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, feedin 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, feedin subsidies and consumption taxes are redundant. The second best requires feedin 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.
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Notes
 1.
E.g. the Presidential Memorandum on December 5, 2013, White House (2013) states a target share schedule for renewables of 20 % as of 2020 up from 10 % for 2015 in the US. COM (2014) sets a similar target of 20 % in 2020 for the whole European Union. According to the same source this level is expected to rise up to 27 % by 2030. Wharton (2013) reports that the Japanese government has set renewable targets of between 25 and 35 % of total power generation by 2030.
 2.
We consider any type of subsidy that raises the productionprice of clean energy as a feedin subsidy and, hence, do not distinguish between feedin tariffs, feedin premiums or tenders.
 3.
According to the database of IEA (2014), more than 60 countries world wide use feedins subsidizing renewables, including the US, Canada, the European countries, Japan, and even China. Under indirect polluter taxes we also count emission trading mechanisms such as the EU ETS.
 4.
RES (2014) reports that these taxes are standard in Europe, including all major countries UK, Germany, France, Italy, and Spain.
 5.
Since we analyze a perfect competitive market, the ownership of these technologies is immaterial. They can be owned by the dirty incumbents or new firms.
 6.
We note that the extra instrument might however be useful when the regulator does not have full control over the tax on polluters, due to political economy constraints or international agreements.
 7.
See also IEA (2008), which expresses similar views.
 8.
Carlton and Loury (1980, 1986) point out that there is no Pigouvian tax rate such that the long run competitive equilibrium corresponds to the socially efficient allocation. The rationale suggested (see Carlton and Loury 1980, p. 563) is that there are two targets to fulfill, i.e., production and number of firms, with solely one instrument. Our findings suggest that this is imprecise as we illustrate that despite the introduction of a larger set of policy instruments with respect to the targets we still obtain inefficient outcomes.
 9.
Katsoulacos and Xepapadeas (1995) study entry when the market structure is imperfectly competitive.
 10.
 11.
 12.
See also Burrows (1979) and Collinge and Oates (1982) who argue that taxation of large scale firms may discourage entry as the tax bills paid by the individual firm exceed the damages caused by that firm’s entry. This problem can be solved by the implementation of command and control or nonlinear taxation, while it appears not to be a problem in a general equilibrium framework (see Kohn 1994). Our problem is of a different nature attributed to the fact that clean entrants cannot fully appropriate the benefits from entry.
 13.
Note that production technologies with constant marginal cost up to some fixed capacity are convex.
 14.
Perfect competition is not essential for our arguments. It however presents the natural framework to study the effectiveness of the instruments, since they are not intended for competition policy. In line with Makowski and Ostroy (1995) of an “occupationalchoice equilibrium”, we assume that the firm takes into account changes in the equilibrium price. Contrary to a pure Walrasian analysis of our setup, this approach ensures existence of equilibrium (for more details, see footnote 7 in Makowski and Ostroy 1995).
 15.
Our Inada conditions together with our convexity assumptions imply that the first order conditions are sufficient and lead to welldefined and positive demand and supply for any \(p>0\).
 16.
 17.
E.g., the result rationalizes COM (2013, §108): “the EU ETS and national CO2 taxes internalise [...] may not (yet) ensure the achievement of the related, but distinct EU objectives for renewable energy [...] the Commission therefore presumes that a residual market failure remains, which aid for renewable energy can address.”
 18.
Recall from Lemma 1 that there is also no role for the consumption tax to finetune the budget.
 19.
This view may change as countries start to auction off rather than give away these permits for free with “grandfathering” schemes.
 20.
Clearly, first best only obtains when the transfer is not financed by a distortionary consumption tax.
 21.
Our result that feedin subsidy and consumer taxes are redundant would however still remain valid.
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We thank Stefan Ambec, Panos Hatzipanayotou, Carsten Helm, Matti Liski, Marco Runkel, Ronnie Schöb, Anastasios Xepapadeas, and two anonymous referees. Financial support from the German Research Foundation (SFB/TR 15) and the BMBF is gratefully acknowledged.
Appendix: Formal Proofs
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
Note that \(x_{d}^{*}=\tilde{x}_{d}(x_{c}^{*})\) and \(\hat{x}_{d}^{*} =\tilde{x}_{d}(0)\). By the implicit function theorem it follows
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
“\(\Rightarrow \)” 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). Nonnegativity 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
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 feedin budget balanced. This follows from \(t_{c}^{\prime }x_{c} =(t_{c}t_{\psi }+t_{\psi }^{\prime })x_{c}=(t_{\psi }^{\prime }x_{d}/x_{c} +t_{\psi }^{\prime })x_{c}=t_{\psi }^{\prime }x_{d}+t_{\psi }^{\prime }x_{c} =t_{\psi }^{\prime }(x_{d}+x_{c})\). \(\square \)
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Antoniou, F., Strausz, R. Feedin Subsidies, Taxation, and Inefficient Entry. Environ Resource Econ 67, 925–940 (2017). https://doi.org/10.1007/s1064001600128
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Keywords
 Taxation
 Feedin tariffs
 Externalities
 Renewables
 Entry
 Pollution
JEL Classification
 D21
 D61
 H23