Control theory applied to natural and environmental resources an exposition

JOURNAL

OF ENVIRONMENTAL

Control

Theory

ECONOhfICS

Applied

AND

MANAGEMENT

to Natural

and

4, 1-24

( 1977)

Environmental

Resources

An Exposition VERNON Department

of Economics,

of Arizona,

University Received

SMITH

L.

April

Tucson,

Arizona

85761

3, 1976

Four control theory models of natural and environmental resource use, drawn from the existing literature, are developed in a manner to emphasize their technical and decentralized interpretive similarity. Renewable, nonrenewable, and amenity resource use are treated as closely related problems of optimal (biological, earth material, ecological, or environmental) capital allocation over time. Thus nonrenewable resources, and the problem of exhaustion, are just limiting (zero growth) cases of renewable resources, and the problem of species extinction. Just as exhaustion can be optimal, extinction can be optimal. Waste recycling is treated as part of the problem of optimal regeneration of “clean” environmental capital; wilderness use as a problem of managing the regeneration of ecological capital.

1.

IXTRODUCTIOS

This paper provides expository and interpretive applications of control thcor) to the dynamic economic analysis of natural and environment,al resources. The analysis of such resources is essentially a part of capital theory, broadly defined, and by providing parallel treat,ment,s of replenishable, exhaustible, recreational, and environment,al resources, the unit,y of concepts and techniques in rcsourw capital theory is emphasized. No atkmpt is made t,o survey in dept,h the littrat,ure of natmural resources, OI even t#hat portion of the literature using control theory methods. The recent’ survey by Fisher and Peterson [9] provides a comprehensive review of t,his literatuw. The purpose here will be t,o provide a rat.her detailed exposition of the assumptions, the modeling techniques, and hhr reprwentat,ion of solutions using phase diagrams. The emphasis is on the t,echnology, production, and consumption of natural resource products in simple two-commodity dynamic economic models. Wtr a.bst,ract c&rely from nonresource capital inputs (and the Ramsey problem of capital accumulation) so extensively t#reat(ed in thr: literature of growth theory [ZS]. The reader is referred to the Symposium volume (Review of Ecor~mr~ic Studies, 1975) especially the papers by Heal and Dasgupta [12], Solow [2X], and Stiglitz [29] for analysis of the implications of introdwing natural rwnnws as an input into models of capital accumulation. Although t,he techniques are those of control theory, the point, of view is not) t,hat, of a central planning board except, as an expository device. Indeed the emphasis is almost entirely upon decentralized int,crprctations of the control theory

Copyright All rights

0 1977 by Academic Press, Inc. of reproduction in any form reserved.

ISSh I 0095-0696

2

VERNON

L.

SMITH

models. In this way an attempt is made to relate some of the implications and functioning of the models to some qualit,ative properties and observations of particular natural resource industries. The decentralized interpretation of control theory models has been criticized on grounds of nonglobal st,ability, the problem of choosing the right path without perfect future markets, or of the need for perfectly realized expectations [ll, 241. But these criticisms merely accentuate the imcompleteness of dynamic theory, and apply equally to centrally controlled economies. The information conditions under which centrally planned programs converge to stable equilibria, are also those likely to produce no difficulties for the decentralized economy. Unless one simply asserts an omniscient government “somehow” able to ensure motion along the optimal trajectory [ll, pp. 645-6461, decentralized and centralized interpretations of optimal control models seem to be equally good (or bad). 2. REPLENISHABLE

RESOURCES

A. A Biological Capital Model Let Q(t) be the biomass at time t of a homogeneous species of trees, fish, or cattle which in the natural state grows according to the law & = IcG(Q). The growth function G(Q), Q 2 0, is concave with G(0) = G(&) = 0, G’(QO) = 0, G”(Q) < 0, where 8 is the natural state equilibrium biomass.’ The (average) growth intensity of the resource is defined by the function kA(Q), where A(Q) = G(Q)/& is monotone decreasing. The parameter k characterizes the growth potential of the species and therefore it’s “vulnerability” to predation or harvesting by man. The economy produces two products with ql(t) the output rate at t of an “ordinary” commodity such as corn and qz(t) the harvest output rate, e.g., meat, from the biological resource. Labor, available in fixed supply i; over time, is allocated in the amount Lj(t) 2 0 to product i with L = L,(t) + L,(t). The concave production functions are pl = fl(L~), qz = fz(Lz, Q), and the growth rate of the resource net of harvest becomes $ = kG(Q) - qz. If 6 is the discount rate that characterizes the society, and the social evaluation criteria is u = ul(ql) + uz(qz) + u3(Q), then the allocation problem over time is to co maximize ue-%t Ll(t). Lz(t). m(t), m(t) i o subject to Ql

=

qz =

(2.1)

fl(L), fiW2,

Q>,

(2.2)

E = L1 f L2,

(2.3)

& = kG(Q) - qt.

(2.4)

1 We emphasize growth in the biological sense, but growth laws of the form & = F(Q) may apply to any replenishable resource such as an air mass, a water reservoir, or a ground water supply. For example, a reservoir or ground water supply replenished by run-off water at a constant rate, T, and subject to evaporative and/or absorption losses proportional to the stock, say kQ, gives a growth law &, = F(Q) = r - ,Q. For additional dynamic treatment of the replenishable resource problem the reader may wish to refer to [4, 5, 10, 18, 21, 227.

CONTROL

AND

NATURAL

RESOURCES

3

Substitute from (2.1) and (2.2) and let the control variables be L,(t), L(t) ; t,hc: auxiliary cont,rol associated with (2.3) is w(t) ; the state variable is Q(t) ; and t#hc auxiliary st’at#e variable is E(1). Then t!he Hamihon-Lagrange current, value criterion [l] can be writt,en

H = w[f~(L>l + dfz(L,

Cc?)] + G(Q)

,211 opt,imal t’raject’ory [L,(t), L(t), w(t), Q(t), t(t)] must sat,isfy (2.3), (2.4), aH/aLi 2 0 (if < holds, then Li = 0), $ = St - (aH/aQ), and the transversality conditions : lim,,, e-6Lt(t)Q(t) = 0, lim,.,, e-*‘l(t) >_ 0. (Refer to [l, Proposition 6; Propositions 7, 8, and 9 also apply]). c?H/~Q, thcsc conditions can be nritt.m in the form u1’ 5 lP/jlt (212’ - 0 I 2L’l(df?lam g = (6 - k(;‘)(

Evaluating

(if <, L1 = 0, I,2 = i;), (if <, L2 = 0, A1 = L). - Ua’ -

(,uy’ - 6) (c?“f,/l3Q).

dH/dL;

:tntl (2..i) (2.6) (,“.7)

In the analysis of this section we will assume t,hat the stock itself yields no public good utility so that uar = 0. In interpreting these conditions, w is t,he wage earned by scarce labor, 7~~’ the price of corn, and uz’ t#he price of a harvested unit of t#he resource such as lumber, beef, or codfish (all valued in utility units). The interpretat,ion of (2.6) is particularly important to an underst,anding of the functioning of markets in natural resources. i(t) is the marginal value, or price, of an unharvested unit of the resource, i.p., the price of living biological capital. In t,hc case of domesticat,cd animals such as cattle and hogs, where animal propert?- right’s are secure and well defined, it is the going market price of such meat on t,he hoof. In the case of frctc standing Cmber in the National Forest Preserve, offered for competitive bidding to lumbering firms, it is t,he competitive bid price for live trees. Where there is no property right or market for the unharvested resource, as in an open-accws common propert#y ocean fishery, optimal conservation of t#hc resource over time requires an implicit price C;(t) to be associated with a live fish at 1. Hence> (2.6) requires a utilized natural resource to be harvested at a rate such t’hat unit harvest value ,net of unharvested value (ua’ - t) be t>qual t,o marginal harvesting c,ost.2 Fut,ure harvest values arc carried to the current, harvest decision by the price E which determines in turn the extent of “conservation” of the stock. It, is t,hc fact that beef commands a price on the hoof as well as in the supermarket, that stays the slaught,er; while it is the fact t’hat t#he King Crab commands a posilive price only in t,he supermarket t,hat endangers t,hc spwics. Of course this int’rrpretat,ion requires t,hat all value associat.td with a unit of biological capital be private and carried wit#h the marketable property right, in t.he capital. This condit,ion requires u3’ = 0 in t,hc above model, i.cb., t.htk st,cwk has no public good value, as in the case of forests with scenic values not, reflwt,ed in the market for live t,rew. 2 This formula is consistent with an incentive explanation of the shutdown of the packing houses during the go-day price freeze in 1972. The freeze fixed the price of beef at retail, but did not change expectations about future beef prices and the control of inflationary conditions. Hence the price ( of beef on the hoof continued to rise and this had the immediate effect of choking off the flow of animals to the packing houses.

VERNON

L. SMITH

/ 45

line

3

FIGURE 1

Equation (2.7) is the intertemporal efficiency condition familiar from ordinary capital theory: It requires the efficient price of biological capital to rise at a rate equal to the difference between the net interest cost of a unit of capital, (6 - kG’) f, and the net value of a unit of capital, us’ + (u2’ - ,$)(dfJaQ). The int.erest rate, 6 - ICC’, in this calculation is reckoned net of biological growth, IcG’, which serves as a capital appreciation rate (deduction from interest) when G’ > 0 and a capital depreciation rate (addition to interest) when G’ < 0. In order to characterize solutions to conditions (2.3)-(2.7) in the state space ([, Q), we introduce two simplifying assumptions which permit this system to be reduced (by inversion) to two differential equations explicit in (E, Q) : 1. u = q1 + 2)q2, i.e., ul’ = 1, uq’ = v, us’ = 0. 2. Constant returns to scale in both production subscript on Lz, the production functions are written

fl(L

- Lz) = r(E - L),

fz(-L, Q) = &4(x>, 4’(O) = 40’ i Q),

technologies.

Dropping

fl’ = Y, aj2 z = 4'(x), lim 4’ (x) = 0,

5-w

ah = 4(x) - N'(x), aQ

x = L/Q.

the

CONTROL

AND

NATURAL

RESOURCES

Gb /-

7-.

-. . . 2

FIGI.RE 2

4 is the output intensity (the harvest per ton of living biomass) of thr natural resource indust,ry, w&h C#B > 0, 4” < 0, and :c is the input, intensit’y (labor hours per ton of biomass). Using these assumptions (2.3)-(2.7) can be stated in the more compact, form I, = 0,

OlL 1 6 = [S - kG'(Q)][ - (v - f)[c#a(.c)- .t$'(,r)], (u -

EWb)

2 7,

Q,= ICC;(Q)- Q+(x) = Q[kA(Q) - +(a,)].

(2.8) (2.9) (2.10)

From (2.8) the state space ({, Q) can be partitioned into three part,s corrrsponding to an economy (see Figs. 1 and 3) that specializes in q1 (I, = 0), a mixed economy (0 < I, < L), or an economy that specializes in q2 (L = L). From (2.8), define where b’(Q) > 0, limv +0b(Q) Eu = v - Y/40’, and b(Q) = u - Y/~‘(E/Q), = --30 , limQ+, b(Q) = &. Then, I. L = 0 for all [ 2 to, Q > 0, and in t,his region the economy specializes in producing ql.

VERNON

6

/

’

L. SMITH

i’ /S(Q) FIGURE 3

II. 0 < L < z for all 50 > ,$ > b(Q), Q > 0, yielding a mixed economy with ~1 > 0, 92 > 0. In this region (2.8) has the inverse I(: = +‘(-l)[r/(~ - i)]. III. L = ,? for all 5 I b(Q), Q > 0, and in this region the economy specializes in producing q2. By making the substitutions (2.8)-(2.10) reduce to

i = 18- ~G'(Q)lt,

for L that apply to each of these three partitions,

5 1 Co,Q > 0.

= [s - kG'(Q)][ - (v - [) 4 t#~'(-l) ( [ (2.11)

= D - kG'(Q)IE- b - r,[m($) - $#@1,

& = kG(Q),in2 to = Q[WQ)

Q>

B. Derivation

0.

b(Q) < cf<

- +#+&)j},

= QIkA(Q) - ~(~/&)I,

t 5 b(Q),Q > 0.

to, Q > 0.

(2.12)

5 I b(Q), Q > 0.

of the Phase Diagram

Consider the set of points ( (F, Q) 1Q = O] defined by (2.12). These points represent biological equilibria, where the biomass harvest intensity is just equal to

CONTROL

AND

NATURAL

FIGURE

t,he biological growt,h intensity st#ationary state supply function Q = S(E) = 0, E 2 50,

4

of the resource, and can be interpret,ed of capital, Q = S(E), given by

as t,he

Q > 0, b(Q) > t < to,

= +f4'(-"(2)]}, = &I, t = b(Q), = Qz, 5 < b(Q),

RESOURCES

Q > 0,

(2.13)

Q > 0, Q > 0,

where &I (Qz) is the smallest (largest) Q such that k A (Q) = 4(5/Q). The derivation of X(t) is illustrated graphically in Fig. 1. The output intensity funct’ion c#J(~#J’(-~)[~/(v- {)]I is strictly decreasing in { (- 00 < C;5 &J as shown and assumes the value zero for all .$ 2 lo. The growth intensit,y function k A(Q) is maximum at kA (0) = kGo', decreases with Q, and becomes zero at, &. Consoquent,ly, for each [ in t#he int,erior region [b(Q) < { < to, Q > 01, we associate a corresponding Q (in quadrant 4, Fig. 1) such that output intensity (in quadrant 1) equals growth intensity (in quadrant 2). If +(L/Q) > kA (Q) for all Q > 0, t,hen S(E) lies entirely in t,he interior region II. If +(E/Q) intersects kA(Q) it must do so in at least two point’s since +(L/Q) is decreasing and limQ+a 4 (A/Q) = 00, lim Q+- 4(E/Q) = 0. Let the intersection points be (Q1, Q2) at corresponding prices (41, .$2). Then S(t) will be discontinuous at &, as shown in Fig. 1. For any .$in the interval & < E < &, and Q in the interval Q1 < Q < Qz, we have +(z;/CJ)

VERNON

I,

-

L. SMITH

I i

d FIGURE

5

kA(&). Hence, the resource grows until biological diminishing returns reduces the growth intensity to kA(QJ = 4(L/Q), the harvest intensity. It should be noted that for given technology, +(E/Q), and species growth behavior, kA(Q), there is always an economy (i.e., an L) small enough so that c$(~/Q) < kA(Q) over some interval Q1 < Q < Q2. Hence, a discontinuity in S(t) at some .$1will be a characteristic feature of sufficiently small economies. From (2.12) and (2.13) it is easy to confirm that Q $0 according as Q 5 S(l). Hence when the biomass is below the supply function there is net biomass growth, and when it is above the supply function there is a net decline. The function S(t) is sufficient to describe an equilibrium path in t,he case of a free-access resource. For such a resource there is no market for live units of the stock, E = 0 for the natural resource industry, and starting at any initial point on the Q axis (Figs. 1 and 3) the resource stock tends to the level Q** = A-‘{ (l/k)$&$‘(-‘)

(y/v)]}.

If fos > 0, where S(&“) = 0, then Q** = 0 and the free-access solution terminates with species extinction (as in Fig. 5). Now, using (2.11), consider the set { (t, &I f = 0} which defines a stationary state demand funct,ion for biological capital, Q = D(l), representing points for which t’here are no capital gains or losses in biomass investiment. In region II this function is given in closed form by

where

(2.14)

CONTROL

The

AND

NATURAL

RESOURCES

0

derivation

of D(E) is illustrated graphically in Fig. 2. The funct’ion re resenting interest net of Dhe relative marginal valut 6 - C(v- t>/tl o(U, P product of capital, begins at .$ with (8, 0) simuhaneously satisfying (2.14) and iv - G$'i~/Q> = Y, (i.e., where 5 = 0 at, t.he boundary separating regions II and III), and increases with E t,o the value 6 at, lo since TJ(&) = 0 (WC quadrant 1, Fig. 2). The funcQion kG’(Q), representing thr marginal biological growt,h ratr, is maximum at h-Go’, decreases with Q, and becomrs zero at Q” (scle quadrant 2. Fig. 2). For each E, we associate a corresponding Q (in quadrant 4) such that interest, net of t,he relative value marginal product, of capital (in quadrant 1) equals the marginal growth rat,e of t,he resource (in quadrant 2). If 6 > kc;,,’ as in the illustration of Fig. 2, D(t) = 0 for all i >_ to”. But if 6 < kG,’ the reader ma> verify that D(4) = &a for all t: > &,, where kG’(Q~) = 6. Hrncc at, zero not, intwest,, 6 - liCi’(Q~), the demand for biological capital is &a, independent of the print. Setting 5 = 0 in (2.11) for region III, D(t) is dcfincd implicit.ly b)

[S - kG'(Q)][ - (u - 6) [$I (i

) - ( $) 4 ($1

= 0.

In this region D(E) begins at t,he point, (f, 0) in Figs. 2 and 3 and dccrcascs

asymptotically to zero as shown in Fig. 3. Figure 3 illust’rates an optimal pat#h passing through the point, ($, 0) and t#hta stationary state equilibrium (p, Q*) for the case in which S(t) lies entirc>ly in the interior region II. An intersection point, Q* = D( [*) = N( $*) 2 0 alwaJ,s exists in this case. However, we could have { gd 5 bB, whe (0” 5 to, f I to” 5 b. If Eod < [OSthe opt#imal pat,h terminates wit,h resource cxtinct,ion, Q* = 0.

C. SomeExtinction Economics The conditions for resource extinction arc part8icularly interesting in viw of t,he current, widespread concern for t,hreatened species such as t#ho blue whale. It has been argued correctly that ext,inct,ion can occur in the abscncc of common property free-access harvesting, but the inference that this is undesirable [ti] has been based on ext,ra bheoretical crit’eria. These issues can be treated formall? wi-it,h the above model if WC examine t#he condit,ions defining the intcwept,s l,,” and to8. From (2.13) w-e have lo8 defined implicit,ly bs

k/i(o) = kGo'= 4[@

(,')I.

Now regard (2.15) as defining EoSas a function of the resource growth k, say to8 = to”(k). Applying the inverse operation to (2.15) we gc% .co”

=

.&s(k)

E

2, -

(2.15) potrrkial,

---?

c$[q’(kGo’)]’ &l*(k) = --2. &,d(k) is dccrcasing and thrwwit#h limk,o &l(k) = 50, and limk fore its graph will appear as sho\Gz in Fig. 4. Similarly, from (2.14), Eodis determined implicitly bs (2.16)

VERNON

10

L. SMITH

which defines a function tad(k). At Ic = 0, let 0 < EOd(0) = .$+< to as can seen from Fig. 2. tOa is increasing in k and when k = 6/Go’ we have lad = with 77(&J = 0 in (2.16). Hence, the graph of EOd(k)will appear as shown in Fig. It is obvious that EOd(k)and to*(k) must have an intersection at some k = IcO> thus proving,

be .& 4. 0,

PROPOSITION 1. Given a society’s technology 4 and y, values v, and impatience rate 6, there exists a species growth potential, k (given Go’), low enough for the optimal trajectory to end with extinction, i.e., a k low enough to yield to8 2 Eod. For any society satisfying the assumptions of the model, the species harvested can be sufficiently vulnerable for extinction to be optimal. The blue whale could be an extinction candidate even if it were not a common property resource. However, given all other characteristics there is an impatience rate low enough to preclude the optimality of extinction.

PROPOSITION 2. Given k, Go’, 4, y and v, if the impatience rate is low enough (6 5 k/Go’), extinction will never be optimal. This follows immediately from t’he derivation of D(E) when 6 2 kG,,’ in which case D(E) > &a > 0 for all [ < to. In Fig. 5 this means that there is no price in region II at which the demand for biological capital is zero, and therefore the optimal stationary state stock of the resource is positive for all supply functions X(0.

PROPOSITION 3. Given v, 4, y ; if the maximum growth intensity of the resource is large enough the species will be protected from extinction independently of 6, and even under free-access harvesting. From (2.15), if kGd > 4[+‘(-‘)(y/v)], then [OS< 0. Consequently for all D(t) functions (and values of 6) an optimal path will not terminate with extinction. The species is economically protected even if [ = 0. This is illustrated in Fig. 3, and explains why squirrels and rabbits are not likely candidates for a list of endangered species. Comment. An equivalent form of the proposition requires harvesting cost to be high enough, given kGo’. This is because kGi > 4[#/-l)(y/v)] implies and is implied by y/v > +‘[4-‘(kGo’)] where y/ v is resource harvesting cost per unit value under free access. If squirrels are not on the endangered species list we cannot attribute this fact singly either to rapid growth or to hunting cost. D. Resource Stocks with Utility

Value

So far the analysis has assumed that the resource has no utility value. If we let the utility function be written u = 41 + vqz +

us(Q),

the differential equation (2.11) is altered by the addition in region II we now have f = I3 - kG’(Q)It

(public good)

- us’(Q) -

of -us’(Q).

That is,

(v - .h([).

If ~~‘(0) is finite, and 6 > kGo’ - [ua’(O)/{o], then there will be some lad < to such that D([od) = 0, as in Fig. 3. Consequently, Proposition 1 will still hold.

CONTROL AND NATURAL RESOURCES

11

However, if lim Q-0 us’ (Q) -+ e , then D(l) >_ Q’ > 0 for all E < .& whew Q’ satisfies f: = [S - IcC’(Q’)]& - u3’(Q’) = 0, and an optimal pat,h must) twminate short of extinct,ion. Hence, protecting the blue whale, what8ever the cost,, \vould bc justified by societ,y’s urgent desire that, t#he species not bc lost, (SW [30] for a treatment, of the utilit,y value of exhaustible rwourcc stocks). Hut, it is clear that not, everybhing can be saved, and society must, make choices. If all t#hc mammot,hs, mast,odons, bison, and ground sloths were still lumbering about! some would surely have to go .3 A major value to man of preservation [15] is perhaps to retain the gene pool, the genetic intelligence accumulat,ed over millions of years of trial-and-error experience. Thus the American bison, t,hat narrowly escaped cxtincbion, is now being successfully crossed with bwf cattle, and maJ. 1~ come a more efficient source of preferred animal prot,ein. 3. EXHAGSTIBLE

RESOURCES4

Proposition 1 asserts that any replenishable resource can be sufficiently slob\ growing to behave like a fixed-stock exhaust’ible wsourcc. The slow-growing Alaskan King Crab is perhaps such an example. If WC let k approach zero t,hcb replcnishable resource model of Section 2 becomes a model of exhaust’ible r+ sources. The resource diffcrenQia1 equation (2.4) is now & = - J’s(Lz, Q) and Q can only decline if there is extraction. Out#put q1 can be intrrpret,ed as any nonresource-based commodit,y or, more specifically, as a substitute for t,he rcsourw [26] such as an altcrnat#ive source of energy (solar power) if Q is a coal or petroleum stock, or as an alt.ernative souwc of matwial if Q is an ore st&ock. The price E is now interpret,ed as the value of a unit of “reserves,” i.c>.,unmiwcl iron ore or unrecovered petroleum deposit,s. Condition (2.6) requires an active: oil field t’o flow at a rat,c such that the value of a unit of refined petroleum l/et of its reserve value is equal to the marginal cost of recovery and refining. H(wc any increase in l relative to uz’ requires increased conservation of the rcsourw. During t,he 1972-1973 energy “crisis,” pet,roleum product prices nrre controlled, but of course t)he implicit value of reserves increased as f>xpcctations of futuw shortages became widespread. This creat,ed intcnt8iws to hoard petroltwn st,ocks which tended to exaggerate the short-run supply problem. Increased conwrvation of such stocks was entirely consistent bvith optimality, but, only to thcl (‘stent that the current market price (up’) was unconstrained. Using the assumptions of linear utility and constant, returns t’o scalp, conditions (2.S)-(2.10) become L = 0, OlLI:I;, (3.lj (v - EWb-9 2 Y > L = 5, i i

f = Q - OJ- ‘9cde> - +‘(~>I, & = - Qd~b),

(3.2) (3.3)

3 This point, it seem to me, exposes the danger of arguing that preservation is justified because many (naturalists, environmentalists, and earth protectors) find value simply in the knowledge that certain biological species exist whether or not they are “useful” or enjoyed in some direct, sense. Our generation lives after most of the great megafauna have become extinct [27], and preservation does not now seem very costly. But, taking the long view, not everything can be saved, and species death is one of the harsh choices that nature does not hesitate to make. It is doubifld that man can henceforth always avoid making such choices. * On the economics of exhaustible resources the reader is referred to [7, 13, 26, 30, 311.

12

50

,, 1

VERNON

I.L=O

f

L. SMITH

1

--------~~~----------

I

\

/--

a

P

FIQURE 6

and (2.11) and (2.12) can be written

Q > 0. b(Q) < t < to, Q > 0. = Q - (0 - ~h(U,

i = G,

5 2 to,

(3.4)

=s~& = 0,

(v-t+(i)-($)#(~)],

tS~;Q),Q>O.

t 2 60,Q > 0.

= - Q4 [4-l)

( +[)I,

b(Q) < t < to, Q > 0.

E i b(Q), Q > 0. Referring to the phase diagram in Fig. 3, as k is decreased X(t) decreases, i.e., shifts closer to the [-axis, [OSincreases, and in the limit X(t) degenerates to the E-axis as illustrated in Fig. 6. Also in Fig. 3, as k is decreased Eoddecreases, and the segment of D(S) in region II approaches the horizontal line, l = [+, illustrated in Fig. 6, where [+ is defined by

a-

v - t+ ___ q(,$+)=o. ( t+ >

Along an optimal path at ($, &) in Fig. 6 the economy at first specializes in production from the exhaustible resource. The resource stock declines over time

CONTROL

AND

NATURAL

RESOURCES

13

and the rcsorve price, 5, rises. The net harvest value 1) - ,t dcclincs, as output, q2 declines, i.e., Q2 = - & = - &[$J - (i;/Q)#‘] < 0. When t,hc state of t,hc systcm reaches ([+, Q’), t,he substitute technology comes on st,ream. Since t,hc inexhaust#ible substitute bechnology is employed at constant unit and marginal co&, capital appreciat.ion on reserve st,ocks ceasrs. Labor is transferrrd out, of mining at t,he same rate that the resource stock falls so that, unit mining cost, is the same as unit cost in the parallel substit,utc t,rchnology. Hence the systtw moves along the path 4 = .$+in region II with t,hr st)ock declining cxponcnt.iall\toward cxhaust,ion at bhe percentage rat,e 4(+‘(-1’[yJ’(c - [+)I 1. 4. WILDERn’ESS,

OR FRAGILE,

RECREATIONAL

R.ESOURCF,S

In the decade 1960-1970 total visibs to Grand Canyon Xat’ional Park nearly doubled while “white water” Colorado River excursions t,hrough Grand Canyon increased about 50 times. In 1972 an incredible 16,400 users braved the rapids in inflatable rubber boats to enjoy this spect,acular wilderness experience. Similar increases in visitor use in the once remot,e High Sierra, the Salmon and Snake Rivers, and other wilderness areas are causing considerable distress for naturalist,s and for Park and Forest Service Administ,rat,ors attemptming t,o prot,ect such lands from erosive USC. With minor changes and reinterpretation the replcnishable rcsourcc model of %&ion 2 capt,ures crrtain essential characteristics of t’he LLconservat,ion” problem in wilderness economics.5 Consider a scenic or wilderness resource t#o bc rcprcscnted by a stock, Q, of pleasurable homogeneous services reflect,ing the qualit>, of the scenic or wilderness experience. Examples would includa Grand Canyon, Yosemit,e and Glacial Nat,ional Parks, ski areas, or any of the wild arcas in the Nat’ional Park or Forest system. In t,he “natural” state undist8urbrd by man, Q is generat,cd by an rcosystrm somewhat heroically assumed to bc govcrncd l)y a differential equation Q = q(Q), g’(Q) < 0, q”(Q) < 0, for all Q > 0, c,(Q) = 0. This is a bold oversimpliticat~ion of aesthrt,ic rtality, but scrvcs to capture t tic idea that, there exist, nat,ural biological and geological forcw by which a sctnic rcsourcc dcvclops in t,he ‘Lundisturbed” st’atc. This sct,s t,hc st,age for the bcaut? or wildcrncss of the resource to be modified by man’s ut,ilizat8ion of it. The rcsourcc is assumed to reside in an economy t#hat, produces two goods. Thrrc is t(hc ordinary commodity qi, and an exposure rate (visists) qz to the sccnit wsource. Iltility is u(gl, q2, Q) = ul(ql) + ~~(9~) + us(Q). Hcnte, the quality of the rcsourcc, Q, and its ut,ilization rate, 42, are bot,h commodious, ~1’ > 0, ,ug’ > 0, ~3’ > 0. Addit,ivit#y of ut,ilit,y is not’ ncccssary to t’he argument, but. simplifies the derivation. The product,ion functZions arc qj = jj(L,), i = 1, 2, with 5 = Iii + L,. The effect, of using the resource is assumed to cause deteriorat8ion at a t8imc rate which is an increasing convex funct’ion of the use ratr, h (93 wit#h !b’ > 0, A” > 0, h(0) = 0. Hence the number of float, trips per year, 42, on t,he Colorado River through Grand Canyon causes ccosyst,cm damage at, the rate h(&. The net1 rate of natmural ccosystcm development is given by Q = y(Q) - h(q2). This yields a model formally t,he same as that in Section 2 except for the substitution of h (q2) for q2 to represent rcsourcc dcplction, t,he condit8ion that 6 For

background

references

see [2,

15, 16, 191.

VERNON

14

L. SMITH

afJa& = 0, and the condition that the resource growth function be strictly decreasing. For an interior solution L1 > 0, Lz > 0, an optimal trajectory must satisfy

UZ'[fi(L2>1- WCfz(Lz)l Ul'[fl(L

- L2)lF

fl'(L - Ld (4.1)

=

fi’(L2)

’

s'(Q)lE - G'(Q), & = g(Q) - W2(-Wl, .i

=

16

-

and the transversality conditions. Equation (4.1) defines a function

(4.2) (4.3)

LZ = LP(~) with

dLz/d.$ = fi’h’/J

< 0,

J < 0.

(4.4)

As in Section 2, ~2’ - ,$h’ is the marginal value, or price, of resource use net of its unit depletion cost, where ,$is the implicit price of “wilderness” or ecosystem quality. The stationary state supply function X(t) of resource quality is defined by (4.1) and (4.3) for & = 0, i.e.,

Q = S(F)= s-'~h(fiCL(~)l)l,

(4.5)

with dQ/dt = h’fs’L2’/g’ > 0. If it is assumed that lims+, Lz(.$) = 0 in (4.1) which must be the case if ~~‘(0) = cc or h’(0) = 0, then in (4.5)

lim+, s-l{h(fz[L~(t>l>) = &. If we let L2** = Lz(0) in (4.1) where qr2** = fz(Lz**) is resource use under free access, then from (4.5) Q** = g-‘(h(f2[L2**]) 1. These characteristics of S(t) are shown in Fig. 7. Setting $ = 0 in (4.2) defines the stationary state resource demand function B(t), i.e., l

=

&l(Q)

E

u3’(Q)

6-

(4.6)

d(Q)'

with dQ/df = (6 - g’)/(u3” + C;g”) < 0, as shown in Fig. 7. Figure 7 illustrates an optimal trajectory through (E*, Q*), and a nonoptimal free access trajectory through (0, Q**). 5. ENVIRONMENTAL

WASTE

AND

RECYCLING

In this section the problem of waste accumulation is viewed as the joint coni.e., let waste degrade by sequence of household and firm decisions to “litter,” natural biological and chemical processes instead of recycling waste into production.6 The consumption of a commodity is assumed to leave a waste residue (such as beverage containers, banana peels, and derelict automobiles). Although such waste may be littered, compacted, burned, or dumped in the ocean we assume that it ultimately accumulates as a stock of “bad” (a liability) from which there is no escape except through recycling. The diagram in Fig. 8a illustrates the material and labor input flows underlying the model. Labor, 5, is distributed to three activities: (1) commodity production 6 Also see Refs. [S, 14, 17, 21, 25, 311.

CONTROL

AND

NATURAL

RESOURCES

1.5

(e.g., bottled beverage), q1 = fI(L1) ; (2) recycled cont’ainer or material production (e.g., bottles), qz = fz(Lz) ; (3) new container or material production (~.g., 1s bottles), q3 = f3(L3). Each fi(L) . increasing and concave. The container or commodit’y makrial, q3, simply replaces the equivalent quantity discarded into the environment, while households must return for recycling all units, gs, rccycled. Hence we have the makrial balance condit,ion, g1 = qz + qa. CG1it.y is u = ul(gd + uz(q2) + us(W), JVh erc IV(t) is t’hc st,ock of waste at, time t, ul’ > OF ~12’5 0, us’ < 0, and u is concave. The condit,ion that, q2 yield disut,ilit,y to houwholds rcflcctSs t#he trouble and cost to households of retaining commodity waste for recycling purposes. WC assume that there is an intolrrablr level of wast,c acwmulat’ion If7 such that limw+F u3(W) = - 3~. Then the st#ock of cnvironmental capital, rcprcscnting a measure of environmental cleanliness, can brb defined as Q = IT7 - IV. Waste in t’he environment is assumed to degrade naturally at a rate given 1)~ the function h (IV), h’ > 0, h” 5 0. Ket wast,e accumulation is given by t,hc differential equation JQ = jI(LI) - f,(Lz) - h(W). Since & = - I@‘,net] inv&mcnt in environmental capital is & = - [fI(L1) - fz(LZ)] + I~(ll: - 12). The: control probkm is to maximize LI(l), 1,2(f),h(t) s ”

{Ul[fl(Ll)]

+ uz[f2(L2)1

+ u3(W - Q)l+Ldt

(5.1,1

subject, toi (.i.L’)

-L + L2 + L3 = 5, f3 (IJ3)

-

j-1 WI)

+

f2(L2)

=

(3.3)

0,

& = - [fl(L,)

- f2(L2)]

+ k(TT: - Q).

(5.4)

The model can be interpreted as one of pure waste reduction [3, 211 by omitting the constraint (5.3) and set,ting L3 - 0. r\‘ow q1 - q2 units of noxious waste is discharged while q2 units are processed wit’h labor Ls before emission so that, it is no longer noxious. Figure 8b shows the material flow for this special case. The Hamiltonian and the necessary and sufficient conditions for an optimal trajectory (in [l], Proposit.ions 6-9 apply) are H = u~f-f~(-L)]

+ uz[fi(Lz)] +

+ u,(w Au3

CL,>

- Q) + w(E - LI - Lz -

-

fl

+

.iC

>

0,

Lw -

fl(L)

(Ul’

-

E -

/.L) fl’

-

w

=

0,

Ll

(u2’

+

E +

P> fi’

-

w

I:

0

(if < , Ls = 0),

$

+

L3)

fzW1 +

(if <, L8 = 0), --w + Pf3’ I 0 i = [a + h’(W - &)I( + us’@ e@‘[(t)Q(t) = 0, lim ~+~l(t) 2 0. t-xc

f2(L2)

+

NV

-

&>I,

(5.5) (5.6)

(5.7) - Q>,

(5.8) (5.9) (5.lOj

7 This model assumes that new commodity material (iron ore for automobiles, aluminum ore for beverage containers, etc.) is subject only to labor processing cost, Ls. Modifications to account for the depletion effect of earth material stocks can be introduced by assuming ~a = fa(Ln, M), and adding the differential equation A% = - f~(L3, M), where M is the stock of depletable ore from which commodities or containers are manufactured, and La the labor input to mining.

VERNON

16

L. SMITH

Q

FIQURE 7

For an interior written

equilibrium,

by substituting

u1’ = (Wlfl’)

+ (w/f37

for p from

(5.8), (5.6) can be

+ 6

requiring the price (in utility units) of commodity to equal its marginal fabrication cost w/fl’, plus the marginal cost of processing new commodity material, w/f3’, plus the public disposal cost, .$, of a unit of commodity residual. Condition (5.7) can be given a somewhat novel, but enlightening, interpretation by defining s as the unit scrap value of commodity residue, equivalent to the private “deposit” fee in the case of beverage containers. For interior equilibria, we must have s + (wlf2’) = wlfa’, i.e., the deposit fee plus the marginal cost of recycling a container cannot differ from the marginal cost of fabricating a new container (or the scrap price of a junk automobile plus the marginal fabrication cost of an automobile equals the marginal cost of making an automobile out of newly mined metal). Consequently, for interior solutions, we can rewrite (5.7) and (5.8) in the form -uz’ = (w’lfs’) - (wlf2.‘) + E = s + t. The marginal household cost of returning a used container, -uz’, equals the deposit fee plus the public disposal cost of a container. Hence, s + 5 is the opportunity cost of littering. By recycling a unit society recovers the scrap value s, and enjoys a cleaner environment with value .$ at the margin. The implicit price .$ is directly interpreted as the unit effluent charge for decentralized control of waste discharge. In effect, for beverage containers, such a charge will supplement the private deposit fee so that the after tax deposit fee

CONTROL

AND

NATURAL

RESOURCES

17

becomes s + t. It, should be emphasized that, such a “tax” should be levied only on net new container, steel, aluminum, et,c., production, i.e., net of recycled unit’s. This will raise scrap value relative to new fabrication material and encourage recycling. The objective of course is not to encourage recycling for its own sake but t#o permit recycling to compete on comparable cost t,erms 1vit.h nw material production which requires the latter to bear any public littering cost. Equation (5.9) is just another version of t#hc intertemporal equilibrium condition requiring the marginal value of a unit of (environmental) capital, -u3’, t,o equal interest plus depreciation on capital, (6 + /I’){, less capital gains, ,$. This condition provides one differential equation in the two-state variables (E, Q). From (5.9) t’he set ( (.$, Q) / i = 0) d efi ncs the stat’ionary state asset demand funct.ion, D(.$) for environmental capital, i.e., from (5.9) D-Cget thr invrrw form

[=

Q’(@ --- Q) E

-

6 + h’(V

D-l(Q),

- Q) (5.11)

with 2:0-l(Q)

lim

= =,

o-l(Q)

=

-

Q-T

Ilpl(o)-, 6 +

h’(O)

and d{/dQ = [(s + 11’)~~” - ~a’h”/(S + 1~‘)~< 0, as shown in Fig. 9. From (5.9) a&/al = (6 + h’) > 0, and all vertical mot’ion must be in a direction away from D(t). From (5.11) it follows that when the invcrw demand for environmental capital is greater t.he greater the marginal disutility of waste, the smaller t,h(l discount rate,.and the smaller the marginal rat,e at, which refuse decomposes in nat,urr. The stationary st,ate supply function of environmental capit,al is defined b) the conditions (5.2)-(5.4) and (5.6)-(,?A) by sett,ing Q! = 0. There are t,hrcc solutions, dtpending upon whether t’he incqualit,y holds in (;i.7), or (5.X), or neither : 1. When L, > 0, LZ = 0, L, > 0, for some t, w have the polar case of zero recycling. With Lz = 0 the constraints (5.2) and (5.3) determine L, = Z,,‘, L, = L$, for all t. From (5.4), & = 0 implies h (If - Q) - J1(L,l), and Q = TV - h-‘[f,(L,‘)]

(5.12)

= (31,

a constant,. Kow, with the equality holding in (5.8), if we substitute in (.5.6)-(5.8) and solve for .$, then 65 -

i

u*‘(O)

1 1 ___ +[ fl’ Wl’) f3’ (Id) 1

- ul’[fl(L1)]

for p and U.

1 1

- ~ [ fz’ (0) f3’ (W

1 ~ fl’(L1’)

IL4

1 __ + ft’(O>

1

= El.

(5.13)

It, follows that a st,ationary state solution for t.his case must be in t.he set ((4, &>I t I (1, Q = &I}, R.h ere &, given by (5.13), is the largest valuat,ion for a unit of clean environment such that recycling is not economically justified. For all l below ELthe public value of environmental capital is too low (given t,he fi, h, and Ui in (5.12)-(5.13)) to support any recycling, and all commodit,y waste is

VERNON

18

L. SMITH

I L -Households

42 a 91 T

Ll

Commodity Production

92

~

Pecycling Activity A

93

-

L3

New Commodity Naterial 4

FIGURE 8

discharged. An interpretation

of 61 in terms of the scrap value

s1= [Wl/f3’(U)l - cwllf~‘(0)l applying to this case can be obtained (5

f1=

directly from (5.7) and (5.8), i.e., -242’(O)

-s1.

The higher is the scrap value of commodity waste the lower must be the public valuation of environmental capital if recycling is not to be justified. Figure 9 illustrates the case in which (I > 0 but it could well be negative as in Fig. 10. For example, if scrap value s1 is high enough or -u,‘(O) low enough (i.e., households find it easy to recycle waste), we could have E1 < 0, and only a subsidy on waste disposal could prevent some recycling. Finally, from (5.4), a&/&! = - h’ < 0, and & will increase or decrease as Q 2 &I (Fig. 9). 2. When Li > 0, i = 1, 2, 3, some but not all waste is recycled. Equations (5.2), (5.3), and (5.6)-(5.8) can be regarded as defining, implicitly, the functions

20

VERNON

L.

SMITH

Q

FICRJRE~O f,(L2)

-

f2(L2)

= 0, Thus (5.4) becomes & = h(W - Q).

(5.4”)

With complete recycling a&/aQ = - h’ < 0, the stock of waste decays to zero, while environmental capital grows to its maximum, Q = w. From (5.7) and (5.8), uZ’[f2(L2)] + E - ~2/f~‘(L~) = - P 2 - wz/f3’(0), or w2

f‘ 2 -

uz’[fz(L22)]

+ ____ f2’

w22)

w2

- __ = j-3’ (0)

u2’[f2(L22>]

- SP =

62,

(5.15)

where [ 2 t2 defines the set of public good prices for which LS = 0, i.e., for all environmental valuations in excess of (2, optimality requires 100% recycling. This is illustrated in Fig. 9 by the line m = X(l) for .$ 2 &. From (5.15) it is evident that 62 could be negative if scrap values are high enough. Figures 9 through 13 combine the functions D(t) and S(t) into phase diagram representations of optimal and free-disposal equilibrium paths for various possible economies. For an optimal economy in Fig. 9, if the initial state is at (z, v), the stock of capital will decline as the stock of waste grows along the indicated trajectory until, Q = Q*, [ = .$*. Along this path it is optimal to recycle some, but not all, waste. The stock of waste grows until it reaches W* = Jv - Q* where the rate of waste decomposition is balanced by the rate of waste discharge

CONTROL

AND

NATURAL

RESOURCES

21

net of recycling, i.e., h(lF - Q*) = fl(L,*) - S2(Lz*), and interest preciation on environmental capital balances it,s marginal valuation, [*[s + h’(Tj’ - Q*)] = - u,‘(li-

plus tle-

- Q*).

Figure 9 also illustrates the contrast’ing nonoptimal free-disposal economy. II households view waste discharge as a costless act#ivit,y (i.e., the value of a clean environment is not, t,ransmit,ted by the pricing system to the recycling decision), then effect,ivrly [ = 0. For such an economy, beginning at Q = Tr’, { = 0, t,he stock of cnvironment,al capit,al will deteriorate along the Q-axis, ati shop\-11in Fig. 9, urni Q = Q1 < Q* where no waste is recycled and thrt natural rat,e of waste decomposition balances the gross rate of waste discharge, 11(lr - Q1) = J’l(L1l). This case probably applies to most, present-day beverage containers, In the absence of a disposal or littering charge on unrrqclcd beverage containers, private salvage values are insufficient to make recycling profitable. X11 such containers are t,hcn littered or go to the city dump at, cit.y expcnsc (financed out of general revenue). -4 specific disposal chargc (corresponding to t(t) aloq an opt,imal path) collected, say, on the production of new bot’tlw or cans \vould incrcast: the market, value of used containers from

to s(f) + t(t). This would jcct.ory in Fig. 9.

nduce recycling

\\ j, = .

\ -----.--------b

‘Y

as represented

‘-%

FIGUREII

by the optimal

tra-

22

VERNON

L. SMITH

w = Q*

IQ1

FIQURE

= p*

12

S(5)

/ ‘.

F‘

-----_

\

-.

-.

‘T.

1 ‘.

-.

--.$i)

\jj___j -

l

a( 1

1

1

= (i*

:

’

= g**

FIQURE

.

13

:

,

i

Q

CONTROL

AND

NATURAL

RESOURCES

23

Figure 10 illustrates the case of scrap mat,erials such as paper for which private residual values are great enough t,o induce some recycling even in the absence of explicit disposal fees. The free disposal economy follows the t’rajectory from Iv to Q** > Q1 while if the demand is D(t) the optimal economy mows from (E, m> to Cl*, Q*>. It is optimal to recycle all waste if and only if D-l (lr) 2 ,tz. This is represcnwd in Fig. 10, if the demand is D’(t). Ih might be suppowd that, in t,he absence of public disposal charges t,he private competitive cco~~omy would always lit,tcr some commodit,y residuals, but this is wrong. Examples abound such as uwd aluminum beer kegs, and broken gold and silver jewelry. The case is illust,ratcld in Fig. 12. Whatever the demand, t,he private UYJnOmy would not lihtt‘r scrap gold even if lit,tering received a considerable subsidy. Also, it, need not be optimal always t’o recycle some watiitc. Small residual itenls like t’oot,hpicks, refuse with a high decompositNion rate, or residuals that arch consumed quickly by scavengers may not bc rwycled even with an optimal syst,em of charges. In Fig. 13, the optimal stat,ionary st’at,e capital stock is Q1 in both the opt’imal economy and t#he free disposal economy. The optimality of the free access economy in this case is sern by comparing the t.rajrctory beginning at, (E, IT’) with the one init,ially at (0, IT) in Fig. 13. For both trajwtorics, L,(t) = L1l, L(t) = 0, and L(t) = L3’ for all f. Hcnw, for both dw~~lopnwnt~ pat,hs, the present valw of discounted utility is

P= J independent

om {uLf~(L~)l

+ UX[~ - Q(t)])r%t,

of E, since Q(t) is determined & = -

fl(L,‘)

by t,he solution to

+ h[IV - Q(t)].

Furthermore, the path ending at (i*, Q1), and the one ending at (0, Q1) both satisfy t,he t8ransversalit,y conditions. Hence I’ is independent of l and t,hc trio paths are of equal social value, alt,hough the upper trajectory requires t,he auxiliary guiding price l(t) t’o move along the indicat,ed pat,h from $ t,o t*. The simpler, but equivalent,, program is to do nothing and thus allow free disposal. This example illust8rates t,he inefficiencies t,hat might, occur in instit,utionalizing a charge system to induce recycling when, as it turns out,! no recycling is optimal. ,4ny cost in administering the charge system is a dead weight loss with no offscatting improvement in resource allocation. REFEREKCES 1. K. Arrow, Applications of cont,rol theory to economic growth in “&Iathemat,ics of the Decisiou Sciences” (G. Dantzig and 9. Veinott, Eds.), Part 2, Smerican .\Iathematical Society, Providence, It. I. (1968). 2. K. Arrow and A. Fisher, Environmental preservation, uncertainty and irreversibility, Quart. J. Econ. 88, 312-319 (1974). 3. W. Brock, A polluted golden age, University of Rochester, to appear, someday, in “Economics of Natural and Environmental Resources” (V. Smith, PZd.), Gordon and Breach, New York (September 1973). 4. (;. Brown, An optimal program for managing common pr(Jpe!‘t,y resouwes wit.11 congestic)rJ externalities, J. PoZit. Econ. 82, lci3-173 (January/February 1974).

24

VERNON

L. SMITH

5. C. Clark, Profit maximization and the extinction of animal species, J. P&t. &on. 81, 950961 (July/August 1973). 6. C. Clark, The economics of overexploitation, Science 181, 639-634 (August 1973). 7. R. G. Cummings, Some extensions of the economic theory of exhaustible resources, West. Econ. J. 7, 201-210 (September 1969). 8. R. D’Arge and K. Kogiku, Economic growth and the environment, Rev. Econ. Studies 40, 61-77 (January 1973). 9. A. Fisher and F. Peterson, ” Natural Resources and the Environment in Economics,” University of Maryland, 1975; published in part as, The environment in economics : A survey, J. Econ. Liter. 14, l-33 (March 1976). 10. B. Goh, Optimal control of a Iish resource, Malay. Sci. 5, 65-70 (1969/1970). 11. F. Hahn, Equilibrium dynamics with heterogeneous capital goods, Quart. J. Ewn. 80, 633646 (November 1966). 12. G. Heal and P. Dasgupta, The optimal depletion of exhaustible resources, Rev. Econ. Studies, Symposium volume, 3-28 (1975). 13. H. Hotelling, The economics of exhaustible resources, J. Polit. Econ. 39, 137-175 (April 1931). 14. E. Keeler, M. Spence, and R. Zeckhauser, The optimal control of pollution, J. Econ. Theory 4, 19-34 (February 1972). 15. J. Krutilla, Conservation reconsidered, Amer. Econ. Rev. 57, 777-786 (September 1967). 16. J. Krutilla and A. Fisher, “The Economics of Natural Environments,” Resources for the Future, Washington, D.C. (1975). 17. R. Lusky, “Conservation of Natural Resources and the Theory of Recycling,” Ph.D. Thesis, M.I.T., Cambridge, Mass. (1972). 18. P. Neher, Notes on the Volterra-Quadratic Fishery, J. Econ. Theory 8, 39-49 (May 1974). 19. Outdoor Recreation Resources Review Commission, “Wilderness and Recreation-A Report on Resources, Values, and Problems” U.S. Government Printing Office, Washington, D.C. (1962). 20. C. Plourde, Exploitation of common-property replenishable natural resource, w~ltst. Econ. J. 9, 256-266 (September 1971). 21. C. Plourde, A model of waste accumulation and disposal, Canad. J. Econ. 5, 119-125 (February 1972). 22. J. Quirk and V. Smith, Dynamic economic models of fishing, “Economics of Fisheries Management-A Symposium” (A. Scott, Ed.), Institute of Animal Resource Ecology, University of British Columbia, Vancouver (1970). 23. K. Shell “Essays on the Theory of Optimal Economic Growth” M.I.T. Press, Cambridge, Mass. (1967). 24. K. Shell and J. Stiglitz, The allocation of investment in a dynamic economy, Quart. J. Econ. 81, 592-609 (November 1967). 25. V. Smith, Dynamics of waste accumulation: Disposal versus recycling, Quart. J. Econ. 76, 600-616 (November 1972). 26. V. Smith, An optimistic theory of exhaustible resources, J. Econ. Theory 9,384-396 (December 1974). 27. V. Smith, The primitive hunter culture, Pleistocene extinction, and the rise of agriculture, J. Polit. Econ. 83, 727-755 (August 1975). 28. R. Solow, Intergenerational equity and exhaustible resources, Rev. Econ. Studies, Symposium volume 2945 (1975). 29. J. Stiglitz, Growth with exhaustible natural resources: Efficient and optimal growth paths, Rev. Econ. Studies Symposium volume 123-152 (1975). 30. N. Vousden, Basic theoretical issues of resource depletion, J. Econ. Theory, 6, 126-143 (April 1973). 31. RI. Weinstein and R. Zeckhauser, Use patterns for depletable and recyclable resources, Rev. Econ. Studies Symposium volume, 67-88 (1975).