Transcript Chapter 14: Expectations: The Basic Tools

CHAPTER
14
Expectations:
The Basic Tools
Prepared by:
Fernando Quijano and Yvonn Quijano
And Modified by Gabriel Martinez
Nominal Versus Real
14-1
Interest Rates
 Nominal interest rates
are interest rates
expressed in terms of
dollars.
 Real interest rates are
interest rates expressed in
terms of a basket of
goods.
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Nominal Versus Real
Interest Rates
Definition and
Derivation of the Real
Interest Rate
it = nominal interest rate for year t.
rt = real interest rate for year t.
(1+ it): Lending one dollar this
year yields (1+ it) dollars next
year. Alternatively, borrowing one
dollar this year implies paying
back (1+ it) dollars next year.
Derivation
Pt = price this year.
Pet+1= expected price next year.
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Nominal Versus Real
Interest Rates
Pt
We know 1  rt  (1  it ) e
P t 1
Pt
1
Now, e 
P t 1 (1   e t )
 et
P e t 1  Pt

Pt
1  it
Consequently, (1  rt ) 
e , which is the exact relation
1  t
between nominal interest rates, real interest rates, and
inflation.
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Nominal Versus Real
Interest Rates
If the nominal interest rate and the expected rate of
inflation are not too large, a simpler expression is:
rt  it   e t
 The real interest rate is (approximately) equal to the
nominal interest rate minus the expected rate of inflation.
 Notice that higher rates of expected inflation reduce the
real interest rate, given the nominal interest rate.
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Nominal and Real Interest Rates
in the United States Since 1978
Nominal and Real
One-Year T-bill
Rates in the United
States, 1978-2001
Nominal
While the nominal
interest rate has
declined considerably
since the early 1980s,
the real interest rate is
actually higher in 2001
than it was then.
 Despite the large decline in nominal interest rates,
borrowing is actually more expensive in 2001 than it
was in 1981.
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Nominal Versus Real
Interest Rates
 Expected and actual inflation may differ.
 r = i –  is called the ex-post real interest
rate.
 r = i – e is called the ex-ante real interest
rate.
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14-2
Expected Present
Discounted Values
Computing Present
Discounted Values
 The expected present discounted value of a
sequence of future payments is today’s
value of this expected sequence of
payments.
 The term 1/(1+it) is called the discount
factor, and the one-year nominal interest
rate, it, it is often called the discount rate.
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Computing Expected Present
Discounted Values
(a) One dollar this year is
worth 1+it dollars next
year.
(b) If you lend 1/(1+it) dollars
this year, you will receive
1
(1  it )  1
(1  it
dollar next year.
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Computing Expected Present
Discounted Values
(c) One dollar is worth
(1  it )(1  it 1 )
dollars two years from
now.
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(d) The present discounted
value of a dollar two
years from today is equal
to
1
(1  it )(1  it 1 )
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A General Formula
 The present discounted value of a sequence of
payments, or value in today’s dollars equals:
1
1
$Vt  $zt 
$ zt 1 
$ zt  2    
(1  it )
(1  it )(1  it 1 )
 When future payments or interest rates are
uncertain, then:
1
1
e
e
$Vt  $zt 
$ z t 1 
$
z
t 2    
e
(1  it )
(1  it )(1  i t 1 )
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Using Present Values: Examples
1
1
e
e
$Vt  $zt 
$ z t 1 
$
z
t 2    
e
(1  it )
(1  it )(1  i t 1 )
 This formula has these implications:
– Present value depends positively on today’s
actual payment and expected future payments.
– Present value depends negatively on current
and expected future interest rates.
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Constant Interest Rates
 To focus on the effects of the sequence of
payments on the present value, assume that
interest rates are expected to be constant
over time, then:
1
1
e
e
$Vt  $zt 
$ z t 1 
$
z
t 2    
2
(1  i )
(1  i )
– PDV is a weighted sum of the payments.
– The weight of future payments declines
geometrically over time.
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Constant Interest Rates
and Payments
 When the sequence of payments is equal—call
them $z, the present value formula simplifies to:


1
1
$Vt  $zt 1 
  
n 1 
(
1

i
)
(
1

i
)


 The terms in the expression in brackets represent
a geometric series. Computing the sum of the
series, we get:
1  [1 / (1  i ) n ]
$Vt  $zt
1  [1 / (1  i )]
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Constant Interest Rates and
Payments, Going on Forever
 Assuming that
– Interest rates are constant,
– payments are constant and start next year and
– payments go on forever (this is a “consol”), then:
$Vt 
1
1
1 
1

$ zt 
$
z





1





 $z
(1  i )
(1  i )  (1  i )
(1  i ) 2

 Using a property of geometric sums, the present
1
1
value formula above is:
$V 
$z
t
 Which simplifies to:
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1  i [(1  (1 / (1  i ))]
$z
$Vt 
i
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Constant Interest Rates and
Payments, Going on Forever
$z
$Vt 
i
 It’s quite clear from this formula that
– Higher payments increase the value of the
security.
– Higher interest rates reduce the value of the
security.
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Nominal Versus Real Interest Rates,
and Present Values
1
1
e
e
$Vt  $zt 
$ z t 1 
$
z
t 2    
e
(1  it )
(1  it )(1  i t 1 )
 Replacing nominal interest with real interest
rates to obtain the present value of a sequence
of real payments, we get:
1
1
e
e
Vt  $zt 
$zt 
$
z
t  2  ...
e
(1 rt )
(1 rt )(1 rt 1)
 Which can be simplified to:
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$Vt
 Vt
Pt
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Nominal Versus Real Interest Rates,
and Present Values
 Why have two formulae?
– Bonds, for example, promise a nominal
payment, so we need the nominal PDV
equation.
– Expectations about Income, and Consumption
and Investment decisions, depend on real
variables (why bother with the extra uncertainty
about inflation). So we use the real PDV
equation.
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Interest Rates and Investment
1
1
e
e
Vt  $zt 
$zt 
$
z
t  2  ...
e
(1 rt )
(1 rt )(1 rt 1)
 Investment is the purchase of a capital
good.
 If Vt is the value of a capital good, the
amount of investment will depend
 … positively on zt
– I.e., the expected income from investment,
determined by sales and Yt.
Y  C(Y  T )  I (Y , r ) 
 … and negatively on rt
+
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–
Olivier Blanchard
Nominal and Real Interest
Rates, and the IS-LM Model
14-3
 When deciding how much investment to
undertake, firms care about real interest
rates.
 They compare the (real) marginal product of
capital with the real interest rate.
 Then, the IS relation must read:
Y  C(Y  T )  I (Y , r )  G
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Nominal and Real Interest
Rates, and the IS-LM Model
 The interest rate directly affected by monetary
policy—the one that enters the LM relation—is
the nominal interest rate, then:
M
 YL(i )
P
 Recall Md is affected by the opportunity cost
of holding money, which includes inflation.
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Nominal and Real Interest
Rates, and the IS-LM Model
Y  C(Y  T )  I (Y , r )  G
r  i 
e
Y  C(Y T )  I (Y , i   )  G
e
M
 YL(i )
P
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Money Growth, Inflation, and
Nominal and Real Interest Rates
14-4
 This section explains the following
assertions:
After a Monetary Expansion
Nominal interest Lower in the
Higher in the
rates, i
short run
medium run
Real Interest
Rates, r
Lower in the
short run
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Unchanged in
the medium run
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Revisiting the IS-LM Model
Equilibrium Output
and Interest Rates
The equilibrium level of
output and the
equilibrium nominal
interest rate are given
by the intersection of the
IS curve and the LM
curve. The real interest
rate equals the nominal
interest rate minus
expected inflation.
If r  i   e  r  i   e
If  e is constant  e  0  r  i
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If i falls, r falls, I rises, and
equilibrium output rises in the goods
market: the IS curve slopes down.
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Nominal and Real Interest Rates
in the Short Run
The Short-run Effects
of an Increase in
Money Growth
An increase in money
growth increases the real
money stock in the short
run (because P does not
adjust in the short run).
This increase in real
money leads to an
increase in output and a
decrease in both the
nominal and the real
interest rate.
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Notice we are assuming that e does
not change in the short run: this means
the IS curve does not shift.
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Nominal and Real Interest Rates
in the Medium Run
 In the medium run, Y  Yn , then IS becomes:
Yn  C(Yn  T )  I (Yn , r )  G
 The relation between the nominal interest rate
and the real interest rate is:
i  r  e
 In the medium run, the real interest rate
equals the natural interest rate, rn, then:
i  rn   e
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rn is the interest rate
that will make spending
equal to the natural
level of output.
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Nominal and Real Interest Rates
in the Medium Run
 In the medium run, e=, expected inflation is
equal to actual inflation, so
i  rn  
 Recall =gm-gy
 Higher output growth leads to more money
demand. If gm>gy, there must be inflation.
 This implies that =gm.
 Hence, in the medium run:
i  rn  gm
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Nominal and Real Interest Rates
in the Medium Run
i  rn    rn  gm
 In the medium run, the nominal interest rate
increases one for one with inflation. This
result is known as the Fisher effect, or the
Fisher Hypothesis.
– For example, an increase in nominal money
growth of 10% is eventually reflected by a 10%
increase in the rate of inflation, a 10% increase
in the nominal interest rate, and no change in
the real interest rate.
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From the Short Run to
the Medium Run
 In the short run, higher gm leads to lower nominal
interest rates, i.
– Because inflation is pre-determined, real interest rates
fall in the short run.
 Low real interest rates lead to high output, low
unemployment, and eventually high inflation.
 High inflation leads to a decrease in the real
money stock, and to an increase in nominal
interest rates.
 In the long run, higher gm leads to higher nominal
interest rates , i.
– But r rises back to rn.
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From the Short Run to
the Medium Run
 Suppose gm’>gm. Using the Phillips Curve,
r  rn  Y  Yn  u  un   
 Over time, inflation keeps increasing, and
i  rn  gm
keeps increasing…
   Eventually   g' m  ( g' m   )  0  i 
M
P
 When  > gm, real money growth turns
negative, which raises nominal interest
rates.
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From the Short Run to
the Medium Run
 What happens to r? We know two things:
– Initially, r fell.
 Immediately after the monetary expansion, nominal rates fell
while inflation was predetermined.
– In the medium run, r = rn.
 So we know that r must rise back to rn.
 How? When  > gm, real money growth turns
negative, which raises nominal interest rates …
and, at a given level of , it raises real interest
rates.
 As r rises back to rn, I and Y fall back to Yn.
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From the Short Run to
the Medium Run
– Notice inflation must overshoot for r to rise to rn.
 So we know that r goes back up to rn. What
happens to i?
– It must be equal to rn plus the new, higher level
of inflation.
 In the medium run,
r  rn
Y  Yn
u  un
  gm
i  rn  gm
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From the Short Run to
the Medium Run
The Adjustment of the Real
and the Nominal Interest Rate
to an Increase in Money
Growth
An increase in money
growth leads initially to a
decrease in both the real
and the nominal interest
rate. Over time, the real
interest rate returns to its
initial value. The nominal
interest rate converges to a
new higher value, equal to
the initial value plus the
increase in money growth.
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Evidence on the Fisher
Hypothesis
 To see if increases in inflation lead to onefor-one increases in nominal interest rates,
economists look at:
– Nominal interest rates and inflation across
countries. The evidence of the early 1990s
finds substantial support for the Fisher
hypothesis.
– Swings in inflation, which should eventually be
reflected in similar swings in the nominal
interest rate. Again, the data appears to fit the
hypothesis quite well.
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Evidence on the Fisher
Hypothesis
Roughly half of the
points are above the
line, the other half
below. This is
evidence that a 1%
increase in inflation
should be reflected in a
1% increase in the
nominal interest rate.
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Nominal
Nominal Interest Rates
and Inflation Across Latin
America in the Early
1990s
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Evidence on the Fisher
Hypothesis
The Three-Month
Treasury Bill Rate and
Inflation, 1927-2000
The increase in
inflation from the early
1960s to the early
1980s was associated
with an increase in the
nominal interest rate.
The decrease in
inflation since the mid1980s has been
associated with a
decrease in the
nominal interest rate.
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Olivier Blanchard