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YLE13: Hotelling model

YLE13: Hotelling model. Marko Lindroos. 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?. Answers will depend on. Demand Discount rate Known reserves of the resource Price of the subsitute.

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YLE13: Hotelling model

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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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