Economics Overexploitation of Resources Question

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The Economics of Exhaustible Resources
Author(s): Harold Hotelling
Source: Journal of Political Economy , Apr., 1931, Vol. 39, No. 2 (Apr., 1931), pp. 137175
Published by: The University of Chicago Press
Stable URL: https://www.jstor.org/stable/1822328
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THE JOURNAL OF
POLITICAL ECONOMY
Volume 39 APRIL 1931 Number 2
THE ECONOMICS OF EXHAUSTIBLE RESOURCES
I. THE PECULIAR PROBLEMS OF MINERAL WEALTH
C ONTEMPLATION of the world’s disappearing supplies
of minerals, forests, and other exhaustible assets has
led to demands for regulation of their exploitation. The
feeling that these products are now too cheap for the good of
future generations, that they are being selfishly exploited at too
rapid a rate, and that in consequence of their excessive cheapness
they are being produced and consumed wastefully has given rise
to the conservation movement. The method ordinarily proposed
to stop the wholesale devastation of irreplaceable natural resources, or of natural resources replaceable only with difficulty
and long delay, is to forbid production at certain times and in
certain regions or to hamper production by insisting that obsolete
and inefficient methods be continued. The prohibitions against
oil and mineral development and cutting timber on certain government lands have this justification, as have also closed seasons
for fish and game and statutes forbidding certain highly efficient
means of catching fish. Taxation would be a more economic meth-
od than publicly ordained inefficiency in the case of purely commercial activities such as mining and fishing for profit, if not also
for sport fishing. However, the opposition of those who are making the profits, with the apathy of everyone else, is usually suffi-
cient to prevent the diversion into the public treasury of any conI37
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I38 HAROLD HOTELLING
siderable part of the proceeds of the exploitation of natural resources.
In contrast to the conservationist belief that a too rapid exploi-
tation of natural resources is taking place, we have the retarding
influence of monopolies and combinations, whose growth in industries directly concerned with the exploitation of irreplaceable
resources has been striking. If “combinations in restraint of
trade” extort high prices from consumers and restrict production,
can it be said that their products are too cheap and are being sold
too rapidly?
It may seem that the exploitation of an exhaustible natural
resource can never be too slow for the public good. For every pro-
posed rate of production there will doubtless be some to point to
the ultimate exhaustion which that rate will entail, and to urge
more delay. But if it is agreed that the total supply is not to be
reserved for our remote descendants and that there is an optimum
rate of present production, then the tendency of monopoly and
partial monopoly is to keep production below the optimum rate
and to exact excessive prices from consumers. The conservation
movement, in so far as it aims at absolute prohibitions rather
than taxation or regulation in the interest of efficiency, maybe
accused of playing into the hands of those who are interested in
maintaining high prices for the sake of their own pockets rather
than of posterity. On the other hand, certain technical conditions
most pronounced in the oil industry lead to great wastes of material and to expensive competitive drilling, losses which may be
reduced by systems of control which involve delay in production.
The government of the United States under the present administration has withdrawn oil lands from entry in order to conserve
this asset, and has also taken steps toward prosecuting a group
of California oil companies for conspiring to maintain unduly
high prices, thus restricting production. Though these moves
may at first sight appear contradictory in intent, they are really
aimed at two distinct evils, a Scylla and Charybdis between which
public policy must be steered.
In addition to these public questions, the economics of exhaus-
tible assets presents a whole forest of intriguing problems. The
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THE ECONOMICS OF EXHAUSTIBLE RESOURCES I39
static-equilibrium type of economic theory which is now so well
developed is plainly inadequate for an industry in which the in-
definite maintenance of a steady rate of production is a physical
impossibility, and which is therefore bound to decline. How much
of the proceeds of a mine should be reckoned as income, and how
much as return of capital? What is the value of a mine when its
contents are supposedly fully known, and what is the effect of
uncertainty of estimate? If a mine-owner produces too rapidly,
he will depress the price, perhaps to zero. If he produces too slowly, his profits, though larger, may be postponed farther into the
future than the rate of interest warrants. Where is his golden
mean? And how does this most profitable rate of production vary
as exhaustion approaches? Is it more profitable to complete the
extraction within a finite time, to extend it indefinitely in such a
way that the amount remaining in the mine approaches zero as a
limit, or to exploit so slowly that mining operations will not only
continue at a diminishing rate forever but leave an amount in
the ground which does not approach zero? Suppose the mine is
publicly owned. How should exploitation take place for the great-
est general good, and how does a course having such an objective
compare with that of the profit-seeking entrepreneur? What of
the plight of laborers and of subsidiary industries when a mine is
exhausted? How can the state, by regulation or taxation, induce
the mine-owner to adopt a schedule of production more in harmony with the public good? What about import duties on coal
and oil? And for these dynamical systems what becomes of the
classic theories of monopoly, duopoly, and free competition?
Problems of exhaustible assets are peculiarly liable to become
entangled with the infinite. Not only is there infinite time to consider, but also the possibility that for a necessity the price might
increase without limit as the supply vanishes. If we are not to
have property of infinite value, we must, in choosing empirical
forms for cost and demand curves, take precautions to avoid assumptions, perfectly natural in static problems, which lead to
such conditions.
While a complete study of the subject would include semireplaceable assets such as forests and stocks of fish, ranging grad-
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I40 HAROLD HOTELLING
ually downward to such short-time operations as crop carry-
overs, this paper will be confined in scope to absolutely irreplaceable assets. The forests of a continent occupied by a new
population may, for purposes of a first approximation at least,
be regarded as composed of two parts, of which one will be replaced after cutting and the other will be consumed without re-
placement. The first part obeys the laws of static theory; the
second, those of the economics of exhaustible assets. Wild life
which may replenish itself if not too rapidly exploited presents
questions of a different type.
Problems of exhaustible assets cannot avoid the calculus of
variations, including even the most recent researches in this
branch of mathematics. However, elementary methods will be
sufficient to bring out, in the next few pages, some of the principles of mine economics, with the help of various simplifying as-
sumptions. These will later be generalized in considering a series
of cases taking on gradually some of the complexities of the actual
situation. We shall assume always that the owner of an exhaustible supply wishes to make the present value of all his future prof-
its a maximum. The force of interest will be denoted by y, so
that emit is the present value of a unit of profit to be obtained
after time t, interest rates being assumed to remain unchanged
in the meantime. The case of variable interest rates gives rise to
fairly obvious modifications.,
2. FREE COMPETITION
Since it is a matter of indifference to the owner
er he receives for a unit of his product a price
poeYt after time t, it is not unreasonable to expec
will be a function of the time of the form p =
apply to monopoly, where the form of the demand function is
bound to affect the rate of production, but is characteristic of
completely free competition. The various units of the mineral
are then to be thought of as being at any time all equally valuable,
excepting for varying costs of placing them upon the market.
‘ As in “A General Mathematical Theory of Depreciation,” by Harold Hotelling,
Journal of the American Statistical Association, September, I925.
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THE ECONOMICS OF EXHAUSTIBLE RESOURCES I4I
They will be removed and used in order of accessibility, the most
cheaply available first. If interest rates or degrees of impatience
vary among the mine-owners, this fact will also affect the order
of extraction. Here p is to be interpreted as the net price received
after paying the cost of extraction and placing upon the marketa convention to which we shall adhere throughout.
The formula
(I)
P=poeyt
fixes the relative prices at different times under free competition.
The absolute level, or the value po of the price when t = o, will depend upon demand and upon the total supply of the substance.
Denoting the latter by a, and putting
q =ft(pt)
for the quantity taken at time t if the price is p, we have the
equation,
T
T
(2) fq dt=f(poeyt, t)dt= a,
the upper limit T being the time of final exhaustion. Since q will
then be zero, we shall have the equation
(3) f(poe-1 , T)= o
to determine T.
The nature of these solutions will depend upon the function
f (p, t), which gives q. In accordance with the usual assumptions,
we shall assume that it is a diminishing function of p, and depends
upon the time, if at all, in so simple a fashion that the equations
all have unique solutions.
Suppose, for example, that the demand function is given by
q=5-Px (o T, so that T becomes the upper limit. W
f(X, q, t)=pqe-t .
Then the owner of the mine (who is now assumed to have a
monopoly) cannot do better than to adjust his production so that
af d 0f
Ox dt 4q
In case f does not involve x, the first term is zero and the former
case of monopoly is obtained.
In general the differential equation is of the second order in x,
since q = dx/dt, and so requires two terminal conditions. One of
these is x = o for t = o. The other end of the curve giving x as a
function of t may be anywhere on the line x = a, or the curve may
have this line as an asymptote. This indefiniteness will be settled
by invoking again the condition that the discounted profit is a
maximum. The “transversality condition” thus obtained,
f-qf
that is,
Op
q2 49-=0,
clq
is equivalent to the proposition that, if p always diminishes when
q increases, the curve is tangent or asymptotic to the line x =a.
Thus ultimately q descends continuously to zero.
Suppose, for example, that q, x, and t all affect the net price
linearly. Thus
P=a-1q-cx+gt.
Ordinarily a, f3, and c will be positive, but g may ha
The growth of population and the rising prices to consumers of
competing exhaustible goods would lead to a positive value of g.
On the other hand, the progress of science might lead to the gradThis content downloaded from
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I54 HAROLD HOTELLING
ual introduction of new substitutes for the commodity in question,
tending to make g negative. The exhaustion of complementary
commodities would also tend toward a negative value of g.
The differential equation reduces, for this linear demand function, to the linear form
d2x dx
23dt2 2 dt j1–x-yt–y
Since 3, c, and y are positive, the root
are real and of opposite signs. Let m denote the positive and -n
the negative root. Since
mn-n=y,
ni9 is numerically greater than n. The solution is
x=Aem’+Be-n + #1 C – 2 gIC2- glcy+ a/c
whence
q=Amemt-Bn-nt+g/c.
Since x=o when I=o,
A+B- 20gIC2 -g/cy+a/c=o.
Since x = a and q = o at the time T of ultimate exhaustion,
AemT +Be-nT +gT/c – 20gIC2 -g/cy +a/c -a = o,
AmemT _Bne-nT+g/c =0.
From these equations A and B are eliminated by equating to zero
the determinant of their coefficients and of the terms not con-
taining A or B. After multiplying the first column by e-mT and
the second by enT, this gives:
e-mT enT – 20gIC2 – glc+a/c
mI I gT-c- 2ngC2 g/c-y+a/c-a =o.
m

n
g1C
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THE ECONOMICS OF EXHAUSTIBLE RESOURCES I55
Expanding and using the relations mr-n =y, and mn
we have for /v and its derivative with respect to T,
/ = (e-mT -enT)g/c+ (ne-mT+ menT) (gT/c- 2#gC2-g/c+ a/Cc-a)
+ (m+n)(20g1C2+g1Cy – a/c’
1= (enT-emT )[T -i/+ (a-ac)/glg7/2
the last expression being useful in applying Newton’s method to
find T. Obviously, the derivative changes sign for only one
value of T; for this value A has a minimum if g is positive, a
maximum if g is negative.
We may measure time in such units that Ay, the force of interest,
is unity. If money is worth 4 per cent, compounded quarterly,
the unit of time will then be about 25 years and i month.
With this convention let us consider an example in which there
is an upward secular trend in the price consumers are willing to
pay: take a=ioo, =I,c=4,g=i6, anda=io. Thenetamount
received per unit is in this case
P=Ioo-q-4x+I-6t.
Substituting the values of the constants, and noting that m= 2
and n= i, we have
e-2T eT I9
A = I I 4T+9 =(8T+I4)eT+(4T+I3)e 2T_ 57,
2
-I
4
Y= (eT-e-2T)(8T+ 22)
Evidently A < o for T = o, A= + oo for T = oo, and A'>o for all
positive values of T. Hence A= o has one and only one positive
root.
For the trial value T= i we have
A=5.IO, Al -77.5
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I56 HAROLD HOTELLING
Applying to T the correction – /A’= – .07 roughly, we take
T=.g3 as a second approximation. For this value of T,
whence -//A’= ooi.
The most profitable schedule of extraction will therefore exhaust the mine in about 0.93I unit of time, or about 23 years and
4 months, perhaps a surprisingly short time in view of the prospect of obtaining an indefinitely higher price in the future, at the
rate of increase of i6 per unit of time.
In order that the time of working a mine be infinite, it is necessary not only that the price shall increase indefinitely but that
it shall ultimately increase at least as fast as compound interest.
The last two equations for determining A and B now become,
since e2T 6.4366 and e-T .3942,
6.4366A + .3942B+-I2. 724o0
12.8732A-. 3942B+4 =0.
Hence A= -.866, B= -i8.I3; so that
x =-. 866e2t-i8. I3e t+44t+ I9 .
As a check we observe that this expression for x vanishes when I
=0.
Differentiating, we have
q=-I . 732e2t+I8. I3e-t+4,
showing how the rate of production begins at 20.40 and gradually
declines to zero. Substitution in the assumed expression for the
net price gives
p= Ioo-q-4X+i6t
= 20+5 . ig6e2t+54.39e-t,
showing a decline from 79.60 at the beginning to 74.90 at exhaus-
tion, owing to the greater cost of extracting the deeper parts of
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THE ECONOMICS OF EXHAUSTIBLE RESOURCES I57
the deposit. The buyer of course pays an increasing, not a decreasing price, namely,
P+4X= ioo-q+i6t
=96+I. 732e2t- 8. I3e t+I6t.
This increases from 79.60 to II4.90.
9. THE OPTIMUM COURSE
To examine the course of exploitation of a mine which would
be best socially, in contrast with the schedule which a wellinformed but entirely selfish owner would adopt, we generalize
the considerations of ? 3. Instead of the rate of profit pq, we must
now deal with the social return per unit of time,
u= p(x, q, t)dq,
x and t being held constant in the integration. Taking again the
market rate of interest as the appropriate discount factor for
future enjoyments, we set
F = ue-alY
and inquire what curve of exploitation will make the total discounted social value,
V= Fdt
a maximum.
The characteristic equation
aF d aF
ax dt aq
reduces to
op d2x+?p dx p ou 3p
aq dt2 ax dt ox at
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I58 HAROLD HOTELLING
The initial condition is x = o for t = o. The other end-point of the
curve is movable on the line x – a = o, a being the amount originally in the mine. The transversality condition,
OF
F-q a=o,
reduces to
u-pq=o.
This is satisfied only for q=o, for otherwise we should have the
equation
p=IXp dq ,
stating that the ultimate price is the mean of the potential prices
corresponding to lower values of q. Since p is assumed to de-
crease when q increases, this is impossible. Even if 9p/Oq is zero
in isolated points, the equation will be impossible if, as is always
held, this derivative is elsewhere negative. Hence q=o at the
time of exhaustion.
If, as in ? 8, we suppose the demand function linear,
P=a-/3q-cx+gt,
the characteristic equation becomes
d2X dx
_ 0′-SY -d–Cyx= _9’t+g-a-y
This differs from the corresponding equation for monopoly only
in that : is here replaced by f/2. In a sense, this means that the
decline of price, or marginal utility, with increase of supply counts
just twice as much in affecting the rate of production, when this is
in the control of a monopolist, as the public welfare would warrant.
The analysis of ? 8 may be applied to this case without any
qualitative change. The values of en and n depend on A, and are
therefore changed. The time T until ultimate exhaustion will be
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THE ECONOMICS OF EXHAUSTIBLE RESOURCES I59
reduced, if social value rather than monopoly profit is to be maximized. For the numerical example given, T was found to be
0.93I unit of time under monopoly. Repeating the calculation
for the case in which maximum social value is the goal, we find
as the best value only 0.674I unit of time.
For different values of the constants, even with a linear de-
mand function, the mathematics may be less simple. For example, the equation A = o may have two positive roots instead of one.
This will be the case if the numerical illustration chosen be varied
by supposing that the sign of g is reversed, owing to the progres-
sive discovery of substitutes, the direct effect of passage of time
being then to decrease instead of increase the price. In such cases
a further examination is necessary of the two possible curves of
development, to determine which will yield a greater monopoly
profit or total discounted social value, according to our object.
IO. DISCONTINUOUS SOLUTIONS
Even if the rate of production q has a discontinuity, as in the
example of ? 5, the condition that ff dt shall be a maximum requires that each of the quantities
aq

q
must nevertheless be continuous.4 This will be true whether f
stands for discounted monopoly profit or discounted total utility.
The equation (8) on p. I47, may be written
aq
which shows, since the left-hand member is continuous, that X
must have the same value before and after the discontinuity.
When p is a function of q alone, the two continuous quantities
may be written in the notation of ? 4, yfe-‘t and (y-qy’) e-y
4 C. Caratheodory, “Uber die diskontinuirlichen LUisungen in der Variations-
rechnung,” thesis, Gdttingen, I904, p. ii. The condition that the first of these quantities must be continuous is given in the textbooks, but for some reason the second
is generally omitted.
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i6o HAROLD HOTELLING
which shows that y’ and y – qy’ are continuous. Thus the expres-
sion X (q-y/y’) appearing in (13), p. I50, is continuous. Conse-
quently the expressions (I3) or (N’) pertaining to the differe
time-intervals may simply be added to obtain an expression of
the same form. Hence the present value of the discounted future
profits of the mine-and therefore of the mine-is in such cases
the difference between the values of
X(q-y/y’)/’y
at present and at the time of exhaustion.
We are now ready to answer such questions as that raised at
the end of ? 5 as to the location of the discontinuity there shown
to exist in the most profitable schedule of production when the
demand function is
p=b-(q-i)3.
Since in this case
f= pqe-^ t= [bq-q(q-I )3]e-‘ At
the two quantities
b -(4q -i) (q -I)2,
3q2(q )2,
are continuous. Consequently
(4q-I) (q 1)2
and
q2(q- I)2
are continuous. If q1 denote the rate of production just before the
sudden jump and q2 the initial rate after it, this means that
(4 I -I) (q – I)2 =(4q2- i)(q2- I)2
q12(qx -1)2 = q22(q2 -I)2
The only admissible solution is:
qI=(3+1/3)/41. Ii830, q2=(3-V3)/4=-03i699.
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THE ECONOMICS OF EXHAUSTIBLE RESOURCES i6i
I I. TESTS FOR A TRUE MAXIMUM
The equations which have been given for finding the production schedule of maximum profit or social value are necessary,
not sufficient, conditions for maxima, like the vanishing of the
first derivative in the differential calculus. We must also consider
more definitive tests.
The integrals which have arisen in the problems of exhaustible
assets are to be maxima, not necessarily for the most general
type of variation conceivable for a curve, but only for the socalled “special weak” variations. The nature of the economic sit-
uation seems to preclude all variations which involve turning
time backward, increasing the rate of production, maintaining
two different rates of production at the same time, or varying
production with infinite rapidity. Extremely sudden increases in
production usually involve special costs which will be borne only
under unexpected conditions, and are to be avoided in long-term
planning. Likewise sudden decreases involve social losses of great
magnitude such as unemployment, which even a selfish monopolist will often try to prevent. This will be considered further in
the next section. It is indeed possible that in some special cases
these “strong” variations might take on some economic significance, but such a situation would involve forces of a different sort
from those with which economic theory is ordinarily concerned.
The critical tests which must be applied are by the foregoing
considerations reduced to two-those of Legendre and Jacobi.S
The Legendre test requires, in order that the total discounted
utility or social value (? 9) shall be a maximum, that
a2u a (p
aqq2 aq< a condition which is always held to obtain save in exceptional cases. In order that the chosen curve shall yield a genuine maximum for a monopolist's profit, the Legendre test requires that a2(pq) 2 ap+ a2p< aq2 =2q aq~2< 5 A. R. Forsyth, Calculus of Variations (Cambridge, 1927), pp. I7-28. This content downloaded from 129.59.122.16 on Wed, 23 Mar 2022 15:56:36 UTC All use subject to https://about.jstor.org/terms I62 HAROLD HOTELLING This means that the curve of Figure i is convex upward at all points touched by the turning tangent. The re-entrant portions, if any, are passed over, producing discontinuities in the rate of production. When the solution of the characteristic equation has been found in the form x=O(t, A, B), A and B being arbitrary constants, the Jacobi test requires that af/aA shall not take the same value for two different values of t. For the example of ? 8 this critical quantity is simply e(m+n)t, which obviously satisfies the test. The solution represents a real, not an illusory maximum for the monopolist's profit. The like is true for the schedule of production maximizing the total discounted utility with the same demand function. Each case must, however, be examined separately, as the test might show in some instances that a seeming maximum could be improved. I2. THE NEED FOR STEADINESS IN PRODUCTION The demand function giving p may involve not only the rate of production q, but also the rate of change q' of q. Such a condition would display a duality with that considered by C. F. Roos6 and G. C. Evans,7 who hold that the quantity of a commodity which can be sold per unit of time depends ordinarily upon the rate of change of the price, as well as upon the price itself. If p is a function of x, q, q', and t, the maximum of monopoly profit or of social value can only be obtained if the course of exploitation satisfies a fourth-order differential equation. 6 "A Pynamical Theory of Economics," Journal of Political Economy, XXXV (I927), 632, and references there given; also "A Mathematical Theory of Depreciation and Replacement," American Journal of Mathematics, L (1928), 147. 7 "The Dynamics of Monopoly," American Mathematical Monthly, Vol. XXXI (1924); also Mathematical Introduction to Economics (McGraw-Hill Book Co., 1930). This content downloaded from 129.59.122.16 on Wed, 23 Mar 2022 15:56:36 UTC All use subject to https://about.jstor.org/terms THE ECONOMICS OF EXHAUSTIBLE RESOURCES i63 More generally we might suppose that p and its rate of change p' are connected with x, q, q', and t by a relation Purchase answer to see full attachment Explanation & Answer: 3 pages Tags: Overexploitation decline of natural resources natural resuources User generated content is uploaded by users for the purposes of learning and should be used following Studypool's honor code & terms of service.