# Economic Growth in the USA and Germany 1960–2013: The Underestimated Role of Energy

## Abstract

Economic growth in Germany and the USA between the years 1960 and 2013 is analyzed, using two LinEx production functions \(Y_{L1}(K,L,E;t)\) and \(Y_{L11}(K,L,E;t)\), which depend linearly on energy and exponentially on quotients of the production factors capital *K*(*t*), labor *L*(*t*), and energy *E*(*t*). The two production functions result from output elasticities that are the simplest, and next simplest, factor-dependent solutions of a set of partial differential equations and their asymptotic boundary conditions, which reflect technical trends in highly industrialized economies. The production functions depend on time *t* *explicitly*, when innovations and structural changes induce temporal variations of two technology parameters. The latter model capital effectiveness and the energy demand of the capital stock. In this article, we present some econometric evidence that, in conjunction with capital, the production factor energy provides the major contribution, and the temporal variations of the LinEx technology parameters provide the minor contribution to that part of economic growth that is assigned to “technical progress” by neoclassical economics. We also discuss assets and shortcomings of the two LinEx functions, and different aspects of using either primary energy data or exergy data for the energy variable *E*.

## Keywords

Energy Innovation Structural change Output elasticities Economic growth## Introduction

The issue of this article is the economic weight of production factors. To be more specific, the question is how, and how much, instrumental capital *K*(*t*), human labor *L*(*t*), and energy *E*(*t*) of a given economic system at time *t* contribute to the production and growth of the system’s output *Y*, which is the gross domestic product (GDP), or parts thereof. This question matters for growth theory and policies concerning energy and the environment. Looking into it, we use macroeconomic production functions *Y*(*K*(*t*), *L*(*t*), *E*(*t*); *t*), in which capital-activating energy complements capital-handling labor. We follow the usual assumption of mainstream economics that production functions are state functions.

State functions describe system properties and dynamics in the natural sciences and economics. They depend only on the actual magnitudes of the independent variables at time *t*, and not on the history of the system. (Examples from physics are hamiltonians in mechanics and entropy in thermodynamics.) The infinitesimal change of a state function is a total differential, whose line integral in the space of the independent variables does not depend on the chosen path.

The *implicit* time dependence of production functions is the time dependence of the factors *K*(*t*), *L*(*t*), *E*(*t*). We will omit the time argument of the production factors and simply write *Y*(*K*, *L*, *E*; *t*), further on. The *explicit* time dependence of the production function takes care of structural changes and innovations, or, in other words, the impact of human ideas, inventions, and value decisions.

Standard economics considers capital and labor as the factors of production^{1} that are the independent variables of the neoclassical production function *A*(*t*)*f*(*K*, *L*). Here, *A*(*t*) is a measure of innovation, called “technical progress.” In the wake of the first and the second oil-price shock between 1973 and 1981, economists like Hudson and Jorgenson (1974), Berndt and Jorgenson (1978), Berndt and Wood (1979), and Jorgenson (1984) introduced energy as a third factor of production. And so did Nordhaus (2008), when he considered economic problems of climate change. Then, the production function is \(A(t)f_E(K,L,E)\).

In these neoclassical production functions the economic weights of the production factors, which are the factors’ output elasticities defined in Eq. (2) below, are determined by the cost-share theorem. The cost-share theorem, derived from the maximization of profit or time-integrated utility without observing technological constraints on factor combinations, says that the output elasticity of a production factor must be equal to its share in total factor cost. Since in highly industrialized countries this share has been roughly 25% for capital, 70% for labor, and 5% for energy, the direct impact of the production factor energy on economic growth has been considered as marginal, at best, by mainstream economists; see e.g., Denison (1979).

With output elasticities obtained from the cost-share theorem, one has the problem of the Solow residual, which is the big difference between empirical and theoretical growth. Figure 1 shows an example. After its discovery by Solow (1957) the technical-progress functions *A*(*t*) were introduced in order to take care of this residual. Later, Solow (1994) commented that the dominating role of technical progress in standard growth theory “has led to a criticism of the neoclassical model: it is a theory of growth that leaves the main factor in economic growth unexplained.”

The laws of physics, especially the ones concerning energy conversion and entropy production, have led a growing number of scholars to look into energy and economic growth from the perspective of thermodynamic laws. And long ago Tryon (1927) already stated: “Anything as important in industrial life as power deserves more attention than it has yet received from economists\(\dots\) A theory of production that will really explain how wealth is produced must analyze the contribution of the element energy.”

Saunders (2008) carefully and critically discussed energy-dependent production functions that may be appropriate for energy consumption analyses, especially with respect to the rebound effect (Sorrell and Dimitropoulos 2008; Brockway et al. 2017). Stern (2011) proposed “to modify Solow’s growth model by adding an energy input that has low substitutability with capital and labor, while allowing the elasticity of substitution between capital and labor to remain at unity.” He expressed *gross* output by a function that “embeds a Cobb–Douglas function of capital (*K*) and labor (*L*) in a CES function of value added and energy (*E*).” Time-dependent augmentation indices \(A_L\) and \(A_E\) of labor and energy reflect changes in technology and factor quality. The embedded Cobb–Douglas function weighs capital and labor with their cost shares. Since *gross* output is the sum of GDP *plus* intermediate consumption, Stern’s nested CES function is a tool for research on issues different from the question which output elasticities of capital, labor, and energy are appropriate for describing the growth of GDP. Santos et al. (2016) looked into cointegration between output and (qualified) capital, labor, and energy inputs in Portugal. They concluded that “energy essentiality in production\(\dots\) is not *a priory* incompatible with the neoclassical assumptions of the cost share theorem.”

But the cost-share theorem is wrong, because technological constraints on factor combinations do exist. They are the limits to capacity utilization^{2} and automation and result in the destruction of the equality of output elasticities and factor cost shares. This has been shown explicitly by optimizing profit and time-integrated utility subject to these constraints (Kümmel et al. 2010, 2014, 2015).

An early study that determined output elasticities econometrically, instead of fixing them by the cost shares, is that of Tintner (1974), who, by applying an energy-dependent Cobb–Douglas function to the Austrian economy between 1955 and 1972, found energy output elasticities of more than 30%.

A set of differential equations with asymptotic boundary conditions, derived by Kümmel (1982) for output elasticities and production functions, yielded the (first) LinEx function \(Y_{L1}(K,L,E;t)\), given by Eq. (14) below, as the production function with the simplest factor-dependent output elasticities. The explicit time dependence of \(Y_{L1}(K,L,E;t)\) is that of two technology parameters, which formally are integration constants of the differential equations, and whose variations reflect efficiency alterations and structural changes. The requirement that output elasticities must be non-negative constrains these parameters, and possible factor combinations as well. With constant technology parameters, recalibrated once, in 1978, a nearly residual-free description of growth between 1960 and 1990 in the Federal Republic of Germany (FRG), the USA, and Japan was possible; the recessions due to the two oil-price shocks from 1973 to 1975 and 1979 to 1981 were well reproduced (Hall et al. 2001). Modeling continuous temporal changes of the technology parameters by, e.g., logistic functions, improved the agreement between theoretical and empirical growth (Kümmel et al. 2002) and even allowed to describe economic growth in Germany between 1960 and 2000, including the reunification of Eastern and Western Germany in 1990 (Kümmel 2011). From these studies resulted what we now call “the fundamental heresy” with respect to neoclassical economics: Energy’s output elasticities are much larger than its cost shares, and for labor just the opposite is true.

Ayres and Warr (2005) and Ayres and Warr (2009) used the LinEx function with exergy-based useful work data *U* instead of primary energy *E*. Exergy is the useful part of energy, which can be converted into any form of useful work. Calculation of “useful work” is difficult. Brockway et al. (2014) contributed toward a common useful work accounting framework, and Serrenho et al. (2014) calculated *U*, including residential energy needs, for 15 EU countries. Although in principle, all fossil and non-fossil primary energy carriers—except heat-transporting media—are 100% exergy, it was useful work data *U* that allowed Ayres and Warr (2005) to reproduce economic growth in the USA from 1900 to the mid 1970s with constant technology parameters and small residuals. Their useful work data incorporate the efficiency changes that occurred in the US economy during the first two-thirds of the twentieth century. Up to 1998, however, the residuals increased substantially. Then Ayres and Warr (2009) applied the LinEx function with *time-dependent* technology parameters to the USA and Japan from 1900 to 2005 (except for the years of World War II), and the former residuals became very small. In both studies, the output elasticities of useful work significantly exceed those of labor after 1910.

By now, new empirical data for output, capital, labor, and primary energy have become available to us for Germany and the USA. They are presented and discussed in Appendix 3 and cover the time from the base year \(t_0 = 1960\) to the end year 2013. Between these years occurred the dramatic fluctuations of the oil prices with their severe economic repercussions, the reunification of Germany in 1990, with the resulting structural break in the political entity called “Federal Republic of Germany” (FRG), the first global recession of the twenty-first century after the bursting of the US housing bubble in 2007, and considerable outsourcing of energy-intensive industries from the highly industrialized countries to industrially less developed regions of the world. The data have been obtained from the national accounts, labor statistics, and energy statistics; some care had to be dedicated to data consistency and the chaining of inflation-corrected data on output and capital.

In the following, the fundamental heresy is put to test with the new data, employing the first LinEx function \(Y_{L1}\) and a new one, \(Y_{L11}\), whose output elasticity of capital has been modified, and which is invariant against transformations of the base year.

In "Output Elasticities and Production Functions" we briefly review the set of differential equations for production functions and output elasticities. The most general form of the production function, in terms of a line integral in factor space over *any* output elasticities that satisfy the twice-differentiability conditions for state functions, is derived. The general solution, and special solutions of the partial differential equations for the output elasticities, is given, and criticism of the resulting production functions is considered. "Technology Parameters and their Dynamics" sketches the method of determining the time-dependent technology parameters “capital effectiveness” and “energy demand of the capital stock” of LinEx functions by SSE minimization subject to inequality restrictions. The resulting theoretical growth curves and the time-varying and time-averaged output elasticities for Germany and the USA are presented in the "Results" section. Finally, we provide "Summary and Conclusions."

## Output Elasticities and Production Functions

Macroeconomic production functions have been criticized by scholars who argued that aggregation of output and inputs should be in physical terms, exclusively (Robinson 1953, 1971; Fisher 1993; Kurz and Salvadory 1995; Felipe et al. 2003; Silverberg 2007). A response to this criticism is based on the understanding that the capital stock of industrialized economies consists of all energy-converting and information-processing machines in conjunction with all buildings and installations necessary for their protection and operation, and that output *Y* results from work performance and information processing by the combination of such capital with (routine) labor^{3} *L*(*t*) and energy *E*(*t*). Strictly speaking, this understanding excludes the residential sector with its considerable energy consumption for room heating and cooling. Chapter 4, Appendix 3, of Kümmel (2011) outlines the basic *physical* aggregation principles for output and capital in terms of work performance and information processing, and their relations to constant currency.

Occasionally, materials were considered as a fourth factor of production, besides capital, labor, and energy (Hudson and Jorgenson 1974; Berndt and Wood 1979). Although materials are indispensable for production, they are not active factors but rather the passive partners of the production process, during which their atoms and molecules are brought into their proper positions in the products by *K*, *L*, and *E*. Scarcity of key materials, e.g., phosphorus, or rare earths, will hamper production and growth. This has been pointed out in rich details by Valero and Valero (2015). These authors expect that we may enter an “Age of the Periodic Table.” A rough model of limits to growth due to materials scarcity, where the output elasticities in Eq. (1) are multiplied by recycling functions, is disregarded here; and the impact of emissions of particles like SO\(_2\), NO\(_X\), and CO\(_2\) on output elasticities and economic growth has been considered elsewhere (Kümmel 2016).

The three parts of this section concern: (1) The growth equation and its integral. (2) The three partial differential equations, which result from the Cauchy integrability conditions for the growth equation, the *K*, *L*, *E*- dependencies of the output elasticities that solve them, and the corresponding production functions. (3) Discussion of the assets and shortcomings of the LinEx functions \(Y_{L1}\) and \(Y_{L11}\). In order that this article is self-contained, Subsection 2.1 and parts of Subsection 2.2 contain material from prior publications of the authors; since the wording is not identical, quotation marks are omitted.

### Integrating the Growth Equation

*Y*(

*K*,

*L*,

*E*;

*t*) by the production function itself and obtain the growth equation:

*output elasticities*, defined by

*creativity*, write \(\frac{\partial \ln Y}{\partial t}\mathrm{d}t = \delta \frac{\mathrm{d}t}{\Delta t}\), and define the “output elasticity of creativity” as

*Y*(

*K*,

*L*,

*E*;

*t*) must be twice differentiable, i.e., its second-order mixed derivatives with respect to

*K*,

*L*,

*E*must be equal. Calculating these derivatives from the growth equation (1) one obtains the integrability conditions

*t*, when the production factors are \(K=K(t), L= L(t), E=E(t)\). The integral of the left-hand side from \(Y_0(t)\) to

*Y*(

*K*,

*L*,

*E*;

*t*) is \(\ln \frac{Y(K,L,E;t)}{Y_0(t)}\). It is equal to the integral of the right-hand side:

*s*in factor space from an initial point \(P_0\) at \((K_0,L_0,E_0)\) to the final point

*P*at (

*K*(

*t*),

*L*(

*t*),

*E*(

*t*)). A very convenient path consists of three orthogonal straight lines parallel to the Cartesian axes of

*K*,

*L*,

*E*space: \(P_0 = (K_0,L_0,E_0)\rightarrow P_1 = (K,L_0,E_0) \rightarrow P_2 = (K,L,E_0)\rightarrow P=(K,L,E).\) Consequently,

*t*, if it were produced by the factors \(K_0, L_0\), and \(E_0\). If creativity were dormant during the time interval \(|t-t_0|\), \(Y_0(t)\) would be also equal to the production function at time \(t_0\). If creativity is active, on the other hand, \(Y_0\) may change, and so will two technology parameters (

*a*and

*c*) in the output elasticities and the production functions to be introduced in the following section.

### Computing Output Elasticities and Production Functions

*t*the production function

*Y*(

*K*,

*L*,

*E*;

*t*) must be linearly homogeneous, because adding to an existing production system an identical one with the same inputs doubles output. Thus, \(\alpha + \beta + \gamma = 1\). With that the integrability conditions (4) turn into \(K \frac{\partial \alpha }{\partial K} + L \frac{\partial \alpha }{\partial L} + E \frac{\partial \alpha }{\partial E} = 0 ,\quad K \frac{\partial \beta }{\partial K} + L \frac{\partial \beta }{\partial L} + E \frac{\partial \beta }{\partial E} = 0 , \quad L \frac{\partial \alpha }{\partial L} = K \frac{\partial \beta }{\partial K}\). The general solutions of these partial differential equations are the output elasticities

*A*(

*L*/

*K*,

*E*/

*K*) and

*J*(

*L*/

*E*) are arbitrary differentiable functions of their arguments.

The simplest *non-trivial* solutions have been constructed to satisfy two asymptotic boundary conditions, which result from the trend of factor evolution in highly industrialized countries. (1) \(\alpha\) should vanish for \(\frac{L/L_0}{K/K_0} \rightarrow 0\) and \(\frac{E/E_0}{K/K_0} \rightarrow 0\). This incorporates the law of diminishing returns for the combination of labor and energy with instrumental capital. It takes care of the facts that “there are many reasons why a unit of production may not want to work at the maximum capacity” (Georgescu-Roegen 1986) and that strong increases of the capital stock, while labor and energy increase much less, contribute less and less to the increase of output. (2) \(\beta\) should vanish, when (*K*(*t*), *L*(*t*), *E*(*t*)) approach the state of *maximum* automation, characterized by the point (\(K_m(t), L_m(t), E_m(t)=E_0\cdot c(t) \cdot K_m(t)/K_0\)) in factor space, where, by definition, an additional unit of labor does not contribute to the growth of output any more. The technology parameter *c*(*t*) measures the energy demand of the fully utilized capital stock *K*(*t*). \(K_m(t)\) is the fully automated capital stock that could be installed to produce a given output \(Y_{given}(t)\) in combination with the inputs \(L_m(t)<<L(t)\) and \(E_m(t)\), if—within the finite space of the production system—the achievable degree of automation at time *t* were not restricted by the volume, mass, and energy demand (VME) of the available information processors. Progress in transistor technology reduces VME. Welfonder and Frederking (2002) and Welfonder (2007) constructed a global socio-economic dynamic model and looked into technical–economic development. One of their findings is that, during the last four decades of the twentieth century, the degree of automation nearly doubled in Germany and more than doubled in the USA. They conclude that, when considering sustainable global socio-economic evolution, one must take into account further progress in automation and information technologies.

*a*, which complements the energy-demand parameter

*c*, is a measure of capital effectiveness, with somewhat different meanings in (10) and (11). In (10) it indicates the weight, with which the ratio of labor

*plus*energy to capital contributes to the output elasticity of capital, while in (11) the parameter

*a*only affects the ratio of labor to capital (i.e., the inverse capital deepening). Since \(\alpha\) determines the negative part of \(\beta\) according to Eq. (8), the output elasticity of labor in Eq. (11) has the same mathematical form as \(\beta\) in Eq. (10). The meaning of the technology parameters is different, though. We will come back to that in the section “Variations of Technology Parameters and Output Elasticities.”

*a*and

*c*may change over time.

### Assets and Shortcomings of \(Y_{L1}\) and \(Y_{L11}\)

We must do with approximate solutions of the partial differential equations for the output elasticities—and thus with approximate production functions—because the *exact* output elasticities for a given economic system would have to satisfy *exact* boundary conditions. These, however, would require the knowledge of \(\beta\) on a boundary surface in factor space and that of \(\alpha\) on a boundary curve in that space.^{4} This is, and will be, impossible. The LinEx functions result from output elasticities that satisfy the *asymptotic* boundary conditions described above. But being approximations, they have assets and shortcomings, of course.

The principal asset is that LinEx functions reproduce economic growth with rather small residuals and that the growth contribution of their explicit time dependence, which is due to the time dependence of their technology parameters *a*(*t*), *c*(*t*), is much smaller than that of neoclassical technical-progress functions *A*(*t*). The growth contribution from the temporal variations of the technology parameters, *plus* the larger contribution from energy in combination with capital, supersedes the growth contribution from *A*(*t*). This is also indicated by the output elasticities of energy, \(\gamma\), and creativity, \(\delta\), as reported by Kümmel (2011) and calculated in the “Results” section, see below. Furthermore, the temporal changes of the technology parameters point to specific efficiency alterations and structural changes, which can be checked with empirical findings. This is why we think that the LinEx functions are a useful tool for analyzing the role of energy and time in economic growth.

Two shortcomings—from a strictly neoclassical point of view—have been discussed in the literature. Kümmel et al. (1985) point out that in the allowed factor space, defined by the restrictions that output elasticities must be non-negative, see Eq. (19) below, the \(L-K\) and \(E-K\) isoquants are convex; the \(L-E\) isoquant, however, is concave. They explain this concavity by progress in automation. “If initially a fixed quantity of capital *K* of low degree of automation is combined with much labor and little energy, the labor-substituting energy at first takes over essentially the performance of hard physical work like lifting and transportation of cargos, deformation of matter, digging of holes etc. With increasing degree of automation at constant numerical value of *K*, the activity of workers is more and more restricted to information processing, which mainly consists of the handling of machines. In the production of a unit of output the necessary amount of information processing requires less energy than the complementary amount of physical work on matter. Therefore, less and less additional energy is needed in order to substitute the more and more exclusively information-processing workers—thus the \(L-E\) isoquant is concave.”

Saunders (2008) notes that \(Y_{L1}\) does not have decreasing marginal *returns* of labor, i.e., \(\partial ^2 Y_{L1}/\partial L^2\) is not negative. This can be easily seen from the definition (2) of labor’s output elasticity, according to which the marginal product of labor is \(\partial Y/\partial L = Y\beta /L\), so that \(\partial ^2 Y/\partial L^2 = (\partial Y/\partial L)\beta /L + Y\partial (\beta /L)/\partial L = Y\beta ^2/L^2 +Y\partial (\beta /L)/\partial L.\) Inserting \(\beta\) from Eqs. (10) or (11) into this yields \(\partial ^2 Y_{L1}/\partial L^2 = Y\beta ^2/L^2 + 0\); this never becomes negative. This would be a serious violation of the law of diminishing returns for production functions in which *L*/*K* could become arbitrarily large. This is not the case for LinEx functions, which are subject to the restrictions (19) of non-negative output elasticities. Thus, in the accessible factor space, *L* cannot exceed the upper limit imposed by the restriction that \(\gamma\) of Eq. (10) must be non-negative. (And in highly industrialized economies *L*/*K* has been and is decreasing strongly, anyway.)

Stern (2011) also objected to the LinEx function \(Y_{L1}\), whose marginal *product* of labor, \(Y_{L1}\beta /L\), decreases, when *E* increases. He wrote: “the marginal product of labor [...] can become negative when energy is very abundant.” But, again, a restriction, this time that the output elasticity of labor, \(\beta\), must be non-negative, prevents that. It imposes an upper limit on energy *E*. The sector of factor space, where \(\beta\) would be negative, is inaccessible. Furthermore, in highly industrialized economies, where automation of the capital stock increases, demanding increasing amounts of (mostly electrical) energy, society is faced with the problem that more and more (routine) jobs get lost. This is what one expects from a decreasing marginal product of labor.

The energy-dependent Cobb–Douglas production function, \(Y_{CDE}\), Eq. (9), has positive marginal products and negative marginal returns everywhere in factor space. Then, why not use it, just disregarding the cost-share theorem as Tintner (1974) did? Growth analyses with CDE after 1974 confirmed that output elasticities are for energy much larger and for labor much smaller than the cost shares. But the reproductions of the economic downturns and upswings since the 1970s are insufficient. Figure 7 shows the only noteworthy result we could obtain with the new data for the German industrial sector. After 1990, the variations of output are hardly reproduced, and the statistical quality measures are poor. For the total economies of Germany and the USA, and with the new data, CDE just produces a smooth growth of GDP, which follows the growth of capital. Furthermore, the arbitrarily large factor magnitudes that are possible within the domain of the CDE may mislead studies on future economic developments into scenarios that are impossible from an engineering point of view, because the CDE allows for (asymptotically) complete factor substitutability; in other words, at given output it would be possible that capital substitutes for energy until only arbitrarily small energy inputs remain. This contradicts thermodynamics. Real-world production processes have finite minimum energy requirements.

Compared to \(Y_{L1}\), the modified LinEx function \(Y_{L11}(K,L,E;t)\) has the advantage of being invariant under the transformation from a base year \(t_0\), with the inputs \(K_0, L_0, E_0\) , and the technology parameters *a* and *c*, to another base year \(t_1\), with the inputs \(K_1, L_1,E_1\) , and the technology parameters \(a_1\) and \(c_1\). The proof is given in Appendix 1.

The invariance of \(Y_{L11}\) against transformations of the base year is related to the *E* / *c* dependence of its output elasticities. *E* / *c* is primary energy *E* divided by the energy demand *c* of the fully utilized capital stock. If at decreasing *c*(*t*) the *composition* of the output in terms of performed physical work and processed information, and the corresponding monetary valuation, would not change, the variable *E* / *c* would be proportional to the *exergy* that acts on matter via capital activation with an efficiency that increases as *c*(*t*) decreases.

Finally, when wondering, whether to use \(Y_{L1}\) or \(Y_{L11}\), one has to consider that \(Y_{L1}\) is the simplest production function with factor-dependent output elasticities. (Therefore, Occam’s razor had cut it out first.) Because of its relative simplicity, the algorithms for the determination of its technology parameters are simpler and consume less computing time than the ones for \(Y_{L11}\). Furthermore, one has to consider another trade-off, namely that between ambiguity and sub-optimal approach to the state of maximum automation. Maximum automation is defined by the asymptotic boundary condition \(\beta \rightarrow 0\) , when the inputs capital, labor, and energy approach \(K_m/K_0\), \(L_m/L_0\), and \(E_m/E_0=cK_m/K_0\). The first LinEx function \(Y_{L1}\) satisfies this boundary condition. But the factor space accessible to \(Y_{L1}\) *also* includes the range where \(E/E_0=cK/K_0\), and where \(\beta\) vanishes for *any* *K* and *L*. This ambiguity is removed by the modified ansatz for \(\alpha\) in the output elasticities (11) of \(Y_{L11}\): there, \(\gamma\) becomes negative, namely \(-ac\frac{L/L_0}{K/K_0}\), when \(E/E_0\rightarrow cK/K_0\), and the restrictions (19) exclude this region of factor space for \(Y_{L11}\). Of course, the state of maximum automation is excluded, too–strictly speaking. But this state is *almost* accessible to \(Y_{L11}\), because \(L_m/L_0<< K_m/K_0\) so that \(\gamma \rightarrow -0\). Since the state of maximum automation is still not too close for present-day economies, although it is on the horizon of “Industry 4.0,” the disadvantage that it belongs only asymptotically to the domain of \(Y_{L11}\) should not be too serious.

Subsequently, we will see that both LinEx functions reproduce economic growth in Germany and the USA satisfactorily for more than 50 years. Although the overall output elasticities are similar, their temporal variations reveal interesting differences in the details of how energy and creativity—the latter through time-changing technology parameters in the production function—contribute to output growth.

## Technology Parameters and their Dynamics

*a*and

*c*, and \(y_0(t)\equiv Y_0(t)/Y_0(0)\), are determined by minimizing the sum of squared errors

The empirical time series of output \(y_{empirical}(t_i)\) and capital \(k(t_i)\) are provided by the national accounts; labor and energy statistics give the empirical \(l(t_i)\) and \(e(t_i)\). These time series are shown in Figs. 2, 3, and 4.

*p*(

*t*) represent either the capital-effectiveness parameter

*a*(

*t*) or the energy-demand parameter

*c*(

*t*) of the LinEx functions. Modeling it by logistic functions, as Kümmel et al. (2002) did, we have

Minimizing the sum of squared errors (18) subject to the inequality restrictions (19) is a problem of non-linear optimization for which the Levenberg–Marquardt algorithm (Press et al. 1992) has proven to be useful. Section 3 of Winkler (2016) presents numerical details such as: (1) The mathematical structure of the algorithm, which is implemented by the Ceres Solver statistics program; (2) The equations for the statistical quality measures Durbin–Watson coefficient \(d_W\) and the adjusted coefficient of determination \(\bar{R}^2\); (3) How to avoid convergence of SSE minimization in a side minimum via a new procedure of estimating the appropriate start values for the free coefficients \(\{a_i\}, \{c_i\}, i=0,1,2,3\), and \(y_0\); (4) Computation of standard deviations.

*t*to economic growth via innovations and structural change is given by creativity’s output elasticity \(\delta\), defined in Eq. (3), with \(\Delta t = |t-t_0|\). Using the normalized form \(y_L(k,l,e;t)\), Eq. (17), for the LinEx functions one has

## Results

The empirical growth of inputs and output in the FR of Germany (FRG) and the USA from 1960 to 2013, and the output computed with the production functions \(Y_{L1}\), Eq. (14), and \(Y_{L11}\), Eq. (15), are presented in Figs. 2, 3, and 4. As base year, we have chosen \(t_0= 1960\). The reproduction of economic growth in Germany and the USA by both LinEx functions is good. There is no Solow Residual. The time-dependent technology parameters *a*(*t*) and *c*(*t*), which, in conjunction with \(y_0(t)\), are determined by SSE minimization, are given by Figs. 5, 6, and 8. These figures also show the resulting output elasticities. In Fig. 7 it is seen how the energy-dependent Cobb–Douglas function reproduces the growth of the German economic sector Industry. Table 1 lists the time-averaged LinEx output elasticities and the statistical quality measures.

### Variations of Output and Inputs

Overall growth follows the growth of the capital stock, whereas the simultaneous downturns and upswings of outputs and energy inputs occur during the times of severe economic recessions between 1973–1975, 1979–1981, and 2008–2010, triggered by the first and the second oil-price shock and the bursting of the US housing bubble. The relation between output and energy input is bidirectional: When energy input decreases, fewer machines can be activated and output goes down; when demand for goods and services decreases and machines are shut down, less energy has to be inputted into the still active components of the capital stock.^{5}

When East Germans decided to abandon the planned economy of the former German Democratic Republic (GDR) and merge their production system with the market economy of the Federal Republic of Germany, reunification in 1990—an unprecedented structural change and challenge to socio-economic creativity—resulted in the jumps of inputs and outputs between 1990 and 1991 that are shown in Figs. 2 and 3. Trends in the “old” FRG before reunification continue in the “new” FRG: labor decreases, while energy tends to flatten out. In the USA, as shown in Fig. 4, quite to the contrary, labor grows, and so does energy, at least until 2000, and except for the times of the oil-price shocks. Nevertheless, despite the opposite trends in labor evolution, the LinEx functions work well for both systems. The diminishing output growth of the German industrial sector, reproduced in Fig. 3, is due to the diminishing and even negative growth of this sector’s capital stock—a consequence of shifting energy-intensive production from Germany to other countries.

### Variations of Technology Parameters and Output Elasticities

The technology parameters *a*(*t*) and *c*(*t*) have different meanings in \(Y_{L1}\) and \(Y_{L11}\). This leads to the different temporal variations of the technology parameters and output elasticities of these two production functions in Figs. 5, 6, and 8. The difference originates from the different technological modeling of how capital-handling labor and capital-activating energy enter the law of diminishing returns in the output elasticities of capital, given by Eqs. (10) and (11). (1) In the output elasticities (10) of \(Y_{L1}\) the parameter *a*(*t*) is the weight with which the ratio of “labor *plus* energy to capital” contributes to the output elasticity of capital, \(\alpha\), and the parameter *c*(*t*) measures the energy demand of the capital stock *K*(*t*) (at full capacity utilization) in the technology model represented by that \(\alpha\). (2) In the output elasticities (11) of \(Y_{L11}\) the parameter *a*(*t*) gives (only) the weight with which the ratio of “labor to capital” contributes to the output elasticity of capital; the weight of the contribution of the ratio “energy to capital” is given by 1 / *c*(*t*). Here *c*(*t*) measures the energy demand of the (fully utilized) capital stock *K*(*t*) in the technology model represented by the modified \(\alpha\). The degree of automation of *K*(*t*) and the professional qualification of labor *L*(*t*), which handles *K*(*t*), are different in the two models. (To model these differences more in detail, one would need output elasticities that are more complex than the ones in Eqs. (10) and (11).)

The difference in the meanings of the technology parameters of \(Y_{L1}\) and \(Y_{L11}\) shows most clearly in Germany’s total economy before and after reunification in 1990. In Fig. 5a, the energy-demand parameter *c*(*t*) jumps from the constant level at 1.01 before 1990 to the constant magnitude 1.52 since 1991. This jump is due to the incorporation of the capital stock of the former GDR, with its relatively low energy efficiency of factories and energy networks, into the capital stock of the “new” FRG. The logistic increase of *a*(*t*) from 0.23 in 1960 to nearly 0.9 in 2013, with the turning point in the mid 1990s, on the other hand, indicates a gradual increase of capital effectiveness with respect to labor *and* energy and its diffusion from the “old” FRG into the eastern parts of the reunited country. In Fig. 5b, however, *a*(*t*) indicates only capital’s effectiveness with respect to labor. It rises from 0.08 to 0.8, and the turning point is before 1990. The linearly declining *c*(*t*) before and after the jump between 1990 and 1991 indicates efficiency improvements in the activation of the capital stock by energy. Here, the decline before 1990, which starts from 2.45 in 1960, is steeper than that after 1990, when the average *c*(*t*) is close to the constant \(c(t)=1.52\) of \(Y_{L1}\) in Fig. 5a.

The variations of *a*(*t*), *c*(*t*) for \(Y_{L1}\) and \(Y_{L11}\) in the German industrial sector, whose contribution to GDP has nearly been halved during the four decades since 1970, are shown in Figs. 6a, b. Here, *c*(*t*) declining linearly from different initial levels, with upward jumps between 1990 and 1991, in combination with (almost) constant *a*(*t*) before 2000, suffices for good fits. Again, *a*(*t*) is larger for \(Y_{L1}\) than for \(Y_{L11}\), and *c*(*t*) is larger for \(Y_{L11}\) than for \(Y_{L1}\). (One may wonder, whether the jump of *a*(*t*) in Fig. 6a between 2010 and 2011 can be interpreted as a reaction to the burst of the US housing bubble, and whether the increase of *a*(*t*) in Fig. 6b after 2000 may be understood as a response to the burst of the dot.com internet bubble. To answer such questions, one would need a model of consumer and investor responses to bubble bursts.)

For the USA, the logistic variations of *a*(*t*), *c*(*t*) in Figs. 8a, b show that the two technology models yield quantitative differences between the *a*(*t*) and *c*(*t*) of \(Y_{L1}\) and the *a*(*t*) and *c*(*t*) of \(Y_{L11}\) that are similar to the ones in Germany: For \(Y_{L1}\), *a*(*t*) increases from 0.18 to 0.70 while *c*(*t*) decreases from 3.8 to 0.78; for \(Y_{L11}\) the decrease of *c*(*t*) is much stronger, from 7.6 to 1.04, and the increase of *a*(*t*) is much smaller, from 0.10 to 0.43. A significant part of the parameter variations should be related to the first and the second oil-price shock. The growth of the internet- and information-based sectors of the US economy may have contributed as well. Both influences are related to the economic effects of human creativity through ideas, inventions, and value decisions: Investors decided to introduce more energy-efficient technology and innovative information technology into the capital stock, which contributed to an increase of capital effectiveness and a decrease of the capital stock’s energy demand, as reflected by the evolution of the technology parameters *a* and *c*.

The time-varying output elasticities of capital, \(\alpha\), labor, \(\beta\), and energy, \(\gamma\), in Figs. 5, 6, 7, and 8 result from Eqs. (10) and (11) with the empirically given time-varying factor inputs and the computed technology parameters *a*(*t*), *c*(*t*). The output elasticities of creativity, \(\delta\), have been computed from Eqs. (23) and (23). (Narrow, sharp peaks from derivatives of step functions are omitted.)

In FR Germany, Total Economy, \(\alpha\) varies between 0.5 and 0.3; \(\beta\) is below 0.2 between 1960 and 1980 and then rises up to roughly 0.4 in 2013; \(\gamma\) rises from roughly 0.4 in 1960 to roughly 0.6 in 1980, and then decreases—more strongly, down to 0.1, for \(Y_{L1}\) than for \(Y_{L11}\), whose \(\gamma\) finishes somewhat below 0.3 in 2013. The contribution of the service sector to the GDP of the FR of Germany grew from 44.9% in 1970 to 72.0% in 2009 (Kümmel 2011, p. 193). We think that this structural change is represented by the variations of \(\beta\) and \(\gamma\). Furthermore, the rise of renewable energy sources leads to a systematic error in the energy data that can lead to decreasing \(\gamma\).^{6}

Figures 6c, d show the dominating role of energy in industrial production. In the industrial sector of the FRG, energy’s output elasticity \(\gamma\) is much larger than the output elasticities of capital and labor, except for \(Y_{L1}\) in the 1960s.

In the total economy of the USA, on the other hand, capital’s output elasticity \(\alpha\) dominates after 1990, \(\beta\) declines significantly between 1960 and 1990, and then starts to rise again, and \(\gamma\) has its maximum shortly after the two oil-price shocks. These dynamics of the output elasticities, exhibited in Figs. 8c, d, show no significant differences between \(Y_{L1}\) and \(Y_{L11}\).

The result that in the US economy the output elasticity of labor is first declining and then rising indicates that different mechanisms are at work. While the mentioned structural change towards a rising share of services increases labor’s output elasticity, increasing automation, mainly in the industrial part of the economy, decreases it. In the USA the latter effect seems to dominate until the 1990s, while the further gained importance since then. In order to separate the effects of rising shares of services on the one hand and increasing automation on the other, sectorally more disaggregated application of the theory would be desirable. This is left to future research.

Time-averaged output elasticities of capital, \(\bar{\alpha }\), labor, \(\bar{\beta }\), energy, \(\bar{\gamma }\), creativity, \(\bar{\delta }\), adjusted coefficient of determination \(\bar{R}^2\), and Durbin–Watson coefficient \(d_W\) obtained with the LinEx production functions \(Y_{L1}\) and \(Y_{L11}\) for the systems FR Germany Total Economy (FRG TE), FR Germany Industry (FRG I), and USA Total Economy (USA TE).

\(Y_{L1}\), System: | FRG TE | FRG I | USA TE |
---|---|---|---|

\(\bar{\alpha }\) | 0.367 ± 0.006 | 0.248 ± 0.008 | 0.518 ± 0.023 |

\(\bar{\beta }\) | 0.188 ± 0.004 | 0.076 ± 0.008 | 0.188 ± 0.041 |

\(\bar{\gamma }\) | 0.445 ± 0.007 | 0.640 ± 0.011 | 0.294 ± 0.047 |

\(\bar{\delta }\) | 0.217 ± 0.006 | 0.132 ± 0.007 | 0.200 ± 0.023 |

\(R^2\) | 0.999 | 0.989 | 0.998 |

\(d_W\) | 1.650 | 1.747 | 0.715 |

\(Y_{L11}\), System: | FRG TE | FRG I | USA TE |

\(\bar{\alpha }\) | 0.399 ± 0.008 | 0.221 ± 0.020 | 0.533 ± 0.016 |

\(\bar{\beta }\) | 0.236 ± 0.012 | 0.015 ± 0.009 | 0.242 ± 0.035 |

\(\bar{\gamma }\) | 0.365 ± 0.014 | 0.765 ± 0.022 | 0.226 ± 0.039 |

\(\bar{\delta }\) | 0.236 ± 0.032 | 0.27 ± 0.16 | 0.168 ± 0.019 |

\(R^2\) | 0.999 | 0.988 | 0.999 |

\(d_W\) | 1.508 | 1.581 | 0.762 |

## Summary and Conclusions

Capital’s growth determines the overall growth of economic output in highly industrialized countries like Germany and the USA. Economic recessions and recuperations are more strongly correlated with the downturns and upswings of capital-activating energy than with the variations of capital-handling labor.^{7} Between 1960 and 2013, theoretical, inflation-corrected GDP grew without Solow residuals by similar factors in both countries: 4.1 in the FR Germany and 5 in the USA. During the same time, empirical labor inputs decreased somewhat in the FRG before and after reunification, and doubled in the USA.

How much output a given capital stock, at certain degrees of automation and capacity utilization, can produce is only a matter of technology. The same holds for the limits to capacity utilization and automation. (Factor prices, however, influence decisions on future investments.) Therefore, the economic weights of capital, labor, and energy should be independent from the factor cost shares. And precisely this engineering conclusion is confirmed by the output elasticities computed with the two LinEx functions used in the analysis of this work: Those of energy are much larger and those of labor are much smaller than the cost shares of these factors.

Technological and structural changes such as increasing automation, outsourcing of energy-intensive industries, changing shares of the industrial and service sectors in GDP, and the absorption of a centrally planned economy by a market economy, as it occurred in German reunification, show empirically in factor inputs and theoretically in the time-changing technology parameters and output elasticities of LinEx production functions. The growth contribution from the temporal variations of the technology parameters, *plus* the larger contribution from energy in combination with capital, supersedes the growth contribution from neoclassical technological progress functions. We think that LinEx functions can serve as a tool for clarifying the contributions of energy and time to technical progress, and structural changes as well. Of course, every tool can be improved. It would be interesting to see what can be achieved by shaking off the chains of the cost-share theorem from energy-dependent CES and translog production functions.^{8}

We feel that a better understanding of the economic growth observed in highly industrialized economies will foster creative responses to the environmental challenges from emissions, which are intimately coupled to energy conversion by the inevitability of entropy production, whenever something happens. Based on a proper understanding of the past, one should develop scenarios for the future. They concern factor inputs according to entrepreneurial optimizing expectations, models for technology parameters in the output elasticities of capital, labor, and energy, and the resulting economic growth. Then society will gain improved visions of technologically and economically feasible paths into the future.

## Footnotes

- 1.
In highly industrialized countries, the role of the factor “land,” so important for biomass production by photosynthesis in agrarian societies, is reduced to providing sites for the capital stock, essentially.

- 2.
The machines of the capital stock cannot operate with energy inputs that exceed the ones they are designed for.

- 3.
“Routine” labor in the sense that it is measured by hours worked per year.

- 4.
How this results from the theory of partial differential equations is pointed out more explicitly in Appendix 6 of Chapter 4 of Kümmel (2011).

- 5.
The actually observed factor inputs result from entrepreneurial decisions influenced by the prices of inputs and output, and price expectations as well.

- 6.
The energy statistics include a systematic error when it comes to accounting renewable energy generation. For conventional energy sources like coal, gas, oil, and nuclear power, the primary energy used to generate the electricity is reported. For renewables, however, only the generated electricity is reported as primary energy input implicitly assuming 100% efficiency. In reality, renewables have an efficiency below 100%. If the share of renewables increases, energy input decreases, affecting the contribution of the factor energy to growth.

- 7.
In business cycles, legal and social constraints make the firing and hiring of people more difficult than the reduction and increase of energy inputs into the capital stock.

- 8.
In a recent paper by Heun et al. (2017) focusing on the estimation of CES production functions for the UK and Portugal 1960–2009, various modeling choices are analyzed and evaluated, including (a) rejecting (or not) the cost share principle, (b) including (or not) energy, (c) quality-adjusting (or not) factors of production, and (d) CES nesting structure. The effects of those choices on the interpretation of the economy and possible policy recommendations are assessed. The paper concludes that this “raises the possibility that energy-economy modeling with aggregate production functions\(\dots\) may tell us more about theory and modeling than about the economy.” Maybe LINEX functions can contribute to this debate.

- 9.
As Dorfman et al. (1957), and Solow (1957) remark “\(\dots\)there seems to have been a misunderstanding somewhere because the technologists do not take responsibility for production functions either. They regard the production function as an economist’s concept, and, as a matter of history, nearly all the production functions that have actually been derived are the work of economists rather than of engineers.”

- 10.
It may be noted that in case of the most important primary fuels, namely coal, oil, gas, and nuclear fuels, primary energy is nearly 100 % exergy (essentially useful work).

- 11.
Needless to say, if one constructs (educational) indices in order to quality-adjust labor data, additional assumptions are necessary for this construction. Educational index data are provided, e.g., by Barro and Lee (2001).

- 12.
Kander and Stern (2014) consider energy as an intermediate in the production function, and use value added plus the cost of energy as output measure.

- 13.
There is a small difference between GDP and Gross Value Added, where the former includes subsidies and taxes and the latter does not.

- 14.
Index subscripts refer to countries (G = Germany, U = USA) or type of index (Q = Quaas, F = Fischer) applied by those countries.

- 15.
The old data were assembled and updated by one of us (R.K.) during several personal visits to US research facilities between 1981 and 1999, with kind help from colleagues in economics.

## Notes

### Acknowledgements

This work was supported in part by the Mercator Foundation in its project “Transformationsprozesse für nachhaltige und wettbewerbsfähige Wirtschafts- und Industriestrukturen in NRW im Kontext der Energiewende” (Project No. 1413600). This support is gratefully acknowledged. We thank Haye Hinrichsen for discussions and helpful suggestions concerning the computation of technology parameters and standard errors. Marc Lindley provided the paper by Georgescu-Roegen (1986). We thank two anonymous referees for helpful comments.

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