Intertemporal Choice. Chapter Ten. Future Value. Present and Future Values. Intertemporal Choice

Transcription

1 Interteporal Choice Chapter Ten Interteporal Choice Persons often receive incoe in lups ; e.g. onthly salary. How is a lup of incoe spread over the following onth (saving now for consuption later)? Or how is consuption financed by borrowing now against incoe to be received at the end of the onth? Present and Future Values Begin with soe siple financial arithetic. Take just two periods; and 2. Let r denote the interest rate per period. Future Value E.g., if r =. then $ saved at the start of period becoes $ at the start of period 2. The value next period of $ saved now is the future value of that dollar.

2 Future Value Given an interest rate r the future value one period fro now of $ is FV =. Given an interest rate r the future value one period fro now of $ is FV = ( ). Present Value Suppose you can pay now to obtain $ at the start of next period. What is the ost you should pay? $? No. If you kept your $ now and saved it then at the start of next period you would have $(+r) > $, so paying $ now for $ next period is a bad deal. Present Value Q: How uch oney would have to be saved now, in the present, to obtain $ at the start of the next period? A: $ saved now becoes $(+r) at the start of next period, so we want the value of for which (+r) = That is, = /(+r), the present-value of $ obtained at the start of next period. Present Value The present value of $ available at the start of the next period is PV =. And the present value of $ available at the start of the next period is PV =. 2

3 Present Value E.g., if r =. then the ost you should pay now for $ available next period is = $ 9. PV = + And if r =.2 then the ost you should pay now for $ available next period is PV = = $ The Interteporal Choice Proble Let and 2 be incoes received in periods and 2. Let c and be consuptions in periods and 2. Let p and p 2 be the prices of consuption in periods and 2. The Interteporal Choice Proble The interteporal choice proble: Given incoes and 2, and given consuption prices p and p 2, what is the ost preferred interteporal consuption bundle (c, )? For an answer we need to know: the interteporal budget constraint interteporal consuption preferences. To start, let s ignore price effects by supposing that p = p 2 = $. 3

4 Suppose that the consuer chooses not to save or to borrow. Q: What will be consued in period? A: c =. Q: What will be consued in period 2? A: = 2. 2 So (c, ) = (, 2 ) is the consuption bundle if the consuer chooses neither to save nor to borrow. c Now suppose that the consuer spends nothing on consuption in period ; that is, c = and the consuer saves s =. The interest rate is r. What now will be period 2 s consuption level? Period 2 incoe is 2. Savings plus interest fro period su to ( ). So total incoe available in period 2 is 2 + ( ). So period 2 consuption expenditure is c2 = 2 + ( ) 4

5 2 + ( ) the future-value of the incoe endowent 2 + ( ) ( c, c ) = (, + ( ) ) 2 2 is the consuption bundle when all period incoe is saved. 2 2 c c Now suppose that the consuer spends everything possible on consuption in period, so =. What is the ost that the consuer can borrow in period against her period 2 incoe of $ 2? Let b denote the aount borrowed in period. Only $ 2 will be available in period 2 to pay back $b borrowed in period. So b ( ) = 2. That is, b = 2 / ( ). So the largest possible period consuption level is c = 2 + 5

6 2 + ( ) 2 ( c, c ) = (, + ( ) ) 2 2 is the consuption bundle when all period incoe is saved. the present-value of the incoe endowent + 2 c 2 + ( ) 2 ( c, c ) = (, + ( ) ) 2 2 is the consuption bundle when period saving is as large as possible. ( c, c ) 2 2 = +, is the consuption bundle when period borrowing is as big as possible. c + 2 Suppose that c units are consued in period. This costs $c and leaves - c saved. Period 2 consuption will then be which is = 2 + ( )( c) c2 = ( ) c ( ). slope intercept 2 + ( ) 2 ( c, c ) = (, + ( ) ) 2 2 is the consuption bundle when period saving is as large as possible. ( c, c ) 2 2 = +, is the consuption bundle when period borrowing is as big as possible. c + 2 6

7 2 + ( ) 2 c2 = ( ) c ( ). slope = -(+r) Saving Borrowing + 2 c ( ) c + = ( ) + 2 is the future-valued for of the budget constraint since all ters are in period 2 values. This is equivalent to c c + 2 = 2 + which is the present-valued for of the constraint since all ters are in period values. Now let s add prices p and p 2 for consuption in periods and 2. How does this affect the budget constraint? Interteporal Choice Given her endowent (, 2 ) and prices p, p 2 what interteporal consuption bundle (c *, *) will be chosen by the consuer? Maxiu possible expenditure in period 2 is 2 + ( ) so axiu possible consuption in period 2 is r + ( + ) 2 =. p2 7

8 Interteporal Choice Siilarly, axiu possible expenditure in period is + 2 so axiu possible consuption in period is r c + 2 / ( + ) =. p Interteporal Choice Finally, if c units are consued in period then the consuer spends p c in period, leaving - p c saved for period. Available incoe in period 2 will then be so 2 + ( )( pc) p2c2 = 2 + ( )( pc). Interteporal Choice p2c2 = 2 + ( )( pc ) rearranged is ( ) pc + p 2 = ( ) + 2. This is the future-valued for of the budget constraint since all ters are expressed in period 2 values. Equivalent to it is the present-valued for p p c r c = where all ters are expressed in period values. + 2 ( ) p 2 2 /p 2 ( ) pc + p 2 = ( ) + 2 Saving Borrowing Slope = ( ) p p 2 /p c + 2 / ( ) p 8

9 Price Inflation Define the inflation rate by π where p( + π ) = p2. For exaple, π =.2 eans 2% inflation, and π =. eans % inflation. Price Inflation We lose nothing by setting p = so that p 2 = + π. Then we can rewrite the budget constraint as p p c r c = π c r c = + + Price Inflation + π c r c = + + rearranges to c = + + π 2 c ( ) π so the slope of the interteporal budget constraint is. + π Price Inflation When there was no price inflation (p =p 2 =) the slope of the budget constraint was -(+r). Now, with price inflation, the slope of the budget constraint is -(+r)/(+ π). This can be written as + = + r ( ρ) + π ρ is known as the real interest rate. 9

10 gives Real Interest Rate ( + ) = + r ρ + π r π ρ =. + π For low inflation rates (π ), ρ r - π. For higher inflation rates this approxiation becoes poor. Real Interest Rate r π r - π ρ Coparative Statics The slope of the budget constraint is + r ( + ρ) =. + π The constraint becoes flatter if the interest rate r falls or the inflation rate π rises (both decrease the real rate of interest). Coparative Statics 2 /p 2 ( + ) = + r slope = ρ + π If the consuer saves then saving and welfare are reduced by a lower interest rate or a higher inflation rate. /p c

11 Coparative Statics 2 /p 2 ( + ) = + r slope = ρ + π The consuer borrows. A fall in the inflation rate or a rise in the interest rate flattens the budget constraint. /p c Coparative Statics 2 /p 2 ( + ) = + r slope = ρ + π If the consuer borrows then borrowing and welfare are increased by a lower interest rate or a higher inflation rate. /p c Valuing Securities A financial security is a financial instruent that proises to deliver an incoe strea. E.g.; a security that pays $ at the end of year, $ 2 at the end of year 2, and $ 3 at the end of year 3. What is the ost that should be paid now for this security? Valuing Securities The security is equivalent to the su of three securities; the first pays only $ at the end of year, the second pays only $ 2 at the end of year 2, and the third pays only $ 3 at the end of year 3.

12 Valuing Securities The PV of $ paid year fro now is / ( ) The PV of $ 2 paid 2 years fro now is 2 2 / ( ) The PV of $ 3 paid 3 years fro now is 3 3 / ( ) The PV of the security is therefore 2 3 / ( ) + 2 / ( ) + 3 / ( ). Valuing Bonds A bond is a special type of security that pays a fixed aount $x for T years (its aturity date) and then pays its face value $F. What is the ost that should now be paid for such a bond? End of Year Incoe Paid Present -Value Valuing Bonds 2 3 T- T $x $x $x $x $x $F $x $ x $ x 2 3 ( ) ( ) $ x $ F ( r) T + ( ) T Valuing Bonds Suppose you win a State lottery. The prize is $,, but it is paid over years in equal installents of $, each. What is the prize actually worth? x x x F PV = + + Κ T T ( ) ( ) ( ) 2

13 Valuing Bonds $, $, $, PV = + + Κ ( + ) ( + ) = $64, 457 Valuing Consols A consol is a bond which never terinates, paying $x per period forever. What is a consol s present-value? is the actual (present) value of the prize. End of Year Incoe Paid Present -Value Valuing Consols 2 3 t $x $x $x $x $x $x $x $ x $ x 2 3 ( ) ( ) $ x ( ) t x x x PV = + + Κ + + r + r + r t + Κ. 2 ( ) ( ) Valuing Consols x x x PV = Κ 2 3 ( ) ( ) r x x x = Κ + 2 ( ) = [ ] r x + PV. Solving for PV gives + PV = x. r 3

14 Valuing Consols E.g. if r =. now and forever then the ost that should be paid now for a console that provides $ per year is PV x $ = = = $,. r 4