The aggregate expenditure curve represents the output gap as a linear function of the difference between the real interest rate and the long-term level,
Output Gap t a d
 r t r

where a t
is an expenditure shock and d is the interest sensitivity of demand. The monetary policy reaction function is r t r b 
  TGT
The supply curve is given by Output Gap t f 
  t
 t
E
where f is the inflation
. sensitivity of supply. Assume that d is 2 and f is 1. The inflation target is 2% (i.e.

TGT =.02) and the long-term interest rate is r = .04
. a.
Assume that inflation expectations are co-ordinated around the target,
 t
E   TGT
.
Write inflation and the output gap as a function of a t
. Assume a 1% positive demand shock, i.e. a t
= .01
. Solve inflation and the output gap when b = .1
and b = 1000 . Which policy results in a smaller shift in inflation? Which policy results in a smaller shift in the output gap? Demonstrate your results on a graph.
We combine the expenditure curve and the monetary policy curve to get the aggregate demand curve. Output Gap t a t d b 
  t
 TGT a t
2 b
  t

.02

Given coordinated expectations the supply curve is Output Gap t f 
  t
 TGT    t

.02

.
The equilibrium inflation sets supply equal to demand a t
2 b
  t

.02
   t

.02
  a t
{2 b 1}
  t

.02

 t

.02

{2 a t
1}

.02

{2
.01
1}
Output Gap t

{2
 t a t

1} {2
.01
1}
Output Gap b=.1 .02833 b=1000 .020005
.00833
.00005


 t
E
AD b=.1 q
P



AS
 TGT
AD b=1000 q

b.
Set a t
= 0, but assume that inflation expectations are suddenly shifted upward,
 t
E  
TGT  c t
.
Write inflation and the output gap as a function of c t
. Assume a 1% positive expectation shock, i.e. c t
= .01
. Solve inflation and the output gap when b = .1
and b = 1000 . Which policy results in a smaller shift in inflation? Which policy results in a smaller shift in the output gap?
The aggregate supply curve is Output Gap t f 
  t
 TGT    t

.02
  c t
. Then we could see an aggregate demand curve
Output Gap t d b
 
TGT 2 b
  t

.02

This could be solved for
  t

.02
 c t
2 b
  t

.02
   t

.02

2
Output Gap t

2 c t
1 2
2
 b
1 c t
 t
Output Gap c t
1 b=.1 .02833 .0016666 b=1000 .020005

.00999
AD b=.1
AS
 E
 TGT
+c
 E



 TGT
AD b=1000 q

Aggregate demand is given by: q t
  t
   r )
Output is given by q
ˆ t

1 f
 t
  t
E
(1.1)
(1.2) where

E is the growth rate of wages which is given by the expectation of the inflation t rate. Monetary policy is given by
Monetary policy shocks r t
  t
 b
 t

follow persistent AR(1) processes. t
 t
  t

1
  t
M where the innovation terms are white noise, E t

1
(
 t
M
)

0 , so
E t

1
   t

1
(1.3)
Consider that at time 0 there is a shock to monetary policy, raising interest rates temporarily,

M
0

0 but in all future periods
 t
M 
0 for all future periods t= 1,2,.3, …..
Also assume no demand shocks
 
0 Solve for the path of the output gap and the t inflation rate in period 0 through 2 as a function of

0
M , under two scenarios, A)
Adaptive expectations,

E
 t
 t

1
; , B) Rational expectations,
 t
E
Assume


1


E t
0.
Simplify by calibrating d = b = f = 1 and

=1/2 .

1
[
 t
];
A) Adaptive Expectations
π t
Period 0 1 2 q t
B) Rational Expectations
π t
Period 0 q t
1 2

q t
  r r t t
  
; q t t q t
  t
  t

1
;
  t
  t t
  t
  t

1
  t
 
1
2
 t

1
2
 t

1
; q
ˆ t



1
2
 t
 
1
2
 t

1

(1.4)
π
 t t q t
Rational Expectations
Period 0

0
M

1
2

0
M

1
2

0
M
1
1
2
 M
0

1
2

0
M

1
4

0
M
2
1
4

0
M

3
8

0
M

1
8

0
M q
ˆ t


 t
  t
E
(1.5)
ˆ t
 
; t t
   t

; q t t
  t t q t


 t

E t

1
[
 t
] ;
E t

1
[ q
ˆ t
]
  
E t

1
[
 t
]

E t

1
[
 t
]
 q t


 t
 
1
2
 t

1
 
 t

1
2
 t

1

1
2
 t

1

E t

1
[
 t
]

E t

1
[
 t
]

  t t
  t

 q t
{


1
2

 t

1
 

1
2 t
M
 t

1
4
 t

1

1
2


 t
M
}

1 { 1
2 2

1

2 t
M
 t

1
  t
M
}

1
4
 t

1
 
1
2
 t
M

1
2
 t

1
4
 t

1


1
2
 t

1

1 { 1
2 2
 t

1
  t
M
}

1
4
 t

1 q t
 t
π t
Period 0

0
M

1
2

0
M

1
2

0
M
1
1
2

M
0

1
2

0
M
0
2
1
4

0
M

1
4

0
M
0

Aggregate demand is given by: q t
  t
   t r ) (1.6)
Output is given by q
ˆ t
 f
1 
 t

E t

1


(1.7) where

E is the growth rate of wages which is given by the expectation of the inflation t rate. Monetary policy is given by r t
  b (
 t o  
TGT ) (1.8)
Where
 o
is the inflation observed by the central bank. Inflation is observed imperfectly t by the policy maker:
 t o   t
  t
.
Consider
 t
to be an error in measuring inflation.
Treat this as a white noise shock.
Examine the effect of monetary policy when the central bank makes mistakes in measuring inflation. Simplify by assuming f =d = 1 and
 t
= 0. Assume that the central bank overestimates inflation by 1% at time 0:

0
= .01. Consider the impact if b = .1
and b = 1000.
A) Rational Expectations
Period b = .1
π t  
11
 t q t  
11
 t
ˆ t d ( r t r ) d b (
 t o  
TGT
) q t

1 f

 t

E t

1

 
 t

E t

1

E t

1

E t

1 b=1000

1000
1001
  t

1000
1001
  t b (
 t
  t
)


E t

1
  b E t

1
(
 t
  t
)
 q t
  t b (
 t
  t
) (1 b )
 t b
 t
 t
  b
(1
 b )
 t q
ˆ t

Aggregate demand is given by:
ˆ t
 r t
N   t t
ˆ t
Output is given by
 t
  q
ˆ t
 
E t
  t

1

Monetary policy is given by
Monetary policy shocks r t
 r N   t
 b
 t
 t
follow persistent AR(1) processes.
 t
  t

1
  t
M where the innovation terms are white noise, E t

1
(
 t
M )

0 , so
E t

1
   t

1
Consider that at time 0 there is a shock to monetary policy, raising interest rates temporarily,

M
0

0 but in all future periods
Simplify by calibrating θ = 1 and

=1/2.
 t
M 
0 for all future periods t= 1,2,3,…

π t q t t
Period 0

2
2
0
M
4

0
M
4 b

 
4 b
0
M
1
1
2
2
2

0
M
2

2

4 b
4 b

0
M
0
M
2
1
2
2
4

0
M
1
4 b

0
M
1
 
2
4 b

0
M q t
  t
 b
 t
 t
    t
 b
 t

1
2 q t

2
  t
 b
 t

;
(1.9)
(1.10)
(1.11)
 t
  t

E t

 t

1

  t
ˆ t 2 t

2
2
  q
ˆ t q t

ˆ t
2
 t
2
 b
2
2
  q
ˆ t

2
   
4 b
  t
2
 
4 b q t
2
4
4 b

 t
2
 t
Output is given by
 t
  q
ˆ t
 
E t
  t

1

Monetary policy is given by
Monetary policy shocks r t
 r
N   t
 b
 t

follow persistent AR(1) processes. t
 t
  t

1
  t
M
(1.12)
(1.13)
(1.14)

A household lives for 20 periods. The real interest rate is 5%. The household’s income grows at a constant rate, g , in every period. The household begins with 1000 in financial wealth. We do not know the current income level, Y
0
. Assuming the permanent income hypothesis holds (i.e (1+r) ×β =1 ), we can write the current consumption in the
Keynesian form, C
0
= A
0
+ mpc×Y
0
where A
0 is proportional to financial wealth, A
0
= mpcw∙FW
0
.
a. Solve for mpcw.
The households financial and human wealth is equal to
W

F
0
 
0
Y
1
1
 r

Y
2

1
 r

2
Y
T

1
 r

T
With a constant income growth rate g, we can write Y t
= (1+g) t Y
0
.
W

F
0
Y
(1

1
 g Y r
0

(1


1
 g Y r
)

2
2
2
(1
 T g Y
T

1
 r

T
. Define z

(1
 g )
1
 r
. Then
W

F
0

Y
0
(1 z z
2   z
T
...
) . When z = 1 (i.e. g = r ), this is equal to
W

F
0

( T Y
0
Y
0
. When z < 1 , this is equal to W

F
0

1
 z
T

1
1
 z
The present value of consumption is C
0

1
C
1
 r

C
2

1
 r

2
C
T

1
 r

T
Y
0
. Assuming the permanent income hypothesis C t
= C
0
, C
0

C
0
1
 r

C
0

1
 r

2
C
0

1
 r

T
. Define x
C
C
0
0


1

1

1

1
 r x
T
, then the present value of consumption is

1
1

1
 x x x
T

1
. If the present value of consumption equals total wealth, then
W

1

1
 x x
T

1
F
0

1
1

 x x
T

1
1

1
 z
T

1 z
Y
0
C
0
∙ (1 + x + x 2
+ x
3
+ ….+ x
T
) =
1

. So if C = A + mpc Y , then A =
1
 x x
T

1 and mpc =
1

1
1

 x x
T

1

1
 x x
T

1
1

1
0.074282007
 z
T z

1
. When r = .05
, then x =
1
1.05
, if T = 20
, so if F
0
= 1000 , then A = 74.28200678
.
, then
When g = 0 , then x = z , so mpc = 1.
F
0 b.
Solve for mpc when g = .05
. Solve for mpc when the growth rate of income is zero ( g = 0 ). Explain the difference.
If r = g , then mpc =
1
 x
( T

1) = 1.559922142.
1
 x
T

1

A household lives for periods 0 through T = 19 and begins period 0 with financial wealth normalized to FW
0
= 1 and has income in any period, Y t
= 0. The household has a utility function of the form
U
 t
T 

0
( t
) ( t
)

C t
1
 1

1

1


1
Note that the subjective discount factor is set equal to β=1. The household’s wealth follows the following dynamic B
0
=FW
0
–C
0 for period 0 and B t
= (1+r)B t-1
–C t
for period t>0 . Therefore the household has a budget constraint of the form t
T 

0
(1
C t
 r ) t

FW
0 a.
Write the first order conditions of the household optimization problem. Show that we can write the Euler equation of the form C t+1
= (1+r)
γ
C t
. Solve for γ.
First order condition is
C t
 1
 t
 1


1

C t

1
  r )
 
C t b.
Since the above means that we can write C t
= [(1+r)
γ
] t
C
0
. Show that we can

 t
T 

0 write the household budget constraint in the form FW
0
= Г∙C
0 where the parameter
Г
is a function of 1+r and γ.
(1
(1

 r r
)
)
 t t


C
0


 t
T 

0
(1
 r )
 
1 t


C
0

FW
0 c.
Solve for Г when r = 0 . When r = 0, Г = T+1 so mpcw = 1/T+1 = 1/20 = .05
d.
Now, solve for
Г when r = .1 and
ψ = .5, 0, and 2
. What is the impact of a rise in the interest rate on the marginal propensity to consume out of wealth at the different intertemporal elasticity of substitutions.
Define x as x
 
(1
 r )
 
1
. Then we write,
 t
T 

0
  t


. When ψ = 1, x = 1. so Г =
20 and mpcw = 1/20 = .5. Under uuit intertemporal elasticity of substitution, the interest rate has no effect on consumption or mpcw.
Otherwise
 
1

1
 x
20 x