Date post: | 31-Dec-2015 |
Category: |
Documents |
Upload: | jason-bates |
View: | 223 times |
Download: | 2 times |
Economics of exhaustible resources
Economics 331bSpring 2011
1
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!
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
5
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.
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
61
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?
7
Screenshot of simple problem
8
Screenshot of solver for simple model
9
Screenshot of shadow price
10
Solution quantities
11
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
12
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?
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 $100 per barrel
14
Department of Energy, Energy Information Agency, Report #:DOE/EIA-0484(2008)
Estimates of Petroleum in Place
15
Petroleum supply data
Sources: Resource data and extraction from EIA and BP; costs from WN
Source USOther non-
OPECOther OPEC
OPEC Middle
EastInitial volume (billion barrels) 1,100 3,300 2,900 2,900 Recovery factor 60% 50% 50% 50%Recoverable (billion barrels) 660 1,650 1,450 1,450 Cumulative producion (billion barrels) 206 434 207 324 Remaining volume 454 1,216 1,243 1,126 Marginal extraction cost ($ per barrel) 80 50 20 10 Decline rate (per year) 10% 10% 10% 10%
16
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 = $100 per barrel of oil equivalent.Conventional oil and backstop are perfect
substitutes.
17
Picture of spreadsheet
19
0
20
40
60
80
100
120
2005 2015 2025 2035 2045 2055 2065 2075 2085
Price of oil
Supply price US
Supply price non OPEC
Supply price non-ME OPEC
Supply price OPEC Middle East
Results: Price trajectory
20
Shadow prices for oil in 2010*
21
ConstraintsFinal Shadow
Cell Name Value Price$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
22
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
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