This article presents a comparative analysis of possible postponement strategies in a

Transcription

1 : Capacity ad Competitio Ja A. Va Mieghem Maqbool Dada Kellogg Graduate School of Maagemet, Northwester Uiversity, Evasto, Illiois Kraert Graduate School of Maagemet, Purdue Uiversity, West Lafayette, Idiaa This article presets a comparative aalysis of possible postpoemet strategies i a two-stage decisio model where firms make three decisios: capacity ivestmet, productio (ivetory) quatity, ad price. Typically, ivestmets are made while the demad curve is ucertai. The strategies differ i the timig of the operatioal decisios relative to the realizatio of ucertaity. We show how competitio, ucertaity, ad the timig of operatioal decisios ifluece the strategic ivestmet decisio of the firm ad its value. I cotrast to productio postpoemet, price postpoemet makes the ivestmet ad productio (ivetory) decisios relatively isesitive to ucertaity. This suggests that maagers ca make optimal capacity decisios by determiistic reasoig if they have some price flexibility. Uder price postpoemet, additioal postpoemet of productio has relatively small icremetal value. Therefore, it may be worthwhile to cosider flexible ex-post pricig before productio postpoemet reegieerig. While more postpoemet icreases firm value, it is couterituitive that this also makes the optimal capacity decisio more sesitive to ucertaity. We highlight the differet impact of more timely iformatio, which leads to higher ivestmet ad ivetories, ad of reduced demad ucertaity, which decreases ivestmet ad ivetories. Our aalysis suggests appropriateess coditios for simple make-to-stock ad make-to-order strategies. We also preset techical sufficiecy ad uiqueess coditios. Uder price postpoemet, these results exted to oligopolistic ad perfect competitio for which pure equilibria are derived. Iterestigly, the relative value of operatioal postpoemet techiques seems to icrease as the idustry becomes more competitive. (Capacity; Ivestmet; Pricig; Competitio; Productio; Ivetory; Strategy; Demad Ucertaity) 1. Itroductio This article presets a comprehesive aalysis of postpoemet strategies i a two-stage decisio model i which firms make three decisios: capacity ivestmet, productio (ivetory) quatity, ad price. The followig examples motivate the strategies to be examied. 1. Cosider a publisher who is about to release a ew book. Before sales ca be observed, the publisher aouces a price ad determies how may copies to prit. Usold books are destroyed. 2. Cosider a maufacturer itroducig a ew product. After ivestig i capacity, the firm aouces a price. The, i respose to the revealed market demad, a appropriate productio quatity is set. 3. Cosider a traditioal automobile dealer who must decide how may cars to buy ad hold o its /99/4512/1631$ electroic ISSN Maagemet Sciece 1999 INFORMS Vol. 45, No. 12, December 1999 pp

2 premises before market demad is kow. The, the sellig price is egotiated with each customer. 4. Cosider the marketig of fresh produce by a farm cooperative. Capacity is determied i advace of the seaso. The the crop is harvested. Depedig o the yield ad market coditios, either all product is brought to the market, which sets the price, or a portio of the harvest is destroyed to ifluece the market-clearig price. These four examples illustrate a wide rage of operatioal strategies see i the retail ad maufacturig sectors of the ecoomy. They highlight the three essetial decisios that are the focus of this article. The strategic capacity ivestmet decisio is made uder ucertaity i all examples. Recet practices of tyig guaratees i supply cotracts to the ex-ate capacity decisio make this type of ivestmet eve more sigificat. 1 The relative timig of the more tactical pricig ad productio decisios differs however. We refer to a strategy uder which all three decisios are made while demad is ucertai, as i Example 1, as o postpoemet. The ability to set the productio quatity after demad ucertaity is resolved, as i Example 2, is a form of productio postpoemet (ad a simplified model of make-toorder). Similarly, the ability to defer the pricig decisio util demad ucertaity is resolved, as i Examples 3 ad 4, is a form of price postpoemet (ad a simplified model of make-to-stock with ex-post pricig flexibility). While much has bee writte i the recet operatios literature o productio postpoemet (see below), price postpoemet ad its value ad impact compared to productio postpoemet has received little attetio. Summary The itet of this article is to provide isight ito the ecoomic ad operatioal value of price postpoemet versus productio postpoemet strategies by aalyzig a relatively simple two-stage stochastic ivestmet-productio-pricig model, adapted from the 1 Whe istallig productio capacity for its Smart car, Daimler- Chrysler guarateed may parts suppliers a aual productio volume of about 100,000 uits to get them to set up ear the Smart factory i Hambach, Frace (Adrews 1999, p. C8). idustrial orgaizatio literature i ecoomics. Sectio 2 itroduces the moopoly versio of the model ad partially raks the six possible postpoemet strategies that arise from chagig the timig of the resolutio of demad ucertaity. The first four strategies correspod to the examples above, while Strategy 5 assumes that both price ad productio ca be postpoed. Strategy 6, where all three decisios are postpoed ad hece made uder perfect iformatio, yields a coveiet upper boud o the value of the other postpoemet strategies. Our aalysis suggests appropriateess coditios for simple make-tostock ad make-to-order strategies. Sectio 3 shows a equivalece betwee productio postpoemet ad o postpoemet (Strategies 2 ad 1), which both ivolve ex-ate price settig. We preset techical sufficiecy coditios ad show that optimal capacity, price, ad firm value are ot mootoe i variability ad they ca be above or below their determiistic couterparts. Sectio 4 focuses o price postpoemet strategies, which are more ameable to aalysis tha their ex-ate price settig couterparts. Explicit uique solutios ad comparisos are preseted for a wide class of ucertaity distributios. This leads to several maagerial isights. First, price postpoemet strategies make the capacity ivestmet ad productio (ivetory) decisios relatively isesitive to ucertaity. I cotrast, the ivestmet uder productio postpoemet always is sesitive to ucertaity so that price postpoemet seems to result i a more effective hedge i the cotext of our model. This also suggests that maagers ca make good capacity ad productio decisios by determiistic reasoig if levels of demad variability are moderate ad the firm has ex-post price flexibility. Secod, this isesitivity result directly shows that if price postpoemet is possible, additioal productio postpoemet has relatively small icremetal value, especially uder low to moderate demad variability. Hece, before devisig productio postpoemet reegieerig techiques it may be worthwhile to cosider flexible ex-post pricig, which typically falls uder marketig s resposibilities. Third, capacity, productio ad ivetory, ad firm value are weakly icreasig i variability uder price postpoemet Maagemet Sciece/Vol. 45, No. 12, December 1999

3 While the capacity ad ivetory respose may be explaied i terms of usual safety stock reasoig, we believe that this respose ad the icreased firm value simply reflect the icreased optio value of ex-post pricig flexibility, which is abset i traditioal ivetory models with costat price. At the same time, we demostrate that additioal postpoemet or, equivaletly, better iformatio i the sese of earlier resolutio ad observatio of ucertaity also icreases capacity, productio ad ivetory, ad firm value. This is cosistet with the recet fidig of Aad (1999) that ivetory ad better iformatio (i the sese of a less oisy observatio of demad ucertaity) ca be complemets i a multiperiod model. Perhaps couterituitive at first sight, this simply highlights the differet impact of more timely iformatio, which justifies higher ivestmet to better exploit ex-post flexibility, ad of reduced demad variability, which results i lower safety-stock ivestmet. Sectio 5 icorporates competitio by itroducig a arbitrary umber of competitors i the gametheoretic aalysis of price postpoemet strategies. I cotrast to productio postpoemet, we show that a subgame-perfect pure strategy equilibrium exists ad we preset techical coditios o the hazard rate that guaratee uiqueess. We fid that as the umber of competitors icreases, the ucertaity-isesitivity result remais but is more muted. Iterestigly, however, the relative value to the firms of additioal productio postpoemet seems to icrease with competitio. Sectio 6 closes with some cocludig remarks. Fially, the Appedix cotais some techical proofs. Relatioship to the Literature Ispired by the practices of Beetto (Sigorelli ad Heskett 1984), there has bee a growig iterest i the formal study of productio postpoemet ad its operatioal trade-offs. The work of Lee ad coauthors, well summarized by Lee ad Tag (1998), itroduces the otio of operatios reversal as a reegieerig paradigm for productio postpoemet ad accompayig variability reductio. Delayig the poit of differetiatio i a multiproduct firm results i icreased flexibility ad i lower safety stock of commo upstream iputs. This results from statistical poolig beefits ad improved forecastig errors due to reduced forecast horizos (Aupidi et al. 1999). These models ca be used to determie whe productio postpoemet would be viable. As i may operatios models, price ad demad are ofte cosidered exogeous so that such models may uderestimate the beefit of productio postpoemet. I cotrast, we aalyze the impact of productio postpoemet o price ad demad ad compare it with price postpoemet by tracig the impact o capacity ivestmet ad firm value. The moopoly aalysis also relates to the cosiderable literature that cosiders the iteractio of pricig, productio, ad capacity decisios (ofte ivetory ivestmet plays a aalogous costraiig role to that of capacity ivestmet). Whiti s model (1955), aalyzed by Mills (1959), appears to have bee the first to cosider the simultaeous choice of ivetory ad prices, equivalet to our o postpoemet strategy; a itegrative review is provided by Petruzzi ad Dada (1999). We cotribute to this literature by presetig ew sufficiecy coditios o the hazard rate, by aalyzig the impact of zero demad states ad by cosiderig price ad productio postpoemet. I particular, the moopoly model of Padmaabha ad Pg (1997) is a special case of the price postpoemet moopoly model of 3 for a biary demad distributio. Aad ad Medelso (1998) study the icreased flexibility ad poolig beefits of delayed productio i a multiproduct supply chai that is edowed with a oisy iformatio system o the biary demad distributio. I studyig cotractig (rather tha competitio) i a supply chai, Ha (1997) also solves a special case of our moopoly model. The game-theoretical aalysis of competitio is fraught with difficulty because the semial equivalece result of Kreps ad Scheikma (1983) that price ad productio postpoemet strategies ca be equivalet i a duopoly model with determiistic demad curve does ot hold whe demad is ucertai. I fact, pure Nash equilibria for productio postpoemet strategies do ot eve exist for the stochastic game (Hviid 1991). Oe way to overcome this techical difficulty with the competitive produc- Maagemet Sciece/Vol. 45, No. 12, December

4 tio postpoemet model is to assume exogeous prices, as i recet literature o competitio i operatios. Lipma ad McCardle (1997) ad Parlar (1988) study competitive ivetory decisios, which i their most simple form directly correspod to capacity decisios, by aalyzig a competitive ewsvedor model. To aalyze subcotractig ad outsourcig, Va Mieghem (1999) studies the competitive capacity ad productio decisios of a cotractor ad subcotractor. Because prices are exogeous i these articles, the effect of tactical price or productio competitio o the ivestmet decisios caot be aalyzed. A secod approach is to impose istitutioal structures that make prices sticky as applied by Deeckere et al. (1996, 1997), Deeckere ad Peck (1995), ad Peck (1996) ad Butz (1997). Aother alterative is to cosider alterative formulatios of our duopoly model. Gal-Or (1987), Hviid (1990), ad Bashyam (1996) chage the timig of the players relative decisios ad cosider sequetial pricig games to study firstmover advatage. Gal-Or, followed by Bashyam, both add private iformatio to the ivestmet decisio. Arthur (1997) itroduces product differetiatio. I cotrast to these papers, our emphasis is o studyig price postpoemet, a strategy that has received limited attetio i the literature, ad o the impact of the itesity of competitio ad ucertaity o the capacity decisio. 2. Moopoly Model: Six Postpoemet Strategies Cosider a moopolist who makes three decisios. First, the firm makes a strategic ivestmet decisio i productio capacity K, followed by two tactical (operatioal) decisios to set price p ad productio quatity q K, costraied by the earlier capacity decisio. The moopolist faces a ucertai market demad curve. Obviously, the three decisios will deped o the available iformatio set, which i this case depeds o whe the moopolist observes actual demad relative to whe each decisio has to be made. Because ivestmet must precede price ad productio settig, our comprehesive aalysis ivolves the study of six strategies, each specifyig which decisio is postpoed util after ucertaity is realized. Before doig so, let us discuss the model features that are commo to all six strategies. Cost Parameters. The moopolist s total cost C is comprised of three parts. First, a capacity decisio K 0 icurs a ivestmet cost C K (K) c K K. (All our results directly exted to covex ivestmet cost fuctios C K (K).) Secod, the productio quatity decisio q K ivolves a costat margial productio cost c q. Third, c h specifies the costat margial ivetory holdig cost rate of ex-ate productio. Reveue Parameters. To specify reveues R, it is useful to differetiate the actual sales quatity s from the productio quatity q ad the demad D. At uit price p, reveues simply are R ps. Clearly, sales caot exceed demad or productio: s miq, D. (1) Ucertaity i the market demad D is modeled by a radom variable, also called a shock. Specifically, the (iverse) demad curve is assumed to be liear ad we ca always scale uits such that p D 0. (2) The radom variable has mea 1 ad perturbs the determiistic demad curve p 1 D by represetig ucertaity i the itercept or market size or willigess-to-pay. Thus, is oegative ad to avoid techicalities we will assume that its probability measure P has a distributio F() with cotiuous desity f() over ad stadard deviatio. Let F 1 F deote the tail distributio ad h f/f the hazard or failure rate. For our aalysis, it will be useful to partitio the state-space for as follows: 0 1 2,or 012 for short, where 0 p 0, p, 1 p, K p, p K ad 2 p, K p K,. (3) The three domais represet three possible outcomes. Domai 0 represets the udesired outcome where the willigess-to-pay is so low, or equivaletly the price is so high, that there is o demad at the aouced price p: D( p) 0. I domai 1, the moopolist has sufficiet capacity to satisfy demad: D( p) K. Fially, i domai 2 market demad 1634 Maagemet Sciece/Vol. 45, No. 12, December 1999

5 exceeds capacity ad some potetial sales are lost: D( p) K. Objective. We assume that the moopolist is riskeutral ad maximizes expected firm value V, which equals expected reveues mius expected cost: V ER C, (4) where E deotes the expectatio operator. To specify V i terms of the model parameters ad to determie the optimal decisios of ivestmet K, productio q, ad price p we must specify their timig relative to the observatio of. This directly leads us to defie the followig six strategies Strategy 1: No Postpoemet Uder a o postpoemet strategy, the firm must make all three decisios K, q, ad p i Stage 1, before ucertaity is resolved. Clearly, it is suboptimal to ivest i excess capacity so that K q. Productio is made to stock ad the firm icurs ivetory holdig costs. 2 All costs are icurred i the first stage: C (c K ) K. I the secod stage, after demad ucertaity is resolved, all costs are suk ad firm value is maximized by maximizig reveues ad thus sales. From (1) ad (2), it follows that s miq, Dp, mik, p. Cosequetly, if demad is low ( 01 ( p K)), some productio is wasted, while some high demad may be left ufilled ( 2 ( p K)). Hece, expected reveues pes lead to firm value (subscripts specify the strategy): V 1 K, p c K K pe mik, p (5) c K K 1 p p dp 2 pk dp. 2 For simplicity, we assume a uit time lag betwee Stages 1 ad 2. To ivestigate differet time lags t oe would use c h t istead of c h. (6) 2.2. Strategy 2: Productio Postpoemet Uder productio postpoemet, the firm must set K ad p before ucertaity is resolved, but it ca postpoe its productio. Ecoomists might say that the firm acts i price settig mode. I the secod stage we observe ad thus also the demad D ( p). Cosequetly, we will produce oly as much as we ca sell (oe will price above productio cost, p, ad produce up to capacity limits: q s K): with firm value: q s mik, p, V 2 K, p c K K p EmiK, p (7) c K K 1 2 p p dp p KdP. (8) 2.3. Strategy 3: Price Postpoemet with Clearace Uder price postpoemet with clearace, the firm must set K ad q before ucertaity is resolved ad all output q is brought to the market. This is kow as quatity settig i ecoomics. Clearly, there is o eed for excess capacity ad K q. I the secod stage we observe ad the market mechaism sets a price p that clears the market, i.e., sells all output, possibly at zero price (if maximum demad at zero price (max D ) is less tha output). We refer to this pheomeo as clearace sales: s q K ad thus p K. The resultig firm value becomes uivariate: V 3 K c K K KE K (9) c K K 12K K K dp. (10) The market clearig price, however, is ot ecessarily the optimal price. Whe maximum demad is less tha output ( 0 (K)), o reveue is geerated ad it would be better to price higher ad ot sell all output. This leads us to Strategy 4. Maagemet Sciece/Vol. 45, No. 12, December

6 2.4. Strategy 4: Price Postpoemet with Hold- Back Uder refied price postpoemet, the firm must set K ad q before ucertaity is resolved; but it has some pricig power i the secod stage. As before, we have K q, but after observig we set a optimal price. Uder low demad coditios, it is better to hold back some stock (e.g., destroy it) ad sell oly a restricted quatity s q at a higher price tha to sell all stock at a lower market clearig price as i Strategy 3. The optimal ex-post price p s maximizes reveues s( s), where s q K, so that s mi 2, K ad p max 2, K. Thus, uder poor market coditios ( 0 (2K)) we choose to hold back some stock ad sell oly /2, which is less tha productio or stock q K. Agai, the value fuctio becomes uivariate: V 4 K c K K Emi 2, K K 4 (11) c K K 02K 12K dp K K dp. (12) To complete our study of all possible postpoemet strategies, we preset the two remaiig strategies: Postpoe price ad productio (Strategy 5) ad also postpoe capacity (Strategy 6) Strategy 5: Price ad Productio Postpoemet I this case oly K must be selected before ucertaity is resolved. I the secod stage we observe ad will price optimally as i Strategy 4 but ow we also postpoe productio. Hece, we oly produce the sold quatity (q s) if the price p s exceeds the margial productio cost (which was suk i Strategy 4) to maximize operatig profits s( s ) so that s mi 1 2, K ad p max 1 2, K. Thus, we do ot produce uder dire market coditios ( 0 (c q )) while we produce ad sell less tha full capacity uder poor market demad ( 1 (c q, 2K)). The value fuctio becomes: V 5 K c K K 1cq,2K 2cq,2K dp K K dp. (13) 2.6. Strategy 6: Full Postpoemet Fially, a full postpoemet strategy captures the (ulikely) evet that all decisios ca be postpoed util is observed. Thus, all decisios are made uder perfect iformatio ad the firm ow faces a determiistic decisio problem. It ca elimiate all shortages, wasted productio, excess capacity ad holdig costs. Hece: q K s p, ad capacity maximizes V 6 () ( K) K (c K ) K, with solutio: ad expected value: K 1 2 c K ad V c K 2 (14) V E c K 2 1 c K 2 2. (15) Domiat Strategies ad the Value of Iformatio Our detailed descriptio of the various postpoemet strategies ehaces ituitio ad directly allows a partial rakig of the value of the strategies. Ideed, postpoig a decisio util after ucertaity is realized yields a perfect iformatio set whe makig that decisio. Postpoemet the directly iduces the followig weak domiace o optimal strategies: 1636 Maagemet Sciece/Vol. 45, No. 12, December 1999

7 Propositio 1. The optimal values of the differet postpoemet strategies rak as: V 1 V 2, V 4 V 3 V 4 V 5 V 6 1 c K (16) Proof. Clearly, full postpoemet domiates all other strategies ad it gives a simple upper boud to performace that oly depeds o the first two momets of the demad distributio. The relative rakig of Strategies 4 ad 5 is also obvious: Postpoig productio i additio to price (Strategy 5) domiates postpoig price oly (Strategy 4) through elimiatio of holdig costs ad suitable make-to-order productio so that o hold-back is ecessary, thereby reducig productio costs. By postpoig pricig i additio to productio, Strategy 5 also domiates the productio postpoemet Strategy 2. The productio postpoemet Strategy 2 weakly domiates Strategy 1 by savig o productio costs uder low demad scearios ad always o ivetory holdig costs: V 2 K, p V 1 K, p E p K 0. (17) Similarly, the price postpoemet with hold-back Strategy 4 weakly domiates Strategy 1 through a optimal state-depedet price choice. It also domiates Strategy 3 because the market clearig price is ot always the optimal price. The relative rakig of the productio postpoemet Strategy 2 ad the refied price postpoemet Strategy 4, however, is uclear. Similarly, the rakig of the o postpoemet Strategy 1 versus the simple price postpoemet Strategy 3 is uclear because Strategy 3 does ot use optimal decisios (rather, it uses market-clearig pricig). If, however, productio is costless (c q 0), productio postpoemet does ot yield ay savigs ad (17) yields: V 1 V 2 V 4 V 5 V 6. (18) I geeral, however, the presece of productio ad holdig costs may break this rakig (while the icorporatio of shortage costs would stregthe it). To gai more isight ito the optimal ivestmet, price, ad productio/ivetory decisios, we ow proceed to a aalysis of the two mai strategies i the ext two sectios. This will allow us to partially rak optimal ivestmet ad productio/ivetory levels uder the various strategies ad highlight their depedece o variability. While we will be able to clarify the value rakig uder moderate ucertaity, a geeral defiitive rakig of V 2 versus V 4 will remai elusive. Equatio (17) suggests that the value of productio postpoemet V 2 V 1 rises as variability ad margial productio ad holdig cost c q ad c h rise. Give that price postpoemet Strategy 4 domiates Strategy 1, this implies that Strategy 4 will domiate 2 for low levels of variability ad low margial productio ad holdig costs. This suggests that simple make-to-stock with price-flexibility strategies are preferred over simple make-to-order strategies if variability levels ad margial productio ad holdig costs are low. I geeral, however, their relative rakig reflects the trade-off betwee savig o productio ad holdig cost versus beefittig from price flexibility. The optimal trade-off will deped o the parameters c q ad c h versus the variability i (Propositio 2 will show that the traslated distributio comes ito play). Fially, 5 exteds our moopoly model ad aalysis to ivestigate the value of postpoemet uder competitio. 3. Aalysis of Productio ad No Postpoemet Strategies 2 ad 1 While it is evidet that productio postpoemet domiates o postpoemet by savig o productio costs uder low demad scearios ad always o ivetory holdig costs, it is aalytically equivalet to o postpoemet: Propositio 2. Strategies 1 ad 2 are equivalet i the sese that the optimal positive decisios K 2 ad p 2 for Strategy 2 ca be obtaied from solvig Strategy 1 with modified parameters (ad vice versa): K 2 K* 1 ad p 2 p* 1, (19) where K* 1 ad p* 1 are the optimal decisios uder Strategy 1, calculated usig distributio f *() f( ) ad zero margial productio ad holdig cost (c* q c* h 0). Productio postpoemet is equivalet to a o postpoemet Strategy 1 with a traslated decreased Maagemet Sciece/Vol. 45, No. 12, December

8 demad distributio (demad shocks decreased by c q ) but without ay holdig or productio costs; we just mark up its associated price by the margial productio cost. Because both strategies are aalytically equivalet, we will focus i the remaider of this sectio o the aalysis of Strategy 1 where we will set c c K. Firm value V 1 equals expected reveues E( p, K) pe mi(k, ( p) ) mius all costs ck. The expected reveue fuctio E( p, K) is a weighted liear superpositio of ( p, K, ) with weight factor f(). For a give K ad a specific realizatio of (that is, a specific sample path), the reveue fuctio ( p, K, ) as a fuctio of price p has three possible shapes depedig o the value of as show i Figure 1. Each ( p, K, ) has a uique maximal p(k, ) ad is uimodal cocave-covex i p ad cocave i K. As a weighted liear superpositio, the expected reveue fuctio E( p, K) may iherit some structural properties from ( p, K, ). Examples. First cosider the determiistic problem, which serves as a good base case to study the effect of ucertaity. The optimal capacity-costraied price p det (K) equals max{1 K, 1 2 }, as directly follows from Figure 1. The associated price-optimized reveue fuctio is cocave: It equals (1 K) K if K 1 2 ad 1 4 elsewhere. For a stochastic example, assume is expoetially distributed with f() e. Its expected reveue fuctio is E pe p (1 e K ) ad, as show i Figure 1, is uimodal ad cocave-covex with optimal capacity-costraied price p exp (K) 1. Its price-optimized reveue fuctio e 1 (1 e K ) is cocave icreasig i K. Explicit calculatios i Va Mieghem ad Dada (1998, Appedix) show that the uiform ad left-trucated ormal distributios also have a uimodal expected reveue fuctio E( p, K) with uique capacity-costraied price p(k). The fact that E( p, K) is ot joitly cocave for may distributios prevets simple coditios that guaratee the uiqueess of the optimal ( p, K) solutio. A traditioal approach (Mills 1959, Petruzzi ad Dada 1999, Ha 1997) is to simply assume that P( 0 ) 0, which esures the uiqueess of the costraiedmoopoly price p(k). I geeral, however, P( 0 ) eed ot be zero ad there appears to be o simple, geeral characterizatio of the class of distributios for which p(k), let aloe the solutio ( p, K), is uique. 3 We kow that a maximizig oegative solutio ( p, K) exists because V 1 (K, p) is cotiuous, V 1 (0, p) 0 ad bouded by (16). Such a solutio must solve the ecessary first-order coditios ad we ca add a simple sufficiecy coditio ad some comparative statics: 3 A simple but restrictive sufficiet coditio is that f( x) is odecreasig (e.g., uiform; see Appedix). Figure 1 The Sample Path of ( p, K, ) as a Fuctio of p for Three Represetative Values of : Low ( 1 ), Medium ( 2 ), ad High ( 3 ) Note. O the right, we have E( p, K, ) whe is expoetially distributed Maagemet Sciece/Vol. 45, No. 12, December 1999

9 Propositio 3. There exists a cost threshold c() 0 for the optimal solutio K ad p uder Strategy 1: If c c(), K 0 ad p is arbitrary, otherwise K 0 ad p 0 satisfy: expoetial ucertaity has a threshold c e (1K) K0 e 1 ad if c e 1 : K exp 1 l c, p exp 1 ad V exp e 1 c l c e 1. pp 2 p, K c p K 1p,K 2p dp (20) ad the optimal firm value V 1 ad K 1 are decreasig i capacity costs (dv 1 /dc K) while sig(dp/dc) sig( ph( p K) 1). If, i additio, the hazard rate satisfies h( p) 1/p 2/p h( p K), the coditios (20) are sufficiet ad the optimal price is icreasig i c. Notice that the hazard rate coditios require that the distributio is locally IFR (icreasig failure rate) at the optimal p ad K. The propositio shows that uder the optimal Strategy 1 there will always be a positive probability of havig isufficiet capacity leadig to lost sales (0 P( 2 ) c/p 1), ad of havig excess capacity (0 P( 1 ) 1 P( 2 ) 1). While this is the familiar result of the ewsvedor model, we will show i the ext sectio that this is ot true whe the firm follows ay of the price postpoemet Strategies 3, 4, or 5. The fact that firm value ad capacity levels K are decreasig i margial ivestmet costs, while p is icreasig, is ot surprisig. Because E( p, K) is cocave odecreasig i K while costs C(K) ck are covex icreasig, the optimal ivestmet strategy follows a critical umber c policy, which ca be evaluated at the optimal capacitycostraied price p(k) if E( p(k), K) iherits the cocavity property: c d dk EpK, K K0. (21) Examples. The determiistic base-case has a threshold cost c (1 2K) K0 1 ad if c 1: K det 1 c 2, p det 1 c c V det ad More iterestig is the effect of ucertaity. As a start, With expoetial ucertaity, the moopolist charges a price p 1 idepedet of the ivestmet cost. This price icludes a mark-up of (1 c)/ 2 compared to the determiistic moopoly price of (1 c)/ 2. Iterestigly, this mark-up is decreasig i cost, perhaps because the total exposure to ucertaity has decreased (K is decreasig i cost). While the elegat explicit solutios for the expoetial distributio give us some first isights ito the effects of variability o price ad capacity, this sigle-parameter distributio has the disadvatage that we caot chage the level of variability. Aalytic comparative statics o the first order optimality equatios as a fuctio of variability yield very complex equatios that caot be siged i geeral. Therefore, we explicitly solved the capacity-pricig problem for the family of uiform distributios with mea 0 1 ad appropriately chose stadard deviatio to ivestigate the impact of various levels of variability o the pricig ad ivestmet decisios (explicit results are reported i Va Mieghem ad Dada 1998, Appedix). Because is oegative, the support iterval [ 0 3, 0 3 ] ad its relative amout of variability as measured by the coefficiet of variatio are bouded whe distributed uiformly: / 0 3 1/2. For compariso, the oe-parameter expoetial distributio has coefficiet of variatio / 0 1. Figure 2 shows the optimal moopoly capacity ivestmet, price, ad firm value uder strategy 1 assumig uiform ad expoetial ucertaity. We highlight three fidigs. First, the threshold c is decreasig i the level of variability : As ucertaity icreases, the firm requires a larger cost break before it is willig to ivest. Secod, the optimal solutio ( p, K) is ot mootoe i variability. While for moderate ad high capacity costs c the ivestmet level K decreases as variability icreases, the reverse is true for low capacity costs: Capacity is so iexpesive that Maagemet Sciece/Vol. 45, No. 12, December

10 Figure 2 The Optimal Moopoly Capacity K, Price p, ad Firm Value V as a Fuctio of the Margial Cost c c K ad Variability (As Measured by the Coefficiet of Variatio / 0 ) Uder the No Postpoemet Strategy 1 Note. The cost threshold c() is i bold. Capacity Ivestmet K: top left Price p: top right Firm Value V: bottom oe ivests i more excess capacity as variability icreases. Similarly, the optimal price is below the determiistic price for low variability levels, while it is icreasig i variability for high variability levels. From a techical perspective, our aalysis highlights the role of, ad the added complicatio due to, the zero demad outcome 0. Disregardig the possibility of 0 as i the classical aalysis of Mills (1959) leads to his well-kow result that the optimal price uder ucertaity is below the correspodig determiistic price. This, however, requires relatively modest variability ad distributios that are bouded from below. Otherwise, there is a positive probability that there is zero demad i some states (P( 0 ) 0), i which case the optimal price ca be higher tha the determiistic price. Third, ucertaity does ot ecessarily result i decreased expected capacity-costraied reveues. Ideed, icreased variability icreases reveues for high capacity levels. Effectively, the demad distributio is cesored ad the effective mea demad ad associated mea reveues are thus icreasig i variability. (Domai 0 cesors demad to zero, while i domais 12 the coditioal mea demad is icreasig i variability.) 1640 Maagemet Sciece/Vol. 45, No. 12, December 1999

11 4. Aalysis of Price Postpoemet Strategies 3, 4, ad 5 Istead of settig price ad capacity ex-ate (Strategies 1 ad 2), the firm ow sets productio quatity ad capacity (Strategies 3 ad 4) or oly capacity (Strategy 5) before ucertaity is realized. These quatity settig problems are sigificatly easier to aalyze tha price settig problems because their expected firm values are uivariate. The optimal ivestmet level K i uder strategy i solves the ecessary firstorder coditio dv i (K)/dK 0, where d dk V 3K 2K dp c K, 12K (22) d dk V 4K 2K dp c K, 122K (23) d dk V 5K 2K dp c K. (24) 12c q2k To uify the aalysis of the strategies i 3, 4, ad 5 (ad of the competitive strategies later), it is useful to first itroduce the series of fuctios k ad some of its properties. The defiitio of k 1 exactly expresses the first order coditio of V 3, while those of k will express coditios uder -firm competitio ad k k is equivalet to the first-order coditios of V 4 ad V 5. Defiitio 1. Each term i the series of fuctios k (c), where ad c, is defied as the smallest positive solutio of g ( x) c if c 1, where 1 g x x dp, (25) 12x ad k (c) 0ifc 1. Lemma 1. If the hazard rate is appropriately bouded i the sese that h( x) ( 1)/x has at most oe zero ad lim x20 xh( x) 1, the g ( x) as a uique positive solutio for c 1. I geeral, the series is icreasig: ad k det c k 1 k 2 k k k, (26) d dc k c 1 ad 1 1 c k c kc. (27) 1 The hazard rate coditio characterizes a large class of distributios, which icludes all icreasig failure rate (IFR) distributios (because the h( x) ( 1)/x is mootoe ad h(0) fiite) as well as ot-too-strogly decreasig failure rate (DFR) distributios. Clearly, k k is always uique, regardless of the distributio. Examples. For the determiistic base case, we have k det c 1 1 c m k det c 1 c. (28) If is expoetially distributed, the k is the uique positive solutio to e k (1 k /) c. If is uiformly distributed over the iterval [1 3, 1 3 ], (28) holds uder moderate ucertaity (3 (1 c)/(1 )) so that k k det, otherwise ((1 c)/(1 ) 3 1): k m k c. 43 2c This example shows that if is bouded from below with probability oe by ad if c is ot too small so that k ad thus P( 12 (k )) 1, the itegrates out ad k equals the determiistic solutio k det (c) (/( 1))(1 c). Formally: Lemma 2 (Ucertaity Isesitivity). If is bouded from below with probability oe by ad capacity is ot too iexpesive c 1 (( 1)/), the k is idepedet of variability ad equal to the determiistic k det (/( 1))(1 c). The lemmas show that k k ad k(c K ) k(c K ) so: Propositio 4. If the failure rate h satisfies the codi- Maagemet Sciece/Vol. 45, No. 12, December

12 tio i Lemma 1, the the value fuctios V i (K) of strategy i 3, 4, ad 5 are uimodal ad the associated uique optimal capacity levels K i rak as: q 3 K 3 k 1 c K q 4 K kc K K kc K, 29 where k k. The optimal ivestmet levels ad firm values are decreasig i c K : K i /c K 0, V i /c K K i 0 ad, evaluatig k 1 ad k at the costs as i (29): V 3 k 1 2 F k 1 V V 5 1 4cq k k 2 df k 2 F k 2 df k 2 F k. (30) (Notice that if h( x) ( 1)/x, the V i is strict cocave, a more striget property tha uimodality.) Thus, there exists a clear rakig amog the price postpoig Strategies 3, 4, ad 5: The moopolist fids it optimal to icrease ivestmet as it has more ex-post flexibility i price ad/or productio (ivetory) decisios. It is somewhat surprisig that this is provable for such a large class of distributios. More importatly, it leads to a crucial distictio betwee iformatio ad ucertaity if oe has ex-post price flexibility: Better iformatio (i the sese that oe observes earlier ad has ex-post flexibility) actually icreases capacity ad productio/ivetory levels, so that both are complemets. This, however, does ot go agaist covetioal wisdom i terms of safety stocks: More ucertaity (or worse ex-ate iformatio i the sese of high variability i the ex-ate forecast of ) still iduces the firm to carry capacity ad productio/ivetory levels above the determiistic level. Ideed, by Lemma 1, the optimal price postpoemet capacity levels are ever lower tha uder certaity: c K K 3 K 4 ad c K K 5. (31) The propositio yields additioal iterestig facts. The ucertaity-isesitivity lemma applies to all price postpoemet strategies so that uder moderate variability the equality sig holds i (31). Cotrary to ituitio, more postpoemet icreases the sesitivity to ucertaity: The ivestmet level uder Strategy 5 is more sesitive to variability tha uder Strategy 4, which is more sesitive tha uder Strategy 3. Ideed, Lemma 2 yields that K 3 equals the determiistic det solutio if P( k (1 c K )) 0, which is less striget to variability tha P( k det 1 c K ) 0, the coditio for K 4 to equal the determiistic solutio. Similarly, K 5 equals its determiistic solutio if P( k det 1 c K ) 0. Thus, while more postpoemet icreases value, it also icreases sesitivity to ucertaity. For example, isesitivity uder the uiform distributio with 1 3 requires 23 c K 1, 3 c K ad 3 c K, for Strategies 3, 4, ad 5, respectively. Thus, moderate levels of ucertaity ( 1/23) ever impact the capacity ivestmet uder Strategy 3. (While its optimal expected value of V 3 equals the determiistic value 1 4 (1 c) 2, the value obviously exhibits variability with stadard deviatio V3 K 3.) Higher variability levels or more postpoemet requires higher margial costs for capacity decisios to remai isesitive. Notice that this isesitivity result ever holds for the ex-ate price settig strategies 1 ad 2 aalyzed i the previous sectio. Also, if variability is moderate i the sese that P( 1 2 (1 c K )) 0 (or 3 c K uder uiform ucertaity), the the simple price postpoemet strategy 3 equals the more refied Strategy 4. Because there is zero probability of zero market-clearig price (P( 0 (K det )) 0) ad the probability of smarter pricig is zero (P( 0 (2K det )) 0), the optimal ex-post price equals the market-clearig price. If, i additio holdig costs are isigificat, price ad productio postpoemet (Strategy 5) is equivalet to Strategies 3 ad 4. This formally proves ad quatifies the ituitio that uder moderate variability there is o icremetal value to holdig back output or postpoig productio, or equivaletly, to receivig more timely iformatio. Moreover, this shows that the value of additioal productio postpoemet may be relatively low if oe ca reduce 1642 Maagemet Sciece/Vol. 45, No. 12, December 1999

13 demad ucertaity (through market demad maagemet, for example) so that V 5 V 4 V 3. Fially, price postpoemet strategies have a cost threshold c 1 that is idepedet of demad variability because k is positive if c 1 ad zero otherwise. With uiform ucertaity, the cost threshold uder o or productio postpoemet decreases as demad becomes more variable ( 2). Hece, with uiform as with expoetial ucertaity a price postpoig moopolist is willig to ivest at higher costs tha a o postpoemet Strategy 1 moopolist. I additio, the ivestmet differs: it is lower at low costs ad higher at high cost as is evidet from comparig Figures 2 ad 3. Oly i the determiistic limit ( 3 0) does the classical ecoomics result that price ad quatity settig give the same outcome hold. Figure 3 shows the optimal moopoly capacity ivestmet ad firm value for the simple price postpoemet Strategy 3 assumig uiform ad expoetial ucertaity. I the zoe c 1 23 of low variability levels or high costs, the capacity ivestmet level ad firm value are idepedet of variability ad equal to their determiistic values. For higher levels of variability, both the capacity level ad firm value icrease. (Effectively, demad distributio trucatio, aalogous to that discussed i the previous sectio, occurs ad the effective mea demad is icreasig i variability.) 5. Price Postpoemet Strategies Uder Competitio 5.1. Competitive Price Postpoemet Model Whe firms compete i a determiistic settig, Kreps ad Scheikma have idetified coditios uder which the productio postpoemet (Bertrad price competitio) ad price postpoemet (Courot quatity competitio) ivestmet decisios coicide. Ufortuately, i the presece of ucertaity the productio postpoemet model is ot well posed as Hviid (1991) showed that o pure strategy equilibria exist uder stochastic price competitio. We will show ext that uder ucertaity the simultaeous price postpoemet duopoly i the competitive versio of our Strategies 3, 4, 5, ad 6 does have a pure equilibrium strategy. The, we will geeralize to oligopoly ad perfect competitio, ad coclude with the value of differet price postpoemet strategies uder competitio. I competitive models oe must specify the timig ad ature of the observability of competitors actios, i additio to ucertaity. Cosider, for example, the -firm competitive versio of the price postpoemet Strategies 3 ( 2). First, each firm i simultaeously ivests i capacity K i. The, ivestmet levels K are observed by all players ad each firm i simultaeously Figure 3 The Optimal Moopoly Capacity K (Left) ad Firm Value V (Right) as a Fuctio of the Margial Cost c c K ad Variability (As Measured by the Coefficiet of Variatio / 0 ) Uder the Price-Postpoemet Strategy 3 Note. The boudary of the ucertaity-isesitive zoe c 1 23 is i bold. Maagemet Sciece/Vol. 45, No. 12, December

14 aouces the quatity q i K i that it will produce ad brig to the market. Fially, ucertaity is resolved ad the market mechaism determies the market clearig price p q for the supplied market quatity q q i. As i 3, this market clearig price is zero with oversupply q. (The competitive versios of Strategies 4, 5, ad 6 are defied similarly.) All firms make their decisios to maximize expected profits, takig ito accout the other firm s likely decisios. Thus, we have a two stage ocooperative game that is solved by workig backwards: First solve the capacity-costraied productio subgame for a give capacity vector K, ad the solve for the capacity decisios. Ulike uder price competitio, the reveue fuctios are cotiuous i the actios (i.e., i the quatities q) ad a pure strategy equilibrium for the full price postpoemet game exists. We will start with the duopoly case ad the exted to oligopoly The Capacity-Costraied Productio Duopoly Subgame Uder Strategy 3. Our questio here is: Give capacity vector K (K 1, K 2 ), what are the (subgame perfect) productio quatity decisios for both competitors uder price postpoemet with clearace? Let c deote the relevat margial cost i this subgame. We will show that there exists a pure strategy equilibrium by showig that the firms reactio curves itersect i a stable maer. Deote firm i s reactio fuctio by R i ( K), where q i R i (q j K) deotes firm i s optimal quatity respose whe firm j chooses quatity q j, with associated first order coditios (FOC) for a iterior ucostraied maximum: 4. I the appedix we show that R i /q j 1 if the failure rate h is appropriately bouded, so that both reactio curves itersect ad a pure strategy equilibrium exists. The resultig ucostraied duopoly equilibrium is ( 1 2 k 2, 1 2 k 2 ), the symmetric itersectio of the reactio curves if K is large. If there is sufficiet capacity, K i k 2, the q(k) k 2, which is idepedet of the capacity vector K. Otherwise the equilibrium is o the itersectio of a reactio curve with a costrait of the form q i K i. Propositio 5. If the hazard rate h satisfies the coditio of Lemma 1 ad h(k 2 ) 2/k 2, the q(k) ( 1 2 k 2, 1 2 k 2 ) is a pure strategy subgame equilibrium for all K q(k) uder price postpoemet. If, i additio, h( x y) x 1 for all 0 x, y k, there exist a uique pure strategy equilibrium q(k) for ay capacity vector K, which is idepedet of K i if firm i has excess capacity: K i q i K: K i qk 0. (34) Compared to the moopoly, existece of a duopoly equilibrium requires oly oe additioal coditio o the hazard rate at the poit k 2. Uiqueess of a competitive equilibrium is typically hard to prove; Figure 4 Reactio Curves for Productio Decisios whe the Duopolists are Costraied by Capacity Vector K R i q j K arg max q i q j q i f d cq i 0q ik i q iq j (32) f 2q i q j f d c. (33) FOCqiqj The strategy space of iterest is the rectagle [0, K 1 ] [0, K 2 ] ad the axis-crossigs of R i ( K) ca be specified i terms of the k series evaluated at c, as show for a represetative situatio i Figure Note. All k i are evaluated at c q for Strategy Maagemet Sciece/Vol. 45, No. 12, December 1999

15 surprisigly, it oly ivolves a rather loose additioal boud of h( x) byx 1 for small x. Examples. For the determiistic example, the reactio curves are R i (q j K) mi( 1 2 (1 q j ), K i ), with a uique solutio (either iterior at q ( 1 3, 1 3 ) if K 1 3, or at the boudary q i K i or q i K j ). With expoetial ucertaity, the reactio curves are trivial ad have a uique itersectio: R i (q j K) mi(1, K i ) The Price Postpoemet Full Game Uder Strategy 3. From Propositio 5, it follows that, similar to the moopoly case, ay excess capacity level K i q i (K) is a suboptimal ivestmet (V i /K i c K 0) provided both firms ivest, i which case each will produce up to its capacity: q K ad the relevat margial cost becomes c c K. The capacity reactio curves become: max 0K i V i K K K K i f d ck i f 2K i K j f d c, FOCK 35 where K i K i deotes the total idustry ivestmet level. Clearly if V i /K i Ki 0 Kj ( K j ) f() d c 0, firm i will ot ivest. Thus, as before, there is a maximal cost-threshold c E 1, which is idepedet of ucertaity ad above which o firm will ivest. If capacity is ot too expesive (c c), both firms ivest (K 0) ad a similar argumet as i the capacity-costraied productio subgame shows that K 1 2 (k 2, k 2 ) is a symmetric duopoly equilibrium ivestmet (which is uique uder the additioal coditio of Propositio 5). The symmetric duopoly result directly geeralizes to a oligopoly with firms (which may have additioal equilibria): Propositio 6. If the failure rate h satisfies the coditio of Lemma 1 ad h(k ) /k, the q K 0 if c c K 1 ad, if c 1, q K (1/)(k,..., k ) is a pure strategy -oligopoly equilibrium uder price postpoemet Strategy 3 with idustry value V ( firms) : V firms, Strategy 3 1 k 2 F k ck 1 1 ck. (36) 1 Examples. The duopoly capacity ivestmet reactio curves for the determiistic referece case are K i (K j ) 1 2 (1 K i c), with a uique iterior equilibrium (for all c 1)q i K i 1 3 (1 c) ad V i 1 9 (1 c) 2. With expoetial ucertaity, the idustry ivestmet K solves (1 1 2 K ) exp(k ) c for c c 1. Three iterestig isights follow from the oligopoly extesio of the price postpoig Strategy 3. First, the qualitative results of a price postpoig moopoly exted to oligopolistic ad perfect competitio uder ucertaity: The fuctioal depedece o cost ad ucertaity is similar to Figure 3. Secod, Lemma 1 ( firms) shows that the idustry ivestmet K k is icreasig i the idustry size while idustry firm values (or profits) are decreasig: K firms K 1 firms K perfect competitio k, (37) V firms V 1 firms V perfect competitio 0. (38) Not oly is this i lie with ecoomic ituitio, it also provides a ice iterpretatio of the referece ivestmet k that defied optimal moopoly capacity levels uder the postpoemet strategies of 4: k is the idustry ivestmet that would obtai uder perfect competitio. Thus, i the cotext of our model, a moopolist adoptig the refied price postpoemet Strategy 4 ivests i exactly half the perfect competitio idustry capacity. Third, the isesitivity result of Lemma 2 that price postpoig firms uder moderate levels of ucertaity ivest exactly like determiistic firms remais valid, but is more subdued, i a oligopoly with firms. Ideed, the optimal idustry ( firms) ivestmet K equals the determiistic ivestmet (/( 1))(1 c) if(/( 1))(1 c) (for the uiform distributio: (c 1)/( 1)3). Oligopoly firms facig moderate levels of ucertaity ( 1/( 1)3) ivest exactly like determiistic firms regardless of the capacity cost c. As competitio itesity rises, however, ucertaity Maagemet Sciece/Vol. 45, No. 12, December

16 becomes more importat because the isesitivity zoe shriks. Yet it ever disappears: Isesitivity to ucertaity remais at higher levels of ucertaity ad uder perfect competitio, provided capacity costs are high (1 c 3 ) The Sales Subgame i Competitive Postpoemet Strategies 4, 5, ad 6 The competitive aalysis of Strategies 4, 5, ad 6 ivolves a additioal ex-post subgame where firms simultaeously brig a sales quatity s to the market after observig. This is a determiistic subgame with reactio curves of the form R i : arg max s iq i s i s ji j s i cs i FOC f 2si s j c if s i q i, (39) ji where c 0 uder Strategy 4 (productio is suk), c uder Strategy 5 (ex-post productio) ad c c K uder full postpoemet Strategy 6. These reactio curves have a uique itersectio (the determiat of the liear system equals 1 0) so that: Lemma 3. The sales subgame has a uique (ucostraied) equilibrium: c s i 1 ad p c 1, (40) which, together with q i K i s i, is the uique equilibrium uder total postpoemet Strategy 6 with correspodig value 2 c V i 1 ad V firms, Strategy 6 E c 2 V det Similar reasoig as before shows that uder Strategy 4 each firm will set q K ad agai there exists a symmetric pure equilibrium that satisfies: 1K i 1K i f d c K, so that, usig k evaluated at c c K : K firms, Strategy 4 k, (41) 1 k firms, Strategy V 4 2 df k 2 F 1 k. 2 0 (42) Similarly, there exist a symmetric pure equilibrium for Strategy 5: 1K ic q 1K i f d c K, so that, agai usig k but ow evaluated at c c K : K firms, Strategy 5 1 k, (43) k firms, Strategy V 5 c 1 2 q 2 df c q k 2 F k. (44) 5.3. The Value of Postpoemet Uder Competitio Postpoemet ad competitio yield two additioal iterestig isights. First, postpoemet is clearly more profitable to the firm ad justifies a higher ivestmet (Corollary 1). I additio, the impact of ucertaity is higher uder postpoemet ad idepedet of competitio itesity. Ideed, the isesitivity zoe shriks from (/( 1))(1 c) to 1 c. Secod, the competitive model allows us to ivestigate the relative value to the firms of the postpoemet strategies as a fuctio of competitio itesity. Figure 5 shows the value of price postpoemet V 3, the value of additioal hold-back strategy (V 4 V 3 ), ad the value of total postpoemet (V 6 V 4 ) relative to the value of total postpoemet (V 6 100%) as a fuctio of. (While the results are show for the uiform distributio with 0.5 with c K 0.4 ad c h 0 thus, V 5 V 4, similar treds were observed for other parameter values.) The 1646 Maagemet Sciece/Vol. 45, No. 12, December 1999