Green fresh product cost sharing contracts considering freshness-keeping effort

Nowadays, along with increased public demands on the high-quality green fresh product, the downstream retailer has to spend in the packaging and cold chain transportation and the upstream farmer needs to invest more in green fresh products producing. In this paper, we investigate a green fresh product supply chain problem, in which the retailer transports and sells green fresh products to the ultimate consumer while the farmer produces green fresh product through spending a greenness improvement investment and sells green fresh products to the retailer. Since the fresh product is perishable, the retailer needs to make a costly freshness-keeping effort. Obviously, the freshness-keeping effort, the price, and the greenness improvement level will affect the demand. Thus, to demonstrate the game structure between the retailer and the farmer, a decentralized model without cost sharing, a decentralized Stackelberg cost sharing model, and a Nash bargaining model with cost sharing are formulated. Results show that: (1) The equilibrium decisions under Stackelberg model with cost sharing are larger than that of the Nash bargaining model with cost sharing, while equilibrium decisions in the decentralized model without cost sharing are the least. (2) Both greenness improvement levels in Stackelberg cost sharing contract and Nash bargaining are greater than that in decentralized model without cost sharing. (3) The retailer’s profits in Stackelberg cost sharing contract and Nash bargaining are larger than that in decentralized case without cost sharing, while the farmer’s profits in Stackelberg cost sharing contract and Nash bargaining are larger than in decentralized model without cost sharing in certain conditions. Meanwhile, a numerical example is given to illustrate the results we obtained in the theoretical analysis process.

1. http://zfs.mee.gov.cn/fl/201809/t20180907_549845.shtml.

This work was supported by Philosophy and Social Science Foundation of Tianjin (No. TJGL16-009Q).

Authors and Affiliations

1. College of Economics and Management, Tianjin University of Science and Technology, Tianjin, 300222, China

   Guoli Wang & Jing Mu

2. School of Management, Northwestern Polytechnical University, Xi’an, 710129, China

   Peiqi Ding

3. College of Mathematics and Statistics, Huanggang Normal University, Huanggang, 438000, Hubei, China

   Huiru Chen

Corresponding authors

Correspondence to Peiqi Ding or Huiru Chen.

Conflict of interest

The authors declare that they have no conflict of interest.

Human and animal rights

This study does not contain any studies with human participants or animals performed by any of the authors.

Communicated by Y. Ni.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

1.1 Appendix A: Summary of abbreviations

1.2 Appendix B: Model solving process and proofs of Propositions

The centralized channel case. The whole channel supply chain profit function can be rewritten as

The first-order condition

and

The second-order condition

and

Moreover, we can also obtain

Thus, the Hessian matrix is

Through computing the Hessian Matrix, when the conditions \(2bL-\beta ^2>0\) and \(-2\,bLK+{\beta }^{2}K+{\alpha }^{2}L<0\) are satisfied, the supply chain profit function is joint concave with respect to the decision variables. Thus, equating the first-order condition to 0 and solving the equations, we obtain

and

Subsequently, the optimal profit of the centralized supply chain is

Proof of Proposition1

Proof

We can easily get the results from the computational process. \(\square \)

Proof of Corollary 1

Proof

According to the analytical expressions given above, we can easily get the results by implementing differentiation. \(\square \)

The decentralized channel case. Under this environment, we solve the farmer’s optimization problem first:

Rewriting as

The Hessian matrix is

Through computing the Hessian matrix, the farmer’s profit function is joint concave with respect to the decision variables when \(2bK-\alpha ^2>0\). The first-order condition is

and

Thus, equating the first-order condition to 0 and solving the equations, we get

and

We substitute Eqs. (21)–(22) into the retailer’s profit function (1) and derive

By computing the first-order condition

and the second-order condition

we can write the Hessian matrix of \(\pi _\mathrm{R}\) as

Through computing the Hessian matrix, the retailer’s profit function is joint concave with respect to the decision variables when \(\frac{b^2K(4LbK-\beta ^2K-2L\alpha ^2)}{(2Kb-\alpha ^2)^2}>0\), i.e., \(4LbK-\beta ^2K-2L\alpha ^2>0\). Let the first-order condition be 0. We get

and

Then, we have

Proof of Proposition2

Proof

We can easily get the results from the computational process. \(\square \)

Proof of Proposition 3

Proof

The above relationships can be proved easily through algebraic comparison. \(\square \)

The model with cost sharing. Due to the game structure, we solve for the farmer’s profit function first:

The first-order condition

and the second-order condition

We can write the Hessian matrix as

Since

then the Hessian matrix is negative definite if

Thus, the farmer’s profit function is jointly concave in w and \(\theta \). Let the first condition be zero, we obtain

Substituting Eqs. (24–25) into (5), we can derive

According to the first-order condition

and the second-order condition

Then, the Hessian matrix is negative definite if

and

Equivalent to

and

which implies

and

By the first-order condition, we have

Substituting m and \(\zeta \) into Eq. (26), we can obtain the retailer’s profit function with respect to \(\psi \)

Then, the first-order condition

and the second-order condition

The second-order condition is negative if \(A_9>0.\) Thus, \(\pi _\mathrm{R}(\psi )\) is concave in \(\psi \) for the above condition. Solving the first-order condition for \(\psi \) gives \(\psi =\frac{1}{3}\) or \(\psi =1\).

When \(\psi =1\), \(A_9>0\) can be reduced to \(-L\alpha ^2>0\), which is a contradiction since \(L,K, \alpha ,\beta >0\).

Logically, the equilibrium \(\psi ^{\mathrm{CS}^{*}}=\frac{1}{3}\). Substituting the value of \(\psi ^{\mathrm{CS}^{*}}=\frac{1}{3}\) into the expressions accordingly, we get

Proof of Proposition4

Proof

We can easily get the results from the computational process. \(\square \)

Proof of Corollary 2

Proof

During the computation process, the

By computing the first-order derivative, we can obtain the result. \(\square \)

Proof of Proposition 5

Proof

We prove the result about \(\zeta \); the result about w can be proved analogously. During the computation process, the

By computing the first-order derivative with respect to \(\psi \) and making analysis, we can obtain the result. \(\square \)

Proof of Proposition 6

Proof

By computing the first-order derivative with respect to \(\psi \) and making analysis, we can obtain the result. \(\square \)

Proof of Proposition 7

Proof

The result can be proved analogously with Proposition 5. \(\square \)

Bargaining cost sharing contract model. Since the farmer and the retailer decide the \(\psi \) via bargaining, we formulate the objective function as

Due to the game structure, we solve the farmer’s optimization problem first:

The first-order condition

and the second-order condition

We can write the Hessian matrix as

The Hessian matrix is negative definite if

Under this assumption, the farmer’s profit function is joint concave in p and \(\zeta \). Let the first-order condition be zero and solve equations for p and \(\zeta \). We can obtain

and

Substituting Eqs. (29–30) into the retailer’s profit function, we get

The first-order condition

and the second-order condition

Then, the Hessian matrix is negative definite if

and

Equivalent to

and

which implies

and

Let the first-order condition be zero and solve equations for m and \(\zeta \). We can get

Substituting m and \(\zeta \) into Eqs. (29–30) and the Nash bargaining function (27). By computing the first-order condition and the second-order condition of the bargaining model

The first-order condition

The second-order condition

The second-order condition is negative if \(A_2<0.\) Thus, the objective function is concave in \(\psi \) for the above condition. Let the first-order condition be 0. We obtain \(\psi =1\),

or

When \(\psi =1\), \(2Kb(1-\psi )-\alpha ^2>0\) can be deduced to \(-\alpha ^2>0\), which is a contradiction. Meanwhile, if

then the \(2Kb(1-\psi )-\alpha ^2>0\) is not satisfied. When

If \(12KLb^2-8L\alpha ^2b-\alpha ^2\beta ^2<0\), then we can get \(2Kb(1-\psi )-\alpha ^2<0\); if \(12KLb^2-8L\alpha ^2b-\alpha ^2\beta ^2>0\), we only need to check \((12KLb^2-8L\alpha ^2b-\alpha ^2\beta ^2)^2-(2bA_3)^2\) is greater than or less than 0. By algebra,

The problem transfers to verify whether \(4KLb^2-2L\alpha ^2b-\alpha ^2\beta ^2\) is greater or less than 0. Since we have \(2KLb-\beta ^2K-L\alpha ^2>0\) and \(2Kb-\alpha ^2>0\), \(2b(2KLb-\beta ^2K-L\alpha ^2)>0\) and \(4KLb^2-2L\alpha ^2b-2\beta ^2Kb>0\). Owing to \(2Kb-\alpha ^2>0\), we can derive \(4KLb^2-2L\alpha ^2b-\alpha ^2\beta ^2>0\). Thus, \(2Kb(1-\psi )-\alpha ^2<0\) when

Obviously, the

should be omitted. Analogously, we can also prove that \(2Kb(1-\psi )-\alpha ^2>0\) can be satisfied when

Thus, the equilibrium

The equilibrium decisions and profit functions are

Proof of Proposition 8

Proof

We can get the results from the computational process. \(\square \)

Model comparison.

Proof of Proposition 9

Proof

We need to prove \(10KLb+2K\beta ^2-9L\alpha ^2-3A_3<0\) which is equivalent to \((10KLb+2K\beta ^2-9L\alpha ^2)^2-(3A_3)^2<0\). By algebra, we need to prove

which is true since \(K>0\) and \(16KLb-4K\beta ^2-9L\alpha ^2>0\). \(\square \)

Proof of Proposition 10

Proof

We prove \(w^{\mathrm{CS}^{*}}>w^{\mathrm{D}^{*}}\) for example. The proofs of \(\theta ^{\mathrm{CS}^{*}}>\theta ^{\mathrm{D}^{*}},\,p^{\mathrm{CS}^{*}}>p^{\mathrm{D}^{*}},\zeta ^{\mathrm{CS}^{*}}>\zeta ^{\mathrm{D}^{*}}\) can be done analogously.

The \(w^{\mathrm{CS}^{*}}=\frac{12KLbc-4K\beta ^2c-9L\alpha ^2c+4KLa}{16KLb-4\beta ^2K-9L\alpha ^2}\) and the \(w^{\mathrm{D}^{*}}=\frac{[L(3bc+a)-c\beta ^2]K-2Lc\alpha ^2}{4LbK-\beta ^2K-2L\alpha ^2}\). It is equivalent to prove \(\frac{w^{\mathrm{CS}^{*}}}{w^{\mathrm{D}^{*}}}>1\).

Obviously, \(a-bc>0\) holds true. \(\square \)

Proof of Proposition 11

Proof

The proof can be obtained with the first-order derivative. \(\square \)

Proof of Proposition 12

Proof

The proof of (1) can be derived from Propositions 7 and 9; the proof of (2) is analogous to Proposition 10. \(\square \)

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