Energy use, population and growth, 1800–1970

Abstract

In this paper, the role of energy use is incorporated into unified growth theory. The paper presents some interesting evidence about the evolution of energy in the transition from stagnation to growth, and it subsequently develops a growth model where the observed increase in conversion efficiency in the coal energy sector is explicitly modelled and calibrated to existing data over the period 1800–1970. The quantitative analysis sheds light on the impact of energy use on the transition from stagnation to growth.

Appendices

Appendix 1: Construction of data series for primary energy production from biomass burning

Biofuel combustion:

Combustion of wood, wood waste, charcoal, dung, crop, crop residue, bagasse, ethanol, black liquor, non-solid fuels, industrial and municipal waste (van Aardenne et al. 2001).

Biomass burning:

Savannah burning, deforestation, organic waste burning and residential biofuel burning (van Aardenne et al. 2001).

Coal:

See Etemad et al. (1991).

Non-renewable energy:

Depletable energy (fossil fuels; Tahvonen and Salo 2001).

Primary energy:

Energy in the form that it is first accounted for in a statistical energy balance, before any transformation to secondary or tertiary forms of energy (EIA 2009).

Renewable energy:

Non-depletable energy, including hydropower, wind energy, solar energy, biomass and geothermal energy (Tahvonen and Salo 2001).

Secondary (derived) energy:

The energy produced by the conversion of primary energy (Energy.EU 2009).

TJ:

Unit of energy, 1 TJ = 10¹² J = 10¹² J.

Data on carbon dioxide emissions from biofuel combustion over the period 1890–1990 are available in van Aardenne et al. (2001). There, biofuel combustion is a separate source of emissions, different from biomass burning. The category biofuel combustion is subdivided into domestic/residential and industrial biofuel burning. van Aardenne et al. (2001) defines biomass burning as savannah burning and deforestation. These activities are not related to the energy sector and are not considered in this study. Throughout this paper, we use the term biomass energy, or energy production from biomass burning, for the class that van Aardenne et al. (2001) denotes biofuel combustion.

It must be clarified that biomass burning in this paper is set equal to biofuel combustion in the glossary above. In the van Aardenne et al. (2001) data set, biofuel encompasses both traditional and modern biofuels, where the latter mainly are regarded as secondary energy forms (Karekezi et al. 2004). Karekezi et al. (2004) report on the deficiencies of existing data sets when it comes to differentiating between traditional and modern biofuel energy use. Hall et al. (1994) describes a lack of data on biofuel production and use. Traditional biofuel is the part of biofuels that is mainly a primary energy source. Since modern biofuels account for a small share in global biofuel energy supply (Karekezi et al. 2004; Tahvonen and Salo 2001) we use the van Aardenne et al. (2001) data set without modifications and let the derived data represent primary energy production from biomass burning.

In the previous paragraphs, we defined biomass energy as we use the term in this paper. van Aardenne et al. (2001) provide data on carbon dioxide emissions and not on energy production. We calculate the corresponding energy production figures using that the emission factor for biomass combustion is 3 × 10^− 8 kgC/J (van Aardenne et al. 2001). Absent other information, we assume that the same emission factor is valid from year 1800 and onwards. Thus, we can divide the data series of emissions with this emission factor and get the corresponding data series for energy production. We then extrapolate this data series back to year 1800 by using the assumption that biomass use per person was not considerably higher or lower in the past than in the present. A similar assumption is used in van Aardenne et al. (2001) when extrapolating the emissions data series back to year 1890. The difference is that the work of van Aardenne et al. (2001) focuses on the rural population, whereas we in this paper utilise the total population. In van Aardenne et al. (2001), it is assumed that this relation holds for the past century. Thus, one can divide the biomass use in year 1990 with the level of population and then multiplying the resulting figure with the levels of population in years 1800 and 1870.

For all data series, the data are presented for years 1800, 1870, 1900, 1910 and 1950–1970. This is done for the sake of consistency. In the cases where data are not available at these years, we have chosen data corresponding to years that are closest to the actual years. We find this approach appropriate for a very long run analysis of the broad trends in the data. For more information, we refer to the original data sources.

Appendix 2: Derivation of the static equilibrium quantities

Maximising profits according to Eq. 17 yields

$$ \label{eq27} p_{{\rm{E}}t} \varphi _1^\xi \left( {\frac{E_t }{E_{{\Pr \rm{B}}t} }} \right)^{1-\xi }=p_{{\Pr \rm{B}}t} , $$

(27)

and

$$ \label{eq28} p_{{\rm{E}}t} A_{{\rm{EC}}t}^\xi \left( {\frac{E_t }{E_{{\Pr \rm{C}}t} }} \right)^{1-\xi }=p_{{\Pr \rm{C}}t} . $$

(28)

Since the energy-producing firms spend all their income on labour costs (Eqs. 18 and 19), we can write Eqs. 27 and 28 as

$$ \label{eq29} p_{{\rm{E}}t} \varphi _1^\xi \left( {\frac{E_t }{E_{{\Pr \rm{B}}t} }} \right)^{1-\xi }=w_{{\Pr \rm{B}}t} \frac{L_{{\Pr \rm{B}}t} }{E_{{\Pr \rm{B}}t} }, $$

(29)

and

$$ \label{eq30} p_{{\rm{E}}t} A_{{\rm{EC}}t}^\xi \left( {\frac{E_t }{E_{{\Pr \rm{C}}t} }} \right)^{1-\xi }=w_{{\Pr \rm{C}}t} \frac{L_{{\Pr \rm{C}}t} }{E_{{\Pr \rm{C}}t} }. $$

(30)

Substituting Eqs. 29 and 30 into Eq. 13 gives

$$ \label{eq31} w_{{\Pr \rm{B}}t} \frac{1}{\varphi _1^\xi }\left( {\frac{E_{{\Pr \rm{B}}t} }{E_t }} \right)^{1-\xi }\frac{L_{{\Pr \rm{B}}t} }{E_{{\Pr \rm{B}}t} }E_t =\pi _{\rm E} Y_t , $$

(31)

and

$$ \label{eq32} w_{{\Pr \rm{C}}t} \frac{1}{A_{{\rm{EC}}t}^\xi }\left( {\frac{E_{{\Pr \rm{C}}t} }{E_t }} \right)^{1-\xi }\frac{L_{{\Pr \rm{C}}t} }{E_{{\Pr \rm{C}}t} }E_t =\pi _{\rm E} Y_t . $$

(32)

Equating wages between Eq. 14 and Eqs. 31–32 yields

$$ \label{eq33} \left( {1-\pi _{\rm A} -\pi _{\rm{EA}} -\pi _{\rm E} } \right)\frac{1}{L_{{\rm{Y}}t} }\frac{1}{\varphi _1^\xi }\left( {\frac{E_{{\Pr \rm{B}}t} }{E_t }} \right)^{1-\xi }\frac{L_{{\Pr \rm{B}}t} }{E_{{\Pr \rm{B}}t} }E_t =\pi _{\rm E} , $$

(33)

and

$$ \label{eq34} \left( {1-\pi _{\rm A} -\pi _{\rm{EA}} -\pi _{\rm E} } \right)\frac{1}{L_{{\rm{Y}}t} }\frac{1}{A_{{\rm{EC}}t}^\xi }\left( {\frac{E_{{\Pr \rm{C}}t} }{E_t }} \right)^{1-\xi }\frac{L_{{\Pr \rm{C}}t} }{E_{{\Pr \rm{C}}t} }E_t =\pi _{\rm E} . $$

(34)

For the economy as a whole, we have that

$$ \label{eq35} \frac{\left( {1-\pi _{\rm A} -\pi _{\rm{EA}} -\pi _{\rm E} } \right)Y_t }{Y_t }=\left( {1-\pi _{\rm A} -\pi _{\rm{EA}} -\pi _{\rm E} } \right)=\frac{wL_{{\rm{Y}}t} }{wL_t }=\frac{L_{{\rm{Y}}t} }{L_t }. $$

(35)

Substituting Eq. 35 into Eqs. 33 and 34 gives

$$ \label{eq36} \frac{1}{L_t }\frac{1}{\varphi _1^\xi }\left( {\frac{E_{{\Pr \rm{B}}t} }{E_t }} \right)^{1-\xi }\frac{L_{{\Pr \rm{B}}t} }{E_{{\Pr \rm{B}}t} }E_t =\pi _{\rm E} , $$

(36)

and

$$ \label{eq37} \frac{1}{L_t }\frac{1}{A_{{\rm{EC}}t}^\xi }\left( {\frac{E_{{\Pr \rm{C}}t} }{E_t }} \right)^{1-\xi }\frac{L_{{\Pr \rm{C}}t} }{E_{{\Pr \rm{C}}t} }E_t =\pi _{\rm E} . $$

(37)

We can rewrite Eqs. 36 and 37 to

$$ \label{eq38} \frac{L_{{\Pr \rm{B}}t} }{L_t }=\varphi _1^\xi \pi _{\rm E} \left( {\frac{E_{{\Pr \rm{B}}t} }{E_t }} \right)^\xi , $$

(38)

and

$$ \label{eq39} \frac{L_{{\Pr \rm{C}}t} }{L_t }=A_{{\rm{EC}}t}^\xi \pi _{\rm E} \left( {\frac{E_{{\Pr \rm{C}}t} }{E_t }} \right)^\xi . $$

(39)

Substituting Eqs. 38 and 39 into Eqs. 8 and 7, respectively, yields

$$ \label{eq40} E_{{\Pr \rm{B}}t} =\left( {\theta \varphi _1^\xi \pi _{\rm E} } \right)^{\frac{1}{1-\xi }}\left( {\frac{1}{E_t }} \right)^{\frac{\xi }{1-\xi }}L_t ^{\frac{1}{1-\xi }}, $$

(40)

and

$$ \label{eq41} E_{{\Pr \rm{C}}t} =\left( {\Lambda A_{{\rm{EC}}t}^\xi \pi _{\rm E} } \right)^{\frac{1}{1-\xi }}\left( {\frac{1}{E_t }} \right)^{\frac{\xi }{1-\xi }}L_t^{\frac{1}{1-\xi }} . $$

(41)

We continue by substituting Eqs. 40 and 41 into Eq. 16 and solve for E _t

$$ \label{eq42} E_t =\left( {\varphi _1^\xi \Omega _{1t}^{\frac{\xi }{1-\xi }} +\Omega _{2t}^{\frac{\xi }{1-\xi }} A_{{\rm{EC}}t}^{\frac{\xi }{1-\xi }} } \right)^{\frac{1-\xi }{\xi }}L_t , $$

(42)

where

$$ \label{eq43} \Omega _{1t} \equiv \theta \varphi _1^\xi \pi _{\rm E} , $$

(43)

and

$$ \label{eq44} \Omega _{2t} \equiv \Lambda \pi _{\rm E} . $$

(44)

Next, we combine Eq. 25 with Eqs. 2 and 3 which yields

$$ \label{eq45} \left( {\alpha \left( {1-\ell _t } \right)-\overline b } \right)^\eta =\frac{\alpha \mu }{1-\mu }\frac{\left( {w_t \ell _t -\overline c } \right)^\gamma }{w_t }. $$

(45)

Substituting Eq. 35 into Eq. 14 and using the productivity measure from Section 3, we arrive at

$$ \label{eq46} w_t =\frac{a_t }{\ell _t^{1-\beta -\psi } }. $$

(46)

Combining Eqs. 45 and 46 and omitting the time subscripts, we could define

$$ \label{eq47} F\left( \ell \right)\equiv \left( {\alpha \left( {1-\ell } \right)-\overline b } \right)^\eta -\frac{\alpha \mu }{1-\mu }\frac{1}{a}\left( {a\ell ^{\beta +\psi }-\overline c } \right)^\gamma \ell ^{1-\beta -\psi }. $$

(47)

The equilibrium satisfies F(ℓ^∗) = 0. The other equilibrium quantities could be obtained through solving the above equations in a recursive manner, once the equilibrium value ℓ^∗ is obtained.

Appendix 3: Calibration of the utility function

Below we describe how the parameters of the utility function (Eq. 4) could be estimated. We start from Eq. 2 and substitute Eq. 3 into this equation. Then we get

$$ \label{eq48} c_t =w_t \left( {1-\frac{b_t }{\alpha }} \right). $$

(48)

Solving for w _t yields

$$ \label{eq49} w_t =\frac{c_t }{\left( {1-\frac{b_t }{\alpha }} \right)}. $$

(49)

Substituting Eq. 49 into Eq. 25 yields

$$ \label{eq50} b_t -\overline b =\left( {\frac{\alpha \mu }{1-\mu }\left( {1-\frac{b_t }{\alpha }} \right)\frac{\tilde {c}_t^\gamma }{c_t }} \right)^{1 \mathord{\left/ {\vphantom {1 \eta }} \right. \kern-\nulldelimiterspace} \eta }. $$

(50)

In the quantitative analysis (Section 4), $\overline b =0$. Thus,

$$ \label{eq51} b_t =\left( {\frac{\alpha \mu }{1-\mu }\left( {1-\frac{b_t }{\alpha }} \right)\frac{\tilde {c}_t^\gamma }{c_t }} \right)^{1 \mathord{\left/ {\vphantom {1 \eta }} \right. \kern-\nulldelimiterspace} \eta }. $$

(51)

Solving for $\frac{\tilde {c}_t^\gamma }{c_t }$ yields

$$ \label{eq52} \frac{\tilde {c}_t^\gamma }{c_t }=b_t^\eta \frac{\left( {1-\mu } \right)}{\mu }\frac{1}{\left( {\alpha -b_t } \right)}. $$

(52)

Suppose that γ is known, then we can define a new variable, $\hat {c}_t \equiv \frac{\tilde {c}_t^\gamma }{c_t }.$ Substituting this variable into Eq. 52 gives

$$ \label{eq53} \hat {c}_t =b_t^\eta \frac{\left( {1-\mu } \right)}{\mu }\frac{1}{\left( {\alpha -b_t } \right)}. $$

(53)

The population growth rate, n _t , is equal to b _t  − d _t . Solving for b _t from this expression and substituting into Eq. 53 yields

$$ \label{eq54} \hat {c}_t =\left( {n_t +d_t } \right)^\eta \frac{\left( {1-\mu } \right)}{\mu }\frac{1}{\left( {\alpha -n_t -d_t } \right)}. $$

(54)

Given the mortality function (Eq. 21) and the parameter values in Table 1, we can estimate η and μ for each initial guess of γ, given observations on per capita consumption and population growth rates.

Table 3 presents observations on per capita consumption and population growth rates. The first and two last observations are taken directly from Jones (2001). The second to fourth observations correspond to average levels of per capita consumption and average annual growth rates of the population over the periods 1870–1900, 1960–1970 and 1980–1990, respectively. The data are taken from Maddison (2007). Using a non-linear least squares method, we obtain the parameter values reported in Table 4.

Table 3 Observations on per capita consumption and population growth rates

Table 4 Estimated parameters of the utility function