Feed-in Subsidies, Taxation, and Inefficient Entry

Abstract

We study (energy) markets with dirty production and lumpy entry costs of clean production (renewables). For intermediate entry costs, markets yield inefficient production and inefficient entry. A mix of three popular regulatory instruments—polluter taxation, feed-in subsidies for renewables, and consumption taxation—cannot correct these market failures for larger entry costs. The instruments are imperfect because they affect marginal incentives, whereas entry is a lumpy fixed cost problem. Whenever the first best is implementable, feed-in subsidies and consumption taxes are redundant. The second best requires feed-in subsidies or consumption taxes in addition to a pollution tax and overshoots first best levels. Given production levels, the instruments do not affect the regulator’s budget.

Appendix: Formal Proofs

This appendix collects the formal proofs of our propositions and lemma.

Proof of Proposition 1

The first statement follows from a straightforward comparison of $W(x^{*}_{d},x^{*}_{c})$ and $W(\hat{x}^{*}_{d},0)$. In order to show $x_{d}^{*}<\hat{x}_{d}^{*}<x^{*}$, define $\tilde{x}_{d}(a)$ implicitly by

$$\begin{aligned} \Psi ^{\prime }(\tilde{x}_{d}(a)+a)=C_{d}^{\prime }(\tilde{x}_{d}(a))+E^{\prime }(\tilde{x}_{d}(a)). \end{aligned}$$

(7)

Note that $x_{d}^{*}=\tilde{x}_{d}(x_{c}^{*})$ and $\hat{x}_{d}^{*} =\tilde{x}_{d}(0)$. By the implicit function theorem it follows

$$\begin{aligned} \Psi ^{\prime \prime }(\tilde{x}_{d}(a)+a)(\partial \tilde{x}_{d}/\partial a+1)=C_{d}^{\prime \prime }(\tilde{x}_{d}(a))\partial \tilde{x}_{d}/\partial a+E^{\prime \prime }(\tilde{x}_{d}(a))\partial \tilde{x}_{d}/\partial a \end{aligned}$$

(8)

so that ${\partial \tilde{x}_{d}}/{\partial a}={\Psi ^{\prime \prime }}/ {(C_{d}^{\prime \prime }+E^{\prime \prime }-\Psi ^{\prime \prime })}<0$, where the inequality follows because $C_{d}$ and E are convex and $\Psi$ is concave. Hence, $\tilde{x}_{d}$ is strictly decreasing so that $x_{d}^{*}=\tilde{x}_{d}(x_{c}^{*})<\tilde{x}_{d}(0)=\hat{x}_{d}^{*}$. Note that since $C_{d}^{\prime \prime }>0$, $E^{\prime \prime }>0$ and $\partial \tilde{x} _{d}/\partial a<0$ the right hand side in (8) is negative. Since $\Psi ^{\prime \prime }<0$ we must have $\partial \tilde{x}_{d}/\partial a+1>0$. Hence, the term $\tilde{x}_{d}(a)+a$ is increasing in a so that it follows $\hat{x}_{d}^{*}=\tilde{x}_{d}(0)+0<\tilde{x}_{d}(x_{c}^{*})+x_{c} ^{*}=x^{*}$. $\square$

Proof of Proposition 2

We first show $x_{d}^{m}>x_{d}^{*}$. We distinguish two cases: 1. If $x_{c}^{m}>x_{c}^{*}$, then by applying (3), convexity of $C_{c}(x_{c})$, and (1), we obtain the chain of inequalities $C_{d}^{\prime }(x_{d}^{m})=C_{c}^{\prime }(x_{c}^{m})>C_{c}^{\prime }(x_{c} ^{*})=C_{d}^{\prime }(x_{d}^{*})+E^{\prime }(x_{d}^{*})>C_{d}^{\prime }(x_{d} ^{*}).$ This inequality implies by the convexity of $C_{d}(x_{d})$ that $x_{d} ^{m}>x_{d}^{*}$. 2. If, instead, $x_{c}^{m}\le x_{c}^{*}$, then $x_{d} ^{m}\le x_{d}^{*}$ would imply $x^{m}\le x^{*}$, by which we obtain the contradiction $0=2\Psi ^{\prime }(x_{d}^{*}+x_{c}^{*})-C_{c}^{\prime }(x_{c} ^{*})-C_{d}^{\prime }(x_{d}^{*})-E^{\prime }(x_{d}^{*})<2\Psi ^{\prime }(x_{d} ^{*}+x^{*}_{c})-C_{c}^{\prime }(x_{c}^{*})-C_{d}^{\prime }(x_{d}^{*})\le 2\Psi ^{\prime }(x^{m})-C_{c}^{\prime }(x_{c}^{m})-C_{d}^{\prime }(x_{d}^{m})=0$, where the last equality follows from the FOCs which define $(x_{d}^{m} ,x_{c}^{m})$.

To show $F^{m}<F^{*}$ define $p^{*}\equiv \Psi ^{\prime }(x^{*} )=C_{d}^{\prime }(x_{d}^{*})+E^{\prime }(x_{d}^{*})$. Since $\hat{x} ^{*}>x_{d}^{*}$ and $x_{d}^{*}$ is such that $C_{d}^{\prime } (x_{d}^{*})+E^{\prime }(x_{d}^{*})=p^{*}$, we have $C_{d}^{\prime }(x)+E^{\prime }(x)>p^{*}$ for all $x\in (x_{d}^{*},\hat{x}^{*})$ due to convexity. Moreover, since $\hat{x}^{*}<x^{*}$ and $x^{*}$ is such that $\Psi ^{\prime }(x^{*})=p^{*}$, we have $\Psi ^{\prime }(x)>p^{*}$ for all $x\in (\hat{x}^{*},x^{*})$, due to the concavity of $\Psi$. Consequently $\int _{\hat{x}^{*}}^{x^{*}}[\Psi ^{\prime }(x)-p^{*}]dx+\int _{x_{d}^{*}}^{\hat{x}^{*}}[C_{d}^{\prime }(x)+E^{\prime }(x)-p^{*}]dx>0.$ Using the former inequality, it then follows $F^{m} =p^{m}x_{c}^{m}-C_{c}(x_{c}^{m})<p^{*}x_{c}^{m}-C_{c}(x_{c}^{m})\le p^{*}x_{c}^{*}-C_{c}(x_{c}^{*})=p^{*}x_{c}^{*}-\int _{0} ^{x_{c}^{*}}C_{c}^{\prime }(x)dx=p^{*}(x^{*}-x_{d}^{*})-\int _{0}^{x_{c}^{*}}C_{c}^{\prime }(x)dx<p^{*}(x^{*}-x_{d}^{*} )-\int _{0}^{x_{c}^{*}}C_{c}^{\prime }(x)dx+\int _{\hat{x}^{*}}^{x^{*} }[\Psi ^{\prime }(x)-p^{*}]dx+\int _{x_{d}^{*}}^{\hat{x}^{*}} [C_{d}^{\prime }(x)+E^{\prime }(x)-p^{*}]dx=\int _{\hat{x}^{*}}^{x^{*} }\Psi ^{\prime }(x)dx+\int _{x_{d}^{*}}^{\hat{x}^{*}}[C_{d}^{\prime }(x)+E^{\prime }(x)]dx-\int _{0}^{x_{c}^{*}}C_{c}^{\prime }(x)dx=\int _{0}^{x^{*}}\Psi ^{\prime }(x)dx-\int _{0}^{x_{d}^{*}}[C_{d}^{\prime }(x)+E^{\prime }(x)]dx-\int _{0}^{x_{c}^{*}}C_{c}^{\prime }(x)dx-\int _{0}^{\hat{x}^{*}}\Psi ^{\prime }(x)dx+\int _{0}^{\hat{x}^{*}} [C_{d}^{\prime }(x)+E^{\prime }(x)]dx=W^{*}-\hat{W}^{*}=F^{*},$ where the first two inequalities follow from $p^{*}>p^{m}$ and revealed preferences, respectively. $\square$

Proof of Lemma 1

By definition, the instruments $(t_{d},t_{c},t_{\psi })$ implement an allocation $(x_{d} ,x_{c})$ with $x_{c}>0$ if and only if there exist some price $p^{r}$ such that (4) holds. The conditions in (4) are equivalent to the set of conditions

$$\begin{aligned} \Psi ^{\prime }(x_{d}+x_{c})&=p^{r}+t_{\psi } \end{aligned}$$

(9)

$$\begin{aligned} C_{d}^{\prime }(x_{d})&=p^{r}-t_{d} \end{aligned}$$

(10)

$$\begin{aligned} C_{c}^{\prime }(x_{c})&=p^{r}+t_{c}, \end{aligned}$$

(11)

$$\begin{aligned} \Pi _{c}^{r}(x_{c})&=(p^{r}+t_{c})x_{c}-C_{c}(x_{c})-F\geqslant 0. \end{aligned}$$

(12)

“$\Rightarrow$” Suppose $(t_{d} ,t_{c},t_{\psi })$ implements the allocation $(x_{d},x_{c})$. Then there exists a $p^{r}$ such that (9)–(11) holds from which it follows:

• Statement (i) follows from rewriting and combining (9) and (10) as follows $t_{\psi }=\Psi ^{\prime }(x_{d}+x_{c} )-p^{r}=\Psi ^{\prime }(x_{d}+x_{c})-C_{d}^{\prime }(x_{d})-t_{d}$ and rewriting and combining (9) and (11) as follows $t_{\psi } =\Psi ^{\prime }(x_{d}+x_{c})-p^{r}=\Psi ^{\prime }(x_{d}+x_{c})-C_{c}^{\prime }(x_{c})+t_{c}$.

• Statement (ii) follows because entrant’s profits are $\Pi _{c}^{r}=(p^{r}+t_{c})x_{c}-C_{c}(x_{c})-F=C_{c}^{\prime }(x_{c})x_{c}-C_{c} (x_{c})-F\geqslant 0$, where the equality follows from (11). Non-negativity follows by (12).

• Statement (iii) follows because consumer’s net surplus is $\Phi ^{r}=\Psi (x_{d}+x_{c})-(p^{r}+t_{\psi })(x_{d}+x_{c})=\Psi (x_{d}+x_{c} )-\Psi ^{\prime }(x_{d}+x_{c})(x_{d}+x_{c})$.

• Statement (iv) follows because profits of the dirty sector are $\Pi _{d}^{r}=(p^{r}-t_{d})x_{d}-C_{d}(x_{d})=C_{d}^{\prime }(x_{d})x_{d} -C_{d}(x_{d})$, where the last equality follows from (10).

• Statement (v) follows from using statement (i), since it thereby follows $B=t_{\psi }(x_{d}+x_{c})+t_{d}x_{d}-t_{c}x_{c}=[t_{\psi }+t_{d} ]x_{d}+[t_{\psi }-t_{c}]x_{c}=[\Psi ^{\prime }(x_{d}+x_{c})-C_{d}^{\prime } (x_{d})]x_{d}+[\Psi ^{\prime }(x_{d}+x_{c})-C_{c}^{\prime }(x_{c})]x_{c}$.

“$\Leftarrow$” Let (i)–(v) hold and define $p^{r}=\Psi ^{\prime }(x_{d}+x_{c})-t_{\psi }$, then (9)–(11) follow directly from (i). Moreover, (ii) implies (12). Since (9)–(12) are equivalent to (4), the result follows. $\square$

Proof of Proposition 3

The chain of inequalities in the proof of Proposition 2 implies $F^{m}<F^{r}<F^{*}$, since $F^{r}=p^{*}x_{c}^{*}-C_{c}(x_{c}^{*})$. Statement (i) follows directly from Lemma 1(ii). Statement (ii) follows directly from Lemma 1(i) and (ii) combined with the conditions for efficiency described in (1 ).$\square$

Proof of Proposition 4

The first statement follows directly, because of the binding constraint at the optimum and the first order condition of maximizing $W(x_{d},x_{c}^{sb})$ with respect to $x_{d}$. The second statement follows because for $F>F^{r}$ the constraint at $x_{c}=x_{c}^{*}$ is violated, which implies $x_{c}^{sb}>x_{c}^{*}$, because the left hand side of the constraint is increasing in $x_{c}$, due to $C_{c}^{\prime \prime }(x_{c})x_{c}+C_{c}^{\prime }(x_{c})-C_{c}^{\prime } (x_{c})=C_{c}^{\prime \prime }(x_{c})>0$. It follows moreover $x_{d}^{{sb} }=\tilde{x}_{d}(x_{c}^{sb})<\tilde{x}_{d}(x_{c}^{*})=x_{d}^{*}$ with $\tilde{x}_{d}(.)$ as defined in (7), where we showed that $\tilde{x}_{d}(.)$ is decreasing but by less than 1. As a result $x^{sb}=x_{d}^{sb}+x_{c}^{sb}>x_{d}^{*}+x_{c}^{*}=x^{*}$. Combining $\Psi ^{\prime }(x_{d}^{sb}+x_{c}^{sb})=C_{d}^{\prime }(x_{d}^{sb})+E^{\prime }(x_{d}^{sb})$ with Lemma 1(i) yields the last statement, since they imply $t_{d}^{sb}=\Psi ^{\prime }(x_{d}^{sb}+x_{c} ^{sb})-C_{d}^{\prime }(x_{d}^{sb})-t_{\psi }^{sb}=E^{\prime }(x_{d}^{sb} )-t_{\psi }^{sb}$ and $t_{c}^{sb}=C_{c}^{\prime }(x_{c}^{sb})-\Psi ^{\prime }(x_{d}^{sb}+x_{c}^{sb})+t_{\psi }^{sb}$. Hence, by Lemma 1, the set of instruments $(t_{d}^{sb},t_{c} ^{sb},t_{\psi }^{sb})$ implements $(x_{d}^{sb},x_{c}^{sb})$. Recall that $x_{c}^{sb}>x_{c}^{*}$ and $x^{sb}>x^{*}$ and $x_{c}^{*}$ is such that $C_{c}^{\prime }(x_{c}^{*})=\Psi ^{\prime }(x^{*})$. Convexity of $C_{c}$ and concavity of $\Psi$ then imply $C_{c}^{\prime }(x_{c}^{sb} )>\Psi ^{\prime }(x^{sb})$ so that $t_{c}^{sb}\ge t_{\psi }^{sb}$. $\square$

Proof of Proposition 5

Maximum welfare is $W(\hat{x}_{d}^{*},0)$ without entry and $W(x_{d}^{sb},x_{c} ^{sb})-F$ with entry. Hence, optimal regulation involves entry when $F<F^{sb}$ with $F^{sb}=W(x_{d}^{sb},x_{c}^{sb})-W(\hat{x}_{d}^{*},0)=W(x_{d} ^{sb},x_{c}^{sb})-W(x_{d}^{*},x_{c}^{*})+F^{*}<F^{*}$, as $W(x_{d}^{sb} ,x_{c}^{sb})<W(x_{d}^{*},x_{c}^{*})$. $\square$

Proof of Proposition 6

Suppose $(t_{d},t_{c},t_{\psi })$ implements the outcome $(x_{d},x_{c})$ with net consumer surplus $\Phi$, profit levels $\Pi _{c}$, $\Pi _{d}$, and a budget B. By Lemma 1 any other set of instruments $(t_{d}^{\prime },t_{c}^{\prime },t_{\psi }^{\prime })$ that implements $(x_{d},x_{c})$ must also yield the same net consumer surplus $\Phi$, profit levels $\Pi _{c}$, $\Pi _{d}$, and a budget B. We therefore only have to show that the specific set of instruments exist.

To see instruments exist that satisfy (i). Set $t_{\psi }^{\prime }=0$, $t_{d}^{\prime }=\Psi ^{\prime }(x_{d}+x_{c})-C_{d}^{\prime }(x_{d})$ and $t_{c}^{\prime }=C_{c}^{\prime }(x_{c})-\Psi ^{\prime }(x_{d}+x_{c})$. It then follows by Lemma 1 that also $(t_{d}^{\prime } ,t_{c}^{\prime },0)$ implements $(x_{d},x_{c})$.

To see instruments exist that satisfy (ii). Set $t_{c}^{\prime }=0$, $t_{\psi }^{\prime }=\Psi ^{\prime }(x_{d}+x_{c})-C_{c}^{\prime }(x_{c})$, and $t_{d}^{\prime }=\Psi ^{\prime }(x_{d}+x_{c})-C_{d}^{\prime }(x_{d})-t_{\psi }^{\prime }=C_{c}^{\prime }(x_{c})-C_{d}^{\prime }(x_{d})$. It then follows by Lemma 1 that also $(t_{d}^{\prime },0,t_{\psi }^{\prime })$ implements $(x_{d},x_{c})$. $\square$

Proof of Proposition 7

Since $(t_{d},t_{c},t_{\psi })$ implements $(x_{d},x_{c})$, it follows by the definition of $(t_{d}^{\prime },t_{c}^{\prime },t_{\psi }^{\prime })$ and Lemma 1 that

$$\begin{aligned} t^{\prime }_{d}= & {} t_{d}+t_{\psi }-t_{\psi }^{\prime }=\Psi ^{\prime }(x_{d} +x_{c})-C_{d}^{\prime }(x_{d})-t_{\psi }^{\prime } \text{ and } t^{\prime }_{c}=t_{c}-t_{\psi }+t_{\psi }^{\prime }=C_{c}^{\prime }(x_{c}) \\&-\Psi ^{\prime }(x_{d}+x_{c})+t_{\psi }^{\prime }. \end{aligned}$$

It then follows from Lemma 1 that also $(t_{d}^{\prime },t_{c}^{\prime },t_{\psi }^{\prime })$ implements $(x_{d},x_{c} )$. It remains to show that $(t_{d}^{\prime },t_{c}^{\prime },t_{\psi }^{\prime })$ is feed-in budget balanced. This follows from $t_{c}^{\prime }x_{c} =(t_{c}-t_{\psi }+t_{\psi }^{\prime })x_{c}=(t_{\psi }^{\prime }x_{d}/x_{c} +t_{\psi }^{\prime })x_{c}=t_{\psi }^{\prime }x_{d}+t_{\psi }^{\prime }x_{c} =t_{\psi }^{\prime }(x_{d}+x_{c})$. $\square$