1. Economics of exhaustible resources Economics 331b Spring 2011 1

2. 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

3. 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

4. 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. 5 Outcome of efficient competitive market (however complex but finite time) Maximization of weighted utility function: Economic Theory Behind Modeling = 1. Basic theorem of “markets as maximization” (Samuelson, Negishi) 2. This allows us (in principle) to calculate the outcome of a market system by a constrained non-linear maximization.

6. Linear programming problem as applied to exhaustible resources 6

7. 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

8. Screenshot of simple problem 8

9. Screenshot of solver for simple model 9

10. Screenshot of shadow price 10

11. Solution quantities 11

12. 12 Solution prices 12 Backstop cost Royalty Market price

13. 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

14. 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

15. Department of Energy, Energy Information Agency, Report #:DOE/EIA-0484(2008) Estimates of Petroleum in Place 15

16. Petroleum supply data Sources: Resource data and extraction from EIA and BP; costs from WN 16

17. 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

18. Solution technique 18

19. Picture of spreadsheet 19

20. Results: Price trajectory 20

21. Shadow prices for oil in 2010* 21 *Interpretation: what you would pay for 1 barrel of oil in the ground.

22. Results: Price trajectory: actual and model 22

23. Results: Output trajectory How differs from Hubbert theory: 1. Much later peak 2. Not a bell curve; slower rise and steeper decline 23

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25. 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