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Intertemporal Choice

Intertemporal Choice. Intertemporal Choice. Persons often receive income in “lumps”; e.g. monthly salary. How is a lump of income spread over the following month (saving now for consumption later)?

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Intertemporal Choice

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  1. Intertemporal Choice

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

  3. Present and Future Values • Begin with some simple financial arithmetic. • Take just two periods; 1 and 2. • Let r denote the interest rate per period. e.g., if r = 0.1 (10%) then $100 saved at the start of period 1 becomes $110 at the start of period 2.

  4. Future Value • The value next period of $1 saved now is the future value of that dollar. • Given an interest rate r the future value one period from now of $1 is • Given an interest rate r the future value one period from now of $m is

  5. Present Value • Suppose you can pay now to obtain $1 at the start of next period. • What is the most you should pay? • Would you pay $1? • 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.

  6. Present Value • Q: How much money would have to be saved now, in the present, to obtain $1 at the start of the next period? • 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) = 1That is, m = 1/(1+r),the present-value of $1 obtained at the start of next period.

  7. Present Value • The present value of $1 available at the start of the next period is • And the present value of $m available at the start of the next period is • E.g., if r = 0.1 then the most you should pay now for $1 available next period is $0.91

  8. The Intertemporal Choice Problem • Let m1 and m2 be incomes received in periods 1 and 2. • Let c1 and c2 be consumptions in periods 1 and 2. • Let p1 and p2 be the prices of consumption in periods 1 and 2.

  9. The Intertemporal Choice Problem • The intertemporal choice problem:Given incomes m1 and m2, and given consumption prices p1 and p2, what is the most preferred intertemporal consumption bundle (c1, c2)? • For an answer we need to know: • the intertemporal budget constraint • intertemporal consumption preferences.

  10. The Intertemporal Budget Constraint • Suppose that the consumer chooses not to save or to borrow. • Q: What will be consumed in period 1? A: c1 = m1/p1. • Q: What will be consumed in period 2? A: c2 = m2/p2

  11. The Intertemporal Budget Constraint c2 So (c1, c2) = (m1/p1, m2/p2) is the consumption bundle if theconsumer chooses neither to save nor to borrow. m2/p2 0 c1 m1/p1 0

  12. Intertemporal Choice • Suppose c1 = 0, expenditure in period 2 is at its maximum at since the maximum we can save in period 1 is m1 which yields (1+r)m1 in period 2 • so maximum possible consumption in period 2 is

  13. Intertemporal Choice • Conversely, suppose c2 = 0, maximum possible expenditure in period 1 is since in period 2, we have m2 to pay back loan, the maximum we can borrow in period 1 is m2/(1+r) • so maximum possible consumption in period 1 is

  14. The Intertemporal Budget Constraint c2 m2/p2 0 c1 m1/p1 0

  15. Intertemporal Choice • Finally, if both c1 and c2 are greater than 0. Then the consumer spends p1c1 in period 1, and save m1 - p1c1. Available income in period 2 will then beso

  16. Intertemporal Choice • Rearrange to get the future-value form of the budget constraint since all terms are expressed in period 2 values. • Rearrange to get the present-value form of the budget constraint where all terms are expressed in period 1values.

  17. The Intertemporal Budget Constraint • Rearrange again to get c2 as a function of other variables intercept slope

  18. Slope = Saving Borrowing The Intertemporal Budget Constraint c2 m2/p2 0 c1 m1/p1 0

  19. The Intertemporal Budget Constraint • Suppose p1 = p2 = 1, the future-value constraint becomes • Rearranging, we get

  20. The Intertemporal Budget Constraint • If p1 = p2 = 1 then, c2 slope = – (1+ r) m2 0 c1 m1

  21. + + Slutsky’s Equation Revisited • Recall that Slutsky’s equation is ∆xi∆xis(ωi – x i)∆xim∆pi∆pi∆m • An increase in r acts like an increase in the price of c1. If p1 = p2 = 1, ω1 = m1and x1 = c1.In this case, we write Slutsky’s equation as ∆c1∆c1s(m1 – c1)∆c1m ∆r∆r∆m

  22. + Slutsky’s Equation Revisited ∆c1∆c1s(m1 – c1)∆c1m ∆r∆r∆m • If r decreases, substitution effect leads to an …………….. in c1 • Assuming that c1 is a normal good then • if the consumer is a saver m1 – c1> 0 then income effects leads to a …... in c1 and total effect is ……... • if the consumer is a borrower m1 – c1< 0 then income effects leads to a ….. in c1 and total effect must be …………….

  23. Slutsky’s Equation Revisited:A fall in interest rate r for a saver c2 Þ Pure substitution effect Þ Income effect m2 m1 c1

  24. Price Inflation • Define the inflation rate by p where • For example,p = 0.2 means 20% inflation, andp = 1.0 means 100% inflation.

  25. Price Inflation • We lose nothing by setting p1=1 so that p2 = 1+ p • Then we can rewrite the future-value budget constraintas • And rewrite the present-value constraint as

  26. Price Inflation rearranges to intercept slope

  27. Price Inflation • When there was no price inflation (p1=p2=1) the slope of the budget constraint was -(1+r). • Now, with price inflation, the slope of the budget constraint is -(1+r)/(1+ p). This can be written asr is known as the real interest rate.

  28. Real Interest Rate gives For low inflation rates (p» 0), r » r - p .For higher inflation rates thisapproximation becomes poor.

  29. Real Interest Rate

  30. Budget Constraint c2 slope = m2/p2 0 c1 m1/p1 0

  31. Budget Constraint • The slope of the budget constraint is • The constraint becomes flatter if the interest rate r falls or the inflation rate p rises (both decrease the real rate of interest).

  32. Comparative Statics • Using revealed preference, we can show that • If a saver continue to save after a decrease in real interest rate , then he will be worse off • A borrower must continue to borrow after a decrease in real interest rate ,and he must be better off

  33. c2 slope = The consumer ………….. m2/p2 0 c1 m1/p1 0 Comparative Statics:A fall in real interest rate for a saver

  34. Comparative Statics:A fall in real interest rate for a saver c2 An increase in the inflation rate or a decrease in the interest rate ……..…… the budget constraint. m2/p2 0 c1 m1/p1

  35. Comparative Statics:A fall in real interest rate for a saver c2 If the consumer still saves then saving and welfare are ………….. by a lower interest rate or a higher inflation rate. m2/p2 0 c1 m1/p1 0

  36. c2 slope = The consumer ………… m2/p2 0 c1 m1/p1 Comparative Statics:A fall in real interest rate for a borrower

  37. Comparative Statics: A fall in real interest rate for a borrower c2 An increase in the inflation rate or a decrease in the interest rate …………..… the budget constraint. m2/p2 0 c1 m1/p1 0

  38. Comparative Statics: A fall in real interest rate for a borrower c2 The consumer must continue to borrow Borrowing and welfare are …………..… by a lower interest rate or a higher inflation rate. m2/p2 0 c1 m1/p1 0

  39. Valuing Securities • A financial security is a financial instrument that promises to deliver an income stream. • E.g.; a security that pays $m1 at the end of year 1, $m2 at the end of year 2, and $m3 at the end of year 3. • What is the most that should be paid now for this security?

  40. Valuing Securities • The security is equivalent to the sum of three securities; • the first pays only $m1 at the end of year 1, • the second pays only $m2 at the end of year 2, and • the third pays only $m3 at the end of year 3.

  41. Valuing Securities • The PV of $m1 paid 1 year from now is • The PV of $m2 paid 2 years from now is • The PV of $m3 paid 3 years from now is • The PV of the security is therefore

  42. Valuing Bonds • 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. • What is the most that should now be paid for such a bond?

  43. Valuing Bonds

  44. Valuing Bonds • 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?

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

  46. Valuing Consols • A consol is a bond which never terminates, paying $x per period forever. • What is a consol’s present-value?

  47. Valuing Consols

  48. Valuing Consols Solving for PV gives

  49. 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

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