Inventory and Pricing Decisions Under Wholesale Price Contract with Social Preferences
-
Shuren Liu
,
Huina Chen
Abstract
This paper introduces the other-regarding preferences coefficients and studies the impact of social preferences on supply chain performance in the price-setting newsvendor setting. It is assumed that the stochastic demand is multiplicative. The manufacturer and retailer play a Stackelberg game. We analyze the impact of the decision-maker’s social preferences on the manufacturer’s optimal wholesale price, the retailer’s optimal retail price and order quantity, the supply chain member’s profits and utilities, and the supply chain system’s profits and utilities under three different cases that only the retailer, only the manufacturer and both are with social preferences. We show that a manufacturer, as a leader, should find a spiteful retailer, while a retailer, as a follower, should find a manufacturer with generous liability, to improve the entire supply chain. Finally, numerical examples are given to illustrate these results.
1 Introduction
A great deal of attention has been paid to supply chain management for over a decade, where supply chain coordination is mainly discussed due to double marginalization in decentralized chains. There is a good deal of analytical modelling literature in operating, starting with Spengler[1], that deals with designing contracts to align incentives (Cachon[2]). However, although the wholesale price contract can not coordinate the supply chain, we focus our analysis on it because it is heavily used in practice and, in fact, may be preferred over more complex contracts due to bounded rationality (Kalkanci, et al.[3,4]). Furthermore, most supply chain contracting models are based on the assumption of self-interested, rational agents and exclude social considerations, such as reciprocity, fairness, and status seeking. However, recent developments in behavioral economics suggest that decision makers may care about reciprocity, fairness, and status in addition to economic benefits (see, for example, Rabin[5], Fehr and Schmidt[6], Charness and Rabin[7], Loch and Wu[8], Urda and Loch[9]). Our work serves to contribute to the recent stream of literature incorporating social preferences into decision making and supply chain interactions[10–12]. Based on the price-setting newsvendor model, this paper studies the coexistence of competition and cooperation in a two-echelon supply chain with wholesale price contract where pricing and ordering decisions are considered simultaneously. On the other hand, we introduce social preferences into decision maker’s utility to study the coopetition in the supply chain.
Fairness has been long recognized as one of the most important factors guiding human interactions in everyday life. It is closely related to the other-regarding preferences, such as status, altruism, reciprocity, so common in the everyday life of individuals, which also plays an important role in the corporate environment. Liu, et al.[13] identified four dimensions of fairness relevant in supplier-buyer relationships: Distributional, procedural, interpersonal, and informational. On the distributional aspect of fairness, Fehr and Schmidt[6] defined fairness as a 50/50 split of profits to study behavior in the fixed-pie ultimatum and dictator games. Decisions which result in advantageous allocations incur disutility with one rate, while disadvantageous allocations incur disutility at a higher rate. Ideal allocation reference points on the fairness preference of inequity aversion are 50/50 or more generally
Based on [6], Cui, et al.[10] first modeled fairness concerns in the context of channel coordination. They develop a model where both parties care about fairness in a bilateral monopoly setting with a supplier and a retailer, and find that the supplier can coordinate the channel using wholesale price contract under sufficient concerns for fairness and linear price-sensitive demand. Demirag, et al.[14] extended [10] to other nonlinear demand functions that are commonly used in the literature, and reveal that the exponential demand function requires less stringent conditions to achieve coordination when only the retailer is fairness-concerned. Katok, et al.[15] extended the model to include incomplete information on the fairness preference coefficients. Ding, et al.[16] extended the model to a quantity discount mechanism, and show that channel coordination can be achieved by setting a simple wholesale price and fixed cost as long as the degree of attention of retailer to supplier’s profit and the retailer’s fairness preference coefficients satisfy certain conditions. Yang, et al.[17] studied cooperative advertising in a distribution channel with fairness concerns and show that channel coordination can be achieved if the retailer has fairness concerns and model parameters satisfy certain condition. Wu and Niederhoff[18] reformulated the existing inequity averse utility function as a piecewise function with clearly comparable parameters and study the impact of fairness concerns on supply chain performance in the newsvendor setting. Recently, Ho, et al.[19] studied a model with a supplier and two-retailer supply chain setting, and investigate the interaction between distributional and peer-induced fairness. On the other hand, by social welfare or competitive preferences proposed in [7––8], Du, et al.[11] discussed how the retailer’s behavior of competitive preferences influences the coordination of supply chain, including the wholesale price contract, the buyback contract and the revenue-sharing contract. They find that the retailer’s behavior of competitive preferences don’t change the state of supply chain coordination. Ge and Hu[20] studied the firms’ altruistic incentives in a supply chain, and showed that the performance in the supply chain in consideration of altruism is between those of scenarios under decentralization and under integration. Shi, et al.[21] studied the effect of altruism on retailer’s and manufacturer’s pricing strategy in two classic dual-channel supply chains by presenting Stackelberg game models.
In this paper, we introduce the other-regarding preferences coefficients and study the impact of social preferences on the inventory and pricing decisions in a two-echelon supply chain with wholesale price contract. It is assumed that the stochastic demand is multiplicative. The manufacturer and retailer play a Stackelberg game. We analyze the impact of the decision maker’s social preferences on the manufacturer’s optimal wholesale price, the retailer’s optimal retail price and order quantity, the supply chain member’s profits and utilities, and the supply chain system’s profits and utilities under three different cases that only the retailer, only the manufacturer and both are with social preferences. We show that a manufacturer, as a leader, should find a spiteful retailer, while a retailer, as a follower, should find a manufacturer with generous liability, to improve the entire supply chain. Finally, numerical examples are given to illustrate these results.
The paper is organized as follows. Section 2 establishes two models of decision-making channel under traditional case. One is the integrated channel and the other is independent channel. Section 3 studies a supply chain selling to the price-setting newsvendor with social preferences. Section 4 gives some numerical examples, and Section 5 is the conclusion.
2 The Integrated Channel and Independent Channel
In order to facilitate the research, we consider a simple distribution channel consisting of one manufacturer and one retailer. The manufacturer is an upstream monopolist in the distributional channel and the retailer is a monopolist seller in the consumer market. The manufacturer and retailer play a Stackelberg game. The manufacturer produces a product at a marginal cost c and offers a wholesale price w to the retailer. The retailer determines the retail price p and order quantity q, and suffers no additional selling costs or charges except the wholesale price. Assume that stochastic demand for the product has the following multiplicative functional form:
where y(p) is a deterministic and decreasing function of the product’s selling price p, and ε is a random factor with distribution function F(.), density function f(.) and a mean value of μ. It is assumed that the probability distribution has support on [A,B] with B > A ≥ 0 and so μ > 0. We let y(p) take the form of y(p) = ap− b, where a > 0 and b > 1. The demand function form is one of the few models that have often been adopted in the literature for studying joint pricing-inventory management in supply chain context (see, for example, Petruzzi and Dada[22], Wang, et al.[23], Song, et al.[24]). In this formulation, the parameter b is the price-elasticity index of demand. The larger the value of b is, the more sensitive the demand is to a change in price. If the price-elasticity index is 1 or less, then a product is defined as price-inelastic; Otherwise, a product is defined as price-elastic. We focus on price-elastic products by assuming b > 1.
Before proceeding to the detailed analysis, we present the following mild assumption about ε. As in [25], we define g(x) = xh(x) as the generalized failure rate of the demand distribution for ε, where
Assumption 1
g(x) is increasing for x ∈ [A,B].
The above increasing generalized failure rate (IGFR) property of ε is indeed satisfied by most of the theoretical distribution used in the literature (see [24] and [25] for details).
2.1 The Integrated Channel
As the baseline of comparison, we first analyze the distribution channel decisions. If the distribution channel is integrated, the manufacturer and the retailer cooperate to maximize the profit of channel. Let ΠI(p,q) denote the expected channel profit for any chosen price p and production quantity q. We have
As in [22], we define
For any fixed z, it follows from (2) that
Let
Note that ap−(b + 1) > 0 due to a > 0. Let G(p) = bcz − (b − 1)[z−Λ(z)]p, then G’(p) = −(b − 1)[z−Λ(z)] < 0, so G(p) is decreasing in p. Therefore, if p > pI(z), we have G(p) < G(pI(z)) = 0, which indicates that
By substituting (3)) into (2), we have by the chain rule
where we use the fact that
in (4). Let H(z) = z − bzF(z) + (b − 1)Λ(z), then H′(z) = [1 − F(z)][1 − bzh(z)], H″(z) = −h(z)H′(z) − b[1 − F(z)][h(z) + zh′(z)]. From Assumption 1, we have g’(z) = h(z) + zh′(z) > 0, then H″(z) < 0 at H′(z) = 0, which implies that H(z) is a unimodal function. On the other hand, it is easy to see that H(A) = A > 0, H(B) = −(b − 1)μ < 0. Thus, we obtain an unique zI in the interval [A, B]. Therefore, from H(z) = 0, we have
In summary, we have the following optimal retail price pI and production quantity qI under the integrated channel case:
2.2 The Independent Channel
If the channel members are independent, the manufacturer and the retailer play a Stackelberg game. The manufacturer first offers a wholesale price w. According to the wholesale price, if the retailer rejects the sale contract, it will lead to zero profits, and the game is over. However, if the retailer accepts the offer, he or she will determine the order quantity q and retail price p to maximize his or her profits according to the given wholesale price w. Under the independent channel case, the retailer’s problem is analogous to the integrated channel’s expected profit except that he or she orders goods at price w instead of producing it at price c. His or her expected profit is
Similar to the analysis under the integrated channel case, the profit function (7) can be written as
Further, we can obtain
Denoting the manufacturer’s profit function by Πm(w), we have
By substituting
where
From
Let
In summary, we have the following optimal wholesale price w0 for the manufacturer, the optimal retail price p0 and order quantity q0 for the retailer under the independent channel case:
3 The Wholesale Price Contract Selling to the Pricing-Setting Newsvendor with Social Preferences
According to [8], social preferences refer to intrinsic concerns for the other party’s welfare, reciprocating a history of a positive relationship, and intrinsic desires for a higher relative payoff compared with the other party’s when status is salient. Intrinsic preferences regarding relationships and status both take into consideration the welfare of the other parties involved in supply chain interactions. In real-life interactions, their effects are usually mixed with each other and with the economically rational pursuit of payoffs, all effects together influencing observed behavior and actions. As in [8], we define the manufacturer and the retailer’s utility functions as follows:
ξ and η are called the “other-regarding parameters” of the two players, which reflects the overall concern that each party has for the other party in his/her utility function. In (12), ξ represents the other-regarding parameter of the manufacturer. If ξ = 0, the manufacturer behaves in a way that is consistent with one not having social preferences and is called traditional. If ξ < 0, the manufacturer incurs disutility as the retailer’s expected profit increases and is called spiteful (or competitive); the larger the absolute value of ξ is, the more spiteful he is. If ξ > 0, the manufacturer increases utility as the retailer’s expected profit increases and is called generous (or altruistic), which implies that the manufacturer cares about improving his or her own expected profit as well as the retailer’s. For convenience, we assume that
Note that if the manufacturer holds fairness preference that ideal allocation reference points on the fairness preference of inequity aversion are
When both the manufacturer and the retailer are with social preferences, it is easy to see the corresponding nine combinations: (ξs,ξt,ξg) × (ηs,ηt,ηg), where s, t and g represent spiteful, traditional and generous preference respectively, and
Next, we prove Lemma 1.
Lemma 1
Assume that
Proof
To prove that ξ + b − bξη − 1 > 0 is equivalent to prove that
Since b > 1 and
Note that ηs > − 1, then −ξsηs > ξs, i.e., 1 − ξsηs > 1 + ξs > 0, thus
When both the manufacturer and the retailer are with social preferences, the retailer’s utility function is
and the manufacturer’s utility function is
For any fixed z, it follows from (14) that
Let
Note that ap−(b + 1) > 0 due to a > 0. Let R(p) = (1 − b)[z−Λ(z)]p + b(1 − η)zw + bηzc, then R′(p) = (1 − b)[z−Λ(z)] < 0, so R(p) is decreasing in p. Therefore, if
Thus, by (16), we have
By substituting (17) into (15), we have
For any fixed z, by (18), we have
Let
Note that ap−(b + 1) > 0 due to a > 0. Let
By Lemma 1, we have
Further, by substituting (20) into (18), we have
Thus,
Note that
On the other hand, by substituting (20) into (17), we have
In summary, we have the following optimal wholesale price w* for the manufacturer, the optimal retail price p* and order quantity q* for the retailer when both the manufacturer and the retailer are with social preferences:
3.1 Only the Retailer Is with Social Preferences
Let ξ = 0 in (23), then, we have the following optimal wholesale price w* for the manufacturer, the optimal retail price p* and order quantity q* for the retailer when only the retailer is with social preferences:
Compared to the independent channel case, it is easy to see that that F(z*) = F(z0), p*(z) = p0(z), q*(z) = q0(z). Therefore, the only influential decision variable is the manufacturer’s wholesale price when only the retailer is with social preferences.
Next, we study how the retailer’s other-regarding parameter η affects the manufacturer’s wholesale price. Note that
If η = 0, we have
If 0 < η < 1, we have w* − w0 > 0, i.e., w* > w0, which implies that the manufacturer’s optimal wholesale price is greater than that in the traditional one when the retailer is generous. We explain this phenomenon as follows: The generous retailer orders more than a traditional retailer when the manufacturer sets the same wholesale price. With this issue, the manufacturer facing a generous retailer will set a wholesale price larger than that facing a traditional retailer. The decreasing trend of the retailer’s order quantity on the wholesale price is moderated by the retailer’s generosity. It is then optimal for the manufacturer to increase the wholesale price to the point at which the generous retailer will order exactly the traditional quantity. So, when only the retailer is with social preferences, the manufacturer’s optimal wholesale price is affected, but the optimal retail price and order quantity for the retailer are the same as those in traditional one. That is to say, the retailer’s other-regarding parameter affects the profit allocation between the manufacturer and the retailer.
If − 1 < η < 0, we have w* − w0 = < 0, i.e., w* < w0, which implies that the manufacturer’s optimal wholesale price is smaller than that in the traditional one when the retailer is spiteful. Under this case, we can implement the analogous analysis as 0 < η < 1.
In the following, we study the influence of the retailer’s other-regarding parameter on the profits and utilities of the supply chain member and the whole system.
Theorem 1
When only the retailer is with social preferences, the retailer’s profit decreases as η increases, the manufacturer’s profit increases as η increases, and the system’s profit is independent of η.
Proof
By substituting (24) into (8), we have
then,
By substituting (24) into (9), we have
then,
Moreover, the supply chain system’s profit is
Note that the above forma is dependent on p and z, but independent of w. Furthermore, by (24), it is easy to see that p and z are independent of η. Thus, the system’s profit of the supply chain is independent of η. That is to say, the retailer’s other-regarding parameter has no effects on the supply chain system’s profit.
Theorem 2
When only the retailer is with social preferences, the retailer’s utility is independent ofη, the supply chain system’s utility increases asηincreases, thus the manufacturer’s utility increases asηincreases.
Proof
By (14), the retailer’s utility function is
By substituting (24) into the above forma, we have
This implies that the retailer’s utility function is independent of η. That is to say, the retailer’s utility remains unchanged when only the retailer is with social preferences.
The supply chain system’s utility is
By substituting (24) into the above forma, we have
then,
Thus, the supply chain system’s utility increases as η increases. This, in conjunction with that the retailer’s utility remains unchanged, shows that the manufacturer’s utility increases as η increases.
3.2 Only the Manufacturer Is with Social Preferences
Let η = 0 in (23), then, we have the following optimal wholesale price w* for the manufacturer, the optimal retail price p* and order quantity q* for the retailer when only the manufacturer is with social preferences:
Obviously, compared with the case that only the retailer is with social preferences, the manufacturer’s other-regarding parameter ξ has effects on the optimal wholesale price for the manufacturer, the optimal retail price and order quantity for the retailer.
Note that F(z*) = F(z0), thus z* = z0. Furthermore,
Next, we study how the degree of the manufacturer’s other-regarding parameter ξ affects the manufacturer’s wholesale price, the retailer’s optimal order quantity and retail price. We consider the following three cases:
If ξ = 0, i.e., the manufacturer is traditional, we have w* = w0; p* = p0; q* = q0.
If 0 < ξ < 1, i.e., the manufacturer is generous, we have w* − w0 < 0, i.e., w* < w0, which implies that the manufacturer sets the wholesale price smaller than the traditional one; p* − p0 < 0, i.e., p* < p0, which implies that the retailer’s optimal retail price is smaller than the traditional one;
If
In the following, we study the influence of the manufacturer’s other-regarding parameter on the profits and utilities of the supply chain member and the whole system.
Theorem 3
When only the manufacturer is with social preferences, the retailer and the whole system’s profits increase as ξ increases; the manufacturer’s profit first increases, then decreases asξincreases, and the manufacturer’s profit achieves the maximization as ξ = 0.
Proof
By substituting (25) into (8) and (9) respectively, we have
Then,
Moreover, the system’s profit is
Note that
then
which implies that u(ξ) is increasing in ξ. Thus, Π* is increasing in ξ, which implies that the system’s profit increases as ξ increases.
Theorem 4
When only the manufacturer is with social preferences, the manufacturer’s, the retailer’s and the system’s utilities of the supply chain all increase asξincreases.
Proof
By (15), the manufacturer’s utility function is
By substituting (25) into the above forma, we have
Then,
which implies that the manufacturer’s utility increases as ξ increases. Similarly, the retailer’s utility function is
By substituting (25) into the above forma, we have
Then,
Moreover, the system’s utility function is
Note that
then
which implies that v(ξ) is increasing in ξ. Thus, U* is increasing in ξ, which implies that the system’s utility increases as ξ increases.
By Theorems 1 and 3, the retailer’s other-regarding parameter η does not affect the supply chain system’s profit; but the manufacturer’s other-regarding parameter ξ has significant effects on the supply chain system’s profit. The more generous the manufacturer is, the more the system’s profit is, vice versa. The essential reason is that the manufacturer is a leader in this Stackelberg game. We will explore further this point in Subsection 3.3.
3.3 Both the Manufacturer and the Retailer Are with Social Preferences
When both the manufacturer and the retailer are with social preferences, let pij, qij and Πij (i, j = s, g, t) denote the retailer’s optimal retail price, order quantity and the system’s profit given that the manufacturer’s and the retailer’s other-regarding parameter preferences are i and j, respectively.
Theorem 5
When both the manufacturer and the retailer are with social preferences, the optimal retail prices under nine combinations satisfy the following relations: pgs < pgt < pgg < ptg = ptt = pts < psg < pst < pss; the optimal order quantities under nine combinations satisfy the following relations: qgs > qgt > qgg > qtg = qtt = qts > qsg > qst > qss.
Proof
First, we consider the relationships between the optimal retail prices under nine combinations. By (23), we have
Note that
i.e.,
Note that
i.e.,
Note that
i.e.,
Note that
i.e.,
Note that
i.e.,
Note that
i.e.,
Moreover, by Subsection 3.1, we have ptg = ptt = pts when only the retailer is with social preferences. Thus, pgs < pgt < pgg < ptg = ptt = pts < psg < pst < pss.
Next, we consider the relationships between the optimal order quantities under nine combinations. By (23), q = ap−bz, then
Theorem 6
When both the manufacturer and the retailer are with social preferences, the optimal system’s profits under nine combinations satisfy the following relations: Πgs > Πgt > Πgg > Πtg = Πtt = Πts > Πsg > Πst > Πss.
Proof
By (8) and (9), the system’s profit under the independent channel case is
Let
then
By (23), (b − 1)Λ(z) + z = bzF(z). Thus, r′(p) = ap−bbz[F(z)−
In the following, we prove that F(z) −
This, in conjunction with r′(p) = ap−bbz[F(z) −
By Theorem 6, the supply chain suffers the most when the manufacturer and the retailer are spiteful. However, the supply chain performance is not the best when both players are generous, but instead when the manufacturer is generous and the retailer is spiteful. Therefore, to improve the entire supply chain (i.e., with larger profit), it is necessary that the manufacturer is generous. Otherwise, the system’s profit diminishes whether the retailer is generous or not. The essential reason is that the manufacturer is a leader in this Stackelberg game. Thus, we obtain the following important insights: The manufacturer, as the leader, should be in charge of the supply chain. The retailer, as the follower, is always in passive status no matter his or her type is. That is, the retailer should do his or her best work without considering the whole supply chain. Therefore, the manufacturer behaves like a government with considering the total social utility of the nation; the retailer behaves like persons in a nation: Believing that doing his or her best will benefit the total social utility of the nation, as explained by Adam Smith’s theory of the invisible hand in the market.
In addition, the relationships are complex on the manufacturer’s optimal wholesale price, the supply chain member and system’s profits and utilities under nine combinations. We will illustrate them in the next section.
At last, by (23), as ξ approaches to 1, the retailer’s optimal order quantity and retailer price are the same as (6) under the integrated channel case, and the manufacturer’s optimal wholesale price equals to marginal product cost c. The extremely generous manufacturer coordinates the supply chain.
4 Numerical Examples
For price-dependent stochastic demand function D(p, ε) = y(p)ε, it is assumed that ε follows the uniform distribution on the interval [0, h], then
Further, let h = 2, and assume that y(p) = 1000p−2, i.e., a = 1000 and b = 2. Moreover, the manufacturer’s marginal cost c equals to 2. By (11), we can obtain that the manufacturer’s optimal wholesale price is w0 = 4, the retailer’s optimal order quantity q0 = 9.259 and optimal retail price p0 = 12, and the system’s optimal profit is 55.556 under the independent channel case.
Let ξ = 0 in (26), we study that the retailer’s other-regarding parameter η has effects on the supply chain system. By (26), we can obtain that the manufacturer’s optimal wholesale price w*, the retailer’s optimal order quantity q* and optimal retail price p*; Furthermore, we can obtain the retailer’s profit
The supply chain system when only the retailer is with social preferences
| η | w* | q* | p* | Π* | U* | |||
|---|---|---|---|---|---|---|---|---|
| −0.8 | 3.111 | 9.259 | 12.000 | 45.268 | 37.037 | 10.288 | 55.556 | 47.325 |
| −0.6 | 3.25 | 9.259 | 12.000 | 43.982 | 37.037 | 11.574 | 55.556 | 48.611 |
| −0.4 | 3.429 | 9.259 | 12.000 | 42.328 | 37.037 | 13.228 | 55.556 | 50.265 |
| −0.2 | 3.667 | 9.259 | 12.000 | 40.124 | 37.037 | 15.432 | 55.556 | 52.469 |
| 0 | 4.000 | 9.259 | 12.000 | 37.037 | 37.037 | 18.519 | 55.556 | 55.556 |
| 0.2 | 4.500 | 9.259 | 12.000 | 32.407 | 37.037 | 23.148 | 55.556 | 60.185 |
| 0.4 | 5.330 | 9.259 | 12.000 | 24.691 | 37.037 | 30.864 | 55.556 | 67.901 |
| 0.6 | 7.000 | 9.259 | 12.000 | 9.259 | 37.037 | 46.296 | 55.556 | 83.333 |
| 0.8 | 12.000 | 9.259 | 12.000 | −37.037 | 37.037 | 92.593 | 55.556 | 129.630 |
Similarly, let η = 0 in (26), we study that the manufacturer’s other-regarding parameter ξ has effects on the supply chain system. Note that we assume that
The supply chain system when only the manufacturer is with social preferences
| ξ | w* | q* | p* | Π* | U* | |||
|---|---|---|---|---|---|---|---|---|
| −0.8 | 20.000 | 0.370 | 60.000 | 7.407 | 6.667 | 0.741 | 14.074 | 8.148 |
| −0.6 | 10.000 | 1.481 | 30.000 | 14.815 | 11.852 | 2.963 | 26.667 | 17.778 |
| −0.4 | 6.667 | 3.333 | 20.000 | 22.222 | 15.556 | 6.667 | 37.778 | 28.889 |
| −0.2 | 5.000 | 5.926 | 15.000 | 29.630 | 17.778 | 11.852 | 47.407 | 41.482 |
| 0 | 4.000 | 9.259 | 12.000 | 37.037 | 18.519 | 18.519 | 55.556 | 55.556 |
| 0.2 | 3.333 | 13.333 | 10.000 | 44.444 | 17.778 | 26.667 | 62.222 | 71.111 |
| 0.4 | 2.857 | 18.148 | 8.571 | 51.852 | 15.556 | 36.297 | 67.407 | 88.149 |
| 0.6 | 2.500 | 23.704 | 7.500 | 59.259 | 11.852 | 47.407 | 71.111 | 106.667 |
| 0.8 | 2.222 | 30.000 | 6.667 | 66.667 | 6.667 | 60.000 | 73.333 | 126.667 |
Furthermore, let ξ = −0.2 and ξ = 0.5 in (26), we study that the retailer’s other-regarding parameter η has effects on the supply chain system, respectively. Under both cases, we obtain Tables 3 and 4, respectively.
The supply chain system as ξ = −0.2
| η | w* | q* | p* | Π* | U* | ||||
|---|---|---|---|---|---|---|---|---|---|
| −0.8 | 4.778 | 3.023 | 21.000 | 27.883 | 21.164 | 8.398 | 2.822 | 36.281 | 23.986 |
| −0.6 | 4.679 | 3.750 | 18.857 | 29.595 | 23.569 | 10.044 | 4.125 | 39.639 | 27.694 |
| −0.4 | 4.679 | 4.481 | 17.250 | 30.566 | 25.765 | 12.002 | 5.889 | 42.568 | 31.654 |
| −0.2 | 4.778 | 5.208 | 16.000 | 30.671 | 27.778 | 14.468 | 8.333 | 45.139 | 36.111 |
| 0 | 5.000 | 5.926 | 15.000 | 29.630 | 29.630 | 17.778 | 11.852 | 47.407 | 41.482 |
| 0.2 | 5.409 | 6.629 | 14.182 | 26.819 | 31.339 | 22.600 | 17.237 | 49.419 | 48.576 |
| 0.4 | 6.167 | 7.316 | 13.500 | 20.729 | 32.922 | 30.483 | 26.337 | 51.212 | 59.259 |
| 0.6 | 7.769 | 7.984 | 12.923 | 6.756 | 34.392 | 46.060 | 44.709 | 52.816 | 79.101 |
| 0.8 | 12.714 | 8.632 | 12.429 | −38.226 | 35.760 | 92.483 | 100.128 | 54.256 | 135.888 |
The supply chain system as ξ = 0.5
| η | w* | q* | p* | Π* | U* | ||||
|---|---|---|---|---|---|---|---|---|---|
| −0.8 | 2.242 | 24.991 | 7.304 | 65.676 | 60.847 | 6.036 | 38.874 | 71.712 | 99.721 |
| −0.6 | 2.298 | 24.162 | 7.429 | 64.144 | 59.829 | 7.191 | 39.263 | 71.335 | 99.092 |
| −0.4 | 2.376 | 23.212 | 7.579 | 62.133 | 58.642 | 8.726 | 39.793 | 70.859 | 98.435 |
| −0.2 | 2.490 | 22.115 | 7.765 | 59.407 | 57.239 | 10.841 | 40.544 | 70.248 | 97.783 |
| 0 | 2.667 | 20.833 | 8.000 | 55.556 | 55.556 | 13.889 | 41.667 | 69.444 | 97.222 |
| 0.2 | 2.962 | 19.319 | 8.308 | 49.783 | 53.498 | 18.576 | 43.467 | 68.359 | 96.965 |
| 0.4 | 3.515 | 17.506 | 8.727 | 40.316 | 50.926 | 26.524 | 46.682 | 66.840 | 97.608 |
| 0.6 | 4.778 | 15.306 | 9.333 | 22.109 | 47.619 | 42.517 | 53.571 | 64.626 | 101.191 |
| 0.8 | 9.143 | 12.603 | 10.286 | −28.807 | 43.210 | 90.021 | 75.617 | 61.214 | 118.827 |
Obviously, Table 1 shows that Theorems 1–2 hold, and the optimal retail price and order quantity for the retailer are independent of the retailer’s other-regarding parameter η, but the manufacturer’s optimal wholesale price is affected by η. Table 2 shows that Theorems 3–4 hold, and the manufacturer’s other-regarding parameter ξ has effects on the optimal wholesale price for the manufacturer, the optimal retail price and order quantity for the retailer. Further, we can conclude from Tables 1, 3 and 4 that: pgs < pgt < pgg < ptg = ptt = pts < psg < pst < pss; qgs > qgt > qgg > qtg = qtt = qts > qsg > qst > qss; Πgs > Πgt > Πgg > Πtg = Πtt = Πts > Πsg > Πst > Πss.
5 Conclusions
In this paper, we introduce the other-regarding preferences coefficients and study the impact of social preferences on supply chain performance in the price-setting newsvendor setting. It is assumed that the stochastic demand is multiplicative. The manufacturer and retailer play a Stackelberg game. We analyze the impact of the decision-maker’s social preferences on the manufacturer’s optimal wholesale price, the retailer’s optimal retail price and order quantity, the supply chain member’s profits and utilities, and the supply chain system’s profits and utilities under three different cases that only the retailer, only the manufacturer and both are with social preferences. One main insight of this paper is that a supply chain will benefit when the manufacturer is generous, having concern for the total profit of the supply chain, and the retailer is spiteful, only having concern for herself.
There are several directions deserving future research. First, we study a typical channel consisting of one manufacturer and one retailer in this paper. Thus, it is more interesting to study a channel with a single manufacturer and multiple retailers. Second, we assume the manufacturer to be the Stacklberg leader in the game, but there are practical examples of large retailers (e.g., Walmart) as channel leaders. Thus, it is another interesting direction that the retailer acts as the Stacklberg leader of the channel.
References
[1] Spengler J J. Vertical integration and antitrust policy. The Journal of Political Economy, 1950, 58(4): 347–352.10.1086/256964Search in Google Scholar
[2] Cachon G P. Supply chain coordination with contracts. The Handbook of Operations Research and Management Science: Supply Chain Management. Amsterdam, Netherlands: Kluwer Academic Publishers, 2003.Search in Google Scholar
[3] Kalkanci B, Chen K Y, Erhun F. Contract complexity and performance under asymmetric demand information: An experimental evaluation. Management Science, 2011, 57(4): 689–704.10.1287/mnsc.1110.1318Search in Google Scholar
[4] Kalkanci B, Chen K Y, Erhun F. Complexity as a contract design factor: A human-to-human experimental study. Production and Operations Management, 2014, 23(2): 269–284.10.1111/poms.12067Search in Google Scholar
[5] Rabin M. Incorporating fairness into game theory and economics. American Economic Review, 1993, 83(5): 1281–1302.10.2307/j.ctvcm4j8j.15Search in Google Scholar
[6] Fehr E, Schmidt K M. A theory of fairness, competition and cooperation. The Quarterly Journal of Economics, 1999, 114(3): 817–868.10.2307/j.ctvcm4j8j.14Search in Google Scholar
[7] Charness G, Rabin M. Understanding social preferences with simple tests. The Quarterly Journal of Economics, 2002, 117(3): 817–869.10.1162/003355302760193904Search in Google Scholar
[8] Loch C H, Wu Y. Social preferences and supply chain performance: An experimental study. Management Science, 2008, 54(11): 1835–1849.10.1287/mnsc.1080.0910Search in Google Scholar
[9] Urda J, Loch C H. Social preferences and emotions as regulators of behavior in processes. Journal of Operations Management, 2013, 31: 6–23.10.1016/j.jom.2012.11.007Search in Google Scholar
[10] Cui T H, Raju J S, Zhang Z J. Fairness and channel coordination. Management Science, 2007, 53(8): 1303–1314.10.1287/mnsc.1060.0697Search in Google Scholar
[11] Du S F, Du C, Liang L, et al. Supply chain coordination considering fairness concerns. Journal of Management Sciences in China, 2010, 13(11): 41–48.Search in Google Scholar
[12] Katok E, Pavlov V. Fairness in supply chain contracts: A laboratory study. Journal of Operations Manangement, 2013, 31: 129–137.10.1016/j.jom.2013.01.001Search in Google Scholar
[13] Liu Y, Huang Y, Luo Y, et al. How does justice matter in achieving buyer-supplier relationship performance? Journal of Operations Manangement, 2012, 30(5): 355–367.10.1016/j.jom.2012.03.003Search in Google Scholar
[14] Demirag C O, Chen Y H, Li J B. Channel coordination under fairness concerns and nonlinear demand. European Journal of Operational Research, 2010, 207(3): 1321–1326.10.1016/j.ejor.2010.07.017Search in Google Scholar
[15] Katok E, Olsen T, Pavlov V. Wholesale pricing under mild and privately known concerns for fairness. Production and Operations Management, 2014, 23(2): 285–302.10.1111/j.1937-5956.2012.01388.xSearch in Google Scholar
[16] Ding C, Wang K, Lai S. Channel coordination mechanism with retailers having fairness preferences. Journal of Industrial and Management Optimization, 2013, 9(4): 967–982.10.3934/jimo.2013.9.967Search in Google Scholar
[17] Yang J, Xie J X, Deng X, et al. Cooperative advertising in a distribution channel with fairness concerns. European Journal of Operational Research, 2013, 227(2): 401–407.10.1016/j.ejor.2012.12.011Search in Google Scholar
[18] Wu X, Niederhoff J. Fairness in selling to the newsvendor. Production and Operations Management, 2014, 23(11): 2002–2022.10.1111/poms.12222Search in Google Scholar
[19] Ho T H, Su X, Wu Y. Distributional and peer-induced fairness in supply chain contract design. Production and Operations Management, 2014, 23(2): 161–175.10.1111/poms.12064Search in Google Scholar
[20] Ge Z H, Hu Q Y. Who benefits from altruism in supply chain management? American Journal of Operations Research, 2012, 2: 59–72.10.4236/ajor.2012.21007Search in Google Scholar
[21] Shi K R, Jiang F, Ouyang Q. Altruism and pricing strategy in dual-channel supply chains. American Journal of Operations Research, 2013, 3: 402–412.10.4236/ajor.2013.34038Search in Google Scholar
[22] Petruzzi N C, Dada M. Pricing and the newsvendor problem: A review with extensions. Operations Research, 1999, 47(2): 183–194.10.1287/opre.47.2.183Search in Google Scholar
[23] Wang Y Z, Li J, Shen Z J. Channel performance under consignment contract with revenue sharing. Management Science, 2004, 50(1): 33–47.10.1287/mnsc.1030.0168Search in Google Scholar
[24] Song Y Y, Ray S B, Li S L. Structural properties of buyback contracts for price-setting newsvendors. Manufacturing and Service Operations Management, 2008, 10(1): 1–18.10.1287/msom.1060.0140Search in Google Scholar
[25] Lariviere M A, Porteus E L. Selling to the newsvendor: An analysis of price-only contracts. Manufacturing and Service Operations Management, 2001, 3(4): 293–305.10.1287/msom.3.4.293.9971Search in Google Scholar
© 2016 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Decision-Making of an Integrated Supply Chain with Hybrid Sales Channels to Cope with Dual Cost Disruptions
- Impacts of Hyperbolic Discounting on Inventory Replenishment Policy Under Inflation
- Method and Application of Operational Architecture Modeling Based on Information Flow Analysis
- Impact of Repatriate’s Knowledge Transfer on Enterprise Performance: The Mediating Effect of Ambidexterity Innovation
- Inventory and Pricing Decisions Under Wholesale Price Contract with Social Preferences
- Smoothing Approximation to the Square-Root Exact Penalty Function
Articles in the same Issue
- Decision-Making of an Integrated Supply Chain with Hybrid Sales Channels to Cope with Dual Cost Disruptions
- Impacts of Hyperbolic Discounting on Inventory Replenishment Policy Under Inflation
- Method and Application of Operational Architecture Modeling Based on Information Flow Analysis
- Impact of Repatriate’s Knowledge Transfer on Enterprise Performance: The Mediating Effect of Ambidexterity Innovation
- Inventory and Pricing Decisions Under Wholesale Price Contract with Social Preferences
- Smoothing Approximation to the Square-Root Exact Penalty Function
