Optimal renewable-energy promotion: Capacity subsidies vs. generation subsidies

Resource and Energy Economics 45 (2016) 144–158

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Optimal renewable-energy promotion: Capacity subsidies vs. generation subsidies夽 Mark Andor a,∗ , Achim Voss b,1 a Division of Environment and Resources, Rheinisch-Westfälisches Institut für Wirtschaftsforschung (RWI), Hohenzollernstr. 1–3, 45128 Essen, Germany b Department of Economics, University of Hamburg, Department of Economics, Von-Melle-Park 5, 20146 Hamburg, Germany

a r t i c l e

i n f o

Article history: Received 19 December 2014 Received in revised form 5 April 2016 Accepted 2 June 2016 Available online 26 June 2016 JEL classification: Q41 Q48 H23 Keywords: Peak-load pricing Capacity investment Demand uncertainty Renewable energy sources Optimal subsidies Feed-in tariffs

a b s t r a c t We derive optimal subsidization of renewable energies in electricity markets. The analysis takes into account that capacity investment must be chosen under uncertainty about demand conditions and capacity availability, and that capacity as well as electricity generation may be sources of externalities. The main result is that generation subsidies should correspond to externalities of electricity generation (e.g., greenhouse gas reductions), and investment subsidies should correspond to externalities of capacity (e.g., learning spillovers). If only capacity externalities exist, then electricity generation should not be subsidized at all. Our results suggest that some of the most popular promotion instruments cause welfare losses. We demonstrate such welfare losses with data from the German electricity market. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Over the last two decades, many governments have introduced support schemes for electricity from renewable energy sources. According to the International Energy Agency’s (IEA) World Energy Outlook 2013, $82 billion were used for these subsidies in 2012. In many countries, it is a declared political goal to further raise renewables’ market share, so the total amount of subsidies will probably rise. In the “New Policies Scenario” projection, which assumes that the governments stick with their plans and serves as the baseline scenario, the IEA expects subsidies to reach almost $180 billion per year in 2035 (IEA, 2013).

夽 Thanks to Lucas Davis, Manuel Frondel, Corina Haita, Jörg Lingens, Mark Schopf, Michael Simora, Stephan Sommer, Wolfgang Ströbele and Colin Vance for very valuable comments. We give particular thanks to Kai Flinkerbusch who co-authored an earlier working paper whose ideas we develop and extend in the current article. Friederike Blönnigen, Maja Guseva and Frank Undorf provided excellent research assistance. This work has been partly supported by the Collaborative Research Center “Statistical Modeling of Nonlinear Dynamic Processes” (SFB 823) of the German Research Foundation (DFG), within the framework of Project A3, “Dynamic Technology Modeling”. ∗ Corresponding author. Tel.: +49 201 8149 216. E-mail addresses: [email protected] (M. Andor), [email protected] (A. Voss). 1 Tel.: +49 40 42838 4529. http://dx.doi.org/10.1016/j.reseneeco.2016.06.002 0928-7655/© 2016 Elsevier B.V. All rights reserved.

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Without questioning whether this level of support is justified, the aim of this article is to derive optimal subsidy policies, so as to analyze whether currently popular promotion schemes are efficient. Our starting point is to recognize that justifications for promoting renewable energies can be divided into two categories: some justifications derive from capacity production and installation and others from electricity generation. Of the most prominent economic rationales, internalizing learning spillovers in manufacturing belongs to the former category, and second-best abatement of greenhouse gas emissions belongs to the latter.2 Likewise, renewable-energy promotion instruments can be distinguished by whether they target capacity, through for example investment tax reductions, or electricity generation, through for example feed-in tariffs, quota systems or renewable portfolio standards. Despite its centrality to welfare analysis, the economic literature has almost completely neglected the difference between capacity and electricity generation when discussing optimal promotion schemes. In this article, we analyze how the source of externalities should shape promotion and we demonstrate the implications of efficient policies for the supply behavior of renewable-energy capacity owners and for welfare. We derive optimal subsidies for electricity generation technologies, taking external benefits of capacity and of electricity generation into account, of which zero externalities of either capacity or generation are special cases. Our model uses the framework of a competitive peak-load pricing model – that is, the decision variable is the supply quantity, but in situations of high demand, supply can be limited by capacity. We are not aware of any literature that explicitly analyzes optimal subsidies in such a framework.3 The model’s unique characteristic is the separation of capacity and electricity generation as targets of subsidies. To focus on the basic principles shaping optimal subsidies, we assume that there is only one moment in which electricity generation and consumption take place and that there is only one electricity generation technology. Additionally, we assume that the government has full information and it has access to non-distortive means of financing the subsidies (i.e., lump-sum taxes). While we think that neither of these assumptions changes the general insights of the model, we recognize that the full implications of these subsidies in a dynamic multi-technology market would have to be modeled explicitly. We find that marginal subsidies for electricity generation should equal its marginal external benefits, and marginal subsidies for capacity should cover marginal external benefits of capacity. This rule allows derivation of a number of policyrelevant conclusions. For example, it implies that if there are only externalities of capacity (for instance, knowledge spillover effects of manufacturing photovoltaic modules), then electricity generation should not be subsidized at all. Furthermore, only under very specific circumstances do optimal promotion schemes for renewable energy resemble the demand-independent “fixed feed-in tariffs” that are popular in many countries. While there is a large literature on renewable-energy promotion, as far as we know, only Bläsi and Requate (2010) and Reichenbach and Requate (2012) consider the implications of distinguishing between capacity and electricity generation. In these papers, learning spillover effects of capacity production are taken into account, making output subsidies for renewableenergy capacity producers optimal. However, in these models, electricity demand is deterministic and “capacity” is the number of firms, so that the distinction between capacity and electricity generation requires increasing marginal generation costs. By contrast, we model capacity as an explicit limit to electricity generation, which also allows to incorporate the case of constant marginal generation cost. In particular, this includes technologies like wind and solar power for which zero generation costs can be assumed. Moreover, our model takes into account that at the moment of investment, demand and capacity availability are uncertain. Implicitly, Newbery (2012) also distinguishes the different sources of positive externalities by stating that capacity rather than electricity generation should be promoted for the case of wind energy. We analyze this point in a general way using a formal model. The paper proceeds as follows. We describe the model setting in Section 2.1, derive a social planner’s solution in Section 2.2, and a decentralized solution in Section 2.3. Section 2.4 defines the optimal subsidies. In Section 3, we use our framework to assess the welfare effects of the widespread policy instrument of fixed feed-in tariffs. A basic assessment of empirical welfare losses is provided in Section 4. Finally, Section 5 discusses the results.

2 The learning-spillover argument states that productivity gains from learning how to produce renewable-energy capacity partly have a public-good character, so that too little renewables capacity, such as wind turbines, is built. The second-best abatement argument is based on the idea that, for whatever reasons, carbon prices cannot be set high enough to be optimal so that renewable energy sources should be subsidized to replace fossil-fuel electricity. See, for example, Rasmussen (2001), Jaffe et al. (2005), Bennear and Stavins (2007), Kverndokk and Rosendahl (2007), Fischer and Newell (2008), Helm and Schöttner (2008), Gerlagh et al. (2009), Kalkuhl et al. (2013). Other alleged reasons to promote renewable energies are fostering employment and increasing the security of energy supply. The employment objective can be interpreted as strategic trade policy (cf. Spencer and Brander, 2008), implying that manufacturing firms are subsidized to oust foreign competition. According to the energy security rationale (which includes reducing the dependence on foreign energy supply), the government should ensure that a high electricity generation capacity is not only produced but kept ready. Thus, both reasons hinge on capacity. For a comprehensive review of renewables-support rationales and instruments, see Fischer and Preonas (2010). For this article, it is irrelevant whether such policy objectives are sensible from an economic point of view; our model is independent of the exact interpretation of externalities. 3 For an excellent survey of the theory of peak-load pricing, see Crew et al. (1995). For current applications of this model framework to electricity markets, see for instance, Borenstein and Holland (2005) or Joskow and Tirole (2007). Chao (2011) models renewable energy sources in a market with multiple intermittent electricity generation technologies. The focus is on socially optimal investment and different pricing schemes; externalities or their internalization are not part of the analysis.

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2. Model 2.1. The market environment We consider a partial-equilibrium model of a market in which a good in the amount of q is traded. We assume that this good is electricity generated by some renewable energy source like biofuel, wind or photovoltaics.4 Supply is limited by available capacity ak, where capacity is denoted by k and the capacity availability variable 0 ≤ a ≤ 1 takes into account that at least some renewable energy technologies, in particular wind and solar energy, are not always completely available owing to intermittency. Thus, a = 0 means that built capacity cannot be used, and a = 1 implies complete availability. The model consists of two stages; the market stage follows the investment stage. In the investment stage, capacity k is built while the state of electricity demand, z, and the availability of capacity, a, are unknown. Thus, we have a setting of peak-load pricing under supply and demand uncertainty. In the market stage, a and z are drawn from a random distribution F(a, z), with a continuous density function f(a, z). z is bounded between zl and zh , where zh > zl . Thus, zh is the highest and zl is the lowest state of demand. The price p is determined by electricity generation q and an inverse demand function p ≡ p(q, z).5 For higher values of z, the price is higher for every quantity q, that is, the demand curve is shifted upwards and therefore, ∂p(q, z)/∂z > 0. There are two kinds of costs. Capacity k is built in the investment stage with an investment cost function C(k). Generation costs of electricity are c(q) and accrue in the market stage. Generation is limited by available capacity, q ≤ ak. In both stages, there may be positive externalities. Capacity externalities are denoted by B(k), and electricity generation externalities are b(q); they are measured in money terms. Both are assumed to be concave functions, so they have non-increasing marginal effects. B(k) may, for example, represent learning spillovers, and b(q) may be the carbon abatement gained by replacing fossil-fuel electricity.6 However, these interpretations are just examples, while the implications of the model are general. For instance, the implications also hold if there are no externalities from electricity generation, but only from capacity investment. This is discussed more concretely in Sections 4 and 5. In the second stage, consumer surplus v and producer surplus s are given by

 v(q, z) ≡

q

p(˜q, z)d˜q − p(q, z)q,

(1)

0

s(q, z) ≡ p(q, z)q − c(q).

(2)

The total market-stage welfare w is the sum of consumer surplus v, producer surplus s, and external benefits b of electricity generation:

 w(q, z) ≡ v(q, z) + s(q, z) + b(q) =

q

p(˜q, z)d˜q − c(q) + b(q).

(3)

0

Letting E denote expectations, total expected welfare reads

 W ≡ E[w(q(a, z), z)] + B(k) − C(k) = zl

zh



1

f (a, z)w(q(a, z), z) da dz + B(k) − C(k),

(4)

0

where q(a, z) is the electricity generated in state (a, z). In this market, capacity investment takes place before demand and capacity availability are known, which in turn determine optimal generation. An alternative would be to think of the different (a, z) realizations as sub-periods of the investment horizon, and f(a, z) then describes how often a certain (a, z) combination takes place. What is important here is that – independent of the concrete interpretation – the model allows us to distinguish between capacity and supply, and thus to analyze the incentives of different promotion schemes on the capacity investment and electricity supply decisions.

4 The model is fairly general. It should be applicable to any sector in which a distinction between capacity and its usage is relevant and either of these can be the source of externalities. An important example is agriculture. 5 We assume that the functions have the typical characteristics, like a negative slope of the demand function, a non-negative slope of the marginal electricity generation costs function etc., and mention those that we deem important for clarification. 6 Carbon abatement by replacing fossil-fuel electricity is not a direct positive externality, but a reduction of a negative externality from a different source. In our model, we would understand carbon abatement as a positive externality because it indeed represents benefits that are external to renewable-energy capacity operators. However, the size of these external benefits depends on the extent to which renewable-energy electricity replaces fossil-fuel electricity and, thus, on the price elasticity of demand.

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2.2. Socially optimal electricity generation and investment decisions In this section we derive how a social planner, who can directly choose an optimal allocation of investment and generation, would maximize welfare. We maximize the total expected welfare, (4), by maximizing the following Lagrangian:

 Lw = zl

zh





1

f (a, z)w(q(a, z), z) da dz + B(k) − C(k) + 0

zh



1

w (a, z)[ak − q(a, z)] da dz.

(5)

0

zl

Capacity k is chosen before the state (a, z) is realized. Electricity generation q is chosen afterwards.7 Thus, the optimal program consists of a supply rule q(a, z) so as to maximize welfare given a, z and capacity k, and of an optimal investment decision. The optimal investment maximizes expected welfare, anticipating that q will be chosen optimally in state (a, z). w (a, z) is the Kuhn–Tucker multiplier for the capacity constraint in state (a, z), and thus the shadow price of capacity in that state. Generation cannot be negative, q ≥ 0. Taking this non-negativity condition into account, the first-order conditions for the optimal electricity generation q in each state (a, z) are8 :

∂Lw ∂q(a, z)

= f (a, z)

∂w(q∗ (a, z), z) − ∗w (a, z) ≤ 0, ∂q

q∗ (a, z) ≥ 0,

q∗ (a, z)

∂Lw = 0. ∂q(a, z)

(6a)

The first-order conditions for the optimal choice of capacity are:

∂Lw = B (k∗ ) − C  (k∗ ) + ∂k



zh

zl



1

a∗w (a, z) da dz ≤ 0,

k∗ ≥ 0,

0

k∗

∂Lw = 0, ∂k

(6b)

and, finally, the Kuhn–Tucker conditions for each state,

∂Lw = ak∗ − q∗ (a, z) ≥ 0, ∂w (a, z)

∗w (a, z) ≥ 0,

∗w (a, z)

∂Lw = 0, ∂w (a, z)

(6c)

represent the capacity constraint. The asterisks denote (social) optimality of the choice variables q, k, and w . Firstly, we can derive an optimal supply rule. For this, we can distinguish three cases, consisting of combinations of the state of demand and available capacity. Depending on the level of demand, it may either be optimal to generate at the capacity limit, q = ak, or to generate a positive amount below the capacity limit, 0 < q < ak, or to generate nothing at all, q = 0. In the following, we derive the conditions for the respective decision to be optimal. In doing so, we assume that k* > 0 to simplify the discussion. In the first case, using all available capacity is optimal, q* (a, z) = ak > 0. By condition (6a), this implies ∗w (a, z) = f (a, z)

∂w(ak, z) , ∂q

(7)

which must be non-negative by (6c), i.e., ∂w(ak, z)/∂q ≥ 0. Thus, all available capacity is used if marginal welfare of generation is still non-negative at the capacity limit, and the shadow price of capacity for the respective state equals marginal welfare weighted by the state’s density. Marginal welfare of generation is

∂w(q, z) = p(q, z) − c  (q) + b (q), ∂q

(8)

so q* (a, z) = ak would imply that the price of electricity (that is, the marginal gross consumer surplus) and marginal externalities of generation at the capacity limit together are at least as high as marginal generation costs. In the second case, a positive generation below the capacity limit is optimal, ak > q* (a, z) > 0. By conditions (6a) and (6c), ∗w (a, z) = f (a, z)

∂w(q∗ (a, z), z) = 0. ∂q

(9)

Thus, generation is stopped below the capacity limit if marginal welfare is zero, and, hence, welfare cannot be increased by extending generation. In other words, if capacity is not scarce, its shadow price is zero. The third and final case is that in which zero generation is optimal. Again, (6c) tells us that the shadow price of capacity in such states is zero. By (6a), we can see that zero generation is optimal if even the first unit of electricity does not yield positive marginal welfare.

7 Equivalently, we can say that in the investment period, a schedule q(a, z) is chosen so that optimal generation is determined conditional on the state of demand and capacity availability. 8 For the methods of non-linear optimization see, for example, Chiang and Wainwright (2005).

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Summarizing, we have the following optimal supply rule: Proposition 1 (Optimal supply). The welfare-maximizing supply and the implied shadow value of (available) capacity are given as follows: Condition

(a)

∂w(ak,z) ≥0 ∂q ∂w(q∗ (a,z),z) = ∂q ∂w(0,z) ≤ 0 ∂q

(b) (c)

Implied ∗w (a, z) =

q* (a, z) =

Case

f (a, z) ∂w(ak,z) ∂q

ak 0 for 0 < q* (a, z) < ak

Defined implicitly by 0

∂w(q∗ (a,z),z) ∂q

=0

0 0

In the following, we just denote the optimal supply quantity and the optimal shadow price by q∗ , ∗w in order to economize on notation. We can determine the impact of changes in the state of demand by differentiating the optimality condition for state (a, z) and rearranging:

⎧

2

∂p(q∗ , z)/∂z ⎨ ∂ w(q∗ , z)/(∂q∂z) = ∂q∗ 2 2 = ∗ 2 −∂ w(q∗ , z)/∂q2 ∂z ⎩ −∂ w(q , z)/∂q 0

if q∗ (a, z) ∈ (0, ak),

(10)

else,

where the first case derives from differentiating the first-order condition (9). The denominator in (10) is positive due to the usual assumptions about cost and benefit functions. Thus, the higher the demand, the more electricity should be generated as long as the capacity limit is not reached. Similarly,

∂q∗ = ∂a



k

if q∗ (a, z) = ak,

0

else.

(11)

This means that higher capacity availability raises electricity generation if capacity is the relevant limit. By contrast, if generation below the capacity limit is optimal, then an increase in available capacity does not change optimal generation. We complete the socially optimal allocation by deriving the optimal investment rule. The first-order conditions for optimal capacity choice, (6b), can be summarized as follows: Proposition 2 (Optimal investment). Capacity is chosen so that marginal investment costs equal the sum of its direct marginal externalities and the sum of shadow prices of available capacity in all states, weighted by the share of capacity that is available,

 C  (k∗ ) = B (k∗ ) + zl

zh



1

a∗w (a, z) da dz,

(12)

0

given that there is a non-negative amount of capacity for which this equation is fulfilled. Otherwise, zero capacity is built. Proof.

Evaluate (6b) for k* > 0. 

From the discussion above, we know that the shadow price of capacity is positive only in those states in which available capacity limits generation: q* = ak. Eq. (12) shows that marginal investment costs must pay off in these states. We can see in (7) and (9) that the sum of shadow values is the expected marginal welfare contribution of capacity. Moreover, if only a fraction a of capacity k is available in state (a, z), the shadow price is weighted by a – low availability makes it more expensive to have available capacity. For example, photovoltaic capacity is less valuable if sunshine is rare. Furthermore, states of high demand justify more capacity investment if it is probable that they actually occur. This follows from the fact that, by (7) and (9), the shadow price for a state equals marginal welfare weighted by the density of that state.9 Note two implications of the welfare-maximizing allocation. Firstly, the larger capacity externalities B(k), the more capacity is built, and the more states (a, z) in which there is idle capacity. If there are very large benefits of building capacity, it may even be true that full capacity should never be used. This can be seen by recognizing that (12) could be fulfilled even if the shadow price is zero for all states. Marginal capacity externalities B (k* ) must then equal marginal investment costs C (k* ). Secondly, based on the characterization of optimal supply in Proposition 1, we can briefly describe which market price will be observed in state (a, z): Corollary 1 (Optimal market price of electricity). The observed prices in optimum can be characterized as follows: (i) Suppose that the optimal supply is given by case a) of Proposition 1, so that q* = ak. Then it holds in optimum that p(ak, z) ≥ c (ak) − b (ak).

9 The density f(a, z) does not influence the supply decision in (6a) and so does not change the scarcity of given capacity in state (a, z). The density function instead changes the expected value of marginal capacity – the shadow price – which influences optimal investment in (6b) and (12). Thus, it changes expected scarcity given optimal investment.

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∗ (ii) Suppose that the optimal supply is given by case b) of Proposition 1, so that 0 ≤ q* ≤ ak is defined implicitly by ∂w(q ,z) = 0.

∂q

Then it holds in optimum that p(q* , z) = c (q* ) − b (q* ). (iii) Suppose that the optimal supply is given by case c) of Proposition 1, so that q* = 0. Then it holds in optimum that p(0, z) ≤ c (0) − b (0). Thus, it can be optimal that a positive amount of electricity is supplied even though the optimal electricity price is below marginal generation costs, but only if there are positive externalities from electricity generation (Cases i–ii). Consequently, it can even be optimal that electricity is generated while the price is negative (see the discussion in Section 4). However, if there are no externalities from generation, positive supply requires that the price is at least as high as marginal generation costs. 2.3. Decentralized allocation We now consider the behavior of profit-maximizing firms under perfect competition. The aim is to derive how much they invest and generate, given a certain structure of subsidies, so that we can later derive optimal subsidies. More precisely, we consider the possibility of a generation subsidy (q, a, z) as well as a capacity subsidy (k). Thus, we can define the firms’ second-stage profits by (q, a, z) = p(a, z)q + (q, a, z) − c(q),

(13)

which differs from producer surplus (2) by the generation subsidy . The latter is a payment from the government to the firms, which may depend on the generated quantity, the state of demand, and the availability of capacity – this allows derivation of whether it is actually optimal to condition subsidies on these variables. A proxy for conditioning subsidies on the state of demand z or the availability of capacity a would be to say that subsidies are higher or lower during different times of the day or year, or to condition them on weather conditions. We write p(a, z) instead of p(q, z) because firms take the price in state (a, z) as given; they are assumed to behave competitively and thus do not take their influence on the price into account. Total expected profit is given by

  ≡ E[(q(a, z), a, z)] + (k) − C(k) = zl

zh



1

f (a, z)(q(a, z), a, z) da dz + (k) − C(k),

(14)

0

where (k) is a subsidy for capacity installation. We assume for (q, a, z) and (k) that they are continuous, concave functions of q and k, respectively – that is, marginal subsidies are constant or decreasing.10 The investors’ Lagrangian is

 L = zl

zh





1

f (a, z)(q(a, z), a, z) da dz + (k) − C(k) +

zh

zl

0



1

 (a, z)[ak − q(a, z)] da dz,

(15)

0

and the first-order conditions are

∂L ∂(q# (a, z), a, z) = f (a, z) − #  (a, z) ≤ 0, ∂q(a, z) ∂q ∂L =0 q# (a, z) ∂q(a, z)

q# (a, z) ≥ 0, (16a)

for profit-maximizing electricity generation in each state (a, z),

∂L =   (k# ) − C  (k# ) + ∂k



zh



zl

1

a#  (a, z) da dz ≤ 0,

k# ≥ 0,

0

k#

∂L =0 ∂k

(16b)

for the firms’ choice of capacity, and, finally, the Kuhn–Tucker conditions of each state (a, z),

∂L = ak# − q# (a, z) ≥ 0, ∂ (a, z)

#  (a, z) ≥ 0,

#  (a, z)

∂L = 0, ∂ (a, z)

(16c)

10 This assumption includes practically all realistically conceivable generation-dependent subsidies that are paid in addition to the electricity price. Receiving a fix subsidy for every installed Watt of photovoltaics is consistent with this and so is a fixed subsidy per unit of electricity, even if its level depends on a and z. We allow for strict concavity to keep the analysis general. For instance, US tax credits for wind energy may be granted for only a limited amount of capacity, like wind energy with a nameplate generating capacity of at most 100 kW (IRC §48, c.4.B). This leads to a concave function if it is possible to split a planned wind park.

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where # denotes profit maximization. The interpretation is exactly the same as in the case of the social planner’s optimality conditions (6), except that marginal profit takes the place of marginal welfare. For equilibrium generation q# (a, z), marginal profit is given by

∂(q# (a, z), a, z) ∂(q# (a, z), a, z) = p(q# (a, z), z) + − c  (q# (a, z)). ∂q ∂q

(17)

We can solve for the firms’ supply behavior along the lines of solving (6), and summarize: Proposition 3 (Profit-maximizing supply). The profit-maximizing supply and the implied shadow value of (available) capacity are given as follows: Condition

(a)

∂(ak,a,z) ≥0 ∂q ∂(q# (a,z),a,z) = ∂q ∂(0,a,z) ≤ 0 ∂q

(b) (c)

Implied # (a, z) =

q# (a, z) =

Case

f (a, z) ∂(ak,a,z) ∂q

ak 0 for 0 < q# (a, z) < ak

Defined implicitly by 0

∂(q# (a,z),a,z) ∂q

=0

0 0

The supply rule says that the firms use all available capacity, q# = ak, if the market’s willingness to pay for electricity plus the marginal generation subsidy is at least as large as marginal generation costs at the capacity limit. A positive amount of electricity below the capacity limit, ak > q# > 0, is generated if this sum is zero, and none is generated if even the first generated unit of electricity does not yield a profit. Likewise, we can solve (16b) for the firms’ investment rule and summarize: Proposition 4 (Profit-maximizing investment). Capacity is chosen so that marginal investment costs equal the sum of marginal capacity subsidies and the sum of shadow prices of available capacity in all states, weighted by the share of capacity that is available,



C  (k# ) =   (k# ) +

zh



1

a#  (a, z) da dz,

(18)

0

zl

given that there is a non-negative amount of capacity for which this equation is fulfilled. Else, zero capacity is built. Thus, capacity subsidies imply that more capacity is built, which means that capacity will be scarce in fewer states. Electricity generation subsidies may in turn also make capacity more valuable: a higher marginal subsidy in a state (a, z) in which capacity is scarce leads to higher marginal profit at the capacity limit and thus a higher shadow value of capacity – see (14) and (16a). This in turn implies that a larger amount of capacity is chosen – see (16b). The difference is that electricity generation subsidies imply a change of electricity supply behavior. 2.4. Decentralizing the optimal allocation with subsidies A welfare-maximizing government can use subsidies to reproduce the optimal allocation. Firms choose the socially optimal allocation if the following conditions are met. Proposition 5 (Optimal subsidies: necessary conditions). (i) Consider the states (a, z) for which the optimal supply rule from Proposition 1 implies using all available capacity, q* = ak. For these states it must hold that p(ak, z) +

∂(ak, a, z) − c  (ak) ≥ 0. ∂q

(19)

(ii) Consider the states (a, z) for which the optimal supply rule from Proposition 1 yields an interior solution, q* ∈ (0, ak). For these states it must hold that

∂(q∗ , a, z) − b (q∗ ) = 0. ∂q

(20)

(iii) Consider the states (a, z) for which the optimal supply rule from Proposition 1 yields zero generation, q* = 0. For these states it must hold that p(0, z) +

∂(0, a, z) − c  (0) ≤ 0. ∂q

(21)

(iv) For firms’ investment (as given in Proposition 4) to be socially optimal as described in Proposition 2, it must hold that



B (k∗ ) −   (k∗ ) + zl

zh



1

a[∗w (a, z) − #  (a, z)] da dz = 0. 0

(22)

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Proof. We start with the inner solution, case ii. For a positive generation below capacity, q = q* < ak, to be both socially optimal and profit-maximizing, we need p(q∗ , z) − c  (q∗ ) + b (q∗ ) = 0, p(q∗ , z) +

∂(q∗ , a, z) − c  (q∗ ) = 0 ∂q

from (6) and (8), (16) and (17). Solving these equations, we see that (20) must hold. For i and iii, the exact level of generation subsidies is not important because firms cannot use more than all available capacity or less than none – see the firms’ optimal supply rule in Proposition 3. (iv) is implied by equating (12) and (18) and rearranging.  In particular, the following subsidy scheme fulfills these necessary conditions11 : Proposition 6 (Optimal subsidies: sufficient conditions). The subsidy scheme (consisting of generation subsidies and capacity subsidies) is optimal if for all states (a, z)

∂(q∗ , a, z) = b (q∗ ), ∂q

(23)

  (k∗ ) = B (k∗ ).

(24)

and if

Proof. If the shadow prices in the first-order condition sets of welfare maximization, (6), and profit maximization, (16), are identical for the same capacity level, then generation decisions must be identical as well. This, in turn, is true if marginal welfare (8) and marginal profit (17) of q* coincide for the same available capacity. Substituting (23) in these equations shows that they always do. (22) then implies (24).  We can summarize this as a rule for optimal subsidies. Firstly, marginal generation subsidies should equal marginal external benefits of generation in every state of demand and capacity availability. Secondly, marginal capacity subsidies should equal marginal external benefits of capacity. In the following, we use the optimality conditions derived in this section to assess one popular support scheme, namely, fixed feed-in tariffs. In Appendix A.1, we show in a more general way what the rule in Proposition 6 implies for the dependence of optimal subsidies on the state and discuss how such a conditioning could take place. 3. Fixed feed-in tariffs as a renewable-energy promotion instrument One of the most widely applied instruments to promote renewable energies is a fixed feed-in tariff (cf. IEA/IRENA, 2014). With such a system, an operator of renewable-energy capacity receives a certain amount of money for every unit of electricity that is generated and fed into the grid. In this section, we analyze this policy instrument. To do so, we first derive how private operators react to a fixed feed-in tariff. Afterwards, we analyze under which conditions such a tariff can be used to implement an optimal policy. If the instrument to promote renewables is a fixed tariff  per unit of electricity, then the tariff  itself does not depend on the generated quantity q. The firms’ second-stage profit (13) becomes: (q) = q − c(q).

(25)

The firms’ Lagrangian is



L = zl

zh





1

f (a, z)[q(a, z) − c(q(a, z))] da dz − C(k) + zl

0

zh



1

 (a, z)[ak − q(a, z)] da dz,

(26)

0

so that the first-order conditions corresponding to (16) are

∂L (a, z) ≤ 0, = f (a, z)[ − c  (q# (a, z))] − #  ∂q(a, z)

q# (a, z) ≥ 0,

q# (a, z)

∂L =0 ∂q(a, z)

(27a)

for profit-maximizing electricity generation in each state (a, z),

∂L = −C  (k# ) + ∂k



zl

zh



0

1

a# (a, z) da dz ≤ 0, 

k# ≥ 0,

k#

∂L =0 ∂k

(27b)

11 There can be other subsidy schemes fulfilling the necessary conditions. Yet, these schemes are complex and, presumably, unrealistic for a real-world application. For example, for states in which all capacity is to be used, subsidies can be arbitrarily high as long as (19) is fulfilled. However, this implies that the shadow price of capacity for the firms exceeds the social shadow price. To still induce optimal investment, (22) then implies that marginal capacity subsidies must be lower than marginal capacity externalities to counterbalance this difference.

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for the firms’ choice of capacity, and, finally, the Kuhn–Tucker conditions of each state (a, z),

∂L = ak# − q# (a, z) ≥ 0, ∂ (a, z)

# (a, z) ≥ 0, 

# (a, z) 

∂L = 0. ∂ (a, z)

(27c)

The following proposition summarizes what this optimization implies for the ability of a fixed feed-in tariff to implement socially optimal electricity generation and investment. Proposition 7 (Optimality of fixed feed-in tariffs). For k* > 0, fixed feed-in tariffs can implement optimal generation and investment as summarized in Propositions 1 and 2 if and only if q* (a, z) = ak* for all a. Proof. To prove the proposition, we first demonstrate that it is impossible to impose a fixed feed-in tariff that induces optimal generation q* (a, z) if there is at least one state in which generation below the capacity limit would be optimal. Afterwards, we show that full capacity utilization can be induced. Suppose that there is a state (a0 , z0 ) for which q* (a0 , z0 ) < a0 k* . To induce optimal generation, it follows from (27c) that #  (a0 , z0 ) = 0. If marginal generation costs are constant, (27a) then implies that the fixed feed-in tariff has to be at most

equal to marginal cost. By (27a) and (27c), for  ≤ c (q# (a, z)), # (a, z) would be zero for all states (a, z), which is, by (27b), 

incompatible with positive capacity. By contrast, if  exceeds marginal generation cost, (27a) and (27c) imply that we have (a, z) for all (a, z). All # (a, z) then increase in , so that the optimal amount of capacity full usage of capacity and a positive #   can be induced by varying the fixed feed-in tariff. Now suppose that marginal generation costs are increasing. Inducing q* (a0 , z0 ) < a0 k* again requires # (a , z ) = 0, so  0 0

that the tariff must fulfill  = c (q* (a0 , z0 )) for an interior solution. By the continuity of f(a, z), if q* (a0 , z0 ) ∈ (0, a0 k* ) is optimal, there must be other states (a1 , z0 ) for which q* (a1 , z0 ) would be interior as well. Given that there is only one fixed feed-in tariff ,  = c (q) cannot be fulfilled for more than one value of q, which makes it impossible that a fixed feed-in tariff is optimal in this case. Finally, if q* (a0 , z0 ) = 0 < ak* is optimal, this can be induced by  ≤ c (0); however, by increasing marginal costs, this would again imply # (a, z) = 0 in all states and thus imply zero capacity. By contrast, for  ≥ c (k* ), we have full  capacity utilization in all states, which by (27b) induces a positive amount of capacity. Again, varying the fixed feed-in tariff induces different amounts of capacity.  Thus, fixed feed-in tariffs can only be optimal if it is always welfare-maximizing to use all available capacity; this in turn will be the case if the optimal amount of capacity is not too large. In the case of constant marginal costs, the intuition for this result is that a fixed feed-in tariff can only induce positive generation if the tariff exceeds marginal costs; but then, generating at the capacity limit always pays off. In the case of increasing marginal cost, fixed feed-in tariffs can only induce one interior amount of generation. This cannot be optimal if there is more than one potential state of demand in which generation below the capacity limit is optimal. This in turn will be the case for the continuous demand-state distribution which we have assumed and which represents the fluctuations in real-world electricity markets. Hence, a fixed feed-in tariff will almost never induce the correct electricity generation if capacity should react to demand. Using (27a) for full capacity utilization in (27b), the optimal  must fulfill



zh

C  (k∗ ) = zl



1

af (a, z)[ − c  (ak∗ )] da dz,

(28)

0

or, using (8) and Propositions 1 and 2 and simplifying,



zh



zl



1

af (a, z) da dz = B (k∗ ) +

 0

zl

zh



1

af (a, z)[p(ak∗ , z) + b (ak∗ )] da dz.

(29)

0

Thus, if there are no capacity externalities,  must on average – and taking availability into account – equal the marginal social benefit of electricity generation. If capacity has positive externalities,  must be larger. Note, however, that if capacity externalities are very large relative to costs, then by (6) it would be welfare-maximizing to expand capacity so much that it should sometimes be left idle. But as we have seen above, fixed feed-in tariffs cannot induce optimal capacity usage in this case. The bottom line is that fixed feed-in tariffs are equivalent to the generation and capacity subsidy schemes analyzed in Section 2.4 if and only if the welfare-maximizing capacity level k* is so small that it is always optimal to generate electricity at the capacity limit. By contrast, inducing a large capacity investment with a fixed feed-in tariff will imply welfare losses. We can analyze these welfare losses more generally by comparing the welfare levels that the promotion schemes provide for any level of capacity, not just for optimal levels. This is relevant, for example, if capacity is politically determined. To do so, we compare the expected welfare for any amount of capacity k for the two different promotion schemes.

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153

We first consider the social-planner solution of Section 2.2. We can use the first-order conditions of (6) without (6b) and thereby evaluate expected welfare for an amount of capacity k while maintaining the assumption that this amount is used optimally. We thus write total expected welfare (4) as a function of k, W(k). Marginal expected welfare then is



zh

W  (k) =



1

a∗w (a, z) da dz + B (k) − C  (k).

(30)

0

zl

Now consider fixed feed-in tariffs. For simplicity, we focus on the case of constant marginal generation costs, for which fixed feed-in tariffs will always induce full utilization of all available capacity. Expected welfare then is



W (k) =

zh



1

f (a, z)w(ak, z) da dz + B(k) − C(k).

zl

(31)

0

Substituting (8), marginal welfare of capacity is



W (k) =

zh



zl

1

af (a, z)(p(ak, z) − c  (ak) + b (ak)) da dz + B (k) − C  (k).

(32)

0

Comparing the two derivatives, the marginal welfare loss of using a fixed feed-in tariff (or of any instrument that always induces full capacity utilization) is



W



(k) − W (k)

= zl

zh



1

a[∗w (a, z) − f (a, z)(p(ak, z) − c  (ak) + b (ak))] da dz,

(33)

0

The shadow price ∗w (a, z) equals marginal welfare whenever using all available capacity is optimal, so that the difference in square brackets vanishes for these states. In all other states, we have ∗w (a, z) = 0. The loss in expected welfare of the marginal unit of capacity thus equals the average excess of marginal costs over price and marginal externality at the capacity limit, weighted by availability, because this unit of capacity would be used whenever it is available. A further conclusion from these considerations can be drawn for the optimal amount of installed capacity under different promotion schemes. The amount of capacity that a social planner would choose with optimal subsidies will be at least as high with optimal subsidies as it would be if he were constrained to use fixed feed-in tariffs. This is because the marginal expected welfare of capacity with an optimal promotion scheme is at least as high as it is with a fixed feed-in tariff system, and possibly higher. Thus, the amount of capacity for which (30) equals zero is never lower than the amount of capacity for which (32) equals zero. 4. Identification of welfare losses We have theoretically derived that the widely applied system of fixed feed-in tariffs provides wrong incentives if capacity is so large that it is not always optimal to use all of it. Now we apply our insights to a real-world electricity market that applies a fixed feed-in tariff system. To estimate the total welfare loss, we would need a simulation of the market and of counterfactual market outcomes. As this is beyond the scope of this article, we focus on identifying welfare losses instead of quantifying them. More concretely, we use electricity prices and supplied quantities to identify times in which welfare losses occur, and to quantify them at the margin. These data are publicly available. Due to its large amount of installed renewables capacity and the application of fixed feed-in tariffs, Germany is a natural candidate for such an analysis. Its cumulative capacity of photovoltaics and wind energy, given full availability, account for about 77 GW in 2014 (BMWi, 2016), while demand usually is between 30 GW and 80 GW (see ENTSO-E, 2014). Given these relations, we would expect that a full utilization of renewable-energy capacity would sometimes meet insufficient demand and, given the applied fixed feed-in tariff system, that welfare losses occur. Marginal welfare of electricity generation is

∂w(q, z) = p(q, z) − c  (q) + b (q). ∂q Following the discussion in the previous section (see Eqs. (31) and (33)), we can identify a welfare loss of wrong dispatch incentives in a given hour if marginal welfare is negative and supply is still positive. Consequently, we can identify a welfare loss whenever we observe a price below the difference between marginal generation costs and marginal generation externalities, and a positive electricity generation of renewables at the same time. We can quantify the marginal welfare loss by the excess of marginal generation costs over the sum of the market price and marginal generation externalities – put another way, this would be the welfare gain of dispatching a marginal unit of available capacity efficiently in the respective hour. While the price is observable, we need assumptions about generation costs and externalities that we discuss in the following. Considering the potential marginal generation externalities of electricity from renewables, one might argue that the substitution of electricity from fossil fuels constitutes an indirect positive generation externality. However, in the European

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Table 1 Negative day-ahead prices at the European Energy Exchange for the German/Austrian market. Year

I

II

Negative prices

2009 2010 2011 2012 2013 2014 2015

III

IV

Wind

Solar

Number of hours

Avg. price (D )

Avg. MWh

Avg. MWh

71 12 15 56 64 64 126

−44.10 −4.99 −10.10 −60.51 −14.17 −15.55 −9.00

14 935.96a 12 094.68 15 653.08 16 526.05 16 618.55 22 180.56 23 033.49

25.65a 292.22 847.49 236.23 5306.24 3680.00 2809.06

Sources: EPEX (2016) and EEX (2016). Column I shows the number of hours with negative prices in the respective year. The average day-ahead price per MWh in these hours is given in column II, while columns III and IV show the average electricity generation in these hours. a Data on renewables generation is available from 2009/10/25 onwards.

Union, the European Emission Trading Scheme (EU-ETS) places a binding cap on carbon emissions. Electricity generation from renewables might substitute electricity from fossil fuels in specific hours, but this does not have an effect on the emission cap. Accordingly, the reduced emissions will be emitted in other hours or in other countries in Europe. Therefore, it is plausible to assume that the marginal generation externalities of electricity from renewables are zero (see e.g., Frondel et al., 2010). Regarding the marginal generation costs, we distinguish two cases. First, we derive the welfare losses for renewableenergy technologies that have zero marginal generation costs; this is a common assumption for wind and photovoltaic energy (e.g., Schmidt and Marschinski, 2009). Even in the case of zero marginal generation costs and externalities, fixed feed-in tariffs induce welfare losses whenever generation is positive while the price is negative, i.e., if we observe p(q, z) < 0 while q > 0.12 The welfare gain of reducing electricity generation by a marginal unit is then −p(q, z). In Germany, we occasionally observe hours with negative electricity prices, while renewables still generate electricity. Table 1 gives an overview of the number of hours with negative prices, the average price as well as the average amount of electricity generated from wind and solar in these hours since 2009. In total, there were 408 h in which the price turned negative. Fig. 1 illustrates the distribution of these negative prices. The graph shows in how many hours the price was below a certain level. For instance, a price of −500 D /MWh occurred once, while the price was below −20 D /MWh for 101 h. Fig. 2 plots these prices and also shows the amount of electricity from wind and solar energy that was generated corresponding to different prices.13 Each dot in the left-hand side figure shows the electricity price in the respective hour and the amount of renewable-energy electricity with zero marginal generation costs that are still fed into the grid. The righthand side figure shows wind and solar energy separately. In every hour with negative prices, wind-energy electricity was generated, and often photovoltaics were active as well. In short, even for renewables with marginal generation costs of zero, welfare losses currently occur on a regular basis. Furthermore, it is worth noting that the welfare losses in the German market will most probably increase substantially in the future. Philipp et al. (2014) forecast that without any adjustments, the number of hours with negative electricity prices will increase in the German market until 2022 to over 1000 h per year. A tendency already seems to be observable in the empirical data: in 2015, the number of hours with negative prices exceeds the number of all previous years. We now turn to the second case of marginal generation costs. For technologies with positive marginal generation costs, welfare losses can also occur when the price is positive: whenever the price falls below marginal generation costs minus marginal generation externalities, generation implies welfare losses. Given again that we assume that there are no marginal generation externalities, welfare losses occur whenever the price falls below marginal generation costs but electricity is generated. The presumably most important example in this category is biomass, with a capacity of 6.9 GW in Germany in 2014 (BMWi, 2016). Marginal generation costs of biomass are estimated to be around 30 D per MWh (see, for instance, Kost et al., 2013; Ea Energy Analyses, 2009). Thus, there is a welfare loss whenever biomass plants feed in while the electricity price is below 30 D /MWh. Given that biomass is operated continually in the German market like a base-load technology, i.e., running the whole year (see, for instance, Philipp et al., 2014), welfare losses seem to occur very often for these technologies

12 The price can become negative because of two intertemporal restrictions. Firstly, all generated electricity has to be used in the same moment and cannot be stored in relevant amounts. Secondly, inflexible power plants have a willingness to pay to continue electricity generation to avoid costly ramping up and down. For illustrations and discussions of negative-price occurrences in Germany see, for example, Andor et al. (2010) or Brandstätt et al. (2011). If such intertemporal costs were also relevant for renewables, they would more often make it optimal to use all capacity. To our knowledge, there are no published data about these costs for renewables. Nonetheless, wind and photovoltaics, in particular, are certainly very flexible technologies with negligible intertemporal costs. Biomass plants may be slightly less flexible, but still much more flexible than coal or nuclear plants. We thus find it justified to leave this aspect out of the model. 13 Note that we have to restrict the time period slightly: the amounts of electricity generated by solar and wind are published only since October 25th, 2009; see EEX (2016).

155

300 200 100 0

Cumulative Occurences

400

M. Andor, A. Voss / Resource and Energy Economics 45 (2016) 144–158

−500

−400

−300

−200

−100

0

Price (Euro/MWh) Fig. 1. Negative day-ahead prices at the European Energy Exchange for the German/ Austrian market, 2009/01/01–2015/12/31. Sources: EPEX (2016) and EEX (2016).

Fig. 2. Negative day-ahead prices at the European Energy Exchange for the German/Austrian market, 2009/10/25–2015/12/31. The left-hand side figure shows the electricity price in the respective hour and the respective generated amount of electricity from wind and solar together. Each dot refers to a period of 15 min, because the data for the amount of generation has this frequency. Prices change hourly, however. The right-hand side figure shows wind and solar energy separately. Black: Wind Energy, Gray: Solar Energy. Sources: EPEX (2016) and EEX (2016).

because operators have no incentive to use less than their available capacity. Table 2 shows the number of hours per year with electricity prices under 30 D /MWh and the estimated average electricity generated by biomass in these hours since 2009. Both the number of hours with an electricity price below 30 D /MWh and the biomass electricity in these hours have increased over the years. In 2015, the electricity price was below 30 D /MWh in almost every second hour. Summarizing, we have seen that welfare losses induced by the fixed feed-in tariff system already occur quite often in the German market due to inefficiencies described by our model. Without adjustments, it is likely that they will occur more often in the future, given the political aim to further increase renewables capacity. An interesting avenue for future research would be to estimate the total welfare losses based on a simulation model of the German electricity market; this would also take the welfare losses into account that are implied by a suboptimal amount of capacity. 5. Discussion This article has shown that defining the rationale to support renewable energies is crucial to identify the optimal promotion instrument. If positive externalities arise from the production and installation of renewable energies capacity, but not from the generation of electricity, then generation-based instruments (like all kinds of feed-in tariffs as well as renewable

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Table 2 Day-ahead prices below the estimated marginal generation costs of biomass of 30 D /MWh at the European Energy Exchange for the German/Austrian market. Year

I

II

Prices below 30 D /MWh

2009 2010 2011 2012 2013 2014 2015

III Biomass

Number of hours

Avg. price (D )

Avg. MWh

2218 1115 569 1426 2427 3658 4143

18.52 20.28 18.60 17.81 19.54 21.92 21.67

2997.15a 3374.54a 3749.77a 4517.08a 4812.33a 4908.68a 5057.08a

Sources: EPEX (2016) and BMWi (2016). Column I shows the number of hours with prices below 30 D in the respective year. The average day-ahead price per MWh in these hours is given in column II, while column III shows the estimated average electricity generation in these hours. a Average electricity generation of biomass estimated by the total yearly amount of electricity from biomass divided by the number of hours in each year.

portfolio standards) are suboptimal. They distort the supply decision, and thus cause welfare losses. Instead, a capacity subsidy that is equivalent to the externality would maximize welfare. If, however, externalities arise from the generation of electricity, but not from capacity, a generation-based subsidy should be paid on top of the market price and reflect this benefit. Finally, if there are externalities from both renewable capacity and electricity generation by renewables, then a combination of a generation subsidy and a capacity subsidy is optimal. At the margin, the subsidies should correspond to the respective externalities. Summing up, economists should carefully identify the reasons for and the aims of promoting renewable energies. Once they are clear, we can systematically derive the optimal promotion of renewables. To be concrete, we briefly describe normative conclusions that arise from this model based on assumptions that we deem plausible. Basic economic theory suggests direct caps on or prices for emitting greenhouse gases as first-best instruments for internalizing this negative externality of fossil fuel. What is more, if an emissions-trading system already fixes the total amount of carbon emissions (no matter whether this amount is chosen optimally), there is no additional climate benefit of electricity generation from renewable energies. Therefore, we do not see carbon emissions as a reason to promote electricity generation from renewable energies. In contrast, learning spillovers seem to be a plausible reason (see, for instance, Fischer and Newell, 2008; Gerlagh et al., 2009). Yet, while the production and installation of renewable capacity will very likely engender beneficial external learning effects,14 it is difficult to conceive of such positive externalities from electricity generation. Based on these assumptions, the optimal promotion scheme for renewable energies is a capacity subsidy.15 In contrast, pursuing the same objective of additional capacity investment by means of a generation-based instrument leads to welfare losses if the desired amount of capacity is so large that generation should react to demand. Effectively, this means that the German system and many others like it, which subsidize generation, but not capacity, are producing welfare losses. However, some authors argue for subsidizing electricity from renewable-energy sources as a second-best policy (cf. Bennear and Stavins, 2007; Kalkuhl et al., 2013). According to this line of argument, the implementation of an efficient first-best instrument – e.g., a correctly adjusted carbon tax – is impossible (or at least very unrealistic) due to political (or other) constraints, but alternative second-best approaches can be a pragmatic solution. This argument may be valid if there is no – or no binding – emissions trading system. The positive externalities of renewables then arise from the generation of electricity because renewables substitute fossil fuels. If this is deemed to be the real cause to promote renewables, a generation-based subsidy may be welfare-maximizing. If the short-run marginal benefits of abating carbon emissions are about constant (cf. McKibbin and Wilcoxen, 2002), and if one additional kWh of electricity from renewable energy sources approximately replaces 1 kWh of electricity from fossil fuels (because the elasticity of electricity demand is low), then the marginal positive externality of electricity from renewables is about constant, and we would suggest a constant per-kWh subsidy. Again, if a direct internalization instrument is already in use, the interactions have to be analyzed carefully, so as to avoid that generation subsidies are rendered ineffective. In contrast, a fixed feed-in tariff may be a viable policy instrument only if the targeted renewables capacity is so small that it is not necessary to make them react to demand. If such a promotion instrument is used for whatever reasons, there should be built-in precautions so that it is possible to change the promotion instrument when capacity is larger. Our analysis is based on expected market surplus. This standard welfare measure for policy analysis (see e.g., Huisman and Kort, 2015), however, implies income-risk neutrality and, thus, quasi-linear utility functions. Policy recommendations

14 Positive externalities of capacity manufacturing arise from spillovers of learning-by-doing. Additional positive externalities may stem from research and development (R&D) spillovers. However, these would be independent of the amount of manufactured capacity and are therefore not part of our model. If these exist, an additional subsidy for R&D could be optimal, see, for instance, Fischer and Newell (2008). 15 The level of optimal subsidies would depend on estimates of learning-spillover effects and, thus, is an empirical question. Such estimates would have to be conducted separately for each technology.

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157

based on the maximization of expected total surplus can lead to wrong conclusions if people are risk-averse (see e.g., Helms, 1985; Stennek, 1999) and, in particular, if insurance markets are incomplete (Gouel, 2013). While it is beyond the scope of this article to explicitly model these features, it might be an interesting avenue for future research. Based on our results and the existing literature, we expect the following. If consumers are risk-averse with respect to their income, the literature would suggest to reduce price-volatility, because such price volatility increases income volatility as well. This would suggest that there are additional benefits of renewable energies capacity, because capacity may serve as an insurance against such fluctuations; if there is more capacity, the price rises less in high-demand situations. If, however, investors are risk-averse with respect to the return on their investment and this risk is not insurable, the effect of different promotion schemes on the distribution of risk should be taken into account. For instance, a capacity subsidy may add a certain payoff to the marginal unit of capacity; generation-based subsidies and fixed feed-in tariffs yield an income only to the extent to which the capacity is available. The risk induced by price volatility, by contrast, is deleted by fixed feed-in tariffs. As the welfare implications are not straightforward, future research in this direction could be insightful. We conclude by noting some further worthwhile extensions of the model. Firstly, it might be interesting to analyze intertemporal linkages. This encompasses, on the one hand, the effects of different degrees of power-plant flexibility and, on the other hand, electricity storage. As of yet, storage is only relevant for hydropower electricity and, to a limited extent, in pumped storage. Given the technological development of batteries, this may change in the years ahead. Secondly, the residual demand for electricity from renewables should be endogenized by explicitly analyzing other electricity generation technologies. Finally, it would be useful to get a clearer understanding why fixed feed-in tariffs seem to be a widely accepted policy model, their unclear theoretical foundations notwithstanding. Appendix A. A.1. Dependence of optimal subsidies on the state of demand and the availability of supply In the following proposition, we derive whether the optimal (marginal) subsidy should change with the state of demand and capacity availability: Proposition A.1 (Dependence of optimal subsidies on the state). Consider the states (a, z) for which the optimal supply rule from Proposition 1 yields an interior solution, q* (z, ak) ∈ (0, ak), or assume that the subsidy scheme has the form of Proposition 6. Then it must hold that 2



2



∂ (a, z, q∗ ) ∂ (a, z, q∗ ) ∂q∗ − b (q∗ ) + = 0, ∂z ∂q ∂q2 ∂z 2



2



∂ (a, z, q∗ ) ∂ (a, z, q∗ ) ∂q∗ − b (q∗ ) + = 0. ∂a∂q ∂q2 ∂a Proof.

(A.1)

(A.2)

Differentiate (20) with respect to a and z, respectively. 

To illustrate, suppose that subsidies take a particular (but typical) form: Proposition A.2 (Per-unit subsidies). Suppose that subsidies are a fixed (but possibly state-dependent) per-unit payment on top of the market price: (a, z) =

∂(a, z, q) . ∂q

(A.3)

Then the dependence of subsidies on the state is given as follows:

∂ (a, z) ∂q∗ = b (q∗ ) , ∂z ∂z Proof.

∂ (a, z) ∂q∗ = b (q∗ ) . ∂a ∂a

(A.4)

Substitute (A.3) in (A.1) and (A.2). 

Thus, if marginal benefits are decreasing in q, (10) and (A.4) imply that an increase in the strength of demand should lead to a lower per-unit subsidy if the capacity limit is not binding. Likewise, by (11) and (A.4) an increase in capacity availability should lower the per-unit subsidy if the capacity limit is binding. In such cases, realistic conditioning on a and z will likely take the form of a conditioning on proxy variables, as already indicated in Section 2.3. For example, with lower demand at night, (A.4) implies that it may be sensible to have higher per-kWh wind-energy subsidies at night as compared to those during the day. However, in the short run marginal benefits will likely be constant. For example, if they indeed refer to carbon emission abatement, it does not matter whether the abatement of some ton of carbon takes place on a particular day. The per-unit subsidy should then be a state-independent subsidy.

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