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Lecture 9
The efficient and optimal use of
non – renewable resources
What are non – renewable resources?
- fossil fuel energy supplies, such as oil, gas and coal;
-minerals, such as copper and nickel.
They are formed by geological processes over millions of years
 once extracted cannot be immediately be renewed,
 what is the optimal path for extraction?
Are non-renewable resources exhaustible?
Are non-renewable resources exhaustible?
Take the example of Aluminium (year 1991, quantities in million of metric tons):
Production:
Reserves, Qty.:
Reserves life, y.:
World reserve base
- base life
Consumption:
Resource potential:
Base resource (crustal mass):
112
23 000
222
28 000
270
19
3 519 000
1 990 000 000 000
Base resource: mass that is thought to exist in the earth crust.
Resource potential: estimates of the upper limits on resource extraction
possibilities given current and expected technologies.
World reserve base: estimates of the upper bounds of resource stocks , that are
economically recoverable under ‘reasonable expectations’.
Reserves: quantities available under current costs and prices.
A non-renewable multi period resource model
t 
MaxW 
We start with the cake-eating model:

U  Ct  e   t dt
t 0
with Ct  Rt
subject to St   Rt
We write the current-valued Hamiltonian and get as the necessary conditions for the
maximum social welfare:
H C  U  Rt   Pt   Rt 
necessary conditions: Pt   Pt
H
dU
  Pt 
0
R
dR
dU
 Pt 
dR
A non-renewable multi period resource model
H C  U  Rt   Pt   Rt 
necessary conditions: Pt   Pt
H
dU
  Pt 
0
R
dR
dU
 Pt 
dR
We now know the rate at which the resource net price (royalty) must rise. This does not
fully characterize the solution to the optimizing problem:
1. We need to know the optimal initial value of the resource net price.
2. We need to know the extraction period T - the optimal value of T.
3. What is the optimal rate of resource extraction at each point in time?
4. What should be the values of P and R at the end of the extraction period?
A non-renewable multi period resource model
We now know the rate at which the resource net price (royalty) must rise. This does not
fully characterize the solution to the optimizing problem:
1. We need to know the optimal initial value of the resource net price.
2. We need to know the extraction period T - the optimal value of T.
3. What is the optimal rate of resource extraction at each point in time?
4. What should be the values of P and R at the end of the extraction period?
We can’t answer these question without knowing the particular form of the resource
demand function. Suppose the resource demand function is:
P  R   Ke aR
P ( R = 0) = K. K is the so-called choke price for this resource. Demand is driven to
zero, ‘choked off’ at this price.
P
K
U(R) = shaded area
Ke-aR
0
R
Quantity of resource extracted, R
Figure 15.2 A resource demand curve, and the total utility from consuming
a particular quantity of the resource (Perman et al.: page 513)
A non-renewable multi period resource model
Given knowledge of:
• a particular resource demand function.
• Hotelling’s efficiency condition,
• an initial value for the resource stock, and
• and a final value for the resource stock,
it is possible to obtain expressions for the optimal initial, interim, and final resource net
price (royalty) and resource extraction rates.
We know already: RT = 0, ST = 0.
For demand to be zero at time T, the net price P must reach the choke price at time T.
That is: PT = K. This implies:
K  P0 e T
A non-renewable multi period resource model
We know:
From that follows:
P  R   Ke aR
Pt  P0 e  t , K  P0e T
P0 e t  Ke  aR
P0 e t  P0 e
 aR   t 
  t  aR  T
 Rt 

a
T  t 
The result gives an expression for the rate at which the resource
should be extracted along the optimal path.
A non-renewable multi period resource model
t T
To find the optimal extraction period T, recall:
  R  dt  S
t
0
t 0
Substituting in the previous result:
t T


T

t
t 0  a    dt  S0

T
t 
1 2
 Tt   
T  S0
a
2 0 2 a
or T 
2
2 S0 a

A non-renewable multi period resource model
Using the result, we can find the initial royalty level P0:
P0  Ke T  Ke
Considering:
 2  S0 a
Pt  P0 e T
We find the price for the resource
royalty at time t:
The optimal initial extraction level is:
Pt  Ke
R0 

a
  t T 
T  0  
T
a

2  S0
a
A non-renewable multi period resource model
Optimality conditions for the multi-period model
Initial (t=0)
 2  S0 a
Royalty, P
P0  Ke
Extraction, R
2  S0
R0 
a
Depletion time
Interim (t=t)
  t T 
Pt  K
T  t 
Rt  0
Pt  Ke
Rt 

a
Final (t=T)
T
2S0 a

The same results are obtained for resource extraction in perfectly competitive
markets. By assuming that the area under the demand function, the gross benefits,
are quantities of utility, we impose the condition,  = r, the social discount rate
equals the social consumption discount rate. Additionally, we assume the private
market interest rate equals the social consumption discount rate.
Net price Pt
PT = K
Demand
P0
R
Pt
45o
R0
Time t
T
Rt
Area = S
= total resource stock
T
45º
Time t
Figure 15.3 A Graphical representation of solutions to the optimal
resource depletion model (Perman et al.: page 517)
A non-renewable multi period resource model
Results under a monopolistic market:
The results are represented in the following figure. The key result is, that profitmaximizing extraction programmes will be different in perfectly competitive and
monopolistic resource markets.
Net price Pt
Perfect competition
P T = PT M = K
Monopoly
Demand
P0M
P0
R
R0
R0M
T
Time t
TM
T
Area = S
TM
45º
Time t
Figure 15.4 A comparison of resource depletion in competitive and
monopolistic markets (Perman et al.: page 519)
Extension of the non-renewable multi period resource model
The following simplifying assumptions are made:
• utility discount rate and market interest rate are constant over time,
• fixed stock of known size non-renewable natural resource,
• demand curve is identical at each point in time,
• no taxation or subsidy is applied to the extraction or use of the resource,
• marginal extraction costs are constant,
• there is a fixed ‘choke price’ (backstop technology exists),
• no technological change occurs,
• no externalities are generated in the extraction or use of the resource.
P
K
A
C
B
P0
T
Time
Figure 15.5 The effect of an increase in the interest rate on the optimal
price of the non-renewable resource (Perman et al.: page 520)
Net price Pt
K
Demand
P0
P’0
R
R’0
T’
R0
Time t
T
T’
T
Time t
45º
Figure 15.6 An increase in interest rates in a perfectly competitive market
(Perman et al.: page 521)
Net price Pt
K
Demand
P0
P’0
R
R’0
T
T’
Time t
T
T’
Time t
45º
Figure 15.7 An increase in the resource stock (Perman et al.: page 521)
Pt
Net price path with no
change in stocks
Net price path with
frequent new discoveries
t
Figure 15.8 The effect of frequent new discoveries on the resource
net price or royalty (Perman et al.: page 522)
Net price Pt
K
P’0
D’
P’0
D
R
R’0
R0
T’
Time t
T
T’
T
45º
Time t
Figure 15.9 The effect of an increase in demand for the resource
(Perman et al.: page 522)
Net price Pt
K
Backstop
price fall
PB
P0
P’0
D
R
R’0
T’
R0
Time t
T
T’
T
45º
Time t
Figure 15.10 A fall in the price of a backstop technology
(Perman et al.: page 523)
K
Figure 15.11
(a) An increase in extraction
costs: deducing the effects on
gross and net prices;
Original gross price
New gross price
(a)
C
L
CH
New net price
Original net price
T
Time
K
Original gross price
(b)
(b) An increase in
extraction costs: actual
effects on gross and net
prices
New gross price
(Perman et al.: page 523)
New net price
Original net price
T
T’
Time
Net price Pt
K
Original gross price path
New gross
price path
P’0
P0
R
R0
R’0
T
T’
Time t
T
T’
Time t
45º
Figure 15.12 A rise in extraction costs (Perman et al.: page 524)
Impact of taxes and subsisdies
Notation:
• pt: net – price (royalty) at time t.
• Pt: gross – price at time t.
• ct: extraction cost at time t, assumed to be constant.
Effect of tax or subsidy on royalty pt:
1  a  pt  1  a  p0eit  pt  p0eit
Hotelling’s efficiency rule continues to operate unchanged in the presence of a royalty
tax (subsidy). It is simply a redistribution of the rent from the non-renewable resource
to the government in the case of a tax.
Impact of taxes and subsidies
Effect of tax or subsidy on revenue Pt. Before tax (subsidy)
pt  p0eit   Pt  c    P0  c  eit
After tax (subsidy):
1    Pt  c   1    P0  c  eit
c  
c  it

  Pt 

P

  0
e
1  
1 

 0    1, tax;
-1    0, subsidy 
Since c/(1-)>c, an imposition of a revenue tax is equivalent to an increase in the
resource extraction cost. Similarly, a subsidy is equivalent to a decrease in extraction
costs.
Tax: higher c => increase P0, decrease growth of P => increase T.
Subsidy: lower c => decrease P0, increase growth of P => decrease T.