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.

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Fig. 1
Fig. 2

Notes

  1. 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. 2.

    We consider any type of subsidy that raises the production-price of clean energy as a feed-in subsidy and, hence, do not distinguish between feed-in tariffs, feed-in premiums or tenders.

  3. 3.

    According to the database of IEA (2014), more than 60 countries world wide use feed-ins 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. 4.

    RES (2014) reports that these taxes are standard in Europe, including all major countries UK, Germany, France, Italy, and Spain.

  5. 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. 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. 7.

    See also IEA (2008), which expresses similar views.

  8. 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. 9.

    Katsoulacos and Xepapadeas (1995) study entry when the market structure is imperfectly competitive.

  10. 10.

    Although not mentioned explicitly studies by Jaffe et al. (2005), Helm and Schöttner (2008) and Ambec and Crampes (2012) also support promoting policies for similar reasons.

  11. 11.

    Using data from OECD countries, Nesta et al. (2014) find empirical support for the hypothesis that current renewable energy policies foster green innovation. In a Nordic case study based on wind power, Boomsma et al. (2012) verify that a feed-in subsidy encourages earlier investment.

  12. 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 non-linear 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. 13.

    Note that production technologies with constant marginal cost up to some fixed capacity are convex.

  14. 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 “occupational-choice 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. 15.

    Our Inada conditions together with our convexity assumptions imply that the first order conditions are sufficient and lead to well-defined and positive demand and supply for any \(p>0\).

  16. 16.

    In a different setup, this degree of freedom is also noted in Palmer and Walls (1997), Schöb (1997), Walls and Palmer (2001), and Fullerton and Wolverton (2000, 2005).

  17. 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. 18.

    Recall from Lemma 1 that there is also no role for the consumption tax to fine-tune the budget.

  19. 19.

    This view may change as countries start to auction off rather than give away these permits for free with “grandfathering” schemes.

  20. 20.

    Clearly, first best only obtains when the transfer is not financed by a distortionary consumption tax.

  21. 21.

    Our result that feed-in subsidy and consumer taxes are redundant would however still remain valid.

References

  1. Acemoglu D, Aghion P, Bursztyn L, Hemous D (2012) The environment and directed technical change. Am Econ Rev 102:131–166

    Article  Google Scholar 

  2. Acemoglu D, Hanley D, Akcigit U, Kerr W (2016) Transition to clean technology. J Polit Econ 124:52–104

  3. Ambec S, Crampes C (2012) Electricity provision with intermittent sources of energy. Resour energy Econ 34:319–336

    Article  Google Scholar 

  4. Ambec S, Crampes C (2015) Decarbonizing electricity generation with intermittent sources of energy. TSE Working Paper, No. 15–603

  5. Bennear LS, Stavins RN (2007) Second-best theory and the use of multiple policy instruments. Environ Resour Econ 37:111–129

    Article  Google Scholar 

  6. Bläsi A, Requate T (2010) Feed-in-tariffs for electricity from renewable energy resources to move down the learning curve? Public Finance Manag 10:213–250

    Google Scholar 

  7. Boomsma TK, Meade N, Fleten S-E (2012) Renewable energy investments under different support schemes: a real options approach. Eur J Oper Res 220:225–237

    Article  Google Scholar 

  8. Burrows P (1979) Pigouvian taxes, polluter subsidies, regulation, and the size of a polluting industry. Can J Econ 12:494–501

    Article  Google Scholar 

  9. Carlton D, Loury G (1980) The limitations of Pigouvian taxes as a long-run remedy for externalities. Q J Econ 45:559–566

    Article  Google Scholar 

  10. Carlton D, Loury G (1986) The limitations of Pigouvian taxes as a long-run remedy for externalities: an extension of results. Q J Econ 101:631–634

    Article  Google Scholar 

  11. Collinge RA, Oates WE (1982) Efficiency in pollution control in the short and long runs: a system of rental emission permits. Can J Econ 15:346–354

    Article  Google Scholar 

  12. COM (2008) European Commission staff working document 57, European Commission, Brussels. 23 January 2008. http://iet.jrc.ec.europa.eu/remea/system/tdf/files/documents/sec_2008_57_support_res_electricity.pdf?file=1&type=node&id=86

  13. COM (2013) Draft guidelines on environmental and energy aid for 2014–2020. European Commission, December 2013. http://ec.europa.eu/competition/consultations/2013_state_aid_environment/index_en.html

  14. COM (2014) Renewable energy—what do we want to achieve? European Commission. Retrieved Dec 2015. http://ec.europa.eu/energy/renewables/index_en.htm

  15. Currier K (2015) Some implications of investment cost reduction policies in energy markets employing green certificate systems. Environ Resour Econ 60:317–323

    Article  Google Scholar 

  16. Eichner T, Pethig R (2015) Efficient management of insecure fossil fuel imports through taxing domestic green energy? J Public Econ Theory 17:724–751

    Article  Google Scholar 

  17. Eichner T, Runkel M (2014) Subsidizing renewable energy under capital mobility. J Public Econ 117:50–59

    Article  Google Scholar 

  18. European Wind Energy Association (2009) The economics of wind energy. Retrieved Dec 2015. http://www.ewea.org

  19. Fischer C, Newell RG (2008) Environmental and technology policies for climate change mitigation. J Environ Econ Manag 55:142–162

    Article  Google Scholar 

  20. Fullerton D, Wolverton A (2000) Two generalizations of a deposit–refund systems. Am Econ Rev 90:238–242

    Article  Google Scholar 

  21. Fullerton D, Wolverton A (2005) The two-part instrument in a second-best world. J Public Econ 89:1961–1975

    Article  Google Scholar 

  22. Garcia A, Alzate JM, Barrera J (2012) Regulatory design and incentives for renewable energy. J Regul Econ 41:315–336

    Article  Google Scholar 

  23. Gillis J (2015) A path for climate change, Beyond Paris. New York Times, 1 December 2015. Retrieved Dec 2015. http://www.nytimes.com/2015/12/01/science/beyond-paris-climate-change-talks.html

  24. Goulder LH, Parry IWH (2008) Instrument choice in environmental policy. Rev Environ Econ Policy 2:152–174

    Article  Google Scholar 

  25. Grafton Q, Kompas T, Van Long N (2012) Substitution between biofuels and fossil fuels: is there a green paradox? J Environ Econ Manag 64:328–341

    Article  Google Scholar 

  26. Helm C, Schöttner A (2008) Subsidizing technological innovations in the presence of R&D spillovers. Ger Econ Rev 9:339–353

    Article  Google Scholar 

  27. Hintermann B, Lange A (2013) Learning abatement costs: on the dynamics of the optimal regulation of experience goods. J Environ Econ Manag 66:625–638

    Article  Google Scholar 

  28. International Energy Agency (IEA) (2008) Deploying renewables: principles for effective policies. ISBN:978-92-64-04220-9

  29. International Energy Agency (IEA) (2014) Global renewable energy. Retrieved Dec 2015. http://www.iea.org/policiesandmeasures/renewableenergy/

  30. Jaffe AB, Newell RG, Stavins RN (2005) A tale of two market failures: technology and environmental policy. Ecol Econ 54:164–174

    Article  Google Scholar 

  31. Johnstone N, Hascic I, Popp D (2010) Renewable energy policies and technological innovation: evidence based on patent counts. Environ Resour Econ 45:133–155

    Article  Google Scholar 

  32. Katsoulacos Y, Xepapadeas A (1995) Environmental policy under oligopoly with endogenous market structure. Scand J Econ 97:411–420

    Article  Google Scholar 

  33. Kohn RE (1994) Do we need the entry-exit condition on polluting firms? J Environ Econ Manag 27:92–97

    Article  Google Scholar 

  34. Lehmann P (2013) Supplementing an emissions tax by a feed-in tariff for renewable electricity to address learning spillovers. Energy Policy 61:635–641

    Article  Google Scholar 

  35. Makowski L, Ostroy J (1995) Appropriation and efficiency: a revision of the first theorem of welfare economics. Am Econ Rev 85:808–827

    Google Scholar 

  36. Nesta L, Vona F, Nicolli F (2014) Environmental policies, competition and innovation in renewable energy. J Environ Econ Manag 67:396–411

    Article  Google Scholar 

  37. Palmer K, Walls M (1997) Optimal policies for solid waste disposal taxes, subsidies, and standards. J Public Econ 65:193–205

    Article  Google Scholar 

  38. Petrakis E, Rasmusen E, Roy S (1997) The learning curve in a competitive industry. RAND J Econ 28:248–268

    Article  Google Scholar 

  39. Reichenbach J, Requate T (2012) Subsidies for renewable energies in the presence of learning effects and market power. Resour Energy Econ 34:236–254

    Article  Google Scholar 

  40. RES (2014) Legal sources on renewable energy. Retrieved Dec 2015. http://www.res-legal.eu/en/search-by-country/

  41. Schneider SH, Goulder LH (1997) Achieving low-cost emissions targets. Nature 389:13–14

    Article  Google Scholar 

  42. Schöb R (1997) Environmental taxes and pre-existing distortions: the normalization trap. Int Tax Public finance 4:167–176

    Article  Google Scholar 

  43. Spulber DF (1985) Effluent regulation and long-run optimality. J Environ Econ Manag 12:103–116

    Article  Google Scholar 

  44. Walls M, Palmer K (2001) Upstream pollution, downstream waste disposal, and the design of comprehensive environmental policies. J Environ Econ Manag 41:94–1082

    Article  Google Scholar 

  45. Wharton (2013) Renewable energy for Japan: a post-Fukushima quest, Wharton School, University of Pennsylvania. Retrieved Dec 2015. https://knowledge.wharton.upenn.edu/article/renewable-energy-japan-post-fukushima-quest/

  46. White House (2013) Presidential memorandum—federal leadership on energy management. Retrieved Dec 2015. https://www.whitehouse.gov/the-press-office/2013/12/05/presidential-memorandum-federal-leadership-energy-management

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Correspondence to Roland Strausz.

Additional information

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

$$\begin{aligned} \Psi ^{\prime }(\tilde{x}_{d}(a)+a)=C_{d}^{\prime }(\tilde{x}_{d}(a))+E^{\prime }(\tilde{x}_{d}(a)). \end{aligned}$$
(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 \(t_{c}^{\prime }x_{c} =(t_{c}-t_{\psi }+t_{\psi }^{\prime })x_{c}=(t_{\psi }^{\prime }x_{d}/x_{c} +t_{\psi }^{\prime })x_{c}=t_{\psi }^{\prime }x_{d}+t_{\psi }^{\prime }x_{c} =t_{\psi }^{\prime }(x_{d}+x_{c})\). \(\square \)

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Antoniou, F., Strausz, R. Feed-in Subsidies, Taxation, and Inefficient Entry. Environ Resource Econ 67, 925–940 (2017). https://doi.org/10.1007/s10640-016-0012-8

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Keywords

  • Taxation
  • Feed-in tariffs
  • Externalities
  • Renewables
  • Entry
  • Pollution

JEL Classification

  • D21
  • D61
  • H23