mentioned above, and we see no problem in mak- ing this assumption for expository purposes. As a
matter of fact, it facilitates the analysis of several issues that are, in our opinion, pertinent to sus-
tainability. First of all, natural capital is a factor of production. Production can be used for con-
sumption purposes and also for investments, di- rected towards improving the quality of natural
capital, or for enlarging the available stock by exploration or for the build-up of backstop tech-
nologies or for the preservation of biodiversity see on the latter Weitzman, 1998. On the other
hand, excessive use of natural capital decreases its ‘value’ but may increase instantaneous welfare
here the examples of water and exhaustible re- sources come to mind. So, there is a trade-off
between natural capital as a factor of production and alternative uses. The question arises which is
the optimal use of natural capital. With a Rawl- sian objective will correspond constant instanta-
neous
‘happiness’ over
time, whereas
it is
generally argued that with a utilitarian criterion with discounting, natural capital will decrease
over time and future generations are doomed or are at least victimised by the greediness of the
present generation. We will show that the latter statements are incorrect in their generality. More
specifically, it will be shown that under a utilitar- ian regime natural capital will steadily increase
over time if it is small initially; furthermore, if initial natural capital is large it will decrease, but,
and this is perhaps surprising, on this trajectory there will nevertheless be investments in improv-
ing natural capital. Hence, the initial abundance of natural capital does not justify only exploiting
it. It is necessary from the beginning to manage it properly by investing in it. The intuition behind
this result is that investments in natural capital increase its capability to produce desirable com-
modities and are therefore to be undertaken un- less natural capital is extremely abundant. The
policy recommendation following from these ob- servations is that even in a ‘neoclassical’ perspec-
tive natural capital should be managed carefully by investing in it. Natural capital as an aggregate
has certain properties of a backstop technology.
Before proceeding to the analysis we wish to mention two serious caveats of our approach.
First we assume that natural capital as such does not have an amenity value: only consumption
yields utility. We therefore abstract from issues such as beauty of, for example, mountains and
forests. Second, working with an aggregate ig- nores that certain constituent parts are possibly
subject to irreversible processes. Exploitation of a given mine is irreversible, loss of biodiversity is
irreversible and many other examples can be given. However, the neglect of irreversibilities
strengthens our result: even if they do not exist a careful management of natural capital is necessary
along an optimum.
The outline of the paper is as follows. The model is presented in Section 2. There we also
derive some preliminary results. The main out- comes of the analysis are in Section 3. Section 4
concludes. The proofs of the two main theorems are given in Appendix A.
2. The model
In the economy under study two production processes or sectors can be distinguished. The first
sector produces a homogeneous commodity that can be used for consumption C
1
and for invest- ment purposes I. These include backstop devel-
opment, exploration,
restoration, etc.
The function G describes the production process hav-
ing natural capital K and a reproducible com- modity V, think of labour, as inputs. The initial
stock of natural capital is given by K . The second
sector is the resource sector. The function F de- scribes the process of improving, enlarging or
recycling natural capital; inputs are investments from the first sector and some reproducible com-
modity W, possibly identical to the other repro- ducible commodity. The consumption of natural
capital is denoted by C
2
, which can be thought of as the direct use made of the stock of natural
capital, for example, for recreational purposes thereby causing pollution or water extraction.
In the economy there is a representative infi- nitely lived consumer or dynasty see Gerlagh
1999 for an overlapping generations approach to sustainability. Aggregate instantaneous utility is
assumed to be strongly additively separable in
consumption and the reproducible inputs. Total instantaneous welfare in the economy depends in
a positive way on the two types of consumption introduced above. It is negatively affected by the
amount of reproducible inputs. In order to keep the model tractable it is assumed that total wel-
fare is quasi-linear, meaning in the model at hand that it is linear in the reproducible inputs with
constant negative coefficients − 6 and − w, re- spectively. These coefficients may represent the
prices of these inputs on outside markets or the opportunity costs in terms of utility, of leisure.
If the relationships introduced here are cast into a formal model, the following optimal control
problem can be stated. Find piece-wise continu- ous positive functions C
1
, C
2
, I, V, W, defined for all positive instants of time t, and piece-wise
continuously differentiable positive K, defined also for all positive instants of time, such that
total welfare
e
− r t
[U
1
C
1
t + U
2
C
2
t − 6Vt − wWt] dt is maximised, subject to
C
1
+ I = GK, V
1 K
: =FI, W−C
2
, K0 = K
2 A dot above a variable denotes its time derivative.
In the intertemporal welfare function r is the positive constant rate of time preference. We are
aware of the fact that in view of the sustainability issue it would be more appropriate to have a
declining rate of time preference, but we assume constancy here for the sake of simplicity.
Note that several existing models can be consid- ered as special cases of the model outlined above.
If there is only one consumer commodity C
2
0, if K is interpreted as physical capital, which accumulates through investments, so that FI,
W
I, and if V is interpreted as labour, which is assumed to be a positive exogenous constant, then
we are in the standard Ramsey – Cass – Koopmans model of optimal economic growth. In that case
the economy will, under some mild assumptions with regard to the production function, converge
to the so-called modified golden rule path, in which the rate of consumption and capital are
constant over time. Alternatively, K can be interpreted as a purely
exhaustible natural resource. To see this, assume that FI, W
0 no regeneration and that only
the commodity extracted from the exhaustible resource is consumed: C
1
0. Then, under the usual assumptions with regard to the instanta-
neous utility function, and if the rate of time preference is positive, the rate of consumption will
necessarily approach zero.
The case of K being a renewable resource like fish or forest with the possibility of growth is
covered if W = K, w = 0, C
1
0. Then C
2
denotes the catch or cut, which yields utility or revenues
U
2
C
2
In the case of a renewable resource there will also occur a long-run steady state, under the
appropriate assumptions with respect to the func- tions involved.
The Solow – Dasgupta – Heal model is, however, not a special case of the model outlined here
because in that model there are two stock vari- ables, namely, physical capital and an exhaustible
natural resource. We only have one state variable. We recall that in the Solow – Dasgupta – Heal
model without technical progress the rate of con- sumption will eventually approach zero if the rate
of time preference is positive. It is therefore opti- mal to leave no substantial consumption opportu-
nities for future generations. This occurs in spite of the fact that positive maintained consumption
is feasible in some circumstances.
With respect to the functions involved the fol- lowing assumptions are made.
Assumption 1. For both i, U
i
is a strictly increas- ing and strictly concave function. Moreover, the
Inada conditions are satisfied: U
i
C
i
0 as C
i
and U
i
as C
i
0.
Assumption 2. F is a strictly increasing, strictly
quasi-concave and linearly homogeneous function for strictly positive arguments. Both inputs are
necessary and F1, z as z Finally, the Inada conditions hold: F
1
z 1 0 and F
2
1 z 0 as z ; F
1
z 1 and F
2
1 z as z 0.
Assumption 3.
G has the same properties as F.
In assumption 2 the indices indicate partial derivatives with respect to first or second argu-
ments of the functions. It should be stressed that assumptions 2 and 3
allow for substitution between factors of produc- tion. In fact, by using more labour to maintain
the stock of natural capital the regeneration ca- pacity is increased, even if investments are re-
duced a bit. However, per se this is not a very strong assumption because due to the concavity
of the regeneration function and the linear cost of labour, substitution will not occur to a large
degree. There is also a substitution assumption in the production function G. Production can be
kept intact using less natural capital and more labour. In the conclusions we will return to the
issue of substitutability.
As a preliminary result we have that along an optimal program the rates of consumption, capi-
tal input K and the other input V are strictly positive. This follows from assumption 1, in par-
ticular from the property that marginal utility goes to infinity when consumption goes to zero,
and from assumption 3, in particular necessity and the fact that marginal product of an input
approaches infinity when the input tends to zero.
Existence of optimal programs in infinite hori- zon control problems is not a trivial matter at all.
In order to guarantee existence boundness condi- tions are to be imposed on state variables as well
as controls and also certain convexity properties have to be satisfied see Baum 1976, Toman
1985. It can be shown that for the model at hand all the requirements are met. Formally how-
ever, existence of a solution to the problem posed above merely guarantees that there are measur-
able controls whereas the commonly applied max- imum
principle assumes
the existence
of piece-wise continuous controls. Fortunately this
problem need not bother us because, mainly due to strict quasi- concavity of the functions in-
volved, the controls will indeed be piece-wise con- tinuous. Moreover, we show that an optimum
exists with the desired properties by actually con- structing it.
The Lagrangian of the problem reads L = e
− r t
[U
1
C
1
+ U
2
C
2
− 6V − wW] + l
1
[GK, V − C
1
− I] + l
2
[FI, W − C
2
] Let C
1
, C
2
, K, I, V, W constitute an optimal program. Then, according to the maximum prin-
ciple see Seierstad and Sydsaeter, 1987 there exist l
1
and l
2
such that, omitting the time argu- ment when there is no danger of confusion,
e
− r t
U
1
C
1
= l
1
e
− r t
U
2
C
2
= l
2
− l :
2
= l
1
G
1
K, V 6
e
− r t
= l
1
G
2
K, V I, W maximises − e
− r t
wW − l
1
I + l
2
FI, W The intuition behind these necessary conditions
is straightforward. The Lagrangian multiplier variable l
1
can be interpreted as the shadow price of the first sector’s product. Then the first
necessary condition states that in an optimum the present value of marginal utility of the first con-
sumption commodity equals the value of produc- ing it. The fourth condition requires that the
profits of the first sector should be maximised with respect to the variable input. The co-state
variable l
2
represents the value in terms of present utility of a marginal additional unit of
the stock of natural capital. According to the second necessary condition it should equal the
present value of marginal utility derived from consuming a marginal unit of natural capital. The
third necessary condition is an arbitrage relation- ship. It boils down to requiring that the rate of
return on natural capital should be equal as a provider of renewable and nonrenewable re-
sources on the one hand and as an input in the first sector of the economy on the other hand. The
final necessary condition says that the second sector maximises its profits.
It will turn out to be helpful for expository purposes to introduce several new variables. In
view of the homogeneity of the functions F and G, their partial derivatives depend on the ratio of the
inputs only. We therefore introduce x
WI and y
VK. Without loss of generality we set w
equal to unity. Furthermore, we define pt
1 e
− r t
l
2
t, the real shadow price of W in terms of
natural capital, and rt
l
1
tl
2
t, the real shadow price of investments I in terms of natural
capital. Then the necessary conditions become: U
1
C
1
= rp, U
2
C
2
= 1p 3
p ; p=rG
1
1y, 1, G
2
1, y = 6pr 4
I, W maximises FI,W − rI − pW 5
In the sequel these necessary conditions are stud- ied in detail in order to give a full characterisation
of the optimal trajectory.
3. The optimal trajectory
