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Optimal resource extraction and uncertainty about new technology
JOURNAL
OF ENVIRONMENTAL
ECONOMICS
AND MANAGEMENT
11,
39-54 (1984)
Optimal Resource Extraction and Uncertainty about New Technology’ GAUTAM BHATTACHARYA Department
of Economics,
Universig
of Kansas,
Lawrence,
Kansas
66045
Received January 11,1982 The optimal extraction policy of a known resource stock in an economy which has imperfect information about existence of a new technology is discussed. The optimal program is shown to depend on the remaining stock and the current information about existence of the new technology (which is updated every period). For two special forms of social welfare function, the optimal extraction policy is shown to be more conservative than in the situation of perfect information. The effects of improved information and greater availability of the known stock on the optimal extraction policy are also investigated.
I. INTRODUCTION Most modem economies currently use a number of natural, exhaustible resources in their production and consumption processes. The prospect of exhaustion of their resources in the future has generated a wide interest in research for determination of the path of depletion of these resources in economies with different types of market structures.* However, improvements in technology in the future are likely to reduce or eliminate the dependence on the services of exhaustible resources. Therefore, the current state of knowledge about future advances of technology will affect the nature of the current extraction policy for these resources. As the types of technical progress that might occur in the future are varied and complex, the nature of present knowledge about future technological progress is difficult to model accurately in a simple framework. Some papers [4-6, 10, 161 take the approach of not specifying exactly the nature of the new technology, assuming instead that the new technology, when available, will provide a perfect flow substitute for the services of the exhaustible resource presently in use. It is also assumed that the date of availability of the new technology is a random variable, with a known probability distribution. This paper differs from the papers mentioned above in its description of the current knowledge about the existence and availability of the new technology itself. It will be
‘This paper contains material from Chap. 2 of my Ph.D. dissertation [2] submitted to the University of Rochester. I am indebted to L. W. Mckenzie, J. W. Friedman, M. Majumdar, and an anonymous referee for helpful comments and suggestions. An earlier version of this paper was presented at the TIMS/ORSA Conference in Colorado Springs, November 1980. Part of the research was supported by University of Kansas Grant 3625-20-003 for Summer 1981. *The Symposium issue of the Review of Economic Studies published in 1974 contains papers by Dasgupta and Heal [4], Solow [21], Koopmans [14], Stiglitz [22], Manne [18] and others dealing with this problem. Current research in this Geld has progressed in many directions. An incomplete list of papers that consider resource extraction problems not related to uncertainty about new technology might include Arrow and Chang [I], Deshmukh and Pliska [7], Gilbert [S], Kemp (111, Pindyck [19], Smith [20], Loury [16], Kemp and VanLong [12], Stiglitx and Dasgupta [23], etc. 39 0095-06%/&1 $3.00 Copyright 0 1984 by Academic Press, Inc All rights of reproduction in any form reserved
40
GAUTAM
BHATTACHARYA
seen that these differences lead to conclusions about the effects of information about new technology that are somewhat different from the earlier papers. Uncertainty about new technology is treated in a different way in this paper. As in the other papers, it is assumed that the,new technology, when available, will provide a perfect flow substitute for the resource. However, the nature of the substitute or the rate at which the flow of the substitute product may be produced by the new technology is assumed to be unknown. A second type of uncertainty, arising from our ignorance about natural and scientific phenomena, is whether a new technology exists at all. Finally, another source of uncertainty is that even if a new technology exists, the time required to make the technology available (through scientific research, etc.) is unknown. If our current knowledge about the new technology is characterized by these types of uncertainty, is it still possible to determine the optimal intertemporal strategy for depletion of a known stock of resource? To answer this question, in Section II of this paper, the resource extraction problem of a socially managed economy in this framework is set up as a discrete-time dynamic programming model with incomplete information. It is shown that the optimal extraction policy exists in each period and depends only on the probability of arrival of the new technology (if it exists) in that period, the remaining stock of the resource, and the current probability of existence of a new technology which is obtained by updating past beliefs using the Bayesian method. It is also shown that the optimal policy is not influenced by our knowledge about what the character of the new technology will be once it is available (provided it is known that the new technology will make the existing technology obsolete). In Section III, two special forms of the social welfare function (logarithmic and constant relative risk-aversion) are used to derive explicit closed-form solutions for the optimal extraction policy. The optimal policy is seen to be more conservative than the policy derived in other papers (for instance, Dasgupta and Heal [4], Dasgupta et al. [5]). The effects of improved information and larger current resource stock on the optimal extraction policy are also analyzed. Further, in spite of existence of different types of uncertainty, it is shown that a certainty-equivalent resource stock exists for this problem. Finally, the properties of prices in a competitive economy whose allocations are the same as the optimal allocations are described in Section IV. II. EXISTENCE
OF AN OPTIMAL’ POLICY
The initial known stock of the resource is s,,. The resource is required for use in production and/or consumption. In the absence of any technological breakthrough, this resource is to be extracted and used over an infinite horizon (although in principle, the possibility of exhausting the resource in finite time is not ruled out). To simplify the model, the production process using the resource and other factors is not considered explicitly. Instead, it is assumed that the instantaneous social welfare function, u(a) is known for every possible amount of extraction, a, of the resource. Consider first, for future reference, the simple problem of resource extraction under certainty when sO is the initial known stock. It is well-known [4] that if society’s problem is to choose an extraction path a,, t = 0, 1, . . . to maximize m such that c a, 4 s,, ii swat> t=o
t=o
RESOURCE
FiXTRACTION
AND
UNCERTAINTY
41
when/3 is the social discountfactor, then thereexistsa function G(s,) the value of which gives the total discountedwelfare accruingto the societyfrom extracting a known stock s0optimally over an infinite horizon. The problem consideredin this paper is in the samespirit as the “cake-eating” problem describedabove,although it, is considerablymore complex. Here, apart from the known resourcestock, there is a possibility that a new technologymay exist. If sucha technologyis available,the flow of substitutesproducedeveryperiod will make it possible to maintain a steady-statelevel of consumptionevery year giving rise to a social welfare of U everyyear. The value of U is assumedto be unknown,but the distribution of U is known and EU = U. One way to incorporate the notion that the technologicalbreakthroughis a desirableeventis to assumethat
i.e., the availability of the new technologymakesthe existing technologyobsolete. Under this assumptionthe amountof the remainingstockwill not influencethe level of social welfare after the new technologybecomesavailable.This is a standard assumptionabouttechnologymade,for instance,in Dasguptaand Heal [4], Dasgupta et al. [5], Kamien and Schwartz[lo], Lehman [15], etc., mainly to facilitate characterizationsof current optimal extraction policy. The concluding Sectionbriefly indicatesthe consequences of relaxingthis assumption. The essenceof the current formulation of the cake-eatingproblem is that thereis imperfect information about whetherthe new technologyexists.Initially, let p. be the probability that such a technologyexists.Assume also that if the technology exists,there is a known probability, q,, that it will arrive in period t, t = 1,2,. . . . Every period,if the technologyis not discovered,an extractiondecisiona, has to be taken (such that 0 G a, < s, when s, is the remainingstock).On the other hand, if the technologyis discovered,U is the level of utility enjoyedeveryperiod from that point onwards.The socially optimal extractionprogram is {a,, t = 0, 1,2,. . . } such that the following is maximized
(2.1) where 0 6 a, d s,
forallt = 0,1,2,...
(2.2)
and Gtl
= s, - a,
forallt=0,1,2
,....
(2.3)
The problem above can be set up as a dynamic programming problem with incompletestate observationsfor a socially managedeconomywherethe decisions aremadeby a centralplanner.This is doneas follows. (A) Let b be a randomvariable,6 = 1 be the eventthat the newtechnologyexists, and b = 0 be the eventthat the new technologydoesnot exist.The plannerdoesnot
42
GAUTAM
BHATTACHARYA
know whether he is in a state where b = 1 or b = 0. Thus, the unobservable state space is B = {l,O}. (B) Let m be a random variable that takes only two values, m = 0 and m = 1. Let m = 1 be the event that the technology is available, m = 0 be the event that it is not available (i.e., not discovered yet). m E M = { 1, O}. Here, s is the remaining stock of the resource. s is assumed to lie in the interval, [0, so] = S. Thus, the observable state is the pair (s, m) such that (s, m) E S X h4. The observable state space is S x M. (C) The amount of extraction, a, is the action variable, a E A = S. (D) The correspondence of feasible actions is a(s,, m,) = [0, sI] for all (s!, m,) E s x M. (E) The transition probability function g, assigns a probability distribution on (b,+19 mf+19 s,+l ) E B x S x M for every value of (b,, m,, sr) and a,. The marginal distributions obtained from g are g’, gz,,, and g” for bt+l, m,+l, and s,+~, respectively. If the technology exists (b, = l), and is not available in the last period (m, = 0), the probability that it is available in the current period is qr+l. Thus, for all s,, a,
gr”,l(m,+, = 114= 1, m, = 0, st, a,>= q,+l gtm+l(m,+l = Olb,= 1, m, = 0, s,, a,) = 1 - qr+l.
(2.4) (2.5)
On the other hand, if the technology does not exist, it cannot be discovered. Thus, for all m,, s,, a,, and for all t, g:l(ml+l
= W, = 0, m,, s,, a,) = 0
(2.6)
gzl(m,+l
= O(b, = 0, m,, s,, a,) = 1.
(2.7)
Similarly, if the technology exists (b, = 1) and was available in the last period (m, = l), it will continue to remain available in the current period. Thus for all t, s,, a, dz*h+1
= lib, = 1, m, = 1, st, a,) = 1.
(2.8)
gh is obtained by the obvious relation that if the technology exists in period t, it will continue to exist in period t + 1. If it does not exist in period t, it will not exist in period t + 1. Thus for all ml, s,, a, &b,+l gb(4+,
= 116, = 1, m,, s,, a,) = 1 = O(b, = 0, m,, s,, a,) = 1.
(2.9) (2.10)
Finally, for every b,, m, @(St+1 = s, - a,(b,, m,, s,, a,) = 1. (F) The initial conditions are m, = 0 and prob. (b, = 1) = pO. (G) The return function v: M
X
A + R is defined as
(2.11) (2.12) I
v(0, u) = u(u)
(2.13)
v(1, a) = ii.
(2.14)
RESOURCE
EXTRACTION
AND
43
UNCERTAINTY
We assume u is a continuous and bounded function and ii < co. Hence u is continuous and bounded. The objective is to choose {a,, t = 0, 1,2,. . . } such that 0 c a, G s,, to maximize (2.15)
when E denotes expectation with respect to (2.2)-(2.12). The problem described by (A)-(G) (problem Z) is a nonstationary dynamic programming problem with incomplete state observations. Yushkevich [24] gives a method of transforming a stationary problem of this type to a stationary problem with observable states. Following his approach, it is possible to show that the problem considered in this paper can also be transformed to a nonstationary problem with observable states. Consider a problem with observable state variables ( p, m, s) when (m, s) is interpreted as before and pt is the probability belief in period t that b, = 1, i.e., the technology exists. Ifp, = p, and m, = 0, then prob. (m,,, = lip, m, from (2.4) and prob. (mt+l = O]p, m, = 0) = p(1 - qr+l) from (2.5). = 0) = P&+1 Further, if m t + 1 = 0, the planner can revise pt and obtain pt + i by using the Bayesian rule. For instance, pr+i = prob. (b,,, = l]m,+, = 0, pt = p) = p(1 - qt+i)J1 So, with the appropriate transition mechanism set up, problem 2 can be P4t+l* transformed to problem 2 as follows. The observable state is (p, m, s) E I x M x S, when I = [0, 11. The action variable is a E A and the correspondence of feasible actions is (Y(S) = [0, s] for all ( p, m, s). The transition function is s,+~ with marginal distributions gp,“, for ( p, m) and g” for s. For all (s,, a,), %(Pt+l
= PO - 4,+1)/l
- P%+A
mt+1
=
OlPt = P, mt = 0, St, at)
= 1 - P4t+1 icl(PI+l
+ PO - 4t+1)/1
- P4t+A
mt+1
=
OlPt = PT mt = 09 sr, at)
= 0 KI(Pt+l
= 1,
izY(Pt+l
= 19 mt+1
mt+1
(2.16)
(2.17)
= lip, = P, m, = 0, St, a,> = p4r+l
(2.18)
l]p, = 1, m, = 1, st, a,) = 1.
(2.19)
=
Also, for every ( pr, m,, s,) and a,
iTh+1 = St- arlpt, m,, St,a,) = 1.
(2:20)
The initial conditions are ( po, m,, so) when p. is given by (2.12). The return function is u(m,, a,) as defined by (2.13) and (2.14). The objective is to choose {G,, t = 0, 1,. . . } such that a, E [0, s,] to maximize
Co if? c 4mt,at> t=o when the expectation is taken with respect to the distributions
(2.21) in (2.16)-(2.20).
44
GAUTAM
BHATTACHARYA
Following the proof-givenby Yushkevich[24, pp. 154-561,for stationaryproblems, problems2 and Z can be shownto be equivalentin the following sense. LEMMA
2.1. If{a:,
t
= &I,... > is an optimal policy for problem 2, it is also an
optimal policy for problem Z with E~u(m,,aj+)=B~u(m,,a:). t-0
t-0
Define the t-periodoptimal value function as Ft(~~~m,,s,)=E
(2.22)
iB’u(m,,a:)lp,,m,,s,
7=t
The right-hand side of (2.22)is the expectedreturn from following an optimal policy from period t onwards.Fo(po, mo, so) is the expectedsocial welfare from following an optimal policy for problem ?! (hencefor problem Z). In fact, the optimal policy a: dependson the statevariablesin period t (p,, m,, s,) and qt, the probability of the technologybeing discoveredin period t. THEOREM 2.1. For problem 2, there exists a function a: = a: (p,, m,, so such that a: is the optimal policy for period t, given pt, m,, and s,. Further, {F,, t = 0, 1,2,. . . } satisfies the following optimality equations.
If m, = 0, I’;(P,t mt9 4) = m~o
I
u(aA + a0
- P,q,+J
(2.23) Ifm, = Lp, = 1, and1;;(1,1,st) = +.
(2.24)
Proof. First, problem z is transformed to a stationary problem Z* in the following way. Let the statevariablefor Z* be ( p, m, s, t) E I X M X S x T when T = { 0, 1,2,. . . }. The actionvariableis a E A = S as in problem2. The correspondenceof feasibleactions is cw(p, m, s, t) = [0, s] as before, the return function is u(m, a) as definedby (2.13)and (2.14),and the initial conditionsare ( po, 0, so, 0). The transition probability function is h with marginal distributions hpmr and h” defined as hPm’(-lp,
h’(-lp,
m, s, t, a) = &‘m(-l~,
m, s, a)
m, s, t, a) = g”( -Ip, m, s, a),
when#‘” and g” aredefinedby (2.16)-(2.20).
RESOURCE
EXTRACTION
AND
45
UNCERTAINTY
If I and S are endowed with relative topology as subsets of the real line and M and T are endowed with discrete topology, we can see that problem Z* is a stationary dynamic programming problem with complete, separable metric spaces as state and action spaces. The correspondence of feasible action is compact valued and upper-semicontinuous and the return function is bounded and continuous. Further, the transition probability function h gives a probability measure on the state space for every (p, m, S, t), is a measurable function of (p, m, s, t) and is weakly continuous in (p, m, S, t). Thus all the conditions for applying the theorem in Maitra [17, p. 2161 are satisfied. Hence there exists an optimal policy for Z*, a*( P, m, s, t).
As problems Z and Z* are equivalent (in the sense of Lemma 2.1), writing a:( p, m, S) = a*( p, m, S, t), we get an optimal policy for problem 2. Further, by Theorem 6 (f) of Blackwell [3], the return function to a*( p, m, s, t) will satisfy the “optimality equation.” Let V( p, m, S, t) be the optimal value function (the expected return from following a*( p, m, S, t)). Writing F,( p, m, s) = V( p, m, s, t) for all t, we get the following. If m, = 0, F,(p,, %+d/U - Pt%+d, 09 St 1, F,(L 1,~~) = m=oda,ss, this proves { 4, t = 1,2,. . . } satisfies the functional
Eq. (2.23) and (2.24).
Q.E.D.
Lemma 2.1 extends the following well-known result in statistical decision theory to the context of optimal resource extraction. The statement of problem Z does not indicate how the initial probability of existence of new technology is affected when information about arrival of the technology becomes available. Problem 2, however, has p,, the current probability of existence of new technology as a state variable and the transition mechanism of this problem specifies a Bayesian revision of pr in every period. Therefore, the equivalence of problems Z and 2 (in the sense of Lemma 2.1) as shown by Yushkevich [24] and others, establishes that revision of the initial probability p. (and all subsequent probabilities) by the Bayesian formula is necessary for deriving the socially optimal extraction policy. This result also indicates the desirability of using Bayesian posteriors for making rational decisions in similar situations characterized by incomplete information. Further, the optimal policy depends only onp,, m,, and q, (also on j3 and the parameters of the utility function). It does not depend on U, the expected value of social welfare obtained after the technology becomes available. It is also clear that more information about the distribution of U (for instance, information about which particular value of U will prevail when the new technology becomes available) will not alter the optimal policy. Although this result seems rather counterintuitive, it is easily explained by the fact that in this model, the stock of resource that remains after the technology becomes available does not influence the level of social welfare that can be maintained from that point of time onwards. Therefore, information about availability and existence of the technology is important for determination of current extraction policy, while information about the nature of the new technology is irrelevant for this purpose.
III. PROPERTIES
OF THE OPTIMAL
POLICY
The explicit forms of optimal extraction policies are found for two special forms of social welfare functions u(u) = log a and u(u) = a*. Assume q. = 0 and define,
GAUTAM BHATTACHARYA
46 foraIlt=1,2,...
PO0 - d(l - q2)*--0 - 4r)+o-Po) yt=PO0 - q,)(l- q2) ***0 - cl-l>+(1-Po) (1- 41N1 - q2) ***0 - 41)Po pz=(1- 41)(1 - q2)***(1- 41)Po +(1- PO) .
(3.1) (3.2)
Here, y, is the probability that the technology will be available in period t (given that it has not arrived yet) and pt is the updated probability at period t of the existence of a new technology given that it has not been available until period t - 1. Given (3.1) and (3.2), the functional E!q. (2.23) can be rewritten as I;;(p,, 0, sI) = fth 4), as f,(p,,
SA = m~OSa,qs,[ ub,)
+ BY,+~~,+~(P,+~~ s, - a,>
+so
- Yc+1F4
- #o-q.
(3.3)
PROPOSITION 3.1. Zf u(a,) = log a,, the optimal policy a: and the optimal value function f,, for the functional Eq. (3.3) are given for t = 0, 1,2,. . . as
(3.4) f,(PV Sr> = ( 1 + i$l(i,,)lOgSt-(l
+ i$lGi,)lOg(l
+ jzlGit)
+i=!+lGitlOgGit +(l- 8)-’ whenforalli=
t+
(3.5)
l,...
Gi, = PiptYt+lYt+2
(3.6)
* * * Yip
4, = Bi-‘Yr+lY1+2
. ’ ’ Yj-It1
(3.7)
- Vi)*
Proof It suffices to check that ( f,, t = 0, 1,2,. . . } as given by (3.5), satisfies the functional Eq. (3.3) for u(a) = log a and a:, as given by (3.4), is the optimal policy obtained from (3.3). Construct H,: Z X S X A --, R as Z-4( Pi, 4, a,) = log a, + B~,+~fr+d P~+~, 4 - a,) + PO - u,+dV(l
- a>
- 1+ it %+Ilog1+ 5 Gi,r+l 1( i=[+2
i=t+2
+
2 i-1+2
+m
Gi,r+llOgGi,1+1
- Yt+W(l
+o
- l-9
-
s>
-‘(
iz24.t+l)E]
1 (3.8)
47
RESOURCEEXTRACTION AND UNCERTAINTY
Thus, from (2.27) and (2.29)
ft(Pt9St)= ~~o~a,&4h~ sr,4 H, is concave in a,. Thus, a; is the unique solution aHpa, = 0 gives
(3.9)
to aHt/aat
= 0. Setting
1+i=t+z. f -$=PYt+1
01
1+ BY,+, + BY,+I
(3.10)
or a:
=
(l
+
ij?ZIGit)~‘st~
The above verifies (3.4). From (3.4) and (3.9) we get, ft(P,t
St) = Ht( ptv stv a:)
-
i
+
E r=t+2
Gi,t+llogGi,t+l
Writing b(Py,+~ + BY,+IZZ,+~ Gi,r+l) = 1% BY~+I + lo&l +
CZ+2
(3.11) is simplified as
+
+PYr+l
2 i=t+2
G,,t+llOgGi,t+l
+(l
-
B)-’
Bv,+l
f r=t+2
%,+I
Gi,r+l),
48
GAUTAM
or, from(3.6)
BHATTACHARYA
and as By,+llog,8y,+l = G,+l,,log Gl+l, t, we can write, f,h
St) = (1 + ;~lc,+g~,
-(l
+
i~lGit)‘og(l
+
iglGil)
+ i=.+lGitlOgC,t +(l - a>-’ This verifies that { f,; t = 0, 1,. . . } satisfies the functional Eq. (3.3) and hence a:, as given by (3.4), is the optimal policy for period t for t = 0, 1, . . . . Q.E.D. PROPOSITION3.2. If u(a) = as, the optimal extraction policy a: and the optimal value function f, for Eq. (3.3) are given for t = 0, 1,2,. . . as -1
ii, =
1 + i
i (G,,)“(‘-“) i=r+l
s,
(3.12)
1 l-6
(Gir)iA1-‘)
+(1 - p)-‘U
2 i
i=f+l
s,!
4,
(3.13) i
when Gi, and 4, are as defined in (3.6) and (3.7), respectively.
Proposition 3.2 can be proven by using the same techniques as used in Proposition 3.1. Hence the proof is omitted. Features of Optimal Extraction Policy
Optimal extraction policies for different situations are described below. a: and &, denote initial extraction for u(a) = log a and u(a) = as, respectively. When p0 = 1, the new technology is known to exist (although it has not been discovered yet). This situation is one of perfect information (PI) and is similar to the one considered by other articles (e.g., Dasgupta and Heal [4]; Kamien and Schwartz [9]; Dasgupta et al. [5]; Dasgupta and Stiglitz [6]; Lehman (151; etc.). In this case
a& so)= (1 + S(l - 4J + S”(l - ql)(l - q2)+ .-.)-lso ?&#,s,)
= (1 +(p(1
- q1))1n1-8)+(p2(l
- ql)(l
- q2)1A1-6)+
(3.14) .**)-$).
(3.15) When 0 -C p0 -C 1, we have the situation of imperfect information (IM) about the new technology (this is the general case analyzed in this paper-the other two
49
RESOURCE EXTRACTION AND UNCERTAINTY
situations arise as special cases of this). Here from (3.4) and (3.12) we have
a,*(po,so>= (1 + BYI+ B2Y1Y2 + *. . )-lso a,(p,,s,)
= (1 +(py1)1/(1-8)+(B2~~Y2)1/(1-b)+
(3.16) *+lql.
(3.17)
When p0 = 0, the new technology does not exist and we have the standard problem of extraction of a known resource stock over an infinite horizon. This is described as the situation with nonexistence (NE) of a new technology. Here,
PROPOSITION3.3.
am so)= (1- B)s,
(3.18)
i&(0, so) = (1 - pi/(*-Q).
(3.19)
For both types of social werfare functions, the initial extraction in
IM will be more conservative than in PI but less conservative than in NE. The initial extraction policy is also a concave increasing function of the initial information about new technology and a linear, increasing function of the initial resource stock. Further, if u(a) = as, an increase in the social index of relative risk-aversion (1 - Q) will reduce the initial extraction.
Proof of Proposition 3.3 is routine and is omitted. It can be checked that the change in initial extraction as a result of a change in initial information (i.e., %&-,/~3p,) or a change in initial stock (i.e., &&/c?s,) is a decreasing function of the index of relative risk-aversion (1 - 8). In other words, the more risk-averse the society is, the less it will respond (regarding current extraction) to an improvement in current resource stock or in current information about future technological progress. The following propositions compare the intertemporal paths of resource extraction for IM and PI situations. PROPOSITION3.4. For every p,,, (i) there exists T* such that for t < T*, a:(p,, sO) -C a:(l, s,,) and for t >/ T*, a:( ~0, so) 2 a:(& so). (ii) there exists T such that for t < T, G,(p,, so) < a,(l, sO) and for t a c q( PI), SIJ 2 a,u, so). Further, 7 is a decreasing function of 6 and p,,. PROPOSITION3.5. For every p0 and E > 0 (i) there exists T** such that for t > T**, la:(p,, st) - a;(O, s,)j c E (ii) there exists T’ such that for t > T’, In,(p,, sr) - a,(O, sr)l < E. Further, T’ is increasing in 6.
In Propositions 3.4 and 3.5 (the proofs of which are omitted from this paper), a:( po, so) = a:( p,, s,) when s, and p, are derived from the following recursive relations St+1
=
St
- 4? Pi, sr)
or
and Pt-10 pt = 0 - %)Pt-1
- 4r) +(I -PI-I).
&+I = s, - “API,
sJ
50
GAUTAM
BHATTACHARYA
Initially, accordingto Proposition3.4,extractionon the PI path will be greaterthan that on the IM path. However,after a point of time, if the technologyis still not available,the IM path will show more rapid extractionthan the PI path. The time where this switch occurswill be deferredif the society becomesmore risk-averse. Accordingto Proposition3.5,as the probability of existenceof new technologyfalls over time in the IM path (if the technologyfails to appear),the IM path, after a period of time, will come close to a NE path having the same remaining stock of resource(this is not the NE path given by (3.4)or (31.2)).Time period T’ will be a decliningfunction of the socialindex of risk-aversion.Thus, Propositions3.4 and 3.5 explain the effectsof revision of initial information about the technologyon the extractionpathsover time. Apart from the information about existenceof a new technology,anothersource of uncertaintyis about the time of discoveryof the new technology.In both PI and IM cases,if {q,} (the set of conditional probabilities of availability of the new technologyat time t) is suchthat q1 # 0 for all t, then the resourceis neveractually Lexhausted if the technologyis not available.Howeverif { qJ} = { ql,. . . ,qT}, thenin the PI casethe resourcewill be exhaustedat time T. In caseof other papersin this areathat assumePI but wherep dependson the stockof the remainingresource,the resourcewill be exhaustedsometime after T. On the other hand,in the IM case,the resourcewill not be exhaustedat a finite time if { qT} = { ql,. . . , qT} and 0 < p. < 1. This is becauseif p. = 1, the technologyis believedto be definitely availableby T; but if p. < 1, the technologyis assumedto be discoveredby T only if it exists. So, if the technologyfails to appear,p, is reviseddownwardsover time and for t 2 T, pr = 0. In fact, the lower p. and the higher the index of risk-aversion,the largerwill be the remainingresourcestock at Tin the IM case(if the technologyis not discovered by 0. The effect of an increasein the conditional mean arrival time of the technology may be examinedin the following simplified situation.Let qt = q for all t. Thus, for instance,if p. = 1, the probabilities of arrival at time t = 1,2,. . . are given by (q, (1 - q)q, (1 - q)‘q, . . .).3 Here an increasein q will imply an increasein mean arrival time of the technologyboth in PI and IM cases.Here, from (3.4)and (3.12), a$( po, so, q) = (1 + B(po(l +/3’(po(l
- 4) + 1 -PO) - q)3 +(1 -po)
+ . ..>-‘so
- PO))
(3.20)
p()(l - q) + (1 - p0))1f11-6)
ao(po, so, q) = (1 + jF-“)( +gz/(l+(
+ B2( Po(l - q)2 +(I
po(l
- q)2
+(l - P~)~‘~-‘) + . 1.)-lso.
(3.21)
The following propositionmay be proven. PROPOSITION3.6. Anti increase in conditional mean arrival time of the new technology will increase init&l extraction. The amount of change in initial extraction corresponding to a change in q will be higher in the case of perfect information ( p. = 1) than 3Lehman [15] has derived two results comparable to this paper’s PI situation when q, = q for all t. Considering a general social welfare function, he shows (in bis Theorems 1 and 2, respectively) that the initial extraction policy for PI is less conservative than that for NE and an increase in q will increase the initial extraction in the PI case.
RESOURCE
EXTRACMON
AND
51
UNCERTAINTY
in the case of imperfect information ( p0 < I). This rate of change will also be a decreasing function of the index of relative risk-aversion. Finally, an attempt is made here to answer the following questions. In spite of different kinds of uncertainties involved in the framework considered here, is it possible to define a certainty-equivalent resource stock? Further, is there any single number that can, be interpreted as an index of the amount of uncertainty in the system? For every pO, sO, and E, the certainty-equivalent resource stock s’ is defined implicitly by fAO7 s’> = fo( PO, %I.
(3.22)
Thus, keeping the social welfare at the same level, the current resource stock and all the uncertainties involved in the IM case may be replaced by a situation with no uncertainty and the certainty-equivalent stock s’. If u(a) = as, (1 +(/3yJ1A1-6)
+ . . .)Y
+(1 - /9-G-4
s’( PO, +J, a = (1
-
2 JIO 1’6 i i=l 1).
p1/(1-8’)!+
(3.23)
However, if so is replaced by s’, the initial extraction of the resource will be h,(O, s’) = (1 - pi~1-8))s’.
(3.24)
Comparing (3.23) with (3.24), it can be checked that a,( po, so) -C a,(O, s’). Similarly a$( po, so) < a,*(O, s’). So, if the certainty-equivalent stock is introduced, keeping the social welfare at the same level, the amount of initial extraction will increase. The amount of uncertainty in the system may be measured by the difference between s’ and so. This measure (s’ - so) is higher in case of PI than in case of IM. Thus, we may conclude that there is actually less uncertainty when the information about existence of new technology is not perfect. The reason for this is that in case of IM, the hope for achieving the new technology in the future is not as strong as in PI, so the amount of uncertainty is less. For the same reason, as p. falls, i.e., initial information becomes more imperfect, the amount of uncertainty decreases. It should also be noted that (s’ - so) is a decreasing function of so. The amount of uncertainty in the economy depends on the initial resource stock and is smaller for a larger initial stock. Thus for large stocks of resource the optimal plan is not very sensitive to the assumption of perfect or imperfect information about the new technology. However, as stocks become smaller, the nature of current information about existence of a new technology becomes more important. IV. THE COMPETITIVE
ECONOMY
The well-known Hotelling’s rule states that price of a unit of stock of resource (net of extraction costs) will increase at a rate equal to the rate of interest in equilibrium in a competitive economy. In papers that consider the’ situation of perfect informa-
52
GAUTAM
BHATTACHARYA
tion (Dasgupta and Heal [4], etc.), the rate of increase in this price was found to be larger than the rate of interest. In this paper we will consider equilibrium conditions for some competitive economy whose allocations correspond to the optimal allocation of the last section. Consider a competitive economy where every producer or consumer has identical information about the new technology (i.e., pa and { qr } as defined earlier). Thus, for each agent, the probability that the technology will not be available in period t is yr as defined by (3.1). In this economy, suppose contingent markets for delivery of the resource for all future dates exist at least for those events where the technology is not available. Let r be the equilibrium interest rate and rl be the equilibrium current price of a unit of resource at period t if the technology is not available yet. Then the arbitrage condition that will hold in any dynamic equilibrium and that will make producers indifferent between extracting the resource between one period and another is given by, for all t = 0, 1,2,. . . (1 +
d?
=
(3.25)
Yt+1?+1*
Further, if Gt is the equilibrium amount extracted in period t and.D-‘(a) inverse market demand function, then for all t = 0, 1, . . .
is the
D-l@,)
(3.26)
In particular, if D-'(a)
= 17,.
= Sag-‘, then the total instantaneous surplus will be p-‘(a)
da = as.
(3.27)
Finding a program that maximizes EX/~'Q~ will lead to the competitive librium allocations for this economy, if p is taken as (l/(1 + y)). a, =
(
1 + p(y,+*)1A1-6)
+(/32yt+*yt+z)1A1-8)
+ . * *)-ls,
equi-
(3.28)
and lr, = 6 ( 1 + ( B(y,+$@@
+ * * .)1-8(S,)6-1.
(3.29)
Equation (3.29) or (3.25) shows that the resource price V~in fact increases at the rate ((l/By,) - 1) in period t. In case of PI, this rate will be higher, equal to ((l//?(l - qr)) - 1). When the information about the technology is imperfect, the rate of increase in price is lower than in PI although it is still higher than the rate of interest (r = ((l//3) - 1)). Further, this rate of growth of price falls over time as yr + 1 and gradually approaches the rate of interest if the technology fails to appear for a long period of time. This is the kind of modified Hotelling’s rule implied by the analysis of this paper. It is still true that if all contingent markets exist, the expected rate of increase in price will be equal to the rate of interest, but here we are interested only in the actual rate of increase in price if the new technology is not discovered. Further, if the inverse market demand function is D-'(a) = &as-l, the absolute value of elasticity of demand is constant and given by l/(1 - 6). Using this fact,
RESOURCE
EXTRACTION
Proposition 3.3 and the following proposition can be easily proven.
AND UNCERTAINTY
remarks and Proposition
53
3.6, the following
PROPOSITION 4.1, The initial equilibrium price will be more sensitive to a change in initial information about new technology and initial stock if the demand is less elastic. An increase in mean arrival time of the new technology will have a larger effect on the initial price if the demand is more elastic.
V. CONCLUDING
REMARKS
Throughout this paper, it was assumed that after the technology is available, the resulting steady state is not affected by the remaining stock of resource. Relaxing this assumption will maintain the validity of existence of an optimal stationary policy depending on p,, m,, and s, (Theorem 2.1). However, explicit solutions for the extraction policy are difficult to obtain in this case. Finally, it is possible to treat technological change as endogenous in this framework by making the distribution of {q,, t = 1,2,. . . } depend on the research and development expenditures incurred by the society. Several authors (Kemp and VanLong [13]; Stiglitz and Dasgupta [23]; Kamien and Schwartz [lo]) examine the nature of optimal extraction policy and optimal policy for research and development expenditures in the context of perfect information about existence of new technology. The framework in this paper is likely to indicate a more conservative extraction policy and a larger expenditure on research to be socially optimal. REFERENCES 1. K. Arrow and S. Chang, “Optimal Pricing, Use, and Exploration of Uncertain Natural Resource Stocks,” Discussion Paper No. 675, Harvard Institute of Economic Research, Harvard Univ. (1978). 2. G. Bhattacharya, “Two Essays in the Economics of Uncertainty: The Importance of Learning in Planning with Exhaustible Resources and Optimality of Equilibrium with Incomplete Markets,” Ph.D. Dissertation, Univ. of Rochester (1980). 3. D. Blackwell, Discounted dynamic programming, Ann. Math. Statis. 226-235 (1965). 4. P. Dasgupta and G. Heal, The optimal depletion of an exhaustible resource, Reu. Econ. Stud. (Symposium) 3-28 (1974). 5. P. Dasgupta, G. Heal, and M. Majumdar, Resource depletion and research and development, in “Frontiers of Quantitative Economics,” Vol. 3 (M. Intrilligator, Ed.), North-Holland, Amsterdam (1977). 6. P. Dasgupta and J. Stiglitz, “Uncertainty and the Rate of Extraction under Alternative Institutional Arrangements,” Technical Report No. 179, Institute of Math. Studies in the Social Sciences, Stanford Univ. (1976). 7. S. Deshmukh and S. Pliska, Optimal consumption and exploration of nonrenewable resources under uncertainty, Econometric4 48, 177-200 (1980). 8. R. Gilbert, Optimal depletion of an uncertain stock, Reu. Econ. Stud. 47-57 (1979). 9. M. Hoel, Resource extraction when a future substitute has an uncertain cost, Rev. Econ. Srud. 637-644 (1978). 10. M. Kamien and N. Schwartz, Optimal exhaustible resource depletion with endogenous technical change, Reo. Econ. Stud. 179-l% (1977). 11. M. C. Kemp, How to eat a cake of unknown size, in “Three Topics in the Theory of International Trade,” (M. C. Kemp, Ed.), North-Holland, Amsterdam (1976). 12. M. C. Kemp and Ngo VanLong, “Exhaustible Resources, Optimality and Trade,” North-Holland, Amsterdam (1980).
54
GAUTAM
BHATTACHARYA
13. M. C. Kemp and Ngo VanLong, On the development of a substitute for an exhaustible natural resource, in “Economic Theory of Natural Resources,” (W. Eicehorn, et al., Edc ), Physica-Verlag, Vienna (1982). 14. T. Koopmans, Proof for a case where discounting advances the doomsday, Rev. Econ. Stud. (Symposium) 117-120 (1974). 15. D. E. Lehman, “Doomsday” reconsidered, Resources and Energy 3, 337-357 (1981). 16. G. Loury, The optimum exploitation of an unknown reserve, Rev. Econ. Stud., 621-636 (1978). 17. A. Maitra, Discounted dynamic programming on compact metric spaces, Sunkhya 30, Part 2, 211-216 (1%8). 18. A. Manne, Waiting for the breeder, Rev. Econ. Srud. (Symposium) 47-66 (1974). 19. R. Pindyck, Optimal exploration and production of a non-renewable resource, J. Pol. Econ., 86, 841-862 (1978). 20. V. Smith, An optimistic theory of exhaustible resources, J. Econ. Theor. 9, 384-396 (1974). 21. R. Solow, Intergenerational equity and exhaustible resources, Rev. Econ. Stud. (Symposium) 29-45 (1974). 22. J. Stightz, Growth with exhaustible resources: Efficient and optimal growth paths, Rev. Econ. Stud. (Symposium) 139-152 (1974). 23. J. Stiglitz and P. Dasgupta, “Market Structure and Resource Depletion,” Research Memorandum No. 261, Econometric Research Program, Princeton Univ. (1980). 24. A. A. Yushkevich, Reduction of a controlled Markov problem with incomplete data to a problem with complete information with Bore1 state and control spaces, Theor. Probub. 153-158 (1976).
OF ENVIRONMENTAL
ECONOMICS
AND MANAGEMENT
11,
39-54 (1984)
Optimal Resource Extraction and Uncertainty about New Technology’ GAUTAM BHATTACHARYA Department
of Economics,
Universig
of Kansas,
Lawrence,
Kansas
66045
Received January 11,1982 The optimal extraction policy of a known resource stock in an economy which has imperfect information about existence of a new technology is discussed. The optimal program is shown to depend on the remaining stock and the current information about existence of the new technology (which is updated every period). For two special forms of social welfare function, the optimal extraction policy is shown to be more conservative than in the situation of perfect information. The effects of improved information and greater availability of the known stock on the optimal extraction policy are also investigated.
I. INTRODUCTION Most modem economies currently use a number of natural, exhaustible resources in their production and consumption processes. The prospect of exhaustion of their resources in the future has generated a wide interest in research for determination of the path of depletion of these resources in economies with different types of market structures.* However, improvements in technology in the future are likely to reduce or eliminate the dependence on the services of exhaustible resources. Therefore, the current state of knowledge about future advances of technology will affect the nature of the current extraction policy for these resources. As the types of technical progress that might occur in the future are varied and complex, the nature of present knowledge about future technological progress is difficult to model accurately in a simple framework. Some papers [4-6, 10, 161 take the approach of not specifying exactly the nature of the new technology, assuming instead that the new technology, when available, will provide a perfect flow substitute for the services of the exhaustible resource presently in use. It is also assumed that the date of availability of the new technology is a random variable, with a known probability distribution. This paper differs from the papers mentioned above in its description of the current knowledge about the existence and availability of the new technology itself. It will be
‘This paper contains material from Chap. 2 of my Ph.D. dissertation [2] submitted to the University of Rochester. I am indebted to L. W. Mckenzie, J. W. Friedman, M. Majumdar, and an anonymous referee for helpful comments and suggestions. An earlier version of this paper was presented at the TIMS/ORSA Conference in Colorado Springs, November 1980. Part of the research was supported by University of Kansas Grant 3625-20-003 for Summer 1981. *The Symposium issue of the Review of Economic Studies published in 1974 contains papers by Dasgupta and Heal [4], Solow [21], Koopmans [14], Stiglitz [22], Manne [18] and others dealing with this problem. Current research in this Geld has progressed in many directions. An incomplete list of papers that consider resource extraction problems not related to uncertainty about new technology might include Arrow and Chang [I], Deshmukh and Pliska [7], Gilbert [S], Kemp (111, Pindyck [19], Smith [20], Loury [16], Kemp and VanLong [12], Stiglitx and Dasgupta [23], etc. 39 0095-06%/&1 $3.00 Copyright 0 1984 by Academic Press, Inc All rights of reproduction in any form reserved
40
GAUTAM
BHATTACHARYA
seen that these differences lead to conclusions about the effects of information about new technology that are somewhat different from the earlier papers. Uncertainty about new technology is treated in a different way in this paper. As in the other papers, it is assumed that the,new technology, when available, will provide a perfect flow substitute for the resource. However, the nature of the substitute or the rate at which the flow of the substitute product may be produced by the new technology is assumed to be unknown. A second type of uncertainty, arising from our ignorance about natural and scientific phenomena, is whether a new technology exists at all. Finally, another source of uncertainty is that even if a new technology exists, the time required to make the technology available (through scientific research, etc.) is unknown. If our current knowledge about the new technology is characterized by these types of uncertainty, is it still possible to determine the optimal intertemporal strategy for depletion of a known stock of resource? To answer this question, in Section II of this paper, the resource extraction problem of a socially managed economy in this framework is set up as a discrete-time dynamic programming model with incomplete information. It is shown that the optimal extraction policy exists in each period and depends only on the probability of arrival of the new technology (if it exists) in that period, the remaining stock of the resource, and the current probability of existence of a new technology which is obtained by updating past beliefs using the Bayesian method. It is also shown that the optimal policy is not influenced by our knowledge about what the character of the new technology will be once it is available (provided it is known that the new technology will make the existing technology obsolete). In Section III, two special forms of the social welfare function (logarithmic and constant relative risk-aversion) are used to derive explicit closed-form solutions for the optimal extraction policy. The optimal policy is seen to be more conservative than the policy derived in other papers (for instance, Dasgupta and Heal [4], Dasgupta et al. [5]). The effects of improved information and larger current resource stock on the optimal extraction policy are also analyzed. Further, in spite of existence of different types of uncertainty, it is shown that a certainty-equivalent resource stock exists for this problem. Finally, the properties of prices in a competitive economy whose allocations are the same as the optimal allocations are described in Section IV. II. EXISTENCE
OF AN OPTIMAL’ POLICY
The initial known stock of the resource is s,,. The resource is required for use in production and/or consumption. In the absence of any technological breakthrough, this resource is to be extracted and used over an infinite horizon (although in principle, the possibility of exhausting the resource in finite time is not ruled out). To simplify the model, the production process using the resource and other factors is not considered explicitly. Instead, it is assumed that the instantaneous social welfare function, u(a) is known for every possible amount of extraction, a, of the resource. Consider first, for future reference, the simple problem of resource extraction under certainty when sO is the initial known stock. It is well-known [4] that if society’s problem is to choose an extraction path a,, t = 0, 1, . . . to maximize m such that c a, 4 s,, ii swat> t=o
t=o
RESOURCE
FiXTRACTION
AND
UNCERTAINTY
41
when/3 is the social discountfactor, then thereexistsa function G(s,) the value of which gives the total discountedwelfare accruingto the societyfrom extracting a known stock s0optimally over an infinite horizon. The problem consideredin this paper is in the samespirit as the “cake-eating” problem describedabove,although it, is considerablymore complex. Here, apart from the known resourcestock, there is a possibility that a new technologymay exist. If sucha technologyis available,the flow of substitutesproducedeveryperiod will make it possible to maintain a steady-statelevel of consumptionevery year giving rise to a social welfare of U everyyear. The value of U is assumedto be unknown,but the distribution of U is known and EU = U. One way to incorporate the notion that the technologicalbreakthroughis a desirableeventis to assumethat
i.e., the availability of the new technologymakesthe existing technologyobsolete. Under this assumptionthe amountof the remainingstockwill not influencethe level of social welfare after the new technologybecomesavailable.This is a standard assumptionabouttechnologymade,for instance,in Dasguptaand Heal [4], Dasgupta et al. [5], Kamien and Schwartz[lo], Lehman [15], etc., mainly to facilitate characterizationsof current optimal extraction policy. The concluding Sectionbriefly indicatesthe consequences of relaxingthis assumption. The essenceof the current formulation of the cake-eatingproblem is that thereis imperfect information about whetherthe new technologyexists.Initially, let p. be the probability that such a technologyexists.Assume also that if the technology exists,there is a known probability, q,, that it will arrive in period t, t = 1,2,. . . . Every period,if the technologyis not discovered,an extractiondecisiona, has to be taken (such that 0 G a, < s, when s, is the remainingstock).On the other hand, if the technologyis discovered,U is the level of utility enjoyedeveryperiod from that point onwards.The socially optimal extractionprogram is {a,, t = 0, 1,2,. . . } such that the following is maximized
(2.1) where 0 6 a, d s,
forallt = 0,1,2,...
(2.2)
and Gtl
= s, - a,
forallt=0,1,2
,....
(2.3)
The problem above can be set up as a dynamic programming problem with incompletestate observationsfor a socially managedeconomywherethe decisions aremadeby a centralplanner.This is doneas follows. (A) Let b be a randomvariable,6 = 1 be the eventthat the newtechnologyexists, and b = 0 be the eventthat the new technologydoesnot exist.The plannerdoesnot
42
GAUTAM
BHATTACHARYA
know whether he is in a state where b = 1 or b = 0. Thus, the unobservable state space is B = {l,O}. (B) Let m be a random variable that takes only two values, m = 0 and m = 1. Let m = 1 be the event that the technology is available, m = 0 be the event that it is not available (i.e., not discovered yet). m E M = { 1, O}. Here, s is the remaining stock of the resource. s is assumed to lie in the interval, [0, so] = S. Thus, the observable state is the pair (s, m) such that (s, m) E S X h4. The observable state space is S x M. (C) The amount of extraction, a, is the action variable, a E A = S. (D) The correspondence of feasible actions is a(s,, m,) = [0, sI] for all (s!, m,) E s x M. (E) The transition probability function g, assigns a probability distribution on (b,+19 mf+19 s,+l ) E B x S x M for every value of (b,, m,, sr) and a,. The marginal distributions obtained from g are g’, gz,,, and g” for bt+l, m,+l, and s,+~, respectively. If the technology exists (b, = l), and is not available in the last period (m, = 0), the probability that it is available in the current period is qr+l. Thus, for all s,, a,
gr”,l(m,+, = 114= 1, m, = 0, st, a,>= q,+l gtm+l(m,+l = Olb,= 1, m, = 0, s,, a,) = 1 - qr+l.
(2.4) (2.5)
On the other hand, if the technology does not exist, it cannot be discovered. Thus, for all m,, s,, a,, and for all t, g:l(ml+l
= W, = 0, m,, s,, a,) = 0
(2.6)
gzl(m,+l
= O(b, = 0, m,, s,, a,) = 1.
(2.7)
Similarly, if the technology exists (b, = 1) and was available in the last period (m, = l), it will continue to remain available in the current period. Thus for all t, s,, a, dz*h+1
= lib, = 1, m, = 1, st, a,) = 1.
(2.8)
gh is obtained by the obvious relation that if the technology exists in period t, it will continue to exist in period t + 1. If it does not exist in period t, it will not exist in period t + 1. Thus for all ml, s,, a, &b,+l gb(4+,
= 116, = 1, m,, s,, a,) = 1 = O(b, = 0, m,, s,, a,) = 1.
(2.9) (2.10)
Finally, for every b,, m, @(St+1 = s, - a,(b,, m,, s,, a,) = 1. (F) The initial conditions are m, = 0 and prob. (b, = 1) = pO. (G) The return function v: M
X
A + R is defined as
(2.11) (2.12) I
v(0, u) = u(u)
(2.13)
v(1, a) = ii.
(2.14)
RESOURCE
EXTRACTION
AND
43
UNCERTAINTY
We assume u is a continuous and bounded function and ii < co. Hence u is continuous and bounded. The objective is to choose {a,, t = 0, 1,2,. . . } such that 0 c a, G s,, to maximize (2.15)
when E denotes expectation with respect to (2.2)-(2.12). The problem described by (A)-(G) (problem Z) is a nonstationary dynamic programming problem with incomplete state observations. Yushkevich [24] gives a method of transforming a stationary problem of this type to a stationary problem with observable states. Following his approach, it is possible to show that the problem considered in this paper can also be transformed to a nonstationary problem with observable states. Consider a problem with observable state variables ( p, m, s) when (m, s) is interpreted as before and pt is the probability belief in period t that b, = 1, i.e., the technology exists. Ifp, = p, and m, = 0, then prob. (m,,, = lip, m, from (2.4) and prob. (mt+l = O]p, m, = 0) = p(1 - qr+l) from (2.5). = 0) = P&+1 Further, if m t + 1 = 0, the planner can revise pt and obtain pt + i by using the Bayesian rule. For instance, pr+i = prob. (b,,, = l]m,+, = 0, pt = p) = p(1 - qt+i)J1 So, with the appropriate transition mechanism set up, problem 2 can be P4t+l* transformed to problem 2 as follows. The observable state is (p, m, s) E I x M x S, when I = [0, 11. The action variable is a E A and the correspondence of feasible actions is (Y(S) = [0, s] for all ( p, m, s). The transition function is s,+~ with marginal distributions gp,“, for ( p, m) and g” for s. For all (s,, a,), %(Pt+l
= PO - 4,+1)/l
- P%+A
mt+1
=
OlPt = P, mt = 0, St, at)
= 1 - P4t+1 icl(PI+l
+ PO - 4t+1)/1
- P4t+A
mt+1
=
OlPt = PT mt = 09 sr, at)
= 0 KI(Pt+l
= 1,
izY(Pt+l
= 19 mt+1
mt+1
(2.16)
(2.17)
= lip, = P, m, = 0, St, a,> = p4r+l
(2.18)
l]p, = 1, m, = 1, st, a,) = 1.
(2.19)
=
Also, for every ( pr, m,, s,) and a,
iTh+1 = St- arlpt, m,, St,a,) = 1.
(2:20)
The initial conditions are ( po, m,, so) when p. is given by (2.12). The return function is u(m,, a,) as defined by (2.13) and (2.14). The objective is to choose {G,, t = 0, 1,. . . } such that a, E [0, s,] to maximize
Co if? c 4mt,at> t=o when the expectation is taken with respect to the distributions
(2.21) in (2.16)-(2.20).
44
GAUTAM
BHATTACHARYA
Following the proof-givenby Yushkevich[24, pp. 154-561,for stationaryproblems, problems2 and Z can be shownto be equivalentin the following sense. LEMMA
2.1. If{a:,
t
= &I,... > is an optimal policy for problem 2, it is also an
optimal policy for problem Z with E~u(m,,aj+)=B~u(m,,a:). t-0
t-0
Define the t-periodoptimal value function as Ft(~~~m,,s,)=E
(2.22)
iB’u(m,,a:)lp,,m,,s,
7=t
The right-hand side of (2.22)is the expectedreturn from following an optimal policy from period t onwards.Fo(po, mo, so) is the expectedsocial welfare from following an optimal policy for problem ?! (hencefor problem Z). In fact, the optimal policy a: dependson the statevariablesin period t (p,, m,, s,) and qt, the probability of the technologybeing discoveredin period t. THEOREM 2.1. For problem 2, there exists a function a: = a: (p,, m,, so such that a: is the optimal policy for period t, given pt, m,, and s,. Further, {F,, t = 0, 1,2,. . . } satisfies the following optimality equations.
If m, = 0, I’;(P,t mt9 4) = m~o
I
u(aA + a0
- P,q,+J
(2.23) Ifm, = Lp, = 1, and1;;(1,1,st) = +.
(2.24)
Proof. First, problem z is transformed to a stationary problem Z* in the following way. Let the statevariablefor Z* be ( p, m, s, t) E I X M X S x T when T = { 0, 1,2,. . . }. The actionvariableis a E A = S as in problem2. The correspondenceof feasibleactions is cw(p, m, s, t) = [0, s] as before, the return function is u(m, a) as definedby (2.13)and (2.14),and the initial conditionsare ( po, 0, so, 0). The transition probability function is h with marginal distributions hpmr and h” defined as hPm’(-lp,
h’(-lp,
m, s, t, a) = &‘m(-l~,
m, s, a)
m, s, t, a) = g”( -Ip, m, s, a),
when#‘” and g” aredefinedby (2.16)-(2.20).
RESOURCE
EXTRACTION
AND
45
UNCERTAINTY
If I and S are endowed with relative topology as subsets of the real line and M and T are endowed with discrete topology, we can see that problem Z* is a stationary dynamic programming problem with complete, separable metric spaces as state and action spaces. The correspondence of feasible action is compact valued and upper-semicontinuous and the return function is bounded and continuous. Further, the transition probability function h gives a probability measure on the state space for every (p, m, S, t), is a measurable function of (p, m, s, t) and is weakly continuous in (p, m, S, t). Thus all the conditions for applying the theorem in Maitra [17, p. 2161 are satisfied. Hence there exists an optimal policy for Z*, a*( P, m, s, t).
As problems Z and Z* are equivalent (in the sense of Lemma 2.1), writing a:( p, m, S) = a*( p, m, S, t), we get an optimal policy for problem 2. Further, by Theorem 6 (f) of Blackwell [3], the return function to a*( p, m, s, t) will satisfy the “optimality equation.” Let V( p, m, S, t) be the optimal value function (the expected return from following a*( p, m, S, t)). Writing F,( p, m, s) = V( p, m, s, t) for all t, we get the following. If m, = 0, F,(p,, %+d/U - Pt%+d, 09 St 1, F,(L 1,~~) = m=oda,ss, this proves { 4, t = 1,2,. . . } satisfies the functional
Eq. (2.23) and (2.24).
Q.E.D.
Lemma 2.1 extends the following well-known result in statistical decision theory to the context of optimal resource extraction. The statement of problem Z does not indicate how the initial probability of existence of new technology is affected when information about arrival of the technology becomes available. Problem 2, however, has p,, the current probability of existence of new technology as a state variable and the transition mechanism of this problem specifies a Bayesian revision of pr in every period. Therefore, the equivalence of problems Z and 2 (in the sense of Lemma 2.1) as shown by Yushkevich [24] and others, establishes that revision of the initial probability p. (and all subsequent probabilities) by the Bayesian formula is necessary for deriving the socially optimal extraction policy. This result also indicates the desirability of using Bayesian posteriors for making rational decisions in similar situations characterized by incomplete information. Further, the optimal policy depends only onp,, m,, and q, (also on j3 and the parameters of the utility function). It does not depend on U, the expected value of social welfare obtained after the technology becomes available. It is also clear that more information about the distribution of U (for instance, information about which particular value of U will prevail when the new technology becomes available) will not alter the optimal policy. Although this result seems rather counterintuitive, it is easily explained by the fact that in this model, the stock of resource that remains after the technology becomes available does not influence the level of social welfare that can be maintained from that point of time onwards. Therefore, information about availability and existence of the technology is important for determination of current extraction policy, while information about the nature of the new technology is irrelevant for this purpose.
III. PROPERTIES
OF THE OPTIMAL
POLICY
The explicit forms of optimal extraction policies are found for two special forms of social welfare functions u(u) = log a and u(u) = a*. Assume q. = 0 and define,
GAUTAM BHATTACHARYA
46 foraIlt=1,2,...
PO0 - d(l - q2)*--0 - 4r)+o-Po) yt=PO0 - q,)(l- q2) ***0 - cl-l>+(1-Po) (1- 41N1 - q2) ***0 - 41)Po pz=(1- 41)(1 - q2)***(1- 41)Po +(1- PO) .
(3.1) (3.2)
Here, y, is the probability that the technology will be available in period t (given that it has not arrived yet) and pt is the updated probability at period t of the existence of a new technology given that it has not been available until period t - 1. Given (3.1) and (3.2), the functional E!q. (2.23) can be rewritten as I;;(p,, 0, sI) = fth 4), as f,(p,,
SA = m~OSa,qs,[ ub,)
+ BY,+~~,+~(P,+~~ s, - a,>
+so
- Yc+1F4
- #o-q.
(3.3)
PROPOSITION 3.1. Zf u(a,) = log a,, the optimal policy a: and the optimal value function f,, for the functional Eq. (3.3) are given for t = 0, 1,2,. . . as
(3.4) f,(PV Sr> = ( 1 + i$l(i,,)lOgSt-(l
+ i$lGi,)lOg(l
+ jzlGit)
+i=!+lGitlOgGit +(l- 8)-’ whenforalli=
t+
(3.5)
l,...
Gi, = PiptYt+lYt+2
(3.6)
* * * Yip
4, = Bi-‘Yr+lY1+2
. ’ ’ Yj-It1
(3.7)
- Vi)*
Proof It suffices to check that ( f,, t = 0, 1,2,. . . } as given by (3.5), satisfies the functional Eq. (3.3) for u(a) = log a and a:, as given by (3.4), is the optimal policy obtained from (3.3). Construct H,: Z X S X A --, R as Z-4( Pi, 4, a,) = log a, + B~,+~fr+d P~+~, 4 - a,) + PO - u,+dV(l
- a>
- 1+ it %+Ilog1+ 5 Gi,r+l 1( i=[+2
i=t+2
+
2 i-1+2
+m
Gi,r+llOgGi,1+1
- Yt+W(l
+o
- l-9
-
s>
-‘(
iz24.t+l)E]
1 (3.8)
47
RESOURCEEXTRACTION AND UNCERTAINTY
Thus, from (2.27) and (2.29)
ft(Pt9St)= ~~o~a,&4h~ sr,4 H, is concave in a,. Thus, a; is the unique solution aHpa, = 0 gives
(3.9)
to aHt/aat
= 0. Setting
1+i=t+z. f -$=PYt+1
01
1+ BY,+, + BY,+I
(3.10)
or a:
=
(l
+
ij?ZIGit)~‘st~
The above verifies (3.4). From (3.4) and (3.9) we get, ft(P,t
St) = Ht( ptv stv a:)
-
i
+
E r=t+2
Gi,t+llogGi,t+l
Writing b(Py,+~ + BY,+IZZ,+~ Gi,r+l) = 1% BY~+I + lo&l +
CZ+2
(3.11) is simplified as
+
+PYr+l
2 i=t+2
G,,t+llOgGi,t+l
+(l
-
B)-’
Bv,+l
f r=t+2
%,+I
Gi,r+l),
48
GAUTAM
or, from(3.6)
BHATTACHARYA
and as By,+llog,8y,+l = G,+l,,log Gl+l, t, we can write, f,h
St) = (1 + ;~lc,+g~,
-(l
+
i~lGit)‘og(l
+
iglGil)
+ i=.+lGitlOgC,t +(l - a>-’ This verifies that { f,; t = 0, 1,. . . } satisfies the functional Eq. (3.3) and hence a:, as given by (3.4), is the optimal policy for period t for t = 0, 1, . . . . Q.E.D. PROPOSITION3.2. If u(a) = as, the optimal extraction policy a: and the optimal value function f, for Eq. (3.3) are given for t = 0, 1,2,. . . as -1
ii, =
1 + i
i (G,,)“(‘-“) i=r+l
s,
(3.12)
1 l-6
(Gir)iA1-‘)
+(1 - p)-‘U
2 i
i=f+l
s,!
4,
(3.13) i
when Gi, and 4, are as defined in (3.6) and (3.7), respectively.
Proposition 3.2 can be proven by using the same techniques as used in Proposition 3.1. Hence the proof is omitted. Features of Optimal Extraction Policy
Optimal extraction policies for different situations are described below. a: and &, denote initial extraction for u(a) = log a and u(a) = as, respectively. When p0 = 1, the new technology is known to exist (although it has not been discovered yet). This situation is one of perfect information (PI) and is similar to the one considered by other articles (e.g., Dasgupta and Heal [4]; Kamien and Schwartz [9]; Dasgupta et al. [5]; Dasgupta and Stiglitz [6]; Lehman (151; etc.). In this case
a& so)= (1 + S(l - 4J + S”(l - ql)(l - q2)+ .-.)-lso ?&#,s,)
= (1 +(p(1
- q1))1n1-8)+(p2(l
- ql)(l
- q2)1A1-6)+
(3.14) .**)-$).
(3.15) When 0 -C p0 -C 1, we have the situation of imperfect information (IM) about the new technology (this is the general case analyzed in this paper-the other two
49
RESOURCE EXTRACTION AND UNCERTAINTY
situations arise as special cases of this). Here from (3.4) and (3.12) we have
a,*(po,so>= (1 + BYI+ B2Y1Y2 + *. . )-lso a,(p,,s,)
= (1 +(py1)1/(1-8)+(B2~~Y2)1/(1-b)+
(3.16) *+lql.
(3.17)
When p0 = 0, the new technology does not exist and we have the standard problem of extraction of a known resource stock over an infinite horizon. This is described as the situation with nonexistence (NE) of a new technology. Here,
PROPOSITION3.3.
am so)= (1- B)s,
(3.18)
i&(0, so) = (1 - pi/(*-Q).
(3.19)
For both types of social werfare functions, the initial extraction in
IM will be more conservative than in PI but less conservative than in NE. The initial extraction policy is also a concave increasing function of the initial information about new technology and a linear, increasing function of the initial resource stock. Further, if u(a) = as, an increase in the social index of relative risk-aversion (1 - Q) will reduce the initial extraction.
Proof of Proposition 3.3 is routine and is omitted. It can be checked that the change in initial extraction as a result of a change in initial information (i.e., %&-,/~3p,) or a change in initial stock (i.e., &&/c?s,) is a decreasing function of the index of relative risk-aversion (1 - 8). In other words, the more risk-averse the society is, the less it will respond (regarding current extraction) to an improvement in current resource stock or in current information about future technological progress. The following propositions compare the intertemporal paths of resource extraction for IM and PI situations. PROPOSITION3.4. For every p,,, (i) there exists T* such that for t < T*, a:(p,, sO) -C a:(l, s,,) and for t >/ T*, a:( ~0, so) 2 a:(& so). (ii) there exists T such that for t < T, G,(p,, so) < a,(l, sO) and for t a c q( PI), SIJ 2 a,u, so). Further, 7 is a decreasing function of 6 and p,,. PROPOSITION3.5. For every p0 and E > 0 (i) there exists T** such that for t > T**, la:(p,, st) - a;(O, s,)j c E (ii) there exists T’ such that for t > T’, In,(p,, sr) - a,(O, sr)l < E. Further, T’ is increasing in 6.
In Propositions 3.4 and 3.5 (the proofs of which are omitted from this paper), a:( po, so) = a:( p,, s,) when s, and p, are derived from the following recursive relations St+1
=
St
- 4? Pi, sr)
or
and Pt-10 pt = 0 - %)Pt-1
- 4r) +(I -PI-I).
&+I = s, - “API,
sJ
50
GAUTAM
BHATTACHARYA
Initially, accordingto Proposition3.4,extractionon the PI path will be greaterthan that on the IM path. However,after a point of time, if the technologyis still not available,the IM path will show more rapid extractionthan the PI path. The time where this switch occurswill be deferredif the society becomesmore risk-averse. Accordingto Proposition3.5,as the probability of existenceof new technologyfalls over time in the IM path (if the technologyfails to appear),the IM path, after a period of time, will come close to a NE path having the same remaining stock of resource(this is not the NE path given by (3.4)or (31.2)).Time period T’ will be a decliningfunction of the socialindex of risk-aversion.Thus, Propositions3.4 and 3.5 explain the effectsof revision of initial information about the technologyon the extractionpathsover time. Apart from the information about existenceof a new technology,anothersource of uncertaintyis about the time of discoveryof the new technology.In both PI and IM cases,if {q,} (the set of conditional probabilities of availability of the new technologyat time t) is suchthat q1 # 0 for all t, then the resourceis neveractually Lexhausted if the technologyis not available.Howeverif { qJ} = { ql,. . . ,qT}, thenin the PI casethe resourcewill be exhaustedat time T. In caseof other papersin this areathat assumePI but wherep dependson the stockof the remainingresource,the resourcewill be exhaustedsometime after T. On the other hand,in the IM case,the resourcewill not be exhaustedat a finite time if { qT} = { ql,. . . , qT} and 0 < p. < 1. This is becauseif p. = 1, the technologyis believedto be definitely availableby T; but if p. < 1, the technologyis assumedto be discoveredby T only if it exists. So, if the technologyfails to appear,p, is reviseddownwardsover time and for t 2 T, pr = 0. In fact, the lower p. and the higher the index of risk-aversion,the largerwill be the remainingresourcestock at Tin the IM case(if the technologyis not discovered by 0. The effect of an increasein the conditional mean arrival time of the technology may be examinedin the following simplified situation.Let qt = q for all t. Thus, for instance,if p. = 1, the probabilities of arrival at time t = 1,2,. . . are given by (q, (1 - q)q, (1 - q)‘q, . . .).3 Here an increasein q will imply an increasein mean arrival time of the technologyboth in PI and IM cases.Here, from (3.4)and (3.12), a$( po, so, q) = (1 + B(po(l +/3’(po(l
- 4) + 1 -PO) - q)3 +(1 -po)
+ . ..>-‘so
- PO))
(3.20)
p()(l - q) + (1 - p0))1f11-6)
ao(po, so, q) = (1 + jF-“)( +gz/(l+(
+ B2( Po(l - q)2 +(I
po(l
- q)2
+(l - P~)~‘~-‘) + . 1.)-lso.
(3.21)
The following propositionmay be proven. PROPOSITION3.6. Anti increase in conditional mean arrival time of the new technology will increase init&l extraction. The amount of change in initial extraction corresponding to a change in q will be higher in the case of perfect information ( p. = 1) than 3Lehman [15] has derived two results comparable to this paper’s PI situation when q, = q for all t. Considering a general social welfare function, he shows (in bis Theorems 1 and 2, respectively) that the initial extraction policy for PI is less conservative than that for NE and an increase in q will increase the initial extraction in the PI case.
RESOURCE
EXTRACMON
AND
51
UNCERTAINTY
in the case of imperfect information ( p0 < I). This rate of change will also be a decreasing function of the index of relative risk-aversion. Finally, an attempt is made here to answer the following questions. In spite of different kinds of uncertainties involved in the framework considered here, is it possible to define a certainty-equivalent resource stock? Further, is there any single number that can, be interpreted as an index of the amount of uncertainty in the system? For every pO, sO, and E, the certainty-equivalent resource stock s’ is defined implicitly by fAO7 s’> = fo( PO, %I.
(3.22)
Thus, keeping the social welfare at the same level, the current resource stock and all the uncertainties involved in the IM case may be replaced by a situation with no uncertainty and the certainty-equivalent stock s’. If u(a) = as, (1 +(/3yJ1A1-6)
+ . . .)Y
+(1 - /9-G-4
s’( PO, +J, a = (1
-
2 JIO 1’6 i i=l 1).
p1/(1-8’)!+
(3.23)
However, if so is replaced by s’, the initial extraction of the resource will be h,(O, s’) = (1 - pi~1-8))s’.
(3.24)
Comparing (3.23) with (3.24), it can be checked that a,( po, so) -C a,(O, s’). Similarly a$( po, so) < a,*(O, s’). So, if the certainty-equivalent stock is introduced, keeping the social welfare at the same level, the amount of initial extraction will increase. The amount of uncertainty in the system may be measured by the difference between s’ and so. This measure (s’ - so) is higher in case of PI than in case of IM. Thus, we may conclude that there is actually less uncertainty when the information about existence of new technology is not perfect. The reason for this is that in case of IM, the hope for achieving the new technology in the future is not as strong as in PI, so the amount of uncertainty is less. For the same reason, as p. falls, i.e., initial information becomes more imperfect, the amount of uncertainty decreases. It should also be noted that (s’ - so) is a decreasing function of so. The amount of uncertainty in the economy depends on the initial resource stock and is smaller for a larger initial stock. Thus for large stocks of resource the optimal plan is not very sensitive to the assumption of perfect or imperfect information about the new technology. However, as stocks become smaller, the nature of current information about existence of a new technology becomes more important. IV. THE COMPETITIVE
ECONOMY
The well-known Hotelling’s rule states that price of a unit of stock of resource (net of extraction costs) will increase at a rate equal to the rate of interest in equilibrium in a competitive economy. In papers that consider the’ situation of perfect informa-
52
GAUTAM
BHATTACHARYA
tion (Dasgupta and Heal [4], etc.), the rate of increase in this price was found to be larger than the rate of interest. In this paper we will consider equilibrium conditions for some competitive economy whose allocations correspond to the optimal allocation of the last section. Consider a competitive economy where every producer or consumer has identical information about the new technology (i.e., pa and { qr } as defined earlier). Thus, for each agent, the probability that the technology will not be available in period t is yr as defined by (3.1). In this economy, suppose contingent markets for delivery of the resource for all future dates exist at least for those events where the technology is not available. Let r be the equilibrium interest rate and rl be the equilibrium current price of a unit of resource at period t if the technology is not available yet. Then the arbitrage condition that will hold in any dynamic equilibrium and that will make producers indifferent between extracting the resource between one period and another is given by, for all t = 0, 1,2,. . . (1 +
d?
=
(3.25)
Yt+1?+1*
Further, if Gt is the equilibrium amount extracted in period t and.D-‘(a) inverse market demand function, then for all t = 0, 1, . . .
is the
D-l@,)
(3.26)
In particular, if D-'(a)
= 17,.
= Sag-‘, then the total instantaneous surplus will be p-‘(a)
da = as.
(3.27)
Finding a program that maximizes EX/~'Q~ will lead to the competitive librium allocations for this economy, if p is taken as (l/(1 + y)). a, =
(
1 + p(y,+*)1A1-6)
+(/32yt+*yt+z)1A1-8)
+ . * *)-ls,
equi-
(3.28)
and lr, = 6 ( 1 + ( B(y,+$@@
+ * * .)1-8(S,)6-1.
(3.29)
Equation (3.29) or (3.25) shows that the resource price V~in fact increases at the rate ((l/By,) - 1) in period t. In case of PI, this rate will be higher, equal to ((l//?(l - qr)) - 1). When the information about the technology is imperfect, the rate of increase in price is lower than in PI although it is still higher than the rate of interest (r = ((l//3) - 1)). Further, this rate of growth of price falls over time as yr + 1 and gradually approaches the rate of interest if the technology fails to appear for a long period of time. This is the kind of modified Hotelling’s rule implied by the analysis of this paper. It is still true that if all contingent markets exist, the expected rate of increase in price will be equal to the rate of interest, but here we are interested only in the actual rate of increase in price if the new technology is not discovered. Further, if the inverse market demand function is D-'(a) = &as-l, the absolute value of elasticity of demand is constant and given by l/(1 - 6). Using this fact,
RESOURCE
EXTRACTION
Proposition 3.3 and the following proposition can be easily proven.
AND UNCERTAINTY
remarks and Proposition
53
3.6, the following
PROPOSITION 4.1, The initial equilibrium price will be more sensitive to a change in initial information about new technology and initial stock if the demand is less elastic. An increase in mean arrival time of the new technology will have a larger effect on the initial price if the demand is more elastic.
V. CONCLUDING
REMARKS
Throughout this paper, it was assumed that after the technology is available, the resulting steady state is not affected by the remaining stock of resource. Relaxing this assumption will maintain the validity of existence of an optimal stationary policy depending on p,, m,, and s, (Theorem 2.1). However, explicit solutions for the extraction policy are difficult to obtain in this case. Finally, it is possible to treat technological change as endogenous in this framework by making the distribution of {q,, t = 1,2,. . . } depend on the research and development expenditures incurred by the society. Several authors (Kemp and VanLong [13]; Stiglitz and Dasgupta [23]; Kamien and Schwartz [lo]) examine the nature of optimal extraction policy and optimal policy for research and development expenditures in the context of perfect information about existence of new technology. The framework in this paper is likely to indicate a more conservative extraction policy and a larger expenditure on research to be socially optimal. REFERENCES 1. K. Arrow and S. Chang, “Optimal Pricing, Use, and Exploration of Uncertain Natural Resource Stocks,” Discussion Paper No. 675, Harvard Institute of Economic Research, Harvard Univ. (1978). 2. G. Bhattacharya, “Two Essays in the Economics of Uncertainty: The Importance of Learning in Planning with Exhaustible Resources and Optimality of Equilibrium with Incomplete Markets,” Ph.D. Dissertation, Univ. of Rochester (1980). 3. D. Blackwell, Discounted dynamic programming, Ann. Math. Statis. 226-235 (1965). 4. P. Dasgupta and G. Heal, The optimal depletion of an exhaustible resource, Reu. Econ. Stud. (Symposium) 3-28 (1974). 5. P. Dasgupta, G. Heal, and M. Majumdar, Resource depletion and research and development, in “Frontiers of Quantitative Economics,” Vol. 3 (M. Intrilligator, Ed.), North-Holland, Amsterdam (1977). 6. P. Dasgupta and J. Stiglitz, “Uncertainty and the Rate of Extraction under Alternative Institutional Arrangements,” Technical Report No. 179, Institute of Math. Studies in the Social Sciences, Stanford Univ. (1976). 7. S. Deshmukh and S. Pliska, Optimal consumption and exploration of nonrenewable resources under uncertainty, Econometric4 48, 177-200 (1980). 8. R. Gilbert, Optimal depletion of an uncertain stock, Reu. Econ. Stud. 47-57 (1979). 9. M. Hoel, Resource extraction when a future substitute has an uncertain cost, Rev. Econ. Srud. 637-644 (1978). 10. M. Kamien and N. Schwartz, Optimal exhaustible resource depletion with endogenous technical change, Reo. Econ. Stud. 179-l% (1977). 11. M. C. Kemp, How to eat a cake of unknown size, in “Three Topics in the Theory of International Trade,” (M. C. Kemp, Ed.), North-Holland, Amsterdam (1976). 12. M. C. Kemp and Ngo VanLong, “Exhaustible Resources, Optimality and Trade,” North-Holland, Amsterdam (1980).
54
GAUTAM
BHATTACHARYA
13. M. C. Kemp and Ngo VanLong, On the development of a substitute for an exhaustible natural resource, in “Economic Theory of Natural Resources,” (W. Eicehorn, et al., Edc ), Physica-Verlag, Vienna (1982). 14. T. Koopmans, Proof for a case where discounting advances the doomsday, Rev. Econ. Stud. (Symposium) 117-120 (1974). 15. D. E. Lehman, “Doomsday” reconsidered, Resources and Energy 3, 337-357 (1981). 16. G. Loury, The optimum exploitation of an unknown reserve, Rev. Econ. Stud., 621-636 (1978). 17. A. Maitra, Discounted dynamic programming on compact metric spaces, Sunkhya 30, Part 2, 211-216 (1%8). 18. A. Manne, Waiting for the breeder, Rev. Econ. Srud. (Symposium) 47-66 (1974). 19. R. Pindyck, Optimal exploration and production of a non-renewable resource, J. Pol. Econ., 86, 841-862 (1978). 20. V. Smith, An optimistic theory of exhaustible resources, J. Econ. Theor. 9, 384-396 (1974). 21. R. Solow, Intergenerational equity and exhaustible resources, Rev. Econ. Stud. (Symposium) 29-45 (1974). 22. J. Stightz, Growth with exhaustible resources: Efficient and optimal growth paths, Rev. Econ. Stud. (Symposium) 139-152 (1974). 23. J. Stiglitz and P. Dasgupta, “Market Structure and Resource Depletion,” Research Memorandum No. 261, Econometric Research Program, Princeton Univ. (1980). 24. A. A. Yushkevich, Reduction of a controlled Markov problem with incomplete data to a problem with complete information with Bore1 state and control spaces, Theor. Probub. 153-158 (1976).