1 Intertemporal Choice

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

9 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 )? For an answer we need to know:  the intertemporal budget constraint  intertemporal consumption preferences.

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

11 c1c1 c2c2 So (c 1, c 2 ) = (m 1 /p 1, m 2 /p 2 ) is the consumption bundle if the consumer chooses neither to save nor to borrow. m 2 /p 2 m 1 /p The Intertemporal Budget Constraint

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

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

14 c1c1 c2c2 m 2 /p 2 m 1 /p The Intertemporal Budget Constraint

15 Intertemporal Choice Finally, if both c 1 and c 2 are greater than 0. Then the consumer spends p 1 c 1 in period 1, and save m 1 - p 1 c 1. Available income in period 2 will then be so

16 Intertemporal Choice where all terms are expressed in period 1values. 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

17 Rearrange again to get c 2 as a function of other variables slope intercept The Intertemporal Budget Constraint

18 c1c1 c2c2 m 2 /p 2 m 1 /p Slope = The Intertemporal Budget Constraint Saving Borrowing

19 Suppose p 1 = p 2 = 1, the future-value constraint becomes Rearranging, we get The Intertemporal Budget Constraint

20 c1c1 c2c2 m2m2 m1m1 0 slope = – (1+ r) The Intertemporal Budget Constraint If p 1 = p 2 = 1 then,

21 Slutsky’s Equation Revisited An increase in r acts like an increase in the price of c 1. If p 1 = p 2 = 1, ω 1 = m 1 and x 1 = c 1. In this case, we write Slutsky’s equation as ∆c 1 ∆c 1 s (m 1 – c 1 ) ∆c 1 m ∆r ∆r ∆m + Recall that Slutsky’s equation is ∆x i ∆x i s (ω i – x i ) ∆x i m ∆p i ∆p i ∆m +

22 Slutsky’s Equation Revisited ∆c 1 ∆c 1 s (m 1 – c 1 ) ∆c 1 m ∆r ∆r ∆m If r decreases, substitution effect leads to an …………….. in c 1 Assuming that c 1 is a normal good then  if the consumer is a saver m 1 – c 1 > 0 then income effects leads to a …... in c 1 and total effect is ……...  if the consumer is a borrower m 1 – c 1 < 0 then income effects leads to a ….. in c 1 and total effect must be ……………. +

23 Slutsky’s Equation Revisited: A fall in interest rate r for a saver c1c1 m2m2 m1m1 c2c2 Pure substitution effect   Income effect

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

25 Price Inflation We lose nothing by setting p 1 =1 so that p 2 = 1+  Then we can rewrite the future-value budget constraint as And rewrite the present-value constraint as

26 Price Inflation rearranges to intercept slope

27 Price Inflation 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.

28 Real Interest Rate gives For low inflation rates (  0),  r - . For higher inflation rates this approximation becomes poor.

29 Real Interest Rate

30 Budget Constraint c1c1 c2c2 m 2 /p 2 m 1 /p slope =

31 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). Budget Constraint

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 Comparative Statics: A fall in real interest rate for a saver c1c1 c2c2 m 2 /p 2 m 1 /p slope = The consumer …………..

34 An increase in the inflation rate or a decrease in the interest rate ……..…… the budget constraint. c1c1 c2c2 m 2 /p 2 m 1 /p 1 0 Comparative Statics: A fall in real interest rate for a saver

35 If the consumer still saves then saving and welfare are ………….. by a lower interest rate or a higher inflation rate. c1c1 c2c2 m 2 /p 2 m 1 /p Comparative Statics: A fall in real interest rate for a saver

36 c1c1 c2c2 m 2 /p 2 m 1 /p 1 0 slope = The consumer ………… Comparative Statics: A fall in real interest rate for a borrower

37 c1c1 c2c2 m 2 /p 2 m 1 /p An increase in the inflation rate or a decrease in the interest rate …………..… the budget constraint. Comparative Statics: A fall in real interest rate for a borrower

38 The consumer must continue to borrow Borrowing and welfare are …………..… by a lower interest rate or a higher inflation rate. c1c1 c2c2 m 2 /p 2 m 1 /p Comparative Statics: A fall in real interest rate for a borrower

39 Valuing Securities A financial security is a financial instrument that promises to deliver an income stream. 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. 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 $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.

41 Valuing Securities The PV of $m 1 paid 1 year from now is The PV of $m 2 paid 2 years from now is The PV of $m 3 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