)=}'+CYo-r)G(ryx+ n
I;
G(t)(n-1)Xdt-(n-1)
t
G(ttxdt. (11 )
The buyer wants to minimize EBC(r, n, x) by selecting r, n, and possibly x. In general, the buyer may require each active firm to invest more than it wants to. This implies that there exists a moral hazard problem if the buyer is unable to observe or verify the R&D decision, x. He has to take each firm's R&D decision as a constraint and should then minimize his costs (11) subject to the R&D constraint (4) and the participation constraint (7). Suppose (rb' n b, Xb) is the solution to the buyer's optimization problem and Xs is the investment level that minimizes the total expected social costs. PROPOSITION
6.
(a)Ennb(xb,rb)=O, and (b)
if n(yo»O then rb
and Xb < Xs'
Proposition 6 implies that, at the buyer's optimum, the firm's nonnegative profits constraint (7) is binding. Therefore, under free entry the buyer is able to control the number of active firms and the R&D investments by offering an appropriate reservation price tb' Given the selection of rb , the number of active firms and the R&D investments are simultaneously determined as the free-entry perfect equilibrium characterized in Section 4. The buyer will select a reservation price, r b , lower than the opportunity cost, Yo, and extract surplus from firms whenever the surplus exists. As a result, each firm invests less aggressively, which causes underinvestment in the industry relative to the social optimum. 12 As Tan [17] has shown, the social planner who minimizes the total expected social costs of procurement prefers only one firm to conduct all of the R&D and production when R&D technology is subject to constant returns to scale on expenditure. Since CRS R&D activity is an independent search process, the R&D outcomes of n firms are equivalent to the outcome of one firm that invests the same amount as all n firms. However, because of fixed costs of investing, more firms participating in 12 See Tan [17] for diSCUSSIOns on a comparison between the buyer's optimal strategy and the social optimum.
54
GUOFU TAN
R&D activities imply higher total costs of R&D. For society, one firm conducting all of the R&D and production is more efficient than several firms and competitive bidding is not necessarily socially efficient. This result contrasts with the conventional view that it is always more efficient to award procurement contracts through a very competitive procedure (see Dasgupta and Spulber [2], Riordan and Sappington [13], among others for more discussions on the advantage of competitive procedures). The fact that it is costly for bidders to acquire similar information before bidding reduces the attractiveness of competitive mechanisms. In many cases of defense procurements, Congress requires the DOD to award the contract in very competitive ways. The above analysis implies that the optimal procurement and R&D policies from the point of view of the buyer and society can be very different under two types of R&D technology. When deciding to choose auction rules and to promote more bidders, both Congress and the DOD should consider the objectives they pursue, the types of projects they procure, and the types of R&D technologies potential firms face.
6.
CONCLUSION
Most results in the theory of auctions and procurement contracting depend on the assumption that there is an exogenously fixed number of well-informed bidders. But potential bidders are often not well informed initially. Different auction mechanisms will in general provide potential bidders different incentives to acquire private information and to enter the competitive bidding. This fact will in turn affect the design of the auction mechanisms. In this paper, I present a model of procurement contracting with entry and private R&D investments. I find that the nature of the private R&D processes plays an important role in determining both the performance of the different auction rules and the optimal procurement policies for the buyer. When the R &D process is subject to diminishing returns to scale in expenditure, FPA and SPA are equivalent as in earlier work. But the Revenue Equivalence Theorem breaks down when the R&D process is subject to constant returns to scale. This is true because multiple equilibria arise when SPA is used; the firms do not select the equilibrium that the buyer prefers. The buyer as a profit maximizer prefers free entry to the industry in general. However, underinvestments in the industry relative to the social optimum usually arise due to the monoposony power of the buyer. While FPA has an advantage over SPA when potential bidders are ex ante homogeneous, SPA works very well when the bidders are ex ante heterogeneous. The bidder with a better R&D technology invests more
ENTRY IN CONTRACT AUCTIONS
55
and is more likely to observe a better R&D outcome (low production costs). Each firm also bids its true production cost in SPA and, in this sense, SPA works efficiently. The interesting issue is then the optimal mechanism design when potential bidders are not well informed ex ante and each may decide to acquire more information before the bidding. The number of actual bidders should be endogeneously determined with the optimal mechanism.
ApPENDIX
Proof of Proposition 1. I prove (b) in the case n = 2. Consider a differentiable equilibrium bidding strategy B, (y,) of firm i, i = 1, 2. It is easy to establish that, B,(y,) is monotonically increasing in y" BI(r) = B 2(r) = r, and B I (1)=B 2(1). Let l!=B,(l) and ¢J,(b)=B,-I(b) VbE[l!,r). Then ¢Ji(b) is also increasing and differentiable, ¢JI(r) = ¢J2(r) = r, and ¢JI(l!) = ¢J2(l!) = }'- The profit for firm 1 of type y I to bid b is
The first order condition yields (12) where I(y) = G(y)/f(y}. Similarly, firm 2 bids b such that (13 )
Since ¢J:(b) >0, (12) and (13) imply that, VYE(l,r), ¢J,(b)
(14 )
Claim 1. Suppose XI
56
GUOFU TAN
The same argument shows: Suppose x, > x 2, then B,(y) > B 2(y) Vy E (y, r). Now, using the Envelope Theorem, I obtain that, VYE (y, r),
Claim 2.
(15) Given the bidding equilibrium (B,(yd, B 2 (Y2)), the ex ante expected profit for firm 1 is En,(x"x 2)=
=
r
n,(B,(y,), yddH(y,lx,)-C(xd-K
r
[1- G(tyl] G(¢J2(B,(t))Y2 dt - c(xd - K,
}'
where the second equality holds because of (15), n,(r, r) = 0, and integration by parts. The first order condition with respect to x, yields
-r
G(tYIG(¢J2(B 1 (t))yZin G(t) dt- C(xd =0.
(16)
y
Similarly, firm 2 chooses X2 such that
If x,
< -G(t)XIG(¢JAB,(t))Y2In G(t), VtE (y, r). (18)
Integrating both sides of (18) violates the equilibrium conditions (16) and (17). Similarly, x, >x 2 also leads to a contradiction. Therefore, x, =x 2 , which also implies B,(y) = B 2 (y) Vy E (y, r]. I Proof of Lemma 1. The expression of En;(x" ... , x n ) is easy to establish and I only derive EBC(x I, ... , x n ) here. Let yl and y2 be the lowest and the second lowest order statistics of (YI, ... , Yn). It can be shown that the joint distribution of yl and y2 has a density function
if yl ~ y2, otherwise,
57
ENTRY IN CONTRACT AUCTIONS
EBC(xj, ... , xn) =
I; t
f'
y2p(y\ y2) di dy2 + r (
p(yl, y2) di dy2 + yoG(r)X,
where the first term can be written as
I; ,t!
(rlJf,(r)- yllJf,(yl)-
= =
rI rf
yllJf;(i)H(ilx,)di
!
,~I
y
1=
=y-r
(1 1Jf,(y2) dy2 ) h(i Ix,) di
i·x_,G(yl),,-,-lf(i)di-(n-l)x
r
iG(yly,-lf(yl)di
r
1
I
G(r)~-'+
r f G(i)~-'di+(n-l)rG(r)X }'
,~I
,~I
and the second term is r
n
n
,~l
,~l
L H(rlx,)G(r)L'=r L G(r)~-'-nrG(r)~.
Combine these three terms of EBC(x 1 ,
... ,
x n ) and Lemma 1 follows.
I
Proof of Proposition 3. It is easy to establish (a) and the first part of (b). I prove the second part of (b) here. From Lemma 1, given X, minimizing EBC(x I ' ... , xn) subject to L7~ I x, = x is equivalent to minimizing
Ln fr G(tt-x'dt ,~I
subject to
L7~
I
(19)
!
x, = x. The first order conditions yield - ( G(tY-x, in G(t) dt - JI. = 0,
"liEN,
(20)
where JI. is the Lagrange multiplier. Equation (20) implies x, = Xl Vi,} EN. Since (19) is a convex function of (XI' ... , Xn), EBC(x 1 , ... , x n) is minimized at the symmetric equilibrium Xl = x/n V} EN. I
58
GUOFU TAN
Proof of Proposition 4.
En,(x I ,
••• ,
xn) =
r
Similar to Lemma 1, firm i has a payoff function
H,(t I·')
}'
TI
[1- H;(t 1x)] dt - C(xJ - K.
}#l
Since H~,\,
(21 )
where ¢(tlxl, ... ,xn)=Il;~1 [1-Hj (tlx)J. The Jacobian of the system (21) is symmetric, and negative definite Vx, ~ 0, i EO N under the assumption (a) or (b), i.e., Vv EO R n and v # 0
"
L.
',jEN
I -(L, O v, OP,(t XJ)2) dt - LV,C"(X,) X, , r
(12En, (" Jo2p,(tlxJ t',v]-;;--;-= ,¢(tlxl, ... ,xn) L. v; 0. 2 uX, uX j
X,
'EN
}
1
<0. By Rosen (1965) Theorem, there exists a unique Nash equilibrium of the R&D game. Now, if firm i is a dominant firm, at the equilibrium (x I, ... , x n ), it cannot be xj=O Vi because of the assumption C(O)< -g, ((oH,(tIO))/ox) dt. If x] > 0 for some i, then (21) holds for i and hence -
I
r oEn, (OP,(tIXJ -",-=/fi(tIXl""'X n ) 0 ux, } x,
OPj(tIX)) i3 dt-C(xJ+C(x) Xj I
I
is positive at x, = x)' Thus, at the equilibrium, x, > Xj ~ 0 Vi # i. Under condition (c), let i be the dominant firm and (XI' ... , xn) be an equilibrium, then x, > x; ~ 0 Vi # i. Suppose Xj > 0 for some i, then (21) holds for i and hence
r oEn'=f d.( 1 ) (OP,(tIXJ 'f' t Xl' ... , Xn '" ox; }' uX,
(1Pj (t IX)) d '" UX;
0 t> ,
i.e., x, cannot be an equilibrium strategy for firm i. This contradiction shows that x; = 0 Vi # i, x, is the unique response and is determined by
Irop,(tlxJdt_C(xJ=O. ox, y
I
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ENTRY IN CONTRACT AUCTIONS
Proof of Proposition 6. I treat n E [1, + 00) as a continuous variable. Since E7r n(r, x) is continuously differentiable with respect to r E [y, y], nE [1, +00), and xER~, the constraints (4) (and (7) form a compact set S of (r, n, x). EBC(r, n, x) is continuous with respect to (r, n, x) E Sand hence there exists a solution (rb' n b, Xb) to the buyer's optimization problem. The constraint (7) and K> 0 imply X h > O. Let L== -EBC(r,n,X)+AE7r n(r,X)+,u( where Then
A~O
-L
G(ttxlnG(t)dt-C'(X)),
and,u are the Lagrange multipliers for constraints (7) and (4).
oL = nxf(r) G(r) G(rt x - 1 or
lyO _(r + (1 _~) H(r Ix) _ t: HAr IX))]. n h(rlx) n h(rlx)
Since Yo>}', limr~y+.(oL/or»O for small B>O. Then rb>Y' The first order conditions with respect to n and x yield oL on = -xCvo - r) G(rt' In G(r)
-r
G(t)(n-l)x[l_ G(ty + xG(tY In G(t)] dt
-'
+ X(A - n) - x,u
r
r
G(t)(1l-1)XH(t I x) In G(t) dt
y
G(tr X In 2 G(t) dt
}'
:( O.
(22)
oL ox = -n(yo- r) G(r)nx In G(r) + (n -l)(A -n)
x
r
G(t)(n-l)x H(t Ix)ln G(t) dt - n,u
!
r
G(tr x In 2 G(t) dt - ,uC"(x)
!
= O. If rb=YO'
(23)
then
(22)
and
(23)
imply
,u~O,
A
hence
< 0 for small B > 0 because of yo:(.v. Thus rb < y. The first order condition with respect to r becomes limr~v_e(oL/or)
yo=r+(l-~) H(rlx) _t:H,(r1x). n h(rlx)
n h(rlx)
(24)
60
GUOFU TAN
Suppose A= 0, then (23) and (24) imply J.1 >
-aL = an
fr G(t)(n-l)x[l -
°
and J.1X
G(tY + x in G(t)] dt + - C(x) > 0,
n
!
which contradicts (22). Therefore A> 0, which implies ETCnb(r b, Xb) = 0. Now suppose rb~Yo, then (23) and (24) imply A~nb and J1~0. Then (22) holds with inequality, which implies nb = 1. But n(yo) = ETCt(Yo, Xb) = that violates the assumption. Therefore, rb < Yo. I
°
REFERENCES
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