Modelling depletion of nonrenewable resources

Mathl Comput. Modelling Vol. 15, No. 6, pp. 91-95, 1991 Printed in Great Britain. All rights reserved

MODELLING

DEPLETION

OF

0895-7177/91 $3.00 + 0.130 Copyright@ 1991 Pergamon Press plc

NONRENEWABLE

RESOURCES

MICHAEL OLINICK Department of Mathematics and Computer Sciences, Middlebury College Middlebury, VT 05753

TABLE 1. 2. 3. 4. 5. 6. 7. 8. 9.

OF CONTENTS

Introduction The Static Index The Exponential Index Comparing the Two Indices Changes in the Known Global Reserve Limitations and Lessons References Exercises Solutions 1. I N T R O D U C T I O N

B y definition, a nonrenewable n a t u r a l resource is one whose s u p p l y over t i m e m u s t decrease. Once we use up a t o n of a v a i l a b l e tin or a b a r r e l of p e t r o l e u m , it is gone forever: we have d i m i n i s h e d t h e e a r t h ' s supply. Since t h e t o t a l reserve of a n y m i n e r a l or fuel is finite, we will e v e n t u a l l y r u n o u t of a n o n r e n e w a b l e resource unless we can cut c o n s u m p t i o n to zero or c o m p l e t e l y recycle t h e product. How c a n we c o m p a r e t h e reserves of two m i n e r a l s ? How long will our c u r r e n t s u p p l y of a n o n r e n e w a b l e r e s o u r c e last? T h e two q u e s t i o n s are i n t e r r e l a t e d . M a n y different u n i t s are used to m e a s u r e c o m m o n resources: tons, p o u n d s , t r o y ounces, flasks, b a r r e l s a n d cubic feet are a few. I t ' s h a r d t o c o m p a r e 4.55 x 109 b a r r e l s of p e t r o l e u m with 1.14 x 1015 cubic feet of n a t u r a l gas. In search of a c o m m o n unit, we switch to time a n d m e a s u r e a v a i l a b i l i t y b y the n u m b e r of y e a r s it will t a k e to e x h a u s t the t o t a l supply. In this note, we discuss a simple d e t e r m i n i s t i c r n a t h e m a t i c a l m o d e l for r e s o u r c e d e p l e t i o n t h a t e m p l o y s f a m i l i a r t e c h n i q u e s from i n t r o d u c t o r y calculus. 2. T H E

STATIC

INDEX

T h e r e are s e v e r a l ways to m e a s u r e t h e l e n g t h of t i m e a n o n r e n e w a b l e resource will be available. A s s u m i n g t h a t t h e initiM s u p p l y c a n n o t be i n c r e a s e d , the t i m e it t a k e s to d e p l e t e it d e p e n d s on how fast we c o n s u m e it. We can s i m p l y ask "tIow long will t h e r e s o u r c e last if we keep using it at t h e current r a t e o f c o n s u m p t i o n ? " A m o r e s o p h i s t i c a t e d q u e s t i o n would b e "IIow long will it last if c o n s u m p t i o n r a t e s increase at a constant percentage rate?" If K r e p r e s e n t s t h e k n o w n reserves of t h e resource, a n d if we use it at the c o n s t a n t r a t e of C u n i t s p e r year, t h e n t h e f r a c t i o n s = K/C gives us the n u m b e r of years we can e x p e c t the resource to last. T h e n u m b e r s is called t h e static index. Typeset by A MS-TEX 91

92

M . OLINIOK

For example, the United States Bureau of Mines estimates that the known global reserve of copper is about 340 million tons. We currently use about 9.5 million tons of copper annually. If we continue to consume copper at this same rate, then the static index is s = 340,000,000/9,500,000 = 36 years. As another example, there is an estimated 4.8 billion pounds of cobalt as yet unused. At the present annual rate of usage of 44 million pounds, we would use all of the reserves of cobalt in 4,800,000,000/44,000,000 = 110 years. 3. T H E E X P O N E N T I A L

INDEX

Although the static index is frequently used to determine how long a resource will last, its major drawback as a realistic measure is the underlying assumption that future consumption will remain constant at its current level. For most resources, this is an unwarranted assumption. Even if per capita consumption of a resource remains constant, the overall rate of consumption will increase as the population increases. For some resources (aluminum, copper and natural gas are examples), per capita consumption has also been increasing. Not only are more people consuming resources each year, but the average person is consuming more each year than we did in the past. The rate of consumption is increasing faster than the growth of population for many resources. For some resources, the per capita consumption rate is growing at the same pace or even more slowly than the increases in population. Recent data show that this is true for such resources as cobalt, iron, lead and tin. The exponential index is a way of measuring how long a resource will last if the rate of consumption grows at a constant percentage rate. If rate of consumption is currently C units per year and is changing at a constant rate of r percent per year, then the consumption rate is experiencing exponential growth. As a function of time t, the consumption rate would then be expressed by Consumption Rate = Ce rt To determine the total amount A(t) consumed over a period of t years is just a bit more involved. The rate of change of A is equal to the Consumption Rate. Thus

d A / d t = Cc rt and if we add the initial condition that A(0) = 0

we obtain, after integrating,

A(t) = (C/r)[e È t -

1]

We can now easily find how long it will take to consume the total known global reserve K. We need to solve the equation I ( = (C/r)[e rT - 1] for T. This yields

er T = ( r K / c ) + l : r s + l

since s = K / C

or

T-

ln(sr+l) 7"

This value for T is called the exponential indcx. As an example, we can estimate the exponential index for copper by using the Bureau of Mines projection of an annual 2.7 percent growth rate in the demand for copper. For a static index of 36 years, we compute the exponential index to be ln(36 * .027 + 1)/.027 = ln(1.972)/.027 = .679/.027 = 25 years The static index for coal, whose global reserve is about 5 trillion tons, is 2300 years. The Bureau of Mines predicts an annual growth rate in consumption of 4.1 percent. At such a rate, the exponential index plummets to 111 years.

M o d e l l i n g d e p l e t i o n of n o n r e n e w a b l e re s ourc e s

4. C O M P A R I N G

93

THE TWO INDICES

How much faster will we use up a resource if consumption grows exponentially than if we continue to deplete the resource at a constant rate? Table 1 shows some illustrative examples: T a b l e 1. A t a b u l a t i o n of e x p o n e n t i M i n d i c e s (in ye a rs ) for different va l ue s of s, t h e s t a t i c i n d e x (also in years) mad v a r y i n g r a t e s of g r o w t h r of c o n s u m p t i o n . r

.01

.02

.03

.04

.05

.06

.07

.08

.09

.10

20

18

17

16

15

14

13

13

12

11

11

30

26

24

21

20

18

17

16

15

15

14

40

34

29

26

24

22

20

19

18

17

16

50

41

35

31

27

25

23

21

20

19

18

60

47

39

34

31

28

25

24

22

21

19

70

53

44

38

33

30

27

25

24

22

21

80

59

48

41

36

32

29

27

25

23

22

90

64

51

44

38

34

31

28

26

25

23

100

69

55

46

40

36

32

30

27

26

24

110

74

58

49

42

37

34

31

29

27

25

120

79

61

51

44

39

35

32

30

27

26

130

83

64

53

46

40

36

33

30

28

26

140

88

67

55

47

42

37

34

31

29

27

150

92

69

57

49

43

38

35

32

30

28

I t ' s instructive to examine graphically the relationship between the exponential index and the s t a t i c index for different values of r. Figure 1 shows such plots. Note t h a t a resource with a 200 year s t a t i c index is depleted in a b o u t 55 years under a 4 percent growth rate. If we can decrease the growth rate to 3 percent, then we extend the time to 65 years. A n o t h e r one percentage point drop to 2 percent extends the period to about 80 years while a decrease to a 1 percent annual growth rate in consumption yields an exponential index of 110 years. 110.00

J

<

r= .01

exponential in cliees 82.50

I

r=

,02

r= .03 - r :.04

55.00

27.50

0.0

0.0 ...... ' ..... 50 '.00 ...... ' . . . . .I.O0.00 .........

1'50.00 .....

' ...... 200 .CO

static index s

F i g u r e 1.

5. C H A N G E S

IN T H E K N O W N

GLOBAL RESERVE

In our calculation of the exponential index, we assumed we knew the total global reserve of our resource. As technology improves and the d e m a n d to find new sources of a dwindling resource grows, we may indeed discover new quantities of a resource t h a t were overlooked in earlier exploration. It may also turn out t h a t methods of extracting a resource which were previously thought to be uneconomical now became more a t t r a c t i v e as decreasing supplies and increasing d e m a n d s raise the price of the resource.

M . OLINICK

94

Estimates of the global reserve can change quite dramatically and quickly. In 1970, for example, the Bureau of Mines put the global reserve of copper at 308 million tons. During the next 15 years, the world consumed at least 120 million tons. Yet in 1985, the Bureau estimated a global reserve of 340 million tons! The actual total reserve of the resource might then be many times greater than our current estimate. It's easy to modify the equations for the static and exponential indices to reflect this new knowledge. If the known reserves are multiplied by a factor of n, then so is the static index, while the exponential index becomes ln(nsr + 1) T'

A resource with a static index of 200 years and a known reserve of K has an exponential index of 55 years under a yearly increase in consumption rate of 4 percent. If the actual reserve turns out to be twice as large, 2K, then the exponential index increases to 71 years. For an actual reserve of 4K, the index becomes 87 years and if the reserve is actually 8 times the initial estimate, the exponential index rises to 104 years. In this example, each doubling of the total reserve extends the date of depletion by only about 16 years. Again, it's useful to examine the situation graphically. Figure 2 shows the exponential indices, under a four percent growth rate, for a range of static index between 0 and 200 years. The graph labelled with 2's shows the exponential index with an initial total reserve equal to twice the original estimate of K units. The graph with 3's does the same with an initial reserve of 4K while the 4's graph assumes an initial reserve of 8K. ExponentiM Indices 10.00

4._____.4-----~-sK

82.50

55.00

27 5 0

0.0

0.0

50.00

100.00

150.00

00.00

Static Index

F i g u r e 2.

6. L I M I T A T I O N S A N D L E S S O N S The simple models presented here cannot capture all of the complex dynamics involved in the consumption of natural resources in the real world. As available reserves of a resource shrink, its price will rise. As price climbs, the demand for the resource may fall as cheaper substitutes are found or people decide on a more modest standard of living. An increase in price may also lead to renewed efforts to explore for new sources of the resource or the development of new technologies that will lower the cost of extraction or make possible the use of lower grade quality. But these in turn may boost the demand. We should also consider possibilities of recycling resources. The possible interactions become too complex for simple mathematical models to handle. Analysts may then turn to more complicated computer sinmlation models. The paper by Naill and Behrens listed in the References introduces this approach. W h a t we can learn from the simple model are some qualitative conclusions about the impact of exponential growth of demand on the availability of nonrenewable resources. W h a t appears to be a modest 4 percent annual growth rate in demand can shorten the "lifetime" of the resource dramatically: a mineral which can last 200 years under a constant consumption rate, for example, will be gone in 55 years. Decreasing the percentage growth rate modestly or discovering new reserves can delay the "day of reckoning" but perhaps only marginally. Although predictions of

M o d e l l i n g d e p l e t i o n of n o n r e n e w a b l e resources

95

future behaviour can often be wrong, these simple models help us understand better happen if we continue behaving in the future as we have been behaving recently.

what

will

REFERENCES 1. D o n e l l a H. M e a d o w s et hi., The L i m i t s To Growth, Second Edition, U n i v e r s e Books, New York, p p . 54-63, (1974). 2. R o g e r F. Naill a n d W i l l i a m W . B e h r e n s III, N o n r e n e w a b l e R e s o u r c e Sector, D y n a m i c s of Growth in a Finite World, D e n n i s L. M e a d o w s et aL, W r i g h t - A l l e n Press, C a m b r i d g e , (1974). 3. U n i t e d S t a t e s B u r e a u of Mines, M i n e r a l F a c t s a n d P r o b l e m s , W a s h i n g t o n : D e p a r t m e n t of t h e Interior, (1985).

EXERCISES 1. If t h e k n o w n global reserve of l e a d is 91 million t o n s a n d we c o n t i n u e to c o n s u m e 3.5 million t o n s annually, w h a t is t h e s t a t i c i n d e x ? 2. E s t i m a t e t h e k n o w n global reserve of t u n g s t e n if t h e s t a t i c i n d e x is 40 years a n d t h e a n n u a l c o n s u m p t i o n is 72.5 m i l l i o n p o u n d s . 3. W h a t is t h e a n n u a l c o n s u m p t i o n of p e t r o l e u m if t h e s t a t i c i n d e x is 31 y e a r s a n d t h e global r e s e r v e is believed to be 455 billion barrels? 4. D e t e r m i n e t h e e x p o n e n t i a l index for lead if t h e p r o j e c t e d r a t e of g r o w t h of c o n s u m p t i o n is two p e r c e n t p e r year. 5. E s t i m a t e s for t h e a n n u a l r a t e of g r o w t h of p e t r o l e u m c o n s t u n p t i o n are b e t w e e n four a n d six p e r c e n t . W h a t r a n g e of values does t h i s give for t h e e x p o n e n t i a l i n d e x ? 6. A s s u m i n g t h a t r s is large in c o m p a r i s o n 1, we c a n a p p r o x i m a t e ln(1 + r s ) w i t h l n ( r s ) . U n d e r this a s s u m p t i o n , s h o w t h a t e a c h d o u b l i n g of t h e global reserve e x t e n d s t h e e x p o n e n t i a l i n d e x b y a b o u t (In 2 ) / r years. 7. Develop a m o d e l for resource d e p l e t i o n a s s u m i n g t h a t t h e rate of c o n s u m p t i o n i n c r e a s e s by a fixed c o n s t a n t a m o u n t e a c h year. Show t h a t t h e a m o u n t c o n s u m e d in t y e a r s is a q u a d r a t i c f u n c t i o n of t. How long does it take to u s e u p t h e k n o w n r e s e r v e / ~ ? 8. C o n s i d e r i n g t h e e x p o n e n t i a l i n d e x as a f u n c t i o n of two variables r a n d s, u s e p a r t i a l differentiation to assess t h e i m p a c t on the i n d e x of a u n i t c h a n g e in r or a u n i t c h a n g e in s. 9. If r s is less t h a n one, t h e n we e x p a n d In(1 + r s ) into its Taylor Series a n d , b y t r u n c a t i n g a f t e r a s u i t a b l e n u m b e r of t e r m s , o b t a i n a p o l y n o m i a l e x p r e s s i o n (in r a n d s) for t h e e x p o n e n t i a l index. C a r r y o u t t h i s analysis for c o p p e r w i t h r = .027 a n d s = 36. W h a t conclusions c a n y o u o b t a i n ? 10. ( P r o j e c t A) Develop a n d a n a l y z e a m o d e l of resource depletion in which t h e price of t h e r e s o u r c e is considered. S u p p o s e t h a t d e m a n d d e c r e a s e s w i t h t h e price while t h e price rises as t h e reserve of t h e r e s o u r c e s declines. 11. ( P r o j e c t B) E x t e n d y o u r m o d e l f r o m Exercise 10 to i n c o r p o r a t e a n increase in t h e reserve t h a t will be discovered w h e n prices are sufficiently h i g h to m o t i v a t e n e w efforts to locate a n d e x t r a c t t h e resource. 12. ( P r o j e c t C) T h e B u r e a u of M i n e s p r e p a r e s t h e v o l u m e M i n e r a l Facts and P r o b l e m s at 5 y e a r intervals. It c o n t a i n s a set of e s s a y s on each of a large n u m b e r of m i n e r a l s , i n c l u d i n g e s t i m a t e s o n global reserve, c u r r e n t a n d p r o j e c t e d d e m a n d s , a n d technological changes. C h o o s e s o m e m i n e r a l a n d u s e t h e d a t a f r o m t h e 1970 e d i t i o n to calculate t h e e x p o n e n t i a l index. C o m p a r e t h e p r o j e c t i o n s of y o u r m o d e l w i t h t h e d a t a a p p e a r i n g in t h e 1985 or 1990 edition. W h a t h a s h a p p e n e d to c h a n g e e s t i m a t e s of reserves a n d of d e m a n d ?

SOLUTIONS 1. ~ = (91 × 1 0 6 ) / ( 3 . 5

× 1 0 8 ) = 26 years

2. K = s C -- (40)(72.5 million) = 2.9 billion p o u n d s 3. C = K / s = (455 billion)/31 -- 14.7 billion balTels 4. W i t h r = .02 a n d s -- 26 ( f r o m Exercise 1), we h a v e a n e x p o n e n t i a l i n d e x of fin(1 + r s ) ] / r = [ln(1.52)]/.02 = .4187/.02 = 21 years. 5. We h a v e s = 31. W i t h a low value of r = .04, t h e e x p o n e n t i a l i n d e x is 20 y e a r s while a h i g h e r value of r = .06 gives a n e x p o n e n t i a l i n d e x of 17.5 years. 6. U s e t h e fact t h a t l n ( 2 r s ) -- l n 2 + l n r s . 7. If t h e increase in c o n s m u p t i o n in each year is I, t h e n t h e C o n s u m p t i o n R a t e is e q u a l to C + It. T h e A m o u n t C o n s u m e d , A ( t ) , t h e n satisfies d A / d t -- C + I T w i t h A(0) = 0. T h u s A ( t ) = C t + ( I / 2 ) t 2. T h i s r e a c h e s a value of K for T s a t i s f y i n g K ----C T + ( I / 2 ) T 2 or T = - ~ c + ~ 2It c I 8. For u ( r , s ) = ln(l+rs)

wehaveur

=

1-~r* - l n ( l + r s )

r2

1

r

a n d u~ = r l + r

1

= l+rs