Sequential Prourement Autions and Their Effet on Investment Deisions Gonzalo isternas Niolás Figueroa November 2007 Abstrat In this paper we haraterize the optimal prourement mehanism and the investment level for an environment where two projets must be adjudiated sequentially, and the winner of the first projet has the opportunity to invest in a distributional upgrade for its osts in the seond projet. We study 4 ases, based on the ommitment level of the seller and the observability of the investment deision. We find that with ommitment, the seond period mehanism gives an advantage to the first period winner, and indues an investment level that is larger than the effiient one. With non-ommitment, the seond period mehanism gives a disadvantage to the first period winner, and indues an investment level that is smaller than the effiient one. Observability is irrelevant in the ommitment ase, but makes the effets more pronouned in the non-ommitment ase. Keywords: Prourement Autions, Sequential Mehanisms, Mehanism Design, ost Reduing Investment JEL D44, 7, 72. 1. Introdution During the last deades prourement autions have been widely used as mehanisms to assign high ost projets onerning goods and servies. By the year 1998, the sum of all governments expenditures in prourements (exluding defense and labor ompensation) was estimated as 7.1% of the worldwide GDP 1. The repeated utilization of prourement autions by speifi by governments and private institutions, whih ontrat with the same pool of firms over time, has made the study of ost redution investment by these firms speially relevant. The objetive of this paper is twofold. First, to haraterize the ost-minimizing prourement mehanisms in an environment with repeated interation between a buyer and multiple sellers, where investment an be undertaken as a ost-redution devie, and also as an strategi ation devised to obtain advantages in future prourement autions. Seond, to analyze the effets these ost-minimizing mehanisms have on the sellers investment deisions. This is partiularly relevant if the relative size of expeted expenditures We are speially grateful to Soledad Arellano and Hétor hade for their onstrutive omments and suggestions. We also reeived many useful omments from partiipants in LAMES 2007 and seminars held at the enter for Applied Eonomis at Universidad de hile. This researh was partially supported by the omplex Engineering Systems Institute and by Fondeyt Grant N o 1107059. entro de Eonomía Apliada, DII, Universidad de hile, Repúblia 701, Santiago, hile; gistern@dim.uhile.l entro de Eonomía Apliada, DII, Universidad de hile, Repúblia 701, Santiago, hile; niolasf@dii.uhile.l 1 Soure: OED, The Size of Government Prourement Markets, Journal of Budgeting Vol.1, No.4, Marh 2002, ISBN 9264196307. 1
is big ompared with the value of just one partiular purhase, sine in that ase the use of tools that may redue future osts, even at the ost of raising today s expenditures, an be partiularly profitable. We onsider a buyer who wants to proure two onseutive projets and faes n 2 potential suppliers. osts for performing both tasks are distributed independently aross time and ompetitors, and they are private information to eah firm. We assume, though, that the first prourement winner (whose identity is publi information for the seond prourement) aquires an intrinsi advantage haraterized by a distributional upgrade. This reflets a knowledge aquired onerning the task performed, to whih the rest of the firms have no aess. Moreover, this upgrade an be influened by a ostly investment that the winner may arry out between the two prourements. In suh a ontext, seond period rules, through the degree of advantage they give to a first period winner, affet the benefits of suh an investment, and therefore determine the investment arried out by the first period winner. As a onsequene, the seond period ost distribution of a first period winner, and the asymmetry between him and the first period losers, are determined endogenously. We first onsider the ex-post effiient benhmark: eah period the projet is assigned to the lowest ost supplier, and the investment level, I e is suh that the marginal ost of investment equals the marginal benefit (in expeted terms) of ost redution. We show that, independent of investment observability, suh an inentive ompatible mehanism exists and an be implemented through two seond prie sealed bid prourements. We then use the mehanism design approah to haraterize the ost-minimizing mehanism, whih we determine under the assumptions of full ommitment and non-ommitment of the buyer. In the first ase, we assume that the rules for both mehanisms are deided at t=0, before any ost realization ours. We show that the optimal mehanism is independent of investment observability, and gives an advantage gap to the first period winner in the seond period. That is, he an win the seond prourement aution even if his ost is higher than the minimum of the other ompetitors. Moreover, in this ontext, the investment level indued, I, is higher than the effiient level I e, so over-investment ours. This result an seem ounterintuitive, sine an advantage gap an make the first period winner relax, knowing he owns a big advantage over ompetitors and is likely to win anyway. However, there is a seond effet that dominates: investment, by dereasing osts, inreases the expeted profits in ase the firm gets the ontrat. Sine with a bigger advantage gap this happens more often, the expeted benefits due to ost redution are higher, thus investment beomes more profitable. In the non-ommitment ase, the seond period rules are determined after the winner of the first period is announed and he has invested in ost redution. We find that it is optimal for the buyer to give a disadvantage to the first period winner, who holds a distributional advantage over its ompetitors. Now, a first period winner does not neessarily wins the seond period prourement even if he has the lowest ost among all ompetitors. This fat is antiipated by the first period winner, making investment less attrative and therefore leading to investment levels of ost-redution below the effiient one. Here, the observability of investment makes a differene. When investment is not observable, the interation between the buyer and the first period winner is equivalent to both agents simultaneously hoosing mehanisms and investment levels, respetively. On the ontrary, when investment an be monitored, the buyer optimally reats with more disadvantageous mehanisms as the first period winner distribution improves. Thus, the first period winner invests less in this last ase. Denoting by Î and Î the investment levels when investment is observable and non-observable we get Î Î < I e < I These results have important impliations. First, ost-minimizing buyers, with the ability to ommit, and working in a dynami environment, indue investment levels above the effiient level. This is beause 2
the seond period advantages are not only introdued to enourage investment, but also to inrease ompetition in the first period. In this respet, the buyer introdues ineffiienies to inrease ompetition, in a similar way a stati buyer introdues ineffiienies setting up a reservation ost. However, a buyer s lak of ommitment indues under-investment, sine the disadvantages to first period winners make ost redution investment less attrative. Seond, investment observability does not play a role in the ase of full ommitment, neither for effiieny nor for ost minimization. Here, the intuition from the moral hazard literature reappears: with risk-neutral agents, the first best level of investment (effort) an be indued. But now, this has a double interesting impliation sine the first best level of effort an be the effiient one and the one that minimizes ost for the seller, and both an be ahieved. However, this result only holds at the optimum. We show in an example that, in the full ommitment ase, a sub-optimal seond period mehanism leads to a seller s investment deision different from the one the seller would have hosen. Finally, we derive some omparative statis with respet to the number of players. As it inreases, all the investment levels mentioned above derease, but weakly preserving the order among them. For a number of firms suffiiently large, the investment level in all environments ollapses to zero: if the number of ompetitors is large enough, induing investment is too expensive for the buyer relative to the marginal benefit in ost redution. This is so beause the added probability of the first period winner getting a ost lower than the minimum of all other ompetitors, dereases with the number of ompetitors. Nevertheless, under full ommitment, although the advantage given to the first period winner at the last ompetition also dereases with the number of firms, it never disappears. This ours beause introduing sequentiality enables the buyer to distribute inentives inter-temporally in suh a way that more ompetition is indued in the first prourement aution, whih out-weights the seond period ineffiienies regardless of the number of players. Our work is related to the literature in several ways. With the methodology of mehanism design, Pesendorfer and Jofre-Bonnet in [9], derive the ost minimizing mehanism for an exogenous level of omplementarity (whih is understood as an exogenous distributional hange between both prourement autions), full ommitment and two players. We extend this result to allow for investment (and therefore endogenous omplementarity), non-ommitment and multiple players. In a non-ommitment setting, Luton and MAfee [5] determined the optimal mehanism for a partiular type of exogenous distributional upgrade. The possibility of ahieving effiieny has been analyzed for the ase when investment an be arried out before a prourement aution with a fixed mehanism (first or seond prie) takes plae. An affirmative answer is found in Piione and Tan [10] when firms are ex-ante symmetri, they an all invest non observably and the tehnology presents diminishing returns to sale. Dasgupta [2], in a similar model (investment stage and non observability), but with a buyer with lak of ommitment, shows that investment level is below the effiient one. Moreover, ommitment rises the investment level, but always below effiieny. Finally, Arozamena and antillon in [1], analyze the effets of allowing only one firm to invest before a first prie sealed bid prourement aution, and make this ation observable by ompetitors. Their main result is the existene of under-investment: as a response to the ost-redution investment, rival firms will bid more aggressively, therefore reduing the investment inentives. In our paper, we also get under-investment, but for a different reason: under non-ommitment, it is the mehanism designer (the buyer) that hanges behavior, giving an advantage to worse firms, and therefore dereasing the inentives to invest. 3
2.1 The Environment 2. The Model onsider a risk-neutral buyer who wants to proure two projets, one at t = 1 and the other at t = 2. The set of ompeting sellers is N = {1,..., n}, n 2, all of whih are risk-neutral and live for the two periods. The buyer is ompelled to proure the two goods or servies 2. In eah period, the ost of undertaking the projet for a seller is drawn from the interval = [, ], and it is private information. At t = 1, these osts are independently distributed aording to a distribution F ( ), differentiable, that satisfies f() F () > 0 if (so sellers are ex-ante symmetri). At t = 2, the ompetitors osts are drawn independently from those drawn in period 1, and independently aross sellers as well. The osts of the first period losers are taken from the same distribution F ( ). Instead, the winner of the first prourement (from now on the winner) has the option of investing an amount I between autions, at a monetary ost Ψ(I), and hanges his distribution to G(, I), differentiable with (, I) > 0 and with the same support. Assumption 1 below implies a distribution improvement for the winner as a funtion of investment: investment implies an inrease in the hane of obtaining lower osts relative to higher ones. As a onsequene, higher investment indues a better ost distribution in the usual sense of first order stohasti dominane. The formal result is in lemma 9 in the next setion. Assumption 1 G 2 ( IR + ). For all 0 I < I IR and <, f( ) f() (, I ) (, I ) < (, I) (, I) The first inequality, when evaluated at I = 0 indiates that there an exist an exogenous improvement for the first period winner. He aquires a knowledge onerning the task performed (the know-how), whih is not available to the losers. The seond inequality shows that the final degree of omplementarity will depend on the amount the seller invests in developing this know-how. We also impose that the marginal benefit of investment is dereasing: Assumption 2 For all I IR +, 2 G I 2 (, I) < 0 in (, ). We now state two tehnial assumptions, the first one is the inreasing hazard rate, and the seond is a ondition needed for integrability. Assumption 3 F () differentiable). f(), G(,I) F () are inreasing in. Also, (,I) f() is differentiable (in partiular, F is twie Assumption 4 There exists f L 1 (IR) suh that (, I) I = (, I) < f(), I IR I Finally, for the investment tehnology we assume, as usual, that is inreasing and onvex. Assumption 5 Ψ( ) is twie differentiable and satisfies Ψ ( ) > 0, Ψ ( ) 0. 2 This is equivalent to assume that the ost for the buyer of arrying out the projet himself is 0 = +. 4
The previous assumptions are not hard to satisfy. For example, the family of distributions introdued in Piione and Tan [10], whih they argue is a way of modeling investment in ost redution of R&D tehnologies, satisfies them. Example 6 Suppose that F ( ) is a twie differentiable onave distribution. Then, it is straightforward that verifies the regularity assumption. The family of distributions given by and G(, 0) = F () η with 0 < η < 1 G(, I) = 1 (1 G(, 0)) γi+1 with γ > 0 satisfies assumptions 1 and 2, 3, and 4 (see Appendix for a proof). For notation purposes, we denote the joint density of the first-period distribution as f n () = n f( j ) and f n 1 ( i ) = j i f( j ), with i = ( 1,..., i 1, i+1,..., n ). j=1 2.2 The Mehanisms The fat that osts are drawn independently aross time enables the buyer to pay attention only to inentive ompatible mehanisms beause the revelation priniple holds. We will fous on two types of environments: full ommitment and non-ommitment of the buyer. In the first ase, the buyer an ommit to the first and seond period mehanisms before any ost realization takes plae. In the seond, he annot, and he hooses the seond period rules after the identity of the first period winner is known and the investment level has been seleted. In eah ase, we analyze the ases when investment is observable and when it is not. The differene is that in the observable ase, the investment, and therefore the winner s ost distribution for the seond period is publi information. Thus, in this setting the mehanisms used by the buyer at t = 2 may depend on the level of investment hosen by the winner of the first prourement aution. We onsider seond period prourement mehanisms that are history dependent only to the extent that they depend on the identity of the first period winner, but not on the ost realization (revelation) of that first period. Moreover, we do not allow the buyer to exlude sellers in the seond period if they do not partiipate in the first. Notie, however, that more general mehanisms an derease the seller s expeted ost. An extreme example of this is the threat of non-partiipation in the seond period. With this strategy, the first period payments an be inreased in the amount of the expeted utility in the seond period mehanism, allowing the seller to fully extrat the seond period surplus. Suh mehanisms seem unrealisti and seldom used in pratie, probably beause of the inability of sellers to ommit to suh a strong and ex-post ineffiient punishments. { } n From now on, we define n = (x 1,..., x n ) IR n + x i = 1 the unit simplex. If investment is observable and the amount of it hosen by the first period winner is I, we will use subsript (w, I) when we refer to this player at t = 2. Analogously, we will use subsript (l, I, i) at t = 2 if player i N was a first period loser (if investment an t be observed, we drop the subsript I). Using the previous notations, we have: Definition 7 A diret mehanism, when investment is not observable, is given by the tuple i=1 5
Γ no = (t 1, q 1, t 2 w, q 2 w, t 2 l, q 2 l ) where t 1 : n IR n, q 1 : n n, t 2 w : n IR, qw 2 : n [0, 1], t 2 l : n IR n 1, ql 2 : n [0, 1] n 1, suh that qw() 2 + ql,i 2 () = 1 for all n. i w Definition 8 A diret mehanism, when investment is observable, is given by the tuple Γ = (t 1, q 1, {t 2 w,i}, {q 2 w,i}, {t 2 l,i}, {q 2 l,i} ) where t 1 : n IR n, q 1 : n n, t 2 w,i : n IR, qw,i 2 : n [0, 1], t 2 l,i : n IR n 1, ql,i 2 : n [0, 1] n 1, suh that qw,i 2 () + ql,i,i 2 () = 1 for all n and I 0. i w When investment is not observable, t s () = (t s 1(),..., t s n()), and t s i () orresponds to the payment to seller i N at time s = 1, 2, onditional on the vetor report = ( 1,..., n ). Analogously, q s () = (q1(), s..., qn()), s with qi s () the probability that ompetitor i N wins the prourement aution at time s = 1, 2 onditional on the same ost report. Finally, when investment an be monitored, the funtions are essentially the same, but now the seond period rules may depend on the investment level arried out by the first period winner. A natural question that arises is whether the mehanism designer an improve by using mehanisms with additional features. For instane, the buyer an make use of seond period rules whih depend on the first period winner s identity. In a ompanion paper ([3]) we show that there is no improvement, in terms of reduing buyer s expeted expenditures, when suh history-dependent mehanisms are taken into aount (beause of ex-ante symmetry). We therefore restrit our analysis to the mehanisms onsidered in definitions 7 and 8. 3. Preliminary Results We now state a well-known result involving some onsequenes of assumption 1. In partiular, that the monotone likelihood ratio property implies first order stohasti dominane as the investment level dereases. Lemma 9 Suppose that assumption 1 holds, then: (i) (, I) 1 G(, I) < (, I ) 1 G(, I ),, 0 I < I. (ii) G(, I) (, I) < G(, I ) (, I ),, 0 I < I. (iii) For every fixed, the funtion G(, ) is inreasing. This is equivalent to first order stohasti dominane as I dereases in the family of distributions {G(, I) I 0}. Proof : Standard. 6
We denote by Q 1 i ( i ), player i s expeted probability of winning the first period prourement aution if his announement is i and the other players are telling the truth. Analogously we denote by T i 1( i ), player i s expeted transfer if his announement is i and the other players are telling the truth: Q 1 i ( i) = T 1 i ( i) = q 1 i ( i, i )f n 1 ( i )d i, i N. (1) t 1 i ( i, i )f n 1 ( i )d i, i N. (2) In the ase of investment observability, expeted probabilities, if the investment level arried out by the first period winner is I, are given by: Q 2 w,i( w) = qw,i( 2 w, w)f n 1 ( w)d w (3) Q 2 l,i,i( i) = n 1 ql,i,i( 2 i, i)f n 2 ( w,i) ( w, I)d i, i w. (4) n 1 with expeted transfers Tw,I 2, T l,i,i 2 defined analogously. We denote by Π 2 w,i ( w, w) the expeted utility at t = 2 of a first period winner with real ost w that delares w. Analogously, Π 2 l,i,i ( i, i ) is the expeted utility of a first period loser with real osts i and delaration i : Π 2 w,i( w, w) = T 2 w,i( w) w Q 2 w,i( w) Ψ(I) (5) Π 2 l,i,i( i, i) = T 2 l,i,i( i) i Q 2 l,i,i( i), i w (6) Finally, we denote by Π 1 i,i ( i, i ) the disounted expeted utility at t = 1 for seller i with ost i that delares i, onditional on truth-telling at t = 2. Π 1 i,i( i, i) = Ti 1 ( i) i Q 1 i ( i) + βq 1 i ( i) Π 2 w,i(, ) (, I)d + β[1 Q1 i ( i)] Π 2 l,i,i(, )f()d. (7) This last expression onsists in expeted payments and osts of the first prourement (the first two terms), and the ones related to the seond period expeted utility (whih depends on being the winner or a loser in the first period). As we said before, we an restrit the analysis to diret mehanisms. Truth-telling in the seond period an be written as: Π 2 w,i ( w, w ) Π 2 w,i ( w, w), w, w, I 0. I 2 o Π 2 l,i,i ( i, i ) Π 2 l,i,i ( i, i ), i, i, i w, I 0 And for the first period it an be written as: I 1 o : i N and I 0, Π 1 i,i ( i, i ) Π 1 i,i ( i, i), i, i. 7
Observation: We have defined seond period rules (transfers and probabilities) and inentive ompatibility onstraints for any possible investment level I 0. However, the only relevant investment level is the one the buyer wants to indue. The rules for any other I an be set-up suh that those investment levels are never seleted, and I is satisfied trivially. This allows us to see mehanisms when investment is observable as a tuple (Γ, I), where I is the investment level fored upon the first period winner and Γ speifies seond period rules for that partiular I (and also first period rules). When investment is not observable, seond period rules an t depend on investment level I. In this setting, the rules (qw, 2 ql,i 2, t2 w, t 2 l,i ) omit the variable I and all the above expressions an be obtained with this notational hange. Nevertheless, inentive ompatibly implies that the investment level I (hosen by the seller) and both period rules must simultaneously satisfy I no I arg max K 0 Π 2 w(, ) (, K)d Π 2 w( w, w ) Π 2 w( w, w), w, w Π 2 l,i ( i, i ) Π 2 l,i ( i, i ), i, i, i w Π 1 i ( i, i ) Π 1 i ( i, i), i, i, i N. The first ondition requires that the the investment level hosen by the first period winner must maximize his utility level onditional on seond period rules. Although we do not write subsript I in all the inequalities, it impliitly appears in the seond period expeted utility of the first period losers and in the first period utility of all players, through the funtion G(, I). We omit the mentioned variable just to emphasize that the buyer an make no use of this more general mehanisms when investment is non-observable. In the next lemma we state the usual haraterization of inentive ompatible mehanisms: Lemma 10 (Inentive ompatibility): Γ is inentive ompatible if and only if (i) For all i N and I 0, Q 1 i ( ) is non inreasing Π 1 i,i ( i, i ) = Π 1 i,i (, ) + (ii) For all I 0, i Q 1 i (s)ds for all i Q t k ( ) is non inreasing, k = (w, I), (l, I, i), i w, i N. Π 2 k ( k, k ) = Π 2 k (, ) + Q 2 k (s)ds for all k, k = (w, I), (l, I, i), i w, i N. k Proof: See Appendix. Partiipation onstraints depend on the level of ommitment and investment observability analyzed, and will be disussed in the setions where optimal mehanisms are haraterized. 8
4. Effiieny An ex-post effiient mehanism assigns, in eah period, the projet to the lowest ost supplier and indues an investment level suh that the marginal ost of investment equals the marginal benefit (in expeted terms) of ost redution. We prove the existene of suh a mehanism regardless of investment observability. This is not obvious: sine the mehanism must assign eah projet to the lowest ost supplier, the investment level (in the non-observable ase) is automatially defined by equation (8) and not neessarily the buyer s desired one. We prove nonetheless, that the level hosen by the seller in suh a mehanism is exatly the one the buyer would like to indue. We will use this investment level as a benhmark to be ompared with the investment levels indued by the ost minimizing mehanisms in the different environments. Sine the effiient mehanism, that we denote by Γ e, must assign eah projet to the ompetitor with the lowest ost, the assignment rules are given by: { q t,e 1 i < i () = j, j N 0 (8) for t = 1, 2, i N. Given this rules, and if the first period winner invests a quantity I, the expeted soial ost is [ n ] (Γ e, I) = i q 1,e i () f n ()d + β w qw 2,e () + i q 2,e i () f n 1 ( w ) ( w, I)d + βψ(i) n i=1 w n i w = n [1 F ()] n 1 n 1 f()d + β [1 F ()] (, I)d +β(n 1) [1 F ()] n 2 [1 G(, I)]f()d + βψ(i) Therefore, the effiient investment level, I e, is the solution to min (Γe, I) In the next result we haraterize I e and find a mehanism that ahieves effiieny regardless of investment observability. Proposition 11 The effiient investment level I e is the solution to min Ψ(I) [1 F ()] n 1 G(, I)d (9) Moreover, it an be indued regardless of investment observability using seond prie sealed bid prourement autions. Proof : See Appendix. 9
In equation (9) we an see that there are osts and benefits assoiated to investment. On the one hand, investment must be paid for (first term). On the other, there is an expeted ost redution due to one ompetitor having a better distribution (seond term) 3. To onlude this setion, notie that the irrelevane of investment observability is due to risk neutrality. The hoie of rules and investment level just desribed is equivalent to the one made by a buyer who is able to observe suppliers osts (that is, a situation where adverse seletion effets are shut down). From this perspetive, the investment level deision orrespond to the effort level hosen by the agent in a risk neutral prinipal-agent model, and it is widely known that the first best solution an be ahieved when effort is not observable under the risk neutrality assumption. 5. Revenue Maximization Under Full ommitment In this environment we assume the existene of institutions that may enfore the ontrats established by the buyer. He ommits, before any ost realization, to rules for the first and seond period prourement mehanisms. Seond period rules an give different onditions to the first period winner and losers, and an depend on the investment level when it is observable. However, if a seller deides not to partiipate in the first period, we assume the buyer does not have enough ommitment power to guarantee he will not have aess to the seond period mehanism and treated just as a first period loser. This last point is important (as it will be explained when we disuss the partiipation onstraints), and very plausible. Even if a buyer ould ommit to this rule, a first period buyer ould hange the name of his firm and ask for aeptane in the seond period, making it impossible for a seller to leave him out without being aused of partiality. We first haraterize the ost minimizing mehanism with investment observability. We find that it gives an advantage to the first period winner and that this advantage is independent of the level of investment required. This, in turn, indues over-investment by the first period winner. Then we turn to the ase of non-observable investment. We find that the optimal mehanism remains unhanged: the level of investment hosen by the first period winner is the same the buyer would have hosen if he ould observe it. 5.1 Investment Observability and Full ommitment In this ontext, sine investment is observable, the buyer an indue any amount of investment he wants. This an be done by setting transfers low enough (even payments to the buyer) so that any other level hosen by the first period winner is unprofitable for him. Under this shemes, histories assoiated to other investment levels never happen. Therefore, we assume the buyer hooses the investment level I 0. Partiipation in the seond period is ensured by Π 2 P o 2 w,i ( w, w ) 0, w (I) Π 2 l,i,i ( i, i ) 0, i, i w. At t = 1, we require partiipation in both prourements autions to be more profitable (in expeted terms) than doing it only in the last one (as in Pesendorfer and Jofre-Bonet [9]). The justifiation for this omes from the fat the buyer annot ommit to exlude a player who did not partiipate in the first 3 Reall that for eah, I G(, I) is inreasing, thanks to Lemma 9, (iii) 10
period mehanism from the seond period one. 4 Therefore, the first period partiipation onstraint an be written as 5 : P o 1 (I) : Π 1 i,i( i, i ) β Π 2 l,i,i(, )f()d, i, i N We are now ready to state the optimization problem faed by the mehanism designer. Denote by (Γ, I) the expeted prourement ost when the buyer hooses a mehanism Γ and an investment level I. It orresponds to: n = i=1 Ti 1 ()f()d + β Tw,I() 2 (, I)d + j w l,i,j()f()d (10) T 2 Therefore, this buyer solves: P o s.t. min Γ,I (Γ, I) I 1 o, I 2 o P 1 o (I), P 2 o (I) The following result shows that the ost minimizing mehanism under investment observability satisfies: (i) In the first period it assigns the ontrat to the lowest virtual ost seller, (ii) Seond period rules give an advantage to the first period winner, but that advantage does not depend on the investment level hosen by the buyer. Theorem 12 Under full ommitment and investment observability, the ost minimizing mehanism, Γ, does not depend on the investment level hosen by the buyer, and it is haraterized by { qi 1 1 i + F (i) ( 1,..., n ) = f( < i) j + F (j) f( j) j i (11) 0 { 1 qw 2 w < k( ( w, w ) = i ) i w 0 (12) ( ) with k() = + 1 + 1 F () n 1 f(),. { 1 ql,i( 2 k(i ) = min{ i, i ) = w, k( j ); j w} 0 (13) Proof : See Appendix. 4 If the seller ould do it, he ould extrat all the expeted seller s rent from the seond period. He ould ask for them as extra payments in the first period and threaten with the impossibility of partiipation in the seond period for sellers who are not willing to omply. 5 By subsript o we denote the investment observability setting. 11
At t = 1 the optimal rules orrespond to the ones derived by Myerson in [8], and are effiient beause of the ex-ante symmetry of ompetitors and assumption 3. However, the first period winner obtains an advantage gap for the seond prourement, that is, he an obtain the seond ontrat even when some rivals have lower osts, so the ost minimizing rule in the seond period sarifies effiieny to redue expeted osts. This advantage gap dereases with the number of sellers: as the number of ompetitors inreases, giving an advantage to the first period winner is more expensive, sine it is more likely that one of the first period losers has the lowest ost. Nevertheless, this gap never disappears, showing that sequentiality introdues memory in the optimal ontrat, expressed in the aforementioned advantage gap. Observation 1: Sine these rules do not depend on the level of investment I, they are also feasible when this variable is not observable. Observation 2: The rule defined in the previous theorem, and the proedure to obtain it, does not rely on the ex-ante degree of omplementarity (given by G(, 0)) nor on the ex-post degree of omplementarity (given by G(, I)). This ours beause the buyer is able to inter-temporally distribute inentives aross time in a better way than performing two independent prourements. In order to inrease the ompetition in the first period, rules for the seond period are modified. Notie that this happens even when the number of seller goes to infinity. To omplete our haraterization of the ost-minimizing mehanism, we now haraterize the investment level hosen by the buyer, I. Theorem 13 Under full ommitment of the buyer and investment observability, the ost minimizing investment level I is the solution to min Ψ(I) [1 F (k 1 ())] n 1 G(, I)d (14) Proof: See Appendix. In the previous expression we an see the the two effets of investment. The first term is the ost of investment (at t = 2 the buyer must ensure partiipation for the first period winner), the seond one reflets the advantages of it: as investment goes up G(, I) goes up, sine more weight is put in low-ost realizations of the first period winner. However, when ompared to the effiieny ase (equation 9), this last effet is stronger. Beause now the first period winner is granted an advantage gap at the seond prourement, it happens that k 1 () < whih implies that [1 F (k 1 ())] n 1 > [1 F ()] n 1. The benefits of investment for the buyer are now larger, sine the first period winner is assigned the ontrat more often, due to the advantage gap. From now on we assume that the first order ondition in (14) is satisfied. 5.2 Investment Non-Observability and Full ommitment As we said before, in this ase rules annot be funtions of the investment level hosen by the first period winner. He deides, knowing the seond period rules, how muh to invest. Therefore, the seond period rules hosen by the buyer influene the investment deision, and this effet must be taken into aount by the buyer. With all this in mind, we an write the buyer s problem as 6 : 6 By subsript no we denote the investment non-observability environment. Reall that in this ase the rules hosen by the buyer are not funtions of investment. 12
P no s.t. I arg max J 0 min (Γ no, I) Γ no,i Π 2 w(, ) (, J)d I 1 no, I 2 no P 1 no(i), P 2 no(i) The first restrition must be added beause the investment level is hosen by the first period winner. It speifies that the ost minimizing investment level that the buyer wishes to be implemented must be profit maximizing under mehanism Γ no for the first period winner. Denote by (Γ no, I no) the solution to P no. It is lear that (Γ, I ) (Γ no, I no) due to the additional restritions in P no. The next result establishes that the mehanism (Γ, I ), optimal when investment is observable, is also feasible when investment is non-observable. This is so not only beause Γ has seond period rules that do not depend on investment, but also beause I, under this rules, is profit maximizing for the first period winner. So, it satisfies I no. Proposition 14 Under full ommitment of the buyer and investment non-observability, the solution to P no is Γ no = Γ, I no = I, with (Γ, I ), the solution with investment observability. Proof: See Appendix. Therefore, as (Γ, I ) is the solution in both settings, observable and non-observable investment, we all it the full ommitment ost-minimizing solution. We an see that when the buyer set rules aording to Γ, he provides the right inentives to indue the first period winner to invest I, the same level that the buyer would have hosen himself. Therefore, the intuition from the moral hazard literature reappears even with all the elements of adverse seletion present in this environment: with risk-neutral agents, the buyer an indue exatly the level of investment desired, without hanging the mehanism he would have hosen if he himself had to selet the investment level. 5.3 Over-Investment Under full ommitment, the buyer gives an advantage to the first period winner in the seond period. Surprisingly, this indues an investment level that is above the effiient one, as shown in the next result. Proposition 15 Assume that I e and I satisfy the first order onditions of problems (9) and (14), respetively. Thus, if the buyer is fully ommitted over-investment ours, that is, I e < I. Proof: See Appendix. The intuition for this result depends on the degree of investment observability. If investment is nonobservable, two effets are present. On the one hand the seond period advantage gap gives the first 13
period winner inentives to relax, and to invest less in a distributional upgrade, sine he does not really need a low ost to win and investment is ostly. On the other hand, investment generates lower expeted osts, inreasing his profit margin in ase of winning. As the advantage gap inreases, winning beomes more likely and therefore investment beomes more attrative. This seond effet dominates the first. To see this, we an rewrite the onditions that define I e and I ((9) and (14)) as I e arg max I arg max [1 F ()] n 1 G(, I)d Ψ(I) [1 F (k 1 ())] n 1 G(, I)d Ψ(I) The first term in both equations orresponds to the fration of the first period winner s expeted payoff in the seond period that depends on investment. In both ases, this term inreases with I, beause G(, ) is inreasing. But sine k 1 () <, this effet is bigger in the full-ommitment ost-minimizing ase. This shows how the inentives to relax are dominated by the inentives to invest and inrease the profit margins. Moreover, this effet is quite strong. onsider two mehanisms Γ and Γ, not depending on investment, suh that the orresponding seond period expeted winning probability funtions for the first period winner satisfy Q 2 w() Q 2 w(),, with strit inequality on a positive-measure subset of. That is, Γ gives more advantage to the first period winner at t = 2 than Γ. Therefore, Q 2 w()g(, I)d Ψ(I) > Q 2 w()g(, I)d Ψ(I) whih allow us to onlude that Ĩ > Ī, with Ĩ, Ī the orresponding indued investment levels (assuming that the first order onditions hold). This shows that the most ineffiient mehanism, Q 2 w() 1, indues the largest amount of investment possible. If investment is observable, and thus hosen by the buyer, the result is the same but the intuition is different. If the first period winner is going to have an advantage in the seond period, and therefore he is likely to win anyway, it is good for the buyer to inrease the likelihood of him having low osts. This effet makes investment more attrative than in the ase of effiient seond period rules (whih give no advantage), and therefore over-investment ours. 6. Revenue Maximization Under Non-ommitment In this environment we assume that the buyer annot ommit to the seond period rules before the investment stage, and this is known by the sellers. This fat indues, through a seond period mehanism that is disadvantageous to the first period winner, an investment level below the effiient one. Also, in this setting, the observability of investment will make a differene (unlike the ommitment ase). To begin with, suppose that the first period winner invests an amount I before the seond prourement. Now, the buyer has inentives to hange rules based on the investment level, and onsiders the investment expenditures, Ψ(I), as sunk osts. We start with the ase of observable investment. 14
6.1 Investment Observability and Non-ommitment Beause investment is observable, the buyer an make use of mehanisms of the form Γ 2 = ({t 2 w,i}, {q 2 w,i}, {t 2 l,i}, {q 2 l,i} ) It is worth to emphasize that beause of the buyer s inability to ommit to mehanisms, he annot deide the investment level even though it is observable. Sine partiipation in the seond period must give non-negative profits, it must be the ase that T 2 w,i() Q 2 w,i() 0, Remembering that the seond period expeted utility for the first period winner is expressed by the partiipation onstraint an be written as: Π 2 w,i(, ) = T 2 w,i() Q 2 w,i() Ψ(I) (P 2 n) Π 2 w,i(, ) Ψ(I),. (15) Finally, as in the previous setions, the seond prourement expeted ost orresponds to 2 (I) = Tw,I() 2 (, I)d + Tl,I,i()f()d 2 i w If the first period winner has already hosen an investment level I (observable), sequential rationality implies that the buyer solves, for the seond period: min 2 (I) Γ 2 P o (I) s.t Π 2 w,i (, ) Ψ(I), Π 2 l,i,i (, ) 0,, i w, i N I 2 o The following result shows that, given any investment level arried out by the first period winner, the buyer gives disadvantage to this agent at t = 2. Sine the buyer annot ommit to ontrats and the first period winner has improved his distribution, the buyer has no inentive to ontinue giving the mentioned advantage gap of the full ommitment solution. As usual, informational asymmetries harm the ompetitor with the best distribution in one-shot autions. Lemma 16 In the absene of buyer s ommitment, the seond period ost minimizing mehanism Γ 2 (I), when the investment observed is I, orresponds to { } q w,i( 2 1 w + G(w,I) < min w, w ) = (w,i) j + F (j) j w f( j) (16) 0 { q l,i,i( 2 1 i + F (i) i, i ) = f( < min i) w + G(w,I) 0 (w,i), min j i,w { } } j + F (j) f( j) (17) 15
Proof : See Appendix. The seond period mehanism gives a disadvantage to the first period winner, sine + F () f() < + G(,I) (,I), I 0 (Lemma 9, (ii)). Thus, it an our that the first period winner loses even if he has the lowest ost. We an see how an opportunisti behavior appears, exatly as in a hold-up situation. The first period winner ommits to a sunk ost investment to improve his ost distribution, allowing the seller to take advantage of this speifi investment. The buyer does so by hanging the rules against the first period winner, and therefore extrating more rent from him. The next result haraterizes the optimal first period mehanism under non-ommitment. The assumption of sequential rationality diretly implies that these rules must be optimal in a one-shot prourement aution setting. Therefore, the buyer should ompare virtual osts in order to assign the first period projet. orollary 17 In the absene of buyer s ommitment, the first period ost minimizing assignment rules orrespond to { qi 1 1 i + F (i) ( 1,..., n ) = f( < i) j + F (j) f( j) j i (18) 0 Proof : Diret. We now turn to the investment level indued in this environment. A first period winner, antiipating the buyer s behavior in the seond period (the seletion of mehanisms of the form Γ 2 (I)), determines how muh to invest. This allows us to fully haraterize the investment level: Proposition 18 With non-ommitment and investment observability, the investment level hosen by the first period winner, Î, solves V (I) [1 F (J 1 (J I ())] n 1 G(, I)d Ψ(I) (19) max with J I () = + G(,I) F () 2 and J() = + (,I) f(). Thus, the optimal mehanism for the buyer is Γ (Î). Proof : See Appendix. We an see how investment affets V ( ) through three hannels. The first term in the integral, [1 F (J 1 (J I ())] n 1, dereases with I, sine as investment inreases, the seond period mehanism gives a bigger disadvantage to the first period winner. The last term, Ψ(I), also reflets a negative effet of investment, sine it is ostly. However, the seond term in the integral, G(, I) is inreasing in I for eah, and it reflets the inreased probability of winning due to a better ost distribution. If we ompare ondition (19) with ondition (9), that defines the effiient level of investment, we observe that the positive effet of investment (given by an inrease in G(, I)) is weaker in the ase of non-ommitment, sine 7 [1 F (J 1 (J I ())] n 1 < [1 F ()] n 1 7. Using assumption 1, lemma 9 (ii), and assumption 3 we have that J 1 (J I ()) > 16
Intuitively, in an ex-post effiient mehanism, the probability of a first period winner sueeding at t = 2 is the probability that all first period losers have higher osts ([1 F ()] n 1 ). On the other hand, his probability of winning in a non-ommitment environment is the probability that all losers have higher virtual osts than him ([1 F (J 1 (J I ())] n 1 ). This last probability is lower sine virtual osts are higher when the ost distribution is improved. The fat that the absene of ommitment redues the advantage granted to the first period winner at the seond prourement (when ompared to the effiient mehanism) will indue a lower investment level hosen by this agent, as shown in the next result: Proposition 19 Assume that I e, I and Î satisfy the first order onditions of problems, (9), (14)and (19), respetively. Then, under non-ommitment of the buyer and investment observability, investment falls below effiieny, that is Î < Ie < I. Proof : See Appendix. Here, a hold-up intuition appears: antiipating that the rules will be biased against him in the seond prourement mehanism, and knowing this effet will be more pronouned the more he invests, the seller invests less than the effiient level. In other words, he does not internalize all the benefits from investment, sine part of it is appropriated by the seller as rent extration. 6.2 Investment Non-Observability and Non-ommitment We now onsider the ase when investment is not observable. If the buyer ould monitor the investment level hosen by the first period winner, I, he would reat optimally imposing Γ 2 (I), as in the previous subsetion. Now, sine investment annot be observed, the buyer annot hoose a ontingent plan with one mehanism for eah level I seleted by the first period winner. We model this situation as a simultaneous-move game between the first period winner and the buyer where the former hooses an investment level and the latter a seond period mehanism. In suh a game, the buyer hooses just one mehanism, whih is a best response to the investment level hosen by the first period winner. Symmetrially, the investment level is a best response to the mehanism hosen by the buyer. Rigourously, we define the ation spae for the first period winner to be A w = [0, + ) For the buyer, the strategy spae ould be the set of inentive ompatible mehanisms that satisfy partiipation onstraints. However, by a rationalizability argument, we onsider only the mehanisms that are a best response for some investment level of the first period winner, that is A b = { Γ(I) I 0} where Γ(I) is the sequentially optimal mehanism given an investment level I defined in lemma 16. In this ontext we an define a pure strategy equilibrium: Definition 20 A pure strategy equilibrium under non-ommitment and investment non-observability is a tuple (Γ, I) A b A w suh that (i) Γ = Γ(I) 17
(ii) I arg max Π 2 w,γ K 0 (, ) (, K)d Ψ(K) where Π 2 w,γ (, ) is the seond period expeted utility of a first period winner with ost that reveals truthfully under the mehanism Γ. ondition (i) implies that the mehanism hosen by the buyer is a best response to the investment level hosen by the first period winner. ondition (ii), respetively, implies that given the mehanism Γ, the first period winner is hoosing his investment level optimally. We now state a simple lemma that allows to proeed with the haraterization of the investment level and the optimal mehanism. Lemma 21 A pure strategy Nash equilibrium is given by ( Î, Γ( Î)) where Î arg max U(K, Î) [1 F (J 1 (J Î()))] n 1 G(, K)d Ψ(K) (20) K 0 Proof : See Appendix. The funtion U(K, I) orresponds to the seond period expeted utility for the first period winner if he invests an amount K and faes the mehanism Γ 2 (I). If we ompare the funtion U(, I) with the funtion V ( ) defined in proposition 18, whih haraterizes investment for the ase of observability, we an see that the negative effet of investment due to more disadvantageous mehanisms is fixed. The reason is that the buyer annot observe the investment level hosen by the first period winner. However, the positive effet orresponding to an inreased probability of winning and the negative effet orresponding to the ost of investment are still present. Sine one of the negative effets of investment is fixed, the inentives to invest are now bigger than in the investment observability ase. The following proposition ensures the existene of an equilibrium, under the assumption that the marginal benefit due to a distributional improvement disappears when the investment level arried out is large enough. It also states that under non-ommitment of the buyer, the possibility to monitor investment indues levels below the ones hosen when investment annot be observed. This ours beause, when investment is observable, the buyer an reat optimally with more disadvantageous mehanisms as investment inreases. Proposition 22 Suppose that the following ondition holds 8 : lim [1 F (J 1 (J I ())] I n 1 (, I) = 0 (21) I Then, a unique equilibrium exists and orresponds to the tuple ( Î, Γ( Î)) where the investment level Î is given by ondition (20). Also, Î Î, with strit inequality if Î satisfies the first order ondition of problem (19). Proof : See Appendix. We an also ompare this investment level with the effiient one. 8 The family of distributional upgrades introdued in example 1 satisfies this ondition. 18
orollary 23 If I e satisfies the first order ondition of problem (9), then, Proof : See Appendix. Î Î < I e The intuition of the result is the following: disadvantageous mehanisms disinentive ost-redution investment. This is so beause the marginal benefit of investment (whih omes from the expeted ost redution in the seond period) dereases when the seond period rules beome less attrative (sine they give a disadvantage to the first period winner). This ours in both the ase of investment observability and non-observability. However, this effet is less marked when the buyer annot observe the investment level and must hoose a mehanism guessing the first period winner s deision. If he an observe the investment level, he an reat optimally and hoose even more disadvantageous mehanisms in the seond period. This fat is antiipated by the first period winner, whih results in lower investment levels. To onlude, note that the assumption of sequential rationality implies that the optimal first period rules in this environment orrespond to the ones defined in orollary 17. As before, the first period transfers are set-up suh that partiipation is ensured for all sellers at both ompetitions. As a onsequene, the ost minimizing mehanisms under non-ommitment (for observable and non-observable investment) are feasible for a fully ommitted buyer. Therefore, the expeted ost in an environment without ommitment is neessarily higher than in an environment with ommitment orollary 24 The expeted ost of both prourements is lower under full ommitment than under nonommitment of the buyer (whether investment is observable or not). Proof : See Appendix. 7.1 Investment Observability Irrelevany 7. Disussion The fat that investment observability annot improve the buyer s ability to redue expeted osts (in the ase of full ommitment) is interesting. Moreover, this is also the ase when the buyer s objetive is ex-post effiieny, sine the ex-post effiient stage mehanisms indue the effiient investment level. A natural question that arises is whether this is true for any mehanism used by the buyer. The answer is no, as shown in the next result. When the buyer sets, for instane, a mehanism that gives less advantage to the first period winner than the one in the ost-minimizing mehanism, the investment level hosen by the buyer (in the observable ase) is higher than the one hosen by the first period winner (in the non-observable ase). The intuition is as follows: with a smaller seond period advantage, seond period rules are loser to a sequentially optimal mehanism. Therefore the buyer aptures more of the surplus generated by a ost distribution improvement 9 and, onsequently has a bigger marginal benefit from investment. It is not surprising, then, that if he hooses investment, he will selet a level above the one a first period winner would hoose. 9 Whih is also bigger, sine suh a mehanism is also loser to an effiient one. 19