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Hotelling Model for Non-renewable Resources: Optimization and Economic Analysis

Learn about the Hotelling model for non-renewable resources, including optimal extraction rates, market prices, and depletion time frames. Explore the impact of demand, discount rates, and known reserves on resource management. Discover the application of the model in maximizing net present value and interpreting scarcity and backstop prices.

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Hotelling Model for Non-renewable Resources: Optimization and Economic Analysis

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  1. YLE13: Hotelling model Marko Lindroos

  2. Questions to be answered wrt non-renewable resources • What is the optimal extraction rate q? • Market prices p in time? • When do we run out x(T)=0, T?

  3. Answers will depend on • Demand • Discount rate • Known reserves of the resource • Price of the subsitute

  4. Hotelling model (JPE 1931) • Initial stock size is x(0), the resource stock decreases in time when it is used. (1) x(t) = x(0) -

  5. Equation of motion

  6. The resource is used untill it is exhausted at time T

  7. Competitive market

  8. Objective function • Maximise the Net Present Value by choosing the extraction q(t) • Max J= • St equation of motion • c = (constant) unit cost of extraction

  9. Optimal control problem • q(t) control variable • x(t) state variable • equation of motion • x(0) initial state

  10. Current value Hamiltonian

  11. Maximun principle

  12. FOC

  13. Interpretation • Net revenue= scarcity price of the resource

  14. Comparison to regular market • Non-renewable resource price is higher than ”normal” competitive market. • Scarcity price measures the difference.

  15. Dynamic condition

  16. Hotelling rule

  17. Interpretation • Scarcity price increases according to the discount rate. The resource should yield the same rate of interest than any other (risk-free) investment

  18. 1.3 Scarcity price in time • Let us solve the differential equation (Hotelling’s rule)

  19. … • On the LHS the time derivative of the discounted scarcity price • integrate

  20. ...

  21. Timepath of scarcity (shadow) price (0 = 10; r = 0.05;)

  22. Backstop-price and optimal price of the resource • Assume c=0.

  23. Initial price • At T, price of the resource is equal to the backstop-price. • At t=0 price can be computed since price increases now with the rate of discount.

  24. Optimal price

  25. 1.5 Optimal rate of extraction and the optimal time to exhaustion • Assume the following demand: • The whole resource stock is exhausted

  26. Optimal extraction rate

  27. Integrate to yield the time to exhaustion

  28. Solving the time to exhaustion • Needs to be computed numerically from the previous equation. It will be affected by backstop-price, discount rate, initial stock and demand. • Note that this needs to be computed first before moving on to computing the optimal extraction and optimal price

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