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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.

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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.
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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).
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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
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