Hotelling v. Hubbert:
How (if at all) can economics and peak oil be
reconciled?
Economics 331b
What is the issue?
The Hubbert peak-oil theory posits that for any given
geographical area, from an individual oil-producing
region to the planet as a whole, the rate of petroleum
production tends to follow a bell-shaped (normal) curve.
There is no explicit economics in this approach.
Hotelling theory
Oil is developed and produced to meet the
arbitrage condition for assets:
ri,j,t < rt* ; for grade i, location j, time t.
Or the rate of return on oil-in-the-ground (Gi,j,) has
a risk-and-tax adjusted rate of return equal to
that of comparable assets as long as Gi,j, > 0.
Approach for Hotelling
Let’s construct a little oil model and see whether the properties
look Hubbertian.
Technological assumptions:
– Four regions: US, other non-OPEC, OPEC Middle East, and
other OPEC
– Ultimate oil resources (OIP) in place shown on next page.
– Recoverable resources are OIP x RF – Cumulative extraction
– Constant marginal production costs for each region
– Fields have exponential decline rate of 10 % per year
Economic assumptions
– Oil is produced under perfect competition  costs are
minimized to meet demand
– Oil demand is perfectly price-inelastic
– There is a backstop technology at $100 per barrel
How to calculate equilibrium
1. We can do it by bruit force by constructing many
supply and demand curves. Not fun.
2. Modern approach is to use the “correspondence
principle.” This holds that any competitive equilibrium
can be found as a maximization of a particular system.
Economic Theory Behind Modeling
1. Basic theorem of “markets as maximization” (Samuelson, Negishi)
Maximization of weighted
utility function:
Outcome of efficient
competitive market
(however complex
but finite time)
n
=
W   i [U i (c ki ,s ,t )]
i 1
for utility functions U; individuals i=1,...,n;
locations k, uncertain states of world s,
i
time periods t; welfare weights  ;
and subject to resource and other constraints.
2. This allows us (in principle) to calculate the outcome of a market
system by a constrained non-linear maximization.
6
Specific Tools
1. Some kind of Newton’s method.
-
Start with system z = g(x). Use trial values until converges
(if you are lucky and live long enough).
2. EXCEL “Solver,” which is convenient but has relatively
low power.
- I will use this for the Hotelling model.
3. GAMS software. Has own language, proprietary
software, but very powerful
- This will be used later in global warming models,
specifically Yale-DICE model.
Estimates of Petroleum in Place
Department of Energy, Energy Information Agency, Report #:DOE/EIA-0484(2008)
Petroleum supply data
Source
Initial volume (billion barrels)
Recovery factor
Recoverable (billion barrels)
Cumulative producion (billion barrels)
Remaining volume
Marginal extraction cost ($ per barrel)
Decline rate (per year)
US
1,100
60%
660
206
454
80
10%
Other nonOPEC
3,300
50%
1,650
434
1,216
50
10%
Other
OPEC
2,900
50%
1,450
207
1,243
20
10%
Sources: Resource data and extraction from EIA and BP; costs from WN
OPEC
Middle
East
2,900
50%
1,450
324
1,126
10
10%
Demand assumptions
Historical data from 1970 to 2007
Then assumes that demand function for oil grows at 2
percent for year (3 percent output growth, income
elasticity of 0.67).
Price elasticity of demand = -0.5
Conventional oil and backstop are perfect substitutes.
Solution technique
min
2200

t  2010
[c i , j ,t x i , j ,t  c B Bt ](1  r )t
subject to
2200

t  2010
x i , j ,t  Ri , j , resource constraints, all regions and grades
 xi , j ,t  Bt  Dt , must meet demand for all time periods
i, j
x i , j ,t = oil production of grade i and j and time t
c i , j ,t = cost per barrel oil production
Ri , j = recoverable oil of grade i and j
Bt = production of backstop technology
Dt = demand for oil
Picture of spreadsheet
Elasticity
total cost
3,613.9
3,613.9
-
806.4
1,158.1
871.4
(0.5)
Disc cost
172.9
US
Source
Initial volume (billion barrels)
Recovery factor
Recoverable (billion barrels)
Cumulative producion (billion barrels)
Remaining volume
Marginal extraction cost ($ per barrel)
Decline
rate (per
year)
Sum
Excess supply
2%
Demand (bbls per
5 years)
Year
1970
1975
1980
1985
1990
1995
2000
2005
2010
2015
2020
2025
2030
2035
2040
2045
2050
2055
2060
2065
2070
2075
2080
2085
2090
2095
2100
2105
2110
2115
2120
2125
2130
2135
2140
5,210.0
1,171.6
4,038.4
4,025.3
1,100.0
0.6
660.0
206.0
454.0
80.0
0.1
444.0
9.9823
605.0
Other high costMedium cost Low cost
Backstop
Other non-OPEC
Other OPEC OPEC Middle East
3,300.0
2,900.0
2,900.0 unlimited
0.5
0.5
0.5
1,650.0
1,450.0
1,450.0
5,210.0
434.5
207.1
324.1
1,171.6
1,215.5
1,242.9
1,125.9
4,038.4
50.0
20.0
10.0
100.0
0.1
1,215.2
0.3370
0.1
1,241.2
1.7295
0.1
1,125.0
0.9861
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
#DIV/0!
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
0.00%
Price
"Royalty"
oil
-
(0.0)
241.7
83.0
49.0
28.9
17.1
10.1
6.0
3.5
2.1
1.2
0.7
0.4
0.3
0.1
-
210.7
239.8
272.8
299.1
79.0
46.6
27.5
16.3
9.6
5.7
3.3
2.0
1.2
0.7
0.4
0.2
0.1
0.1
-
74.7
138.5
185.6
217.3
239.0
254.2
54.0
31.9
18.8
11.1
6.6
3.9
2.3
1.4
0.8
0.5
0.3
0.2
0.1
0.1
0.0
0.0
0.0
0.0
151.9
198.3
206.7
213.9
145.0
85.6
50.6
29.9
17.6
10.4
6.1
3.6
2.1
1.3
0.7
0.4
0.3
0.2
0.1
0.1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Other high cost
Medium cost
Low cost
200.0
Total supply
109.5
109.5
114.6
109.6
120.2
128.5
139.8
151.9
167.7
185.2
204.5
225.8
249.3
275.2
303.8
335.5
370.4
408.9
451.5
498.5
550.4
607.6
670.9
740.7
817.8
902.9
996.9
1,100.7
1,215.2
1,341.7
1,481.4
1,635.5
1,805.8
1,993.7
2,201.2
US
0.05
7,032.6
(2,994.2)
demand error
109.5
109.5
114.6
109.6
120.2
128.5
139.8
151.9
198.3
206.7
213.9
219.7
224.1
236.2
247.1
256.7
264.6
270.8
275.4
293.8
311.5
328.0
343.0
370.4
408.9
451.5
498.5
550.3
607.6
670.9
740.7
817.8
902.9
996.9
1,100.6
Price
interest rate =
209.0
291.3
362.2
423.9
482.2
540.7
601.9
667.5
738.7
816.6
902.2
996.5
1,100.4
Excess supply
US
oil+back
109.5
109.5
114.6
109.6
120.2
128.5
139.8
151.9
198.3
206.7
213.9
219.7
224.1
236.2
247.1
256.7
264.6
270.8
275.4
293.8
311.5
328.0
134.0
79.1
46.7
27.6
16.3
9.6
5.7
3.4
2.0
1.2
0.7
0.4
0.2
109.50
109.50
114.57
109.56
120.19
128.49
139.77
151.93
198.3
206.7
213.9
219.7
224.1
236.2
247.1
256.7
264.6
270.8
275.4
293.8
311.5
328.0
343.0
370.4
408.9
451.5
498.5
550.3
607.6
670.9
740.7
817.8
902.9
996.9
1,100.6
Other high cost
Medium cost
Low cost
Supply price
history
-
17.128028
33.677859
58.555964
43.337541
27.841267
22.831712
23.895319
43.250069
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
20.1
22.8
26.4
30.9
33.9
37.8
42.7
49.0
57.0
67.2
72.0
78.0
85.8
95.7
100.0
100.0
100.0
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
$ 0.09 $
$ 0.12 $
$ 0.15 $
$ 0.19 $
$ 0.25 $
$ 0.32 $
$ 0.40 $
$ 0.52 $
$ 0.66 $
$ 0.84 $
$ 1.07 $
$ 1.37 $
$ 1.74 $
$ 2.23 $
$ 2.84 $
$ 3.63 $
$ 4.63 $
$ 5.91 $
$ 7.54 $
$ 9.62 $
$ 12.28 $
$ 15.67 $
$ 20.00
0.27
0.35
0.44
0.57
0.72
0.92
1.18
1.50
1.92
2.44
3.12
3.98
5.08
6.49
8.28
10.57
13.49
17.21
21.97
28.04
35.78
45.67
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
0.75
0.95
1.22
1.55
1.98
2.53
3.23
4.12
5.25
6.71
8.56
10.92
13.94
17.79
22.71
28.98
36.99
47.21
$
$
$
$
$
$
$
$
$
$
$
$
1.43
1.82
2.33
2.97
3.79
4.84
6.18
7.89
10.06
12.85
16.39
20.92
US
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
80.1
80.1
80.2
80.2
80.2
80.3
80.4
80.5
80.7
80.8
81.1
81.4
81.7
82.2
82.8
83.6
84.6
85.9
87.5
89.6
92.3
95.7
Other high cost
Medium cost
Low cost
$ 50.3 $ 20.7 $ 11.4
$ 50.3 $ 21.0 $ 11.8
$ 50.4 $ 21.2 $ 12.3
$ 50.6 $ 21.6 $ 13.0
$ 50.7 $ 22.0 $ 13.8
$ 50.9 $ 22.5 $ 14.8
$ 51.2 $ 23.2 $ 16.2
$ 51.5 $ 24.1 $ 17.9
$ 51.9 $ 25.3 $ 20.1
$ 52.4 $ 26.7 $ 22.8
$ 53.1 $ 28.6 $ 26.4
$ 54.0 $ 30.9 $ 30.9
$ 55.1 $ 33.9
$ 56.5 $ 37.8
$ 58.3 $ 42.7
$ 60.6 $ 49.0
$ 63.5 $ 57.0
$ 67.2 $ 67.2
$ 72.0
$ 78.0
$ 85.8
$ 95.7
Results: Price trajectory
120
Price of oil (2008 prices)
100
80
60
Efficiency price of oil
Supply price US
40
Supply price non OPEC
Supply price non-ME OPEC
20
Supply price OPEC Middle East
History
0
1970 1980 1990 2000 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100 2110
Results: Output trajectory
Oil Production (billion barrels per 5 years)
500
450
Conventional oil
400
Oil and backstop
350
History
300
250
200
150
100
50
0
1970 1980 1990 2000 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100
How differs from Hubbert theory:
1. Much later peak
2. Not a bell curve; slower rise and steeper decline
Rate of price change
Rate of increase in real oil prices
15.0%
History
10.0%
Efficiency
5.0%
-5.0%
-10.0%
2090
2085
2080
2075
2070
2065
2060
2055
2050
2045
2040
2035
2030
2025
2020
2015
2010
2005
2000
1995
1990
1985
1980
0.0%