Scand. J. of Economics 106(2), 361–384, 2004
DOI: 10.1111/j.1467-9442.2004.00362.x
A General Approach to Welfare
Measurement through National
Income Accounting*
Geir B. Asheim
University of Oslo, NO-0317 Oslo, Norway
g.b.asheim@econ.uio.no
Wolfgang Buchholz
University of Regensburg, DE-93040 Regensburg, Germany
wolfgang.buchholz@wiwi.uni-regensburg.de
Abstract
A framework is developed for analyzing national income accounting using a revealed welfare
approach that is sufficiently general to cover, both the standard discounted utilitarian and
maximin criteria as special cases. We show that the basic welfare properties of comprehensive
national income accounting, previously ascribed only to the discounted utilitarian case, extend
to this more general framework. In particular, under a wider range of circumstances, it holds
that real NNP growth (or, equivalently, a positive value of net investments) indicates welfare
improvement. We illustrate the applicability of our approach in the Dasgupta–Heal–Solow
model of capital accumulation and resource depletion.
Keywords: National income accounting; dynamic welfare
JEL classification: C43; D6; O47; Q1
I. Introduction
Net national product (NNP) represents the maximized value of the flow of
goods and services that are produced by the productive assets of society. If
NNP increases, then the capacity of society to produce has increased, and—
one might think—society is better off. Although such an interpretation is
* We have benefited from discussions with Kenneth Arrow and Martin Weitzman. We also
thank Finn Førsund, Lawrence Goulder, Peter Hammond, Geoffrey Heal and David Miller as
well as three anonymous referees for helpful comments. An earlier version was circulated
under the title ‘‘Progress, Sustainability, and Comprehensive National Accounting’’. Asheim
appreciates the hospitality of the research initiative on The Environment, the Economy and
Sustainable Welfare at Stanford University, where much of this work was done. We gratefully
acknowledge financial support from the Hewlett Foundation through the above-mentioned
research initiative (Asheim), CESifo Munich (Asheim) and the Research Council of Norway
(Ruhrgas grant, both authors).
# The editors of the Scandinavian Journal of Economics 2004. Published by Blackwell Publishing, 9600 Garsington Road,
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362
G. B. Asheim and W. Buchholz
often made in public debate, the assertion has been subject to controversy in
the economic literature. While Samuelson (1961, p. 51) writes that ‘‘[o]ur
rigorous search for a meaningful welfare concept has led to a rejection of all
current income concepts. . .’’, Weitzman (1976), in his seminal contribution,
shows that greater NNP indicates higher welfare if
(i) dynamic welfare equals the sum of utilities discounted at a constant rate, and
(ii) current utility equals the market value of goods and services consumed.
Weitzman’s result is truly remarkable—as it implies that changes in the
stock of forward-looking welfare can be picked up by changes in the flow
of the value of current production—but, unfortunately, very strong
assumptions are invoked. Recently, Asheim and Weitzman (2001) have
established that assumption (ii) can be relaxed in the context of whether
welfare is increasing locally over time. Real NNP growth corresponds to
welfare improvement even when current utility does not equal the market
value of current consumption, as long as NNP is deflated by a Divisia
consumption price index. It is the purpose of the present paper to show
how Weitzman’s assumption (i) can also be relaxed, and that a ‘‘snapshot’’
of the change in society’s current performance still indicates change in
dynamic welfare.
Why relax the assumption of discounted utilitarianism? First of all, such
an assumption restricts the use of NNP comparisons for indicating welfare
changes to situations where it can readily be determined that society maximizes the sum of utilities discounted at a constant rate. Moreover, there is a
contradiction between having welfare correspond to discounted utilitarianism and being concerned with welfare improvement. For example, in the
Dasgupta–Heal–Solow model of capital accumulation and resource depletion, eventually the welfare of society is optimally decreasing along the
discounted utilitarian path; see Dasgupta and Heal (1974, 1979) and Solow
(1974). Increasing welfare over time is not of independent interest when
society implements a path that maximizes the sum of discounted utilities. In
contrast, real-world societies care about whether welfare is improving, both in
terms of what proponents of economic growth may refer to as ‘‘progress’’ and
in terms of what environmentalists call ‘‘sustainability’’.
To incorporate concerns regarding progress and sustainability, we extend
Weitzman’s (1976) result by developing a framework for national accounting that is sufficiently general to include, in addition to discounted utilitarianism, cases like (i) maximin, (ii) undiscounted utilitarianism and (iii)
discounted utilitarianism with a sustainability constraint. In these cases,
non-decreasing current welfare does entail that current utility can be sustained indefinitely, as opposed to the case of unconstrained discounted
utilitarianism; cf. Asheim (1994) and Pezzey (1994). Within this general
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framework we demonstrate how changes in dynamic welfare are revealed by
current NNP and hence by observable current prices and quantities.
A prerequisite for the positive results on the relationship between welfare
improvement and change in current NNP—from Weitzman (1976) to the
current paper—is that the list of goods and services included in NNP be
comprehensive. National accounts are ‘‘comprehensive’’ if all variable determinants of current productive capacity are included in the vector of capital
stocks, and if all variable determinants of current well-being are included in
the vector of consumption flows. In other words, one has to ‘‘green’’ national
accounts by (i) including depletion and degradation of natural capital as
negative components to the vector of investment goods, and (ii) adding flows
of environmental amenities to the vector of consumption goods.
In this paper we show first that, even outside the realm of discounted
utilitarianism, the value of net investments has the following welfare significance: welfare is increasing if and only if the value of net investments is
positive.1 Thus, in an economy with natural capital, welfare is increasing if
and only if the accumulation of man-made capital (including stocks of
knowledge) in value more than compensates for natural resource depletion
and environmental degradation. Then, using the analysis of Asheim and
Weitzman (2001), we establish that a positive value of net investments
corresponds to real NNP growth. Hence, under quite general assumptions,
welfare improvement is indicated by increasing real NNP or by the value of
consumption falling short of NNP, so that the value of net investments is
positive.
The empirical challenges of making national accounts comprehensive are
the same in this more general framework. Forward-looking terms to capture
technological progress and changing terms-of-trade can also be incorporated
here; cf. Aronsson and Löfgren (1993) and Sefton and Weale (1996). Moreover, Asheim (2004) shows how the framework can be extended to accommodate exogenous population growth.
After introducing the basic model in Section II, we consider national
income accounting in the special cases of discounted utilitarianism and
maximin in Section III. Then, in Section IV, we turn to our general framework for revealed welfare analysis, and show in Section V how this framework implies that real NNP growth can indicate welfare improvement in a
general setting. The applicability of our framework is illustrated in Section
VI by considering progress and sustainability in the Dasgupta–Heal–Solow
model. Section VII concludes.
1
Under discounted utilitarianism, this is proven by Weitzman (1976, eq. (14)), and reported
by, e.g., Hamilton and Clemens (1999), Dasgupta and Mäler (2000) and Pemberton and Ulph
(2001).
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II. The Model
Consider the model used by Weitzman (1970, 2001) and Asheim and Weitzman
(2001), who generalize Weitzman (1976) by allowing for multiple consumption
goods.
Let C represent an m-dimensional consumption vector that includes environmental amenities and other externalities. (Supplied labor corresponds to
negative components.) Let U be a given concave and non-decreasing utility
function with continuous partial derivatives that assigns instantaneous utility
U(C) to any consumption vector C. Assume an idealized world where C
contains all variable determinants of current instantaneous well-being,
implying that society’s instantaneous well-being is increased by moving
from C0 to C00 if and only if U(C0 ) < U(C00 ).
Let K denote an n-dimensional capital vector that includes not only the
usual kinds of man-made capital stocks, but also stocks of natural resources,
environmental assets, human capital (such as education and knowledge
capital accumulated from R&D-like activities), and other durable productive
assets. Moreover, let I(¼ K_ ) stand for the corresponding n-vector of net
investments. The net investment flow of a natural capital asset is negative if
the overall extraction rate exceeds the replacement rate.
Assume again an idealized world where K contains all variable determinants of current productive capacity, implying that the set of attainable
(m þ n)-dimensional consumption–investment pairs is a function S only of
the available capital stocks K, not of time. Hence, (C, I) is attainable given
K if and only if (C, I) 2 S(K), where S(K) is a convex and smooth set that
constitutes current productive capacity.
It holds that NNP is the maximized market value of current productive
capacity in a perfect market economy where C and I are included in NNP
and valued at market prices (cf. Section V). As time passes, NNP changes both
because K, and thus productive capacity S(K), change due to a non-zero vector
of net investments, and because market prices of consumption and investment
flows change. Since NNP is used for (a) consumption now and (b) accumulation of capital goods yielding increased future consumption, relating NNP
growth to welfare improvement requires a notion of dynamic welfare. Welfare
judgments should not only take into account the utility derived from current
consumption, but should also reflect the utility possibilities inherent in future
consumption. For this purpose, we assume that the welfare judgments of
society are described by complete and transitive social preferences on the set
of utility paths. However, these underlying social preferences are assumed not
to be directly observable by the national accountant.
What the national accountant can observe at any point in time is how the
agents in society make decisions according to a resource allocation mechanism that assigns an attainable consumption–investment pair (C(K), I(K)) to
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Welfare measurement through national income accounting
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any vector of capital stocks K.2 We assume that the functions C and I are
continuous everywhere and differentiable almost everywhere, and that there
exists a unique solution {K*(t)} to the differential equations K*(t) ¼ I(K*(t))
that satisfies the initial condition K*(0) ¼ K0, where K0 is given. Hence,
{K*(t)} is the capital path implemented by the resource allocation mechanism.
Write C*(t):¼ C(K*(t)) and I*(t):¼ I(K*(t)).
In this manner, the resource allocation mechanism implements a utility
path {U(C*(t))} for any vector of initial capital stocks K0. Hence, the social
preferences yield a complete and transitive binary relation on the set of
capital vectors, under the presumption that paths are implemented by the
resource allocation mechanism. Assume that, for given social preferences
and resource allocation mechanism, this binary relation can be represented
by an ordinal welfare index, W, that is unique up to a monotone transformation, signifying that society’s dynamic welfare is increased by moving from
K0 to K00 if and only if W(K0 ) < W(K00 ). Moreover, assume that W is continuous and differentiable everywhere. To retain focus, in the following we
refer to optimal control theory and the maximum principle under standard
assumptions, without explicitly stating what these assumption are.
III. Discounted Utilitarianism and Maximin
A main motivation for this analysis is that it applies to a variety of methods for
aggregating the interests of different generations in social evaluation. Discounted utilitarianism is the conventional example of social preferences in an
intertemporal context. A prime example of an alternative welfare criterion is
maximin—i.e., the ranking of paths according to the utility of the worst-off
generation—as proposed by Rawls (1971) and Solow (1974). Analysis of these
often applied kinds of social preferences and the corresponding resource
allocation mechanisms points to properties that will ensure welfare significance
of national income accounting also for a wider class of social preferences and
resource allocation mechanisms.
In the special cases of discounted utilitarianism and maximin, we will
identify the level of the welfare index, W(K), with the utility level that—if
held constant—is equally as good as the implemented utility path, given K
as the vector of initial stocks. This makes W(K) a stationary equivalent of
future utility. This is in line with a standard constructive technique for preference representation in consumer theory, as in e.g. Mas-Colell, Whinston
and Green (1995, pp. 47–8) and Varian (1992, p. 97), and inspired by Hicks
(1946, Ch. 14) and Weitzman (1970, 1976) in the present context.
2
This is inspired by Dasgupta (2001, p. C20) and Dasgupta and Mäler (2000).
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G. B. Asheim and W. Buchholz
Discounted utilitarianism. Social preferences are represented by
Z 1
et UðCðtÞÞdt;
ð1Þ
0
where is a positive and constant utility discount rate. Assume that the
resource allocation mechanism, for any vector of initial capital stocks K0,
implements a path {C*(t), I*(t), K*(t)} that maximizes (1) over all
feasible consumption paths. By the maximum principle there exists a
path {C(t)} of investment prices in terms of utility such that (C*(t),
I*(t)) maximizes U(C) þ C(t)I subject to (C, I) 2 S(K*(t)) at each t.
Associate welfare W(K0) with the utility level, that if held constant, is
equally as good as the implemented path:
R1
0
WðK Þ ¼
0
et UðC ðtÞÞdt
R1
¼
t dt
0 e
Z
1
et UðC ðtÞÞdt:
0
A main result of Weitzman (1970, eqs. (5) and (16)) (reported in Weitzman
(1976) in the case where C is one-dimensional and U(C) ¼ C) is that
UðC ð0ÞÞ þ Cð0ÞI ð0Þ ¼ Z
1
et UðC ðtÞÞdt:
ð2Þ
0
Hence, W(K0) ¼ U(C*(0)) þ C(0)I*(0) under discounted
R 1 utilitarianism.
Since C(0) is the vector of partial derivatives of 0 et UðC ðtÞÞdt w.r.t.
the initial stocks, we obtain that the vector of partial derivatives of W,
rW(K0), equals C(0). By the maximum principle it now follows that
(C*(0), I*(0)) maximizes U(C) þ rW(K0)I subject to (C, I) 2 S(K0),
since U(C) þ rW(K0)I ¼ (U(C) þ C(0)I) and > 0.
Maximin. Social preferences are represented by inftU(C(t)). Assume that the
resource allocation mechanism implements maximin and that this leads to an
efficient path with constant utility; formally, Burmeister and Hammond
(1977) and Dixit, Hammond and Hoel (1980) call this a regular maximin
path. Then with K*(0) ¼ K0 as the initial condition, there exists a path of
utility discount factors {(t)} such
R 1 that it is as if the implemented path
{C*(t),I*(t), K*(t)} maximizes 0 ðtÞUðCðtÞÞdt over all feasible
conR1
sumption paths. Since U(C*(t)) is constant, it follows that 0 ðtÞdt is
finite. This holds if the supporting utility discount rates, _ ðtÞ=ðtÞ, are
positive and do not decrease too fast. Again, by the maximum principle,
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there exists a path {C(t)} of investment prices in terms of utility such that (C*(t),
I*(t)) maximizes U(C) þ C(t)I subject to (C, I) 2 S(K*(t)) at each t.
Associate welfare W(K0) also in this case with the utility level that, if held
constant, is equally as good as the implemented path:
0
R1
WðK Þ ¼ UðC ðtÞÞ ¼
0
ðtÞUðC ðtÞÞdt
R1
:
0 ðtÞdt
By the converse of Hartwick’s rule, we have that C (t)I*(t) ¼ 0 at each t; cf.
Hartwick (1977), Dixit et al. (1980), Withagen and Asheim (1998) and Mitra
(2002). Hence, W(K0) ¼ U(C*(0)) þ C(0)I*(0) even
R 1 under maximin.
Since C(0) is the vector of partial derivatives of 0 ððtÞ=ð0ÞÞUðC ðtÞÞdt
w.r.t. the initial stocks, by invoking the envelope
theorem,
we obtain that
R 1
rW(K0) equals *C(0), where :¼ ð0Þ 0 ðtÞdt is the infinitely longterm supporting utility discount rate at time 0 (i.e., the discounted average of
the instantaneous discount rates, _ ðtÞ=ðtÞ, from time 0 onward). By the
maximum principle it follows that
(C*(0), I*(0)) maximizes *U(C) þ rW(K0)I subject to (C, I) 2 S(K0),
since *U(C) þ rW(K0)I ¼ *(U(C) þ C(0)I) and * > 0.3
Two observations follow from the cases of discounted utilitarianism and
maximin:
(i) Refer to U(C*(0)) þ C(0)I*(0) as net national product in terms of utility
or ‘‘utility NNP’’. For both cases, utility NNP represents dynamic welfare globally; i.e., welfare is greater if and only if utility NNP is greater.
(ii) Interpret and * as Lagrangian multipliers for the constraint that
U(C) 5 U(C*(0)). For both cases, welfare improvement at time 0,
rW(K0)I, is maximized subject to (a) (C, I)2 S(K0), and (b)
U(C) 5 U (C*(0)).
The analysis below will show how observation (ii) can be used as the basis for
revealed welfare analysis where welfare significance of national accounting
aggregates is obtained. In contrast, we show that observation (i) cannot be
generalized; it does not, for example, apply to the case of undiscounted utilitarianism (as will be shown in Section VI). We now turn to the general analysis.
3
The observation that rW(K*(t)) is proportional to C(t) constitutes a simple proof of the
converse of Hartwick’s rule: constant utility implies that 0 ¼ dW(K*(t))/dt ¼ rW(K*(t))I*(t),
which due to proportionality of rW(K*(t)) and C(t) yields C(t)I*(t) ¼ 0. See also Cairns
(2000).
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G. B. Asheim and W. Buchholz
IV. Resource Allocation and Welfare Improvement
Fix the underlying, but unobservable, social preferences used to rank utility
paths, and consider a resource allocation mechanism. What properties of the
resource allocation mechanism are both (i) strong enough for the underlying
welfare concerns to be revealed through national income accounting and (ii)
weak enough to hold for a wide range of circumstances? We answer this
question by imposing two properties that hold if the most preferred paths
under discounted utilitarianism and maximin are implemented, but, as illustrated in Section VI, have a broader application. The first of these properties
is the following.
Property 1 (Implementation of an efficient path). Let {C*(t), I*(t), K*(t)}
be the path implemented by the resource allocation mechanism with
K*(0) ¼ K0 as the initial condition. There exists a continuous path of
positive supporting utility discount factors {(t)}, with corresponding
discount rates _ ðtÞ=ðtÞ being positive
R 1at almost every t, such that it is
as if {C*(t), I*(t), K*(t)} maximizes 0 ðtÞUðCðtÞÞdt over all feasible
consumption paths with K*(0) ¼ K0 as the initial condition.
This property is clearly satisfied when discounted utilitarianism is implemented, as well as for maximin when implementation of this criterion leads
to
R 1a regular maximin path (cf. Section III). The maximization is as if since
0 ðtÞU ðCðtÞÞdt is not necessarily the primitive objective of society. For
example, in the maximin case, {(t)} simply characterizes the implemented
path without having any intrinsic welfare significance.
If Property 1 holds, then the maximum principle yields efficiency prices
which support the efficient path. In particular, there exists a continuous path
{C(t)} of investment prices in terms of utility such that, at each t,
ðC ðtÞ; I ðtÞÞ maximizes UðCÞ þ CðtÞI subject to ðC; IÞ 2 SðK ðtÞÞ:
This yields the maximized current-value Hamiltonian:
H ðtÞ ¼ HðK ðtÞ;CðtÞÞ :¼
max
ðC;IÞ2SðK ðtÞÞ
UðCÞ þ CðtÞI ¼ UðC ðtÞÞ þ CðtÞI ðtÞ:
Refer to CI* as the value of net investments. Furthermore,
rK HðK ðtÞ; CðtÞÞ ¼ _ ðtÞ
_ ðtÞ;
CðtÞ C
ðtÞ
where r denotes a vector of partial derivatives.
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The following basic result—which is at the heart of the analyses of e.g.
Weitzman (1976), cf. eq. (14)) and Dixit et al. (1980, cf. Theorem 1)—can
now be established.
Lemma 1. If Property 1 holds, then U(C*(t)) þ C(t)I*(t) is continuous and
_ ðtÞ
CðtÞI ðtÞ
rUðC ðtÞÞC_ ðtÞ þ d ðCðtÞI ðtÞÞ=dt ¼ ðtÞ
holds at almost every t.
Proof: H* ¼ U(C*) þ CI* is continuous since K* and C are continuous.
Otherwise, adapt the proof of Asheim and Weitzman (2001, Lemma 1). &
This result says that change in utility NNP equals the supporting utility
discount rate times the value of net investments.
Let, as in Section II, the binary relation over vectors of stocks, induced
from the social preferences for a given resource allocation mechanism, be
represented by a welfare index, W, that is unique up to a monotone transformation. To ensure that the underlying welfare concerns can be revealed
through national income accounting, we make, in addition to Property 1, the
following assumption: the resource allocation mechanism and the accompanying welfare index satisfy that welfare improvement is maximized subject to (C,I) being attainable and utility being at least U(C(K)). This can be
stated by the following property, where (K) is formally a Lagrangian
multiplier on the lower bound for utility.
Property 2 (No waste of welfare improvement). For every K, there exists
(K) > 0 such that
ðCðKÞ; IðKÞÞ maximizes ðKÞUðCÞ þ rWðKÞI subject to ðC; IÞ 2 SðKÞ:
We have observed in the preceding section that Property 2 holds when discounted utilitarianism and (under regularity conditions) maximin are implemented; (K) can be interpreted as a supporting utility discount rate in these cases.
In all our examples, we show that Properties 1 and 2 hold for resource
allocation mechanisms that are optimal in the sense that, for any initial
stocks, they implement paths that are weakly preferred to any feasible
path according to the social preferences. We conjecture that Properties 1
and 2 are necessary for optimal resource allocation if the social preferences
and the technological environment satisfy the following condition: there
does not exist an alternative path that, compared to an optimal path, has
higher utility in an initial period, at the end of which the alternative path is
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G. B. Asheim and W. Buchholz
deemed as good as the optimal path.4 The investigation of such a primitive
condition on preferences and technology seems, however, to require a discrete
time framework and, thus, falls outside the scope of the present analysis.
By writing W*(t): ¼ W(K*(t)) for the welfare level along {C*(t),I*(t),K*(t)},
so that W*(t) ¼ rW(K*(t))I*(t), we obtain the following result.
Lemma 2. If Properties 1 and 2 hold, then at every t,
W_ ðtÞ ¼ ðK ðtÞÞCðtÞI ðtÞ:
Proof: Since U is concave and S(K) is convex and smooth, there is a unique
n-dimensional hyperplane that supports the set of feasible (n þ 1)-dimensional utility-investment vectors. By comparing the maximum principle with
Property 2, we can conclude that
rWðK ðtÞÞ ¼ ðK ðtÞÞCðtÞ
and, thus, W_ *(t) ¼ rW(K*(t)I*(t)) ¼ (K*(t))C(t)I*(t)) hold at every t.
&
Lemma 2 shows that the sign of the value of net investments along the
implemented path indicates whether welfare is increasing.5
The main result of this section follows from Lemmas 1 and 2.
Proposition 1. If Properties 1 and 2 hold, then dynamic welfare is increasing if and only if there is growth in U(C*(t)) þ C(t)I*(t).
Thus, changes in dynamic welfare according to the unspecified aggregation of
the interests of different generations are revealed through changes in utility NNP.
Proposition 1 is a result for local-in-time comparisons along the implemented path. It does not imply that utility NNP represents dynamic welfare
4
In the case of maximin, this condition holds if maximin paths are regular, while it can fail otherwise, for instance in a one-sector model where the initial capital stock exceeds the golden rule level.
The maximin criterion illustrates that Property 2 does not necessarily imply a linear trade-off
between current utility and welfare improvement.
Properties 1 and 2 can hold even if the resource allocation mechanism does not implement an
optimal path. For example, suppose that society adheres to discounted utilitarianism in a
technology where implementation of discounted utilitarianism would have led to non-constant
utility, but, in fact, a regular maximin path is implemented. Both Properties 1 and 2 hold under
such circumstances.
5
Onuma (2003) has independently derived a result that is similar to our Lemma 2. The result
in his Proposition 3 is based on a property called generation rationality under a bequest
constraint, which corresponds to our Property 2.
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Welfare measurement through national income accounting
371
globally (i.e., welfare is greater if and only if utility NNP is greater), which
would entail that
WðK ðtÞÞ ¼ UðC ðtÞÞ þ CðtÞI ðtÞ
holds at each t. To shed light on the problems involved, note that
W_ * ¼ (K*)CI* (by Lemma 2), and d ðUðC Þ þ CI Þ=dt ¼ ð_ =ÞCI
(by Lemma 1). Hence, the combination of ðK Þ ¼
6 _ = and CI* ¼
6 0
precludes that utility NNP can represent welfare globally. As shown in Section
III, it works for discounted utilitarianism because ðK Þ ¼ _ =, and it
works for maximin because CI* ¼ 0. However, in general we must allow
for cases where ðK Þ ¼
6 _ = is combined with CI* ¼
6 0 and, thus,
W(K*) ¼ U(C*) þ CI* cannot hold at each t. Indeed, Proposition 3 in Section
VI introduces a case where global representation of welfare by means utility
NNP is not possible.
V. Real NNP Growth and Local Comparisons
So far we have considered NNP and the value of net investments in utility
terms. Utility, however, is not observable directly, while market prices in
principle are. Comprehensive NNP that is measurable by market prices is
often identified in the literature by the ‘‘linearized’’ Hamiltonian as the sum
of the value of consumption and the value of net investments, measured in
monetary units; cf. Hartwick (1990). To demonstrate that welfare is increasing locally over time along the implemented path if and only if such
measurable NNP is also increasing, we now adapt the analysis of Asheim
and Weitzman (2001) to the present more general setting.
What may be observed directly at each time t are nominal prices for
consumption goods and investment flows, p(t) ¼ rU(C*(t))/(t) and
q(t) ¼ C(t)/(t), where (t) > 0 is the continuous not-directly-observable
marginal utility of current expenditures. Comprehensive NNP in nominal
prices, y(t), is then defined by:
yðtÞ :¼ pðtÞC ðtÞ þ qðtÞI ðtÞ:
If Property 1 holds, then y(t) ¼ max(C,I)2S(K*(t))p(t)C þ q(t)I.
By Lemma 2, dynamic welfare is increasing if and only if NNP in
nominal prices exceeds the value of consumption. However, since NNP in
nominal prices at t depends on (t), and (t) is arbitrary, y_ (t) > 0 need not
signify welfare improvement. In order for a change in NNP to indicate a
change in welfare, NNP must be measured in real prices. The application of
a price index {p(t)} turns nominal prices {p(t), q(t)} into real prices {P(t),
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G. B. Asheim and W. Buchholz
Q(t)} by imposing P(t) ¼ p(t)/p(t) and Q(t) ¼ q(t)/p(t) at each t. Following
Asheim and Weitzman (2001), we use a Divisia consumption price index,
p_ ðtÞ p_ ðtÞC ðtÞ
¼
;
pðtÞ pðtÞC ðtÞ
implying that p(t) is continuous and P_ C* ¼ 0 holds. Comprehensive NNP in
real Divisia prices, Y(t), is then defined by:
YðtÞ :¼ PðtÞC ðtÞ þ QðtÞI ðtÞ:
Lemma 3. If Property 1 holds, then Y(t) is continuous and
Y_ ðtÞ ¼ RðtÞðYðtÞ PðtÞC ðtÞÞ
holds at almost every t, where the real interest rate, R(t), at time t is given by
RðtÞ ¼ _ ðtÞ=ðtÞ _ ðtÞ=ðtÞ p_ ðtÞ=pðtÞ.
Proof: It follows from the continuity of U(C*) þ CI* (cf. Lemma 1) that Y is
continuous since U has continuous partial derivatives and both and p are
continuous. Otherwise, adapt the proof of Asheim and Weitzman (2001,
Proposition 3). &
Since Lemma 3 entails that
change in real NNP ¼ real interest rate value of net investments
and, by Lemma 2, a positive value of net investments indicates welfare
improvement, we obtain the main result of Asheim and Weitzman (2001) in
our generalized setting.
Proposition 2. Provided that Properties 1 and 2 hold and the real interest
rate is positive, dynamic welfare is increasing if and only if there is growth
in measurable NNP in real Divisia prices.
Proof: The result follows from Lemmas 2 and 3 since:
ðK ÞCI ¼ ðK ÞpQI ¼ ðK ÞpQðY PC Þ;
where (K*), and p are all positive.
&
As noted by Asheim and Weitzman (2001), real NNP growth indicates welfare
improvements locally over time. Unless real NNP grows in a monotone
manner between t0 and t00 , it does not necessarily follow that a higher real
NNP at t00 than t0 indicates that welfare is higher at t00 compared to t0 .
# The editors of the Scandinavian Journal of Economics 2004.
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373
VI. Progress and Sustainability in a Resource Model
We now use the Dasgupta–Heal–Solow (DHS) model of capital accumulation and resource depletion to illustrate the applicability of our framework.
In the DHS model, a stock of man-made capital (KM) is combined with
extracted raw material from a stock of a natural resource (KN) to produce
output that can be split between consumption and investment. For tractability, we assume that the production function is Cobb–Douglas and exhibits
CRS, implying that the consumption–investment pair (C, IM, IN) is attainable
given (KM, KN) if and only if
a
C þ IM 4KM
ðIN Þb ; b < a < a þ b ¼ 1;
where C 5 0, IN 4 0, KM 5 0 and KN 5 0. The assumption that b < a is
required to ensure that progress and sustainability are feasible in the present
setting.
Consider paths for which C, IN, KM and KN remain positive throughout,
so that smoothness of the attainable set is satisfied. Let the ratio between
man-made capital and output be denoted by :
¼
KM
a
KM
ðIN Þb
¼
KM
IN
b
:
If the implemented path satisfies Property 1, then the real interest rate
along the path measures the marginal productivity of KM and is given by
R(t) ¼ a/*(t), where *(t) is the capital–output ratio along the
implemented path at time t. Moreover, the real investment prices are
given by
QM ðtÞ ¼ 1
a
and QN ðtÞ ¼ b ðtÞb ;
ð3Þ
since, with output as numéraire, QN(t) measures the marginal productivity of
IN. The Hotelling rule for short-run efficiency yields Q_ N(t)/QN(t) ¼ R(t),
implying
_ ðtÞ ¼ b:
ð4Þ
If, in addition to (4), the following transversality conditions are satisfied,
lim
KM
ðtÞ
t!1 b
a
ðtÞb
¼ 0 and
lim KN ðtÞ ¼ 0;
t!1
ð5Þ
# The editors of the Scandinavian Journal of Economics 2004.
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G. B. Asheim and W. Buchholz
then routine calculations show that, by
U(C) ¼a C,
Property 1 holds.
R 1setting
The implemented path maximizes 0 1= b ðtÞb CðtÞdt over all
0
feasible paths, for any initial stocks ðKM
; KN0 Þ 0.
Progress Paths
Combine efficiency (Property 1) with an exogenous investment rule,
a
IM ¼ KM
ðIN Þb ; b < < a;
ð6Þ
which leads to sustained progress; cf. Hamilton (2002) for an independent
and similar investigation. We now apply our revealed welfare analysis (by
invoking Property 2) and show how real NNP growth picks up the welfare
improvement entailed by sustained consumption growth. We confirm this
welfare result by establishing that the investment rule is optimal under
undiscounted utilitarianism. We observe that—even though growth in measurable real NNP measures welfare improvement locally over time—utility
NNP cannot represent welfare globally.
a
Note that bKM
ðIN Þb equals resource rents; i.e., the share of output
that is attributable to extraction of raw material. According to Hartwick’s
rule, reinvesting resource rents forever leads to constant consumption; cf.
Hartwick (1977) and Dixit et al. (1980). Since > b, more than resource
rents are reinvested by following (6), thereby leading to progress in the sense
that consumption increases in a sustained manner. Since < a, feasibility of
the implemented path is ensured.
The efficiency conditions (4)–(5) and the investment rule (6) determine a
resource allocation mechanism specifying C, IM and IN as functions of capital
stocks, (KM, KN). It follows from the definition of and the investment rule
(6) that these functions can, for positive capital stocks, be described by
KM
CðKM ; KN Þ ¼ ð1 Þ KKMN
ð7Þ
KM
IM ðKM ; KN Þ ¼ KKMN
ð8Þ
KM
IN ðKM ; KN Þ ¼ 1 ;
b
KKMN
# The editors of the Scandinavian Journal of Economics 2004.
ð9Þ
Welfare measurement through national income accounting
375
where, by imposing (4) and the KN part of (5) as efficiency conditions, we
can calculate the capital–output ratio as an explicit function of KM/KN,
ba
KM
ba KM
;
¼ ða Þ
KN
KN
ð10Þ
and check that the KM part of (5) is satisfied. For given initial stocks
0
ðKM
; KN0 Þ 0 at time 0, equations (8)–(10) determine the implemented
path of capital stocks, fKM
ðtÞ; KN ðtÞg, which in turn yields the implemented
paths of consumption and investment flows: C ðtÞ ¼ CðKM
ðtÞ; KN ðtÞÞ,
IM ðtÞ ¼ IM ðKM ðtÞ; KN ðtÞÞ, and IN ðtÞ ¼ IN ðKM ðtÞ; KN ðtÞÞ. By combining
(7) and (8) with (4), we can establish that consumption grows at a positive
(but decreasing) rate since > b:
C_ ðtÞ b
¼
> 0:
C ðtÞ ðtÞ
This is consistent with the results obtained by Mitra (1983) in a general
discrete-time DHS model; see in particular his Example 3.A. Moreover, by
combining (3) with (10), it follows that the relative price of natural capital in
terms of man-made capital is positively related to , the parameter that
indicates society’s emphasis on progress:
QN ðtÞ
b
K ðtÞ
M
¼
:
QM ðtÞ a KN ðtÞ
ð11Þ
By assuming that the implemented path does not waste opportunity for
welfare improvement—i.e., by adding Property 2—Proposition 2 implies that
welfare is increasing if and only if there is real NNP growth, where real NNP
can be written as
C ðtÞ þ QM ðtÞIM
ðtÞ þ QN ðtÞIN ðtÞ ¼ a KM
ðtÞ
ðtÞ
due to the constant factor shares. It follows from (4) and (8) that the growth
rate of NNP equals that of consumption. Thus, the revealed welfare analysis
captures that consumption increases in a sustained manner.
By Lemma 3, increased welfare can also be indicated by a positive value
of net investments: QM ðtÞIM
ðtÞ þ QN ðtÞIN ðtÞ > 0. Since (8)–(10) imply
IN ðKM ; KN Þ
a KN
;
¼
IM ðKM ; KN Þ
KM
ð12Þ
# The editors of the Scandinavian Journal of Economics 2004.
376
G. B. Asheim and W. Buchholz
and > b, (11) implies that welfare is improving along the implemented
path:
QM ðtÞIM
ðtÞ þ QN ðtÞIN ðtÞ ¼ IM
ðtÞð1 b=
Þ > 0:
Property 2 (cf. the proof of Lemma 2) entails that any welfare index
W(KM, KN) satisfies
@WðKM ; KN Þ
CN
QN
b
KM
@KN
¼
¼
¼
:
@WðKM ; KN Þ CM QM a KN
@KM
ð13Þ
A direct consequence of (13) is that welfare can be represented by
a
b
KN :
WðKM ; KN Þ ¼ KM
ð14Þ
Moreover, it follows from (12) that the implemented path in (KM, KN)a
space is described by KM
KN being constant, with K_ M ¼ IM > 0 and
_
K N ¼ IN < 0. That welfare is improving along the implemented path can now be
seen alternatively by comparing the iso-welfare contours given by (14) with the
contour that describes the implemented path in (KM, KN)-space.
Are there explicitly specified social preferences such that this resource
allocation mechanism, for any vector of initial stocks, implements a most
preferred path? This question may be answered by observing that the
resource allocation mechanism can be derived from the utilitarian problem
of maximizing, without discounting,
Z
1
1
CðtÞ
b dt
ð15Þ
0
over all feasible paths. It follows from Dasgupta and Heal (1979, pp.
303–308) that an undiscounted utilitarian optimum exists if (1 )/(
b) >
(1 a)/(a b), which implies b < < a. This is exactly the assumption
we have made.
Now, rank positive initial stocks, (KM, KN), by the maximum value of the
integral they give rise to in (15). We find that this ranking can be represented
by (14). This confirms the result already derived through our revealed
welfare analysis: that welfare is increasing along the implemented path.
Under discounted utilitarianism, utility NNP represents dynamic welfare
globally; cf. equation (2) in Section III as well as Weitzman (1970, 1976).
Moreover, under maximin, the converse of Hartwick’s rule implies that
# The editors of the Scandinavian Journal of Economics 2004.
Welfare measurement through national income accounting
377
utility NNP is equal to the constant utility level and therefore represents
welfare globally. For the progress paths that we analyze here in the context
of the DHS model, however, it turns out that utility NNP cannot represent
welfare globally.
Proposition 3. Consider the resource allocation mechanism determined by
(7)–(10) in the context of the DHS model. There exists no utility function
such that net national product in terms of utility represents dynamic welfare
globally.
Proof: It follows from (6) and the constant factor shares that
QM ðtÞIM
ðtÞ þ QN ðtÞIN ðtÞ ¼ ð
bÞC ðtÞ=ð1 Þ. Since CM (t) ¼ U0 (C*(t))
QM (t) and CN (t) ¼ U0 (C* (t))QN (t), this implies
Utility NNP ¼ UðCðKM ; KN ÞÞ þ U 0 ðCðKM ; KN ÞÞ
b
CðKM ; KN Þ
1
for an arbitrary U function, when the pair of capital stocks is (KM, KN).
Assume that utility NNP represents welfare globally. Then utility NNP must
be invariant when moving along any iso-welfare contour defined by (14):
b
b
@C
@C
0
0
00
U ðCÞ þ U ðCÞ
þ U ðCÞ
C
dKM þ
dKN ¼ 0:
1
1
@KM
@KN
Since C(KM, KN) increases when moving along an iso-welfare contour by
increasing KM and decreasing KN, the second parenthesis is non-zero and
U0 (C)a/(
b) þ U00 (C)C ¼ 0 must hold. Hence, U is in the class of affine
transformations of
1
UðCÞ ¼ C
b :
ð16Þ
Since the implemented path maximizes (15), the supporting utility discount
rate is zero throughout for any utility function in this class. Under these
circumstances, Lemma 1—extrapolated to the case with a zero utility discount rate—implies that utility NNP does not change as a consequence of
non-zero value of net investments. Indeed, it follows that utility NNP is zero
for any pair of initial stocks if the utility function is given by (16). Hence,
utility NNP has no welfare significance within the class of utility functions
that are affine transformations of (16). &
Property 1 does not hold for any utility function in the class considered in
the proof of Proposition 3, i.e., that is an affine transformation of (16). If we
instead use a utility function so that Property 1 holds for a path of supporting
utility discount factors {(t)} with discount rates _ ðtÞ=ðtÞ that are
# The editors of the Scandinavian Journal of Economics 2004.
378
G. B. Asheim and W. Buchholz
positive (U(C) ¼ C is an example), it follows from the analysis of Sefton and
Weale (1996) that
Utility NNP ¼
Z
0
1
_ ðtÞ ðtÞ
UðC ðtÞÞdt:
ðtÞ ð0Þ
Hence, utility NNP is a weighted average of future utility. However, by
following a given iso-welfare contour defined by (14) as KM ! 1 and KN !
0, and considering the consumption paths that would be implemented for
these initial conditions, it can be shown that the minimal consumption (which
occurs at time 0) along these paths goes to infinity. This implies that
the constant welfare along such an iso-welfare contour cannot be expressed
as a weighted average of future utility.
However, even though utility NNP cannot represent dynamic welfare
globally, Proposition 2 implies that growth in measurable NNP in real prices
measures welfare improvement locally over time, even in the current
setting. This illustrates the generality of the positive result that we report
in Proposition 2.
Sustainability as a Constraint
Consider a society that deems unsustainable development unacceptable, and
which adopts a resource allocation mechanism that, among the acceptable
sustainable paths, implements the path that maximizes the sum of discounted
utilities
Z
1
et UðCðtÞÞdt;
ð17Þ
0
where is a positive and constant utility discount rate.6 We confirm that
Properties 1 and 2 hold and show that measurement of welfare improvement
through real NNP growth can be useful for managing the assets of society.
Real NNP growth approaching zero indicates that unconstrained development is no longer sustainable.
A consumption path is said to be sustainable if, at all times, current
consumption does not exceed the maximum sustainable consumption level
given the current capital stocks. Since unconstrained maximization of (17) in
the DHS model leads to consumption converging to zero as time goes to
6
Asheim, Buchholz and Tungodden (2001) present ethical axioms under which only
sustainable paths are acceptable in the DHS model. Discounted utilitarian paths under a
sustainability constraint in the DHS model are analyzed in continuous time by Asheim
(1986) and Pezzey (1994), and in discrete time by Asheim (1988).
# The editors of the Scandinavian Journal of Economics 2004.
Welfare measurement through national income accounting
379
infinity, the sustainability constraint imposed on the implemented path is
binding. Since, by (4), the real interest rate R(t) ¼ a/*(t) is decreasing along
any efficient path, the sustainability constraint binds in an eventual phasewith constant consumption, which can possibly be preceded by an unconstrained utilitarian phase with increasing consumption. For given initial stocks
0
ðKM
; KN0 Þ 0, the implemented path fC ðtÞ; IM
ðtÞ; IN ðtÞ; KM
ðtÞ; KN ðtÞg
can be determined by maximizing (17) subject to the constraint that
consumption is non-decreasing; standard arguments imply that such a path
exists.
Therefore, if, in this example, social preferences over paths are represented by (17) on the set of non-decreasing consumption paths, while paths
that are not non-decreasing are strictly less preferred, then it follows that the
resource allocation mechanism defined above implements a most preferred
path also in this case. Since implemented paths are non-decreasing, welfare
0
WðKM
; KN0 Þ can be associated with the utility level that, if held constant, is
equally as good when evaluated by (17):
0
; KN0 Þ
WðKM
R1
¼
0
et UðC ðtÞÞdt
R1
¼
t dt
0 e
Z
1
et UðC ðtÞÞdt:
0
For simplicity, assume constant elasticity of marginal utility; i.e., for all
C > 0, (U00 (C)C)/U0 (C) ¼ > 0. Then the resource allocation mechanism
becomes homogeneous of degree 1 since production exhibits CRS. In particular, the capital–output ratio is a function of KM/KN, and the dividing line
between the sustainability unconstrained and constrained regimes is a ray in
(KM, KN)-space. Consumption increases if and only if the infinitely long-term
real interest rate exceeds . This rate equals the inverse of the value of a
perpetual bond; hence, it is the discounted average of the instantaneous real
interest rate, a/* (s), from time t onward:
R 1
t
a
1= b ðsÞb ða= ðsÞÞds a b
¼ :
R1 a
ðtÞ
ðsÞb ds
1=
b
t
Note that the infinitely long-term interest rate (a b)/*(t) is smaller than
the instantaneous rate a/*(t) since the latter is decreasing throughout.
In the eventual sustainability constrained phase, the resource allocation
mechanism implements efficient paths with constant consumption, implying
that the resource allocation mechanism is described by (7)–(10)
with ¼ b. Since this phase is entered when the infinitely long-term interest
# The editors of the Scandinavian Journal of Economics 2004.
380
G. B. Asheim and W. Buchholz
rate (a b)/*(t) equals , it follows from (10) that paths are in the unconstrained discounted utilitarian phase if
1
KM
ab b
<
;
KN
and in the eventual sustainability constrained phase otherwise.
For tractability, assume also that U (C) ¼ Cb so that ¼ a. Then the
unconstrained discounted utilitarian phase is characterized by (7)–(9) with
¼ ðÞ ¼ 1 a
a2
b2
e
ðabÞ
ab
;
where (4) implies that is an increasing function of KM/KN for KM/KN <
((a b))1/b since K_ M ¼ IM > 0 and K_ N ¼ IN < 0, and continuous at
KM/KN ¼ ((a b))1/b since *(t) is differentiable w.r.t. time. Note that is a decreasing function of (and by (4) of time) and converges to b as approaches the value (a b)/ at which time paths enter into the sustainability
constrained phase. Hence, output, C, IM and IN are throughout continuous
functions of (KM, KN) and time.
0
Fix the initial stocks ðKM
; KN0 Þ 0 and consider the implemented path
determined by the resource allocation mechanism described above. Let denote the time at which the implemented path enters the eventual sustainability constrained phase. Set ¼ 0 if the path starts in this phase, i.e., if
0
KM
=KN0 5ðða bÞ=Þ1=b . We can now verify that this path does indeed
satisfy Properties 1 and 2 and maximize (17) subject to consumption being
non-decreasing.
Property 1 holds for the continuous path of supporting utility discount
factors {(t)} determined (up to the choice of numéraire) by
ðtÞ
ðtÞ
ab
R1
for t 2 ½; 1Þ; ð18Þ
¼ for t 2 ½0; Þ and R 1
¼ ðtÞ
t ðsÞds
t ðsÞds
implying that utility discount rates are positive: _ ðtÞ=ðtÞ ¼ for t 2 (0, )
and _ ðtÞ=ðtÞ ¼ a= ðtÞ for t 2 (,1). This can be seen by choosing a
path of investment prices in terms of utility {M (t), N(t)} so that the
current-value Hamiltonian is maximized at any point in time:
CM ðtÞ ¼ U 0 ðC ðtÞÞ QM ðtÞ ¼ bC ðtÞa
a
CN ðtÞ ¼ U 0 ðC ðtÞÞ QN ðtÞ ¼ bC ðtÞa b ðtÞb :
Then the co-state differential equations hold,
# The editors of the Scandinavian Journal of Economics 2004.
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381
_ ðtÞ
a
_ M ðtÞ
CM ðtÞ C
CM ðtÞ ¼ ðtÞ
ðtÞ
_ ðtÞ
_ N ðtÞ;
CN ðtÞ C
0¼
ðtÞ
and the consumption path satisfies Ramsey’s rule:
a
ðtÞ
¼
_ ðtÞ
C_ ðtÞ
þa :
ðtÞ
C ðtÞ
ð19Þ
Since _ ðtÞ=ðtÞ jumps from ¼ (a b)/*() to a/*() when the sustainability constrained phase is entered, it follows from (19) that the rate of
consumption growth decreases abruptly from b/(a*()) to 0 at that time.
The following result (the proof of which is available from the authors)
establishes formally that (17) is maximized subject to consumption being
non-decreasing.
0
Lemma 4. For any initial stocks ðKM
; KN0 Þ 0, the path implemented
by the
R1
resource allocation mechanism described above maximizes 0 et CðtÞb dt
over all feasible non-decreasing consumption paths.
To apply the revealed welfare analysis of Sections IV and V, we have to
show that Property 2 holds. It follows from (18) that
ðtÞ
R1
¼ et for t 2 ½0; Þ and
ðsÞds
0
R1
Z 1
ðsÞds
R1
es ds:
¼
ðsÞds
0
Hence, since consumption is constant in the eventual sustainability con0
strained phase, WðKM
; KN0 Þ can be rewritten as follows:
Z
R1
ðtÞC ðtÞb dt
R1
:
0
0 ðtÞdt
R1
Since (CM(0),CN(0)) is the vector of partial derivatives of 0 ðtÞC ðtÞb dt
w.r.t. the initial stocks, we obtain by invoking the envelope theorem that
0
; KN0 Þ
WðKM
¼
1
t
e
b
C ðtÞ dt ¼
0
0
0
@WðKM
; KN0 Þ
ð0Þ
@WðKM
; KN0 Þ
ð0Þ
¼ R1
¼ R1
CM ð0Þ and
CN ð0Þ:
@KM
@KN
0 ðtÞdt
0 ðtÞdt
R 1
0
; KN0 Þ ¼ ð0Þ 0 ðtÞdt ,
Hence, Property 2 holds since, by setting ðKM
the maximum principle implies that ðC ð0Þ; IM
ð0Þ; IN ð0ÞÞ maximizes
# The editors of the Scandinavian Journal of Economics 2004.
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G. B. Asheim and W. Buchholz
0
ðKM
; KN0 ÞCb þ
0
0
@WðKM
; KN0 Þ
@WðKM
; KN0 Þ
IM þ
IN
@KM
@KN
over all attainable consumption–investment pairs.
Proposition 2 is therefore applicable and welfare is increasing if and only
if there is growth in real NNP, aKM
ðtÞ= ðtÞ. Since (4) implies that the
growth rate of real NNP equals (
() b)/, welfare is increasing as long as
the path remains in the unconstrained utilitarian phase, during which
() > b. Since () reaches b at the point in time at which the sustainability
constraint becomes binding, the observation that the growth rate of real
NNP decreases towards zero indicates that unconstrained development is no
longer sustainable. Hence, the information about welfare changes offered by
the growth rate of real NNP is useful for managing the assets of society,
given that unsustainable paths are deemed socially unacceptable. Note that
consumption yields no such indication, since the rate of consumption growth
falls discontinuously to zero at the time the path enters the sustainability
constrained phase.
By Lemma 3, increased welfare can also be indicated by a positive value
of net investments,
QM ðtÞIM
ðtÞ þ QN ðtÞIN ðtÞ ¼ IM
ðtÞð1 b=
ð ðtÞÞÞ:
Again, () > b during the unconstrained utilitarian phase implies that welfare is increasing, while the observation that the value of net investments
decreases towards zero as () approaches b indicates that unconstrained
development is no longer sustainable. Thus, the sign of the value of net
investments is also useful for asset management.
NNP growth (and the value of net investments) indicate when the sustainability constraint becomes binding precisely because policies that implement
sustainable development are expected and, hence, reflected in the ratio of
investment prices. Sustainability cannot be indicated in this way if, instead,
an unconstrained utilitarian path is expected to be followed throughout; cf.
Asheim (1994) and Pezzey (1994).
VII. Concluding Remarks
We have established that real NNP growth—or equivalently, a positive value
of net investments—can be used to indicate welfare improvement, independently of the welfare criterion adopted, in a constant population society with
comprehensive national accounting. Provided it holds that the implemented
policies lead to an efficient path that does not waste opportunity for welfare
improvement, the underlying—but unspecified and unobservable—welfare
# The editors of the Scandinavian Journal of Economics 2004.
Welfare measurement through national income accounting
383
judgments are revealed through prices and quantities that are available in a
perfect market economy. In such a revealed welfare approach, we do not
need to know the actual social preferences when drawing welfare conclusions on the basis of national accounting aggregates.
We have thus shown that the result of Asheim and Weitzman (2001)—
namely that increasing measurable NNP in real Divisia prices indicates
welfare improvement in a multiple consumption good setting even when
utility itself is not measurable—holds even in situations where society does
not subscribe to discounted utilitarianism. Our analysis has also covered
circumstances where, for example, progress and sustainability are important
concerns. We have exemplified this in Section III by showing that, in
general, maximin is encompassed by the present approach, and in Section
VI by considering two resource allocation mechanisms in the Dasgupta–
Heal–Solow model of capital accumulation and resource depletion, one that
implements undiscounted utilitarianism and one that maximizes the sum of
discounted utilities within the subset of sustainable paths. In the latter case,
real NNP growth approaching zero indicates that unconstrained development
is no longer sustainable.
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First version submitted October 2002;
final version received July 2003.
# The editors of the Scandinavian Journal of Economics 2004.