- Email: [email protected]
Resource extraction and recycling with environmental costs
JOUnNAL
OF ENVIRONMENTAL
Resource
Extraction
ECONOMICS
AND
MANAGEMENT
and Recycling ~IICHAEL
University of Oslo, Institute of Economics, Received
March
with
5, 220-235
( 1978)
Environmental
Costs’
HOEL Box 1095, Hlindern, Oslo 5, ,Vorway
17, 1977; revised November
22 1977
Extraction and use of a natural resource is assumed to affect the environment adversely. A perfect substitute for the resource can be supplied through a recycling process. Recycling may also have harmful effects on the environment, but to a smaller extent than extraction. The optimal path of extraction and recycling is studied under various assumptions about the environmental effects of recycling and the assimilative capacity of the environment. In particular, it is shown how the cost of recycling will affect initial resource extraction as well as the environmental quality at the time of resource exhaustion and in the long-run stationary
state.
1. INTRODUCTIOX Most of the literature treating the economics of exhaustible resources disregards the connection between resource extraction and the disposal of residuals in the environment. However, unless the products made from the resources are completely recycled once they have served their “final” good function, the resources will reappear as residuals in the environment. As long as these residuals are regarded as harmless, an analysis treating resource extraction may as well disregard the disposal of them. HoKevcr, many residuals are harmful, or will become harmful once their stock is sufficiently large. Examples of harmful residuals are carbon dioxide (which can affect the climate), thermal pollution from all types of energy based on exhaustible resources, several t’oxic metals such as cadmium, lead, mercury, and various radioactive elements, and other elements such as sulphur, phosphor, etc. In addition to the types of environmental pollution mentioned above, the extracting activity itself may have environmental effects. Two typical examples are ecological consequences of oil spills from offshore oil derricks, and recreational and agricultural losses from strip mining, Some recent literature has treated various aspects of resource extraction t’aking explicit account of environmental effects. d’hrge and Kogiku [a], Maler [12, Chap. 31, and Lusky [ll] treat macroeconomic optimization problems in which exhaustible resources as well as environmental costs enter the analysis. Smith [19] focuses on the connection between recycling and waste accumulation, but only 1An earlier version of this paper [7] was written during a one-year visit at MIT. Financial support from The Bank of Norway’s Fund for Economic Research, The United States Educational Foundation in Norway, and National Science Foundation Grant SOC 75-14258 is gratefully acknowledged. 220 0095-0696/78/0053-0220$02.00/O Copyright All rights
6 197.8 by Academic Press,‘Inc. of reproduction in any form reserved.
RESOURCE
EXTRACTION
AND
RECYCLING
221
little attention is given to how the environmental costs affect resource extraction. More explicit attention to the consequences of environmental costs for resource extraction in a microeconomic context has been given by Schulze [17]. As we shall see, this part of his analysis will be similar to the special case we treat in Section 4. Schulze also analyzes some consequences for resource extraction of possibilities of recycling, but disregards environmental costs in this part of his work. Finally, a more empirically oriented work in this field which should be mentioned is by Nordhaus [14, 151. H ere the consequences of carbon dioxide pollution for the optimal use of energy resources are studied. Resource extraction may in some cases be substituted by recycling activities or by a product produced by the means of a “backstop technology” (cf. Nordhaus [13]).2 In such cases the scarcit’y of the exhaustible resource will be a less serious problem for the economy than cases in which no substitute exists. The presence of a substitute will also make it possible to avoid the negative environmental effects of resource extraction. On the other hand, recycling or substitute production may itself have negat,ive effects on the environment. Recycled material may, for instance, reappear as a harmful residual in the environment, or the process of recycling may require various chemicals, energy use, etc., which all may have negative effects on the environment. An example of a backstop technology with negative environmental effects is nuclear energy as a substitute for fossil fuels. The present paper is a microeconomic partial analysis of how environmental costs affect optimal resource extraction. Unlike the literature mentioned above, the analysis will give explicit attention to the possibility of recycling or the existence of a backstop technology, allowing for the possibility of recycling or substitute production having negative environmental effects. In particular, we shall investigat’e how the cost of rrcycling or producing the substitute will affect the resource extraction and the development of harmful residuals or other negative impacts on the environment.
2. THE
SOCIAL
OPTIMIZATION
The optimization problem we shall The social welfare funct’ion is3
consider
PROBLEM may
be formalized
as follows:
co
w=
i’
e+[U(x
+
y) - h’(R) - cy]dt,
0
which
will be maximized
subject
(1)
to the constraints
s = -2, i? = x + ay -
S(0) 6R,
= so,
R(O) = Ro,
(2) (3)
and the nonnegativity constraints x(t), y(t), S(t) 2 0. The rate of resource extraction is denoted by z(t), and the rate of recycling or the rate of substitute production is denoted by y(t). The remaining resource stock is S(f), while R(t) is a. measure of the stork of harmful residuals existing in t)ht? Bh considerable literature studying resource extraction with a backstop technology but ~with*i-ut environmental costs has appeared since Nordhaus’ article (see, for instance, C&-C]). 3 Time references will usually be omitted for notational convenience.
222
MICHAEL
HOEL
economy. U(Z + y) is the gross utility for society at any instant of time derived from having the rate of extraction plus recycling equal to 2 + y. We shall assume that U’ > 0, U” < 0, c/(1 - a) < U’(0) < 00 and lim,,, U’(y) = 0. E(R) is the disutility for the society at any instant of time from having a stock of harmful residuals equal to R. We shall assume that E(R) is convex, E'(R)> 0, and E'(R,)/(r + 13) < c/(1- a),where R. is t’he initial stock of harmful residuals. This last inequality will be seen to be necessary in order to have any resource extraction take place in an optimal solution. Except for the special case in Section + S)> U'(0)for a 6, we shall also assume that E"(R) > 0 and that aE'(R)/(r sufficiently large value of R. Notice t’hat the environmental effect given by E(R) only includes the stock of harmful residuals, and not the gross or net increase in these residuals. The term cy takes care of the cost of recycling y units of material, measured in the same units as the utility of extraction plus recycling and disutility of harmful residuals. To focus on the existence of environmental costs, we have assumed that traditional extraction costs are zero.4 A somewhat more general social welfare would be to have o(z + y, R) - cyunder the integral in (1). Our specification is chosen for analytical convenience. Relation (2)shows how the development of the resource stock depends on t’he rate of extraction; the initial stock So is known with certainty. Relation (3) shows how the development of the stock of harmful residuals depends on resource extraction and recycling or substitute production. The term 6R (where 6 is a nonnegative constant) represents the natural degrading of the harmful residual due to the assimilative capability of the natural environment. The same simplified representation of the degrading process is used in [9, 16, 19, 201. The case of 6 = 0 (which impliciUy is assumed in [2, 11, 171) will be treated as a special version of our general optimization problem. In addition to the depreciation process by the natural environment, the stock and flow of harmful residuals can be reduced by various types of pollution abatement activities (cf. [l, 9, 16, 20)). However, the present analysis disregards these types of activit,ies. One interpretation of (3) is that after having served its purpose embodied in a final good, the resource is disposed in the environment such that a share (Y (where 0 < (Y 5 1) is a harmful residual (due to where and in what form this fraction is disposed) while the rest is a harmless residual. We shall disregard the time lag between the date of extraction or recycling and the date of disposal, and denote the stock of harmful residuals by P(1) ( now measuring this stock in the same units as resource extraction and recycling). Since recycled material ends up as a residual in the same way as the virgin resource does, the gross increase in P(t) is given by cr[~(t) + y(2)]. Assume now that each unit of recycled material is based on a fraction p of harmful residuals and 1 - p of harmless residuals. It seems reasonable to expect p s (Y, as it usually is easier to recycle from waste in solid form than from waste in liquid or gas form, which often is the most harmful form of residual. Throughout our analysis, we shall assume that 0 < p 4 a’, and as an important special case we shall consider p = CL This special case can, for instance, arise if the material is collected for recycling before any of it is disposed of in t>he environment. The recycling act,ivity uses harmful residuals at the rate @yCr), while the natural * Alternatively, we could let the cost of extraction and recycling be c, and ~2, respectively, alld the gross utility function of using z + r/be UI (.z + 1~).We would then get (1) by defirling U(.c + y) = U,(Z+?/) -c,snndc=c>-cl.
RESOURCE
degrading of P(t) takes is therefore given by
EXTRACTION
place with
AND
the rate P(t).
RECYCLING
The change
223
in P(t)
P=a(z+y)-py-SP.
over time (4)
By defining a = 1 - p/a! (where 0 5 a < 1 due to our assumption 0 < p 5 a) the measuring unit of the harmful residual, w-e and R = P/a, thus changing obtain (3). Our formulation of the optimization problem disregards any constraint on how large the rate of recycling can be. The reason why such a constraint can exist is that if the duration of the material embodied in a final good is taken into consideration, there will be a limit at any time on how much material is available to recycle into additional goods. Another way of saying this is that the stock of material embodied in durable final goods cannot exceed the total amount of material available. However, such a limit can be disregarded as long as the initial stock of material is high compared with the long-run stationary level of recycling which we shall derive. If we were to consider a growing demand for the recycled material such a constraint would sooner or later become relevant (cf. Weinstein and Zeckhauser [21]). Another property of our specification of the recycling activity is t’hat stationary states of y are feasible. In this sense we have 100% recycling. Notice, however, that if 6 = 0 and a > 0 any constant positive level of recycling cannot continue forever. The reason for this is that some of the material will get lost in the sense that it forever (since 6 = 0) ends up as harmful material, while recycling requires a larger fraction of harmless residuals than the fraction of harmless residuals in the waste (p < (Y, or a > 0). For inst’ance, if some material is partly disposed of in gas form in t’he at’mosphere and recycling only is possible from solid waste, a constant level of recycling will only be possible for a finite time period. A more thorough treatment of such cases is given by Schulze [17], where losses of material to unrecoverable forms of residuals occur both in the USC of the material and from the stock of recoverable residuals. In the following analysis we shall disregard such cases, i.e., we shall not consider the case 6 = 0, a > 0. It is clear that the way we have introduced the recycling activit’y, y(2) could as well be interpreted as a substitute produced by a backstop technology. In this case we would have a = 0 in the case of a “clean” substitute for the resource (for instance, solar energy for fossil fuels). If the substitute also had some negative environmental effect (for instance, nuclear energy), we would have a > 0.5 It is of no practical importance whether one interprets y as recycled material or as a substitute; we shall stick to the recycling interpretation in verbal discussions in the following. We mentioned earlier that the negative environmental effect could also be thought of as a direct consequence of extraction activities, such as strip mining and oil spills. In this case we can have complete irreversibility (i.e., 6 = 0), for instance, when recreational and aest,hetic values associated with undisturbed environments arc lost, (see Rrutilla and Fisher [lo, Chap. 31 for a discussion of whether or not t,hc:rc is full irreversibility in such casts). We can also have 6 > 0 in t,his cnsc. An cxamplr of t,his \sould be oil spill s, where the oil gradually gets l.Jf rithrr 3s recycled material &graded )J.V 111~sea. Like Lrfure,y c-31be tl~uugllt. 01‘as a produced substitulr. III both cases we gel a > 0 or (I = C) depending on 5Obviously, it would be better to specify two different residuals in this case.
MICHAEL
224
HOEL
whether or not there are any negative environmental effects from the recycling or production activity. Although the case 6 = 0, a > 0 excluded the possibility of a constant y forever in the case of harmful residuals and recycling, this is not true if either the environmental effects are a consequence of the extraction act’ivity itself or if y is a substitute produced by a backstop technology. In these two cases there is nothing to prevent y constant forever even if 6 = 0, a > 0. 3. CHARACTERIZATION
OF THE
OPTIMAL
SOLUTION
The current value Hamiltonian associated with the problem (l)-(3), so that the shadow prices ~(1) and g(t) turn out to be nonnegative, is H= The necessary
-
U(S+Y)
conditions
E(R)
-
ey -
X(x + ay -
for an optimal
solution
i = (1’+ S)X -
‘Y(R),
imply
6R) -
px.
that” (5)
/.i = 1^/.L,
(6)
SAX+,
where
< implies
5 = 0,
(7)
+ y) 5 c + UX,
where
< implies
y = 0.
(8)
fj’(r+y) V(z
written
equations Using (7) and (8), Eqs. (2), (3), (5), and (6) give us four differential in R, S, X, and p. In addition to the initial conditions R(0) = Ro and S(0) = So we have lim s(t) t-41
2 0,
where
> implies
I
= 0
(9)
and lim e-r%(t) t+rn
= 0,’
(10)
so that the optimal time paths for R, S, X, p, 2, and y are determined. Solving the differential equations (5) and (6) and using (10) gives us
s m
x(t)
=
e(7+6)t
(11)
E’[R (7)]e-(r+~)rd~
t
and ~(0 where ~(0) is determined
so that
(12)
= e%(O),
(9) holds.
Since E(R) is convex,
a useful
conse-
6For notational convenience, we have not introduced any additional superscript or the like to denote optimal values of R, 8, 2, and y. This should not cause any misunderstanding. 7 Strictly speaking, a transversality condition of the type (10) is in general not necessary for an optimal solution. However, it is easy to show that a solution derived by using (10) must be optimal: First, it is obvious from economic considerations that when aE'(R)/(r + 8)> U'(O) for R sufficiently large, the solution to our optimization problem is not changed by introducing an additional constraint R(t)5 N, where N is a large (but finite) given number, since it cannot be optimal to let R + m with our assumptions. Since our Hamiltonian is concave, a solution (3, 8, 2, g) must be optimal if limr_,, (e-‘% (S - 3) - e-r% (R - 8) 1 L 0 for all feasible paths s(t) and R(t)(cf. Seierstad and Sydsaeter [18, Theorem lo]). But P 2 0, S >= 0 and limr,, p.7 = 0, (10) therefore implies that this inequality holds since R(t) s N.
RESOURCE
EXTRACTION
AND
RECYCLING
225
E’(0) E”(R”) -5 x(t) 5 ~ r+s’ r+6 whore Rm is the highest value R has It can easily be derived from our condition for x and y to be positive a period with both x and y positive period. In the special cases we shall will never occur. From (s)-(7) it, is easy to see that $
[U’(x)]
(13)
along the optimal path. equations [7, pp. 12-131 that a necessary simultaneously is that 6 > 0 and p. > 0. If exists, R(t) will be declining during such a treat in the next sections, such a situation when
= rU’(z)
y = 0 and x > 0 we must -
[E’(R)
have
- 6h].
(14)
Wit’hout any environmental effects, i.e., E’[R(t)] = 0 for all 1, it follows from (11) and (14) that we get the standard Hotelling [S] result, i.e., that the undiscounted marginal utility of using the resource should increase with a rate equal to the discount factor. When resource extraction affects the environment in a negative way, this rule is altered according to Eq. (14), where X may be inserted from (11). Whenever resource extraction takes place in a period when there is no recycling, (7) holds with equality and y = 0. In other words, the marginal utility of using the resource at any time equals the sum of the two shadow prices X(t) and p(t), which are the (undiscounted) shadow prices of reducing the stock of harmful residuals by one unit and of increasing the resource stock by one unit, respectively. Similarly, when x = 0 and y > 0 the marginal utility of using the recycled resource at any time equals the sum of recycling costs and the shadow price ax(t). In some of the cases we shall treat in the following sections the optimal solution will approach a long-run stationary stat’e where x = 0 and y and R are constant, dcnotcd by y* and R”. The values y* and R* arc given by
E’(R*) = c + a--T+&
u’(y*)
(15)
and 6 y*=-R*
(16) a
a > 0. Notice that when a > 0 and 6 = 0 we get y* = 0 as long as aE’(R)/(r + 6) > U’(0) for R sufficiently large. In other words, even in cases when
where a constant y is physically possible when a > 0 and 6 = 0 (cf. Section 2), it is optimal to have y* = 0. When a > 0 and 6 > 0 so that (R*, y*) follows from (1.5) and (l(i), implicit] derivation gives us the following results :
dR”/dc < 0,
aR*/aa
$0,
aR*/as
$ 0,
aR*/ar
> 0,
ay*/ac
ay*/aa
< 0,
aye/as
> 0,
ay*/ar
> 0.
-c 0,
It is rather surprising that lower the cost of recycling
(17)
the long-run stock of harmful residuals is higher the is. The reasoning is that the lower this cost is, the
MICHAEL
226
HOEL
C);‘(L) (II’ rcfraitring frc~ni try ((i’(y) lhighrr is Otit? otq)~~rtlurity cost, (Il~lcY~s11rt~c-t adding an additional unit to t(hc stock of lr:un~l’ul rcGduals. WC may also find t,hnt the optimal long-run stock of harmful residuals is Iargcr the larger t’hc assimilative capacity of the environment (mcasurcd by 6) is. The reason for this is similar t,o the reason for dR*/dc< 0.The other results speak for themselves, and need no comment. When a = 0, (15) becomes (15’) u’(y*) = c, i.e., y* is independent determined by earlier zero (for 6 > 0).
of a and 6. In this resource extraction
4. COMPLETE
case we either have R* constant and (for 6 = 0), or R will decline toward
IRREVERSIBILITY
We shall now consider the case where 6 = 0, i.e., where there is no natural depreciation of the harmful residuals in the environment. This is the case which Schulze [17] has treated, but he disregards the possibilit’y of recycling as an alternative to extraction with environmental effects. Defining y = X + ~1, 6 = 0 inserted in (3), (5), and (6) give ?; = ry -
E’(R)
(18)
and & =
f(r)
for y < c + aA,
US(c + ax>
for y > c + aX,
(19)
where f-l(~) = U’(Z). The case in which recycling has no negative environmental effects, i.e., a = 0, is illustrated in Figs. 1 and 2. Figure 1 covers the case in which the resource constraint s(t) 2 0 is binding, while s(t) 2 0 is a nonbinding constraint in Fig. 2. In Fig. 1 y(O) is determined so that R(t)= Ro + So and ~(1) = c are reached simultaneously. In Fig. 2 we have r(t) = A(t), and -r(O) is determined so that simultaneously. In this latter case the optimal stock r(t) reaches c and E'(R)/r
FIGURE 1
RESOURCE
I
EXTRACTION
AND
RECYCLING
:
227
*‘(::=,<
N 0
+‘: 0
-.-:;
II
l!iL)
FIGURE 2
of unextracted resource, S*, is determined by E’(Ro + So - S*)/r = c, which makes the discounted environmental cost of extracting an additional unit of the resource equal to the unit cost of recycling. Formally, this follows from (13), which will hold only if the optimal trajectory is as in Fig. 2. If the constraint S(t) 2 0 is binding like in Fig. 1, it is clear that the long-run stock of harmful residuals R” = Ro + So will be unaffected by the recycling cost. We also see from Fig. 1 that the lower c is, the lower must y (0) be for y(t) to reach c at the same time as R(t) reaches R *. From (3) and (19) we see that this means that the initial resource extraction and the initial growth of harmful residuals is higher the lower is the recycling cost. If R* < Ro + So, like in Fig. 2, we see that R* is lower the lower the cost of recycling is. Notice that this conclusion differs from the conclusion we had when both 6 and a were positive (cf. (17)). Another way of expressing this conclusion is that’ when it is not optimal to exhaust’ the resource completely, the optimal total amount of extraction is lower the lower the cost of recycling is. We also see from Fig. 2 that the lower c is, the lower must y(O) be for r(t) to reach c and E’(R)/r simultaneously. In other words, also when the resource constraint is not binding we find that the initial resource extraction and initial growth of harmful residuals is higher the lower the cost of recycling is. Let us now turn to the case in which a > 0. From (19) we see that in this case d > 0 also after the switch from extraction to recycling, although fi at the time of the switch jumps downwards. Let us first look at the development of x(t) and R(t) after the time to when one switches from resource extraction to recycling. Defining t(t) = c + ax(t), it follows from (5) (with 6 = 0) and (19) (with y > c + ax) that f = r(E -
c) -
d’(R),
A! = af([). Together
with some initial
condition
R(t,)
f’n~ e-+4(t)
(20) (21)
= I? and (lo), which implies = 0,
MICHAEL HOEL
22s
1:
1:’
I!, I,
FIGURE 3
(20) and (21) determine time paths for f(t) and R(t). These paths are illustrated by the phase diagram in Fig. 3. The initial value t(to) must be such that t(t) reaches U’(0) and c + aE’(R)/r at the same time, otherwise (13) will not hold, i.e., e-lt[(t) will not approach zero. If the resource constraint is binding, we must obviously have R = R. + So. The development of r(t) = X(t) + p(t) and R(t) before to in this case follows from (18) and (19) (with y < c + ax). The initial value y(O) is determined so that ~(2) reaches c + aA = [(to) at the same time as R(t) reaches I? = R. + So (cf. the phase diagram in Fig. 4). The case in which the resource constraint is not binding is perhaps best illustrated in a single figure describing the development of X and R for all t: When P(L) = 0, (18) and (19) are simply i = rX -
h”(R), c for X < ___ 1 - a’
f(A) &= af(c
(22)
+ aA>
c
(23)
for X > ~. 1-U
These two differential equations together with R(0) = R. and (10) determine the development of X(1) and R(t). This is illustrated by the phase diagram in Fig. 5. The reason why the curve describing this development is kinked at X = c/(1 - a) is that a < 1, i.e., recycling has less negative impact on the environment than resource extraction has. The initial value X(0) is determined so that X(t) reaches (U’(0) - c)/u and E’(R)/r at the same time. It is easy to see that no other development of A(t) will satisfy (13). Irrespectively of whether or not the resource constraint is binding, we see that the long-run stock of harmful residuals is determined by c + uE’(R*)/r = U’(O), which is identical to (15) when y * = 6 = 0. The results given by (17) are still valid, except that aR*/& I 0 is now excluded. In particular, the long-run st,ock of harmful residuals will be higher the lower the recycling cost is. It is also easy
RESOURCE EXTRACTION
AND RECYCLING
229
FIGURE 4
to see from our differential equations and phase diagrams that the initial resource extraction and initial growth of harmful residuals will be higher (may be lower or higher) the lower the recycling cost is when the resource constraint is binding (nonbinding). Finally, it can be shown that when the resource constraint is nonbinding, the total resource extraction (R - Ro) will be lower the lower the recycling cost is. For details, see Hoe1 [7, pp. 18-191. We have seen that both for a = 0 and a > 0 it can be optimal to extract some, but not all, of the available resource stock. This conclusion holds also when the possibility of recycling is excluded (cf. Schulee [17]). Schulze also finds the possibility of having a positive resource extraction after a period with no extraction. It is easy to verify from our phase diagrams that such a situation can never occur in the present model. The reason for this difference compared with Schulze is not that we have included recycling, but that Schulze assumes a fixed, finite horizon. As one comes close to this horizon, the shadow price of an additional unit of harmful residuals declines, since the disutility of an increased stock of
FIGURE 5
MICHAEL HOEL
230 residuals will find it optimal although this assumed, this
only be integraled over a, shod Io~riotl. 1 tj is thrrcfor~ possi\A~ I o to hava resource ext radian tjo\v:lrtl t,hc cntl of t hr planning pcriotl was not optimal carlicr. Wit,h an infinite horizon, like wc have possibility is excluded. 5. NONBINDING
RESOURCE
CONSTRAINT
We shall now consider the case where 6 > 0, but where the constraint S(t) 2 0 is not binding. In this case the differential equations determining X and R follow from (3), (5), (7), and (8): i = (T + 6)X - E’(R),
f(x) - 6R R=
(24) c for X < 1 - a’ (25)
af(c + ax) - 6R I
forX>+--. -a
Together with the initial value Ro and the condition (10) these two differential equations determine the paths for X and R, provided that’ the corresponding solution for IL:does not violate s(t) 2 0. The solution is illustrated in the phase diagram in Fig. 6. The curve giving A = 0 is as usual given by X = E’(R)/(r + 6) (cf. (24)). The curve giving fi = 0 follows from (25), and is discontinuous at X = c/(1 - a) since a < 1. As “a” approaches zero, the upper part of this broken curve approaches the vertical axis. The heavily drawn curve marked with arrows is th: only development of X and R which satisfies (13). This curve is kinked at 2 (= the stock of harmful residuals at the time of the switch from resources extraction to recycling) because a < 1. The long-run stock of harmful residuals is given from Fig. 6 by U’(6R*/a) = c + uE’(R*)/(r + a), which of course is identical t’o Eq. (15). iSotice that Fig. 6 is drawn so that the curves giving x = 0 and &? = 0 intersect at a value X* > c/(1 - a). If the curve for x = 0 had lain to the right of the upper part of
RESOURCE
EXTRACTION
AND
231
RECYCLING
I III’ I’lll’Vl! I’or I? -=0 (‘III’ dl x 1 I:,‘( I (I), i hc lc~ng-rrm va111(*h’” wo111(1hnvn ilnpliM1 a IOIlg-III11 pOSitiVC! VdW 01’ IY’SOIIIW: (!XtrWCt iOIl. SillCC! f his SOOIN’r Or I:ibT would violat,c the constraint S(t) 2 0, wc can ccmc~ludcthat a nccc~ssary condition for the solution to follow from Fig. 6 is that X* = E’(R*)/(P + 6) 2 G/(1 - a), where R* follows from (15). If a = 0 when 6 > 0, R(t)will approach zero, so that t’hc necessary condition in this case becomes E’(O)/(r + 6) 2 c, which will imply that no resource extraction should ever take place. When a > 0, the solution following from Fig. 6 may or may not imply some resource extract,ion. Notice that the condition E'(Ro)/(r + 6)< c/(1 - a ) is necessary, but’ not sufficient for resource extraction to take place. If the solution does give X(0) < c/(1 - a), implying some resource extraction, the solution will of course only be valid if the conslraint, 8(t) 2 0 is not violated. and X(0) will be affected by the cost of recycling. Let us now seen how R*,l?, If a = 0, our solution will be U’(y*) = c, Z? = Ro,and R(t)--+ 0 whatever value c has. If a > 0, the following conclusions can be drawn (see [7, pp. 22-231 for details) : The lower the cost of recycling, the higher the long-run stock of harmful residuals and the lower this stock at the time when one switches from resource extraction to recycling, provided that some extraction takes place. If no resource extraction takes place, X(0) will bc higher, i.e., the initial growth of harmful residuals will be lower, the lower the cost of recycling. When some resource extraction takes place, the initial extraction and init,ial growth of harmful residuals will be lower, unaffect’ed, or higher the lower the recycling cost is, depending on the functions U(s) and E(R).
6. CONSTANT
XARGINAL
EKVIRONMFXTAL
COST
The last special case WC shall treat is E(R) = bR,i.e., B'(R)= constant. In this case the shadow price x(t) is constant, cf. (11) :
b is a positive
(26) Our assumption b/(r + 6) < c/(1 - a) (cf. the Introduction) implies that cl0 > 0, otherwise we would get a positive and constant resource extraction forever, violating S(2) 2 0. From (14) and (26) we see that when resource extraction is taking place we have
-p(x)=r [U’(x) -
b
;+;
1
,
(27)
which is a standard result in the theory of resource extraction, except that D/(r + S) now represents environmental costs instead of traditional extraction costs. The initial resource extraction z(O) is determined so that the resource is completely exhausted at the same time as U’(z) reaches the total cost of recycling. The time path of U’(J: + y) is illustrated in Fig. 7. From (27) and Fig. 6 it is easy to verify that our conclusion about a lower recycling cost giving a higher initial extraction and higher initial growth of harmful residuals holds also in the present case. It is also easy to see that U’(z(0)) is lower the lower is 6. We can therefore conclude that the initial extraction and growth of harmful residuals is higher the less harmful the environmental effects are regarded.
232
MICHAEL
When recycling takes place, it is constant
u’(y*)
HOEL
and equal to y* (cf. (8) and (26)) :
= c + ;-;+,
(28)
which gives y” > 0 because of our assumption b/(~ + 6) = E’(&)/(r + 6) < c/ (1 - a) < U’(0) and U” < 0. So far, we have said nothing about the optimal development of harmful residuals, except how this stock grows initially. From (3) it is clear that if 6 = 0 we must get R * = Ro + So (disregarding the case where it is optimal not to extract the resource) if a = 0 and l? = ay* > 0, i.e., R(t) -+ 00, if a > 0. The assumption about a constant marginal environmental cost is at best an approximation to some situations. Such an approximation may not be too wrong over certain regions of R, but will clearly be misleading when R -+ m . The approximation E’(R) = b is therefore not suited to treat the case with 6 = 0, a > 0. When 6 > 0, our previous results may be used to find an explicit solution for R(t) : t [X(T) + ay(7)]e6’d7
1 .
There are several possible developments of R(t). We may have R(t) increasing as long as resource extraction takes place. However, it is also possible that as s(t) declines and R(t) grows, we reach a time point when R(t) starts to decline, although resource extraction still is taking place. Once the resource is completely extracted, R(t) will fall toward zero if a = 0. If a > 0, R(t) may rise or decline at first, depending on a, 6, y* and the magnitude of R(t) at the time recycling starts. In any case, R(t) will approach R* = a~*/& where y* is determined by (28). It can easily be verified that R* and y* in this case depends on c, a, 6, and r in the same way as in the general case treated in Section 3 (cf. (17)). Furthermore, since ay*/& < 0 follows from (28), we must have aR*/db < 0. In other words, the less harmful the environmental effects are regarded, the higher is the optimal long-run stock of harmful residuals.
RESOURCE
EXTRACTION TABLE
AND
RECYCLING
233
I
Table Over aR*/dc AS(t) 1 0 nonbinding 6 = 0
a=0 b=O
>
a>0 6>0 a=0
1
S(l) 2 0 binding, E” > 0
s(t) 5 0 binding, E” = 0
Positive
Zero
Zero
Negative
Negative
R(t) +
R(t) -+ 0
K(t) -+ 0
12(t) + 0
Negative
Negative
Negative
+ m
>
6>0 a>0
7. CONCLUSIONS In this section we shall repeat some of the conclusions we found for the cases studied. In these cases resource extraction and recycling will never take place simultaneously. At first only resource extraction takes place, and the optimal rate of extraction declines over time. When recycling starts, it is either constant (for a = 0) or declining (for a > 0). In the latter case the rate of recycling approaches a stationary level, which will be positive if and only if 6 > 0 (i.e., some nat,ural degrading of harmful residuals). , The existence of environmental effects may make it optimal to leave some of a finite resource stock unextracted. Even when it is optimal t’o extract the entire resource stock, in spite of the environmental consequences, the optimal path of extraction will be affected by the existence of environmental effects. The stock of harmful residuals, both in the long-run (R*) and at the time when resource extraction stops (ri) will usually be affected by the recycling costs (c). The same holds for the initial growth of harmful residuals, which is determined by the initial resource extraction (~(0)). Tables I-III summarize how c affects R*,I?,and X(O) for the cases we have treated, cases not treated are blank. The tables do not treat cases in which no resource extraction takes place. From Table II we see that whenever l? is affected by c, this stock is lower the lower is c.
TABLE
II
Table Over ad/&
s(t) 2 0
s(t) 2 0
s(t) L 0
binding, E” > 0
binding, E” = 0
Positive
Zero
Zero
Positive
Zero
Zero
nonbinding 6 = 0 a=0 > 6=0 a>0 > 6>0 a=0 > 6 > 01 a > Ol
No resource extraction Positive
-
-
X34
MICHAEL TABLE
HOEL III
Table Over az(o)/ac s(t) 2 0 nonbinding 6=0 a=0 6=0 a>0 6>0 a=0 6>0 a>0
#S(L)2 0 binding, E” > 0
S(t) 2 0 binding, E" = 0
Negative
Negative
Negative
Undetermined
Negative
Negative
> > >
I-
No resmrce extraction Undetermined
Negative -
Negative
The opposite conclusion holds for R*, except’ for the case where 6 = a = 0, when R* = l? (see Table I). From Table III we see that for most of our cases, the initial resource extraction and growth of harmful residuals will be higher the lower is the recycling cost. Although this may seem somewhat surprising, it corresponds to the usual result that initial extraction is higher the lower the cost of producing a substitute (see Hoe1 [5], where it’is also shown that this result does not necessarily hold when a monopolist owns the resource). Notice, however, that when a > 0 this result does not always hold. In the case of a > 0 we may get a lower initial extraction and growth of harmful residuals the lower the cost is of recycling. We have said nothing about how a competitive economy will perform compared with our optimal solution. In the absence of environmental effects, it is well known that a competitive economy under certain conditions will give optimal rates of extraction and recycling (see, for instance, Weinstein and Zeckhauscr [21]). Even under these conditions the presence of the external environmental effects will in general imply that the competitive solution is nonoptimal. However, from (7) and (8) it can be seen that the competitive solution can be made optimal by introducing Pigouvian taxes X(t) and ax(t) on resource extraction and recycling, respectively, where x(t) is given by (11). Notice that such a tax scheme implies that recycling should not be subsidized, and should be taxed if a > 0. This may seem to be in sharp cont’rast to several policy proposals aiming at reducing resource extraction and/or reducing pollution. However, the contrast’ is only superficial, since the following tax scheme is identical to the one above : Resourcn extraction is not taxed, disposal of used material (virgin or recycled) in t’hc environment is taxed with the rate X(t), and recycling is subsidized with the rate (1 - a)x(t). Like before, this gives net tax rates equal to x(t) and ax(t) on the whole process of extraction and disposal and the whole process of recycling and disposal, respectively. Notice that t’he special case in which material is collected for recycling before any of it is disposed of in the environment (i.e., (Y = p, or a = 0) implies that the net tax on t>he recycling process is zero. If the param&rs CYand p from Srction 2 arc not, complotcly technically d&rmined, a good tax schcmc ought t,o give the agents in the economy an incentive to make (Y low ulnl /j high (cf. Scct.ion 2). Tile tax ~c~hcmes above do no1 have this property, since the tax or subsidy, does not distinguish between where and in what form the material is disposed or collected for recycling. The following modification gives this inccntivc: Instead of taxing total disposal, only harmful
RESOURCE
EXTRACTION
AND RECYCLING
2%
disposal is taxed at the rate x(t)/w 8 Similarly, only harmful mat’erial used for recycling is subsidized at the rate (1 - a)x(t)/P = x(t)/(~. The discussion above on tax schemes to improve a competitive economy disregards all problems of what kind of taxes and subsidies are feasible from a technical, administrative, and political point of view. An extension to such a discussion is beyond the scope of the present analysis. REFERENCES quality, Swedish J. Ewn. 73, (1971). R. C. d’Arge and K. C. Kogiku, Economic growth and the environment, Rev. Economic Studies 40, 61-77 (1973). P. Dasgupta and J. E. Stiglitz, “Uncertainty and the Rate of Extraction under Alternative Institutional Arrangements,” Technical Report No. 179, Institute for Mathematical Studies in the Social Sciences, Stanford Univ., 1976. M. Hoel, Resource Extraction under Some Alternative Market Structures, Mathematical Systems in Economics 39, Verlag Anton Hain, Meisenheim am Glan, 1978. M. Hoel, Resource extraction, substitute production and monopoly, J. .&on. Theory, to appear. M. Hoel, Resource extraction when a future substitute has an uncertain cost, Rev. Economic Studies, to appear. M. Hoel, “Resource Extraction and Recycling with Environmental Costs,” Working Paper No. 197, Department of Economics, MIT, 1977. H. Hotelling, The economics of exhaustible resources, J. Political Econ. 39, 137-175 (1931). E. Keeler, M. Spence, and R. Zeckhauser, The optimal control of pollution, J. Ewn. Theory 4, 19-34 (1972). Studies in the J. V. Krutilla and A. C. Fisher, “The Economics of Natural Environments: Valuation of Commodity and Amenity Resources,” John Hopkins Press, Baltimore, Md., 1975. 1~. Lusky, Optimal taxation policies for conservation and recycling, J. Econ. Theory 11, 315328 (1975). K. G. Maler, “Environmental Economics: A Theoretical Inquiry,” John Hopkins Press, Baltimore, Md., 1974. W. D. Nordhaus, The allocation of energy resources, Brookings Papers, No. 3, 529-570 (1973). W. D. Nordhaus, “Strategies for the Control of Carbon Dioxide,” Cowles Foundation Discussion Paper, mimeo, 1976. W. D. Nordhaus, Economic growth and climate: The carbon dioxide problem, Amer. Econ. Rev. 67 (Papers and Proceedings), 341-346 (1977). C. G. Plourde, A model of waste accumulation and disposal, Canud. J. Econ. 5, 119125 (1972). W. D. Schulze, The optimal use of non-renewable resources: The theory of extraction, J. Environ. Econ. Manag. 1, 53-73 (1974). A. Seierstad and K. Sydsaeter, Sufficient conditions in optimal control theory, Internal. Econ. Rev. 18, 367-391 (1977). V. L. Smith, Dynamics of waste accumulation : Disposal versus recycling, Quart. J. Econ. 86, 600-616 (1972). S. StrGm, Economic growth and biological equilibrium, Swedish J. Econ. 75, 164-175 (1973). M. C. Weinstein and R. J. Zeckhauser, Use patterns for depletable and recyclable resources, Rev. Econ. Studies Symposium on the Economics of Exhaustible Resources, 67-88 (1974).
1. R. C. d’Arge, Essay on economic growth and environmental 25-41
2. 3.
4. 5. 6. 7. 8. 9. 10.
11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
seems to imply that the tax rate X(t)/a is higher the lower is 01,i.e., the less is disposed of in a harmful way. However, the function E(R) is given by E(R) = J?(&) = E(P), so that for a given function e(P), the function E(R), and therefore also x(t) (cf. (1 I)), depends on a. For irrstanc*e, if our original disutility function in Section 6 is fl(I’) = i;P, it. follows from (26) and K = P/u that h(t)/u = i;l(r + 6). i.e., incleprndrilt of LY.
8 This modification of the used material
OF ENVIRONMENTAL
Resource
Extraction
ECONOMICS
AND
MANAGEMENT
and Recycling ~IICHAEL
University of Oslo, Institute of Economics, Received
March
with
5, 220-235
( 1978)
Environmental
Costs’
HOEL Box 1095, Hlindern, Oslo 5, ,Vorway
17, 1977; revised November
22 1977
Extraction and use of a natural resource is assumed to affect the environment adversely. A perfect substitute for the resource can be supplied through a recycling process. Recycling may also have harmful effects on the environment, but to a smaller extent than extraction. The optimal path of extraction and recycling is studied under various assumptions about the environmental effects of recycling and the assimilative capacity of the environment. In particular, it is shown how the cost of recycling will affect initial resource extraction as well as the environmental quality at the time of resource exhaustion and in the long-run stationary
state.
1. INTRODUCTIOX Most of the literature treating the economics of exhaustible resources disregards the connection between resource extraction and the disposal of residuals in the environment. However, unless the products made from the resources are completely recycled once they have served their “final” good function, the resources will reappear as residuals in the environment. As long as these residuals are regarded as harmless, an analysis treating resource extraction may as well disregard the disposal of them. HoKevcr, many residuals are harmful, or will become harmful once their stock is sufficiently large. Examples of harmful residuals are carbon dioxide (which can affect the climate), thermal pollution from all types of energy based on exhaustible resources, several t’oxic metals such as cadmium, lead, mercury, and various radioactive elements, and other elements such as sulphur, phosphor, etc. In addition to the types of environmental pollution mentioned above, the extracting activity itself may have environmental effects. Two typical examples are ecological consequences of oil spills from offshore oil derricks, and recreational and agricultural losses from strip mining, Some recent literature has treated various aspects of resource extraction t’aking explicit account of environmental effects. d’hrge and Kogiku [a], Maler [12, Chap. 31, and Lusky [ll] treat macroeconomic optimization problems in which exhaustible resources as well as environmental costs enter the analysis. Smith [19] focuses on the connection between recycling and waste accumulation, but only 1An earlier version of this paper [7] was written during a one-year visit at MIT. Financial support from The Bank of Norway’s Fund for Economic Research, The United States Educational Foundation in Norway, and National Science Foundation Grant SOC 75-14258 is gratefully acknowledged. 220 0095-0696/78/0053-0220$02.00/O Copyright All rights
6 197.8 by Academic Press,‘Inc. of reproduction in any form reserved.
RESOURCE
EXTRACTION
AND
RECYCLING
221
little attention is given to how the environmental costs affect resource extraction. More explicit attention to the consequences of environmental costs for resource extraction in a microeconomic context has been given by Schulze [17]. As we shall see, this part of his analysis will be similar to the special case we treat in Section 4. Schulze also analyzes some consequences for resource extraction of possibilities of recycling, but disregards environmental costs in this part of his work. Finally, a more empirically oriented work in this field which should be mentioned is by Nordhaus [14, 151. H ere the consequences of carbon dioxide pollution for the optimal use of energy resources are studied. Resource extraction may in some cases be substituted by recycling activities or by a product produced by the means of a “backstop technology” (cf. Nordhaus [13]).2 In such cases the scarcit’y of the exhaustible resource will be a less serious problem for the economy than cases in which no substitute exists. The presence of a substitute will also make it possible to avoid the negative environmental effects of resource extraction. On the other hand, recycling or substitute production may itself have negat,ive effects on the environment. Recycled material may, for instance, reappear as a harmful residual in the environment, or the process of recycling may require various chemicals, energy use, etc., which all may have negative effects on the environment. An example of a backstop technology with negative environmental effects is nuclear energy as a substitute for fossil fuels. The present paper is a microeconomic partial analysis of how environmental costs affect optimal resource extraction. Unlike the literature mentioned above, the analysis will give explicit attention to the possibility of recycling or the existence of a backstop technology, allowing for the possibility of recycling or substitute production having negative environmental effects. In particular, we shall investigat’e how the cost of rrcycling or producing the substitute will affect the resource extraction and the development of harmful residuals or other negative impacts on the environment.
2. THE
SOCIAL
OPTIMIZATION
The optimization problem we shall The social welfare funct’ion is3
consider
PROBLEM may
be formalized
as follows:
co
w=
i’
e+[U(x
+
y) - h’(R) - cy]dt,
0
which
will be maximized
subject
(1)
to the constraints
s = -2, i? = x + ay -
S(0) 6R,
= so,
R(O) = Ro,
(2) (3)
and the nonnegativity constraints x(t), y(t), S(t) 2 0. The rate of resource extraction is denoted by z(t), and the rate of recycling or the rate of substitute production is denoted by y(t). The remaining resource stock is S(f), while R(t) is a. measure of the stork of harmful residuals existing in t)ht? Bh considerable literature studying resource extraction with a backstop technology but ~with*i-ut environmental costs has appeared since Nordhaus’ article (see, for instance, C&-C]). 3 Time references will usually be omitted for notational convenience.
222
MICHAEL
HOEL
economy. U(Z + y) is the gross utility for society at any instant of time derived from having the rate of extraction plus recycling equal to 2 + y. We shall assume that U’ > 0, U” < 0, c/(1 - a) < U’(0) < 00 and lim,,, U’(y) = 0. E(R) is the disutility for the society at any instant of time from having a stock of harmful residuals equal to R. We shall assume that E(R) is convex, E'(R)> 0, and E'(R,)/(r + 13) < c/(1- a),where R. is t’he initial stock of harmful residuals. This last inequality will be seen to be necessary in order to have any resource extraction take place in an optimal solution. Except for the special case in Section + S)> U'(0)for a 6, we shall also assume that E"(R) > 0 and that aE'(R)/(r sufficiently large value of R. Notice t’hat the environmental effect given by E(R) only includes the stock of harmful residuals, and not the gross or net increase in these residuals. The term cy takes care of the cost of recycling y units of material, measured in the same units as the utility of extraction plus recycling and disutility of harmful residuals. To focus on the existence of environmental costs, we have assumed that traditional extraction costs are zero.4 A somewhat more general social welfare would be to have o(z + y, R) - cyunder the integral in (1). Our specification is chosen for analytical convenience. Relation (2)shows how the development of the resource stock depends on t’he rate of extraction; the initial stock So is known with certainty. Relation (3) shows how the development of the stock of harmful residuals depends on resource extraction and recycling or substitute production. The term 6R (where 6 is a nonnegative constant) represents the natural degrading of the harmful residual due to the assimilative capability of the natural environment. The same simplified representation of the degrading process is used in [9, 16, 19, 201. The case of 6 = 0 (which impliciUy is assumed in [2, 11, 171) will be treated as a special version of our general optimization problem. In addition to the depreciation process by the natural environment, the stock and flow of harmful residuals can be reduced by various types of pollution abatement activities (cf. [l, 9, 16, 20)). However, the present analysis disregards these types of activit,ies. One interpretation of (3) is that after having served its purpose embodied in a final good, the resource is disposed in the environment such that a share (Y (where 0 < (Y 5 1) is a harmful residual (due to where and in what form this fraction is disposed) while the rest is a harmless residual. We shall disregard the time lag between the date of extraction or recycling and the date of disposal, and denote the stock of harmful residuals by P(1) ( now measuring this stock in the same units as resource extraction and recycling). Since recycled material ends up as a residual in the same way as the virgin resource does, the gross increase in P(t) is given by cr[~(t) + y(2)]. Assume now that each unit of recycled material is based on a fraction p of harmful residuals and 1 - p of harmless residuals. It seems reasonable to expect p s (Y, as it usually is easier to recycle from waste in solid form than from waste in liquid or gas form, which often is the most harmful form of residual. Throughout our analysis, we shall assume that 0 < p 4 a’, and as an important special case we shall consider p = CL This special case can, for instance, arise if the material is collected for recycling before any of it is disposed of in t>he environment. The recycling act,ivity uses harmful residuals at the rate @yCr), while the natural * Alternatively, we could let the cost of extraction and recycling be c, and ~2, respectively, alld the gross utility function of using z + r/be UI (.z + 1~).We would then get (1) by defirling U(.c + y) = U,(Z+?/) -c,snndc=c>-cl.
RESOURCE
degrading of P(t) takes is therefore given by
EXTRACTION
place with
AND
the rate P(t).
RECYCLING
The change
223
in P(t)
P=a(z+y)-py-SP.
over time (4)
By defining a = 1 - p/a! (where 0 5 a < 1 due to our assumption 0 < p 5 a) the measuring unit of the harmful residual, w-e and R = P/a, thus changing obtain (3). Our formulation of the optimization problem disregards any constraint on how large the rate of recycling can be. The reason why such a constraint can exist is that if the duration of the material embodied in a final good is taken into consideration, there will be a limit at any time on how much material is available to recycle into additional goods. Another way of saying this is that the stock of material embodied in durable final goods cannot exceed the total amount of material available. However, such a limit can be disregarded as long as the initial stock of material is high compared with the long-run stationary level of recycling which we shall derive. If we were to consider a growing demand for the recycled material such a constraint would sooner or later become relevant (cf. Weinstein and Zeckhauser [21]). Another property of our specification of the recycling activity is t’hat stationary states of y are feasible. In this sense we have 100% recycling. Notice, however, that if 6 = 0 and a > 0 any constant positive level of recycling cannot continue forever. The reason for this is that some of the material will get lost in the sense that it forever (since 6 = 0) ends up as harmful material, while recycling requires a larger fraction of harmless residuals than the fraction of harmless residuals in the waste (p < (Y, or a > 0). For inst’ance, if some material is partly disposed of in gas form in t’he at’mosphere and recycling only is possible from solid waste, a constant level of recycling will only be possible for a finite time period. A more thorough treatment of such cases is given by Schulze [17], where losses of material to unrecoverable forms of residuals occur both in the USC of the material and from the stock of recoverable residuals. In the following analysis we shall disregard such cases, i.e., we shall not consider the case 6 = 0, a > 0. It is clear that the way we have introduced the recycling activit’y, y(2) could as well be interpreted as a substitute produced by a backstop technology. In this case we would have a = 0 in the case of a “clean” substitute for the resource (for instance, solar energy for fossil fuels). If the substitute also had some negative environmental effect (for instance, nuclear energy), we would have a > 0.5 It is of no practical importance whether one interprets y as recycled material or as a substitute; we shall stick to the recycling interpretation in verbal discussions in the following. We mentioned earlier that the negative environmental effect could also be thought of as a direct consequence of extraction activities, such as strip mining and oil spills. In this case we can have complete irreversibility (i.e., 6 = 0), for instance, when recreational and aest,hetic values associated with undisturbed environments arc lost, (see Rrutilla and Fisher [lo, Chap. 31 for a discussion of whether or not t,hc:rc is full irreversibility in such casts). We can also have 6 > 0 in t,his cnsc. An cxamplr of t,his \sould be oil spill s, where the oil gradually gets l.Jf rithrr 3s recycled material &graded )J.V 111~sea. Like Lrfure,y c-31be tl~uugllt. 01‘as a produced substitulr. III both cases we gel a > 0 or (I = C) depending on 5Obviously, it would be better to specify two different residuals in this case.
MICHAEL
224
HOEL
whether or not there are any negative environmental effects from the recycling or production activity. Although the case 6 = 0, a > 0 excluded the possibility of a constant y forever in the case of harmful residuals and recycling, this is not true if either the environmental effects are a consequence of the extraction act’ivity itself or if y is a substitute produced by a backstop technology. In these two cases there is nothing to prevent y constant forever even if 6 = 0, a > 0. 3. CHARACTERIZATION
OF THE
OPTIMAL
SOLUTION
The current value Hamiltonian associated with the problem (l)-(3), so that the shadow prices ~(1) and g(t) turn out to be nonnegative, is H= The necessary
-
U(S+Y)
conditions
E(R)
-
ey -
X(x + ay -
for an optimal
solution
i = (1’+ S)X -
‘Y(R),
imply
6R) -
px.
that” (5)
/.i = 1^/.L,
(6)
SAX+,
where
< implies
5 = 0,
(7)
+ y) 5 c + UX,
where
< implies
y = 0.
(8)
fj’(r+y) V(z
written
equations Using (7) and (8), Eqs. (2), (3), (5), and (6) give us four differential in R, S, X, and p. In addition to the initial conditions R(0) = Ro and S(0) = So we have lim s(t) t-41
2 0,
where
> implies
I
= 0
(9)
and lim e-r%(t) t+rn
= 0,’
(10)
so that the optimal time paths for R, S, X, p, 2, and y are determined. Solving the differential equations (5) and (6) and using (10) gives us
s m
x(t)
=
e(7+6)t
(11)
E’[R (7)]e-(r+~)rd~
t
and ~(0 where ~(0) is determined
so that
(12)
= e%(O),
(9) holds.
Since E(R) is convex,
a useful
conse-
6For notational convenience, we have not introduced any additional superscript or the like to denote optimal values of R, 8, 2, and y. This should not cause any misunderstanding. 7 Strictly speaking, a transversality condition of the type (10) is in general not necessary for an optimal solution. However, it is easy to show that a solution derived by using (10) must be optimal: First, it is obvious from economic considerations that when aE'(R)/(r + 8)> U'(O) for R sufficiently large, the solution to our optimization problem is not changed by introducing an additional constraint R(t)5 N, where N is a large (but finite) given number, since it cannot be optimal to let R + m with our assumptions. Since our Hamiltonian is concave, a solution (3, 8, 2, g) must be optimal if limr_,, (e-‘% (S - 3) - e-r% (R - 8) 1 L 0 for all feasible paths s(t) and R(t)(cf. Seierstad and Sydsaeter [18, Theorem lo]). But P 2 0, S >= 0 and limr,, p.7 = 0, (10) therefore implies that this inequality holds since R(t) s N.
RESOURCE
EXTRACTION
AND
RECYCLING
225
E’(0) E”(R”) -5 x(t) 5 ~ r+s’ r+6 whore Rm is the highest value R has It can easily be derived from our condition for x and y to be positive a period with both x and y positive period. In the special cases we shall will never occur. From (s)-(7) it, is easy to see that $
[U’(x)]
(13)
along the optimal path. equations [7, pp. 12-131 that a necessary simultaneously is that 6 > 0 and p. > 0. If exists, R(t) will be declining during such a treat in the next sections, such a situation when
= rU’(z)
y = 0 and x > 0 we must -
[E’(R)
have
- 6h].
(14)
Wit’hout any environmental effects, i.e., E’[R(t)] = 0 for all 1, it follows from (11) and (14) that we get the standard Hotelling [S] result, i.e., that the undiscounted marginal utility of using the resource should increase with a rate equal to the discount factor. When resource extraction affects the environment in a negative way, this rule is altered according to Eq. (14), where X may be inserted from (11). Whenever resource extraction takes place in a period when there is no recycling, (7) holds with equality and y = 0. In other words, the marginal utility of using the resource at any time equals the sum of the two shadow prices X(t) and p(t), which are the (undiscounted) shadow prices of reducing the stock of harmful residuals by one unit and of increasing the resource stock by one unit, respectively. Similarly, when x = 0 and y > 0 the marginal utility of using the recycled resource at any time equals the sum of recycling costs and the shadow price ax(t). In some of the cases we shall treat in the following sections the optimal solution will approach a long-run stationary stat’e where x = 0 and y and R are constant, dcnotcd by y* and R”. The values y* and R* arc given by
E’(R*) = c + a--T+&
u’(y*)
(15)
and 6 y*=-R*
(16) a
a > 0. Notice that when a > 0 and 6 = 0 we get y* = 0 as long as aE’(R)/(r + 6) > U’(0) for R sufficiently large. In other words, even in cases when
where a constant y is physically possible when a > 0 and 6 = 0 (cf. Section 2), it is optimal to have y* = 0. When a > 0 and 6 > 0 so that (R*, y*) follows from (1.5) and (l(i), implicit] derivation gives us the following results :
dR”/dc < 0,
aR*/aa
$0,
aR*/as
$ 0,
aR*/ar
> 0,
ay*/ac
ay*/aa
< 0,
aye/as
> 0,
ay*/ar
> 0.
-c 0,
It is rather surprising that lower the cost of recycling
(17)
the long-run stock of harmful residuals is higher the is. The reasoning is that the lower this cost is, the
MICHAEL
226
HOEL
C);‘(L) (II’ rcfraitring frc~ni try ((i’(y) lhighrr is Otit? otq)~~rtlurity cost, (Il~lcY~s11rt~c-t adding an additional unit to t(hc stock of lr:un~l’ul rcGduals. WC may also find t,hnt the optimal long-run stock of harmful residuals is Iargcr the larger t’hc assimilative capacity of the environment (mcasurcd by 6) is. The reason for this is similar t,o the reason for dR*/dc< 0.The other results speak for themselves, and need no comment. When a = 0, (15) becomes (15’) u’(y*) = c, i.e., y* is independent determined by earlier zero (for 6 > 0).
of a and 6. In this resource extraction
4. COMPLETE
case we either have R* constant and (for 6 = 0), or R will decline toward
IRREVERSIBILITY
We shall now consider the case where 6 = 0, i.e., where there is no natural depreciation of the harmful residuals in the environment. This is the case which Schulze [17] has treated, but he disregards the possibilit’y of recycling as an alternative to extraction with environmental effects. Defining y = X + ~1, 6 = 0 inserted in (3), (5), and (6) give ?; = ry -
E’(R)
(18)
and & =
f(r)
for y < c + aA,
US(c + ax>
for y > c + aX,
(19)
where f-l(~) = U’(Z). The case in which recycling has no negative environmental effects, i.e., a = 0, is illustrated in Figs. 1 and 2. Figure 1 covers the case in which the resource constraint s(t) 2 0 is binding, while s(t) 2 0 is a nonbinding constraint in Fig. 2. In Fig. 1 y(O) is determined so that R(t)= Ro + So and ~(1) = c are reached simultaneously. In Fig. 2 we have r(t) = A(t), and -r(O) is determined so that simultaneously. In this latter case the optimal stock r(t) reaches c and E'(R)/r
FIGURE 1
RESOURCE
I
EXTRACTION
AND
RECYCLING
:
227
*‘(::=,<
N 0
+‘: 0
-.-:;
II
l!iL)
FIGURE 2
of unextracted resource, S*, is determined by E’(Ro + So - S*)/r = c, which makes the discounted environmental cost of extracting an additional unit of the resource equal to the unit cost of recycling. Formally, this follows from (13), which will hold only if the optimal trajectory is as in Fig. 2. If the constraint S(t) 2 0 is binding like in Fig. 1, it is clear that the long-run stock of harmful residuals R” = Ro + So will be unaffected by the recycling cost. We also see from Fig. 1 that the lower c is, the lower must y (0) be for y(t) to reach c at the same time as R(t) reaches R *. From (3) and (19) we see that this means that the initial resource extraction and the initial growth of harmful residuals is higher the lower is the recycling cost. If R* < Ro + So, like in Fig. 2, we see that R* is lower the lower the cost of recycling is. Notice that this conclusion differs from the conclusion we had when both 6 and a were positive (cf. (17)). Another way of expressing this conclusion is that’ when it is not optimal to exhaust’ the resource completely, the optimal total amount of extraction is lower the lower the cost of recycling is. We also see from Fig. 2 that the lower c is, the lower must y(O) be for r(t) to reach c and E’(R)/r simultaneously. In other words, also when the resource constraint is not binding we find that the initial resource extraction and initial growth of harmful residuals is higher the lower the cost of recycling is. Let us now turn to the case in which a > 0. From (19) we see that in this case d > 0 also after the switch from extraction to recycling, although fi at the time of the switch jumps downwards. Let us first look at the development of x(t) and R(t) after the time to when one switches from resource extraction to recycling. Defining t(t) = c + ax(t), it follows from (5) (with 6 = 0) and (19) (with y > c + ax) that f = r(E -
c) -
d’(R),
A! = af([). Together
with some initial
condition
R(t,)
f’n~ e-+4(t)
(20) (21)
= I? and (lo), which implies = 0,
MICHAEL HOEL
22s
1:
1:’
I!, I,
FIGURE 3
(20) and (21) determine time paths for f(t) and R(t). These paths are illustrated by the phase diagram in Fig. 3. The initial value t(to) must be such that t(t) reaches U’(0) and c + aE’(R)/r at the same time, otherwise (13) will not hold, i.e., e-lt[(t) will not approach zero. If the resource constraint is binding, we must obviously have R = R. + So. The development of r(t) = X(t) + p(t) and R(t) before to in this case follows from (18) and (19) (with y < c + ax). The initial value y(O) is determined so that ~(2) reaches c + aA = [(to) at the same time as R(t) reaches I? = R. + So (cf. the phase diagram in Fig. 4). The case in which the resource constraint is not binding is perhaps best illustrated in a single figure describing the development of X and R for all t: When P(L) = 0, (18) and (19) are simply i = rX -
h”(R), c for X < ___ 1 - a’
f(A) &= af(c
(22)
+ aA>
c
(23)
for X > ~. 1-U
These two differential equations together with R(0) = R. and (10) determine the development of X(1) and R(t). This is illustrated by the phase diagram in Fig. 5. The reason why the curve describing this development is kinked at X = c/(1 - a) is that a < 1, i.e., recycling has less negative impact on the environment than resource extraction has. The initial value X(0) is determined so that X(t) reaches (U’(0) - c)/u and E’(R)/r at the same time. It is easy to see that no other development of A(t) will satisfy (13). Irrespectively of whether or not the resource constraint is binding, we see that the long-run stock of harmful residuals is determined by c + uE’(R*)/r = U’(O), which is identical to (15) when y * = 6 = 0. The results given by (17) are still valid, except that aR*/& I 0 is now excluded. In particular, the long-run st,ock of harmful residuals will be higher the lower the recycling cost is. It is also easy
RESOURCE EXTRACTION
AND RECYCLING
229
FIGURE 4
to see from our differential equations and phase diagrams that the initial resource extraction and initial growth of harmful residuals will be higher (may be lower or higher) the lower the recycling cost is when the resource constraint is binding (nonbinding). Finally, it can be shown that when the resource constraint is nonbinding, the total resource extraction (R - Ro) will be lower the lower the recycling cost is. For details, see Hoe1 [7, pp. 18-191. We have seen that both for a = 0 and a > 0 it can be optimal to extract some, but not all, of the available resource stock. This conclusion holds also when the possibility of recycling is excluded (cf. Schulee [17]). Schulze also finds the possibility of having a positive resource extraction after a period with no extraction. It is easy to verify from our phase diagrams that such a situation can never occur in the present model. The reason for this difference compared with Schulze is not that we have included recycling, but that Schulze assumes a fixed, finite horizon. As one comes close to this horizon, the shadow price of an additional unit of harmful residuals declines, since the disutility of an increased stock of
FIGURE 5
MICHAEL HOEL
230 residuals will find it optimal although this assumed, this
only be integraled over a, shod Io~riotl. 1 tj is thrrcfor~ possi\A~ I o to hava resource ext radian tjo\v:lrtl t,hc cntl of t hr planning pcriotl was not optimal carlicr. Wit,h an infinite horizon, like wc have possibility is excluded. 5. NONBINDING
RESOURCE
CONSTRAINT
We shall now consider the case where 6 > 0, but where the constraint S(t) 2 0 is not binding. In this case the differential equations determining X and R follow from (3), (5), (7), and (8): i = (T + 6)X - E’(R),
f(x) - 6R R=
(24) c for X < 1 - a’ (25)
af(c + ax) - 6R I
forX>+--. -a
Together with the initial value Ro and the condition (10) these two differential equations determine the paths for X and R, provided that’ the corresponding solution for IL:does not violate s(t) 2 0. The solution is illustrated in the phase diagram in Fig. 6. The curve giving A = 0 is as usual given by X = E’(R)/(r + 6) (cf. (24)). The curve giving fi = 0 follows from (25), and is discontinuous at X = c/(1 - a) since a < 1. As “a” approaches zero, the upper part of this broken curve approaches the vertical axis. The heavily drawn curve marked with arrows is th: only development of X and R which satisfies (13). This curve is kinked at 2 (= the stock of harmful residuals at the time of the switch from resources extraction to recycling) because a < 1. The long-run stock of harmful residuals is given from Fig. 6 by U’(6R*/a) = c + uE’(R*)/(r + a), which of course is identical t’o Eq. (15). iSotice that Fig. 6 is drawn so that the curves giving x = 0 and &? = 0 intersect at a value X* > c/(1 - a). If the curve for x = 0 had lain to the right of the upper part of
RESOURCE
EXTRACTION
AND
231
RECYCLING
I III’ I’lll’Vl! I’or I? -=0 (‘III’ dl x 1 I:,‘( I (I), i hc lc~ng-rrm va111(*h’” wo111(1hnvn ilnpliM1 a IOIlg-III11 pOSitiVC! VdW 01’ IY’SOIIIW: (!XtrWCt iOIl. SillCC! f his SOOIN’r Or I:ibT would violat,c the constraint S(t) 2 0, wc can ccmc~ludcthat a nccc~ssary condition for the solution to follow from Fig. 6 is that X* = E’(R*)/(P + 6) 2 G/(1 - a), where R* follows from (15). If a = 0 when 6 > 0, R(t)will approach zero, so that t’hc necessary condition in this case becomes E’(O)/(r + 6) 2 c, which will imply that no resource extraction should ever take place. When a > 0, the solution following from Fig. 6 may or may not imply some resource extract,ion. Notice that the condition E'(Ro)/(r + 6)< c/(1 - a ) is necessary, but’ not sufficient for resource extraction to take place. If the solution does give X(0) < c/(1 - a), implying some resource extraction, the solution will of course only be valid if the conslraint, 8(t) 2 0 is not violated. and X(0) will be affected by the cost of recycling. Let us now seen how R*,l?, If a = 0, our solution will be U’(y*) = c, Z? = Ro,and R(t)--+ 0 whatever value c has. If a > 0, the following conclusions can be drawn (see [7, pp. 22-231 for details) : The lower the cost of recycling, the higher the long-run stock of harmful residuals and the lower this stock at the time when one switches from resource extraction to recycling, provided that some extraction takes place. If no resource extraction takes place, X(0) will bc higher, i.e., the initial growth of harmful residuals will be lower, the lower the cost of recycling. When some resource extraction takes place, the initial extraction and init,ial growth of harmful residuals will be lower, unaffect’ed, or higher the lower the recycling cost is, depending on the functions U(s) and E(R).
6. CONSTANT
XARGINAL
EKVIRONMFXTAL
COST
The last special case WC shall treat is E(R) = bR,i.e., B'(R)= constant. In this case the shadow price x(t) is constant, cf. (11) :
b is a positive
(26) Our assumption b/(r + 6) < c/(1 - a) (cf. the Introduction) implies that cl0 > 0, otherwise we would get a positive and constant resource extraction forever, violating S(2) 2 0. From (14) and (26) we see that when resource extraction is taking place we have
-p(x)=r [U’(x) -
b
;+;
1
,
(27)
which is a standard result in the theory of resource extraction, except that D/(r + S) now represents environmental costs instead of traditional extraction costs. The initial resource extraction z(O) is determined so that the resource is completely exhausted at the same time as U’(z) reaches the total cost of recycling. The time path of U’(J: + y) is illustrated in Fig. 7. From (27) and Fig. 6 it is easy to verify that our conclusion about a lower recycling cost giving a higher initial extraction and higher initial growth of harmful residuals holds also in the present case. It is also easy to see that U’(z(0)) is lower the lower is 6. We can therefore conclude that the initial extraction and growth of harmful residuals is higher the less harmful the environmental effects are regarded.
232
MICHAEL
When recycling takes place, it is constant
u’(y*)
HOEL
and equal to y* (cf. (8) and (26)) :
= c + ;-;+,
(28)
which gives y” > 0 because of our assumption b/(~ + 6) = E’(&)/(r + 6) < c/ (1 - a) < U’(0) and U” < 0. So far, we have said nothing about the optimal development of harmful residuals, except how this stock grows initially. From (3) it is clear that if 6 = 0 we must get R * = Ro + So (disregarding the case where it is optimal not to extract the resource) if a = 0 and l? = ay* > 0, i.e., R(t) -+ 00, if a > 0. The assumption about a constant marginal environmental cost is at best an approximation to some situations. Such an approximation may not be too wrong over certain regions of R, but will clearly be misleading when R -+ m . The approximation E’(R) = b is therefore not suited to treat the case with 6 = 0, a > 0. When 6 > 0, our previous results may be used to find an explicit solution for R(t) : t [X(T) + ay(7)]e6’d7
1 .
There are several possible developments of R(t). We may have R(t) increasing as long as resource extraction takes place. However, it is also possible that as s(t) declines and R(t) grows, we reach a time point when R(t) starts to decline, although resource extraction still is taking place. Once the resource is completely extracted, R(t) will fall toward zero if a = 0. If a > 0, R(t) may rise or decline at first, depending on a, 6, y* and the magnitude of R(t) at the time recycling starts. In any case, R(t) will approach R* = a~*/& where y* is determined by (28). It can easily be verified that R* and y* in this case depends on c, a, 6, and r in the same way as in the general case treated in Section 3 (cf. (17)). Furthermore, since ay*/& < 0 follows from (28), we must have aR*/db < 0. In other words, the less harmful the environmental effects are regarded, the higher is the optimal long-run stock of harmful residuals.
RESOURCE
EXTRACTION TABLE
AND
RECYCLING
233
I
Table Over aR*/dc AS(t) 1 0 nonbinding 6 = 0
a=0 b=O
>
a>0 6>0 a=0
1
S(l) 2 0 binding, E” > 0
s(t) 5 0 binding, E” = 0
Positive
Zero
Zero
Negative
Negative
R(t) +
R(t) -+ 0
K(t) -+ 0
12(t) + 0
Negative
Negative
Negative
+ m
>
6>0 a>0
7. CONCLUSIONS In this section we shall repeat some of the conclusions we found for the cases studied. In these cases resource extraction and recycling will never take place simultaneously. At first only resource extraction takes place, and the optimal rate of extraction declines over time. When recycling starts, it is either constant (for a = 0) or declining (for a > 0). In the latter case the rate of recycling approaches a stationary level, which will be positive if and only if 6 > 0 (i.e., some nat,ural degrading of harmful residuals). , The existence of environmental effects may make it optimal to leave some of a finite resource stock unextracted. Even when it is optimal t’o extract the entire resource stock, in spite of the environmental consequences, the optimal path of extraction will be affected by the existence of environmental effects. The stock of harmful residuals, both in the long-run (R*) and at the time when resource extraction stops (ri) will usually be affected by the recycling costs (c). The same holds for the initial growth of harmful residuals, which is determined by the initial resource extraction (~(0)). Tables I-III summarize how c affects R*,I?,and X(O) for the cases we have treated, cases not treated are blank. The tables do not treat cases in which no resource extraction takes place. From Table II we see that whenever l? is affected by c, this stock is lower the lower is c.
TABLE
II
Table Over ad/&
s(t) 2 0
s(t) 2 0
s(t) L 0
binding, E” > 0
binding, E” = 0
Positive
Zero
Zero
Positive
Zero
Zero
nonbinding 6 = 0 a=0 > 6=0 a>0 > 6>0 a=0 > 6 > 01 a > Ol
No resource extraction Positive
-
-
X34
MICHAEL TABLE
HOEL III
Table Over az(o)/ac s(t) 2 0 nonbinding 6=0 a=0 6=0 a>0 6>0 a=0 6>0 a>0
#S(L)2 0 binding, E” > 0
S(t) 2 0 binding, E" = 0
Negative
Negative
Negative
Undetermined
Negative
Negative
> > >
I-
No resmrce extraction Undetermined
Negative -
Negative
The opposite conclusion holds for R*, except’ for the case where 6 = a = 0, when R* = l? (see Table I). From Table III we see that for most of our cases, the initial resource extraction and growth of harmful residuals will be higher the lower is the recycling cost. Although this may seem somewhat surprising, it corresponds to the usual result that initial extraction is higher the lower the cost of producing a substitute (see Hoe1 [5], where it’is also shown that this result does not necessarily hold when a monopolist owns the resource). Notice, however, that when a > 0 this result does not always hold. In the case of a > 0 we may get a lower initial extraction and growth of harmful residuals the lower the cost is of recycling. We have said nothing about how a competitive economy will perform compared with our optimal solution. In the absence of environmental effects, it is well known that a competitive economy under certain conditions will give optimal rates of extraction and recycling (see, for instance, Weinstein and Zeckhauscr [21]). Even under these conditions the presence of the external environmental effects will in general imply that the competitive solution is nonoptimal. However, from (7) and (8) it can be seen that the competitive solution can be made optimal by introducing Pigouvian taxes X(t) and ax(t) on resource extraction and recycling, respectively, where x(t) is given by (11). Notice that such a tax scheme implies that recycling should not be subsidized, and should be taxed if a > 0. This may seem to be in sharp cont’rast to several policy proposals aiming at reducing resource extraction and/or reducing pollution. However, the contrast’ is only superficial, since the following tax scheme is identical to the one above : Resourcn extraction is not taxed, disposal of used material (virgin or recycled) in t’hc environment is taxed with the rate X(t), and recycling is subsidized with the rate (1 - a)x(t). Like before, this gives net tax rates equal to x(t) and ax(t) on the whole process of extraction and disposal and the whole process of recycling and disposal, respectively. Notice that t’he special case in which material is collected for recycling before any of it is disposed of in the environment (i.e., (Y = p, or a = 0) implies that the net tax on t>he recycling process is zero. If the param&rs CYand p from Srction 2 arc not, complotcly technically d&rmined, a good tax schcmc ought t,o give the agents in the economy an incentive to make (Y low ulnl /j high (cf. Scct.ion 2). Tile tax ~c~hcmes above do no1 have this property, since the tax or subsidy, does not distinguish between where and in what form the material is disposed or collected for recycling. The following modification gives this inccntivc: Instead of taxing total disposal, only harmful
RESOURCE
EXTRACTION
AND RECYCLING
2%
disposal is taxed at the rate x(t)/w 8 Similarly, only harmful mat’erial used for recycling is subsidized at the rate (1 - a)x(t)/P = x(t)/(~. The discussion above on tax schemes to improve a competitive economy disregards all problems of what kind of taxes and subsidies are feasible from a technical, administrative, and political point of view. An extension to such a discussion is beyond the scope of the present analysis. REFERENCES quality, Swedish J. Ewn. 73, (1971). R. C. d’Arge and K. C. Kogiku, Economic growth and the environment, Rev. Economic Studies 40, 61-77 (1973). P. Dasgupta and J. E. Stiglitz, “Uncertainty and the Rate of Extraction under Alternative Institutional Arrangements,” Technical Report No. 179, Institute for Mathematical Studies in the Social Sciences, Stanford Univ., 1976. M. Hoel, Resource Extraction under Some Alternative Market Structures, Mathematical Systems in Economics 39, Verlag Anton Hain, Meisenheim am Glan, 1978. M. Hoel, Resource extraction, substitute production and monopoly, J. .&on. Theory, to appear. M. Hoel, Resource extraction when a future substitute has an uncertain cost, Rev. Economic Studies, to appear. M. Hoel, “Resource Extraction and Recycling with Environmental Costs,” Working Paper No. 197, Department of Economics, MIT, 1977. H. Hotelling, The economics of exhaustible resources, J. Political Econ. 39, 137-175 (1931). E. Keeler, M. Spence, and R. Zeckhauser, The optimal control of pollution, J. Ewn. Theory 4, 19-34 (1972). Studies in the J. V. Krutilla and A. C. Fisher, “The Economics of Natural Environments: Valuation of Commodity and Amenity Resources,” John Hopkins Press, Baltimore, Md., 1975. 1~. Lusky, Optimal taxation policies for conservation and recycling, J. Econ. Theory 11, 315328 (1975). K. G. Maler, “Environmental Economics: A Theoretical Inquiry,” John Hopkins Press, Baltimore, Md., 1974. W. D. Nordhaus, The allocation of energy resources, Brookings Papers, No. 3, 529-570 (1973). W. D. Nordhaus, “Strategies for the Control of Carbon Dioxide,” Cowles Foundation Discussion Paper, mimeo, 1976. W. D. Nordhaus, Economic growth and climate: The carbon dioxide problem, Amer. Econ. Rev. 67 (Papers and Proceedings), 341-346 (1977). C. G. Plourde, A model of waste accumulation and disposal, Canud. J. Econ. 5, 119125 (1972). W. D. Schulze, The optimal use of non-renewable resources: The theory of extraction, J. Environ. Econ. Manag. 1, 53-73 (1974). A. Seierstad and K. Sydsaeter, Sufficient conditions in optimal control theory, Internal. Econ. Rev. 18, 367-391 (1977). V. L. Smith, Dynamics of waste accumulation : Disposal versus recycling, Quart. J. Econ. 86, 600-616 (1972). S. StrGm, Economic growth and biological equilibrium, Swedish J. Econ. 75, 164-175 (1973). M. C. Weinstein and R. J. Zeckhauser, Use patterns for depletable and recyclable resources, Rev. Econ. Studies Symposium on the Economics of Exhaustible Resources, 67-88 (1974).
1. R. C. d’Arge, Essay on economic growth and environmental 25-41
2. 3.
4. 5. 6. 7. 8. 9. 10.
11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
seems to imply that the tax rate X(t)/a is higher the lower is 01,i.e., the less is disposed of in a harmful way. However, the function E(R) is given by E(R) = J?(&) = E(P), so that for a given function e(P), the function E(R), and therefore also x(t) (cf. (1 I)), depends on a. For irrstanc*e, if our original disutility function in Section 6 is fl(I’) = i;P, it. follows from (26) and K = P/u that h(t)/u = i;l(r + 6). i.e., incleprndrilt of LY.
8 This modification of the used material