Firms' Strategic Choices under Demand Uncertainty (Preliminary and Incomplete)

Transcription

1 Firms' Strategic Choices uder Demad Ucertaity (Prelimiary ad Icomplete) Walter Beckert Uiversity of Florida December 2002 CORRESPONDENCE should be directed to: Walter Beckert, Departmet of Ecoomics, 224 Matherly Hall, PO Box , Uiversity of Florida, Gaiesville FL , USA; Tel: , Fax: , This work was doe while I ejoyed the hospitality of the Departmet of Ecoomics at UCL. All errors are mie. 1

2 Abstract This paper ivestigates how rms choose prices ad productio capacity whe facig stochastic demad. With demad ucertaity, rms face demad curves for their goods which implicitly subsume the sellig mechaism adopted i the markets for these goods. The sellig mechaisms cosidered i this aalysis are biddig markets or auctios, ad take-it-or-leave-it sales. They have di eret implicatios for the respective expected demad curves the rms face, ad hece for the rms' capacity choices. These implicatios etail cosequeces for welfare. It is show that auctios yield higher expected comsumer welfare tha take-it-orleave-it sales whe productio costs are high, while take-it-or-leave-it sales ehace welfare whe productio costs are low. KEYWORDS: Stochastic Demad, Auctios, Take-It-Or-Leave-It sales, Moopoly. 2

3 1 Itroductio This paper ivestigates the behavior of rms uder itrisic market ucertaity. The source of market ucertaity cosidered i this paper is stochastic demad, arisig from buyers' valuatios for goods beig private iformatio. The paper aalyzes how rms choose capacity whe demad is ucertai. The aalysis of rms' capacity choices builds o the isights provided by Kreps ad Scheikma (1983) ad (...) that rms' capacity choices are best thought of as a two-stage process. I their decisio process, rms face demad fuctios which implicitly subsume the mechaism adopted to sell goods o a secod stage, oce rms have pre-committed themselves to productio capacity o the rst stage. This paper shows that, uder demad ucertaity, the secod-stage sellig mechaism has importat implicatios for the rms' rst stage capacity decisio. Biddig markets, such as a auctio market as the secod stage sellig mechaism, force the rm to make capacity decisios uder price ucertaity, as bids are ex ate ucertai. Take-it-or-leave-it sales, o the other had, i which the seller quotes a price ad potetial buyers decide whether or ot to purchase at this price, itroduce quatity ucertaity ito the rms' decisio process, as potetial buyers willigess to pay is ex ate ukow. The paper shows that, ext to productio costs, the type of ucertaity iduced by the secod stage sellig mechaism determies rms' capacity choices. Ad it ideti es importat welfare implicatios of either mechaism whe demad is ucertai. The results of this aalysis have theoretical ad policy implicatios. The departure from determiistic demad, itroducig stochastic demad, etails some surprisig cosequeces. As already alluded to, demad ucertaity ca emerge i two represetatios: price ucertaity i biddig markets such as auctios, ad quatity ucertaity i take-it-or-leave-it sales. For moopolistic sellers, domiace of auctios i terms of expected reveue is a easy corollary, provided i this paper, to the well-kow result that expected reveue i sigle item auctios exceeds expected reveue i take-it-or-leave-it sales (see, for example, Beckert (2002), ad related Bulow ad Klemperer (...)). Oe might ituitively suspect that biddig markets are better suited to aim at the highest-valuatio buyers, while oce-ad-for-allprice quotes i take-it-or-leave-it sales target a broader group of potetial buyers with a wider rage of valuatios. Hece, this suggests that whe productio costs are su±cietly high, a moopolist istitutioally forced to sell i a take-it-or-leave-it sale may d these costs prohibitive, choosig ot to produce at all. If allowed to sell i auctio, o the other had, the moopolistic rm may expect to catch high valuatio buyers ad, therefore, may establish productio capacity, at least o a small scale. Hece, as the paper demostrates i geeral, with demad ucertaity, biddig markets as the secod stage sellig mechaism ad esuig price ucertaity have obvious welfare bee ts i high productio cost eviromets, as they iduce productio, while take-it-or-leave-it sales may ot sustai its expected pro tablility. As a coverse, cosider as a extreme case the absece of ay productio costs. The, a 3

4 auctio seller will keep capacity restricted because, beyod a certai poit, the margial expected reveue from a additioal successful bidder is egative. O the other had, it turs out that additioal buyers always geerate positive margial expected reveue for the take-it-or-leave-it seller. Therefore, the take-it-or-leave-it moopolistic seller will produce more tha the auctio seller, ad charge a lower price tha what the auctio seller may expect to receive from bids. Hece, as the paper also establishes i some geerality, with demad ucertaity, take-it-or-leaveit sales have cosumer welfare bee ts i low cost eviromets, sice productio ad expected cosumer surplus are higher. To summarize, the rst part of the paper shows that moopolists' capacity choices are ot ivariat to the sellig mechaism whe demad i stochastic, ad it highlights that this ivariace has sigi cat cosumer welfare cosequeces. As a dual aalysis, the paper ivestigates procuremet situatios as well. The importat distictio betwee procuremet decisios ad productio capacity decisios is that, while the producer's problem is a two-stage problem with suk capacity costs o the rst stage, procuremet costs ad bee ts typically arise joitly. Hece, typical procuremet decisios are ot two-stage decisio processes. Distiguishig procuremet auctios ad take-it-or-leave-it cotract awards, welfare results remiiscet to the oes arisig from previously cosidered productio decisios ca be ideti ed. Ii is show that if, for reasos of political credibility or commitmet, a project has to be udertake oce it is tedered, procuremet auctios will oly be held for su±cietly high bee ts of the tedered project; take-it-or-leave-it awards allow the exibility to quote a procuremet cost ceilig that always results i a expected pro t from tederig ad hece always leads to a teder beig put out. The welfare implicatios carry some sigi cace i light of the dual, ofte simultaeous prevalece of these sellig mechaisms i may markets, ad the frequet legal madate to adopt biddig procedures, especially for the procuremet of public goods. Real estate markets, markets for buildig cotracts ad for heavy buildig equipmet, ad markets for airport ladig slots are amog prime examples of markets o which these competig mechaisms co-exist. I the cotext of buildig costructio, the results of this paper are particularly relevat because public costructio cotracts are ofte legally boud to be awarded by holdig procuremet auctios. Biddig i highway procuremet has bee studied ecoometrically by Jofre-Boet ad Pesedorfer (2000), Feistei et al. (1985), Porter ad Zoa (1993), ad Bajari (1997). Bajari et al. (2001) examie the factors that determie whether o-residetial buildig cotracts i the private sector were awarded by auctio or egotiatio, as a more re ed versio of a take-it-or-leave-it sale. O the basis of data from Norther Califoria coverig the period , they observe that auctios ted to be couter-cyclical, while egotiatios appear to be pro-cyclical. Their theoretical iterpretatio is that auctios are favored whe buyers are more cost-sesitive. Procuremet auctios, with a moopsoistic buyer ad the wiig bidder submittig the lowest procuremet cost as his or her bid, are the mirror image of the type of auctios cosidered i this paper. The empirical exploratio by Bajari et al. suggests that auctios are favored whe expected pro t margis are 4

5 critical, ad this seems to coicide with ecoomic dowturs. The aalysis i this paper suggests that, if these projects were relatively low valuatio projects, the awardig them by egotiatios would have created higher procuremet ad would have acted as couter-cyclical public policy measure. The results of this paper are also relevat for the recet exploratios of policy optios to maage ladig slots at cogested airports, whe cosidered i the joit cotext of capacity expasio. Cogestio maagemet essetially always ivolves two compoets: decisios o capacity ad a mechaism to allocate it. Traditioal pricig schemes for allocatig ladig ad take-o capacity are aki to take-it-or-leave-it sales 1. Recet iitiatives by the U.S. Departmet of Justice, however, cosider ladig slot auctios for cogested airports, such as New York's La Guardia Iteratioal Airpot. The results of this paper ideti e the log-term capacity repercussios that such allocatio mechaism ca be expected to etail. The paper proceeds as follows. Sectio two starts by layig out some auxiliary results from mathematical statistics which permit coveiet ormal approximatios to biomial distributios ad to the distributios of order statistics, both of which play importat roles i the developmet of the theory of optimal strategies uder the two mechaisms ad idepedet private valuatios. It proceeds by presetig geeral results for the moopoly case, illustratig all the relevat implicatios of the two sellig mechaism o strategic moopolistic capacity choice i the cotext of a simple example with idepedet, uiform valuatios. Sectio three carries the aalysis further to aalyze procuremet situatios. Sectio four expads the aalysis to dyamic settig. Ad sectio ve cocludes. Proofs are collected i the appedix of the paper. 2 Moopolistic Price ad Capacity Choices 2.1 Prelimiaries This sectio is devoted to the aalysis of a moopolist's capacity choice uder demad ucertaity. The mooplist's decisio problem is recogized as a two-stage process. O the rst stage, the mooplistic producer commits to a capacity level. O the secod stage, uits of the good up to the capacity level are sold, accordig to some sellig mechaism. Potetial buyers' valuatios of uits the good are assumed to be private iformatio. Two mechaisms are cosidered: Auctios, i which potetial buyers submit bids for utis of the good, ad take-it-or-leave-it sales, i which 1 I practice, these are more itricate tha simple price quotes, ivolvig ofte iteratioal agreemets ad so-called \gradfather rights", as well as secodary tradig. It is a ope policy questio how the recet debate of ladig slotauctios might address such itricacies. For a comprehesive descriptio, see documets by the British Civil Aviatio Authority. The two most cogested Lodo airports, Heathrow ad Gatwick, are also curretly subject to a debate o expadig their capacity. 5

6 the moopolistic seller quotes a uit price ad potetial buyers purchase uits of the good if their private valuatios of these uits exceeds the price. To formalize the aalysis, the followig assumptios are maitaied throughout ad commeted o as they become relevat: A1 There are potetial buyers with uit demads; their valuatios X i, i = 1;:::;, for a uit of the good are idepedetly ad idetically distributed with cumulative distributio fuctio F (x), x 2X, iffx : x 2Xg 0, with cotiuous desity f (x) > 0 for all x 2X ; ad E[X] <1. A2 Margial reveue 1 F (x) xf(x) is strictly dowward slopig 2 ; A3 The productio of the good uder cosideratio ivolves costat margial costs c ad o xed costs; productio costs are suk oce cacapity is chose. A4 The item has zero value for the seller if it is ot sold; the seller maximizes expected pro t, ad the buyers maximize expected surplus. A5 The umber of potetial buyers,, is large. The auctio seller faces price ucertaity at the secod stage. Sice from the seller's poit of view o the secod stage all that matters is expected auctio reveue, i light of the Reveue Equivalece Theorem (Vickrey (1961), Riley ad Samuelso (1981), Myerso (1981)) oe ca remai agostic about the format of the auctio. For expositioal purposes, the format is thought of as aalogous to a secod-price sealed-bid auctio i the sigle item case. If k items are sold, the the k + 1st highest bid is paid by each of the k successful bidders. Let X (i), i = 1;:::;, deote theith order statistic; i.e. X (1) = mifx i ;i = 1;:::;g ::: X () = maxfx i ;i = 1;:::;g: The, the expected reveue of the auctio seller of k items is ke[x ( k) ], where the expectatio is take with respect to the distributio of the ( k)th order statistic. This distributio of the kth order statistic has a distribuio with desity f X(k) (x) =! (k 1)!( k)! F (x)k 1 (1 F (x)) k f (x); x2x; see, e.g., Bickel ad Doksum (1977). Hece, the auctio seller's rst-stage capacity choice problem is max k(e[x ( k ] c): k Notice that this embeds the solutio to the secod stage sellig mechaism. 2 I auctio theory, this is sometimes referred to as a regularity coditio; see Klemperer (2000). 6

7 Assumptio A5 premits some coveiet approximatios which are heavily made use of i the further developmet of the theory of capacity choice uder demad ucertaity. For large, the scaled order statistics are asymptotically ormally distributed (compare, e.g., Bickel ad Docksum (1977)), i.e. for k = [t], t2 [0; 1], p ()(X(k) F 1 (t))! d N 0;t(1 t)=(f (F 1 (t))) 2 so that the approximate distributio of X (k) is give by à µ A k X (k)» N F 1 ; 1 k (1 k )! (f (F 1 ( k ; k = 1;:::;; )))2 here, F 1 ( ) deotes the iverse cumulative distributio fuctio. 3 rst-stage decisio problem ca be re-cast as maxk k µ F 1 µ k c : Hece, the auctio seller's Cosiderig, ext, the take-it-or-leave-it seller. O the secod-stage a oce-ad-for-all uit price p is quoted ad buyers subsequetly decide whether or ot to purchase a uit of the good. Buyers are thought of as submittig purchase orders; if more purchase orders are submitted tha permitted by rst-stage capacity, demad is ratioed accordigly. Ratioig is assumed to take the form of each of the purchased orders beig ful lled with equal probability. I this mechaism, the seller faces quatity ucertaity o the secod stage. The umber of purchase orders at price p has a biomial distributio bi(; 1 F (p)), because the probability of a potetial buyer actually buyig at price p equals the probability of this buyer's valuatio exceedig p, Pr(X > p) = 1 F (p). Assumptio A5 permits aother useful approximatio i the case of take-it-or-leave-it sales. For large, the biomial distributio ca be closely approximated by the ormal distributio (see, e.g., Bickel ad Doksum (1977)). Hece, for large, the umber of purchase orders N (;p) submitted is approximately distributed as N(;p)» A N ((1 F (p));f(p)(1 F (p))): Let ¹(;p) = (1 F (p)) ad ¾(;p) = p F (p)(1 F (p), ad deote the cumulative distributio fuctio ad the desity of the stadard ormal distributio by ad Á, respectively. Also, let N(;k;p) deote the umber of items sold at the secod stage. Usig this approximatio, the seller's expected sales, give price p at the secod stage ad capacity k chose o the rst stage, are approximated by Z k µ µ µ 1 k ¹(;p) k ¹(;p) E [N(;k;p)] = u 1 ¾(;p) Á du + k 1 ¾(;p) ¾(;p) µ µ µ µ k ¹(;p) k ¹(;p) k ¹(;p) = ¹(;p) + k 1 ¾(;p)Á : ¾(;p) ¾(;p) ¾(;p) 3 Give that A1 requires f(x)> 0 ox, F(x) has ot at regios ad, therefore, the iverse CDF is well de ed. 7

8 Note that i the case of both ormal approximatios, the approximatio error is of the order of 1. Bickel ad Doksum (1977) provide further details. 2.2 Optimal Capacity Choices This sectio develops some geeral results regardig optimal capacity choices uder the two mechaisms. These are illustrated later i the cotext of a simple example. As a cosequece of A1, pe [N(;k;p)] is cotiuous i p ad ite, sice E[X] < 1 implies that there is o probability mass at i ity if X i ubouded. Therefore, it has a maximum over p o the iterior of X. Let v(;k) = max p pe [N(;k;p)], the value fuctio associated with the secod stage optimizatio problem i the take-it-or-leave-it mechaism. Recall for this iterpretatio that productio costs ck are suk, oce the moopolistic producer has committed capacity k o the rst stage. Although the problem of capacity choice is iheretly discrete uder the assumptio of uit demads of potetial buyers, treatig k as a cotiuous variable greatly facilitates the aalysis. Therefore, k is heceforth treated as a cotiuous variable. A umber of useful properties of this value fuctio are summarized i the followig auxiliary result. Lemma 1: Uder assumptios A1, A2, A4 ad A5, (i) p k = arg max p pe [N (;k;p)] exists ad is uique, ad (ii) the value fuctio v(;k) is mootoically icreasig ad cocave i k. The proofs of this ad some subsequet results are collected i the appedix to the paper. The secod part of this result ca be paraphrased as dimiishig, but uiformly positive margial expected reveue of additioal uits from the perspective of the take-it-or-leave-it seller. It has a immediate cosequece for the optimal capacity choice k? (c) = arg max k v(;k) ck of the take-it-or-leave-it seller. Theorem 1: Uder assumptios A1-A5, k? (c) < if, ad oly if, c > 0. The proof follows directly from Lemma 1 ad the simplifyig treatmet of k as a cotiuous choice variable. Margial expected reveue is ot uiformly positive for the auctio seller. Margial expected reveue for the auctio seller is give by d dk ke d X ( k) = dk kf 1 µ k µ k = F 1 k f µ k 1 ; therefore, f (x) > 0 for all x 0 = iffx : x 2Xg (A1) implies, at k =, F 1 (0) [f (0)] 1 < 0. Lettig ^k(c) = arg max k k(e X ( k) ] c), this proves a couterpart to Theorem 1: Theorem 2: Uder assumptios A1-A5, ^k(c) < for ay c. 8

9 The two results, take together, might lead to the erroeous cojecture that take-it-or-leave-it sellers always commit to higher capacity at the rst stage tha the auctio seller. The reaso why this is ot true is that the margial expected reveue of the auctio seller for small capacities k is higher tha the margial expected reveue of the take-it-or-leave-it seller. Hece for high uit costs, provided they are ot prohibitively high, the auctio seller derives a expected pro t from establishig productio capacity, while the take-it-or-leave-it seller does ot. This is a cosequece of the ext result. Theorem 3: Uder assumptios A1, A2, A4 ad A5, (i) kf 1 k = ke X( k) v(;k), for ay k ^k(0), ad i (ii) ^k(0)e hx ( ^k(0)) p E[N(;;p )] = max p pe[n(;;p)]. The result is prove for the sigle uit case k = 1 i Beckert (2002) ad follows for k > 1 by iductio o k; the proof is provided i the appedix. 4 As already alluded to, it has a importat cosequece. Cosider the case k = 1. The result states that the expected reveue from sellig a sigle uit i a auctio is higher tha the expected reveue from sellig it i take-it-or-leave-it fashio. Hece, there exists a rage of high uit costs of productio for which the auctio seller expects establishig productio capacity to be pro table, while the take-it-or-leave-it seller expects to icur a loss from producig ay positive amout. O the other had, there clearly also exist situatios i which uit costs are prohibitively high for a auctio seller to establish productio capacity. These arise for margial cost c such that E X ( 1) < c. Theorem 3 implies that a take-it-or-leave-it seller, for such c, would ot expect productio to be pro table either. The three theorems, take together, by cotiuity of the respective objective fuctios i c, have the the followig, immediate corollary which summarizes the rst-stage optimal capacity choices, give the secod-stage sellig mechaisms. Corollary 1: Uder assumptios A1-A5, there exists (i) ¹c > 0: c ¹c ) ^k(c) = k? (c) = 0; (ii) c > 0; c < ¹c: c2 (c; ¹c) ) k? (c) = 0; ^k(c) > 0 ad k? (c) = 1 ad ^k(c) 1; (iii) ^c > 0; ^c < c: c ^c ) ^k(c) k? (c). The sigi cace of the corollary lies i idetifyig cost ad mechaism costellatios which, uder the maitaied assumptios, permit uambigous welfare rakigs. For high uit costs, 4 The auctio format i Beckert (2002) is the oe of rst-price sealed-bid auctios; the result applies by virtue of the Reveue Equivalece Theorem. 9

10 the auctio mechaism domiates the take-it-or-leave-it sale mechaism, while for low uit costs the reverse is true. Ad there exists a rage of costs where further welfare aalysis requires a assessmet of optimal price quotes i the take-it-or-leave-it sales, ad expected per-uit bids i the case of auctios, with the aim to deduce expected cosumer surplus. Notice that Theorem 3 already succictly raks expected reveues. The last result of this sectio characterizes optimal price quotes ad expected bids for iterior capacity choices. Theorem 4: Uder assumptios A1-A5, for c ad ^c as i Corollary 1, (i) E (ii) E h i X ( ^k(c)) p k? (c) = p 1, i hx ( ^k(c)) p k? (c) for c ^c. The theorem, prove i the appedix, implies, i light of Corollary 1, that expected cosumer surplus at the threshold cost margi c is higher uder the auctio mechaism: Productio capacity is higher, ad expected uit paymets are lower. By Theorem 3, expected auctio reveue is also higher tha expected sales reveue. Sice capacity is higher uder the auctio mechaism, overall expected welfare is higher. For low uit cost c ^c, expected cosumer surplus is higher uder the take-it-or-leave-it sales mechaism, sice both the productio capacity uder this mechaims is higher ad the sales price falls below the expected wiig bid. Expected reveue is lower, however. As a cosequece of capacity beig higher uder the sales mechaism, expected social welfare is higher. 2.3 A Uiform Example This sectio illustrates the mai ideas of the aalysis of mooploistic capacity choice uder demad ucertaity i the cotext of a simple, tractable example. It uses the previously itroduced approximatios ad specializes the distributioal assumptios. Speci cally, it is assumed that the cumulative distributio of potetial buyers valuatios F is the uiform distributio, so that X = [0; 1] ad F (x) = x, x 2 [0; 1]. All other assumptios are retaied. Cosider, rst, as a bechmark the case c = 0. The take-it-or-leave-it seller chooses k? =, by virtue of Lemma 1. Therefore, at price p2 [0; 1], ¹(;p) = (1 p), ¾ 2 (;p) = p(1 p), ad expected pro ts are " Ã! Ã Ã!! p p pe[n(;;p)] = p (1 p) p + 1 p p(1 p) p(1 p) p Ã!# p p(1 p)á p p(1 p) 10

11 = p 1 p µr p p p µr p p(1 p)á : 1 p 1 p It is easy to see that this expressio is maximized at p? = p = 1 2. To verify, the ormal probability reduces to ( p )! 1 ad the ormal desity reduces to 1 p 2p Á( )! 0 as gets large. Therefore, the etire expressio reduces to p(1 p) which is ideed maximized at the claimed value. Note that, for c = 0, the maximizig price 1 2 is idepedet of. Expected demad at this price is 2. Now cosider the auctio seller. The auctio seller maximizes k k, which is maximizes at ^k = 2 < = k?, as predicted by Corollary 1. The expected wiig bid is ^k = 1 2 = p?, cosistet with Theorem 4. Notice that also the wiig bid for c = 0 is idepedet of. Sice the expected wiig bid equals the optimal price quote, ad expected sales equal the umber of uits for auctio, expected cosumer surplus is equal uder the two mechaisms. This ca also be formally show, as follows. Expected surplus uder the take-it-or-leave-it sale mechaism is Z 1 (E[X;X p? ] p? ) = xdx = 3 1=2 8 : Expected surplus uder the auctio mechaism, give ay k, is Substitutig ^k = 2 yields E X () + + X ( k) = 1 kx ( i) = k 1 i=0 kx i: i=0 E h i X () + + X ( 2 ) =2 = 2 1 X i ¼ 2 1 = 3 8 : i=0 Z =2 0 idi Leavig the bechmark case, cosider the case of uit cost c = 1 ad = 50. The auctio model 5 ca still be solved aalytically, but the take-it-or-leave-it sale model has to be solved umerically. 5 Followig the same steps as above, ^k( 1 5 ) = 20, ad the expected wiig bid is E[X 30] = 3 5. Therefore, expected pro ts are 20( ) = 8. Numerically solvig the take-it-or-leave-it model 5 yields k? ( 1 5 ) = 22, p k? ( 1 5 ) = 0:59, ad expected pro ts of 7:24, illustratig Corollary 1 ad Theorems i 3 ad 4. Sice ow p k?( 1 5 ) < E hx ( ^k( 15) uder the take-it-or-leave-it mechaism. ad k? ( 1 5 ) > ^k( 1 ), expected cosumer surplus is higher 5 For completeess, from the exact distributio 6 of the maximum of idepedetly uiformly distributed radom variables, E X () = 5 The computatios are straightforward, ad a code is available upo request. 6 I the uiform example, X k» (k; k+ 1), for k = 1;:::;. = ¹c; for = 50, this yields ¹c = 0:98. Similarly, the +1 11

12 maximum uit price for the take-it-or-leave-it seller is max p p(1 p ) = = 50, this amouts to c = 0: ( + 1) 1= = c; for 3 Procuremet The previous sectio cosidered auctio sales ad take-it-or-leave-it sales uder demad ucertaity. The results preseted i that sectio have atural couterparts i procuremet. Procuremet auctios have bee studied by [...] i the case of [...]. Here, projetcs are tedered by a procuremet agecy. The procuremet agecy takes the role of the seller, ad the project providers take the role of the buyers. The project's uit value is kow to the procuremet agecy. The cost of procuremet icurred by the potetial provider of the project is private iformatio. Assumptios A1 through A5 have the followig aalogues: B1 There are potetial providers with uit supplies; their productio costs C i, i = 1;:::;, are idepedetly ad idetically distributed with cumulative distributio fuctio F (c), c 2C, iffc : c2cg 0, with cotiuous desity f (c) > 0 for all c2c; ad E[C ] <1. B2 1 F (c) cf() is dowward slopig; B3 the provisio of the project uder cosideratio produces uit value v; B4 the project provisio etails zero cost to potemtial providers they are ot chose i the teder; the procuremet agecy maximizes expected surplus, ad the providers maximize expected pro ts. B5 The umber of potetial providers,, is large. I stadard auctios for k items, the wiig bidders pay the k + 1st highest bid. I the procuremet auctio for k uits, the wiig bidders receive the k + 1st lowest bid. The procuremet agecy maximizes expected surplus k(v E[X (k+1) ]). 12

13 4 Dyamic Choices 5 Appedix 5.1 Proof of Lemma 1 (i) Algebra yields µ µ d k ¹(;p) k ¹(;p) pe[n(;k;p)] = [(1 F (p)) pf(p)] + k(1 F (p)) dp ¾(;p) ¾(;p) 1 µ µ 2 pf(p)(1 2F (p)) 1 k ¹(;p) k ¹(;p) ¾(;p) Á ¾(;p)Á ¾(;p) ¾(;p) µ µ k ¹(;p) +k 1 : (5-1) ¾(;p) Dividig by, µ 1 d k ¹(;p) pe[n(;k;p)] = [1 F (p) pf (p)] + o() + o( p ); dp ¾(;p) where terms of order o() arise from the secod summad i 5 1, ad terms of order o( p ) from the last two terms. For p = iffx : x2xg, this expressio is positive, while for p = supfx : x2xg, it is egative. Hece, assumptio A2 implies that there exists a uique, iterior p k 2 ± X such that d dp pe[n(;k;p)]j p=p k = 0. (ii) Let p k = arg max p pe [N (;k;p)]. Sice v(;k) = max p pe [N(;k;p)], it ER(;k;p p ke [N(;k;p k )] µ k ¹(;pk ) = 1 > 0; ¾(;p k ) where the rst equality follows as a cosequece of the Evelope Theorem. This proves mootoicity. Cocavity follows from the cocavity of ER(;k;p) ³ = pe [N(;k;p)] i k ad p. It follows from the precedig ER(;k;p) = 1 k > 0. Therefore, 2 ER(;k;p) = ³ 2 1 ¾(;p) Á k ¹(;p) < 0. Sice ER(;k;p) has a maximum over p, give k, as argued ER(;k;p) (>)0 wheever p (<)p k, ad as p k maximizes ER(;k;p) < 0. Fially, stadard algebra shows 2 d ER(;k;p) < 0, which, icidetally, implies dk p k < 0. Hece, the Hessia of ER(;k;p) with respect to k ad p is egative semi-de ite, ad therefore the cocetrated value fuctio v(;k) is cocave i k. 2 13

14 5.2 Proof of Theorem 3 The proof proceeds by iductio. Part (i) for k = 1 is prove i Beckert (2002). The iductio step is implied by (ii) ad the mootoicity of v(;k), provided kf 1 k is mootoe for k ^k(0) as well. To see this, observe that d 2 µ k dk 2kF 1 = 2 µ k f 1 k µ k f 2 < 0; so that kf 1 k is see to be cocave. Sice ^k(0), by de itio, is its maximizer, mootoicity for k ^k(0) follows. To prove ow (ii), observe that it follows from the iductio hypothesis for k = 1, the de itios of p ad ^k(0) as well as the respective rst-order coditios that i i 2 ³ h i ^k(0)e hx ( ^k(0)) = E hx ( ^k(0)) f E X ( ^k(0)) ] E X ( 1) 2 f (E[X( 1) ]) v(; 1) v(1; 1) = (1 F (p )) 2 =f (p ) = p (1 F (p )) = p E [N(;;p )]; which completes the proof Proof of Theorem 4 Let à (x) = Pr(X () > x) xp X() (x), where p X() (x) is the ormal desity with mea E[X () ]. For ease of otatio, let ^k = ^k(c) ad p k? = p k? (c) = p 1. By de itio of p k?, à (p k?) = 0. Hece, to prove (i), it su±ces to prove that à (E[X ( ^k) ]) > 0. To see this, observe that à ³ E[X ( ^k) ] = Pr ³X > E[X ] ( ^k) E[X ]p ( ^k) X() ³E[X ] ( ^k) Pr X > E[X () ] E[X () ]p X() E[X() ] = 1=2 + o(1); where the iequality follows from E[X () ] > E[X ( ^k) ] for ^k 1 ad the fact that the ormal desity is maximized at its mea. To prove (ii), usig the abbreviated otatio ^k = ^k(c) ad k? = k? (c) for c ^c, observe that E[X ( ^k) ] E[X ( k? +1)]. For k? + 1 = [t], usig the ormal approximatio to the distributio of the order statistics, à [t] (F 1 (t)) = 1 2 p F 1 (t) s f 2 (F 1 (t)) t(1 t)2¼ 0 for large : 14

15 ³ k Sice E[X ( hatk) ] > F 1 (t) = F 1? +1, the result follows. This completes the proof of the theorem. 2 15

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