INTERACTION BETWEEN A MANUFACTURER AND A RISK-AVERSE RETAILER: A SUPPLY CHAIN PERSPECTIVE ON RISK-AVERSE NEWSVENDOR MODEL

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1 The Journal of Japanese Operations Management and Strategy, Vol. 4, No. 2, pp , 2014 INTERACTION BETWEEN A MANUFACTURER AND A RISK-AVERSE RETAILER: A SUPPLY CHAIN PERSPECTIVE ON RISK-AVERSE NEWSVENDOR MODEL Shota Ohmura Kobe University ABSTRACT To share the risk and return between the supply chain partners is an important theme in supply chain management (SCM). There exists the extensive literature on the supply chain coordination, especially on the design of contracts that align the incentives of supply chain partners. However, few studies address the effect of risk attitude upon the coordination, and most of them are dealing with a risk-averse player within the context of a single player model. In order to coordinate supply chain, it is valuable to investigate how the risk attitudes of supply chain partners affect the supply chain relationships. In this paper, we consider the decentralized supply chain consisting of a risk-averse retailer and a manufacturer who are concerned with only their own objectives. The decentralized supply chain is facing an uncertain market demand and dealing with a single product over a single period. We show complex effects of risk aversion on the manufacturer-retailer interaction numerically. We also drive analytical results as much as possible. The risk models analyzed in this paper are the mean-variance (MV) model, the mean-standard deviation (MS) model, and the conditional value-at-risk (CVaR) model. We show that the dynamics of interaction is different depending on the risk measurement used in the analysis. Keywords: risk-averse newsvendor model, conditional value-at-risk (CVaR), supply chain management INTRODUCTION To share the risk and return between the supply chain partners is an important theme in SCM. There exists the extensive literature on the supply chain coordination, especially on the design of contracts that align the incentives of supply chain partners. However, few studies address the effect of risk attitude upon the coordination, and most of them are dealing with a risk-averse player within the context of a single player model. In order to coordinate supply chain, it is valuable to investigate how the risk attitudes of supply chain partners affect the supply chain relationships. Wholesale price contract is commonly observed in practice as the standard way to govern transaction in supply chain. We consider the decentralized supply chain consisting of a risk-averse retailer and a manufacturer who are operating under the wholesale price contract. Under the wholesale price contract, the risk attitude of the manufacturer does not matter since the manufacturer is not exposed to the risk associated with demand uncertainty. Even if only the retailer s risk attitude is considered, the effect of the risk attitudes on the supply chain 31

2 relationships is intricate. In general, the more risk averse the retailer is, the less inventory is expected to be procured. Facing a more risk-averse retailer than the usual ones, should the manufacturer set the wholesale price lower to induce a larger order, or set it higher to directly increase the revenue? The answer is not straightforward. Ohmura and Matsuo (2012) show the intricate effects of the retailer's risk aversion on the supply chain performance under the wholesale price contract, using the same model in Tsay (2002). In their model, a retailer is facing an uncertain linear demand curve and making decisions to maximize its own mean-standard deviation (MS) value. In this paper, we address the issues with the basic newsvendor model and two other risk models. The risk models analyzed in this paper are the mean-variance (MV) model, the mean-standard deviation (MS) model, and the conditional value-at-risk (CVaR) model. MV and MS models are categorized in the mean-risk approach, which is widely used in practice especially in portfolio management. The mean-risk approach quantifies the problem in a form of the mean which is the expected value of outcome and the risk which is variability of outcome. Although the mean-risk approach is deemed deficient relative to the expected utility approach, it is implementable because only two moments are required (Van Mieghem, 2003). CVaR is categorized in downside risk measurement and also widely used in practice. Downside risk measurement may be more desirable risk measurement than variance and standard deviation, since it captures only the undesirable deviation. It should also be added that CVaR is a coherent measure of risk. Artzner et al. (1999) suggest four axioms that a risk measure should satisfy, and the risk measure is called a coherent measure of risk if it satisfies the four axioms. The axiomatic approach has become the dominant framework in risk analysis, and has been used in the SCM literature. CVaR has computational advantage in the newsvendor model as shown later. We show complex effects of risk aversion on the manufacturer-retailer interaction numerically and drive analytical results as much as possible, using the risk models. We show that the dynamics of interaction is different depending on the risk measurement used in the analysis. Ohmura (2013) addresses the issue under the buyback contract as well as the wholesale price contract. Under the buyback contract, the manufacturer is exposed to risk, because the retailer can return leftover inventory with a pre-defined buyback price to the manufacturer. Thus, the risk attitude of manufacturer affects the supply chain relationships, and the problem becomes more complex than the case of the wholesale price contract. We report our research outcomes in the case of the wholesale price contract in this paper. The reader is referred to Ohmura (2013) for the research outcomes in the case of the buyback contract. LITERATURE REVIEW In the SCM literature, the problem of supply chain coordination is treated as inventory management of one product over one period. The inventory management over one period is often modeled as a newsvendor model. As mentioned above, few studies address the effect of risk attitude upon the coordination, and most of them are dealing with a risk-averse player within the context of a single player model. There are a few exceptional papers considering the interaction between two players (Tsay, 2002; Lau and Lau, 1999; Choi et al., 2008). Cachon (2003) and Tsay et al. (1999) provide comprehensive reviews of contract models in supply chain. Cachon (2003) provides an extensive review covering various contract types, multiple period or location models, asymmetric information models and so on. He mainly considers the contract models to coordinate a supply chain. Tsay et al. (1999) provide a 32

3 classification scheme for the literature on contracts in the supply chain management context. In this paper, we study a wholesale price contract between a manufacturer and a retailer. Lariviere and Porteus (2001) analyze the wholesale contract in detail. This simple contract is commonly observed in practice as the standard way to govern transactions in supply chain. At the same time it is known as a contract that does not coordinate the supply chain. Double marginalization causes this inefficiency in supply chain (Spengler, 1950). Choi et al. (2011) provide a categorization of risk-averse newsvendor models. There are four typical approaches to model a risk-averse newsvendor. They are expected utility theory, mean-risk optimization approach, downside risk, and coherent measure of risk. We review the related literature in this subsection. The expected utility theory of von Neumann and Morgenstern (1944) derives the existence of a non-decreasing utility function from simple axioms about preference relation of the decision maker, which are completeness, transitivity, continuity, and independence. In this model, the risk-averse newsvendor maximizes its expected utility function. Lau (1980) is the early paper considering the risk-averse newsvendor maximizing the expected utility. He numerically shows that the optimal order quantity becomes smaller than the risk-neutral one using the utility function approximated by a polynomial. Eeckhoudt et al. (1995) examine the newsvendor model with general utility function. Wang et al. (2009) analyze the classes of utility functions, which are CARA, IARA, and DARA classes, within the expected utility theory framework. The mean-risk approach is well-known. The mean-variance function categorized in this approach is used in the context of portfolio optimization (Markowitz 1959). A merit of this function is that we can analyze a trade-off between the mean outcome and the variance as its measure of risk. It is also known that the utility function can be approximated by the mean-variance (MV) function if it is quadratic or if the outcome is normally distributed. Chen and Federgruen (2000) model a risk-averse newsvendor in MV model, assuming that the newsvendor's utility function is quadratic. Choi et al. (2008) consider the risk-averse players who maximize the expected profit subject to a constraint on the standard deviation of profit. Lau and Lau (1999) also study the MV model numerically. Wu et al. (2009) analyze the MV model including stockout cost. Anvari (1987) uses the capital asset pricing model (CAPM) to study a newsvendor facing a normal demand distribution. The mean-standard deviation (MS) value function we consider in this paper uses the standard deviation as its risk measure, and it is categorized as a mean-risk approach. Although it is not used in the literature as much as the mean-variance model, Tsay (2002) uses this MS function and refers to Bar-Shira and Finkelshtain (1999) for its justification. They argue that using a value function that increases in mean and decreases in standard deviation is more robust than the approaches based on the expected utility. In addition, since the dimensional units of mean and standard deviation that are added up are the same, Tsay points out that this representation "dollarizes" a party's aversion to risk. Since the MS value function is sometimes more mathematically tractable than the other models, we believe that it should be studied further for its theoretical and practical meanings. Some studies use the downside risk measures. Downside risk measurement may be more desirable risk measurement than variance and standard deviation, since it only captures the undesirable deviation. Chance constrained programming is introduced in the field of stochastic programming (Charnes and Cooper, 1959). The chance constraints are now called as Value-at-Risk (VaR), and are used in newsvendor type formulations as their constraints, limiting the probability of particular events happening (Gan et al., 2005; Tapiero, 2005). 33

4 Recently, some studies use the Conditional Value-at-Risk (CVaR) to model the risk-averse newsvendor. The Conditional Value-at-Risk (CVaR) measures the average value of the profit falling below a certain percentile level. It is also called the mean excess loss or tail VaR or expected shortfall. Artzner et al. (1999) suggest four axioms that a risk measure should satisfy, and the risk measure is called a coherent measure of risk if it satisfies the four axioms. CVaR is a coherent measure of risk and has better computational characteristics than VaR. Ahmed et al. (2007) solve the CVaR maximization problem for the newsvendor model and shows the existence of an optimal solution. Choi et al. (2011) also use the CVaR and show that CVaR actually represents a trade-off between the expected profit and a certain risk measure, and thus can be regarded as a special mean-risk criterion. Chen et al. (2009) provide conditions under which there exists the optimal price and order quantity for both additive and multiplicative demand models. MODEL We consider a two-echelon supply chain that consists of a risk-averse retailer and a manufacturer selling a product over single-period and facing an uncertain market demand. We use the newsvendor model and analyze the case of the wholesale price contract under which the manufacturer determines the wholesale price (i.e., as a Stackelberg leader). As risk-averse models, we use the mean-variance (MV) model, mean-standard deviation (MS) model, and conditional value-at-risk (CVaR) model. The nonnegative random variable representing the demand of a product over a single period is, with pdf and cdf. is differentiable, strictly increasing and 0 0. At the beginning of the selling season, the manufacturer determines the unit wholesale price. And then the retailer chooses order quantity and receives the units with the unit wholesale price from the manufacturer. The retailer sells the product at unit retail price during the selling season. To avoid trivial cases, we assume. We assume that the salvage value and the shortage penalty are equal to zero for simplicity of argument. The model assumes symmetric information. That is, the same information on the demand and the risk attitudes is shared by the both parties at the beginning. The risk attitude is represented by three types of function. The functions,,, and, define the retailer's mean-variance value, mean-standard deviation value, and conditional value-at-risk measure, respectively. The retailer's profit function,, is represented as follows: The risk-averse retailer determines an optimal order quantity that maximizes its objective function. The retailer having the mean-variance criterion maximizes the following objective function. where is the retailer's risk-aversion parameter. and are representing the expected value and the variance of retailer's profit. When 0, the retailer is risk neutral. As increases, the retailer becomes more risk-averse. The retailer having the mean-standard deviation criterion maximizes the following 34

5 objective function. where is the retailer's risk-aversion parameter. When 0, the retailer is risk neutral. As increases, the retailer is more risk-averse. The retailer having the conditional value-at-risk criterion maximizes the following objective function. where is the -quantile of the retailer's profit. Thus, 0,1 and it reflects the degree of risk-aversion for the retailer. When 1, the retailer is risk neutral. As decreases, the retailer is more risk-averse. For further details, see Rockafellar and Uryasev (2000, 2002), Pflug (2006), and Choi et al. (2011). Table 1 summarizes the notation used in this paper. And the decision sequence protocol is shown in Table 2. Stage 0: Stage 1: Stage 2: Stage 3: Stage 4: Table 1 - The notations of the model demand during the selling season ; 0; distribution function of demand ; is differentiable, strictly increasing and 0 0; density function of demand; retail price; retailer's order quantity; unit wholesale price charged by the manufacturer; manufacturer s unit production cost; retailer s profit; manufacturer s profit; retailer s sensitivity to risk for MV value, where 0; retailer s MV value function; retailer s sensitivity to risk for MS value, where 0; retailer s MS value function; retailer s sensitivity to risk for CVaR measure, where 0,1; retailer s CVaR risk measure; Table 2 - Decision sequence protocol The manufacturer and retailer share the information on the demand and the risk attitudes (type of function and risk parameter). The manufacturer determines the unit wholesale price. The retailer determines the order quantity. A demand is realized. The retailer sells products with the unit retail price. 35

6 RISK AVERSION UNDER THE WHOLESALE PRICE CONTRACT The method of backward induction is used to obtain the equilibrium of the game defined in the previous section. We solve the maximization problems faced by the two players sequentially and backwardly in the decision sequence protocol in Table 2. As it becomes clear, the manufacturer s risk attitude does not matter for the wholesale price contract. Mean-variance (MV) model for the wholesale price contract In Stage 2, given the unit wholesale price, the retailer determines its order quantity so that it maximizes its MV value function, argmax. The retailer s optimal order quantity satisfies the following equation. The detail derivation is shown in the proof of Proposition 1 in Appendix /. (1) It is difficult to obtain the closed form of in general. However, we can easily obtain it numerically as shown later. Using (1), we can derive Proposition 1 that shows the properties of. Proposition 1 (The properties of the optimal order quantity of the retailer having MV criterion) : For the wholesale price contract, (a) is less than the optimal order quantity of the risk-neutral retailer, (b) is strictly decreasing with respect to the wholesale price, and (c) is strictly decreasing with respect to the retailer's sensitivity to risk. Proposition1 states that (a) the risk-averse retailer having MV criterion orders less than risk-neutral retailer, (b) as the wholesale price increases, the risk-averse retailer decreases the order quantity, and (c) as the retailer is more risk averse, the order quantity decreases. It is important that Proposition 1 shows intuitive results, because some risk-averse newsvendor models show counter-intuitive results depending on the model settings. Eeckhoudt et al. (1995) examine the newsvendor model with general utility function and show that increasing the retail price and wholesale price can affect the order quantity in both directions. In Stage 1, anticipating the retailer s order quantity in (1), the manufacturer determines the unit wholesale price that maximizes its profit function,. That is, the manufacturer solves the following maximization problem. The maximization problem shows that the manufacturer is facing the trade-off between the marginal profit and the order quantity, when determining. If the wholesale price decreases, then the order quantity increases (Proposition 1(b)) but the marginal profit decreases. Since decreases with respect to (Proposition 1(c)), the degree of trade-off is affected by the retailer s sensitivity to risk. 36

7 The maximization problem is straightforward if it is unimodal. Unfortunately, the obvious condition based of concave function fails to hold for most common distributions. In the case of the risk neutral retailer, it is known that is unimodal if the demand distribution has an IGFR (increasing generalized failure rate) (Lariviere and Porteus, 2001; Lariviere, 2006). The IGFR assumption is not restrictive since it is widely satisfied by common distributions, such as uniform, normal, truncated normal at zero, and exponential distributions. Proposition 2 shows that is also unimodal if the demand distribution has an IGFR for the case of the risk-averse retailer having MV criterion. Proposition 2 (the uniqueness of the wholesale price for the case of the retailer having MV criterion) : If the demand distribution has an IGFR, then there is a unique wholesale price that maximizes the manufacturer's profit for the case of the risk-averse retailer having MV criterion. Although there is a unique wholesale price that maximizes the manufacturer's profit, it is difficult to obtain the equilibrium outcomes with the closed form. In order to analyze the effect of the retailer's sensitivity to risk on the equilibrium outcomes, we show a numerical example in later section. Mean-standard deviation (MS) model for the wholesale price contract For the case of the retailer having MS criterion, it is difficult to prove some properties analytically as in the MV case due to the complexity of the objective function. Although we cannot show the same properties on the optimal order quantity and the optimal wholesale price as in the MV case analytically, such properties hold in our numerical example. The analytical confirmation is required which is left for future research. In Stage 2, given the unit wholesale price, the retailer determines its order quantity that maximizes its MS value function, argmax. It is difficult to obtain the closed form of in general. However, we can easily obtain it numerically as shown later. Proposition 3 shows the properties of. Proposition 3 (The properties of the optimal order quantity of the retailer having MS criterion) : For the wholesale price contract, is less than the optimal order quantity of the risk-neutral retailer. Proposition 3 states that the risk-averse retailer having MS criterion orders less than risk-neutral retailer. In Stage 1, anticipating the retailer s order quantity, the manufacturer determines the unit wholesale price that maximizes its profit function,. That is, the manufacturer solves the following maximization problem. 37

8 It is also difficult to show that there is a unique optimal wholesale price in general. As we mentioned above, there is a unique in our numerical example. We show a numerical example in later section. Conditional Value-at-Risk (CVaR) model for the wholesale price contract In Stage 2, given the unit wholesale price, the retailer determines its order quantity that maximizes its CVaR measurement, It is known that is determined as follow: We can obtain the optimal expression in closed form. This is an advantage of the CVaR in the risk-averse newsvendor model. Using the optimal order quantity, we can easily obtain Proposition 4 about the properties of. Proposition 4 (The properties of the optimal order quantity of the retailer having CVaR criterion) : For the wholesale price contract, (a) is less than the optimal order quantity of the risk-neutral retailer, (b) is strictly decreasing with respect to the wholesale price, and (c) decreases as the retailer is more risk averse (as decreases). Proposition 4 states that (a) the risk-averse retailer having CVaR criterion orders less than risk-neutral retailer, (b) as the wholesale price increases, the risk-averse retailer decreases the order quantity, and (c) as the retailer is more risk averse, the order quantity decreases. It is consistent with Proposition 1 in the case of MV function. Proposition 4 also shows intuitive results. In Stage 1, anticipating the retailer s order quantity, the manufacturer determines the unit wholesale price that maximizes its profit function,. That is, the manufacturer solves the following maximization problem. argmax.. In the case of CVaR, the manufacturer is also facing the trade-off between the marginal profit and the order quantity, when determining. The degree of trade-off is affected by the retailer s sensitivity to risk through. Chen et al. (2009) shows that if the demand distribution has an IGFR, then the above is also unimodal. Thus, we have Proposition 5 about the uniqueness of the wholesale price that maximizes the manufacturer's profit for the case of the risk-averse retailer having CVaR criterion. 38

9 Proposition 5 (the uniqueness of the wholesale price for the case of the risk-averse retailer having CVaR criterion): If the demand distribution has an IGFR, there is a unique wholesale price that maximizes the manufacturer's profit for the case of the risk-averse retailer having CVaR criterion. Analytical results in the MV and CVaR cases show how the risk-averse retailer determines the order quantity. As the wholesale price increases, the risk-averse retailer decreases the order quantity. As the retailer is more risk averse, the order quantity decreases. The manufacturer is facing the trade-off between the marginal profit and the order quantity determining the wholesale price. The degree of trade-off is affected by the retailer s sensitivity to risk through the order quantity. In the case of MS model, although we cannot show the same properties on the optimal order quantity and the optimal wholesale price as in MV and CVaR cases analytically, such properties hold in our numerical example shown in the following section. NUMERICAL ANALYSIS OF THE EFFECT OF THE RISK AVERSION In order to analyze the effect of the retailer's sensitivity to risk on the equilibrium outcomes, we show a numerical example in this section. All numerical outcomes in this paper are calculated using the R language (R Development Core Team, 2010). Using the equations in the previous section, we calculate the equilibrium outcomes with respect to the retailer s sensitivity to risk. We assume that 4,c1 and demand is normally distributed with 400 and 100. We increase the retailer's sensitivity to risk by the point that the equilibrium order quantity becomes almost half of it from risk-neutral retailer. This is because it is difficult to make direct comparison of the magnitude of the three risk measurements. Thus, we conduct the numerical analyses focusing on the retailer's order quantity. Optimal wholesale price with respect to the retailer s risk aversion Figures 1, 2, and 3 show how the equilibrium wholesale price changes with respect to the risk sensitivity of the retailer having MV, MS, and CVaR criterion, respectively. For the MV and CVaR cases, the manufacturer almost maintains the same level of wholesale price even if the retailer's sensitivity to risk increases. Analytical results in the previous section show that if the retailer's sensitivity to risk increases for the same wholesale price, then the order quantity decreases. Thus, Figures 1 and 3 show that even if the order quantity decreases with respect to retailer s sensitivity to risk, the manufacturer does not reduce the wholesale price to induce the retailer to order more. It is because the negative effect of reducing wholesale price on the manufacturer s revenue by decreasing marginal profit ( ) is roughly greater than the positive effect of it on the revenue by increasing order quantity in the numerical examples. On the other hand, the wholesale price decreases with respect to the retailer's sensitivity to risk for the MS case. In the MS case, the negative effect of reducing wholesale price on the manufacturer s revenue by decreasing marginal profit is less than the positive effect of it on the revenue by increasing order quantity. Thus, the manufacturer reduces the wholesale price. The degree of manufacturer s trade-off between the marginal profit and the order quantity depends on the type of demand distribution and the parameter values, as shown in (1). Thus, we should notice that the behavior of equilibrium wholesale price can change with the condition. What is important here is that the behavior of equilibrium wholesale price is different depending on the risk measurement under the same condition. 39

10 Figure 1 - (MV) vs Figure 2 - (MS) vs Figure 3 - (CVaR) vs Order quantity and supply chain performance given the optimal wholesale price Figures 4, 5, and 6 show how the equilibrium order quantity changes with respect to the risk sensitivity of the retailer having MV, MS, and CVaR criterion, respectively. For the MV and CVaR cases, the manufacturer does not change the wholesale price to react to the retailer's sensitivity to risk. Thus, the magnitude of the retailer's sensitivity to risk affects the order quantity directly. For the MS case, since the manufacturer decreases the wholesale price, the magnitude of the retailer's sensitivity to risk affects the order quantity indirectly. Figure 4 - (MV) vs Figure 5 - (MS) vs Figure 6 - (CVaR) vs Figures 7, 8, and 9 show how the equilibrium manufacturer's profit, the retailer's value, and expected profit of supply chain change with respect to the risk sensitivity of the retailer having MV, MS, and CVaR criterion, respectively. The order quantity determines the supply chain expected profit. Thus, in the MS model, it can be said that the manufacturer's action of decreasing the wholesale price mitigates the effect of risk aversion on the supply chain performance. In risk neutral cases (when =0, =0, and =1), Figures 7, 8, and 9 show that the manufacturer takes most of the profit. In the MS case, Figures 8 shows that the manufacturer s share decreases with respect to the retailer s sensitivity to risk. This is primarily because the manufacturer reduces the wholesale price in the MS case. Rtl s value Mfr s profit SC expected profit Figure 7 - (MV) Supply chain relationships w.r.t. Figure 8 - (MS) Supply chain relationships w.r.t. Figure 9 - (CVaR) Supply chain relationships w.r.t. 40

11 The numerical analyses show that the dynamics of interaction is different depending on the risk measurement. This implies that we should recognize not only the magnitude of sensitivity to risk, but also the risk measurement used in the supply chain. From the view of the manufacturer, the numerical analyses might answer the question of how the wholesale price should be set facing the risk-averse retailer. The numerical analyses might suggest that reducing the wholesale price from that of risk-neutral retailer is optimal for the MS model, and that maintaining the wholesale price is optimal for the MV and CVaR models. We conduct the numerical analyses focusing on the retailer's order quantity because of the difficulty in comparing the magnitude of risk parameters. On the other hand, the numerical analyses imply the meaningful range of each risk parameter. As shown in Figure 2, the equilibrium order quantity immediately decreases for 0 and Analytical results focusing on the magnitude of risk parameter are left for future research. CONCLUSION In this paper, we considered the decentralized supply chain consisting of a risk-averse retailer and a manufacturer who are concerned with only their own objectives. We showed complex effects of risk aversion on the manufacturer-retailer interaction numerically and derived analytical results for the three risk models. The numerical analyses showed that the manufacturer s decision about the wholesale price is different depending on the risk measurement. It affected the resulted supply chain relationships. For the MS model, the manufacturer decreases the wholesale price if the retailer's sensitivity to risk increases. The manufacturer's action (i.e., decreasing the wholesale price), mitigates the effect of risk aversion on supply chain. For the MV model and CVaR model, the manufacturer maintains the wholesale price even if the retailer's sensitivity to risk increases. The risk aversion affects directly on supply chain relationships in the cases. This implies that we should recognize not only the magnitude of sensitivity to risk, but also the risk measurement used in the supply chain. Future research in the risk-averse newsvendor model should be conducted in the context of SCM. REFERENCES Ahmed, S., Cakmak, U. and Shapiro, A. (2007), Coherent risk measures in inventory problems, European Journal of Operational Research, Vol. 182, No. 1, pp Anvari, M. (1987), Optimality criteria and risk in inventory models: The case of the newsboy problem, Journal of the Operational Research Society, Vol. 38, No. 7, pp Artzner, P., Delbaen, F., Eber, J. and Heath, D. (1999), Coherent measures of risk, Mathematical Finance, Vol. 9, No. 3, pp Bar-Shira, Z. and Finkelshtain, I. (1999), Two-moments decision models and utility-representable preferences, Journal of Economic Behavior & Organization, Vol. 38, No. 2, pp Cachon, G. P. (2003), Supply chain coordination with contracts, in De Kok, A. G. and Graves, S. C. (Ed.), Handbooks in Operations Research and Management Science, Elsevier, Amsterdam, Vol. 11, pp Charnes, A. and Cooper, W. W. (1959), Chance-constrained programming, Management Science, Vol. 6, No.1, pp Chen, F. and Federgruen, A. (2000), Mean-variance analysis of basic inventory models, Working paper, Division of Decision, Risk and Operations, Columbia University, New York. Chen, Y., Xu, M. and Zhang, Z. G. (2009), Technical note: A risk-averse newsvendor model under the CVaR criterion, Operations Research, Vol. 57, No. 4, pp Choi, S., Ruszczynski, A. and Zhao, Y. (2011), A multiproduct risk-averse newsvendor with law-invariant coherent measures of risk, Operations Research, Vol. 59, No. 2, pp Choi, T. M., Li, D. and Yan, H. (2008), Mean-variance analysis of a single supplier and retailer supply chain under a returns policy, European Journal of Operational Research, Vol. 184, No. 1, pp Eeckhoudt, L., Gollier, C. and Schlesinger, H. (1995), The risk-averse (and prudent) newsboy, Management 41

12 Science, Vol. 41, No. 5, pp Gan, X., Sethi, S. P. and Yan, H. (2005), Channel coordination with a risk-neutral supplier and a downside-risk-averse retailer, Production and Operations Management, Vol. 14, No. 1, pp Lariviere, M. A. and Porteus, E. L. (2001), Selling to the newsvendor?: An analysis of price-only contracts, Manufacturing & Service Operations Management, Vol. 3, No. 4, pp Lariviere, M. A. (2006), A note on probability distributions with increasing generalized failure rates, Operations Research, Vol. 54, No. 3, pp Lau, H. S. (1980), The newsboy problem under alternative optimization objectives, The Journal of the Operational Research Society, Vol. 31, No. 6, pp Lau, H. S. and Lau, A. H. L. (1999), Manufacturer s pricing strategy and return policy for a single-period commodity, European Journal of Operational Research, Vol. 116, No. 2, pp Markowitz, H. M. (1959), Portfolio Selection, Wiley, New York. Ohmura, S. and Matsuo, H. (2012), The effect of retailer's risk aversion on supply chain performance under a wholesale price contract, The Journal of Japanese Operations Management and Strategy, Vol. 3, No. 1, pp Ohmura, S. (2013), The analysis of risk sensitivity in supply chain coordination: The implications for distribution policies, Doctoral Dissertation, Graduate School of Business Administration, Kobe University. Pflug, G. C. (2006), A value-of-information approach to measuring risk in multi-period economic activity, Journal of Banking & Finance, Vol. 30, No. 2, pp R Development Core Team (2010), R: A Language and Environment for Statistical Computing [computer software]. Available from (accessed on November 11, 2013). Rockafellar, R. and Uryasev, S. (2000), Optimization of conditional value-at-risk, Journal of Risk, Vol. 2, No. 3, pp Rockafellar, R. T. and Uryasev, S. (2002), Conditional value-at-risk for general loss distributions, Journal of Banking & Finance, Vol. 26, No. 7, pp Spengler, J. (1950), Vertical integration and antitrust policy, Journal of Political Economy, Vol. 58, No. 4, pp Tapiero, C. S. (2005), Value at risk and inventory control, European Journal of Operational Research, Vol. 163, No. 3, pp Tsay, A. A., Nahmias, S. and Agrawal, N. (1999), Modeling supply chain contracts: A review, in Tayur, S., Ganeshan, R. and Magazine, M. (Eds.), Quantitative Models for Supply Chain Management, Kluwer Academic Publishers, pp Tsay, A. A. (2002), Risk sensitivity in distribution channel partnerships: implications for manufacturer return policies, Journal of Retailing, Vol. 78, No. 2, pp Van Mieghem, J. A. (2003), Commissioned paper: Capacity management, investment, and hedging: Review and recent developments, Manufacturing & Service Operations Management, Vol. 5, No. 4, pp von Neumann, J. and Morgenstern, O. (1944), Theory of Games and Economic Behavior, Princeton University Press, Princeton, New Jersey. Wang, C. X., Webster, S. and Suresh, N. C. (2009), Would a risk-averse newsvendor order less at a higher selling price? European Journal of Operational Research, Vol. 196, No. 2, pp Wu, J., Li, J., Wang, S. and Cheng, T. (2009), Mean-variance analysis of the newsvendor model with stockout cost, Omega, Vol. 37, No. 3, pp APPENDIX Proof of Proposition 1: (a) Let as the optimal order quantity of the risk-neutral retailer. The expectation of the retailer's profit is as follow: 42

13 Since 0, is concave. And from the first order condition, satisfies 1. (2) The variance of the retailer's profit is as follow: 2 2 (3) The first order derivative of with respect to is as follow: From the first order condition, satisfies Since, 0. Thus, From (2) and (4), we have /. (4) 0,1 and 1 is strictly decreasing with respect to (b) 1. Since and 12 1, the left hand side of (4) is strictly decreasing with respect to from 1 to 0. Therefore, we have a unique and is decreasing with respect to (See Figure 10). (c) Combining the proof of (b) with proves that is decreasing with respect to the. 43

14 Figure 10 - The optimal order quantities for the risk-neutral retailer and for the risk-averse retailer having MV criterion (demand is normally distributed). Proof of Proposition 2: From Proposition 1-(b), there is a one-to-one mapping between and. Let be the unique wholesale price that induces the retailer to order. Since, we consider 0, where is the order quantity such that. is represented as follows: where is a revenue function. The first and second order derivatives of are as follows: where. Define as the order quantities such that 1. If 0 for 0,, then 0 for 0, because 0 from Proposition 1-(b) and for 0,. Therefore, is strictly concave for 0,. Since 0 and 0 for,, is maximized at and strictly decreasing for,. Since /, the manufacturer's profit is unimodal for 0, and the optimal order quantity lie in 0,. Thus, there is a unique wholesale price that maximizes the manufacturer's profit. Thus, we show that 0 for 0, when the demand distribution has an IGFR. 0 for 0, is equivalent to obtained from (4). Thus, is as follow: 0 for 0,. is 2. (5)

15 If the demand distribution has an IGFR, then it is known that 1 is increasing with respect to. And the first term of the right hand side of (5) is as follow: Since for 0, from (4), the above is strictly increasing for 0,. Therefore, is strictly decreasing for 0,, that is 0 for 0,. Proof of Proposition 3: (a) From (3), the standard deviation of the retailer's profit is as follow: 2 The first order derivative of with respect to is as follow: From the first order condition, satisfies 2 Since, 0. Thus, 1 1. From (2) and (6), we have /. (6) 45