A NEW
PRODUCTIVITY INDEX TO MEASURE ECONOMIC SUSTAINABILITY OF THE MINING INDUSTRY
UN NUEVO
ÍNDICE DE PRODUCTIVIDAD PARA MEDIR
LA SOSTENIBILIDAD
ECONÓMICA EN LA MINERÍA
XOSÉ A. RODRÍGUEZ
Profesor en la Universidad de
Santiago de Compostela, xoseanton.rodriguez@usc.es
Received for review June 16 th, 2009, accepted December
6 th, 2009, final
version December 17 th, 2009
ABSTRACT: This document
aims to introduce a total productivity index to measure the economic
sustainability of the mining industry. This index will take into account any
technical developments, means of use of productive factors (i.e. inefficiencies
and scale effects) and the effects on the growth of the geological properties
in the resources to be exploited (particularly the effects of resource reduction
or reserves depletion). This new index will then be applied to the example of
the Spanish mining industry, with some interesting findings regarding the
configuration of sustainable mining policies.
KEYWORDS: economic
sustainability, productivity index, technological change, reserves depletion.
RESUMEN: En este trabajo se
propone un índice de productividad total para medir la sostenibilidad económica
en las industrias mineras. En este
índice se tiene en cuenta los avances
técnicos, el modo de utilización de los factores productivos (ineficiencias y
efectos de escala) y los efectos sobre el crecimiento de los
cambios en las características geológicas de los recursos a extraer (en
particular, los efectos de la reducción de recursos o agotamiento de reservas).
Este nuevo índice lo aplicamos al caso de le la minería española, deduciendo de
los resultados interesantes conclusiones en cuanto a la configuración de una
política minera sostenible.
PALABRAS
CLAVE:
sostenibilidad económica, índice de productividad, cambio tecnológico,
agotamiento de reservas.
1. INTRODUCTION
According to Munasinghe [1],
any alternative means of sustainable development needs to adhere to the
following three sustainability criterions: economic, environmental and social.
The criteria regarding economic sustainability aims to maintain the highest
possible rate of economical growth, using the available resources in an optimal
manner from the point of view of the market.
This document aims to
introduce a total productivity index that measures the economic sustainability
of the mining industry. This index will take into account any technical
developments, means of use of productive factors (i.e. inefficiencies and scale
effects) and
the effects on the growth of
the geological properties in the resources to be exploited (particularly the
effects of reserves depletion). Therefore, this index will act as a complete
indicator to measure the economic sustainability of the mining industry, given
that:
- Productivity is an adequate
indicator to gauge economical development, and this growth can be limited by
the effects of changes in the geological characteristics of resources to be
exploited.
- The index takes into account
the way in which productive potentialities are employed: for instance, whether
there are any scale effects, whether there are inefficiencies, or whether the
degree of usage capacity of these factors is appropriate.
- The index allows us to
analyze the dynamics between the effects of technical developments (which tend
to reduce costs) and the effects of resource depletion (which tend to increase
costs). According to Tilton and
Lagos
[2], such dynamics could determine the future availability of mining resources.
- In addition to this, this
index offers an estimate of technological developments, which allows us to
anticipate whether the conditions are optimal for the replacement of natural
capital with artificial capital. Indeed, according to Romer [3], only new
technologies can enable such a replacement and generate extraction processes
that contribute to saving natural resources. If such a replacement takes place,
it is then also possible to fulfil the criteria of social sustainability
(inter-generational equity) from a perspective of weak sustainability [4].
This work is divided into
three main sections. Section 2 describes the methodological development in the
elaboration of the index of decomposition of productivity growth, and section 3
applies the aforementioned index to the example of the Spanish mining industry.
The document then concludes with the most relevant conclusions.
2. INDEX ELABORATION
The starting point is the
traditional Divia index of Total Factor Productivity (TFP). Solow [5] proves
that under constant returns to scale in a long run competitive equilibrium the
index (
) can be interpreted as a measure of technical change. In the
index 
, (
) is a index of aggregate output and can be written as:
where, Qj is the quantity of the j-th output, Pj is
the price of the j-th output, 
is total revenue and 
the growth rate of the j-th output.. Likewise, (
) is a index of aggregate input and can be written as:
where, Xi is the quantity of the i-th input, Wi is
the price of the i-th input, 
is total cost and 
is the growth rate of
the i-th input.
This index , referred to in the literature as the Solow residual [6,7],
measures the changes in the output aggregate not explained by changes in the
input aggregate.
The starting point is a primal
representation of technology such as 
or alternatively their dual cost function 
. Under constant returns to scale in a competitive
equilibrium in the product and factor markets, it can be shown that:,
where, the Xi’s are input, t denotes technology , Q output, C is total cost and the Wi’s
are input prices.
The result in (3) shows that
the Solow Residual can be interpreted as a shift of the production function,
not attributable to changes in inputs but to technical change or can also be interpreted as well as a shift of the cost function, not
attributable to changes in input prices or output quantity but to technical
change.
The use of the Solow Residual
as a measure of technical change relies on a number of simplifying assumptions. If
these assumptions do not hold, the residual has to be corrected accordingly.
The effects of non-constant returns to scale and the violation of the various
conditions of a long run competitive equilibrium have been analyzed empirically
[8-11].
In this paper we derive a multi-decomposition of the Total Factor
Productivity (TFP) index that explicitly takes into account the distinguishing
productive characteristics of mining industries. Our alternative decomposition simultaneously introduces the possible
effects on productivity growth of non constant returns to scale, mark-up
pricing, inefficiencies and sub-equilibrium due to quasi-fixed factors, and the
influence of geological factors.
A
measure of productivity in mining industries starts with the production
function:
where Q’=( Q1,...,Qm) is a
vector of m outputs obtained with r variable inputs, XV=(X1,…,Xr); n-r quasi-fixed inputs, XF=(Xr+1,…,Xn); the technology (t); and a set of s geological characteristics (such as level of reserves, quality of
deposits…) of the natural resource
denoted by G=(G1…Gs).
Under certain regularity conditions of the production
function [12] and under the assumption of cost-minimizing behaviour, there is a
dual variable cost function that contains all relevant information about the
technology and which can be represented as:
where VC denotes
variable cost and WV=(W1,…,Wr) is a vector of prices of the
variable inputs (XV). The
total cost function can be written as:
where Wi is the price of the j-th quasi-fixed input.
Next, we extend previous results [8-11] to take into
account the productive characteristics of mining industries.
Using the index and totally differentiating the
function (6) with respect to time and rearranging, we obtain:
where 
is the corrected index
of Total Factor Productivity growth, 
is the elasticity of
cost with respect to input i, 
is the cost shares of input i, 
denotes aggregate
input growth using cost elasticities, rather that cost shares, as weights, 
is the shadow value
of quasi-fixed input i, 
is the marginal cost
of quasi-fixed input i, 
is the rate of change
of quasi-fixed input i, 
is the elasticity of
cost with respect to output j, 
is the index of
aggregate output, but using cost elasticities, rather than revenue shares, as
weights, 
is the elasticity of
cost with respect to geological characteristic k (for example, level of reserves) and 
is the rate of change
of geological characteristic k.
Following De la Fuente [13] the global cost elasticity
(
) can be written in terms of Capacity of Utilization (CU)
and returns to scale (RS) and which can be represented as:
Therefore, substituting (8) into equation (7) the
corrected index can be written as:
Equation (9) provides a
decomposition of the corrected index of Total Factor Productivity growth (
) into five components. The first component (
) is commonly interpreted as a measure of technical change. The second component (
) can be interpreted as a measure
of allocative inefficiency: it measures the effects of non-optimal allocation
of factors on TFP. If the industry is allocatively efficient, then 
, and this term is equal to zero. The
third component (
) measures the effects on Total Factor Productivity of sub-equilibrium
due to quasi-fixed factors. This term vanishes in a
competitive equilibrium when the shadow value of the fixed factors (
) is equal to its marginal cost (
). The fourth component 
measures the effects of changes in the scale
of production on Total Factor Productivity. Firms might find it difficult to
change some inputs and, as result, they do not necessarily operate at the
optimum level of capacity utilization. This term vanishes when there are
constant returns to scale (RS=1) and the capacity is fully utilized (CU=1).
The last term (
) measures the effects of changes in geological characteristics on Total Factor Productivity.
This term vanishes only if the geological characteristics do not change.
If the geological characteristics do not change, under
constant returns to scale and at a competitive equilibrium in the product and
factor markets, it can be easily shown that:
3. APLICATION TO THE
SPANISH MINING INDUSTRY
In this
section, the corrected measure of TFP (described in expression 9) is applied to
Spanish mining during the period 1974-2004. The
starting point is the following production function:
where Q’≡( Qe, Qm,
Qn, Qq ) is a
vector of outputs –energy mining output (Qe), metal mining output (Qm), non-metal mining output (Qn) and quarry mining output (Qq)– obtained with
three variable inputs: employment (XL), energy (XE)
and materials (XM); a quasi-fixed input, capital (XK);
the technology (t); and the
depletion of mineral reserves denoted by D (the geological
characteristic is the level of reserves).
The
dual variable cost function of (11) can be represented as:
where VC denotes
variable cost and WL, WE and WM are the input prices of labour, energy and materials. The total cost function can be
written as:
where WK is the user cost of capital.
Therefore,
differentiating with respect to time the function (13) and rearranging, we
obtain a decomposition of the corrected index similar to (9):
where 
is the shadow value of Capital, 
is the rate of change of Capital, 
is the rate of change
of mineral reserves and 
measures the effects of reserves depletion on
Total Factor Productivity.
The estimation of (14) requires the estimation of a
Variable Cost Function in which the mineral reserves depletion is included as
an explanatory variable. A central issue of this empirical analysis is the
definition of a variable that measures the quantity of mineral depletion
available for extraction at each period of time. To the best of our knowledge,
there is no annual estimation of mineral reserves in
Spain .
However, we have data on the quantities of mineral
extracted from 1974 to 2004. For this reason, we estimate a proxy variable of mineral reserves depletion in year t (Dt) using data on mineral extracted as:
where (Q1, … , Qt-1)
are the aggregated quantities of mineral extracted in previous periods of time. The basic
simplifying assumption contained in expression (15) is that minerals extraction
is the main force in the evolution of reserves depletion while new discoveries
are not very relevant. This assumption is not unrealistic in the case of
Spanish mining (in fact, the number of active mines went from 4443 in 1974 to 4226 in
2004).
We estimate a variable cost function using data on
minerals production, input prices and quantities reported in Mining Statistics
(Estadística Minera) an annual publication of the Spanish Ministry of Economy
[14]. The dataset contains data of mining industries operation in
Spain aggregated by the four big industries (energy mining, metal mining,
non-metal mining and quarry mining) from 1974 to
2004. The mining industries include government and privately-owned firms
carrying out both surface and underground mining. Some descriptive statistics
of the variables used for the estimation are shown in Table 1.
Table 1. Descriptive statistics (1974-2004)
Some clarifications on the construction of the variables are needed.
Firstly, the hours of Labor (XL) are weighted by the cost
share of each labor qualification. The tons of Materials (XM)
are weighted by the cost share of each type of material. Capital is measured as
hours of machinery use. In this case, the hours are weighted by the power of
each type of machinery. The user cost of Capital (WK) is
defined as:
where WK0 is the user cost of Capital in the base year reported in Gómez [15] and It is an
Index of
price of machinery. Finally, output is
measured in Tons of mineral extracted. The output of each big industry is weighted by the values of the
different minerals included in the industry. Similarly, the aggregated
quantities of mineral extracted used in estimation of Reserves Depletion (D)
are weighted by the value of the different minerals
The Translog functional form has been used in the estimation. This is a
flexible function form used previously by Brown and Christensen [16] and Berndt
and Hesse [17]. The variable cost function can be written as:
where i,j refer to the three variable inputs,
Labor (L), Energy (E) and Materials (M); s,v refer to the four
outputs, energy mining (e), metal mining (m), non-metal mining (n)
and quarry mining (q) while D is the measure of reserves depletion in
expression (15). All variables in the translog cost function are in natural
logarithms with the exception of the time trend (t) and the measure of
reserves depletion (D).
In addition to the variable cost function we estimate the
input equations for the variable factors and the price equations. Using
Shephard’s lemma in equation (17) we have that:
where 
, and Si denotes the variable cost share of input i.
The price equations are
derived using expression 17. Under imperfect competition, which typifies
the mining industries, market
power leads firms to set prices above (or under) marginal costs:
(19)where 
, and 
denotes
the markups on marginal costs.
However,
price markups over marginal costs are not directly observable. Various
approaches to their indirect estimation are suggested in the literature. For
example, Dobringsky et al. [18] propose a direct relationship between the
markup and the returns to scale index
as:
(20)
where 
is the industry’s average profit margin and 
is a returns to scale
index of the s-th industry.
Substituting equation (20) into equation (19), the index 
can be estimated as an additional parameter of the s-th industry production technology.
The
system of equations (17), (18) and (19) is estimated after imposing parametric
restrictions of symmetry and homogeneity of degree one on input prices. The share equation of materials is dropped to
avoid singularity of the system. Prices and Variable Cost have been divided by
the price of Materials to impose homogeneity of degree one on input prices of
the cost function. The resulting system of equations is estimated by Iterative
Three-Stage Least Squares (I3SLS).
I3SLS is an appropriate technique when right-hand side
variables are endogenous. We use this method since the possible endogenously of
the outputs. As instruments of outputs we use their lagged values. Furthermore, we adjust the ISURE estimates
to account for serial correlation by
adding AR(1) terms to the price
equations.
The estimation produced a number of
coefficients not significantly different from zero. Moreover, we could not
reject the null hypothesis that the non significant coefficients were jointly
different from zero. Therefore, in order
to deal with what looks like a multicollinearity problem, we decided to
re-estimate the cost function without the terms whose coefficients were jointly
non significant. The final results of
the estimation are show in Table 2.
Table 2: Restricted
Translog Variable Cost Function
The findings suggest that the model is robust. The
R-squared of the estimated equations indicated good explanatory power. The
values of the Durbin-Watson test indicated that autocorrelation is corrected. The coefficients
have the expected signs and are significantly different from zero at
conventional levels of significance.
The decomposition of the
corrected TFP growth rate following equation (14) is shown in Table 3. The five
components of in equation (14) are
shown in the first five columns of Table 3 while the estimated value of appears in the sixth column. The result´s that TFP grew at an
average annual rate of 1.75%, although there was a great deal of variation
across years (for example, TFP growth has been negative in some years). The
Technical Change (column 1) was a main source of TFP growth gains (this
component grew at an average annual rate of 2.49%). The other four components
had jointly a negative contribution to TFP growth (-0.74%), although the particular effect of each component
was different. The component that measures the subequilibrium due to quasi-fixed capital factor present a average value
positive but quite small (0.09%). This value is positive since the shadow value
of capital is generally larger than its marginal cost (
). The average value of the scale effect is positive but also
small (0.66%). The small average value is driven by the existence of positive
values that are partially balanced by negative values. In most years of the
sample the elasticity of scale (RS) is larger than the index of Capacity
Utilization (CU), with average values of 0.96 and 0.79 respectively. The
average value of the effect of non-optimal allocation of factors is negative
(-0.71%). This result is driven mainly by the existence of negative values that
are partially balanced by positive values.
Table 3: Total
Factor Productivity growth and its components
Finally, the
average value of the component that measures the effect of the reserves
depletion has a negative contribution to the average TFP growth (-0.78%), which
might indicate that the reserves depletion are likely to increase the cost of
extraction.
4. CONCLUSIONS
To take into
account the productive characteristics of mining industries we present a
decomposition of the corrected TFP growth into five components: technical
change, subequilibrium, scale, allocative inefficiency and resource depletion. This index can be
used as an indicator of economic sustainability for mining, and its application
to the example of
Spain ’s
mining industry has brought us to the following conclusions:
- The economic growth of the
sector for the suggested period (in terms of productivity) has proved to be
moderate (an average per annum growth rate of 1.75%). This growth was partly
undermined (by about 0.78%) by the effect caused by reduction in reserves (this
is known as the “depletion effect”).
- The use of production
potential has not proved to be optimal, given that scale economics have not
been used to their full advantage, production capacity has been under-used, and
production factors have not been efficiently assigned.
- The component of
technological change has shown the best behaviour, anticipating that it is
possible to generate a process of replacement of natural capital with
artificial capital in the Spanish mining industry. Should this process take
place, it would help lay out some good groundwork to improve the criteria of
inter-generational equity, a hitherto hard-to-achieve goal in this type of
industry.
- In addition to this, the
effects of technological change on this growth have amply compensated for the
effects of reserves depletion on said growth, which is akin to what occurs with
other minerals [2].
Because of these results, and
with view to establishing some policies to maintain sustainability in
Spain ’s
mining industry, we suggest the following:
- To improve knowledge of
mineral reserves, in order to more accurately measure the effects of its
reduction in economic growth and to enable some planning to optimize its use.
- To continue favouring
technological progress, as this is a safe
way to guaranteeing the growth of artificial capital (given
the evident reduction in natural capital) and to compensate for the possible
effects of resource reduction (and in this way guarantee their better
availability in the future).
- To improve the human capital
that intervenes in the management and production processes within the mining
industry. This seems like the safest alternative to enable its efficient
management and, by default, an optimal use of its productive capabilities.
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