Journal of Economic Dynamics and Control
23 (1998) 159—165
Characterizing sustainability: The converse of
Hartwick’s rule
Cees Withagen!,*, Geir B. Asheim"
! Department of Spatial Economics, Free University Amsterdam, De Boelelaan 1105,
1081 HV Amsterdam, The Netherlands and Tilburg University, Tilburg, The Netherlands
" Department of Economics, University of Oslo, 0317 Oslo, Norway
Received 7 February 1997; accepted 11 September 1997
Abstract
This note offers a general proof of the converse of Hartwick’s rule, namely that — in an
economy with stationary instantaneous preferences and a stationary technology — an
efficient constant utility path is characterized by the value of net investments being zero
at each point in time. In a one consumption economy with two stocks — a stock of
a natural resource and a stock of man-made capital — this means that if consumption
remains constant at the maximum sustainable level, then the accumulation of man-made
capital always exactly compensates in value for the depletion of the natural
resource. ( 1998 Elsevier Science B.V. All rights reserved.
JEL classification: C61; Q20; Q30
Keywords: Optimal control theory; Sustainable development; Hartwick’s rule
1. Introduction
A requirement of sustainability entails that no generation should allow itself
a level of utility that cannot also be shared by all future generations. In the
present paper we pose the following question: What characterizes a development that yields any generation the maximum utility level that can be sustained
also by future generations? This maximum sustainable level is the economywide analogue to the notion of income suggested by Hicks (1946), (p. 174): an
individual’s income “must be defined as the maximum amount of money which
* Corresponding author. E-mail: cwithagen@econ.vu.nl.
0165-1889/98/$ — see front matter ( 1998 Elsevier Science B.V. All rights reserved.
PII S 0 1 6 5 - 1 8 8 9 ( 9 7 ) 0 0 1 0 9 - 7
160 C. Withagen, G.B. Asheim / Journal of Economic Dynamics and Control 23 (1998) 159–165
the individual can spend this week, and still expect to be able to spend the same
amount in real terms in each ensuing week”. So, extending this concept to an
economy as a whole, income would represent the maximum well-being that can
be enjoyed in a given period, leaving the economy with the capacity to generate
the same well-being in each ensuing period. Hence, we seek to characterize
a path along which utility as an indicator of instantaneous well-being remains
equal to Hicksian income in this generalized sense.
Consider an economy with stationary instantaneous preferences and a
stationary technology. That is to say that the representation of the preferences
at each instant of time is invariant with respect to time. Moreover, it is assumed
that exogenous technical change is absent (note that endogenous technical
change through the accumulation of human capital is allowed for). Hartwick’s
rule states that if on an efficient path the value of net investments is zero at each
point in time, then utility is constant. This rule was established for a very general
class of models in an elegant and important piece of work by Dixit et al. (1980).
In a one consumption good economy endowed with two stocks — a stock of an
exhaustible non-renewable resource and a stock of man-made capital —
Hartwick’s rule means that if the accumulation of man-made capital always
exactly compensates in value for the resource depletion, then consumption
remains constant at the maximum sustainable level.
In the seminal work of Solow (1974) such a two-stock economy is analyzed.
The natural resource is exploited at no cost, and the raw material extracted (R)
is, together with capital (K), used as an input in a production process with
a Cobb—Douglas technology (with factor elasticities of a for K and a for R).
1
2
Output of this process is used for consumption purposes (C) and accumulation
of capital (KQ ). The question Solow addresses is whether in this framework there
exists a positive consumption level that can be maintained indefinitely. It is
shown that the answer is in the affirmative if a 'a . The maximal constant
1
2
level of consumption can be derived explicitly. It is easily seen that — along the
path with maximal constant consumption — net investment equals the value of
the ‘revenues’ from exploitation (marginal product of the raw material times the
input of the raw material) at each point in time.
The fact that such exact compensation for resource depletion implies constant
consumption in this setting was first pointed out by Hartwick (1977), after whom
the rule is called Hartwick’s rule. The rule can easily be derived from a general
neoclassical production function F. A necessary condition for efficiency in the
economy is that Hotelling’s rule holds, saying that marginal productivity of
capital equals the rate of change in the marginal productivity of the raw material
(F "FQ /F ; here we assume differentiability of F). It follows from KQ "
K
R R
F(K, R)!C that K$ "F KQ #F RQ !CQ . If KQ "F R, then also K$ "
K
R
R
FQ R#F RQ "F F R#F RQ "F KQ #F RQ . Hence, CQ "0.
R
R
K R
R
K
R
Not only Hartwick’s rule holds in the model considered by Solow (1974), the
converse of Hartwick’s rule holds as well: If consumption remains constant at
C. Withagen, G.B. Asheim / Journal of Economic Dynamics and Control 23 (1998) 159–165 161
the maximum sustainable level, then the accumulation of man-made capital
always exactly compensates in value for the resource depletion. This is also
shown for the Solow model by Hamilton (1995) and for the Ramsey model
(without exhaustible resources and with a non-constant utility discount rate) by
Aronsson et al. (1995). A question that naturally arises is whether the converse of
Hartwick’s rule holds in general in an economy with stationary instantaneous
preferences and a stationary technology: Does an efficient constant utility path
imply that the value of net investments equals zero at each point in time? Dixit et
al. (1980) also attempted to establish the converse of Hartwick’s rule. However,
they succeeded to do so only under an additional assumption that is related to
a ‘capital deepening’ condition employed by Burmeister and Turnovsky (1972).
This assumption is hard to interpret, and it is not an attractive primitive
foundation on which to base the analysis. Hence, it seems worthwhile to offer
a proof that does not rely on it.
Consider an efficient constant utility path that is supported by positive
utility discount factors having the property that the integral of the discount
factors exists. I.e., the path is a regular maximin path in the sense of Burmeister
and Hammond (1977). Then this constant utility path solves the problem of
maximizing the integral of utilities discounted by these discount factors, subject
to the feasibility constraints. Analyze this problem by optimal control theory.
Provided that the Hamiltonian converges to zero as time approaches infinity,
the converse of Hartwick’s rule follows from a result established by Dixit et al.
(1980), (Theorem 1), namely that the value of net investments is constant in
present value prices if and only if utility is constant. Michel (1982) has shown
that the Hamiltonian converges to zero as time approaches infinity if there is
a constant utility discount rate. However, to assume a constant utility discount
rate is too restrictive here; in particular, such an assumption is incompatible
with constant utility (or consumption) in Solow’s (1974) model discussed above.
Hence, one route for proving the converse of Hartwick’s rule would be to
extend Michel’s result to the case without a constant utility discount rate and
combine this extended result with Dixit et al.’s (1980) Theorem 1. In response to
the current note, Seierstad (private communication) has indicated that such an
extension could be shown by using an additional state variable to make the
problem autonomous. Below we have chosen to follow an alternative route. We
establish a direct and comprehensive proof of the converse of Hartwick’s rule in
a very general setting without having to make any additional assumptions.
Thereby our note makes a contribution to the characterization of sustainability.
2. Statement and proof
Consider an optimal control problem with n state variables, denoted
by x:"(x , x ,2, x ), and r instruments, denoted by u:"(u , u ,2, u ).
1 2
n
1 2
r
162 C. Withagen, G.B. Asheim / Journal of Economic Dynamics and Control 23 (1998) 159–165
Obviously n3N and r3N. Let X be an open connected subset of Rn and let º be
a subset of Rr. There are given functions ( f , f )"( f , f , f ,2, f ): X]
0
0 1 2
n
ºPRn`1. These variables and functions are interpreted as follows:
f The vector of state variables (x , x ,2, x ) is interpreted as a vector of stocks,
1 2
n
which consists of different kinds of man-made capital (including accumulated
knowledge) as well as natural and environmental resources.
f The vector of instruments (u , u ,2, u ) determines jointly with the vector of
1 2
r
stocks the output of consumption goods and environmental amenities, the input
of various types of labor, and the accumulation (or depletion) of the stocks.
f f (x, u) is the instantaneous utility that is derived from the vector of stocks
0
x when the vector of instruments equals u.
f f (x, u), 1)j)n, is the time-derivative of stock j when the vector of stocks
j
equals x and the vector of instruments equals u.
f The vector of instruments u is feasible if u3º.
That f is time-independent, means that the instantaneous preferences are
0
stationary. That f for all j"1,2,n are time-independent, means that the
j
technology is stationary. A stationary technology entails that any technological
progress is endogenous, implying that such progress is captured through accumulated stocks of knowledge.
Fix some x03X and ¹*0. The set F(x0,¹) of feasible paths, given that the
vector of stocks at time 0 equals x0 and that the final time is ¹, is defined as
follows: (x, u)3F(x0,¹) if and only if x: [0,¹]PX is absolutely continuous,
x(0)"x0, u: [0,¹]Pº is measurable, and xR (t)"f (x(t), u(t)) for all t3[0,¹]. For
given x03X and ¹*0, we say that (xL , uL ) is optimal w.r.t. the utility discount
factors n: [0,¹]PR
if (xL ,uL )3F(x0,¹) and
``
T
T
n(t) f (xL (t),uL (t)) dt5 n(t) f (x(t),u(t)) dt
0
0
0
0
for all (x,u)3F(x0,¹).
The premise of the converse of Hartwick’s rule entails that there exists
a constant utility path that is supported by positive utility discount factors.
Hence, suppose there exist (a) (xL , uL )3F(x0,R) and a constant w with
f (xL (t), uL (t))"t for all t3[0,R), and (b) n: [0,R)PR , such that (xL , uL ) is
0
``
optimal w.r.t. p given x03X and ¹"R. Note that this implies that :=n(t) dt is
0
finite.
Fix some arbitrary s ('0) and consider the problem of maximizing
P
P
P
T
n(t)[ f (x(t),u(t))!t] dt
0
0
over F(x0,¹) with the additional constraint that x(¹)"xL (q), and where the
maximization takes place with respect to ¹ as well. So, we have a free final time
optimal control problem.
C. Withagen, G.B. Asheim / Journal of Economic Dynamics and Control 23 (1998) 159–165 163
Proposition 1. A solution to the free final time optimal control problem stated
above is: ¹"t and (x(t), u(t))"(xL (t), uL (t)) for t3[0,¹].
Proof. First note that the proposed solution is clearly feasible. Suppose that
there exist ¹* and (x*,u*)3F(x0,¹*) with x*(¹*)"xL (q), which yields an higher
value of the objective functional.
Then
P
P
T*
q
n(t)[ f (x*(t),u*(t)!t] dt' n(t)[ f (xL (t), uL (t))!t] dt"0.
0
0
0
0
Consider the path x(t), (u(t))"(x*(t),u*(t)) for t3[0,¹*] and (x(t),u(t))"
(xL (t!¹*#q), uL (t!¹*#q)) for t3[¹*,R). Due to the fact that f is timeindependent, this path is feasible in the sense that (x,u)3F(x0,R). But
P
=
P
P
=
=
n(t)t dt" n(t) f (xL (t), uL (t)) dt,
0
0
0
0
contradicting that (xL , uL ) is optimal w.r.t. p given x03X and ¹"R.
n(t) f (x), u(t)) dt'
0
h
A more general approach would entail the inclusion in the optimal control
problem of mixed constraints of the form g (x, u)50 for all i"1,2,m, where
i
g"(g , g ,2,g ):X]ºPRm (some m3N). Hence, to be an element of the set
1 2
m
F(x0,¹) of feasible paths, the condition g (x(t), u(t))50 for all t3[0,¹] would
have to be satisfied as well. The inclusion of such mixed constraints would not
change Proposition 1 as long as g for all i"1,2,m are time-independent.
*
Proposition 2. Assume that the maximum principle holds for the infinite horizon
optimal control problem of maximizing :=n(t) f (x(t), u(t)) dt over F(x0,R), and
0
0
let jK "
: (jK , jK ,2, jK ) be the unique vector of co-state variables corresponding to
1 2
n
(xL , uL ). ¹hen jK (t) ) xQ L (t)"0 for all t3(0,R).
Proof. Fix some arbitrary q('0). Then (x(t), u(t))"(xL (t), uL (t)) for t3[0,q] solves
the fixed final time problem of maximizing :q n(t) f (x(t), u(t)) dt (or equivalently,
0
0
:q n(t)[ f (x(t), u(t))!t] dt) over F(x0,q) with s as finite horizon and xL (q) as final
0
0
stock. The unique co-state variables of the infinite horizon problem, restricted to
[0,s], are the unique co-state variables of this problem. Moreover, as established
in Proposition 1, ¹"s and (x(t),u(t))"(xL (t),uL (t)) for t3[0,¹] solve the free final
time problem of maximizing :Tn(t)[ f (x(t), u(t))!t]dt over F(x0,¹) with the
0
0
end point in [¹ , ¹ ], where ¹ (s(¹ , and with xL (q) as the final stock. Since
1 2
1
2
s is an optimal final time, the unique co-state variables of the fixed final time
problem coincide with the co-state variables of this problem. Hence, the unique
co-state variables of the infinite horizon problem, restricted to [0,s], coincide
164 C. Withagen, G.B. Asheim / Journal of Economic Dynamics and Control 23 (1998) 159–165
with the co-state variables of the free final time problem. As a necessary
condition (see Seierstad and Sydsæter, 1987, Chapter 2, Theorem 11) we have
that the Hamiltonian of the free final time problem equals zero at time s:
0"H(xL (q), uL (q), q, jK (t)) :"n(q)[ f (xL (q), uL (q))!t]#jK (q) ) xQ L (q) .
0
Since f (xL (q), uL (q)) equals w, and s can be chosen arbitrary, the result is estab0
lished. h
The uniqueness of the co-state variables of the infinite horizon problem is
assumed to eliminate the possibility that they do not coincide with the co-state
variables of the free final time problem. To illustrate the rationale behind this
assumption, consider an optimal control problem where the state variables do
not appear in the objective functional and the instruments (u) are not present in
the system of differential equations describing the evolution of the state variables. Then there is a continuum of vectors of co-state variables that satisfy the
necessary conditions, and it is not difficult in general to find one which does not
attribute a zero value to net investments. This is also the case if the solution to
the optimal control problem is unique. Note that our uniqueness assumption
does not require uniqueness of a solution. It requires that to any solution there
corresponds a unique vector of co-state variables; stated differently, any other
vector of co-state variables will generate another u as the maximizer of the
Hamiltonian.
It is well-known that with constraints of the form g(x,u)50 the necessary
conditions for optimality presuppose a constraint qualification (see e.g.
Seierstad and Sydsæter, 1987, Chapter 4, Theorem 3). If we are willing to make
this additional assumption Proposition 2 still holds, after the straightforward
modification of the set F(x0,¹). If the constraint qualification is not satisfied
— e.g., in the case of pure state constraints — matters become much more
complicated. The reader is referred to Seierstad and Sydsæter (1987), (pp.
397—398).
The unique vector of co-state variables (jK , jK ,2,jK ) is interpreted as the
1 2
n
vector of present value prices associated with the vector of stocks. Hence,
Proposition 2 states that on a constant utility path that is supported by positive
utility discount factors, the value of net investments equals zero at each point in
time. This means that Proposition 2 is a statement of the converse of Hartwick’s
rule, and that we — by way of the proof of Proposition 2 — have shown this result
in a very general setting.
Note that the assumptions of stationary instantaneous preferences and a
stationary technology are needed in order to establish the converse of Hartwick’s rule. In particular, with exogenous technological progress, the continuation from time ¹* on of the path (x, u) constructed in the proof of Proposition
1 may not be feasible if ¹*(s. Hence, with exogenous technological progress,
C. Withagen, G.B. Asheim / Journal of Economic Dynamics and Control 23 (1998) 159–165 165
the optimal ¹ in the free final time problem may well be earlier than s. This in
turn indicates that — when evaluated along the path (xL , uL ) — the Hamiltonian of
the free final time problem may well be negative at time s, entailing that
jK (q) ) xLQ (q)(0. This is in line with what one would expect.
Acknowledgements
We thank Atle Seierstad for valuable discussions and a referee for helpful
comments.
References
Aronsson, T., Johansson, P.-O., Löfgren, K.-G., 1995. Investment decisions, future consumption and
sustainability under optimal growth, Umea> Economic Studies No. 371, University of Umea> .
Burmeister, E., Hammond, P., 1977. Maximin paths of heterogeneous capital accumulation and the
instability of paradoxical steady states. Econometrica 45, 853—870.
Burmeister, E., Turnovsky, S.J., 1972. Capital deepening response in an economy with heterogeneous capital goods. American Economic Review 62, 842—853.
Dixit, A., Hammond, P., Hoel, M., 1980. On Hartwick’s rule for regular maximin paths of capital
accumulation and resource depletion. Review of Economic Studies 47, 551—556.
Hamilton, K., 1995. Sustainable development, the Hartwick rule and optimal growth. Environmental and Resource Economics 5, 393—411.
Hartwick, J., 1977. Intergenerational equity and investing of rents from exhaustible resources.
American Economic Review 66, 972—974.
Hicks, J., 1946. Value and Capital. 2nd ed., Oxford University Press, London.
Michel, P., 1982. On the transversality condition in infinite horizon optimal control problems.
Econometrica 50, 975—985.
Solow, R.M., 1974. Intergenerational equity and exhaustible resources, Review of Economic Studies
(Symposium), 29—45.
Seierstad, A., Sydsæter, K., 1987. Optimal Control Theory with Economic Applications. NorthHolland, Amsterdam.