VT =
T +
S t = T
ptqt − Hqt 1 + i
t − T
. Some have observed that this forest has analo-
gies to a mine, especially if it is not to be re- planted. Even so, we do not have recourse to
Hotelling’s rule, nor find any use for what some have called ‘Hotelling rent’ for details see Cairns,
2000a. Consider the part of the stand to be cut at time t \ T
. It has present value 6
t
T =
ptqt − Hqt 1 + i
t − T
. During time s B t, there is an appreciation of
the value of this part of the stand, equal to 6
t
s + 1 − 6
t
s = i ptqt − Hqt
1 + i
t − s
. This could be imputed as an appreciation. The
total undiscounted appreciation of the value of this part of the stand to time t is then
A =
T +
t − 1 s = T
[6
t
s + 1 − 6
t
s] = 6
t
T +
t − 6
t
T .
In this case, the depreciation of this part of the stand is
D = ptqt − Hqt = 6
t
T +
t. Therefore, the total net depreciation of the
stand is D − A = 6
t
T ,
and the total over all stands is VT , the value at
the time it is decided to cut the forest. Alterna- tively, the value VT
could be recorded at time T
, and be depreciated over time. Ideally, the depreciation
would be
6
t
T = [ptqt −
Hqt]1 + i
t
, so that the net contribution to NNP from the harvest at time T
+ t would be
N
t
= ptqt − Hqt − 6
t
T =
[ptqt − Hqt] 1 −
1 1 + i
t − T
n
. The net gain, N
t
, is the total of the appreciation above; it arises because interest increases the value
of this part of the stand from 6
t
T at time T
to 6
t
T +
t at time T +
t. In practice, an account- ing formula could be used for the depletion of the
full value VT .
An environmentalist might react to this analysis by remarking that the preference for traditional
over green NNP arises because Faustmann’s forest is a sustained forest, and that a major
environmental concern is deforestation. Suppose that a forest is taken into another use, and in this
use has a different value from that in forestry. Some have argued that a special treatment of this
change in value should be made in the accounts. But, so long as the 6alues in question are all
commercial, the appropriate treatment of the change in value is conceptually not different from
that applied to other assets which change their uses, such as buildings or rezoned urban land.
Indeed, the source of environmentalists’ concern about deforestation is loss of non-commercial
value.
Therefore, as far as the purely commercial as- pects of forestry are concerned, there is no com-
pelling reason to change the current practice of accounting for forests. Even if the forest is ex-
ploited suboptimally, the economic and account- ing implications are comparable with those in
non-resource industries. The equivalency we have noted arises because accounting methods have
been designed over the centuries to provide the maximal obtainable information about the effi-
ciency of commercial activities.
The only exception is the harvesting of a forest directly from its pristine state. In this case, as we
have observed above, depletion should be at- tributed. The reason is that the original planting,
by nature, is not ‘priced’. Non-priced features constitute a systematic problem.
3. Non-priced amenities
Pearce 1994 succinctly discusses environmen- tal amenities which are not mediated by markets,
and hence are not part of private decision making. These amenities, such as maintaining biodiversity,
watershed protection, carbon fixing, aesthetics, etc., are the reasons often cited for shifting from
clear cutting, the implicit perspective of the eco- nomics of point-input, point-output forestry, to
selective cutting from forests with a wide range of ages and species standing together.
For values which are not priced in markets, shadow prices can be determined using a macroe-
conomic model. Let there be a stock R of an environmental asset producing amenities AR and
let social utility U be given by a function of current consumption C and amenities. The aim of the
planner is to maximize
t
U[C, AR]e
− rs − t
ds. Let the produced capital stock be K and
evolve through time through investment I and physical deterioration dK according to the equa-
tion K : =I−dK. Let total output be a function of
the produced capital stock, FK. Let the environ- mental asset evolve according to the equation
R : =M−aFK−gR, with M being the level of
improvements or
maintenance; aFK
being degradation due to economic activity; and gR
being degradation from natural causes. Finally, let the cost of the different types of expenditures
in terms of consumption be given by fI and c
M, so that C = FK − fI − cM. The current-value Hamiltonian of this problem
is H
= UC, A + 6I − dK + u[M − aFK − gR].
If the problem has no exogenous source of time dependence, then Weitzman shows that the
Hamiltonian at any time is equal to interest on the present value of the discounted utility the
objective at that time. Therefore, the Hamilto- nian is a form of generalized income from the
generalized wealth of the economy. If there is time dependence, such as the natural fluctuations in the
climate that have been observed by climatologists studying the history of the earth before the indus-
trial revolution, then to retain this interpretation, a term measuring the ‘drift’ of the system should
be added to the Hamiltonian Cairns, 2000b. Interpreting the Hamiltonian as a measure of
current contributions to the objective amounts to making the reasonable assumption that the drift
from year to year is negligible.
The Hamiltonian can be written analogously to an affine function:
H =
SC, A + CU
C +
AU A
+ 6 I − dK
+ u[M − aFK − gR].
This ‘affine Hamiltonian’ is equal to the Hamil- tonian, and not an approximation from a Taylor’s
series expansion. It consists of 1. a term SC, A, which we identify as the total
consumers’ surplus over goods and amenities in utility terms, plus
2. terms involving quantities times prices of goods or amenities which enter the utility
function, CUC + AUA, plus 3. the net investment terms 6I − dK + u[M −
aFK − gR]. The welfare function U can be scaled such that
UC = pt, the money price of consumption goods. The first-order, necessary conditions for a
maximum are that 6 = fIUC = pfI = p
I
, the market price of investment goods, and that
u = pcM = p
M
, the market price of mainte- nance expenditures. Thus, we can write the affine
Hamiltonian as H
= SC, A + pC +
AU A
+ p
I
I − dK +
p
M
[M − aFK − gR]. We define green NNP by evaluating consump-
tion and amenities at the margin. That is to say, we linearize the ‘affine Hamiltonian’ by purging it
of its intercept, the consumers’ surplus, which depends non-linearly on C and A:
NNP = pC + AU
A +
p
I
I − dK +
p
M
[M − aFK − gR]. Purging the non-linearities permits aggregating
results from small economic units at the econo- my’s shadow prices, as is done in traditional NNP
using market prices as proxies for shadow prices.
6
Consistently, with our analysis of the commercial forest, traditional NNP, pC + p
I
I + p
M
M, cor- rectly attributes commercial values if the market
prices actually are shadow prices. Traditional NNP, however, neglects the terms, AUA and
6
Some of the papers in the influential collection of Ahmad et al. 1989 propose use of marginal values in order to
preserve consistency with traditional national accounting. The present analysis shows the economic meaning of the tradi-
tional approach. It corresponds to a linear, and hence additive, index of intertemporal welfare.
p
M
[aFK + gR], which are to be included in green NNP. Contrary to the intuition of some,
maintenance or defensive expenditures M con- tinue to be included. But green NNP also im-
putes, as a form of depletion, any environmental costs, including natural ones gR.
Some have reasoned that ‘greening’ the na- tional accounts could utilize cost – benefit analysis
to obtain the non-commercial values. Our discus- sion points up a subtle difference between cost –
benefit analysis and both the environmental and traditional accounts. Consumers’ surpluses more
properly, compensating or equivalent variations are a part of the social values of particular assets
and are estimated in a cost – benefit analysis. But NNP abstracts from consumers’ surpluses by
valuing consumption at the margin at market prices, where they exist. The part of consumers’
surplus which is attributable to amenities, should not be part of green NNP, just as the part which
is attributable to marketed goods is not a part of traditional NNP. In general, consumers’ surplus
may not easily be divided among marketed and environmental goods. For example, both food
and some amenities are essential to life.
Some changes in forest use may not, however, be subject to internalization by even this type of
imputation of shadow values. For example, forests may be burnt without regard to interna-
tional transmission of pollutants in air or water, or cut without regard to global warming. Reduc-
ing international pollution is a public good; the global shadow value of reducing it is the sum of
the marginal values over all inhabitants of the globe.
The sum of the marginal values over the citi- zens of even a large industrial country is much
smaller than this global shadow value. Therefore, evaluating preservation at the global shadow
value overstates the value to a home country of control, and entering it into green NNP overval-
ues the contribution to welfare of the given na- tion. But imposing the sum of national marginal
values as a mandated price within a country would imply an internationally suboptimal ratio
of prices.
What can be done in this case is to impute the difference between the shadow value and the do-
mestic willingness to pay as an export to other countries. This shadow export of real value
would have to be compensated by a shadow transfer of, say, some service abroad.
7
By the same token, imports would have to be imputed
for the results of other countries’ efforts which benefit the nationals of the country under discus-
sion, as well as an offsetting shadow transfer of a service from the country. In the real expenditure
accounts, though, such imports could otherwise go unnoticed; they would not affect the country’s
green NNP, as they would offset additions to imputed consumption that they engendered. But
they would be needed to account correctly for trade flows.
4. Conclusion
