Renewable energy hybrid subsidy combining input and output subsidies

Abstract

With respect to sustainable development, how to promote renewable energy is a major issue. Here, we introduce a hybrid subsidy mechanism that considers both input and output subsidies. Hybrid subsidies are analyzed with stochastic optimization approaches. An outstanding advantage of hybrid subsidies is the flexibility to adjust the intensity between the input and output subsidies. Our study shows that input-biased subsidies advance outputs and improve environmental efficiency (EE), while output-biased subsidies reduce risk and boost producer subsidy equivalents (PSEs). Therefore, the policy implication of this research is that different subsidy intensities should be employed according to preferences or social requirements.

Appendix

Proof of Proposition 1

By (10) and the budgeting constraints, we have

$$ {s}_O{\int}_{-\frac{1}{2}}^{\frac{1}{2}}\left(1+\eta \right){\left[\frac{\tau -{s}_I}{\left(1+\eta \right)\alpha \left(1+{s}_O\right)}\right]}^{\frac{\alpha }{\alpha -1}}f\left(\eta \right) d\eta +{s}_I{\int}_{-\frac{1}{2}}^{\frac{1}{2}}{\left[\frac{\tau -{s}_I}{\left(1+\eta \right)\alpha \left(1+{s}_O\right)}\right]}^{\frac{1}{\alpha -1}}f\left(\eta \right) d\eta =B. $$

(16)

For convenience, we denote

$$ {\displaystyle \begin{array}{c}F\left({s}_O,{s}_I\right)=\\ {}{s}_O{\int}_{-\frac{1}{2}}^{\frac{1}{2}}{\left(1+\eta \right)}^{\frac{1}{1-\alpha }}{\left[\frac{\tau -{s}_I}{\alpha \left(1+{s}_O\right)}\right]}^{\frac{\alpha }{\alpha -1}}f\left(\eta \right) d\eta +{s}_I{\int}_{-\frac{1}{2}}^{\frac{1}{2}}{\left(1+\eta \right)}^{\frac{1}{1-\alpha }}{\left[\frac{\tau -{s}_I}{\alpha \left(1+{s}_O\right)}\right]}^{\frac{1}{\alpha -1}}f\left(\eta \right) d\eta -B=0.\end{array}} $$

(17)

$$ {\displaystyle \begin{array}{c}\frac{\partial F\left({s}_O,{s}_I\right)}{\partial {s}_O}={\int}_{-\frac{1}{2}}^{\frac{1}{2}}{\left(1+\eta \right)}^{\frac{1}{1-\alpha }}f\left(\eta \right) d\eta \left\{\right[{}^{\frac{\tau -{s}_I}{\alpha \left(1+{s}_O\right)}}\\ {}+\frac{\alpha {s}_O}{\left(1-\alpha \right)\left(1+{s}_O\right)}\Big[{}^{\frac{\tau -{s}_I}{\alpha \left(1+{s}_O\right)}}\\ {}+\frac{s_I}{\left(1-\alpha \right)\left(1+{s}_O\right)}\left[{}^{\frac{\tau -{s}_I}{\alpha \left(1+{s}_O\right)}}\right\}>0,\end{array}} $$

(18)

$$ {\displaystyle \begin{array}{c}\frac{\partial F\left({s}_O,{s}_I\right)}{\partial {s}_I}={\int}_{-\frac{1}{2}}^{\frac{1}{2}}{\left(1+\eta \right)}^{\frac{1}{1-\alpha }}f\left(\eta \right) d\eta \left\{\right[{}^{\frac{\tau -{s}_I}{\alpha \left(1+{s}_O\right)}}\\ {}+\frac{s_I}{\left(1-\alpha \right)\left(\tau -{s}_I\right)}\Big[{}^{\frac{\tau -{s}_I}{\alpha \left(1+{s}_O\right)}}\\ {}+\frac{\alpha {s}_O}{\left(1-\alpha \right)\left(\tau -{s}_I\right)}\left[{}^{\frac{\tau -{s}_I}{\alpha \left(1+{s}_O\right)}}\right\}>0.\end{array}} $$

(19)

According to the implicit function theorem, we have the following relationship:

$$ \frac{\partial {s}_O}{\partial {s}_I}=-\frac{\frac{\partial F\left({s}_O,{s}_I\right)}{\partial {s}_I}}{\frac{\partial F\left({s}_O,{s}_I\right)}{\partial {s}_O}}<0 $$

(20)

On the other hand, by virtue of (8), we have

$$ {\displaystyle \begin{array}{c}\frac{\partial Exp\left({q}^{\ast}\right)}{\partial {s}_I}=\frac{\alpha }{\left(1-\alpha \right)}{\int}_{-\frac{1}{2}}^{\frac{1}{2}}{\left(1+\eta \right)}^{\frac{1}{1-\alpha }}f\left(\eta \right) d\eta \left\{\frac{1}{\left(\tau -{s}_I\right)}\right[{}^{\frac{\tau -{s}_I}{\alpha \left(1+{s}_o\right)}}\\ {}+\frac{\partial {s}_o}{\partial {s}_I}\frac{1}{\left(1+{s}_o\right)}\left[{}^{\frac{\tau -{s}_I}{\alpha \left(1+{s}_o\right)}}\right\}\\ {}\frac{\alpha {\int}_{-\frac{1}{2}}^{\frac{1}{2}}{\left(1+\eta \right)}^{\frac{1}{1-\alpha }}f\left(\eta \right) d\eta}{\left(1-\alpha \right)\frac{\partial F}{\partial {s}_o}}\left\{\frac{\partial F}{\partial {s}_o}\frac{1}{\left(\tau -{s}_I\right)}\right[{}^{\frac{\tau -{s}_I}{\alpha \left(1+{s}_o\right)}}\\ {}-\frac{\partial F}{\partial {s}_I}\frac{1}{\left(1+{s}_o\right)}\left[{}^{\frac{\tau -{s}_I}{\alpha \left(1+{s}_o\right)}}\right\}\\ {}=\frac{\alpha {\left[{\int}_{-\frac{1}{2}}^{\frac{1}{2}}{\left(1+\eta \right)}^{\frac{1}{1-\alpha }}f\left(\eta \right) d\eta \right]}^2}{\left(1-\alpha \right)\frac{\partial F}{\partial {s}_O}}\left\{\frac{1}{\left(\tau -{s}_I\right)}\right[{}^{\frac{\tau -{s}_I}{\alpha \left(1+{s}_O\right)}}\\ {}-\frac{1}{\left(1+{s}_O\right)}\left[{}^{\frac{\tau -{s}_I}{\alpha \left(1+{s}_O\right)}}\right\}\\ {}=\frac{\alpha {\left[{\int}_{-\frac{1}{2}}^{\frac{1}{2}}{\left(1+\eta \right)}^{\frac{1}{1-\alpha }}f\left(\eta \right) d\eta \right]}^2}{\left(1-\alpha \right)\frac{\partial F}{\partial {s}_O}}\left({}^{\tau}\right({}^1\left({\alpha}^{\frac{2\alpha }{1-\alpha }}-{\alpha}^{\frac{\alpha +1}{1-\alpha }}\right)>0.\end{array}} $$

(21)

The last inequality holds because 0 < α < 1 yields the relationship ${\alpha}^{\frac{2\alpha }{1-\alpha }}>{\alpha}^{\frac{\alpha +1}{1-\alpha }}$. Therefore, higher input subsidy intensity manifests more outputs.

By , $\frac{\partial Exp\left({q}^{\ast}\right)}{\partial {s}_I}=\frac{a}{\left(1-a\right)}{\int}_{-\frac{1}{2}}^{\frac{1}{2}}{\left(1+\eta \right)}^{\frac{1}{1-a}}f\left(\eta \right) d\eta \left\{\frac{1}{\tau -{s}_I}{\left[\frac{\tau -{s}_I}{a\left(1+{s}_O\right)}\right]}^{\frac{a}{a-1}}+\frac{{\partial s}_O}{{\partial s}_I}\frac{1}{\left(1+{s}_O\right)}{\left[\frac{\tau -{s}_I}{a\left(1+{s}_O\right)}\right]}^{\frac{a}{a-1}}\right\}$

the term $\frac{\alpha }{\left(1-\alpha \right)}{\int}_{-\frac{1}{2}}^{\frac{1}{2}}{\left(1+\eta \right)}^{\frac{1}{1-\alpha }}f\left(\eta \right) d\eta \frac{1}{\left(\tau -{s}_I\right)}{\left[\frac{\tau -{s}_I}{\alpha \left(1+{s}_O\right)}\right]}^{\frac{\alpha }{\alpha -1}}$ is the direct increasing effect of the input subsidies and the other term $\frac{\partial {s}_O}{\partial {s}_I}\frac{\alpha }{\left(1-\alpha \right)}{\int}_{-\frac{1}{2}}^{\frac{1}{2}}{\left(1+\eta \right)}^{\frac{1}{1-\alpha }}f\left(\eta \right) d\eta \frac{1}{\left(1+{s}_O\right)}{\left[\frac{\tau -{s}_I}{\alpha \left(1+{s}_O\right)}\right]}^{\frac{\alpha }{\alpha -1}}\Big\}$ is the indirect decreasing impact of the input subsidies. The effects of input subsidy intensity are jointly determined by the direct impact and indirect effect. In the above formulation, the direct increasing effects are larger than the indirect decreasing effects. Thus, the total effects of input subsidy intensity on outputs are positive. Further, by (9), along with the relationship between input subsidy intensity and the expected outputs, we immediately have the conclusion that higher input subsidy intensity manifests in higher profits.

Therefore, conclusions are achieved, and the proof is complete. ■

Proof of Proposition 2

By (11), the lowest risk subsidy aims to solve the following problem:

$$ {\displaystyle \begin{array}{c}\underset{s_O,{s}_I}{\mathit{\operatorname{Min}}}{s}_O{\int}_{-\frac{1}{2}}^{\frac{1}{2}}\left(1+\eta \right){\left[\frac{\tau -{s}_I}{\left(1+\eta \right)\alpha \left(1+{s}_O\right)}\right]}^{\frac{\alpha }{\alpha -1}}f\left(\eta \right) d\eta +{s}_I{\int}_{-\frac{1}{2}}^{\frac{1}{2}}{\left[\frac{\tau -{s}_I}{\left(1+\eta \right)\alpha \left(1+{s}_O\right)}\right]}^{\frac{1}{\alpha -1}}f\left(\eta \right) d\eta =B\\ {}s.t.{s}_O{\int}_{-\frac{1}{2}}^{\frac{1}{2}}\left(1+\eta \right){\left[\frac{\tau -{s}_I}{\left(1+\eta \right)\alpha \left(1+{s}_O\right)}\right]}^{\frac{\alpha }{\alpha -1}}f\left(\eta \right) d\eta +{s}_I{\int}_{-\frac{1}{2}}^{\frac{1}{2}}{\left[\frac{\tau -{s}_I}{\left(1+\eta \right)\alpha \left(1+{s}_O\right)}\right]}^{\frac{1}{\alpha -1}}f\left(\eta \right) d\eta =B\kern0.5em .\end{array}} $$

(22)

The above constraints are the budget constraints. The above problem is equivalent to the following problem:

$$ {\displaystyle \begin{array}{c}\underset{s_O,{s}_I}{\mathit{\operatorname{Min}}}\kern0.5em {\left[\frac{\alpha \left(1+{s}_O\right)}{\left(\tau -{s}_I\right)}\right]}^{\frac{2\alpha }{1-\alpha }}\\ {}s.t.\begin{array}{cc}\begin{array}{c}{s}_O{\int}_{-\frac{1}{2}}^{\frac{1}{2}}\left(1+\eta \right){\left[\frac{\tau -{s}_I}{\left(1+\eta \right)\alpha \left(1+{s}_O\right)}\right]}^{\frac{\alpha }{\alpha -1}}f\left(\eta \right) d\eta \\ {}+{s}_I{\int}_{-\frac{1}{2}}^{\frac{1}{2}}{\left[\frac{\tau -{s}_I}{\left(1+\eta \right)\alpha \left(1+{s}_O\right)}\right]}^{\frac{1}{\alpha -1}}f\left(\eta \right) d\eta =B.\end{array}& \end{array}\end{array}} $$

(23)

Based on the analysis of Proposition 1, under budget subsidy constraints, the above object function also increases with input subsidies but decreases with the output subsidy intensity.

Therefore, the output risks of hybrid subsidies reach minimization under the output subsidy. Conclusions are thus achieved, and the proof is complete.