Chapter Ten
Intertemporal Choice
What Are We Doing in this Chapter?
?We apply our basic framework of
consumer choice to study issues of
choices across different time
periods;
?Again,in terms of theoretical
framework,not much is new!
What Are the Questions?
?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?
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.
Future Value
?Given an interest rate r the future
value one period from now of $m is
FV m r? ?( ).1
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.
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
PV
r
?
?
1
1
.
PV m
r
?
?1
.
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,
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.
The Intertemporal Budget Constraint
?To start,let’s ignore price effects by
supposing that
p1 = p2 = $1.
The Intertemporal Budget Constraint
c1
c2
So (c1,c2) = (m1,m2) is the
consumption bundle if the
consumer chooses neither to
save nor to borrow.
m2
m100
The Intertemporal Budget Constraint
c1
c2
m2
m100
m
r m
2
11
?
?( )
the future-value of the income
endowment
The Intertemporal Budget Constraint
c1
c2
m2
m100
is the consumption bundle when all
period 1 income is saved.
? ?(,),( )c c m r m1 2 2 10 1? ? ?m
r m
2
11
?
?( )
The Intertemporal Budget Constraint
?Now suppose that the consumer
spends everything possible on
consumption in period 1,so c2 = 0.
?What is the most that the consumer
can borrow in period 1 against her
period 2 income of $m2?
?Let b1 denote the amount borrowed
in period 1.
The Intertemporal Budget Constraint
?Only $m2 will be available in period 2
to pay back $b1 borrowed in period 1.
?So b1(1 + r ) = m2.
?That is,b1 = m2 / (1 + r ).
?So the largest possible period 1
consumption level is
The Intertemporal Budget Constraint
?Only $m2 will be available in period 2
to pay back $b1 borrowed in period 1.
?So b1(1 + r ) = m2.
?That is,b1 = m2 / (1 + r ).
?So the largest possible period 1
consumption level is
c m m
r1 1
2
1
? ?
?
The Intertemporal Budget Constraint
c1
c2
m2
m100
is the consumption bundle when all
period 1 income is saved.
? ?(,),( )c c m r m1 2 2 10 1? ? ?m
r m
2
11
?
?( )
m m r1 21? ?
the present-value of
the income endowment
The Intertemporal Budget Constraint
?Suppose that c1 units are consumed
in period 1,This costs $c1 and
leaves m1- c1 saved,Period 2
consumption will then be
c m r m c2 2 1 11? ? ? ?( )( )
The Intertemporal Budget Constraint
?Suppose that c1 units are consumed
in period 1,This costs $c1 and
leaves m1- c1 saved,Period 2
consumption will then be
which is
c m r m c2 2 1 11? ? ? ?( )( )
c r c m r m2 1 2 11 1? ? ? ? ? ?( ) ( ),
? ?? ?? ?? ?
slope intercept
The Intertemporal Budget Constraint
c1
c2
m2
m100
? ?(,),( )c c m r m1 2 2 10 1? ? ?
(,),c c m m
r1 2 1
2
1
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.
m m r1 21? ?
m
r m
2
11
?
?( )
The Intertemporal Budget Constraint
c1
c2
m2
m100
m
( r)m
2
11
?
?
m m r1 21? ?
slope = -(1+r)
c r c m r m2 1 2 11 1? ? ? ? ? ?( ) ( ),
The Intertemporal Budget Constraint
c1
c2
m2
m100
m
( r)m
2
11
?
?
m m r1 21? ?
slope = -(1+r)
c r c m r m2 1 2 11 1? ? ? ? ? ?( ) ( ),
The Intertemporal Budget Constraint
( ) ( )1 11 2 1 2? ? ? ? ?r c c r m m
is the,future-valued” form of the budget
constraint since all terms are in period 2
values,This is equivalent to
c c r m m r1 2 1 21 1? ? ? ? ?
which is the,present-valued” form of the
constraint since all terms are in period 1
values.
The Intertemporal Budget Constraint
?Now let’s add prices p1 and p2 for
consumption in periods 1 and 2.
?How does this affect the budget
constraint?
Intertemporal Choice
?Suppose c1 units are consumed in
period 1 then the consumer spends
p1c1 in period 1,leaving m1 - p1c1
saved for period 1,Available income
in period 2 will then be
so
m r m p c2 1 1 11? ? ?( )( )
p c m r m p c2 2 2 1 1 11? ? ? ?( )( ).
Intertemporal Choice
p c m r m p c2 2 2 1 1 11? ? ? ?( )( )
rearranged is
( ) ( ),1 11 1 2 2 1 2? ? ? ? ?r p c p c r m m
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
p c p r c m m r1 1 2 2 1 21 1? ? ? ? ?
where all terms are expressed in period 1
values.
The Intertemporal Budget Constraint
c1
c2
m2/p2
m1/p100
The Intertemporal Budget Constraint
c1
c2
m2/p2
m1/p100
( )1 1 2
2
? ?r m m
p
The Intertemporal Budget Constraint
c1
c2
m2/p2
m1/p100 m m r
p
1 2
1
1? ?/ ( )
( )1 1 2
2
? ?r m m
p
The Intertemporal Budget Constraint
c1
c2
m2/p2
m1/p100 m m r
p
1 2
1
1? ?/ ( )
( )1 1 2
2
? ?r m m
p
Slope = ? ?( )1 12r
p
p
( ) ( )1 11 1 2 2 1 2? ? ? ? ?r p c p c r m m
The Intertemporal Budget Constraint
c1
c2
m2/p2
m1/p100 m m r
p
1 2
1
1? ?/ ( )
( )1 1 2
2
? ?r m m
p
Slope = ? ?( )1 12r
p
p
( ) ( )1 11 1 2 2 1 2? ? ? ? ?r p c p c r m m
Price Inflation
?Define the inflation rate by p where
?For example,
p = 0.2 means 20% inflation,and
p = 1.0 means 100% inflation.
p p1 21( ),? ?p
Price Inflation
?We lose nothing by setting p1=1 so
that p2 = 1+ p,
?Then we can rewrite the budget
constraint
as
p c p r c m m r1 1 2 2 1 21 1? ? ? ? ?
c r c m m r1 2 1 211 1? ?? ? ? ?p
Price Inflation
c r c m m r1 2 1 211 1? ?? ? ? ?p
rearranges to
?????? ??p??p???? 2112 mr1 m)1(c1 r1c
so the slope of the intertemporal budget
constraint is,
1
r1
p?
??
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 as
r is known as the real interest rate.
? ? ? ? ??( )1 11r pr
Real Interest Rate
? ? ? ? ??( )1 11r pr
gives
r pp? ??r1,
For low inflation rates (p? 0),r ? r - p,
For higher inflation rates this
approximation becomes poor.
Comparative Statics
?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).
.
1
r1)1(
p?
???r??
Comparative Statics
c1
c2
m2/p2
m1/p100
? ? ? ? ??( )1 11r prslope =
Comparative Statics
c1
c2
m2/p2
m1/p100
? ? ? ? ??( )1 11r prslope =
Comparative Statics
c1
c2
m2/p2
m1/p100
? ? ? ? ??( )1 11r prslope =
The consumer saves.
Comparative Statics
c1
c2
m2/p2
m1/p100
? ? ? ? ??( )1 11r prslope =
The consumer saves,An
increase in the inflation
rate or a decrease in
the interest rate
“flattens” the
budget
constraint.
Comparative Statics
c1
c2
m2/p2
m1/p100
? ? ? ? ??( )1 11r prslope =
If the consumer saves then
saving and welfare are
reduced by a lower
interest rate or a
higher inflation
rate.
Comparative Statics
c1
c2
m2/p2
m1/p100
? ? ? ? ??( )1 11r prslope =
Comparative Statics
c1
c2
m2/p2
m1/p100
? ? ? ? ??( )1 11r prslope =
Comparative Statics
c1
c2
m2/p2
m1/p100
? ? ? ? ??( )1 11r prslope =
The consumer borrows.
Comparative Statics
c1
c2
m2/p2
m1/p100
? ? ? ? ??( )1 11r prslope =
The consumer borrows,A
fall in the inflation rate or
a rise in the interest rate
“flattens” the
budget constraint.
Comparative Statics
c1
c2
m2/p2
m1/p100
? ? ? ? ??( )1 11r prslope =
If the consumer borrows then
borrowing and welfare are
increased by a lower
interest rate or a
higher inflation
rate.
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?
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.
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 r2 21/ ( )?
m r3 31/ ( )?
m r m r m r1 2 2 3 31 1 1/ ( ) / ( ) / ( ),? ? ? ? ?
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?
Valuing Bonds
End of
Ye ar
1 2 3 … T- 1 T
Incom e
Pa id
$x $x $x $x $x $F
Pr esent
-Value
$ x
r1 ?
$
( )
x
r1
2
?
$
( )
x
r1
3
?
…
$
( )
x
r
T
1
1
?
?
$
( )
F
r
T
1 ?
PV x
r
x
r
x
r
F
rT T
?
?
?
?
? ?
?
?
??1 1 1 12 1( ) ( ) ( )
.?
Valuing Consols
?A consol is a bond which never
terminates,paying $x per period
forever.
?What is a consol’s present-value?
Valuing Consols
End of
Ye ar
1 2 3 … t …
Incom e
Pa id
$x $x $x $x $x $x
Pr esent
-Value
$ x
r1 ?
$
( )
x
r1
2
?
$
( )
x
r1
3
?
…
$
( )
x
r
t
1 ?
…
PV x
r
x
r
x
r t
?
?
?
?
? ?
?
?
1 1 12( ) ( )
.? ?
Valuing Consols
? ?
PV
x
r
x
r
x
r
r
x
x
r
x
r
r
x PV
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
1 1 1
1
1 1 1
1
1
2 3
2
( ) ( )
( )
.
?
?
Solving for PV gives
PV x
r
?,