ACCOUNTING
METHODOLOGIES FOR CALCULATING THE DEPRECIATION OF NATURAL CAPITAL
Adjustment
in National Accounts
Literature on depreciation accounting
procedures of natural resource exhaustion is still open to debate about the
most appropriate method.7 Economic progress has been measured in
almost every nation world-wide by the system of national accounts using the
Gross National Product (GNP). It uses market prices to measure the aggregate
value of an economy’s output in a given year. Despite being the most used
indicator of economic progress, it is widely known that GNP is a very poor
approximation of economic well being. Another indicator, Net National Product
(NNP), is a much better measure of well being because it accounts for the
depreciation of the wealth of a country. Man-made capital for example,
depreciates through time and this loss in value is captured in the NNP measure.
Nonetheless NNP, in its traditional measure, only takes into account the losses
of man made capital, leaving aside the depreciation of natural capital.8
This depreciation happens when the stock of a natural resource is decreased.
The idea of adjusting the system of
national accounts to incorporate the losses due to the use of natural
resources, is closely related to the concept of sustainability as an indicator
of sustainable development.9 Although there are many definitions of
sustainable development, economic definitions have focused on sustainable
development as non-declining per capita welfare over time [Pearce and
Atkinson, (1995)]. The idea behind it is that every future generation must have
the option of being at least as well off as its predecessor. Hence a
sustainable path has the characteristic that along it, the overall productive
capacity of an economy is not reduced.10
The productive capacity of an economy
depends on the total stock of capital available, as well as on its
productivity. If some capital is used up and not replaced, the possibilities of
future production is decreased. This is not so much of an issue with physical
capital, since it can be produced and replaced. Nevertheless, it is an issue
with natural capital which cannot be created by mankind. Therefore, when
natural resources are used up or destroyed, there is implicitly a question of
substitution.
The question to ask is, at each point
in time, how much of the natural capital productive base can be used up? The
answer given by economic theory is based on two different approaches developed
in a parallel way and yielding theoretically equivalent results. Both aim at
measuring the impacts of resource depletion on long run human welfare, but
while they are equivalent in theory, they will not yield necessarily the same
result in practice.11
The first approach was derived in
Hartwick (1977). He showed that so long as the capital stock of an economy did
not decline over time, non-declining consumption was possible. We can
illustrate Hartwick’s idea with a simple economy consisting of two types of
capital: physical and natural capital denoted by Kt and Rt,
respectively. The net investment in this economy is given by the gross
investment minus the depreciation in each of the form of capital:
. If the net investment is greater than or equal to zero, the
country can at least sustain its actual consumption level. If this country
produces a non-renewable resource, non-declining consumption is possible by
reinvesting all Hotelling rents from the exhaustible resource in physical
capital.
These rents are those resulting from an
intertemporal efficient extraction programme. This is what is known in the
literature as the Hartwick rule. It tell us the amount that we have to
invest in other forms of capital when we extract a natural resource and earn
rents on it. It was shown by Solow (1986) that it also represents a requirement
to "keep capital intact". There are two important assumptions
implicit in Hartwick’s result. First, physical capital and natural capital are
assumed to be close substitutes.12 Second, the model assumes that an
individual only derives utility from the consumption of goods and not directly
from the environment.
The second approach was developed from
the work of Weitzman (1976) on Net Domestic Product (NDP). We can define NDP
for an economy with balanced trade13 as,
. Weitzman (1976) demonstrated that under optimal growth, NDP
should be though of as income in the Hicksian sense14 interpreted as
a long run measure of economic well-being, that is, the stationary equivalent
of future consumption. In this sense, national wealth for a country would be given
by the present value of the best consumption path into the future that society
can afford.15 Weitzman proved that Net National Product (NNP) in any
year t is equal to the discount rate (social rate of time preference)16
multiplied by wealth: NNP(t)=ρ W(t). Based on this result, he argued
that a true measure of NDP should include the value of changes in resource
stocks. Consequently, the net investment should be defined taking into account
the depreciation in all types of capital, not only physical. Weitzman (1976)
result depends on two crucial assumptions: first that a country growth path is
optimal and second that social welfare equals consumption.
In both of the approaches presented
above, we need an estimate of the depreciation of natural capital in order to
subtract it from the value of the capital stock. In this manner we would adjust
the gross investment to net investment. For an exhaustible resource, Hartwick
and Hageman (1993) define depreciation as the change in value of an exhaustible
resource under optimal use.
If optimality conditions do not hold
and substitution possibilities between physical capital and natural capital are
restricted, then, the above approaches face criticisms on the basis of
sustainability views. These approaches are regarded as based on the weak
sustainability principle which sustains that natural capital can be permanently
substituted by physical capital. Consequently, when natural resources are
extracted, we can set aside part of its revenue to invest in physical capital
and maintain intact the economy’s productive capacity.
This idea is not widely accepted. For
some natural resources serving as production factors, there could be
thresholds, carrying or assimilative capacity, causing some types of natural
capital to be hardly substitutable. Many assets are essential to human being
survival in the long run and this point of view is expressed in the strong
sustainability concept, based on the conservation of many types of natural
capital (i.e. maintaining them intact for future generations) or recognition of
their safe minimum standards. Although depreciation methods can be
analysed along these sustainability principles,17 the proposed
depreciation methodologies can be generalized as elaborated in V&H (1997)
and summarized in the following sub-sections.
The
Change in Value Method
Depreciation of an asset over a period
of time can be calculated as the value of the asset in the beginning of the
period minus the value of the same asset in the end of the period. This method
is known as the change in value method. Adopting the definition of income as
the level of consumption that could be enjoyed without diminishing the capital
stock, we can define depreciation as the consumption of the assets in excess of
this amount. More formally, depreciation exists when there is degradation of a
resource. If we have the value of a resource at the end of a period and the
value at the beginning of the period, we can calculate the depreciation of the
resource as the difference between the two values.18 Defining the
value of a resource as the discounted sum of the resource rents that are
generated over time19 and using some algebraic manipulation, we can
express the depreciation of an asset as:20
(1)
The previous expression is known as the
fundamental equation of asset equilibrium. Although in theory it is a good
representation for the depreciation of an asset, carrying out the previous
calculation in practice has many complications. We would need projection of the
future flow of rents, in which case we need prices, quantities extracted and
cost schedules into a future finite stream.
Total
Hotelling Rent as Depreciation
Hotelling rent (HR) is defined as the
rent that exists on the marginal quantity of an exhaustible resource (price
minus marginal cost) and it is considered a measure of the intertemporal
scarcity of that resource. This rent exists because the resource is exhaustible
and consequently, the owners of the resource extract less than the amount that
would equate marginal revenue to marginal cost. Multiplying the HR by the
quantity extracted of the resource, gives the total Hotelling rent. It is
interpreted as the portion of profit that accrue to extractive firms because
they are mining an exhaustible resource.Hartwick (1988) proved that along
dynamically efficient paths of extraction, Hotelling rent and economic
depreciation (with a negative sign) are equals. Using this result, it is
possible to use a short cut to calculate economic depreciation. Nevertheless,
it is important to note that this equivalence is only correct under certain
conditions, namely: a) resource extracted optimally, i.e., Hotelling’s
rule holds; b) price of extracted resource constant over time; c)
marginal cost as an increasing function of the amount extracted, unrelated to
the size of the reserve and constant over time; and d) constant discount
rate over time.
The
Net Price Method (NPM)
As it was mentioned previously, under
special conditions, economic depreciation of natural resources can be
approximated by the Hotelling rent. This last term was defined as price minus
marginal extraction cost (the profit on the marginal ton extracted), multiplied
by the amount extracted. Hence we can use the following expression to calculate
the depreciation of natural capital from the NPM:
(2)
In theory this method would be easier
to apply than the change on value method. Nevertheless in practice, the data on
marginal extraction cost is not generally available. Consequently previous
studies that attempted to calculate depreciation of natural capital used the so
called net price method. It uses data on average cost instead of marginal cost
and calculates the depreciation as:
(3)
It is important to note that, as it was
mentioned by several authors,21 using the average net price instead
of the marginal net price, one is calculating the total resource rent which
overstates net accumulation (unless the MC=AC as in the case of a linear
cost function).
The
El Serafy Method (ESM)
Alternatively to the net price method,
El Serafy (1989) developed a method to calculate depreciation based on the
concept of user cost. He equates the finite stream of rents earned by a
resource to an annuity X earned forever, which can be obtained by
selling the mine and depositing the value of the mine (V) in a bank
account. Mathematically this is given by:
(4)
where r is the discount rate, n+t
is the last year of the extraction, X is the annuity received and R
is the rent defined as
. The expression R-X is a measure of depreciation, and
El Serafy (1989) simplifies further the analysis to obtain a closed form
solution.Assuming that the rent is constant over
time,22 such that
, the equation above is simplified to:
(5)
The previous expression is the user
cost or economic depreciation for the ESM. It is important to note that as
Hartwick and Hageman (1993) pointed out, if R(t) changes over time, the
measure R-X could be a poor approximation to R(t+i)-X in period i,
consequently the ESM would yield a poor approximation of the depreciation
value. In the ESM, n is the remaining
years of life of the resource stock. This is estimated by dividing total
reserves remaining by the year’s quantity extracted. This calculation
implicitly assumes that this year’s extraction will continue into the future at
the same rate. If S(t) is the remaining stock of the resource at period t
and q(t) the quantity extracted of the resource in the same period, we
can express ESM formula to estimate depreciation as:
(6)
Vincent
and Hartwick Approach
(7)
Where
β is the elasticity of the marginal cost curve with respect to the quantity
extracted. It is important to note that on the previous formula, for relative
elastic marginal cost curves (β≥1), unit price has to be at least twice
as large as the unit average cost. Otherwise, one would witness a negative
estimate of depreciation which would not make economic sense in presence of
increasing depletion. This arithmetical trap is only due to the simplification
of the isoelastic assumption of the marginal cost curve.Another
approach used by Vincent (1997), is to use the previous total cost function and
the fact that under an optimal extraction path, the transversality condition
implies that MC=AC when the resource is exhausted. As a result, it is
shown that for the previous functional form, this transversality condition
occurs if and only if q(t)=MC=AC=0 in the final period. Using
the Hotelling rule and the fact that MC(T)=0, Vincent (1997) derives the
following expression for the Hotelling rent:
(8)
The
previous expression is a generalization of the net price and El Serafy methods
in the sense that it embodies both estimations as particular cases. If
, equation (8) simplifies to the ESM and If
, the whole right hand side goes away and the Hotelling rent
would just equal the total rent.The
usefulness of the previous formula is that in practice it is much easier to
find data available for the average cost than for the marginal cost. However,
estimating β can be a practical complication. Obviously the elasticity of
marginal cost will vary depending on the nature of the resource (timber, oil,
etc.) and type of industries.23 Equations
(7) and (8) are presented by Hartwick and Vincent (1997) as being equivalent.
The choice of which one to use depends just on the availability of data. If
data is obtainable on the average extraction cost, then one can use expression
(7) to estimate the depreciation. If instead, data is available on the resource
rent, Vincent (1997) has demonstrated that expression (8) is an alternative
measure of the Hotelling rent. Nevertheless it is noted in Vincent (1997) that
expression (7) would yield less accurate estimates of economic depreciation
than formula (8). This would happen due to the fact that formula (7) is based
on current marginal rent and, on the other hand, expression (8) is based on
long run principles. Although
Vincent (1997) is correct in his concern for estimating long run instead of
short run Hotelling rents, it is not clear in his argument why the formula from
expression (8) would take into account out of equilibrium behaviour in the
timber market. Since it also uses current price and costs of logged wood, any
tendency of resource prices to fluctuate in the short run will be incorporated
into his estimates using expression (8) as well as expression (7).Moreover,
since expression (8) incorporates a discount rate and formula (7) does not, any
bias caused by fluctuation in prices could be compensated by biases caused by
the choice of an incorrect discount rate. Consequently, it does not seem to
exist any a priori reason why biases in the discount rate chosen would
not be even worse than biases from price fluctuations or out of the equilibrium
behaviour in timber markets. In
spite of previous points, our estimates of depreciation values will follow
equation (8) and we will be assuming additionally permanent conversion of
forest land to agriculture purposes. This would imply that we will not be
considering any form of reforestation or second growth forest conversion.
