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Sharing Inventory Information to Manage Supply Chain ...

  1. 1. Sharing Inventory Information to Manage Supply Chain Frictions Anil Arya Ohio State University Brian Mittendorf Ohio State University September 2009
  2. 2. Sharing Inventory Information to Manage Supply Chain Frictions Abstract Supply chains often encounter inefficiencies stemming from self-interested vendor pricing and ensuing distortions introduced by self-interested retailer purchases. This paper adds the issues of information sharing and inventory management to the mix of decisions made by such supply chain participants. Despite each acting in self-interest (and, in fact, due to it), the supply chain parties can be in agreement over formal sharing of inventory information. In particular, by providing information on inventory and sales volume to its vendor, the retailer permits the vendor to engage in usage-contingent pricing contracts. Rather than being exploitative, such usage-contingent pricing can engender the manufacturer to offer rebates to the retailer to encourage early sales. With its attention shifted more to sales, the retailer in turn relies less on carrying inventory as a strategic weapon to pressure the vendor in subsequent periods. In effect, information sharing encourages each party to make concessions to the other, thereby benefiting all supply chain participants (including consumers). The paper also demonstrates that the information-contingent manufacturer rebate pricing scheme can also be equivalently implemented by a simple Vendor-Managed Inventory system, an arrangement commonly associated with retailers both sharing information and ceding inventory control to vendors.
  3. 3. 1. Introduction Formal information sharing mechanisms in supply chains, once a rarity, have become increasingly common in a wide spectrum of industries. The use of such information (via, say, Electronic Data Interchange) in conjunction with Vendor-Managed inventory (VMI) arrangements too has expanded substantially. These arrangements, characterized by a retailer intentionally ceding control (and information advantage) over inventory to vendors are aimed at developing an agile supply chain, one whose operations are responsive to global and local demand conditions. With formal information sharing, vendors, who are already privy to macro-level demand information, are provided the information and wherewithal to quickly adjust supply levels based on demand at specific retail locations. While vendor learning and supply chain responsiveness purportedly form the basis for the ubiquity of information sharing, this paper demonstrates another benefit, one rooted in improving strategic interactions among supply chain parties. In particular, when pricing and inventory decisions are choices made by self-interested vendors and retailers, supply chain coordination is a concern. In light of such frictions, this paper demonstrates the value of a retailer intentionally ceding power by giving the vendor access to its inventory and sales volume data. Such information sharing acts as a goodwill gesture that yields reciprocal action in that the self-interested vendor may naturally respond by using the information to provide rebates to the retailer for early sale of items. This, in turn, incentivizes the retailer to sell more of its purchases to consumers and to retain less in inventory which could otherwise have been used as a strategic weapon to pressure the vendor down-the-road. The net result is that information sharing can benefit the retailer, vendor, and consumers alike. Further, the paper demonstrates that the ideal contractual outcome under information sharing achieved via manufacturer rebates can also be equivalently implemented using a simple Vendor-Managed Inventory (VMI) arrangement wherein the vendor prescribes
  4. 4. 2 inventory levels as a percentage of the retailer's sales, i.e., the vendor sets the required inventory-to-sales ratio. To elaborate, we consider a two-period model of vendor-retailer interactions wherein the vendor sets periodic wholesale prices and the retailer responds by procuring and setting prices in the retail market. In such a setting, the usual problem of double-marginalization threatens to undermine supply chain profitability, and such concerns give rise to inventory as a strategic device. In an effort to convince the vendor to lower its wholesale price in the second period, a retailer may opt to carry additional stocks of inventory after the first period. These extra stocks lower the retailer's marginal benefit of, and thus its willingness to pay for, additional units in the second period. Aware of this retailer tendency to hold inventories for strategic gains, the vendor opts to set an excessive wholesale price in the first period to avert retailer inventory hoarding. Under information sharing, the retailer indirectly gives a measure of inventory control to the vendor at the outset. Such control translates into the vendor's ability to condition wholesale prices on the retailer's (now fully transparent) use of the products. And rather than being exploitative, the vendor's optimal use of contingent pricing can entail offering a discount for early sales which gives the retailer pause in its quest to accumulate excess inventory for strategic purposes. This eases the pressure on the vendor to use a high first-period wholesale price to rein in retailer self-interest. In short, information sharing can cultivate an environment wherein the parties commit to being less exploitative of each other.1 Importantly, information sharing governs the interactions in a specific way: it encourages a modicum of strategic inventories and thus maintains the use of inventory for strategic reasons but restrains the retailer's tendency to be excessive in this regard. This feature means that not only can information sharing be preferred to withholding information, but it 1 In a similar vein, Fellingham and Young (1990) demonstrate that keeping a history of reported costs can discipline a party’s intra-firm behavior in a multi-period context even when intertemporal independence precludes learning effects. In the current paper, the multi-period effects arise in inter-firm relationships, yielding Pareto improvements.
  5. 5. 3 can simultaneously also be preferred to the alternative of a commitment to no inventory. Further, we show that Vendor-Managed Inventory can serve as a means of implementing such outcomes even in the absence of a manufacturer rebate scheme, as VMI gives the vendor an analogous level of control over inventory. In this case, by stipulating inventory levels as a function of retail sales volume, VMI ensures that the only way for the retailer to carry greater strategic inventory stocks is by boosting first-period retail sales (i.e., cutting first-period retail price). Unwilling to cut retail price excessively, the retailer is less aggressive in inventory holdings. This eases the pressure on the vendor to use a high first- period wholesale price to discipline the retailer. To consider variants of the model and test its robustness, we examine the effects of additional tensions that often govern inventory management. In particular, we consider the consequences of inventory holding costs, cost of capital (discounting), supply-side disruptions, and demand-side uncertainty on the value of information sharing. While each added consideration points to subtle new tensions in information sharing and inventory management (detailed in the paper), there is a unifying theme: the stronger the retailer's incentives to hoard inventories for strategic reasons, the more it is willing to formally share information; further, such sharing of information is generally met with approval by the vendor and consumers. Inventory management is a routine topic in managerial accounting. The standard discussion views the issue as primarily a decision problem entailing a tradeoff between inventory carrying to shield from unforeseen supply and demand risks versus ordering costs (see, for example, Horngren et al. 2008, Ch. 20). In contrast, this paper looks at strategic implications of inventory management, and its consequences for information sharing and vendor inventory control. Starting with Spengler (1950), many studies have examined the lack of supply chain coordination due to self-interested pricing (for summaries, see Katz 1989 and Lariviere 2008). The prevalence of such double- marginalization problems has been validated both in experimental and archival settings (e.g.,
  6. 6. 4 Durham 2000; West 2000; Park and Lee 2002). Proposed means of alleviating these supply chain frictions include vertical integration, retail price maintenance, increased upstream competition, expanded contractual terms, and strategic inventory holdings. Most germane to the present study is the notion that a retailer’s inventory stocks can put downward pressure on future wholesale prices and, thus, alleviate frictions in future interactions (Anand et al. 2008). Anand et al. demonstrate that a retailer’s decision to carry inventory, while reducing future wholesale prices, can also intensify near-term double- marginalization. Building on this theme, the present paper demonstrates a role for formal information sharing (and delegated inventory control) in balancing across-period double marginalization concerns. Extant research examines other facets of information sharing in supply chains. The traditional practitioner view of information sharing is that it enables a supply chain to better coordinate inventory and sales flow to meet changing demand. In a setting consistent with this view, Lee et al. (2000) show that benefits of a retailer sharing information with its supplier are most pronounced when demand is volatile, consumer preferences are correlated across time, and/or there are substantial lead times in production. Similarly, Cachon and Fisher (2000) demonstrate efficiencies from sharing sales and inventory data arising from improved replenishment policies and better allocation of products to multiple retailers. A key related advantage is that information sharing can mute the effects of the "bullwhip" effect, wherein demand variability has increasingly large effects as one moves up the supply chain (e.g., Chen et al. 2000). The literature has also documented reasons that discourage information openness by the retailer. For one, concomitant wholesale price effects can dissuade a retailer from sharing its demand information with a supplier (e.g., Li 2002). Baiman and Rajan (2002) show that investment hold-up concerns and strategic appropriation by the supplier can also undermine information sharing. The potential for information leakage to retailer
  7. 7. 5 competitors further undercuts incentives for information sharing (Li 2002).2 Other key considerations are the extent to which the information shared can be deemed reliable by the supplier and the degree to which the shared information gives the supplier either de facto or de jure inventory control (Kulp 2002). These various competing tensions that permeate considerations of supply chain information sharing are also borne out in archival data (e.g. Kulp 2002; Kulp et al. 2004) and controlled experiments (e.g. Croson and Donohue 2005).3 In the present analysis, coordination of inventory levels to balance holding and stockout costs is initially excluded from consideration. Further, information sharing provides no avenue through which the supplier learns about demand so as to exploit the retailer, be it through variation in wholesale prices or reduction in investments. By (intentionally) excluding such extant tensions in our analysis, we are able to isolate the key upside of information sharing that is unique to our analysis – information sharing helps alleviate frictions wrought by excessive accumulation of strategic inventory. In subsequent sections, we layer in other benefits and costs of inventory information sharing to test the robustness of the results and to better identify key determinants of information sharing. The remainder of the paper proceeds as follows. Section 2 describes the basic model. Section 3 demonstrates the key results: section 3.1 derives the equilibrium under information sharing; section 3.2 derives the equilibrium under no information sharing; section 3.3 compares outcomes under each information regime; and section 3.4 examines the use of Vendor-Managed Inventory to implement the desired contractual arrangement. 2 In subsequent work Li and Zhang (2008) demonstrate that the nature and intensity of competition among retailers may actually justify information sharing due to the supplier's desire to make concessions to put its customers on level footing. Importantly, such forces are prominent only when the supplier keeps such information in confidence. 3 Interestingly, Croson and Donohue (2005) note that due to ever-present behavioral biases, the beneficial effects of information sharing on mitigating the bullwhip effect arise in more subtle ways than operational reasons alone would suggest, and that this feature means that more of the benefits of information sharing accrue to upstream parties. Subsequent work by Croson et al. (2009) has detailed the robust nature of these behavioral biases and the associated role of "coordination" by inventory stocks aimed at reducing exposure to risks due to suboptimal behavior by supply chain partners.
  8. 8. 6 Section 4 then addresses the robustness of the results by considering other facets of inventory management: section 4.1 considers the effects of inventory holding costs; section 4.2 examines the consequences of a nontrivial cost of capital; section 4.3 introduces the effects of supply-side disruptions; and section 4.4 includes the possibility of demand-side uncertainty. Section 5 concludes. 2. Model Consider the following baseline model of supply chain inventory management. Over the course of two periods, an upstream firm (vendor) produces and sells products to a downstream firm (retailer) who then markets and sells the products to consumers. The (common knowledge) demand function for the retail product in each period is q( pi ) = [a − pi ] / b , where pi, i = 1, 2, denotes the retail price in period i. The vendor's per- unit production cost is c, and the retailer's sales/marketing cost of each unit is s. At the beginning of each period, the vendor establishes the (per-unit) wholesale price, wi. Given the wholesale price, the retailer sets its retail price, pi. To isolate the forces of interest from traditional considerations in inventory management (e.g., safety stocks, holding costs, cost of capital, uncertainty), for now we presume that the vendor can supply the products as and when needed and that the retailer incurs no cost from carrying inventory. Denoting the inventory carried forward by I, period 1 purchases amount to q( p1 ) + I, and period 2 purchases amount to q( p2 ) - I. As is standard, we assume a > c + s to ensure nontrivial sales choices (we assume analogous conditions, as detailed in the appendix, for subsequent extensions to the baseline model). The focus of this study is on the role of information sharing in supply chains and how it impacts the strategic use of inventory. In particular, we examine the circumstances under which the retailer opts to establish an information system that formally reports its inventory stocks. Such an information system permits the vendor to make its wholesale prices contingent on whether units are initially sold to consumers or kept in inventory. In
  9. 9. 7 effect, while informal channels may allow a vendor to observe retailer inventory and sales levels, only a formal information sharing arrangement permits such information to be contracted upon. To reflect the above feature, we consider both Information Sharing (IS) and No Information Sharing (NI) regimes. Under the IS regime, the vendor can tailor its first- S I S period wholesale price to its use. In particular, the vendor specifies {w1 , w1 }, where w1 I ( w1 ) is the wholesale price for items initially sold to consumers (kept in inventory). In contrast, under NI, the vendor can not fine tune its pricing terms, and is restricted to charging w1 for all purchases. (Throughout, we use "^" to denote IS outcomes and "~" to denote NI outcomes.) The following timeline summarizes the sequence of events. Retailer chooses Vendor sets first- Retailer sets retail Vendor sets w2. Retailer sets p2. information regime period wholesale price, p1 , and Consumer demand S I (IS or NI). price: {w1 , w1 } chooses inventory is satisfied by under IS and w1 level, I. inventory and under NI. additional purchases. FIGURE 1. Timeline. Given this basic setting, we seek to compare the outcomes under IS and NI and identify the retailer's information sharing decision. The ensuing analysis employs backward induction to identify the unique (subgame perfect) equilibrium. 3. Information Sharing and Inventory Management Since the model entails no uncertainty or intertemporal learning, the retailer seemingly has no reason to carry extra inventory let alone share such information with its supplier. Yet, strategic supply chain tensions compel the retailer to carry forward inventory. This is because carrying forward inventory reduces the retailer's subsequent demand for inputs in the second period, which, in turn, convinces the vendor to lower its second-period wholesale price (Anand et al. 2008). In light of this strategic behavior, the question we ask
  10. 10. 8 is whether the retailer is willing to give its vendor a means through which it can regulate inventory levels (in particular, usage-based pricing). To analyze this question, we next examine the equilibrium outcomes under each information regime. 3.1. Outcome under Information Sharing (IS) Working backwards in the game, given I and w2 , the retailer's second-period price, p2 , solves: Max [ p2 − s]q( p2 ) − w2 [q( p2 ) − I]. (1) p2 In (1), the first term reflects the revenue (net of sales cost) for the units sold in period two, and the second term reflects the payment to the vendor in the second period for units purchased. Solving (1) yields the retailer's chosen retail price, p2 (w2 ) = [a + s + w2 ] / 2, and substituting this back into (1) yields Π R2 (w2 , I), the retailer's second-period profit as a function of the wholesale price in the period and the carried-forward inventory. Also, p2 (w2 ) generates the induced second-period demand for the vendor, q( p2 (w2 )) − I . As a result, the vendor sets wholesale price to solve: Max [q( p2 (w2 )) − I][w2 − c]. (2) w2 Solving (2) reveals the second-period wholesale price, w2 (I) = [a + c − s] / 2 − bI , and substituting this back into (2) yields Π V 2 (I), the vendor's second-period profit as a function of the inventory level. In period two, the carried-forward inventory provides the retailer with the benefit of reducing wholesale price. Roughly stated, the retailer can supply high-value customers in period-two using its own stock, thus, reducing its willingness to pay for new purchases; the vendor attempts to offset retailer hesitance by cutting its wholesale price. Knowing such second-period wholesale price effects are on the horizon, the retailer chooses first-period retail price and inventory level to solve:
  11. 11. 9 I S Max [ p1 − s]q( p1 ) − w1 I − w1 q( p1 ) + Π R2 (w2 (I), I). (3) p1 ,I S S The solution to (3) yields p1 (w1 ) = [a + s + w1 ] / 2 and I I I(w1 ) = [3a + c − 3s − 4w1 ] [6b]. Given the induced period-one demand, the vendor sets wholesale price to solve: S S I I I Max q( p1 (w1 ))[w1 − c] + I(w1 )[w1 − c] + ΠV 2 (I(w1 )). I S (4) w1 ,w1 Solving (4) reveals the usage-contingent first-period wholesale prices, 9[a − s] + 7c a−s+c ˆI w1 = ˆS and w1 = ˆI ˆS . Iteratively substituting w1 and w1 into the 16 2 solutions to (4), (3), (2), and (1) reveals the equilibrium outcome in the information sharing case, as presented in Lemma 1. (All proofs are presented in the Appendix.) Lemma 1. The outcome under Information Sharing entails 9[a − s] + 7c a−s+c 3[a − s] + 5c ˆI (i) wholesale prices: w1 = ˆS , w1 = ˆ , and w2 = ; 16 2 8 3a + s + c 11a + 5s + 5c ˆ (ii) retail prices: p1 = ˆ and p2 = ; 4 16 a−s−c 5[a − s − c] ˆ (iii) demand: q( p1 ) = ˆ and q( p2 ) = ; 4b 16b ˆ a−s−c; (iv) inventory: I = 8b ˆ 17[a − s − c]2 ˆ 35[a − s − c]2 (v) profits: Π V = and Π R = . 64b 256b Notice from Lemma 1 how the strategic use of inventory is manifest in wholesale prices under information sharing. Due to the retailer's decision to carry extra stocks in order to reduce its second-period willingness to pay, the wholesale price applied to second- ˆ ˆS period is below that for first period sales: w2 = w1 − [a − c − s] / 8. Further, to dissuade excess inventory and to prop up second-period wholesale price, the vendor embeds a ˆI ˆS premium for units carried forward in inventory: w1 = w1 + [a − c − s] / 16. Equivalently, ˆS ˆI one can view the fact that w1 < w1 as the vendor (manufacturer) offering a rebate to the
  12. 12. 10 retailer to encourage quick turnover of first-period purchases. Given the inherent tension between the supply chain partners, and the fact that the vendor uses the supplied information to charge an inventory-premium to reduce strategic inventories, it might seem that the retailer would be hesitant to share information. We next consider the outcome if the retailer opts not to share information and, thus, prevent usage-contingent pricing by the vendor. 3.2. Outcome under No Information Sharing (NI) When a formal information sharing arrangement is excluded, the second period interactions are again as in (1) and (2). The key difference arises in the first period, where the vendor has no ability to employ usage-contingent pricing to (indirectly) control the retailer's inventory choices. In particular, under NI the retailer chooses first-period price and inventory to solve: Max [ p1 − s]q( p1 ) − w1I − w1q( p1 ) + Π R2 (w2 (I), I). (5) p1 ,I The solution to (5) yields p1 (w1 ) = [a + s + w1 ] / 2 and I(w1 ) = [3a + c − 3s − 4w1 ] [6b]. Given the induced period-one demand, the vendor sets wholesale price to solve: Max [q( p1 (w1 )) + I(w1 )][w1 − c] + ΠV 2 (I(w1 )) . (6) w1 9[a − s] + 8c ˜ Solving (6) reveals the first-period wholesale price under NI, w1 = . 17 ˜ Iteratively substituting w1 into the solutions to (6), (5), (2), and (1) reveals the equilibrium outcome in the no information sharing case, as presented in Lemma 2. Lemma 2. The outcome under No Information Sharing entails 9[a − s] + 8c 6[a − s] + 11c ˜ (i) wholesale prices: w1 = ˜ and w2 = ; 17 17 13a + 4s + 4c 23a + 11s + 11c ˜ (ii) retail prices: p1 = ˜ and p2 = ; 17 34
  13. 13. 11 4[a − s − c] 11[a − s − c] ˜ (iii) demand: q( p1 ) = ˜ and q( p2 ) = ; 17b 34b ˜ 5[a − s − c] ; (iv) inventory: I = 34b ˜ 9[a − s − c]2 ˜ 155[a − s − c]2 (v) profits: Π V = and Π R = . 34b 1156b Note from Lemma 2 that since NI precludes the vendor from charging an inventory premium, the retailer does find it cheaper to carry forward inventory relative to under IS: ˜ ˆI w1 − w1 = −9[a − s − c] / 272 . At the same time, the vendor's desire to discourage inventory carry-forward, and its inability to decouple pricing for sales and inventory under NI, necessarily implies that the retailer's procurement costs for units sold in period one are ˜ ˆS higher relative to that under IS: w1 − w1 = [a − s − c] / 34. Finally, the vendor's weakened ability to limit inventory carry-forward under NI results both in greater inventory levels ˜ ˆ ( I − I = 3[a − s − c] / [136b]) and ensuing lower second-period wholesale prices ˜ ˆ ( w2 − w2 = −3[a − s − c] / 136 ). We next consider how these features affect retail prices, profits of the parties and, critically, the decision to share information. 3.3. Contrasting the IS and NI Regimes In comparing IS and NI, it is helpful to first contrast each with the single-period benchmark. In a single-period interaction (reflected in (1) and (2) with I = 0), the vendor's wholesale price is w = [a − s + c] / 2 and the ensuing retail price is p = [3a + s + c] / 4. Recall that under both IS and NI the retailer opts to hold inventory so as to reduce wholesale price in period two; and, knowing first-period input demand will be higher due to this effect, the vendor can increase wholesale price(s) in period one. So, the multiperiod interaction means that wholesale and retail prices are lower (higher) in period two (one) relative to a single period benchmark. These features point to two distinct supply chain consequences of information sharing for inventory management in the setting. The first effect, the "average pricing
  14. 14. 12 effect", reflects the fact that inventory may decrease the retail price in period two more than it increases it in period one (or vice-versa). That is, the way in which inventory is carried forward can alter the underlying problem of double-marginalization inherent in supply chains. The second effect, the "pricing variation effect", reflects the fact that inventory leads to across-period variability in pricing. Since retail (and supply chain) profits are concave in retail price, such variation represents a deadweight loss. The key question in comparing IS and NI is how pronounced the two (strategic) inventory effects are under each. Proposition 1 performs such a comparison. Proposition 1. (i) The average retail price is lower under IS than under NI. (ii) In period one (two), the retail price is lower (higher) under IS than under NI. As can be gleaned from Proposition 1(i), the benefit of inventories, the average pricing effect, is more pronounced under IS. In fact, comparing prices in Lemma 1(ii) to the single-period benchmark, IS allows for a lower second-period retail price (relative to the single period benchmark) without necessitating any increase in the first-period retail price. Further, from Proposition 1(ii), the cost of strategic inventories, the pricing variation effect is also less pronounced under IS. When prices are lower (in period two), they are not as low under IS; and when prices are high (in period one), they are not as high under IS. Thus, from a supply chain efficiency perspective, IS is sure to be preferred to NI in the baseline case. To elaborate, consider the consequence of providing the vendor an extra measure of S control over inventories due to usage-contingent pricing. When the vendor stipulates w1 it I sets it low enough (relative to w1 ) so as to introduce an added caveat for the retailer seeking to parlay first-period purchases into second-period concessions. This vendor rebate means that not only does the retailer rely less on inventories but also that it is encouraged to
  15. 15. 13 purchase more units for the purpose of a quick sale. In short, the added incentive of manufacturer rebates linked to retailer sales proves quite attractive to the retailer. Taken together, the above considerations point to IS preserving the key benefit of inventory (lower second-period retail price) without the pronounced downside (higher first- period retail price), as reflected in Proposition 1(i). By reducing the magnitude of such price shifts, IS also diminishes the inherent price variability due to inventory, as in Proposition 1(ii). This two-pronged advantage leads to a more efficient outcome for the supply chain. The key question is whether the supply chain gains from IS translate into gains for the retailer. After all, providing the vendor a measure of control over the its inventory choices is a double-edged sword from the retailer's perspective: on one hand, usage- contingent pricing can undercut its strategic use of inventories while, on the other hand, the means through which such usage-contingent pricing arises is in the form of vendor rebates for quick sales. A comparison of profits across the regimes reveals ˆ ˜ Π R − Π R = 195[a − c − s]2 / [73984b] > 0. In other words, the retailer is willing to give its vendor a measure of control over its inventory choices, as the vendor uses such control more as a "carrot" than as a "stick". The next proposition confirms this retailer preference, as well as presents the unanimity for which the preference is held. Proposition 2. (i) Due to strategic repercussions, the retailer opts to share inventory information. (ii) The retailer's decision to share information also benefits the vendor and consumers. From Proposition 2, not only does IS magnify the benefits and diminish the costs of strategic inventory at the supply chain level, but it also yields net benefits for each firm. This is because IS allows the parties to, in a sense, commit to helpful reciprocal actions. In particular, by giving an additional lever of control to the vendor, the retailer "precommits" not to overly exploit the vendor in period two via excessive inventory. The same
  16. 16. 14 interlinkage also enables the vendor to return the favor by being less exploitative of the retailer in period one sales. In contrast, under NI, an excessive period-one wholesale price that is applied to all period one purchases significantly lowers sales to consumers. Put a bit differently, under NI, the only means available to the vendor to curb the retailer's penchant for inventories is by setting a high wholesale price for all period one purchases. Under IS, the wholesale price hikes can be targeted to goods kept in inventory. As a result, the vendor sets a lower wholesale price for units sold by the retailer in period one under IS. Thus, though counterintuitive on its face, the retailer delegating a measure of control over inventory via information sharing helps both the vendor and the retailer. The benefit of IS to consumers follows a related line of thinking. With linear demand, consumer surplus is calculated using the familiar quadratic expression ( ) CS = (b / 2) [q( p1 )]2 + [q( p2 )]2 . Notice the convexity in the expression implies the consumers favor greater pricing variation. Thus, the intertemporal smoothing in retail prices that arises under IS serves as a boon for the firms but not for the consumers. That said, the reduction in average price accompanying IS turns out to be the more pronounced effect for the consumers. Since gains can arise from IS due to its ability to reduce the retailer's aggressiveness in holding inventory for strategic reasons, a natural follow-up question is whether eliminating inventories all together would be preferable for any of the parties. Such a precommitment to just-in-time (JIT) purchases by the retailer, simply amounts to a two-fold replication of the single-period supply chain game.4 Comparing this repeated single-period outcome to that under IS, note that IS permits an equivalent retail price in period one but a lower price in period two. Thus, as discussed above, IS engenders an efficiency-enhancing role of strategic inventories that is disabled under JIT. As it turns out, these gains are also shared by each party. 4 This is also the retailer's preferred inventory level if the retailer were to precommit to a particular inventory level.
  17. 17. 15 Corollary. All parties prefer flexible inventories with IS to a Just-In-Time inventory arrangement. As the corollary confirms, IS permits the retailer to commit to less aggressive use of inventory while maintaining the key advantage to the supply chain of retailer flexibility. It is readily confirmed that this basic notion too applies to the vendor. That is, if the vendor could precommit at the outset to wholesale prices that govern both periods (or a single wholesale price that persists both periods), the outcome is precisely the same as in the Just- In-Time case. Thus, all parties also prefer IS with dynamic pricing and inventory choices to a world in which the vendor precommits to all of its behavior. To summarize, IS harnesses the retailer's use of inventory to influence subsequent wholesale prices. At the same time, IS alters the dynamic evolution of the parties' interactions so as to prevent excessive initial wholesale prices. In effect, IS restrains the retailer while also preserving the advantages of strategic inventories. We next consider implementation of the optimal contract using a common manifestation of information sharing arrangements in supply chains. 3.4 Implementing the IS outcome using Vendor Managed Inventory The above analysis has outlined the benefits of IS presuming implementation of usage-based pricing (contingent manufacturer rebates) is simple and costless. While this presumption provides the optimum contractual outcome, it is worth noting that even if manufacturer rebates were not employed, information sharing permits alternate practical means of implementing the solution. In particular, information sharing and a retailer's associated decision to cede inventory control often takes the form of Vendor-Managed Inventory (VMI). Under VMI, the manufacturer uses the retailer's shared information on retail sales not to adjust wholesale prices but instead to regulate and manage retailer inventory levels. This often takes the form of a vendor stipulated inventory-to-sales ratio
  18. 18. 16 target, wherein the vendor establishes a target level of inventory which depends on the reported level of sales. The question we ask in this section is how such a VMI arrangement fares relative to the ideal contract that explicitly stipulates usage-based pricing. Denoting the vendor's chosen inventory-to-sales ratio by r, the equilibrium outcome under VMI is as follows. Since inventory is established at the end of period one, the parties' second period interactions are precisely as before. The distinction with VMI arises in period one. In this case, the retailer's choice entails only picking p1 , while I is replaced by rq( p1 ), as reflected in (7). Max [ p1 − s]q( p1 ) − w1rq( p1 ) − w1q( p1 ) + Π R2 (w2 (rq( p1 )),rq( p1 ))) . (7) p1 The solution to (7) yields p1 (w1 ,r) = [a + s + w1 ] / 2 − r[3(1 − r)(a − s − w1 ) − (w1 − c)] / [8 + 6r 2 ]. Given the induced period-one demand, the vendor sets wholesale price and an inventory-to-sales ratio to solve: Max [q( p1 (w1,r)) + rq( p1 (w1,r))][w1 − c] + ΠV 2 (rq( p1 (w1,r)). (8) w1 ,r Solving (8) reveals the first-period wholesale price with VMI and the preferred inventory-to-sales ratio. Iteratively substituting each into the solutions to (8), (7), (2), and (1) reveals that information sharing with VMI in lieu of usage-contingent pricing replicates the preferred contractual arrangement.5 Proposition 3. VMI with the vendor's preferred inventory-to-sales ratio replicates the optimal information sharing outcome. By ceding substantial inventory control and information access to vendors, VMI is 5 This arrangement also replicates the solution obtained if VMI is used in conjunction with usage- contingent pricing. That is, VMI and usage-contingent pricing are contractual substitutes under information sharing.
  19. 19. 17 ostensibly aimed at developing supply chain operations that are more responsive to global and local demand conditions. With VMI, vendors, who are already privy to macro-level demand information, are provided the information and wherewithal to quickly adjust inventory levels based on demand at specific retail locations. While vendor learning and supply chain responsiveness purportedly form the basis for VMI's success, the added premise identified herein is that ceding control of inventory also has strategic implications. As confirmed in the proposition, such strategic motives provide further justification for the use of information sharing and VMI arrangements. In particular, with vendor-managed inventory, the retailer willingly gives a measure of inventory control to the vendor that takes the form of an ability to stipulate inventory levels as a function of sales volume. By permitting the vendor to prescribe the sales-to- inventory tie-in, VMI ensures that the only way for the retailer to carry greater strategic inventory stocks is by boosting first-period retail sales (i.e., cutting first-period retail price). Unwilling to cut retail price excessively, the retailer is less aggressive in inventory holdings. This eases the pressure on the vendor to use high first-period wholesale price to rein in the retailer. In short, just as with usage-contingent pricing, VMI cultivates an environment wherein the parties commit to being less exploitative of each other. This less cut-throat arrangement leads to gains for the vendor, retailer, and consumers alike, even in the absence of information or learning effects. The basic analysis herein has demonstrated strategic benefits of information sharing and Vendor-Managed Inventory resulting in better supply chain coordination. Admittedly, (and intentionally) it has done so in a setting wherein other more recognized inventory management tensions are absent. In what follows, we examine some natural ways of appending the analysis to examine if and how other such considerations alter the conclusions.
  20. 20. 18 4. Robustness of Information Sharing In this section, we layer in additional considerations in inventory management both to test the robustness of the main conclusions of the analysis and to examine the key determinants of information sharing and inventory management. 4.1. Cost of Holding Inventory The strategic justification both for holding inventory and for information sharing between the vendor and retailer unfolded in the baseline analysis presuming costless inventory holding. Of course, one oft-discussed reason for reduced inventory levels is that it is costly to maintain inventory (say due to storage costs, spoilage, or opportunity costs of shelf space). To determine the effect of such holding costs on the analysis, say each unit of inventory carried forward compels the retailer to bear a cost h associated with holding the unit. In this case, the analysis follows precisely the logic as before, except the retailer profit calculations in (3) and (5) each is reduced by an extra term of -hI. Repeating the analysis using this variant reveals the following proposition. Proposition 4. (i) The retailer opts to share inventory information if and only if h < [1 / 4][a − s − c]. The retailer's decision to share information also benefits the vendor and consumers. a − s − c − 4h (ii) Under VMI, the vendor's preferred inventory-to-sales ratio is r = . 2[a − s − c] With this choice, VMI replicates the optimal information sharing outcome regardless of how holding costs are split among the parties. The central message of Proposition 4(i) is that holding costs do not derail the strategic benefits of information sharing unless they are sufficiently pronounced so as to preclude inventories in the first place. Proposition 4(ii) introduces an additional wrinkle in that a question arising naturally in VMI arrangements is who should bear the costs of
  21. 21. 19 inventory holdings. As demonstrated in the proposition and its proof, holding costs (and how they are shared), while influencing the wholesale price and preferred inventory-to-sales ratio, do not affect the efficacy of VMI in governing the strategic use of inventory. To see a visualization of the strong connection between benefits of information sharing and its role in reducing (but not eliminating) the strategic control of inventory, Figure 2 plots retailer profits and equilibrium inventory levels as h changes. 5@a-s-cDê34b è I HhL 2 @a-s-cDê8b 35@a-s-cD ê256b 155@a-s-cD2 ê1156b ` P R HhL ` I HhL è P R HhL h h 0 @a-s-cDê4 0 @a-s-cDê4 FIGURE 2. Retailer Profit and Inventories as a function of h. While the central message herein holds for the tangible cost h of holding inventory, another important cost of carrying inventory is the inherent opportunity cost associated with tied-in capital. We next consider if and how this cost alters the underlying premise of the base analysis. 4.2. Cost of Capital (Discounting) In this section, we revisit the model by modeling the opportunity cost of capital as a prominent cost of inventory holding. After all, from the retailer's standpoint even if its shelves permit costless holding of inventory, there is an added cost of having to pay for units now which will not yield cash inflows for some time to come, particularly when a firm is cash constrained. That is, presume there is no exogenous h-cost of holding inventory but
  22. 22. 20 instead that the firms discount future cash flows based on a periodic cost of capital, k, 0 ≤ k ≤ 1. In this case, there is an implicit holding cost of inventory (due to mismatched timing between the cash outflow for the goods and the cash inflow from their later sale). Further, the implicit cost is not just a dead-weight loss (as with h), but instead a transfer of wealth upstream (since early cash outlays by the retailer are a plus for the vendor). The derivation of the equilibrium outcome in this case under each regime mirrors that in sections 3.1 and 3.2. Here, the key difference is that Π R2 (w2 , I) and Π V 2 (I) in (3), (4), (5), and (6) are multiplied by the discount factor 1 [1 + k]. Following this process yields analogous (albeit substantially cumbersome) solutions to those in Lemmas 1 and 2. From this solution (detailed in the appendix), the key tension introduced by discounting is 9[a − s] + 7c + 12ck best seen in the equilibrium wholesale prices under IS: w1 = ˆI and 16 + 16k a−s+c ˆS w1 = . The wholesale price for first-period sales is as before (reflecting the fact 2 that the retailer's cash outflows and inflows associated with such sales are in the first period and thus discounting is a non-issue); the wholesale price for inventory, however, is decreasing in the discount rate, k. Roughly stated, since the retailer is paying up front for inventories it will not see cash flows from until a future period, the vendor must provide a discount in nominal terms in order to keep the wholesale price level in real terms. It is this feature that adds a new wrinkle when time value of money is a prominent consideration. To elaborate, in the absence of discounting (k = 0), the role of strategic inventories as a function of the information environment is as described in section 3.3. With discounting (k > 0), there is an additional divergence between retailer and vendor preferences. In particular, the retailer pays for inventory units upfront but does not glean benefits until down-the-road (in terms of retail sales and reduced subsequent wholesale price). This mismatch of cash flows makes the retailer hesitant to carry inventory. All else equal, the vendor of course gains from the early payment for inventory and, hence is increasingly inclined to support inventory buildup. In fact, as this consideration becomes more prominent, the vendor may actually use information sharing to give a discount not for
  23. 23. 21 quick sale of units but instead as an enticement for the retailer to carry forward additional ˆI ˆS inventory. Recall, as k increases, w1 decreases while w1 is unchanged; in fact, for k > a−s−c ˆI ˆS , w1 < w1 . Since the retailer's underlying reason to share information is to 4[2(a − s) − c] secure discounts for items purchased for quick sale, such a circumstance makes the retailer unwilling to share information. Consistent with the above intuition, the following a−s−c proposition confirms that the retailer only shares information for k < . 4[2(a − s) − c] Proposition 5. a−s−c (i) The retailer opts to share inventory information if and only if k < . The 4[2(a − s) − c] retailer's decision to share information also benefits the vendor and consumers. a − s − c[1 + 4k] (ii) Under VMI, the vendor's preferred inventory-to-sales ratio is r = . 2[a − s − c] With this choice, VMI replicates the optimal information sharing outcome for any k. The key aspect of the proposition is that strategic inventories continue to support information sharing provided the divergence of preference induced by cost of capital is not a−s−c too severe. Once k > the ramifications of a decision to share information lie 4[2(a − s) − c] not in the realm of better management of strategic inventories but instead in the realm of the vendor exploiting the retailer to accelerate payments for units not sold until future periods. As in previous sections, the advantage of information sharing to the retailer manifests in its ability to reduce (but not eliminate or magnify) the use of strategic inventories. This feature is demonstrated in Figure 3.
  24. 24. 22 5@a-s-cDê34b è I HhL 35@a-s-cD2 ê256b ` P R HkL 155@a-s-cD2 ê1156b ` I HhL è @a-s-cDê8b P R HkL a-s-c k a-s-c k 0 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅ ÅÅ ÅÅÅÅÅÅÅÅ 0 ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ÅÅÅÅÅÅ ÅÅÅÅÅÅÅÅ 4@2 Ha - sL - cD 4@2 Ha - sL - cD FIGURE 3. Retailer Profit and Inventories as a function of k. As noted in Proposition 5(ii), the underlying key feature of the baseline result remains even in the presence of discounting. That is, the ideal contractual arrangement under IS is replicated using a simple VMI implementation. In this case, discounting again does not affect the efficacy of VMI, only the ideal inventory-to-sales ratio. 4.3. Supply-side Disruptions While the focus thus far has been on testing the robustness of the main results when layering in additional tensions that represent costs of inventory, it is worth noting there are also benefits of carrying inventory beyond strategic reasons. Namely, an inventory buffer helps insure a retailer from future shocks to its supply line and/or unexpected shifts in demand, either of which can lead to potential stockout. In this subsection, we hone in on the first such benefit; the subsequent subsection will consider model the case of demand uncertainty. To reflect possible disruptions in the future supply which may necessitate (or discourage) inventory stocks, consider the following parsimonious adjustment to the model. Say at the outset of the parties' interactions, it is recognized that the current vendor may be unable to provide inputs in the second period . In particular, the first-period input supply
  25. 25. 23 source may cease with (common knowledge) probability d. This reflects the possibility that the vendor departs/exits the industry, runs into capacity constraints, or faces labor or other disputes that stall production. Each of these circumstances are often suggested as reasons for a retailer to shield itself via "safety stocks" of inventory. The flip side of the equation is that there is also an (independent) probability e that an entrant to the supply market will introduce competition for the incumbent supplier in the second period. This reflects that from the retailer's perspective, some supply disruptions are welcome. For simplicity, presume the entrant vendor produces inputs at the same cost as the incumbent. Using this succinct characterization of supply disruptions, equilibrium outcomes are derived as follows. With probability (1-d)(1-e), the initial vendor is the retailer's only supplier in the second period (i.e., d = e = 0 corresponds to the baseline analysis). With probability d(1-e), the retailer is faced with no additional supplies in period two and can only sell its inventory stocks (i.e., q( p2 ) = I , or p2 = a − bI ). With probability (1-d)e, the retailer is faced with two suppliers who, due to price competition, sell inputs at marginal cost. Finally, with probability de, the retailer faces only one supplier in period two who is the new entrant; while this scenario yields an identical outcome in period two for the retailer as the baseline analysis, it clearly alters the incumbent vendor's choices in period one. Given these four possibilities, under IS the retailer chooses first-period price and inventory level to solve: I S Max [ p1 − s]q( p1 ) − w1 I − w1 q( p1 ) + (1 − d)(1 − e)Π R2 (w2 (I), I) p1 ,I (9) + d(1 − e)[a − bI − s]I + (1 − d)eΠ R2 (c, I) + deΠ R2 (w2 (I), I). S The solution to (9) yields retail price and inventory level denoted p1 (w1 ;d,e) and I I(w1 ;d,e), respectively. Given the induced period-one demand, and cognizant that supply uncertainties reduce its future potential profit, the vendor sets wholesale price to solve: S S I I I Max q( p1 (w1 ;d,e))[w1 − c] + I(w1 ;d,e)[w1 − c] + (1 − d)(1 − e)ΠV 2 (I(w1 ;d,e)). I S (10) w1 ,w1
  26. 26. 24 ˆI Solving (10) reveals the usage-contingent first-period wholesale prices, w1 (d,e) and ˆS w1 (d,e). In the NI case, events proceed in a similar fashion, except the vendor is unable to condition first-period wholesale prices on their use. An alternate interpretation is that the I S vendor's problem in (10) entails an additional constraint that w1 = w1 . We relegate the details of this derivation to the appendix; here, we get to the brass tacks. Recall, the underlying motivation for information sharing is that it engenders a mutually-beneficial give and take between the supply chain partners. Importantly, this give and take entails the vendor softening its input price for goods sold in the first period while having an opportunity to raise its second period price. The more likely it is that the beneficiary of increased second period wholesale price is a new vendor, the less likely the incumbent vendor is to provide initial softening of input prices. Thus, the higher e, the more information sharing is viewed by the vendor as an opportunity to employ usage-contingent pricing not as a precursor to a mutually-beneficial long-term relationship but instead as an invitation to take advantage of the retailer in the short-run (not internalizing the subsequent supply chain consequences). For this reason, the retailer adopts IS only if e is not too high. Similar logic indicates that if d too is large, the likely-to-exit period-one vendor again resorts to short-term thinking, thereby transforming information sharing into an invitation for abuse. The following proposition formally confirms this intuition. Proposition 6. (i) The retailer opts to share inventory information if and only if * 5 − d − 2d 2 − 2 4 + 5d − 14d 2 + 6d 3 e < e (d) = . The retailer's decision to share [3 − 2d]2 information also benefits the vendor and consumers.
  27. 27. 25 1 − e + 3d (ii) Under VMI, the vendor's preferred inventory-to-sales ratio is r = . 2[1 − e] + d[2 + e] With this choice, VMI replicates the optimal information sharing outcome for any d and e. Consistent with the above intuition, from Proposition 6(i), information sharing is associated with a lower-tail of e-values. And, since it is readily confirmed that e * (d) is decreasing in d, lower d-values too are linked with information sharing. The basic intuition from before continues to apply: information sharing is beneficial to all parties if and only if it results in less aggressive inventory holding. This view is evidenced in Figure 4 which plots retailer profits and inventory levels as a function of e for a given d value (d = 1/5). For e < e * (d), inventory levels are lower under IS, and it is precisely then that information sharing is in the forefront. 11@a-s-cDê62b è I H1ê5,eL ` P R H1ê5,eL ` è I H1ê5,eL 97@a-s-cD2 ê720b P R H1ê5,eL 257@a-s-cD2 ê1922b @a-s-cDê6b e e 0 e* H1ê5Lº0.07 0 e* H1ê5Lº0.07 FIGURE 4. Retailer Profit and Inventories as a function of e. As confirmed in Proposition 6(ii), the result that VMI can replicate the optimal information sharing outcome is also robust to the nature of supply disruptions. By judiciously setting the inventory-to-sales ratio, the vendor ensure that VMI replicates the usage-contingent contract despite the potential for supply disruptions.
  28. 28. 26 4.4. Demand-side Uncertainty Another commonly-discussed tension in inventory management is the role of demand uncertainty in boosting (or softening) the preferred inventory cushion. That is, inventory is often held to accommodate unexpected spikes in demand so as to avoid stockout costs. To reflect this notion most succinctly in our setup, consider the possibility that demand is uncertain in each period, and when choosing inventory levels the retailer is as of then unsure of its subsequent demand. In particular, suppose the parameter of consumer demand, ai , is independently and identically distributed in each period: ai ∈[a − ∆, a + ∆], with mean a and standard deviation σ . Demand is (publicly) observed only at the beginning of each period, and the decision to share information is made upfront by the retailer (before learning a1 or a2). This formulation permits a simple characterization of the role of uncertainty. In particular, the second-period interactions are as before, whereas the added caveat in the first period is that pricing and inventory decisions are made in expectation of period two conditions. As in the previous section, we will relegate details to the appendix, and focus discussion here on the underlying added forces. Recall, there are two key strategic effects of inventory to consider: the average pricing effect and the pricing variation effect. For σ = 0 , each of these effects is as under the demand certainty case. Further, as confirmed in the appendix, uncertainty does not alter either the average retail price or the average inventory level. As such, the only role of uncertainty is in altering pricing variation. In the demand uncertainty case, price variation entails an added dimension, that of variation in prices within each period. In particular, an extreme demand realization (be it high or low) for period 1 has less of an effect on wholesale (and thus retail) prices in period one under NI. This is because the wholesale price under NI serves the dual role of a price for units sold in period one and units kept in inventory, and only the former role suggests fluctuations with the extreme realization. In
  29. 29. 27 contrast, since the supplier's wholesale price is decoupled based on use under IS, wholesale prices for period one sales are more responsive to fluctuations in demand. This feature can create a demand for NI for the retailer in that NI entails less intra-period (stochastic) volatility in wholesale and retail prices. Proposition 7. 65[a − s − c]2 (i) The retailer opts to share inventory information if and only if σ 2 < . 2368 The retailer's decision to share information benefits the vendor; it also benefits 3[a − s − c]2 consumers if and only if σ 2 < . 1984 1 a1 − a (ii) Under VMI, the vendor's preferred inventory-to-sales ratio is r = − . 2 2[a1 − s − c] With this choice, VMI replicates the optimal information sharing outcome for any σ. An alternative way of viewing Proposition 7(i) is that the conclusion of the baseline analysis is that when inventories are used to secure lower subsequent wholesale prices, information sharing acts as a natural salve on supply chain frictions. However, as demand uncertainty increases, a different justification for inventory becomes paramount: under NI, the retailer can boost (reduce) inventories when its first period demand is low (high) given that the common period-one supplier price is lower (higher) when demand is low (high). As uncertainty becomes increasingly prominent, this feature becomes more vital to the retailer and leads the retailer to shy away from information sharing which permits the vendor to prescribe different wholesale prices for inventory and sales. Proposition 7(ii) is consistent with another running theme of the analysis: though additional tensions affecting inventory management stand to alter the chosen inventory-to- sales ratio, VMI continues to implement the ideal contractual arrangement under information sharing.
  30. 30. 28 5. Conclusion Common wisdom posits that the prevalence of formal information sharing arrangements and vendor-managed inventory (VMI) systems are rooted in a spirit of cooperation in supply chains. That is, when parties exhibit substantial cooperation, information sharing and the use of VMI can benefit the parties by improving vendor responsiveness to rapidly changing needs of the supply chain. In contrast, this paper demonstrates that information sharing benefits can also arise due to strategic frictions that underlie supply chain relationships. We demonstrate that when a retailer uses inventory as a strategic tool to force lower future wholesale prices only to face the downside of retaliatory higher near-term wholesale prices, information sharing can provide an efficient means of restraining the parties' behavior. When the retailer opts to provide information on inventory and sales levels, it permits a formal means through which the vendor can price discriminate based on retailer usage. This translates into the vendor providing rebates to the retailer for early sales; the retailer responds in kind by reducing its use of inventories and thereby increasing its willingness to pay in the future. These gestures arise from a spirit of self-interest yet benefit the retailer, vendor, and consumers. We further demonstrate that such information sharing contractual outcomes can be implemented using a standard VMI relationship wherein the vendor sets inventory-to-sales targets. This paper also examines how these key results are impacted by additional inventory management tensions including holding costs, cost of capital, supply disruptions, and demand-side uncertainty. While each added tension represents subtle incremental considerations in information sharing, the underlying theme of analysis remains unaltered: the more prominent is the role for strategic and self-interested behavior in inventory management, the more effective is information sharing in reducing frictions in the supply chain. Future work could consider potential reporting distortions, and the accompanying
  31. 31. 29 role for audits, given strategic considerations in managing inventories. Extant work has stressed that information sharing may be derailed by incentives for information distortion (e.g., Mishra et al. 2007). While such issues may point to the necessity of frequent audits, multi-period considerations may naturally curb distortions. In particular, the natural linking of unaudited data to future inventory and pricing decisions may create countervailing incentives which might add credence to disclosures. So, for example, with demand correlated across time, a retailer who wants to convey low demand to secure a low wholesale price does so at potential cost since the vendor responds by requiring lower (and, thus, insufficient) inventory stocks. In effect, delegated inventory decisions can create a bond between supply chain partners across periods, a tie-in that can serve to discipline reporting.
  32. 32. 30 A PPENDIX Proof of Lemma 1. Consider the outcome under IS. Working backwards, given I and w2, the first-order condition of (1) yields the second-period retail price: a + s + w2 p2 (w2 ) = . (A1) 2 Using (A1), the vendor's problem in (2) yields the second-period wholesale price: a−s+c w2 (I) = − bI . (A2) 2 The retailer's problem in period 1 is presented in (3). In choosing p1 and I, the retailer accounts for the fact that p2 depends on w2 (as noted in (A1) and w2 in turn depends on I (as noted in (A2)). The solution to (3) yields: S I S I a + s + w1 S I 3[a − s] + c − 4w1 p1 (w1 , w1 ) = and I(w1 , w1 ) = . (A3) 2 6b S Using (A1), (A2), and (A3), the vendor's problem is written as a function of w1 and I w1 in (4). Solving the first-order condition of (4) with respect to these variables yields: a−s+c 9[a − s] + 7c ˆS w1 = ˆI and w1 = . (A4) 2 16 Substituting (A4) into (A3) yields the first-period retail price and inventory; substituting these, in turn, into (A2) and (A1), yields second-period wholesale and retail ˆ prices; the demand in period i is q( pi ). Substituting this solution into (3) and (4) yields retailer and vendor profit, respectively. Finally, note that the condition a > c + s ensures ˆ ˆ ˆ ˆ ˆ that demand ( q( p1 ) and q( p2 )), the inventory level ( I ), and purchases ( q( p1 ) + I and ˆ ˆ q( p2 ) − I ) are each positive. This completes the proof of Lemma 1. Proof of Lemma 2. In the NI case, the second-period problem for the retailer and vendor are as in (1) and (2), respectively, so (A1) and (A2) again represent the second-period retail and wholesale prices. Given the choice of w1 by the vendor, the retailer chooses p1 and I to solve (5). The first-order condition of (5) yields: a + s + w1 3[a − s] + c − 4w1 p1 (w1 ) = and I(w1 ) = . (A5) 2 6b Using (A1), (A2), and (A5), the vendor's problem is written as a function of w1 in (6). The first-order condition of (6) yields: 9[a − s] + 8c ˜ w1 = . (A6) 17 Substituting (A6) into (A5) yields the first-period retail price and inventory; substituting these into (A2) and (A1) yields second-period wholesale and retail prices; the
  33. 33. 31 ˜ demand in period i is q( pi ). Substituting this solution into (5) and (6) yields retailer and vendor profit, respectively. Again, the condition a > c + s ensures demand, the inventory level, and purchases are each positive. This completes the proof of Lemma 2. Proof of Proposition 1. (i) From Lemma 1(ii) and Lemma 2(ii), the average retail price is lower under IS than under NI: ˆ ˆ ˜ p1 + p2 p1 + p2 ˜ a−s−c − =− < 0. 2 2 544 (ii) Again, from Lemma 1(ii) and Lemma 2(ii), the retail price in period 1 is lower under IS than under NI: a−s−c ˆ ˜ p1 − p1 = − < 0. 68 However, in period 2, the retail price is higher under IS than under NI: 3[a − s − c] ˆ ˜ p2 − p2 = > 0. 272 This completes the proof of Proposition 1. Proof of Proposition 2. (i) The retailer's preference for IS follows from Lemma 1(v) and Lemma 2(v): 2 ˆ ˜ 195[a − s − c] ΠR − ΠR = > 0. 73984b (ii) The vendor's preference for IS also follows directly from Lemma 1(v) and Lemma 2(v): 2 ˆ ˜ [a − s − c] ΠV − ΠV = > 0. 1088b Turning to consumers, using the demand expressions derived in Lemma 1(iii) and Lemma 2(iii), the difference in consumer surplus is as follows: b⎛ [q( p1 )]2 + [q( p2 )]2 ⎞ b⎛ [q( p1 )]2 + [q( p2 )]2 ⎞ ˆ ˆ ˜ ˜ ^ ~ ⎝ ⎠ ⎝ ⎠ 9[a − s − c]2 CS − CS = − = > 0. 2 2 147968b This completes the proof of Proposition 2. Proof of the Corollary. In the absence of inventory, the solution is merely a two-fold replication of the standard single-period supply chain pricing problem under both IS and NI. Formally, this can be verified by repeating the backward induction process employed in Lemmas 1 and 2 with the added constraint that I in (3) is not a choice variable for the retailer but is fixed at I = 0. With I = 0, (A2) and (A1) reveal that, in each period, the
  34. 34. 32 wholesale price is [a − s + c] / 2 and the retail price is [3a + s + c] / 4 . Using this solution, the two-period retailer profit is [a − s − c]2 / [8b], vendor profit is [a − s − c]2 / [4b], and consumer surplus is [a − s − c]2 / [16b]. From Lemma 1, under IS, retailer profit is 35[a − s − c]2 / [256b], vendor profit is 17[a − s − c]2 / [64b], and consumer surplus is 41[a − s − c]2 / [512b]. Using the above expressions, the retailer profit's under IS less its profit under just- in-time inventory is: 35[a − s − c]2 [a − s − c]2 3[a − s − c]2 − = > 0. 256b 8b 256b The vendor's profit's under IS less its profit under just-in-time inventory is: 17[a − s − c]2 [a − s − c]2 [a − s − c]2 − = > 0. 64b 4b 64b Finally, the consumer surplus difference is: 41[a − s − c]2 [a − s − c]2 9[a − s − c]2 − = > 0. 512b 16b 512b This completes the proof of the Corollary. Proof of Proposition 3. Under VMI, the second-period problem for the retailer and vendor are as in (1) and (2), respectively, so (A1) and (A2) again represent the second- period retail and wholesale prices. Given the choice of w1 and r by the vendor, the retailer chooses retail price p1 to solve (7). The first-order condition of (7) yields: a[4 − 3r + 6r 2 ] − r[c − 3s] + 4[(1 + r)w1 + s] p1 (w1 ,r) = . (A5) 2(4 + 3r 2 ) Using (A1), (A2), and (A5), the vendor's problem is written as a function of w1 and ∗ r in (8). The first-order condition of (8) yields wholesale price w1 and inventory-to-sales ratio r ∗ : ∗ 25[a − s] + 23c 1 w1 = and r ∗ = . (A6) 48 2 Substituting (A6) into (A5) yields the first-period retail price of [3a + s + c] / 4 , ˆ which is the same as p1 in Lemma 1; using this price, the inventory level is ∗ ˆ ˆ r q( p1 ) = [a − s − c] / [8b], which is the same as I in Lemma 1; from (A1) and (A2) it follows that the second-period wholesale price and retail price under VMI is equal to w2 ˆ ˆ and p2 , respectively. Finally, using the VMI solution in (7) and (8) yields retailer and ˆ ˆ vendor profits of Π R and Π V , respectively. This completes the proof of Proposition 3.
  35. 35. 33 Proof of Proposition 4. (i) With holding costs, in the IS case, the analysis proceeds in the same way as in the proof of Lemma 1 except that the retailer's profit in (3) is reduced by the holding cost hI . With this change, the retailer's first-order condition yields: S I S I a + s + w1 S I 3[a − s] + c − 4[w1 + h] p1 (w1 , w1 ;h) = and I(w1 , w1 ;h) = . (A7) 2 6b Using (A1), (A2), and (A7) in (4), , and solving the vendor's first-order conditions S I with respect to w1 and w1 yields: a−s+c 9[a − s] + 7c − 4h ˆS w1 (h) = ˆI and w1 (h) = . (A8) 2 16 Using (A8) in (A7), in the h-adjusted (3), and in (4) yields the optimal inventory level and profits listed below: ˆ a − s − c − 4h ˆ 17[a − s − c]2 − 8h[a − s − c − 2h] I (h) = ; Π V (h) = ; and 8b 64b ˆ 35[a − s − c]2 − 24h[a − s − c − 2h] Π R (h) = . (A9) 256b In the NI case, the analysis proceeds in the same way as in the proof of Lemma 2 except that the retailer's profit in (5) is reduced by the holding cost hI . With this change, the retailer's first-order condition yields: a + s + w1 3[a − s] + c − 4[w1 + h] p1 (w1 ;h) = and I(w1 ;h) = . (A10) 2 6b Using (A1), (A2), and (A10) in (6), and solving the vendor's first-order condition with respect to w1 yields: 9[a − s] + 8c − 2h ˜ w1 (h) = . (A11) 17 Using (A11) in (A10), in the h-adjusted (5), and in (6) yields the optimal inventory level and profits listed below: 2 ˜ (h) = 5[a − s − c − 4h] ; Π (h) = 9[a − s − c] − 4h[a − s − c − 2h] ; and I ˜ V 34b 34b ˜ 155[a − s − c]2 − 2h[59(a − s − c) − 152h] Π R (h) = . (A12) 1156b ˆ ˜ From the inventory expressions in (A9) and (A12), both I (h) and I (h) equal 0 when h = [1 / 4][a − s − c]. Hence, for h ≥ [1 / 4][a − s − c], the solution under both IS and NI entail a two-fold repetition of the single-period solution, and so information sharing is
  36. 36. 34 moot. Using the retailer profit expressions in (A9) and (A12), it follows that for h < [1 / 4][a − s − c], the retailer strictly prefers to share information: ˆ ˜ [a − s − c − 4h][195(a − s − c − 4h) + 2176h] Π R (h) − Π R (h) = > 0. 73984b As noted below, when the retailer shares information (i.e., a − s − c − 4h > 0), it also benefits the vendor and consumers: 2 ˆ ˜ [a − s − c − 4h] Π V (h) − Π V (h) = > 0 and 1088b ^ ~ [a − s − c − 4h][9(a − s − c − 4h) + 544h] CS(h) − CS(h) = > 0. 147968b (ii) With holding costs, in the VMI case, the analysis proceeds in the same way as in the proof of Proposition 3 except that the retailer's profit in (7) is reduced by [1 − f ]hI and the vendor's profit in (8) is reduced by fhI . With this change, the retailer's first-order condition yields: a[4 − 3r + 6r 2 ] − r[c − 3s − 4h(1 − f )] + 4[(1 + r)w1 + s] p1 (w1 ,r;h, f ) = . (A13) 2[4 + 3r 2 ] Using (A1), (A2), and (A13), the vendor's first-order condition yields: ∗ a − s + c [a − s − c − 4h][a − s − c − 4h(1 − 4 f )] w1 (h, f ) = + and 2 16[3(a − s − c) − 4h] a − s − c − 4h r ∗ (h, f ) = . (A14) 2[a − s − c] Using (A1), (A2), (A13) and (A14), and noting I = rq( p1 ), provides the solution to all variables of interest, and using these in (7) and (8) yields retailer and vendor profits. These profit numbers, along with the inventory level, are listed below: a − s − c − 4h 17[a − s − c]2 − 8h[a − s − c − 2h] I ∗ (h, f ) = , Π ∗ (h, f ) = V , and 8b 64b 35[a − s − c]2 − 24h[a − s − c − 2h] Π ∗ (h, f ) = R . 256b From the above, notice the inventory level and profits are free of f and equal to the corresponding expressions under IS derived in (A9). This completes the proof of Proposition 4. Proof of Proposition 5. As in the base k = 0 setting, here we presume a is sufficiently large so as to ensure prices, quantities, and inventory derived using first-order conditions are c[5 + 16k + 8k 2 ] positive; this requires a > + s. With discounting, in the IS case, the 5 − 4k
  37. 37. 35 retailer's and vendor's second-period profits in (3) and (4), respectively, are multiplied by the Π R2 (w2 (I), I) term 1/[1+k]. That is, Π R2 (w2 (I), I) in (3) is replaced by and 1+ k I I Π (I(w1 )) Π V 2 (I(w1 )) in (4) is replaced by V 2 . In the NI case, the same change is made 1+ k in (5) and (6). In the VMI case, the same change is made in (7) and (8). Given these changes, the rest of the backward induction process proceeds as in the proofs of Lemma 1 (for the IS case), in Lemma 2 (for the NI case), and Proposition 3 (for the VMI case). Rather than provide tedious details of these intermediate steps, we next present the solution in each case as a function of the cost of capital. IS Outcomes 9[a − s] + c[7 + 12k] S a−s+c (a) wholesale prices: ˆI w1 (k) = ˆ , w1 (k) = , and 16[1 + k] 2 3[a − s] + c[5 + 4k] ˆ w2 (k) = ; 8 3a + s + c 11a + 5s + c[5 + 4k] (b) retail prices: ˆ p1 (k) = ˆ and p2 (k) = ; 4 16 a−s−c 5[a − s] − c[5 + 4k] (c) demand: ˆ q( p1 (k)) = ˆ and q( p2 (k)) = ; 4b 16b ˆ a − s − c[1 + 4k] (d) inventory: I (k) = ; 8b ˆ [a − s]2 [17 + 8k] − 2[a − s]c[17 + 12k] + c 2 [17 + 16k(1 + k)] (e) profits: Π V (k) = and 64b[1 + k] ˆ [a − s]2 [35 + 16k] − 14[a − s]c[5 + 4k] + c 2 [35 + 40k + 48k 2 ] Π R (k) = . 256b[1 + k] NI Outcomes 9[a − s] + 2c[4 + 3k] (a) wholesale prices: ˜ w1 (k) = and 17 + 8k 6[a − s][1 + k] + c[11 + 12k + 4k 2 ] ˜ w2 (k) = ; 17 + 8k a[13 + 4k] + 4s[1 + k] + c[4 + 3k] (b) retail prices: ˜ p1 (k) = and 17 + 8k a[23 + 14k] + s[11 + 2k] + c[11 + 12k + 4k 2 ] ˜ p2 (k) = ; 2[17 + 8k] 4[a − s][1 + k] − c[4 + 3k] (c) demand: ˜ q( p1 (k)) = and b[17 + 8k]
  38. 38. 36 [a − s][11 + 2k] − c[11 + 12k + 4k 2 ] ˜ q( p2 (k)) = ; 2b[17 + 8k] ˜ [a − s][5 − 4k] − c[5 + 16k + 8k 2 ] (d) inventory: I (k) = ; 2b[17 + 8k] ˜ 9[a − s]2 [1 + k] − 2c[a − s][9 + 11k + 2k 2 ] + c 2 [9 + 13k + 12k 2 + 4k 3 ] (e) profits: Π V = and 2b[17 + 25k + 8k 2 ] ˜ 1 ⎧[a − s]2 [155 + 230k + 220k 2 + 64k 3 ] − 2c[a − s][155 + Π R (k) = 2 ⎨ 4b[1 + k][17 + 8k] ⎩ 289k + 158k 2 + 24k 3 ] + c 2 [155 + 348k + 400k 2 + 228k 3 + 48k 4 ]⎫. ⎬ ⎭ VMI Outcomes ∗ [a − s]2 [25 + 16k] − 2[a − s]c[1 + 12k] − c 2 [23 + 56k + 48k 2 ] (a) wholesale prices: w1 (k) = 16[1 + k][3(a − s) − c(3 + 4k)] ∗ 3[a − s] + c[5 + 4k] and w2 (k) = ; 8 ∗ 3a + s + c ∗ 11a + 5s + c[5 + 4k] (b) retail prices: p1 (k) = and p2 (k) = ; 4 16 ∗ a−s−c ∗ 5[a − s] − c[5 + 4k] (c) demand: q( p1 (k)) = and q( p2 (k)) = ; 4b 16b a − s − c[1 + 4k] a − s − c[1 + 4k] (d) inventory: r ∗ (k) = and I ∗ (k) = ; 2[a − s − c] 8b [a − s]2 [17 + 8k] − 2[a − s]c[17 + 12k] + c 2 [17 + 16k(1 + k)] (e) profits: Π ∗ (k) = V and 64b[1 + k] [a − s]2 [35 + 16k] − 14[a − s]c[5 + 4k] + c 2 [35 + 40k + 48k 2 ] Π ∗ (k) = R . 256b[1 + k] ˆ ˜ (i) Comparing Π R (k) with Π R (k), it follows that the retailer shares information if and only a−s−c c[5 + 16k + 8k 2 ] if k < ; note this cutoff k-value is in (0,1) since a > + s. 4[2(a − s) − c] 5 − 4k ˆ ˜ Comparing Π V (k) with Π V (k), the vendor is always better off under IS: 2 ˆ ˜ Π V (k) − Π V (k) = [(a − s)(8k − 1) − c(4k − 1)] > 0. 64b[1 + k][17 + 8k] Turning to consumers, the consumer surplus difference is: ^ ~ [(a − s)(1 − 8k) − c(1 − 4k)][9(a − s)(1 + 24k) − c(9 − 292k − 192k 2 )] CS(k) − CS(k) = . 512b[17 + 8k]2 ^ ~ a−s−c Some tedious algebra verifies that CS(k) − CS(k) > 0 for k < . 4[2(a − s) − c]

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