### Introduction on Energy Policy

```Economics of exhaustible resources
Economics 331b
Spring 2011
1
Why we will learn numerical optimization
1. You will use to build a little economy-climate change
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!
2
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.
3
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.
4
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.
5
Linear programming problem as applied to exhaustible resources
min V  min cost 
T
  c 1x1 (t )  c 2 x 2 (t )  (1  r ) t
t 0
subject to
x 1 (t )  x 2 (t )  D(t )
T
 x i (t )  Ri
t 0
x i (t) = oil production of grade i at time t
c i = cost per unit oil production
Ri = recoverable oil of grade i
D(t )= demand for oil
x i (t)  0
Results:
Vˆ = minimum cost
 i (t) = "shadow price" on resources (opportunity cost)
xˆ 1 (t ) = optimal path of extraction
6
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 oil
unlimited amount of “backstop oil” at \$100 per unit
Discount rate = 5 % per year
Questions:
1. What is efficient price and quantity?
7
Screenshot of simple problem
8
Screenshot of solver for simple model
9
10
Solution quantities
12
10
8
Low cost
6
Backstop
4
2
0
1
3
5
7
9 11 13 15 17 19 21 23 25 27 29
11
Solution prices
140
120
Backstop cost
100
80
Market price
60
supply price
Backstop
Price
royalty
40
Royalty
20
0
1
3
5
7
9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
12
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?
13
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
14
Estimates of Petroleum in Place
Department of Energy, Energy Information Agency, Report #:DOE/EIA-0484(2008)
15
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
16
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.
17
19
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
20
Shadow prices for oil in 2010*
Constraints
Cell
\$E\$16
\$G\$16
\$H\$16
\$F\$16
Name
Sum US
Sum Other OPEC
Sum OPEC Middle East
Sum Other non-OPEC
Final
Value
454.0
1,242.9
1,125.9
1,215.5
Price
-0.66
-5.25
-9.02
-1.92
*Interpretation: what you would pay for 1 barrel of oil in the ground.
21
Results: Price trajectory: actual and model
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
22
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
23
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%
24
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?
25
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