Hotelling v. Hubbert:

How (if at all) can economics and peak oil be reconciled?

Economics 331b

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Hubbert concerns the QHotelling concerns the P

Can they be married into a happy P-Q couple?

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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.

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Hubbert curve for US

0

400

800

1,200

1,600

2,000

2,400

2,800

3,200

3,600

00 10 20 30 40 50 60 70 80 90 00 10

US crude oil productionHubbert curve

(peak 1976; total = 222; cum to date = 197)

Pro

duct

ion (ba

rrel

s/ye

ar)

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)

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Data source: Oil price data from EIA and BLS. Price deflation by CPI from BLS.

200

100

80

6050

40

30

20

1050 55 60 65 70 75 80 85 90 95 00 05 10

Real crude oil prices (2010 $ per barrel)

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200

100

80

6050

40

30

20

101975 1980 1985 1990 1995 2000 2005 2010

Hotelling line Real oil price

Hotelling growth at r = 3% per year

Real oil price

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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

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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.

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Outcome of efficientcompetitive market(however complex

but finite time)

Maximization of weighted utility function:

Economic Theory Behind Modeling

1

and subject to resource and other constraints.

for utility functions U; individuals i=1,...,n;

locations k, uncertain states of world s,

time periods t; welfare weights

ni i i

k,s,ti

i;

W [U (c )]=

1. Basic theorem of “markets as maximization” (Samuelson, Negishi)

2. This allows us (in principle) to calculate the outcome of a marketsystem by a constrained non-linear maximization.

Specific Tools for Finding Solution

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 is used in many economic integrated

assessment models of climate change.

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Department of Energy, Energy Information Agency, Report #:DOE/EIA-0484(2008)

Estimates of Petroleum in Place

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Petroleum supply data

Sources: Resource data and extraction from EIA and BP; costs from WN

Source USOther non-

OPECOther OPEC

OPEC Middle

EastInitial volume (billion barrels) 1,100 3,300 2,900 2,900 Recovery factor 60% 50% 50% 50%Recoverable (billion barrels) 660 1,650 1,450 1,450 Cumulative producion (billion barrels) 206 434 207 324 Remaining volume 454 1,216 1,243 1,126 Marginal extraction cost ($ per barrel) 80 50 20 10 Decline rate (per year) 10% 10% 10% 10%

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Demand assumptions

Historical data from 1970 to 2008Then 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 = 0Backstop price = $100 per barrel of oil equivalent.Conventional oil and backstop are perfect

substitutes.

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Solution technique

2200

, , , ,2010

2200

, , ,2010

, ,,

, ,

min [ ](1 )

subject to

, resource constraints, all regions and grades

, must meet demand for all time periods

= oil productio

ti j t i j t B t

t

i j t i jt

i j t t ti j

i j t

c x c B r

x R

x B D

x

, ,

,

n of grade i and j and time t

= cost per barrel oil production

= recoverable oil of grade i and j

= production of backstop technology

= demand for oil

i j t

i j

t

t

c

R

B

D

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Picture of spreadsheet

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0

20

40

60

80

100

120

2005 2015 2025 2035 2045 2055 2065 2075 2085

Price of oil

Supply price US

Supply price non OPEC

Supply price non-ME OPEC

Supply price OPEC Middle East

Results: Price trajectory

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Results: Price trajectory and actual

0

20

40

60

80

100

120

1970 1980 1990 2000 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100 2110

Pric

e of

oil

(200

8 pr

ices

)

Efficiency price of oil

Supply price US

Supply price non OPEC

Supply price non-ME OPEC

Supply price OPEC Middle East

History

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Results: Output trajectory

0

50

100

150

200

250

300

350

400

450

500

1970 1980 1990 2000 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100

Oil

Prod

ucti

on (b

illio

n ba

rrel

s per

5 y

ears

) Conventional oil

Oil and backstop

History

How differs from Hubbert theory: 1. Much later peak 2. Not a bell curve; slower rise and steeper decline 19

-15.0%

-10.0%

-5.0%

0.0%

5.0%

10.0%

15.0%

20.0%

1980

1985

1990

1995

2000

2005

2010

2015

2020

2025

2030

2035

2040

2045

2050

2055

2060

2065

2070

2075

2080

2085

2090

Rate of increase in real oil prices

History

Efficiency

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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?

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