Economics of exhaustible resources

Economics 331bSpring 2011

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Why we will learn numerical optimization

1. You will use to build a little economy-climate change model and optimize your policy.

2. You have learned the theory (Lagrangeans etc.), so let’s see how it is applied

3. Optimization is extremely widely used in modern analysis:- statistics, finance, profit maximization, engineering

design, sustainable systems, marketing, sports, just everywhere!

4. It is fun!

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Standard Tools for Numerical Optimization in Economics and Environment

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). [For picture, see http://en.wikipedia.org/wiki/File:NewtonIteration_Ani.gif]

2. EXCEL “Solver,” which is convenient but has relatively low power.- I will use this for the Hotelling model. [proprietary version is

better but pricey and I sometimes use (Risk Solver Platform.]

3. GAMS software (LP and other) . Has own language, proprietary software, but very powerful.- This is used in many economic integrated assessment models of climate

change. GAMS software. Has own language, proprietary software, but very powerful.

4. MATLAB and similar.

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How to calculate competitive 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

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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.

Linear programming problem as applied to exhaustible resources

1 1 2 20

min min cost ( ) ( ) (1 )T

t

tV c x t c x t r

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Results:ˆ = minimum cost

(t) = "shadow price" on resources (opportunity cost)ˆ ( )= optimal path of extractioni

V

x t

1 2

0

subject to

( ) ( ) ( )

( )

(t) = oil production of grade i at time t

= cost per unit oil production

= recoverable oil of grade i

( )= demand for oil

(t) 0

T

i it

i

i

i

i

x t x t D t

x t R

x

c

R

D t

x

Simple Example of Hotelling theory

Let’s work through an example

Assume demand = 10 per year (zero price elasticity)Resources:

201 units of $10 per unit oilunlimited amount of “backstop oil” at $100 per unit

Discount rate = 5 % per year

Questions:1. What is efficient price and quantity?

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Screenshot of simple problem

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Screenshot of solver for simple model

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Screenshot of shadow price

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Solution quantities

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0

2

4

6

8

10

12

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Low cost

Backstop

12

0

20

40

60

80

100

120

140

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

supply price Backstop

Price royalty

Solution prices

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Backstop cost

Royalty

Market price

Important question to think about

Recall idea of shadow price.Defined as change in objective function for unit change

in a constraint.LP and “shadow price” idea were invented by a Russian

mathematician studying how to set efficient prices under Soviet central planning.

He argued (and it was later proved) that economic efficiency comes when market prices = shadow prices

This is used in environmental problems and global warming.

Important question in this context: Why does efficiency price of exhaustible resource rise at discount rate over time?

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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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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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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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Shadow prices for oil in 2010*

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ConstraintsFinal Shadow

Cell Name Value Price$E$16 Sum US 454.0 -0.66$G$16 Sum Other OPEC 1,242.9 -5.25$H$16 Sum OPEC Middle East 1,125.9 -9.02$F$16 Sum Other non-OPEC 1,215.5 -1.92

*Interpretation: what you would pay for 1 barrel of oil in the ground.

Results: Price trajectory: actual and model

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 23

-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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