1. © 2009 Pearson Education Canada 5/1 Chapter 5 Intertemporal Decision Making and Capital Values

2. © 2009 Pearson Education Canada 5/2 Intertemporal Decision Making  Intertemporal resource allocation relates to allocation of resources to present and future uses.  The rate of interest paid to borrow money is the borrowing rate (i b ).  The rate of interest earned on savings is the deposit rate (i d ).

3. © 2009 Pearson Education Canada 5/3 Intertemporal Value Comparisons  Future value (FV) is the value that a sum of money invested today will turn into at a point in the future.  Suppose $1000 is invested for one year at an interest rate (i d ) of 10%.  The FV = $1000 (1+i d ) = $1100.  $1000 invested for 1 year at an interest rate of 10% will turn into $1100 in one year.

4. © 2009 Pearson Education Canada 5/4 Future Value Rule  From the example: 1. Choose $1000 received today if i d exceeds 10%. 2. Choose $1100 received 1 year from now if i d is less than 10%. 3. Choose either if i d equals 10%.

5. © 2009 Pearson Education Canada 5/5 Present Value  The present value (PV) is the value today of a sum of money received in the future.  The PV of $1100 received in one year is the sum of money you could exchange today for a payment of $1100 one year from now (i b =10%).  PV = $1100/(1+i b ) = $1000

6. © 2009 Pearson Education Canada 5/6 Present Value Rule From the previous example: 1. Choose $1000 payable today if i b exceeds 10%. 2. Choose $1100 payable 1 year from now if i b is less than 10%. 3. Choose either if i b equals 10%.

7. © 2009 Pearson Education Canada 5/7 Separation Theorem 1. Individuals will choose among different income streams by choosing the largest present value. 2. They will choose consumption expenditures over time to maximize utility, given the constraint that the PV of income does not exceed the PV of consumption expenditures.

8. © 2009 Pearson Education Canada 5/8 Present Values  M t is the present value of an income stream (M 0, M 1, M 2, … M t ).  If you deposit PV, at the end of t periods you will have PV(1+i) t.  This must equal M t since PV is the present value of M t and (PV(1+i) t =M t ).  Therefore: PV = M t /(1+i) t.

9. © 2009 Pearson Education Canada 5/9 Present Values  The PV of an income stream: PV=M 0 /(1+i)+M 1 /(1+i) 2 +…+M T /(1+i) T  An income stream with equal payments forever is a perpetuity.  The present value of a perpetuity or consol can be approximated as: PV = M/i

10. © 2009 Pearson Education Canada 5/10 Rates of Return R = (p 1 -p 0 )/p 0 Where: R = rate of return p 0 = current selling price p 1 = future selling price

11. © 2009 Pearson Education Canada 5/11 The Demand for Consumer Capital and Complimentary Goods  Capital goods are anything that yields service over time.  Human capital is human endowments (skills) that yield service over time.  Consumer capital - goods valued not for themselves but for the services they provide over time.  Reservation price is the maximum price a person is willing to pay.

12. © 2009 Pearson Education Canada 5/12 Figure 5.1 The demand for film

13. © 2009 Pearson Education Canada 5/13 Figure 5.2 The demand for a camera

14. © 2009 Pearson Education Canada 5/14 Figure 5.3 Reservation prices

15. © 2009 Pearson Education Canada 5/15 Intertemporal Allocation of Nonrenewable Resources  Bagwell’s decisions are how much of his 10 000 barrels of oil to pump (sell) in period zero (z 0 ) and period one (z 1 ).  His goal is to maximize the present value of his oil income where w 0 and w 1 are the price of oil in period zero and period one respectively.

16. © 2009 Pearson Education Canada 5/16 Intertemporal Allocation of Nonrenewable Resources The present value of oil income is: PV = w 0 z 0 + w 1 z 1 /(1+i) Given that z 1 =10 000-z 0 (oil sold in one period reduces oil sold in the other period). Substituting gives: PV = 10 000w 1 /(1+i)+z 0 (w 0 -w 1 /(1+i))

17. © 2009 Pearson Education Canada 5/17 PV = 10 000w 1 /(1+i)+z 0 (w 0 -w 1 /(1+i)) PV = 10 000w 1 /(1+i)+z 0 (w 0 -w 1 /(1+i)) Where:  10 000w 1 /(1+i) is wealth if all oil sold in period 1.  z 0 (w 0 -w 1 /(1+i)) is the rate of change in wealth as one more unit of oil is sold in period 0 and one less in period 1.  If w 0 >w 1 /(1+i), all oil sold in period 0.  If w 0 <w 1 /(1+i), all oil sold in period 1.  If w 0 =w 1 /(1+i), oil may be sold in each.

18. © 2009 Pearson Education Canada 5/18 Hotelling’s Law  Assuming that each of a great number of people own a small portion of total oil supply: PV=w 0 z 0 + w 1 z 1 /(1+i)  If z 0 & z 1 are positive: w 0 = w 1 /(1+i)  Rewriting gives w 1 = w 0 /(1+i)  The price of oil rises from one period to the next at the rate of interest.

19. © 2009 Pearson Education Canada 5/19 Figure 5.4 The optimal time to harvest

20. © 2009 Pearson Education Canada 5/20 The Life-Cycle Model  The total available for consumption in period 1 is: C 1 =M 1 +(1+i)(M 0 -C 0 ).  Rearranging gives the budget line: C 0 (1+i)+C 1 =M 0 (1+i)+M 0  Note that -(1+i) is the slope showing that the opportunity cost of a dollar consumed in period 0 is (1+i) dollars in period 1.

21. © 2009 Pearson Education Canada 5/21 Figure 5.5 An intertemporal budget line

22. © 2009 Pearson Education Canada 5/22 Figure 5.6 The rate of interest and the intertemporal budget line

23. © 2009 Pearson Education Canada 5/23 Figure 5.7 Choosing an intertemporal consumption bundle

24. © 2009 Pearson Education Canada 5/24 Figure 5.8 Comparative statics of a change in income

25. © 2009 Pearson Education Canada 5/25 Initial Savings and a Rise in the Interest Rate (Figure 5.9)  In this case there is savings in period one and initial equilibrium of E.  When i rises, the budget lines swivels around the initial endowment at A.  The new equilibrium is at E’ (consumption in both periods increases).

26. © 2009 Pearson Education Canada 5/26 Figure 5.9 Comparative statics of an increase in i – part 1

27. © 2009 Pearson Education Canada 5/27 Initial Savings and a Rise in the Interest Rate (Figure 5.9)  For consumption in period 1, the income and substitution effect are complementary and C 1 increases.  For consumption in period 0 the income effect leads to an increase in C 0, while the substitution effect leads to a decrease. So C 0 may rise or fall.

28. © 2009 Pearson Education Canada 5/28 Figure 5.10 Comparative statics of an increase in i – part 2

29. © 2009 Pearson Education Canada 5/29 Initial Borrowing and a Rise in the Interest Rate (Figure 5.10)  Here there is borrowing in period one and initial equilibrium of E.  When i rises, the budget lines swivels around the initial endowment at A.  The new equilibrium is at E’ (consumption in period 0 rises and consumption in period 1 falls).

30. © 2009 Pearson Education Canada 5/30 Initial Borrowing and a Rise in the Interest Rate (Figure 5.10)  For consumption in period 0 the income and substitution effect are complementary and C 0 increases.  For consumption in period 1 the income effect leads to a decrease in C 1, while the substitution effect leads to a decrease. So C 1 may rise or fall.