Dynamic Efficiency Hotelling s Rule [adapted from S. Hackett s lecture notes]

1. Dynamic Efficiency & Hotelling’s Rule[adapted from S. Hackett’s lecture notes]

2. Dynamic efficiency Recall static notion of Pareto efficient resource allocation is that one cannot change how resources are split to generate larger gains from trade (without making some one else worse off)

3. Dynamic efficiency The notion of dynamic efficiency is an intuitive concept.

4. Dynamic efficiency Reasons why most people would rather have $10,000 today instead of 10 years from now:

5. Dynamic efficiency Reasons why most people would rather have $10,000 today instead of 10 years from now (continued):

6. Dynamic efficiency Suppose that you have inherited $10,000, which will be held in trust for you for 10 years.

7. Dynamic efficiency As an aside, why might your present discounted value of a $10,000 payment 10 years in the future differ from that of someone else?

8. Dynamic efficiency Note: The discount rate (like an interest rate) reflects the time value of money:

9. Dynamic efficiency Since different people have different discount rates, then at the prevailing market interest rate, some people are lenders (financial investors), while others are borrowers.

10. Dynamic efficiency Finance is an application of economics that focuses on time value of money. We will limit ourselves to an elementary application of the time value of money.

11. Dynamic efficiency Suppose that you will receive a single guaranteed future payment “i” years from the present, and your discount rate (interest rate) is “r”. Then the present discounted value (PV) of that future payment (FP) is given by the following formula:

12. Dynamic efficiency PV example:

13. Dynamic efficiency Based on the preceding example, the person is indifferent between having $8,264.63 right now and getting $10,000 two years from now.

14. Dynamic efficiency Final point on PV: If you will receive a stream of payments over time (e.g., social security payments), then the PV of that stream of payments is found as follows:

15. Dynamic efficiency Moving on…

16. Dynamic efficiency There is a well-functioning competitive market for the nonrenewable resource in question (no monopolies or cartels) Market participants are fully informed of current and future demand, marginal production cost, market discount rate, available supplies, and market price We will look at the most basic dynamic case: two time periods: today (period 0) and next year (period 1)

17. Dynamic efficiency Marginal cost is constant Market demand is “steady state”, meaning that demand in period 1 is the same as in period 0 (no growing or shrinking demand)

18. Dynamic efficiency Model: Demand: P = 200 – Q Supply: P = 10 Discount rate “r” = 10 percent (0.1) Total resource stock Qtot = 100

19. Dynamic efficiency Case 1: Ignore period 1 while in period 0 (“live for today”)

21. PV of total gains from trade over periods 0 and 1:

23. The theory of dynamically efficient resource markets Case 2: Divide Qtot equally over periods 0 and 1:

25. Case 2: Divide Qtot equally over periods 0 and 1:

27. Case 2: Divide Qtot equally over periods 0 and 1:

28. Methods for solving for the dynamically efficient allocation of the fixed stock of resource over time:

29. Hotelling’s rule (P0-MC)/(1+r)0 = (P1–MC)/(1+r)1

30. Hotelling’s rule Less math-intensive solution method:

31. Hotelling’s rule Less math-intensive solution method:

32. Hotelling’s rule Less math-intensive solution method:

33. Optional Hotelling’s rule More math-intensive solution method (optional):

34. Optional Hotelling’s rule The dynamically efficient allocation solves the following maximization problem:

35. Optional Hotelling’s rule Now let’s apply the parameters from our problem (a = 200, b = 1, c = 10, r = 0.1, 2 periods). the dynamically efficient solution satisfies:

36. Optional Hotelling’s rule

37. Optional Hotelling’s rule

38. Dynamically Efficient Market Allocation

39. Dynamically Efficient Market Allocation

40. Dynamically efficient equilibrium Intuition

41. Dynamically efficient equilibrium Intuition

42. Dynamically efficient equilibrium Intuition

43. Dynamically efficient equilibrium Intuition

44. Dynamically efficient equilibrium Intuition

45. Dynamically efficient equilibrium Intuition

46. Dynamically efficient equilibrium Intuition

47. Dynamically efficient equilibrium Further Study

48. Practice Problem – Dynamic Efficiency Demand: P = 200 – Q Supply: P = 10 Discount rate “r” = 20 percent (0.2) Total resource stock Qtot = 100 1. Solve for the dynamically efficient allocation (within $1 of marginal profit) 2. How does this increase in the discount rate affect the dynamically efficient allocation? 3. Now suppose that “r” = 0.1 but Qtot = 60. Solve for the dynamically efficient allocation (within $1 of marginal profit). How does a reduction in resource stock affect the dynamically efficient allocation?