*3.2. Notation*

Based on the problem description and assumptions above, the notations used in this study are described in Table 1 below.


**Table 1.** The description of the notations.

### **4. Model Construction and Solving**

*4.1. Decentralized Decision Model*

According to the above assumptions, the profit function of agricultural producer after implementing quality improvement actions is

$$
\pi\_n = q \left( (1+\alpha)w\_0 - c\_n - \frac{1}{2}zg^2 \right),
\tag{1}
$$

The profit function of the agricultural produce seller is

$$
\pi\_{\mathfrak{s}} = p\_0(1+a)(q - \text{bp}\eta(1+a) + \text{kg}) - q((1+a)w\eta + c\_{\mathfrak{s}}),
\tag{2}
$$

**Proposition 1.** *There exists a quality safety degree g that maximizes the profit of the agricultural producer and a level of price subsidy α that maximizes the profit of the agri-foods seller.*

**Proof.** The profit function *πn* is as follows:

$$
\pi\_n = q \left[ (1+\alpha)w\_0 - c\_n - \frac{1}{2}zg^2 \right],
$$

The partial derivative of *πn* with respect to *g* yields *∂πn ∂g* = −*qzg* = 0, *<sup>∂</sup>*2*πn ∂*<sup>2</sup> *g* = −*qz* < 0, (*q* > 0, *z* > <sup>0</sup>). Therefore, *πn* is a strictly concave function on *g*; i.e., there exists a maximum value of the profit function *πn*. -

The profit function *πs* is as follows:

$$\pi\_s = p\_0(1+\mathfrak{a})(\mathfrak{q} - \mathfrak{b}p\_0(1+\mathfrak{a}) + \mathfrak{k}\mathfrak{g}) - \mathfrak{q}((1+\mathfrak{a})w\_0 + c\_s)$$

The partial derivative of *πs* with respect to *α* yields *∂πs ∂α* = −<sup>2</sup>(<sup>1</sup> + *<sup>α</sup>*)*bp*0<sup>2</sup> + (*kg* + *q*)*p*0 − *qw*0 = 0, *<sup>∂</sup>*2*πs ∂*2*α* = −*bp*0<sup>2</sup> < 0. Therefore, *πs* is a strictly concave function on *α*; i.e., there exists a maximum value of the profit function *πs*.

(1) Producer-led Model (Model 1)

In this case, the producer is the leader of the supply chain, and the seller is the follower, such as the family farm-dominated or cooperative-dominated agricultural supply chain. Hence, this is a Stackelberg game model dominated by agricultural producer. In this model, the producer determines the quality input *u* (or quality safety degree *g*), and then, the seller chooses the optimal price compensation factor *α*<sup>∗</sup> based on the producer's input, and the backward induction method is applied to solve the model.

When *∂πn ∂*g = −*qzg* = 0, the optimal quality safety degree can be obtained as follows:

$$g\_1^\* = 0,\tag{3}$$

Substitute *g*∗1 into Equation (1); when *∂πs ∂α* = *p*0*q* − <sup>2</sup>*bp*02*<sup>α</sup>* − <sup>2</sup>*bp*0<sup>2</sup> − *qw*0 = 0, the optimal price subsidy factor can be obtained as follows:

$$\alpha\_1^\* = \frac{q(p\_0 - w\_0)}{2bp\_0^2} - 1,\tag{4}$$

Obviously, the agricultural producer as the leader lacks the incentive to improve the quality in this case, but the seller still provides price subsidies due to information asymmetry. The market demand is *d*∗1 = *q*(*p*0+*w*0) 2*p*0 . Substituting Equations (3) and (4) into

Equations (1) and (2), the optimal profit of the agricultural producer and the optimal profit of the agri-foods seller are obtained as follows:

$$
\pi\_{\mathfrak{n}}^{1\*} = \frac{q^2(p\_0 - w\_0)w\_0}{2bp\_0^2} - qc\_{\mathfrak{n}}
$$

$$
\pi\_{\mathfrak{s}}^{1\*} = \frac{q^2p\_0^2 - 2q^2p\_0w\_0 + q^2w\_0^2}{4bp\_0^2} - qc\_{\mathfrak{s}}
$$

**Proposition 2.** *When decentralized decision making is dominated by the agricultural producer, the maximum profit of both producer and seller decreases with the increase of price elasticity of consumer demand, and the maximum profit of the agricultural seller increases as the retail price per unit of produce increases. When p*0 < 2*w*0*, the maximum profit of the agricultural producer increases with the increase of the retail price per unit of agricultural produce. Likewise, when p*0 > 2*w*0*, the maximum profit of the agricultural producer decreases with the increase of the retail price per unit of agri-foods.*

**Proof.** From the equations *∂*π1<sup>∗</sup> n *∂b* = −(*p*0−*w*0)*<sup>w</sup>*0*q*<sup>2</sup> <sup>2</sup>*p*<sup>2</sup> 0*b*<sup>2</sup> < 0, *∂*π1<sup>∗</sup> s *∂b* = −(*p*0−*w*0) <sup>2</sup>*q*<sup>2</sup> <sup>4</sup>*p*<sup>2</sup> 0*b*<sup>2</sup> < 0, and *∂*π1<sup>∗</sup> s *∂p*0 = (*p*0−*w*0)*<sup>w</sup>*0*q*<sup>2</sup> <sup>2</sup>*p*<sup>3</sup> 0*b* > 0, it can be obtained that *π*1<sup>∗</sup> n , *π*1<sup>∗</sup> s are negatively correlated with *b*, while *π*1<sup>∗</sup> s is positively correlated with *p*0. When *p*0 < 2*w*0, *∂π*1∗ *n ∂p*0 > 0 can be known, and otherwise, *∂π*1∗ *n ∂p*0 < 0. -

Proposition 2 shows that the maximum profit of agricultural supply chain participants is closely related to the level of economic and social development and the consumption environment. When the income level of consumers is low (high price elasticity of demand), the profit of agricultural operators is also slimmer. The higher the price of the produce before quality improvement, the higher the profit for the produce seller to participate in quality improvement. While the profit of agricultural producers is mainly influenced by the level of their price appreciation in the supply chain before quality improvement, if the retail price of agri-foods exceeds the wholesale price by more than two times, the lower the retail price of agri-foods before the implementation of quality improvement actions, and the greater the producer's profit.

### (2) Seller-led Model (Model 2)

In this case, the seller of agri-foods is the leader of the supply chain, while the producer is the follower, such as the "company + farmer" type of agricultural supply chain. Therefore, this is a Stackelberg game model dominated by agricultural seller. The price subsidy coefficient *α* is determined by the seller of agri-foods firstly, and then, the ideal quality safety degree *g*∗ and the optimal quality input level are determined by the producer according to *α*, and the backward induction method is also used to solve the model.

When *∂πs ∂α* = *p*0*q* + *p*0*kg* − <sup>2</sup>*bp*0 <sup>2</sup>*α* − <sup>2</sup>*bp*0 2 − *qw*0 = 0, the optimal price subsidy system can be obtained as follows: *α*<sup>∗</sup> 2 (*g*) = *p*0*q*<sup>+</sup>*kgp*<sup>0</sup>−*w*0*q* 2*bp*<sup>2</sup> 0 − 1. Substitute *α*<sup>∗</sup> 2 (*g*) into Equation (2); when *∂πn ∂*g = *kw*0 <sup>2</sup>*bp*0 − *zg* = 0, the optimal quality safety degree can be obtained as follows:

$$g\_2^\* = \frac{kw\_0}{2zbp\_0},\tag{5}$$

Substitute Equation (5) into the optimal response function *α*<sup>∗</sup> 2 (*g*) of the produce seller, and then, obtain the optimal price subsidy level for the quality improvement of agri-foods as follows:

$$a\_2^\* = \frac{qzb(p\_0 - w\_0) + 0.5k^2w\_0}{2xb^2p\_0^2} - 1,\tag{6}$$

Thus, the actual market demand is *d*∗2 = *qzb*(*p*0+*w*0)+0.5*k*2*w*0 2zb*p*0 . Based on this, substituting Equations (5) and (6) into Equations (1) and (2), the optimal profit of the agricultural producer and the optimal profit of the agricultural seller are obtained as follows:

$$
\pi\_n^{2\*} = \frac{(bqzp\_0w\_0 - qzbw\_0^2 - 0.25k^2w\_0^2)q}{2zb^2p\_0^2} - qc\_n
$$

$$
\pi\_s^{2\*} = \frac{q^2z^2(p\_0 - w\_0)^2b^2 + k^2qzbw\_0(p\_0 - w\_0) + 0.25k^4w\_0^2}{4z^2b^3p\_0^2} - qc\_s
$$

**Proposition 3.** *When decentralized decision making is dominated by sellers of agri-foods, the maximum profit of both agricultural producer and seller decreases with the increase of production impact factor z. The maximum profit of agricultural producer decreases as the quality elasticity of demand increases, and the price elasticity of demand decreases. The maximum profit of agricultural seller increases as the quality elasticity of demand increases, and the price elasticity of demand decreases.*

**Proof.** From the equations *∂π*2<sup>∗</sup>n *∂k* = − *qkw*20 <sup>4</sup>*zp*20*b*<sup>2</sup> < 0, *∂π*2<sup>∗</sup>n *∂z* = − *qk*2*w*20 8*z*<sup>2</sup> *p*20*b*<sup>2</sup> < 0, and *∂π*2<sup>∗</sup>n *∂b* = −*q*2*zbw*0(*<sup>w</sup>*0−*p*0)−0.25*qk*2*w*20 <sup>2</sup>*zp*20*b*<sup>3</sup> > 0, it can be obtained that *<sup>π</sup>*2<sup>∗</sup>*n* is negatively correlated with *k* and *z* and positively correlated with *b*. Additionally, *<sup>π</sup>*2<sup>∗</sup>*s* is positively correlated with *k* and negatively correlated with *z*, and *b* can be known due to *∂π*2<sup>∗</sup>*s ∂k* = *kqzbw*0(*p*0−*w*0)+0.5*w*20*k*<sup>3</sup> 2*z*<sup>2</sup> *p*20*b*<sup>3</sup> > 0, *∂π*2<sup>∗</sup>*s ∂z* = −*k*2*w*0(*qbz*(*p*0−*w*0)+0.5*k*2*w*0) 4*z*<sup>3</sup> *p*20*b*<sup>3</sup> < 0, and *∂π*2<sup>∗</sup>*s ∂b* = −*q*2*z*2*b*<sup>2</sup>(*p*0−*w*0)<sup>2</sup>−2*qzbk*2*w*0(*p*0−*w*0)−0.75*k*4*w*20 4*z*<sup>2</sup> *p*20*b*<sup>4</sup> < 0. -

Proposition 3 shows that the greater the quality elasticity of consumer demand, the less price sensitive consumers are; and the smaller the quality inputs required to improve the quality and safety of agricultural products, the greater the maximum profit for agricultural sellers is. When consumers pay more attention to quality, the supply chain profit distribution will be more unfavorable to producers, which will then force the producer to improve quality input, and the producer's cost increases, and the profit decreases in a short period of time.

### *4.2. Centralized Decision Model (Model 3)*

In this case, self-interest maximization is no longer the decision-making goal of the participants of agricultural supply chain. Instead, centralized decision making is made through win-win cooperation to maximize the overall interests of the supply chain. At this time, the total profit function of the agricultural supply chain is:

$$\mathcal{L}\pi\_{\mathbb{R}^3} = p\_0(1+a)(q - bp\_0(1+a) + \text{kg}) - q((1+a)w\_0 + c\_s) + q((1+a)w\_0 - c\_\mathbb{R} - \frac{1}{2}zg^2)$$

**Proposition 4.** *When* 2*bqz* − *k*2 > 0, *πns is concave in α and g. In this case, the overall profit function of the supply chain has a maximum value.*

**Proof.** The Hessian matrix of *π*ns is *<sup>∂</sup>*2*π*ns *∂*<sup>2</sup> *g <sup>∂</sup>*2*π*ns *∂α∂g <sup>∂</sup>*2*π*ns *∂g∂α <sup>∂</sup>*2*π*ns *∂*2*α* = −*qz kp*0 *kp*0 −<sup>2</sup>*bp*<sup>20</sup> = 2*bqz* − *<sup>k</sup>*<sup>2</sup>*p*20. When 2*bqz*−*k*2 >0,conditions2*bqz*−*<sup>k</sup>*<sup>2</sup>*p*20>0and<0canbesatisfied,sotheHessian

 −*qz* matrix is negative definite. The profit function *πns* is a joint concave function with respect to the price subsidy coefficient *α* and the quality safety degree *g*. Hence, there exists optimal solutions *α*∗3and *g*∗3to maximize the profit function. -

By combining the equations *∂πns ∂α* = *p*0*q* + *p*0*kg* − 2*bp*20 − 2*bp*20 *α* = 0 and *d<sup>π</sup>ns dg* = *p*0*k* + *p*0*k<sup>α</sup>* − *qzg* = 0, the optimal price subsidy coefficient and quality safety degree are obtained as follows:

$$\mathfrak{a}\_3^\* = \frac{zq^2}{p\_0(2bqz - k^2)} - 1$$

$$\mathfrak{g}\_3^\* = \frac{qk}{2bqz - k^2}$$

At this point, the actual market demand is *d*∗3 = *bzq*<sup>2</sup> <sup>2</sup>*bqz*−*k*<sup>2</sup> . The maximum profit of the agricultural supply chain under centralized decision making is

$$
\pi\_{\rm ns}{}^\* = \frac{0.5zq^3}{2bqz - k^2} - (c\_\* + c\_n)q^3
$$

**Proposition 5.** *In the centralized model, the overall profit of the agricultural supply chain increases with the increase of the quality elasticity of demand k and decreases with the increase of the price elasticity of demand b and the production impact factor z. That is, when consumers are more concerned about quality and less concerned about price, and the smaller the cost of quality inputs required by agricultural producers to implement quality improvement actions, the greater the overall profitability of the supply chain.*

**Proof.** Due to *∂π*ns<sup>∗</sup> *∂b* = − *q*4*z*<sup>2</sup> (<sup>2</sup>*qzbp*<sup>0</sup>−*k*<sup>2</sup>)<sup>2</sup> < 0, *∂π*ns<sup>∗</sup> *∂k* = *kzq*<sup>3</sup> (<sup>2</sup>*qzb*−*k*<sup>2</sup>)<sup>2</sup> > 0, and *∂π*ns<sup>∗</sup> *∂z* = − 0.5*k*2*q*<sup>3</sup> (<sup>2</sup>*qzb*−*k*<sup>2</sup>)<sup>2</sup> < 0, *<sup>π</sup>*ns<sup>∗</sup> is positively correlated with *k* and negatively correlated with *b* and *z*. -

### *4.3. Comparison and Analysis*

According to the calculation results above, summarizing the quality improvement and price subsidy decisions of the agricultural supply chain participants as well as the corresponding changes in market demand, Table 2 can be obtained below.

**Table 2.** Comparison of the three game models.


The following prerequisites can be derived from the results in Table 2:


Based on the discussion above, the following propositions can be known:

**Proposition 6.** *The quality of agri-foods in the centralized decision model is higher than that in the decentralized model when the participants of the agricultural supply chain make decisions to maximize the profit of the supply chain. In addition, the quality of agri-foods is higher when the* *seller is the leader than in the producer-led case. That is, the best quality safety degree satisfies g*1<sup>∗</sup> < *g*2<sup>∗</sup> < *g*3<sup>∗</sup>.

**Proof.** Due to *g*2<sup>∗</sup> = *kw*0 <sup>2</sup>*zbp*0 , and since *k*, *w*0, *z*, *b*, *p*0 are positive, *g*2<sup>∗</sup> > *g*1<sup>∗</sup> = 0 is satisfied. Again, due to *g*3<sup>∗</sup> = *qk* <sup>2</sup>*bqz*−*k*<sup>2</sup> , for *g*3<sup>∗</sup> > *g*2<sup>∗</sup> to be true, condition <sup>2</sup>*bqzk*(*p*0 − *<sup>w</sup>*0) + *<sup>k</sup>*3*w*0 > 0 should be satisfied. Since the selling price is greater than the buying price, i.e., *p*0 − *w*0 > 0, and *k*, *w*0 are positive, *g*3<sup>∗</sup> > *g*2<sup>∗</sup> can be known. That is, *g*1<sup>∗</sup> < *g*2<sup>∗</sup> < *g*3<sup>∗</sup> is proven. -

**Proposition 7.** *In general, the optimal price compensation factor satisfies <sup>α</sup>*1<sup>∗</sup> < *<sup>α</sup>*2<sup>∗</sup> *and <sup>α</sup>*1<sup>∗</sup> < *<sup>α</sup>*3<sup>∗</sup>.

**Proof.** Due to *<sup>α</sup>*1<sup>∗</sup> = *q*(*p*0−*w*0) 2*bp*20 − 1 and *<sup>α</sup>*2<sup>∗</sup> = *qzb*(*p*0−*w*0)+0.5*k*2*w*<sup>0</sup> 2*zb*<sup>2</sup> *p*20 − 1, it can be obtained that *<sup>α</sup>*2<sup>∗</sup> − *<sup>α</sup>*1<sup>∗</sup> = *<sup>w</sup>*0*k*<sup>2</sup> 4*zb*<sup>2</sup> *p*20 > 0. That is, *<sup>α</sup>*1<sup>∗</sup> < *<sup>α</sup>*2<sup>∗</sup>. Again, due to *<sup>α</sup>*3<sup>∗</sup> = *zq*<sup>2</sup> *<sup>p</sup>*0(2*bqz*−*k*<sup>2</sup>) − 1, for *<sup>α</sup>*1<sup>∗</sup> < *<sup>α</sup>*3<sup>∗</sup> to be true, condition *<sup>k</sup>*<sup>2</sup>(*<sup>w</sup>*0 − *p*0) < <sup>2</sup>*bzqw*0 should be satisfied. Because *w*0 − *p*0 < 0, i.e., *<sup>k</sup>*<sup>2</sup>(*<sup>w</sup>*0 − *p*0) < 0, as well as <sup>2</sup>*bzqw*0 > 0, *<sup>α</sup>*1<sup>∗</sup> < *<sup>α</sup>*3<sup>∗</sup> can therefore be proven. -

Proposition 7 shows that in this agricultural supply chain, the price compensation factors both in the Stackelberg model dominated by sellers and the centralized decision model are greater than those in the Stackelberg game model dominated by agricultural producers. That is, in terms of quality and price, the quality improvement actions driven by producers on the supply side are not as effective as the quality improvement actions driven by consumption on the demand side.

**Corollary 1.** *When the initial price of agri-foods p*0 > *k*2 4*b*2*z , the price appreciation of agri-foods satisfies <sup>α</sup>*1<sup>∗</sup> < *<sup>α</sup>*2<sup>∗</sup> < *<sup>α</sup>*3<sup>∗</sup>.

**Proof.** For *<sup>α</sup>*3<sup>∗</sup> > *<sup>α</sup>*2<sup>∗</sup> to be true, condition *zq*<sup>2</sup> *<sup>p</sup>*0(2*bqz*−*k*<sup>2</sup>) > *qzb*(*p*0−*w*0)+0.5*k*2*w*<sup>0</sup> 2*zb*<sup>2</sup> *p*20 should be satisfied, which is equivalent to proving that

$$2.0.5k^2w\_0\left(2bqz-k^2\right)-2z^2b^2q^2w\_0+bqzk^2(w\_0-p\_0)<0\tag{7}$$

Firstly, *bqzk*<sup>2</sup>(*<sup>w</sup>*0 − *p*0) < 0 can be easily known, so to prove (7) is true, that is, the proof of 0.5*k*2*w*02*bqz* − *k*2 − <sup>2</sup>*z*2*b*<sup>2</sup>*q*<sup>2</sup>*w*0 < 0 is satisfied. Secondly, it is clear from the precondition that *zq*<sup>2</sup> *p*0 > 2*bqz* − *k*2, so to prove 0.5*k*2*w*02*bqz* − *k*2 − <sup>2</sup>*z*2*b*<sup>2</sup>*q*<sup>2</sup>*w*0 < 0.5*k*2*w*0 *zq*<sup>2</sup> *p*0 − <sup>2</sup>*z*2*b*<sup>2</sup>*q*<sup>2</sup>*w*0 is as same as to prove *zq*<sup>2</sup>*k*2*w*0 − <sup>4</sup>*z*2*b*<sup>2</sup>*q*<sup>2</sup> *p*0*w*0 < 0, that is, to prove that *zw*0*q*<sup>2</sup>*k*<sup>2</sup> − 4*zb*<sup>2</sup> *<sup>p</sup>*0 < 0, which means *p*0 > *k*2 4*b*2*z* . Hence, when *p*0 > *k*2 4*b*2*z* is satisfied, *<sup>α</sup>*2<sup>∗</sup> < *<sup>α</sup>*3<sup>∗</sup> is true, and *<sup>α</sup>*1<sup>∗</sup> < *<sup>α</sup>*2<sup>∗</sup> < *<sup>α</sup>*3<sup>∗</sup> can be obtained finally. -

Corollary 1 shows that the price compensation factor in the centralized decision model is greater than the price compensation factor under the decentralized decision model dominated by sellers only when the market price of agri-foods is high, and the condition *p*0 > *k*2 4*b*2*z* is met. Combined with Proposition 6, it can be seen that for high-priced agri-foods, supply chain cooperation to improve the quality of agri-foods is most beneficial for price appreciation, and the quality and price of agri-foods can be realized to the greatest extent.

**Corollary 2.** *The actual market demand satisfies d*2<sup>∗</sup> > *d*1<sup>∗</sup>*. When* 0 < *w*0 < *k*2 *p*0 <sup>2</sup>*bzq*−*k*<sup>2</sup> , *d*3<sup>∗</sup> > *d*1<sup>∗</sup>*, and when w*0 > *k*2 *p*0 <sup>2</sup>*bzq*−*k*<sup>2</sup> *, d*3<sup>∗</sup> < *d*1<sup>∗</sup>*. When* 0 < *w*0 < 2*bzqk*<sup>2</sup> *p*0 <sup>4</sup>*b*2*z*2*q*2−*k*<sup>4</sup> *, d*3<sup>∗</sup> > *d*2<sup>∗</sup>*, and when w*0 > *k*2 *p*0 <sup>2</sup>*bzq*−*k*<sup>2</sup> *, d*3<sup>∗</sup> < *d*2<sup>∗</sup>.

**Proof.** Due to *d*1<sup>∗</sup> = *q*(*p*0+*w*0) 2*p*0 and *d*2<sup>∗</sup> = *qzb*(*p*0+*w*0)+0.5*k*2*w*0 <sup>2</sup>*zbp*0 , *d*2<sup>∗</sup> − *d*1<sup>∗</sup> = *<sup>w</sup>*0*k*<sup>2</sup> <sup>4</sup>*zbp*0 > 0, which means *d*2<sup>∗</sup> > *d*1<sup>∗</sup>. Again, due to *d*3<sup>∗</sup> − *d*1<sup>∗</sup> = 0.5(*<sup>w</sup>*0+*p*0)*qk*2−*bzq*<sup>2</sup>*w*<sup>0</sup> *<sup>p</sup>*0(2*bzq*−*k*<sup>2</sup>) , 0.5(*<sup>w</sup>*0 + *<sup>p</sup>*0)*qk*<sup>2</sup> − *bzq*<sup>2</sup>*w*0 < 0 can be known, and then, *d*3<sup>∗</sup> < *d*1<sup>∗</sup> is obtained. Since the condition 0.5(*<sup>w</sup>*0 + *p*0) *qk*<sup>2</sup> − *bzq*<sup>2</sup>*w*0 < 0 is equivalent to *w*0 > *k*2 *p*0 <sup>2</sup>*bzq*−*k*<sup>2</sup> , when 0 < *w*0 < *k*2 *p*0 <sup>2</sup>*bzq*−*k*<sup>2</sup> , *d*3<sup>∗</sup> > *d*1<sup>∗</sup> can be obtained, and when *w*0 > *k*2 *p*0 <sup>2</sup>*bzq*−*k*<sup>2</sup> , *d*3<sup>∗</sup> < *d*1<sup>∗</sup> can be obtained. -

Similarly, from the equation *d*3<sup>∗</sup> − *d*2<sup>∗</sup> = 0.5*bzqp*0*k*2−*b*2*z*2*q*2*w*0+0.25*<sup>w</sup>*0*k*<sup>4</sup> *bzp*0(2*bzq*−*k*<sup>2</sup>) , 0.5*bzqp*0*k*<sup>2</sup> − *<sup>b</sup>*2*z*<sup>2</sup>*q*<sup>2</sup>*w*0 + 0.25*<sup>w</sup>*0*k*<sup>4</sup> < 0 can be known, which proves *d*3<sup>∗</sup> < *d*2<sup>∗</sup>. Since the condition 0.5*bzqp*0*k*<sup>2</sup> − *<sup>b</sup>*2*z*<sup>2</sup>*q*<sup>2</sup>*w*0 + 0.25*<sup>w</sup>*0*k*<sup>4</sup> < 0 is equivalent to *w*0 > 2*bzqk*<sup>2</sup> *p*0 <sup>4</sup>*b*2*z*2*q*2−*k*<sup>4</sup> , *d*3<sup>∗</sup> > *d*2<sup>∗</sup> can be obtained when 0 < *w*0 < 2*bzqk*<sup>2</sup> *p*0 <sup>4</sup>*b*2*z*2*q*2−*k*<sup>4</sup> , and *d*3<sup>∗</sup> < *d*2<sup>∗</sup> can also be obtained when *w*0 > *k*2 *p*0 <sup>2</sup>*bzq*−*k*<sup>2</sup>

.

Corollary 2 shows that in case of decentralized models, a seller-led initiative to improve the quality of agri-foods is more beneficial to increase actual market sales than a producerled one. The centralized decision is more favorable to increase the sales volume of agri-foods when the agricultural producers sell to sellers with low unit price. Furthermore, the actual market demand in the centralized model is less than that in the decentralized decision case when the agricultural producers sell to sellers with high-value agri-foods.

### **5. Contract Coordination Strategy Based on Cost Sharing of Agricultural Quality Improvement**

According to the analysis above, the quality improvement of agri-foods driven by the demand side is more effective. Based on this, referring to the cost-sharing contract model of Yang et al. [31], it is assumed that agri-foods sellers are willing to share the quality improvement cost at a ratio of *β* ∈ (0, <sup>1</sup>), which incentivizes producers to increase inputs on agri-food quality improvement. In the decentralized scenario dominated by sellers, the profit functions of the agricultural producers and sellers are

$$
\pi\_{\mathbf{n}} = q \left( (1+a)w\_0 - c\_n - (1-\beta)\frac{1}{2}zg^2 \right)
$$

$$
\pi\_{\mathbf{s}} = p\_0(1+a)(q - bp\_0(1+a) + k\mathbf{g}) - q \left( (1+a)w\_0 + c\_\mathbf{s} + \frac{1}{2}\beta zg^2 \right)
$$

Extending the solution method in Model 2, the optimal decisions and profits of producers and sellers of agri-foods under cost-sharing contracts can be obtained as follows:

$$\mathcal{g}\_d^\* = \frac{0.5 k w\_0}{(1 - \beta) z b p\_0}$$

$$\alpha\_d^\* = \frac{0.25 \text{k}^2 w\_0}{z b^2 p\_0^2 (1 - \beta)} - 1$$

At this point, the profits of the participants in the agricultural supply chain are:

$$
\pi\_{\mathbf{n}}^{d\*} = \frac{0.5k^2 w\_0^2 q}{4zb^2 p\_0^{-2} (1 - \beta)} - q c\_{\mathbf{n}}
$$

$$
\pi\_{\mathbf{s}}^{d\*} = \frac{((1 - \beta)p\_0 + (0.5\beta - 1)w\_0) w\_0 k^2 q z b + 0.25 w\_0^2 k^4}{4b^3 p\_0^{-2} (1 - \beta)^2 z^2} - q c\_{\mathbf{s}}
$$

**Proposition 8.** *The effect of cost contract coordination strategy of this agricultural supply chain depends on the original parameter. When* <sup>2</sup>*qbzw*0 = *qbzp*0 + 0.75*<sup>w</sup>*0*k*2*, the optimal cost-sharing coefficient β*∗ *can be obtained.*

**Proof.** The second-order derivative of *<sup>π</sup>*s<sup>∗</sup> with respect to *β* can be obtained as follows: *<sup>∂</sup>*2*π*s<sup>∗</sup> *∂*<sup>2</sup>*β* = *<sup>k</sup>*2*w*0(−*qbzβp*0+*qbzp*0+0.5*qbzβ<sup>w</sup>*0−2*qbzw*0+0.75*<sup>w</sup>*0*k*<sup>2</sup>) 2*b*<sup>3</sup> *<sup>p</sup>*0<sup>2</sup>(*β*−<sup>1</sup>)<sup>4</sup>*z*<sup>2</sup> . Considering the numerator −*qbzβp*<sup>0</sup> + *qbzp*0 + 0.5*qbzβ<sup>w</sup>*0 − <sup>2</sup>*qbzw*0 + 0.75*<sup>w</sup>*0*k*<sup>2</sup> as <sup>F</sup>(*β*), *∂*F(*β*) *∂β* = −0.5*qbz*(*p*<sup>0</sup> − 0.5*<sup>w</sup>*0)*<sup>w</sup>*0*k*<sup>2</sup> < 0 can thus be obtained easily, which indicates F(*β*) is a decreasing function about *β* ∈ (0, <sup>1</sup>). When *β* = 0, MaxF(0) = *qbzp*0 − <sup>2</sup>*qbzw*0 + 0.75*<sup>w</sup>*0*k*<sup>2</sup> can be known. When MaxF(0) = 0, *qbz*(<sup>2</sup>*w*0 − *p*0) = 0.75*<sup>w</sup>*0*k*<sup>2</sup> can be obtained. Obviously, when *qbz*(<sup>2</sup>*w*0 − *p*0) = 0.75*<sup>w</sup>*0*k*<sup>2</sup> holds, if *β* > 0, then F(*β*) < F(0) = 0, at which point *<sup>∂</sup>*2*π*s<sup>∗</sup> *∂*<sup>2</sup>*β* < 0; i.e., there exists the optimal cost-sharing coefficient that makes *<sup>π</sup>*s<sup>∗</sup> maximize. Moreover, *β*∗ = *<sup>q</sup>bz*(<sup>2</sup>*p*0−3*<sup>w</sup>*0)+*k*2*w*<sup>0</sup> *qzb*(<sup>2</sup>*p*0−*w*0) can be obtained when *∂π*s<sup>∗</sup> *∂β*= 0. -

**Proposition 9.** *Compared to the decentralized model led by sellers, agri-foods in the supply chain coordination model based on cost-sharing contracts are of higher quality and cheaper price under specific conditions.*

**Proof.** Itis known that *g*∗ <sup>−</sup>*g*<sup>∗</sup>2 = 0.5*kβ<sup>w</sup>*0 *zbp*0(<sup>1</sup>−*β*) > 0 and*α*<sup>∗</sup> <sup>−</sup>*α*<sup>∗</sup>2 = <sup>2</sup>*zb*<sup>2</sup>(<sup>1</sup>−*β*)*p*02−*zbq*(<sup>1</sup>−*β*)*p*0<sup>+</sup>*zbqw*0(<sup>1</sup>−*β*)+0.5*k*2*w*0*β* <sup>2</sup>*zb*2*p*0<sup>2</sup>(<sup>1</sup>−*β*) . Considering the numerator 2*zb*<sup>2</sup>(1 − *β*)*p*0<sup>2</sup> − *zbq*(1 − *β*)*p*0 + *zbqw*0(<sup>1</sup> − *β*) + 0.5*k*2*w*0*β* as *<sup>γ</sup>*(*p*0), *<sup>γ</sup>*(*p*0) is thus a parabola with an opening upward about *p*0. When *p*0 = *q*4*b* , *<sup>γ</sup>*(*p*0) obtains the minimum value *Minγ*(*p*0) = −0.0625(<sup>1</sup> − *β*)*zq*<sup>2</sup> + 0.5*zbqw*0(<sup>1</sup> − *β*) + 0.25*k*2*w*0*β*. Then, if *Minγ*(*p*0) ≥ 0, i.e., *w*0 ≥ 0.25*zq*<sup>2</sup>(<sup>1</sup>−*β*) <sup>2</sup>*zbq*(<sup>1</sup>−*β*)+*k*<sup>2</sup>*β* , *α*<sup>∗</sup> ≥ *α*∗2 holds. Otherwise, when 0 < *w*0 < 0.25*zq*<sup>2</sup>(<sup>1</sup>−*β*) <sup>2</sup>*zbq*(<sup>1</sup>−*β*)+*k*<sup>2</sup>*β* , *α*<sup>∗</sup> < *α*∗2 can be obtained. -

From the equations *<sup>π</sup>d*<sup>∗</sup>*n* − *<sup>π</sup>*2<sup>∗</sup>*n* = (0.5*k*2*w*0(<sup>1</sup>−0.5*β*)+(*qbzw*0−*qbzp*<sup>0</sup>−0.5*k*2*w*0<sup>2</sup>)(<sup>1</sup>−*β*))*qw*<sup>0</sup> <sup>2</sup>(<sup>1</sup>−*β*)*zb*<sup>2</sup> *<sup>p</sup>*0<sup>2</sup> and *<sup>π</sup>d*<sup>∗</sup>s − *<sup>π</sup>*2<sup>∗</sup>s = −(*qzb*(*β*−<sup>1</sup>)(*<sup>w</sup>*0−*p*0)−0.5*βk*2*w*0)<sup>2</sup>+0.5*βk*4*w*0<sup>2</sup> 4*z*2*b*<sup>3</sup> *<sup>p</sup>*0<sup>2</sup>(*β*−<sup>1</sup>)<sup>2</sup> , it can be seen that the positive or negative sign on the right side of the equation cannot be judged directly, which depends on the value of the relevant parameters. Therefore, the basic cost-sharing contract cannot significantlyincreasetheprofitoftheparticipants.

Proposition 9 shows that the introduction of cost-sharing contracts based on the decentralized decision model can significantly improve the quality of products in the agricultural supply chain, and in most cases, the selling price of agricultural products will not be higher than that before the introduction of cost-sharing contracts. At this time, agri-foods in the supply chain based on cost-sharing contracts are of better quality and cheaper price. However, it cannot significantly improve the profit of agricultural supply chain participants.
