Draft May 2012
Induced Biased Innovation and
Exhaustible Resource Prices
John R. Boycea), Daniel V. Gordonb) and Sjak Smuldersc)
Abstract: This paper tests a prediction from endogenous growth theory regarding the rate
at which commodity prices in exhaustible resource markets grow. The hypothesis we test
is that innovation is more likely to occur in large sectors of the economy. This is tested
using a panel of 80 mineral commodities annual price growths over the period 19002006. We find support for this hypothesis.
Keywords: Endogenous Growth, Resource Prices
JEL: Q3, C23, C36
a) Professor of Economics, University of Calgary, 2500 University Drive NW Calgary
Canada T2N 1N4 email: boyce@ucalgary.ca
b)
Professor of Economics, University of Calgary, 2500 University Drive NW Calgary
Canada T2N 1N4 email: gdgordon@ucalgary.ca
c)
Professor of Economics, Tilburg University P.O. Box 90153, 5000 LE Tilburg, The
Netherlands. Email: j.a.smulders@uvt.nl
1 1. Introduction
This paper contributes to our understanding of commodity price formation by empirically
examining the relationship between commodity prices for exhaustible resources and induced
innovation. The underlying hypothesis we test is quite simple. Induced biased endogenous
growth theory (e.g., Kennedy (1964), Acemoglu (2002, 2007), Smulders and de Nooij (2003))
suggests that innovation is more likely to occur in markets in which the gains from innovation
are large. Those tend to be markets for which the size of the market is quite large. Thus, if
endogenous innovation drives economic growth, and if innovation is of the cost-reducing form,
we would expect to see that commodity prices are more likely to decline when the size of the
world value product of the commodity is large.
Economists
have long been interested in natural resource prices. Barnett and Morse (1963)
were the first to draw upon a large data set of U.S. mineral prices to analyze resource scarcity.
This was updated by Smith (1977). Slade (1982) was the first to examine the connection between
innovation and exhaustible resource prices. She focused on the time trend of natural resource
prices, regressing the real price of twelve minerals against time and time squared to determine
whether or not mineral prices followed a U-shaped price path. Her work was criticized by
Mueller and Gorin (1984) for not taking into account specific events concerning market structure
and by Perck and Roberts (1986) for not taking account of the time-series properties of her data.
This literature was reviewed and Slade's empirical work extended by Krautkraemer (1998). More
recently, Kellard and Wohar (2006) examined price trends for twenty-four commodities (some of
which are renewable resources) using time series methods, and Gaudet (2007) examines the rate
of growth in prices in the context of the Hotelling model. None of these papers investigates the
empirical connection between innovation and the rate of growth in prices.
2 This paper is closely related to Popp (2002), who looked at the influence of energy prices
on innovation. While Popp was concerned with how natural resource prices affected innovation
activities, we indirectly examine the flip side of that question. What we wish to explain is
resource commodity price growth rates. Our linkage to innovation is indirect. We expect
innovation to be positively related to the size of the commodity in the economy. Thus,
commodities that are more important in the economy will attract more innovation. Commodities
for which innovation is higher will have higher downward pressure on price growth. In this
sense, our work is in a similar spirit to De Cian (2008), which was developed independently of
our own paper.
De Cian (2008) uses data from fifteen OECD countries over the period 1978-2005 to assess
the impact of factor augmenting technical change in three inputs; labor, capital and energy. The
methodology utilizes a constant elasticity of substitution aggregate production function with
technical change affecting the productivity of each of the inputs. It is not possible to identify
factor augmenting technological change from technological change that affects each input, so the
assumption is made that all technological change is factor augmenting. She estimates two
empirical models. The first assumes that factor augmenting technological change is exogenous,
while the second, which is more closely related to this paper, assumes that factor augmenting
technological change is endogenous. She postulates that endogenous factor augmenting
technological change is driven by R&D expenditures, aggregate imports, imports of machinery,
and that labor productivity is affected by education expenditures. Her empirical results on
endogenous technological change suggest that R&D expenditures and education expenditures
can explain part of the factor augmenting technological change.
3 The paper proceeds as follows. Section 2 presents and reviews the theoretical models on
price formation in exhaustible resource models and how the price of an exhaustible resource will
grow over time. Section 3 describes the panel data available for price growth analysis for 80
minerals with a time series extending in some cases from 1900-2006. Section 4 describes the
econometric modeling and specification of instrumental variables required for empirical
estimation. Section 5 concludes.
2. Theoretical Analysis
The Hotelling (1931) model of exhaustible resources assumes a fixed stock of the resource, 𝑅! ,
cost of extraction, 𝐶 𝑄(𝑡) = 𝑐𝑄 𝑡 , and demand curve, 𝑃 𝑡 = 𝐷 𝑄(𝑡) , in industry extraction,
𝑄(𝑡) where 𝐷 ! 𝑄 < 0. When the stock of reserves is fixed, the equation of motion for reserves
is (dropping time notation)
𝑅≡
!"
!"
= −𝑄, 𝑅 0 = 𝑅!
(1)
Alternatively, it will be convenient to work in cumulative extraction, 𝑋, where
𝑋≡
!"
!"
= 𝑄, 𝑋 0 = 𝑋!
(2)
Feasibility requires that 𝑅(𝑡) ≥ 0 and 𝑥 𝑡 ≤ 𝑋! + 𝑅! for all 𝑡.
2.1 Perfect Competition: Basic results
With competitive firms, each owning an infinitesimal share of the stock, and facing an
exogenous rate of return to capital of 𝑟, Hotelling showed that the equilibrium price path satisfies
𝑃 =𝑐+𝜆
=𝑐+𝜇
(3a)
(3b)
Where 𝜆 > 0 is the current value of the resource stock in situ, and 𝜇 < 0 is the marginal value of
cumulative extraction. With competitive markets for ownership of the stock, 𝜆 = −𝜇 is the
4 scarcity rental price that would be paid for a unit of the stock in the ground. In order that the rate
of return from holding the exhaustible resource asset is equal to the rate of return elsewhere in
the economy, the scarcity rental values must rise over time at the rate of interest, 𝑟,
!
𝜆 ≡ ! = 𝑟
(4a)
𝜇 ≡ ! = 𝑟
(4b)
!
Therefore, time differentiating (3) and using (4), the Hotelling prediction is that the price grows
over time:
!
𝑃! ≡ ! = !(!!!)
!
> 0.
(5)
2.2 Imperfect Competition
Exhaustible resource markets often have few firms producing the resource (e.g., Salant (1974),
Loury (1986), Polasky (1992)). The effect of imperfect competition can be seen by considering a
simple Cournot model of imperfect competition in the extraction industry. Suppose there are 𝑁
firms, each holding 𝑅!! initial reserves and facing costs 𝑐. Then the first order condition for the
𝑖 !! firm is
𝑃 1−
!!
∈
= 𝑐 + 𝜆! , 𝑖 = 1, ⋯ , 𝑁,
where 𝑠! ≡ 𝑞! 𝑄 is firm 𝑖′𝑠 share of production and ∈ ≡ (6)
!"
!"
∙
!
!
is the elasticity of demand.
Holding 𝑠! and ∈ constant, the rate of change of price in an oligopolistic market is
𝑃! = !
! ! !! ! !!
∈
!
!(!! ! )
(7)
∈
Comparing the growth rate of the competitive market with that of the oligopolistic market at a
common starting price, 𝑃 yields that the competitive market price grows more quickly than the
oligopolistic price:
5 0<
𝑃 1−
𝑃! = 𝑠!
∈
𝑠!
𝑠!
− 𝑐 < (𝑃 − 𝑐) 1 −
∈
∈
𝑠!
−𝑐
𝑟(𝑃 − 𝑐)
∈
<
≡ 𝑃! ,
𝑠!
𝑃
𝑃(1 − )
∈
𝑟 𝑃 1−
where the second inequality is obtained by adding -1 to each side of the equation, multiplying by
𝑐, and adding 𝑃(1 −
!!
∈
) to the result; and the second inequality is obtained by dividing each side
of the equation by 𝑃 and then by 1 −
!!
∈
. Loury (1986) and Polasky (1992) obtain further results
by allowing 𝑠! and ∈ to vary over time. However, for our purposes, the main result is that an
increase in market power lowers the rate of growth of prices.
2.3 Effects of Cumulative Extraction
The competitive model was extended by Schulze (1974), Solow and Wan (1976), and Slade
(1982), who considered models where deposits differ in quality and where marginal costs may be
increasing with output. When each deposit is of infinitesimal size and differs by quality, Solow
and Wan showed that the reserves will be used in the order of least costs first. This implies that
we can model the unit costs of extraction as a function of reserves, 𝑅, or, equivalently, of
cumulative extraction, 𝑋. Letting 𝐶(𝑄, 𝑋) denote the costs of extraction, then 𝑐! ≡ 𝜕𝐶 𝜕𝑋 > 0
is implied if resources are used in the order of least cost, and 𝐶! > 0, 𝐶!! ≥ 0, and 𝐶!" ≥ 0. If
the equilibrium price in this model satisfies
𝑃 = 𝐶! 𝑄, 𝑋 − 𝜇
(8)
The equation of motion for 𝜇 is
𝜇=𝑟+
!! (!,!)
!
.
(9)
6 From (8), it follows that the price growth path is
𝑃 = 𝐶!" 𝑋 + 𝐶!! 𝑄 − 𝜇
!
!
= 𝐶!" 𝑄 + !!!
− 𝑟𝜇𝐶!
! (!)
𝑃 = 1 −
!!! !! !!" !!!!
! ! (!)
!
(10)
+
!(!!!! )
!
,
where the second line uses the definition that 𝑃 = 𝐷(𝑄) to show that 𝑄 = 𝑃 𝐷! 𝑄 , and uses
!
𝑋 = 𝑄 from (2); and the third equality divides each side of the equation by 𝑃 and by 1 − !!!!
.
(!)
Thus, again the Hotelling effect is the second expression, and this causes prices to rise over time.
The effect of cumulative extraction depends upon the sign of the numerator of the first
expression, which is 𝐶!" 𝑄 − 𝐶! . This expression is equal to 𝜕(𝐶! 𝑄 − 𝐶) 𝜕𝑋. The term
𝐶! 𝑄 − 𝐶 is positive if 𝐶!! > 0 and zero if 𝐶!! = 0. Thus, in order for there to be an effect on
price from cumulative extraction, it must be that the marginal cost of extraction is increasing in
𝑄. The sign of 𝐶!" 𝑄 − 𝐶! is an empirical question (Farzin, 1992).
2.4 Exogenous Technological Change
Stiglitz (1974) and Slade (1982) each considered models in which exogenous technological
change occurred. In Slade, the technological change was in regard to the cost of extraction, while
in Stiglitz the technological change was a change in the total factor productivity. We focus on the
Slade type of exogenous technological change. Let the cost of production be denoted as 𝐶(𝑄, 𝐵),
where 𝐵 is a measure of technology, where 𝐶! > 0, 𝐶!! ≥ 0, 𝐶! < 0 𝑎𝑛𝑑 𝐶!" < 0 are assumed.
Further, with exogenous technological change, assume that 𝐵 grows according to
!
𝐵 ≡ ! = g > 0.
(11)
Then, the equilibrium equation for the market price is
𝑃 = 𝐶! 𝑄, 𝐵 + 𝜇.
(12)
7 Differentiating this with respect to time yields
!
!!
𝑃 = 1 − !! (!)
!! ! !
!" !
!
+
!(!!!!
!
.
(13)
While the second expression is the Hotelling effect from scarcity, which is positive, the first term
is the effect of technological change. This expression is negative, since 𝐶!" < 0. By observing
that innovation is most valuable when reserves are large, Slade (1982) argued that the innovation
effect should dominate early in the extraction cycle and that the Hotelling scarcity effect should
dominate later in the extraction cycle. Thus, she predicted U-shaped price paths.
2.5 Endogenous Technological Change
Smulders and de Nooij (2003) consider a model in which endogenous technological change
occurs. In their model, the rate of innovation depends upon the size of the economy. Let 𝜃 ≡
!"
!
,
where 𝑌 is gross world GDP, denote the share of a mineral resource in the economy. Then
Smulders and de Nooij (2003) postulated that the rate of technological improvement, 𝐵 is an
increasing function of the size of the sector in the economy:
𝐵 = ℎ(𝑣)
(14)
where ℎ! 𝑉 > 0. Thus, in their model, the growth in the price of an exhaustible
resource is given by
!
𝑃 = 1 − !!!!
(!)
!! ! !!(!) !(!!!
!"
!
!
!
.
(15)
Since 𝐶!" < 0 and ℎ! 𝑉 > 0, it follows that an increase in 𝑉 causes the price growth rate to
decrease. The key empirical difference between the endogenous and exogenous technological
change models is that in the exogenous technological change model, ℎ 𝑉 = 𝑔, which is
independent of 𝑉. Hence, the key empirical test whether the rate of growth of prices is inversely
correlated with the size of the sector in the economy.
8 3 Data Definitions and Summary Statistics
The focus of the empirical work is to empirical test the hypothesis predicting a negative
relationship between the growth rate in mineral price and endogenous technical change as
measured by world value product. The data for analysis includes 80 minerals covering the period
1900-2006. While price and quantity data are available for only 27 of the 80 minerals for more
than 100 years, data are available for 26 years or longer for all 80 minerals. The data consist of a
main database covering all 80 minerals and includes the growth rate in mineral price, world
value product, cumulative production, world population and the U.S. real bond rate. The
secondary data set includes only a subset of the defined minerals and a much-shortened time
series but augments the variables in the main database with proxy measures of monopoly power
on both the demand and supply side of the market.
The primary source for the data is Kelly and Matos (2007), which is an electronic
database maintained by the United States Geological Survey (USGS). This database contains
price and production data for some 90 minerals. We use all minerals for which both price and
quantity data are available. Prices are U.S. prices per tonne. This yielded data for 76 minerals.
This data was supplemented with data for four additional mineral resources: petroleum, coal,
natural gas and uranium. That data comes from a variety of sources. The oil price data is from
the BP Statistical Review of Energy 2008. The natural gas, uranium, and coal price data are from
the United States Energy Information Agency's (EIA) Annual Energy Review. Coal quantity data
is from two sources; from 1965 forward, coal production data is contained in the BP Statistical
Review of Energy 2008, and world coal production data is available back to 1932 from the
USGS publication Minerals Yearbook (various years). World petroleum production data is from
World Oil's Annual Forecast and Review Issue. Uranium consumption data is complied by using
9 the quantities of fuel injected into reactors in the United States, published in the EIA's Annual
Energy Review, and combining this with the electricity production by nuclear energy sources
published in the BP Statistical Review of Energy 2008. Using the U.S. average consumption
rates, world production1 rates were then estimated.
For the 𝑖 !! mineral in the 𝑡 !! period, the growth in mineral price is defined as
𝑃!"# = ln (
𝑃!"
𝑃!"!! )
(1)
where 𝑃!" is real price of the 𝑖 !! mineral in the 𝑡 !! period. Prices are in 2000 U.S. dollars per
tonne. Table 1 reports for each mineral in the data set the mineral id number, time period
available, average price and average growth rate in price. Prices vary dramatically, with
gemstones reaching almost 2 billion dollars per tonne down to $17 per tonne for industrial
gravel.2 On average, mineral prices were declining at rate -0.008 per year in an un-weighted
average over all minerals and years, and almost 3/4 of the minerals experienced declining prices
on average. While 15 mineral prices declined by more than 2% per annum, only two (quartzcrystals and thallium) experience growth rates of more than 2%. Thus, Barnett and Morse
(1963), Slade (1982), Krautkraemer (1998), Kellard and Wohar (2006) and Gaudet (2007), inter
alia, have observed, the preponderance of evidence is that mineral prices have declined in the
twentieth century.
The time series structure of individual prices requires that we investigate the stochastic
properties of each. Undoubtedly individual prices are nonstationary3 implying that the moments
of the distribution are changing over time, but the transformation to the growth in prices is a
form of first differences and likely a transformation to stationary stochastic properties. We test
for stationarity using a standard Dickey-Fuller statistic and report the results in column 5 of
1
These exclude uranium used in nuclear weapons and other non-energy uses.
Natural gas averaged approximately $2 per million cubic feet and uranium averaged about $21 per pound.
3
Dickey-Fuller statistics for individual prices confirms this property.
2
10 Table 1. For each mineral the Dickey-Fuller statistic is significantly important and we can reject
the null hypothesis of non-stationary in the growth in prices. In other words, the transformation
to the growth in prices is in fact a stationary transformation. This is important information for
econometric specification and will allow reduced form modelling with a stationary (i.e.,
integrated of order zero) dependent variable.
Table 2 records for each mineral world production in tonnes, average value of world
value product and cumulative production. To calculate world value product by mineral, we used
data on world GDP compiled by Angus Maddison.4 That data contains both population and GDP
information by country through 2003, with the GDP data in 1990 U.S. dollars. We extended
Maddison's GDP series with data from the World Bank's World Development Indicators data. As
there is no reliable world GDP data existing prior to 1950, we use the Maddison data that existed
for the thirty countries for which data is continuously available back to 1900. Those countries
include the twenty original members of the OECD5, plus seven Latin American countries not
original members of the OECD (Argentina, Brazil, Chile, Columbia, Peru, Uruguay, and
Venezuela), and three Asian countries not original members of the OECD (India, Indonesia, and
Sri Lanka). We call this the ‘Group of Thirty’ countries and their combined GDP30 is used in
calculating our measure of world GDP.
For the 𝑖 !! mineral in the 𝑡 !! period, world value product is defined as
𝑉!" = ln (
𝑃!"∙ 𝑞!"
𝐺𝐷𝑃!!"
)
(2)
where 𝑞!" is world production. From Table 2, the share of minerals in world GDP is quite small.
Petroleum is the largest (accounting for just less than 1/2 of the world value product of the
4
5
Downloaded from http://www.ggdc.net/maddison/.
These are Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Italy, Japan, Mexico,
Netherlands, New Zealand, Norway, Portugal, Spain, Sweden, Switzerland, the United Kingdom, and the United
States. 11 minerals we consider), but even here it only accounts for 2.12% of world GDP. Coal is second in
share accounting for 1.1%. All other minerals are less than 1% of world GDP: cement (0.47%),
iron-ore (0.27%), aluminum (0.15%), nitrogen (0.10%), copper (0.22%) and gold (0.23%) are the
only minerals accounting for more than 1/10th of world GDP. At the other end of the spectrum,
thallium accounted for only 5/1000th of a percent of
world GDP. The average share of all
minerals was about 0.08% of world GDP, and the total for all minerals was 5.63% of world
GDP.6
Cumulative production is calculated from annual world production since 1900. As we
wish to have a measure that varies across time and since cumulative production varies most
dramatically across minerals, we use as our measure of cumulative production the ratio of
current production to cumulative past consumption. This measure is without units, and varies
between zero and one in value, declining as time progresses.
For the 𝑖 !! mineral in the 𝑡 !! period, cumulative production is defined as
𝑄!"# = ln (
𝑞!"
𝑋!" )
(3)
where 𝑋!" is cumulative production. For completeness we report average values of cumulative
production for each mineral in column 4 of Table 2.
Again, we want to characterize the stochastic structure of the individual time series for
both world value product and cumulative production. We report in Table 3 standard DickeyFuller time series statistics for each mineral for the variables of interest. The table tells and
interesting story; for world value product only 16 of the 80 minerals show stationarity in the
defined variable at less than the 5% p-value and this increases to only 22 out of 80 at less than
the 10% p-value. This is fairly strong statistical evidence that for the vast majority of minerals
6
Our group of minerals accounted for 4.63% of total world GDP.
12 the world value product variable in a strictly time series setting is non-stationary and thus
integrated of order one. However, for cumulative production we observe the opposite and here
the vast majority of minerals (73 of 80) are statistically stationary (at less than 5% p-value) and,
consistent with the price growth variable, integrated of order zero.
Econometrically the Dickey-Fuller results for the three main variables of interest cause
serious concern. In a time series perspective, it is in general not possible to form an equilibrium
relationship with variables integrated of different order (Enders, 1994), which is exactly the case
at hand with price growth and cumulative production integrated of order zero and world value
added integrated of order one. However, there is one possibility that may allow us to proceed
with the statistical model. The Dickey-Fuller statistics presented above are a purely time series
evaluation and thus ignore the stochastic panel properties of our data. It is possible that all
variables are stationary in a panel sense.
Here we evaluate the stationary prospects of our panel using two statistics for testing unit
root hypotheses in un-balanced panels. Im, Pesearan and Shan (2003), hereafter the IPS test,
develop a transformation of the standard Dickey-Fuller procedure for testing the null hypothesis
that all panels have unit root against an alternative that at least some panels are stationary. The
test allows for trend, demeaning the data and accounting for autocorrelation structures. Under the
assumption that 𝑇 ⟶ ∞ and N⟶ ∞ the test statistic is asymptotically standard normal. The
second test is a Fisher-type test where the individual Dickey-Fuller statistics obtained above
(Table 1 and 2) are combined using meta-analysis to generate a powerful test for stationary in the
panels. The null and alternative hypotheses are as defined for the IPS test. Choi (2001) shows
that if the time dimension is large and the number of panels finite (less than 100) the test statistic
13 follows an inverse normal. Choi defines the test statistics as a ‘Z-statistic’ and suggests it use
based on power and size.
For the three panel data variables of interest we report in Table 4 both the IPS and Fishertype test statistics. Now we observe for each variable and for each statistical test that the null
hypothesis of unit root in each panel can be rejected. This provides statistical evidence that
within the panel data framework all important variables have stochastic properties amenable to
equilibrium evaluation. The equilibrium relationship can be further evaluated after estimation by
testing the predicted errors for properties of stationarity.
We augment the panel variables above with strictly time series variables that control for
demand and supply shifters that may affect the equilibrium. We include two types of shifters. On
the supply side, we use as a proxy the real U.S. bond rate (Reference, year), which is constant
across commodities but varies by time. In Figure 1, we graphically represent the variation in this
variable overtime- 1900-2006. Notice the relative stability in the bond rate prior to the 1940s but
gradually increasing after this point to reach a maximum of over 12 points in 1981 before
declining to 4.8 points in 2006.
The demand side shifter is world population defined as the combined population of our
group of thirty countries. All else equal, we expect an increase in the population to increase the
demand for the commodity.7 This variable is shown in Figure 2 for the period 1900-2006.
Both the demand and supply shift variables are constant across the different minerals but
vary over time. Figure 1 and 2 suggest trends in each series and standard Dickey-Fuller statistics
7
This variable is used in lieu of world per capita income as the numerator of per capita income is world GDP, which
is used in the denominator of the world value product for each mineral. 14 show that a null hypothesis of non-stationarity cannot be rejected.8 Standard first-differences
techniques transform both series to stationary processes.9
Our secondary data set includes all of the variables defined above for a reduced number
of minerals over the period 1990-2004.10 The minerals and time period are dictated by the
availability of indices of market power.11 The market power variables are based on data
published in the Minerals Yearbook on 49 minerals for which data on the country of production
exits. This data is available from 1990-2004. From this data, we calculate the Herfindahl index,
𝐻!"! =
!
!!!(100
∙ 𝑠!" )! where 𝑠!" is the share of world production of the 𝑖 !! mineral for the 𝑗!!
country 𝑗 = 1 ⋯ 𝐶. We use this data to measure market power among suppliers. This is an
imperfect measure of supplier market power, as it is country based rather than firm based.
However, it is the only measure that is widely available. In addition, Kelly and Matos (2007)
report data for 56 commodities from the United States on which industries consume the
commodity. For example, titanium is used primarily in the aerospace industry. From this data,
we are able to construct a Herfindahl index of the demander's of the resource, 𝐻!"! =
!
!!!(100
∙
𝑠!" )! where 𝑠!" is the share of the 𝑖 !! mineral consumed by the 𝑘!! company 𝑘 = 1 ⋯ 𝐾. This is
our measure of market power on the demand side of the market. Our Herfindahl demand index
cannot measure the number of firms in a sector, but it does measure the number of sectors that
are buying the commodity. We list the minerals available for the market power variables and
both the Herfindahl demand and supply index in Table 5.
8
The DF statistics are -0.79 (0.966) and -1.66 (0.769) for population and real bond rate, respectively. (p-value in
parentheses)
9
The DF statistics are -5.12 (0.000) and -7.74 (0.000) for population and real bond rate, respectively. (p-value in
parentheses) 10
The mineral mercury is included in this data set for the period 1990-2003.
11
Boyce (2009),
John we need some reference to your building the Herfindahl
indices.
15 The Herfindahl market power variables have short time periods but we can evaluate the
stochastic property of each variable within the panel stationary testing procedures. We rely on
the IPS t-bar statistic that allows for both the number of cross-sections and time periods fixed.
These statistics are reported at the bottom of Table 5 and, again, provides statistical support that
in a panel setting both Herfindahl indices can be considered stationary.
In summing up, the main data set available for econometric analysis is an unbalanced
panel on three cross section variables (mineral price growth, world value product and cumulative
production) and two time series variables (population and real U.S. bond rate). The main data set
has 80 panels with individual time periods ranging from 22 to 107 years for a total sample of
6064. A secondary data set augments the variables with proxy measures of market power on both
the demand and supply side of the market. This data set has 22 panels with time periods of 14 or
15 years for a total sample of 307 observations.
4 Econometric Specification and Results
The econometric search is to model the data generating process describing the growth in mineral
prices. Theory suggests a number of variables that impact the moments of this distribution and
our efforts will be to attempt to statistically identify the importance of endogenous technical
change within this stochastic process. We start by specifying a general reduced form equation
including all variables as:
ln(
Pit
Pq
q
) = β v ln( it it30 ) + β x ln( it ) + β n ln Popt30 + β Rb Rbt + β d ln H itd + β s ln H its + ηi + ε it
Pit −1
X it
GDPt
4)
16 where ln(
Pq
Pit
) is annual rate of growth in the real price of mineral i in year t; ln( it it30 ) is
Pit −1
GDPt
world value product of mineral i in year t; ln(
qit
) is the ratio of world production to cumulative
X it
production of mineral i in year t; ln Popt30 is world population in year t; Rbt is the U.S. real bond
rate in year t; ln H itd is Herfindahl index of demand sources for mineral i in year t; ln H its is
Herfindahl index of supply sources for mineral i in year t; ηi is unobserved heterogeneity of
mineral i and ε it is idiosyncratic error term.
Several econometric issues present themselves in equation 4): First, all variables defined
in equation 4) are integrated of order zero except population and bond rate, which are integrated
of order one. In estimation, the log of the first differences of both population and bond rate are
used. This allows us to define equation 4) as an equilibrium expression. Second, we must decide
on an appropriate estimator to account for the unobserved heterogeneity within the panels. And
finally, both world value product and cumulative production are endogenous in the sense that
they are correlated with the idiosyncratic error term. World value product includes the current
price of mineral i, which is in fact part of the endogenous variable that we are trying to explain.
Whereas cumulative production includes current quantity of mineral i, which is a choice variable
to the economic agents we are modelling. For the second issue, we will rely on a standard fixed
effect model using the within estimator. In general, this will account for the unobserved
heterogeneity within the panels and can generate consistent estimates of the parameter
coefficients in our reduced form model. For the third issue, we account for the endogenous
problem using an instrumental variable (IV) procedure that relies on the time series properties of
the variables (hereafter called lagged IV). To make the econometrics tractable and test the
17 robustness of the lagged IV estimator combinations of lags up to four periods will be used. In
addition, we attempt an alternative IV procedure relying on the physical properties of the
minerals investigated. We obtain information on the abundance of each mineral within the
earth’s crust, the minerals are ranked based on abundance and for each mineral we arbitrarily
pick the price of the immediate more abundant and less abundant mineral for use as instrumental
variables (hereafter called Earth IV).
Table 6 shows a ranked list of our minerals that are amenable to the Earth IV estimator.
We list mineral id as defined in Table 1 and the id of the mineral immediate more abundant and
less abundant. The table lists 46 minerals with the most abundant in the earth’s crust being
silicon and aluminum and the least abundant being gold and rhenium. Each of our minerals is
associated with two price instruments except silicon and rhenium which because of end point
restrictions receive one price instrument each.
Table 7 shows some results for equation 4) applied to our main data set using the Fixed
Effect estimator and lagged IV. To start the results we estimate the simple two-variable
regression of price growth on world value product using a two-lagged IV variable for world
value product within the Fixed Effect estimator. Of course, within our reduced form specification
the simple regression model is subject to omitted variable bias but does set a reference point for
comparison with more complete models. In the simple regression, we see a statistically
significant negative relationship from world value product to mineral price growth. In fact, a
literal interpretation would indicate a 1% change in current world value product resulting, on
average, in a -0.053% decline in mineral price growth. Moving to the full-reduced form
specification in regression 2 (again using a two lagged IV for both world value product and
cumulative production) we observe that the negative relationship is maintained between world
18 value product and price growth and not substantially impacted by the introduction of the other
variables. Now, we also observe a positive impact, as expected a prior, of cumulative production,
population and U.S bond rate on price growth. Note, however, the bond rate is not statistically
significant. The positive elasticity result for cumulative production is in agreement with the
observation by Slade (1982) that prices were likely to decline in the early part of the data than in
the later part i.e., prices follow a U-shaped path.
It is important to evaluate how robust our main estimated prediction is on the relationship
between world value product and price growth to the definition of the lagged IV variable for
world value product. In Table 7a we show only the estimated coefficient and p-value for world
value product for four alternative definitions of lagged IV.12 The conclusion is clear and
indicates that actual number of lags defining the lagged IV is not the determining factor in setting
the sign of the coefficient. Parsimony and short time series on some minerals pushes us to define
a two period lag for the lagged IV estimator for further estimation.
We test further the results obtained for the two period lagged IV reported in regression 2
Table 7 by testing the estimated residuals for stable stochastic properties. Our reduced form
model is an equilibrium model so there is no notion of cointegration but rather if the model is
statistically specified correctly the predicted residuals must be stable or stationary. We test
predicted residuals for each mineral and the full panel data set and we find solid evidence of
stable stochastic properties.
Returning to Table 7 columns 4 and 5 re-estimate the reduced form equation using
different truncations of the main data set. Column 4 shows results for regression 3 that uses only
data on minerals each with 107 years of observations. There are 20 minerals in this category for
a total of 2099 observations. Column 5 shows results for regression 4 that uses only data on
12
Other coefficients in the reduced form are not substantially impacted by changes in the lagged IV.
19 minerals with less than 107 years of observations. There are 60 minerals in this category for a
total of 3766 observations. In both regressions, we still predict on average a negative and
statistically important elasticity for our main hypothesis between world value product and
mineral price growth, albeit regression 3 under predicts the magnitude of the effect whereas
regression 4 over predicts. Also, note that in regression 3 world value product is the only variable
statistically significant.
Table 7 certainly provides statistical support of our main hypothesis of endogenous
technical change negatively impacting the growth in prices but does not provide an empirical
measure of the lagged impact13 of innovation on prices. Perhaps one way of looking at this
problem is to approach estimation of the reduced form not with an IV estimator but rather the use
of lagged proxy variables. We do this by re-estimating regression 2 in Table 7 using sequentially,
lags one to six of world value product as proxy for current value of world value product.14 The
results for the coefficients on world value product are displayed figuratively in Figure 3. Upper
and lower confidence limits accompany predicted coefficients. The first five lags result in
statistically significant negative coefficients increasing in value towards zero. The lag six proxy
variable is statistically zero. We interpret this result to indicate that the impact of innovation on
price growth is time dependent but declining in statistical importance over time. For our data set
we observe a statistical window of five years of innovation impacting current mineral price
growth.
Finally, Table 8 shows some alternative regression results with respect to world value
product and price growth. Column 2 shows a re-estimation of our main regression model adding
in dummy variables to account for four major shocks that occurred in the 20th century; First
13
Theory does not provide a prediction about how long before the effect of innovation is seen in prices.
We maintain an IV estimator for cumulative production.
14
20 World War (DWWI-1914-1917), Second World War (DWWII-1939-1945), a world depression
(Ddep-1929-1935) and a major world recession (Dres-1980-1986). From the results displayed,
these shocks although perhaps important in their own right have little or no impact on marginal
effects of interest to this research and are not considered further.
Regressions 2 and 3 in columns 3 and 4 of Table 8 show the reduced form results using
the Earth IV. Regression 2 uses all minerals available for the Earth IV including the end points,
whereas, regression 3 repeats the exercise but drops the end point observations. In general both
sets of regression results confirm our earlier findings on an important negative elasticity between
innovation and price growth but note the comparatively large standard errors on the estimated
world value coefficients in Table 8 compared to regression 2 in Table 7. This might suggest that
the Earth IV is not as statistically efficient as the standard lagged IV. It does however show
robustness in our main result.
Finally, the last regression reported in column 5 augments the main regression
specification by including proxy variables for market power on both the demand and supply side
of the market. Lagged IV procedures for both world value product and cumulative production are
used in combination with the Fixed Effect estimator in estimation. By introducing the power
proxies we suffer a serious reduction in sample size and number of minerals used in the
regression but we do obtain a more complete variable specification of our reduced form model.
Interestingly the Herfindahl specification results in much higher elasticity of world value product
(by almost an order of magnitude) compared to the main reduced form equations. In any case the
Herfindahl specification shows both demand and supply power proxies unimportant in
estimation.
21 Overall the statistical results obtained using many different regression specifications are
consistent in the findings of our main theoretical hypothesis of endogenous innovation impacting
the growth in mineral prices.
5 Conclusions
References
Acemoglu, Daron, “Directed Technical Change” Review of Economic Studies, 2002, 69
(October), 781-810.
-- “Equilibrium Bias of Technology” Econometrica, 2007, 75 (September),1371-1409.
Barnett, Harold J. and Chandler Morse, Scarcity and Growth: The Economics of Natural
Resource Availability, Baltimore: Johns Hopkins University, University Press, 1963.
Choi, I. 2001. “Unit root tests for panel data,” Journal of International Money and Finance 20:
249–272.
De Cian, Enrica, “Factor-Augmenting Technical Change: An Empirical Analysis,” Resource and
Energy Economics, 2008, (forthcoming).
Enders, W. 2010, Applied Econometric Time Series Wiley.
Farzin, Y. H., “The Time Path of Scarcity Rent in the Theory of Exhaustible Resources,” The
Economic Journal, 1992, (July), 813-830.
Gaudet, Gerard, Canadian Journal of Economic, 2007, (November).
Hotelling, Harold, “The Economics of Exhaustible Resources,” Journal of Political Economy,
1931, (April), 137-175.
Im, K. S., M. H. Pesaran, and Y. Shin. 2003. “Testing for unit roots in heterogeneous panels,”
Journal of Econometrics 115: 53-74.
Kellard, Neil and Marc A. Wohar, “On the Prevalence of Trends of Primary Commodities,”
Journal of Development Economics, 2006, 79, 146-167, 12
Kelly, T.D. and G. R. Matos, “Historical statistics for mineral and material commodities in the
United States,” U.S. Geological Survey Data Series 140 (http://pubs.usgs.gov/ds/2005/140/),
2007.
22 Kennedy, Charles, “Induced Bias in Innovation and the Theory of Distribution,” The Economic
Journal, 1964, 74 (September), 541-547.
Krautkraemer, Je_rey A., “Nonrenewable Resource Scarcity," Journal of Economic Literature,
1998, 36, 2065-2107.
Loury, Glenn C., “A Theory of 'Oil'igopoly: Cournot Equilibrium in Exhaustible Resource
Markets with Fixed Supplies,” International Economic Review, 1986, 27, 285-301.
Mueller and Gorin, Journal of Environmental Economics and Management, 1984.
Perck, Peter and Roberts, Journal of Environmental Economics and Management, 1986.
Polasky, Stephen, “Do Oil Producers Act as 'Oil'igopolists?,” Journal of Environmental
Economics and Management, 1992, 22.
Popp, David, “Induced Innovation and Energy Prices,” American Economic Review, 2002, 92,
160-180.
Salant, Stephen W., “Exhaustible Resources and Industrial Structure: A Nash-Cournot Approach
to the World Oil Market,” Journal of Political Economy, 1974, 84 (October), 1079-1093.
Schulze, William, “The Optimal Use of Non-Renewable Resources: The Theory of Extraction,”
Journal of Environmental Economics and Management, 1974, 1, 53-73.
Slade, Margaret M., “Trends in Natural-Resource Commodity Prices: An Analysis of the Time
Domain,” Journal of Environmental Economics and Management, 1982, 9, 122-137.
Smith, V. Kerry, Scarcity and Growth Revisited 1977.
Smulders, Sjak and Michiel de Nooij, “The Impact of Energy Conservation on Technology and
Economic Growth,” Resource and Energy Economics, 2003, 25, 59-79.
Solow, Robert M. and Henry F. Wan, “Extraction Costs in the Theory of Exhaustible
Resources,” Bell Journal of Economics, 1976, pp. 359-370.
Stiglitz, Joseph E., \Growth with Exhaustible Natural Resources: Efficient and Optimal Growth
Paths” Review of Economic Studies, 1974, 41, 123-137.
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