1
ENGS 41
SUSTAINABILITY and NATURAL RESOURCE
MANAGEMENT
Benoit Cushman-Roisin
8 & 10 January 2024
Non-Renewable Resources
(Lynch book, Chapter 1, with additions)
If the resource is not sustainable,
we must at least think of the future in some way…
The major questions facing us when dealing with exhaustible resources:
1. Are there any renewable substitutes? Or, do we have the capability of
recycling what we have previously extracted?
2. How can we stretch the extraction over time in a way that is fair to future
generations?
Anticipated answers: - Price equity over time,
- Stretch for the long run or run out no earlier than
when a substitute becomes available.
coal oil natural gas
2
We do extract a variety of non-renewable (sterile) resources,
and the annual amount of extraction is increasing (~2%/year).
slope = 1/30 years
Rate of extraction (vertical) seems limited by amount of reserves (horizontal).
X
S
30 years
S
X
3
Note that does not mean exhaustion in 30 years.
This is because, as exploitation reduces the remaining stock,
the rate of extraction diminishes, too.
Mathematically:
30 years
S
X
For with T = 30 years,
the resource is being depleted according to
S
X
T
/
0
tT
dS S
XSSe
dt T
At 30 years (t = T), the remaining amount is
1
000
(30 years) 0.368 40% ofSSeSS
Thus, at all times, it remains 40% of what was still there 30 years previously.
Market price is related to scarcity (dilution in environment).
Since gradual depletion leads to increased scarcity, the price has a tendency to rise.
4
← Note the concern
for future generations
well before the word
Sustainability was in
use.
Hotelling’s Rule
(transposed in notation used in Lynch’s book)
X(t) = extraction rate (in tonnes per year)
P(t) = market price (in $/tonne)
r = discount rate, dt = 1 year
at time t
Rent (= money earned) at time t in the future, discounted to put all years
on comparable level, is
() ()
()
(1 )
t
P
tXtdt
Rt
r
Hotelling showed that the socially optimal rate of extraction is the one for
which the price increases at the discount rate:
0
() (1 )
t
P
tP r
In that case, the rent is the same across years per unit extracted:
0
() ()Rt P X t dt
Question: How to make this equitable across generations?
It can be shown that it also maximizes the total rent over the entire extraction time.
()Xtdt
amount extracted in year t
5
Hotelling’s Rule is valid only under the following restrictions:
• No technological development over the years,
• Fixed stock of the exhaustible resource (no new discoveries),
• Constant market conditions,
• Totally competitive market (actors adjust their production until
their marginal production cost + their opportunity cost* reaches
the market price).
* Opportunity cost = loss of future earning due to extracting that portion today,
also called “scarcity rent”.
Obviously, many events and disruptions occur at various times,
even quite frequently (ex. volatility of oil price), but we may assume
that Hotelling’s Rule holds during the stretches between
consecutive events.
Well… Let’s admit that the situation is a little more complicated.
X
X
Fracking
technology
US oil production in lower 48 states
6
Simple Modeling
Variables: S(t) = stock of the resource remaining
X(t) = extraction rate
P(t) = market price per unit production
Mass balance: (resource dwindles as it is being extracted)
Price-sensitive demand: (high price, low demand, low extraction)
Price evolution: (Hotelling’s Rule with continuous time, as default choice)
X
dt
dS
P
a
X
rP
dt
dP
0
0
0
0
()
()
() 1
rt
rt
rt
dP
rP P t P e
dt
aa
XXte
PP
dS a
XStS e
dt rP
at time t
(Lynch, Section 1.1.1)
Solution:
If you don’t like using differential equations,
use discrete math on an Excel spreadsheet.
nnn
n
nnn
XSS
P
a
XPrP
11
,,)1(
7
0
00
() , () 1
rt rt
aa
Xt e St S e
PrP
We note that, depending on the rate of extraction (and thus on the market price),
the amount of resources can reach zero in a finite time or level off at a residual
positive value.
Plotting of solution:
Hitting zero in finite time
Residual resource left untouched
If end value is < 0, it means that the resource is exhausted in a finite time T
If end value is > 0, it means that there is a leftover that will never be extracted:
00
00
() 1 ( ) 0 or 0?
rt
aa
St S e S S
rP rP
1
00
00
() 0
aa
SS P
rP rS
0
0
0
Sr
P
a
X
Terminal condition
0
000
1
() 1 0 ln
rT
aa
ST S e T
rP r a rP S
0
0
()
a
SS
rP
The optimal use of the resource is
because, if so, we use it all across all generations.
This occurs for the initial production rate
1
30 years
r
8
Revenue
Rate of revenue = unit price times the number of units produced:
Cumulated over time, with proper discounting:
Value:
PX
0
rt
R
ePXdt
000
00
max 0 0
()
rt
R
e PXdt P Xdt P S S
RPS
This is called the Rent.
(initial price times total amount extracted)
when nothing left at the end
0
P
Of course, it is hard to achieve this optimal situation because
1. We don’t know how much is there to begin with, and
2. We don’t control the price; the market does.
So, we very much act in the dark.
In many situations, an external pressure not related to near exhaustion
comes to change the dynamics.
Examples:
- Coal extraction much decreased when petroleum became available;
it was just a matter of convenience
(a liquid is more convenient than a solid);
- We are about to switch away from petroleum not because we are
running out but because of the climate impact of carbon dioxide.
- We may be running of rare-earth minerals (needed for electronics)
but, if we do, we can switch from mining new minerals to recycling
our old electronics.
9
Effect of a maximum price: (Lynch, Section 1.1.2)
Consider the case when a high price causes buyers to switch to an alternative.
Extraction X ceases when the price P reaches a maximum P
max
,
which is the price of the alternative.
Assume that this maximum price is known all along.
Thus, to optimize rent (income), producers seek to exhaust the resource
by the time
T when the price P reaches the maximum P
max
:
max 0 0 max
0
0
0() 1
rT rT
rT
PPe PPe
a
ST S e
rP
0max
0max
1
1
ln 1
rT
rS P
e
a
rS P
T
ra
00
0max max
rT
aa a
XerS
PP P
Corresponding initial extraction rate:
Hence, we extract
more now if we know
that we will have to
exhaust the resource
at an earlier time.
new term
The preceding expressions were under the assumption that the maximum
price was known since the beginning of the exploitation.
Should the max price be known at the present moment, that is, if an
alternative emerges at one point along the way, then simply consider the
adjusted current values as new starting values.
Thus,
max
1
max
max
1
ln 1
a
XrS
P
a
P
a
rS
P
rS P
T
ra
adjusted optimal rate of extraction
time remaining
Under these conditions, the rent is:
0000
0
0
0
)0( SPSPdtXPdtXPeR
T
rt
T
unchanged since the entire resource is still produced equitably across the
now-finite number of generations (until time
T instead of ∞).
adjusted current price
10
End of Part 1
Consumer’s Surplus: (Lynch, pages 7-8)
The selling price is P, but consumers may value the resource at value V,
higher than
P : V > P, up to P
max
the price of an alternative.
Now,
V would be set by a supply-demand curve, as on the graph.
The consumer pays P but
was ready to pay up to
P
max
.
Thus, the consumer’s surplus
is
P
max
–P at any given time.
The cumulated surplus is
obtained by integration of
sale over price (yellow area).
11
There may be more of the resource that we could exploit if we had better
technology or there may be more “out there” than we presently know.
Distinction between Reserves and Resource Base:
Resource Base = All that there ever was minus what has been exploited to date
Reserves = Part of Resource Base that has been discovered
and established as economically viable for extraction.
Effect of discovery (Lynch, Section 1.3)
Imagine that the market prompts an effort to discover additional reserves.
Denote the rate of discovery by D and the remaining still undiscovered
resources by
U . The pertinent mass-conservation equations are:
D
dt
dU
DX
dt
dS
These equations state that D takes
away from
U and adds to S, the
known amount resource
(zero-sum because of no creation).
Now, we need to say what controls the rate of discovery.
The reasoning is that it is easier to discover the resource if there is still more
of it “out there.” Thus, the rate
D may be taken proportional to the
undiscovered amount
U:
The coefficient
represents the ease (high
) or difficulty (low
)
with which new discoveries occur for a given undiscovered stock.
It is a function of technology and proclivity to invest in a discovery effort.
UD
12
UD
0
t
dU
D
UUUe
dt
With , the remaining undiscovered amount U is dwindling according to:
dD dU
D
UD
dt dt
Since
tracking
D(t) over time permits the estimation of the parameter
.
Obviously,
U
0
remains a major unknown because we still don’t know
how much was “out there” as we are still in the process of discovering
some of it.
0
t
D
Ue
0
t
D
UUe
To , we now add the remaining dynamics with the existence
of a maximum price at which production will cease:
max
P
a
Sr
P
a
X
t
eU
P
a
Sr
dt
dS
DX
dt
dS
0
max
(...) (...) (...)
trt
See
The algebra to obtain the analytical solution is very cumbersome,
but clearly the solution should look like:
positive contribution due to
discovery, but slowing down
over time as it gets harder to
discover more
negative contribution due to
gradual exhaustion caused by
extraction, like in basic model
13
For all cases:
r = 0.10
= 0.5
No price limit
S
0
= 10
U
0
= 10
= 0.10
peak production at t = 5.75
with S
max
= 11.25
S
0
= 10
U
0
= 50
= 0.10
peak production at t = 11.96
with S
max
= 30.26
Same as above but with
slower rate of discovery
= 0.04
No peak production
“peak oil”
Production with
discovery and effort
(Lynch, Section 1.3.4)
Effort of exploration E is modeled as
()
dE
dt
Coefficient of
proportionality
Profit earned on
production
ED (...)(...)
Rate of
discovery
Effort
Rationale:
Income attracts effort to discover
*
E
DU
E
Discovery D proportional
to effort E
14
Varying Demand (Lynch, Section 1.1.4)
We now consider the fact that the production rate X may be a function
of more than only price
P. In particular, it may also depend on customer
factors unrelated to price, such as availability of alternatives, changing
tastes, or new technologies (ex. less need for gasoline when electric
cars get on the market).
The way to model this is to include a time variation in the numerator of
the expression linking price to production:
With price as before:
P
ta
X
)(
trrt
e
P
ta
XePP
0
0
)(
An interesting case is that of linear growth in demand (Lynch, pages 12-14).
In this case, we take:
The mathematics give sequentially:
trtrtr
tr
teaeaa
Pr
StSe
P
taa
X
dt
dS
e
P
taa
tX
110
0
0
0
10
0
10
1)(
1
)(
)(
taata
10
)(
The interesting aspect is the initial behavior of the production:
tr
e
P
taara
dt
dX
0
101
X
dt
dX
ara
X
dt
dX
ara
t
t
,0
,0
0
01
0
01
grows initially
always decays
15
Costly Production (Lynch, Section 1.2.2)
If we denote the cost of extraction per unit produced as C,
then the profit to the producer is
P – C per unit produced.
The rent is likewise adjusted downward:
Generally, the cost of extraction is related to the abundance of the
resource: The more there is (larger
S), the easier and cheaper it is to
extract some of it, and the cost increases as the resource becomes
more scarce (
C when S). Thus, C = C(S), a decreasing function.
0
()
t
rt
R
tePCXdt
Consider a time interval t, such as one quarter (3 months) or one year.
The present contribution to rent plus that of the next consecutive period
t is:
tX
t
r
SCP
tXSCPRRR
2
22
1112112
1
)(
)(
The producer may ask:
What if I anticipate a small portion
X of production from period 2 to period 1?
To answer this question, we consider the change in the above income when
11 2 2 22
22
,,
( ) ( )
XXXX X XSSXt
dC
CS CS X t
dS
Some algebra shows that the change to the two-period rent above is:
2
2
22
1112
)(
1
1
1
)(
)( tXXX
dS
dC
t
r
tX
t
r
SCP
tXSCPR
16
The point of indifference corresponds to no net change in rent:
0)(
1
1
1
)(
)(
0
2
22
11
12
tXX
dS
dC
trtr
SCP
SCP
R
Higher-order differential
Re-arranging, we obtain sequentially:
211
1122
21122
)(
)()(
)()1()(
X
dS
dC
SCPr
t
SCPSCP
tX
dS
dC
SCPtrSCP
In the limit of a short time interval:
X
dS
dC
SCPr
dt
SCPd
)(
)(
Time discounting (inflation)
called “stock” effect
Finally, recalling the mass balance
we have
and the point of indifference corresponds to:
This is the decision rule under costly production.
We note that the cost of production C slows down the rate of price increase.
dt
dC
dt
dS
dS
dC
X
dS
dC
X
dt
dS
)(
)(
)(
SCPr
dt
dP
dt
dC
SCPr
dt
SCPd
17
Closing remarks
(Lynch, page 42)
The only possible fate of a non-renewable resource is EXHAUSTION.
But exhaustion may take three different forms:
1. Physical exhaustion – We just run out of the resource.
2. Economic exhaustion – The resource can no longer be produced economically.
(ex. there is a maximum price above which people switch to an alternative.)
3. Political exhaustion – Extraction of the resource has been made illegal.
More closing remarks for Chapter 1
(Lynch, page 32)
A sterile resource, like petroleum, has only one fate, exhaustion.
Time to exhaustion is likely to be finite
(as when there is a maximum price above which demand vanishes).
Implication for sustainability:
- A substitute needs to be found during the extraction process before
exhaustion occurs.
- Some of the accumulated income (rent) needs to be invested in the
search for the alternative, or one has to learn “how to do without.”
Either way, there is a learning curve.
- The cumulative learning ought to be the sustainable aspect.
Ultimately, however, substitution alone is inadequate if it progresses
through a series of finite resources exhaustions.
There must be a “Sustainable Finale”!
