Japan and the World Economy
16 (2004) 331–335
Discussion
Perspectives on Weitzman’s stationary equivalent
Discussion of Kazuo Mino’s ‘‘Weitzman’s
Rule with Market Distortions’’1
Geir B. Asheim*
Department of Economics, University of Oslo, P.O. Box 1095 Blindern, NO-0317 Oslo, Norway
1. Introduction
Weitzman’s Rule—as established by Martin Weitzman in his seminal contributions
(Weitzman, 1970, 1976)—is the point of departure for the well-written and interesting
paper by Kazuo Mino. Weitzman’s Rule can be stated as follows. Under
a stationary technology, in the sense that the set of feasible utility–investment–capital
triples2
ðu; z; kÞ;
does not depend directly on time t, an assumption that is associated with the term
‘‘green’’ accounting, signifying that the vector of current capital stocks captures all
variable determinants of current productive capacity;
discounted utilitarianism, in the sense that the dynamic welfare of the economy at
time t is given by
Z 1
erðstÞ uðsÞ ds;
t
where r is a positive utility discount rate;
optimality, in the sense that the economy has an optimal resource allocation mechanism which implements a path that maximizes dynamic welfare;
1
Based on comments given at the Technical Symposium ‘‘Economic Conservation Laws and Optimizing
Behavior of Individuals: Conventional and Differential Geometric Approaches,’’ held at Center for Japan–US
Business and Economic Studies, Stern School of Business, New York University, 4 April 2003.
*
Tel.: þ47-22855498; fax: þ47-22855035.
E-mail address: g.b.asheim@econ.uio.no (G.B. Asheim).
2
Weitzman (1976) as well as Mino assume that u (which Weitzman, 1976, refers to as composite
‘‘consumption’’) is an observable measure of cardinal utility.
0922-1425/$ – see front matter # 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.japwor.2003.12.002
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G.B. Asheim / Japan and the World Economy 16 (2004) 331–335
it holds that
YðtÞ ¼ uðtÞ þ pðtÞzðtÞ ¼ r
Z
1
erðstÞ uðsÞ ds;
(1)
t
where p(t) is the vector of investment prices in terms of utility. Eq. (1) states the
Hamiltonian, Y(t)—consisting of utility plus the value of net investments in utility
terms—equals the utility discount rate, r, times dynamic welfare under discounted
utilitarianism. Hence,
‘‘utility income’’¼‘‘utility interest rate’’ ‘‘utility wealth’’:
Mino’s paper poses the following questions. How to adjust Weitzman’s Rule if
1. the assumption of a stationary technology is relaxed due to exogenous technical
change; or
2. the assumption of optimality is relaxed due to externalities, arising either from
spillovers from knowledge capital, or from constrained policy intervention.
2. A fundamental conservation law
Following the work of Paul Samuelson, Ryuzo Sato has spurred an interest in uncovering
economic conservation laws or invariance principles in dynamic models (see Sato, 1981,
1985, and some of his later contributions). Let me use this opportunity to suggest that such
conservation laws or invariance principles are most easily uncovered if they are expressed
in terms of present value prices. In particular, under a stationary technology, discounted
utilitarianism, and optimality it holds that
_ þ
ert uðtÞ
d rt
ðe pðtÞzðtÞÞ ¼ 0;
dt
(2)
i.e., the change in utility valued in present value prices plus the change in the present
value of net investments is equal to zero. By integrating, we get that the present value of
utility changes from 0 to t plus the present value of net investments at t does not vary
with time, and is thus conserved. It is straightforward to see that Eq. (2) is identical
to Weitzman (1976) Eq. (14), which has also appeared prominently in Ryuzo Sato’s
work.
Through their Theorem 1, Dixit et al. (1980) show that discounted utilitarianism is not
needed for Eq. (2). Rather, it holds along any efficient path that is supported by a path of
rt
present value prices of utility, fmðtÞg1
in Eq. (2), even
t¼0 , that m(t) can be substituted for e
when m(t) is not an exponentially decreasing function. Then it follows directly that utility,
u(t), is constant if and only if the present value of net investments, m(t)p(t)z(t), is constant, a
result that Dixit et al. (1980) refer to as a ‘‘generalized Hartwick’s Rule’’ (since it
generalizes the result of Hartwick, 1977).
Furthermore, in Ramsey (1928) analysis of optimal saving, then—since his application
of undiscounted utilitarianism means that utility discount rate is zero—it follows directly
from Eq. (2) that the Hamiltonian is constant.
G.B. Asheim / Japan and the World Economy 16 (2004) 331–335
333
In the present setting, it is important to notice how Eq. (2) is the key to establishing
Weitzman’s Rule. First, differentiate the present value of utility w.r.t. time:
d rt
_
ðe uðtÞÞ ¼ r ert uðtÞ þ ert uðtÞ:
dt
By integrating on both sides of Eq. (3), and rearranging terms, we obtain that
Z 1
Z 1
_ dt;
r
ert uðtÞ dt ¼ uð0Þ þ
ert uðtÞ
0
(3)
(4)
0
a result that depends only on the constant utility discount rate that discounted utilitarianism
entails.
R 1Byrtalso invoking a stationary technology and optimality, it follows from Eq. (2) that
_
uðtÞ—the
present value of future changes in utility—is measured by p(0)z(0)—the
0 e
value of net investments—provided that limt!1 ert pðtÞzðtÞ ¼ 0 holds as a investment
value transversality condition. Hence, in combination with Eq. (4), this leads to Weitzman’s Rule:
Z 1
r
ert uðtÞ dt ¼ uð0Þ þ pð0Þzð0Þ:
(WR)
0
However, if both the assumptions of Ra stationary technology and optimality do not
1
_
apply, then p(0)z(0) will not fully reflect 0 ert uðtÞ.
So in models where one or both of
these
assumptions
cannot
be
made,
what
must
be
added
to p(0)z(0) in order to measure
R 1 rt
_
uðtÞ?
This is the main question posed by Mino in his paper.
0 e
3. Comments and questions
This is a nicely written paper that raises important questions. However, similar kinds of
forward-looking terms have also been calculated in some earlier contributions, including
Aronsson et al. (1997) book and a series of articles by Aronsson and Löfgren as well as,
e.g., the contribution by Vellinga and Withagen (1996). On this background, it is helpful for
the reader that the author signals in his introduction what the original results of the present
paper are. Still, an even more detailed comparison with the relevant literature would have
been interesting.
Mino’s paper contains an intriguing numerical example, showing that a technical
progress premium, as calculated by Weitzman (1997) in the case of exogenous technical
change, can also be estimated when technical progress is caused by spillovers from
knowledge capital. It seems that the size of the premium does not hinge on whether
technical progress is exogenous or endogenous; rather, it depends in a non-trivial way on
preferences and capital depreciation.
In the part on optimal policy intervention, q(t), the social value of investment
flows, is different from p(t), the private value of investment flows, due to policy
constraints. The analysis thereby combines in an interesting manner an inefficient
resource allocation mechanism (in the sense of Arrow et al., 2003) with constrained
optimality.
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G.B. Asheim / Japan and the World Economy 16 (2004) 331–335
The serious empirical problems associated with measuring utility are not addressed.
However, the paper’s lack of emphasis on this issue can be defended, as the paper here
follows a tradition that started with Weitzman (1976) seminal article. Recently, Weitzman
(2001) has provided methods for satisfying the informational demands that the assumption
of measurable utility involves; see also my discussion of this topic in Asheim (2003,
Section 6).
4. Sustainability
It follows from Eq. (4) that
Z 1
Z 1
Z
rt
rt
e uðtÞ dt ¼
e
uð0Þ þ
0
0
0
0
1
e
rs
_ ds dt:
uðsÞ
(5)
0
R1
_ ds is a Hicks–Weitzman stationary equivalent. Moreover, we
Hence, uð0Þ þ 0 ers uðsÞ
have that if fuðtÞg1
t¼0 is (constrained) optimal under discounted utilitarianism, then
Z 1
Z 1
ert uðtÞ dt ert uð0Þ dt;
(6)
where uð0Þ denotes the maximum sustainable utility, given the vector of capital stocks
that are available at time zero. Because otherwise, the optimality of fuðtÞg1
t¼0 would have
been contradicted (cf. Pezzey, 2004). Hence, under (constrained) discounted utilitarian
optimality,
Z 1
_ ds
uð0Þ þ
ers uðsÞ
0
is an upper bound for sustainable utility at time zero. In particular, it need not itself be
sustainable
R 1 (cf. Asheim, 1994).Therefore, it is nice that Mino attempts to avoid referring to
_ ds as ‘‘sustainable utility,’’ as done by other contributors to this
uð0Þ þ 0 ers uðsÞ
literature. Rather, the terminology
adopted better reflects the fact that it might not be
R1
_ ds indefinitely.
feasible to maintain uð0Þ þ 0 ers uðsÞ
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