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
Along an optimal trajectory there are essen- tially two regimes possible, according to invest-
ments I in natural capital being zero or positive. What we wish to investigate is when each of these
regimes will occur and whether switches from one regime to another are feasible or not. To this end
both regimes will be described in detail.
If, at some instant of time, I, W \ 0 then it follows from Eq. 5 and the homogeneity of F
that r = F
1
1x, 1 and p = F
2
1, x. It follows from the implicit function theorem that r and p
are on a decreasing curve in r, p-space, which we shall denote by r = ap. This curve is called the
factor price frontier because it contains the shadow factor prices of the inputs in the produc-
tion process described by F for which shadow profits are zero at the maximum. Along an opti-
mum we obviously have rt ] apt for all t, because otherwise no maximum exists in Eq. 5.
Now fix positive levels for natural capital and the price of the reproducible input: K \ 0 and
p \ 0. We vary r to see what happens to invest- ments I while keeping these variables positive at
the given levels. It follows from the homogeneity of F that I = KG1, y − C
1
. Consumption C
1
de- creases monotonically as a function of r see Eq.
3. Moreover, if r 0 then C
1
and if r
then C
1
0. The ratio y is increasing monotoni-
cally in r. Moreover, if r 0 then y 0 and if r then y . So, I seen as a function of r
vanishes at a unique r, denoted by r = Rp, K to stress that p and K were given in the exercise we
performed. In general we must have r ] Rp, K. In particular r ] max ap, Rp, K. Summaris-
ing, we have r = ap, whenever investments are positive and r = Rp, K, whenever investments
are zero.
Let g be the curve in p, K-space given by a
p = Rp, K, see Fig. 1. It divides the space into two regions. For p, K above the curve
region G investments are obviously positive. For values below the curve region G investments are
zero.
Since the initial stock of natural capital is given, the input price p has to be chosen optimally.
Therefore, the question addressed next is what will happen if the initial values of p are chosen in
either of these two regions.
We first consider the positive investment region G.
Take a
fixed price
p \ 0. Let
C
1
p, C
2
p, yp solve Eqs. 3 and 4 with r = ap inserted. All these functions are well
defined by virtue of the properties of the utility functions and the production functions. It follows
from Eq. 4 that p ; p=0 if rpG
1
1yp, 1 = r. Define p as the unique solution of this equa-
tion. Furthermore, it follows from Eqs. 1 and 2, and the fact that F and H are homogeneous
that K : =0 if
K: = Kp = C
1
p G1, yp
+ C
2
p G1, ypF1, xp
What are the properties of the curve p, Kp? It obviously lies in G, because KpG1, yp\
C
1
p so that investments in natural capital are positive. Also, Kp 0 as p 0 because if p 0
Fig. 1. Phase diagram for K B
K.
then C
1
p 0, C
2
p 0, G1, yp and F1, xp . We conclude that there exists a
stationary point in G, denoted by p, K. In Fig. 1 the arrows indicate the directions in
which K and p go from any point in G. Obviously K
: \0 if and only if K\Kp. Hence, the picture strongly suggests that for any K
B K, one can
find p0 such that p0, K lies in G and such
that the resulting path converges to p, K, This is indeed the case, as is shown in the first theorem.
Theorem 1. Let C
1
, C
2
, K, I, V, W constitute an optimal trajectory. Suppose K
B K. Then
i pt, Kt G for all t. ii pt, Kt p, K as t , where con-
6 ergence is monotonic.
iii The rates
of consumption
monotonically decrease.
iv It \ 0 and Wt \ 0 for all t.
Proof 1. The proof of the first two statements is given in the Appendix A. The proof of iii fol-
lows immediately from the fact that p is increas- ing. The final statement follows from i.
The theorem essentially states the following. If the initial stock of natural capital is sufficiently
small then in a utilitarian optimum there will always be investments to manage natural capital
in a proper way. Moreover, natural capital monotonically increases to a steady state and
rates of consumption decrease monotonically, also to positive steady states. It is interesting to
perform a sensitivity analysis with respect to the rate of time preference. Clearly the steady state p
decreases as r increases. Therefore, a larger rate of time preference reduces the steady-state rates of
consumption, as well as the steady-state stock of natural capital. This conforms with intuition.
The case K \
K is less straightforward to analyse. The proof of Theorem 1 uses the exis-
tence of a so-called stable branch in the region G defined for all p, K 5 p, K. This branch is a
curve in G, denoted by, Kp, having the property that it is followed along an optimal trajectory
Kt = K
pt. The curve Kp is also defined for p \ p but it might be the case that the curve
Fig. 2. Phase diagram for K \
K.
intersects the curve g and hence lies in G for p sufficiently large. This, for example, occurs if the
elasticities of marginal utility denoted by h
1
and h
2
are constant and satisfy 1 \ h
2
\ h
1
. For a proof of this assertion the reader is referred to the
appendix. In this example, which is depicted in Fig. 2 below, K
p intersects g only once, say, but it cannot be excluded that there is more than one
point of intersection. However, that is not rele- vant for the main purpose of the present paper,
namely, to show that also for K \
K invest- ments in natural capital are positive from some
moment in time on. The optimum is characterised in the following
theorem.
Theorem 2. Let C
1
, C
2
, K, I, V, W constitute an optimal trajectory. Suppose K
\ K. Then
i pt, Kt G for all t]t for some t
] 0.
ii pt, Kt p, K as t , where con- 6
ergence is monotonic. iii The
rates of
consumption monotonically
decrease. iv It \ 0 and Wt \ 0 for all t ] t
.
Proof 2. Parts i and ii are proven in the Appendix A. Parts iii and iv follow immedi-
ately from i and ii.
This theorem states that if the initial stock of natural capital is very large it is likely that there
will be no investments directed towards the man- agement of natural capital. But eventually natural
capital will approach a positive steady state. Along the approach path near the steady state
there will be positive investments. Investments are also positive in the steady state and are just
enough to keep natural capital intact given the rate of exploitation. The rates of consumption are
high initially but they will decrease to a positive steady state level.
4. Conclusions
