Report

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 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 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 Screenshot of shadow price 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 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 $100 per barrel 14 Estimates of Petroleum in Place Department of Energy, Energy Information Agency, Report #:DOE/EIA-0484(2008) 15 Petroleum supply data Source Initial volume (billion barrels) Recovery factor Recoverable (billion barrels) Cumulative producion (billion barrels) Remaining volume Marginal extraction cost ($ 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 Picture of spreadsheet 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 Shadow 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