Optimization of a Two-Echelon Supply Chain Considering Consumer Low-Carbon Preference

Abstract

This paper considers a fresh food supply chain with a supplier who takes responsibility for the cold chain and a retailer who needs to reprocess the fresh food. Carbon emissions will be produced in the processes of production, transportation, processing, etc. We consider the four-stage game, obtain the function expressions of optimal market prices with respect to carbon emission reduction level (CERL), analyze the best responses of the supplier and the retailer regarding their CERLs, and obtain the 25 optimal CERLs under competitive equilibrium. In 24 of the 25 equilibrium cases, the supplier or the retailer either do nothing to reduce carbon emissions, or make the most effort to reduce carbon emissions. Excluding these special cases, we focused on a non-trivial case where the increasing consumer preferences for low-carbon products will encourage the supplier and the retailer to reduce carbon emissions. Interestingly, we find that when the consumer preference for low-carbon products is low, the retailer’s and supplier’s equilibrium carbon reduction levels are low, so that the potential market size is small such that the competition for two kinds of customers is fierce. Then, an increase in the sale cost will reduce the retail price. However, when the consumer preference for low-carbon products is high, the potential market size is large such that the competition is not fierce. Then, an increase in the sale cost will advance the retail price.

1. Introduction

Fresh food is necessary for people. The data from the National Bureau of Statistics of China shows that the consumption of fresh food by Chinese residents reached 3.1 trillion tons in 2019. Along with an enhancement in consumption, people have an unceasingly increasing demand for fresh food. In January 2021, the World Economic Forum (WEF) released a report “Net Zero Carbon Emission Challenge: Opportunities for Supply Chain”, which pointed out that the carbon emissions of the food industry ranked first among the eight major industries, even exceeding the manufacturing industry, accounting for 25% of global carbon emissions. The International Research Institute of Green Finance at the Central University of Finance and Economics points out that farming, animal feeding, and food processing will all generate a large amount of carbon emissions. Among the kinds of food that people often eat, beef ranks first in carbon emissions. Fresh food is perishable and must be transported and stored at low temperatures. In this process, the use of energy contributes a lot of carbon emissions. The carbon emissions of 1 ton of fresh meat in its life cycle are 166.97 kg, of which the carbon emissions in the production, transportation, storage, and sales account for 28.68%, 52.37%, 8.90%, and 10.04% of the total supply chain, respectively. Note that transport produces the highest carbon emissions [1]. The cold chain emissions of fruits and vegetables increased from 3.2157 million tons in 2007 to 4.11 million tons in 2016, with an average annual increase of 9.84 million tons of carbon emissions [2].

With the gradual promotion of a low-carbon lifestyle and the requirement for continuous improvements in people’s quality of life, more and more people are inclined to buy low-carbon and green food, and even willing to pay higher prices. According to iResearch’s 2017 China Online Fresh Food Consumption Research Report, 57% of consumers first consider freshness, green safety certification, and other product quality and safety factors when purchasing fresh agricultural products online. The 2021 China Sustainable Consumption Report shows that more than 90% of consumers are willing to pay a premium for low-carbon food, and more than half of consumers are willing to pay a premium of more than 10%. Wang et al. [3] show that consumers’ low-carbon preferences increase manufacturers’ investment in carbon reduction. Zhou and Wu [4] consider carbon emission reduction decisions and supply chain profit in a retailer-led low-carbon supply chain. On the one hand, the increase in the investment in preservation will increase energy consumption and generate more carbon emissions [5]; on the other hand, consumers’ high requirements for freshness and gradually increasing low-carbon consumption preference are also important considerations in the decision-making process of enterprises, which have an important impact on stimulating enterprises to reduce the level of cold chain and carbon emissions. Therefore, it is of great significance to consider the consumers’ preference for freshness and low carbon at the same time in the fresh food supply chain decision-making problem.

Freshness is one of the most important factors for consumers to consider when purchasing fresh food, which will affect the decisions of enterprises and supply chains. Therefore, food safety experts and supply chain experts have carried out a lot of research. Wang and Dan [6] take into account the different preferences of different types of consumers for the freshness of fresh agricultural products, construct a consumer time-varying utility function of freshness and price changing with time, and analyze the retailer’s optimal fresh-keeping decision and optimal pricing decision. Cao et al. [7] consider the optimal pricing strategy and supplier freshness effort selection problem in a supply chain consisting of two suppliers and a retailer. Wang et al. [8] consider the joint influence of the suppliers’ preservation effort levels and the time required to transport the products on the freshness of the products, and further study the optimal preservation effort level and the optimal pricing cost-sharing contract in the centralized and decentralized supply chain, on the basis of considering the impact of freshness on consumer behavior. Gao et al. [9] consider the effects of the freshness-keeping effort on the coordination of a fresh agricultural product supply chain. Dan et al. [10] studies the information sharing strategy among supply chain members. Ye et al. [11] constructs a model of fresh food consumption rate on the degree of cold chain logistics efforts, and analyzes the optimal decisions of supply chain members in the degree of cold chain logistics efforts and pricing under different trade models. Cai et al. [12] studies the optimal ordering strategy, optimal freshness maintenance effort level, and optimal sales price strategy of fresh food distributors. Bo et al. [13] analyzes the value loss of fresh agricultural products under different transportation modes, studies the conditions for suppliers to choose cold chain transportation or ordinary transportation, and the decisions of retailers’ revenue-sharing contract or cost-sharing contract under different transportation modes.Cao et al. [14] takes the supplier-led dual-channel supply chain of fresh agricultural products into consideration, considers the impact of freshness decay on consumer demand, and discusses the supply chain coordination problem in centralized and decentralized models.

With the growing awareness of low carbon and food health and safety, more and more consumers will pay attention to whether the food package is marked with low-carbon information when purchasing food [15,16]. Especially, well-educated people with fixed income are more willing to buy low-carbon food [17]. Singh et al. [18] point out that beef has the highest carbon footprint among all agricultural foods, and most of its emissions are generated in beef farms. It is necessary to strengthen the vertical coordination of the supply chain between farmers, slaughterhouses, and processors. Wang et al. [19] take the fresh food supply chain composed of a large supplier and several small retailers into consideration, and consider the replenishment and pricing strategies of suppliers and retailers in the context of carbon trading.

Recently, researchers have noticed that freshness and low carbon are both considerations when consumers buy fresh food. In a dual-channel supply chain of fresh food, Xie et al. [16] considers the impact of carbon emissions and the freshness of fresh food on consumer behavior, and studies the optimal decision for carbon emission reduction and pricing in the supply chain under three dual-channel models. Dan et al. [10] takes the three-level fresh food supply chain composed of suppliers, third-party logistics, and retailers into consideration, and analyzes the cold chain preservation, carbon emission reduction, and pricing strategies. However, the two papers assume that the freshness is constant, and do not consider the effect of time on the decay of fresh food. Yang and Yao [20] considers that freshness and the carbon emissions of fresh food have an impact on consumers’ purchasing behavior, builds a mathematical model to describe the relationship between the efforts of fresh food producers to maintain freshness and low carbon and consumer demand, and points out that the cost-sharing contract and two-part pricing strategy can coordinate the fresh food supply chain. In terms of freshness, when consumers choose fresh food, they directly perceive the freshness of the food itself rather than the efforts of the enterprise. Therefore, based on the above considerations, this paper establishes a time-varying demand model. The demand changes with the decay of freshness, as well as the low-carbon level of the product.

To sum up, the existing relevant research mainly focuses on the impact of fresh food’s level of freshness on the demand and supply chain, and the research literature from a low-carbon perspective is not deep and comprehensive. The papers consider freshness and low carbon at the same time, ignoring the influence of the decay of freshness with time. Therefore, this paper investigates a three-level fresh food supply chain composed of a fresh food supplier, a retailer, and consumers. Two types of consumers are considered, one type of consumer is only sensitive to price and freshness, the other type of consumer is also sensitive to the low-carbon level. The paper studies the pricing and carbon reduction level, and the freshness maintenance level strategy. Compared with the existing research, the innovation of this paper lies in: (1) considering the impact of different types of consumers on the fresh food supply chain, because consumers will be affected by their education level, wage level, and other factors when purchasing low-carbon food or non-low-carbon food, so it is of practical significance to explore the impact of consumer preferences on supply chain decision making; (2) considering the influence of time on the decay rate of fresh food’s freshness; (3) studying a four-stage game model in a fresh food supply chain and obtaining the optimal market pricing, freshness decay control level, and carbon emission reduction strategies.

The rest of the paper is organized as follows. In Section 2, we introduce the model in detail. In Section 3, we analyze a four-stage game among the supplier, the retailer, and the consumers, and obtain the optimal market prices for low-carbon and non-low-carbon fresh food, and the competitive equilibrium of carbon emission levels. Finally, we conclude the paper in Section 4. All of the proofs of propositions can be found in Appendix A.

2. Model

We consider decision optimization of a two-echelon fresh food supply chain consisting of a supplier and a retailer during the sales period $[0,T]$ (see Figure 1). The supplier directly sells the food to the retailer and takes responsibility for the transportation, storage, and cold chain services. At time $t=0$, the retailer places a purchase order with the supplier, and then the supplier delivers the fresh food to the retailer. At moment $t=t_1$, the retailer finishes the processing and starts selling until the end of the sales period T.

Fresh food is naturally perishable and the freshness deteriorates faster over time, which affects the fresh food market. The freshness of the fresh food $\theta \left(t\right)$ decays over time, i.e., ${\theta }^{\prime }\left(t\right)<0$. Assume that $\theta \left(t\right)={\theta }_0-\eta {\left(\frac{t}{T}\right)}^2$ ($t\in [0,T]$), where ${\theta }_0$ is the initial freshness at $t=0$ [21,22,23]. $\eta \in (0,\overline{\eta }]$ is the freshness decay rate coefficient. $\eta \to 0$ means that the cold chain level is so high that the food hardly deteriorates; $\eta =\overline{\eta }$ is the decay value of the natural deterioration of the fresh food without taking any cold chain measures. The supplier can control the decay rate of fresh products by controlling the temperature. $\tau \left(\eta \right)$ is the cold chain cost related to the decay rate (see Figure 2). The freshness decay rate is affected by temperature, humidity, etc. $\eta \to 0$ means the freshness does not change under the best freezing environment. In this case, the cost is very high and assumed to be infinity; $\eta \to \overline{\eta }$ means there is no cryopreservation whatsoever, so no cost is incurred. Usually, because of the huge investment in $R\&D$, the breakthroughs in technology, such as refrigeration technology, always come with higher marginal costs. Therefore, we assume ${\tau }^{\prime }\left(\eta \right)>0$ and ${\tau }^{″}\left(\eta \right)>0$. To make the results trackable, we use a simple function to evaluate the cold chain cost and assume it to be $\tau \left(\eta \right)=\rho (\frac{1}{\eta }-\frac{1}{\overline{\eta }})$.

With the gradual enhancement of people’s awareness of low-carbon products and the continuous improvement of living standards, some consumers with a good education, stable wages, and the pursuit of a high quality of life are willing to buy low-carbon food [15,17]. We assume there are two kinds of consumers in the market. One kind of consumer also has a certain preference for low carbon and likes to buy low-carbon fresh food, and the demand of the kind 1 consumers is denoted $D_1$; another kind of consumer is only sensitive to the self-price, the cross-price, and the freshness, and the demand of the kind 2 consumer is denoted $D_2$. According to the classical Bertrand model that is commonly used in the literature (Gurnani et al. [24], Yan et al. [25], Yuan et al. [26], Biswas et al. [27]), the demand functions of low-carbon preference and non-low-carbon preference customers is given by

$$\begin{array}{cc}\hfill D_1& =a+k{\int }_{t_1}^T\theta \left(t\right)dt-p_1+\beta p_2+\gamma (e_r+e_s).\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill D_2& =a+k{\int }_{t_1}^T\theta \left(t\right)dt-p_2+\beta p_1.\hfill \end{array}$$

(2)

where $a>0$ is the potential market size and k is denoted as consumers’ sensitivity to food freshness at time t. $p_1$ and $p_2$ are the market price of low-carbon and non-low-carbon fresh food charged by the retailer. $\beta \in (0,1)$ is the substitute factor between the two kinds of demands. $e_r\in [0,{\overline{e}}_r]$ and $e_s\in [0,{\overline{e}}_s]$ represent the retailer’s and supplier’s carbon reduction level, respectively, ${\overline{e}}_r$ and ${\overline{e}}_s$ are the maximal reduction level of the retailer and the supplier, respectively. $\gamma >0$ is the assumed low-carbon preference coefficient of consumers. The meanings of each parameter are shown in

Table 1. Model parameters and variables.

Type	Parameters	Description
Decision variables	$e_s$	The supplier’s carbon emission reduction level
	$e_m$	The retailer’s carbon emission reduction level
	$\eta$	The freshness decay rate coefficient
	$p_1$	The market price of the low-carbon fresh food
	$p_2$	The market price of the non-low-carbon fresh food
Cost parameters	$c_s$	The supplier’s unit producing cost
	$c_m$	The retailer’s unit processing cost
	$w_1$	The wholesaling price of the low-carbon fresh food
	$w_2$	The wholesaling price of the non-low-carbon fresh food
Cost parameters	k	Sensitivity coefficient of consumers to the freshness of fresh products
	$\beta$	Cross price elasticity coefficient of two fresh food
	$\gamma$	Sensitivity coefficient of consumers with low-carbon preferences to the carbon emission reduction level
	${\xi }_s$	The supplier’s low-carbon production cost coefficient
	${\xi }_m$	The retailer’s low carbon processing cost coefficient
	$\rho$	Cold chain cost coefficient
	${\overline{e}}_s$	The supplier’s maximum carbon emission reduction level
	${\overline{e}}_m$	The retailer’s maximum carbon emission reduction level
	$\overline{\eta }$	Maximum Attenuation coefficient of freshness
	I	Cold chain preservation efforts level

.

3. Decision Sequence in the Game

We consider a four-stage game (see Figure 3): in the first stage, the supplier and the retailer determine their individual carbon emission reduction levels, $e_s$ and $e_r$, according to their current carbon emission reduction technology; in the second stage, the supplier controls the freshness decay rate $\eta$; in the third stage, each retailer decides its retail price $p_1$ and $p_2$; in the last stage, the market is cleared according to the Bertrand model (1) and (2). The supplier and the retailer are risk neutral and completely rational, and both of them make decisions to maximize their profits.

Carbon reduction is an important national strategy. The supplier and the retailer take carbon emission reduction decisions as long-term decisions in the next few years. Price can be seen as a short-term tactical strategy.

3.1. Objective Function

The supplier’s and the retailer’s profit functions can be expressed by

$$\begin{array}{cc}\hfill {\pi }_s& =(w_1-c_s)D_1+(w_2-c_s)D_2-\frac{1}{2}{\xi }_s{e}_s^2-\tau \left(\eta \right),\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill {\pi }_r& =(p_1-w_1-c_r)D_1+(p_2-w_2-c_r)D_2-\frac{1}{2}{\xi }_r{e}_r^2.\hfill \end{array}$$

(4)

where $w_1$ and $w_2$ ($w_1>w_2$) are the wholesaling price, $\frac{1}{2}{\xi }_s{e}_s^2$ and $\frac{1}{2}{\xi }_r{e}_r^2$ denote the supplier’s and the retailer’s costs for reducing carbon emissions, and ${\xi }_s$ and ${\xi }_r$ are the low-carbon cost coefficients for the supplier and retailer, respectively. $c_s$ is the unit producing cost that satisfies $w_1>c_s$ and $w_2>c_s$. $c_r$ is the unit retailing cost. Therefore, $(w_1-c_s)D_1$ and $(p_1-w_1-c_r)D_1$ are the revenue from selling low-carbon food, $(w_2-c_s)D_2$ and $(p_2-w_2-c_r)D_2$ are the revenue from selling non-low-carbon food, while $\frac{1}{2}{\xi }_s{e}_s^2$ and $\frac{1}{2}{\xi }_r{e}_r^2$ are the costs incurred by reducing carbon emissions. $\tau \left(\eta \right)$ is the cold chain cost.

3.2. The Optimal Decisions in Stages 2–4

In the second stage, given the supplier’s and the retailer’s carbon emission efforts $e_s$ and $e_r$, the supplier controls the freshness decay rate $\eta$; in the third stage, each retailer decides their retail price $p_1$ and $p_2$, respectively. With backward induction, we obtain the equilibrium prices in the third stage and the optimal freshness decay rate in the second stage and show them in Proposition 1.

Proposition 1.

(1) if $\overline{\eta }⩽\sqrt{\frac{6T^2\rho }{k(T^3-t_1^3)(w_1+w_2-2c_s)}}$,

$$\begin{array}{cc}\hfill {\eta }^{*}& =\overline{\eta },\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill p_1^{*}& =\frac{1}{2(1-\beta )}\left[a+k\left({\theta }_0(T-t_1)-\frac{{\eta }^{*}(T^3-t_1^3)}{3T^2}\right)\right]+\frac{c_r+w_1}{2}+\frac{\gamma (e_r+e_s)}{2(1-{\beta }^2)},\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill p_2^{*}& =\frac{1}{2(1-\beta )}\left[a+k\left({\theta }_0(T-t_1)-\frac{{\eta }^{*}(T^3-t_1^3)}{3T^2}\right)\right]+\frac{c_r+w_2}{2}+\frac{\beta \gamma (e_r+e_s)}{2(1-{\beta }^2)}.\hfill \end{array}$$

(7)

(2) if $\overline{\eta }>\sqrt{\frac{6T^2\rho }{k(T^3-t_1^3)(w_1+w_2-2c_s)}}$,

$$\begin{array}{cc}\hfill {\eta }^{*}& =\sqrt{\frac{6T^2\rho }{k(T^3-t_1^3)(w_1+w_2-2c_s)}},\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill p_1^{*}& =\frac{1}{2(1-\beta )}\left[a+k\left({\theta }_0(T-t_1)-\frac{(T^3-t_1^3)}{3T^2}{\eta }^{*}\right)\right]+\frac{c_r+w_1}{2}+\frac{\gamma (e_r+e_s)}{2(1-{\beta }^2)},\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill p_2^{*}& =\frac{1}{2(1-\beta )}\left[a+k\left({\theta }_0(T-t_1)-\frac{(T^3-t_1^3)}{3T^2}{\eta }^{*}\right)\right]+\frac{c_r+w_2}{2}+\frac{\beta \gamma (e_r+e_s)}{2(1-{\beta }^2)}.\hfill \end{array}$$

(10)

As shown in Proposition 1, the equilibrium price and the optimal freshness decay rate depend on the threshold of the maximum decay rate, which can be the food’s decay rate under the best freezing and preservation technology. When the best freezing and preservation technology is low enough such that the natural freshness decay rate is less than the threshold, i.e, $\overline{\eta }⩽\sqrt{\frac{6T^2\rho }{k(T^3-t_1^3)(w_1+w_2-2c_s)}}$, the optimal freshness decay rate ${\eta }^{*}$ equals $\overline{\eta }$, which means the supplier does not need to take any cold chain measures. However, when the best freezing and preservation technology is high enough such that $\overline{\eta }>\sqrt{\frac{6T^2\rho }{k(T^3-t_1^3)(w_1+w_2-2c_s)}}$, the supplier will use the freezing and preservation technology and the optimal freshness decay rate is related to factors such as the length of the sales period, the wholesale price, and the production cost. In this case, we will analyze the impacts of the length of the sales period, wholesale price, and production cost on the optimal freshness decay rate. The results are shown in Proposition 2.

Proposition 2.

When $\overline{\eta }>\sqrt{\frac{6T^2\rho }{k(T^3-t_1^3)(w_1+w_2-2c_s)}}$,

(1) $\frac{d{\eta }^{*}}{d{t}_1}>0$, $\frac{d{\eta }^{*}}{dT}<0$;

(2) $\frac{d{\eta }^{*}}{d\rho }>0$, $\frac{d{\eta }^{*}}{dk}<0$, $\frac{d{\eta }^{*}}{d{w}_1}<0$, $\frac{d{\eta }^{*}}{d{w}_2}<0$, $\frac{d{\eta }^{*}}{d{c}_s}>0$.

Under the condition that the sales period of the fresh food is fixed, the shorter the sales period (T decreases or $t_1$ increases), the impact of the freshness decay rate coefficient on the total sales is weaker, so the supplier should reduce the cold chain input ($\eta$ increases). The higher the cold chain cost coefficient, or the higher the production cost per unit product, the more the supplier needs to reduce the cold chain input. If consumers are more sensitive to the freshness of fresh products, or the wholesale price, the amount the supplier sells to retailers is higher, the supplier should increase the investment in the cold chain to improve the fresh products for consumers and retailers.

3.3. Analysis of Stage 1

In this section, we will analyze the supplier’s and the retailer’s optimal carbon emission reduction levels in the first stage. We first derive the best responses of the supplier and the retailer, respectively, and show them in the following two Lemmas.

Let $l_1=\frac{(c_r+w_1)(1-{\beta }^2)-M^{*}(1+\beta )}{\gamma }-\frac{[{\gamma }^2-2{\xi }_r(1-{\beta }^2)]{\overline{e}}_r}{{\gamma }^2}$, $l_2=\frac{(c_r+w_1)(1-{\beta }^2)-M^{*}(1+\beta )}{\gamma }$, $l_3=\frac{(c_r+w_1)(1-{\beta }^2)-M^{*}(1+\beta )}{\gamma }-\frac{[{\gamma }^2-2{\xi }_r(1-{\beta }^2)]{\overline{e}}_r}{2{\gamma }^2}$, $l_4=\frac{(w_1-c_s)\gamma }{2{\xi }_s}$ and $M^{*}\left({\eta }^{*}\right)=a+k\left({\theta }_0(T-t_1)-\frac{{\eta }^{*}(T^3-t_1^3)}{3T^2}\right)$, where ${\eta }^{*}$ is expressed as in Proposition 1.

Lemma 1.

Given the supplier’s carbon emission reduction level $e_s$, the retailer’s best response $e_r^{*}$ can be divided into the following three cases. To better illustrate the retailer’s best responses, we plot them in Figure 4.

(1) When $0<{\xi }_r<\frac{{\gamma }^2}{2(1-{\beta }^2)}$ (Figure 4(1)),

$$e_r^{*}=\left\{\begin{array}{c}0,e_s<l_3,\hfill \\ {\overline{e}}_r,e_s>l_3,\hfill \\ 0or{\overline{e}}_r,e_s=l_3.\hfill \end{array}\right.$$

(11)

(2) When ${\xi }_r=\frac{{\gamma }^2}{2(1-{\beta }^2)}$ (Figure 4(2)),

$$e_r^{*}=\left\{\begin{array}{c}0,e_s<l_2,\hfill \\ {\overline{e}}_r,e_s>l_2,\hfill \\ \forall e_r\in [0,{\overline{e}}_r],e_s=l_2.\hfill \end{array}\right.$$

(12)

(3) When ${\xi }_r>\frac{{\gamma }^2}{2(1-{\beta }^2)}$ (Figure 4(3)),

$$e_r^{*}=\left\{\begin{array}{c}0,e_s\le l_2,\hfill \\ {\overline{e}}_r,e_s\ge l_1,\hfill \\ \frac{[(c_r+w_1)({\beta }^2-1)+M^{*}(\beta +1)]\gamma }{2{\xi }_r(1-{\beta }^2)-{\gamma }^2}+\frac{{\gamma }^2}{2{\xi }_r(1-{\beta }^2)-{\gamma }^2}{e}_s,l_2<e_s<l_1.\hfill \end{array}\right.$$

(13)

By analyzing the first-order derivative of the retailer’s profit function with respect to their carbon emission reduction level $e_r$, we obtain three thresholds: $l_1$, $l_2$, and $l_3$. Depending on the value of the retailer’s low-carbon cost coefficients ${\xi }_r$ and the value $e_s$, there are five possible best responses for $e_r^{*}$. When the retailer’s low-carbon cost coefficient is low (${\xi }_r<\frac{{\gamma }^2}{2(1-{\beta }^2)}$), if $e_s^{*}<l_3$, the best response of the retailer is $e_r^{*}=0$, otherwise, the best response of the retailer is $e_r^{*}={\overline{e}}_r$ (see Figure 4(1)).

When ${\xi }_r=\frac{{\gamma }^2}{2(1-{\beta }^2)}$, whether the best response of the retailer is $e_r^{*}=0$ or $e_r^{*}={\overline{e}}_r$ depends on the comparison result of $e_s$ and $l_2$ (see Figure 4(2)).

When the retailer’s low-carbon cost coefficient is high (${\xi }_r>\frac{{\gamma }^2}{2(1-{\beta }^2)}$), if $l_2<e_s<l_1$, the retailer’s best response linearly increases with the supplier’s carbon emission reduction level; if $e_s<l_2$, the retailer will do nothing to reduce carbon emissions ($e_r^{*}=0$); if $e_s>l_1$, the retailer will reduce emissions to the maximum level ($e_r^{*}={\overline{e}}_r$) (see Figure 4(3)).

In most cases, the best response of the retailer is on the boundary point, 0 or ${\overline{e}}_r$. Interestingly, when ${\xi }_r>\frac{{\gamma }^2}{2(1-{\beta }^2)}$ and $l_2<e_s<l_1$, the value of $e_r^{*}$ increases in $e_s$.

Next, we show the best response of the supplier in the next Lemma.

Lemma 2.

Given the retailer’s carbon emission reduction level $e_r$, the supplier’s best response $e_s^{*}=min({\overline{e}}_s,l_4)$, which is illustrated in Figure 5.

The supplier’s optimal carbon emission reduction level will not be affected by the retailer. Specially, if ${\overline{e}}_s\le l_4$, $e_s^{*}={\overline{e}}_s$; otherwise, $e_s^{*}=l_4$.

When the retailer and the supplier make their best response simultaneously, there are equilibria. Based on Lemmas 1 and 2, we can derive 25 equilibria, and we summarize them in

Table 2. The retailer’s and supplier’s equilibrium carbon reduction levels.

${\mathit{\xi }}_{\mathit{r}}$	${\mathit{\xi }}_{\mathit{s}}$	${\overline{\mathit{e}}}_{\mathit{s}}$	$({\mathit{e}}_{\mathit{s}}^{*},{\mathit{e}}_{\mathit{r}}^{*})$
$0<{\xi }_r<\frac{{\gamma }^2}{2(1-{\beta }^2)}$	$\frac{(w_1-c_s)\gamma }{2l_3}<{\xi }_s<\frac{(w_1-c_s)\gamma }{2{\overline{e}}_s}$	——	$(e_s^{*},e_r^{*})=({\overline{e}}_s,0)$
	${\xi }_s>\frac{(w_1-c_s)\gamma }{2{\overline{e}}_s}$		$(e_s^{*},e_r^{*})=(\frac{(w_1-c_s)\gamma }{2{\xi }_s},0)$
$l_5<{\xi }_r<\frac{{\gamma }^2}{2(1-{\beta }^2)}$	${\xi }_s<\frac{(w_1-c_s)\gamma }{2l_3}$	——	$(e_s^{*},e_r^{*})=({\overline{e}}_s,0)$
${\xi }_r=l_5$		——	$(e_s^{*},e_r^{*})=({\overline{e}}_s,0)$ or $({\overline{e}}_s,{\overline{e}}_r)$
${\xi }_r<l_5$	${\xi }_s⩽\frac{(w_1-c_s)\gamma }{2{\overline{e}}_s}$	——	$(e_s^{*},e_r^{*})=({\overline{e}}_s,{\overline{e}}_r)$
	${\xi }_s>\frac{(w_1-c_s)\gamma }{2{\overline{e}}_s}$	——	$(e_s^{*},e_r^{*})=(\frac{(w_1-c_s)\gamma }{2{\xi }_s},{\overline{e}}_r)$
$l_5<{\xi }_r<\frac{{\gamma }^2}{2(1-{\beta }^2)}$	${\xi }_s=\frac{(w_1-c_s)\gamma }{2l_3}$	——	$(e_s^{*},e_r^{*})=({\overline{e}}_s,0)$
${\xi }_r⩽l_5$			$(e_s^{*},e_r^{*})=(\frac{(w_1-c_s)\gamma }{2{\xi }_s},0)$ or $(\frac{(w_1-c_s)\gamma }{2{\xi }_s},{\overline{e}}_r)$
${\xi }_r=\frac{{\gamma }^2}{2(1-{\beta }^2)}$	${\xi }_s>\frac{(w_1-c_s)\gamma }{2{\overline{e}}_s}$	——	$(e_s^{*},e_r^{*})=(\frac{(w_1-c_s)\gamma }{2{\xi }_s},0)$
	$l_6<{\xi }_s⩽\frac{(w_1-c_s)\gamma }{2{\overline{e}}_s}$	——	$(e_s^{*},e_r^{*})=({\overline{e}}_s,0)$
	${\xi }_s=l_6$	${\overline{e}}_s<l_2$	$(e_s^{*},e_r^{*})=({\overline{e}}_s,0)$
		${\overline{e}}_s⩾l_2$	$(e_s^{*},e_r^{*})=(\frac{(w_1-c_s)\gamma }{2{\xi }_s},e_r)$, $e_r\in [0,{\overline{e}}_r]$
	${\xi }_s<l_6$	${\overline{e}}_s<l_2$	$(e_s^{*},e_r^{*})=({\overline{e}}_s,0)$
		${\overline{e}}_s=l_2$	$(e_s^{*},e_r^{*})=({\overline{e}}_s,e_r)$, $e_r\in [0,{\overline{e}}_r]$
	$\frac{(w_1-c_s)\gamma }{2{\overline{e}}_s}<{\xi }_s<l_6$	${\overline{e}}_s⩾l_2$	$(e_s^{*},e_r^{*})=(\frac{(w_1-c_s)\gamma }{2{\xi }_s},{\overline{e}}_r)$
	${\xi }_r⩽\frac{(w_1-c_s)\gamma }{2{\overline{e}}_s}$		$(e_s^{*},e_r^{*}={\overline{e}}_s,{\overline{e}}_r)$
${\xi }_r=\frac{{\gamma }^2}{2(1-{\beta }^2)}$	${\xi }_s>\frac{(w_1-c_s)\gamma }{2{\overline{e}}_s}$	——	$(e_s^{*},e_r^{*})=(\frac{(w_1-c_s)\gamma }{2{\xi }_s},0)$
	$l_6<{\xi }_s⩽\frac{(w_1-c_s)\gamma }{2{\overline{e}}_s}$	——	$(e_s^{*},e_r^{*})=({\overline{e}}_s,0)$
	$\frac{(w_1-c_s)\gamma }{2l_1}⩽{\xi }_s⩽l_6$	${\overline{e}}_s⩽l_2$	$(e_s^{*},e_r^{*})=({\overline{e}}_s,0)$
	$\frac{(w_1-c_s)\gamma }{2l_1}⩽{\xi }_s⩽\frac{(w_1-c_s)\gamma }{2{\overline{e}}_s}$	${\overline{e}}_s>l_2$	$(e_s^{*},e_r^{*})=({\overline{e}}_s,\frac{[(c_r+w_1)({\beta }^2-1)+M^{*}(\beta +1)+\gamma {\overline{e}}_s]\gamma }{2{\xi }_r(1-{\beta }^2)-{\gamma }^2})$
	$\frac{(w_1-c_s)\gamma }{2{\overline{e}}_s}⩽{\xi }_s⩽l_6$		$(e_s^{*},e_r^{*})=(\frac{(w_1-c_s)\gamma }{2{\xi }_s},\frac{2{\xi }_s[(c_r+w_1)({\beta }^2-1)+M^{*}(\beta +1)]\gamma +(w_1-c_s){\gamma }^2}{2{\xi }_s[2{\xi }_r(1-{\beta }^2)-{\gamma }^2]})$
	${\xi }_s⩽\frac{(w_1-c_s)\gamma }{2l_1}$	${\overline{e}}_s⩽l$	$(e_s^{*},e_r^{*})=\left({\overline{e}}_s\right),0$
${\xi }_r⩾l_7$		$\overline{e_s}>l_2$	$(e_s^{*},e_r^{*})=({\overline{e}}_s,\frac{[(c_r+w_1)({\beta }^2-1)+M^{*}(\beta +1)+\gamma {\overline{e}}_s]\gamma }{2{\xi }_r(1-{\beta }^2)-{\gamma }^2})$
${\xi }_r<l_7$	$\frac{(w_1-c_s)\gamma }{2{\overline{e}}_s}<{\xi }_s<\frac{(w_1-c_s)\gamma }{2l_1}$	——	$(e_s^{*},e_r^{*})=(\frac{(w_1-c_s)\gamma }{2{\xi }_s},{\overline{e}}_r)$
	${\xi }_s⩽\frac{(w_1-c_s)\gamma }{2{\overline{e}}_s}$	——	$(e_s^{*},e_r^{*})=({\overline{e}}_s,{\overline{e}}_r)$

. In addition, in order to show the equilibria clearly, we show all of the possible equilibria in Figure A2.

Among the 25 cases, we mainly focus on a non-trivial case, presented in Proposition 3.

Proposition 3.

When ${\xi }_r>\frac{{\gamma }^2}{2(1-{\beta }^2)}$, and ${\overline{e}}_s>l_2$, $\frac{(w_1-c_s)\gamma }{2{\overline{e}}_s}⩽{\xi }_s⩽l_6$, the retailer’s and supplier’s equilibrium carbon reduction levels are

$$\begin{array}{c}\hfill (e_s^{*},e_r^{*})=(\frac{(w_1-c_s)\gamma }{2{\xi }_s},\frac{2{\xi }_s[(c_r+w_1)({\beta }^2-1)+M^{*}(\beta +1)]\gamma +(w_1-c_s){\gamma }^3}{2{\xi }_s[2{\xi }_r(1-{\beta }^2)-{\gamma }^2]}.\end{array}$$

(14)

The 25 equilibrium cases in the proof of Proposition 3 induce 10 optimal price strategies (refer to Appendix B). We only show the representative one in the following proposition, which is the market prices corresponding to the equilibrium (14).

Proposition 4.

When ${\xi }_r>\frac{{\gamma }^2}{2(1-{\beta }^2)}$, and ${\overline{e}}_s>l_2$, $\frac{(w_1-c_s)\gamma }{2{\overline{e}}_s}⩽{\xi }_s⩽l_6$, the market prices corresponding to the equilibrium (14) are

$$\begin{array}{cc}\hfill p_1^{*}& =\frac{M^{*}}{2(1-\beta )}+\frac{c_r+w_1}{2}+\frac{{\gamma }^2\{{\xi }_r(1-{\beta }^2)(w_1-c_s)+{\xi }_s[(c_r+w_1)({\beta }^2-1)+M^{*}(\beta +1)]\}}{2{\xi }_s(1-{\beta }^2)[2{\xi }_r(1-{\beta }^2)-{\gamma }^2]},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill p_2^{*}& =\frac{M^{*}}{2(1-\beta )}+\frac{c_r+w_1}{2}+\frac{\beta {\gamma }^2\{{\xi }_r(1-{\beta }^2)(w_1-c_s)+{\xi }_s[(c_r+w_1)({\beta }^2-1)+M^{*}(\beta +1)]\}}{2{\xi }_s(1-{\beta }^2)[2{\xi }_r(1-{\beta }^2)-{\gamma }^2]}.\hfill \end{array}$$

(16)

In many cases, the equilibrium values of $e_s^{*}$ and $e_r^{*}$ are on the boundary points, which is not affected by the exogenous parameters, such as when ${\xi }_r<\frac{{\gamma }^2}{2(1-{\beta }^2)}$, the equilibrium value of $e_r^{*}$ is 0 or ${\overline{e}}_r$. However, when ${\xi }_r>\frac{{\gamma }^2}{2(1-{\beta }^2)}$, and ${\overline{e}}_s>l_2$, $\frac{(w_1-c_s)\gamma }{2{\overline{e}}_s}⩽{\xi }_s⩽l_6$, the best responses of the retailer and the supplier intersect at the inner point. Then, the equilibrium values of $e_s^{*}$ and $e_r^{*}$ depend on the exogenous parameters. Next, we will analyze the impact of the exogenous parameters on the retailer’s and supplier’s equilibrium carbon reduction levels, $e_s^{*}$ and $e_r^{*}$.

Proposition 5.

When ${\xi }_r>\frac{{\gamma }^2}{2(1-{\beta }^2)}$, and ${\overline{e}}_s>l_2$, $\frac{(w_1-c_s)\gamma }{2{\overline{e}}_s}⩽{\xi }_s⩽l_6$ and $\overline{\eta }>\sqrt{\frac{6T^2\rho }{k(T^3-t_1^3)(w_1+w_2-2c_s)}}$,

(1) $\frac{d{e}_s^{*}}{d{w}_1}>0$, $\frac{d{e}_s^{*}}{d\gamma }>0$, $\frac{d{e}_s^{*}}{d{\xi }_s}<0$;

(2) $\frac{d^2{e}_r^{*}}{d{w}_1^2}<0$, $\frac{d{e}_r^{*}}{d{w}_2}>0$, $\frac{d{e}_r^{*}}{d\gamma }>0$, $\frac{d{e}_r^{*}}{d{\xi }_s}<0$, $\frac{d{e}_r^{*}}{d{\xi }_r}<0$, $\frac{d{e}_r^{*}}{d{c}_s}<0$;

(3) $\frac{d{p}_1^{*}}{d{w}_2}>0$, $\frac{d{p}_1^{*}}{d{\xi }_s}<0$, $\frac{d{p}_1^{*}}{d{\xi }_r}<0$, $\frac{d{p}_1^{*}}{dk}>0$, $\frac{d{p}_1^{*}}{d\gamma }>0$, $\frac{d{p}_1^{*}}{d{c}_s}<0$;

(4) if ${\gamma }^2\in ({\xi }_r(1-{\beta }^2),2{\xi }_r(1-{\beta }^2))$, $\frac{d{p}_1^{*}}{d{c}_r}<0$, while if ${\gamma }^2⩽{\xi }_r(1-{\beta }^2)$, $\frac{d{p}_1^{*}}{d{c}_r}⩾0$;

(5) $\frac{d{p}_2^{*}}{d{\xi }_s}<0$, $\frac{d{p}_2^{*}}{d{\xi }_r}<0$, $\frac{d{p}_2^{*}}{dk}>0$, $\frac{d{p}_2^{*}}{d\gamma }>0$, $\frac{d{p}_2^{*}}{d{c}_s}<0$;

(6) if ${\gamma }^2\in ({\xi }_r(1-\beta ),2{\xi }_r(1-{\beta }^2))$, $\frac{d{p}_2^{*}}{d{c}_r}<0$, while if ${\gamma }^2⩽2{\xi }_r(1-\beta )$$\frac{d{p}_2^{*}}{d{c}_r}⩾0$.

An increase in the wholesale price of low-carbon food will stimulate suppliers to increase their emission reduction level. In addition, an increase in the wholesale price of non-low-carbon food will actually encourage the retailer to increase their emission reduction level. In addition, increasing consumer preference for low-carbon food effectively motivates the supplier and the retailer to reduce carbon emissions at the same time. Therefore, it is necessary to strengthen low-carbon publicity for consumers. At the same time, the supplier and the retailer should increase technology research and development to reduce the implementation cost of carbon emission reduction.

If consumers are more sensitive to the freshness of fresh products, the supplier is likely to increase investment in the cold chain, which will inevitably lead to an increase in wholesale prices. Therefore, the retailer should also increase the market prices of both products. The increasing consumer preference for low-carbon products will also prompt the retailer to increase the prices of both kinds of food. As ${\xi }_s$ and ${\xi }_r$ increase, the supplier and the retailer are unwilling to bear the excessive costs for carbon reduction, therefore, in order to attract more consumers, retailers can only adopt a price reduction strategy.

Interestingly, when the consumer preference for low-carbon products is low (${\gamma }^2\in ({\xi }_r(1-\beta ),2{\xi }_r(1-{\beta }^2))$), $e_r^{*}$ and $e_s^{*}$ are low, the potential market size is small, such that the competition for the two kinds of customers is fierce. Then, an increase in the sales cost will reduce the retail price. However, when the consumer preference for low-carbon products is high (${\gamma }^2⩽2{\xi }_r(1-\beta )$), the potential market size is large, such that the competition is not fierce. Then, an increase in the sales cost will advance the retail price.

Based on Proposition 3, the equilibrium of $e_s$ and $e_r$ is not related to the substitution factor $\beta$. However, based on Proposition 4, the impact of the substitution factor $\beta$ is related to the equilibrium price. The price is so complicated that we cannot obtain the analytical result. Then, we perform a lot of numerical studies and use the most representative one to show the impact of the substitution factor $\beta$ on the equilibrium prices $p_1^{*}$ and $p_2^{*}$.

In the case of ${\xi }_r>\frac{{\gamma }^2}{2(1-{\beta }^2)}$, ${\xi }_s\in (\frac{(w_1-c_s)\gamma }{2l_1},l_6)\cap (\frac{(w_1-c_s)\gamma }{2{\overline{e}}_s},\infty )$, and $\overline{\eta }>\sqrt{\frac{6T^2\rho }{k(T^3-t_1^3)(w_1+w_2-2c_s)}}$, we assume that $T=1$, $t_1=0.1$, $\rho =1$, $k=1$, $w_1=12$, $w_2=8$, $c_s=c_r=2$, $\gamma =1$, ${\theta }_0=1$, $a=10$, ${\xi }_s=1$, ${\xi }_r=4$, ${\overline{e}}_r=2$, then, we have ${\eta }^{*}=0.61$, $M^{*}=10.70$, $l_1=-30{\beta }^2-10.7\beta +17.3$, $l_2=-14{\beta }^2-10.7\beta +3.3$, $l_4=5$. Since ${\xi }_r>\frac{{\gamma }^2}{2(1-{\beta }^2)}$ and $l_2<l_4<l_1$, we have $\beta \in (0,0.49)$.

As shown in Figure 6, the more sensitive consumers are to the price difference between two products, the larger the potential market size, and retailers can increase the prices of both products to earn more profit simultaneously.

4. Conclusions

This paper studies a two-echelon fresh food supply chain composed of a supplier and a retailer. Considering that many consumers prefer low-carbon food, the supplier sells both low-carbon fresh food and non-low-carbon fresh food to the retailer. A four-stage game among the supplier, the retailer, and consumers is considered. Interestingly, in the second stage we find that the shorter the sales period (T decreases or $t_1$ increases), the impact of the freshness decay rate coefficient on the total sales is weaker, so the supplier should reduce the cold chain input ($\eta$ increases). In addition, in the first stage, when the consumer preference for low-carbon products is low (${\gamma }^2\in ({\xi }_r(1-\beta ),2{\xi }_r(1-{\beta }^2))$), $e_r^{*}$ and $e_s^{*}$ are low, and the potential market size is small such that the competition for two kinds of customers is fierce. Then, an increase in the sales cost will reduce the retail price. However, when the consumer preference for low-carbon products is high (${\gamma }^2⩽2{\xi }_r(1-\beta )$), the potential market size is large, such that the competition is not fierce. Then, an increase in the sales cost will advance the retail price. In addition, the research results show that the higher the preference of consumers for the low-carbon food, the more incentive there is for the supplier and the retailer to increase their carbon emission reduction efforts. In addition, as consumers become more sensitive to freshness, the retailer needs to increase the market prices of both products.