1 intertemporal choice. 2 persons often receive income in “lumps”; e.g. monthly salary. how is a...

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.

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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

FV r 1 .

FV m r ( ).1

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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.

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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

r1

1PV

r1

mPV

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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.

The Intertemporal Choice Problem

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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.

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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

The Intertemporal Budget Constraint

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c1

c2

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

m2/p2

m1/p100


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

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c1

c2

m2/p2

m1/p100


2

2

2

1

p

m

p

m)r1(

)r1(p

m

p

m

1

2

1

1

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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 be

so

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( ) ( ) .1 11 1 2 2 1 2 r p c p c r m m

p cp

rc m

mr1 1

22 1

21 1

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

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Rearrange again to get c2 as a function of other variables

12

1

2

2

2

12 c

p

p)r1(

p

m

p

m)r1(c

slopeintercept


18

c1

c2

m2/p2

m1/p100

)r1(p

m

p

m

1

2

1

1

Slope = 2

1

p

p)r1(

( ) ( )1 11 1 2 2 1 2 r p c p c r m m


2

2

2

1

p

m

p

m)r1(

Saving

Borrowing

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Suppose p1 = p2 = 1, the future-value

constraint becomes

Rearranging, we get

1212 c)r1(mm)r1(c


2121 m+m)r+1(=c+c)r+1(

20

c1

c2

m2

m10

21 mr)m1(

mm

r12

1

slope = – (1+ r)

1212 c)r1(mm)r1(c

The Intertemporal Budget Constraint If p1 = p2 = 1 then,

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Slutsky’s Equation Revisited

An increase in r acts like an increase in the price of c1. If p1 = p2 = 1, ω1 = m1 and x1 = c1. In this case, we write Slutsky’s equation as

∆c1 ∆c1s

(m1 – c1) ∆c1m

∆r ∆r ∆m +

Recall that Slutsky’s equation is∆xi ∆xi

s (ωi – x i) ∆xi

m ∆pi ∆pi ∆m +

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Slutsky’s Equation Revisited ∆c1 ∆c1

s (m1 – c1) ∆c1

m

∆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 …………….

+

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Slutsky’s Equation Revisited:A fall in interest rate r for a saver

c1

m2

m1

c2Pure substitution effect

Income effect

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Price Inflation

Define the inflation rate by where

For example, = 0.2 means 20% inflation, and = 1.0 means 100% inflation.

p p1 21( ) .

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Price Inflation We lose nothing by setting p1=1 so that p2 = 1+ Then we can rewrite the future-value budget

constraintas

And rewrite the present-value constraint as

cr

c mm

r1 2 121

1 1

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Price Inflationrearranges to

121

2 c1

r1

)1(

mm)r1(c

intercept slope

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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+ ). This can be written as

is known as the real interest rate.

( )111

r

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Real Interest Rate

( )1

11

r

gives

r1

.

For low inflation rates ( 0), r - .For higher inflation rates thisapproximation becomes poor.

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Real Interest Rate

r 0.30 0.30 0.30 0.30 0.30

0.0 0.05 0.10 0.20 1.00

r - 0.30 0.25 0.20 0.10 -0.70

0.30 0.24 0.18 0.08 -0.35

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Budget Constraint

121

2 c1

r1

)1(

mm)r1(c

c1

c2

m2/p2

m1/p100

r1

r1)1(slope =

)1(

mm)r1( 21

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The slope of the budget constraint is

The constraint becomes flatter if the interest rate r falls or the inflation raterises (both decrease the real rate of interest).

.1

r1)1(

Budget Constraint

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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

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Comparative Statics: A fall in real interest rate for a saver

c1

c2

m2/p2

m1/p100

( )111

r

slope =

The consumer …………..

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An increase in the inflation rate or a decrease in the

interest rate ……..…… the budget constraint.

c1

c2

m2/p2

m1/p1

0


35

If the consumer still saves then saving and welfare are ………….. by a lower interest rate or

a higher inflation rate.

c1

c2

m2/p2

m1/p100


36

c1

c2

m2/p2

m1/p1

0

( )111

r

slope =

The consumer …………

Comparative Statics: A fall in real interest rate for a borrower

37

c1

c2

m2/p2

m1/p100

An increase in the inflation rate or a decrease in the

interest rate …………..… the budget constraint.


38

The consumer must continue to borrow

Borrowing and welfare are …………..… by a lower interest rate or a higher inflation rate.

c1

c2

m2/p2

m1/p100


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?

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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.

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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

m r1 1/ ( )

m r221/ ( )

m r331/ ( )

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?

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Valuing Bonds

PVx

rx

r

x

r

F

rT T

1 1 1 12 1( ) ( ) ( ).

End of Year

1 2 3 … T-1 T

Income Paid

$x $x $x $x $x $F

Present -Value

$xr1 $

( )

x

r1 2

$

( )

x

r1 3

… $

( )

x

r T1 1

$

( )

F

r T1

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

PV

$100, $100,

( )

$100,

( )

$614,

0001 0 1

000

1 0 1

000

1 0 1

457

2 10

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 ConsolsEnd ofYear

1 2 3 … t …

IncomePaid

$x $x $x $x $x $x

Present-Value

$xr1

$

( )

x

r1 2

$

( )

x

r1 3… $

( )

x

rt1…

PVx

rx

r

x

r t

1 1 12( ) ( ).

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Valuing Consols

PVx

rx

r

x

r

rx

xr

x

r

rx PV

1 1 1

11 1 1

11

2 3

2

( ) ( )

( )

.

Solving for PV gives

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Valuing ConsolsE.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

PVxr

$1000$10, .

0 1000

1 intertemporal choice. 2 persons often receive income in “lumps”; e.g. monthly salary. how is a...

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start of period

period ise

present valuethe present

rate r

future valuethe value

prices of consumption

futurevalue form

consumption prices p1