Value and Income: An Eisegesis of
Environmental Accounting
Robert D. Cairns∗
Department of Economics, McGill University
May 6, 2004
∗
I am grateful to Geir Asheim and Ngo Van Long for conversations that have cleared some
of the dust from my theoretical spectacles and to FCAR and SSHRCC for financial support.
Value and Income: An Eisegesis of Environmental Accounting
Abstract. The objective in an economic model induces a concept of value and
thereby of income. Herein, two types of program are distinguished qualitatively by
the forms of their objectives. Utilitarian programs in which utility or the value of
consumption is discounted, be they optimal or distorted, and maximin programs
are distinguished by the forms of their objectives. Welfare statistics are based on
only the first type of program, and practical statistics are based on non-optimal
programs. There is no practical statistical indicator of whether an economy is
being sustained.
2
1. Introduction
Dasgupta and Mäler (2000: 70) list three reasons for an interest in the national
accounts. 1) Society requires an index of aggregate economic activity. 2) It is
desirable to have an index of social welfare for spatial and temporal comparisons
and for evaluating policy. 3) Academics have been seeking an index of sustainable
income. Devising the first index, gross national product (GNP), was a timely
response by the economics profession to a devastating social problem in the 1930s.
A practical, summary statistic of macroeconomic performance, it remains vital for
formulating policy toward unemployment and inflation.
The last two indices have been subjects of research for several years. Ideally,
an outcome would be statistics as practical as GNP and summarizing the values of the society in question. Sometimes it is thought that both purposes can
be achieved with a single statistic, net national product (NNP). The present paper will argue, however, that the two objectives apply to qualitatively different
problems and imply qualitatively different statistics. The argument depends on
discussing social objectives that are not utilitarian. In particular, in a sustainability (or intergenerational-equity) problem, maximin is the appropriate objective.
3
2. Summary of the Argument
Objectives (preferences) are primitive in economics. Any economic objective expresses a concept of value. It incorporates the society’s trade-offs among economic
goods in any period and among present and future times. Since income is a flow
of value, what qualifies as income is determined by the objective postulated by
the analyst.
Two distinct types of objective, each legitimate subjects of study, are utilitarian and maximin. Since alternative objectives are not commonly compared in
economics, it is worthwhile to review some features of the dominant, utilitarian
paradigm. By a utilitarian economy is meant an economy in which value–the
objective, implicit or explicit–is expressed as an integral of an instantaneous aggregator function discounted using given, non-negative, forces of interest at each
instant (or the analogous sum in discrete time). The instantaneous aggregator
function and the forces of interest express social preferences; they are further aggregated into value. Utilitarian problems can be posed as theoretical idealizations
or as evaluations of actual realizations.
There are two idealized types of utilitarian problem. (a) The instantaneous
aggregator can be a non-linear, usually strictly concave function, utility, and the
4
value integral can be maximized. The integral is usually called welfare. (b)
The aggregator can be the sum of products of prices and quantities of consumed
goods and services and value can be maximized, usually in a competitive economy.
Since the value functional is a resultant of the values of the individual actors, it
is implicit. The value integral is usually called (market) wealth. The solution to
each type of problem is Pareto optimal. By the two theorems of welfare economics,
there is a one-to-one, onto mapping between the sets of solutions. The force of
interest may be constant in some members of the set, especially the first, but
there is no compelling reason to require that it be constant.
Corresponding to idealization (a), a utilitarian value (welfare) can also be an
integral of discounted utility for a non-optimal economy. And, as in idealization
(b), market or other non-optimal prices, quantities and forces of interest may be
incorporated into an implicit value functional (wealth), not necessarily maximized.
‘The market’ is sometimes held to maximize implicitly some fictitious or virtual
value functional that is a resultant of individuals’ optimizing choices. Solutions
to idealized utilitarian problems provide guides to the development of theory for
non-optimal utilitarian problems. Since it is collected for real economies, NNP as
a real statistic finds its application in the last type of problem. It is argued below
that NNP is a theoretically meaningful income statistic solely in the context of a
5
utilitarian economy.
Nothing has to now been said of sustaining the economy or of intergenerational
equity. If sustaining an economy is a value of a society then sustainment is a social
objective and must be expressed in the value function.
A consistent, theoretical way to incorporate intergenerational equity (or sustainability) as an objective is through a maximin program. For many maximin
programs the optimal path, on which some aggregator such as utility is sustained,
is a Pareto optimum. By the theorems of welfare economics, the solution path
can be put into one-to-one correspondence with a competitive path. In turn, that
competitive path corresponds to a particular, utilitarian path. On the three solution paths, quantities and prices are the same. The solutions are not equivalent,
however: each has a distinct value, or objective functional of the paths of quantities, and induces a particular concept of (mathematical expression for) income. In
the maximin (sustainment) problem, value is not an integral of a discounted aggregator function. Consequently, NNP is not a meaningful statistic for evaluating
the sustainment of an economy. Income is not net national income.
That maximin programs have thus far been applied only to ideal, optimal
economies and not to real economies is a severe limitation on the notion of economic sustainment. Still, the non-applicability of a statistic (NNP) derived for a
6
utilitarian objective holds when sustainment is the objective.
We begin with a review of recent thought on these two distinct forms of social
value.
3. Different Objectives
Nordhaus and Tobin (1972) argued that a general measure of economic welfare
would be a more comprehensive version of a statistic already being gathered by
many national agencies, viz. net national expenditure at market prices. In addition to flows of consumption, investment, governmental services and trade, the
measure would impute values of non-marketed flows, including flows from the
environment.
In a theoretic approach to dynamic welfare measurement, Weitzman (1976)
postulated a utilitarian problem of maximizing the integral of discounted consumption, produced using several capital stocks including environmental and other
non-marketed types of capital. He showed that the Hamiltonian of the problem
corresponded to net national expenditure, the sum of current consumption and
total, net, current investment evaluated at the shadow values of investment in consumption terms. Weitzman’s contribution was to show that, because the Hamiltonian was derived from a utilitarian objective, the dominating paradigm of value
7
in economic analysis, net national product had profound economic significance.
In a parallel development responding to the energy crisis, Solow (1974) and
others considered how to maintain what they called intergenerational equity in
the face of natural constraints. Solow extended to a dynamic problem Rawls’s
(1971) maximin criterion of maximizing the utility of the individual or group
with minimum utility,
max min u (cs ) , or max ū s.t. u (cs ) ≥ ū for all s ≥ t.
s∈[t,∞)
(3.1)
Unless there was a constraint preventing the smoothing of utility over time, all
generations’ utilities would be equal in the solution. A decade before sustainability
became a popular term, Hartwick (1977) showed that an economy following a
maximin path (in which utility was being sustained) would maintain the algebraic
sum of the values of net investments in all types of capital (natural, human,
manufactured, etc.) equal to zero.
Burmeister and Hammond (1977) find sufficient conditions in a quite general
maximin problem. A so-called regular problem achieves a Pareto optimum with
constant utility for all generations. Dixit, Hammond and Hoel (1980) base further
analysis of the maximin program on the second theorem of welfare economics, that
8
any Pareto optimum can be achieved by some competitive equilibrium. Since by
the definition of a Pareto optimum an increase in utility at one instant cannot be
achieved without reducing utility at another, the optimum generates a continuum
of constraints that have shadow values. These shadow values, [λ (s)]∞
s=t , correspond to the Lagrange multipliers in Burmeister and Hammond’s model (Cairns
and Long 2001).
The shadow values are the attained marginal rates of transformation in the
maximin problem (3.1). They are not knowable without solving that problem.
They are not defined off the maximin path. Although sometimes imprecisely
called utility-discount factors and expressed in a way that looks like a utilitarian
welfare maximization (cf. Dixit et al. 1980: 552),
Ut = max
Z
t
∞∙
¸
λ (s)
u (cs ) ds,
λ (t)
(3.2)
the factors should, strictly speaking, not be viewed as time preferences. The utilitarian objective (3.2), which provides for trading utilities u (cs ) at prices (proportional to) λ(s), is incompatible with the maximin goal (3.1).
In a special sense, however, the shadow values λ (s) can be viewed as tantamount to utility-discount factors. The objective,
9
R∞
t
[λ (s) /λ (t)] u (cs ) ds, which
is a linear functional of the utilities u (cs ), corresponds to the infinite-dimensional
hyperplane separating the utility-possibilities set from the maximin welfare function (3.1). If these shadow values are applied to utility as if they were discount
factors, the maximin path coincides with the path that maximizes the utilitarian
objective. The shadow values can be considered virtual utility-discount factors,
then, for a virtual utilitarian problem. Virtual discount factors can take the form
∙ Z s
¸
λ (s) /λ (t) = exp −
ρ (τ ) dτ
t
for a force of interest ρ (τ ) that usually varies through time. In Solow’s problem,
for example, the virtual force of interest decreases through time.
Analyst and reader must be wary of identifying the two types of program. By
writing down the virtual program and analyzing it using optimal-control theory,
an analyst adopts the virtual, utilitarian objective as the formal objective of
analysis, and thereby the utilitarian concepts of value and of income.
10
4. Value and Income in Ideal Economies
4.1. Discounted Consumption or Utility
Consider a utilitarian problem of maximizing discounted utility given stocks of
non-renewable, natural capital S, of manufactured capital K, and of labor N.
(All of the analysis of the present paper can be generalized to incorporate vectors
of any number of different types of stock.) Often, the force of interest is assumed
to be constant in a utilitarian problem, and the assumption is accorded an axiomatic basis by Koopmans (1960). Apart from Koopmans’s axioms, there is no
compelling reason so to restrict it, however. De gustibus non est disputandum:
in economics preferences need not obey axioms. In recent years in utilitarian
problems, more general (hyperbolic and other) discount factors have been applied
to problems affecting the far future. Let the force of interest ρ (t) be possibly
non-constant, so that the objective is to
max U (S, K, N, t) =
Z
t
∞
∙ Z s
¸
u (cs ) exp −
ρ (τ ) dτ ds.
(4.1)
t
The argument t in the objective function U (S, K, N, t) arises because the problem
is not stationary if the force of interest is not constant.
11
The transition equations governing this system are
Ṡ (t) = −R (t) ;
K̇(t) = F (K, R, N) − c − δK; and
Ṅ = 0.
The current-value Hamiltonian for the problem is
H U = u (c) − λU R + µU [F − c − δK] .
Necessary conditions for a maximum include:
u0 − µU = ∂H U /∂c = 0;
µU FR − λU = ∂H U /∂R = 0;
λ̇
U
= ρλU − ∂H U /∂S = ρλU ;
µ̇U = ρµU − ∂H U /∂K = µU (ρ + δ − FK ) .
Algebraic manipulation of these conditions yields Ramsey’s rule, linking the rate
of utility discount ρ, the rate of change of marginal utility u̇0 /u0 , and the net (of
12
deterioration) productivity of capital,
FK − δ = ρ − u̇0 /u0 .
It also yields the macroeconomic rendering of Hotelling’s rule, relating the rate of
change of the marginal productivity of the resource and the net marginal productivity of capital,
ḞR
= (FK − δ) .
FR
(4.2)
Consider now a perfectly competitive economy. The implicit objective is to
maximize the integral of consumption evaluated at present-value prices [ps ]∞
t
(prices into which the market discount factors are already incorporated), i.e. to
1
max V (S, K, N, t) =
pt
Z
∞
ps cs ds,
t
subject to the same conditions. The present-value Hamiltonian is
H V = pc + µV [F − c − δK] − λV R.
13
(4.3)
Necessary conditions include:
p − µV
µV FR − λV
λ̇
V
µ̇V
= ∂H V /∂c = 0;
= ∂H V /∂R = 0;
= −∂H V /∂S = 0;
= −∂H V /∂K = −µV (FK − δ) .
These conditions imply that
ḞR = (FK − δ) FR ,
as in Hotelling’s rule (4.2) above, and that
ṗ = −p (FK − δ) ,
h R
i
t
so that pt = p0 exp − 0 (FK − δ) τ dτ . Therefore, the force of interest used to
discount consumption is (FK − δ) . Let r (t) = (FK − δ)t .
By the first and second theorems of welfare economics, there is a bijection
between these two classes of economy. A planner seeking to attain an optimum in
14
h R
i
t
problem (4.1) can decentralize present-value prices pt = u (ct ) exp − 0 r τ dτ =
h R
i
h R
i
t
t
µU exp − 0 r τ dτ = µV and (pFR )t = λVt = λUt exp − 0 r τ dτ . At these
0
prices, ρ − u̇0 /u0 = FK − δ = r. (If at any time u is changing, then r 6= ρ and if
u is constant then r = ρ.) The two economies follow the same path. Since there
are economies with objective (4.3) that correspond to economies with objective
(4.1), there is good reason to let the force of interest vary in the latter.
Along the optimal path that is the explicit solution for problem (4.1) and
implicit for problem (4.3), differentiation in two ways yields
ρU − u (c) = U̇ = K̇∂U/∂K + Ṡ∂U/∂S + ∂U/∂t = µK̇ + λṠ + ∂U/∂t.
The term ∂U/∂t arises because of non-stationarity, i.e., exogenous changes of
preferences, technology or other underlying conditions. In the present problem,
the force of pure time preference, ρ, is allowed to change; if ρ is constant then
∂U/∂t = 0. Let u0 (c) + cu0 (c) = u0 (c) + pc = u (c) at time t, and identify u0 (c)
as aggregate consumers’ surplus. Then
ρU − ∂U/∂t = H = u + µK̇ + λṘ = u0 + pc + pK̇ + (pFR ) Ṡ.
15
Except for the term ∂U/∂t, interest (at the pure rate of time preference ρ) on
wealth U is equal to NNP at market prices, pc + pK̇ + (pFR ) Ṡ, plus aggregate
consumers’ surplus, u0 . The last is not easily measured and depends on the point
on the utility surface.
(One can also, purely formally, define time t as a state variable of the problem
with ṫ = 1 and shadow price q = ∂U/∂t. In this case the Hamiltonian of the
problem would be u + µK̇ + λṘ + q = H + q; it would be exactly interest on
wealth. This approach is not pursued herein.)
One can also write, for the implicit objective of problem (4.3),
rV − pc = dV /dt = K̇∂V /∂K + Ṙ∂V /∂R + ∂V /∂t = pK̇ + pFR Ṡ + ∂V /∂t. (4.4)
Therefore,
rV − ∂V /∂t = pC + pK̇ + pFR Ṡ.
(4.5)
Net national expenditure at market prices, pC + pK̇ + pFR Ṡ, is equal to interest,
at the consumption rate of discount r, on value (or market wealth) V , minus the
drift ∂V /∂t.
Both problems illustrate Weitzman’s contribution. (See also Weitzman 2000.)
The Hamiltonian is (or approximates if the problem is not stationary so that
16
∂U/∂t 6= 0 or ∂V /∂t 6= 0) the current return on value. That is to say, it is (or
approximates) the force of interest applied to the value of the objective.
In defining NNP the statistician neglects u0 + ∂U/∂t, or neglects ∂V /∂t. If
time is truly shifting the value function, then the effect of time should be evaluated
and appear in statistics; a rower’s rate of progress in a stream depends on both
the level of exertion and the current. In reality there may be some drift. Whether
it is important is an empirical question. If the problem is comprehensive, in that
all sources of change are incorporated into some form of capital, then the term
∂U/∂t or ∂V /∂t occurs only because of changes in the force of interest. It may
be small, as the term structure of interest rates has remained in a narrow band
through time.
The neglect of aggregate consumers’ surplus u0 is the reason that (real) NNP
is only a first-order approximation to the Hamiltonian of problem (4.1). On a
personal note, I am sure that my own consumer’s surplus is finite, but have no
inkling of its level, let alone the world’s aggregate consumers’ surplus. I am
sympathetic to neglecting it in an economic statistic, especially in view of the
correspondence between problems (4.1) and (4.3) and of the practicality of the
resulting statistic.
A much more fundamental objection to NNP is that one may not concur
17
with the value implied by using a utilitarian objective in general, or with the set
of preferences U (S, K, N, t) and {ρt }∞
t=0 that give rise to the particular implicit
objective to which a given competitive outcome corresponds.
4.2. Wealth vs. Income Accounting
When the force of utility discount is a constant, γ, Weitzman (2000) notes that
Ut =
Ht
,
γ
so that if Ht were available at all future times s, the value of Ut would be the
same:
Z
∞
−γ(s−t)
u(c)e
ds = Ut =
t
Z
∞
Ht e−γ(s−t) ds.
t
Weitzman calls Ht , which is the net national product, the stationary equivalent.
In the general case, the stationary equivalent is not attainable (lies outside the
utility-possibilities frontier). Even though Weitzman was at pains to stress that
the stationary equivalent may not be sustainable, occasionally it is misinterpreted
as a measure of sustainable consumption. It is a mathematical curiosum that
holds only for a constant force of interest. The expression becomes messy if the
rate of utility discount is not constant. Let the force of utility discount be, as
18
above, ρt . The interpretation of Weitzman’s finding depends on the fact that
Ht = ρt Ut ,
so that the Hamiltonian (NNP) is interest on, and hence income from, value Ut .
This formulation is consistent with a Hicksian interpretation of NNP; the following
asset-equilibrium conditions for total value are exact:
rV
= pc + dV /dt;
ρU = u (c) + dU/dt.
Interest, at the appropriate rate, on value (be it V or U depending on the objective
being studied) is equal to the current dividend (pc or u (c)) plus the capital gain
(dV /dt or dU/dt). Interest on the economy’s value is its income.
Thus, value and income are sides of the same coin (Samuelson 1961). If consumption pc or utility u (c) is equal to income, then value does not change. Hicks
(1946) assumes that an individual’s goal is to sustain consumption; the individual
has the maximin objective. In problems (4.1) and (4.3), however, maximin is not
the goal. If value is changing (if dV /dt or dU/dt is changing) then the dividend
19
that is being (optimally) consumed is greater or less than income.
Hicks’s notion of income can be applied in the context of a utilitarian economy
if one observes that the definition of income is not dependent on sustaining utility
or consumption. NNP is not an index of welfare or value; rather, it measures the
flow emanating from the stock of welfare.
4.3. Sustained Consumption or Utility
If an analyst wishes to study sustainability as a social value, then sustaining the
economy belongs in the objective. The solution (Cairns and Long 2001) of a
stationary maximin program is a level of utility υ that is sustained through time
and that is equal to an aggregate of the stocks,
υ = φ (S, K, N) .
The function φ indexes value, which is the minimum attained utility over all
time. Unlike in a utilitarian problem, in a maximin problem value is a flow, not
an aggregate on which interest is earned. Such an aggregate is not defined in a
maximin problem. Indeed, there is no interest rate. (There is a virtual interest
rate, but that rate is a derived, technical variable, not a preference using which
20
the analyst aggregates levels attained at different times into a measure of value.)
Society’s income at every point in time is also the minimum attained level of
utility, in this case also its value. This observation coincides with Hicks’s for an
individual maximizing sustained utility.
If φ is differentiable then
∂φ
∂φ
∂φ
∂φ
∂φ
dφ
dυ
Ṡ +
K̇ +
Ṅ =
Ṡ +
K̇ =
=
= 0.
∂S
∂K
∂N
∂S
∂K
dt
dt
(4.6)
Hartwick’s rule (4.6) (Hartwick 1977) is that at any time total net investment,
evaluated at the shadow prices of a maximin problem, is zero. The expression
qS Ṡ + qK K̇ is the Hamiltonian in a maximin problem, and in the solution qS =
∂φ/∂S and qK = ∂φ/∂K.
In the competitive economy postulated by Dixit et al., Hartwick’s rule is said
to hold. This is not to say, however, that the rule is defined in a market economy.
In their model the maximin problem has implicitly been solved to determine the
shadow values of equation (4.6) and then those (relative) shadow values, incorporating the appropriate virtual discount factors, have been decentralized. In this
specific competitive economy, the net value of investment is equal to zero at all
times.
21
If sustaining the economy is the social value (if the objective is to sustain
consumption or utility), net national product is not a useful statistic. The virtual
Hamiltonian from the virtual utilitarian problem does not have the meaning that it
has in an explicit utilitarian problem. A willingness to exchange utility at different
times at shadow prices λ (s), which is implied by writing down the utility integral
or the Hamiltonian, is incompatible with the analyst’s postulated preferences.
A maximin problem gives rise to two distinct statistics, the level of utility,
which remains constant, and the values of net investments in all stocks, which
are such that total net investment remains equal to zero. There is no theoretical
basis to add them to form NNP as in a discounting problem. Computing such a
sum is pointless. On the optimal maximin path, since aggregate net investment
is always equal to zero, NNP never changes as it is always equal to consumption.
Off the optimal path, the shadow values ({pt } and {λt }) have no interpretation.
4.4. NNP and Sustainment
The concept of income is at the heart of green accounting. Studying the problem of
sustaining an economy “as if” it were a utilitarian problem like problem (3.2) has
provided many insights. But the two problems are not identical. They express
different values, have different objectives, and hence have different concepts of
22
income.
Sustaining an economy is a long-run objective: the economy’s value must be
maintained forever. Solving a maximin problem gives the short-run conditions,
including Hartwick’s rule, for sustainment. The method of solution guarantees
that these conditions hold at each point on the maximin path. Solving a general,
utilitarian problem gives short-run conditions that may look like Harwick’s rule
but may not hold forever along the optimal path.
By choosing an objective to study, the analyst imposes a concept of value.
The main problem with trying to apply extended NNP to measure sustainment of
an economy is that use of NNP implies a social objective or value that is different
from sustainment. The difference has implications for the usefulness of NNP.
NNP is the (linearized) Hamiltonian of a utilitarian optimization problem
(evaluated with given discount factors). Even if sustainment is the analyst’s intent
and a discounted-utility problem is used as a vehicle for deriving properties of the
solution, the utilitarian problem is written as in equation (3.2). The discussion of
the maximin program above, however, indicates that the discount factors, λ (t),
are defined only on the maximin path. In order to find the discount factors, the
maximin problem must be solved. Moreover, writing down equation (3.2) implies
that any point on the hyperplane separating the production-possibilities frontier
23
from the objective makes the same contribution to value. When u (s) 6= ū, s ≥ t,
it is inappropriate to write
R ∞ λ(s)
t
u (s) ds. It is appropriate to write this integral
λ(t)
only in a utilitarian problem in which {λ (s)}∞
t is predetermined (exogenous to
the problem).
On the maximin path (the only path on which {λ (s)}∞
t is defined),
Z
∞
t
λ (s)
u (s) ds =
λ (t)
Z
t
∞
λ (s)
ūds = ū
λ (t)
Z
t
∞
λ (s)
ds = ū;
λ (t)
the integral collapses to the maximin objective, ū = max u such that u (s) ≥ ū.
By Hartwick’s rule, net investment is always zero; there is no point in reporting
net investment as a statistic or part of a statistic (NNP). Off the optimal path,
the shadow values have no interpretation: The inappropriateness of writing the
integral carries over to the Hamiltonian and hence to NNP. In short, when the
prices have meaning, nothing is gained by computing the investment component;
only consumption is a useful statistic.
If society’s real objective is not to sustain utility, it is well-known that keeping
net investment at zero (interpreted as following Hartwick’s rule, which usually
implies moving off the path that is optimal for the utilitarian objective) at the
given point in time does not imply sustainability. That is to say, for a general
24
utilitarian objective, setting net investment to zero is myopic. Hartwick’s rule
has to apply at each future point for utility to be sustained. The only utilitarian
objective for which this property holds is the one that coincides with the maximin
solution. As we have noted, even for this problem, NNP is not a useful statistic.
An optimal utilitarian path is Paretian. One criticism of maximin programs is
that, if a problem is not regular, if it is not possible to “spread” utility equally to
all points on the maximin path, the problem is not Paretian. That is to say, the
maximin value is the minimum level of utility on the path, and the greater levels
of utility at other points do not ‘count’ and the excess can be thrown away. The
reason, of course, is that a maximin program treats equality among generations
as the social value. No matter how defined, however, even with use of a utilitarian
welfare functional, attempting to impose a condition of intergenerational equity
must throw up non-Paretian cases. The criterion for sustainment on a utilitarian
path is often held to be whether utility is increasing (whether u̇ ≥ 0) or else
whether value is increasing (whether V̇ ≥ 0 ). If there are constraints preventing
the spreading of consumption equally in an economy, it may not be possible
always to have u̇ ≥ 0 or V̇ ≥ 0. The imposition of one of these conditions as a
constraint renders the sustainment problem non-Paretian as well. Furthermore,
the utilitarian formulation with the constraint is not a complete set of preferences,
25
as it does not distinguish among paths that have the same present value, in which
value is not decreasing but in which the rate of change of value is not the same.
Consider a fishery with a stock S. Let the harvest be H, and let Ṡ =
S (1 − S) − H be the transition equation. The maximin level of consumption
is S (0) if S (0) ≤ 1/2, and 1/2 if S (0) ≥ 1/2. Let the initial stock be S0 > 1/2.
Imposing the constraint that u̇ (0) ≥ 0 or that V̇ (0) ≥ 0 (with any set of discount
factors) requires that H = 1/2, when it could be greater early in the program. The
level of harvest is not Pareto optimal. Maximin does not have this requirement.
Consider a path on which consumption (be it optimal according to some criterion or not) fluctuates about a rising, linear trend,
C (t) = A [1 + α sin (ωt + β)] + Bt,
(4.7)
so that the change in consumption,
Ċ (t) = Aαω sin (ωt + β) + B,
has the sign of B − αωA sin (ωt + β). If B < Aαω then Ċ (t) < 0 on regular
intervals, over which Aαω sin (ωt + β) falls at a rate greater than B. Is this type
26
of path an “unsustained development”?
Now let this consumption path be discounted at some discount rate, r. Without loss of generality we can let ω = 1 and β = 0. Present value is
V (t) =
Z
∞
[A (1 + α sin s) + Bs] e−rt dt
t
A
Aα
B
+
(cos t + r sin t) + 2 (1 + rt) .
=
2
r
1+r
r
V̇ (t) =
Aα
B
(− sin t + r cos t) + .
2
1+r
r
If t = ±2nπ,
V̇ (t) =
B
Aα
−
r
1 + r2
and so if
B < Aα
r
( < Aα),
1 + r2
(4.8)
there are regularly repeated intervals on which V̇ (t) < 0. The value function,
however, never falls below
A
Aα
B
−
+ 2,
2
r
1+r
r
and we can make this arbitrarily close to A/r (and A/r arbitrarily large) while
satisfying condition (4.8). Value eventually grows like (B/r) t. There is an ε
27
(small) for which V (t) >
A
r
− ε for all t, even if V̇ (t) < 0 for some values of t,
perhaps the current value. Is this an unsustained economy?
The problems just raised are why the literature has derived many contradictory and conflicting definitions and concepts of sustainable utility starting from a
utilitarian value function. The only advantage of this objective function over the
direct formulation of maximin is its familiarity.
Thus far, discussion has been limited to optimal programs. The fundamental
reason why accounts are needed is that the economy is not optimal. A real
economy does not maximize an explicit objective. What are to be used as value
and income in a real economy, for which we must produce real statistics?
5. Value and Income in a Real Economy
5.1. Ideal Valuation
One is entitled, if not obliged, to challenge the outcome of any real economy
in terms of values that are of interest to the analyst. Such an evaluation is a
possible interpretation of Dasgupta and Mäler’s (2000: 83 ff.) discussion of social
well being and the concept of sustainability. They present a model of a real
economy following some observed path through time. The economy’s institutions
28
and policies are represented by a vector, α. We can generalize their postulated
expression of value as
R
V (K, t |α) =
Z
∞
δ (s) u α (cs ) ds.
(5.1)
t
(In a completely general accounting system, the institutions and policies could
form a sub-vector of the vector of the economy’s capital stocks, K.) Dasgupta
and Mäler make no mention of maximizing value, V R (K, t |α ); they derive no
Ramsey condition related to α. A small policy change Oα is contemplated, but it
need not be optimal, nor indeed even an improvement.
The problem expresses what the authors wish to examine as value; they could
as reasonably have chosen as the measure of value the level of utility of the generation living at some time t0 , or the level of utility of the generation with lowest (infimum) utility. In the context of a value function having the form of V R (K, t |α ),
it is still possible to challenge the values inherent in the utility function u α (c)
or the discount factors δ (t). The main property of NNP is that its form is the
same, whatever the particular choices of u α or δ. If both sides of equation (5.1)
29
are differentiable we get
Ã
δ̇
−
δ
!
∙
¸
∂V
∂V
0
V − u α (c) +
+
= ∇u α · C + ∇VK · K̇.
∂t
∂α
(5.2)
The first term on the left-hand side is income, i.e., interest on value at the pos³
´
tulated force of interest −δ̇/δ . In the square brackets are two terms we have
discussed above as being neglected, consumers’ surplus and drift, plus the net
benefit of any postulated policy change, another term for which there will be no
statistic. On the right-hand side is the familiar expression for NNP. A similar
exercise could be performed for discounted market values, using
R
U (K, t |α ) =
Z
∞
t
γ (s) ps · cs ds,
in which the consumers’ surplus would not appear.
There is no need for optimization, and hence no need for constructing a Hamiltonian, in defining NNP as a measure of income. The formula for income is similar
so long as the form of the value function involves an integral of a discounted flow.
For different utility functions and discount factors, there are different value functions, and hence different levels on the RHS of equation (5.2). Evaluating NNP
30
requires shadow values, ∇u α and ∇VK , and hence some at least implicit knowledge of V (K, t |α ), as well as a fore-knowledge of how any increase in a component
of the vector K will be deployed.
5.2. NNP at Market Prices
Even a distorted economy may be viewed as behaving as if having some implicit objective similar to that of the hypothetical perfectly competitive economy
(Cairns 2002b). According to the separation theorem, which separates decisions
on investment from those on allocation of consumption, the objective, which need
not be viewed as being maximized, is market wealth. Let the technology be as
in problems (4.1) and (4.3). At time s, let the market force of interest be represented by rs and the present-value price of the consumption good by πs (so that
³R
´
t
π̇ s /πs = rs and hence πt = π0 exp 0 rs ds ). Market wealth is
W (S, K, t) =
Z
t
31
∞
πs
cs ds.
πt
(5.3)
Let Y represent current NNP and let Ṡ∂W/∂S + K̇∂W/∂K, net investment, be
represented by I.1 If W (S, K, t) is differentiable, then differentiating it in two
ways and a then further differentiation yield
Ẇ (S, K, t) = Ṡ∂W/∂S + K̇∂W/∂K + ∂W/∂t;
(5.4)
rW = c + Ẇ = Y + ∂W/∂t;
(5.5)
Ẏ = ṙW + rẆ − d/dt (∂W/∂t) .
(5.6)
If the exogenous drift in the economy (accounting for ∂W/∂t) and in the force
of interest (ṙ) are equal to zero–heroic assumptions in a real economy–equations
(5.4) and (5.6) imply that
sgnẎ = sgnẆ = sgnI:
1
(5.7)
The asset value of the resource is the discounted value (at rate rt ) of its earnings,
µ Z s
¶
Z ∞
πRFR exp −
r τ dτ ds.
At =
t
t
The depreciation of this value is −Ȧ = πRFR − rA; it is the change in income from the resource,
and is net investment on the income side of the green accounts. On the expenditure side,
net investment is −πRFR , the component of the Hamiltonian corresponding to changes in the
quantity of the resource. Cairns (2002a) presents a detailed discussion. The expression Ṡ∂W/∂S
corresponds to πRFR .
32
At any time, each of market income or green NNP (Y ) and market wealth (W )
moves in the direction of the market value of net investment. Equation (5.7) is a
net-investment rule in a market economy. (If the problem is non-stationary, the
rule is only approximately true.) This net-investment rule uses observed or at
least potentially observable statistics. The statistic, NNP at market prices, is a
measure of achieved performance, not optimal or potential performance.
Manipulation of equations (5.5) and (5.4) yields that
c = rW − Ẇ = rW − I − ∂W/∂t.
By differentiation and then substitution from equation (5.4),
d ∂W
ċ = ṙW + rẆ − I˙ −
dt ∂t
µ
¶
³
´
d ∂W
˙
= rI − I + ṙW + r −
.
dt ∂t
If ṙ = 0 and ∂W/∂t = 0, then (cf. Hartwick 2003)
³
´
˙
sgn ċ = sgn r − I/I .
(5.8)
Equation (5.8) provides a local indicator of the rate of change of consumption. It
33
is not a measure of sustainability of consumption, nor even of local sustaining of
consumption, as these require global solutions.
5.3. Sustaining a Decentralized Economy
Consider an ideal, competitive economy or a real, market economy. For appro∞
priate present-value prices [ps ]∞
s=t , or for appropriate current-value prices [vs ]s=t
and discount factors [βs ]∞
s=t , the economy is implicitly maximizing a utilitarian
objective,
Φt =
Z
t
∞
ps · cs ds =
Z
t
∞
(βs vs ) · cs ds.
Let the prices satisfy all the conditions enunciated by Dixit et al. for a regular
path,2 and also let it be feasible for the economy to follow a sustainable path.
Suppose that an onlooker wishes to evaluate whether the economy is being
sustained. (The onlooker, then, rejects the implicit, utilitarian objective of the
economy and has the maximin objective.) Unless the prices [ps ] or [βs vs ] are such
that this economy happens to be following a constant-utility path for all time,
the prices are not relevant for the onlooker’s evaluation. Observed, market prices
are not “right” for determining whether an alternative objective is being followed;
2
Regularity narrows things down but not very far. There are an uncountably infinite number
of regular paths, only one of which is a constant-utility path.
34
rather, if there is an alternative objective to “what the economy is implicitly
maximizing” (for which market prices are the appropriate shadow values), the
prices used should be the shadow prices for that objective. Conceptually, the
onlooker can determine whether the economy is on the sustainable path at time
t, given its endowment of stocks, by solving the maximin problem (3.1) to find
the shadow values of the stocks at time t. These shadow values can be applied
to the net investments, according to Hartwick’s rule (4.6). The sign of total, net
investment is descriptive of the economy: If the sum is zero, then the economy is
being sustained at time t.
If the sum is negative, the economy is not being sustained. If the sum is
positive, the economy is “more than” being sustained. If the sum is non-zero on
an interval, however, the economy is taken off the maximin path and the prices
no longer have the interpretation of shadow prices for the maximin problem. The
evaluation can no longer be made.
Mathematically, there do exist current prices for evaluating whether the economy is being sustained at time t. They are the right prices for time t, irrespective
of whether the decision makers of the present or the future decide to sustain or
not to sustain the economy. It is impractical to think that the shadow prices could
be determined for a real economy.
35
For the economy actually to be sustained for all future time from time t,
Hartwick’s rule must be followed at all times s ≥ t, using the shadow prices
for those times. The maximin solution prescribes Hartwick’s rule, and so the
net-investment rule for a sustained economy is prescriptive.
In a steady state, since the rate of change of each stock is zero, Hartwick’s rule
is trivially satisfied. A steady state is perhaps a naive ecologist’s dream. It implies,
however, ecological stagnation. An ecologically vibrant, sustained economy must
have positive shadow prices, so that some stocks are rising and some falling. Stocks
are substituting for each other in producing the constant level of utility through
time. Ecological sustainment is not possible without substitution.
6. Conclusion
The objective in a theoretical model is the theorist’s postulate of the meaning
of value, and the outcome of the model yields a representation of value and of
income. It is not consistent to use that model to consider other concepts of value
or income. The net-investment rule is no guide for sustaining an economy when
market prices are the basis of evaluating net investments. Rather, if the objective
is to maximize sustainable consumption, the analyst must solve an overwhelmingly
complex maximin problem to obtain the shadow prices of investments for use in
36
Hartwick’s rule. Theoretically, in both coverage and relative prices, sustainability
accounts entail a complete break from the national accounts. Consequently, they
are impractical.
It is vital to understand the meaning of environmental accounting and what it
can accomplish for society. As Nordhaus and Tobin observed over thirty years ago,
using techniques of non-market valuation to incorporate unpriced environmental
flows can improve NNP as an indicator of utilitarian value. The interpretation is
founded on discounting, which is an unappealing feature to many. No explicit optimization of a value function is required or contemplated, however. If there are
important divergences from perfect competition, market prices are still shadow
prices for a virtual value function, for whatever “the market is implicitly maximizing,” subject to all the practical constraints facing a real economy. Extending
NNP at market prices in the intent of providing an income-based indicator of
welfare implies acceptance of that virtual value. Acceptance of the market outcome may be even less appealing than acceptance of discounting. Still, economists
can make the case that if growth, defined as increasing green NNP consistently
through time, is successful, then environmental quality and other features of the
quality of life will almost surely continue to improve, because they are normal
goods.
37
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