Green tradable certificates versus feed-in tariffs in the promotion of renewable energy shares

Abstract

The paper analyzes the relationship between CO₂ mitigation policy and promotion policies designed to deploy renewable energy sources for electricity production (RES-E). If an emission cap is the only policy target, an optimal mix consisting of high and low carbon use of fossil fuels, deployment of RES-E, and energy savings can best be achieved by either setting a uniform carbon tax or by implementing a cap-and-trade system covering all CO₂ sources. An additional RES-E share target causes higher costs in achieving the cap. Conversely, a more ambitious emission target automatically increases the RES-E share. In a second step, we investigate different policies for inducing an RES-E quota. Such a quota can be efficiently achieved either by a system of tradable green certificates, budget-balanced FIT system, or budget-balancing premium system. We also show that differentiated, technology-specific FITs are not efficient.

Appendix

Proof of Proposition 2

Differentiating (7)–(10) with respect to $\bar{E}$, we can write the resulting equation system in matrix form (omitting the function arguments) as

$$\left[ {\begin{array}{*{20}c} {P^{{\prime }} - C_{\text{b}}^{{\prime \prime }} } & {P^{{\prime }} } & {P^{{\prime }} } & { - \alpha_{\text{b}} } \\ {P^{{\prime }} } & {P^{{\prime }} - C_{\text{r}}^{{\prime \prime }} } & {P^{{\prime }} } & 0 \\ {P^{{\prime }} } & {P^{{\prime }} } & {P^{{\prime }} - C_{\text{f}}^{{\prime \prime }} } & { - \alpha_{\text{f}} } \\ {\alpha_{\text{b}} } & 0 & {\alpha_{\text{f}} } & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\text{d}}Q_{\text{b}} /{\text{d}}\bar{E}} \\ {{\text{d}}Q_{\text{r}} /{\text{d}}\bar{E}} \\ {{\text{d}}Q_{\text{f}} /{\text{d}}\bar{E}} \\ {{\text{d}}\lambda /{\text{d}}\bar{E}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 1 \\ \end{array} } \right]$$

(62)

Let ${\text{Det}}(M) = \left[ {\alpha_{\text{f}}^{2} C_{\text{b}}^{{\prime \prime }} + \alpha_{\text{b}}^{ 2} C_{f}^{{\prime \prime }} } \right]\left[ {C_{\text{r}}^{{\prime \prime }} - P^{{\prime }} } \right] - C_{\text{r}}^{{\prime \prime }} P^{{\prime }} \left[ {\alpha_{\text{b}} - \alpha_{\text{f}} } \right]^{2} > 0$ be the determinant of the matrix in (62). Solving (62), we obtain

$$\frac{{{\text{d}}Q_{\text{b}} }}{{{\text{d}}\bar{E}}} = \frac{{\left[ {\alpha_{\text{f}} - \alpha_{\text{b}} } \right]C_{\text{r}}^{{\prime \prime }} P^{{\prime }} + \alpha_{\text{b}} C_{\text{f}}^{{\prime \prime }} \left[ {C_{\text{r}}^{{\prime \prime }} - P^{{\prime }} } \right]}}{{{\text{Det}}(M)}} > 0$$

(63)

$$\frac{{{\text{d}}Q_{\text{f}} }}{{{\text{d}}\bar{E}}} = \frac{{\left[ {\alpha_{\text{b}} - \alpha_{\text{f}} } \right]C_{\text{r}}^{{\prime \prime }} P^{{\prime }} + \alpha_{\text{f}} C_{\text{b}}^{{\prime \prime }} \left[ {C_{\text{r}}^{{\prime \prime }} - P^{{\prime }} } \right]}}{{{\text{Det}}(M)}}$$

(64)

$$\frac{{{\text{d}}Q_{\text{r}} }}{{{\text{d}}\bar{E}}} = \frac{{\alpha_{\text{f}} C_{\text{b}}^{{\prime \prime }} + \alpha_{\text{b}} C_{\text{f}}^{{\prime \prime }} }}{{{\text{Det}}(M)}} < 0$$

(65)

$$\frac{{{\text{d}}\lambda }}{{{\text{d}}\bar{E}}} = \frac{{P^{{\prime }} \left[ {C_{\text{b}}^{{\prime \prime }} C_{\text{r}}^{{\prime \prime }} + C_{\text{b}}^{{\prime \prime }} C_{\text{f}}^{{\prime \prime }} + C_{\text{f}}^{{\prime \prime }} C_{\text{r}}^{{\prime \prime }} } \right] - C_{\text{b}}^{{\prime \prime }} C_{\text{r}}^{{\prime \prime }} C_{\text{f}}^{{\prime \prime }} }}{{{\text{Det}}(M)}} < 0.$$

(66)

While (64) is ambiguous as to the sign, we see immediately by adding (63) and (64) that

$$\frac{{{\text{d}}\left[ {Q_{\text{b}} + Q_{\text{f}} } \right]}}{{{\text{d}}\bar{E}}} = \frac{{\alpha_{\text{f}} C_{\text{b}}^{{\prime \prime }} \left[ {C_{\text{r}}^{{\prime \prime }} - P^{{\prime }} } \right]}}{{{\text{Det}}(M)}} > 0.$$

(67)

Finally, for the share of renewable energy we obtain

$$\frac{{{\text{d}}[Q_{r} /(Q_{b} + Q_{f} + Q_{r} )]}}{{{\text{d}}\bar{E}}} = \frac{P'}{{C_{r} ''}} < 0.$$

(68)

That the sign of $\frac{{{\text{d}}Q_{\text{f}} }}{{{\text{d}}\bar{E}}}$ is indeed ambiguous can be shown by example. Choose $P(Q) = A - BQ$ and $C_{i} (Q_{i} ) = \frac{{c_{i} }}{2}Q_{i}^{2}$ and let A = 100.0, B = 1.0, c _b = 0.1, c _f = 0.3, c _r = 1.0, α _b = 1.0, α _f = 0.5. If we now tighten the emission cap from $\bar{E} = 50$ to $\bar{E} = 40$, we will find that Q _f increases from 16.67 to 37.78. If we choose c _f = 0.9, keeping all other parameters as before and tightening the emission cap from $\bar{E} = 50$ to $\bar{E} = 40$, we will find that Q _f decreases from 18.13 to 17.10.¹⁴

Proof of Proposition 3

Differentiating (12)–(14) with respect to ζ yields in matrix form

$$\left[ {\begin{array}{*{20}c} {P^{{\prime }} - C_{\text{b}}^{{\prime \prime }} } & {P^{{\prime }} } & {P^{{\prime }} } \\ {P^{{\prime }} } & {P^{{\prime }} - C_{\text{f}}^{{\prime \prime }} } & {P^{{\prime }} } \\ 0 & 0 & {C_{\text{r}}^{{\prime \prime }} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\text{d}}Q_{b} /{\text{d}}\zeta } \\ {{\text{d}}Q_{f} /{\text{d}}\zeta } \\ {{\text{d}}Q_{r} /{\text{d}}\zeta } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 1 \\ \end{array} } \right]$$

(69)

Let ${\text{Det}}(M) = C_{\text{r}}^{{\prime \prime }} \left[ {C_{\text{b}}^{{\prime \prime }} C_{\text{r}}^{{\prime \prime }} - P^{{\prime }} (C_{\text{b}}^{{\prime \prime }} + C_{\text{f}}^{{\prime \prime }} )} \right] > 0$ be the determinant of the matrix in (69). Solving this equation, we obtain

$$\frac{{{\text{d}}Q_{\text{b}} }}{{{\text{d}}\zeta }} = \frac{{C_{\text{f}}^{{\prime \prime }} P^{{\prime }} }}{{{\text{Det}}(M)}} < 0$$

(70)

$$\frac{{{\text{d}}Q_{f} }}{{{\text{d}}\zeta }} = \frac{{C_{\text{b}}^{{\prime \prime }} P^{{\prime }} }}{{{\text{Det}}(M)}} < 0$$

(71)

$$\frac{{{\text{d}}Q_{\text{r}} }}{{{\text{d}}\zeta }} = \frac{{C_{\text{b}}^{{\prime \prime }} C_{\text{f}}^{{\prime \prime }} - P^{{\prime }} [C_{\text{b}}^{{\prime \prime }} + C_{\text{f}}^{{\prime \prime }} ]}}{{{\text{Det}}(M)}} > 0.$$

(72)

From this we derive, after simplification,

$$\frac{{{\text{d}}Q}}{{{\text{d}}\zeta }} = \frac{{C_{\text{b}}^{{\prime \prime }} C_{\text{f}}^{{\prime \prime }} }}{{{\text{Det}}(M)}} > 0$$

(73)

$$\frac{\text{d}}{{{\text{d}}\zeta }}\left( {\frac{{Q_{\text{r}} }}{Q}} \right) = \frac{{ - P^{{\prime }} Q\left[ {C_{\text{f}}^{{\prime \prime }} + C_{\text{b}}^{{\prime \prime }} } \right] + C_{\text{b}}^{{\prime \prime }} C_{\text{f}}^{{\prime \prime }} \left[ {Q_{\text{b}} + Q_{\text{f}} } \right]}}{{{\text{Det}}(M)}} > 0.$$

(74)

Proof of Proposition 4

Differentiating (17)–(20) with respect to $\zeta$ yields in matrix form

$$\left[ {\begin{array}{*{20}c} {P^{{\prime }} - C_{\text{b}}^{{\prime \prime }} } & {P^{{\prime }} } & {P^{{\prime }} } & { - 1} \\ {P^{{\prime }} } & {P^{{\prime }} - C_{\text{f}}^{{\prime \prime }} } & {P^{{\prime }} } & { - 1} \\ 0 & 0 & {C_{\text{r}}^{{\prime \prime }} } & 0 \\ {t + P^{{\prime }} Q_{\text{r}} } & {t + P^{{\prime }} Q_{\text{r}} } & {p + P^{{\prime }} Q_{\text{r}} - \zeta } & {Q_{\text{b}} + Q_{\text{f}} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\text{d}}Q_{b} /{\text{d}}\zeta } \\ {{\text{d}}Q_{f} /{\text{d}}\zeta } \\ {{\text{d}}Q_{r} /{\text{d}}\zeta } \\ {{\text{d}}t/{\text{d}}\zeta } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 \\ 0 \\ 1 \\ {Q_{\text{r}} } \\ \end{array} } \right].$$

(75)

By stability of the competitive equilibrium the determinant of the matrix ${\text{Det}}(M) = C_{\text{r}}^{{\prime \prime }} \left[ {\left[ {C_{\text{f}}^{{\prime \prime }} + C_{\text{b}}^{{\prime \prime }} } \right]\left[ { - P^{{\prime }} \; \times \;Q - t} \right] + C_{\text{f}}^{{\prime \prime }} C_{\text{b}}^{{\prime \prime }} \left[ {Q_{\text{b}} + Q_{\text{f}} } \right]} \right]$ must be positive. Solving (75), we obtain

$$\frac{{{\text{d}}Q_{\text{b}} }}{{{\text{d}}\zeta }} = \frac{{C_{\text{f}}^{{\prime \prime }} [P^{{\prime }} Q] - C_{\text{r}}^{{\prime \prime }} Q_{\text{r}} + p - \zeta }}{{{\text{Det}}(M)}} < 0$$

(76)

$$\frac{{{\text{d}}Q_{f} }}{{{\text{d}}\zeta }} = \frac{{C_{b} ''[P'Q] - C_{r} ''Q_{r} + p - \zeta }}{{{\text{Det}}(M)}} < 0$$

(77)

$$\frac{{{\text{d}}Q_{\text{r}} }}{{{\text{d}}\zeta }} = \frac{1}{{C_{\text{r}}^{{\prime \prime }} }} > 0$$

(78)

$$\frac{{{\text{d}}Q}}{{{\text{d}}\zeta }} = \frac{{C_{\text{b}}^{{\prime \prime }} C_{\text{f}}^{{\prime \prime }} \left[ {Q_{b} + Q_{f} } \right] + \left[ {C_{\text{b}}^{{\prime \prime }} + C_{\text{f}}^{{\prime \prime }} } \right]\left[ {p - \zeta - C_{\text{r}}^{{\prime \prime }} Q_{\text{r}} - t} \right]}}{{{\text{Det}}(M)}}$$

(79)

$$\begin{gathered} \frac{\text{d}}{{{\text{d}}\zeta }}\left( {\frac{{Q_{\text{r}} }}{Q}} \right) = \frac{{Q\left[ { - C_{\text{f}}^{{\prime \prime }} P^{{\prime }} \times Q + t} \right]] + C_{\text{b}}^{{\prime \prime }} [C_{\text{f}}^{{\prime \prime }} [Q_{\text{b}} + Q_{\text{f}} ] - P^{{\prime }} \times Q - t]}}{\det (M)} \hfill \\ \quad + \frac{{Q_{\text{r}} \left[ {C_{\text{b}}^{{\prime \prime }} + C_{\text{f}}^{{\prime \prime }} } \right]\left[ {C_{\text{r}}^{{\prime \prime }} Q_{\text{r}} + t + \zeta - p} \right]}}{\det (M)} > 0 \hfill \\ \end{gathered}$$

(80)

$$\frac{{{\text{d}}t}}{{{\text{d}}\zeta }} = \frac{{ - C_{\text{b}}^{{\prime \prime }} C_{\text{r}}^{{\prime \prime }} \left[ {p - \zeta + \left[ {P^{{\prime }} - C_{\text{r}}^{{\prime \prime }} } \right]Q_{\text{r}} } \right] + \left[ {C_{\text{b}}^{{\prime \prime }} + C_{\text{f}}^{{\prime \prime }} } \right]P^{{\prime }} \left[ {p - \zeta - C_{\text{r}}^{{\prime \prime }} Q_{\text{r}} - t} \right]}}{\det (M)} > 0$$

(81)

To show that the sign of ${\text{d}}Q/{\text{d}}\zeta$ is ambiguous, we again choose linear (inverse) demand $P(Q) = A - BQ$ and cost functions of the type $C_{j} (q) = c_{j0} q + c_{j1} q^{2} /2$ for j = b, f, r. Parameters are selected according to A = 100.0, B = 1.0, $c_{b0} = 0.1$, $c_{b1} = 0.1$, $c_{f0} = 0.2$, $c_{f1} = 0.3$, $c_{r0} = 1.0$, $c_{r1} = 0.05$.

Then for ζ = 8 (ζ = 10, ζ = 12) we obtain Q = 380 (Q = 385, ζ = 380).

Proof of Proposition 6

By differentiating (26)–(29) with respect to β we can write the resulting equation system in matrix form (omitting the function arguments) as

$$\left[ {\begin{array}{*{20}c} {P^{{\prime }} - C_{\text{b}}^{{\prime \prime }} } & {P^{{\prime }} } & {P^{{\prime }} } & { - \beta } \\ {P^{{\prime }} } & {P^{{\prime }} - C_{\text{f}}^{{\prime \prime }} } & {P^{{\prime }} } & { - \beta } \\ {P^{{\prime }} } & {P^{{\prime }} } & {P^{{\prime }} - C_{\text{r}}^{{\prime \prime }} } & {1 - \beta } \\ \beta & \beta & {1 - \beta } & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\text{d}}Q_{b} /{\text{d}}\beta } \\ {{\text{d}}Q_{r} /{\text{d}}E\beta } \\ {{\text{d}}Q_{f} /{\text{d}}E\beta } \\ {{\text{d}}\mu /{\text{d}}E\beta } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} \mu \\ \mu \\ \mu \\ Q \\ \end{array} } \right].$$

(82)

Writing the determinant of the matrix in (82) as ${\text{Det}}[M] = - C_{\text{b}}^{{\prime \prime }} C_{\text{f}}^{{\prime \prime }} [1 - \beta ]^{2} + \beta^{2} [C_{\text{b}}^{{\prime \prime }} + C_{\text{f}}^{{\prime \prime }} ][P^{{\prime }} - C_{\text{r}}^{{\prime \prime }} ] < 0$ for short and solving (82), we obtain

$$\frac{{{\text{d}}Q_{\text{b}} }}{{{\text{d}}\beta }} = \frac{{C_{\text{f}}^{{\prime \prime }} [\mu (1 - \beta ) + Q[\beta C_{\text{r}}^{{\prime \prime }} - P^{{\prime }} ]}}{{{\text{Det}}[M]}} < 0$$

(83)

$$\frac{{dQ_{f} }}{d\beta } = \frac{{C_{b} ''[\mu (1 - \beta ) + Q[\beta C_{r} '' - P']}}{Det[M]} < 0$$

(84)

$$\frac{{{\text{d}}Q_{\text{r}} }}{{{\text{d}}\beta }} = \frac{{\left[ {C_{\text{f}}^{{\prime \prime }} P' + C_{\text{b}}^{{\prime \prime }} \left[ {P' - C_{\text{f}}^{{\prime \prime }} } \right]} \right]Q + \beta \left[ {\mu \left[ {C_{\text{b}}^{{\prime \prime }} + C_{\text{f}}^{{\prime \prime }} } \right] + C_{\text{b}}^{{\prime \prime }} C_{\text{f}}^{{\prime \prime }} Q} \right]}}{{{\text{Det}}[M]}}$$

(85)

$$\frac{{{\text{d}}\mu }}{{{\text{d}}\beta }} = \frac{{C_{\text{r}}^{{\prime \prime }} \left[ {C_{\text{b}}^{{\prime \prime }} + C_{\text{f}}^{{\prime \prime }} } \right]\beta \mu + C_{\text{r}}^{{\prime \prime }} \left[ {C_{\text{f}}^{{\prime \prime }} + C_{\text{b}}^{{\prime \prime }} } \right]P^{{\prime }} Q - C_{\text{b}}^{{\prime \prime }} \left[ {C_{\text{f}}^{{\prime \prime }} (1 - \beta )\mu + \left[ {C_{\text{r}}^{{\prime \prime }} - P^{{\prime }} } \right]Q} \right]}}{{{\text{Det}}[M]}}$$

(86)

$$\frac{{{\text{d}}Q}}{{{\text{d}}\beta }} = \frac{{C_{\text{f}}^{{\prime \prime }} \left[ {\mu + \beta C_{\text{r}}^{{\prime \prime }} Q} \right] + C_{\text{b}}^{{\prime \prime }} \left[ {\mu - \left[ {(1 - \beta )C_{\text{f}}^{{\prime \prime }} + \beta C_{\text{b}}^{{\prime \prime }} } \right]Q} \right]}}{{{\text{Det}}[M]}}.$$

(87)

To show the ambiguity of $\frac{{{\text{d}}Q_{\text{r}} }}{{{\text{d}}\beta }}$, $\frac{{{\text{d}}\mu }}{{{\text{d}}\beta }}$, and $\frac{{{\text{d}}Q}}{{{\text{d}}\beta }}$ we take the functional forms as in the proof of Proposition 2 and choose A = 100.0, B = 1.0, $c_{b} = 0.1$, $c_{f} = 0.3$, $c_{r} = 1.0$. Increasing $\beta$ leads to strictly increasing Q _r, decreasing total output and increasing shadow cost of the RES-E. Taking a less elastic inverse demand function by selecting B = 0.2, we can show Q _r and the shadow cost μ to be inverted U-shaped. Choosing B = 1.5, we can see that for small but binding β, total output is first increasing then decreasing when β is increased.