Understanding Intertemporal Choice and Present and Future Values in Microeconomics

1. Hal Varian Intermediate Microeconomics Chapter Ten Intertemporal Choice

2. Intertemporal Choice u Persons often receive income in lumps; e.g. monthly salary. u How is a lump of income spread over the following month (saving now for consumption later)? u Or how is consumption financed by borrowing now against income to be received at the end of the month?

3. Present and Future Values u Begin with some simple financial arithmetic. u Take just two periods; 1 and 2. u Let r denote the interest rate per period.

4. Future Value u E.g., if r = 0.1 then $100 saved at the start of period 1 becomes $110 at the start of period 2. u The value next period of $1 saved now is the future value of that dollar.

5. Future Value u Given an interest rate r the future value one period from now of $1 is u Given an interest rate r the future value one period from now of $m is

6. Present Value u Suppose you can pay now to obtain $1 at the start of next period. u What is the most you should pay? u $1? u No. If you kept your $1 now and saved it then at the start of next period you would have $(1+r) > $1, so paying $1 now for $1 next period is a bad deal.

7. Present Value u Q: How much money would have to be saved now, in the present, to obtain $1 at the start of the next period? u A: $m saved now becomes $m(1+r) at the start of next period, so we want the value of m for which m(1+r) = 1 That is, m = 1/(1+r), the present-value of $1 obtained at the start of next period.

8. Present Value u The present value of $1 available at the start of the next period is u And the present value of $m available at the start of the next period is

9. Present Value u E.g., if r = 0.1 then the most you should pay now for $1 available next period is u And if r = 0.2 then the most you should pay now for $1 available next period is

10. The Intertemporal Choice Problem u Let m 1 and m 2 be incomes received in periods 1 and 2. u Let c 1 and c 2 be consumptions in periods 1 and 2. u Let p 1 and p 2 be the prices of consumption in periods 1 and 2.

11. The Intertemporal Choice Problem u The intertemporal choice problem: Given incomes m 1 and m 2 , and given consumption prices p 1 and p 2 , what is the most preferred intertemporal consumption bundle (c 1 , c 2 )? u For an answer we need to know: the intertemporal budget constraint intertemporal consumption preferences.

12. The Intertemporal Budget Constraint u To start, lets ignore price effects by supposing that p 1 = p 2 = $1.

13. The Intertemporal Budget Constraint u Suppose that the consumer chooses not to save or to borrow. u Q: What will be consumed in period 1? u A: c 1 = m 1 . u Q: What will be consumed in period 2? u A: c 2 = m 2 .

14. The Intertemporal Budget Constraint c 1 c 2 m 2 m 1 0 0

15. The Intertemporal Budget Constraint c 1 c 2 So (c 1 , c 2 ) = (m 1 , m 2 ) is the consumption bundle if the consumer chooses neither to save nor to borrow. m 2 m 1 0 0

16. The Intertemporal Budget Constraint u Now suppose that the consumer spends nothing on consumption in period 1; that is, c 1 = 0 and the consumer saves s 1 = m 1 . u The interest rate is r. u What now will be period 2s consumption level?

17. The Intertemporal Budget Constraint u Period 2 income is m 2 . u Savings plus interest from period 1 sum to (1 + r )m 1 . u So total income available in period 2 is m 2 + (1 + r )m 1 . u So period 2 consumption expenditure is

18. The Intertemporal Budget Constraint u Period 2 income is m 2 . u Savings plus interest from period 1 sum to (1 + r )m 1 . u So total income available in period 2 is m 2 + (1 + r )m 1 . u So period 2 consumption expenditure is

19. The Intertemporal Budget Constraint c 1 c 2 m 2 m 1 0 0 the future-value of the income endowment

20. The Intertemporal Budget Constraint c 1 c 2 m 2 m 1 0 0 is the consumption bundle when all period 1 income is saved.

21. The Intertemporal Budget Constraint u Now suppose that the consumer spends everything possible on consumption in period 1, so c 2 = 0. u What is the most that the consumer can borrow in period 1 against her period 2 income of $m 2 ? u Let b 1 denote the amount borrowed in period 1.

22. The Intertemporal Budget Constraint u Only $m 2 will be available in period 2 to pay back $b 1 borrowed in period 1. u So b 1 (1 + r ) = m 2 . u That is, b 1 = m 2 / (1 + r ). u So the largest possible period 1 consumption level is

23. The Intertemporal Budget Constraint u Only $m 2 will be available in period 2 to pay back $b 1 borrowed in period 1. u So b 1 (1 + r ) = m 2 . u That is, b 1 = m 2 / (1 + r ). u So the largest possible period 1 consumption level is

24. The Intertemporal Budget Constraint c 1 c 2 m 2 m 1 0 0 is the consumption bundle when all period 1 income is saved. the present-value of the income endowment

25. The Intertemporal Budget Constraint c 1 c 2 m 2 m 1 0 0 is the consumption bundle when period 1 borrowing is as big as possible. is the consumption bundle when period 1 saving is as large as possible.

26. The Intertemporal Budget Constraint u Suppose that c 1 units are consumed in period 1. This costs $c 1 and leaves m 1 - c 1 saved. Period 2 consumption will then be

27. The Intertemporal Budget Constraint u Suppose that c 1 units are consumed in period 1. This costs $c 1 and leaves m 1 - c 1 saved. Period 2 consumption will then be which is slope intercept

28. The Intertemporal Budget Constraint c 1 c 2 m 2 m 1 0 0 is the consumption bundle when period 1 borrowing is as big as possible. is the consumption bundle when period 1 saving is as large as possible.

29. The Intertemporal Budget Constraint c 1 c 2 m 2 m 1 0 0 slope = -(1+r)

30. The Intertemporal Budget Constraint c 1 c 2 m 2 m 1 0 0 Saving Borrowing slope = -(1+r)

31. The Intertemporal Budget Constraint is the future-valued form of the budget constraint since all terms are in period 2 values. This is equivalent to which is the present-valued form of the constraint since all terms are in period 1 values.

32. The Intertemporal Budget Constraint u Now lets add prices p 1 and p 2 for consumption in periods 1 and 2. u How does this affect the budget constraint?

33. Intertemporal Choice u Given her endowment (m 1 ,m 2 ) and prices p 1 , p 2 what intertemporal consumption bundle (c 1 *,c 2 *) will be chosen by the consumer? u Maximum possible expenditure in period 2 is so maximum possible consumption in period 2 is

34. Intertemporal Choice u Similarly, maximum possible expenditure in period 1 is so maximum possible consumption in period 1 is

35. Intertemporal Choice u Finally, if c 1 units are consumed in period 1 then the consumer spends p 1 c 1 in period 1, leaving m 1 - p 1 c 1 saved for period 1. Available income in period 2 will then be so

36. Intertemporal Choice rearranged is This is the future-valued form of the budget constraint since all terms are expressed in period 2 values. Equivalent to it is the present-valued form where all terms are expressed in period 1 values.

37. The Intertemporal Budget Constraint c 1 c 2 m 2 /p 2 m 1 /p 1 0 0

38. The Intertemporal Budget Constraint c 1 c 2 m 2 /p 2 m 1 /p 1 0 0

39. The Intertemporal Budget Constraint c 1 c 2 m 2 /p 2 m 1 /p 1 0 0

40. The Intertemporal Budget Constraint c 1 c 2 m 2 /p 2 m 1 /p 1 0 0 Slope =

41. The Intertemporal Budget Constraint c 1 c 2 m 2 /p 2 m 1 /p 1 0 0 Saving Borrowing Slope =

42. Price Inflation Define the inflation rate by where For example, = 0.2 means 20% inflation, and = 1.0 means 100% inflation.

43. Price Inflation We lose nothing by setting p 1 =1 so that p 2 = 1+ . u Then we can rewrite the budget constraint as

44. Price Inflation rearranges to so the slope of the intertemporal budget constraint is

45. Price Inflation u When there was no price inflation (p 1 =p 2 =1) the slope of the budget constraint was -(1+r). Now, with price inflation, the slope of the budget constraint is -(1+r)/(1+ ). This can be written as is known as the real interest rate .

46. Real Interest Rate gives For low inflation rates ( 0), r - . For higher inflation rates this approximation becomes poor.

47. Real Interest Rate

48. Comparative Statics u The slope of the budget constraint is The constraint becomes flatter if the interest rate r falls or the inflation rate rises (both decrease the real rate of interest).

49. Comparative Statics c 1 c 2 m 2 /p 2 m 1 /p 1 0 0 slope =

50. Comparative Statics c 1 c 2 m 2 /p 2 m 1 /p 1 0 0 slope =

51. Comparative Statics c 1 c 2 m 2 /p 2 m 1 /p 1 0 0 slope = The consumer saves.

52. Comparative Statics c 1 c 2 m 2 /p 2 m 1 /p 1 0 0 slope = The consumer saves. An increase in the inflation rate or a decrease in the interest rate flattens the budget constraint.

53. Comparative Statics c 1 c 2 m 2 /p 2 m 1 /p 1 0 0 slope = If the consumer saves then saving and welfare are reduced by a lower interest rate or a higher inflation rate.

54. Comparative Statics c 1 c 2 m 2 /p 2 m 1 /p 1 0 0 slope =

55. Comparative Statics c 1 c 2 m 2 /p 2 m 1 /p 1 0 0 slope =

56. Comparative Statics c 1 c 2 m 2 /p 2 m 1 /p 1 0 0 slope = The consumer borrows.

57. Comparative Statics c 1 c 2 m 2 /p 2 m 1 /p 1 0 0 slope = The consumer borrows. A fall in the inflation rate or a rise in the interest rate flattens the budget constraint.

58. Comparative Statics c 1 c 2 m 2 /p 2 m 1 /p 1 0 0 slope = If the consumer borrows then borrowing and welfare are increased by a lower interest rate or a higher inflation rate.

59. Valuing Securities u A financial security is a financial instrument that promises to deliver an income stream. u E.g.; a security that pays $m 1 at the end of year 1, $m 2 at the end of year 2, and $m 3 at the end of year 3. u What is the most that should be paid now for this security?

60. Valuing Securities u The security is equivalent to the sum of three securities; the first pays only $m 1 at the end of year 1, the second pays only $m 2 at the end of year 2, and the third pays only $m 3 at the end of year 3.

61. Valuing Securities u The PV of $m 1 paid 1 year from now is u The PV of $m 2 paid 2 years from now is u The PV of $m 3 paid 3 years from now is u The PV of the security is therefore

62. Valuing Bonds u A bond is a special type of security that pays a fixed amount $x for T years (its maturity date ) and then pays its face value $F. u What is the most that should now be paid for such a bond?

63. Valuing Bonds

64. Valuing Bonds u Suppose you win a State lottery. The prize is $1,000,000 but it is paid over 10 years in equal installments of $100,000 each. What is the prize actually worth?

65. Valuing Bonds is the actual (present) value of the prize.

66. Valuing Consols u A consol is a bond which never terminates, paying $x per period forever. u What is a consols present-value?

67. Valuing Consols

68. Valuing Consols Solving for PV gives

69. Valuing Consols E.g. if r = 0.1 now and forever then the most that should be paid now for a console that provides $1000 per year is