A Theory of Intermeiation in Supply Chains Base on Inventory Control Zhan QU Department of Economics, Technical University Dresen Horst RAFF Department of Economics, Kiel University, Kiel Institute for the Worl Economy, an CESifo Nicolas SCHMITT Department of Economics, Simon Fraser University, an CESifo August 06 We woul like to thank the Social Sciences an Humanities Research Council of Canaa for nancial support. Corresponing author: Horst Ra, Department of Economics, Kiel University, 48 Kiel, Germany, Email: ra @econ-theory.unikiel.e
Abstract The paper shows that taking inventory control out of the hans of retailers an assigning it to an intermeiary increases the value of a supply chain when eman volatility is high. This is because an intermeiary can help solve two incentive problems associate with retailers inventory control an thereby improve the intertemporal allocation of inventory. Aing an intermeiary as a new link in a supply chain is also shown to reuce total inventory, to make shipments from the manufacturer less frequent an more variable in size, as well as to reuce social welfare. JEL classi cation: L, L, L, L8 Keywors: intermeiation, inventory, eman volatility, supply chain
Introuction The optimal control of inventory is one of the greatest challenges face by rms in a supply chain. Consier, for instance, a supply chain, in which a manufacturer istributes goos through retailers who hol inventory because they have to orer an take elivery of goos before observing the state of eman an selling to consumers. A key problem in this setting, as explaine by Krishnan an Winter (007), is that the retailers incentives to hol inventory are generally not aligne with the manufacturer s interests. Hence the challenge is how to solve such incentive problems so that the supply chain s value can be maximize. In the current paper, we o er two main insights into this issue. First, we ientify two speci c incentive problems associate with retailers inventory control that arise from the intertemporal allocation of inventory. Secon, we show how these incentive problems may be mitigate by taking inventory control out of the hans of retailers an assigning it to an intermeiary. In particular, we erive conitions uner which aing an intermeiary in the supply chain to control inventory raises the chain s value. Our moel of intermeiation base on inventory control allows us then to erive preictions about the use of intermeiaries in supply chains an about the associate pattern of shipments an inventories, as well as to examine the normative implications of using intermeiaries to control inventory. Stuying the role of intermeiaries in controlling inventory is relevant an interesting, not least because we observe a general tren towar shifting inventory control from retailers to intermeiaries in many retail inustries. Three evelopments are particularly noteworthy in this respect, namely rop-shipping, inventory consignment an venor-manage inventory. Drop-shipping is an arrangement where (mostly internet) retailers forwar buyers orers to a wholesaler who then ships the prouct from its own inventory; this arrangement allows retailers to avoi holing any inventory while proviing a large choice of varieties to consumers. Inventory consignment allows an upstream rm to own inventories hel by ownstream rms while venor-manage inventory (VMI), an increasingly popular arrangement thanks in part to electronic sales an inventory tracking technologies, allows an upstream rm to control these inventories. See Ranall, Netessine an Rui (006) for a recent comparison of rop shipping with the classic case of retailer-controlle inventory. Companies engage in this practice inclue Alliance (www.aent.com), the largest istributor of home entertainment auio an vieo in the US; Baker an Taylor (www.baker-taylor.com), a leaing istributor of books, vieos an music proucts; ChemPoint (chempoint.com) in the chemical inustry, an Garen.com in the garen supply retail inustry; see Netessine an Rui, 004. Retailers such as Staples are both traitional retailers holing inventories an engage in rop-shipping for out-of-stock items; see Ranall et al. 00. See Govinam (0) for a survey, an Mateen an Chatterjee (05). Wal-Mart pioneere VMI with Procter an Gamble in the early 980s; early aopters inclue KMart
These practices are part of a more general tren to take inventory control out of the hans of ownstream rms an to eliver goos to them just in time. This tren is re ecte in the growing use of thir party logistics (PL) proviers an their increasingly important role in hanling inventory. Warehousing an istribution activities represente.7% of the US PL market of $58 billion (PL gross revenue) in 04, an the importance of these activities is likely to increase in the future. Not only is the PL market expecte to grow at an annual rate of 4.4% between 05 an 0, 4 but the number of supply chains wanting PL proviers to eal with their warehousing an istribution nees has been growing as well year after year. 5 In e ect PL proviers have mae themselves inispensable to supply chains, transforming themselves from being use on eman to establishing stable an multi-year contractual arrangements with them. As a result the PL inustry has experience signi cant consoliations, especially concerning the most successful value-ae warehouse istribution proviers (Excel, UPS SCS, Kenco, Genco, Jacobson an DSC) which were growing especially fast in the early 000s (Foster an Armstrong, 004) an which have now been largely absorbe by large PL proviers. 6 To construct a theory of intermeiation focusing on the incentive issues associate with inventory control one nees to put asie a variety of other aspects that have unoubtely also contribute to the success of ropshipping, inventory consignment, VMI an, more generally, the inventory management activities of intermeiaries like the PL proviers. In particular, we ignore economies of scale in inventory management that might give intermeiaries an avantage over retailers because they pool proucts from i erent rms. We also ignore avantages coming from complementarities between inventory management an other specialize services such as transportation, as well as technological avantages that intermeiaries may have, for instance, in inventory tracking technology that may lower their variable an JCPenney. Procter an Gamble now uses this practice worlwie; see atalliance.com. Some of the intermeiaries mentione in the previous footnote also use VMI. Of course, omestic an international transportation management still represents the main activity of logistics proviers with 66% of US PL s gross revenue; see http://www.plogistics.com/pl-market-info-resources/pl-market-information/us-plmarket-size-estimates/. On the size of PL s market, see also Worl Bank (04). 4 See https://www.gminsights.com/inustry-analysis/thir-party-logistics-pl-marketsize. 5 For instance, 5% of users wante their PL proviers to hanle inventory management in 06 compare to % in 0, while orer management an ful llment represente respectively 9% in 06 an 4% in 0; see Thir- Party Logistics Stuy 0 an 06. For the 06 report, see https : ==www:kornferry:com=meia=siebar ownloas=06 P L stuy:pf. 6 Excel has become DHL Supply Chain in 06; Genco has been absorbe by FeEx in 05; Jacobson was absorbe by Norbert Dentressangle in 04 which in turn was absorbe by XPO Logistics in 05; Kenco, a North American provier, has partnere with Hermes, a large European logistics provier for its international business.
costs of managing inventory relative to retailers. Each of these features has the potential of reinforcing the role playe by intermeiaries. By ignoring them we choose to stack the moel against intermeiation in orer to isolate how intermeiaries may help solve the incentive issues associate with inventory control. The moel use for this purpose is a stanar moel of a supply chain, in which a manufacturer istributes goos through retailers. What is new is that the manufacturer has to ecie whether to assign inventory control to retailers or to an intermeiary. A key aspect of our theory rests on the assumption that the intermeiary possesses market power relative to retailers. This is consistent with Spulber (999) who argues that any theory of intermeiation requires intermeiaries to have market power relative to other market participants so that they can set prices an balance supply an eman across time by holing inventory. Balancing eman an supply by staning reay to buy goos an sell them at i erent points in time, is exactly the role we assign to the intermeiary. We moel the asymmetry in market power between retailers an intermeiary in the simplest possible way, namely by assuming that retailers are perfectly competitive, whereas the intermeiary is a monopolist. 7 We show that market power gives the intermeiary two avantages over competitive retailers in controlling inventory. The rst avantage has to o with the intertemporal allocation of inventory. Competitive retailers allocate inventory so that toay s retail price is equalize with tomorrow s expecte retail price. We refer to this as intertemporal market integration. An intermeiary with market power, by contrast, allocates inventory to equalize toay s marginal revenue with tomorrow s expecte marginal revenue. This intertemporal market segmentation implies that retail prices can ajust more reaily when eman conitions i er across perios; that is, it improves intertemporal price iscrimination. The secon avantage of the intermeiary arises when goos can be reorere. Competitive retailers are reay to sell inventory carrie over from the past at any positive retail price, simply because the cost of these units has alreay been sunk. This means that the resiual eman face by the manufacturer toay is reuce an so is the manufacturer s price. In e ect, the manufacturer competes with the inventory carrie over by retailers from the past. To limit this inventory competition the manufacturer woul have to keep shipments to retailers small in each perio, but by oing so he runs the risk of losing sales ue to stockouts. We show that by assigning inventory control to an intermeiary the manufacturer may limit inventory competition, while at the same time avoiing stockouts. Simply put then, the intermeiary s incentives to control inventory are 7 See Deneckere et al. (996) for an in uential paper that aopts a similar vertical structure without intermeiation to stuy incentive issues in inventory control.
better aligne on two counts with the interests of the manufacturer. Our moel preicts that the intertemporal misallocation associate with intertemporal market integration an with inventory competition becomes worse the higher is the variance of eman. Thus, from the manufacturer s point of view an intermeiary is especially useful in markets where nal eman is very volatile. We also show that, consistent with the intermeiary s role in reucing inventory competition, a shift in inventory control from retailers to the intermeiary reuces total inventory holings an ecreases social welfare. By constructing a theory of intermeiation base on inventory control our paper irectly contributes to the market microstructure literature that seeks to unerstan the role of intermeiaries in market clearing. While this literature has recognize the role that inventory plays in helping intermeiaries match supply an eman intertemporally, it oes not examine the incentive problems that inventory control entails. 8 Stuying these incentive problems is the omain of the literature on vertical control in inustrial organization an the management literature on supply chain coorination. 9 In particular, Krishnan an Winter (007) an Deneckere et al. (996) explain that the price system generically fails to align retailers incentives with those of the manufacturer as soon as inventory control is involve. Using a vertically integrate supply chain as benchmark, an thus one in which all incentive problems are solve, the approach of Krishnan an Winter (007, 00) is to ientify contract forms, such as vertical price controls an buyback policies, that woul permit a ecentralize supply chain to achieve the vertically integrate solution. Importantly, the structure of the supply chain is hel xe by assuming that inventory can only be hel by retailers. Thus this literature oes not examine how, within a supply chain, an intermeiary may help solve incentive problems associate with inventory. Our paper i ers from the literature on at least two counts: rst, the intertemporal incentive problems we stuy are i erent from those examine in the papers above. In Krishnan an Winter (007) an Deneckere et al. (996) inventory perishes after one perio, so intertemporal problems o not even arise. In Krishnan an Winter (00) inventory oes not perish so quickly, which allows them to explicitly stuy intertemporal incentives associate with retailer controlle inventory. But the incentive problems in their paper arise from the assumption that the level of inventory hel by a retailer irectly raises consumer eman, because a big inventory signals 8 See Spulber (999) for a survey. The optimal control of inventory in a supply chain is a classic problem of both economics an management science. The analysis of optimal orer policies an inventory levels goes back to the seminal contribution of Arrow, Harris an Marshak (95). Clark an Scarf (960) were the rst to establish an optimal inventory policy in a multi-echelon supply chain. 9 See Krishnan an Winter (0) for a synthesis of the theory of contracts in supply chains. 4
to consumers that the retailer is more likely to have the prouct in stock; hence retailers may hol too little inventory from the manufacturer s point of view. In our moel, by contrast, the manufacturer is mostly concerne about retailers holing, if anything, too much inventory, both because intertemporal market integration implies they may carry more inventory forwar than woul be require for optimal price iscrimination, an because unsol goos create inventory competition. Secon, rather than holing the structure of the supply chain xe an looking for contracts that achieve the vertically integrate solution, we ask what happens if inventory control is passe from retailers to an intermeiary. By showing that assigning inventory control to an intermeiary may raise the aggregate pro t of the supply chain, our moel may explain why, increasingly, intermeiaries act as an important link in supply chains, especially if, in aition to inventory control, they o er complementary services such as transportation an logistics coorination. One of the intertemporal incentive problems ienti e by our paper, namely the inventory competition problem, is closer in spirit to that face by a storable goos monopolist (Dunine et al., 006): if consumers anticipate an increase in future eman an thus higher future prices, they stockpile goos toay, which reuces future resiual eman an forces the monopolist to cut future prices. 0 The consequences of strategic consumer behavior like this for supply chains have been investigate by a number of papers in the management science literature (see Krishnan an Winter, 0, for a iscussion). But, unlike in our paper, their focus is neither on the intertemporal incentive issues involve in inventory control, nor on the potential bene ts of shifting inventory control from retailers to an intermeiary. The rest of the paper is organize as follows. In Section we present our moel an the benchmark equilibrium for the case in which inventory is controlle by retailers. We go through two versions of the benchmark case. In the rst, restricte version retailers orer an take possession of goos only once an then allocate goos across time. This restricte version allows us to highlight the problem of intertemporal market integration. In the secon version, we allow retailers to orer goos in each perio which leas to the problem of inventory competition. We mainly use the restricte version as a evice to istinguish the two problems highlighte by the analysis: intertemporal market integration an inventory competition. In Section inventory control is passe to an intermeiary. In Section 4 we show how the intermeiary helps solve these problems an what this implies for the use of intermeiaries, orer an inventory patterns, as well as social welfare. 0 Notice also the connection with the urable goos monopoly problem (Bulow, 98) where the monopolist has an incentive to cut price in the future, once consumers with the highest willingness to pay have been serve. 5
Section 5 contains conclusions, an the Appenix collects the proofs of our results. Moel an Benchmark Cases Consier a manufacturer proucing an selling goos before eman is known. The manufacturer receives orers from an ships proucts either irectly to competitive retailers (in the absence of intermeiation), or to a single intermeiary. Once eman has been reveale, the retailers then sell to consumers if they hol the proucts, or in the case of intermeiation, the intermeiary sells to retailers who then sell to consumers. This highlights two key i erences between the case with an without intermeiation. First, in the absence of intermeiation, the retailers hol inventories, that is to say the units receive from the manufacturer an store before they are sol; otherwise inventory is hel by the intermeiary. Secon, intermeiation involves a single inventory holer, whereas in the absence of intermeiation, inventories are hel by many retailers. There are two sales perios, t = ;, which means that the prouct uner consieration loses its value after two perios. Deman at time t = ; is given by the linear inverse eman function: p t = A s t +" t, where p t is the retail price an s t enotes nal sales. The ranom variables " t [ ; ] are intertemporally inepenent an uniformly istribute with ensity =. The orer of moves when the manufacturer sells irectly to retailers is as follows. At the beginning of perio, the manufacturer announces a proucer price P, retailers orer an take possession of q units of goos before eman in perio is known; then perio-one eman is reveale an the retailers sell s q in perio, holing unsol units in inventory for perio. In perio, the manufacturer sets proucer price P, an retailers orer quantity q, again before perio-two eman is known. Deman in perio is then reveale an retailers sell s q + (q s ). When ealing with an intermeiary the manufacturer may use two-part tari s, consisting of a proucer price, P t, an a xe payment (or transfer), T t for t = ;. The timing of moves is then as follows: at the beginning of perio the manufacturer sets a two-part tari (P ; T ), the intermeiary orers an takes possession of quantity q. Deman in perio is then reveale, the intermeiary chooses wholesale price w, retailers purchase from the intermeiary an sell to consumers a quantity s q. In perio the manufacturer chooses the two-part tari (P ; T ), an the intermeiary orers a quantity q. Then eman in perio is reveale, the intermeiary We also consier the restricte case with a single orer in perio covering both sales perios; see below. In principle the manufacturer coul also use two-part tari s when it sells irectly to retailers, as coul the intermeiary in its ealings with retailers. But perfect competition in retailing implies that the transfer in each case woul be equal to zero in equilibrium. 6
sets wholesale price w, retailers orer an sell s q + (q s ). Notice our implicit assumption that the intermeiary only ships to retailers in each perio exactly what they sell, i.e., s t in perio t = ;. Any inventory of unsol goos in perio, q s, is thus hel by the intermeiary. This feature of the moel mirrors the arrangements uner rop-shipping an venor-manage inventory, where retailers o not exercise any control over inventory. Our assumptions about pricing strategies (two-part tari s when the manufacturer eals with the intermeiary, simple unit prices otherwise) imply that there is no ouble marginalization an that the entire expecte pro t generate by the supply chain goes to the manufacturer. They also imply that the manufacturer has nothing to gain from having more than one intermeiary. 4 Our assumptions about the prouction an istribution technologies are as simple as possible. The manufacturer incurs a constant unit cost of prouction c. We let c w enote the per-unit cost of intermeiation. The marginal cost of retailers is normalize to zero, as is the cost of holing inventory. There is also no iscounting between perios. All market participants are risk neutral. Hence importantly, intermeiation is not associate with economies of scale an even involves a cost isavantage relative to irect sales to retailers. We also assume A > c + c w, an that the eman shock is not too big,, so that equilibrium prices an quantities in each perio are always non-negative in all the environments consiere in the analysis. 5 Importantly, the latter assumption implies that in equilibrium all inventory remaining in perio is being sol at a positive price an thus that estructive competition (Deneckere et al., 996, 997), another source of (static) incentive problems arising from retailer controlle inventory, is assume away. Allowing for these aitional problems woul obscure the intertermporal incentive issues that we focus on. We now consier the equilibrium in which inventory is controlle by retailers. We n it useful to start the analysis with the restricte case in One way to think further about intermeiation which is implicit in the timing of eliveries is that the intermeiary is physically locate closer to the retailers than it is to the manufacturer. 4 The assumption of perfectly competitive retailers is also not overly restrictive in the sense that one can view the competitive outcome as the limit of a a sequential game among oligopolistic retailers as the number of retailers gets large. In stage one retailers orer inventory, each taking the quantity of the other retailers as given (Cournot competition). In stage two, after observing the true realization of eman, retailers simultaneously announce retail prices. The outcome of the subgame perfect equilibrium of this game converges to the perfectly competitive outcome as the number of retailers goes to in nity. See Tirole (988), ch. 5 an the references cite there for the relevant convergence results. 5 Of course, epens on the other parameters of the moel, A, c an cw, in a way that is speci c to each environment; see Appenix. 7
which retailers make a single orer an take elivery at the beginning of perio only, an then analyze the case where retailers are free to orer an to receive elivery at the start of each perio. Investigating these two cases separately allows us to isolate the two incentive issues that arise when retailers manage inventory, namely intertemporal market integration an inventory competition.. Intertemporal Market Integration Consier rst the case in which retailers may orer goos only once. Thus at the beginning of perio the manufacturer announces a proucer price P. Retailers orer an take possession of q units of goos before eman in perio is known; then perio-one eman is reveale an the retailers sell s q, holing unsol units in inventory for perio. In perio, retailers sell s q s. To erive the equilibrium suppose the eman shock " has been observe by retailers in perio. Retailers then sell in perio as long as the rst perio retail price, p, excees the expecte secon-perio retail price, E (p ); otherwise, they hol on to goos for sale in perio, where all remaining inventory is sol. Hence, in equilibrium, p = E (p ), or A s + " = E (A s + " ) = A s. This is what we mean by retailers engaging in intertemporal market integration. Given that s + s = q, we have s = (q + " ) = an s = (q " ) =. The rst-perio retail price an expecte secon-perio retail price are then both equal to p(q; " ) = A q+". This has the following implications for the manufacturer s expecte equilibrium output, prices an pro ts, where enotes equilibrium values an the superscript r refers to retailer controlle inventory: Proposition Suppose inventory is controlle by retailers an there is no re-orering. Intertemporal market integration implies that the proucer price an the expecte retail prices in both perios are equal to the static monopoly price, P ~ r = E(~p r ) = (A+c). The manufacturer s equilibrium output, ~q r = A c, an expecte equilibrium pro t, E (~ r (A c) ) =, are equal to the sum over two perios of the static monopoly outputs, respectively static monopoly pro ts. Proof: see Appenix. Competition among retailers ensures that retailers equalize rst-perio retail price with secon-perio expecte retail price. Thus with a single orer to cover both sales perios an inventory controlle by retailers, the manufacturer completely gives up control of the intertemporal allocation of the units he sells an there is nothing he can o to exploit eman i erences across perios: the market in perio one an two are ex ante ientical from 8
the manufacturer s point of view. Not surprisingly then the outcome is equivalent to the static monopoly situation.. Inventory Competition Consier now the case where retailers are free to orer goos in each perio. The rst thing we show is that it makes a i erence for the proucer price an expecte retail price in perio whether retailers carry unsol goos into the perio (s < q ), or whether they have sol all their initial inventory (s = q ). In the former case the manufacturer faces inventory competition an is force to reuce proucer price P below the static monopoly price. In particular, we can show: Proposition Suppose inventory is controlle by retailers. The seconperio proucer price an expecte retail price are lower than the static monopoly price if there is inventory competition, an is equal to the static monopoly price otherwise. Speci cally ( A+c (q s ) P = E(p ) = if s < q, A+c otherwise. Proof: see Appenix. How oes the manufacturer optimally reuce inventory competition? To unerstan this notice how much retailers with a given amount of inventory in perio choose to sell in perio an how much inventory they keep for perio, after they have observe ". As we know from the previous subsection, retailers engage in intertemporal market integration by selling a quantity s so that the retail price in perio equals the expecte retail price in perio, that is, p = E(p ), or p = A s + " = A + c q + s = E (p ): () Given the level of q hel by retailers, () o ers two possible outcomes for s. Either s < q so that there is inventory competition in perio, or s = q meaning that retailers stock out an o not carry any unsol goos into perio. Which solution is relevant epens on the value of ". Speci cally, s = A c+" + q < q if " < ^", q otherwise, () where A c ^" = q : () This shows how the manufacturer may reuce the likelihoo of inventory competition, namely by reucing q, which ecreases ^". But obviously 9
there is a trae-o, since by reucing q the manufacturer not only limits inventory competition, but also raises the likelihoo that a stockout occurs an sales are lost. The following proposition shows how the manufacturer optimally eals with this trae-o, where ^ enotes equilibrium values: Proposition Suppose inventory is controlle by retailers. The manufacturer reuces inventory competition by selling in perio more than the one-perio static monopoly output, namely q r (A c) = + 5, an by incurring a 40% probability that retailers stock out. Proof: See Appenix. To unerstan the intuition behin the manufacturer s choice of q, it is useful to rewrite the manufacturer s rst-orer conition erive in Appenix (see (4)) as follows: Z (A c q ) + 7 ^" A c q + 8 7 " " = 0: (4) The rst term represents the maximization conition for expecte pro ts assuming there is no possibility of a stockout. In this situation the unsol inventory carrie by retailers into perio causes inventory competition, leaing to a lower secon-perio proucer price an to lower retail prices. This term woul be equal to zero at the static monopoly output of q = (A c)=. That is, if there were no possibility of a stockout, the manufacturer woul simply eliver the unconstraine optimal monopoly quantity for perio, because shipping so little reuces inventory competition. The secon term re ects the ajustment in the maximization conition that has to be mae to account for the possibility of a stockout. Here it is clearly seen that the static monopoly output is insu cient to maximize pro t. Instea the manufacturer woul want to choose a higher output in perio. The probability of a stockout being less than 50% is then the by-prouct of shipping more than the static monopoly quantity in perio. For perio, however, we can show that the expecte output is smaller than the static monopoly output. Overall, we obtain the following result for the manufacturer s expecte total output an total pro t: Proposition 4 Suppose inventory is controlle by retailers. The manufacturer s total expecte equilibrium output, E(q r ) = q r + E(qr ) = (A c) + (A c) an expecte equilibrium pro t, E ( r ) = + 5, excee the sum over two perios of the static monopoly output, respectively static monopoly pro t. Proof: See Appenix. Re-orering allows the manufacturer to set P after observing the eman in perio. Since P = E(p ) = p ue to retailers competition, the 5, 0
manufacturer s price in perio is i erent than his perio price which satis es P = E(p ). This means that the two perios are not the same for the manufacturer, leaing to intertemporal price iscrimination an higher expecte pro ts than the sum over two perios of the static monopoly profits. This section has shown that, when retailers control inventory, intertemporal market integration an inventory competition place speci c constraints on a manufacturer in a two-perio environment. In the next Section, we consier the case where the manufacturer uses an intermeiary with the manate to control inventory an to respon to orers from the retailers. Inventory Control by an Intermeiary Like in our previous analysis, it is useful to start with the restricte case where the intermeiary may orer only once at the beginning of perio (no-re-orering case) before turning to the unrestricte case where the intermeiary is free to orer in both perios (re-orering case).. The No-Re-orering Case In this restricte case the manufacturer sets a two-part tari (P; T ) at the beginning of perio, an the intermeiary orers an takes possession of quantity q. Deman in perio is then reveale, the intermeiary chooses wholesale price w, retailers purchase from the intermeiary an sell to consumers a quantity s q. In perio the intermeiary sets wholesale price w, retailers orer from the intermeiary an sell s q s. The key i erence compare to the case of retailer controlle inventory lies in the way the intermeiary allocates q across the two perios. To see this consier perio. After the realization of " has been reveale an given a wholesale price w, retailers orer an then sell an amount s so that the secon-perio retail price equals the marginal cost face by retailers; that is (A s + " ) = w. The intermeiary s expecte perio-two revenue is thus equal to E [(A s + " ) s ] = (A s ) s, an the expecte marginal revenue is E (MR ) = A s. Similarly, in perio, after " has been reveale, the intermeiary sets a wholesale price w an retailers purchase an sell quantity s, such that the rst-perio retail price equals w, (A s + " ) = w. The intermeiary s revenue hence is equal to (A s + " ) s, an the corresponing marginal revenue is MR = A s + ". In perio, given the reveale eman shock ", the intermeiary allocates output across perios until the marginal revenue in perio is equal to expecte marginal revenue in perio, that is until A s + " = A (q s ) : (5)
This is what we mean by intertemporal market segmentation: by equalizing marginal revenues, the intermeiary generally causes retail prices to i er across perios. In other wors, the intermeiary engages in intertemporal price iscrimination inepenently of re-orering. This has the following consequences for the manufacturer s equilibrium output an expecte pro t: Proposition 5 Suppose inventory is controlle by an intermeiary an there is no re-orering. Intertemporal market segmentation implies that the manufacturer s equilibrium output, ~q i = A c c w, is equal to the sum over two perios of the static monopoly output, an expecte equilibrium pro t, E ~ i = (A c cw) + 4 Proof: see Appenix., excees the sum of static monopoly pro ts. Several comments are in orer. First, the fact that the manufacturer s equilibrium output is the same (at least net of c w ) with an without intermeiation when a single orer is involve is not surprising. The intermeiary is exactly in the same ex ante position as the manufacturer, an the two-part tari ensures there is no ine ciency associate with the vertical structure. Secon, the manufacturer s expecte pro t is greater with intermeiation (again at least if c w is low enough) because the intermeiary is able to price iscriminate across perios. 6 Thir, the bene t from price iscrimination increases with an thus with the variance of eman. To see why this is so recall how retailers woul allocate goos intertemporally. After observing eman in perio, they woul sell the quantity that equalizes the rst-perio consumer price with the expecte secon-perio consumer price. When eman turns out to be low in perio, this intertemporal market integration implies that retailers sell very little an thus hol a large inventory of goos for perio ; but when rst-perio eman is high, retailers sell a lot an hol very little inventory for perio. In fact, when retailers control the inventory, rst-perio sales i er from expecte secon-perio sales by the amount of the eman shock: s s = ". By contrast, when an intermeiary controls the inventory, sales in perio i er from expecte sales in perio by only half the amount of the shock: s s = " (see (5)). Hence the intertemporal misallocation of inventory when it is controlle by retailers increases with the size of the eman shock; a bigger implies that big eman shocks are more likely to occur.. The Re-orering Case In this subsection we want to show that by using an intermeiary to control inventory the manufacturer is better able to suppress inventory competition. 6 With linear eman the comparison is also similar to the one between uniform pricing an price iscrimination when all markets (here both perios) are serve: total expecte output is the same with an without price iscrimination while expecte pro ts are not. See Tirole (988).
To see why inventory competition is a potential concern, notice that the intermeiary in perio can sell any inventory left over from perio, q s. If it oes not orer any aitional goos, it can sell these units at an expecte pro t margin of [A (q s ) c w ] an thus guarantee itself a pro t in perio of at least out [A (q s ) c w ] (q s ): (6) As a result the manufacturer has to either reuce the transfer it charges the intermeiary in perio by this amount, or else ecrease its secon-perio proucer price. The manufacturer s optimal solution for this problem is to ship exactly the same quantity in perio as in the case of no re-orering. That is, it ships in perio the static monopoly output for two perios. The reason for this result, as we show in the proof of the following proposition, lies in the fact that the intertemporal arbitrage conition that governs the intermeiary s allocation of rst-perio orers between perios an is the same as (5), so that the manufacturer can trust the intermeiary to correctly allocate goos across perios: Proposition 6 Suppose inventory is controlle by an intermeiary. The manufacturer optimally sells in perio the same quantity as without reorering, namely q i = ~qi = A c c w. There is hence no possibility of a stockout. Proof: see Appenix. Now that we have etermine how much the manufacturer will ship in perio, we can etermine how much will be shippe in equilibrium in perio. Because expecte sales in perio are equal to A c cw an the optimal allocation between perios an ictates that s = q + " 4 (see (8) in Appenix ), then q = s (q s ) = A c c w (q s ) = " 4 : (7) It follows that q > 0, precisely if " > 0. That is, goos are orere in perio only if eman in the rst perio turns out to be greater than expecte. When " < 0, the intermeiary oes not orer any goos in perio an is content with the initial orer. Accoringly, the expecte secon-perio shipment by the manufacturer is only Z E q i = 0 h " i 4 " = 6 : (8)
The manufacturer s total expecte output an total expecte pro t can now be compute. They are as follows: Proposition 7 Suppose inventory is controlle by an intermeiary. The manufacturer s equilibrium output, E q i q i + E(qi ) = A c c w + 6, is greater than the sum over two perios of the static monopoly output, an the expecte equilibrium pro t, E i = static monopoly pro ts. Proof: see Appenix. (A c cw) + 5 96, excees the sum of Clearly, an intermeiary has the potential to a more value to the supply chain when he engages in intertemporal market segmentation an when he controls inventory competition. 4 Implications We are now in a position to raw several implications from the analysis by comparing the equilibrium where retailers control inventories with the equilibrium where an intermeiary oes so. We focus most of our attention on the cases where agents have the ability to place orers in every perio. The rst implication concerns the circumstances uner which the manufacturer woul use an intermeiary. Recall that in the restricte case where re-orering is not allowe the pro t associate with intermeiation is increasing in the variance of eman an thus, simply re ecting the fact that intertemporal market segmentation an therefore price iscrimination become more important the greater is the potential for eman i erences across perios. A comparison of Propositions an 5 reveals that the manufacturer woul prefer inventory to be controlle by an intermeiary if is su ciently big to compensate for the resource cost of intermeiation. The following proposition shows that this result also hols in the more general case where we allow intermeiaries an retailers to orer goos in both perios (Propositions 4 an 7). We hence obtain the following implication: Proposition 8 The manufacturer uses an intermeiary to control inventory if the variance of eman (an thus ) is su ciently big. Thus whether through the ability to price iscriminate or to control inventory competition, the opportunity cost of not using an intermeiary is higher for the manufacturer the bigger is the variance of eman. To see why, consier the impact of a big negative eman shock on inventory competition. In this case retailers are likely to carry a lot of unsol inventory into the secon perio, even if the manufacturer only shippe a small quantity in perio, thus exposing the manufacturer to inventory competition 4
an forcing him to cut the proucer price. In the case of a big positive eman shock the manufacturer faces another problem, namely that retailers are likely to stock out an sales are lost. Another implication concerns the total expecte output of the manufacturer an hence total inventory: Proposition 9 Total expecte output an hence total inventory is smaller when inventory is controlle by an intermeiary rather than retailers. Notice that this result hols, even if c w = 0. The reason has to o with how the manufacturer eals with the problem of intertemporal competition. In the case of retailer-controlle inventory the manufacturer wants to keep rst-perio output own so that retailers o not carry too much inventory into perio. But the less it ships, the bigger is the risk of a stockout. In the case of intermeiary-controlle inventory the manufacturer is able to keep overall output small without running the risk of a stockout simply by shipping a bigger quantity in perio, namely the unconstraine optimal monopoly quantity for both perios, an shipping aitional units in perio only if eman in perio is higher than expecte. Clearly the i erent strategy the manufacturer employs to limit inventory competition when inventory is controlle by an intermeiary rather than retailers also has implications for the pattern of per-perio shipments by the manufacturer. We can show: Proposition 0 If inventory is controlle by an intermeiary rather than retailers, then (i) per-perio shipments by the manufacturer occur on average less frequently, an (ii) the variation in the size of expecte shipments over the two perios is greater. As explaine above, half of the time there is no shipment between the manufacturer an the intermeiary in perio. By contrast, when retailers control inventory, secon perio shipments by the manufacturer, as is easily ascertaine, are positive for every realization of ". A simple comparison of shipments in perio an expecte shipments in perio proves (ii). Also notice that the intermeiary s expecte shipments to retailers are the same in both perios, as retailers sales satisfy E(s ) = E(s ) = (A c c w). An interesting implication of Proposition 0 is thus that the intermeiary smooths the shipments to the retailers as compare to those between the manufacturer an the retailers, or to those between the manufacturer an the intermeiary. By itself, being able to engage in intertemporal market segmentation reuces expecte consumer surplus but it oes not reuce expecte social welfare when there is no impact on expecte output (i.e. for ~q r = ~q i when c w = 0). However, since an intermeiary reuces expecte output even for 5
c w = 0 when it engages in inventory control (i.e., E(q i ) < E(q r )), it is primarily through this e ect that the use of an intermeiary has an anticompetitive e ect an negative impacts on both expecte consumer surplus an expecte social welfare. Inee, we show that, even if c w = 0, Proposition Inventory control by an intermeiary reuces expecte consumer surplus an expecte social welfare. Proof: See Appenix. 5 Conclusions This paper shows that shifting inventory control from retailers to an intermeiary, thereby aing a link in a supply chain, may be an optimal strategy to follow for manufacturers in an environment in which orers have to be place before eman is known. This is the case even if aing an intermeiary is costly an ecreases the overall volume of sales. The reason is that an intermeiary brings two avantages to inventory control, both of which stem from better incentives to allocate inventory over time. First, an intermeiary can help a manufacturer price iscriminate across perios by intertemporally segmenting markets. Secon, he can play a role in reucing inventory competition, precisely because an intermeiary is able to segment markets intertemporally an can hence be truste to allocate inventory optimally. These avantages are shown to be especially big in markets where eman is very volatile. A number of implications ow from the analysis. The rst one is that using intermeiaries to control inventory tens to be anticompetitive because it reuces total sales an thus total inventory. This is because an intermeiary takes over the manufacturer s monopoly position once orers have been shippe by the manufacturer. The secon implication of the analysis is that the shipments between the manufacturer an the intermeiary ten to be less frequent an their sizes more volatile than those between the manufacturer an the retailers. This is interesting because it says that the lumpiness an volatility of shipments may have a lot to o with who the buyer is an his role in the supply chain an not only, as it is usually assume, with prouct characteristics or the existence of xe costs per shipment (see, for instance, Hornok an Koren, 05 an Kropf an Sauré, 04). In fact the analysis reveals that the use of intermeiation an shipment lumpiness an volatility often go han in han but without clear-cut causal links. On the one han, when shipment lumpiness an volatility come from exogenous constraints (as implicitly assume in the restricte no-reorering case), our results show that an intermeiary may still increase the value of a supply chain, although less so than when no such exogenous constraint 6
exists. On the other han, when re-orering is possible an intermeiation is optimal, shipments to an intermeiary may still be as lumpy an as volatile, this time because the optimal shipments an their timing ictate it. Hence, intermeiation an shipment lumpiness/volatility ten to go together whether by choice or by constraint. 7 Showing that intermeiation is able to enhance the value of a supply chain oes not imply that intermeiation is necessarily the best tool to o so. In fact it can be shown that intermeiation oes not succee in achieving maximal vertical value. An since the literature on the theory of contracts in supply chains shows that vertical restraints an other policies can help achieving, at least in principle, maximal vertical value, one coul conclue that aing intermeiation as a new link in a supply chain may well be a secon-best tool. This suggests that the choice between vertical contracting arrangements an intermeiation very much epens on the speci c conitions uner which supply chains operate. In that regar the rapi growth of PL rms note in the introuction, especially their increase global reach as warehousing an istribution proviers, as well as other new arrangements aime at shifting inventory control away from retailers, emonstrate that intermeiation might be especially useful in international markets. 8 This shoul not come as a surprise. Global supply chains eal with long istance, multiple markets, i erent legal jurisictions an cultural environments, all of which likely increase the cost of vertical contracting as compare to the cost of elegating inventory control an associate ecisions to an intermeiary. Although it is well beyon the scope of the present article, our results are potentially testable, especially in an international trae context as shipments, prouct characteristics, an the buyer s ientity are often recore. But it is interesting to note that some of our theoretical preictions are consistent with the empirical results about rop-shipping provie by Ranall, Netessine an Rui (006). The authors compare rop-shipping to the more traitional arrangement where retailers hol their own inventories. This is a similar structure to ours in so far as the rop-shipping arrangement correspons to the case where an intermeiary takes over inventory control from retailers. The authors n empirical evience that traitional retailers who manage their own inventories face lower eman uncertainty than the retailers that rely on rop-shippers to control inventory. This is consistent with our result that using intermeiaries to control inventories is optimal 7 Choice an constraint can obviously operate both at the same time as is likely the case for Welspun, Inia s biggest manufacturer of towels, when it ships 40-50 containers of towels at once to its warehouse in the US, a voyage taking at least ays to reach its estination an when the shipment stays for another 0-5 ays as inventory before retailers orers arrive (Economist, 05). 8 Interestingly, wholesale rop-shipping an other innovative wholesaling practices often have an important international component; see PRWeb, 006, 0 for examples. 7
when there is high eman uncertainty. 9 They also n that the greater the number of retailers, the greater the use of rop-shipping. Although our retailers are perfectly competitive an thus we have no particular result on that imension, it is interesting to note that the funamental reason why intermeiaries might be neee is because retailers, as price takers, o not have the same incentives as a manufacturer or as an intermeiary. In that sense this empirical ning is also consistent with our theoretical results. 6 Appenix The upper limit is etermine as follows. First, consier the case of inventory control by retailers an no re-orering. In equilibrium, q = A c. Thus s = q+" = A c+" an s = A c ". Thus, s an s are positive when < A c: It is easy to establish that p an p are both positive if < A+c. Secon, consier the case of inventory control by retailers with re-orering. In equilibrium, q = (A c) + 5. Thus s = A c+" + q = (A c) + " + 5 an s = A c+(q s ). We know that if " < 5 (see Appenix ), then q > s ; otherwise there is stockout. It is apparent then that s is always larger than zero. To ensure s > 0; we require < 5 6 (A c). Along with < A+c, this conition guarantees that q, p an p are positive irrespective of ". Thir, consier the case of inventory control by an intermeiary an no re-orering. In equilibrium, q = A (c + c w ). Thus s = [A (c+cw)]+" 4 an s = [A (c+cw)] " 4. Thus, to make sure s ; s > 0, we nee < [A c c w ]. An intermeiary controlling inventory also controls how much to sell in the secon perio. In particular, one nees to make sure that all the inventory from perio, q s = [A (c+cw)] " 4 is sol in perio. This implies MR = A [A (c+cw)] " 4 + " > 0 an thus that < (c+cw). It can then be checke that this conition also makes sure that MR, p an p are positive as long as s ; s > 0. Finally, the last case is inventory control by intermeiary with re-orering. In equilibrium, q = A c c w, s = A c cw + " 4. If " > 0, then q = " 4 > 0; otherwise q = 0. s = q + (q s ) = q + A c cw " 4 is obviously always larger than zero. To make sure s > 0, we nee < [A c c w ], which also guarantees that there is no stockout in the rst perio (i.e., q > s ). It can also be checke that for the marginal revenue to be greater than zero in each perio, then MR = A s + " > 0 requires < (c+cw), which in turn makes sure that MR > 0, p > 0 an p > 0 as long as s,s > 0: 9 Belavina an Girotra (0) argue that intermeiaries help rms aapt to a volatile environment even if they are much larger than the intermeiaries they typically use. 8
It follows that (c + cw ) < = min ; (A c c w ) ; 5 6 for all the prices an quantities to be positive. A + c (A c) ; (9) 7 Appenix 7. Proof of Proposition Consier how much retailers orer before observing ". Given perfect competition the equilibrium orers satisfy that the expecte retail price Ep(q; " ) is equal to the marginal cost face by retailers, namely the proucer price P, Z (A q + " ) " = P: c)q]. Solving the cor- The manufacturer maximizes expecte pro t E [(P responing rst-orer conition, Z A q + " " c = 0; yiels the esire result for expecte output. The result for expecte prices an pro t follows immeiately. 7. Proof of Proposition In perio retailers sell all of the proucts on han, because they have alreay pai for these goos an, since, the retail price is positive; hence s = q + q s. Retailers orer goos in perio until the expecte consumer price in perio equals the proucer price P : E (A s + " ) = A q q + s = P : (0) The manufacturer chooses P, respectively q, that maximizes its perio- expecte pro t (P c)q. This expecte pro t is maximize for q = (A c q + s ) =. Using this output in (0) gives the result. 7. Proof of Proposition Being perfectly competitive, retailers orer goos in perio until the expecte retail price is equal to marginal cost, which in this case is the proucer price P : Z^" A + c q + s Z " + ^" (A q + " ) 9 " = P : ()