Chapter Ten
Intertemporal Choice
（ 跨时期选择）
Structure
Present and future values
Intertemporal budget constraint
Preferences for intertemporal
consumption
Intertemporal choice
Comparative statics
Valuing securities
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?
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
E.g.,if r = 0.1 then $100 saved at the
start of period 1 becomes $110 at the
start of period 2.
The value next period of $1 saved
now is the future value of that dollar.
Future Value
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 r1,
FV m r( ).1
Present Value （ 现值）
Suppose you can pay now to obtain
$1 at the start of next period.
What is the most you should 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.
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
.
Present Value
E.g.,if r = 0.1 then the most you
should pay now for $1 available next
period is
And if r = 0.2 then the most you
should pay now for $1 available next
period is
PV?
1
1 0 1
91$0,
PV?
1
1 0 2
83$0,
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
Suppose that the consumer chooses
not to save or to borrow.
Q,What will be consumed in period 1?
A,c1 = m1.
Q,What will be consumed in period 2?
A,c2 = m2.
The Intertemporal Budget Constraint
c1
c2
m2
m100
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
Now suppose that the consumer
spends nothing on consumption in
period 1; that is,c1 = 0 and the
consumer saves
s1 = m1.
The interest rate is r.
What now will be period 2’s
consumption level?
The Intertemporal Budget Constraint
Period 2 income is m2.
Savings plus interest from period 1
sum to (1 + r )m1.
So total income available in period 2 is
m2 + (1 + r )m1.
So period 2 consumption expenditure
is
The Intertemporal Budget Constraint
Period 2 income is m2.
Savings plus interest from period 1
sum to (1 + r )m1.
So total income available in period 2 is
m2 + (1 + r )m1.
So period 2 consumption expenditure
is
c m r m2 2 11( )
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 1m
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 1m
r m
2
11
( )
m m r1 21
the present-value of
the income endowment
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
r m
2
11
( )
m m r1 21
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 bec m r m c
2 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
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 2r 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 toc c
r m
m
r1
2 1 2
1 1
which is the,present-valued” form of the
constraint since all terms are in period 1
values.
As r rises,it is as if period 1 consumption is
more expensive or period 2 consumption
cheaper.
The Intertemporal Budget Constraint
If we allow for inflation or deflation,
then prices of consumption goods
may be different in two periods,
Use p1 and p2 to denote prices for
consumption in periods 1 and 2.
How does this affect the budget
constraint?
Intertemporal Choice
Given her endowment (m1,m2) and
prices p1,p2 what intertemporal
consumption bundle (c1*,c2*) will be
chosen by the consumer?
Maximum possible expenditure in
period 2 is
so maximum possible consumption
in period 2 is
m r m2 11( )
c m r m
p2
2 1
2
1( ),
Intertemporal Choice
Similarly,maximum possible
expenditure in period 1 is
so maximum possible consumption
in period 1 is
m m
r1
2
1
c m m r
p1
1 2
1
1/ ( ),
Intertemporal Choice
Finally,if 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 2r 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” formp c p
r c m
m
r1 1
2 2 1 2
1 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
1
2
r pp
( ) ( )1 11 1 2 2 1 2r 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
1
2
r pp
( ) ( )1 11 1 2 2 1 2r p c p c r m m
Intertemporal Preferences
Extreme cases
–Perfect substitute
–Perfect complement
Intermediate case of well-behaved
preferences
Intertemporal Preferences
c1
c2
Intertemporal Choice
c1
m2/p2
m1/p100
The consumer saves.
c2
c1
m2/p2
m1/p1
The consumer borrows.
c2
Comparative Statics
Interest rate rises or falls
Price inflation
Comparative Statics
The slope of the budget constraint is
The constraint becomes flatter if the
interest rate r falls.
The constraint becomes steeper if
the interest rate r rises.
( )1 1
2
r pp
Comparative Statics
c1
c2
m2/p2
m1/p1
slope =( )1
1
2
r pp
Comparative Statics
c1
c2
m2/p2
m1/p100
slope =
The consumer saves.
( )1 1
2
r pp
Comparative Statics
c1
c2
m2/p2
m1/p100
slope =
A decrease in the interest rate
“flattens” the
budget constraint.
The consumer saves.
( )1 1
2
r pp
Comparative Statics
c1
c2
m2/p2
m1/p100
slope =
If the consumer choose to
remain a lender then her
welfare is reduced by a lower
interest rate,Sign
on saving is
ambiguous.
( )1 1
2
r pp
The consumer initially saves.
Comparative Statics
c2
c1
m2/p2
m1/p100
slope =
The consumer borrows.
( )1 1
2
r pp
Comparative Statics
c1
c2
m2/p2
m1/p100
slope =
A fall in the interest rate
“flattens” the
budget constraint.
The consumer borrows.
( )1 1
2
r pp
Comparative Statics
c1
c2
m2/p2
m1/p100
slope =
The consumer will remain a
borrower,Welfare is
increased by a lower
interest rate,Will
borrow more.
( )1 1
2
r pp
The consumer borrows.
Borrow or Lend after a Change in
Interest Rate?
Use WARP.
Different effects for rising interest
rates (see textbook for analysis of
rising interest rates).
Effect on Consumption (Saving)?
Slutsky Equation
1 1 1
11()
t s mc c c
mc
r r r
(?) (-) (?) (+)
If lender,(m-c)<0,total effect is
negative.
If borrower,(m-c)>0,total effect is
ambiguous.
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 1p
Price Inflation
c r c m m r1 2 1 211 1p
rearranges to
pp 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 （ 实际利率）,
r is called the nominal interest rate （ 名义利率）,
( )1 11r pr
Real Interest Rate
( )1 11r pr
gives r p
p?
r
1,
For low inflation rates (p? 0),r? r - p,
For higher inflation rates this
approximation becomes poor.
Real Interest Rate
r 0,30 0,30 0,30 0,30 0,30
p 0,0 0,05 0,10 0,20 1,00
r - p 0,30 0,25 0,20 0,10 -0.70
r 0,30 0,24 0,18 0,08 -0.35
Comparative Statics
The slope of the budget constraint is
The constraint becomes flatter if the
inflation rate p rises.
The effect is the same as falling (nominal)
interest rate,Both have the effect of
decreasing the real rate of interest.
Earlier comparative static analyses apply.
.
1
r1)1(
p?
r
Valuing Securities （ 证券）
A financial security （ 金融证券） 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 （ 债券）
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 （ 面值） $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 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?
Valuing Bonds
PV?
$100,$100,
( )
$100,
( )
$614,
000
1 0 1
000
1 0 1
000
1 0 1
457
2 10
is the actual (present) value of the prize.
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
,
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
PV xr$1000 $10,.0 1 000
Intertemporal Choice
（ 跨时期选择）
Structure
Present and future values
Intertemporal budget constraint
Preferences for intertemporal
consumption
Intertemporal choice
Comparative statics
Valuing securities
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?
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
E.g.,if r = 0.1 then $100 saved at the
start of period 1 becomes $110 at the
start of period 2.
The value next period of $1 saved
now is the future value of that dollar.
Future Value
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 r1,
FV m r( ).1
Present Value （ 现值）
Suppose you can pay now to obtain
$1 at the start of next period.
What is the most you should 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.
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
.
Present Value
E.g.,if r = 0.1 then the most you
should pay now for $1 available next
period is
And if r = 0.2 then the most you
should pay now for $1 available next
period is
PV?
1
1 0 1
91$0,
PV?
1
1 0 2
83$0,
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
Suppose that the consumer chooses
not to save or to borrow.
Q,What will be consumed in period 1?
A,c1 = m1.
Q,What will be consumed in period 2?
A,c2 = m2.
The Intertemporal Budget Constraint
c1
c2
m2
m100
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
Now suppose that the consumer
spends nothing on consumption in
period 1; that is,c1 = 0 and the
consumer saves
s1 = m1.
The interest rate is r.
What now will be period 2’s
consumption level?
The Intertemporal Budget Constraint
Period 2 income is m2.
Savings plus interest from period 1
sum to (1 + r )m1.
So total income available in period 2 is
m2 + (1 + r )m1.
So period 2 consumption expenditure
is
The Intertemporal Budget Constraint
Period 2 income is m2.
Savings plus interest from period 1
sum to (1 + r )m1.
So total income available in period 2 is
m2 + (1 + r )m1.
So period 2 consumption expenditure
is
c m r m2 2 11( )
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 1m
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 1m
r m
2
11
( )
m m r1 21
the present-value of
the income endowment
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
r m
2
11
( )
m m r1 21
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 bec m r m c
2 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
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 2r 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 toc c
r m
m
r1
2 1 2
1 1
which is the,present-valued” form of the
constraint since all terms are in period 1
values.
As r rises,it is as if period 1 consumption is
more expensive or period 2 consumption
cheaper.
The Intertemporal Budget Constraint
If we allow for inflation or deflation,
then prices of consumption goods
may be different in two periods,
Use p1 and p2 to denote prices for
consumption in periods 1 and 2.
How does this affect the budget
constraint?
Intertemporal Choice
Given her endowment (m1,m2) and
prices p1,p2 what intertemporal
consumption bundle (c1*,c2*) will be
chosen by the consumer?
Maximum possible expenditure in
period 2 is
so maximum possible consumption
in period 2 is
m r m2 11( )
c m r m
p2
2 1
2
1( ),
Intertemporal Choice
Similarly,maximum possible
expenditure in period 1 is
so maximum possible consumption
in period 1 is
m m
r1
2
1
c m m r
p1
1 2
1
1/ ( ),
Intertemporal Choice
Finally,if 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 2r 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” formp c p
r c m
m
r1 1
2 2 1 2
1 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
1
2
r pp
( ) ( )1 11 1 2 2 1 2r 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
1
2
r pp
( ) ( )1 11 1 2 2 1 2r p c p c r m m
Intertemporal Preferences
Extreme cases
–Perfect substitute
–Perfect complement
Intermediate case of well-behaved
preferences
Intertemporal Preferences
c1
c2
Intertemporal Choice
c1
m2/p2
m1/p100
The consumer saves.
c2
c1
m2/p2
m1/p1
The consumer borrows.
c2
Comparative Statics
Interest rate rises or falls
Price inflation
Comparative Statics
The slope of the budget constraint is
The constraint becomes flatter if the
interest rate r falls.
The constraint becomes steeper if
the interest rate r rises.
( )1 1
2
r pp
Comparative Statics
c1
c2
m2/p2
m1/p1
slope =( )1
1
2
r pp
Comparative Statics
c1
c2
m2/p2
m1/p100
slope =
The consumer saves.
( )1 1
2
r pp
Comparative Statics
c1
c2
m2/p2
m1/p100
slope =
A decrease in the interest rate
“flattens” the
budget constraint.
The consumer saves.
( )1 1
2
r pp
Comparative Statics
c1
c2
m2/p2
m1/p100
slope =
If the consumer choose to
remain a lender then her
welfare is reduced by a lower
interest rate,Sign
on saving is
ambiguous.
( )1 1
2
r pp
The consumer initially saves.
Comparative Statics
c2
c1
m2/p2
m1/p100
slope =
The consumer borrows.
( )1 1
2
r pp
Comparative Statics
c1
c2
m2/p2
m1/p100
slope =
A fall in the interest rate
“flattens” the
budget constraint.
The consumer borrows.
( )1 1
2
r pp
Comparative Statics
c1
c2
m2/p2
m1/p100
slope =
The consumer will remain a
borrower,Welfare is
increased by a lower
interest rate,Will
borrow more.
( )1 1
2
r pp
The consumer borrows.
Borrow or Lend after a Change in
Interest Rate?
Use WARP.
Different effects for rising interest
rates (see textbook for analysis of
rising interest rates).
Effect on Consumption (Saving)?
Slutsky Equation
1 1 1
11()
t s mc c c
mc
r r r
(?) (-) (?) (+)
If lender,(m-c)<0,total effect is
negative.
If borrower,(m-c)>0,total effect is
ambiguous.
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 1p
Price Inflation
c r c m m r1 2 1 211 1p
rearranges to
pp 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 （ 实际利率）,
r is called the nominal interest rate （ 名义利率）,
( )1 11r pr
Real Interest Rate
( )1 11r pr
gives r p
p?
r
1,
For low inflation rates (p? 0),r? r - p,
For higher inflation rates this
approximation becomes poor.
Real Interest Rate
r 0,30 0,30 0,30 0,30 0,30
p 0,0 0,05 0,10 0,20 1,00
r - p 0,30 0,25 0,20 0,10 -0.70
r 0,30 0,24 0,18 0,08 -0.35
Comparative Statics
The slope of the budget constraint is
The constraint becomes flatter if the
inflation rate p rises.
The effect is the same as falling (nominal)
interest rate,Both have the effect of
decreasing the real rate of interest.
Earlier comparative static analyses apply.
.
1
r1)1(
p?
r
Valuing Securities （ 证券）
A financial security （ 金融证券） 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 （ 债券）
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 （ 面值） $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 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?
Valuing Bonds
PV?
$100,$100,
( )
$100,
( )
$614,
000
1 0 1
000
1 0 1
000
1 0 1
457
2 10
is the actual (present) value of the prize.
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
,
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
PV xr$1000 $10,.0 1 000
