As we proceed it will become clear that there is a dichotomy of results between the model with class I u-functions on the one hand and the model with class II and III u-functions on the other. We will begin by looking at the most basic case, that of a zero discount rate.
^ See Arrow and Kurz [3], Proposition 5, part (g) and preceding discussion.
^ Condition (3.7) is only a necessary condition when optimal T turns out to be finite. If T is infinite then we cannot use (3.8) as a necessary condition (see Arrow and Kurz [3], p.46).
Case 1: 6 = 0 :
(a) Class I u-function:
It follows from (3.5) and (3.6) that the rate of exploitation is constant throughout the programme. Because of the necessary
continuity of we can easily deduce from (3.6) that it is never optimal to jump from a strictly positive E to E = 0 for X > 0 (it would lead to an upward jump in . For X = 0 however such a jump in the E variable is permissible since the multiplier Xi can become positive and preserve the continuity of \p (see Arrow [2] p.9, discussion prior to statement of Proposition 4). (3.7) tells us that when T is finite X(T) = 0 (since ^(t) > 0 V^). All of this gives us a clear picture of the optimal plan when T is any finite number. E will be set at some constant positive level until the resource is exhausted when it will fall to zero. What will be the optimal level of E? This may be determined by inspecting
the present value integral:
P E u(E)dt
0
T'
u(E)dt + u(0)dt , 0 T'
where T' is the time at which X is exhausted and therefore equals ^^^^.
u(E) - u(0) Tu(0) + X(0)
{ ^ J
and so maximizing P entails choosing E to maximize ^ "(0) ^
a class I function the relevant value of E is zero. Accordingly P will be maximized by choosing E to have as small a value as is consistent with resource exhaustion within the time available — this will be the
value of E which spreads exploitation of the resource over the whole planning period from time 0 to time T, just exhausting the resource at
time T. As the planning period, T, is lengthened, the optimal value of E will fall (see Figure 3.3) and as T ^ optimal E 0. So that in the limiting case, T = there is no optimal solution. Any positive extraction path is dominated by a path with lower extraction while a zero extraction path (the limiting case) is dominated by any positive extraction path. This is of course a result which emerged from the Gale "cake-eating" example in [13].
Figure 3.3
(b) Class II and III u-functions:
As before, when T is finite X(T) = 0. Now, we define E to be the value of E which maximizes
u(E) - u(0) E
(in particular E will satisfy u'(E) = " )• Because the Hamiltonian is non-concave for some values of E in this case the
Pontryagin necessary condition that the constrained Hamiltonian be maximized at each point in time must be approached from first
principles. Simply, we wish to
Max H(E) = u(E) - ipE E
s.t. E > 0 . 4
It is easily shown that when \p < u'(E), optimal E will be the •s
solution to (jj = u'(E) with E > E (it can be seen diagrammatically with the aid of Figure 3.4 drawn for a class II u-function: in the diagram ip^ > u'(E), ip^ = u'(E) and ip^ < u'(E)). When ^ > u'(E),
H(E) < u(0)V^ > 0, H(0) = u(0), and E = 0 consequently maximizes H. E
UTien ip = u' (E) the maximum value of H is u(0) and this is attained when /N
either E = 0 or E = E. Thus it is clearly never optimal to operate in the interval 0 < E < E. Moreover E cannot jump from a value greater than E to zero for X > 0 for this would require a jump in so for paths which involve E > E at some stage, E will be constant at that
5 /N level until X = 0. It may be shown that for E ^ E paths associated
If il; < u'(E), and H(E) - u(E) - i|;E, then E f^ 0, since if it were: H(E) > u(E) - u'(E).E = u(E) - (u(E) - u(0)) = u(0) = H(0). Also, for E e (0, E),
H(E) - H(E) = (u(E) - u(0)) - (u(E) - u(0)) - i|;(E - E) > (u(E) - u(0)) - (u(E) - u(0)) - u'(E)(E - E) > (u(E) - u(0)) - I (u(E) - u(0)) - u'(E)(E - E) = [(u(E) - u(0)) - E.u'(E)] = 0.
E
So we can assume an interior solution and differentiate H to obtain u'(E) = ip < u'(E)^and E > E (since H'(E) = u'(E) - > 0 in some neighbourliood of E) .
For E > E, [u(E) - u(0)]/E is a decreasing function of E. Since the present value integral equals
'u(E) - u(0) Tu(0) + X(0)
E
Figure 3.4
with lower values of E have a higher present value integral than paths for which E is high. Since X(T) is necessarily zero, for low values of T (T < X(0)/E) the optimal path will entail E being set at that
constant level consistent with exhausting the resource at time T. When T = X(0)/E, then E will be set at E for the whole programme just running X to zero at time T. For longer planning horizons (T > X(0)/E) clearly
/N
the appropriate positive level of extraction is E (since E dominates all higher values of E when it is feasible, and lower values of E have
already been shown to be non-optimal). This will mean however, that for some of the programme E should be zero. Moreover, since when = u (E), H(0) = H(E) = u(0) both E = 0 and E = E will maximize H and there is
t h e r e f o r e , nothing to stop E jumping to zero before X is e x h a u s t e d , later j u m p i n g back up to E , e t c . until X = 0 . Clearly all such paths
(and there are an infinite number of them) yield the same present value for the stream of utility and are all optimal solutions to (3.4). There is no unique optimal p a t h . H o w e v e r , the path for which E = E until
exhaustion w i l l do as w e l l as a n y .
Before proceeding to the case of a positive discount rate let us briefly assess the results obtained above:
The non-existence of an optimal solution for a class I u-function when T = is the result of two factors:
(i) The type of u-function involved prevents us from finding a positive E which maximizes
u(E) - u(0) E
Because this e x p r e s s i o n , which we w i l l term "average excess utility" (AEU),^ is always greater than marginal u t i l i t y , it is a decreasing function of E so that the same number of total units of extraction would yield more "total excess utility" (TEU)
TEU E (u(E) - u(0))dt 0
^ This concept of average excess utility has significance here for two reasons:
(i) We are interested in the excess of utility from extraction over the utility derived when all consumption is derived from the alternative source (C) ;
(ii) Because we are trying to find the intertemporal allocation of resource extraction which gives us the highest total utility summed over all periods and because all periods are valued e q u a l l y , we would like to know whether each unit of extraction is yielding as much excess utility as possible — i.e. is
u(E) - u ( O l maximized? E
if s p r e a d as thinly as p o s s i b l e over the given time than if c o n c e n t r a t e d in a s h o r t e r i n t e r v a l of time (the AEU b e i n g higher in the former c a s e ) . T h e TEU is t h e r e f o r e seen to rise as E m o v e s closer to the E w h i c h m a x i m i z e s AEU (E 0) . Wlien a p o s i t i v e E can b e found to m a x i m i z e AEU (as in the
case of class II and III u - f u n c t i o n s ) it c o n s t i t u t e s the o p t i m a l e x t r a c t i o n level for long time h o r i z o n s .
(ii) T h e a b s e n c e of a discount rate w i t h the p r e s e n t g e n e r a t i o n w a n t i n g the same b e n e f i t s for all f u t u r e g e n e r a t i o n s as for
itself m e a n s that w h e n a class I u - f u n c t i o n is involved the TEU over all p e r i o d s w i l l b e greater the lower the l e v e l of e x t r a c t i o n at any point in time to the point w h e r e e x t r a c t i o n is zero in a l l p e r i o d s . Given the scarcity of the r e s o u r c e , a p o s i t i v e discount rate may w e l l be n e e d e d to help r a t i o n the limited stock of the resource among g e n e r a t i o n s .
T h e a b s e n c e of d i s c o u n t i n g is also r e s p o n s i b l e for the
i n d e t e r m i n a c y a r i s i n g in the case of class II and III f u n c t i o n s . Q u i t e s i m p l y , w i t h o u t a discount rate w e h a v e no criterion for c h o o s i n g
b e t w e e n points in time and therefore no criterion for e s t a b l i s h i n g one policy as u n i q u e l y o p t i m a l . W e now turn to the case of a p o s i t i v e d i s c o u n t r a t e .
C a s e 2: 6 > 0:
As w e w o u l d expect w e again find that the class I and class II-III f u n c t i o n s y i e l d results w h i c h a r e s u p e r f i c i a l l y different but w h i c h a r e n e v e r t h e l e s s r e l a t e d . In case 1 a b o v e , b e c a u s e a community
m a x i m u m f o r a v e r a g e e x c e s s u t i l i t y , in t h e i n f i n i t e h o r i z o n c a s e it a t t e m p t e d to a p p r o a c h t h a t i d e a l s i t u a t i o n b y r u n n i n g t h e l e v e l of e x t r a c t i o n to z e r o (a b o u n d a r y m a x i m u m f o r A E U ) a n d t h e r e w a s n o o p t i m a l s o l u t i o n . F o r a d i f f e r e n t r e a s o n t h e r e w e r e a n i n f i n i t e n u m b e r o f o p t i m a l s o l u t i o n s for c l a s s I I - I I I f u n c t i o n s . N a t u r a l l y t h e i n t r o d u c t i o n of a p o s i t i v e d i s c o u n t r a t e c h a n g e s t h i s s i t u a t i o n . (a) C l a s s I u - f u n c t i o n : T h e i n t e r i o r s o l u t i o n to ( 3 . 5 ) , ( 3 . 6 ) a n d (3.8) is c h a r a c t e r i z e d b y t h e f o l l o w i n g r e l a t i o n s h i p s : E > 0 = ^ A i = A 2 = 0 li^ = a n d ip = u ' (E) . H e n c e E = ^ ^ < 0 for 6 > 0 . u (L) T h u s E f a l l s o v e r t i m e a s t h e r e s o u r c e is d e p l e t e d . F o r a l l f i n i t e T X ( T ) = 0 . It t u r n s o u t to b e o p t i m a l for l a r g e T to run E ( t ) c o n t i n u o u s l y to z e r o so that it r e a c h e s z e r o at t h e t i m e at w h i c h X is e x h a u s t e d . In t h e ip-X p h a s e - p l a n e ( F i g u r e 3 . 5 ( a ) s h o w s s o m e of t h e p a t h s s a t i s f y i n g t h e n e c e s s a r y c o n d i t i o n s in i p - X s p a c e ) t h e o p t i m a l t r a j e c t o r y is t h e h i g h e s t f e a s i b l e p a t h on or b e l o w p a t h a in F i g u r e 3 . 5 ( a ) . O p t i m a l i t y is e s t a b l i s h e d u s i n g t h e f o l l o w i n g c o m p a r i s o n of i n t e g r a l s p r o o f : L e t a s t e r i s k s u p e r s c r i p t s d e n o t e t h e p a t h c l a i m e d to b e o p t i m a l ( v i z . t h e l o n g e s t e x h a u s t i o n p a t h f e a s i b l e in t h e t i m e a v a i l a b l e ) . T h e n w e c o m p a r e t h e p r e s e n t v a l u e i n t e g r a l a l o n g t h i s p a t h
(P*) with the present value along any other feasible Pontryagin path (i.e. some lower path in Figure 3.5(a)).
P* - P E T _ r [u(E*) - u(E)]e '^^dt 0 u'(E*)(E* - E)e" ^dt
(since u is strictly concave in E)
u' (E*) (E* - E)e"'^'^dt
0
(where T* is the time at which X is exhausted along the path claimed to be optimal; since this is the longest feasible exhaustion path
E*(t) = E(t) = 0 V^ e (T*, T]) T
r _ r
ij^*(t)e (E'^ - E)dt 0
- X*) = 0 since X(T) = X^(T) = 0 and X(0) = X*(0) = 0 .
Thus, in terms of Figure 3.5(a) the optimal trajectory will be the
highest feasible path on or below path a. For a time horizon of Ti say, (Figure 3.5(b)) path y will be optimal. For a longer time horizon, say T2 it will be feasible to spread extraction more over time so that path 3 is optimal. For time horizons greater than or equal to T3, path a is optimal. In particular as T ^ path a remains optimal. Provided u'(0) < °° (which is the likely situation when C > 0) the resource will be exhausted in a finite time. This follows because
Figure 3.5(a) Figure 3.5(b)
and w h e n
i>(t) = liJoe^^ = u'(E) + A2 - Aj E > 0 = u ' ( E ) ;
t h e r e f o r e E cannot be p o s i t i v e for an infinite period of time since lim ip(t) =
t;->oo
Finally it is noted again that for long time h o r i z o n s o p t i m a l E w i l l fall c o n t i n u o u s l y to zero reaching zero at the time (finite) at