April 28, 2006
Natural Capital Index of Canada
: A barometer of the stock of natural resources
Kazi N. Islam
Economist
Statistics Canada
7-E, R. H. Coats Bldg
120 Parkdale Avenue
Ottawa, ON K1A OT6
Kazi.islam@statcan.ca
ii
Table of Contents
Abstract
iii
Section 1: Introduction
1
Section 2: Similar Studies
2
Section 3: Data
4
Section 4: Methodology
6
Section 5: Natural Capital Index of Canada
9
Section 6: Natural Capital Index for the Provinces and Territories
18
Section 7: Conclusion
20
Appendix A: Figures—Index for the Provinces and Territories
22
Appendix B: Reserve and Present Value
32
Appendix C: Example of Index Construction
34
References
36
iii
Abstract
Canada, the second largest country in the world by landmass after Russia, is endowed with a huge
amount of natural capital--which refers to the land, the ecosystem and the natural resources such
as oil and gas and gold. Conservation and replacement or regeneration of this natural capital play
a crucial role for sustainable development—development of the current generation without
compromising the needs for future generations. This paper has developed an index that can be
used to track the trend in natural capital.
Currently Statistics Canada’s natural capital basket consists of 16 resources. The stock of the
resources and the present value of the stream of rents that may be generated from the stocks are
the basis of the newly developed natural capital index. The index is necessary to gauge the trend
in aggregate natural capital stock. Because the resources are measured using different units—gold
is measured in kilograms and oil is measured in cubic meters. Even if the resources could be
converted to the same unit they cannot be added as long as their net prices/rents differ. Therefore,
physical quantities or resources cannot be added together. The index can not only overcome this
measurement problem but also works like a barometer that can be used to track the state of
natural capital.
First, the paper has modified the standard Laspeyres, Paasche and Fisher quantity index formulas
to apply the stock and the present value of the rent data. Data for the period from 1980 to 2005
are used to calculate the chained version of each of these indexes. The chain-Fisher index is
mainly used for explaining the trends in the natural capital index as it is superior to the other
formulas. From 1990 to 2005, Canada’s natural capital had depleted by 13%, which can be a
source of concern for sustainable development. During the same period, Canada’s population
increased by 18%--from 27 million to 32 million. Further analysis reveals that much of this
depletion is due to the drop in the stocks of minerals and timber.
This paper has also provided the natural capital index for the provinces and territories—as they
are the owners and stewards of much of the natural capital in Canada. Thus, policymakers as well
as politicians may find the index useful for monitoring changes in resource stocks and developing
appropriate sustainable development strategies for their jurisdictions.
1
Natural Capital Index of Canada
: A barometer of the stock of natural resources
1. Introduction
Canada, the second largest country in the world by landmass after Russia, is endowed
with a large amount of natural resources such as oil and gas and gold. These resources are
the major components of natural capital--which refers to natural resources, the land
which provides space on which to live and work, and the ecosystems that maintain clean
water, air and a stable climate. In the past, natural capital was considered a free gift of
nature, in abundant supply. With industrialisation and increased world population, the
world demand for natural capital has increased rapidly. As a result, in recent years
resource based economies such as Canada have been extracting and exporting natural
capital at an increasing rate. By doing so, Canada--especially its provinces such as
Alberta and Newfoundland--are collecting substantial natural resource royalties. Most of
these royalties are spent to increase the welfare of the current generation.
However, rapid depletion of natural capital might be detrimental for sustainable
development--development of the current generation without compromising needs of
future generations. From a capital perspective, sustainable development requires nondeclining per capita national wealth by replacing or conserving the sources of that
wealth—the stocks of produced, human, social and natural capital.
For conservation, it is essential to have time series data on depletion as well as addition
of the natural capital. Statistics Canada, through its natural resources and stock accounts
(NRSA) initiative, has been gathering time series data on extraction and reserve for
natural gas, crude oil and timber since 1961. In later years, reserves of other resources
such as gold and coal were added to the NRSA as they became economically and
technologically feasible to extract: currently there are 16 categories of resources. These
reserves, known as established reserves, are expected to generate a stream of future
rents—the difference between the market value of a resource and its extraction costs.
The present value1 of potential rents that can be derived from the reserves are often used
by nations to gauge their natural capital riches or wealth. In 2005, per capita net worth of
Canadians stood at $166,000 of which more than $66,000 or 40% were derived from
natural capital, and the remaining 60% were derived from produced assets such as
machinery and equipment, and buildings and bridges.2 In recent years, the natural
resource wealth has been growing faster than the other types of wealth. This is mainly
due to the higher resource prices stemming from the increased world demand. The
1
The present value (PV) is the product of rent and reserve life, which is the ratio of reserve to production.
Thus, the PV is affected either by a change in rent or reserve or both. A note on the derivation of present
value of rents is shown in Appendix B.
2
In 2005, per capita net worth excluding natural resource wealth was $137,300. Natural capital wealth is
the sum of natural resource wealth and wealth generated from the land. For details, see, The Daily
<http://dissemination.statcan.ca/Daily/English/060317/d050317a.htm>
2
growth in price immediately affects the growth in rents and thereby the present value of
the resource stock. However, the impact of price on the stock of the resource is often
inconclusive. The price increase is likely to have dual effects on the stock: a) a higher
price would provide an incentive for exploration and drilling activities that might result in
opening new mine sites, thereby increasing the level of reserves, and b) on the other
hand, the higher price may increase profit and accelerate the level of extraction, which
would reduce the reserve. The following example demonstrates the relationship among
price, present value and reserve growth rates.
Figure 1: Crude oil--growth rates of price, present value and reserve
120
100
80
growth %
60
40
20
20
05
20
04
20
03
20
02
20
01
20
00
19
99
19
98
19
97
19
95
19
96
19
94
19
93
19
91
19
92
19
90
19
89
19
88
19
87
19
86
19
85
19
83
19
84
19
82
19
81
19
80
0
-20
-40
-60
Price
PV
Reserve
Figure 1 indicates that the relationship between crude oil price and its present value is
highly positive, whereas the relationship between the growth rate of price and the growth
rate of reserve is negative. From 1998 to 2001, all the three growth rates experienced
high degrees of volatility due to a sharp increase in reserve resulting from offshore oil in
Hibernia, located 315 kilometres east-southeast of St. John's, Newfoundland. Although the
Hibernia site was discovered in 1979, the resource was not considered a reserve until
1998, when extraction became economically and technologically feasible. Similar
patterns have been often observed across the natural resources.
As a result, the trend in the nominal value of the wealth inadequately reflects the trend in
the quantity stock, and cannot be used as a yardstick for conservation or sustainable
development. Although the reserve data would reflect the change in net stocks due to
3
price as well as technological changes, these data are often confidential at the provincial
and territorial level. Also, when looking at each resource individually, it is hard to
appraise the overall change when the reserve of one resource such as gold declines and
the reserve of another resource such as natural gas increases. Mainly because gold is
measured in kilograms and natural gas is measured in cubic meters. Even if two or more
resources are measured using the same unit, their physical stocks cannot be added unless
their average rents are identical.
Thus, an index showing the physical quantity of the remaining stock is needed for
tracking the natural resources stocks. The natural capital index (NCI), like other quantity
indexes such as the GDP volume index, is a weighted average of the remaining quantities
of natural resources, where the PV of rents is used as weights. The index would reflect
period-over-period change in the stock without revealing the stock levels. Clearly the
natural capital index, a unit-free measure, is the best choice in tackling both the
confidentiality and comparability concerns.
Certainly, the index will enrich the existing information about individual resource stocks,
and provide a snapshot of Canada’s natural capital stock for the policymakers and
politicians as well. In particular, the index can be used for the following purposes:
1) to track year-over-year change in aggregate natural capital both at
the federal as well as provincial and territorial levels,
2) to pin down which resource is depleting faster than it is being
replenished, and therefore conservation, regeneration or
replacement are warranted,
3) to compare inter-regional performance, i.e., which provinces or
territories are depleting or replenishing their resource stocks at what
rate?
In a word, the index would act like a barometer, which can be used without knowing
much about its construction. Notably, the marginal cost of updating the index would be
very low, since the index can be considered a by-product of NRSA, which compiles the
ingredients of the index: resources reserves and their present values.
The index is expected to generate renewed enthusiasm among Canadians, especially
those who are concerned about sustainable development. With this aim in mind, the rest
of the paper is organized as follows: Section 2 briefly reviews the experiences of
Australia and the Netherlands; Section 3 looks at the availability of data; Section 4
derives and discusses the methodology of constructing the NCI; Section 5 shows the NCI
of Canada and its implications; Section 6 examines the NCI for the provinces and
territories; and the last section makes concluding remarks.
4
2. Experiences of Australia and the Netherlands
The number of empirical studies related to the natural capital index is very limited: so far,
the literature suggests that Australia and the Netherlands have conducted empirical
studies in this regard. In 2001, the Australian Bureau of Statistics (ABS) introduced an
experimental real national balance sheet that excludes the effects of price change.3 The
ABS has derived chain volume estimates for subsoil assets, timber and land for the period
1992 to 2000. The constant price balance sheet was then used to evaluate the
composition of non-financial assets over time and to construct an index of real growth of
different assets in the balance sheet. In Australia, the chain volume estimates of subsoil
assets such as oil and gas and gold increased close to 50% from 1992 to 2000.This was
due to new discoveries exceeding extractions. Standing timber fell by 3.8% over the same
period.
The Netherlands, through its critical natural capital indicator initiative, has attempted to
develop three types of indicators: the indicator for stocks of natural capital, general
natural capital index and a protected areas index.4 Indicators for stocks of natural capital
are developed to identify and analyse the most important and critical functions of the
selected stocks of natural capital--fresh water, soil, wetlands, forest, fish and biodiversity.
Indicators are developed based on a set of operational criteria for the selection of the
“most important” functions. Also, an aggregated “total indicator” for the whole stock of
natural capital is produced.
The second set of indicators, known as the general natural capital index, is defined as the
product of the remaining area (ecosystem quantity) and its quality (ecosystem quality).
Ecosystem quantity refers to the percentage of remaining ecosystem in a particular region
and ecosystem quality is the ratio between current state of the ecosystem and the baseline
state. The natural capital index ranges from 0% to 100%: 0% means that the entire
ecosystem has deteriorated either because there is no area left or that the quality is 0%, or
both, while 100% means the entire ecosystem is intact and is at its maximum value.
Finally, an index to capture the protected areas was developed by using a number of
criteria. These criteria, however, are partly based on scientific grounds and partly on
policy targets. Thus, what certain countries may define as “critical” on a national level
may not be “critical” on a global/universal scale.
The approaches taken by the two countries are quite different. The ABS index is based on
the theory of quantity index, whereas the Netherlands approach is based on a pre-defined
scale or criterion. The ABS approach is similar to other quantity indexes such as the GDP
volume index. The ABS report, however, has not published the natural capital index as
3
4
th
For details, see Developments in Australian Wealth Statistics, 30 Annual Conference of Economists, Perth.
Australian Bureau of Statistics, 2001.
For details, see Towards a Method to Estimate Critical Natural Capital, Discussion Paper for the second meeting
of the CRITINC-project, France, Wageningen University & Research Centre, Department of Environmental Sciences,
2000.
5
such. The index has been used to calculate real growth rate of natural resources. Given
the availability of data, as shown in the next section, Statistics Canada is better positioned
to produce various quantity indexes, which can be used independently to track the trend
in natural capital stock.
3. Data
Ideally the natural capital basket should include all non-produced assets such as those
identified in the System of Integrated Environmental and Economic Accounting
(SEEA)—the internationally agreed-upon handbook of national accounting--as shown in
the following table.
Table 1: Components of Natural Capital
1.Natural resources
o Mineral and energy resources
o Soil resources
o Water resources
o Biological resources
ƒ Timber resources
ƒ Crop and plant resources other than timber
ƒ Aquatic resources
ƒ Animal resources other than aquatic
2.Land and surface water
o Land underlying buildings and structures
o Agricultural land and associated surface water
o Wooded land and associated surface water
o Major water bodies
o Other land
3.Ecosystems
o Terrestrial ecosystems
o Aquatic ecosystems
o Atmospheric systems
Source: United Nations, 2003, Handbook of National Accounting, Integrated and Environmental Economic Accounting,
SP/ESA/STAT/SER.F/61/REV.1 (final draft), table 7.2.
Several of the aforementioned items such as ecosystems and water are yet to be included
in the Natural Resources and Stock Accounts (NRSA) 5—Statistics Canada’s natural
capital database. The primary reason is the lack of data, which are hard to collect if a
resource is not economically and technologically feasible to extract. Conceptual
5
Detailed discussion of the derivation of the NRSA is beyond the scope of this paper. For further information on the
methodology, please refer to Statistics Canada, 1997, Econnections, Catalogue no. 16-505-GPE, Ottawa.
6
complexities as well as financial constraints are also major obstacles for bringing
additional items into the NRSA. Currently the NRSA provides the physical and monetary
data on energy, mineral resource, agricultural land and timber as shown in the following
table.
Table 2: Available Data
Category
Name
Year
Energy Resource
Natural Gas
Crude Oil
Crude Bitumen
Bituminous Coal
Sub-Bitumen Coal and Lignite
1961 to 2005
1961 to 2005
1967 to 2005
1975 to 2005
1975 to 2005
Mineral Resource
Gold-Silver
Nickel-Copper
Copper-Zinc
Lead-Zinc
Iron Ore
Molybdenum
Uranium
Potash
Diamonds
Timber
Agricultural Land
1978 to 2005
1976 to 2005
1975 to 2005
1978 to 2005
1975 to 2005
1977 to 2005
1975 to 2005
1975 to 2005
1998 to 2005
1961 to 2005
1920 to 2005
Timber
Land
Note: For most resources, the reserve and the corresponding present value data for 2004 and 2005 are
estimated, as there are more than two years of time lag in getting the actual data. Data for some mineral
resources such as copper and zinc are not available individually as these minerals are often produced from
the same mine site, and the collected data only represent combined extraction costs. Agricultural land data
are collected every five years, through the census of agriculture, and estimated for the interim period.
Similarly, timber data are also estimated for most of the years, as they are not collected annually.
Table 2 shows that all the data series except diamonds are available from 1978 to 2005.
Diamonds became an established reserve in October 1997, and annual data are available
since 1998. Although offshore oil and gas also came on stream in 1998 or later, these
resources are merged with onshore oil and gas respectively as they are measured using
the same units.
Also, in a few cases, the present value of resources in 1978 and 1979 is negative, which
are not unusual because the extraction costs in the initial years often exceed the total
revenue. As a result, the natural capital index is calculated using the quantity (reserve)
and present value data from 1980 to 2005. There are several widely-used quantity indexes
such as the Laspeyres, the Paasche and the Fisher. The next section discusses and
modifies these indexes so that the available data can be applied directly.
7
4. Methodology
Theories of index numbers are well-established in economic literature and widely used to
measure various economic phenomena such as the consumer price index and the GDP
deflator. 6 The natural capital index is also based on the same theory, and therefore, the
index would have sound theoretical ground. However, there are different types of
indexes, and each one of them has pros and cons. Using the most important ones-namely the Laspeyres, Paasche, and Fisher quantity index--would enable us to understand
the trend of Canada’s natural capital.
4.a. The Laspeyres quantity index:
For the Laspeyres quantity index, a pre-selected base year’s prices are used as weights.
Symbolically the index can be written as follows:
n
Q
L
q ti p ti
∑
i
=
=1
n
q
∑
i
=1
i
t
p
i
t
L L L L L L L L L L L L L1
where
Q L = the Laspeyres quantity index for the period t using the base period 0,
p = the price series, and
q = the quantity series.
i = refers to the ith item, and n is the total number of items in the basket.
For simplicity , ' i' would be dropped from here on.
Since the NRSA data are available in quantity and value (V=pq, or price multiplied by
quantity) forms, through algebraic manipulation the above formula can be rewritten in a
more usable form.
Q
L
∑ qt p
=
∑q p
0
0
where V0
⎡q ⎤
0
=
∑ ⎢ q t ⎥q
⎣
0
∑q
⎦
0
0
p0
⎡q ⎤
p0
=
∑ ⎢ q t ⎥V
⎣
0
⎦
∑V
0
LLL 2
0
= present value of the resource in the base year
Equation 2 is used to calculate the fixed-base Laspeyres index. This formula is easy to
interpret and additive—if separate indexes are calculated for subgroups such as minerals
and energy, then they can be added to get the aggregate natural capital index. However,
6
For details, see, Allen, R.G.D. 1975, Index Numbers in Theory and Practice, Aldine Publishing Company,
Chicago.
8
the formula is highly dependent upon the selection of the base year. A selected base year
loses its relevance for the distant years as the prices change over time. Setting base year
in the middle changes the meaning of the index as the pre-base year period index
becomes the Paasche type, which uses the current year as a weight, and the post-base
year period index becomes the Laspeyres type.
A more appropriate version is the chained Laspeyres index, where every preceding year
is used as a base year for the current year. To calculate the chained index, first the yearover-year index (also known as unchained index) is calculated using the following
formula:
the unchained Laspeyers quantity index,
QucL =
∑ qt pt
∑ qt pt
−1
−1
−1
=
⎡q ⎤
⎥ qt −1 pt −1
⎣ t −1 ⎦
∑ ⎢q t
∑ qt
−1 p t −1
=
⎡q ⎤
⎥Vt −1
⎣ t −1 ⎦
LLLL3
∑ ⎢q t
∑Vt
−1
In other words, to calculate the unchained Laspeyres quantity index, one needs to
calculate the weighted average of the quantity relatives where the previous period’s price
(pt-1) is assigned as the weight. Using the value from formula 3, the chained index can be
calculated as follows:
The chained Laspeyres index
QcL = 1 × (QucL )period 2 × (QucL ) period 3 L (QucL )periodt LLLLLLLLLLLL 4
The chained Laspeyres index is simply the product of unchained Laspeyres indexes,
where the index for the starting period (also known as the reference period) is assumed to
be 1 or 100%. Because of its multiplicative form, it is easy to shift the reference year
without losing the trend. Even in the chained form, the Laspeyres index usually
overestimates the actual trend. To avoid this shortcoming, Paasche proposed an index
which uses the value of current period as weight.
4.b: The Paasche quantity index
In a time series, if the final or end year’s price is considered as weight, then the index
becomes the Passche quantity index as shown in the following formula.
9
=
QP
∑ qt pt
∑ q pt
∑ qt pt
=
⎡q ⎤
∑ ⎢ q0 ⎥ qt pt
⎣ t⎦
0
∑Vt
=
⎡q ⎤
∑ ⎢ q0 ⎥Vt
⎣ t⎦
LL 5
where
Q P is the Paasche quantity index, and
Vt = present value of the resource in the current period
The unchained version of equation 5 is,
QucP =
∑ qt pt = ∑ qt pt
∑ qt pt ∑ ⎡ qt ⎤ q p
t t
⎢
q ⎥
−1
−1
⎣
t
⎦
=
∑Vt
⎡q ⎤
∑ ⎢ qt −1 ⎥Vt
⎣ t ⎦
LLLLLL 6
Similar to the chained Laspeyres index, the chained Paasche index can be calculated
using the following formula:
QcP = 1 × (QucP )period 2 × (QucP ) period 3 L (QucP )periodt
LLLLLLLLLLLL 7
Although the chained Laspeyres and the chained Paasche indexes are more relevant than
the fixed based ones, both of them typically suffer from a bias. If the price relatives
(pt/pt-1) and the quantity relatives (qt/qt-1) are negatively correlated, then the underlying
change is overstated in the Laspeyres index and understated in the Paasche index, i.e.
creates upward and downward biases respectively. The opposite happens if the price and
quantity relatives are positively correlated. To resolve this problem, Irving Fisher
proposed an index that averages the two indexes.
4.c: The Fisher quantity index:
The Fisher quantity index is the geometric mean of the Laspeyres and Paasche indexes,
and can be written as follows:
10
⎡q ⎤
Q
F
= Q ×Q =
L
P
∑ ⎢ q t ⎥V
⎣
0
⎦
∑Vt
0
∑V
0
×
⎡q ⎤
∑ ⎢ q0 ⎥Vt
⎣ t⎦
LLLLL8
or the unchained version
F
QuC
= QCL × QCP =
⎡q ⎤
⎥Vt −1
⎣ t −1 ⎦
×
∑ ⎢q t
∑Vt −1
∑Vt
⎡q ⎤
∑ ⎢ qt −1 ⎥Vt
⎣ t ⎦
LLLLL 9
The chained Fisher index
QcF = 1 × (QucF ) period 2 × (QucF )period 3 L (QucF ) periodt LLLLLLLLL10
The Fisher index not only removes the bias but also fulfills the time and factor reversal
tests of index numbers—important criteria for an ideal index formula. The time reversal
test requires that if the base and current periods are interchanged, the formula would
produce the reciprocal of the original index, i.e. 1/time-interchanged-index = original
index. The factor reversal test needs that the product of price and quantity indexes will
produce the value index, i.e., P01Q01=V01.
The main weakness of the chained Fisher index, like other chained indexes, is its nonadditivity, i.e., index calculated using components do not add up to the aggregate index.
Despite this limitation the Canadian System of National Accounts is currently based on
the chained Fisher index because of its superiority over the other index. In order to
produce a consistent and coherent natural capital index, the chained Fisher index is
mainly referred in this paper. The next section exhibits the natural capital index of
Canada.
11
5. Natural Capital Index
As mentioned in section 3, Canada’s natural capital basket is based on the NRSA data on
reserves and their present value of the reserve. For a number of reasons, the PV data
sometimes become zero, which makes it difficult to select an appropriate reference year.
If price drops below a threshold level, rent could become zero, and thereby the PV. In
fact, a few zero values are observed in the NRSA data set for 1978 and 1979. In 1980,
however, all the series had positive numbers; therefore, 1980 is used as the reference
year.
Figure 1: The Natural Capital Index of Canada
120
100
%
80
60
40
20
4
3
2
1
0
5
20
0
20
0
20
0
20
0
20
0
20
0
8
9
7
6
Chained Paasche
19
9
19
9
19
9
19
9
5
4
Chained Laspeyers
19
9
3
19
9
19
9
1
0
2
19
9
19
9
9
Laspeyers_1980=100
19
9
19
8
7
6
5
4
3
2
1
8
19
8
19
8
19
8
19
8
19
8
19
8
19
8
19
8
19
8
0
0
Chained Fisher
Figure 1 indicates that the Laspeyres index overestimates the trend whereas the Paasche
index underestimates the trend, especially when the indexes diverge from each other. For
almost a decade starting from 1981, the natural capital index had been steady around
97%, implying that the extractions and additions had been canceling each other. The
index dropped slightly in 1998, and bounced back to 100% again in 1989 mainly due to a
huge increase in the physical stock of potash. Since 1990, the index had steadily declined,
and in 2005 the index stood at 87%, or a 13% drop from the 1990 level.
This drop implies that Canada’s natural capital stock has been declining almost 1% each
year since 1990. In other words, addition or regeneration of natural capital is about 1%
less than the rate of depletion, and this gap can be a source of concern. In order to unearth
12
the underlying reasons for this drop, the various subcomponents of the natural capital are
examined.
5.1 Subindexes
As mentioned in the data section, Canada’s natural capital basket consists of four major
components--energy, minerals, timber and agricultural land. Individual index for each of
these components would provide further insights about the drop in the overall index. The
following graph depicts the relative importance of the components in the creation of
expected wealth overtime.
Figure 2: The Percentage Share of Each Component—Present Value ($)
120
100
Agricultural land
80
%
Minerals
60
Timber
40
20
Energy
20
05
20
04
20
03
20
02
20
01
20
00
19
99
19
97
19
98
19
96
19
95
19
94
19
93
19
92
19
91
19
90
19
89
19
87
19
88
19
86
19
85
19
84
19
83
19
81
19
82
19
80
0
Figure 2 shows that in 2005, the energy resource wealth accounted for around 50% of
natural resources wealth followed by timber (29%), minerals (11%) and land (9%). In
the mid-1980s, energy accounted for around 60% of Canada’s natural resource wealth.
The oil crisis of the 1970s had increased energy prices, which created incentives for
prospecting for new sites. During the 1990s, however, timber’s share exceeded that of
energy perhaps due to reduced energy prices. The relative share of agricultural land, as
expected, had not changed much. The relative share of mineral resources had fluctuated
13
quite a bit. This is mainly because of volatility of mineral prices. Also, the opening and
closing of mine sites occur more frequently than the other components of the natural
resources. The natural capital index of each of the components is depicted below.
5.1 Energy Index: The NRSA consists of non-renewable energy resources: natural gas,
coal, crude oil and crude bitumen. Excessive consumption of these fossil fuels has serious
implications for sustainable development as the combustion of fossil fuel emits much of
the greenhouse gases that are partly responsible for global warming and climate change.
Figure 3: Energy Index
120
100
80
60
40
20
4
3
2
1
0
9
8
7
5
20
0
20
0
20
0
20
0
20
0
20
0
19
9
19
9
6
Chained Paasche
19
9
4
3
5
19
9
19
9
19
9
2
19
9
1
0
9
Chained Laspeyers
19
9
19
9
19
9
19
8
7
6
5
4
3
2
0
8
19
8
19
8
19
8
19
8
19
8
19
8
19
8
19
8
19
8
1
0
Chained Fisher
Figure 3 indicates that the energy index had slightly increased until the mid-1980s, and
had started to decrease after 1985 and reached its nadir in 1997, at close to 84%. Much of
this can be attributed to the lower energy prices in the early 1990s, as it not only
discouraged exploration and drilling in Canada but also made the already discovered
resources unprofitable to extract thereby decreasing the stock.7
There was a jump in the index in 1998, and by 2000 the index reached its peak at 107%.
In the recent years, the index has been hovering around 100%. Thus, the energy resource
reserve did not become a significant factor to the 13% drop in the aggregate index. The
7
In the mid-1990s, the price of crude oil was hovering US$18 per barrel as compared to US$30 per barrel
in the late 1970s, and US$60 per barrel in the recent years.
14
recent increases in price as well as technological improvement have enabled businesses to
tap additional resources, especially crude bitumen (tar sands) as shown in the following
figure.
Figure 4: Energy index by resource type
To calculate the index for one item, the PV series is irrelevant as there is no need to
assign any weight. The item itself will have the full weight; therefore all the index
number formulas mentioned in section 3 converge to a simple quantity ratio (qt/q80).
Although these simple ratios cannot be added together, they can be used to identify the
depletion rate of the resource.
600
500
400
300
200
100
4
3
2
1
0
9
5
20
0
20
0
20
0
20
0
20
0
20
0
8
19
9
7
Crude bitumen
19
9
5
4
6
19
9
19
9
19
9
3
1
2
Crude oil
19
9
19
9
19
9
0
9
Natural gas
19
9
19
9
7
6
5
4
3
2
0
8
19
8
19
8
19
8
19
8
19
8
19
8
19
8
19
8
19
8
19
8
1
0
Coal
The above chart indicates that the biggest jump occurred in crude bitumen. In 1999, the
crude bitumen index reached at 566%. Alberta’s oil sands are the main source of this
reserve. The crude oil and natural gas stock has been steadily declining, while the coal
index has been hovering slightly above 100 since 2002. After several years of steady
depletion, the reserve increased rapidly in 1998, with the addition of offshore oil in
Hibernia, when the index reached 78 from 62 in the previous year.
15
5.2 Mineral Index: The NRSA data indicate that the present value of minerals such as
gold and molybdenum became zero few times. Much of this has occurred due to volatile
resource prices and short life expectancy of mine sites. As a result, the mineral index is
more volatile than any other components as shown in the following figure.
Figure 5: Mineral Resource Index
120
100
%
80
60
40
20
Chained Laspeyers
Chained Paasche
20
05
20
03
20
04
20
02
20
01
19
99
20
00
19
98
19
97
19
95
19
96
19
94
19
93
19
91
19
92
19
90
19
89
19
87
19
88
19
86
19
85
19
83
19
84
19
82
19
81
19
80
0
Chained Fisher
Figure 4 shows that from 1980 to 1988, the index had decreased by 12%. In 1989, the
index jumped to 105, mainly because of a rapid increase of the potash reserve. The index
remained around 100 until 1996. It started to decline thereafter, and in 2005 the index
dropped to 65%. This 35% drop may have contributed significantly to the overall drop in
the index, as the mineral resource share in the overall index is around 11%8. Due to the
high commodity prices in the recent years, the depletion rate became high; however,
there was no big discovery except diamonds in the recent years. The following chart and
table indicate the index for each item in the mineral resource basket.
8
Although the chain index is non-additive, the approximation indicates that the mineral index might have
contributed around 4% (.35*.11) of the drop in the overall index.
16
Gold-Silver
Iron ore
Year
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
GoldSilver
100
112
118
201
211
246
284
331
355
321
304
278
268
264
303
312
325
277
267
254
224
202
197
186
169
142
NickelCopper
100
95
92
92
90
88
84
82
78
77
72
71
70
67
67
73
70
64
71
62
60
54
51
49
41
39
Nickel-Copper
Molybdenum
CopperZinc
100
93
101
97
93
85
77
77
75
72
68
69
62
54
50
48
54
51
42
40
37
32
28
24
22
18
Lead
-Zinc
100
97
94
94
94
89
83
77
75
74
64
58
54
48
47
45
42
31
24
21
18
13
10
8
7
6
Iron
ore
100
76
76
81
81
82
82
82
82
83
83
83
83
83
83
83
83
83
21
21
21
21
21
21
21
20
Copper-Zinc
Uranium
Molybd
enum
100
92
85
80
66
60
57
42
38
38
36
32
30
30
27
24
25
26
23
22
22
18
16
14
20
18
Uranium
100
77
85
75
59
59
60
58
56
56
66
68
69
70
68
102
105
95
99
95
99
103
100
98
95
92
20
04
20
02
20
00
19
98
19
96
19
94
19
92
19
90
19
88
19
86
19
84
19
82
500
450
400
350
300
250
200
150
100
50
0
19
80
%
Figure 6: Individual Mineral Resource Index
Lead-Zinc
Potash
Potash
100
100
100
100
100
100
100
100
115
338
338
338
338
338
338
338
338
338
338
338
338
338
431
431
430
429
Diamon
ds
100
96
80
76
68
250
231
213
17
The above figure and the associated table show the trend in individual mineral resource
reserve. No wonder why the minerals resource index had dropped by 35%, as most of the
mineral reserves except gold and potash had dropped markedly—for example, Lead-Zinc
dropped 84%, and Molybdenum dropped 82%. In 1998, with the activation of the
initiation of the Ekati diamond mine-- located about 300 kilometers northeast of
Yellowknife, Northwest Territories--Canada became a diamond producer country. Until
2003, the reserve continued to decline because of extraction. In 2003, the Diavik mine,
located 300 km northeast of Yellowknife, gave a boost to the reserve. However, the share
of diamonds in the total mineral resource basket is insignificant.
5.3 Land Index
Ideally land assets should include land associated with residential and non-residential
buildings, agricultural land and land used for recreation or environmental protection such
as parkland. Currently, only the agricultural land is included in the NRSA data base, and
the calculated index is shown below.
Figure 7: Land Index
110
100
90
80
70
60
50
40
30
20
10
20
05
20
04
20
03
20
02
20
01
20
00
19
99
19
98
19
97
19
96
19
95
19
94
19
93
19
92
19
91
19
90
19
88
19
89
19
87
19
86
19
85
19
84
19
83
19
82
19
81
19
80
0
Agricultural land
The quantity of agricultural land is the only item in this basket; thus, the index reflects
the quantity relatives. The index remained close to 100% till 1996. In 2005, the index
dropped to 97%. This 3% loss may be attributed to the increased level of urbanization in
18
recent years. Since the relative share of agricultural land in the overall natural capital is
only 9%, the impact of this small decline of land on the overall index is insignificant—
perhaps around 0.27%. If the residential and urban lands were added the index might
remain the same–close to 100%, as the geographical boundary of a country remains the
same unless there is a serious land erosion or land is taken by or given away to the
another nation.
5.4 Timber Index
Timber plays a significant role in sustaining environment and ecosystems. Timber is
conducive in maintaining a stable climate; therefore, special attention should be given to
this capital for sustainable development. According to the NRSA timber assets are those
timber stocks that are capable of producing a merchantable stand within a reasonable
period of time, that are physically accessible and that are not reserved for purposes other
than harvesting.
Figure 8: Timber Index
110
100
90
80
70
60
50
40
30
20
10
20
05
20
04
20
03
20
02
20
01
20
00
19
99
19
98
19
97
19
96
19
95
19
94
19
93
19
92
19
91
19
90
19
88
19
89
19
87
19
86
19
85
19
84
19
83
19
82
19
81
19
80
0
Timber
Figure 6 represents the index for timber, where timber is the only item in the basket. The
index indicates a relatively high rate of depletion of timber. By the end of 2005, the index
dropped by 11% from the 1980 level. Given the importance of timber in Canada’s natural
19
capital basket, this 11% drop may have contributed significantly to the overall drop in the
index.9
From the above discussion it is clear that Canada’s natural capital index has been
declining since 1990. Clearly per capita natural capital has been decreasing even faster:
Canada’s population in 1989 was around 27 million, and in 2005 the population reached
above 32 million. Thus, for sustainable development, more conservation or replacement
or regeneration may be needed.
In Canada, the provincial and territorial authorities are the owners and stewards of much
of the natural resources. Also, the changes in the stocks of natural capital tend to be
specific or localized. With this in mind, the provincial and territorial natural capital
indexes are explored in the next section.
9
In 2005, the share of timber was 29% of the total PV of resources. Thus, the drop in timber index might
have contributed 3.2% of the drop in the overall index, which dropped by 13% in 2005 from 1990.
20
6. Provinces and Territories
Canada is constituted of 10 provinces and 3 territories, and resource endowments are
quite divergent in these 13 regions. The regional data are scarcer than the national data.
For some metals such as iron and zinc, data are available only at the national level
although the resources belong to several regions. Also, 2004 and 2005 estimates are
unavailable for the provinces and territories. As a result the latest data point is 2003, as
shown in the following table.
Table 2: Data for the Provinces and Territories
N
F
L
D
P
E
I
N
S
N
B
Q
U
E
O
N
T
Nat Gas
1961 to 2003
X
M S
A A
N S
K
X
Crude Oil
Crude Bitumen
Bituminous Coal
Sub-bitumen Coal
and Lignite
Gold-Silver
Nickel-Copper
Copper-Zinc
Lead-Zinc
Iron Ore
Molybdenum
Uranium
Potash
Diamonds
Timber
Agricultural land
1961 to 2003
1967 to 2003
1975 to 2003
1975 to 2003
X
X
1978 to 2003
1976 to 2003
1975 to 2003
1978 to 2003
1975 to 2003
1977 to 2003
1975 to 2003
1975 to 2003
1998 to 2003
1961 to 2003
1921 to 2003
X
X
X
X
X
X
X
*
X
X
*
X
X
*
A
L
B
B
C
X
X
X
X
X
X
X
Y
U
K
N N
W U
T N
X
X
X
*
*
*
*
*
*
*
*
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
P
P
P
Legend: X indicates the availability of data. * indicates that the regional data are unavailable, but the
national data are derived from these regions resources. P indicates partial--data are available since 2001.
Table 3 shows that the availability of provincial and territorial data are even more
restrictive than the national data in terms of commodity and time coverage. In some cases
only one series is available for a region. For example, the natural capital index for Prince
Edward Island (PEI) is essentially an agricultural land index. This is not because land is
the only natural capital found in the province, but because there are currently no available
data for the other types of natural capital in PEI such as timber. For the territories,
agricultural land data are available only since 2001; therefore land is excluded for the
territories. Similarly, timber data for Manitoba are also unavailable. Despite these
limitations, the provincial index shown in appendix A would be useful in tracking the
trend in resource stocks.
The provinces and territories that have experienced significant improvements over the
years are as follows: Saskatchewan mainly due to increase in the potash and uranium
21
reserves; Nova Scotia and Newfoundland primarily because of offshore oil and gas; and
the Northwest Territories chiefly due to the addition of diamonds. On the other hand, the
provinces that have experienced substantial declines over the years are the following:
Manitoba and New Brunswick due to decrease in energy reserves, Ontario because of
reduced mineral reserves, and Prince Edward Island due to shrinkage of agricultural land.
The natural capital index for Alberta and British Columbia, and Quebec has slightly
decreased over this period.
The indexes show the overtime change of a region’s natural capital stock. Each region
can use the respective index to formulate their strategies for conservation or regeneration.
Although the index does not capture the inter-regional variation, it can be used to identify
which regions are performing better than others over time.
7. Conclusion
Although produced capital such as buildings and bridges are often considered substitutes
of natural capital, there is a limit to substitutability as many of the ingredients of
produced capitals are derived from natural capital. To construct a building, for example,
in addition to human capital, natural capital such as rods and cements are essential. Thus,
it is important to conserve, replace and regenerate natural capital for sustainable
development.
With the increase in incomes of the world’s populous economies such as India and China,
the demand for, and hence the prices of resources are likely to go up. As a result,
extraction as well as addition of natural resources may increase and the net change in
quantity stock should be recorded to pursue sustainable development policies, which are
important for a resource-based economy such as Canada.
When resource prices go up, the value of the Canadian dollar as compared to the US
dollar tends to go up. Following the 1997/98 East Asian financial crisis, commodity
prices in the world market dropped and so did the Canadian dollar. In recent years, the
resource prices as well as the value of the Canadian dollar have gone up. Resources that
are viewed free today may become a scarce capital tomorrow. Also a resource that may
not look feasible for extraction today may become a major source later. A decade ago, for
instance, extraction of offshore oil and gas was not technologically and economically
feasible. Today, however, offshore oil and gas has become a major source of Canada’s
energy supply; in 2005, offshore oil accounted for 30% of conventional crude oil supply.
Molybdenum, on the other hand, has started to disappear, with 80% depleted over the
years.
The disappearance and reappearance of resources or the addition of new resources creates
a complication, if one wants to monitor the trend in the natural capital. This is because
different resources are measured in different units. Also, in some cases, the stock data
may not be available for monitoring as they are confidential at the regional level. In order
to resolve both the comparability and confidentiality issues, this paper has proposed a
22
natural capital index based on well established theories of index numbers including the
chain Fisher index—an ideal index number, which has been used in the Canadian System
of National Accounts since 2001.
Like other quantity indexes, the natural capital index is a weighted average of the
physical quantity of the resources where the present value of expected rents is used as
weights. Occasionally the weight can become zero, especially if the price drops
significantly, then rent and thereby the PV become zero, which may distort the natural
capital index. Also, the reserve data are subject to revision.
Despite these limitations, the index seems useful and the best method to get an overall
picture of the health of natural capital of a country or a region. From 1990 to 2005,
Canada’s natural capital had fallen by 13%, which should be taken seriously. Much of
this is attributed to the drop in minerals and timber, which plays a vital role in
maintaining a stable climate and sustaining sound ecosystems.
Certainly this drop in the natural capital stock is discouraging as Canada’s population has
been growing. In 1990, Canada’s population was close to 27 million, and the figure
increased to above 32 million in 2005--more than 13% growth for the period. Thus the
per capita natural capital index has deteriorated further. These trends have highlighted the
importance of conservation and regeneration of natural capital. To this end, the proposed
natural capital index would be helpful, as it has enables us to pin down the source of
depletion as well as addition. Also noteworthy is the marginal cost of producing the index
would be very small, since the required data are already compiled by Statistics Canada
for other purposes such as updating the national balance sheet.
For conservation and regeneration, provincial and territorial authorities need to be
involved since the provinces and territories are the owners and stewards of much of
Canada’s natural capital. This paper has provided a picture of natural capital for these
jurisdictions. Thus, policymakers as well as politicians may find the index useful for
monitoring changes in resource stocks and developing appropriate sustainable
development strategies for their jurisdictions. Finally, the index is expected to ignite
further debate and discussion including the construction and signaling capacity of this
new barometer.
23
Appendix A
Natural Capital Index for the Provinces and Territories10
Figure 1A: Relative Share of Natural Capital and the NCI of British Columbia
120
100
Land
80
%
60
Timber
40
20
Mineral
Energy
0
1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
Year
Energy
Mineral
Timber
Land
120.00
100.00
80.00
%
60.00
40.00
20.00
0.00
1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
Year
Laspeyres_80_based
10
CL
CP
CF
I am currently updating the (new time frame 1980-2003) provincial and territorial indexes, and the
updated version will be ready soon.
24
Figure 2A: Relative Share of Natural Capital and the NCI of Alberta
120
100
Land
Timber
80
%
60
Energy
40
20
0
1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
Year
Energy
Timber
Land
120.00
100.00
80.00
%
60.00
40.00
20.00
0.00
1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
Year
Laspeyeres_80_based
Chained Laspeyres
Chained Paasche
Chained Fisher
25
Figure 3A: Relative Share of Natural Capital and the NCI of Saskatchewan
120
100
Land
80
Timber
%
60
Minerals
40
20
Energy
0
1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
Year
250.00
200.00
150.00
%
100.00
50.00
0.00
1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
Year
Laspeyres_80_based
Chained Laspeyres
Chained Paasche
Chained Fisher
26
Figure 4A: Relative Share of Natural Capital and the NCI of Manitoba
120
100
%
80
Land
60
40
Energy
20
0
1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
Year
Energy
Land
120.00
100.00
80.00
%
60.00
40.00
20.00
0.00
1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
Year
Laspeyres_80_based
CL
CP
CF
27
Figure 5A: Relative Share of Natural Capital and the NCI of Ontario
120
100
Land
%
80
60
Timber
40
Minerals
20
0
1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001
Year
Energy
Minerals
Timber
Land
120.00
100.00
80.00
%
60.00
40.00
20.00
0.00
1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
Year
Laspeyres_85_based
Chained Laspeyres
Chained Paasche
Chained Fisher
28
Figure 4: Relative Share of Natural Capital and the NCI of Quebec
120
100
Land
80
%
60
Timber
40
20
0
1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
Year
Minerals
Timber
Land
120.00
100.00
60.00
40.00
20.00
Year
Laspeyers_83_based
Chained Laspeyers
Chained Paasche
Chained Fisher
20
02
20
01
19
99
20
00
19
98
19
97
19
96
19
94
19
95
19
93
19
92
19
91
19
89
19
90
19
88
19
87
19
86
19
84
19
85
19
83
19
82
19
81
19
80
19
79
0.00
19
78
%
80.00
29
Figure 5: Relative Share of Natural Capital and the NCI of New Brunswick
120
100
Land
%
80
60
Timber
40
20
Energy
0
1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
Year
Energy
Timber
Land
120.00
100.00
80.00
%
60.00
40.00
20.00
0.00
1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
Year
Laspeyres_80_based
Chained Laspeyres
Chained Paasche
Chained Fisher
30
Figure 8A: Relative Share of Natural Capital and the NCI of Nova Scotia
120
100
Land
80
%
Timber
60
40
20
Energy
0
1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
Year
Energy
Timber
Land
600.00
500.00
400.00
%
300.00
200.00
100.00
0.00
1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
Year
Laspeyres_80_based
Chained Laspeyres
Chained Paasche
Chained Fisher
31
Figure 9A: Relative Share of Natural Capital and the NCI of Newfoundland
NFLD
120
100
Land
%
80
60
Timber
40
20
0
Minerals
Energy
1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
Year
Energy
Minerals
Timber
Land
160.00
140.00
120.00
100.00
80.00
60.00
40.00
20.00
0.00
1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
Laspeyres_2000_based
Chained Laspeyres
Chained Paasche
Chained Fisher
32
Figure 10A: NCI of PEI—only land
102.00
100.00
98.00
96.00
94.00
%
92.00
90.00
88.00
86.00
84.00
82.00
80.00
1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003
Year
Laspeyers_78_based
Chained Laspeyers
Chained Paaasche
Chained Fisher
Figure 11A: NCI of Territories—only gold
250.00
200.00
150.00
%
100.00
50.00
0.00
1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002
Year
Laspeyres_80_based
33
Figure12A: NCI of the North West Territories—only diamonds
300.00
250.00
200.00
%
150.00
100.00
50.00
0.00
1998
1999
2000
2001
Year
Laspeyres_98_based
2002
2003
2004
34
Appendix B
The two ingredients of natural capital index are: 1) q--reserve or quantity data, and 2)
V—present value data.
1. Reserve or Quantity Data
Each year, various government departments such as Natural Resources Canada, Statistics
Canada, and Alberta Energy Board gather production and reserve data by surveying the
relevant businesses that are involved in exploration, drilling and extraction of the natural
capital. Based on the survey results, the closing stock or reserve data of each of the
resources are compiled. Agricultural land data are collected through the census of
agriculture in every five years, and estimates are made for the intercensal periods.
The survey also collects data on total revenue and extraction costs incurred by the
establishment engaged in the production of the resource. Based on these data, the present
value of the reserve of the resource is computed. The following formulas and
hypothetical example can be used to understand the present value calculation process.
2. Rent and Present Value (PV) of Rent
Rent, the value of a resource, is the difference between the value of current production
and the extraction cost of the resource.
⎤
t ⎥
t =1 ⎣ (1 + r ) ⎦
RL
⎡ Rt
PV = ∑ ⎢
where
R = rent , r = discount rate
t = current year , RL = reserve life =
Stock
Pr oduction
Hypothetical example
Suppose the established reserve of a mineral is 12 units, and each year 4 units are
extracted. Then the reserve life would be 3 years. If the revenue generated from the
extracted units are $20, and the extraction costs are $15, then the resource rent would be
$5. Assuming that everything will remain the same for the next three years, the site will
generate $5 worth of rent by the end of each year. However, $5 at the end of each year
would worth less now, and therefore, all the future rents need to be discounted to derive
the present value. Finally, by summing the discounted present value, the total PV of the
resource can be calculated as shown in the following formula.
Total PV = ∑ PV =
5
5
5
+
+
1
2
(1 + .10) (1 + .10) (1 + .10)3
= $8.26
35
Appendix C
Hypothetical Example of Calculating Quantity Index Number
0
A
Q (kg)
3
1
2
Period
QL =
QP
QF
∑q p
∑q p
=
=
1
0
0
0
∑q p
∑q p
1
1
0
1
p ($/kg)
4
B
q (meter)
1
p ($/meter)
8
5
2
7
2 * 4 + 2 * 8 24
=
= 1.2 = 120%
3 * 4 + 1* 8
20
=
= QL ×QP
2 * 5 + 2 * 7 24
=
= 1.09 = 109%
3 * 5 + 1* 7
22
=
24 24
*
= 114.42 %
20 22
Therefore, Q L (120%) f Q F (114.4%) f Q P (109%)
In other words, the Laspeyers index is upward bias and the Paasche index is downward
bias because the price and quantity relatives in the above example are negatively
correlated. The opposite occurs if these two relatives are positively correlated.
36
Hypothetical Example--Chain Indexes
Period
0
A
Q (kg)
3
1
2
$/meter
8
2
5
2
7
1
6
3
6
∑ qt pt
∑ qt pt
∑q p
QL =
∑q p
−1
QtL/ t −1 =
−1
or
$/kg
4
B
q (meter)
1
−1
2
1
1
1
2 /1
∑ qt pt
∑ qt pt
∑q p
QL =
∑q p
=
1 * 5 + 3 * 7 26
=
= 1.08
2 * 5 + 2 * 7 24
=
1 * 6 + 3 * 6 24
=
= 1.00
2 * 6 + 2 * 6 24
QtP/ t −1 =
−1
or
2
2
1
2
2 /1
QtF/ t −1 = QCL × QCP
Unchain
Index Laspeyers
Period
0
Unchain
Paasche
Chain
Laspeyers
%
Chain
Paasche
%
Chain
Fisher
%
--
--
1.00 = 100%
1.00 =100%
100
1.00*1.20 = 120%
1.00*1.09 =109%
114.42
1.20
1.09
1.08
1.00
1*1.20*1.08 =
130%
1.00*1.09*1.00
=109%
118.85
1
2
As stated in the methodology section, the chain Fisher index fulfills the criteria of an
ideal index. Hence, the chained Fisher is used for the Natural Capital Index of Canada.
37
References:
Allen, R.G.D. (1975), Index Numbers in Theory and Practice, Aldine Publishing
Company, Chicago.
Bartelmus, P. et al. (1991), “Integrated Environmental and Economic accounting:
framework for SNA satellite system,” Review of Income and wealth, ser. 37, no. 2, pp.
111-148.
Born, A. (1992), “Development of Natural Resource Accounts,” discussion paper number
11, EASD, Statistics Canada.
Bortkiewicz, L. (1923), “Purpose and structure of price Index-number, First Article”
Nordisk Statistics Tidscrift, 1.2, 369-408.
Fisher, I. (1922), The Making of Index Numbers, Boston.
Hotelling, H., (1931), “The Economics of Exhaustible Resources,” Journal of Political
Economy, vol. 39, No. 2, pp. 137-175.
Laspeyres, E. (1864), Hamburger Warenprecise 1850-1863’, Jahrbucher fur
Nationalokonomie und Statistik, 3, 81 and 209.
Paasche, H. (1874), Uber die Preisentwicklung der letzten Jahre,’ Jahrbucher fur
Nationalokonomie und statistic, 23, 168.
Statistics Canada, (1982), The Consumer Price Index reference paper, concepts and
procedures, Ottawa.
Statistics Canada, (1997), Econnections, Catalogue no. 16-505-GPE, Ottawa