Entry timing in a secondary market: When to trade?

ARTICLE IN PRESS Int. J. Production Economics 124 (2010) 62–74

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Entry timing in a secondary market: When to trade? Tingting Yan , Kevin J. Dooley Department of Supply Chain Management, W.P. Carey School of Management, Arizona State University, Campus Box 874706, Tempe, AZ 85287-4706, USA

a r t i c l e in f o

a b s t r a c t

Article history: Received 2 February 2009 Accepted 24 September 2009 Available online 27 October 2009

In volatile, long-lead time and short selling-season markets, a secondary market enables buyers to update their inventory during the selling season. The decision of when to update involves complicated trade-offs between forecast accuracy, expected lost sales, and average purchasing cost. We use a twostage inventory replenishment model to identify the optimal trading time across different scenarios. Across most scenarios the optimal trading time is around the midpoint of the season and is sensitive to the expected profit margin and demand forecast errors. We discuss the impacts of the timing decision on upstream suppliers’ sales and channel performance in terms of sales revenue and supply-demand mismatch costs. Specifically, secondary market trading always reduces upstream suppliers’ sales no matter when trading occurs, but trading at an optimal time can maximize a secondary market’s profit gain over the no-trade scenario through reducing supply-demand mismatch. Published by Elsevier B.V.

Keywords: Supply chain management Electronic surplus market Information learning Entry timing Two-stage inventory replenishment

1. Introduction Secondary markets, such as Converge.com and Virtual Chip Exchange, enable firms to update their inventory of products during a short selling season, thus allowing them to adapt to volatile demand and/or inaccurate demand forecasts (Lee and Whang, 2002). Such secondary markets are especially important for products with long production lead-times and short sellingseasons, such as electronic components and computer products (Lee and Whang, 2002; Dong and Rudi, 2004; Dong and Durbin, 2005; Zou, 2005). Whereas the primary market involves transactions between producers and buyers, the secondary market involves transactions between buyers and other buyers (sometimes facilitated by the producer). An effective secondary market connects firms with a shortage of certain components (buyers) with firms that find themselves with surplus inventory (sellers) during the selling season. When a firm only makes a single purchase, the order quantity which fulfills the demand for the entire selling season must be determined a priori. Because products with short selling seasons tend to have more unpredictable demand (Fisher et al., 1997; Lee and Whang, 2002; Tang et al., 2004; Chew et al., 2009), the demand estimate, made long before the selling season begins, may be highly inaccurate and result in exceedingly large under or over stock. The value of a secondary market is that it allows a firm to purchase enough products for part of the selling season, and

 Corresponding author. Tel.: + 1 831 295 4120.

E-mail addresses: [email protected] (T. Yan). [email protected] (K.J. Dooley). 0925-5273/$ - see front matter Published by Elsevier B.V. doi:10.1016/j.ijpe.2009.09.007

then adjust the inventory level through either making additional orders or selling on the secondary market based on revised, more accurate estimates of product demand. In addition to the benefit of being able to ‘‘learn’’ from current demand, purchases in a secondary market will cost less in situations where there is a decreasing spot price, i.e. purchase cost is less than original wholesale price. The decision of when to trade involves complex tradeoffs, involving both the amount of demand information one can accumulate and trading price dynamics. If the buyer trades relatively later in the selling season and demand between purchases are correlated, its updated estimate of demand will be more accurate and it will minimize mismatch costs (either overstock or stock-out cost) in the subsequent period. The disadvantage of late trading is that it requires a good initial demand forecast, without which the first purchase supplydemand mismatch cost will be high. An early trader, on the other hand, reduces its risk of initial losses due to poor demand information, but makes forecasts of future demand with relatively poorer data than the late trader. If spot prices in the secondary market are continuously decreasing because the item has zero or little salvage value at the end of the season, the late trader is benefited; if spot prices are increasing because the item is more valuable at the end of the season because of its rarity, then the early trader is benefited. Our study aims to examine these tradeoffs and identify the optimal time to trade for different types of products and markets. While Lee and Whang (2002), Dong and Rudi (2004), Dong and Durbin, 2005, and Shi et al. (2004) have examined the value of secondary market in terms of increasing channel sales and upstream suppliers’ sales, and reducing channel excessive

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inventory, this study is the first to examine the entry timing decision in secondary market using a continuous time inventory management model. In order to simplify the study of this complex issue, we examine a basic two-period model where buyers make a single purchase or sell in the secondary market, thus splitting the season into two demand periods, and we assume that demand in the two periods are correlated, which is a commonly used assumption in two-stage inventory models (Lee and Whang 2002; Dong and Rudi, 2004; Shi et al., 2004; Milner and Kouvelis, 2002; Barnes-Schuster et al., 2002). Lee and Whang (2002) we concentrate on scenarios with a decreasing spot price, i.e. item has zero or little salvage value, as is typical in many consumer goods such as electronics, food and fashion products (Zou, 2005; Helo, 2004; Frazier, 1986; Hammond, 1990; Fisher et al., 1994). Secondary markets for after-season sale of unsold products, such as outlets, and those for used durable products, such as Ebay, are not within the scope of this study. Our study contributes to the secondary market literature by characterizing what are the optimal trading times during the selling season on a secondary market given various market and product characteristics. Furthermore it is shown that the timing decision significantly affects the value of secondary market in terms of higher expected profit for traders and smaller supply-demand mismatch. Relative to the literature concerning electronic markets (Johnson and Seungjin, 2002; Jishnu and Mahadevan, 2006, et al.), our results suggest that products that differ in forecast accuracy and retail margin should have different optimal bidding mechanisms in terms of bidding time. Thus literature on price quotation and electronic market design (e.g., Yang and Max, 2007, et al.) could be extended by taking products’ heterogeneity into consideration. In terms of model generalizability to other contexts, our model can be extended to inventory and supplier portfolio management research (Qizhi and Robert, 2006, et al.). The entry timing in our model could be treated as the portfolio weight in a purchasing channel selection model including both electronic markets purchases/sales and traditional purchases from upstream suppliers. The optimal time or channel portfolio weight is determined to maximize the total expected profit. Thus the secondary market could be treated as a secondary supplier which provides lower-price products during the selling season. The timing decision becomes a purchasing portfolio decision. The paper is organized as follows. We will present an overview of relevant literature and then present a model that addresses some of the limitations of existing work in the area. A two-period inventory model is presented and details are given concerning its implementation as a simulation model. Experimental design and results are given, and the most significant insights are provided. Finally, we discuss some general principles which underlie these insights, and discuss how our results might change given variations in the baseline model.

2. Literature review A variety of mechanisms have been suggested for dealing with the inherent uncertainty associated with markets which have long lead times, short selling seasons, and volatile demand (Kanchanasuntorn and Techanitisawad, 2006; Li et al., 2007). Some studies suggest multi-supplier purchasing as an adaptive mechanism. For example, Yan et al. (2003) found that if buying firms can dynamically shift between costly, rapid response suppliers and inexpensive, slow response suppliers, they can reduce supply-demand gap and improve performance. Others suggest inventory management methods; for example, Ferguson and Koenigsberg (2007) find that carrying over unsold products to

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the next season as lower-quality products can also improve overall profit if firms take into account the effect of first period decisions on second period profits. However, for products which are only available from one source during the season and whose end-of-season value is zero, surplus market trading may be a better way to hedge risks and increase profits because buyers cannot buy from the original supplier during the season due to the long lead time and products are worthless when carried over to the next period. Studies of surplus markets are closely related to the vast literature on inventory/capacity pooling and lateral transshipments, which are used as ways to redistribute inventory or capacity within a distribution network. Dong and Durbin (2005) have provided a detailed review of these studies. For centralized distribution systems, research has focused on finding the optimal/ heuristic inventory/transshipment policies for distribution networks of various structures (e.g. Eppen and Schrage, 1981; ¨ Robinson, 1990; Herer et al., 2006; Lee, 1987; Axsater, 1990, 2003, and references therein). For decentralized distribution systems, research has focused on finding the best incentive alignment mechanism to enable inventory/transshipment policies to be conducted in a cooperative way to realize system optima (e.g. Rudi et al., 2001; Anupindi et al., 2001). A few studies have compared local and global optimization strategies in two-stage supply chains for production planning (Saharidis et al., 2006, 2009; Gnoni et al., 2003; Chen and Chen, 2005). They conclude that coordinated policies are at least as good as decentralized policies in reducing supply chain costs. In the economics and marketing/production interface literature, researchers have examined secondary market strategies for durable products. A recent example is Desai et al. (2007). One key difference between our study and this existing stream of research is that in our study, the products on the secondary market are treated as the same as new products, instead of being considered used or deteriorated; we are not considering cases where the spot price is lower because quality has degraded. Another difference is we do not assume internal competition between end consumers and manufacturers, which is a common assumption in durable products literature. For example, Virtual Chip Exchange’s membership enrollment website (http://www.virtualchip.com/vc?/CO/ JOIN_PREP), states ‘‘Membership is free and restricted to Original Equipment Manufacturers, Contract Manufacturers, Franchised Distributors and Component Manufacturers’’; note the list does not include end consumers. Several studies are particularly relevant to ours. Lee and Whang (2002) consider a two-period inventory replenishment model with a single manufacturer and many resellers in a decentralized system. Starting with the assumption that each buyer purchases from the same supplier at the beginning of period one and trades amongst all the traders at the beginning of period two, they derive optimal ordering strategies. They show that if the first period order quantity is within a certain range, which includes the standard newsvendor ordering quantity for the whole period, then channel sales could be increased and excessive inventory and stock out could be reduced simultaneously. Furthermore, if compared with the unit overage cost the unit underage cost is small enough, manufacturers’ sales could be increased at the same time. Dong and Rudi (2004) studied the impact of a surplus market in an integrated system in which buyers belong to the same firm. They show that the supplier often benefits from transshipment because it makes buyers less sensitive to the wholesale price, considering both exogenous and endogenous wholesale prices. Dong and Durbin (2005) extend the surplus market model from final products to components by developing a one-period model in which a monopolist supplier sells components to a number of

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independent manufacturers who are uncertain about demand for final goods. They derive conditions regarding different types of demand uncertainty that determine whether a surplus market will increase or decrease supplier profits. The surplus market in their study may decrease overall efficiency of the supply chain, since benefit of better allocation of components may be outweighed by an increased double-marginalization effect. Finally, Zou (2005) demonstrates that the secondary market indeed plays an informational role to aggregate and communicate demand information to market participants. All these studies assume a pre-determined arbitrary trading time and do not study the impact of the trading time on the value of the secondary market. One of the few studies which addressed the timing issue is Shi et al. (2004), who show that the value of trading options in a secondary market is an inverse U-shape function of trading time, i.e., the optimal trading time is somewhere towards the middle of the selling season. They derive optimal replenishment policies for the buyers and discuss how options enhance information flow, encourage risk sharing, and improve supply chain efficiency. However, there are two major differences between Shi et al.’s study and ours. First, their model does not consider the effect of different market and product factors on the optimal trading time. The U-shape function for the optimal trading time is only shown for a single scenario. Second, options are traded in their model as opposed to products, which is a less risky context to the buyer. Our study is most closely related with Milner and Kouvelis (2002). They study the value of information, production flexibility and supplier flexibility for a product for which an initial and a subsequent order may be placed. They setup three models: static (both the time and volume of the second purchases from an upstream supplier are determined a priori), partially dynamic (only time is fixed) and fully dynamic (both time and volume are not restrained). The optimal policy is determined by minimizing total cost. They observed that as the difference between high and low demand cases increase, the value of information increases, though for long lead time, production flexibility is required to take advantage of the updated information. Also greater uncertainty within each prior distribution leads to greater value of information relative to value of production flexibility. However, the greater uncertainty around the mean demand leads to lower value of information because this uncertainty cannot be resolved through observation. Our model is similar to their partial dynamic model but is different in the following ways. First, they do not study the impacts of different factors on the optimal trading time, which are studied in this research. In a secondary market trading context, it is important to know when a firm should enter the market given different levels of retail margin and prior information quality. Second, decision makers could only buy after the season begins in Milner and Kouvelis’ model, but can either buy or sell in our model. Allowing both buying and selling activates is a realistic assumption for the secondary market context. Third, unit purchasing cost is not related to the in-season purchasing time in their model, while the in-season purchasing/selling price is a function of the trading time in our model. Thus the trading time not only affects information quality but also determines spot price in our model. For short-life cycle products, it is reasonable to assume that spot price decreases with time due to reduced probabilities of selling such products when approaching the end of the season. Finally, we do not consider inventory holding cost, while Milner and Kouvelis (2002) include it as part of their model. It is realistic to assume that holding costs, compared with purchasing costs, are negligible for electronic components with a short life cycle and small volume.

3. The model 3.1. Problem description We consider a two-period inventory replenishment model with the time dividing the two periods as the decision variable. Even though most real scenarios may not constrain traders to just a single buy/sell during the season, this simplification allows us to model the problem reasonably; and compare our results to others who have used a two-period inventory model (Lee and Whang, 2002; Milner and Kouvelis, 2002; Shi et al., 2004). Finally, it is not unreasonable to believe that buyers or sellers would attempt to minimize the actual number of trades, as each trade yields transaction costs which may not be trivial (Jishnu and Mahadevan 2006, et al). Thus constraining traders to a single buy/sell is both economically reasonable and is a necessary first step towards understanding the multi-trading scenario. First we shall provide a narrative overview of how we determine the optimal trading time for a given scenario. In our model the trading time is determined before the selling season begins. The firm then trades on the secondary market using updated demand information at this pre-determined trading time. There are two possible critiques of the model—that trading time is predetermined, and that it is inflexible. First, days-of-inventory is usually used by firms to measure their inventory level. In order to sell products to end consumers in the first period, firms have to buy non-zero days of inventory before the selling season begins. Thus a pre-determined trading time is embedded in firms’ first purchase decision. Second, there are several reasons why trading time may be decided a priori versus in real-time. Committing to a pre-determined trading time simplifies managerial decision and complexity. Even if inventory, secondary market spot price and end market demand information are monitored and updated continuously and automatically, it is still difficult and resource-consuming for firms to make real-time trading time decisions every day. Committing to a pre-determined trading time simplifies the decision process and reduces managerial costs. Secondary experiments we performed also demonstrated that this approach was beneficial compared to a model whereby the purchasing decision occurs when inventory level drops to zero (the best alternative choice when assuming zero purchasing lead time). The event sequence of the model is as follows: for any given t, an initial order of product is made based on the newsvendor problem formulation (Lee and Whang, 2002) for demand up to time t; after actual demand is realized through time t, the trader then makes a buy or sell in the secondary market based on its newly forecasted demand. Then actual demand in period two is then realized. Total profit is expected profit obtained in both periods. The optimal trading time t* for each scenario is that which provides maximum total profit. Each scenario is defined a priori by a spot price function, a retail margin, demand forecast bias (i.e. accuracy), and demand volatility. The spot price is a strictly decreasing function of time and defines the price that a trader can either buy or sell inventory after the selling season has begun. We assume that a large number of firms trade on the secondary market (Converge. com, for example, has more than 6500 trading partners globally), thus the impact of an individual firm’s trading quantity on the spot price is negligible. The retail margin is defined as the difference between the retail and the wholesale price, expressed as a ratio. The demand forecast accuracy measures the ratio of forecasted mean over the true mean, while demand volatility refers to the coefficient of variation (CV) of the true demand distribution which generates daily demand. We will discuss what happens if some of these assumptions are changed or relaxed in the discussion section.

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Under certain assumptions the model can be examined analytically and we shall discuss such results as appropriate. More generally we determine the optimal trading time for a given scenario via numerical simulation, where we do a brute force search of the parameter space of trading time (t) in order to find the optimal trading time (t*).

3.2. Analytical model Major notations: t

the trading day, ranging from 0 to T, the total number of days for the selling season pðtÞ the total expected profit obtained over the two periods, given t Q1 ðtÞ optimal order-up-to inventory level for first period (0, t) optimal order quantity for the whole season (0, T) Q0 without secondary markets Q2 ðtÞ optimal order-up-to inventory level for second period (t, T) the wholesale price charged by the upstream supplier pw pðtÞ the spot price function on the secondary market at time t retail price to end consumer pr a demand volatility ratio, ratio of standard deviation over mean for daily demand distribution m demand forecast accuracy; i.e., m ¼ forecasted mean demand=true mean demand, m A ð0; þ1Þ s1 ðtÞ expected end-market sales in period (0,t) s2 ðtÞ expected end-market sales in period (t, T) f1 ðx; tÞ true probability density function (PDF) for total demand in period (0, t) f2 ðx; tÞ true PDF for total demand in period (t, T) F^ 1 ðxÞ initial estimated cumulative probability function (CDF) for total demand in period (0, t) F^ 2 ðxÞ updated estimated CDF for total demand in period (t, T)

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information available (ergo we take a frequentist perspective as opposed to a Bayesian perspective). The coefficient of variation (CV) a is known, thus the standard deviation of the estimated daily demand distribution is calculated, instead of being known, as the product of the volatility ratio a and estimated mean. Formulae of estimated means and standard deviations for F^ 1 ðxÞ and F^ 2 ðxÞ are shown in Table 1. The firm uses the two estimated demand distributions F^ 1 ðxÞ and F^ 2 ðxÞ to determine the two order-up-to inventory levels Q1 ðtÞ and Q2 ðtÞ, which are the optimal order-up-to levels for period (0, t) and (t, T) respectively. Newsvendor solutions are used to maximize the estimated expected profit based on estimated demand distributions. But the true demand distribution is unknown to the firm. After the firm updates inventory level to either Q1 ðtÞ or Q2 ðtÞ, actual daily demand happens according to the true demand distribution. The optimal trading time t is chosen to maximize the EðpðtÞÞ. Given different demand distribution assumptions (normal or uniform distributions), with the objective function either becomes a neither convex nor concave objective function or a sixth-order polynomial function. Thus closed-form formula for the optimal time cannot be obtained (more details in Appendix A). Therefore numerical simulation is used to derive the bulk of our results. For numerical simulations we used MATLAB 7.0.4 (R14, Service Pack 2). The validity of our model and its embodiment was checked by (a) verifying that hand calculations and computer calculations were equivalent, (b) performing independent, parallel code reviews, and (c) verifying that cases at the boundary of our model matched expected results. In order to test the effect of margin and forecast accuracy on optimal trading time, we examined all combinations of the pair, ranging from a forecast accuracy of 10% to 300% (with a step of 20%), and retail margins ranging from 1.1 to 3 (with a step of 0.2). Four values of the demand volatility a are run in the simulation: 0.1, 0.3, 0.5 and 0.8. Thus we have 15  10  4= 600 scenarios in total for the baseline model.

4. Results The objective function is defined as max EðpðtÞÞ ¼  Q1 ðtÞpw  pðtÞ½Q2 ðtÞ  ðQ1 ðtÞ  s1 ðtÞÞ þ  þ pr ðs1 ðtÞ þ s2 ðtÞÞ

4.1. Optimal trading time-general findings

t

¼  Q1 ðtÞpw  pðtÞ½Q2 ðtÞ  "Z Z Q1 ðtÞ

0

Z

xf2 ðx; tÞ dx þ 0

Z

þ1

ðQ1 ðtÞ  xÞf1 ðx; tÞ dx Q1 ðtÞf1 ðx; tÞ dxÞ

Q1 ðtÞ

Q2 ðtÞ

Q1 ðtÞ

0 þ1

xf1 ðx; tÞ dxþ

þpr ð þð

Z

#

Q2 ðtÞf2 ðx; tÞ dxÞÞ

ð1Þ

Q2 ðtÞ

where E(*) is the expected value function. The true daily external demand is assumed to be independently and identically distributed (i.i.d.) with a normal distribution, whose mean is m and standard deviation is am. The coefficient of variation a is maintained in a range that ensures a positive daily demand with above 97.5% probability. Thus total demand from day 0 to t is the sum of t identically and independently distributed prandom numbers, which is also ffiffi normally distributed Nðt m; t amÞ with f1 ðx; tÞ as the PDF. The same is true for demand from day t to T: total demand for this pffiffiffiffiffiffiffiffiffiffiffi period is distributed according to NððT  tÞm; T  t amÞ with f2 ðx; tÞ as the PDF. The true mean of daily demand m is unknown and needs to be estimated by the trader. Both initial and updated estimates of m are ‘‘best-guesses’’ of the true mean based on

Fig. 1 depicts the optimal trading time as a function of retail margin and forecast accuracy; a trading time of (e.g.) 0.40 corresponds to trading 40% of the way through the selling season. Under all the 600 scenarios defined by retail margin n, forecast accuracy m and demand volatility a, the optimal trading time is between 0.47 and 0.67, 335 of which are 0.47, 120 of which are 0.5, 44 of which are 0.53, 57 of which are 0.57, 31 of which are 0.6, 11 of which are 0.63 and 2 of which are 0.67. Thus 83.17% of the optimal trading times are within the interval [0.47, 0.53]. Also the profit gained when trading at the middle point 0.5 is always within about 83% of the maximum profit gained for all the 600 scenarios. These findings suggest that buyers should roughly buy products not less than those the first half of the selling season and no more than what is needed for the first two-thirds of the season. Thus the simplest timing strategy which is not far away from the optimal solution in general is trading around the middle of the season. Thus, given the specific assumptions of our model, we propose the following. Proposition 1. If a single purchase or sale is made on a secondary market during a trading season, trading near the midpoint of the selling season will produce near-optimal expected profits across

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Table 1 Estimated mean and standard deviations.

m^ (Estimated mean)

s^ (Estimated standard

Note

deviation) Initial daily demand forecast

mm

am m

m is the true mean of demand. An m greater (smaller) than 100% corresponds to overestimation(underestimation) of demand

Initial total demand forecast for period (0, t) PDF: F^ 1 ðx; tÞ

tm m

pffiffi t am m

Updated daily demand forecast

½1 þ ðm  1Þf ðtÞm

a½1þ ðm  1Þf ðtÞm

Updated total demand forecast for period (t, T) PDF: F^ 2 ðx; tÞ

ðT  tÞ½1 þ ðm  1Þf ðtÞm

pffiffiffiffiffiffiffiffiffiffiffi T  t a½1 þ ðm  1Þf ðtÞm

Total demand for (0, t) is the sum of t identically and independently distributed random variables with mean mm and standard deviation amm f(t) is a decreasing function with respect to t and is used to model the decreasing gap between estimated mean and true mean Total demand for (t, T) is the sum of (T t) identically and independently distributed random variables with mean ½1 þ ðm  1Þf ðtÞm and standard deviation a½1 þ ðm  1Þf ðtÞm

Fig. 1. Optimal trading time.

a range of different profit margins and demand forecast accuracy levels. 4.2. The impact of forecast accuracy Fig. 1 also depicts that in general, the worse the forecast accuracy (the absolute value of m), the farther away the optimal trading time is from 0.5 (i.e. midway through the selling season). This pattern holds for all the 40 experimental conditions, which are defined by different values of retail margins and demand volatility. The effect is asymmetric however, so we shall discuss both cases separately. Proposition 2. Ceteris paribus, firms with worse prior demand information for a product should trade farther away from the middle of the season. 4.2.1. Demand underestimation bias Our results suggest that the greater the potential underestimation bias, the more the trader should consider a later trading date. This pattern is consistent in results generated from all the 40 experimental conditions defined by retail margin and demand volatility. Underestimating demand implies that inventory may reach a stock-out condition before arriving at the trading time. Our results, however, suggest that there are cases where

short-term loss of sales is less important than demand information gained through waiting. In situations where a lost sale meant a significantly dissatisfied customer, then our finding would be tempered because the cost of a single lost sale would have to also consider future lost sales. But in contexts where a ‘‘lost sale’’ does not incur other long term costs, this finding contradicts common practice. A numerical example demonstrates the trade-off between short-term loss of sale and long-term gain through information learning. Consider the case where demand accuracy m= 10% (the estimated mean demand level is only 10% of the true mean m), coefficient of variance a = 0.5, retail margin n = 2 (the retail price is two times the wholesale price), true daily mean demand m is 10, and the total number of days T=30. In this case, the optimal trading time in the base model is t ¼ 18, where average sales revenue is 209.29, the supply-demand mismatch cost in period 1 (from day 1 to 18) is 161.32 and that in period two (from day 19 to 30) is 54.525, and total profit is 156.16. The total mismatch cost is 215.845. If the initial purchase Q1 is used to account for 18 days of demand, however, a stock-out, due to the bad initial information, will happen at day 2. According to traditional news vendor models (Silver et al., 1998), trading should happen when inventory level drops to zero, which is day 2. Then the total sales revenue is 110.73, the mismatch cost in period one (from day 1 to 2) is 0 and in period two is 259.54, and total profit is 57.813.

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The total mismatch in this case is 259.54, which is 4215.845 in the base case. Even though the mismatch cost is higher in period one when trading on day 18, a much lower mismatch cost in period two leads to an overall more profitable situation. Thus we propose that: Proposition 2a. Ceteris paribus, higher prior underestimation bias leads to a later optimal trading time. 4.2.2. Demand overestimation bias Overestimating demand always has smaller impact on the optimal trading time than underestimating demand, which is a consistent pattern in across all experimental conditions. As Fig. 1 depicts, the optimal trading time is relatively insensitive to retail margin and demand accuracy in cases of overestimation. Different combinations of retail margin and estimation bias only change the height of the response surface, not its flatness. This finding is due to the fact that when demand accuracy (m) is well beyond 200%, the last component in the objective function, total expected sales revenue pr ðs1 ðtÞ þ s2 ðtÞÞ, approaches the product of the constant true mean demand and retail price, which is not a function of trading time t. The other two parts of the objective function, cost of initial purchasing volume and cost/revenue of trading volume on the secondary market, approach linear functions of time t, because the distribution functions play a diminishing role as m increases. This finding suggests that overestimation bias has little impact on the timing decision. Waiting longer to learn market information is more costly when demand is overestimated than underestimated. In all the experiments we ran, the optimal trading time with demand overestimation is no later 0.5 (i.e. halfway through the season). The results also indicate that profit margin does not significantly affect the trading time decision when overestimation bias is big.

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Proposition 3. Ceteris paribus, optimal trading time increases as profit margin increases.

4.3.2. Spot price In the base model discussed in Section 4.3.1 we assume that spot price is a linear function of t:pðtÞ ¼ ð1  t=TÞpw . In order to study the impact of the spot price function on the optimal trading time, we examined two different spot price functions, concave ðpðtÞ ¼ ð1  t=TÞ0:5 pw Þ and convex ðpðtÞ ¼ ð1  t=TÞ2 pw Þ, for all the 600 scenarios defined by forecast accuracy, retail margin and demand volatility (see Fig. 2). Results consistently indicate that when the spot price function is concave (decreasing at a lower speed), the optimal trading times all shift downwards compared to the base case, suggesting that buyers should trade relatively earlier; this scenario also increases the impact of demand forecast inaccuracy on optimal trading time. When the spot price function is convex (decreasing at a higher speed), optimal trading times increase and is more robust to changes in prior information quality. Together, the two results imply that there are two forces, the spot price function and information gained from demand learning. When spot prices decrease at a faster rate, the value of demand information dominates in determining the optimal trading time, which leads to higher sensitivity to forecast inaccuracy. When spot price decreases at a slower rate, demand information plays a smaller role and obtaining a lower spot price becomes more important. In sum, these two forces give birth to the total value of secondary market trading, which we will discuss in more details in Section 5.2. Proposition 4. Ceteris paribus, as spot prices decrease less rapidly, optimal trading time increases.

4.4. Quantity effect and allocation effect Proposition 2b. The optimal trading time is less sensitive to overestimation bias than to underestimation bias. Proposition 2c. The optimal trading time is not sensitive to profit margin changes when overestimation bias is huge. 4.3. The impact of margin Margin is determined by retail price, wholesale price and spot price. The margin for products purchased for the first period is retail minus wholesale price, while the margin for products purchased in the second period is retail minus spot price. Since we assume that wholesale price is constant in all scenarios, we vary retail price and the spot price function in order to study the impact of margin on the optimal timing decision. 4.3.1. Retail price Across all 60 experimental conditions with different values of forecast accuracy and demand volatility we consistently found that retail profit margin is negatively related with the optimal trading time. This suggests that procurement of higher-margin products should rely more on secondary markets than that of lower-margin products, given the two markets are fully substitutable (See Section 5.1.2 for situations when the secondary market transaction is not perfect and the two markets are not fully substitutable). We can infer from this finding that, for higher margin products, minimizing the supply-demand mismatch produced by prior bad demand information is more important than a favorable spot price and better demand information for the second period.

The first secondary market model established by Lee and Whang (2002) assumes that all the traders order the same amount of products in the first period from the same supplier, then trade at the same time and do not update information across periods. In our model, we allow varying levels of forecast accuracy and margins, and assume that the mean demand for any given time period is proportional to the length of the period, and forecast accuracy increases over time. These differences in assumptions lead to different findings regarding the ‘‘quantity effect’’ and ‘‘allocation effect’’ defined in Lee and Whang (2002).

Pw

t p (t ) = (1 − ) pw T

t p (t ) = (1 − )0.5 pw T

t p (t ) = (1 − ) 2 pw T 0

T Fig. 2. Spot price functions.

t

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Excess(no-trading)-Excess(trading)

4.4.1. Quantity effect Quantity effect is defined as the effect of secondary market trading on the sales of upstream suppliers (Lee and Whang, 2002). As shown in Appendix B, if we assume demand follows a uniform distribution, there is a sufficient condition under which a secondary market always reduces purchases from upstream suppliers. pffiffiffi Lemma 1. Suppose a o t= 3, when pr  pw =pr 4 1=2, if daily demand is independently and uniformly distributed with mean of m and standard deviation of am, then a secondary market always reduces purchases from upstream suppliers, i.e. Q1 ðtÞ  Qo o 0; 8t A ð0; TÞ. The Proofs are provided in Appendix B. pffiffiffi A sufficient condition for demand volatility a o t= 3 to hold is pffiffiffi a o 1= 3 ¼ 0:577 (t is the number of days when trading occurs, thus t Z 1), which is a realistic and common assumption to ensure the lower bound of the uniformly distribution demand to have positive lower bound or normally distributed demand to be positive with a probability of over 95%. Lemma 1 suggests that as long as the Newsvendor fractile pr  pw =pr is not too low, the secondary market always reduces purchases from upstream suppliers compared to the case where there is no secondary market. This is the same conclusion as that obtained in Lee and Whang (2002). However, when we use numerical methods to identify the applicability of Lemma 1 across different scenarios, we found out some different results. Using more general demand distributions such as normal distributions, numerical simulation indicates that Q1 ðtÞ  Qo , the difference between purchase from upstream suppliers with and without trading, is negative for all the trading times t and for all the scenarios, and Q1 ðtÞ is always an increasing function with respect to t no matter how small the fractile pr  pw =pr is (note that Q1 ðTÞ ¼ Q0 , where T is the end of the season). Thus a secondary market reduces initial purchases from upstream suppliers no matter when the trading time is in all the numerical experiments. The reason for this negative quantity effect is: there are two factors affecting the shape of Q(t) with respect to t, the Newsvendor fractile Pr  Pw =Pr  PðtÞ(service level factor) and the proportion of total demand for period (0, T) that Q(t) is used to account for (demand factor). When t increases, the fractile decreases from 1 to Pr  Pw =Pr , which lead Q1 ðtÞ to decrease. As the forecasted mean and standard deviation of demand increase proportionally with t, Q1 ðtÞ should increase. When the two factors act together, the demand factor dominates, which leads Q1 ðtÞ to increase with t. This suggests that unless upstream suppliers strategically react to buyers’ ordering behaviors, they will be hurt

by the existence of secondary markets due to the fact that buyers will buy less from them. Suppliers may react by using incentives or contracts to encourage or force buyers to buy more; see Lee and Whang (2002) for a broader discussion. Proposition 5. Secondary market trading de facto reduces a firms’ initial purchases from upstream suppliers no matter when the trading time is. 4.4.2. Allocation effect According to Lee and Whang (2002), allocation effects include changes in channel sales, stock-outs and excessive inventory, all of which focus on supply chain performance, with the existence of secondary market trading. We found out that a secondary market almost always enables traders to realize equal or higher expected sales, and equal or lower stock-out and excessive inventory compared to the no-trading case. Improvements in these 3 dimensions are all inverse U-shape functions with respect to the trading time t. Thus the effect of the secondary market on supply chain performance is moderated by the trading time. Similar with Lee and Whang (2002), we also identified scenarios where a secondary market can worsen supply chain performance in terms of increasing excessive inventory. Fig. 3 shows the scenarios with different values of m smaller than 100% (underestimation) when coefficient of variation a=0.5 and retail margin n=3 (these patterns also hold for combinations of other values of demand volatility and retail margin). When buyers underestimate demand and trade relatively late on the secondary market, the total excessive inventory is increased by secondary market trading. Interestingly, better prior information worsens this negative impact by expanding the ranges of trading time t which lead to higher excessive inventory compared with that in the no-trading case. When prior information is very good (e.g. demand forecast accuracy m ¼ forecasted mean demand=true mean demand ¼ 90%), buyers will end up with higher channel excessive inventory no matter when they trade. Similar results are found for channel stock-out and sales performance. Fig. 4 shows the scenarios with different values of m4100% (overestimation) when a = 0.5 and n =3. Furthermore, prior information quality is consistently negatively correlated with secondary market’s potential in improving supply chain performance for all the scenarios. Retail margin and demand volatility are both positively related with the allocation effect. Thus for products with higher margin and more volatile daily demand, secondary market trading potentially have greater impacts on changing channel sales, excessive inventory

2 0 -2

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

-4 -6

m=10%

-8

m=30%

-10

m=50%

-12

m=70%

-14

m=90%

-16 Trading day

Fig. 3. Note: daily mean demand m = 10 and the total season is 30 days. Excess inventory is increased by secondary market trading.

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69

Stockout(no-trading)-stockout(trading)

0.05 0

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

-0.05 -0.1 m=110%

-0.15

m=130%

-0.2

m=150%

-0.25

m=170%

-0.3

m=190%

-0.35 -0.4 Trading day Fig. 4. Channel sales (stock-out) is decreased (increased) by secondary market trading.

Perfect learning f (t ) = 1 −

t T

Imperfect learning f (t ) = 1 −

t 2T

Low volatility (a=0.1)

High volatility (a=0.5)

Fig. 5. Impact of demand learning on the optimal trading time.

and stock-out. Whether the changes are positive or negative depends on forecast accuracy and trading time. Proposition 6. Trading at the optimal time maximizes the allocation effect of secondary markets, in terms of increasing channel sales and reducing excessive inventory/stock-out. Proposition 7. A secondary market’s potential in increasing channel sales and reducing excessive inventory/stock-out is negatively related with prior information quality.

5. Discussion 5.1. Sensitivity analysis of the optimal trading time 5.1.1. The impact of demand learning In the base model we assume that by the end of the season, the buyer obtains perfect information about the mean of the daily

demand distribution. In reality buyers may differ in the speed at which they learn about demand which in our model would manifest itself as higher forecast inaccuracy throughout the selling season, even at its end. To study the impact of this demand learning rate on the optimal trading time, we ran experiments with different demand updating functions f(t) to compare to the base case where perfect demand learning is assumed. When f ðtÞ ¼ 1  t=2T in the slow demand learning case, forecast inaccuracy (no matter over-or underestimation bias) is always higher than the high speed case where f ðtÞ ¼ 1  t=T. The low speed may be due to error in measuring actual demand or lack of demand information. Results across all experimental conditions (Fig. 5) indicate that as learning becomes more imperfect, the optimal trading time increases, primarily because there is a greater need to wait longer in order to achieve the same level of demand forecast accuracy. We also note that overall expected profit decreases with greater demand inaccuracy, thus implying the value of IT and market research related investments

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Fig. 6. (A) Impact of imperfect market trading on the optimal trading time (p =0.5). (B) Profit difference between the adjustment model and the base model.

which improve demand forecast accuracy, similar to findings in (e.g.) Melville et al. (2004), Fisher et al. (1997), et al. Proposition 8. Ceteris paribus, lower demand learning speed increases the optimal trading time. 5.1.2. The impact of unreliable secondary market purchases In the base model, we assume that buyers can buy or sell the exact quantity they desire. In reality, this is not necessarily true. Thus we use a new parameter p to represent the proportion of products that are actually delivered to or sold by the firm at the beginning of period two, compared to what was intended to be purchased or sold; when p is o1 it models situations such as supply-demand mismatch, inferior quality of secondary market products, or situations involving non-zero purchasing lead-time. In this model, the actual inventory level after trading is no longer Q2 ðtÞ but ðQ1 ðtÞ  s1 ðtÞÞ þ þp½Q2 ðtÞ  ðQ1 ðtÞ  s1 ðtÞÞ þ , the difference between the two terms is the amount of products that firms want

to sell on or buy on the secondary market but could not do so due to transaction unreliability. Fig. 6 shows the optimal trading time plot for p =0.5 (we also ran other values of p ranging from 0.1 to 0.9, with a step of 0.1, for all the 600 scenarios). Consistent findings from comparing these results with the base model could be identified. First, in most scenarios, except those with very large overestimation bias, firms should trade relatively later when secondary market supply becomes unreliable. This finding is reasonable in the sense that lower reliability drives traders to rely more on the primary market (plan to use initial purchases to account for a greater portion of total demand). Second, secondary market unreliability drives the timing decisions in scenarios with very large overestimation bias to be earlier, instead of later, than those in a more reliable market. The reason is that market unreliability corrects the overestimation bias in a way that the ‘‘overoptimistic’’ firms are forced to buy less, which reduces the need to wait longer to learn market information for such traders.

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71

135 130

VOS

Profit

125

(learning)

VODI

120

VOS(no-

115

learning)

110 105 100 1.1

1.3

1.5

1.7

1.9 2.1 retail margin

2.3

2.5

2.7

2.9

200 180 160 140 VODL

120 100 80 60 40 20 0 1.1

1.3

1.5

1.7

1.9 retail margin

2.3

2.5

a=0.1 m=10%

a=0.1 m=100%

a=0.1 m=300%

a=0.5,m=10%

a=0.5,m=100%

a=0.5,m=300%

2.7

2.9

Fig. 7. Note: coefficient of variation a =0.1, forecast accuracy m= 130%, and daily mean demand m = 10. (A) VODL, VOSP and VOS. (B) VODL with respect to retail margin, forecast accuracy and demand Volatility.

Regarding the impacts of unreliability on buyers’ profits, in all the scenarios buyers’ maximum expected profits (obtained at the optimal trading time) are reduced by market unreliability. This is intuitive because the unreliability reduces buyers’ capability to fully realize the value of secondary market trading. A counterintuitive finding is that, given the same retail price and low market reliability level (po0.8), poorer demand information always pushes buyers to trade earlier, instead of later. This suggests that there is an interaction effect between market reliability and prior information quality on the optimal trading time. When market reliability is high, buyers with better prior information should trade more around the middle of the season; when reliability is low, however, buyers with better information should trade more towards the end of the season. Proposition 9. Ceteris paribus, secondary market transaction unreliability almost always increases the optimal trading time, compared with that in more reliable market, except for cases with very large demand forecast overestimation bias.

Proposition 10. Ceteris paribus, when supplied quantities from a secondary market are not reliable, better prior demand information leads the optimal trading time to be later. 5.2. Value of demand learning In Section 4.3, we mentioned that spot price and demand learning contribute to the total value of the secondary market. In order to parse the total value of secondary market into these two components, we modified the base model to forbid learning, i.e., the update function f ðtÞ  1; 8t A ð0; TÞ. The total value of secondary market trading (VOS) is composed by value of spot price (VOSP) and value of demand learning (VODL). They are obtained by using the following equations: Profitðt ¼ t Þlearning  Profitno-market ¼ VOSðtotal value of secondary maket tradingÞ Profitðt ¼ t  Þno-learning  Profitno-market ¼ VOSPðvalue of spot priceÞ VOS2VOSP ¼ VODLðvalue of demand learningÞ

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Table 2 Heuristics for choosing the optimal trading time. Scenarios

Optimal trading time

On average High retail margin Bad prior information quality High spot price decreasing speed

Middle point Later than the middle point Farer away from the middle point Earlier than the middle point

Fig. 7A shows the relationship among these three values. Fig. 7B shows the impacts of retail margin, forecast accuracy and demand volatility on VODL. From 7B we can observe several interesting findings regarding the value of demand learning. First, the value of demand learning increases with respect to forecast bias. The better the initial demand forecast is the smaller the value of information learning. Secondly, retail margin is positively related with the value of learning–higher margin products can earn higher benefits from information learning if they are traded at the right time. Thirdly, demand volatility interacts with demand forecast accuracy to impact the value of learning. When forecast bias is small (the two lines at the bottom in Fig. 7B), demand volatility is negatively correlated with VODL ceteris paribus. When forecast bias is large (the two pairs of lines at the top in Fig. 7B), however, demand volatility is positively correlated with VODL. Intuitively we may think that learning is less useful for more volatile demand distribution because there is too much systematic variance. For example Milner and Kouvelis (2002) observed a negative relationship between true demand variance and the value of information in numerical experiments. But our results show that this negative relationship only holds when the prior forecast bias is not too large, the situation where the value of information is small to begin with. When prior forecast bias is significant, higher demand volatility leads to greater value of demand learning if products are traded at the right time. This finding is consistent with the logic of the (real) option theory (Huchzermeier and Christoph, 2001 et al.) which suggests a positive relationship between uncertainty and (real) options’ value. Proposition 11. When prior demand information quality is not too low, high demand volatility leads to a lower value of demand learning; when prior information quality is low enough, high demand volatility leads to higher value of demand learning.

5.3. Managerial insights For firms who want a general guidance in secondary market trading time decision, trading once at the midpoint of a selling season is a simple heuristic to implement. Given specific information concerning expected retail margin, prior information quality and spot price function, our model could be used to determine an actual optimal trading time. Ceteris paribus, products with higher retail margin and products with relatively higher spot prices should be traded relatively later, while products with better prior demand information should be traded at the midpoint of the season. A summary of the heuristics could be seen in Table 2. If a firm wants to maximize the secondary market’s potential in increasing channel sales and reducing channel excessive inventory/stock-out, the timing decision is even more critical. After controlling for the impacts of trading time, the secondary market’s potential in improving channel performance is higher for components of which the firm has worse prior demand information. But no matter when the trading time is, secondary market trading always reduces initial purchases from upstream suppliers.

Thus supply chain coordination mechanisms should be designed to help improve upstream suppliers’ benefits. 5.4. The need to study the timing issue in supply chain management field Research into issues of timing has traditionally represented a very small niche of activity in supply chain management research. Some recent timing research includes work on the timing of IT upgrades (Mukherji et al., 2006), market entry timing (Miller and Timothy, 2002), new product entry timing (Prasad et al., 2004), environmental policy adoption timing (Pindyck, 2002), investment timing (De´camps and Thomas, 2004), and capacity investment timing (Uiku et al., 2005). In these studies factors such as costs, incentives and risks (Uiku et al., 2005) tend to lead to different timing decisions. The common theme across these timing studies is the trade-off between value of waiting longer to get more information and risks/costs of waiting longer, which has to be considered explicitly or implicitly in many supply chain management decisions. For example, managerial flexibility, which is extensively discussed recently in supply chain management literature (Santiago and Bifano, 2005; Huchzermeier and Christoph, 2001), can be viewed as a timing decision of when to abandon this flexibility. Similarly, in contract management, the decision of whether to sign the current best-offer or wait for the next offer can be viewed as a timing problem of when to forgo the flexibility to choose contracts. Along similar lines, the decision concerning immediate termination of a contract because of poor performance versus providing a second chance to an agent can be viewed as another timing decision concerning when to abandon the flexibility inherent in maintaining the relationship. If such problems were commonly framed as timing issues, we may gain more insight into this illusive problem. 5.4. Limitations and future directions Our model has several limitations that could be examined in future research. First, we assume that the upstream supplier passively accepts what ever amount the buyers order without strategically adjusting wholesale price accordingly. In reality, quantity discount contracts usually exist and can affect the traders’ timing decisions. Thus a more accurate (but complex) model could take into account upstream suppliers’ strategic behavior when finding the optimal entry time for traders. Second, we do not consider the case where there are a limited number of traders in the secondary market and individual trading and bidding behavior may change the spot price irregularly, including cases where the spot price may be higher than wholesale price. Further studies are needed to examine the underlying mechanism that drives spot prices to go higher and its impacts on timing decisions. Third, we did not model the dynamic behavior of traders. In reality, traders may decide an initial trading time which yields a first period order quantity, and then change their mind during the season because of new information and trade at another time. An extension to our model would take such dynamic behaviors into consideration. Loch and Christian (2005) demonstrate a generalized model of real-time decision-making which enables decision-makers to modify decisions according to preliminary information. A possible way to integrate their model with our problem is to setup a two-stage decision model and use backward induction to derive the optimal (t1, t2), where t1 is the initially determined trading time and t2 is the adjusted final trading time. Given t1, t2 could be solved using dynamic programming to find out the best adjusted trading time according to new information. Finally multiple trading times could be studied in future studies. In this model, we only assume one

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trading time, which leads to a two-stage inventory model. But if transaction cost and managerial complexity is low enough, multiple trading time decisions are possible in reality. Thus future studies could be done to examine the trade-off between multiple trading opportunities during the selling season and transaction costs. 6. Conclusion In summary, our results suggest that traders who have access to secondary markets should determine the relative optimal trading time for each type of products based on their relative prior information quality, retail margin and demand volatility. In general, the optimal trading time is never too early or too late in the selling season. Trading around the middle of the season is a robust heuristic rule, especially for products with relatively good prior demand information. As a rule of thumb, buyers should always trade higher value products earlier. Also they should postpone the second purchase to learn more demand information when serious underestimation bias is found in period one. Investing in demand information collection technology is beneficial in terms of increasing overall profit through improved demand learning, regarding upstream suppliers, if they do not intervene in the secondary market trading, then their sales will probably be reduced by the existence of secondary sources of products. For the whole supply chain, the question of whether secondary market trading can increase channel sales, reduce stock-out and reduce excess inventory through secondary market trading depends on the chosen trading time, forecast accuracy, retail margins and demand variability. Trading at the right time could maximize the potential of secondary markets in improving channel performance. Appendix A The objective function is max EðpðtÞÞ ¼  Q1 ðtÞpw  pðtÞ½Q2 ðtÞ  ðQ1 ðtÞ  s1 ðtÞÞ þ  þ pr ðs1 ðtÞ þ s2 ðtÞÞ t

¼  Q1 ðtÞpw  pðtÞ½Q1 ðtÞ  "Z Z Q1 ðtÞ

0

ðQ1 ðtÞ  xÞf1 ðx; tÞ dx Q1 ðtÞf1 ðx; tÞdxÞ

Q1 ðtÞ

Q2 ðtÞ

xf2 ðx; tÞ dx þ

þð

Q1 ðtÞ

0 þ1

xf1 ðx; tÞ dxþ

þpr ð Z

Z

0

Z

þ1

#

Q2 ðtÞf2 ðx; tÞdxÞ

Q2 ðtÞ

@pðtÞ ¼ ½pw þpðtÞFt ðQ1 ðtÞ; tÞ þ pr ð1 @t  F1 ðQ1 ; tÞÞQ10 ðtÞ þ ½pðtÞ þpr ð1  F2 ðQ2 ; tÞÞQ20 ðtÞ

where @GðQ1 ; tÞ @GðQ2 ; T  tÞ ¼ F1 ðQ1 ; tÞ; ¼ F2 ðQ2 ; tÞ; Q10 ðtÞ @Q1 @Q2 ðpr  pw Þp0 ðtÞ @F^ 1 ðQ1 ; tÞ  2 @t ðpr  pðtÞÞ ¼ ^f ðQ ; tÞ 1 1 and p0 ðtÞ @F^ 2 ðQ2 ; tÞ   p @t r : Q20 ðtÞ ¼ f^ 2 ðQ2 ; tÞ

For normal distributions, @F^ 1 ðQ1 ; tÞ=@t, @F^ 2 ðQ2 ; tÞ=@t, @GðQ1 ; tÞ=@t and @GðQ2 ; T  tÞ=@t make the analytical form for the first and second derivative condition impossible to derive in closed form. Furthermore, in all the numerical simulations we run, it is shown that the objective function is even neither convex nor concave function with respect to the trading time when demand is normally distributed. Even if we use the simplest continuous distribution (a uniform distribution on (0, 1), for daily demand) and set both pðtÞ and f ðtÞ to their simplest linear forms (pðtÞ ¼ pw ð1  t=TÞ and f ðtÞ ¼ 1  t=T), the objective function is still a sixth degree polynomial with respect to t, thus we rely on numerical methods for the bulk of our results.

Appendix B Proof. Let F^ T ðxÞ be the initial estimated CDF for total demand in the pffiffiffi whole period (0, T), with mean Tmm and standard deviation T mam.     1 1 Pr  Pw Pr  Pw  F^ T let gðtÞ ¼ Q1 ðtÞ  Qo ¼ F^ 1 Pr  PðtÞ Pr It is easy to show that gð0Þ o0 and gðTÞ ¼ 0.Thus in order to determine the sign of gðtÞ for t A ð0; TÞ, we only need to know the sign of g 0 ðtÞ. When demand distribution type is not specified, we can obtain: ðpr  pw Þp0 ðtÞ g 0 ðtÞ ¼ ð1þ mÞm

ðpr  pðtÞÞ2



R Q1 ðtÞ d ^ 1 dt ðf 1 ðx; tÞÞ dx

f^ 1 ðQ1 ðtÞ; tÞ

It can be shown that the sign is not determinable even if we assume a known demand. If daily demand is uniformly distributed with mean as m and standard deviation as am, the daily pffiffiffi pffiffiffi demand is uniformly distributed over ðm  3am; m þ 3amÞ. Total demand for (0, t) is also uniformly distributed over pffiffiffiffiffi pffiffiffiffiffi ðt m  3t am; t m þ 3t amÞ. Because t 41 and a 5 1, we could see pffiffiffiffiffi that the lower bound t m  3t am is positive.  pffiffiffiffiffi pffiffiffiffiffi Pr  Pw þ t  3t a gðtÞ ¼ Q1 ðtÞ  Qo ¼ mm 2 3t a Pr  PðtÞ pffiffiffiffiffiffi  pffiffiffiffiffiffi Pr  Pw  T þ 3T a 2 3T a Pr we could see that g(T)=0. pffiffiffiffiffiffi    pr  pw T gð0Þ ¼ mm 3T a 1  2 pr

It follows that the first derivative of EðpðtÞÞ is

@GðQ1 ; tÞ  ½Q2 ðtÞ  GðQ1 ; tÞp0 ðtÞ þ ½pðtÞ  pr  @t @GðQ2 ; T  tÞ  pr @t

73

g 0 ðtÞ ¼

ðB:1Þ

" #   pffiffiffi pffiffi pr  pw pffiffi ðpr  pw Þp0 ðtÞ 1 3amm 1 þ1= t  2 t ðpr  pðtÞÞ 2 ðpr  pðtÞÞ2 ðB:2Þ

ðA:1Þ

Because p ðtÞ o0 and pðtÞ 40, we could see that pr  pw =pr 4 1=2 is a sufficient condition for gð0Þ o 0 in (B.1) and g 0 ðtÞ 40; 8t A ð0; TÞ in (B.2). Combined with g(T)= 0, we know that pr  pw =pr 41=2 is a sufficient condition for gðtÞ o 0; 8t A ð0; TÞ. 0

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