How energy conversion drives economic growth far from the equilibrium of neoclassical economics

A production function $Y(K,L,E;t)$ expresses the value added Y of the goods and services produced in an economic system as a function of the production factors and time. The output Y is the gross domestic product (GDP) of a national economy, or a part of the GDP produced by an economic subsector. Y is a state function⁸ of the inputs $K,L,E$ in the same sense as the potential energy of a particle in a conservative force field, or thermodynamic potentials like Gibbʼs free energy, are state functions of their spatial or thermodynamic variables.

The growth equation

$$\begin{eqnarray}&&\frac{{\rm d}Y}{Y}=\alpha \frac{{\rm d}K}{K}+\beta \frac{{\rm d}L}{L}+\gamma \frac{{\rm d}E}{E}+\delta \frac{{\rm d}t}{t-{{t}_{0}}},\qquad \delta \equiv \frac{t-{{t}_{0}}}{Y}\frac{\partial Y}{\partial t},\end{eqnarray} \tag{ 1 }$$

which is just the total differential of the production function $Y(K,L,E;t)$, divided by Y, is governed by the output elasticities of capital, α, labor, β, and energy, γ, defined as

$$\begin{eqnarray}&&\alpha \equiv \frac{K}{Y}\frac{\partial Y}{\partial K},\ \beta \equiv \frac{L}{Y}\frac{\partial Y}{\partial L},\ \gamma \equiv \frac{E}{Y}\frac{\partial Y}{\partial E}.\end{eqnarray} \tag{ 2 }$$

$\alpha ,\beta$, and γ measure, how the growth rate of output, ${\rm d}Y/Y$, depends on the growth rates ${\rm d}K/K$, ${\rm d}L/L$, ${\rm d}E/E$ of the inputs $K,L,E$. From equations (1) and (2) we see that the output elasticity of a production factor measures the productive power of the factor in the sense that (roughly speaking) it gives the percentage of output change when the factor changes by 1%, while the other factors stay constant. Since they give the economic weights (productive powers) of the production factors, the output elasticities are of fundamental importance for the theory of production and growth. δ in equation (1) describes the influence of creativity, with t₀ being the base year of the time-series analysis performed in section 3; we will see that creativity manifests itself in time-changing technology paramenters.

At any fixed time t an increase of all inputs by the same factor κ must increase output by κ, because at the fixed state of technology that exists at the given time t a, say, doubling of the production system doubles output; in other words: two identical factories with identical inputs of capital, labor and energy produce twice as much output as one factory. Thus, the production function must be linearly homogeneous in $K,L,E$. As a consequence, the Euler relation, in combination with equation (2), yields the so-called 'constant returns to scale' relation:

$$\begin{eqnarray}&&\alpha +\beta +\gamma =1.\end{eqnarray} \tag{ 3 }$$

Capital, labor, and energy can be treated as independent variables within a certain region of positive $K,L,E$-space. This region is constrained technologically by the limit to the degree of capacity utilization $\eta (K,L,E)$ and the limit to the degree of automation $\rho (K,L,E)$. As pointed out in [1, 3], we have

$$\begin{eqnarray}&&\eta (K,L,E)=\eta _{0}^{*}{{\left( \frac{L}{K} \right)}^{\lambda }}{{\left( \frac{E}{K} \right)}^{\nu }},\quad \rho (K,L,E)=\eta \frac{K}{{{K}_{m}}(Y)}.\end{eqnarray} \tag{ 4 }$$

The parameters $\eta _{0}^{*}$, λ, and ν can be determined from empirical data on capacity utilization. An example is given in section 4. K_m(Y) is the capital stock that would be required for maximally automated production of a given output Y; in this state of the economy, an additional unit of labor would contribute no longer to the growth of output.

Obviously, $\eta (K,L,E)$ cannot exceed unity, because the capital stock cannot work at more than full capacity. (Energy inputs that exceed the power limits the machines are designed for, do not make more productive use of the capital stock. Rather, they may damage the machines; and too much heating or cooling of rooms is also counterproductive. Neither does it make sense to employ more workers than a production system, working at full capacity, requires for its operation and maintenance.) Furthermore, in a given state of technology at time t, the degree of automation of the capital stock has a limit ${{\rho }_{T}}(t)$ at $\eta =1$. This limit depends on the mass and the volume of the energy-conversion devices and information processors the production system can accomodate when producing the output Y. The outmost limit to automation is 1, when $K={{K}_{m}}(Y)$ and $\eta =1$. Thus, the technological constraints on the combinations of capital, labor, and energy are

$$\begin{eqnarray}&&\eta (K,L,E)\leqslant 1,\qquad \rho (K,L,E)\leqslant {{\rho }_{T}}(t)\leqslant 1.\end{eqnarray} \tag{ 5 }$$

The behavioral assumptions that firms maximize profit and individuals maximize welfare originate from microeconomics. But they are also applied to macroeconomic systems [4, 8]. Much more important than the difference between these two assumptions is the difference between neglect and non-neglect of the technological constraints (5) when calculating the equilibria in which macroeconomic systems supposedly operate at a given time t. In order to elaborate that, it is sufficient to look into the necessary conditions for profit and welfare maximization.

For notational convenience we identify $K,L,E$ with the components ${{X}_{1}},{{X}_{2}},{{X}_{3}}$ of the vector

$$\begin{eqnarray}&&{\bf X}=({{X}_{1}},{{X}_{2}},{{X}_{3}})\equiv (K,L,E).\end{eqnarray} \tag{ 6 }$$

In this notation, and with the help of the slack variables ${{{\bf X}}_{\eta }}$ and ${{{\bf X}}_{\rho }}$, the constraints (5) can be brought into the form of equations,

$$\begin{eqnarray}&&{{f}_{\eta }}({\bf X},t)=0,\quad {{f}_{\rho }}({\bf X},t)=0.\end{eqnarray} \tag{ 7 }$$

The slack variables for labor, energy, and capital are ${{L}_{\eta }},{{E}_{\eta }}$ and ${{K}_{\rho }}$. They define the range in factor space within which the factors can vary independently at time t. Inserting them into equation (4) yields the the explicit form of equation (7) as

$$\begin{eqnarray}&&{{f}_{\eta }}({\bf X},t)\equiv \eta _{0}^{*}{{\left( \frac{L+{{L}_{\eta }}}{K} \right)}^{\lambda }}{{\left( \frac{E+{{E}_{\eta }}}{K} \right)}^{\nu }}-1=0,\quad {{f}_{\rho }}({\bf X},t)\equiv \frac{K+{{K}_{\rho }}}{{{K}_{m}}(Y)}-{{\rho }_{T}}(t)=0.\end{eqnarray} \tag{ 8 }$$

We assume that the three production factors $({{X}_{1}},{{X}_{2}},{{X}_{3}})$ have the exogeneously given prices per factor unit $({{p}_{1}},{{p}_{2}},{{p}_{3}})\equiv {\bf p}$, so that total factor cost is ${\bf p}({\bf t})\cdot {\bf X}({\bf t})=\sum _{i=1}^{3}{{p}_{i}}(t){{X}_{i}}(t)$. Economic equilibrium is defined as the state in which profit

$$\begin{eqnarray}&&G({\bf X},{\bf p},t)\equiv Y({\bf X},t)-{\bf p}\cdot {\bf X}\end{eqnarray} \tag{ 9 }$$

is maximum.

The necessary condition for a maximum of profit $G\equiv Y-{\bf p}\cdot {\bf X}$, subject to the technological constraints (8), is:

$$\begin{eqnarray}&&\vec{\nabla }\left[ Y({\bf X};t)-\mathop{\sum }\limits_{i=1}^{3}{{p}_{i}}(t){{X}_{i}}(t)+{{\mu }_{\eta }}{{f}_{\eta }}({\bf X},t)+{{\mu }_{\rho }}{{f}_{\rho }}({\bf X},t) \right]=0,\end{eqnarray} \tag{ 10 }$$

where $\vec{\nabla }\equiv (\partial /\partial {{X}_{1}},\partial /\partial {{X}_{2}},\partial /\partial {{X}_{3}})$ is the gradient in factor space; ${{\mu }_{\eta }}$ and ${{\mu }_{\rho }}$ are Lagrange multipliers. (The sufficient condition for profit maximum involves a sum of second-order derivatives. One assumes that the extremum of profit at finite X_i is the maximum.) Equation (10) yields the three equilibrium conditions

$$\begin{eqnarray}&&\frac{\partial Y}{\partial {{X}_{i}}}-{{p}_{i}}+{{\mu }_{\eta }}\frac{\partial {{f}_{\eta }}({\bf X},t)}{\partial {{X}_{i}}}+{{\mu }_{\rho }}\frac{\partial {{f}_{\rho }}({\bf X},t)}{\partial {{X}_{i}}}=0,\quad i=1,2,3.\end{eqnarray} \tag{ 11 }$$

Multiplication of (11) with ${{X}_{i}}/Y$, and writing the output elasticities $\alpha ,\beta ,\gamma$, defined in equation (2), as

$$\begin{eqnarray}&&{{\epsilon }_{i}}\equiv \frac{{{X}_{i}}}{Y}\frac{\partial Y}{\partial {{X}_{i}}},\quad i=1,2,3,\end{eqnarray} \tag{ 12 }$$

brings the equilibrium conditions into the form

$$\begin{eqnarray}&&{{\epsilon }_{i}}\equiv \frac{{{X}_{i}}}{Y}\frac{\partial Y}{\partial {{X}_{i}}}=\frac{{{X}_{i}}}{Y}\left[ {{p}_{i}}-{{\mu }_{\eta }}\frac{\partial {{f}_{\eta }}}{\partial {{X}_{i}}}-{{\mu }_{\rho }}\frac{\partial {{f}_{\rho }}}{\partial {{X}_{i}}} \right],\quad i=1,2,3.\end{eqnarray} \tag{ 13 }$$

One can write these equilibrium conditions as

$$\begin{eqnarray}&&{{\epsilon }_{i}}=\frac{{{X}_{i}}[{{p}_{i}}+{{s}_{i}}]}{\mathop{\sum }\limits_{i=1}^{3}{{X}_{i}}[{{p}_{i}}+{{s}_{i}}]},\quad i=1,2,3,\end{eqnarray} \tag{ 14 }$$

by summing the left and right hand sides of (13) over $i=1,2,3$, observing that $\sum _{i=1}^{3}{{\epsilon }_{i}}=1$ acording to (3), and expressing Y by the other terms in the resulting relation. The s_i are

$$\begin{eqnarray}&&{{s}_{i}}\equiv -{{\mu }_{\eta }}\frac{\partial {{f}_{\eta }}}{\partial {{X}_{i}}}-{{\mu }_{\rho }}\frac{\partial {{f}_{\rho }}}{\partial {{X}_{i}}}.\end{eqnarray} \tag{ 15 }$$

They map the constraints into monetary terms in our nonlinear optimization. We call them 'generalized shadow prices'—'generalized,' in order to avoid confusion with the concept of 'shadow prices,' which refers to terms of the same form one obtains in linear optimization⁹ . The Lagrange multpliers ${{\mu }_{\eta }}$ and ${{\mu }_{\rho }}$ depend on the factor prices p_i, the production function Y, and the derivatives of ${{f}_{\eta }}({\bf X},t),{{f}_{\rho }}({\bf X},t)$. Y cannot be known without knowledge of the output elasticities ${{\epsilon }_{i}}$. Equations (14) would only be suited for computing output elasticities, if the neoclassical optimum would lie within the region of $K,L,E$ space that is accessible in the presence of the technological constraints (5). This is not the case for the prices of capital, labor, and energy we have known so far. Figure 6 shows an example how the constraint from capacity utilization prevents an economic system from 'rolling downhill' in its cost mountain—toward the region where profit would be maximum.

But in the absence of technological constraints, all of positive $K,L,E$ space would be accessible to the economic system, the Lagrange multipliers ${{\mu }_{\eta }}$, ${{\mu }_{\rho }}$ and the generalized shadow prices s_i would be zero, and the equilibrium conditions (14) would turn into the cost-share theorem: on the rhs of equation (14) the numerator would be the cost ${{p}_{i}}{{X}_{i}}$ of the factor X_i, the denominator would be the sum of all factor costs, and the quotient, which is equal to the output elasticity ${{\epsilon }_{i}}$, would represent the cost share of X_i in total factor cost.

This would also justify the neoclassical duality of production factors and factor prices, according to which all relevant economic information on production is in the prices. This duality, which is often used in orthodox growth analyses, is a consequence of the Legendre transformation that results from the requirement that profit $G({\bf X},{\bf p})$ is maximum without any constraints on ${{X}_{1}},{{X}_{2}},{{X}_{3}}$. Then equation (11) would hold with ${{\mu }_{\eta }}=0={{\mu }_{\rho }}$ and yield equilibrium values ${{X}_{1M}}({\bf p}),{{X}_{2M}}({\bf p}),{{X}_{3M}}({\bf p})$. With ${{{\bf X}}_{M}}({\bf p})$ the profit function would turn into the price function

$$\begin{eqnarray}&&G({{{\bf X}}_{M}}({\bf p}),{\bf p})=Y({{{\bf X}}_{M}}({\bf p}))-{\bf p}\cdot {{{\bf X}}_{M}}({\bf p})\equiv g({\bf p}).\end{eqnarray} \tag{ 16 }$$

The essential information on production would be contained in the price function $g({\bf p})$, which is the Legendre transform of the production function $Y({\bf X})$. (This is in formal analogy to the Hamilton function being the Legendre transform of the Lagrange function in classical mechanics, or to enthalpy and free energy being Legendre transforms of internal energy in thermodynamics.) However, because of the technological constraints and the resulting generalized shadow prices, the cost-share theorem and equation (16) are not valid in general. For an understanding of the economy, prices are not enough.

Maximization of overall welfare is an alternative to the derivation of equilibrium conditions from profit maximization. It tests the sensitivity of the equilibrium relations between output elasticities, on the one hand, and factor quantities and prices, on the other hand, to modified behavioral assumptions. The optimization calculus is performed in analogy to the derivation of the Lagrange equations in classical mechanics; for more details see [1].

In welfare optimization one assumes that society maximizes (undiscounted) time-integrated utility U [8]. The integral W is called (overall) welfare. We take it between the times t₀ and t₁, during which the relevant variables evolve along a curve symbolized by $[s]$. We limit the calculation to the simplest case that the utility function U only depends on consumption C: $U=U[C]$. A simple function with the property of decreasing marginal utility¹⁰ is, for instance,

$$\begin{eqnarray}&&U(C)={{C}_{0}}{\rm ln} C/{{C}_{0}}+{{U}_{0}}.\end{eqnarray} \tag{ 17 }$$

Macroeconomic consumption is the difference between output and capital formation.

Thus, the optimization problem is: maximize overall welfare

$$\begin{eqnarray}&&W[s]=\int _{{{t}_{0}}}^{{{t}_{1}}}U[C]{\rm d}t,\end{eqnarray} \tag{ 18 }$$

subject to the technological constraints (8) and the additional economic constraint that the total cost ${\bf p}\cdot {\bf X}$ of producing consumption C by means of the factors $({{X}_{1}},{{X}_{2}},{{X}_{3}})\equiv {\bf X}$ must not diverge. Rather, its magnitude c_f (t) must be finite at all times t, where each price per factor unit, p_i, is exogeneously given:

$$\begin{eqnarray}&&{{c}_{f}}(t)-\mathop{\sum }\limits_{i=1}^{3}{{p}_{i}}(t){{X}_{i}}(t)=0.\end{eqnarray} \tag{ 19 }$$

Therefore, besides ${{\mu }_{\eta }}$ and ${{\mu }_{\rho }}$, the variational formalism of intertemporal welfare optimization includes the Lagrange multiplier μ, which takes care of (19). The curve $[s]$, of which $W[s]$ is a functional, depends on the variables that enter consumption C. Output (per unit time) is described by the macroeconomic production function $Y({\bf X};t)$. Part of Y goes into consumption C and the rest into new capital formation ${{\dot{X}}_{1}}\equiv {\rm d}{{X}_{1}}/{\rm d}t$ plus replacement of depreciated capital. Economic research institutions provide the price of capital utilization p₁ as the sum of net interest, depreciation and state influences. We use this price. Since it already includes depreciation, we can omit explicit reference to the depreciation rate, so that consumption is given by

$$\begin{eqnarray}&&C({\bf X},{{\dot{X}}_{1}})=Y({\bf X};t)-{{\dot{X}}_{1}}.\end{eqnarray} \tag{ 20 }$$

All taken together we have to optimize

$$\begin{eqnarray}&&W[s]=\int _{{{t}_{0}}}^{{{t}_{1}}}{\rm d}t\left\{ U[C({\bf X},{{{\dot{X}}}_{1}})]+\mu [{{c}_{f}}(t)-{\bf p}\cdot {\bf X}]+{{\mu }_{\eta }}{{f}_{\eta }}({\bf \vec{X}},t)+{{\mu }_{\rho }}{{f}_{\rho }}({\bf \vec{X}},t) \right\}.\end{eqnarray} \tag{ 21 }$$

Thus, $W[s]$ is a functional of the curve $[s]=\{t,{\bf X}:\quad {\bf X}={\bf X}(t),\ {{t}_{0}}\leqslant t\leqslant {{t}_{1}}\}$.

Consider another curve $[s,{\bf h}]=\{t,{\bf X}:\quad {\bf X}={\bf X}(t)+{\bf h}(t),\ {{t}_{0}}\leqslant t\leqslant {{t}_{1}}\}$ close to $[s]$, which goes through the same end points so that ${\bf h}({{t}_{1}})=0={\bf h}({{t}_{0}})$. Its functional $W[s,{\bf h}]$ is obtained from $W[s]$ by changing ${\bf X}$ to ${\bf X}+{\bf h}$ and ${{\dot{X}}_{1}}$ to ${{\dot{X}}_{1}}+{{\dot{h}}_{1}}$ everywhere in the integrand of equation (21). Since ${\bf h}$ is small, the integrand can be approximated by its Taylor expansion up to first order in ${\bf h}$ and ${{\dot{h}}_{1}}$.

The necessary condition for the maximum of overall welfare is that the variation $\delta W\equiv W[s,{\bf h}]-W[s]$ vanishes. $\delta W$ contains the time integral of

$$\begin{eqnarray}\begin{array}{rcl} \delta U & \equiv & U[C({\bf X}+{\bf h},{{{\dot{X}}}_{1}}+{{{\dot{h}}}_{1}})]-U[C({\bf X},{{{\dot{X}}}_{1}})] \\ {} & = & \frac{{\rm d}U}{{\rm d}C}{\rm d}C=\frac{{\rm d}U}{{\rm d}C}\left[ \mathop{\sum }\limits_{i=1}^{3}\frac{\partial C}{\partial {{X}_{i}}}{{h}_{i}}+\frac{\partial C}{\partial {{{\dot{X}}}_{1}}}{{{\dot{h}}}_{1}} \right]. \\ \end{array}\end{eqnarray} \tag{ 22 }$$

Partial integration shifts the time derivative from ${{\dot{h}}_{1}}$ to the functions multiplying it. Then, the integrand in $\delta W$ is a sum of terms that multiply ${{h}_{1}},{{h}_{2}},{{h}_{3}}$. The condition

$$\begin{eqnarray}&&\delta W\equiv W[s,{\bf h}]-W[s]=0\end{eqnarray} \tag{ 23 }$$

is satisfied, if these terms vanish. Observing (20) one sees that they do, if

$$\begin{eqnarray}&&\frac{{\rm d}U}{{\rm d}C}\frac{\partial Y}{\partial {{X}_{i}}}=\mu {{p}_{i}}-{{\mu }_{\eta }}\frac{\partial {{f}_{\eta }}}{\partial {{X}_{i}}}-{{\mu }_{\rho }}\frac{\partial {{f}_{\rho }}}{\partial {{X}_{i}}}-\frac{{\rm d}}{{\rm d}t}\left( \frac{{\rm d}U}{{\rm d}C} \right){{\delta }_{i,1}},\quad i=1,2,3;\end{eqnarray} \tag{ 24 }$$

the Kronecker delta ${{\delta }_{i,1}}$, is 1 for i = 1 and 0 otherwise¹¹ .

Equations (24) are the general equilibrium conditions for an economic system subject to cost limits and technological constraints, if the behavioral assumption is that society optimizes time-integrated utility. They correspond to the equilibrium conditions (11) from profit optimization. The difference between the two equilibrium conditions becomes negligible under the conditions indicated below equation (25).

Dividing equations (24) by ${\rm d}U/{\rm d}C$ and multiplying them by ${{X}_{i}}/Y$produces the output elasticities ${{\epsilon }_{i}}$ on the lhs, and the rhs involves terms that, after some manipulations, can be brought into the form of (modified) generalized shadow prices

$$\begin{eqnarray}&&{{s}_{i}}\equiv -\frac{{{\mu }_{\eta }}}{\mu }\frac{\partial {{f}_{\eta }}}{\partial {{X}_{i}}}-\frac{{{\mu }_{\rho }}}{\mu }\frac{\partial {{f}_{\rho }}}{\partial {{X}_{i}}}-{{\delta }_{i,1}}\frac{1}{\mu }\frac{{\rm d}}{{\rm d}t}(\frac{{\rm d}U}{{\rm d}C}),\quad i=1,2,3.\end{eqnarray} \tag{ 25 }$$

With these modified s_i the equilibrium conditions (24) assume the form of equation (14).

The shadow prices (25) differ from those of (15) in two aspects. First, there is the term ${{\delta }_{i,1}}\frac{1}{\mu }\frac{{\rm d}}{{\rm d}t}(\frac{{\rm d}U}{{\rm d}C})$. This term originates from equation (20) and is due to taking capital formation into account in intertemporal utility optimization, whereas capital formation is no issue in profit optimization. It vanishes, if one can disregard decreasing marginal utility and approximate the utility function $U[C]$ in (17) by its Taylor expansion up to first order in $C/{{C}_{0}}-1$. Second, the ratios ${{\mu }_{\eta }}/\mu$, ${{\mu }_{\rho }}/\mu$ of Lagrange multipliers take the positions of the Lagrange multipliers ${{\mu }_{\eta }}$, ${{\mu }_{\rho }}$ in (15). If one does profit maximization subject to the additional constraint ${\bf \vec{p}}\cdot {\bf \vec{X}}={{c}_{f}}$, which fixes factor cost, one gets the quotients of the Lagrange multipliers as in (25). Thus, the second difference is rather a formal one.

As in profit maximization, because of the technological constraints the cost-share theorem is not generally valid in intertemporal utility maximization.

From here on we consider economies that cannot operate in the neoclassical maximum of profit or welfare, because the barriers from the technological constraints and the associated generalized shadow prizes block the access to this maximum. Consequently, the equilibrium point may be either in the interior or at the boundary of the accessible region in $K,L,E$ space.

Suppose an economy has the maximum of its objective function right at a barrier, where the '=' sign holds in (5). Nevertheless, it might not operate there either, because of several reasons not taken into account so far. For instance, entrepreneurs prefer combinations of $K,L,E$, in which the quantities of labor L and energy E, which handle and power the capital stock K of a certain degree of automation ρ, produce an output $Y(K,L,E)$ that meets demand. Since demand is fluctuating, entrepreneurs have to adjust the degree of capacity utilization η, defined in (4), accordingly. This cost-saving flexibility requires operation in points of factor space that are positioned at some distance from the barrier at $\eta (KL,E)=1$. In such a situation, the technological constraint on the degree of capacity utilization acts as a 'virtually binding' constraint.

Conceivably, there are yet other aspects that keep the economy at some distance from the constraint barrier(s). For instance, entrepreneurs may not wish to automate the capital stock as much as the technological limit ${{\rho }_{T}}(t)$ would allow, because they would have to pay compensations to laid-off workers according to contracts and negotiations with labor unions; legal and social obligations may matter as well. We call these aspects 'soft' constraints.

Thus, realistic economic decision makers operate in regions of $K,L,E$ space, where the '<' sign holds in equations (5) and where $K,L$, and E are independent variables.

Production functions $Y(K,L,E;t)$, which describe economic evolution, are state functions and integrals of the growth equation (1) along a convenient path in factor space from an initial point $({{K}_{0}},{{L}_{0}},{{E}_{0}})$ with Y₀ at time t₀ to the point $(K,L,E)$ at the time of interest t. It is convenient to work with quantities that are normalized to their magnitudes in the base year t₀. Thus, we work with the dimensionless variables

$$\begin{eqnarray}&&k(t)\equiv \frac{K(t)}{{{K}_{0}}},\quad l(t)\equiv \frac{L(t)}{{{L}_{0}}},\quad e(t)\equiv \frac{E(t)}{{{E}_{0}}},\quad y[k,l,e;t]\equiv \frac{Y[k{{K}_{0}},l{{L}_{0}},e{{E}_{0}};t]}{{{Y}_{0}}}.\end{eqnarray} \tag{ 26 }$$

The growth equation (1) and the output elasticities (2) are invariant under the transformations (26).

The production function $y[k,l,e;t]$ as a state function of $k,l,e$ must be twice differentiable with respect to $k,l,e$. This results in the partial differential equations

$$\begin{eqnarray}&&k\frac{\partial \alpha }{\partial k}+l\frac{\partial \alpha }{\partial l}+e\frac{\partial \alpha }{\partial e}=0,\quad k\frac{\partial \beta }{\partial k}+l\frac{\partial \beta }{\partial l}+e\frac{\partial \beta }{\partial e}=0,\quad l\frac{\partial \alpha }{\partial l}=k\frac{\partial \beta }{\partial k}\quad \end{eqnarray} \tag{ 27 }$$

for the output elasticities, where (3) has been taken into account. These equations correspond to the Maxwell relations in thermodynamics. The most general solutions of (27) are

$$\begin{eqnarray}&&\alpha =A\left( \frac{l}{k},\frac{e}{k} \right),\ \beta =B\left( \frac{l}{k},\frac{e}{k} \right)={{\int }^{k}}\frac{l}{k^{\prime} }\frac{\partial A}{\partial l}{\rm d}k^{\prime} +J\left( \frac{l}{e} \right),\ \gamma =1-\alpha -\beta ;\end{eqnarray} \tag{ 28 }$$

here $A(l/k,e/k)$ and $J(l/e)$ are any differentiable functions of their arguments. The output elasticities, and thus the combinations of $k,l,e$, must satisfy the restrictions

$$\begin{eqnarray}&&\alpha \geqslant 0,\quad \beta \geqslant 0,\quad \gamma =1-\alpha -\beta \geqslant 0,\end{eqnarray} \tag{ 29 }$$

which result from the technical-economic requirement that all output elasticities must be non-negative. Otherwise, the increase of an input would result in a decrease of output—a situation the economic actors will avoid.

It is not hard to verify with the help of equations (1) and (28) that the corresponding general form of the twice-differentiable, linearly homogeneous production function is given by the rhs of

$$\begin{eqnarray}&&y=e\mathcal{F}\left[ \frac{l}{k},\frac{e}{k};t \right]\equiv e\mathcal{F}\left[ u,v;t \right],\end{eqnarray} \tag{ 30 }$$

where $u\equiv l/k,\ v\equiv e/k$.

We insert (30) into the growth equation that results from (1) by the transformation (26) to dimensionless variables. This yields

$$\begin{eqnarray}&&\frac{{\rm d}\mathcal{F}}{\mathcal{F}}=\beta (u,v)\frac{{\rm d}u}{u}-\left[ \alpha (u,v)+\beta (u,v) \right]\frac{{\rm d}v}{v}+\delta \frac{{\rm d}t}{t-{{t}_{0}}};\ \delta \equiv \frac{t-{{t}_{0}}}{\mathcal{F}}\frac{\partial \mathcal{F}}{\partial t}.\end{eqnarray} \tag{ 31 }$$

The economic boundary conditions that would allow to determine the output elasticities (28) and the production function (30) exactly for a given economic system are not known, and never will be, because—according to the theory of partial differential equations—they would require knowledge of β on a surface and of α on a curve in $k,l,e$ space. In this sense, all output elasticities and production functions are approximations.

The simplest solutions of equation (27) are constant output elasticities ${{\alpha }_{0}}$, ${{\beta }_{0}}$, and ${{\gamma }_{0}}$. The corresponding integral of the growth equation is the energy-dependent version of the well-known Cobb–Douglas production function:

$$\begin{eqnarray}&&{{y}_{{\rm CDE}}}=y_{{\rm CDE}}^{0}{{k}^{{{\alpha }_{0}}}}{{l}^{{{\beta }_{0}}}}{{e}^{1-{{\alpha }_{0}}-{{\beta }_{0}}}}=y_{{\rm CDE}}^{0}e{{\left( \frac{k}{e} \right)}^{{{\alpha }_{0}}}}{{\left( \frac{l}{e} \right)}^{{{\beta }_{0}}}}.\end{eqnarray} \tag{ 32 }$$

The simplest factor-dependent solutions are the output elasticities ${{\alpha }_{L1}}=a(l+e)/k$, ${{\beta }_{L1}}=a\left( cl/e-l/k \right)$, and ${{\gamma }_{L1}}=1-ae/k-acl/e$. With these elasticities integration of the growth equation yields the (first) LinEx production function,

$$\begin{eqnarray}&&{{y}_{L1}}=y_{L1}^{0}e{\rm exp} \left[ a\left( 2-\frac{l+e}{k} \right)+ac\left( \frac{l}{e}-1 \right) \right],\end{eqnarray} \tag{ 33 }$$

which depends linearly on energy and exponentially on factor quotients. Here $a,c$, and $y_{L1}^{0}$ are technology parameters: a indicates capital efficiency, and c indicates the energy demand of the fully utilized capital stock. Innovations may change them in time and contribute to δ in equation (1). The output elasticity of capital, ${{\alpha }_{L1}}$, vanishes for vanishing ratios of labor and energy to capital and thus reflects the law of diminishing returns. The output elasticity of labor, ${{\beta }_{L1}}$ vanishes, when the capital stock approaches the magnitude k_m required for maximum automation, and when simultaneously the energy input approaches the quantity e_m = ck_m that is demanded by the fully utilized capital stock k_m.

The technology parameters $a,c$ and y₀ of the LinEx function are determined from the empirical time series of output y_empirical(t). We allow for time dependencies of the technology parameters and model them by logistic functions [10], which are typical for growth in complex systems and innovation diffusion. Alternatively, we have used Taylor expansions of a(t) and c(t) in $(t-{{t}_{0}})$. The free coefficients of the logistics, or of the Taylor expansions, are determined by minimizing the sum of squared errors

$$\begin{eqnarray}&&{\rm SSE}=\mathop{\sum }\limits_{i}{{\left[ {{y}_{{\rm empirical}}}({{t}_{i}})-{{y}_{L1}}({{t}_{i}}) \right]}^{2}}.\end{eqnarray} \tag{ 34 }$$

The sum goes over all years t_i between the initial and the final observation time. It contains the empirical time series of output ${{y}_{{\rm empirical}}}({{t}_{i}})$, and the LinEx function ${{y}_{Lt}}({{t}_{i}})$ with the empirical time series of k, l and e as inputs at times t_i. Minimization is subject to the restrictions (29). Methodological details and the empirical time series of $y;k,l,e$ are given in [1].

The reproduction of economic growth in Germany, Japan, and the USA during the second half of the 20th century by the LinEx function is shown in figures 2–5. These figures are taken from [11]. Their left parts exhibit the empirical growth (squares) and the theoretical growth (circles) of the dimensionless output $y=Y/{{Y}_{0}}(\equiv q=Q/{{Q}_{0}})$, and the right parts present the empirical time series of the dimensionless factors capital $k=K/{{K}_{0}}$, labor $l=L/{{L}_{0}}$, and energy $e=E/{{E}_{0}}$. The base year t₀ is 1960 for Germany and the USA, and 1965 for Japan. Note the variations of inputs and outputs in conjunction with the oil price explosions. The price of a barrel of crude oil in inflation-corrected ${\rm US}{{\$ was driven by the OPEC boycott in response to the Yom-Kippur war from 15$ in 1973 to 53$ in 1975, and by the war between Iraque and Iran from 48$ in 1979 to 100$ in 1981. Then the oil price plummeted to 30$ in 1986 (with dramatic consequences for the Soviet Union). Between 1997 and 2011 it has risen again, from 18$ to 110$.

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The time-averaged output elasticities of capital, $\bar{\alpha }$, labor, $\bar{\beta }$, energy, $\bar{\gamma }$, and creativity, $\bar{\delta }$, are presented in table 1 for the economic systems: FRG TE (total economy of the Federal Republic of Germany before and after reunification), FRG I (German industrial sector 'Warenproduzierendes Gewerbe' (GWG)), Japan I (Japanese sector 'Industries'), and USA TE (total economy of the USA). R² is the adjusted coefficient of determination, and d_W is the Durbin–Watson coefficient (whose best value is 2). Both statistical quality measures are quite good.

Table 1.  Time-averaged LinEx output elasticities and statistical quality measures. Reproduced from [1], chapter 4, pp 206–208, with kind permission of Springer Science + Business Media.

	FRG TE	FRG I	Japan I	USA TE
System	1960–2000	1960–99	1965–92	1960–96
$\bar{\alpha }$	0.38 ± 0.09	0.37 ± 0.09	0.18 ± 0.07	0.51 ± 0.15
$\bar{\beta }$	0.15 ± 0.05	0.11 ± 0.07	0.09 ± 0.09	0.14 ± 0.14
$\bar{\gamma }$	0.47 ± 0.1	0.52 ± 0.09	0.73 ± 0.16	0.35 ± 0.11
$\bar{\delta }$	0.19 ± 0.2	0.12 ± 0.13	0.14 ± 0.19	0.10 ± 0.17
R2	>0.999	0.996	0.999	0.999
dW	1.64	1.9	1.71	1.46

We note that for all the systems considered the output elasticity of energy is much larger than energyʼs cost share of roughly 5%, and the output elasticity of labor is much smaller than laborʼs cost share of about 70%. The economic weight $\bar{\delta }$ of creativity, computed from the time changes of a(t) and c(t), is comparable to that of labor, $\bar{\beta }$.

Ayres and Warr have analyzed economic growth in the USA during the 20th century, using 'useful work' as the energy variable in the LinEx function. 'Useful work' is defined as exergy, multiplied by appropriate conversion efficiencies, plus physical work by animals. The 'useful work' data [12] already include most of the efficiency improvements that have occurred in the energy converting systems of the USA during the 20th century. In this case two constant technology parameters suffice to reproduce well the gross domestic product of the US economy between 1900 and 1998 [13, 14]. The time averages of the corresponding output elasticities of capital, labor and 'useful work' are similar to the ones in table 1.

These findings are supported by cointegration analyses [15]. The energy-dependent Cobb–Douglas function (32), whose constant output elasticities are similar to the time-averaged LinEx output elasticities, also reproduces economic growth more or less satisfactorily, albeit with larger discrepancies between theoretical and empirical growth during and after the 1973–1981 oil-price shocks [1, 14].

The reproduction of economic growth in Germany, Japan, and the USA during up to four decades with small residuals, and output elasticities that are for energy much larger and for labor much smaller than the cost shares of these factors, are the principal results of the KLEC-model applied in this section. The contribution of human creativity to growth appears as comparable to that of labor¹² . The fundamental presumption of the model is, that the macroeconomic production function $Y(K,L,E;t)$ (and its dimensionless version $y(k,l,e;t)$ as well) is a state function of the economic system, so that its infinitesimal change is a total differential and the integral of this differential between ${{K}_{0}},{{L}_{0}},{{E}_{0}}$ and $K,L,E$ does not depend on the path in factor space. Such $Y(K,L,E;t)$ must be twice differentiable with respect to $K,L,E$. From this property follow the growth equation (1) and the partial differential equations (27) for the output elasticities¹³ . Essential is that the output elasticities are not set equal to the cost shares, as it is done in mainstream economics, but that instead they are determined by fitting the production functions to the time series of output, where the restrictions (29)—output elasticities must be non-negative—have to be observed.

A number of analyses of past growth with alternative energy-dependent production functions, whose output elasticities satisfy (27), have yielded results similar to the ones presented here [1, 7]. Cobb–Douglas functions are the work-horses of mainstream econometrics. Unfortunately, they ignore the physical restrictions on factor substitution. But since the economic actors are aware of these restrictions, Cobb–Douglas is not too bad for reproductions of past economic growth, despite the insensitivity of the constant output elasticities to input fluctuations. (For scenarios of the future, however, the law of diminishing returns and the approach to the state of maximum automation should be incorporated in the output elasticities, as it is exemplified by the LinEx function (33).) The choice of the base year and the methods to minimize the sum of squared errors SSE (34) have had little influence on the econometric findings so far¹⁴ . Energy-dependent service production functions also reproduce the growth of the (West) German service sector quite well, yielding output elasticities of energy and labor that deviate from these factors' cost shares not quite as much as in the total economy [7, 16]. The increasing automation in agriculture and industries shifts more and more of the labor force into the less energy-intensive service sector. Nevertheless, computer- and electricity-based automation is progressing in that sector, too, especially in finance, logistics, and retail business.

Whatever the details of modeling energyʼs working as a factor of production, one result is common to all models:

If one foregoes cost-share weighting and determines the output elasticities of capital, labor, energy, and creativity econometrically, one gets for energy economic weights that exceed energyʼs cost share by up to an order of magnitude, and the Solow residual disappears. The production factor energy accounts for most, and creativity for the rest of the growth that neoclassical economics attributes to 'technological progress'.