### Hotelling theory - Department of Economics

```Hotelling v. Hubbert:
How (if at all) can economics and peak oil be
reconciled?
Economics 331b
1
Hubbert concerns the Q
Hotelling concerns the P
Can they be married into a happy P-Q couple?
2
Hubbert theory
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.
3
Hubbert curve for US
3,600
US crude oil production
Hubbert curve
Production (barrels/year)
3,200
2,800
2,400
2,000
1,600
1,200
800
(peak 1976; total = 222; cum to date = 197)
400
0
00
10
20
30
40
50
60
70
80
90
00
10
Data source: Oil production for EIA. Hubbert curve fit by Nordhaus.
4
Hotelling theory
• Let rt = net price of oil in ground = pt – et
= price of oilt – extraction costt
• Oil is developed and produced to meet the arbitrage
condition for assets:
rt* = market rate of return on assets = ri,j,t = return on oil in
the ground for grade i, location j, time t.
• Note that arbitrage condition holds only when
production is positive (price-quantity duality condition)
5
Real crude oil prices (2010 \$ per barrel)
200
100
80
60
50
40
30
20
10
50
55
60
65
70
75
80
85
90
95
00
05
10
Data source: Oil price data from EIA and BLS. Price deflation by CPI from BLS.
6
200
Real oil price
100
80
60
50
40
30
20
10
Hotelling growth at
r = 3% per year
1975
1980
1985
1990
Hotelling line
1995
2000
2005
2010
Real oil price
7
Arranged marriage of Hotelling and Hubbert
Let’s construct a little Hotelling-style 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
8
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.
9
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.
10
Specific Tools for Finding Solution
1. Some kind of Newton’s method.
-
(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 is used in many economic integrated assessment
models of climate change.
11
Estimates of Petroleum in Place
Department of Energy, Energy Information Agency, Report #:DOE/EIA-0484(2008)
12
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%
OPEC
Middle
East
2,900
50%
1,450
324
1,126
10
10%
Sources: Resource data and extraction from EIA and BP; costs from WN
13
Demand assumptions
Historical data from 1970 to 2008
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
Backstop price = \$100 per barrel of oil equivalent.
Conventional oil and backstop are perfect substitutes.
14
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
15
16
Results: Price trajectory
120
100
80
Price of oil
Supply price US
60
Supply price non OPEC
Supply price non-ME OPEC
40
Supply price OPEC Middle East
20
0
2005
2015
2025
2035
2045
2055
2065
2075
2085
17
Results: Price trajectory and actual
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
18
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
19
Rate of increase in real oil prices
20.0%
15.0%
History
10.0%
Efficiency
5.0%
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%
-5.0%
-10.0%
-15.0%
20
Further questions
Why are actual prices above model calculations?
Why is there so much short-run volatility of oil prices?
Since backstop does not now exist, will market forces
induce efficient R&D on backstop technology?
21
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