ERE6: Non-Renewable
Resources
• Resources and Reserves
• Social optimum and a model for a perfectly
competitive market
• Sensitivity analysis
– Increase in interest rate and resource stock
– Change in demand and extraction costs
• Market failure
– Monopoly
• Taxes and subsidies
• Reality
Last week
• A simple optimal depletion model
–
–
–
–
Resource substitutability
Static and dynamic efficiency
Hotelling‘s rule
Optimality
• Extraction costs
• Renewable resources
• Complications
Potential, Resources and Reserves
Gesamtpotential (Mrd. toe)
687
5.537
Öl
1.507
Gas
Bis Ende 2000 gefördert
Kohle
Verbleibendes Potenzial
125
552
57
1.450
100
3.243
Reserven
219
123
3.343
236
287
316
1.163
Ressourcen
152/66
122/1
469
Source: RWE Weltenergiereport 2004
334
1.327
2.774
84/250
165/1.162
Resources and Reserves
Demonstrated
Inferred
Hypothetical Speculative
Indicated
marginal
marginal
Para-
Reserves
Sub-
Subeconomic
Economic
Measured
Undiscovered
McKelvey classification
Increasing degree of geological assurance
Increasing degree of economic feasibility
Identified
Potential for oil
Source: Bundesamt für Geowissenschaften und Rohstoffe (BGR)
Oil production
Source: BGR
Availability
Source: BGR
Mineral Reserves
Mineral
Prod.
Cons.
Econ. Res.
Exp. Res.
Tech. Res.
Life
112
19
23000
28000
3519000
222
930
960
150000
230000
2035000
161
Manganese
25
22
800
5000
42000
32
Chromium
13
13
419
1950
3260
32
Zinc
7.1
7.0
140
330
3400
20
Nickel
.92
.88
47
111
2590
51
Copper
9.3
10.2
310
590
2120
33
Lead
3.4
5.3
63
130
550
18
Tin
.18
.22
8
10
68
45
Tungsten
.041
.044
3.5
?
51
80
Mercury
.003
.005
0.13
0.24
3.4
43
Aluminium
Iron
Million metric tons
Social optimum: Two-periods
Demand function:
Net social benefits:
Welfare function:
Pt  a  bRt
b 2
NSBt  Bt  Ct  aRt  Rt  cRt
2
U
NSB1
maxW  U0  1  NSB0 
R0 ,R1
1 p
1 p
Constraint:
R0  R1  S
Langrange:
L W   (S  R0  R1 )
Necessary conditions:
L
 a  bR0  c    0
R0
L a  bR1  c

 0
R1
1 p
 P1  c  (1  p )(P0  c )
Social Optimum: Multi-periods
Social welfare function:
maxW 
Rt
Necessary conditions:
 t
U
(
R
)
e
dt
t

t=0
Equations of motion:
Hamiltonian:
T
S  Rt
H  U (Rt )  Pt (Rt )
H dU
dU

 Pt  0 
 Pt
R
dR
dR
H
Pt
Pt  Pt 
 Pt   
S
Pt
Demand function:
P (R )  Ke aR
Demand goes to zero if price exceeds the choke price (K): PT  K
Optimality has that the stock is zero too:
ST  0
RT  0
Net price Pt
Graphical
solution
PT =K
Demand
P0
Pt
45°
R0
R
Rt
Area = S
= total resource stock
T
Time t
T
Time t
Perfect Competition
Identical firms:
j  
Firms objective function:
Perfect competition:
T

maxWj 
Pj  P
 j (Rj ,t )e it dt
t=0
Equations of motion:
S   Rj ,t
Hamiltonian:
H j   j (Rj ,t )  Pj ,t (  Rj ,t )
Necessary conditions:
H j
j
j
Rj

d j
dRj
Pj ,t  iPj ,t
Intertemporal efficiency:
i 
 Pj ,t  0 
d j
dRj
 Pj ,t
Pj ,t
H

 iPj ,t 
i
S
Pj ,t
Increase
in demand
Net price Pt
K
P0/
D/
P0
D
R
R0/
T/
R0
T
Time t
T/
T
Time t
45°
Increase in
interest rate
P
A
C
B
K
P0
Time
T
Net price Pt
Increase in
interest rate (2)
K
Demand
P0
P0/
R
R0/
T/
R0
T
Time t
T/
T
45°
Time t
Net price Pt
Increase in
stock size
K
Demand
P0
P0/
R
R0/
T
R0
T/
Time t
T
T/
Time t
45°
Frequent new
discoveries
Pt
Net price path with no
change in stocks
Net price path with
frequent new discoveries
t
Backstop technology
becomes cheaper
Net price Pt
K
Backstop
price fall
PB
P0
P0/
D
R
R0/
R0
R*
T/
T
Time t

T/
T
Time t
45°
Results of the sensitivity
analysis so far
• Higher demand: Higher initial price, higher initial
extraction; price increase unaffected, so choke
price reached earlier
• Higher interest rate: Initial price will be lower,
but price increase faster, and choke price reached
earlier; overall higher extraction
• Greater resource stock: Initial price goes down,
initial extraction goes up; growth unaffected;
exhaustion postponed
• Lower choke price: Final price lower, but price
increase unaffected, so initial price must be lower;
overall higher extraction
Extraction costs
Gross price:
Pt  pt  c
Hoteling rule required:
pt  p0e t
Original gross price
Resource
price
New gross price
Original net price
New net price
cL
cH
Time
T
Extraction costs (2)
Resource
price
Original gross price
K
Original net price
New gross price
New net price
T
T/
Time
A rise in
extraction costs
Gross price Pt
Original gross price path
K
New gross
P0/
price path
P0
R
R0
T
R0/
T/
Time t
T
T/
Time t
45°
Sum up: Extraction costs
• Gross price increases slower
• Final gross price is choke price
• If the new gross price starts lower, it
never picks up with the old; resource
extraction must be greater during the
entire period; this cannot be optimal
• Therefore, new gross price starts higher,
extraction is lower, and exhaustion is
reached later
Monopoly
Firms objective function:
T

(Rt )e it dt with   P (Rt )Rt
t=0
Equations of motion:
S  Rt
Hamiltonian:
H  (Rt )  Pt * (Rt )
Necessary conditions:
H d
d

 Pt *  0 
 Pt *
R dR
dR
H
Pt *
Pt *  iPt * 
 iPt * 
i
S
Pt *
Marginal profit function:
t
 aRt Ke aRt  Ke aRt  Ke ahRt
Rt
Prefect Competition
Initial Royalty
Royalty Path
Initial Extraction
Extraction Path
P0  Ke 
2iSa
Pt  P0e it
R0 
2iS
a
i
Rt  (T  t )
a
Exhaustion Time T 
2Sa
i
Monopoly
P0  Ke

2iSa
h
Pt  P0e (it / h )
R0 
2iS
ha
i
Rt 
(T  t )
ha
T 
2Sah
i
Net price Pt
Monopoly
and perfect
competition
Perfect competition
PT = PT = K
M
Demand
Monopoly
P0M
P0
R
R0
T
R0M
TM
T
Area = S
TM
45°
Time t
Time t
Royalty and Revenue Taxes
• A royalty tax does not change extraction
(1   ) pt  (1   ) p0e t
• A royalty tax does redistribute revenue
from firms to the government
• Subsidies are negative taxes
• A revenue tax is equivalent to increasing
the extraction cost, that is, higher initial
gross price, slower growth, exhaustion
postponed
c  
c  t

pt  (1   )Pt  c   Pt 
   P0 
e
1   
1  

Further issues
• Private and social extraction costs
might differ
• Private and social discount rates might
differ
• Absence of forward markets and
expectations
• Differences in risk perception
• Uncertainty
How Real is Hotelling?
• Hotelling‘s rule has been derived for very
simple economies
• So, either the analysis has to be made more
complicated, or the data have to be
manipulated before we can subject Hotelling
to an empirical test
• Studies that have done either or both are
inconclusive; some say, Hotelling is real, others
say not so
• It may be that markets assume that resource
stocks are infinite, until they are almost
depleted