The extraction of natural resources: The role of thermodynamic efficiency

Ecological Economics 68 (2009) 2594–2606

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Ecological Economics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e c o n

Analysis

The extraction of natural resources: The role of thermodynamic efficiency Antonio Roma a,b,⁎, Davide Pirino c a b c

Dipartimento di Economia Politica, Universitá degli Studi di Siena, Italy London Business School, United Kingdom Dipartimento di Fisica “Enrico Fermi”, Universitá degli Studi di Pisa, Italy

a r t i c l e

i n f o

Article history: Received 14 July 2008 Received in revised form 1 March 2009 Accepted 9 April 2009 Available online 28 May 2009 Keywords: Production inputs substitutability Choice of numeraire Energy consumption Energy market

a b s t r a c t The modelling of production in microeconomics has been the subject of heated debate. The controversial issues include the substitutability between production inputs, the role of time and the economic consequences of irreversibility in the production process. A case in point is the use of Cobb–Douglas type production functions, which completely ignore the physical process underlying the production of a good. We examine these issues in the context of the production of a basic commodity (such as copper or aluminium). We model the extraction and the refinement of a valuable substance which is mixed with waste material, in a way which is fully consistent with the physical constraints of the process. The resulting analytical description of production unambiguously reveals that perfect substitutability between production inputs fails if a corrected thermodynamic approach is used. We analyze the equilibrium pricing of a commodity extracted in an irreversible way. We force consumers to purchase goods using energy as the means of payment and force the firm to account in terms of energy. The resulting market provides the firm with a form of reversibility of its use of energy. Under an energy numeraire, energy resources will naturally be used in a more parsimonious way. © 2009 Elsevier B.V. All rights reserved.

1. Introduction The central role of energy as an input in the production process of economically valuable goods is difficult to dispute. Yet in economics a consensus on how it should be modelled has not yet emerged. Intensive energy use creates environmental damage through the release of residual heat. As discussed in the seminal contribution of GeorgescuRoegen (1971), the second law of thermodynamics, which governs energy utilization and degradation, is a key source of negative externalities. The production literature has only dedicated a limited amount of attention to the concept of waste as an unavoidable joint product of any production process (Ayres and Kneese, 1969; Ethridge, 1973; Kummel, 1989, 1991). Traditional economic analysis of production generally avoids thermodynamic considerations. Typical production models require substitutability between all inputs, none of which, including energy, has a special role. On these premises, economics literature dealing with the use of energy has largely focused on the possibilities of substituting it as a factor of production in the presence of energy price shocks or energy shortages. Factors of production (as well as consumption goods) are interchangeable if they provide the same functionality. However, the complete substitutability between natural resources (including energy), labor and capital leads to paradoxical consequences. Daly (1997) ⁎ Corresponding author. Dipartimento di Economia Politica, Universitá degli Studi di Siena, Italy. E-mail address: [email protected] (A. Roma). 0921-8009/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolecon.2009.04.013

observes that, if labor and natural resources are substitutes and not complements then it would be possible to make a cake with “[…]only the cook and his kitchen. We do not need flour, eggs, sugar, etc., nor electricity or natural gas, nor even firewood. If we want a bigger cake, the cook simply stirs faster in a bigger bowl and cooks the empty bowl in a bigger oven that somehow heats itself […]”. A dazzling example of this paradox can be found in the standard representation of funds and flows in the Cobb–Douglas production function model, that is: α

α

α

Q = K 1L 2R 3;

ð1:1Þ

where Q is the output of the process per unit of time, K represents the stock of capital, L the labor supply and R the flow of natural resources. From expression (1.1) it is evident that we can obtain a fixed amount of output Q0 and even if R → 0, it is sufficient to choose an amount of K such that:
K=

Q0 L Rα 3 α2

1

α1

Y + ∞:

ð1:2Þ

Although it is well understood that the Cobb–Douglas production function is only an abstract concept and not an actual description of a production process, this very concept has a pervasive influence on economic modelling. Substitutability plays a critical role in Neoclassical general equilibrium construction. It underlies the view that there are no real limits to economic growth as, in the extreme, “the world can, in effect, get along without natural resources” (Solow, 1974). In point of

A. Roma, D. Pirino / Ecological Economics 68 (2009) 2594–2606

fact, Neoclassical general equilibrium also implies the substitutability of every good as a numeraire and means of payment. Still, this is a useful framework that yields insight on the value and optimal exploitation in time of non-renewable resources (Hotelling, 1931). It may sensibly be argued, as Ayres and Miller (1980) put it, that there are definite and well-known limits on physical performance in almost every field, which derive from the unique role of some specific factor of production. Energy is a case in point. See Cleveland and Ruth (1997) for a survey of the literature covering this alternative perspective. However, such analyses are often of a qualitative nature. The case for (the lack of) substitutability can often be argued either way, due to the vagueness of the approach. As a contribution to this debate, we provide a robust theoretical foundation for the lack of substitutability of the energy input in the extraction and refinement of a commodity. Specifically, we propose an analytic model for a mining operator who refines a mineral from its natural concentration to a strike concentration that defines the product.1 The production process model is consistent with actual physical constraints and follows the thermodynamic theory of an irreversible separation process. 2 All real world transformations involving energy are, in fact, irreversible. A thermodynamic transformation or cycle is said to be reversible if it is carried out by making infinitesimal changes to a state variable, which allow the system to be at rest throughout the entire process (Fermi, 1956). Such a transformation is impossible because it would require an infinite amount of time. All production processes are thus irreversible in nature, because they have to be carried out in a finite time-period in order to bring production to the market. They therefore involve a strictly positive increase in entropy ΔSN 0. An increase in entropy means a reduction in “useful” energy, that is the part of energy that can be converted into work by an engine. Thus, an entropy increase can be interpreted as waste or resource degradation. If the production function does not accommodate the real thermodynamic process that leads to the final output, the impact of the producer's choices on resource depletion and the waste released into the environment will not be evident. Despite attempts to minimize its use as an input, a positive energy amount is always necessary to extract the natural resource. This minimal requirement provides an answer to all issues raised by recent and past literature about substitutability between production process inputs. Although limited substitution of energy is possible, total substitution is impossible because there is a physical energy threshold, for a given quantity of raw material input, below which no production exists. This emphasises the importance of a physics-based approach to production modelling as a correct methodological way of resolving the substitutability issue. Except for a few papers, microeconomic analysis completely avoids a detailed consideration of the physical constraints that are the essence of every production process.3 Krysiak and Krysiak (2003) show that general equilibrium theory is consistent with the mass and energy conservation laws. Krysiak (2006) analyzes the consequences of the second law of thermodynamics on economic equilibrium in a general framework.4 A production function derived from finite time 1 A number of papers deal with the decisions concerning the exploitation of physical resources and the extraction/production of commodities (Brennan and Schwartz, 1985; Coratzar et al., 1998; Hartwick, 1978; Stiglitz, 1976) but none describe the thermodynamics of extraction. 2 Separation processes are at the heart of many production processes and include distillation, evaporation, drying, deep freezing, centrifugation, membrane separation etc. 3 Sav (1984) uses a micro-engineering production function derived from physical laws to model the exploitation of solar energy for domestic water heaters. Substitution elasticities between nonrenewable fuel inputs (oil or natural gas) and capital-intensive solar-produced heat are investigated. 4 On the debate about the impact of the entropy law on economic equilibrium see also Young (1991) and Daly (1992), who lead to two completely opposite conclusions. Many authors have investigated Nicholas Georgescu-Roegen’s paradigm of ecological economics, again obtaining conflicting conclusions, such as those reported by Khalil (1990), subsequently criticized by Lozada (1991) and finally re-stated by Khalil (1991).

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thermodynamic constraints can be found in Roma (2000, 2006). The production process we analyze here from a physical point of view differs from the thermal production process in Roma (2000, 2006) in that the final product obtained will be of homogeneous quality and unused energy can be stored. An important precedent for our commodity extraction problem is found in Ruth (1995). This paper contains explicitly thermodynamic constraints on production, in the form of lower bounds on the inputs to be fed into a Cobb–Douglas production function. The lower energy bound is calculated for a reversible separation process. As mentioned above, this approach does not reflect the reality of production, which involves only irreversible processes. We then turn to an analysis of the rational use of energy from an economics point of view. Berry et al. (1978) highlight the difference between the concepts of thermodynamic optimality of the use of energy in production and overall economic (cost) efficiency. They assume that the substitution of energy by other inputs along a production isoquant amounts to an increase in thermodynamic efficiency. As thermodynamic efficiency cannot be increased beyond the reversible limit, full substitutability of energy is prevented, i.e. it is impossible to move beyond a certain point on the isoquant in the energy dimension. The economic notion of Pareto optimality for resource allocation (where no individual can be better off without making at least one other worse off) will not in general amount to the optimal use of available energy from a thermodynamic point of view. We advance the analysis proposed in Berry et al. (1978). We can analytically derive the production isoquant under the assumption of maximum finite time thermodynamic efficiency (a possibility dismissed by Berry et al. (1978) as optimistic, see p.133). This is still compatible with some (quite limited) substitutability between energy and raw material. We analyse the problem of the optimal scale of production and the consequent exploitation of natural resources, including energy, under the finite time thermodynamic foundation adopted. Even if the most efficient thermodynamic technology is used, higher production in the same amount of time implies greater deviation from reversible efficiency and higher energy waste. Hence it is the scale and speed of production that ultimately determine thermodynamic efficiency. We incorporate the production model in a simple general equilibrium framework in which we analyze inter-temporal production decisions. We derive the equilibrium in a two-period economy in which our good is produced and consumed. In a Neoclassical equilibrium, efficiency in the use of available energy will not, on its own, drive economic decisions. Under a Neoclassical model, the amount of thermodynamic waste will be irrelevant and natural resources will be fully exploited. On the other hand, the negative externalities associated with the degradation of energy, if taken into account, would lead to a thermodynamic efficiency criterion in the use of this scarce resource. However, thermodynamic efficiency would have to be imposed on the decision makers (by way, for example, of a “green tax”). We find, similarly to Roma (2006), that a competitive economic equilibrium will be twisted towards higher thermodynamic efficiency if energy is forced to be the numeraire and means of payment. This creates a market in which the energy price of production is established. When energy is the only scarce input, the producer will compare the energy cost of production with the energy value of the firm's sales, determined by supply and demand. The lack of substitutability between input factors and the decreasing return to scale feature of irreversible technology will prevent the complete use of available energy in production. This will in turn decrease energy degradation and thermodynamic waste. The neoclassical solution, where a change in numeraire and means of payment would not alter the resource allocation, is finally obtained if the production process is carried out over an infinite time, i.e. if it is reversible. Reversibility is equivalent to constant returns to scale technology. This paper is organized as follows: Section 2 is dedicated to a description of the physical process underlying the production of the commodity. We derive the reversible limit of the production technology

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and provide a brief comparison between our model and that of Ruth (1995) in order to highlight the differences between the two approaches. In Section 3 we introduce our concept of the thermodynamic production function. An empirical model for the degradation of the resource due to cumulative extraction is described and incorporated in a two-period production framework. In Section 4 the problem of a producer who uses a thermodynamic production function and faces the problem of the scarcity of energy is presented. Energy is used as the numeraire and means of exchange in our economy. It is shown that, when energy is scarce and cannot be substituted, the producer's decision varies drastically according to whether the accounting is done in terms of energy or in terms of another numeraire (such as the product itself). The producer's choices under a reversible technology are discussed in Section 5. We show that in this case the choice of the numeraire and means of payment is indeed irrelevant. In Section 6 we present our conclusions. 2. The physical model Industrial activity is subject to the thermodynamic energy limits of the physical and chemical processes through which production is obtained. The thermodynamic optimization of such processes is a key goal in industrial engineering. Advancement in production efficiency can be achieved by improving upon the actual thermodynamic limits of processes and machinery such as heat pumps, thermal engines, chemical and electrochemical systems, separation processes and so on. Given the inputs to the industrial process, the higher the thermodynamic efficiency the higher the output. Industries must obviously produce in a finite time and with finite facilities. Hence theoretical reversible efficiencies can never be achieved. In most cases process optimization problems are solved from a thermodynamic point of view, using one of two alternative approaches: finite time thermodynamics (Andersen et al., 1977; Berry et al., 2000) or entropy generation minimization (Bejan, 1996; Nummedal et al., 2005; Salamon et al., 1980), which are usually referred to in the literature as FTT and EGM respectively. The idea underlying these approaches is that available energy is a scarce resource and its use must be optimized in industrial processes. Sieniutycz (2003) provides an exhaustive review of the most commonly used methods for computing the thermodynamic energy limits of common industrial components.5 It is interesting to note that improvement in the design of industrial plants is unambiguously identified with an effort to improve current thermodynamic limits in a vast portion of engineering literature while in general economics literature such concepts are seldom mentioned. There is strong evidence, however, that improvement in thermodynamic efficiency is a key driver of economic growth (Ayres et al., 2003; Ayres and Warr, 2005; Ayres, 2008; Warr et al., 2008). We describe below how the engineer would go about describing and optimizing a specific production process of economic interest: the extraction of a basic commodity that is only available in nature mixed with other waste material. The essence of such production is a separation process, for which a full thermodynamic description is available. From this description we can derive an analytical production function consistent with physical laws. For the benefit of readers with an economics background, we will present details of the underlying physics which may sound obvious to readers with experience in this field. The physical model we adopt was developed by Tsirlin and Titova (2004). They compute (via EGM) the minimum energy required to achieve the separation of an ideal mixture of components with a specified output. Fig. 1 from Tsirlin and Titova (2004) illustrates the scheme used to model the process. A subsystem to be separated out is in contact with an infinite reservoir characterised by a temperature T and a pressure P, 5 Energy limits in industrial processes are investigated from a classical thermodynamic point of view by Denbigh (1956), Kim and Loungani (1985) and Forland et al. (1988).

Fig. 1. The scheme used by Tsirlin and Titova (2004) to model the separation of a finite subsystem from an infinite reservoir. The thermodynamic state of the reservoir is identified by the temperature T, the pressure P, and the composition vector C0. The pumps g0 and g1 are the coefficients of mass-transfer from the reservoir to the working medium and from the working medium to the reservoir, respectively. The working medium has a chemical potential µP0 at the contact point with the reservoir and a chemical potential µP1 at the contact point with the subsystem.

which do not vary during the transformation. This may be interpreted as a stylized production plant in which valuable substances with prespecified purity must be refined and extracted from the original mixture of substances which are found in a natural reservoir. The substances extracted are accumulated in the subsystem. The mining operator faces an initial mixture of useful minerals and waste rock. We suppose that the minerals are initially a minority presence with respect to the waste rock and that the mining operator tries to purify the mixture and extract highly concentrated mineral. At time t = 0 the composition vector of the reservoir, which describes the (percentage) concentrations of the different mixture components, is (C01,…,C0n). The subsystem is initially at equilibrium with the reservoir and is therefore characterised by the same thermodynamic variables T, P, and by the same composition vector (C01,…,C0n), as the reservoir. We indicate with N0 the number of moles in the subsystem at t = 0. Separation processes and chemical reactions are usually described by means of a thermodynamic quantity known as chemical potential, which can be defined as follows. Suppose that our system is a homogeneous substance in a specified thermodynamic state and that its volume and entropy do not vary. If we add an infinitesimal quantity of any substance (the i-th substance) to the system we will observe a variation of the system's internal energy.6 The chemical potential of the i-th substance in the system is given by the ratio between the increment of internal energy observed and the quantity of substance added. In other words, the chemical potential of a substance in a system represents the variation of the system's internal energy when a unit of substance is added, keeping the system's volume and entropy constant. Each component of the mixture has a chemical potential µ0i (T, P) in the reservoir, which depends on its concentration C0i and is given by: μ 0i ðT; P Þ = μ 0 ðT; P Þ + RT log C0i ;

ð2:1Þ

where R is the universal gas constant and µ0 (T, P) represents the reservoir's chemical potential that does not vary during the transformation. We can interpret µ0 (T, P) as the ground level from which all other chemical potentials are measured. As it is only a scale

6 The internal energy of a thermodynamic system is defined as the sum of translational, rotational and vibrational energies of its components plus the interaction energy. Internal energy is a state function of the system and can be expressed as a function of its pressure P , temperature T and molar composition (N1, ..., Nk), where Ni is the number of moles of the i-th component.

A. Roma, D. Pirino / Ecological Economics 68 (2009) 2594–2606

variable, we could impose µ0 (T, P) ≡ 0. However, we choose to leave µ0 (T, P) unspecified for the sake of generality. The subsystem and the reservoir are in contact with a third medium, a working medium, which has the chemical potentials µP0 and µP1 at the contact points with the reservoir and the subsystem, respectively. This is a shorthand notation to indicate that the i-th component has a chemical potential µP0i at the contact point with the reservoir and µP1i at the contact point with the subsystem. A working medium is simply defined as a system whose chemical potentials are under control. When in contact with the reservoir and the subsystem, the working medium generates chemical potential drops between the contact points. Particles tend to diffuse from regions of high chemical potential to regions of low chemical potential, allowing for the mass transfer. Examples of mixture separation with the formalism of chemical potentials can be found in Wijmans and Baker (1995) and Vallieres et al. (2003). As observed by Wijmans and Baker (1995) a chemical potential variation dµi of the i-th mixture component across a membrane of thickness dx produces a mass flux, given by: dμ Ji = − Li i ; dx

ð2:2Þ

where Li is a coefficient of proportionality. All driving forces such as gradients in pressure, temperature, concentration and electromotive forces can be reduced to chemical potential gradients and, therefore, generate mass fluxes which can be expressed in the form (2.2). Through this mechanism the working medium creates a mass flux g0i of the substance i between the reservoir and the working medium, and a mass flux g1i of the substance i between the working medium and the subsystem. All these fluxes have the dimension of mole per time: [g] = [mol] [time]−1. Following the approach of Onsager (1930), Onsager (1931) and Miller (1994), it is assumed that the system is close to equilibrium. More precisely, as described in Miller (1994), “one divides the system into small subsystems and assumes that each subsystem is in local equilibrium, i.e., it can be treated as an individual thermodynamic system characterized by the small number of equilibrium variables.” This common assumption leads to linear irreversible thermodynamics, in which the relation between chemical drops and mass transfer coefficients is linear (Onsager's kinetics). At the final instant t = τ the subsystem has a new specified composition vector (Cτl,…,Cτn) and contains a number of moles Nτ. Following the reasoning in Tsirlin and Titova (2004), the work of separation in an isothermic process (i.e. with temperature kept constant) for an adiabatically insulated system (i.e. heat exchanges are not allowed) is given by the Gouy–Stodola formula: E = E0 + T ΔS;

ð2:3Þ

where E0 is the reversible work and ΔS the entropy increment. The reversible work is computed by summing the total internal energy variation of the reservoir to that of the subsystem7:

E0 = ΔURes + ΔUSub = Nτ RT

n X

½Cτi logCτi − C0i log C0i ;

ð2:4Þ

i=1

and it is independent of N0. Note that, as expected for a statefunction, formula (2.4) only depends on the initial and final states of the system, Cτi, C0i i.e. it is path-independent.8 The minimum value for E in Eq. (2.3) is achieved when the entropy increment ΔS is

7 8

A full derivation is available from the authors. See Fermi (1956).

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minimized. The entropy increment depends on the mass transfer coefficients g0i9: ΔS = =

1 T 1 T

Z Z

τ

n h    i X P P g0i μ 0i − μ 0i + g1i μ 1i − μ 1i dt

0 i=1 n τX 0 i=1

ð2:5Þ

½g0i Δμ 0i + g1i Δμ 1i dt;

where μli is the chemical potential of the i-th component in the subsystem. In the case of an Onsager kinetics (see above) we have that gij = αijΔμij, and it can be shown that, under this additional assumption, (see Tsirlin and Kazakov, 2004; Tsirlin et al., 1998) optimal solutions to the minimization of Eq. (2.5) are constant mass transfer coefficients: j = 0; 1: i = 1; N ; n:

gji =

Nτ Cτi − N0 C0i : τ

ð2:6Þ

In order to put the formulas in a simpler form, we define the total mass variation of the i-th component: def

ΔðNCi Þ u Nτ Cτi − N0 C0i ;

ð2:7Þ

and introduce the equivalent mass-transfer coefficient: def

αi u

α 0i α 1i : α 0i + α 1i

ð2:8Þ

Substituting the optimal solutions (2.6) in expression (2.5), we obtain the minimum entropy variation. It follows that the minimum energy required to carry out the transformation [N0, (C0l,…,C0n)]→ [Nτ, (Cτl,…,Cτn)] in a finite time τ is: Emin = R T Nτ

n X i=1

½Cτi logCτi − C0i log C0i  +

n 1X ½ΔðNCi Þ2 : τ i=1 αi

ð2:9Þ

In the simplest case the valuable substance may be assumed to be mixed with only one other, so the initial vector of concentrations is (C0,1 − C0). For example, C0 can be the concentration of a mineral in the mineral ore and thus 1 − C0 can be interpreted as the waste rock concentration. Differently from Ruth (1995), who adopts the extreme assumption that the mineral can be completely separated from the waste rock, we generalize the definition of the final product and define it more realistically as mineral extracted with a given strike concentration k N C0, but still mixed with a residual concentration 1 − k of waste rock. With n = 2, (C01,C02) = (C0,1 − C0) and (Cτl,Cτ2) = (k,1 − k). Therefore Eq. (2.9) takes the form: E = R T Nτ ½k logk + ð1 − kÞ log ð1 − kÞ − C0 log C0 −ð1 − C0 Þ log ð1 − C0 Þ + i 1 h 2 2 + ðNτ k− N0 C0 Þ + ðNτ ð1 −kÞ − N0 ð1− C0 ÞÞ ; τα ð2:10Þ

where we have imposed, for simplicity, ᾱl = ᾱ2 ≡ α. The second term on the right hand side of expression (2.10) denotes the irreversible use of energy. It involves the square of the difference Nτ k − N0 C0 between the refined moles extracted and the moles of pure material in the subsystem, and the corresponding difference Nτ (1 − k) − N0 (1 − C0) for the waste material. Finally, note that the optimal coefficients g0i, gli define the speed at which the extraction plant must be operated in order to waste the minimum possible amount of useful energy. The irreversible energy 9 Entropy can be defined as the ratio of energy to temperature. Here energy is defined as change in chemical potential over each instant.

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input (2.5) implies a path dependency: production depends on the evolution over time of the mass transfer coefficients gji, rather than on only the initial and final states of the system. Mirowski (1989) notes that the absence of path dependency in production is an essential feature that allows Neoclassical production theory to be symmetric to the consumer's theory.10 This is precisely what the second law of thermodynamics rules out. In our model a specific path of production is chosen that minimizes the energy wasted. This type of efficiency is clearly desirable for any production plant. However, it limits the speed of production, although it is conceivable that the plant could be operated faster if energy were not a scarce resource. On the other hand, the scale of production, which is given by Nτ, is a free choice. The energy required for separation is a monotonically increasing function of Nτ. For a given level of output Nτ the production technique defined by Eq. (2.10) leaves a choice between extracting the actual amount at the desired level of concentration or, where possible, “averaging up” the concentration of an amount N0 of the original mixture already contained in the subsystem. This is the source of the limited substitutability between N0 and E. As a special case of Eq. (2.9) the minimum energy required to completely separate N0 moles of a binary mixture into its pure components is (see Tsirlin and Titova, 2004): bin

Em = − R T N0 ½C0 lnC0 + ð1 − C0 Þ lnð1 − C0 Þ  ! N02 C02 ð1−C0 Þ2 ; + + τ α1 α2

ð2:11Þ

where ᾱl and ᾱ2 are the parameters defined in Eq. (2.8). 2.1. The reversible limit

R

ð2:12Þ

−ð1 − C0 Þ logð1 − C0 Þ

 γ1


Jt −Jt JA −JA

 γ2

Et −Et EA −EA

;

ð2:15Þ

where t denotes time, Jt is the raw material input, Et is the energy input and the starred quantities Jt⁎ and Et⁎ are the minimum material input and minimum energy input required for the production of Yt, respectively. With JA and EA (JA⁎ and EA⁎), Ruth (1995) indicates the same quantities in a base year. The Cobb–Douglas production function (2.15) tries to capture the limits to the substitution between energy and raw material that are necessary to produce the quantity of the final good Yt. The lower limit Et⁎ is derived, consistently with Berry et al. (1978), as the reversible limit of Eq. (2.11) for τ → +∞ (and time varying concentration C0, see Section 3.3). Still, Et⁎ and Jt⁎ may be partially substituted. Our approach will be completely different: we will derive the production function Yt from finite thermodynamic considerations alone. In this approach the limits to the substitution between energy and raw material are represented by an analytical thermodynamic constraint. This production function has very different properties from a Cobb–Douglas function.

Eq. (2.10) implicitly defines a production function Nτ (N0, E), where the inputs are energy, E, and raw material, N0, and where production time, τ, also determines the amount of final product that can be obtained. We can re-arrange expression (2.10) as:

2

n0 uk log k + ð1 − kÞ logð1 − kÞ − C0 log C0 −ð1 − C0 Þ logð1 − C0 ÞN 0: ð2:13Þ We assume that our mineral is initially available at a minority concentration, namely C0 b 12. We try to purify the substance in order to obtain a strike concentration k such that 1 − C0 b k, which is greater than the initial concentration of the waste rock. In this framework ξ0 N 0 and the number of moles produced when τ → +∞ is: E · RTn0


Yt =

Nτ

where NR is the number of moles produced reversibly. Eq. (2.12) says that the transformation [C0,1 − C0] → [k,1 − k] requires a positive amount of energy for the pairs (C0, k) such that:

NR =

Ruth (1995) presents the problem of the mining operator as the maximisation of a value function essentially defined as the timeintegral of production growth minus the growth in energy expenditure for production. Hence his model assumes that the energy input is the only cost of production. To model production, Ruth (1995) uses a Cobb–Douglas type function given by:

3. The production model

The reversible increment in the internal energy of the system that is required for production can be computed by taking the limit for τ → +∞ in Eq. (2.10): Em = R T NR ½k log k + ð1 − kÞ logð1 − kÞ − C0 log C0

2.2. The Ruth (1995) model of mining technology

  i 1 h 2 N ψ 2 k + ð1 −kÞ + Nτ RTn0 − 2 0 0 τα τα +

ð3:1Þ

i N02 h 2 2 C0 + ð1− C0 Þ − E = 0; τα

where the constants ξ0 ≡ k log k + (1−k) log (1–k) –C0 log C0 – (1–C0) log (1–C0) and ψ0 ≡kC0 + (1–k) (1–C0) depend only on the initial and final concentrations. Only the solution of Eq. (3.1) that increases in the energy input has a physical meaning. Hence we define our production function: Nτ ðE; N0 Þ u

−RTn0 + 2Qτ N0 ψ0 +

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðRTn0 Þ2 − 4Qτ2 ðk−C0 Þ2 N02 − 4RTQτ N0 ψ0 n0 + 4Gk Q E 2Qτ Gk

;

ð3:2Þ

ð2:14Þ

Tsirlin and Titova (2004) note that the actual energy used for separation exceeds its reversible limit by a factor of hundreds of thousands in specific cases. 10 “[In the Neoclassical model] output is a path-independent phenomenon which implies that the manner of the combination of the inputs does not affect the magnitude of the final output. […] Otherwise, merely slowing down the production process would alter the resulting output.” See Mirowski (1989) p.314.

where we have introduced the notation Qτ = τ1α and Gx =x2 + (1–x)2. Expression (3.2) explicitly gives the number of moles of the highly refined mixture (k, 1 –k) we obtain, starting from N0 moles of raw material and spending energy E for production in a finite time τ. A solution to the problem exists if, and only if, the quantity under the square root of Eq. (3.2) is greater than or equal to zero: 2

2

2

2

δ = ðRTn0 Þ − 4Qτ ðk− C0 Þ N0 − 4RTQτ N0 ψ0 n0 + 4Gk Q E z 0: ð3:3Þ

A. Roma, D. Pirino / Ecological Economics 68 (2009) 2594–2606

This condition implies that there is a limited substitutability between process inputs and defines a lower bound on the energy input:

Ez

i 1 h 2 2 2 2 N0 4Qτ ðk −C0 Þ + N0 ð4 RT n0 ψ0 Qτ Þ −ðRT n0 Þ : 4Qτ Gk

ð3:4Þ

Inspecting Eq. (3.4), we can see that if k ≠C0 and N0 N 0, a lower bound on energy always exists for a high enough positive value of N0. In our model of production k N C0, because we want to improve the quality of the valuable substance, and N0 N 0 because we assume that our basin

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Table 1 A set of chosen parameters. Parameter

Value

k C0 T τ α

80% 30% 1250 K 3600 s 10− 4 mol2/(J s)

is not empty at the initial time.11 This means that we start production only if the energy is over the threshold given by Eq. (3.4). As pointed out by Islam (1985), such restrictions on production inputs drastically modify the isoquants with respect to those of a Cobb–Douglas function. Fig. 2 shows the isoquants of Nτ (E, N0) for a chosen specification of the model parameters (reported in Table 1). The chosen values have no particular significance having been chosen only for the purpose of graphical representation. The isoquants of Eq. (3.2) display an “economically inefficient” portion, where the same level of production is obtained with more N0 than necessary. We can disregard this portion in our microeconomic analysis. Isoquants derived from a Cobb–Douglas type production function are also reported as thin dotted lines. plotted Cobb–Douglas function has E γ1The γ the form NC − D ðE; N0 Þ = A RT N0 2 and A, γ1 and γ2 are chosen to obtain nearly the same levels of Nτ (E, N0) isoquants. Using NC − D (E, N0) as a production function would generate perfect substitutability between (1) the two production inputs. That is, given a point (N(1) /(RT)) that 0 ,E belongs to an isoquant NC − D (N0, E) =K1, it is possible to obtain the same level of production K1 by arbitrarily reducing the quantity of energy entering the process and by increasing the number of moles N0 of raw material. This property disappears when the production function is obtained via the real thermodynamic process. The constraint (3.4) does not allow perfect substitutability between inputs. This gives a more realistic description of the process and recognizes the different nature of energy as a production input. Expression (3.2) provides a theoretically consistent production function for the extraction of refined mineral from a low grade mixture. It naturally incorporates a lower bound on the energy input in the spirit of Ruth (1995) and, as anticipated, makes the lack of substitutability between inputs completely clear. The lower bound on the quantity of low grade mixture N0 is zero. In what follows, we assume that there is an unlimited quantity of the raw material N0 and it is therefore freely available to the producer, while energy is scarce. In this we follow Ayres and Miller (1980), who argue that basic materials will always be available in some concentration in the earth's crust, but the availability of energy (work) needed to extract them is, in fact, the only limit to economic growth.12 3.1. The minimum energy production frontier

Fig. 2. (Top) The dotted lines are isoquants of a Cobb–Douglas type production function E γ1 γ NC − D (N0, E) = A RT N0 2, with A = 100 and γ1 = γ2 = 0.1. Continuous lines represent isoquants of the proposed production function Nτ (E, N0), where the model parameters are those reported in Table 1. (Bottom) The plot shows the efficient portion of the production function Nτ (E, N0) isoquants as black continuous lines. The parameters chosen are those reported in Table 1. The thin dotted lines are the economically inefficient isoquants of Nτ (E, N0). The bold dotted line represents the production frontier (given by equation 3.4), which defines the minimum energy required to carry out production for a given value of N0. The continuous bold line plots the optimal choice of the freely available raw material N0 for each value of the energy E (see Eq. (3.5)). The intersection between an isoquant and this line gives the number of moles N0 that should be used to achieve the chosen level of production with the minimum energy expenditure. The area colored in black is a zone where, despite energy being over the depicted threshold, it is not enough to have positive solutions of Eq. (3.1).

If energy is the only scarce input, it is rational to minimize its use and substitute it with the freely available raw material as far as possible. Inspecting Fig. (2) it can be seen that, for a given level of output, there is a value of the input raw material N0 that minimizes the energy required to obtain such a level of production. This value can be computed analytically. From Eq. (2.10) it can easily be seen that the 11 As we will show later, energy-efficient production requires a strictly positive value of N0 that is proportional to the level of production (see Eq. (3.5)). 12 Georgescu-Roegen (1979) argued that the degradation of quality materials is, in fact, the real limit to the economic growth, rather than the scarcity of energy itself. This approach is criticised by Ayres and Miller (1980), who argue that technical progress can overcome the scarcity of physical resources. They point out that a finite quantity of resources must always be embodied in capital and a limit on economic growth and technical efficiency always exists, due to the finite availability of renewable resources. Their point is that all resources, no matter how they are distributed on the earth, can be extracted if enough energy is available. Thus they conclude that energy is, in fact, the only resource that could ultimately limit economic growth.

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level of N0 that minimizes the energy input for a fixed level Nr of production is given by: N0 ðNτ Þ = Nτ

1 − C0 − k + 2C0 k : 1 − 2C0 + 2C02

ð3:5Þ

By substituting Eq. (3.5) into the Eq. (2.10) and solving it for Nτ, we obtain the production frontier at which every level of production is achieved at an absolute minimum energy cost: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi Nτ ðEÞ =

4 EðC0 −kÞ2 Qτ GC0 + GC0 RTn0 2ðC0 −kÞ Qτ 2

2

− GC0 RTn0

:

ð3:6Þ

Eq. (3.6) defines the maximum amount of mineral refined to the given concentration k that can be extracted in a period of time τ given the energy input E. Under our assumptions, a fixed proportion of raw material and energy is required for each level of production, which only depends on the scale of production and time available for the production process. Solving relation (3.6) for E as a function of production we find that: 2

EðN Þ =

2

ðC0 −kÞ Qτ N + GC0 RTn0 N GC0

:

ð3:7Þ

The energy E (N) in Eq. (3.7) can be interpreted as a conditional energy demand (input) when the other factor is at the optimal level. From expression (3.7) we can compute the marginal and average costs AE of production in terms of energy, AN and NE : 2ðC0 −kÞ2 Qτ N + GC0 RTn0 AE = ; AN GC0

ð3:8Þ

ðC0 −kÞ2 Qτ N + GC0 RTn0 E = : N GC0

ð3:9Þ

C (t) (of the mineral to be refined) at time t and the cumulative quantity of extracted material J (t) is: log J ðt Þ = η − ρ log C ðt Þ;

ð3:11Þ

where η reflects the relative abundance of the metal considered and ρ is the degradation velocity. Chapman and Roberts (1983) report different values of ρ, which vary from ρ=1.62 for copper to ρ=17.34 for chromium. Values estimated on historical data for the constant η can be found in Nguyen and Yamamoto (2007). Values for the constants ρ and η are always estimated when J (t) is expressed in metric tons. In our case we measure the quantity of material extracted in moles. Suppose that we have extracted N moles of a substance with atomic mass MA. Then we have extracted MA ×N grams of the substance. This means that J=103 ×MAN is the quantity of material extracted expressed in metric tons. From now on it will be useful to use the convention ϑ≡103 ×MA, namely J=ϑN. 3.3. Two-period production technology We assume that the production process by which the mineral is extracted occurs over two periods of time. First we allow our extraction plant to work in a basin, for a period τ1, with a number of initial moles N0 of a mixture (C0,1−C0), where the first component of the mixture is the useful mineral and the second component is waste rock. The concentration of the useful material is lower than that of the waste rock, that is C0 b1−C0. The final consumption good is defined by the mixture (k,1−k), with kN 1−C0. If the mining operator produces N1 moles of the consumption good in a time τi he needs the energy given by Eq. (3.7). In the first time period the producer does not completely deplete the basin, allowing for additional production to be carried out in a second time period, τ2, which we assume to be equal to τ1. After the extraction of N1 moles at time τ1, the new concentration C1 of the useful substance is given by:
 η −1 log ðϑN1 Þ = η − ρ log C1 YC1 = exp ðϑN1 Þ ρ : ρ

ð3:12Þ

By taking the limit τ → + ∞ we have that Qτ → 0 and then: lim

τY + ∞

AE E = lim = RTn0 : τY + ∞ N AN

ð3:10Þ

This simple result could also be obtained by taking the limit τ → +∞ in Eq. (2.10). Eq. (3.10) means that if the process is reversible then the technology is linear, that is marginal and average costs are the same technological constant RTξ0. 3.2. Quality degradation of the resource In practice the natural concentration C0 of the mineral to be refined is not a constant during extraction. Suppose that the extraction is carried out for an initial period of time τ and then stopped. We assume that when we restart the extraction we will have to deal with a new natural concentration C0′ b C0 because extraction depletes the resource. Then we need more energy than in the first period of extraction to obtain the same strike concentration k. A quantitative approach based on empirical data can be found in Mudd (2007). This paper reports Australian data on the copper ore grade during the period 1842–1995. The ore grade of the copper was about 25% in the year 1842 and decreased to a value b5% in the year 1995. Simultaneously the quantity of ore milled started from nearly ≈0 Mt and reached the value of ≈80 Mt (1 Mt = 103 kg). Resource depletion is then strictly correlated with the cumulative quantity of extracted material. To model resource depletion we follow the approach of Chapman and Roberts (1983) and Ruth (1995) and assume that the relation between the concentration

In the second production period the plant works in a new basin where the natural concentration C1 is lower than the initial one C0. We denote with E2 and N2 the energy used and the number of moles produced in the second period. From the previous discussion, it follows that the quantity N2 depends not only on E2 but also on N1 (E1) through C1: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   N2 ðE2 ; N1 ðE1 ÞÞ =

4E2 ðC1 − kÞ2 Qτ GC1 + GC1 RTn1

2

− GC1 RTn1

2ðC1 −kÞ2 Qτ

;

ð3:13Þ where ξ1 =k log k +(1 −k) log (1−k) –C1 log C1 – (1–C1) log (1 – C1) and C1 is given by Eq. (3.12). 4. Absence of substitutability and economic equilibrium In what follows we sketch a simple general equilibrium framework in which the qualitative effect of the irreversibility and of lack of substitutability of energy can be easily analysed. To keep the structure of the model as simple as possible, we assume that the production described in this paper is the only industry sector present in the economy. This may sound at odds with the basic resource nature of our output, which may itself be an input in other industries. However, as we only need to highlight the role of the consumer's preferences in the pricing of the output, we avoid any specific connotation of the good in this section. This will avoid a more complex description of the

A. Roma, D. Pirino / Ecological Economics 68 (2009) 2594–2606

productive structure of the economy, which would obscure appreciation of the effects we will be emphasizing. General equilibrium theory combines the optimal allocation of individual consumers' budgets with the production decisions of profit-maximizing firms that produce and supply the consumption good to the economy. By matching supply and demand a market clearing mechanism determines values (i.e. market prices) according to which resources are allocated. A basic requirement of the general equilibrium analysis is that all values should be comparable, that is expressed in the same unit of account. This is achieved by normalizing prices in terms of one of the goods available in the economy, the numeraire. In a Neoclassical equilibrium, in which every good can be bartered against any other good, the choice of the numeraire is irrelevant as it will not alter the quantities finally produced and consumed. However, a key feature of Neoclassical general equilibrium models is also the perfect substitutability, on the production side, between factors of production. The lack of substitutability of energy affects the equivalence between different factors of production and introduces extra rigidity in the firm's decisions. We analyse here the problem of a value-maximising producer operating over two periods under the irreversible technology we have described in the previous sections. The producer is only constrained by scarcity of energy, which does not allow full exploitation of the natural resources available in the time frame provided. The problem is that of allocating the use of available energy, given the prices at which the commodity produced can be sold in the two periods. We will assume that the firm can store energy but not the finished product.13 Given that the value of the goods produced in the different periods must be expressed in a common numeraire, we take it to be energy rather than the good produced in one of the two periods. That is, we assume that when the producer sells the final product he receives energy in exchange. It is assumed (following Ruth, 1995) that no capital enters the production process and that the raw material (mixture of mineral and rock) is freely available in an unlimited quantity.14 Under this choice of the unit of account for the producer, the profitability of the firm will differ from that achievable when the unit of account is a finished good. The idea that accounting in terms of energy rather than in standard monetary terms may modify the attractiveness of an investment strategy may seem counterintuitive. However, this concept is firmly established within the literature concerning the concept of EROEI (Energy Return On Energy Investment) of power plants. EROEI is defined as the ratio of gross energy produced by an energy supply process to the total (direct plus indirect) energy cost of its production (Cleveland, 1992, 2005; Gately, 2007). Thus the EROEI approach provides the net energy analysis of an energy-based production process and can lead to completely different results from a pure financial assessment. Financial evaluations of investments deal only with market prices for the construction of the power plant plus maintenance costs and subsequent financial revenues (Kaldellis et al., 2005; Tian and Wang, 2001; Tsoutsos et al., 2003).15 13 This is not an essential assumption but avoids any potential problem arising from the ad-hoc adjustment to the concentration C1. The amount of the reduction in concentration over time is not necessarily consistent with thermodynamic theory. The presence of an infinite reservoir may give the firm an incentive to produce during the first period the output to be delivered in the second. 14 Intertemporal production functions that do not explicitly involve the use of capital are common in the financial literature, see for example Cox et al. (1985), Fama (1972) and Hartwick (1970). 15 A comparative analysis between financial evaluation of power plants and EROEI can be found in Hall et al. (1979). The authors deal with a problem that arose in the NYSEG (New York State Electric and Gas Corporation) service area in the late 70s, when there was a proposal for a new 870MW coal-fired generating station. The authors compare the energy (with the EROEI approach) and the dollar cost of building the power plant with the cost of a comprehensive regional program of insulation. Their results show that the insulation program would be more efficient in conserving energy than the power plant would be in providing it, by a factor of at least 4 in economic terms and a factor of 15 in EROEI terms. This reinforces the idea that the decisionmaking strategy can be quite different if an energy approach is considered.

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In our model we recognize that, if energy is taken to be the numeraire and means of exchange, energy itself represents “value”, rather than the product. This implies that unused energy balances will enter the producer's value function. These specific features of the model force the producer to account for the profit of his investment in terms of energy. If the additional energy spent on production has a negative return it will not be used but rather retained at its face value as an inventory of the firm. The limit to the substitutability of the energy input will play a central role in this result. The total energy available for production is E T. Energy is transferred to the firm before the production process begins. We assume, for simplicity, that a representative consumer exists who only derives utility from the finished good. The amount of scarce energy ET is in the hands of the consumer, who exchanges it with the firm (in a market) against rights to the consumption good produced. This will determine the spot price of the consumption good in terms of energy, Pe. This arrangement may appear to give energy a special role in the exchange mechanism, by requiring that every good can only be purchased by exchanging it for a prespecified good, rather than by bartering. This is the essence of a cash in advance constraint (Clower, 1967), which is used to provide a role for fiat money (paper money without any intrinsic value forced onto the economy by a Central Bank) that would otherwise have no economic role in the economy. This imposition should have no effect in a barter economy in which only physical goods with a direct economic use are present. Nevertheless, if energy is exchanged for the availability of the good produced then a price of the good in terms of energy will be defined. This will allow the use of energy as a numeraire. Furthermore, it will provide information for its efficient allocation in the economy. 4.1. The consumer's problem The consumer's problem is: max uðc1 ; c2 Þ;

c1 ;c2

sub

Pe ðc1 + Bc2 Þ V ET ;

ð4:1Þ

ð4:2Þ

where c1 and c2 denote consumption of the good at the end of the time intervals 1 and 2, respectively, and B is the real discount factor. The consumer trades his endowment of energy ET with the firm before production begins, in exchange for future production. As energy does not enter the standard utility function u(·,·), there will be no incentive for the consumer to avoid spending some of the available energy in exchange for the good, thus pushing the energy price of the good to the highest level allowed by the budget constraint. This will in turn provide the producer with an incentive to produce more. Hence Eq. (4.2) will hold as an equality. The consumer's first order conditions imply that:

B=

Au Ac2 Au Ac1

;

ð4:3Þ

and Pe =

ET : c1 + Bc2

ð4:4Þ

4.2. The producer's problem Given that in a competitive market the price of the good in terms of energy and the real discount factor are both outside the firm's control,

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the producer's problem under the energy numeraire can be written as: max E1 ;E2

N1 Pe + BN2 Pe − E1 − E2 ;

1 sub N1 = f ðE1 Þ ; 2 N2 = f ðE1 ; E2 Þ; E1 + E2 V ET ; E1 N 0 ; E2 N 0;

ð4:5Þ

5. Reversible technology

where E1 denotes the energy used to produce N1 and E2 the energy used to produce N2. Functions f 1 (E1) and f 2 (E1, E2) are defined by expressions (3.6) and (3.13) respectively. The firm can carry energy forward to the second period without cost. Apart from this possibility there is no inter-temporal energy market. Hence a unit discount factor is applied to E2 in the net present value of the firm. In problem (4.5) the producer, who may use up to the amount of energy ET, may choose not to use part of the available energy for production and retain it as an asset of the firm. The unused energy will be ET − E1 − E2 and will add to the value of the firm. This setting describes an energy input that is transformed into a different output, which is then valued in terms of energy in the market. In equilibrium c1 = N1 and c2 = N2. Hence: Pe =

ET : N1 + BN2

ð4:6Þ

4.3. Use of energy by the producer In the setting described above, where energy is the firm's unit of account, it will not generally be optimal for the producer to use all the available energy. Production will not exceed the level where the marginal energy cost is equal to the price set by the market-clearing condition (4.6). This result derives from the inherent non-linearity of the irreversible production function, which displays increasing marginal cost, and from the limit to the substitutability of energy by other factors of production. There is no alternative technology in the economy under which unused energy balances may efficiently enter production. The producer will not be able to exchange unused energy with a competitor, so at the equilibrium allocation some energy resources will remain unused. This would not be the case under perfect substitutability of the two production inputs (e.g. a Cobb– Douglas production function). The general solution of problem (4.5) is obtained by maximizing the Lagrange function: LðE1 ; E2 ; λÞ = N1 ðE1 ÞPe + B N2 ðE1 ; E2 ÞPe − E1 − E2 + λðET − E1 − E2 Þ

ð4:7Þ w.r.t. its arguments. Maximization w.r.t. E1 and E2 gives the following pair of equations if the energy bound holds as a strict inequality in an internal solution (namely E1 + E2 b ET) and λ = 0: Pe

AN1 AN2 + B Pe − 1 = 0; AE1 AE1

B Pe

vanishing energy slack), for specification of the model parameters.16 Slack arises because additional use of energy for production does not increase the value of the firm.17 The use of energy in production generates entropic waste. The decrease in energy consumption implied in the solution may be welfare-improving if such an externality is taken into account (Roma, 2006).

ð4:8Þ

AN2 − 1 = 0: AE2

AE2 The second condition of Eq. (4.8) implies that AN = B Pe , that is 2 the marginal cost of production in the second period is equal to the discounted price B Pe. It can be verified that for typical preferences a solution to the problem (4.5) involves a non-zero energy slack. The Appendix A provides an analytical solution to the problem (4.5) when depletion is neglected (i.e. AN2 AE1 = 0) with positive energy slack. It is not possible to find an analytical solution to the producer problem (4.5) when depletion is accounted for. However we find, numerically, an internal solution (i.e. with a non-

The solution to problem (4.5) differs from the Neoclassical solution in which all available energy would be used for production. This result has to do with role that the price Pe plays under irreversible technology. In equilibrium the price Pe represents the amount of energy that must be spent to produce a marginal amount of the consumption good. We can think of energy as being physically transformed into the produced good (or, more precisely, used to physically transform the raw material into the finished good). The reverse physical transformation of the produced good into energy is, however, severely limited under irreversible technology. The price Pe is therefore the physical cost of a transformation in one direction-energy to the finished good. The same transformation can also be achieved in the market, by purchasing the good against energy, other than in a physical sense by spending energy to actually produce marginal good. However, once spent, energy cannot be physically extracted from the product.18 The reverse transformation of the produced good into energy can be only achieved by the firm in the market by selling the produced good against energy. Such a market alone determines the level of energy cost that can be recovered and provides the only form of energy reversibility available to the firm. Only if production is reversible, meaning that it takes place over a time period τ → ∞, is it possible to recover the energy that has been used in production. As a consequence, under reversible technology the energy market price of the good produced must be equal to the marginal cost of production (which, in this case, is a constant independent from the production level) to prevent arbitrage opportunities. Under reversible technology, energy and goods can be physically transformed into one another both physically and in the market. As energy spent to produce a given quantity of the good can be extracted 16 The chosen depletion parameters ϑ, η and ρ are those of copper (see Nguyen and Yamamoto, 2007). 17 An extension of the production side of the economy to include additional production processes (i.e. other firms producing additional goods) is unlikely to eliminate the potential slack in the use of energy if each additional process is subject to a similar minimum energy threshold. An average minimum energy bound would arise for the production side of the economy, which would prevent the substitution of energy by other factors. Similarly, the possibility of including energy as an argument of the consumer’s utility function would not rule out slack. If Ec is the portion of the consumer’s initial endowment that is consumed directly, and therefore not exchanged against the goods produced, the resulting energy price Pe = NE1T +−BENc 2 will in general be lower, decreasing the incentive to produce even further. 18 Even if we try to extract the infinitesimal energy amount dE held in the quantity of good dN (a transformation so small that the thermo-dynamic system remains in equilibrium), we will only be able to extract the reversible energy dN R T ξ0, which is not greater than the marginal cost under irreversible technology. Suppose that we have irreversible technology and we achieve economic equilibrium at a production level N⁎ AE1 (without loss of generality in the first production period). If Pe N AN j N4 , we can produce 1 and sell an extra infinitesimal quantity of the  final good dN, spending an energy AE1 AE1 AE1 dE = AN j N4 dN, with a net energy gain dN Pe − AN jN4 N 0. If Pe b AN jN4 , the 1 1 1 producer can buy an extra quantity of refined material dN at a price dNPe and reversibly extract the energy dNRT ξ0. Upon inspection of expression (3.8), it can be seen that AE1 AE1 AN1 jN4 z RTn0 ; 8N4 . Thus the condition Pe b AN1 jN4 is not sufficient to establish that the quantity of energy dN R T ξ0 − dN Pe delivered by this strategy is positive and therefore there is an arbitrage opportunity. To find an arbitrage opportunity in the irreversible case, the price Pe must ibe lower than the reversible marginal cost of production. In the range h AE1 Pe a RTn0 ; AN j N4 we have no arbitrage restrictions on the price Pe. The same type of 1 arbitrage argument is used in Roma (2006) to determine the no-arbitrage price in terms of energy for a more elementary product (hot fluid). This case is quite different from ours and leads to a complete identification of the energy price Pe of the good through no-arbitrage considerations, because the product itself is thermal energy (stored in a fluid) and there is no need to operate any thermodynamic engine backward to extract infinitesimal energy quantities.

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from it at any time, the inter-temporal market for the finished good is indistinguishable from the inter-temporal market for energy. Under reversible technology the energy market is a derivative market. It does not alter the consumer's possibilities of allocating wealth over time. Given Pe2 the price in terms of energy of the good produced at time 2, which must again be equal to the constant marginal cost to prevent arbitrage opportunities, we can define the energy discount factor

Z=B

Pe Pe2

ð5:1Þ

which must be used to discount the value of energy available in the second period to the present. The value of the firm to be maximized in Eq. (4.5) must then be defined as: Pe N1 + B Pe N2 − E1 − Z E2 ;

ð5:2Þ

where − (E1 + ZE2) is the present value of the energy used by the firm. The firm's objective function can equally be expressed in terms of energy or finished product, dividing by Pe throughout. As opposed to the irreversible case, under this technology E1/Pe (and ZE2/Pe) is in fact equivalent to the amount of the finished good obtained through a physical transformation. It can be verified that the first order condition (4.8) for the firm's profit maximization problem with the inequality constraint E1 + E2 b ET cannot be satisfied for the linear technology. The constraint E1 + E2 ≤ ET will be binding. All available energy will be used up (E2 = ET − E1), and the firm will have zero profit. The consumer's budget constraint can also be expressed equivalently: C1 + BC2 = N1 +BN2. Hence, as expected, under the assumption of reversible path-independent production we return to the Neoclassical model: the proposed change in the means of payment and unit of account will be irrelevant.

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inputs for every level of production will be combined with fixed coefficients which depend only on the scale of production and the time available for the production process. The derived production function displays increasing marginal cost in terms of energy as long as the process occurs in a finite time. Our assumptions lead to a oneto-one relationship between energy employed and production of the basic commodity considered. We note here that a vast body of literature devoted to the successful estimation of aggregate production, using electric energy as the key explanatory variable, assumes the existence of such a one-to-one relationship (Bodo and Signorini, 1987, 1991). This technological description is used within the optimisation problem of a profit-maximising mining operator who produces over two periods and sells the product in a competitive market. We impose that consumers can only purchase the produced good against energy and that the representative firm accounts for its profit in energy terms. Such an arrangement creates an energy market through which the producer can transform its output back into energy. This reverse transformation would otherwise be prevented by the second law of thermodynamics. Energy that cannot be recovered through the energy market will not be spent in production. Contrary to the conclusions of a Neoclassical barter model, the equilibrium allocation will leave some energy unused. This will in turn increase the thermodynamic efficiency of production and decrease entropic waste. In a Neoclassical model with reversible production the energy market would not provide any additional trading opportunity due to the equivalence between goods and energy. Hence its effect on the equilibrium allocation would disappear. We have tried to reconcile the key role of energy, as a scarce non-substitutable resource that underlies thermodynamic optimization in the engineering literature, with economic optimization in a microeconomic model. Whether our results may be considered a reasonable approximation for what will be obtained in the case of more complex production processes, or possibly an abstraction that may apply to the description of aggregate production in a macroeconomic sense, should be the subject of further research.

6. Conclusions Acknowledgement In this paper we have addressed the issue of substitutability between inputs in a production function and, in particular, the existence of a lower bound on the amount of useful energy necessary for production to take place, as in Berry et al. (1978). The hypothesis of perfect substitutability has convenient implications for general equilibrium modelling, but does not plausibly represent many actual production processes. Our methodological approach to providing a clear cut answer to this much debated question is to resort to finite time thermodynamics and model the input requirements for a given production based on physical principles. Economic theory has long drawn on physics, by exploiting the formal analogies between the two fields. This has been particularly fruitful for the development of consumer's theory in pure exchange economies, as discussed by Mirowski (1989). Furthermore, recent advances in pure exchange general equilibrium theory are based on such formal analogies (Smith and Foley, 2008). We believe that there is a lot to be gained by also applying physics to the proper modeling of production. In this realm physics can be applied directly rather than by analogy. In our paper the production process is modelled analytically, following the thermodynamic theory of an irreversible separation process. The physical model confirms the existence of an energy bound below which production is impossible. This prevents full substitutability between factors of production and contradicts the standard Cobb–Douglas representation of production. If we assume, following the view in Ayres and Miller (1980), that the original mixture from which the commodity has to be extracted is a freely available natural resource, while energy is in aggregate scarce, the

The authors would like to thank Richard Brealey, Alberto Dalmazzo, Roberto Renó and Stephen Schaefer for useful comments on an early draft of the manuscript. Appendix A A.1. Analytical and numerical solutions A.1.1. Analytical case: No-depletion and positive energy slack The consumer–producer problem is: max uðc1 ; c2 Þ; c1 ;c2

sub Pe ðc1 + B c2 Þ V ET ;

ð1:1Þ

and: max E1 ;E2

N1 Pe + B N2 Pe − E1 − E2 ;

1 sub N1 = f ðE1 Þ; 2 N2 = f ðE1 ; E2 Þ ; E1 + E2 V ET ; E1 N 0; E2 N 0:

ð1:2Þ

With market clearing conditions given by c1 = N1, c2 = N2. We found analytical solutions for the consumer–producer's prob2 lem (1.1)–(1.2) when depletion is neglected (i.e. AN AE1 = 0) and

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when u(c1,c2) = c1β c12 − β, with β ∈ (0,1). In this situation B = γ NN12 where γ = 1 −β β and the solutions are: E14 =

This means that the fractions of the energy spent in the two production periods depend only on the consumer's preference γ. Moreover:

f

  1 2 2 2 4 ET k Qτ + 2C0 2 ET Qτ −ð1 + γÞðRTn0 Þ 8ðC0 −kÞ2 Qτ ð1 + γÞ   2 −2C0 4 ET k Qτ −ð1 + γÞðRTn0 Þ  qffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −RTn0 RT ð1 + γÞn0 − GC0 1 + γ 8 ET ðC0 −kÞ2 Qτ + GC0 ð1 + γÞðRTn0 Þ2 ;

g

ð1:3Þ E24 =

f

  1 2 2 2 4 ET k Qτ γ + 2C0 2 ET Qτ γ −ð1 + γ ÞðRTn0 Þ 8ðC0 −kÞ2 Qτ ð1 + γ Þ   −2C0 4 ET k Qτ γ −ð1 + γÞðRTn0 Þ2  qffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi −RTn0 RT ð1 + γ Þn0 − GC0 1 + γ 8 ET ðC0 − kÞ2 Qτ γ + GC0 ð1 + γ ÞðRTn0 Þ2 ;

g

ð1:4Þ

E14 + E24 1 = : + ∞ 2 ET

ET Y

That is, only half of the available energy is used for production. A.1.2. Numerical solutions in the case of depletion We now investigate the solutions to problem (1.2) when depletion is taken into account. As in Appendix (1.1), we model the consumer's utility function choosing u(c1,c2) = c1β c12 − β, with β ∈ (0,1), so that B = γ NN12 where γ = 1 −β β. It is more convenient, in the case under analysis, to re-write problem (1.2) in terms of N1 and N2. We then use the following equivalent formulation: N1 Pe + BN2 Pe − E1 − E2 ;

max It is possible to verify that E1 + E2 b ET, that is Eqs. (1.3)–(1.4) actually correspond to a solution with an energy slack regardless of the value of ET. In order to see this, consider the fraction of the total energy spent in the production:

f

  E14 + E24 1 2 2 = 2ð1 + γ Þ 2 ET ðC0 −kÞ Qτ − GC0 ðRTn0 Þ 2 ET 8 ET ðC0 −kÞ Qτ ð1 + γÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + GC0 ð1 + γÞRTn0 8 ET ðC0 −kÞ2 Qτ + GC0 ð1 + γ ÞðRTn0 Þ2

½

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + 8 ET ðC0 −kÞ2 Qτ γ + GC0 ð1 + γÞðRTn0 Þ2

g



E1 = f 14 ðN1 Þ ; E2 = f 24 ðN1 ; N2 Þ; E1 + E2 V ET ; N1 N 0 ; N2 N 0;

sub

14

E1 = f ðN1 Þ =

ð1:12Þ

ðC0 −kÞ2 Qτ N12 + GC0 RTn0 N1 GC0

:

We will show that:

24

E2 = f ðN1 ; N2 Þ =

ð1:6Þ

for all ET. Straightforward computations show that inequality (1.6) holds if and only if:   2 2 WðET Þu − 2ð1 + γÞ 2ET ðC0 −kÞ Qτ + GC0 ðRTn0 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + GC0 ð1 + γÞ RT n0 8 ET ðC0 − kÞ2 Qτ + GC0 ð1 + γÞðRTn0 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi + 8 ET ðC0 −kÞ2 Qτ γ + GC0 ð1 + γÞðRTn0 Þ2  b 0:

½

ð1:7Þ It is easy to see that Ψ(ET) → − ∞ when ET → +∞. Moreover, the equation Ψ(ET) = 0 has only complex solutions, given by: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h iffi

2 2 2 2 −GC0 ðC0 −kÞ Qτ ð1 + γ Þ ðRTn0 Þ F − G2C0 ðC0 −kÞ4 Qτ2 γ2 −1 ðRTn0 Þ4 2ðC0 −kÞ4 Qτ2 ð1 + γ Þ2

;

ð1:8Þ i.e. the function Ψ(ET) does not intercept the real axis and then Ψ(ET) b 0, ∀ET. This proves inequality (1.6). It is interesting to analyze the producer's choice when energy available is very large (i.e. ET → +∞). From expressions (1.3)–(1.4) it is easy to see that:

E14 1 ; = ET Y + ∞ ET 2ð1 + γ Þ

Pe −

GC1

:

ð1:14Þ

AE1 AE2 − = 0; AN1 AN1

B Pe −

ð1:15Þ

AE2 = 0: AN2

ð1:16Þ

Using expressions (1.13)–(1.14) and the explicit dependence of C1, GC1 and ξ1 on N1, the system of Eqs. (1.15)–(1.16) becomes: " # 2 ET 2ðC0 −kÞ N1 Qτ − N1 + RTn0 1+γ 1 + 2 ðC0 − 1Þ C0 3η h η   i 1 2 2 2e ρ N2 Qτ e ρ ð4 k − 2Þ + 1 − 2 k ðN1 ϑÞ ρ + + h 2η i η 1 2 2 2 e ρ −2 e ρ ðN1 ϑÞ ρ + ðN1 ϑÞ ρ ρ

" −

!

η

e ρ N2 RT log

eρ 1

ðN1 ϑÞ ρ

!#

η

− log 1 −

eρ 1

ðN1 ϑÞ ρ

1 ρ

ðN1 ϑÞ ρ

= 0;

h η i 1 2 2 N2 Qτ e ρ −kðN1 ϑÞ ρ ET γ − 2η + ðk − 1Þ RT log ð1 − kÞ − k RT log k N2 ð1 + γ Þ 2 e ρ − 2 e ρη ðN ϑÞ 1ρ + ðN ϑÞ ρ2 1 1

ð1:9Þ

( +

!

η

η

ð1:10Þ

ð1:17Þ

h η i η 1 2 2 e ρ N2 Qτ e ρ ð1 − 2kÞ + ðk − 1ÞkðN1 ϑÞ ρ h 2η i + η 1 2 2 e ρ − 2 e ρ ðN1 ϑÞ ρ + ðN1 ϑÞ ρ ρ

RT e ρ log

E24 γ lim : = ET Y + ∞ ET 2ð1 + γ Þ

ð1:13Þ

ðC1 − kÞ2 Qτ N22 + GC1 RTn1 N2

η

lim

;

If we are looking for an internal solution (E1 + E2 b ET) then the first order conditions of problem (1.12) are:

E14 + E24 b 1; ET

F

E1 ;E2

The functions f 1⁎ and f 2⁎ are explicitly given by:

ð1:5Þ

ET =

ð1:11Þ

lim

eρ ðN1 ϑÞ

1 ρ

h ηi 1 + ðN1 ϑÞ ρ − e ρ log 1 − 1

ðN1 ϑÞ ρ

!)

η

eρ ðN1 ϑÞ

1 ρ

= 0:

ð1:18Þ

A. Roma, D. Pirino / Ecological Economics 68 (2009) 2594–2606

Fig. 3. Numerical estimate of the total percentage of energy spent for production as a function of the strike concentration and for different values of the total energy ET. The set of chosen parameters is reported in Table 1.

There are no analytical (closed form) solutions to the system (1.17)–(1.18). From Eq. (1.18) it is possible to derive N2 as a function of N1 and then substitute into Eq. (1.17) that becomes an equation in N1 only. With this substitution (1.17) can be easily solved numerically. We obtain an equation of the form F (N1) = 0. We choose to solve this equation by computing the function F (N1) on a lattice of the type Ni1 = iΔN1, with i = 0,1,2,…. We take as an approximation for the solution the point Ni1⁎ such that |F (Ni1⁎)| = mini |F (Ni1)|. The smaller ΔN1, the more accurate the estimation of the solution. We choose a value of ΔN1 such that ΔN1 ≈10 − 3 . N1i⁎ We solve systems (1.17)–(1.18) with the parameter set in Table 1 and for different values of k and ET. Specifically, we choose k =[75%, 80%, 85%, 90%, 95%, 99%] for the strike concentration and ET = [108 J, 1010 J, 1012 J, 1014 J, 1016 J] for the total energy available. We focus on the case of copper, for which η = 7.84 and ρ =1.80 (see Nguyen and Yamamoto, 2007). Fig. 3 shows the numerical estimate for the percentage Pu100 × E1 + E2 of energy spent in the whole production process as a function of ET the strike concentration and for different values of ET . The producer tends to spend half of his available energy, provided that he has a sufficient energy supply (ET ≥ 1014), no matter what the strike concentration k is. If less energy is available, the higher the strike concentration, the higher is the fraction of the total energy used by the producer to carry out the whole production process. References Andersen, B., Salamon, P., Berry, R.S., 1977. Thermodynamics in finite time: extremals for imperfect heat engines. Journal of Chemical Physics 66 (4), 1571–1577. Ayres, R.U., 2008. Sustainability economics: where do we stand? Ecological Economics 67 (2), 281–310. Ayres, R.U., Kneese, A.V., 1969. Production, consumption, and externalities. American Economic Review 59 (3), 282–297. Ayres, R.U., Miller, S.M., 1980. The role of technical change. Journal of Environmental Economics and Management 7, 353–371. Ayres, R.U., Warr, B., 2005. Accounting for growth: the role of physical work. Structural Change and Economic Dynamics 16 (2), 181–209. Ayres, R.U., Ayres, L.W., Warr, B., 2003. Exergy, power and work in the US economy, 1900–1998. Energy 28 (3), 219–273. Bejan, A., 1996. Entropy Generation Minimization: The Method of Thermodynamic Optimization of Finite-size Systems and Finite-time Processes. CRC Press, Boca Raton. Berry, R.S., Salamon, P., Heal, G., 1978. On a relation between economic and thermodynamic optima. Resource and Energy 1 (2), 125–137. Berry, R.S., Kazakov, V., Tsirlin, A.M., Sieniutycz, S., Szwast, Z., 2000. Thermodynamic Optimization of Finite-time Processes. Wiley. Bodo, G., Signorini, L.F., 1987. Short-term forecasting of the industrial production index. International Journal of Forecasting 3 (2), 245–259. Bodo, G., Signorini, A.C.L.F., 1991. Forecasting the Italian industrial production index in real time. Journal of Forecasting 10 (3), 285–299.

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