Théophile T Azomahou

Raouf Boucekkine
email: raouf.boucekkine@uclouvain.be

Phu Nguyen Van

Phu Nguyen-Van

Nicolas Boccard

Cecilia Garcia-Penalosa

Cuong Le Van

Fabio Mariani

Paul Pezanis-Christou

Thi Kim

Cuong Pham

Aude Pommeret

Luis Puch

Régis Renault

Yves Smeers

Vintage capital and the diffusion of clean technologies

Keywords: JEL Classification codes: O40, E22, Q40 Energy-saving technological progress, vintage capital, market imperfections, natural monopoly, investment subsidies

We develop a general equilibrium vintage capital model with energy-saving technological progress and an explicit energy sector to study the impact of investment subsidies on equilibrium investment and output. Energy and capital are assumed to be complementary in the production process. New machines are less energy consuming and scrapping is endogenous. Two polar market structures are considered for the energy market, free entry and natural monopoly. First, it is shown that investment subsidies may induce a larger equilibrium investment into cleaner technologies either under free entry or natural monopoly. However in the latter case, this happens if and only if the average cost is decreasing fast enough. Second, larger diffusion rates do not necessarily mean lower energy consumption at equilibrium, which may explain certain empirical observations.

Introduction

Vintage capital models are increasingly used to investigate several key economic issues ranging from optimal investment policies [START_REF] Jovanovic | When Should Firms Invest in Old Capital?[END_REF] to economic growth and fluctuations [START_REF] Kobayashi | Boom and Bust Cycle of the Stock Market, and Economic Growth in a Vintage Capital Model[END_REF]. We shall argue in this paper that these models are also most convenient to study technology adoption problems, particularly adoption of clean technologies, one of the hottest topics in many agendas lately. Indeed, since a significant part of pollution can be attributed to energy consumption, the development of cleaner technologies has often consisted in searching for more energy-saving technologies. Essentially, the development of less energy-consuming technologies corresponds to the conception of new capital goods, new vintages, requiring less energy to be operated than older vintages. Therefore the vintage capital framework is natural to study the diffusion of clean technologies and to desgin the policies allowing to fasten it. Beside realism, there are at least three reasons to use these models:

1. First of all, in such a setting technological progress is embodied in capital goods so that switching to cleaner technologies amounts to investing in new machines, implying that there is no need to distinguish between technology adoption and investment.

In short, investment subsidies can be roughly interpreted as technology adoption subsidies without any additional specifications increasing the size of the model.

2. Second, a nice property of this kind of models (see in particular, [START_REF] Boucekkine | Replacement Echoes in the Vintage Capital Growth Model[END_REF]1998) is that an investment subsidy does also induce firms to shorten the lifetime of operating capital goods, therefore inducing scrapping of the less profitable machines. Thus, within such a set-up, there is no need to distinguish between investment subsidies and scrapping subsidies.

3. Last but not least, another sensitive property of this class of models connects the optimal scrapping time with the cost (or price) of the production inputs. A machine or technology is thrown out once its profitability drops to zero, and of course profitability depends on the operation cost of the capital good involved (see the seminal [START_REF] Solow | Neoclassical Growth with Fixed Factor Proportions[END_REF][START_REF] Malcomson | Replacement and the Rental Value of Capital Equipment Subject to Obsolescence[END_REF], and again Boucekkine et al., 1997). Therefore, the efficiency of investment subsidies should tightly depend on the price formation of inputs, like energy, that is on the market structure of the associated inputs.

In this paper, we do not inquire whether a substantial part of the gains in energy efficiency is due or not to induced-innovation-like mechanisms, technological progress is exogenous. Numerous papers have been already devoted to this issue (see for example [START_REF] Jaffe | Dynamic Incentives of Environmental Regulations: The Effects of Alternative Policy Instruments on Technology Diffusion[END_REF]. One of the most recent is due to [START_REF] Acemoglu | The Environment and Directed Technical Change[END_REF] who consider a unique final good produced by combining inputs from two sectors, one using dirty machines and the other operating clean machines (with no negative impact on the environment). Research can be directed to improving the technology of machines in either sector. Optimal reserach strategy and fiscal incentives are then designed to ensure sustainable growth. This idea is rather old and has been investigated in many previously published papers, including [START_REF] Hart | Growth, environment and innovation-a model with production vintages and environmentally oriented research[END_REF]. In the latter, an explicit vintage capital formulation is adopted.

However, the number of existing vintages is fixed exogenously, and not determined by an optimal scrapping condition in the tradition of [START_REF] Solow | Neoclassical Growth with Fixed Factor Proportions[END_REF]. Therefore, the essence of vintage modelling is not really accounted for. To our knowledge, the unique paper combining vintage modelling (in the proper sense) and endogenous technical progress, and henceforth studing the induced-innovation hypothesis in a proper vintage model, is due to [START_REF] Boucekkine | Scarcity, regulation, and endogenous technical progress[END_REF]. However, the latter suffers from a major drawback: it involves a dramatically high computational cost, which disables a deep analytical characterization in general equilibrium. [START_REF] Boucekkine | Scarcity, regulation, and endogenous technical progress[END_REF] only solve a partial equilibrium version (that is, the firm problem) of the model.

In this paper, we are exclusively concerned about the effectiveness of fiscal instruments in favoring investment in the new and cleaner technologies, and about their impact on GDP: under a given pace for energy-saving technical progress, do investment (in new capital goods) subsidies and/or scrappage subsidies have ultimately a positive impact on investment and output? This question is far from obvious in a general equilibrium framework where energy suppliers may also react to such policies. This paper highlights the crucial role of market structure in this respect, in particular the energy market.

Few papers have been devoted to analyze the energy questions outlined above within a vintage structure, probably due to the mathematical sophistication implied by this structure (compared to the homogenous capital structure). In addition to the paper by [START_REF] Boucekkine | Scarcity, regulation, and endogenous technical progress[END_REF] cited above, Pérez [START_REF] Barahona | A Comparative Study of Energy-Saving Technical Progress in a Vintage Capital Model[END_REF] and [START_REF] Bertinelli | Economic Development and Environmental Quality: a Reassessment in Light of Nature's Self-Regeneration Capacity[END_REF] are among these few devoted to the analysis of long-term consequences of exogenous energy-saving technological progress, highlighting some non-standard implications of vintage models.

In this paper, we keep energy-saving technological progress exogenous but we depart from the standard assumptions in the vintage literature by introducing an explicit energy market.

The modelling of this market is simple: producing energy requires only the final good in our model. However, such a benchmark framework allows to analytically study how the structure of energy markets does affect the pace of energy-saving technology diffusion. 1 We shall highlight how the specific mechanisms operating within a vintage capital model shape the effect of investment subsidies on energy-use and macroeconomic performance 1 One should note that the effect of investment subsidies on energy use as analyzed in this paper should be distinguished from the well-known rebound effect. Indeed, the rebound effect, which is recognized in the literature as closely related to the use of energy-saving technologies (see, e.g., [START_REF] Berkhout | Defining the rebound effect[END_REF][START_REF] Khazzoom | Economic implications for mandated efficiency in standards for household appliances[END_REF][START_REF] Sorrell | The rebound effect: Microeconomic definitions, limitations and extensions[END_REF][START_REF] Wirl | The Economics of Conservation Programs[END_REF], is entirely isolated from the analysis because technological progress is exogenous.

given the market structure of the energy market and for a given pace of energy-saving technical progress.

It is nowadays widely admitted that imperfect competition (externalities, barriers, market power, etc.) may explain the observed energy-efficiency gap or the slow diffusion of energysaving technologies, and that public intervention is a necessary condition for organizing the markets and promoting energy efficiency (see, e.g., [START_REF] Almeida | Energy Efficiency and the Limits of Market Forces: The Example of the Electric Motor Market in France[END_REF][START_REF] Brown | Market Failures and Barriers as a Basis for Clean Energy Policies[END_REF].

Our point is however on the energy market, which has received much less attention in the related macroeconomic literature. 2 To highlight the crucial importance of energy market structure in the performance of energy-saving technologies' subsidies, we consider two polar market structures for this market: free entry and natural monopoly. These two cases are not only interesting for tractability but also to partially assess the recent restructuring and regulatory reforms that have targeted the energy sector, particularly electricity, in the USA and Europe towards more competition in energy markets to achieve a higher energy efficiency. Natural monopoly is a plausible assumption as energy markets generates enormous fixed costs and economies of scale. Water, electricity, and natural gas utilities are typically cited as examples of natural monopolies. In fact, recent deregulation policies observed in several countries (e.g. Argentina, England, New Zealand, Europe, the USA, and Japan) has aimed to encourage a competitive energy generation sector, energy transmission and distribution remaining close to a regulated monopoly situation (Joskow, 1997, Crampes and[START_REF] Crampes | Water Resource and Power Generation[END_REF]. Studying the two extreme cases pointed out above, while certainly insufficient to reflect the complexity of actual energy markets, sounds as a desirable benchmark analysis though.

In the environmental literature, the role of subsidies was analyzed in several studies. Based on US data regarding the adoption of thermal insulation technology in new home construction, [START_REF] Jaffe | Dynamic Incentives of Environmental Regulations: The Effects of Alternative Policy Instruments on Technology Diffusion[END_REF] found that technology adoption subsidies have positive effect on the energy efficiency of new homes. De [START_REF] Groot | Energy Saving by Firms: Decision-Making, Barriers and Policies[END_REF] also observed for a survey of Dutch firms that cost savings are the most important driving force for investing in energysaving technologies, which suggests an effective role of policy measures like subsidies and fiscal arrangements in promoting for higher energy efficiency. However, possible adverse effects of subsidies were also pointed out. For example, [START_REF] Verhoef | The Adoption of Energy-Efficiency Enhancing Technologies: Market Performance and Policy Strategies in Case of Heterogenous Firms[END_REF] found in another heterogenous firms modeling that the promotion of energy-efficiency enhancing technologies by means of subsidies may be counter-productive because it could actually increase energy use. De [START_REF] Groot | Subsidising the Adoption of Energy-Saving Technologies: Analyzing the Impact of Uncertainty, Learning and Maturation[END_REF] suggested that investment subsidies for energysaving technologies can be also counter-productive as they may favor a lock-in into relatively inferior technologies. [START_REF] Bjørner | Energy Taxes, Voluntary Agreements and Investment Subsidies -A Micro-Panel Analysis of the Effect on Danish Industrial Companies' Energy Demand[END_REF] found in a panel of Danish industrial firms that subsidies in energy efficiency have no significant effect on energy use. crease the price of energy either under free entry or natural monopoly. This property comes directly from a specific "vintage" mechanism, that's property 3 outlined above: lifetime of capital goods and energy prices are tightly related via the scrapping conditions inherent to vintage models. Such a canonical relationship is a at the start of a fully articulated and original mechanism affecting both equilibrium energy use, diffusion of new (and cleaner) technologies and macroeconomic performances. Indeed, we show that whence the energy price increases either under free entry or natural monopoly, energy output increases under free entry, and falls down under monopoly. Given the complementarity between energy and capital, the latter effect may even end up pushing investment level down in the latter case! Applied to the debate of promoting energy-saving technologies, our paper brings out several findings. First, we show that while subsidies may induce a larger investment into cleaner technologies either under free entry or natural monopoly, this is not always granted. For example, in the natural monopoly case, this happens if and only if the average cost is decreasing fast enough. Second, larger diffusion rates do not necessarily mean lower energy consumption at equilibrium, which may explain certain empirical observations. We also provide for comparison an analysis of the implications of a carbon tax policy.

The related empirical literature is obviously much thinner than the one devoted the study of the efficiency of investment subsidies. The absence of international agreements on the implementation of a carbon tax makes it difficult to implement it at a national level (see Hoel, 1993, for a discussion). This said, some national plans to implement a carbon tax are under way. 3 For example, Ireland has implemented a carbon tax since 2009. A key aspect is the way the resources collected by the tax are used. This point is made, among others, by Coneferey et al. (2008): they find that a double dividend exists if the carbon tax revenue is recycled through reduced income taxes. If the revenue is recycled by giving a lumpsum transfer to households, a double dividend is unlikely to occur. In this paper, we do not tackle this question, we mainly aim to identify the impact of a carbon tax in a simple vintage model like ours, abstracting away from second round effects coming from the further use of taxes collected. The idea is to compare with the impact of investment subsidies, which are so far a much more popular policy instrument.

The paper is organized as follows. Section 2 presents the vintage model with energy-saving technical progress. Section 3 provides the balanced growth path where all endogenous variables growth at the same constant rate. Section 4 discusses the impacts of investment subsidies on the economy. Section 5 describes briefly the impact of a carbon tax in our model. Section 6 concludes the study.

A vintage capital model with energy-saving technical progress

Relying on [START_REF] Boucekkine | Replacement Echoes in the Vintage Capital Growth Model[END_REF], we build a decentralized vintage capital model with energy-saving technological progress where the energy sector is either governed by a natural monopoly or under free entry. This model has some salient characteristics. First of all, the production function is linear in vintage capital, following the traditional specification of [START_REF] Solow | Neoclassical Growth with Fixed Factor Proportions[END_REF]. Second, to guarantee the existence of a balanced growth path (see [START_REF] Solow | Neoclassical Growth with Fixed Factor Proportions[END_REF], for an illuminating assessment of this question), we will assume that the successive vintages only differ in their (decreasing) energy requirement, and not in their productivity. Thirdly, growth is exogenous. We start by a detailed exposition of the structure of the model and its properties. 4

Households

Let us assume that the representative household solves a maximization problem with nonlinear instantaneous utility function:

max {c(t),a(t)} ∞ 0 u[c(t)] e -ρt dt, (1)
subject to the budget constraint

ȧ(t) = r(t)a(t) -c(t) -τ (t),
with initial wealth a 0 given; c(t) and a(t) represent per capita consumption and per capita asset holden by household respectively. The interest rate r(t) is taken as given by the household. τ (t) is per-capita lump-sum taxes. In the model, investment subsidies are entirely financed through this type of taxes. This is the simplest way to disentangle the role of the latter subsidies. For simplification, we shall consider a logarithmic utility function. This optimization problem is very standard, and the corresponding necessary conditions are: ċ c = r(t) -ρ, with lim t→∞ φ(t)a(t) = 0, where φ(t) is the co-state variable associated with the wealth accumulation equation. 5

4 An earlier version of this paper includes a more sophisticated 3-sector model. In this version, we remove the intermediate inputs sectors (and the associated monopolistic competition) to drastically simplify the algebra and to focus on the main contribution of the paper: how does the energy market structure affect the impact of investment subsidies in an otherwise standard vintage capital growth model? 5 We shall abstract hereafter from the transversality conditions involved in the optimization work along the paper, and assume convergence to well-defined balanced growth paths granted. More mathematical literature about this specific issue can be found in [START_REF] Boucekkine | Replacement Echoes in the Vintage Capital Growth Model[END_REF][START_REF] Boucekkine | Creative Destruction, Investment Volatility, and the Average Age of Capital[END_REF].

Final good sector

The final good is produced competitively and the representative final firm is price-taker and solves the following problem: max

y(t),i(t),T (t)} ∞ 0 [y(t) -p e (t)e(t) -(1 -s q (t))i(t)] R(t) dt (2)
subject to

y(t) = b t t-T (t) i(z) dz (3) e(t) = t t-T (t)
q(z)i(z) dz ( 4)

q(t) = e -γt (5)
with initial conditions i(t) given ∀ t < 0. i(t) is investment in capital of vintage t, y(t) is production of the final good at t, and e(t) is energy use at the firm level. p e (t) and s q (t)

denote energy price and subsidies devoted to the purchase of new equipment respectively.

The final good is the numeraire. b is a fixed productivity parameter. Equation ( 3), ( 4) and

(5) describe the technology used at the firm level in the final good sector. The production function ( 3) is Leontief, capital and energy are assumed to be gross complements. It is widely admitted that there exist at least a certain degree of complementarity between these two inputs, so that the Leontief technology used here is in the worse case a worthwhile benchmark. Moreover, as explained in [START_REF] Boucekkine | Replacement Echoes in the Vintage Capital Growth Model[END_REF], such a complementarity is needed to have finite time scrapping at equilibrium. In particular, (4) gives total energy demand at the firm level, which depends on the energy requirements of all active machines.

Recall that in this framework, technical progress is assumed to make machines (equipment) less energy-consuming over time. In equation ( 3) and ( 4), it is modeled via the variable q(t):

a machine of vintage z requires q(t) = e -γt units of energy, γ > 0 is therefore the given rate of energy-saving technical chance. Finally, government subsidizes the acquisition of new machines via s q (t) following from taxes τ (t). The discount factor R(t) takes the form:

R(t) = e -t 0 r(z) dz
Following [START_REF] Malcomson | Replacement and the Rental Value of Capital Equipment Subject to Obsolescence[END_REF], after changing the order of integration and applying some algebra, the problem can be rewritten as max

{y(t),i(t),J(t)} ∞ 0 [(1 -λ(t)) y(t) -(1 -s q (t))i(t)] R(t) dt + ∞ 0 i(t) t+J(t) t [bλ(z) -p e (z)q(t)] R(z) dz dt + 0 -T (t) i(t) t+J(t) 0 [bλ(z) -p e (z)q(t)] R(z) dz dt
where λ(t) denotes the shadow value of y(t) and J(t) = T (t + J(t)). Notice that T (t) = J(t -T (t)). J(t) is the optimal life of machines of vintage t. The first order conditions (for an interior maximum) with respect to y(t), i(t) and J(t) are respectively, ∀ t ≥ 0:

λ(t) = 1 R(t)(1 -s q (t)) = t+J(t) t b -p e (z) e -γt R(z) dz b = p e (t) e -γ(t-T (t))
where now q(t) = e -γt is explicitly replaced. The first condition non-surprisingly stipulates that the shadow price of output should be equal to 1. The second equation gives the optimal investment rule equalizing the marginal cost of acquiring one unit of (new) capital goods at t and the marginal benefit which amounts to the actualized sum of net benefits over the expected lifetime of the acquired good (that is from t to t + J(t)). The last equation is the typical scrapping condition, mentioned repeatedly in the introduction section, it corresponds to the optimality condition with respect to J(t), and can be rewritten as:

p e (t) = b e γ(t-T (t)) .
This is the counterpart of the classical scrapping condition in Leontief vintage capital models, with energy playing the role of labor in the early vintage models à la [START_REF] Solow | Neoclassical Growth with Fixed Factor Proportions[END_REF].

The marginal value of energy, the price p e (t) at the decentralized equilibrium, should be equal to the marginal productivity of energy, here equal to b e γ(t-T (t)) , where e γ(t-T (t)) is the inverse of the energy requirement of the oldest vintage still in use at t. Therefore, as announced before, the scrapping condition induced by our vintage structures does connect tightly energy price with the optimal lifetime of machines. This connection is key in the main results produced in this paper.

Energy sector

In the energy sector, we assume that the production function only uses the final good according to:

f (h t ) = h(t) A(t) α , (6)
where h(t) denotes the quantity of final goods devoted to energy production, and A(t) is an exogenous variable intended to capture the difficulty or complexity to produce energy.

Indeed, the specified production function implies that to produce one unit of energy, A(t) units of the final good are needed, A(t) could be therefore interpreted as a marginal cost.

As it will be clear later, our model requires A(t) to be growing over time (or energy to be increasingly difficult to produce) for a regular balanced growth path to arise. In this sense, our specification is close in spirit to the models incorporating complexity to guarantee balanced growth paths (like [START_REF] Segerstrom | The Long-Run Effects of R&D Subsidies[END_REF]. The profit of a firm in the energy sector is:

π(t) = p e (t)f (h(t)) -h(t) (7)
where we remind that p e (t) denotes the energy price. We shall distinguish two market structures:

1. The natural monopoly: This is the case of decreasing average cost, typically implied by the existence of fixed costs. This structure is obtained when setting α > 1. Hereafter we refer to it as the NM structure.

2. Free entry: This is the case of increasing average cost and free entry that is typically obtained under decreasing returns, α < 1. We refer to it as the FE structure (FE for free entry).6

In both cases, the pricing of energy will correspond to the zero profit condition:

p e (t) = h(t) 1-α A(t) α . ( 8
)
While the condition is formally the same in both cases, it does not cover the same kind of equilibrium concept. In the natural monopoly case, it corresponds to the well-known second-best Ramsey-Boiteux pricing (see, e.g., Sherman 1989, Carlton and Perloff 2005). 7 This paper will show clearly that the economic implications of investment subsidies strongly depend on the market structure considered for the energy sector.

Beside the choice of the polar market structures considered, which are justified more extensively below, our modelling of the energy sector requires some further comments. First of all, we assume that the sole input in production is the final good. This is made for simplicity: we could have considered another good, non-renewable natural resources (with the corresponding extraction sector), in addition to the final good (representing renewable resources in our model). This would have rather obscured the mechanisms inherent to vintage models we want to isolate. Second, it is important to note that no optimization (by energy producers) is explicitly considered here: for example in the free entry case, we only pick the ultimate zero-profit condition. Third, we do not introduce fixed costs in the natural monopoly case to avoid redundancy with the increasing returns assumption through parameter α, which is enough to generate decreasing average production costs. Fourth, assuming A increasing is not only needed for balanced growth paths to exist, it also seems to be compatible with the recent observations of the IEA ( 2008), notably for the electricity generating sector. According to the IEA (2008) there have recently been sharp increases in construction costs of new power plants, particularly in OCDE countries (i.e. in the United

States, the construction costs of nuclear power plants are at least 50% higher than previously expected). Factors cause these higher costs are: increases in the costs of materials used in power plants (iron, steel, aluminium, copper), increases in energy costs, tight manufacturing capacity, increases in labor costs, etc. Also, the IEA (2008) predicts that electricity prices are expected to increase in the future, mainly due to fuel price increase and higher construction costs. In particular, projected electricity prices are on average about 11% higher in 2030 than in 2006 in OECD countries.

Finally, and as briefly mentioned in the introduction, the two market structures considered here can be motivated by numerous studies. For example, [START_REF] Crampes | Water Resource and Power Generation[END_REF] underlined that transmission and distribution of electricity have common features of natural monopoly while competition may work for generation. This observation is consistent with the empirical results of [START_REF] Christensen | Economies of Scale in U.S. Electric Power Generation[END_REF] who found that the U.S. electric power generation sector was governed by scale economies in 1955 while almost all firms were operating in 1970 in the flat area of the average cost curve and a non trivial amount of electricity was generated by a firm with diseconomies of scale. However, in a recent study [START_REF] Hisnanick | Modeling Economies of Scale: The Case of US Electric Power Compagnies[END_REF] reached a different conclusion: for them, increasing returns to scale are prevailing in US electric power companies for the period 1957-1987. In the case of Japan, [START_REF] Hosoe | The Deregulation of Japan's Electricity Industry[END_REF] observed that natural monopoly prevails in the electricity industry except the generation sector where there is no definite (or very weak) evidence of scale economy. Burns and Weyman-Jones (1998) found that gas marketing and customer service costs of the British gas sector represent a constant returns to scale when domestic and non-domestic outputs (in terms of British Gas regions) rise by the same proportion.

There were however economies of scale when one output is held fixed and the other is kept expending. [START_REF] Isoard | Technical Change Dynamics: Evidence from the Emerging Renewable Energy Technologies[END_REF] found that the European emerging renewable energy sector (namely photovoltaic and wind technologies) has a decreasing returns to scale production (the coefficient of returns to scale ranged from 0.8 to 1) in the short run but would not diverge from a constant returns to scale production in the long run.

By studying the two polar market structures, we aim to exemplify some key economic mechanisms which are relevant in the performance of clean energies promotion policies, which would be otherwise hidden in a model with a more complex (realistic) picture for the energy market. To make things even simpler, we focus on the steady state decentralized equilibrium.

Decentralized equilibrium

From previous sections, the equilibrium of this economy is characterized by the following system, ∀ t ≥ 0:

ċ c = r -ρ (9) y(t) = b t t-T (t)
i(z) dz ( 10)

R(t)(1 -s q (t)) = t+J(t) t b -p e (z) e -γt R(z) dz (11) b = p e (t) e -γ(t-T (t)) (12) f (h(t)) = t t-T (t)
i(z) e -γz dz ( 13)

y(t) = i(t) + c(t) + h(t) (14)
J(t) = T (t + J(t)) (15)
with initial conditions i(t), ∀ t ≤ 0 given. Equation ( 13) represents the equilibrium in the energy market where here f (h(t)) denotes the energy supply and where the parameter γ represents (Harrod neutral) technical progress. Equation ( 14) represents the equilibrium in the goods market. All others equations were previously derived from agents' problems.

Equations ( 9)-( 15) allow us to solve the endogenous variables y(t), c(t), r(t), i(t), J(t), T (t)

and p e (t) given the exogenous technological process. We also assume balanced budget, that is investment subsidies are entired financed by lump-sum taxes: s q (t)i(t) = τ (t). The latter equation is residual, it allows to compute the amount of lump-sum taxes needed to balance the budget, for given investment subsidy rates and the induced investment pattern.

Balanced growth paths

Let us define the environment for balanced growth path (BGP). We assume that at the stationary equilibrium, c(t) = c e γt , p e (t) = p e e γt , y(t) = y e γt , i(t) = i e γt . Accordingly, we set τ (t) = τ e γt and A(t) = A e γt , for the BGP to exist.

Definition.-The BGP equilibrium is a situation where all endogenous variables grow at the same constant rate γ except J(t) = T (t) = T .

We obtain:

r = γ + ρ (16) y = c + i + h (17) y = b i γ (1 -e -γT ) (18) 1 -s q b = t+T t
1 -e γ(z-T ) e -γt e -r(z-t) dz ( 19)

p e = b e -γT ( 20
) t t-T i(z)e -γz dz = h A α , and then iT = h A α ( 21
)

p e = h 1-α A α (22)
Finally, setting u = z -t we can compute the stationary value for the scrapping age:

1 -s q b = T 0 1 -e -γ(T -u) e -(γ+ρ)u du ≡ F (T, γ, ρ) (23)
which defines function F (T, γ, ρ). This integral function can also be rewritten as

F (T, γ, ρ) = T 0 T τ γ exp{-ρz -γσ} dσ du (24)
Along the balanced growth path, the optimal investment rule simplifies to (23). In particular, F (T, γ, ρ) provides a measure of the marginal return from investment in the long run.

Using (24), we can derive the necessary and sufficient conditions for a balanced growth path (defined above) to exist. Indeed, the stationary system above has a clear recursive structure. This nice configuration is mainly due to the Leontief technology used by the intermediate inputs producers. 8 Once T computed, all the other unknowns can be recovered immediately from the system ( 16)-( 22). For example, equilibrium energy price level can be recovered from (20) given T , and once this price computed, one can use equation ( 22)

to calculate the long-term energy sector input h. And so on. The existence of a long run scrapping age along a balanced growth path is settled in the next proposition.

Proposition 1 A balanced growth path (BGP) exists if and only if ρ + γ < b 1-sq . If γ tends to zero, T tends to infinity.

Proof. Proposition 1 states a necessary and sufficient condition for a unique long-run (positive) scrapping value T to exist, that is such that F (T, .) = which gives the parametric condition of the proposition. Notice that when γ tends to zero (no energy-saving technological progress), the integrand appearing in (24) tends to zero, and T should consequently be infinite for the optimal investment rule to hold.

The parametric condition identified for the existence of balanced growth paths is formally identical to the one exhibited in [START_REF] Boucekkine | Creative Destruction, Investment Volatility, and the Average Age of Capital[END_REF] (with s q = 0). The productivity parameter b should be large enough to "compensate" for depreciation and impatience. The larger the subsidy, the easier the condition will be. Note also that the introduction of an energy sector does not matter in the existence of the balanced growth paths. The main reason behind this property is the Leontief technology adopted in the final good sector, which induces that scrapping is independent of the energy production function.

Comparative statics show further interesting results from the economic point of view. Consistently with [START_REF] Boucekkine | Creative Destruction, Investment Volatility, and the Average Age of Capital[END_REF], it is effectively possible to say more about the scrapping behavior in terms of the parameters of the problem, using equations ( 23) and ( 24).

Proposition 2 Assuming that conditions in Proposition 1 hold, the following properties hold:

(i) T is a decreasing function of b and s q . It is increasing in ρ.

(ii) T does not depend on the parameters of the energy sector production function, f (h).

(iii) T is decreasing with respect to γ provided T is lower than 1 γ .

Proof. The proof of (iii) is quite hard given the complicated nature of the integral equation ( 23). We report its demonstration in the Appendix. The first properties are trivial mathematically speaking.

The depicted properties are mostly easy to get and to understand economically. For example, notice that an increase in b decreases the left hand side of (23). Hence, F (T, γ, ρ)

should decrease for the optimal investment rule to be still valid. As function F (.) is strictly increasing in T , the scrapping age should go down to keep on moving on the balanced growth path. In economic terms, this outcome means that as capital goods become more productive (whatever the vintage), the incentives to scrap old capital and to switch to clear technologies become more important.

The same general argument would a priori apply to γ. However in our model, an increase in γ raises the equilibrium interest rate by equation ( 16), which diminishes the marginal return from investing. As in [START_REF] Boucekkine | Creative Destruction, Investment Volatility, and the Average Age of Capital[END_REF], and more recently in [START_REF] Boucekkine | Technological Progress, Obsolescence and Depreciation[END_REF], this negative effect is more than compensated by the positive one as long as the interest burden is bounded over the lifetime of machines, for example when γT ≤ 1 (see the Appendix). Hereafter, we shall assume that we are only considering the parameterizations such that the latter property holds.9

Concerning the subsidy variable, the outcomes are rather clear and intuitive as far as scrapping is concerned. For example, an increase in the investment subsidy decreases the marginal cost of acquiring new machines, which accelerates scrapping and boosts new investment.

More intriguingly, notice that since equation ( 23) does not depend neither on the energy production function f (h), the long-term optimal scrapping will neither. Indeed as one can see from ( 21), a change in f (h) affects the optimal level of investment but not its lifetime. This is a sensitive property of the model, and we shall use it intensively later on.

We now come to an important property which is crucial to understand why the energy market structure is so important for the efficiency of subsidies. The following proposition

shows up some properties of energy supply and energy price, which are fundamental to understand the mechanisms operating in our model.

Proposition 3 Assuming that conditions in Proposition 2 hold, the following properties hold:

(i) p e = p e (γ, b, s q ) decreases with γ, but increases with b and s q .

(ii) Under the NM structure, h = h(γ, b, s q , A) has the opposite comparative statics of the energy price p e , it is increasing in A.

(iii) Under the FE structure, h = h(γ, b, s q , A) has the same comparative statics as the energy price p e , it is decreasing in A.

The proof is trivial. Using (20) and Proposition 2, one gets immediately that p e is increasing in b directly, and via the scrapping variable T which goes down when each of these parameters increases. More straightforwardly, p e is an increasing function of the subsidy rate s q exclusively via the scrapping variable. The effect of a technological acceleration through the rate γ on p e is much harder to disentangle since p e is proportional to e -γT in the long-run, and the scrapping time is shortened when γ is raised. The Lemma in the appendix solves the problem. Actually, the product γT is an increasing function of γ, or in other terms T is less than a linear function of γ. This establishes the properties (i) of the Proposition.

Properties (ii) and (iii) are obvious consequences of (i) and the relationship depicted in equation ( 22), that's:

p e = h 1-α A α , or h = p 1 1-α e A α α-1 .
As mentioned just above, Proposition 3 is important to get through the mechanisms of the model. In particular, notice that in our model, a rise in investment subsidy does increase the price of energy either under free entry or natural monopoly. This property comes from the scrapping condition (12) (giving rise to the long-run relationship in equation ( 20)). This key relationship is specific to vintage capital models with endogenous capital lifetime. As the increase in the subsidy rate leads to shorten the scrapping time, the total marginal operation cost of the oldest vintage still in use goes down while the marginal productivity of any vintage is kept constant, equal to b. The price of energy on the cost side of equation ( 12) should go up to re-establish the optimality condition. Broadly speaking, it appears clearly that a scrapping condition like the typical rule in (12) necessarily generates a negative correlation between energy price and scrapping time for any shock which does not affect the productivity parameter, b. Some observations are in order here. First of all, a negative correlation between energy prices and lifetime of capital goods is a fact which has been at the heart of a highly interesting discussion for decades. For example, in [START_REF] Baily | Productivity and the Services of Capital and Labor[END_REF], a higher energy price is associated with a shorter capital lifetime, and this argument is quite central in his interpretation of the productivity slowdown. While this view has been challenged in several directions (see for example [START_REF] Gordon | Comments and Discussion of Baily's Paper[END_REF], it is commonly shared, and it can therefore be used to validate the benchmark analysis we are performing in this paper. Second, it is also absolutely clear that this negative correlation property is obtained so clearly here because of the gross complementarity assumed between energy and capital: thanks to the Leontief specification, the price of energy and scrapping time are not simultaneously determined in the long-run, the investment rule ( 19) determines the scrapping time, and equation ( 20) determines the energy price given the scrapping time. With a general production function, things would have been much more complicated, but one can always claim that if energy and capital are close to gross complements,10 our results should still hold. In any case, the results obtained so far are anyway valuable as a benchmark.

While the subsidy rise increases energy price, its effect on energy supply depends on the market structure of the energy sector: it raises the quantity of energy under free entry but pushes it down under monopoly by equation ( 22). The latter generic equation implies a negative correlation between energy price and supply under natural monopoly, while the latter variables move in the same direction under free entry.11 Therefore, at equilibrium, energy consumption will increase under free entry, and will decrease under natural monopoly.

Henceforth, the latter seems to be better adapted to reduce energy use. Nonetheless, given the complementarity between energy and capital, the latter supply effect may be paradoxically accompanied by a slower diffusion of clean technologies under natural monopoly.

This is exactly what we will study in the next section.

The impact of investment subsidies on investment and output

In this section, we study the effects of subsidies on investment and the output-maximizing subsidies.

Impact of subsidies on investment level

Let us start with investment response to an increment in the subsidy rate s q . From (21), one gets:

i = 1 T h A α .
Notice that an increase in s q has a priori an ambiguous effect on investment. On one hand, it shortens scrapping (Proposition 2), inducing a more intense investment effort in the new and cleaner technologies (demand effect), but one the other hand, it also affects investment in the energy sector (variable h) and therefore the energy supply (supply effect). By Proposition 3, we know that such an effect dramatically depends on the market structure of the energy sector. It follows that the overall effect of larger investment subsidies on the investment level is unclear and mainly depends on whether the energy market is under FE or NM structures.

We can go a step further and bring an analytical solution to the ambiguity problem stated just above. One can use equations ( 20) and ( 22) to write i as a function of T . One gets:

i = (b) α 1-α A α α-1 e αγT α-1 T . ( 25
)
We shall denote by Θ(T ) the function:

Θ(T ) = e αγT α-1
T . Under the structure F E, that is when α < 1, function Θ(T ) is decreasing as the product of two positive decreasing functions.

Therefore, i investment is boosted by investment subsidies in such a situation since they lower equipment lifetime. Actually, using our interpretation just above, a larger subsidy will yield both positive demand and supply effects in such a case: not only investment is boosted by the typical demand effect inherent to vintage models, it is also stimulated by the rise of energy supply as depicted in Proposition 3, property iii), due to gross complementarity between energy and capital. Therefore under (FE), we get the paradoxical property that subsidizing clean technologies speeds up diffusion as expected but this success is paid at equilibrium by a rise in energy use! Things are much more complicated in the NM case where the supply effect lowering energy use pushes investment level down, and can offset the positive demand effect induced by the investment subsidy. We show hereafter that the result depends on the strength of the natural monopoly in a very concrete sense.

To clarify the latter concept, let us start with some trivial algebra. Clearly, the impact of subsidies depends algebraically on the properties of functions Θ(T ). Differentiating it yields:

Θ (T ) = e αγT α-1 T 2 αγT α -1 -1 .
Suppose α > 1 and γT < 1. Recall that the latter condition is sufficient to guarantee the realism of the model, and in particular that T is decreasing under technological accelerations.

The main trick which allows to be conclusive is the observation that T is independent of α (property (ii) of Proposition 2). Therefore, one can "play" on α without affecting the longrun equilibrium value of T . Since α α-1 is a strictly decreasing function of α, the outcome is clear. For α > α 0 = 1 1-γT , Θ (T ) < 0, and investment, being a decreasing function of scrapping, is boosted by subsidies. In such a case, the NM structure yields the same prediction as the FE structure. However, when 1 < α < α 0 = 1 1-γT , we get Θ (T ) > 0, and investment gets depressed by subsidies! Therefore, under the NM structure, investment is stimulated by subsidies if and only if the natural monopoly is strong enough in the sense that returns to the production function in the energy sector are large enough (or equivalently, if and only if the average cost in the energy sector is decreasing rapidly enough). Below the α-threshold value, α 0 , the reverse happens. We summarize the results in the following proposition: Proposition 4 Assuming that conditions in Proposition 1 hold, and provided γT < 1, the following properties hold:

(i) Under the FE structure, an increase in the investment subsidy s q raises the investment level in the long-run.

(ii) Under the NM structure, an increase in investment subsidy stimulates long-run investment if and only if returns to the production function in the energy sector are large enough, i.e. if and only

if α > α 0 = 1 1-γT . Otherwise, either investment is depressed (1 < α < α 0 = 1 1-γT ) or insensitive to fiscal stimulus (α = α 0 = 1 1-γT ).
Henceforth, our model shows clearly that the market structure of the energy sector does matter as to the efficiency of investment subsidy. The interpretation of the previous proposition is quite neat. As mentioned above, raising the investment subsidy rate s q definitely stimulates investment, but induces a supply effect which depends on the market structure of the energy sector. Under an FE structure for the energy sector, energy use goes up, thus reinforcing the former demand effect, and boosting investment given complementarity between energy and capital. Under an NM structure for the energy sector, energy use goes down, and can eventually offset the positive demand effect again given complementarity between capital and energy. Proposition 4 shows that this happens under weak enough increasing returns in the production technology of energy. In such a case, one gets the paradoxical property that while investment subsidies lower energy use, they do slowdown investment and therefore the diffusion of clean technologies. Clearly the strength of the supply effect depends on the shape of the (decreasing) average cost in the energy sector: if it is decreasing fast enough, then the supply effect will be limited, energy supply will fall but the magnitude of the drop is limited, and so will be the decline in the investment level involved. The positive demand effect will dominate. Only in such a case, we get the virtuous simultaneous occurrence of lower energy use and faster clean technologies diffusion.

Thus, in general one can see that an increase in investment subsidies generally triggers a higher diffusion of energy-saving technologies as new capital embodies energy-saving technological change. However, our analysis of subsidies within the general equilibrium vintage setting considered brings out two important results. First of all, and consistently with several empirical studies (listed in the introduction section), our paper shows that while investment subsidies may speed up the diffusion of clean technologies, this need not be associated with a lower energy consumption at equilibrium. Second, it could even be the case that investment subsidies do not speed up the diffusion of clean technologies: this is clearly the case under natural monopoly in the energy sector with weakly increasing returns and Ramsey-Boiteux pricing. This new result points at an intermediate energy market configuration which is definitely bad for clean technology diffusion.

How does this affect output response? Before getting to the algebraic developments, a few comments are in order. By construction, the production function of the final good (which is used for consumption, investment and production of energy) is a vintage capital Leontief technology. It depends on two ingredients: investment and lifetime of machines. The larger investment and the longer the lifetime of machines, the larger output. When the investment subsidy is raised, the lifetime of machines always drops, but not necessarily investment. Under an FE structure in the energy sector, investment does increase, and it is also the case under an NM structure with large enough increasing returns. In these two cases, the overall impact of rising investment subsidies is ambiguous and will be tackled in the next section. Note however that if we retain an NM configuration with low enough increasing returns, the overall effect of subsidies on output is already clear: both the lifetime of machines and investment drop, which unambiguously and markedly depresses output.

Henceforth, the latter case is clearly identified as the case against investment subsidies in terms of investment and output impact though it pushes energy consumption down. Let us summarize this property in the following Corollary to Proposition 4.

Corollary Assuming that conditions in Proposition 1 hold, and provided γT < 1, long-run output level declines in response to rising investment subsidies under the NM structure for the energy sector with low enough increasing returns.

Impact of subsidies on output level

Using equations ( 18) and ( 25), one can readily express detrended output y as a function of T , precisely:

y = Ψ e αγT α-1 T 1 -e -γT , (26)
where Ψ is a constant independent of s q , implying that the impact of investment subsidies on y exclusively depends on the shape of its relationship with T . The first T -function, e αγT α-1

T , comes from long-term investment level as given in equation ( 25). It is a decreasing function of T , and notice that it goes to infinity when T goes to zero. The second T -function, 1-e -γT , measures directly the impact of capital lifetime on output: a longer lifetime implies a larger output level (since firms will operate a wider range of machines). Notice that this term goes to zero when T tends to infinity. How does output behave when T tends to infinity given that the investment effect goes to infinity and the scrapping time effect goes to zero? A trivial computation leads to the result that output will tend to a constant Ψγ when T goes to zero. This happens when the subsidy rate s q tends to 1: output is still defined in the limit and equal to a well-identified constant. Nonetheless, such a situation violates the positivity of consumption level in the long-run. By equation ( 17), since either y and h are finite when T goes to zero while i becomes infinite, consumption must go to -∞. We shall therefore disregard this limit situation as economically irrelevant.

Let us dig deeper. Differentiating output as given by the previous equation with respect to T , one ends up finding that the sign of the derivative depends on the sign of the following difference:

e -γT 1 - γT α -1 -1 - α α -1 γT ,
which is by no means trivial and depends, among others, on the position of α with respect to 1. The following proposition states that output level is a monotonic function of the subsidy in both remaining cases: α < 1 or α > α 0 , that is either under the FE or NM structures provided the increasing returns are large enough in the latter configuration. Beside this property, the FE and NM structures produce opposite results, as stated in the next and final proposition:

Proposition 5 Assuming that conditions in Proposition 1 hold, and provided γT < 1:

(i) Under the FE structure, long-run output is an increasing function of the subsidy rate, s q .

(ii) Under the NM structure with large enough increasing returns, long-run output is a decreasing function of the subsidy rate, s q .

The proof is a bit tricky, we report it in detail in the appendix. Two remarks are in order here. First of all, the mechanisms underlying the properties highlighted just above are clear. Under either an FE or NM structure (with large enough increasing returns), rising the subsidy rate increases investment, which raises output, but lowers the lifetime of machines, which reduces output. Property i) above means that under the FE configuration, the first effect always dominates. In the alternative case, the opposite happens. That is in the NM case, the increase in investment following the rise in the subsidy rate is not large enough to compensate for the output loss due to the larger fraction of capital scrapped. Secondly, the proposition tends to confirm that the NM structure for the energy sector eliminates the potential advantages of investment subsidies in terms of output gains, whatever the extent of increasing returns in that sector. One would conclude from this property that such subsidies would be welfare-worsening under the NM structure as a decrease in output level is likely to induce a drop in consumption level, therefore driving welfare down. This is not that trivial within our benchmark set-up as one can infer from equation ( 17), and it is even less in more general frameworks attributing to energy consumption a welfare loss associated to the induced pollution increment.Indeed, one has to keep in mind that while the NM structure may not be the best market structure to raise investment in cleaner technologies, it does allow to reach a lower energy consumption. Ultimately, the arbitrage between NM and FE structures would thus depend on how consumption and pollution (or environmental quality) are weighted in the preferences of the economic agents. This issue is beyond the scope of this paper.

The impact of carbon taxes

We shall briefly discuss here the impact of carbon taxes in the diffusion of clean technologies. More specifically, we aim to compare the efficiency of the latter with the properties of investment subsidies studied before. Let us make the following observation before getting to the analytics. In order to disentangle the pure effects due to the carbon tax, we will not account for second-round effects possibly occuring if the resources collected by the tax are further used to reinforce the intended diffusion of clean technologies (for example if they go to finance the investment subsidies studied above).

Suppose the government want to permanently tax energy consumption (carbon tax) at rate τ e . As one can check, in the steady state only equation ( 20), that is the scrapping condition, will get altered, becoming:

p e (1 + τ e ) = b e -γT .
Importantly enough, the optimal investment decision (19), which gives the steady state value of scrapping, does not change. As a result, an increase in the tax τ e will not affect scrapping time in contrast to an increase in investment subsidy. Therefore, the impact of a carbon tax is much simpler compared to the previous policy. 12 We summarize here after the sequence of effects following the recursive structure of the steady state system ( 16)-( 22).

1. First of all, an increase in τ e increases the operation cost of machines (which detrended form is the left-hand side of equation ( 20)) without affecting their marginal productivity (which detrended form is given by the right hand-side of ( 20)). As a consequence, for the scrapping condition to still hold, the (detrended) energy price should fall.

2. In virtue of the zero profit condition ( 22), the impact on energy production depends on the energy sector market structure. If α < 1, that is under the FE structure, energy production will be cut. In contrast, under the NM structure, if α > 1, energy production will be increased.

3. Now, we get to the impact on investment in clean technologies as given by equation ( 21):

i = 1 T h A α .
Because the carbon tax does not modify the value of the scrapping time (unlike the investment subsidy policy), its impact on investment will track its impact on production in the energy sector. In other words, a carbon tax will speed up the diffusion of clean technologies under the NM structure, and will decrease it under the FE structure. However, given the point above, this means that we have again, as in the subsidy case, an ambiguous outcome: at long-run equilibrium, a carbon tax will end up increasing energy consumption but speeding up diffusion of new technologies under natural monopoly, while in the free entry case, we have the opposite picture. Finally notice also by the (detrended) production function ( 18), that the comparative statics of output level are the same as those of investment level.

So in contrast to the investment subsidy, carbon taxes cannot generate a double-dividend outcome: either diffusion of clean technologies is fastened but total energy consumption increases (NM structure) or diffusion slowed down with energy consumption going down (FE structure). Needless to say, this result should be taken as it is: a comparison with the investment subsidy policy. As mentioned in the introduction, the picture is incomplete and the efficiency analysis of carbon taxes should incorporate the way resources collected are used: for example carbon taxes may be used to finance investment subsidies, which might be a way to ensure double-dividend outcomes.

Conclusion

In this paper, we develop a general equilibrium vintage capital model with energy savingtechnological progress, endogenous scrapping and an explicit energy market. Because of the scrapping condition inherent to vintage capital models, the price of energy is tightly connected with the (optimal) age structure of the operating capital stock. The impact of imperfect competition on the outcomes of the decentralized equilibria are deeply characterized along the paper. In particular, we show that investment subsidies designed to fasten the diffusion of cleaner technologies may not always achieve this objective due a a well-identified general equilibrium effect. Such a result is rather consistent with the highly conflicting related empirical reports. More specifically, increasing investment subsidies do not only generate the typical positive demand effect on investment, often pointed out in partial equilibrium studies, they also affect energy supply and equilibrium energy price, which affects again investment via the scrapping mechanism repeatedly invoked along this paper. Under a free entry structure for the energy sector, the latter effect is positive, thus reinforcing the former demand effect, and boosting investment. Under a natural monopoly structure for the energy sector, the supply effect is negative, and can eventually offset the positive demand effect, which does happen when increasing returns in the production technology in the energy sector are not strong enough. We have got more results on the impact of investment subsidies on output level.

Of course, the mechanisms and results identified in this paper deserve further empirical and theoretical analysis. It goes without saying that our results are extracted under linear production functions in the final good sector, and this linearity simplifies our study to a certain extent. In particular, it allows to solve for the balanced growth paths following a straightforward recursive scheme. Such a scheme, in turns, has allowed for a neat identification of the demand and supply effects described along the paper. We are currently studying another version of the model with a more general production function which breaks down partially the above-mentioned recursivity, therefore only allowing for numerical analysis. Another useful complementary study concerns the empirical testing of the theory developed in this paper, which requires in particular an accurate appraisal of the characteristics of energy markets. This looks like a daunting task but it is certainly a necessary step to take to understand the diffusion factors of clean technologies. Finally, our analysis calls for a further investigation on the welfare implications of investment subsidies. As mentioned in the previous section, while the NM structure may not be the best market structure for investment subsidies to speed up diffusion of cleaner technologies, it does lower energy consumption in the long-run equilibrium. Therefore, the welfare implications of our analysis are far from obvious and would deserve a closer appraisal taking into account the welfare loss due to pollution. We are currently working along this line.

Lemma Assuming that conditions in Proposition 1 hold, the product γT is an increasing function of γ. (ρz+γσ) dσ du a sufficient condition for T to decrease when γ rises is the positivity of function 1 -γσ on the integration domain. This is checked if only if the line σ = 1 γ is above the integration domain. This is the case if T ≤ 1 γ . Now, note that, using the integral function defined in (23), the condition T ≤ 1 γ is equivalent to the inequality For any fixed γ , one can find the values of x (x > γ ) such that the above inequality holds.

Proof of

Note that this inequality is very easy to tabulate for function in x and γ on both sides. In particular, the inequality holds for γ < x < 1 4 . Such a sufficient condition ensures that T is decreasing with respect to γ and is consistent with parameterizations usually adopted in empirical studies.

Proof of Proposition 5: Recall that the sign of the derivative of output with respect to scrapping time T is the sign of the difference

e -γT 1 - γT α -1 -1 - α α -1 γT ,
which we may write ψ 1 (T ) -ψ 2 (T ) with obvious notations.

Consider the case α < 1. We have to study both functions ψ 1 (T ) and ψ 2 (T ) for 0 ≤ T ≤ 1 γ . ψ 2 (T ) is an affine function increasing from 1 to 1 1-α . Differentiating ψ 1 (T ) one gets:

ψ 1 (T ) = γ e -γT α -γT 1 -α .

Therefore, ψ(T ) is increasing on the interval [0 α γ ], from ψ 1 (0) = ψ 2 (0) = 1 to ψ 1 α γ , then decreasing on the interval ( α γ 1 γ ]. On the other hand, one can readily prove that ψ 1 (T ) is strictly concave on the whole interval [0 1 γ ]. Indeed:

ψ 1 (T ) = γ e -γT - 2γ 1 -α + γ 2 T 1 -α ,
and since T ≤ 1 γ , we get ψ 1 (T ) < 0 on the interval [0 1 γ ]. Notice now that ψ 1 (0) = ψ 2 (0) = 1 and that ψ 1 (0) = ψ 2 (0) = αγ 1-α . Hence the two functions start at the same point at T = 0 and with the same slope (tangency). Since ψ 1 (T ) is strictly concave while ψ 2 (T ) is affine increasing, it follows that the two functions can not intersect in the interval (0 α γ ], and ψ 2 (T ) > ψ 1 (T ) on this interval. This establishes the first part of Proposition 5.

Let us consider now the case α > α 0 = 1 1-γT > 1. In such a case, ψ 2 (T ) is an affine function decreasing from 1 to 1 1-α . The crucial thing with respect to the case α < 1 is that ψ 1 (T ) is now strictly decreasing and strictly convex on the interval [0 1 γ ]. It is enough to have a look at the expressions of the first and second order derivatives of this function displayed just above. Further given that ψ 1 (0) = ψ 2 (0) = 1 and that ψ 1 (0) = ψ 2 (0), the two functions cannot intersect, and ψ 2 (T ) < ψ 1 (T ) on (0 1 γ ].

  , by (24), F (T, .) is strictly increasing in T . It should be noticed that F (T, .) is the integral value of a positive function for which the integration support increases with T . Since F (0, γ, ρ) = 0, a positive long run value for T exists if and only if lim T →∞ F (T, ρz-γσ dσ du = 1 ρ + γ 8 Removing this specification breaks down recursivity and makes the model analytically intractable.

  24), the function F can be rewritten as F (T, γ, ρ) = T 0 e -ρz e -γz -e -γT du the required partial derivatives can be obtained after some algebraic operations: (ρ+γ)z du which is positive. From Proposition 1, we know that ∂F ∂T > 0, we deduce that γT is an increasing function of γ.It is now possible to prove Property (iii) of Proposition 2. Consistently with[START_REF] Boucekkine | Creative Destruction, Investment Volatility, and the Average Age of Capital[END_REF], we will show that a sufficient condition for T to decrease with γ is T ≤ 1 γ . The latter property is satisfied if ρ + γ < b 4(1-sq) . In fact, the total differentiation of the equation F (T, γ, .) = 1 leads to ∂T > 0 (Proposition 1), T is a decreasing function of γ if and only if the partial derivative of F with respect to γ is positive. Given that γ + σ) e -

  In terms of parameters' expressions of Proposition 1, denote x = ρ+γ b , with b = 1-sq b . Observe that x > γ ≡ γ b . Elementary algebraic operations allow us to write the following inequalityx 2 + e -1 -1 -γ x + γ < γ e -x γ

For example, Pérez[START_REF] Barahona | A Comparative Study of Energy-Saving Technical Progress in a Vintage Capital Model[END_REF] assume an exogenously given energy supply.

Some have been simpy abandoned as in France where the carbon tax plan, included in the so called Grenelle environmental package led by the French environment minister Borloo, has been roughly abandoned in 2010.

We shall exclude the case α = 1 in our study, it will be crystal clear in the next section that a balanced growth path cannot exist under this zero-measure parameterization.

It should be noted that while the Ramsey-Boiteux pricing is profusely studied in theoretical models concerning the energy sector it is few observed in practice (despite its desirable properties) because of some concerns on ethical consideration and practical implementation[START_REF] Netz | Price Regulation: A (Non-Technical) Overview[END_REF].

Notice that this is the realistic case. For γ around 2.5% per year, we restrict T to be lower than 40 years, which covers by far the typical figures.

Some complementarity is anyway needed to ensure a finite scrapping time as mentioned in earlier sections.

We insist that these relationships are generic, at least for the free entry configuration. Under natural monopoly, our properties derive immediately from the Ramsey-Boiteux pricing. Other pricing rules are possible under natural monopoly but we prefer to focus on the latter pricing for its simplicity and the ease of comparison with the free entry case within our benchmark analysis.

This simplification occurs because the steady state optimal scrapping does not depend on the characteristics of the energy market, which is ultimately due to the linearity of the production function.

We are thankful to the editor and an anonymous referee for their helpful comments. Financial support from the project ANR CEDEPTE n • ANR-05-JCJC-0134-01 is gratefully acknowledged. Boucekkine also ac-

Appendix : Proofs

Proof of Proposition 2: As already mentioned, Properties (i) and (ii) are trivial. Let us prove Property (iii). To this end, we need the following Lemma.