Policy of Government Subsidy for Supply Chain with Poverty Alleviation

Abstract

Government subsidy is a common practice in poverty alleviation. We used game theory and the mathematical model of operations management to investigate the efficiency of subsidy with different poverty scales when the firm owns the decision power of the wholesale price. Comparative analysis of the equilibrium solutions demonstrated the following results: Exclusive subsidy has a significant effect on the payoff of the poor farmer, but the dilemma is that the increase in the payoff of the poor farmer is against the payoff decrease of the regular farmer. Sharing subsidy has a counterbalancing effect on the payoff of the poor and regular farmers. Co-subsidy is the best for consumer surplus and social welfare, but it has little effect on improving the poor farmer’s payoff. Generally, when the poor farmers are in the majority, sharing subsidies or co-subsidy is more reasonable than exclusive subsidy. When the poor farmers are in the minority, exclusive or sharing subsidies will be more economical for the government than co-subsidy. Our research helps the government recognize that spending more money may achieve a poor result in poverty alleviation and help the firm realize that it is better to give more subsidies to the poor farmer than to itself. The highlights of the paper are as follows. Firstly, our work provides a new perspective in supply chain operations management with poverty alleviation by considering the participation of the poor and regular farmers together; secondly, the poverty scale is introduced into the mathematical model; thirdly, we pay attention to the impact of government subsidy to enterprise on the payoff of the poor farmer.

1. Introduction

Poverty has become one of the most sharp-pointed social problems in the world. Focusing on poverty has shifted from food security during the last half of the 20th century to an emerging strategy of helping small-scale farm households increase their income to reduce rural poverty (Swanson, 2006) [1]. Many companies have developed supply chains with the poor as suppliers or distributors to seek both profit and poverty alleviation in some developing countries, so it has gradually received attention regarding the operations management of the supply chain with poverty alleviation (Sodhi and Tang, 2015) [2].

Government subsidy is a common method in the operation of the supply chain within poverty alleviation in China. For example, in the beef cattle breeding project established by the China Resources Group cooperating with poor households in Haiyuan county, the government gives every poor household an extra subsidy of CNY 2000 for a head of cattle they raise, and the cattle is purchased exclusively by China Resources Group. In the mushroom planting project, companies receive a subsidy of CNY 1 from the government for one mushroom bag purchased from poor households at the agreed wholesale price.

When considering the special phenomenon that the company purchase from both the poor and regular farmers in the same rural areas, some interesting questions arise. Firstly, what role does government subsidy play in supply chain operation involving the poor and regular farmers together? Secondly, in what ways do government subsidies achieve a better and improved payoff of the poor farmer?

To answer the above questions, we focus on a supply chain consisting of an enterprise and the poor and regular farmers in the same rural areas. The enterprise purchases the farmers products at a given wholesale price and resells them at the market clearing price. We use the proportion of the poor farmers as the poverty scale to describe the relative degree of poverty, and we incorporate the government subsidy as an exogenous variable in our models. A series of supply chain game models with no subsidy, exclusive subsidy for the poor farmer, sharing subsidy between the poor farmer and the firm, and subsidy together between the poor and the regular farmer (hereinafter referred to briefly as no subsidy, exclusive subsidy, sharing subsidy, and co-subsidy) are built to reveal the effect of government subsidy on poverty alleviation and social welfare.

This paper is among the few papers using operations management models to investigate the role of government subsidies in poverty alleviation. Our work contributes to the literature as follows. Firstly, our work provides a new perspective by considering the participation of the poor and regular farmers together in supply chain. Secondly, we introduce the poverty scale in our model. Thirdly, our research pays attention to the effect of government subsidy for enterprise on the payoff of the poor farmer, and it is rare in the literature.

The remainder of this paper is organized as follows. Section 2 provides a general review of the pertinent literature. Section 3 describes the model development and introduces a game decision case with no subsidy as a benchmark. Section 4 proposes the game models with exclusive subsidy, sharing subsidy, and co-subsidy. Section 5 provides the comparative analysis of equilibrium solutions. Section 6 summarizes this study and proposes future research directions.

2. Literature Review

Literature related to this paper comes mainly from two streams, i.e., (i) supply chain with poverty alleviation, and (ii) supply chain operation optimization under government subsidy.

2.1. Supply Chain with Poverty Alleviation

Most previous literature on supply chains with poverty alleviation involves a case study and empirical research, and tries to indicate how to seek both profit and poverty alleviation through operations management of the supply chain with the poor as suppliers or distributers (Swanson, 2006; Sodhi, 2015) [1,2]. Some papers focus on reducing intermediate echelons to obtain a higher selling price by purchasing the poor output directly (Cameranesi et al., 2010) [3]. For example, Wal-Mart purchases the crops directly from the farmers in China, therefore the farmers can obtain a higher price and Wal-Mart can reduce its procurement cost and obtain fresher produce (An et al., 2013) [4]. In developing countries, the logistics infrastructure is weak and distribution cannot reach remote rural areas, so the company hires the rural villagers to deliver goods. This can overcome the high cost of “last-mile” distribution for the company while increasing the income of the rural villagers (Tang and Zhou, 2012; Rangan and Rajan, 2007) [5,6]. Tan et al. (2022) [7] proposed a generalized blockchain-based supply chain management platform for targeted poverty alleviation. By taking advantage of blockchain, this platform could effectively help establish trust to enhance the sustainability of poverty alleviation.

In addition to case and empirical studies, there are also a few papers using operations management (OM) models to help decision optimization and coordinate supply chain with poverty alleviation (Wu and Li, 2017) [8]. Kang et al. (2019) [9] focused on a supply chain with the poor raw material suppliers and analyzed the cooperation behavior through a microfinance game model. Zhao and Sun (2020) [10] constructed a nonlinear integer programming model for poor people’s Medicare decisions with opportunity constraints and tried to handle the storage of medicines and related medical devices. Wan and Wan (2020) [11] used an evolutionary game model to explore the behavioral strategies of the cooperative poverty alleviation ecosystem in the smart supply chain platform. Ye and Deng (2021) [12] explored the optimal decision-making problem of the three-level (government + enterprises + farmers) poverty alleviation supply chain under asymmetric cost information. Barman et al. (2021) [13] focused on the impact of government subsidy and tax policy on the optimal pricing and greening strategy in a competitive green supply chain.

Our work is in line with the latter stream by using OM models to investigate operation optimization of the supply chain with poverty alleviation, but existing studies rarely pay attention to the strategy selection of government subsidy, and do not take both the poor and regular farmers into account in OM models. Our work differs from the above literature by placing a new perspective on poverty alleviation of rural farmers in China by considering the participation of the poor and regular farmers together in supply chain operation. Furthermore, we introduce the poverty scale in our OM models.

2.2. Supply Chain Operation Optimization under Government Subsidy

Government subsidy is usually used as the encouragement for renewable energy (Chen et al., 2021) [14], technology innovation (Nie et al., 2022) [15], and environment-friendly logistics and supply chain management (Ma et al., 2018; Wu et al., 2022) [16,17]. Game theory is a common approach in supply chain operations management with government subsidy. Previous literature used game theory to design an allocation mechanism and disclose the impact of government subsidy on supply chain operation. For example, Wang et al. (2019) [18] investigated the strategies for the allocation of government subsidies among the parties in a reverse supply chain and the results suggested that the remanufacturing utilization rate had a great influence on the allocation strategy of government subsidies. Cao et al. (2019) [19] studied the selection of the channel structure in a closed-loop supply chain with government subsidy. They found that government was able to encourage the manufacturer to adopt the desired channel structures by setting appropriate subsidy levels. Chen et al. (2020) [20] found that government subsidies would coordinate the supply chain only when its target collection level was high. Li et al. (2019) [21] found that government subsidies effectively stimulated the demand for remanufactured products, reducing the price of both new and remanufactured products. Liu et al. (2019) [22] pointed out that a certain range of government subsidies could promote the supply chain members to undertake corporate social responsibility (CSR) and improve the overall performance of the supply chain. Some literature focuses on the impact of government subsidies on food safety investment and green technology investment in the supply chain (Yu et al., 2021; Ma et al., 2021) [23,24]. Javad et al. (2022) [25] focused on subsidized and unsubsidized price competition in a multi-echelon natural gas supply chain with governmental and private members.

There are also a few papers focusing on the optimal policy of government subsidy. Yang et al. (2021) [26] applied a principal-agent contract in a reverse supply chain under asymmetric recycling cost information of the recycler, and proposed that the government should subsidize the manufacturer rather than the recycler.

Our work is in line with the latter stream by using OM models to investigate the optimal policy of government subsidy in supply chain operation, but the existing studies rarely paid attention to the impact of government subsidies to enterprise on the payoff of the poor farmer. Our research focuses on poverty alleviation and considers the impact of government subsidies to enterprise on the payoffs of the poor and regular farmers together.

3. Game Decision Case with No Subsidy: A Benchmark

In this section, we consider the case without subsidy as a benchmark for comparison.

3.1. Model Development

Consider that two types of farmers, poor farmers and regular farmers, live in the same rural area, and produce the same type of agricultural product, and then sell them to a firm M at the wholesale price $w$. Let $i=1$ or $2$ be an index of the two types of farmers where $i=1$ indicates the poor type and $i=2$ indicates the regular type. Let $\lambda \left(\lambda \in \left(0,1\right)\right)$ denote the proportion of poor farmers, it implies the poverty scale, and let $\varphi =1-\lambda$ be that of regular ones. Farmers of the same type are assumed homogeneous. This means that the same type of farmers will exert the same effort level $e_i$ and yield the same production quantity $q_i\left(i=1,2\right)$. Moreover, for both types of farmers, we assume that: (i) the relationship between their production quantity and their effort level keeps the same linear relationship, i.e., $q_i=k{e}_i$, where $k$ is a positive parameter ($k>0$); (ii) the effort cost is a convex function of the effort level, i.e., $C_i\left(e_i\right)=h_i{e}_i{}^2/2$; (iii) Considering poor farmers are in a weak position, such as low education, lack of funds, and limited access to information, and so on, they face high barriers to access production, and they need to make extra investments to overcome the disadvantages linked to these. For this reason, we assume poor farmers have to face a higher quadratic cost factor at the same effort level, i.e., $h_1>h_2>0$. Given $\xi =h_2/h_1\left(0<\xi \le 1\right)$, i.e., $h_1=h_2/\xi$. The less $\xi$ means the greater gap between the poor farmer and the regular farmer in the unit cost of production effort, and $\xi =1$ means the poor farmer is on an equal basis with the regular farmer in production effort.

Then, the payoff of an individual farmer is

$$\begin{array}{c}{\pi }_i=w{q}_i-C_i\left(e_i\right)\hfill \\ =wk{e}_i-\frac{h_i{e}_i{}^2}{2},i=1,2\hfill \end{array}$$

(1)

We normalize the number of all farmers to 1, then the entire production quantity received by the firm M is $q=\lambda q_1+\varphi q_2$. As for the retail market, we suppose the firm’s retail price follows the Cournot model, i.e.,

$$\begin{array}{c}p=a-q\hfill \\ =a-\lambda q_1-\varphi q_2\hfill \end{array}$$

(2)

where $a (a>0)$ is the maximum potential sales, which means the product quantity that the consumer can obtain at a zero market price. When $q=0$, it means that the maximum market price is $a$. Hence, we can obtain the consumer surplus

$$CS=q^2/2.$$

(3)

Then, the firm’s profit is

$$\begin{array}{c}\pi =\left(p-w\right)q\hfill \\ =\left(a-\lambda q_1-\varphi q_2-w\right)\left(\lambda q_1+\varphi q_2\right)\hfill \end{array}$$

(4)

The total profit of the supply chain is

$$\pi =\lambda {\pi }_1+\varphi {\pi }_2+{\pi }_m.$$

(5)

The social welfare is

$$SW=\pi +CS.$$

(6)

The decision sequence is shown in Figure 1. Before the selling season, the firm M firstly declares its wholesale price $w$, and then the farmers determine their efforts $e_i$. At the beginning of the selling season, the firm M sells the product at the clearing price $p$ basis of the yields of the farmers $q$.

For easy reference, we list all notations in

Table 1. The description of the notation.

Notation Definition
$i$	A subscript symbol to indicate the type of the farmer, $i=1$ indicates the poor farmer and $i=2$ indicates the regular type
$m$	A subscript symbol to indicate the firm M
$\lambda$	The proportion of the poor farmer, $0<\lambda <1$
$\varphi$	the proportion of the regular farmer, $\varphi =1-\lambda$
$e_i$	The effort level of the farmers, farmer’s decision variable
$k$	The impact factor for efforts, $k>0$
$q_i$	The yields of the farmer
$h_i$	The cost factor of farmer’s effort, $h_i>0$
$\xi$	The gap between $h_1$ and $h_2$,$\xi =h_2/h_1$,$0<\xi \le 1$
$g$	The government subsidy, $g\ge 0$
$w$	The wholesale price, the firm M’s decision variable
$p$	The market clearing price
${\pi }_i$	The payoff of the farmer
${\pi }_m$	The profit of the firm M
$NS$	A superscript symbol to indicate the case of no subsidy
$S$	A superscript symbol to indicate the case of exclusive subsidy
$SS$	A superscript symbol to indicate the case of sharing subsidy
$ST$	A superscript symbol to indicate the case of co-subsidy

.

3.2. Decision Analysis

In this subsection, we consider no subsidy as a performance benchmark. We remark this case as superscript symbol $NS$. We utilize a backward induction approach to derive the equilibrium decisions of the three channel members and obtain the equilibrium solutions as in Proposition 1. In addition, for the simplicity of formula, we give $\beta =1-\lambda +\lambda \xi$.

Proposition 1.

When the government does not subsidize the poor farmer, the firm’s equilibrium wholesale price is

$$w^{NS\ast }=\frac{a{h}_2}{2\left(k^2\beta +h_2\right)}$$

(7)

The equilibrium effort levels of the individual poor and regular farmer are respectively

$$e_1{}^{NS\ast }=\frac{ak\xi }{2\left(k^2\beta +h_2\right)}$$

(8)

$$e_2{}^{NS\ast }=\frac{ak}{2\left(k^2\beta +h_2\right)}$$

(9)

The equilibrium market price of the supply chain is

$$p^{NS\ast }=\frac{a\left(k^2\beta +2h_2\right)}{2\left(k^2\beta +h_2\right)}$$

(10)

the optimal payoffs of the individual poor and regular farmer are respectively

$${\pi }_1{}^{NS\ast }=\frac{a^2{k}^2{h}_2\xi }{8{\left(k^2\beta +h_2\right)}^2}$$

(11)

$${\pi }_2{}^{NS\ast }=\frac{a^2{k}^2{h}_2}{8{\left(k^2\beta +h_2\right)}^2}$$

(12)

the optimal profit of the firm M is

$${\pi }_m{}^{NS\ast }=\frac{a^2{k}^2\beta }{4\left(k^2\beta +h_2\right)}$$

(13)

the optimal profit of the whole supply chain is

$${\pi }^{NS\ast }=\lambda {\pi }_1{}^{NS\ast }+\varphi {\pi }_2{}^{NS\ast }+{\pi }_m{}^{NS\ast }$$

(14)

The consumer surplus is

$$C{S}^{NS\ast }=\frac{a^2{k}^4{\beta }^2}{8{\left(k^2\beta +h_2\right)}^2}$$

(15)

The social welfare is

$$S{W}^{NS\ast }={\pi }^{NS\ast }+C{S}^{NS\ast }$$

(16)

From Proposition 1, we have Corollary 1 and Corollary 2.

Corollary 1.

$\partial w^{NS\ast }/\partial \lambda >0,\partial w^{NS\ast }/\partial \xi <0$.

Corollary 1 demonstrates the fact that if the government does not subsidize the poor farmer, the firm has to supply a higher wholesale price when the poor farmers are involved. This means the firm will spend more money doing business with the poor farmer. Therefore, the best choice for the firm is to only do business with regular farmers. As a result, the poor farmer will be in a particularly weak position without outside help.

Corollary 2.

$e_1{}^{NS\ast }<e_2{}^{NS\ast },{\pi }_1{}^{NS\ast }<{\pi }_2{}^{NS\ast }$.

Corollary 2 shows that the effort and payoff of the poor farmer are less than the regular farmer’s without government subsidies.

4. Game Decision Cases with Subsidy

In this section, we investigate the three cases: exclusive subsidy for the poor farmer, sharing subsidy between the poor farmer and the firm, and subsidy together between the poor and the regular farmer. For the sake of expression, we describe them as exclusive subsidy, sharing subsidy and co-subsidy hereinafter.

4.1. Game Decision Case with Exclusive Subsidy

In this subsection, we consider that the government gives subsidy $g\left(g\ge 0\right)$ per unit product to the poor farmers, and $g$ is an exogenous variable. We remark this case as superscript symbol $S$. We have the payoff of the individual poor farmer and regular farmer as follows.

$${\pi }_1{}^S=\left(w+g\right)k{e}_1-h_1{e}_1{}^2/2$$

(17)

$${\pi }_2{}^S=wk{e}_2-h_2{e}_2{}^2/2$$

(18)

The profit of the firm M is as follows.

$${\pi }_m{}^S=\left(a-\lambda k{e}_1-\varphi k{e}_2-w\right)\left(\lambda k{e}_1+\varphi k{e}_2\right)$$

(19)

The profit of the whole supply chain is as follows.

$${\pi }^S=\lambda {\pi }_1{}^S+\varphi {\pi }_2{}^S+{\pi }_m{}^S$$

(20)

In this case, we can easily find the equilibrium solutions as follows in Proposition 2.

Proposition 2.

When the government gives subsidy $g$per unit product to the poor farmers, (1)$g$should be$g\in (0,\frac{a{h}_2\beta }{\left(2k^2\beta +h_2\right)\lambda \xi })$, and (2) the firm’s equilibrium wholesale price is

$$w^S{}^{\ast }=\frac{a{h}_2\beta -\left(2k^2\beta +h_2\right)\lambda \xi g}{\left(2k^2\beta +h_2\right)\beta }$$

(21)

the equilibrium effort levels of the individual poor and regular farmer are, respectively,

$$e_1{{}^S}^{\ast }=\frac{\left[\left(a+g\right)h_2\beta +\left(1-\lambda \right)\left(2k^2\beta +h_2\right)g\right]k\xi }{\left(2k^2\beta +h_2\right)h_2\beta }$$

(22)

$$e_2{{}^S}^{\ast }=\frac{\left[a{h}_2\beta -\left(2k^2\beta +h_2\right)g\lambda \xi \right]k}{\left(2k^2\beta +h_2\right)h_2\beta }$$

(23)

the equilibrium market price of the supply chain is

$$p^S{}^{\ast }=\frac{a\left(k^2\beta +2h_2\right)-k^{2}g\lambda \xi }{\left(2k^2\beta +h_2\right)}$$

(24)

the optimal payoffs of the individual poor and regular farmer are, respectively,

$${\pi }_1{{}^S}^{\ast }=\frac{{\left[\left(a+g\right)h_2\beta +\left(1-\lambda \right)\left(2k^2\beta +h_2\right)g\right]}^2{k}^2\xi }{8{\left(k^2\beta +h_2\right)}^2{h}_2{\beta }^2}$$

(25)

$${\pi }_2{{}^S}^{\ast }=\frac{{\left[a{h}_2\beta -\lambda \xi g\left(2k^2\beta +h_2\right)\right]}^2{k}^2}{8{\left(k^2\beta +h_2\right)}^2{h}_2{\beta }^2}$$

(26)

the optimal profit of the firm M is

$${\pi }_m{{}^S}^{\ast }=\frac{{\left(a\beta +\lambda \xi g\right)}^2{k}^2}{4\left(k^2\beta +h_2\right)\beta }$$

(27)

the optimal profit of the whole supply chain is

$${\pi }^S{}^{\ast }=\lambda {\pi }_1{{}^S}^{\ast }+\varphi {\pi }_2{{}^S}^{\ast }+{\pi }_m{{}^S}^{\ast }$$

(28)

At this time, the total subsidy $G^S{}^{\ast }$input by the government is

$$G^S{}^{\ast }=\frac{\left[\left(a+g\right)h_2\beta +\left(1-\lambda \right)\left(2k^2\beta +h_2\right)g\right]k^2\lambda \xi g}{2\left(k^2\beta +h_2\right)h_2\beta }$$

(29)

The consumer surplus is

$$C{S}^S{}^{\ast }=\frac{{\left(a\beta +g\lambda \xi \right)}^2{k}^4}{8{\left(k^2\beta +h_2\right)}^2}$$

(30)

The social welfare is

$$S{W}^S{}^{\ast }={\pi }^S{}^{\ast }+C{S}^S{}^{\ast }-G^S{}^{\ast }$$

(31)

From Proposition 2, we have Corollary 3 and Corollary 4.

Corollary 3.

$\partial w^{S^{\ast }}/\partial g<0,\partial e_1{}^{S\ast }/\partial g>0,\partial e_2{{}^S}^{\ast }/\partial g<0,\partial p^S{}^{\ast }/\partial g<0$.

Corollary 3 shows that the higher per unit subsidy the government gives, the lower wholesale price the farmers get, and that the higher subsidy the government gives, the more efforts of the poor farmer input and the less efforts of the regular farmer input. Corollary 3 demonstrates that the market price will decrease with the increase of per unit subsidy.

Corollary 4.

When the government subsidizes the poor farmer$g$per unit product, and $g\in (0,\frac{a{h}_2\beta }{(2k^2\beta +h_2)\lambda \xi })$, there exists a $g^{\ast }=\frac{a{h}_2\left(1-\xi \right)\beta }{\left(2k^2+h_2\right)\xi \beta +h_2\xi }$to make $e_1{{}^S}^{\ast }=e_2{{}^S}^{\ast }$.

Corollary 4 shows that when the government subsidy is equal to $\frac{a{h}_2\left(1-\xi \right)\beta }{\left(2k^2+h_2\right)\xi \beta +h_2\xi }$, the poor farmer will input the same efforts as the regular farmer.

Corollary 4 suggests that it is difficult to make decision on $g^{\ast }$ because $a$ is not easy to know exactly. For some ordinary agricultural product, we can forecast $a$ from the historic datum, but for some emerging product, forecasting $a$ is very difficult. In the meanwhile, $\xi$ is difficult to quantify, in other words, it is very hard to measure the distance of effort cost between the poor farmer and regular farmer.

4.2. Game Decision Case with Sharing Subsidy

Consider sharing subsidies between the firm and the poor farmer, and the proportion $\theta \left(0\le \theta \le 1\right)$ for the poor farmer and $1-\theta$ for the firm. We remark this case as superscript symbol $SS$. In this case, the payoffs of the poor and regular farmer, and the profits of the firm and the whole supply chain are as follows.

$${\pi }_1{}^{SS}=\left(w+\theta g\right)k{e}_1-h_1{e}_1{}^2/2$$

(32)

$${\pi }_2{}^{SS}=wk{e}_2-h_2{e}_2{}^2/2$$

(33)

$${\pi }_m{}^{SS}=\left(a-\lambda k{e}_1-\varphi k{e}_2-w\right)\left(\lambda k{e}_1+\varphi k{e}_2\right)+\left(1-\theta \right)g\lambda k{e}_1$$

(34)

$${\pi }^{SS}=\lambda {\pi }_1{}^{SS}+\varphi {\pi }_2{}^{SS}+{\pi }_m{}^{SS}$$

(35)

We can easily find the equilibrium solutions as follows in Proposition 3.

Proposition 3.

When government subsidy $g$for the poor farmer is shared with the firm, (1) $g$should be $g\in (0,\frac{a{h}_2\beta }{\left(2k^2\beta +h_2\right)\lambda \xi })$, and (2) the firm’s equilibrium wholesale price is

$$w^{SS\ast }=w^S{}^{\ast }+\frac{\left(1-\theta \right)g\lambda \xi }{\beta }$$

(36)

the equilibrium effort levels of the individual poor and regular farmer are, respectively,

$$e_1{}^{SS\ast }=e_1{{}^S}^{\ast }-\frac{\left(1-\theta \right)\left(1-\lambda \right)gk\xi }{h_2\beta }$$

(37)

$$e_2{}^{SS\ast }=e_2{{}^S}^{\ast }+\frac{\left(1-\theta \right)\lambda gk\xi }{h_2\beta }$$

(38)

The equilibrium retail price of the supply chain is

$$p^{SS\ast }=p^S{}^{\ast }=\frac{a\left(k^2\beta +2h_2\right)-k^{2}g\lambda \xi }{2\left(k^2\beta +h_2\right)}$$

(39)

The optimal payoff of the poor and regular farmer is as follows

$${\pi }_1{}^{SS\ast }=\frac{{\left[a{h}_2\beta +h_{2}g\lambda \xi +2\left(1-\lambda \right)\theta g\left(k^2\beta +h_2\right)\right]}^2{k}^2\xi }{8{\left(k^2\beta +h_2\right)}^2{\beta }^2{h}_2}$$

(40)

$${\pi }_2{}^{SS\ast }=\frac{{\left[a{h}_2\beta +h_{2}g\lambda \xi -2\lambda \theta g\left(k^2\beta +h_2\right)\xi \right]}^2{k}^2}{8{\left(k^2\beta +h_2\right)}^2{\beta }^2{h}_2}$$

(41)

The optimal profit of the firm M is,

$${\pi }_m{}^{SS\ast }={\pi }_m{{}^S}^{\ast }+\frac{\theta \left(1-\theta \right)\lambda \left(1-\lambda \right)k^2{g}^2\xi }{h_2\beta }$$

(42)

The optimal profit of the whole supply chain is

$${\pi }^{SS\ast }={\pi }^S{}^{\ast }-\frac{{\left(1-\theta \right)}^2\left(1-\lambda \right)\lambda k^2{g}^2\xi }{2h_2\beta }$$

(43)

at this time, the total subsidy$G^{SS\ast }$input by the government is

$$G^{SS\ast }=G^S{}^{\ast }-\frac{\lambda \left(1-\lambda \right)\left(1-\theta \right)k^2{g}^2\xi }{h_2\beta }$$

(44)

The consumer surplus is

$$C{S}^{SS\ast }=C{S}^S{}^{\ast }=\frac{{\left(a\beta +g\lambda \xi \right)}^2{k}^4}{8{\left(k^2\beta +h_2\right)}^2}$$

(45)

The social welfare is

$$S{W}^{SS\ast }={\pi }^{SS\ast }+C{S}^{SS\ast }-G^{SS\ast }$$

(46)

From Proposition 3, we have Corollary 5 and Corollary 6.

Corollary 5.

$\partial w^{SS\ast }/\partial \theta <0,\partial e_1{}^{SS\ast }/\partial \theta >0,\partial e_2{}^{SS\ast }/\partial \theta <0$.

Corollary 5 indicates that the more subsidies the firm gets, the higher wholesale price the firm will give, and the more efforts the regular farmer will input. At the same time, it means that the less subsidies the poor farmers get, the more the efforts of the poor farmer will decrease.

Corollary 6.

When $\theta =0.5$, the enterprise achieves maximized profits; when $\theta =1$, the whole supply chain realizes the maximized profit.

Corollary 5 shows that sharing subsidy will increase the profit of the firm, but the profit will not always increase with the increase of the sharing proportion, the best situation is that the firm gets a half of the subsidy. Corollary 6 indicates that the sharing subsidy will reduce the profit of the whole supply chain.

4.3. Game Decision Case with Co-Subsidy

In this subsection, we consider that the government gives subsidy $g\left(g\ge 0\right)$ per unit product to the poor and the regular farmer together. We remark this case as superscript symbol $ST$. We have the payoff of the individual poor and regular farmer as follows

$${\pi }_1{}^{ST}=\left(w+g\right)k{e}_1-h_1{e}_1{}^2/2$$

(47)

$${\pi }_2{}^{ST}=\left(w+g\right)k{e}_2-h_2{e}_2{}^2/2$$

(48)

The profit of the firm M is as follows

$${\pi }_m{}^{ST}=\left(a-\lambda k{e}_1-\varphi k{e}_2-w\right)\left(\lambda k{e}_1+\varphi k{e}_2\right)$$

(49)

The profit of the supply chain system is as follows

$${\pi }^{ST}=\lambda {\pi }_1{}^{ST}+\varphi {\pi }_2{}^{ST}+{\pi }_m{}^{ST}$$

(50)

In this case, the equilibrium solutions are given in Proposition 4.

Proposition 4.

When the government subsidizes per unit product $g$to the poor and regular farmers together, (1) $g$should be $g$  $\in$  (0,$\frac{a{h}_2}{2k^2\beta +h_2}$), (2) the firm’s equilibrium wholesale price is

$$w^{ST\ast }=\frac{a{h}_2-\left(2k^2\beta +h_2\right)g}{2\left(k^2\beta +h_2\right)}$$

(51)

the equilibrium effort levels of the individual poor and regular farmer are, respectively,

$$e_1{}^{ST\ast }=\frac{\left(a+g\right)k\xi }{2\left(k^2\beta +h_2\right)}$$

(52)

$$e_2{}^{ST\ast }=\frac{\left(a+g\right)k}{2\left(k^2\beta +h_2\right)}$$

(53)

the equilibrium retail price of the supply chain is

$$p^{ST\ast }=\frac{a\left(k^2\beta +2h_2\right)-k^{2}g\beta }{2\left(k^2\beta +h_2\right)}$$

(54)

the optimal payoffs of the individual poor and regular farmer are, respectively,

$${\pi }_1{}^{ST\ast }=\frac{{(a+g)}^2{h}_2{k}^2\xi }{8{(k^2\beta +h_2)}^2}$$

(55)

$${\pi }_2{}^{ST\ast }=\frac{{(a+g)}^2{h}_2{k}^2}{8{(k^2\beta +h_2)}^2}$$

(56)

the optimal profit of the firm is

$${\pi }_m{}^{ST\ast }=\frac{{(a+g)}^2{k}^2\beta }{4\left(k^2\beta +h_2\right)}$$

(57)

the optimal profit of the whole supply chain is

$${\pi }^{ST\ast }=\lambda {\pi }_1{}^{ST\ast }+\varphi {\pi }_2{}^{ST\ast }+{\pi }_m{}^{ST\ast }$$

(58)

at this time, the total subsidy G^ST* input by the government is

$$G^{ST\ast }=\frac{\left(a+g\right)g{k}^2\beta }{2\left(k^2\beta +h_2\right)}$$

(59)

The consumer surplus is

$$C{S}^{ST\ast }=\frac{{\left(a+g\right)}^2{k}^4{\beta }^2}{8{(k^2\beta +h_2)}^2}$$

(60)

The social welfare is

$$S{W}^{ST\ast }={\pi }^{ST\ast }+C{S}^{ST\ast }-G^{ST\ast }$$

(61)

5. Comparative Analysis of Equilibrium Solutions

In this section, we carry on the comparative analysis of equilibrium solutions for the above cases.

5.1. Exclusive Subsidy and No Subsidy

Based on the comparison between Proposition 1 and Proposition 2, we can obtain Proposition 5 as follows.

Proposition 5.

When $g\in (0, \frac{a{h}_2\beta }{\left(2k^2\beta +h_2\right)\lambda \xi })$, (1) $w^{S\ast }<w^{NS\ast }$; (2) $e_1{}^{S\ast }>e_1{}^{NS\ast },e_2{}^{S\ast }<e_2{}^{NS\ast }$; (3) $C{S}^{S\ast }>C{S}^{NS\ast }$; (4) ${\pi }_1{}^{S\ast }>{\pi }_1{}^{NS\ast },{\pi }_2{}^{S\ast }<{\pi }_2{}^{NS\ast }$, ${\pi }_m{}^{S\ast }>{\pi }_m{}^{NS\ast }$,${\pi }^{S\ast }>{\pi }^{NS\ast }$; (5) $\lambda \left({\pi }_1{}^{S\ast }-{\pi }_1{}^{NS\ast }\right)+\varphi \left({\pi }_2{}^{S\ast }-{\pi }_2{}^{NS\ast }\right)>0$; (6) $S{W}^{S\ast }>S{W}^{NS\ast }$.

Proposition 5 shows that when the government provides subsidies to the poor farmer, the firm will give a lower wholesale price. Therefore, it will lead to the regular farmers decreasing their efforts. We easily know $\Delta w=w^{NS\ast }-w^{S\ast }=\frac{\left(2k^2\beta +h_2\right)\lambda \xi g}{2\left(k^2\beta +h_2\right)\beta }$ and $g-\Delta w=\frac{\left[\left(1-\lambda \right)\left(2k^2\beta +h_2\right)+h_2\beta \right]g}{2\left(k^2\beta +h_2\right)\beta }>0$. This means that the subsidy the poor farmers get exceeds the decrease of the wholesale price. Therefore, it will encourage the poor farmer to input more efforts.

Proposition 5 indicates that the government subsidy improves the payoff of the poor farmer, the profit of the firm, and the whole supply chain. On the other hand, the subsidy leads to the decrease in the payoff of the regular farmer, but the total payoff increase of the poor farmer is more than the total payoff decrease of the regular farmer. Generally, government subsidy is beneficial to the whole supply chain and the social welfare. Given $\lambda \Delta {\pi }_1{}^{SN}=\lambda {\pi }_1{}^{S\ast }-\lambda {\pi }_1{}^{NS\ast },\varphi \Delta {\pi }_2{}^{SN}=\varphi {\pi }_2{}^{S\ast }-\varphi {\pi }_2{}^{NS\ast },\Delta {\pi }_m{}^{SN}={\pi }_m{}^{S\ast }-{\pi }_m{}^{NS\ast },\Delta {\pi }^{SN}={\pi }^{S\ast }-{\pi }^{NS\ast }$, and $\Delta S{W}^{SN}=S{W}^{S\ast }-S{W}^{NS\ast }.$ From Equations (11)–(14), (16), (25)–(28), and (31), we find that the mathematical formulas above are very complex. To provide a direct view for the reader, we use Maple 13 to simulate the mathematical formulas with the $\lambda$. Give $a=\mathrm{50,000},h_2=0.1,\xi =0.1,k=1,g=2$ and substitute $\varphi$ with $\lambda \left(\lambda \in \left(0,1\right)\right).$ We have Figure 2 as follows.

It is obvious from Figure 2 that the incremental profits of the firm and the whole supply chain increase with the increase of the poor proportion, and the increment of social welfare increases with the increase of the poor proportion under government subsidy. This means that the increased profits of the whole supply chain are not just the transfer of government subsidies. Figure 2 indicates the whole increase payoff of the poor farmer is more increased than the whole decrease payoff of the regular farmer with the increase of the poor proportion. Although subsidy may lead to the payoff decrease of the regular farmer, it has a positive effect on alleviating poverty and increasing social welfare, especially when the poor farmer is in majority.

5.2. Exclusive Subsidy and Sharing Subsidy

Section 5.1 shows that the firm will benefit from subsidy by lowering wholesale prices, and it will hurt the regular farmer. We question whether the situation can be improved by letting the firm share subsidy with the poor farmer. Based on the comparison analysis between Proposition 2 and Proposition 3, we can obtain Proposition 6 as follows.

Proposition 6.

When$g\in (0,\frac{a{h}_2\beta }{\left(2k^2\beta +h_2\right)\lambda \xi })$, (1) $p^{SS\ast }=p^{S\ast },C{S}^{SS\ast }=C{S}^{S\ast }$; (2) $w^{SS\ast }\ge w^{S\ast },e_1{}^{SS\ast }\le e_1{}^{S\ast },e_2{}^{SS\ast }\ge e_2{}^{S\ast }$; (3) ${\pi }_1{}^{SS\ast }\le {\pi }_1{}^{S\ast },{\pi }_2{}^{SS\ast }\ge {\pi }_2{}^{S\ast },{\pi }_m{}^{SS\ast }\ge {\pi }_m{}^{S\ast },{\pi }^{SS\ast }\le {\pi }^{S\ast }$; (4) $\lambda \left({\pi }_1{}^{SS\ast }-{\pi }_1{}^{S\ast }\right)+\varphi \left({\pi }_2{}^{SS\ast }-{\pi }_2{}^{S\ast }\right)\le 0$; (5) $G^{SS\ast }\le G^{S\ast }$; (6) $S{W}^{SS\ast }\ge S{W}^{S\ast }$.

Proposition 6 indicates that, compared with exclusive subsidy, subsidy sharing will not change the consumer surplus, and it will lead to the payoff of the poor farmer decrease and that of the regular farmer increase, and the profit of the firm increase. Moreover, the whole payoff decrease of the poor farmer is greater than the whole payoff increase of the regular farmer before and after subsidy sharing, and the total profits of the supply chain decreases, but the social welfare increases. Actually, subsidy sharing is not beneficial for the whole supply chain.

To help understand the change with $\theta$ and $\lambda$ before and after subsidy sharing, we give a numerical simulation through Maple 13. We make $\lambda \Delta {\pi }_1{}^S=\lambda {\pi }_1{}^{SS\ast }-\lambda {\pi }_1{}^{S\ast },\varphi \Delta {\pi }_2{}^S=\varphi {\pi }_2{}^{SS\ast }-\varphi {\pi }_2{}^{S\ast },\Delta {\pi }_m{}^S={\pi }_m{}^{SS\ast }-{\pi }_m{}^{S\ast },\Delta {\pi }^S={\pi }^{SS\ast }-{\pi }^{S\ast },$ and $\Delta S{W}^S=S{W}^{SS\ast }-S{W}^{S\ast }.$ Give $a=\mathrm{50,000}$, $h_2=0.1,\xi =0.1,k=1,g=2$, we obtain Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8.

Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 indicate that whether the poor farmer is in minority or majority, the sharing subsidy has a great influence on the payoffs of the poor and the regular farmer, and has a small influence on the profits of the enterprise and the whole supply chain, and the social welfare. The results reveal that when the enterprise has the decision-making power of the wholesale price, subsidy sharing cannot resolve the dilemma that the payoff of the poor farmer increases with the payoff decrease of the regular farmer, and its role is a counterbalance between the payoff of the poor and the regular farmer.

5.3. Comparative Analysis of Equilibrium Solutions among Four Cases

Based on the comparative analysis among Proposition 1, Proposition 2, Proposition 3, and Proposition 4, we can obtain Proposition 7 as follows.

Proposition 7.

When$g\in (0,\frac{a{h}_2}{2k^2\beta +h_2})$, (1) $p^{ST\ast }<p^{SS\ast }=p^{S\ast }<p^{NS\ast }$; (2) $C{S}^{ST\ast }>C{S}^{SS\ast }=C{S}^{S\ast }>C{S}^{NS\ast }$; (3) when $\theta \in (\frac{0.5h_2}{k^2\beta +h_2},1],w^{ST\ast }<w^{S\ast }\le w^{SS\ast }<w^{NS\ast }$, when $\theta \in [0,\frac{0.5h_2}{k^2\beta +h_2}],w^{ST\ast }<w^{S\ast }<w^{NS\ast }\le w^{SS\ast }$; (4) ${\pi }_1{}^{NS\ast }<{\pi }_1{}^{ST\ast }<{\pi }_1{}^{SS\ast }\le {\pi }_1{}^{S\ast }$, (5) when $\theta \in (\frac{0.5h_2}{k^2\beta +h_2},1],{\pi }_2{}^{ST\ast }>{\pi }_2{}^{NS\ast }>{\pi }_2{}^{SS\ast }\ge {\pi }_2{}^{S\ast }$, when $\theta \in [0,\frac{0.5h_2}{k^2\beta +h_2}]$, ${\pi }_2{}^{ST\ast }>{\pi }_2{}^{SS\ast }\ge {\pi }_2{}^{NS\ast }>{\pi }_2{}^{S\ast }$; (6) ${\pi }_m{}^{ST\ast }>{\pi }_m{}^{SS\ast }\ge {\pi }_m{}^{S\ast }>{\pi }_m{}^{NS\ast }$; (7) ${\pi }^{ST\ast }>{\pi }^{S\ast }\ge {\pi }^{SS\ast }>{\pi }^{NS\ast }$; (8) $G^{ST\ast }>G^{S\ast }\ge G^{SS\ast }$; (9) $S{W}^{ST\ast }>S{W}^{SS\ast }\ge S{W}^{S\ast }>S{W}^{NS\ast }$.

Proposition 7 indicates that co-subsidy is the best for improving consumer surplus and social welfare, and it can increase the payoffs of both the poor and regular farmer, but it means the government has to spend more money; on the other hand, Formula (4) indicates that it has the worst effect on poverty alleviation in the above three subsidy ways, and the best way for alleviating poverty is exclusive subsidy; Formula (5) indicates that sharing subsidy can mitigate the negative effects of exclusive subsidy on the regular farmer.

To further understand the impact of subsidy with different $\lambda$, we make $\Delta {\pi }_1{}^{SN}={\pi }_1{}^{S\ast }-{\pi }_1{}^{NS\ast }$, $\Delta {\pi }_1{}^{SSN}={\pi }_1{}^{SS\ast }-{\pi }_1{}^{NS\ast }$, $\Delta {\pi }_1{}^{STN}={\pi }_1{}^{ST\ast }-{\pi }_1{}^{NS\ast }$; $\Delta {\pi }_2{}^{SN}={\pi }_2{}^{S\ast }-{\pi }_2{}^{NS\ast }$, $\Delta {\pi }_2{}^{SSN}={\pi }_2{}^{SS\ast }-{\pi }_2{}^{NS\ast }$, $\Delta {\pi }_2{}^{STN}={\pi }_2{}^{ST\ast }-{\pi }_2{}^{NS\ast }$; $\Delta {\pi }_m{}^{SN}={\pi }_m{}^{S\ast }-{\pi }_m{}^{NS\ast }$, $\Delta {\pi }_m{}^{SSN}={\pi }_m{}^{SS\ast }-{\pi }_m{}^{NS\ast }$, $\Delta {\pi }_m{}^{STN}={\pi }_m{}^{ST\ast }-{\pi }_m{}^{NS\ast }$; $\Delta {\pi }^{SN}={\pi }^{S\ast }-{\pi }^{NS\ast }$, $\Delta {\pi }^{SSN}={\pi }^{SS\ast }-{\pi }^{NS\ast }$, $\Delta {\pi }^{STN}={\pi }^{ST\ast }-{\pi }^{NS\ast }$; $\Delta S{W}^{SN}=S{W}^{S\ast }-S{W}^{NS\ast }$, $\Delta S{W}^{SSN}=S{W}^{SS\ast }-S{W}^{NS\ast }$, $\Delta S{W}^{STN}=S{W}^{ST\ast }-S{W}^{NS\ast }$. We use Maple 13 to simulate above-increment change with $\lambda$. We give $a=\mathrm{50,000}$, $k=1$, $g=2$, $h_2=0.1$, and we give a fraction of subsidy to the firm, and give $\theta =0.8$. Furthermore, considering deep poverty, we give $\xi =0.1$, and $h_1=h_2/\xi =1$, then we get Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14.

Figure 13 shows that exclusive subsidy and sharing subsidy is better for poverty alleviation than co-subsidy, and when the poor farmer is in the minority, such as λ < 0.5, co-subsidy makes little impact on poverty alleviation. Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14 show co-subsidy is the best for the regular farmer, the firm, and social welfare.

Furthermore, we show the increase value with different λ and θ in the

Table 2. The increase value with different $\lambda$ and $\theta$.

		$\mathit{\lambda }=0.8$		$\mathit{\lambda }=0.2$
θ	$0.80$	$0.50$	$0.13$	$0.80$	$0.50$	0.05
$\Delta {\pi }_1{}^{SN}$	9894	9894	9894	5311	5311	5311
$\Delta {\pi }_1{}^{SSN}$	8014	5194	1716	4250	2658	272
$\Delta {\pi }_1{}^{STN}$	1731	1731	1731	295	295	295
$\Delta {\pi }_2{}^{SN}$	−32,646	−32,646	−32,646	−1253	−1253	−1253
$\Delta {\pi }_2{}^{SSN}$	−25,127	−13,850	59	−988	−590	6
$\Delta {\pi }_2{}^{STN}$	17,313	17,313	17,313	2953	2953	2953
$\Delta {\pi }_m{}^{SN}$	10,526	10,526	10,526	1086	1086	1086
$\Delta {\pi }_m{}^{SSN}$	10,526	10,527	10,526	1086	1086	1086
$\Delta {\pi }_m{}^{STN}$	36,842	36,842	36,842	44,566	44,566	44,566
$\Delta C{S}^{SN}$	3878	3878	3878	484	484	484
$\Delta C{S}^{SSN}$	3878	3878	3878	484	484	484
$\Delta C{S}^{STN}$	13,573	13,573	13,573	19,860	19,860	19,860
$G^{S\ast }$	10,528	10,528	10,528	1087	1087	1087
$G^{SS\ast }$	10,528	10,527	10,526	1087	1087	1087
$G^{ST\ast }$	36,843	36,843	36,843	44,567	44,567	44,567

.

6. Conclusions

Poverty is a worldwide problem, and government subsidy is a common factor in poverty alleviation. This paper focuses on the impact of government subsidy on poverty alleviation from the perspective of supply chain operations management and investigates the optimal policy of government subsidy in poverty alleviation and social welfare improvement. Considering the actual situation that a company usually buys from both the poor and regular farmer, we focus on a supply chain consisting of the poor, the regular farmer, and a firm, and the farmers are the suppliers. The firm determines its wholesale price and resells the farmer’s product in the market, and the farmers determine their production effort level with a given wholesale price. We introduce the government subsidy as an exogenous variable and build a series of decision game models with no subsidy, exclusive subsidy, sharing subsidy, and co-subsidy to discuss the impact of government subsidies on poverty alleviation and social welfare improvement with different scales of poverty.

Compared with the existing literature, the main theoretical contribution of this paper lies in the following. Firstly, this paper provides a new perspective by considering the participation of the poor and regular farmers together in the supply chain operation. Secondly, this paper introduces the poverty scale in OM models. Thirdly, this study pays attention to the effect of government subsidy to enterprise on the payoff of the poor farmer, and it is rare in literature.

Our research leads to the following findings. ① Exclusive subsidy for the poor farmer has a significant effect on poverty alleviation, but a dilemma is that the payoff of the poor farmer increases with the payoff decrease of the regular farmer. Especially when the poor farmers are in the vast majority, exclusive subsidy will lead to the large payoff loss of the regular farmer; ② sharing subsidy between the poor farmer and the firm has a counterbalancing effect on the payoff of the poor and regular farmers, and when transferring a vast proportion, even all subsidies to the firm, government subsidy will lead to little loss to the regular farmer. ③ One interesting finding is that this does not mean that the firm will gain more profits when it gives more subsidies to itself, and the best situation for the firm is one half of the subsidy for the poor farmer and the other half for itself. ④ Co-subsidy is the best for improving consumer surplus and social welfare, and it can increase the payoff of the regular farmer, but it makes little impact to improving the poor farmers’ payoff, especially when the poor farmers are in the minority, moreover, it means the government will spend more money. ⑤ Relatively speaking, when the poor farmers are in the majority, sharing subsidy or co-subsidy is more reasonable than exclusive subsidy, and sharing subsidy is more economical for the government than co-subsidy. When the poor farmer is in the minority, exclusive subsidy or sharing subsidy will be a wiser choice for the government than co-subsidy. In fact, the firm is the biggest beneficiary of the government subsidy when it has the decision power of the wholesale price.

A practical value of our paper is that although government subsidy is considered a common practice in poverty alleviation, our models prove the efficiency of subsidy when the firm owns the decision power of the wholesale price. Our research can help the government recognize that spending more money may achieve a poor result in poverty alleviation. Our model can also help the firm to make a decision, and realize it is better to give more money to the poor farmer than to itself in a supply chain operation with the participation of the poor.

Limitations and possible future research of this study include the following. Firstly, we consider that the firm can make profits at a clear price in a supply chain with the poor farmer. It is worth further studying the special circumstance when the participation of the poor farmer will lead to making a firm unprofitable. Secondly, we consider the market demand is constant and the market is risk free. It is worth further studying the impacts of government subsidy on the operation of the supply chain with poverty alleviation in a high market risk. Thirdly, we consider that the firms are greedy and always pay attention to their own interests. In fact, increasingly, enterprises have poverty concerns, so it is worth to further study the impacts of poverty concerns on poverty alleviation and supply chain operation.