A Depletion Protocol for Non-Renewable Natural Resources

Natural Resources Research, Vol. 15, No. 3, September 2006 (



2006)

DOI: 10.1007/s11053-006-9018-1

A Depletion Protocol for Non-Renewable Natural

Resources: Australia as an Example

Albert A. Bartlett

1,2

Received 5 April 2006; accepted 18 May 2006

Published online: 29 November 2006

This paper examines the implications of statements by Australia’s Minister of... Resources

that Australia’s exports of coal are growing rapidly and that Australia’s coal will last “110

years at current rates of production.” If one assumes that coal production P(t), follows a

Gaussian curve (similar to a Hubbert curve) one can construct a family of Gaussian curves

showing possible future paths of P(t) which are consistent with the cited “110 years.” Each

curve reaches a maximum after which P(t) declines toward zero. Knowledge of the present

value of dP/dt allows one member of the family to be identiﬁed as the most probable fu-

ture path of P(t). Families of curves and tabular data are presented for resource quantities

that would last 50, 100 and 200 years “at current rates of production.” If, instead, Australia’s

P(t) follows a declining exponential curve (exp( −kt)) with k = (1/110) per year, the stated

quantity of coal will allow production to continue forever, with P(t) declining with a half life

of 76 y. This and more rapidly declining exponential paths are the only paths that can be

said to be sustainable. The envelope of the family of Gaussian curves divides the (P, t) plane

into “allowed” and “forbidden” areas. The declining exponential curve divides the “allowed”

area into an upper area that is “terminal” and a lower area that is “sustainable.” These facts,

coupled with Australia’s expectations of rapid growth of its population, suggest that Aus-

tralia’s present resource policies are “anti-sustainable” and that the people of Australia need

to rethink their present policy of rapidly exporting their fossil fuels.

KEY WORDS: Australia, energy policy, fossil fuels, greenhouse gases, Hubbert curve, population.

INTRODUCTION

The Australian Minister of Industry, Tourism,

and Resources recently wrote:

“Australia is the world’s largest exporter of coal...

Our demonstrated recoverable economic resource

for black coal is estimated to be more than 39 bil-

lion tones, or more than 110 years’ worth of supply

at current rales of production... The Australian ex-

port coal industry... has expanded production at the

rate of around 5 per cent per annum over the last

20 years... In 2003, Australia was the fourth-largest

liqueﬁed natural gas (LNG) exporter in the Asia-

Paciﬁc region and the seventh-largest in the world,

Department of Physics, University of Colorado, Boulder, CO

80309-0390, USA.

To whom correspondence should be addressed; e-mail: albert.

[email protected].

exporting 8 million tonnes of LNG. Australia’s LNG

export capacity could potentially rise further to

more than 50 million tonnes per annum (Mtpa) by

early next decade...The reserve-to-production ratio

[for LNG] in Australia delivers an expected resource

lifetime of approximately 77 years.” (All emphasis

has been added) (Macfarlane, 2005)

This quotation is representative of the optimistic

view of the future of Australia and its energy re-

sources as expressed by the Minister. The editor

of the journal in which the Minister’s article ap-

peared wrote: The Minister’s “focus is the growth and

sustainability of Australian industries...” (emphasis

added).

The goal of this paper is to make a mathe-

matical examination of the hypothetical possible

future paths of the rate of production vs. time of

151

1520-7439/06/0900-0151/0

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2006 Springer Science+Business Media, LLC

152 Bartlett

a nonrenewable resource to see if the Minister’s

optimism is warranted by the facts.

DEFINITION OF TERMS

We deﬁne needed quantities:

t = time in years; the present time is designated as

t = 0.

R(i) = the quantity of the resource initially available

before any of it was extracted.

R(0) = the quantity of the resource remaining at

time t = 0 in units such as tonnes. The value of

R(0) for speciﬁc resources is debatable and cannot

be known exactly.

R(t) = an estimate of the quantity of the resource

remaining as a function of time, in units such as

tonnes.

P(t) = the annual rate of production of the resource

as a function of time in units such as tonnes/y

P(0) = the annual rate of production at the time

t = 0 in tonnes/y.

k = the fractional annual growth rate of P(t) in units

of t

−1

; if k is positive and constant, one has expo-

nential growth. If k is negative and constant one

has exponential decay.

t(E) = is the abrupt expiration time of the reserves

R(0), “at current rates of production.”

σ = the standard deviation of the Gaussian Error

Curve.

R/P = is the reserves to production ratio (R/P ra-

tio). It is the hypothetical estimate of the life-

expectancy (years) of the reserves of a resource “at

current rates of production.” The ratio will change

as resources are consumed or as new reserves be-

come available.

R(0)/P(0) = the value of the R/P ratio at time t = 0

R(0)/P(0) = t(E)

THE RESERVE TO PRODUCTION

RATIO; R/P

When people wish to convey an idea of the life

expectancy of reserves of nonrenewable resources

they may give the life expectancy of the reserves “at

current rates of production” which is determined by

dividing the current estimate of the reserves R(0) (in

tonnes) by the current rate of production P(0) (in

tonnes/y). This is termed the “Reserve-to-Production

Ratio,” R/P which has the units of time (years).

The life expectancy given by the R/P ratio is what

one would calculate if P(t) remains unchanged from

its present value P(0) until R(0) is completely con-

sumed (Bartlett, 1978). If P(t) is growing, the life-

expectancy of a resource will be less than it would be

if P(t) continued unchanged “at current rates of pro-

duction.” Yet even when

P(t) is growing, this “zero

growth” parameter, R/P, is frequently cited and ac-

cepted as the life-expectancy of a resource. When

P(t) is growing, this use of the R/P ratio conveys

the optimistic expectation of a life-expectancy that

is longer than is warranted by the facts. This serious

misrepresentation of the facts is widespread. It gives

rise to unwarranted optimism on the part of political

leaders, journalists and others.

For given values of R(0)/P(0) of a resource, we

want to examine, in generic terms, the limited scenar-

iosoftheP(t)vs.t that are allowed by the numbers,

and the even more limited scenarios that are sustain-

able.

HUBBERT CURVES

M. King Hubbert observed that the long-term

history of the production P(t) (tonnes/y) of a re-

source vs. time starts at zero, rises to one or more

maxima, and then returns to zero. Hubbert (1974)

The area under the complete curve represents R(i)

the total tonnes of the resource before any of it was

extracted. Hubbert used the derivative of the Logis-

tic function as a model for the rise and fall of P(t). In

a curve-ﬁtting analysis of U.S. and world oil produc-

tion data, Bartlett (2000) reported:

“The analysis... was done with both curves (the

derivative of the Logistic curve and the Gaussian

curve) to allow comparison of the results. The dif-

ferences in the results were smaller than the root

mean square deviations of the ﬁts, so that the results

did not indicate a clear preference for either curve.

Both curves are widely understood, but the Gaus-

sian curve was used because the analysis seemed

simpler in execution and interpretation.”

Technically then, only the derivative of the Lo-

gistic function should be termed a Hubbert curve.

Because the two curves are indistinguishable at the

level of precision of the analysis of typical production

data, the Gaussian curve is a good approximation to

a Hubbert curve and is used here.

The results of calculations will be presented

numerically and graphically for three represen-

tative values, 50, 100, and 200 years, of the R/P

A Depletion Protocol for Non-Renewable Natural Resources: Australia as an Example 153

life expectancy of resources “at current rates of

production.” Interpolation or extrapolation then can

be done to estimate the hypothetical future scenarios

of P(t) for resources for which the current value of

R/P has been estimated, including the two examples

from Australia.

CALCULATIONS

We assume that the Gaussian Error Curve is

a reasonable approximate scenario for the curve of

P(t)vs.t for ﬁnite nonrenewable resources. We will

refer to the Gaussian scenarios as the “business as

usual” scenarios. For each of the three assumed val-

ues of the R(0)/P(0) ratio (50, 100, and 200 years) a

generic family of Gaussian curves of differing widths

is created which all start with P(0) = 100 tonnes per

year at the present time, t = 0. Each curve rises to

a maximum and then declines toward P(t) = 0. The

curves differ from one another by having different

standard deviations, σ, (widths). The area under the

curve of P(t)vs.t from t = 0 to t approaching ∞

is the quantity of the resource remaining at t = 0,

so all members of a family of curves must have the

same area between t = 0 and t =∞. Each mem-

ber of a family of curves has its own fractional rate of

growth; k = ((1/P)dP/dt), at t = 0. From the family,

one picks the curve whose initial value of k equals the

reported current fractional rate of increase of P(t).

This curve then can be considered to be the best esti-

mate of the probable future path of P(t) for the given

(0)/P(0) ratio and for the “business as usual” sce-

nario (Gaussian Curve).

The scenarios are modeled on an Excel spread-

sheet which was used for all of the calculations and

plotting.

UNCERTAINTIES IN THE

CALCULATIONS

The largest uncertainty in these results arises

from the uncertainty of our knowledge of the value

of R(0). Smaller uncertainties arise from the good-

ness of the ﬁt of a Gaussian Curve to data for P

from the real world. As an example, using data for

the P(t)vs.t of U.S. petroleum production, the root

mean square (RMS) deviation between the data and

the best-ﬁt Gaussian was 3.2% of the height of the

Gaussian maximum. (Bartlett, 2000)

EXPONENTIAL GROWTH OF P(t)

“At the current rate of production” implies the

unrealistic scenario of k = 0, in which P(t) = P(0)

until the last tonne of the resource is produced,

whereupon P(t) falls abruptly to zero at the time t(E)

which is equal to the ratio R(0)/P(0). The graph of

the P(t) vs. time is a rectangle whose height is P(0);

whose length is R(0)/P(0) and whose area is R(0).

However, if the rate of production P(t)isgrow-

ing at some fractional rate k > 0, such as 5% per

year, P(t) = P(0) exp(kt), then it is enlightening to

see what happens if one assumes that the fractional

growth continues at an unchanged rate until all of

the reserves R(0) have been consumed at an expira-

tion time t(k), whereupon P(t

) falls abruptly to zero.

This also is an unrealistic scenario, but it must be ex-

amined because our national and global economies

are predicated on unending steady growth of our

economies and of our rates of consumption of re-

sources. The expiration time t(k) is given by Robis-

coe (1973),

t(k) = (1/k)ln((kR(0)/P(0)) +1) (1)

At the peak, just before expiration, the rate of

production P(t) is larger than the initial rate P(0) by

a factor:

P(t(k))/P(0) = ((kR(0)/P(0)) +1) (2)

Representative expiration times for ﬁve val-

ues of R(0)/P(0) and for different constant rates of

growth of P(t) are given in Table 1.ForP(0) = 100

tonnes/y, the corresponding peak heights just before

expiration are given in Table 2. For example, if a re-

source would last 200 years at present rates of pro-

duction, Table 1 shows that with a constant growth

of 5% per year, the resource will expire in 48 years.

Table 2 shows that just before the resource expires

P(t)is11timesP(0). Several such scenarios P(t)vs.t

for R(0)/P(0) = 200 y are shown in Figure 1

On the subject of LNG, the Minister writes op-

timistically that the R(0)/P(0) ratio for Australia’s

(LNG) gives an “expected resource lifetime of ap-

proximately 77 years,” and he reports that LNG ex-

ports are expected to grow from 8 million tonnes/y

to 50 million tonnes/y by early in the next decade. If

one assumes the expected growth takes place over a

period of 10 years, then the average rate of growth

of exports of LNG projected by the Minister is:

(Bartlett, 1993)

k = (1/10) ln(50/8) = 0.18 or 18% per year (3)

154 Bartlett

Table 1. Expiration Times of a Nonrenewable Resource, Ex-

pressed in Years, for a Series of Reserve to Production Ra-

tios, R(0)/P(0), for Each of Several Constant Annual Growth

Rates from 1 to 10% per Year, in Accord with the Patterns

of Figure 1 and Equation (1)

Annual Percent Growth Rates

R(0)/P(0)

(Years) 1 3 5 7 10

10 9.5 8.7 8.1 7.6 6.9

20 18.2 15.7 13.9 12.5 11.0

30 26.2 21.4 18.3 16.2 13.9

50 40.5 30.5 25.1 21.5 17.9

70 53.1 37.7 30.1 25.4 20.8

100 69.3 46.2 35.8 29.7 24.0

200 109.9 64.9 48.0 38.7 30.4

300 138.6 76.8 55.5 44.2 34.3

500 179.2 92.4 65.2 51.2 39.3

If one imagined that this growth rate could be

continued until the cited amount of Australia’s nat-

ural gas was completely extracted, Equations (1) and

(2) show that the resource would expire in 15 years

rather than 77 years cited by the Minister, and the

peak production P(t) would be 15 times P(0).

CLOSER TO THE REAL WORLD

Even though it is consistent with accepted eco-

nomic goals, it is unrealistic to imagine that an econ-

omy could maintain a constant unchanging P(t) until

the last bit of a resource has been extracted. To be

more realistic we note that positive rates of growth of

P(t) will cause the graph of P(t) vs. t to pass through

the maximum after which P(t) will decline and ap-

Table 2. Peak Heights as Multiples of P(0) for a Series Re-

serve to Production Ratios, R(0)/P(0), for Each of Several

Steady Annual Growth Rates from 1 to 10% per Year from

Equation (2) and Figure 1

Annual Percent Growth Rates

R(0)/P(0)

(Years) 1 3 5 7 10

10 1.1 1.3 1.5 1.7 2.0

20 1.2 1.6 2.0 2.4 3.0

30 1.3 1.9 2.5 3.1 4.0

50 1.5 2.5 3.5 4.5 6.0

70 1.7 3.1 4.5 5.9 8.0

100 2.0 4.0 6.0 8.0 11.0

200 3.0 7.0 11.0 15.0 21.0

300 4.0 10.0 16.0 22.0 31.0

500 6.0 16.0 26.0 36.0 51.0

proach zero. We can model this behavior mathemat-

ically by assuming that P(t) follows a Gaussian and

that dP/dt > 0att = 0 so that the maximum is in the

future. The curve of P(t)vs.tisassumedtobegiven

by:

P = P(0)exp(−0.5(t/σ)

)(4)

where P(0) will be assumed to be 100 tonnes/y, the

time t is expressed in years, and σ is the standard de-

viation of the Gaussian curve in years.

RESULTS THAT CAN BE APPLIED

TO THE REAL WORLD

In order to provide a basis for applying these re-

sults to a variety of resource situations, families of

Gaussian curves for values of R(0)/P(0) = 50 y, 100

y, and 200 y are shown respectively in Figures 2–4.

In addition to the family of curves, each ﬁgure has

a rectangular curve that ends at the year R(0)/P(0)

which represents P continuing “at current rates of

production,” i.e. k = 0. The individual members of

each family of curves are characterized by different

standard deviations σ, and each curve in a given fam-

ily has its own initial rate of growth of P(t)att = 0.

All of the curves have the same area, R(0) which is

equal to 100(R(0)/P(0)) tonnes. Data from the calcu-

lations for each of these situations are summarized in

Tables 3–8.

EXAMPLE

For the situation of R(0)/P(0) = 200 y, Figure

4 shows the family of curves of P(t)vs.t. The pa-

rameters of these curves are summarized in Tables

7 and 8.Forσ = 30 y, Table 7 shows that at t = 0,

P(t) is growing 4.8 percent per year, the peak will be

reached in year 44 and its height will be 2.86 times

P(0). In year 87, P(t) will have fallen to its initial

value of 100 tonnes/y and by symmetry, it will be

falling at 4.8 percent per year. Table 8 shows that for

σ = 30 years, it will take 47 years to consume 50%

of R(0). At the peak of the curve, the R/P ratio for

the then remaining resource will be 38 years and in

year 87 when P has dropped to its initial value of 100

tonnes/y, the R/P ratio will be 16 years. This exam-

ple describes a rapid extraction and depletion of the

resource.

Now let’s ﬁnd a curve that approximately ﬁts

the data given for Australia’s LNG: R(0)/P(0) = 77

A Depletion Protocol for Non-Renewable Natural Resources: Australia as an Example 155

Figure 1. Generic graph of rate of production P(t)vs.t. For all curves initial production

at time t = 0 is 100 tonnes/y. If Reserves to Production ratio R(0)/P(0) = 200 years, then

implied graph of P(t) vs. time is given by lower horizontal line graph labeled “0%.” If rate

of growth of P(t) is constant 5% per year until resource is consumed completely, then P(t)

vs. time is shown by sharply rising upper curve where peak rate of production is 11 times

initial P(0), and P(t) then falls abruptly to zero in just under 50 years. Similar scenarios are

shown for steady growth of P(t) of 2% and 1% per year. Four curves all have same area of

20,000 tonnes. Data from these curves are included in Tables 1 and 2. These scenarios are

illustrative but completely unrealistic.

Figure 2. Generic graph of curves of P(t)invs.t for R(0)/P(0) = 50 years. Horizontal

straightline shows a constant P(t) of 100 tonnes/y for 50 years at which point the resource

is consumed completely and P(t) falls abruptly to zero. Other curves are “business as

usual” Gaussian scenarios with values of the standard deviation σ of curves of (from top

to bottom) 10 y, 15 y, 20 y, 25 y, and 30 y. Curves all have same area of R(0) = 5000

tonnes. Numerical data from this family of curves are given in Tables 3 and 4.

156 Bartlett

Figure 3. Generic graph of curves for P(t) vs. t for case of R(0)/P(0) = 100 years. Horizontal

curve shows constant P(t) of 100 tonnes/y for 100 years at which point resource is consumed

completely and P(t) falls abruptly to zero. Other curves are “business as usual” Gaussian

scenarios for values of standard deviation σ of curves of (from top to bottom) 10 y, 15 y, 20 y,

30 y, 40 y, 50 y, and 60 y. Curves all have same area of R(0) = 10,000 tonnes. Numerical data

from this family of curves are given in Tables 5 and 6.

years and ((1/P)dP/dt) = 18% per year. An approx-

imate estimate of the future production scenario can

be determined by using Figure 3 and Tables 5 and 6

which are for the R(0)/P(0) of 100 years. In Table 5

we will use the numbers for σ = 10 y (initial slope =

20 percent per year) in place of the 18 percent per

year from the Minister’s statement. The tables in-

dicate the following approximate results: the peak

Figure 4. Generic graph of curves of P(t) vs. t for case of R(0)/P(0) = 200 years. Horizontal

curve shows constant P(t) of 100 tonnes/y for 200 years at which point resource is consumed

completely and P(t) falls abruptly to zero. Other curves are “business as usual” Gaussian sce-

narios for values of standard deviation σ of curves of (from top to bottom) 10 y, 15 y, 20 y, 30 y,

40 y, 50 y, 60 y, 70 y, and 80 y. Curves all have same area of R(0) = 20,000 tonnes. Numerical

data from these curves are given in Tables 7 and 8.

A Depletion Protocol for Non-Renewable Natural Resources: Australia as an Example 157

Table 3. For the Initial Reserve to Production Ratio, R(0)/P(0)

of 50 Years, and P(0) = 100 Tonnes/y, as Shown in Figure 2,This

Table Gives the Expected Values of Four Features of the Gaussian

Curves that Represent the “Business as Usual” Scenarios of P(t)

vs. t for Different Values of σ

σ Sigma

in Years

Initial

Slope at

t = 0% Per

Year (%)

Year of

Gaussian

Peak

Height of

the

Gaussain

Peak

Tonnes/Year

Year of

Return to

Original

P(0) 100

tonnes/y

10 14.3 13 220 25

15 6.8 14 158 29

20 3.7 14 129 29

25 2.1 13 114 25

30 1.1 10 105 19

35 0.42 5 101 10

For example, for σ = 20 years, the initial slope (1/P)dP/dt =

+3.7% per year. The Gaussian peak will be reached in 14 years

at which time P(14) = 129 tonnes/y. At t = 29 y, P(29) = 100

tonnes/y and by symmetry (1/P)dP/dt =−3.7% per year.

for Australian LNG would be reached in 17 years at

which time P would be 4.16 P(0) and the R/P ratio

would then be 13 years. In 34 years P would fall to

its initial value of 100 tonnes/y at which time the R/P

ratio would be about 5.2 years. The curve for σ = 10

years in Figure 3 is a graphic portrayal of the approx-

imate expected “business as usual” scenario for the

future production of Australia’s LNG, based on the

numbers given by the Minister.

CALCULATION OF R/P RATIOS

AS A FUNCTION OF TIME

The R/P ratio can be calculated as a function of

time for every point on each of the Gaussian curves

Table 4. For the Reserve to Production Ratio R(0)/P(0) = 50

Years, as shown in Figure 2, This Table Gives, for Several Values

of σ, the Number of Years Needed to Consume Half of R(0), the

R/P Ratio at the Peak of the Gaussian Curve, the R/P Ratio after

the Peak has been Passed and P(t) has Declined Back to its Initial

Value of 100 Tonnes/Year

σ Sigma in

Years

Years to

Consume Half

of R(0)

R/P in Years

at Peak

R/P at in

Years When

P(t) = P(0)

10 14 12.6 6.32

15 18 19.7 10.6

20 21 26.1 16.0

25 23 31.9 22.9

30 25 38.1 30.7

35 26 44.9 40.3

Table 5. For the Initial Reserve to Production Ratio, R(0)/P(0) of

100 Years, and P(0) = 100 Tonnes/y, as Shown in Figure 3,This

Table Gives the Expected Values of Four Features of the Gaussian

Curves that Represent the “Business as Usual” Scenarios of P(t)

vs. t for Different Values of σ

σ Sigma in

Years

Initial

Slope at

t = 0% Per

Year (%)

Year of

Gaussian

Peak

Height of

the

Gaussain

Peak

Tonnes/Year

Year of

Return to

Original

Rate of

Production

10 20.1 17 416 34

15 10.7 22 286 44

20 6.72 25 221 50

30 3.32 29 159 58

40 1.84 29 130 58

50 1.04 26 114 51

60 0.55 20 106 39

70 0.23 11 101 23

For example: for σ = 20 years, the initial slope

(1/P)dP/dt =+6.72% per year. The Gaussian peak will be

reached in 25 years at which time P(25) = 221 tonnes/y. At t = 50

y, P(50) = 100 tonnes/y and by symmetry (1/P)dP/dt =−6.72%

per year.

of P(t)vs.t and, as expected, the R/P ratio, based on

the current estimate of R(0), decreases continuously

with increasing time as is shown in Figure 5 for the

case of R(0)/P(0) = 200 years. These curves are sim-

ilar to those shown by Bartlett (2000, p. 14) for the

curves for the R/P vs. time for Gaussian curves for

world oil. The reasonableness of the values of R/P is

indicated by the fact that the current R/P for world

oil is often stated to be approximately 40 years which

is in good agreement with the value from Bartlett’s

R/P curve for R(i) = 2 × E(12) barrels.

Table 6. For the Reserve to Production Ratio R(0)/P(0) = 100

Years, as Shown in Figure 3, This Table Gives, for Several Values

of σ, the Number of Years Needed to Consume Half of R(0), the

R/P Ratio at the Peak of the Gaussian Curve, and the R/P Ratio

After the Peak Has Been Passed and P(t) Has Declined Back to

its Initial Value of 100 Tonnes/Year

σ Sigma in

Years

Years to

Consume Half

of R(0)

R/P in Years

at Peak

R/P in Years

When

P(t) = P(0)

10 18 13 5.2

15 23 19 8.5

20 28 26 12

30 35 38 21

40 41 51 31

50 45 63 44

60 49 75 59

70 51 88 77

158 Bartlett

Table 7. For the Initial Reserve to Production Ratio, R(0)/P(0) of

200 Years, and P(0) = 100 Tonnes/y, as Shown in Figure 4,This

Table Gives the Expected Values of Four Features of the Gaussian

Curves that Represent the “Business as Usual” Scenarios of P(t)

vs. t for Different Values of σ

σ Sigma in

Years

Initial

Slope at

t = 0%

Per Year

(%)

Year of

Gaussian

Peak

Height of

the

Gaussian

Peak in

Tonnes/Year

Year of

Return to

Original

P(0)

10 20.5 20 812 41

15 12.3 28 548 55

20 8.4 34 417 68

30 4.8 44 286 87

40 3.2 51 222 101

50 2.2 55 184 110

60 1.6 58 159 116

70 1.2 59 142 117

For example: for σ = 20 years, the initial slope

(1/P)dP/dt =+8.4% per year. The Gaussian peak will be

reached in 34 years at which time P(34) = 417 tonnes/y. At t = 68

y, P(68) = 100 tonnes/y and by symmetry (1/P)dP/dt =−8.4%

per year.

ENVELOPE OF CURVES OF RATES

OF PRODUCTION

On each of the Figures 2–4, a smooth enve-

lope curve tangent to the right sides of the family

of peaks can be drawn. Such an envelope curve for

R(0)/P(0) = 200 years is shown in Figure 6 as the up-

per curve AA. Similar curves BB and CC are shown

for R(0)/P(0) values respectively of 100 years and 50

years. All of the Gaussian (business as usual) tra-

jectories of P(t)vs.t for a given R(0)/P(0) are con-

Table 8. For the Reserve to Production Ratio R(0)/P(0) = 200

Years, as Shown in Figure 4, This Table Gives, for Several Values

of σ, the Number of Years Needed to Consume Half of R(0), the

R/P Ratio at the Peak of the Gaussian Curve, and the R/P Ratio

After the Peak Has Been Passed and P(t) has Declined Back to its

Initial Value of 100 Tonnes/Year

σ Sigma in

Years

Years to

Consume Half

of R(0)

R/P in Years

at Peak

R/P in Years

When

P(t) = P(0)

10 21 13.5 4.65

15 29 19 7.28

20 35 25.4 9.94

30 47 38.1 16.3

40 56 50.5 23.5

50 64 63.4 31.7

60 71 71.8 40.5

70 77 88 50.6

ﬁned to the region below the appropriate envelope

curve. The envelope curve divides the plane of the

graph into two areas, an “allowed” area below and

the “forbidden” area above the envelope curve. The

separation of the graph into two areas is not absolute.

One could gerrymander a hypothetical curve of P(t)

vs. time, such as those shown in Figure 1, that rises

rapidly and suddenly collapses to zero. Such a curve

could rise brieﬂy into the forbidden area, but the

“business as usual curves” won’t have any excursions

above the envelope curve. For example, one can look

at the envelope curve in Figure 6 and say that for

R(0)/P(0) = 200 years (curve AA), the “business as

usual scenarios” will not permit the resource to be

produced at a rate of 200 tonnes/y at t = 100 years.

SUSTAINABILITY

The deﬁnition of sustainable development (sus-

tainability) is taken from Bruntdland (1987):

Sustainable development is development that meets

the needs of the present without compromising

the ability of future generations to meet their own

needs.

Sustainability has to imply “for periods of time

long compared to a human lifetime.” It is clear that

nonrenewable resources that we consume today will

not be available for the use of future generations.

This has the clear consequence that consumption of

nonrenewable resources is not sustainable, a fact that

is rarely acknowledged by sustainability “experts.”

Figure 1 shows that steady growth in the rate of pro-

duction of a resource is not sustainable. It follows

that when one is dealing with real material things

the term “sustainable growth”isanoxymoron.Itis

clear from Figures 2–4 that the scenario of produc-

ing the resource “at the present rates of production”

is not sustainable. It is also clear from Figures 2–4

that the “business as usual” scenarios of P(t)vs. t all

rapidly approach zero asymptotically, so these sce-

narios leave little production for future generations.

Is there any way we can consume resources to-

day without seriously compromising the ability of fu-

ture generations to meet their own needs? Can we

eat our cake and have it too?

The mathematics provides a qualiﬁed “yes.”

Suppose P(t) declines steadily according to the rela-

tion:

P(t) = P(0) exp(−kt). (5)

A Depletion Protocol for Non-Renewable Natural Resources: Australia as an Example 159

Figure 5. Graph showing evolution with time of Reserve to Production ratio, R/P, for scenario

“at current rates of production” (declining straightline) and “business as usual” hypothetical

Gaussian curves, for values of standard deviation σ of (from top to bottom) 80 y, 60 y, 40 y, 20

y, and 10y, all for initial value of R(0)/P(0) = 200 years. The smaller the value of sigma, the

higher and narrower is the Gaussian peak and the more rapid is the decline of the R/P Ratio.

The total quantity of the resource remaining at

t = 0isR(0) which must be equal to all the resource

that will be extracted in all future time. All future

consumption is given by the integral from zero to in-

ﬁnity P(0)



exp(−kt) dt. This integral has the value

P(0)/k which must equal R(0). From this we can de-

ﬁne the “sustainability coefﬁcient” k(s) to have the

value:

k(s) = P(0)/R(0) (6)

Figure 6. Curves showing envelopes of “business as usual” Gaussian curves for three values

of ratio R(0)/P(0), 50, 100, and 200 y. For given value of R(0)/P(0) area above corresponding

curve is “Forbidden” and area below curve is “Allowed.” Looking at curves one can say that

if present value of R(0)/P(0) is 100 years, then “business as usual” scenarios can not produce

a rate of production of 200 tonnes/y 50 years from now but could produce 100 tonnes/y.

160 Bartlett

The sustainability coefﬁcient k(s)isthein-

verse of the R(0)/P(0) ratio. When P(t) declines ac-

cording to Equation (5) with k having the value

given by Equation (6), P(t) can continue forever

on this constantly declining curve! For example, if

R(0)/P(0) = 200 years, then the sustainability coef-

ﬁcient k(s) = P(0)/R(0) = 1/200 = 0.005 per year

(half a percent per year). If R(0)/P(0) = 100 years,

then a curve of P(t) that declines 1% per year is suf-

ﬁcient to allow the resource to last forever! If P(t)

declines more rapidly than given by this sustainabil-

ity coefﬁcient, then P(t) can continue forever without

consuming all of the resource.

If a resource will last R(0)/P(0) years at present rates

of production, and if the rate of production of the re-

source follows the curve that starts at P(0) and has

a constant fractional decrease per unit time whose

magnitude is greater than or equal to P(0)/R(0) per

year, then the production can be truly said to be sus-

tainable.

This is probably as close as it is possible to come

to the Brundtland deﬁnition of sustainability for the

use of a nonrenewable resource. This path provides a

measure of real, but declining, intergenerational eq-

uity in the use of ﬁnite reserves of nonrenewable re-

sources for all future generations.

This concept has been given the name “Sus-

tained Availability” (Bartlett, 1986) and it has other

interesting characteristics:

(1) At the time R(0)/P(0), the rate of pro-

duction has declined to 1/e = 0.368...

of its initial value P(0). For example, for

R(0)/P(0) = 200 years, P would decline to

36.8% of its initial value in 200 years.

(2) For any current value of R(0)/P(0) one can

calculate the sustainability coefﬁcient k(s)

needed to deﬁne the sustainable curve from

that point forward. If changes are made in

R(0) through new discoveries or through cor-

rections of earlier estimates of R(0), one can

calculate the new value of k(s) and thereafter

proceed down the new declining exponential

curve.

(3) At all points on a declining exponential

curve, the R/P ratio is unchanged from its ini-

tial value of R(0)/P(0).

The mathematics of this curve and the options

it presents for a producing country that consumes

part of its production and exports the balance of its

production, have been explored in detail (Bartlett,

1986).

FORBIDDEN, TERMINAL, AND

SUSTAINABLE PRODUCTION PATHS

We now can add the declining exponential curve

to the graph of Figure 6 as is shown in Figure 7 for the

case of R(0)/P(0) = 100 years. The two lines divide

the plot into three areas. The area in the upper right

above the curve AA is “F” for “forbidden,” because

the “business as usual” scenarios of P(t)vs.t cannot

appear in this area. Between the envelope curve AA

and the declining exponential curve BB is an area la-

beled “T” for “terminal,” in which all of the “business

as usual” Gaussian scenarios of P(t) vs. t are possible

but they approach zero more rapidly than the declin-

ing exponential curves, BB, and hence are effectively

terminal. Below the curve BB is the area “S” for “sus-

tainable.” Any scenario of P(t) vs. t that starts at P(0)

and stays below curve BB is sustainable.

Figure 7 shows that the options for sustainability

are limited. This fact is not appreciated by govern-

mental, industrial, educational, scientiﬁc, environ-

mental, and policy leaders, many of whom seem to

be dedicated to unending economic growth and sus-

tainability, hoping perhaps that marvels of technol-

ogy will arrive in time to remove the limits dictated

by nature that are outlined here. Indeed, a friend told

me of attending an international conference on sus-

tainability where he heard people saying words to the

effect that “We’re tired of hearing of limits. We’re

going to grow the limits!”

AUSTRALIA’S ENERGY POLICIES

AND THE LIMITATIONS IMPOSED

BY NATURE

A number of troubling things come to mind

when one reads the article written by the Australia’s

Minister of Industry, Tourism and Resources from

which the opening statement was taken.

(1) The Minister’s optimistic assessment of

Australia’s energy situation is based on the

Government’s major long-range energy plan.

Thus, the misrepresentations set forth by the

Minister and cited here, are probably built

into Australia’s National Energy Plan.

A Depletion Protocol for Non-Renewable Natural Resources: Australia as an Example 161

Figure 7. (P, t) plane is divided into three areas by two lines, AA and BB as shown here for

R(0)/P(0) = 100 years. Line BB is curve of sustainability in which P(t) decreases at constant

fractional rate of 1% per year. If P(t) starts at 100 tonnes/y and stays on or below this curve,

resource can be produced theoretically at declining rates forever. Line AA is envelope of

“business as usual” Gaussian curves. Region above line AA is “F” for “forbidden.” Region

between lines is “T” for “terminal” region. All “business as usual” Gaussian curves of P(t)vs.

t whose initial slope is positive will start in “T” region and will ultimately fall below line BB

and will approach zero more rapidly than “sustainable” declining exponential curve BB.

(2) The Minister made no mention of the growth

of Australia’s population. This suggests that

the Government’s long-range energy plan

does not concern itself with the size of

Australia’s population or with its rate of

growth. If this is the situation, the omis-

sion dooms the Government’s entire plan.

Any responsible governmental policy for

“Australia’s long-term development, secu-

rity and environmental needs as they relate

to energy” has to take full account of Aus-

tralia’s continuing population growth and of

the predictable effects this growth will have

in forcing the future growth of Australia’s

domestic energy consumption. In the face

of the acknowledged ﬁnite nature of fos-

sil fuels and the predictable growing do-

mestic demand for energy, the Minister re-

counts with pride that Australia is export-

ing its fossil fuels as rapidly as possible, pre-

sumably in accord with the Government’s

plan.

“I am expecting and planning for Australia to fur-

ther expand its energy export base over the coming

decade and beyond.”

It is probable that before mid-century the people

of Australia will be needing the fossil fuels that their

government is so eagerly exporting.

(3) In something like a century Australia’s pop-

ulation has increased by a factor of 5 from

about 4 million to 20 million. Equation (3)

shows that this enormous increase repre-

sents an average growth rate of only 1.6%

per year! Another century of 1.6% per year

growth would cause Australia’s population

to increase by another factor of 5 to 100 mil-

lion!

(4) Malkin (2005) describes the City of Cities

plan for the City of Sydney, reporting that

in the next 25 years Sydney will construct

447,000 new dwellings and create almost

300,000 new jobs. Even though this projected

growth violates the Laws of Sustainability

(Bartlett, 1994), Sydney’s Planning Minister

makes the amazing suggestion that this will

be done carefully “to ensure the city grows

in a sustainable way.” Other Australian cities

may have similarly plans for enormous pop-

ulation expansions.

(5) Contemporary reports tell of the shortage

of potable water in Australian cities which

162 Bartlett

has led to plans to build large plants for

the desalination of sea water. These plants

are just one of the many predictable costly

infrastructure developments that will be

needed to accommodate the projected large

increases in population, all of which will re-

quire substantial annual consumption of en-

ergy. Here is a Greek tragedy unfolding. The

Government’s policy is one of getting rid of

(exporting) the nation’s fossil fuels as rapidly

as possible and at the same time the Govern-

ment seems to be promoting, or accepting,

long-term rapid growth of Australia’s popu-

lation. Any conservative plan for Australia’s

future would recognize that this projected

population growth will require the fossil fuels

that now are being exported. Yet the Minis-

ter writes that the “government’s white paper

on energy” from which these policies ﬂow,

“is a substantive and forward-thinking doc-

ument that allows us to maintain some of the

world’s lowest energy costs while also reduc-

ing our greenhouse signature.” This policy is

not “forward-thinking” for the reason that it

ignores the simple conﬂuence of the arith-

metic of population growth and the arith-

metic of resource reserves and consump-

tion. The Government’s policy is not even

“backward-thinking” because it ignores the

recent history which makes clear the grow-

ing magnitude of the quantitative problems

that ﬂow from continued population growth.

And, as is shown here, the policies described

by the Minister do not decrease Australia’s

“greenhouse signature,” they increase it.

(6) The Minister opens his article by accepting

the premise that. “The demand for world en-

ergy is growing and will continue to grow

for the foreseeable future.” Demand may

grow, but there are many indications that the

world’s fossil fuel resources will, in the near

future, fall short of meeting this growing de-

mand. This raises a fundamental question: Is

it the obligation of countries that presently

have reserves of fossil fuels, to export these

fuels to the importing countries where the

demand is already larger than the importing

countries can meet from their own domestic

supplies?

(7) The title of the Minister’s article, “A

Growing Role for Australia in Meeting

the World’s Energy Needs” suggests that

Australia is accepting an obligation to export

its fossil fuels as rapidly as possible. This vi-

olates the Brundtland statement of sustain-

ability because it is clear that this cannot

be done without “compromising the ability

of future generations of Australians to meet

their own needs.” The Government’s policy

set forth by the Minister is not sustainable;

quite the opposite, it’s “anti-sustainable.”

The policy is the equivalent of shooting your-

self in the foot.

(8) The life-expectancy of Australian coal of

110 years “at current rates of production”

is alarmingly short compared to the life-

expectancy of a major nation. A thoughtful

response to this datum by the Government

would be alarm, coupled with a call for mea-

sures to reduce the rate of production of Aus-

tralian coal well below the present level so that

reserves of the resource will be available for

Australians beyond 110 years. A responsible

government policy would be to call for a halt

to the export of Australian energy resources

so that the coming generations of Australians

will be able to have the beneﬁts of the use of

these precious resources.

(9) The Minister is proud to boast of how long

the resources would last “at current rates of

production,” but it makes clear that the Gov-

ernment’s policies are not to maintain the

“current rates of production” but rather to

have rapid growth of the rates of production.

This is misleading in the extreme.

(10) There seems to be no recognition of the ob-

vious, but inconvenient, truth that growth

of the rates of production reduces the life-

expectancy of ﬁnite reserves of nonrenew-

able resources far below the ﬁgure given by

the frequently cited R(0)/P(0) ratio.

(11) The Government professes a devotion to sus-

tainability but its policies violate the First

and Second Laws of Sustainability:

First Law: “Population growth and/or growth in the

rates of consumption of resources cannot be sus-

tained.”

Second Law: “In a society with a growing pop-

ulation and/or growing rates of consumption of re-

sources, the larger the population and/or the larger

the rates of consumption of resources, the more dif-

ﬁcult it will be to transform the society to the condi-

tion of sustainability.” (Bartlett, 1994)

A Depletion Protocol for Non-Renewable Natural Resources: Australia as an Example 163

WHO SHOULD BE RESPONSIBLE

FOR GREENHOUSE GASES?

On two counts, it is simply not true that the Gov-

ernment’s policies are “reducing our greenhouse sig-

nature.” By accepting domestic population growth,

Australia will almost certainly be increasing its domes-

tic production of greenhouse gases. By exporting coal

and LNG to be burned in other countries, Australia

is making a signiﬁcant further contribution to the in-

crease of global atmospheric greenhouse gases.

This raises a fundamental policy question. If

country A annually exports a quantity of fossil fuels

which are burned in country B to produce C tonnes

of greenhouse gases, which country should be held

responsible for the introduction of these C tonnes

of greenhouse gases into the Earth’s atmosphere? A

good case can be made for holding each country re-

sponsible for the full amount of C. Afﬁrmative deci-

sions are required by both countries A and B in order

to introduce the greenhouse gases into the global at-

mosphere. Country A can decide not to export the

fossil fuels, and country B can decide not to import

them. Consequently, both countries A and B should

be held responsible for the C tonnes/y of greenhouse

gases.

SHORT-TERM POLITICIANS

AND LONG-TERM PROBLEMS

Politicians live for the short term, and one can

see on Figures 2–4 that for a decade or two it will

be possible to have Australia’s production of fossil

fuels follow the growth curves that make business

people and politicians happy. In business and gov-

ernmental circles, growth is the universally accepted

measure of success, and for this reason politicians can

follow the “business as usual” scenarios in these early

growth periods and can later claim that they have had

great success in boosting the growth of their coun-

try’s economy. But governmental leaders seem not

to recognize that in the near future, growth will lead

to the peak of the rate of production. Government ac-

tions can delay the peak but there are no governmental

actions that can eliminate or avoid the peak. After the

peak, there will be hardships.

Although the example cited here is from

Australia, similar examples abound in all of the

industrialized nations of the world. Breathtaking

innumeracy is an integral part of governmental

planning and actions throughout the “educated”

world. Journalists rarely point out this innumeracy

because, for the most part, they too are innumerate.

INTERNATIONAL TRADE AGREEMENTS

International trade agreements are supported

ardently by the governments of the developed na-

tions which largely have consumed their domestic

fuels and which are consequently dependent on im-

ports of fossil fuels. One can reasonably presume that

a major motivation for these currently popular inter-

national trade agreements is to let the developed na-

tions get their hands on other nations’ resources be-

fore the people of the other nations develop to the

point of needing their own resources. This contempo-

rary colonialism has led to widespread environmen-

tal degradation and it has contributed to the grow-

ing economic disparity between the rich and the poor

(Eherenfeld, 2005). These trade agreements need to

be re-examined. The international trade agreements

are an impediment to sustainability if they prohibit a

nation from following the conservative steps of tak-

ing its exportable natural resources off of the interna-

tional market to save the resources for future domestic

use.

POLICY RECOMMENDATIONS

It is essential that the people of Australia and of

other nations similarly situated;

(1) Stop their population growth and encourage

their populations to decline to sustainable

levels;

(2) Stop the export of their nonrenewable natu-

ral resources, especially fossil fuels;

(3) Put the production of domestic resources

on the declining exponential “sustainability

curves.”

(4) Improve the efﬁciency with which all re-

sources are used;

(5) Develop their domestic renewable energy re-

sources as rapidly as possible.

All of these policy recommendations should

be central to any governmental plan that pur-

ports to “Secure [the] Energy Future” of its

people.

164 Bartlett

ACKNOWLEDGMENTS

The author wishes to thank Alison (Pete)

Palmer. Robert Ristinen. Walter Youngquist and

anonymous reviewers for many constructive sugges-

tions.

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