Strategic inventories with quality deterioration

Accepted Manuscript

Strategic Inventories with Quality Deterioration Benny Mantin, Lifei Jiang PII: DOI: Reference:

S0377-2217(16)30701-9 10.1016/j.ejor.2016.08.062 EOR 13948

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European Journal of Operational Research

Received date: Revised date: Accepted date:

17 July 2015 22 June 2016 19 August 2016

Please cite this article as: Benny Mantin, Lifei Jiang, Strategic Inventories with Quality Deterioration, European Journal of Operational Research (2016), doi: 10.1016/j.ejor.2016.08.062

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Jiang and Mantin: Strategic Inventories with Quality Deterioration

Highlights • We study strategic inventories subject to quality deterioration

• We find that prices may increase over time

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• We characterize conditions under which such inventories are kept by the retailer

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• We highlight implications due to the ignorance of quality deterioration

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Strategic Inventories with Quality

Benny Mantin∗

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Deterioration

Department of Management Sciences

University of Waterloo, 200 University Avenue West, Waterloo, ON, Canada N2L 3G1

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[email protected] Lifei Jiang

Department of Management Sciences

University of Waterloo, 200 University Avenue West, Waterloo, ON, Canada N2L 3G1

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jlf [email protected]

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August 27, 2016

∗

Corresponding author: tel 1-519-888-4567 ext. 32235; fax 1-519-746-7252

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Jiang and Mantin: Strategic Inventories with Quality Deterioration

Strategic Inventories with Quality Deterioration

Abstract

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Retailers may carry inventories strategically in order to induce the suppliers to lower the wholesale price in their future negotiations. Earlier literature on strategic inventories assumed that units carried from one period over to the next were perfect substitutes to newly procured ones. However, in practice, when inventories are

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carried their quality is subject to deterioration over time. Thus, inventoried units become vertically differentiated from newer units. Such quality deterioration affects the supplier-retailer interaction: first, it significantly limits the retailer’s ability to carry strategic inventories (only when both their deterioration and the cost of holding them is not too high). This occurs as the deterioration enables the supplier to set a wholesale

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price such that the retailer is discouraged from holding strategic inventories. Yet, the supplier is worse off in the presence of product deterioration; second, despite the car-

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rying of strategic inventories, it may result with a wholesale price, as well as a selling price, that increase, rather than decrease, over time; third, some product deterioration

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may be beneficial for the supply chain. We further characterize differences that occur as a consequence of the consideration inventory deterioration.

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Key words: strategic inventories; quality deterioration; supply chain coordination;

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contracts

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Introduction

Inventory is commonly carried for a variety reasons, such as dealing with uncertainty in demand or supply, achieving economies of scale, or hedging against price fluctuation. Importantly, inventory may also be carried to play a strategic role in multi-period vertical competition environments. Specifically, buyers may strategically purchase an excess amount 1

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of inventory in one period only to be carried over to the next period, while incurring holding cost, in order to weaken the suppliers’ power in future contract negotiation. By carrying this inventory, the buyer signals to the supplier that if negotiations break, then the buyer can still sell these carried over units. Hence, the negotiations are over the incremental quantity to

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be purchased by the buyer (to be sold in the current period and possibly carried over to the ensuing period). Given the presence strategic inventories, the supplier is forced to lower the wholesale price (with respect to the first period wholesale price) to stimulate purchases from the buyer. The buyer expects the wholesale price to drop from the first to the second period, and accordingly limits the amount of strategic inventory. The holding cost further limits the

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buyer’s incentive to stock units. Thus, the buyer faces a delicate trade-off when choosing the optimal level of strategic inventories to be carried: increasing the amount of strategic inventories, increases the holding cost but reduces the future wholesale price charged by the supplier. Conversely, when no inventories are carried, no holding cost is incurred, resulting

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with the supplier charging monopoly wholesale price in both periods of interaction. Anand et al. (2008) were the first to identify the strategic role of inventories in a multi-

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period supply chain setting. Their results show that the buyer’s optimal strategy is to carry inventories1 and the supplier is unable to prevent this, although in equilibrium the wholesale

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price is lower in the second period. Hence, optimal vertical contracts must take the possibility of strategic inventories into account.

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The literature on strategic inventories is rather limited. Only a few other contributions were dedicated to analysis of strategic inventories. Arya et al. (2012) used a two-period

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strategic inventory model to study the decentralized decision making in procurement and inventory control. Hartwig et al. (2015) presented the first empirical study to test the effect of strategic inventories on supply chain performance based on the theoretical models of Anand et al. (2008). They found that the positive effect of strategic inventories in practice is even more pronounced than it is in theory, because with strategic inventories, buyers have 1

Technically, they assume that the holding cost is lower than a certain threshold required to make inventories feasible in the first place. They find that in this range the buyer always holds strategic inventories.

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power to reduce payoff inequalities, and suppliers are willing to reduce inequalities as long as their payoff remains above a certain threshold. Arya and Mittendorf (2013) explored the role of rebates offered by manufacturers directly to consumers in mediating the effect of strategic inventories. Essentially, they found that manufacture-to-consumer rebates make

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the retailer less aggressive in carrying inventories and the manufacturer less exploitative in setting wholesale prices. Also, the retailer, manufacturer and consumers are all better off due to manufacture-to-consumer rebates. Recently, Gerchak (2015) extended the basic strategic inventory framework to address demand uncertainty.

All of the above papers adopt the assumption that goods, once carried over from one

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period to the next, are perfect substitutes to newly purchased goods from the supplier. This assumption, in our view, is quite strong. Goods are generally subject to some degree of deterioration over time. For example, consumers usually devalue the old products due to a variety of reasons. If the goods are perishable, then inventoried goods are not as fresh as

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new products, if the goods are durable, often inventories are older versions relative to new products, so they are less attractive to consumers and therefore are offered at a lower price.

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There is a substantial literature covering deteriorating inventories. The notion of inventory deterioration has been formalized by Whitin (1953), and since then many papers have

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accounted for different aspects and impacts induced by deterioration. The extensive reviews by Goyal and Giri (2001) and by Bakker et al. (2012) overview the body of literature ded-

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icated to inventory deterioration. The former considers papers prior to 2001, whereas the latter focuses on papers that appeared after 2001. These reviews classify deterioration into

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three categories: (i) fixed lifetime, (ii) age dependent deterioration rate (implying the deterioration is associated with a probability distribution), and (iii) time or inventory (but not age) dependent deterioration rate. Despite the spread and breadth of analysis of inventory deterioration, the fundamental concept of strategic inventory is missing from the discussion on inventory deterioration. Hence, we believe that the proper way to explore the aspect of strategic inventories is

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through consideration of quality degradation (which corresponds to the third category in the above classification). When a retailer has carried-over inventories which compete with the new products, it may affect the channel decisions and profits. Deterioration of products is relevant for many products ranging from fresh food, text-

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books, to automotive as well as electronics. For many of these quality deteriorating products, a cycle may be observed. New products are being restocked at rather predictable intervals and new models (or versions) are being regularly introduced. With deterioration taking place in many industries, the relevance of strategic inventories is being challenged in this paper. Additionally, the agents along the supply chain may be able to control the degree

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of deterioration (e.g., via product design, advertising, storage), which raised an interesting question: are the retailer and the supplier in agreement with respect to the optimal level of product deterioration?

Our focus is on strategic inventories with quality deterioration. We explore the following

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questions. With product quality differentiation, what are the equilibrium decisions and profits for the retailer and supplier? Is carrying strategic inventories still the optimal choice

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for the retailer? Also, how do the strategic inventories affect consumer surplus? To address these questions, we build a two-period model exploring the interaction between

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a single retailer and a single supplier. Our modeling framework essentially revisits that of Anand et al. (2008) by accounting for the presence of inventory deterioration. This is

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achieved by assuming units experience a constant deterioration rate per stocked item (the third category in Bakker et al., 2012). We find that when strategic inventories are kept in

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equilibrium, then they always benefit the supplier and the consumers while the retailer is almost always better off (unless the deterioration is minimal and the holding cost is high). The analysis of the equilibrium gives rise to an important distinction from Anand et al. (2008) who stated that “the buyers optimal strategy in equilibrium is to carry inventories, and the supplier is unable to prevent this”. Indeed, inventory deterioration can harm the supplier, however, the supplier can respond by charging the monopoly wholesale price thereby

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discouraging the retailer from carrying any strategic inventories. Thus, the supplier will induce the retailer to carry such inventories when he—the supplier—is better off with such inventories. This difference is critical: when deterioration is ignored, the initial wholesale price is higher than the monopoly wholesale price, and this wholesale price drops in the

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second period. When deterioration is present, the opposite may occur. Specifically, the initial wholesale price may be lower than monopoly wholesale price, and then in subsequent period it increases (and it may or may not exceed the monopoly wholesale price). Thus, when the deterioration is not too high and the holding cost is sufficiently low, the retailer will optimally carry strategic inventories. Otherwise, the supplier will raise the wholesale

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price to that charged in the single period setting to eliminate such inventories, effectively making the retailer worse off.

Regardless, even if the supplier does not discourage the retailer from stocking strategic inventories, we find that such inventories may be completely eliminated when deterioration

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takes place. Namely, the carried over units, which are perceived as lesser quality by consumers, may simply cannibalize demand for new units, and coupled with the holding cost

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to carry these units, practically eliminate the benefit from holding them. Accordingly, the threat to the supplier is removed, and the supplier can return to practicing the single period

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wholesale pricing.

Further, in the absence of deterioration there exists a holding cost threshold above which

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(about 0.19 in our and in Anand et al.’s, 2008, models) strategic inventories are detrimental to the supply chain. By contrast, we find that with some deterioration, strategic inventories can

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benefit the supply chain even for higher holding costs (up to 0.25, depending on the degree of deterioration). With perfect substitution the retailer suffers from strategic inventories when the holding cost is high, and this profit loss exceeds the profit gain made by the supplier. With some deterioration, the retailer’s profit loss is reduced, and even reversed, while the supplier’s profit gain slightly diminishes, overall resulting with an improved profit from a supply chain perspective.

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The paper is organized as follows. We introduce the model framework in §2, and provide the results and analysis in §3, where we study two contractual formats—commitment and dynamic—which are then in §4 compared and further contrasted with the results of Anand

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et al. (2008). Conclusions and further research are provided in §5.

Model Framework

Similar to Anand et al. (2008) and Yin et al. (2010)2 , we consider a dynamic two-period model of full information and no uncertainty, which helps us to focus on the interactions be-

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tween the retailer and supplier via strategic inventories. In the model, the profit-maximizing supplier and retailer are vertical channel partners selling products to end consumers in two periods. A new set of consumers shows up at the beginning of each period and their valuation, v, of the good has a uniform distribution from 0 to 1. The retailer can purchase

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more than the demand in the first period and hold the unsold goods to sell in the second period. If the retailer chooses to carry inventories over from one period to the next, then it

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will cost him h per unit for holding the inventory. Also, the retailer will choose the quantity of the inventories to sell in the second period. On top of the above setting which is also used

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by Anand et al. (2008), we are possibly the first to consider the quality difference between strategic inventories and new products. Specifically, due to quality deterioration, consumers’

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valuation for the strategic inventories is affected by a discount factor δ ∈ [0, 1]. That is, while a new good will have a perceived valuation v, an older unit will have a valuation δv.

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As in Anand et al. (2008), we consider two types of contracts: a dynamic contract and a commitment contract. Under the dynamic contract, the sequence of events of our model

is as follows. In period 1, the supplier first sets the unit wholesale price w1 and then the retailer determines the unit retail price p1 as well as the quantity of inventories I he carries 2 Yin et al. (2010) take a different approach to a related problem: rather than assuming the good deteriorates over time, they let the supplier determine the improvement in quality of new products that are released to the market. They show that the emergence of a secondary market results in vertical competition, which reduces the frequency of product upgrades, reduces temporal retail price increase and can be either beneficial or harmful for both channel members. However, strategic inventory is missing in their model.

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to the next period. The corresponding demand in the first period is q1 and, hence, the total quantity of products purchased by the retailer in the first period is Q1 = q1 + I. In period 2, a new set of consumers shows up. The supplier sets the new wholesale price w2 , and then the retailer sets the retail price for new products, p2n , and for inventories carried over from the

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first period, p2o . The corresponding quantities of new and old products are denoted as q2n and q2o (with q2o ≤ I), respectively. On the other hand, under the commitment contract, the supplier commits to a price sequence at the beginning of the horizon. That is, the supplier announces wholesale prices w1 and w2 at the beginning of the first period, and the retailer responds in the first and second periods as before.

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As inventories may be carried over, giving rise to vertical differentiation, we adopt the Mussa-Rosen (1978) consumer utility model. We let U1 , U2o and U2n denote consumers’ expected utility of buying a new product in period 1, an old product in period 2 and aa new product in period 2, respectively, and we let v1 , vo and vno denote the indifference points

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between buying a new product and not buying in period 1, between buying an old product and not buying in period 2, and between buying an old and a new product in period 2,

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respectively.

The inverse demand function for new products in period 1 is given by p1 = 1 − q1 .

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We illustrate the consumer choice model in the second period in Figure 1. In period 2, a consumer has three different options: buying a new product, buying an old product which is

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carried over from period 1, and not buying at all. The utility of buying new a new product is U2n = v − p2n . And the utility of buying an old one is U2o = δv − p2o . The indifference

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point between buying a new product and an old one is at v no = (p2n − p2o )/(1 − δ). The indifference point between buying an old product and not buying anything is at v o = p2o /δ. Thus, consumers whose valuations are in [v no , 1] will buy the new product while those with valuations in [v o , v no ] will buy the old products from the retailer. Therefore, the demand functions for new products and old products in period 2 are given by q2n = 1 − q2o =

p2n −p2o 1−δ

−

p2o , δ

p2n −p2o 1−δ

and

respectively. The inverse demand functions for new products and old

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products, respectively, in period 2 are: p2n = 1 − q2n − δq2o and p2o = δ(1 − q2n − q2o ). new utility

vo

vno

0

1

p2o

q2n

q2o

valuation

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p2n

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old

Figure 1: Consumer choice model

Model analysis

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The models are solved backwards to yield the subgame perfect Nash equilibria. The equi-

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librium decisions, profits and consumer surplus under each of the contracts are provided in

The two contracts

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3.1

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Table 1.

We first consider the dynamic contract. In equilibrium, the retailer’s decision on whether

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to carry strategic inventories or not depends on the values of the discount factor, δ, and the unit holding cost, h. Specifically, we have the following proposition. All proofs are provided in the Appendix. ˜≡ Proposition 1. Only when h ≤ h

√

4δ 3 −7δ 2 +5δ−(1−δ) δ(δ 3 −8δ 2 +8δ+16) , 4(δ 2 +1)

the retailer will carry

strategic inventories. Otherwise, the retailer will not hold strategic inventories. ˜ = 1 . Thus, the Note that in the absence of quality deterioration δ = 1 and hence h 4 8

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˜ Dynamic (h > h);

˜ Dynamic (h ≤ h) ∗ ) (q1∗ + I ∗ , q2n

I∗ ∗ ∗ (q1∗ , q2n , q2o ) ∗ ∗ ∗ (p1 , p2n , p2o ) Π∗R Π∗S

CS ˜≡ Notes: h

4δ 3 −7δ 2 +5δ−(1−δ) δ(δ 3 −8δ 2 +8δ+16) ; 4(δ 2 +1)

( 12 , 12 ) ( 14 , 14 ) 0 ( 14 , 14 , 0) ( 34 , 34 , −) 1 8 1 4 1 16

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(w1∗ , w2∗ )

Commitment

3 2 +36δ+12hδ−16h 1−2δI ∗ ( δ −19δ , 2 ) 2(δ 3 −8δ 2 +8δ+16) ∗ ∗ 1−w1∗ ∗ 1−w2 −2δq2o ( 2 +I , ) 2 3δ−4h−4w1∗ 2δ(4−δ) ∗ 1−w∗ 1−w2∗ −2δq2o ( 2 1, , I ∗) 2 ∗ ∗ ∗ ∗ (1 − q1∗ , 1 − q2n − δq2o , δ(1 − q2n − q2o )) ∗ ∗ + p∗2o q2o p∗1 q1∗ − w1∗ (q1∗ + I ∗ ) − hI ∗ + (p∗2n − w2∗ )q2n ∗ w1∗ (q1∗ + I ∗ ) + w2∗ q2n R1 R vno R1 U dv + vo U2o dv + vno U2n dv v1√ 1

Π∗R and Π∗S are the supplier’s and retailer’s profit; CS is the

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˜ are provided in the Appendix. consumer surplus; complete expressions for the dynamic contract when h ≤ h

Table 1: Optimal decisions under the two contracts ˜ coincides with Anand et al.’s (2008) assumption that h < condition h ≤ h

1 . 4

In their

case this condition is required to guarantees that carrying inventories are feasible and they

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find that under this condition strategic inventories are always carried. We find a range of parameters where inventories are not carried strategically. Specifically, in the presence of 1 , 4

then in equilibrium the retailer will not carry such

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˜ < h < quality deterioration, if h

inventories. However, the source of this threshold is different than in Anand et al. (2008).

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While in their analysis—in the absence of inventory deterioration—the supplier is always better off due to strategic inventories, the supplier has no incentive to discourage the retailer

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from holding them. However, due to inventory deterioration, the supplier can be worse off , and therefore decides to eliminate such inventories completely by committing to the

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the commitment contract when the holding cost exceeds a certain threshold. We elaborate further on this threshold later in this section. The following corollary identifies the range of δ values such that no inventories are carried

regardless of the holding cost. Corollary 1. When the discount factor for the inventories is sufficiently low, δ ≤ δ˜ ≡ 89+46K−K 2 45K

≈ 0.592, no strategic inventories are kept regardless of the value of the holding 9

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√ 1 cost, where K = (6209 + 135 2154) 3 . Figure 2 shows the region where strategic inventories are carried. The intuitive interpretation for strategic inventories still holds if both the degree of deterioration and the unit holding cost are sufficiently low. The retailer expects the supplier to set the second-period

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wholesale price as a strategic response to his first-period purchasing actions. Hence, the

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benefit of carrying strategic inventories overrides the drawback of having holding cost.

݄෨

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h

‫ݓ‬ଵ ൏ ‫ݓ‬ଶ ‫݌‬ଵ ൏ ‫݌‬ଶ௡

݄෨ᇱ

‫ݓ‬ଵ ൒ ‫ݓ‬ଶ ‫݌‬ଵ ൒ ‫݌‬ଶ௡

ߜሚ ൎ0.592

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Figure 2: Equilibrium decisions (Strategic inventories only exist in the grey area)

Anand et al. (2008) show that the supplier anticipates the retailer to purchase strate-

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gic inventories which curtail the supplier’s monopoly power in the second period, so he encourages the retailer to purchase less by raising the wholesale price in the first period.

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Interestingly, this result (w1 ≥ w2 ) does not hold universally in our model. We have the following theorem which is different from Anand et al. (2008). Theorem 1. In the subgame-perfect equilibrium when strategic inventories are carried, (i) ˜0 ≡ w1 ≥ w2 if h ≤ h

3δ 3 −3δ 2 −22δ+16 4(δ 2 −δ−6)

and w1 < w2 otherwise; (ii) Q1 > q2n . Also, (iii) p1 ≥ p2n

˜ 0 and p1 < p2n otherwise; (iv) q1 ≥ q2n . if h ≤ h

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The intuition is that raising the wholesale price in the first period also lowers the quantities sold in this period, hence the supplier needs to balance these two considerations. Depending on the value of the discount factor and unit holding cost, these two effects can both be dominating in certain cases. As a result, the retail price in the second period can

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also be higher or lower than that in the first period. This result is different from Anand et al. (2008) and important, because it shows that carrying strategic inventories does not necessarily force the supplier to lower the second period wholesale price with respect to the ˜ 0 < h < h. ˜ This is illustrated in Figfirst period wholesale price, that is, w2 > w1 when h ure 3a. The case where δ = 1 corresponds to the scenario analyzed by Anand et al. (2008).

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Evidently, when δ = 1 we have w1 > 0.5 > w2 . That is, the supplier sets the initial wholesale price higher than the static solution (single period) and the supplier cannot possibly switch to the commitment contract as this will require him to drop the wholesale price to 0.5, which may induce the retailer to actually increase the amount of inventories stocked (and in fact, in

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that case the supplier has no incentive to deviate from the dynamic contract). Importantly, when δ is low (that is, the units deteriorate a lot), the supplier sets the initial wholesale

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price lower than under commitment contract (i.e., less than 0.5). This is the core difference: ˜ he switches to the since the supplier is worse off under the dynamic contract when h > h,

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commitment contract by raising the wholesale price. The low wholesale price allows the retailer to keep a certain amount of strategic inventory; yet, by raising the initial wholesale

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price to the static level (i.e., to 0.5), the incentive to stock strategic inventories is eliminated. An interesting question that may emerge is what happens when the supplier does not (or

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cannot) switch from the dynamic to the commitment contract? We revisit this question in §3.2

Proposition 2. When carrying strategic inventory is preferred, for all δ ∈ [0, 1] and h ∈ ˜ (i) [0, h]), dq2n dδ

dw1 dδ

< 0 ; (iii)

> 0, dΠS dδ

dp1 dδ

> 0,

>0,

d 2 ΠR dδ 2

dw2 dδ

< 0,

dp2n dδ

< 0,

dp2o dδ

> 0; (ii)

dQ1 dδ

> 0,

dq2o dδ

> 0, and there exists a value of δ ∈ [0, 1] where

> 0,

dq1 dδ

dΠR dδ

= 0.

< 0,

The above proposition is illustrated in Figures 3a, 3b, and 3c, showing the changes in 11

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prices and wholesale prices, quantities, as well as profits and CS. Proposition 2 reveals that as the value of δ increases, the supplier benefits from higher first period wholesale price and greater first period ordering quantity, which compensates him for the loss in the second period because of lower wholesale price and lower new products’ demand. At the same time,

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as the value of δ increases, the retailer benefits from selling old products in the second period, but the shrinking first period demand and lower profit margin harm his first period profit. As a result, under the dynamic contract, a lower value of δ is mostly preferred by the retailer. p1

q1

I

p 2n p 2o

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q1

S

q 2n

w1

R

q 2o

w2

(b) comparative quantities

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(a) comparative prices

CS

(c) comparative profits, CS

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Figure 3: Equilibrium decisions when strategic inventories are carried, h = 0 and δ˜ > 0.592

We note the the supplier is the agent who induces the switch between the dynamic

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and commitment contracts. Hence, the supplier is indifferent between the two contracts ˜ but always better off under the dynamic contract setting when it is employed. when h = h

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Further, the supplier’s profit is increasing in δ which implies that the supplier prefers a lower deterioration level. Namely, if the supplier were able to control the deterioration level of the

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product, he would design the product such the no value is lost over time (i.e., δ = 1). ˜ At this point To evaluate the impact on the retailer, we first estimate the profit at h = h.

the profit under the dynamic contract (strictly) exceeds the profit under the commitment contract for any δ ≤ 1 (when δ < 1).3 Since the retailer profit is convex in δ, he would 3

√

(δ 4 −6δ 3 +15δ 2 +4δ−16)

ΠR |h=h˜ = shown to be strictly greater than

δ(δ 3 −8δ 2 +8δ+16)(δ−1)2 +(5δ 6 −43δ 5 +77δ 4 +39δ 3 −100δ 2 +42δ+48) 8(δ 3 −8δ 2 +8δ+16)(δ 2 +1)2 1 for any δ ≤ 1 and equal to 81 when δ = 1. 8

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which can be

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be best off either under the smallest value of δ (that is, just when the supplier switches between the two contracts) and the largest value of δ (that is, δ = 1). Numerically, we find the former is preferred by the retailer. This implies that if the retailer were able to control the deterioration level of the product, he would manage the product such the value would

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sufficiently decrease over time to induce the supplier to offer the dynamic contract. The range of parameters where the retailer is better off or worse off is illustrated in Figure 6a. As can be noticed from this figure, except a small range of parameters values where both δ is very high (representing a value loss due to deterioration of less than 2%) and the holding cost is rather high (greater than about 0.14), the retailer prefers the dynamic contract.

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Observation 1. The supplier is best off with δ = 1 whereas the retailer is best off with the lowest value of δ that results with strategic inventories (this would be the value of δ that ˜ solves h = h).

This observation highlights the conflict between the retailer and his supplier with respect

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to the choice of deterioration when strategic inventories are carried. While the supplier

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prefers to have the product designed or be perceived as a highly durable, the retailer will seek to limit the product’s durability.

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What is the broader impact of deterioration on the channel profit and what is the optimal choice of quality deterioration for the product that is best for the channel if the channel

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partners were able to control its level? Figure 4 reveals that switching from the commitment contract (no strategic inventories) to the dynamic contract (when strategic inventories are

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˜ carried) is generally profit improving from the channel’s perfective. Specifically, once h = h the transition to carry strategic inventories spikes the channel’s profit upwards (from 0.375 to a profit that is greater than 0.375 when δ < 1). At this point, all the profit increase

is gained by the retailer. As the deterioration of the product diminishes (i.e., δ increases), there is a minimal change in the total profit (which appears to be convex with respect to δ); yet, importantly, the distribution of profit between the retailer and the supplier changes. As was demonstrated earlier (see Figure 3c), the profit of the supplier increases while that 13

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of the retailer mostly decreases. Figure 4 suggest that the most profit to be gained by the channel is just when the ˜ At the transition, transition from commitment to dynamic contract occurs (i.e., at h = h). however, all the gains are made by the retailer, to share the gains the retailer and supplier

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might need to negotiate an alternative contract that will redistribute the gains differently

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when strategic inventories are carried.

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Figure 4: The channel profit as a function of δ and h: the contours depict different levels of combined retails and supplier profit

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We now consider briefly the commitment contract. We have the same result as in Anand et al. (2008).

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Proposition 3. Under the commitment contract, strategic inventories are always eliminated, in which case it has the same equilibrium decisions and profits as the case under dynamic contract without strategic inventories. With the wholesale price committed at the beginning of the horizon, the retailer cannot force the supplier to lower his second-period wholesale price, hence there is no incentive for the retailer to carry any strategic inventories. Also, under this commitment contract the 14

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equilibrium outcomes are the same as in the dynamic contract when no strategic inventories are carried.

3.2

What if the supplier does not commit

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Thus far, we have assumed that the supplier can easily switch from the dynamic to the commitment contract by raising the wholesale price. In this subsection we revisit this assumption to understand its robustness. We have the following statement.

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ˆ≥ Proposition 4. Under the dynamic contract, no strategic inventories are carried if h ≥ h 2 ˆ ˜ where h ˆ ≡ (3δ 2−14δ+6)δ . That is, strategic inventories are carried whenever h < h. h 4(δ −4 δ−2) This is a fundamental result as it reveals that even if the supplier cannot credibly switch from the dynamic to the commitment contract, the nature of the deterioration will facilitate

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this transition anyway. Although the “new” threshold is somewhat higher (in terms of the

evident from Figure 5a.

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holding cost) than the previously derived threshold, they are very close to each other as is

What happens when the holding cost equals this threshold? we demonstrate the behavior

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of the wholesale price and quantities purchased in Figure 5b. One can observe that the second period wholesale price is constant at 0.5, which is exactly the single period value. This is

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because no inventories are carried and hence the supplier can price as in the static case. However, the initial wholesale price is less than static value, 0.5. Thus, as discussed before,

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the wholesale price increases over time. Since the initial wholesale price is low, the retailer orders a larger amount than in the static case: unless δ = 1 the order quantity is greater

than 0.25. All these units are sold during the first period. Note, in that case the supplier is worse off as surplus is shifted from the supplier to the retailer. For any holding cost larger ˆ the retailer holds no strategic inventories and hence cannot threat the supplier any than h, longer.

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ˆ (b) w and q estimated at h

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(a) The thresholds

Figure 5: The thresholds above which no strategic inventories are carried as well as the ˆ wholesale price and quantities estimated at the threshold, h In the previous section strategic inventories were discouraged by the supplier who switched

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from a dynamic to commitment contract. However, even if he does not switched between contract, we observe that when the degree of deterioration is sufficiently high, strategic in-

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ventories lose their appeal. Due to the low profit margin of selling those inventories, the differentiation harms the retailer more than it benefits. Recognizing that the presence of

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strategic inventories may result with the retailer having a bargaining chip in future negotiations, the supplier may have the incentive to control the degree of product deterioration to a

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level such that the retailer does not hold any strategic inventories. If, indeed, suppliers are worse off in the presence of strategic inventories, then this may suggest that suppliers can

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benefit from an increased rate of product deterioration (i.e., low δ), which, in turn, explains the high speed of versioning observed for several electronic goods and textbooks. In §4, we explore whether suppliers are worse off or better off in the presence of strategic inventories.

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4

Comparisons

In this subsection we carry out two comparisons. First, we contrast the dynamic contract with the commitment contract to evaluate the effect of strategic inventories on the various agents in the supply chain. Then, we compare our analysis with quality deterioration to

proposition and depicted graphically in Figure 6a.

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the results of Anand et al. (2008). The former comparison is summarized in the following

Proposition 5. When strategic inventories exist, the supplier is always better off, and there exist hr and h0r as well as hc and h0c such that the retailer is better off unless hr < h < h0r and

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the channel profits are higher unless hc < h < h0c . Additionally, consumer surplus is greater with strategic inventories.

The values hr , h0r , hc , and h0c are defined in the proof of the proposition. As Figure 6a reveals, the retailer is generally better off due to strategic inventories except for a small range

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of h values when the discount factor is sufficiently high. In most of this range, the gain of supplier exceeds that of the loss of the retailer and hence the supply chain is still better off,

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except for an even smaller range where the loss of the retailer is larger than the supplier’s gain.

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We now contrast our results to those of Anand et al. (2008), and both are overlaid in Figure 6b, to reveal the performance of the retailer, the supplier and the channel profit

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for the different parameter values. Anand et al. (2008) show that under the dynamic contract, carrying strategic inventories is the equilibrium decision (under the assumption

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that h < 0.25) with the following insights: the supplier is always better off when h < hA S = 0.25 (Region ABKI), where the superscript A is for Anand; the retailer is better off when

A h < hA R ≈ 0.14 (Region FHKI); and the channel profit is higher when h < hC ≈ 0.19 (Region

CEKI).4 ˜ compared to Anand et al. (2008), Corollary 2. When strategic inventories exist (h < h), 4

A Note: hr |δ=1 = hA R and hc |δ=1 = hC .

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h 'c h

' r

h 'c

hc

h SA

h 'r

0.25

h c' h c hr

B

S↑, R↓, C↓

h 'r hr

A

S↑, R↓, C↓

h CA

Zoom in

0.19

C

hc

D

E

S↑, R↓, C↑ h

h

h RA

0.14

F

~ h

G

H

S↑, R↓, C↑

~ h

S↑, R↑, C↑

S↑, R↑, C↑

~

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hr

I

(a) Our result

J

K

(b) Comparison with Anand et al. (2008)

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Figure 6: Gains and losses due to strategic inventories. Notes: S, R , and C denote the supplier, the retailer and the channel profit in our paper, respectively. The superscript A refers to the corresponding results from Anand et al. (2008). our analysis in the presence of quality deterioration suggests that: (i) the retailer is better off in Region B(h0r hr )HG (rather than being worse off as in Anand et al. (2008)) with a

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dynamic contract, and (ii) the channel is better off in Region B(h0c hc )ED (rather than being worse off as in Anand et al. (2008)) with a dynamic contract.5

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This corollary reveals that in the presence of quality deterioration, strategic inventories can benefit the retailer beyond the threshold hA R . Namely, even for higher unit holding costs

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the retailer may be better off as long as it is in Region B(h0r hr )HG. Similarly, the channel’s range for which it benefits from strategic inventories extends beyond hA C to include the range

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B(h0c hc )ED. Additionally, Anand et al. (2008) suggest that the channel is better off under the dynamic contract in the range CDJI, however, we find that no strategic inventories are

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kept in this range and, hence, the channel is indifferent to the contract type. From managerial perspective, we find that when strategic inventories are kept under

the dynamic contract, such inventories benefit the channel partners for most of the feasible range of parameter values. Importantly, it is sufficient to have some minimal level of product 5 Note, capital letters refer to vertices and lowercase letters refer to edges. Thus, the shape B(h0r hr )HG refers to the triangle BHG where the straight edge BH is replaced by the curved edge h0r hr that connects B and H. Similarly, B(h0c hc )ED is the triangle BED with the line BE replaced by h0c hc .

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deterioration to observe differences in between the results of our model and those of Anand et al. (2008). To conclude, we summarize the differences between our model and the one by Anand et al. (2008) that emerge due to the inclusion of product deterioration.

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1. We generalize the holding cost threshold to show that strategic inventories are held ˜ This result reveals that strategic inventories are less common as might be when h < h. implied by Anand et al.’s (2008) result when we consider the h − δ space. Specifically, our result reveals that holding strategic inventories requires not only the holding cost

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to be sufficiently low, but also that the deterioration of the products to be within certain values.

˜ 0 < h < h) ˜ such that the 2. Due to deterioration, there exists a range or parameters (h wholesale price as well as the price of the new product increases rather than decreases

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over time.

3. While Anand et al. (2008) suggest that the channel is worse off with strategic invento-

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ries when h is greater than about 0.19, we find that the channel can be better off even with higher values of h. Channel profits under the dynamic contract are higher almost

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everywhere when strategic inventories are carried. There exists only a sliver area— a combination of parameters where δ is larger than 0.99 and h is larger than about

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0.19—where the channel is actually worse off with strategic inventories. Elsewhere,

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˜ the channel is better off. when h < h, 4. Similarly, Anand et al. (2008) suggest that the retailer is worse off with strategic inventories when h is greater than about 0.14, whereas we find that the retailer can be better off even with higher values of h. Except for a small area—a combination of parameters where δ is larger than 0.98 and h is larger than about 0.14—the retailer’s profit under the dynamic contract is higher.

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5

Concluding remarks

The concept of strategic inventories has only recently been defined and has already attracted increasing attention to the various emerging implications. The literature thus far has assumed the inventories are perfect substitutes to new goods, thereby ignoring the fact the

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inventories are generally subject to quality deterioration. In our two-period model we explore the implications of strategic inventories when they are sold as inferior substitutes in the second period. Due to quality deterioration, we find that the range of parameter values for which the retailer keeps strategic inventories under the dynamic contract is limited. However,

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for most of this range of parameter values all agents in the supply chain are better off. Interestingly, we also find that due to this deterioration, the wholesale price under the dynamic contract may increase, rather than decrease, over time. Hence, our results complements earlier insights derived by the literature on the implications of strategic inventories. As quality deterioration can benefit the supply chain partners, an important extension

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that can be considered in the future is the extent to which the supplier and retailer may

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wish to control the degree of product deterioration. This could be achieved, for example, by adopting a modeling framework similar to Yin et al. (2010) where the supplier can increase the quality of the new good (offered in the second period) by making the appropriate

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investment. In the absence of cost associated with controlling the degree of durability, our

CE

analysis reveals that the retailer and the supplier are in disagreement with respect to the optimal level

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Our modeling framework can be extended to include other important features. For instance, our current model is deterministic. Integrating demand uncertainty as in Gerchak (2015) or in Lee (2015) can facilitate analysis of risk—faced with uncertain demand, the retailer is not in full control of the amount of inventory carried over from one period over to the next. One can also assume that some of the consumers behave strategically. Strategic consumers time their purchase to take advantage of lower future prices while giving up some of their utility (see for example Lee (2015) and discussion in Mantin and Rubin (2016)). 20

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On the one hand, such behavior may decrease the retailer’s incentive to stock strategic inventories; yet, on the other hand, our analysis reveals that in certain settings the future price may actually be higher thereby suppressing strategic consumer behavior.

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Acknowledgments This manuscript was partially funded by an NSERC Discovery Grant.

References Science 54(10) (2008) 1792-1804.

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Anand, K., Anupindi, R., Bassok, Y. Strategic inventories in vertical contracts. Management

Arya, A., Frimor, H., Mittendorf, B. Decentralized procurement in light of strategic inventories. Management Science, 61(3) (2014) 578-585.

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Arya, A., Mittendorf, B. Managing strategic inventories via manufacturer-to-consumer rebates. Management Science 59(4) (2013) 813-818.

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Bakker, M., Riezebos, J., Teunter, R. H. Review of inventory systems with deterioration

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since 2001. European Journal of Operational Research, 221(2) (2012) 275-284. Gerchak, Y. Strategic Inventory with Uncertain Demands. Working Paper, Tel-Aviv Uni-

CE

versity (2015).

Goyal, S.K., Giri, B.C. Recent trends in modeling of deteriorating inventory, European Jour-

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nal of Operational Research 134 (1) (2001) 116.

Hartwig, R., Inderfurth, K., Sadrieh, A., Voigt, G. Strategic inventory and supply chain behavior. Production and Operations Management 24 (8) (2015) 1329-1345.

Lee, C. H., Choi, T.-M., Cheng, T. C. E. Selling to strategic and loss-averse consumers: Stocking, procurement, and product design policies. Naval Research Logistics 62 (6) (2015) 435-453. 21

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Mantin, B., Rubin, E. Fare Prediction Websites and Transaction Prices: Empirical Evidence from the Airline Industry. Marketing Science forthcoming (2016). Mussa, M., Rosen, S. Monopoly and product quality. Journal of Economic Theory 18(2) (1978) 301-317.

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Yin, S., Ray, S., Gurnani, H., Animesh, A. Durable products with multiple used goods markets: Product upgrade and retail pricing implications. Marketing Science 29(3) (2010) 540-560.

Whitin, T.M. The Theory of Inventory Management, Princeton University Press, Princeton,

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CE

PT

ED

M

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NJ, USA, 1953.

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Appendices Complete expressions: dynamic contract ˜ when h ≤ h

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Appendix A

3 2 2 +8δh+4h+8 , 2δ −11δ δ+7δ−2hδ ) 3 −8δ 2 +8δ+16 2(δ 3 −8δ 2 +8δ+16) δ 4 −3δ 3 +8δ 2 +20δ−4hδ 2 −16δh−16h 2δ 3 −11δ 2 +7δ−2hδ 2 +8δh+4h+8 ∗ ∗ ∗ Purchase quantities (q1 + I , q2n ): ( , ) 4δ(δ 3 −8δ 2 +8δ+16) 2(δ 3 −8δ 2 +8δ+16) 3 2 2 −6δ+4hδ −16δh−8h Inventory I ∗ : −3δ +14δ 2δ(δ 3 −8δ 2 +8δ+16) ∗ ∗ Sales quantities (q1∗ , q2n , q2o ): δ 3 +3δ 2 −20δ−12δh+16h+32 2δ 3 −11δ 2 +7δ−2hδ 2 +8δh+4h+8 −3δ 3 +14δ 2 −6δ+4hδ 2 −16δh−8h , , ) ( 4(δ 3 −8δ 2 +8δ+16) 2(δ 3 −8δ 2 +8δ+16) 2δ(δ 3 −8δ 2 +8δ+16) Retail prices (p∗1 , p∗2n , p∗2o ): 3 2 +52δ+12δh−16h+32 3δ 3 −19δ 2 +15δ−2hδ 2 +8δh+4h+24 −2δ 3 −5δ 2 +30δ+2hδ 3 −12hδ 2 +12δh+8h ( 3δ −35δ , , ) 4(δ 3 −8δ 2 +8δ+16) 2(δ 3 −8δ 2 +8δ+16) 2(δ 3 −8δ 2 +8δ+16) ∗ Retailer’s profits ΠR : 1 [−7δ 7 + 110δ 6 − 519δ 5 + 944δ 4 − 308δ 3 − 880δ 2 + 1280δ + h(24δ 6 − 328δ 5 16δ(δ 3 −8δ 2 +8δ+16)2 4 3 2 2 5 4 3 2 3 −19δ 2 +36δ+12hδ−16h

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Wholesale prices (w1∗ , w2∗ ): ( δ

Supplier’s profits Π∗S :

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+1240δ − 1184δ − 1632δ + 1408δ) − h (16δ + 192δ − 560δ + 128δ + 1216δ + 256)] 17δ 4 −62δ 3 +65δ 2 +16δ−32hδ 3 +16h2 δ 2 +56hδ 2 −40δh+16h2 8δ(δ 3 −8δ 2 +8δ+16)

1 [31δ 7 32δ(δ 3 −8δ 2 +8δ+16)2 4 3

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Consumer surplus CS:

− 354δ 6 + 1267δ 5 − 1504δ 4 + 28δ 3 + 1072δ 2 − 1280δ − h(80δ 6 − 840δ 5

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+2440δ − 1120hδ − 2464δ 2 + 1152δ) + h2 (48δ 5 − 448δ 4 + 944δ 3 + 384δ 2 − 1088δ − 256)]

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CE

Table 2: Complete expressions of equilibrium decisions, profits and consumer surplus under ˜ the dynamic contract when h ≤ h

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Appendix B

Proofs

Proof of Proposition 1. To establish the equilibrium decisions, we use backward induction. In period 2, the retailer sets (p2n , p2o ) to maximize his profit in period 2. This is

Π2R = (p2n − w2 )q2n + p2o q2o .

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equivalent to choosing (q2n , q2o ) to maximize the profit in period 2

(1)

subject to q2o ≤ I, and q2n , q2o ≥ 0,

Where p2n = 1 − q2n − δq2o and p2o = δ(1 − q2n − q2o ). Π2R is jointly concave in (q2n , q2o ).

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∗ ∗ ∗ Thus, there is a unique global optimal (q2n , q2o ). Solving first for q2n for a given q2o yields ∗ ∗ under the constraint 0 ≤ q2o ≤ I, we have two = [(1 − w2 − 2δq2o )/2]+ . Solving for q2o q2n

cases: (a) I ≤ 1/2 and (b) I ≥ 1/2.

∗ depends on the value of w2 and I. There are two (a) When I ≤ 1/2, the optimal q2o

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cases again. (i) For I ≤ (1 − w2 )/(2δ), or equivalently, w2 ≤ 1 − 2Iδ, because q2o ≤ I ≤ ∗ (1 − w2 )/(2δ), we always have q2n = (1 − w2 − 2δq2o )/2. Substituting it into Π2R yields a

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concave function of q2o with the unconstrained global maximizer q2o = w2 /(2(1 − δ)) ≥ 0. ∗ = min(I, w2 /(2(1 − δ))). (ii) For I ≥ (1 − w2 )/(2δ), or equivalently, w2 ≥ 1 − 2Iδ, Hence, q2o

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∗ = (1 − w2 − 2δq2o )/2 the retailer has two choices for q2o : (1) If q2o ≤ (1 − w2 )/(2δ), then q2n ∗ and since w2 ≥ 1 − 2Iδ ≥ 1 − δ, q2o = min((1 − w2 )/(2δ), w2 /(2(1 − δ)) = (1 − w2 )/(2δ)). (2)

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∗ If (1−w2 )/(2δ) ≤ q2o ≤ I, q2n = 0, which leads to the retailer’s period 2 profit function being ∗ concave in q2o with an unconstrained optimal q2o = 1/2 ≥ I. Hence q2o = I. Because choice

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∗ (1) is a boundary solution of choice (2), we have q2o = I when w2 ≥ 1 − 2Iδ. Combining the

analysis in the two sub-cases (i) and (ii) of (a), we conclude:    ∗  ((1 − w2 − 2δq2o )/2, w2 /(2(1 − δ))) if w2 ≤ 2I(1 − δ)     ∗ ∗ ∗ (q2n , q2o ) = ((1 − w2 − 2δq2o )/2, I) if 2I(1 − δ) ≤ w2 ≤ 1 − 2Iδ       (0, I) if w2 ≥ 1 − 2Iδ 24

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(b) When I ≥ 1/2, we carry out the analysis as in case (a) above, which results with:

∗ ∗ (q2n , q2o )=

   ∗ ((1 − w2 − 2δq2o )/2, w2 /(2(1 − δ))) if w2 ≤ 1 − δ   (0, 1/2)

otherwise

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We now proceed to the supplier’s optimal wholesale price, w2 , which maximizes his profit ∗ in the second period, Π2S = w2 q2n . We have two cases with respect to I.

(a) I ≤ 1/2. Consider three choices as follows: (1) If w2 ≤ 2I(1 − δ), Π2S = w2 (1 − ∗ ∗ w2 − 2δq2o )/2, where q2o = w2 /(2(1 − δ)). The profit function is concave in w2 with the

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unconstrained optimal w2 = (1 − δ)/2. Hence w2∗ = 2I(1 − δ) if I ≤ 1/4, and w2 = (1 − δ)/2 ∗ if I ≥ 1/4. (2) If 2I(1 − δ) ≤ w2 ≤ 1 − 2Iδ, Π2S = w2 (1 − w2 − 2δq2o )/2, which is concave in

w2 with the unconstrained optimal w2 = (1 − 2Iδ)/2. Thus, w2∗ = 2I(1 − δ) if I ≥ 1/(4 − 2δ), ∗ = 0. and w2∗ = (1 − 2Iδ)/2 if I ≤ 1/(4 − 2δ). (3) If w2 ≥ 1 − 2Iδ, Π2S = 0 because q2n

Comparing (1) and (2), we have w2∗ = (1 − 2Iδ)/2 if I ≤ I, and w2∗ = (1 − δ)/2 if I ≥ I,

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where I makes the supplier indifferent between choice (1) and (2) at w2∗ = (1 − 2Iδ)/2 and √ w2∗ = (1 − δ)/2, respectively, and I = (1 − 1 − δ)/(2δ) ∈ (1/4, 1/(4 − 2δ)). (b) I ≥ 1/2. Based on the retailer’s best reaction functions in period 2, it is obvious

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that the supplier will not choose w2 ≥ 1 − δ because he will get a zero profit. Hence, the ∗ ∗ )/2, supplier will set a w2 so that w2 ≤ 1 − δ, for which we have q2n = (1 − w2 − 2δq2o

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∗ where q2o = w2 /(2(1 − δ)). Therefore, Π2S is concave in w2 with the constrained optimal

w2∗ = (1 − δ)/2.

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Combining cases (a) and (b), we conclude that

∗ ∗ (w2∗ , q2n , q2o )=

   ∗ ((1 − 2Iδ)/2, I, (1 − w2 − 2δq2o )/2) if I ≤ I   ((1 − δ)/2, 1/4, 1/4)

otherwise

Accordingly, we can calculate p∗2n and p∗2o .

In period 1, the retailer sets (p1 , I) to maximize his total profit in periods 1 and 2. This 25

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is equivalent to choosing (q1 , I) to maximize ∗ ∗ ΠR = (p∗2n − w2∗ )q2n + p∗2o q2o + (1 − q1 − w1 )q1 − (h + w1 )I.

(2)

subject to q1 , I ≥ 0,

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The retailer has two choices for I: (1) I ≤ I, substituting the corresponding best reaction functions in period 2 into ΠR yields a jointly concave function in (q1 , I). Solving first for q1∗ for a given I results with q1∗ = (1 − w1 )/2, which is always positive. Solving for I ∗ under the constraint 0 ≤ I ≤ I, the unconstrained optimal I = (3δ − 4h − 4w1 )/(2δ(4 − δ)). Hence, depending on where the     0    

if w1 ≥ 3δ/4 − h

√ (3δ − 4h − 4w )/(2δ(4 − δ)) if δ − 1 − h + (4 − δ) 1 − δ/4 ≤ w1 ≤ 3δ/4 − h 1     √   I if w1 ≤ δ − 1 − h + (4 − δ) 1 − δ/4

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I∗ =

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unconstrained optimal I is located along the segment [0, I], we have

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(2) I ≥ I, substituting the corresponding best reaction functions in period 2 into ΠR yields a function concave in q1 with the unconstrained optimal q1 = (1 − w1 )/2 and

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monotonously decreasing in I. Hence, q1∗ = (1 − w1 )/2 and I ∗ = I, which is a boundary solution of case 1.

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Lastly, we consider the supplier’s wholesale price w1 to maximize his total profit in ∗ periods 1 and 2, ΠS = w2∗ q2n + w1 (q1∗ + I ∗ ). Based on the analysis above, the supplier has

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three choices for w1 :

(1) If w1 ≥ 3δ/4 − h, substituting the corresponding best decision variables in the

subsequent stages into ΠS gives a concave function in w1 with the unconstrained optimal w1 = 1/2. Hence, if 1/2 ≥ 3δ/4 − h, or equivalently if h ≥ 3δ/4 − 1/2, w1∗ = 1/2; otherwise, w1∗ = 3δ/4 − h, which is on the boundary of choice (2). √ (2) If δ − 1 − h + (4 − δ) 1 − δ/4 ≤ w1 ≤ 3δ/4 − h, substituting the corresponding 26

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best decision variables in the subsequent stages into ΠS gives a concave function in w1 with the unconstrained optimal w1 =

δ 3 −19δ 2 +36δ+12hδ−16h . 2(δ 3 −8δ 2 +8δ+16)

We can prove that this unconstrained

optimal value of w1 is greater than or equal to the lower bound, hence, when it is lower than or equal to the upper bound 3δ/4 − h, or equivalently h ≤ (δ(3δ 2 − 14δ + 6))/(4(δ 2 − 4δ − 2)), δ 3 −19δ 2 +36δ+12hδ−16h . 2(δ 3 −8δ 2 +8δ+16)

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Otherwise, w1∗ = 3δ/4 − h, which is on the boundary of choice (1). √ (3) If w1 ≤ δ − 1 − h + (4 − δ) 1 − δ/4, substituting the corresponding best decision vari-

w1∗ =

ables in the subsequent stages into ΠS gives a concave function in w1 with the unconstrained optimal w1 =

√ 1+δ− 1−δ . 2δ

We can prove that this unconstrained optimal value of w1 is greater √ √ than the upper bound δ − 1 − h + (4 − δ) 1 − δ/4, so w1∗ = δ − 1 − h + (4 − δ) 1 − δ/4,

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which is on the boundary of choice (2).

Combining (1), (2) and (3), we conclude that when δ ≤ 0.477, w1∗ = 1/2; when δ ≥ 0.477, then if h ≥ (δ(3δ 2 − 14δ + 6))/(4(δ 2 − 4δ − 2)), w1∗ = 1/2; otherwise, we need to compare the supplier’s profit, ΠS , at w1 = 1/2 and at w1 =

We find that when

δ 3 −19δ 2 +36δ+12hδ−16h . 2(δ 3 −8δ 2 +8δ+16)

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˜ w∗ = 1/2; otherwise, w∗ = h ≥ h, 1 1

δ 3 −19δ 2 +36δ+12hδ−16h . 2(δ 3 −8δ 2 +8δ+16)

Proof of Theorem 1. When carrying strategic inventories is preferred under the dynamic

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˜ we compare the equilibrium decision variables contract, that is, when δ ≥ δ˜ and h ≤ h, between the first and the second period. −3δ 3 +(3+4h)δ 2 +(22−4h)δ−24h−16 2(8δ−8δ 2 +δ 3 +16)

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(i) Solving w1 − w2 =

˜0 = = 0 reveals a threshold of h, h

˜ such that w1 ≥ w2 if h ≤ h ˜ 0 and w1 < w2 otherwise. < h,

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3δ 3 −3δ 2 −22δ+16 4(δ 2 −δ−6)

(ii) Since I > 0, q1 ≥ q2n (see the proof of (iv) below), we can derive Q1 = q1 + I > q2n . (iii) Solving p1 − p2n =

−3δ 3 +(3+4h)δ 2 +(22−4h)δ−24h−16 4(8δ−8δ 2 +δ 3 +16)

˜ 0 , such = 0 reveals a threshold of h, h

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˜ 0 and p1 < p2n otherwise. that p1 ≥ p2n if h ≤ h ˜ 1], h ∈ [0, h], ˜ q1 − q2n = (iv) ∀δ ∈ [δ,

−3δ 3 +25δ 2 −34δ+16+4h(δ 2 −7δ+2) 4(8δ−8δ 2 +δ 3 +16)

˜ Proof of Proposition 2. For all δ ∈ [0, 1] and h ∈ [0, h]), (i)

dw1 dδ

dp1 dδ

=

=

11δ 4 +(−24h−56)δ 3 +(184+144h)δ 2 +(−256h−608)δ+576+320h 2(8δ+δ 3 −8δ 2 +16)2

11δ 4 +(−24h−56)δ 3 +(184+144h)δ 2 +(−256h−608)δ+576+320h 4(8δ+δ 3 −8δ 2 +16)2

27

> 0.

> 0.

≥ 0.

ACCEPTED MANUSCRIPT

Jiang and Mantin: Strategic Inventories with Quality Deterioration (−5+2h)δ 4 +(−16h+18)δ 3 +(40+36h)δ 2 −224δ+96h+48 (8δ+δ 3 −8δ 2 +16)2

=

(−5+2h)δ 4 +(−16h+18)δ 3 +(40+36h)δ 2 −224δ+96h+48 2(8δ+δ 3 −8δ 2 +16)2

dp2n dδ

=

dp2o dδ

=

(ii)

dQ1 dδ

dq2o dδ

=

dq1 dδ

=

< 0. < 0.

(21−4h)δ 4 +(−92+8h)δ 3 +(72h+104)δ 2 +(−160−256h)δ+480+128h 2(8δ+δ 3 −8δ 2 +16)2

=

> 0.

−5δ 6 +8δ 5 h+(28+16h)δ 4 +(224−192h)δ 3 +(−32−320h)δ 2 +256δh+256h 4(8δ+δ 3 −8δ 2 +16)2 δ 2

3δ 6 +(−8h−28)δ 5 +(106+80h)δ 4 +(−192−224h)δ 3 +272δ 2 +128δh+128h 2(8δ+δ 3 −8δ 2 +16)2 δ 2

−11δ 4 +(24h+56)δ 3 +(−184−144h)δ 2 +(256h+608)δ−576−320h 4(8δ+δ 3 −8δ 2 +16)2

(−5+2h)δ 4 +(−16h+18)δ 3 +(40+36h)δ 2 −224δ+96h+48

dq2n dδ

=

(iii)

dΠS dδ

2(8δ+δ 3 −8δ 2 +16)2

=

A 4(8δ+δ 3 −8δ 2 +16)3 δ 2

> 0.

> 0.

CR IP T

dw2 dδ

< 0.

< 0.

> 0, where A = 3δ 9 + (−8h + 3)δ 8 + (144 − 12h)δ 7 + (−480h −

848 + 96h2 )δ 6 + (2992h − 64h2 − 264)δ 5 + (2880 + 960h − 1728h2 )δ 4 + (−4352 − 9088h − d2 ΠR dδ 2

=

B 8(8δ+δ 3 −8δ 2 +16)4 δ 3

AN US

128h2 )δ 3 + (5760 + 5888h + 3584h2 )δ 2 − 3072δh2 − 2048h2 .

> 0, where B = (34+24h)δ 1 2+(−641−96h2 −648h)δ 1 1+(7184+

8112h + 2304h2 )δ 1 0 + (−50224 − 65024h − 22880h2 )δ 9 + (228352 + 328000h + 121984h2 )δ 8 + (−718912 − 986368h − 356992h2 )δ 7 + (1462784 + 1738752h + 493568h2 )δ 6 + (−1687552h −

M

192512h2 −1468928)δ 5 +(157696h+131072h2 +505856)δ 4 +(83968+376832h−86016h2 )δ 3 − 65536δ 2 h2 + 262144δh2 + 131072h2 , and solving dΠR dδ

< 0 when δ < δ 0 and

dΠR dδ

= 0 reveals a value of δ 0 ∈ [0, 1] such

≥ 0 otherwise.

ED

that

dΠR dδ

Proof of Proposition 3. Under the commitment contract, we first solve for the retailer’s

PT

optimal decisions to a given w1 and w2 . The retailer sets (q1 , I, q2n , q2o ) to maximize his total

CE

profit in periods 1 and 2.

AC

ΠR = (p2n − w2 )q2n + p2o q2o + (p1 − w1 )q1 − (h + w1 )I.

(3)

subject to q1 , I, q2n , q2o ≥ 0, I ≥ q2o ,

where p1 = 1 − q1 , p2n = 1 − q2n − δq2o , p2o = δ(1 − q2n − q2o ). ΠR is monotonously decreasing

∗ in I, so I ∗ = 0, and since q2o ≤ I, q2o = 0. Substituting the optimal I and q2o into ΠR

yields a jointly concave function in (q1 , q2n ) with the unique global optimal q1∗ = (1 − w1)/2, ∗ q2n = (1 − w2)/2. With the retailer’s optimal decisions, we next solve for the supplier’s

28

ACCEPTED MANUSCRIPT

Jiang and Mantin: Strategic Inventories with Quality Deterioration

optimal decisions by maximizing his total profit in periods 1 and 2 ∗ + w1 (q1∗ + I ∗ ) = ΠS = w2 q2n

w2 (1 − w2 ) w1 (1 − w1 ) + . 2 2

w2∗ = 1/2.

CR IP T

ΠS is a jointly concave function in (w1 , w2 ) with the unique global optimal w1∗ = 1/2,

Proof of Proposition 4. Follows immediately from equating I ∗ to zero.

Proof of Proposition 5. When strategic inventories exist, we prove that the supplier can

hr ≡

gain

a

higher

profit,

√ −176δ+204δ 2 +148δ 3 −155δ 4 +41δ 5 −3δ 6 − M 4(16+76δ+8δ 2 −35δ 3 +12δ 4 −δ 5 )

the

retailer

< h < h0r ≡

that the channel profits are generally higher unless h < h0c ≡

√ −16δ+60δ 2 +84δ 3 +31δ 4 −37δ 5 +5δ 6 + N , 2 3 4 5 4(48+92δ+24δ −17δ −4δ +δ )

is

generally

better

off

unless

√ −176δ+204δ 2 +148δ 3 −155δ 4 +41δ 5 −3δ 6 + M , and 4(16+76δ+8δ 2 −35δ 3 +12δ 4 −δ 5 ) √ 2 +84δ 3 +31δ 4 −37δ 5 +5δ 6 − N < hc ≡ −16δ+60δ 4(48+92δ+24δ 2 −17δ 3 −4δ 4 +δ 5 )

AN US

always

where M = −12288δ −5120δ 2 +26624δ 3 +3584δ 4 −

19072δ 5 +4608δ 6 +4144δ 7 −2836δ 8 +728δ 9 −87δ 10 +4δ 11 , N = −12288δ −19456δ 2 +16384δ 3 +

M

36864δ 4 − 7552δ 5 − 23424δ 6 + 7344δ 7 + 5300δ 8 − 3672δ 9 + 881δ 10 − 96δ 11 + 4δ 12 , and that consumer surplus is greater with strategic inventories.

ED

Denote ∆S as the difference of the supplier’s optimal profit between when carrying strategic inventories is preferred and when it is not. ∆S =

A , 8δ(8δ−8δ 2 +δ 3 +16)

where A =

PT

˜ 1], h ∈ [0, h], ˜ 15δ 4 − 2(16h + 23)δ 3 + (16h2 + 56h + 49)δ 2 − 8(2 + 5h)δ + 16h2 . ∀δ ∈ [δ,

CE

8δ(8δ − 8δ 2 + δ 3 + 16) ≥ 0, A ≥ 0. Denote ∆R as the difference of the retailer’s optimal profit between when carrying strategic inventories is preferred and when it is not. ∆R =

B , 16δ(8δ−8δ 2 +δ 3 +16)2

where B =

AC

−9δ 7 + 2(12h + 71)δ 6 − (328h + 679 + 16h2 )δ 5 + 8(142 + 155h + 24h2 )δ 4 − 4(296h − 19 +

˜ 1], h ∈ [0, h], ˜ 140h2 )δ 3 − 16(102h − 8h2 + 87)δ 2 + 64(22h + 12 + 19h2 )δ + 256h2 . ∀δ ∈ [δ, 16δ(8δ − 8δ 2 + δ 3 + 16)2 ≥ 0. Also, when hr < h < h0r , B < 0, otherwise B ≥ 0. Denote ∆C as the difference of the optimal channel profit between when carrying strategic inventories is preferred and when it is not. ∆C =

C , 16δ(8δ−8δ 2 +δ 3 +16)2

where C = 21δ 7 −10(4h+

19)δ 6 + (296h + 395 + 16h2 )δ 5 − 8(31h − 8 + 8h2 )δ 4 − 4(68h2 + 168h + 89)δ 3 − 16(5 + 30h − 29

ACCEPTED MANUSCRIPT

Jiang and Mantin: Strategic Inventories with Quality Deterioration

˜ 1], h ∈ [0, h], ˜ 16δ(8δ − 8δ 2 + δ 3 + 16)2 ≥ 0. 24h2 )δ 2 + 64(23h2 + 4 + 2h)δ + 768h2 . ∀δ ∈ [δ, Also, when hc < h < h0c , C < 0, otherwise C ≥ 0. Denote ∆CS as the difference of consumer surplus between when carrying strategic inventories is preferred and when it is not. D , 32δ(8δ−8δ 2 +δ 3 +16)2

where D = −33δ 7 + 2(40h + 193)δ 6 − (840h + 1427 + 48h2 )δ 5 +

CR IP T

∆CS =

8(212 + 305h + 56h2 )δ 4 − 4(280h − 89 + 236h2 )δ 3 − 16(154h + 24h2 + 99)δ 2 + 64(18h + 12 +

AC

CE

PT

ED

M

AN US

˜ 1], h ∈ [0, h], ˜ 32δ(8δ − 8δ 2 + δ 3 + 16)2 ≥ 0, D ≥ 0. 17h2 )δ + 256h2 . ∀δ ∈ [δ,

30