Comparing supply-side and demand-side policies in the solar cell supply chain under competitive circumstances: a case study

Abstract

Due to the destructive effects of fossil fuels on the environment, using renewable energies has nowadays been suggested. In addition, because of the increased use of solar energy and the prevention the solar cell supply chain (SCSC), this chain is mainly supported by government funding. In this study, we mathematically model both supply-side and demand-side policies for a two-echelon SCSC, considering the competition between domestic and foreign suppliers as well as government intervention under the supply-side subsidy policies to support SCSC members and finance the customers through demand-side subsidy policies. The aim of this paper is to investigate the effects of government intervention under the supply-side and demand-side policies on supporting the members of SCSC and encouraging customers to increase the use of solar panels. In addition, this study explores the best policy for improving and promoting SCSC. In the real world, both supply-side and demand-side policies will help industrial factories, power plants, and households to enhance the use of solar energy for producing electricity. This study has been investigated using one real example, whose results show that the efficiency under the supply-side policy is about 7% more than the efficiency under the demand-side policy. The price under the supply-side policy is roughly 10% less than the price under the demand-side policy. The members’ profit under the supply-side policy is approximately 5% bigger than the one under the demand-side policy. According to real examples, the government’s utility under the demand-side policy is nearly 1% smaller than the government’s utility under the supply-side policy. Finally, key findings are considered for the proposed model.

Appendices

Appendix 1

Supply-side policy

To prove concavity of the government function in the supply-side policy, the Hessian matrix of ${\pi }_{G,p1}$ is shown as follows:

$$H\left({t}_{a},{t}_{d}\right)=\left[\begin{array}{cc}-2b{c}_{0}^{2}& -\frac{{c}_{0}{cs}_{d}(2b{c}_{d}+\gamma \left(\gamma +{\theta }_{s}\right))}{4{c}_{d}}\\ -\frac{{c}_{0}{cs}_{d}(2b{c}_{d}+\gamma \left(\gamma +{\theta }_{s}\right))}{4{c}_{d}}& -\frac{{cs}_{d}^{2}({\gamma +{\theta }_{s})}^{2}}{2{c}_{d}}\end{array}\right]$$

${H}_{1}\left({t}_{a},{t}_{d}\right)=-2b{c}_{0}^{2}<0, {H}_{2}\left({t}_{a},{t}_{d}\right)=\left(-2b{c}_{0}^{2}\right)\left(-\frac{{cs}_{d}^{2}({\gamma +{\theta }_{s})}^{2}}{2{c}_{d}}\right)-{\left(-\frac{{c}_{0}{cs}_{d}\left(2b{c}_{d}+\gamma \left(\gamma +{\theta }_{s}\right)\right)}{4{c}_{d}}\right)}^{2}>0 \mathrm{if} \left(-2b{c}_{0}^{2}\right)\left(-\frac{{cs}_{d}^{2}({\gamma +{\theta }_{s})}^{2}}{2{c}_{d}}\right)>{\left(-\frac{{c}_{0}{cs}_{d}\left(2b{c}_{d}+\gamma \left(\gamma +{\theta }_{s}\right)\right)}{4{c}_{d}}\right)}^{2},({2b{c}_{d}+\gamma (\gamma +{\theta }_{s}))}^{2}<16b{c}_{d}({\gamma +{\theta }_{s})}^{2}.$

The concavity is proved if a negative value for the second-order condition of domestic and foreign suppliers’ functions is obtained.

$$\frac{{\partial }_{{\pi }_{DS,p1}}^{2}}{\partial {x}_{d,p1}^{2}}=-2{c}_{d}<0.$$

$$\frac{{\partial }_{{\pi }_{FS,p1}}^{2}}{\partial {x}_{f,p1}^{2}}=-2{c}_{f}<0.$$

To show concavity, the Hessian matrix of ${\pi }_{A,p1}$ is seen as follows:

$$H\left({p}_{d,p1},{p}_{f,p1}\right)=\left[\begin{array}{cc}-2b-2\theta & 2\theta \\ 2\theta & -2b-2\theta \end{array}\right]$$

${H}_{1}\left({p}_{d,p1},{p}_{f,p1}\right)=-2b-2\theta <0, {H}_{2}\left({p}_{d,p1},{p}_{f,p1}\right)=4{b}^{2}+8b\theta >0.$

Decision variables and profit function of assembler in supply-side policy are expressed as follows:

$${p}_{d,p1}^{*}=\frac{2b\left(b+2\theta \right){c}_{0}\left(1-{t}_{a}\right)+\frac{\begin{array}{c}{c}_{f}({w}_{d}-{cs}_{d}(1-{t}_{d}))\left(\gamma +{\theta }_{s}\right)\left(\gamma \left(b+\theta \right)+b{\theta }_{s}\right)+{c}_{d}(2{c}_{f}\left(a\left(b\alpha +\theta \right)+b\left(b+2\theta \right){w}_{d}\right)-({cs}_{f}(1\\ +{t}_{f})-{w}_{f})(\gamma +{\theta }_{s})(\gamma \theta -b{\theta }_{s}))\end{array}}{{c}_{f}{c}_{d}}}{4b(b+2\theta )}$$

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$${p}_{f,p1}^{*}=\frac{2b\left(b+2\theta \right){c}_{0}\left(1-{t}_{a}\right)+\frac{\begin{array}{c}{c}_{f}\left({w}_{d}-{cs}_{d}\left(1-{t}_{d}\right)\right)\left(\gamma +{\theta }_{s}\right)\left(\gamma \theta -b{\theta }_{s}\right)+{c}_{d}(2{c}_{f}(a\left(b-b\alpha +\theta \right)+b\left(b+2\theta \right){w}_{f})-({cs}_{f}(1\\ +{t}_{f})-{w}_{f})(\gamma +{\theta }_{s})(\gamma \left(b+\theta \right)+b{\theta }_{s}))\end{array}}{{c}_{f}{c}_{d}}}{4b(b+2\theta )}$$

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$${\pi }_{A,p1}^{*}=\frac{1}{16b\left(b+2\theta \right){c}_{f}^{2}{c}_{d}^{2}}({c}_{f}^{2}({{cs}_{d}\left(-1+{t}_{d}\right)+{w}_{d})}^{2}({\gamma +{\theta }_{s})}^{2}\left({\gamma }^{2}\left(b+\theta \right)+2b{\theta }_{s}\left(\gamma +{\theta }_{s}\right)\right)-2{c}_{d}{c}_{f}\left({cs}_{d}\left(-1+{t}_{d}\right)+{w}_{d}\right)\left(\gamma +{\theta }_{s}\right)\left(2{c}_{f}\left(\gamma \left(-a\left(b\alpha +\theta \right)+b\left(b+2\theta \right)\left(-{c}_{0}\left(-1+{t}_{a}\right)+{w}_{d}\right)+b\left(a-2a\alpha +\left(b+2\theta \right){(w}_{d}-{w}_{f}\right)\right){\theta }_{s}\right)+{(cs}_{f}\left(1+{t}_{f}\right)-{w}_{f}\right)\left(\gamma +{\theta }_{s}\right)\left({\gamma }^{2}\theta -2b{\theta }_{s}\left(\gamma +{\theta }_{s}\right))\right)+{c}_{d}^{2}(4{c}_{f}^{2}({a}^{2}\left(b+2b\left(-1+\alpha \right)\alpha +\theta \right)+b(b+2\theta )(2b{c}_{0}^{2}{\left(-1+{t}_{a}\right)}^{2}+\left(b+\theta \right){w}_{d}^{2}-2{w}_{d}\left(a\alpha +\theta {w}_{f}\right)+{w}_{f}\left(2a\left(-1+\alpha \right)+\left(b+\theta \right){w}_{f}\right)+2{c}_{0}\left(-1+{t}_{a}\right)\left(a-b{(w}_{d}+{w}_{f})))\right)+{\left(-{cs}_{f}\left(1+{t}_{f}\right)+{w}_{f}\right)}^{2}{\left(\gamma +{\theta }_{s}\right)}^{2}\left({\gamma }^{2}\left(b+\theta \right)+2b{\theta }_{s}\left(\gamma +{\theta }_{s}\right)\right)+4{c}_{f}\left({cs}_{f}\left(1+{t}_{f}\right)-{w}_{f}\right)\left(\gamma +{\theta }_{s}\right)\left(ab\left(-1+\alpha \right)\gamma -a\gamma \theta +ab\left(-1+2\alpha \right){\theta }_{s}-b\left(b+2\theta \right)\left(\gamma {c}_{0}\left(-1+{t}_{a}\right)+{w}_{d}{\theta }_{s}-{w}_{f}\left(\gamma +{\theta }_{s}\right)))\right)\right)$$

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Appendix 2

Demand-side policy

To show concavity, the second-order condition of ${\pi }_{G,P2}$ should be negative.

$$\frac{{\partial }^{2}{\pi }_{G,p2}}{\partial {t}_{c,p2}^{2}}=-b-\theta <0$$

To prove concavity, the second-order condition of supplier functions based on decision variables should be negative.

$$\frac{{\partial }_{{\pi }_{DS,p2}}^{2}}{\partial {x}_{d,p2}^{2}}=-2{c}_{d}<0$$

$$\frac{{\partial }_{{\pi }_{FS,p2}}^{2}}{\partial {x}_{f,p2}^{2}}=-2{c}_{f}<0.$$

The Hessian matrix of ${\pi }_{A,p2}$ to prove the concavity is seen as follows:

$$H\left({p}_{d,p2},{p}_{f,p2}\right)=\left[\begin{array}{cc}-2b-2\theta & 2\theta \\ 2\theta & -2b-2\theta \end{array}\right]$$

${H}_{1}\left({p}_{d,p2},{p}_{f,p2}\right)=-2b-2\theta <0, {H}_{2}\left({p}_{d,p1},{p}_{f,p1}\right)=4{b}^{2}+8b\theta >0.$

Decision variables and profit function of assembler in demand-side policy are shown as follows:

$${p}_{d,p2}^{*}=\frac{b\left(b+2\theta \right){c}_{0}+\frac{\begin{array}{c}-{c}_{f}({cs}_{d}-{w}_{d})(\gamma \left(b+\theta \right)+b{\theta }_{s})(\gamma +{\theta }_{s})+{c}_{d}(2{c}_{f}\left(a\left(b\alpha +\theta \right)+b\left(b+2\theta \right)\left({w}_{d}+{t}_{c}\right)\right)-\\ ({cs}_{f}(1+{t}_{f})-{w}_{f})(\gamma +{\theta }_{s})(\gamma \theta -b{\theta }_{s}))\end{array}}{{2c}_{d}{c}_{f}}}{2b(b+2\theta )}$$

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$${p}_{f,p2}^{*}=\frac{b\left(b+2\theta \right){c}_{0}+\frac{\begin{array}{c}-{c}_{f}\left({cs}_{d}-{w}_{d}\right)\left(\gamma +{\theta }_{s}\right)+{c}_{d}(2{c}_{f}\left(a\left(b-b\alpha +\theta \right)+b\left(b+2\theta \right){w}_{f}\right)-({cs}_{f}(1+{t}_{f})-{w}_{f})\\ (\gamma +{\theta }_{s})(\gamma \left(b+\theta \right)+b{\theta }_{s}))\end{array}}{2{c}_{d}{c}_{f}}}{2b\left(b+2\theta \right)}$$

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$${\pi }_{A,p2}^{*}=\frac{1}{16b\left(b+2\theta \right){c}_{d}^{2}{c}_{f}^{2}}\left({c}_{f}^{2}{\left({cs}_{d}-{w}_{d}\right)}^{2}{\left(\gamma +{\theta }_{s}\right)}^{2}\left({\gamma }^{2}\left(b+\theta \right)+2b{\theta }_{s}\left(\gamma +{\theta }_{s}\right)\right)+2{c}_{d}{s}_{f}\left({cs}_{d}-{w}_{d}\right)\left(\gamma +{\theta }_{s}\right)\left(2{c}_{f}\left(-a\gamma \left(b\alpha +\theta \right)+b\gamma \left(b+2\theta \right)\left({c}_{0}-{t}_{d}+{w}_{d}\right)+b\left(a-2a\alpha -\left(b+2\theta \right)\left({t}_{d}-{w}_{d}+{w}_{f}\right)\right){\theta }_{s}\right)+{(cs}_{f}\left(1+{t}_{f}\right)-{w}_{f}\right)\left(\gamma +{\theta }_{s}\right)\left({\gamma }^{2}\theta -2b{\theta }_{s}\left(\gamma +{\theta }_{s}\right)\right)\right)+{c}_{d}^{2}(4{c}_{f}^{2}\left({a}^{2}\left(b+2b\left(-1+\alpha \right)\alpha +\theta \right)+b\left(b+2\theta \right)\left(2b{c}_{0}^{2}+\left(2a\alpha +\left(b+\theta \right)\left({t}_{d}-{w}_{d}\right)+2\left(a\left(-1+\alpha \right)+\theta {t}_{d}-\theta {w}_{d}\right){w}_{f}+\left(b+\theta \right){w}_{f}^{2}-2{c}_{0}\left(a+b{t}_{d}-b\left({w}_{d}+{w}_{f}\right)\right)\right)\right)+4{c}_{f}\left({cs}_{f}\left(1+{t}_{f}\right)-{w}_{f}\right)\left(\gamma +{\theta }_{s}\right)\left(\gamma \left(ab\left(-1+\alpha \right)-a\theta +b\left(b+2\theta \right)\left({c}_{0}+{w}_{f}\right)\right)+b\left(a\left(-a+2\alpha \right)+\left(b+2\theta \right)\left({t}_{d}-{w}_{d}+{w}_{f}\right)\right){\theta }_{s}\right)+{\left(-{cs}_{f}\left(1+{t}_{f}\right)+{w}_{f}\right)}^{2}{\left(\gamma +{\theta }_{s}\right)}^{2}\left({\gamma }^{2}\left(b+\theta \right)+2b{\theta }_{s}\left(\gamma +{\theta }_{s}\right)))\right)\right)$$

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