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Economics 514
Macroeconomic Analysis
Final Exam
December 16, 2004
1. (15 points) There is an economy with a short-run aggregate supply curve, so that
when output is above long run potential, inflation will be above the inflation plans
of the corporate sector, πt – π = θ∙ (qt – q) A government likes price stability
and wants output to be near some target level, q* which is greater than the
potential output, q. The government chooses an inflation policy function πP to
minimize min a×( πP)2 + b×(qt –q* )2 subject to πP – π = θ∙ (qt – q).
a. Draw the first order condition that describes inflation policy, πP as a
linear function of the inflation plans of the corporate sector π . Explain
the intercept and the slope of the line.
πP
π
b. Discuss the equilibrium inflation rate when the corporate sector has
rational expectations. Why is it difficult to build credibility for a zero
inflation policy? [Note: I do not expect an explicit solution. I am
interested in the intuitive explanation using graphs and words]
Under rational expectations, planned inflation must equal inflation policy. This means
that the corporate sector must expect that the government will raise inflation policy to
that point where inflation becomes so costly that they wouldn’t want to push it up farther
even to achieve their high output goal. Any lower expectations would lead to systematic
errors in under-predicting inflation by the corporate sector. Since inflation policy has
little effect on output, the government might like lower inflation. But the government has
a difficult time achieving credibility for a low inflation policy, because if the public ever
did expect low inflation, the government would no longer want to follow a low inflation
policy.
2. (25 points) In an economy, the demand for goods is set by Mt ∙Vt = Pt ∙Qt . In
this economy, money supply is always constant, so we can write the demand
curve with actual inflation as a function of velocity growth, νt, and the output gap,
1) πt = νt – (qt – q).
On the supply side, firms set an inflation plan, π, actual inflation increases as a
function of costs which are determined by the output gap and an oil price shock,
φt.
2) πt = π + θ∙ (qt – q) + φt
Finally inflation plans are set according to rational expectations.
3) π = REt-1[πt ]
Solve for the effect of demand shocks (νt) and supply shocks (φt). Hint: Follow
the following steps. i. Solve for π remembering that REt-1[  t ] = 0; ii. Plug this
formula into equation 2) and solve for qt and πt.
a. Assume that there are no oil shocks, φt = 0. Velocity shocks are a random
walk, i.e. they are a sum of yesterdays output and an unpredictable shock.
4) νt = νt-1 +  tV . REt-1[νt] = νt-1
Solve for qt as a function of q, νt-1 , and
 tV .
First equation 2 along with model consistent expectations suggests REt-1[πt
]= REt-1[π ] + θ∙ (REt-1[qt] – q) implies REt-1[πt ]= REt-1[πt ]+ θ∙
(REt-1[qt] – q) implies REt-1[qt] = q. Equation 1 and model
consistent expectations implies REt-1[πt ]= REt-1[νt ]-( REt-1[qt] -q)
= REt-1[νt] = νt-1 from equation 4. Substitute into equation 2) to get
πt = = νt-1 + θ∙ (qt – q) . Combine with equation 1 to get νt-1 + θ∙ (qt
– q) = νt – (qt – q) which implies (1+ θ)∙ (qt – q)= νt - νt-1 which
implies
 tV
qt  q 
1
b. Assume that there are no demand shocks, νt = 0. Oil shocks follow a
random walk.
5) φt = φt-1 +  tO . REt-1[φt] = φt-1
Solve for qt as a function of q, φt-1, and
 tO .
From equation 1, we write πt = – (qt – q). So, REt-1[πt ]= -RE[(qt – q)]
From equation2), we get πt = π + θ∙ (qt – q) + φt, so
REt-1[πt ]= REt-1[πt ]+ θ∙RE[(qt – q)]+ RE[ φt] implying
RE[(qt – q)] = - φt-1/ θ = REt-1[πt ]
We then write πt = – (qt – q) = - φt-1/ θ + θ∙ (qt – q) + φt-1 +  tO .
This means that qt  q 
 tO
 1

t 1 .
1    1   
c. Do supply or demand shocks have persistent effects on output under
rational expectations?
Only supply shocks have persistent effects.
3. (25 points) Construct a small model of the economy that includes A) a Planned
expenditure curve; B) a Taylor rule; and C) an expectations augmented Philips
curve
A. The planned expenditure curve has two parts. The first part sets domestic
demand as a negative function of the real interest rate and an exogenous demand
shock, αt. The second part includes net exports.
yt  t  b  rt  nxt
Net exports are a function of the real exchange rate, st. The real exchange rate
adjusts so that net exports are equal to outward, capital flows that are
themselves a negative function of the gap between foreign and domestic interest
rates.
nxt  f  (st 1)  cft  g  (r f  rt )
So the expenditure curve can be written as
yt  t  b  rt  g  (r f  rt )  t  g  r f  (b  g )rt
B. The Taylor rule sets the real interest rate as an increasing function of the
inflation rate
rt  r  d  t
C.
The SRAS curve sets inflation acceleration as a function of the output gap.
 t   t 1    yt  y 
Calibrate the parameters of the model equal to r = .1, rf = .1, q = 1, b = 2
, d = .5, f = 2, g = 2 and θ = .5.
 t 1 = 1.2 and that at time t-1, the economy is operating at
potential output, i.e.  t 1   t 2 and yt 1  y  1 . Solve for the inflation
a. Assume that
rate, real interest rate, and real exchange rate in this economy.
Combine the expenditure curve and the Taylor rule,
yt  t  g  r f  (b  g )rt  t  g  r f  (b  g )r  d  (b  g ) t to get the
aggregate demand curve. Fill in the numbers to get
yt  t  .2  2 t
We can write
If  t 1 = 1.2 and yt 1  y  1 then 1  1.2  .2  2 t 1   t 1  0 . Then
rt 1  r  d  0  r  .1 . Then nxt 1  g  (.1  .1)  0 and
0  f  ( st 1  1)  st 1  1
b. Now, assume that at time t, an increase in fiscal spending leads to a shift
out in domestic demand and  t = 1.5. Solve for the new level of output,
real interest rate, inflation, and real exchange rate. Does the increase in
government spending cause an increase or decrease in net exports?
If
 t 1  0 then the supply curve is  t  .5  yt 1  .5 yt  .5 , so solving for
yt we write
Consider an alternate, extremely globalized economy with very strong
yt  t  .2  2 t  1.5  .2  2  .5  yt 1  2.3  yt
 yt  1.15   t  .075  rt  .1375 
nxt  g  (r f  rt )  .075  st  .9625
c. capital movements, g = 10. Now, solve for the increase in output, qt that
occurs when demand shifts from  t 1 = 1.2 to  t = 1.5. Is a globalized
economy, more or less stable in terms of output?
Not covered this year.
Midterm 2
Economics 514
Macroeconomic Analysis
2. Rational Expectations and Persistent Shocks
Demand follows the quantity theory of money
t  vt  yt  yt   t
where mt is (logged) money supply, vt is velocity, pt is the price level, and qt is the
output level. We will assume a fixed money supply , so mt = m. The supply curve is
given by

 t   te   yt  y t

e
t
where p is the expectation of the price level formed with the information available
at time t-1 and yt is the potential output level. Potential output is given by a
constant, q, plus random technology, xt.
yt  q  xt
Both supply shocks (xt) and demand shocks (vt) follow persistent movements.
vt   vt 1   tD
where the innovation terms,  tD
xt   xt 1   tS
and  tS are unpredictable, so
vte   vt 1
xte   xt 1
where vte is the expectation of the time t velocity formed at time t-1 and xte is the
expectation of the time t technology formed at time t-1. Assume that expectations
are formed rationally. Normalize  t =q = 0. Solve for the expectation of the output
level and the price level as a function of vt-1, xt-1, Solve for the actual output and
price level as a function of vt-1 and xt-1  tD and  tS .
Output can deviate from potential output only if actual prices to differ from expected
prices. As we cannot rationally expect the actual prices will be at some different level
from expectation, expected output must be equal to potential output. After the
e
normalization, the expectation of potential output is equal to y t  xte   xt 1 . Given a
constant money stock, the price level is equal to velocity minus output. Thus, the expecte
price level equals expected velocity minus expected output.  te   vt 1   xt 1 .
Substituting this into the supply curve we get
 t   vt 1   xt 1    qt   xt 1   tS 
The equation for the price level is  t  vt  yt   vt 1   tD  yt Substitute this into the
 vt 1   tD  yt   vt 1   xt 1    yt   xt 1   tS  
 tD   yt   xt 1     yt   xt 1   tS  
supply equation to solve for qt.
yt   xt 1 
 t   vt 1

1

1
 tS 


1 D
t 
1
 tD   xt 1 

1


 tS 
3. Baumol-Tobin Model
An econometrician estimates a version of the Baumol Tobin Model of money
demand. Her results suggest that velocity can be written as a function of the nominal
interest rate Vt  14 i . Assume the real interest rate is equal to the real output growth
rate. What is the seignorage maximizing growth rate of the money supply? What
percentage of output can possibly be collected as seignorage?
In the long run, if the real interest rate is equal to the real growth rate, the
nominal interest rate is equal to the money growth rate. Seignorage is equal to
S ( gM ) 
gM
gM M
gM Q
gM Q


4

4
Q
1 g M P 1 g M V
1 g M i
1 g M
1
S '( g M )  0  2
gM
1 g M
4
gM
1  g 
M
2

1
gM

 g M  1  S (1)  2Q
2 1 g M
Practice Problems
4. Rational Expectations and Monetary Policy Shocks
Demand follows the quantity theory of money
t  t   yt  y    t
where mt is (logged) money supply, vt is velocity, pt is the price level, and qt is the
output level. The supply curve is given by
 t  REt 1  t     yt  y 
Assume a constant potential output. To simplify algebra, normalize potential output, q
= 0. Monetary policy follows a random walk.
a. Monetary policy follows a random walk so that the growth rate of money is
unpredictable white noise: t   t . This implies that te  REt 1 ( t )  0 . Assume
that velocity is constant,  t =0. Calculate the expected price level. Calculate actual
levels of output as a function of εt.
Velocity is constant vt = v. So expected prices are
 t  t  v  yt   tE  Et 1  t   Et 1  t  v  yt  
Et 1  t   Et 1  t   v  Et 1  yt   t 1  v  y t
Actual output is
yt  y t 
1
  v  y  v  y    (  { y  y })



1
t
t
t
t
t
t
1
1
 yt  y t    t 
t
1
1


1

b. Assume that velocity follows the process,  t   t 1   tv where ρ < 1. This
implies that  te  REt 1 ( t )   t 1 . Assume that monetary policy systematically
targets velocity, t    t 1  te    t 1 . Calculate output and prices as
functions of  t 1 and  tv . What effect does the choice of λhave on output?
It is still the case that expected output is potential output. Thus, expected prices are
Et 1  t   Et 1  t   Et 1 vt   Et 1  yt   (1   )vt 1  y t
We can also calculate that actual prices are  t  t  vt  yt  (1   )vt 1   t  yt
We can then calculate output
yt  y t 
as
 (1   )v

1
t 1

  t  yt  (1   )vt 1   t  y t 
1

( t  {qt  y t })
1
What effect
1
 yt  y t    t   t  (1   )vt 1 
 t  yt
1
1
1
1


does the choice of λhave on output?The parameter of systematic monetary policy
has no effect on output because it is automatically incorporated into prices.
5. Okun’s Law and Time Consistency
Each % point increase in output above potential output results in θ% extra inflation.
 t  REt 1  t     yt  y 
Each % point increase in output above potential output reduces unemployment below
the natural rate by ψ%.


urt  ur NR   qt  q   t  REt 1  t   

urt  ur NR 


The government is interested in low inflation and pushing unemployment toward
some target level ur*. The government sets inflation policy to minimize
2
2
a  t   b  urt  ur *
subject to  t  REt 1  t   

ur  ur NR  taking expectations as given.
 t
a. Calculate the optimal inflation policy, πP, as a function of ur*, urNR, and
REt 1  t  .
The optimal policy design is to minimize



2
2
min a  t   b  urt  ur *    REt 1  t  
urt  ur NR   t 



FOC



a P
  urt  ur * 


b


 P  REt 1  t     urt  ur NR    urt  ur *  ur * ur NR 


2b  urt  ur *  
2a P   ,

 P  REt 1  t   


P 
1
a
b2
2
ur


NR

a  P

  ur * ur NR 

b



 ur * 

1
REt 1  t 
a 2
1
b2
b. What will equilibrium inflation and unemployment be when   REt 1  t    P
Rational expectations suggests that when inflation is set according to a predictable
policy, that policy will be incorporated into expectations. Thus, inflation will not push
output above potential output. So unemployment will equal the natural rate. Then
equilibrium inflation is


 
P
1
a
b2
2
ur
NR


ur NR  ur *
1
P
P
.
 ur * 
  
a
a 2
1
b
b2

c. If the government can choose a target unemployment rate, ur*, in order to have
zero inflation, what would it be?
Whenever the government has a bias toward unemployment below the natural rate,
inflation policy will require positive inflation. Therefore if ur* = urNR, then
equilibrium inflation is zero.
6. Adaptive Expectations and Dynamics
Construct a small model of the economy that includes A) a Planned expenditure
curve; B) a Taylor rule; and C) an expectations augmented Philips curve
A. The planned expenditure curve sets demand as a negative function of the real
interest rate and an exogenous demand shock, αt.
yt   t  b  rt
B. The Taylor rule sets the real interest rate as an increasing function of the
inflation rate
rt  r  d   t
C. The SRAS curve sets inflation acceleration as a function of the output gap.
 t   t 1    yt  y 
Calibrate the parameters of the model equal to r = .1, q = 1, b = .5 , d = .5 and θ = .5.
We might first solve these three equations for qt , rt , and πt as a function of exogenous
variables. First combine the planned expenditure curve and the Taylor rule to develop
the AD curve. yt  t  b  r  d   t   t  b  r  b  d   t . Combine the SRAS
 t   t 1    yt  y  with the AD curve to solve for output.


yt   t  b  r  b  d   t   t  b  r  b  d   t 1   yt  y 


1  b  d    yt   t  b  d   y  b  r  b  d  t 1
 t  b  d    y  b  r  b  d  t 1
1  b  d  
Then plugging into an inverted expenditure curve
 t  d   y  r  d
d    t  d   y  r  d t 1
t 1


t
t
1
Then
rt 

yt 
 b

b
b
b
1  b  d  
1  b  d  
   t    y  b   r   t 1
using the inverted Taylor rule  t   r  1 rt 
.
d
d
1  b  d  
For part a)-e), we will keep to r = .1, y = 1, b = .5, and θ = .5 constant. For a)-d) we
  ..075  .25 t 1
keep d = .5 . For simplicity, fill in these numbers. yt  t
amd
1.125
.25  t  .15  .5 t 1
.5  t  .525   t 1
and  t 
rt 
1.125
1.125
yt 
a. Assume that at time t-1, the economy was in a steady state where  t 1   t 2 and
 t 1 =1.10. Solve for output, yt-1, the interest rate, rt-1, and inflation πt-1.
.5  t 1  .525   t 2
Start with  t 1 
. If  t 1   t 2 and  t 1 =1.10. then
1.125
.025   t 1
.025
1.175  .25  .2
 t 1 
  t 1 
 .2   t 2 . Then qt 1 
 1  q . Then
.125
1.125
1.125
.225
rt 1 
 .2
1.125
b. Assume that unexpectedly  t =1.10 drops to 1.05. Using the answer from 3a for
πt-1, solve for the new real interest rate output, qt, the interest rate, rt, and inflation
πt.
c. Using πt from 3b, solve for yt+1 and πt+1. Using πt+1 solve for yt+2 and πt+2. Using
πt+2 solve for qt+3 and πt+3. Graph the dynamic response of inflation and output to
this demand shock.
d. In the far future, inflation will again reach a constant level. Solve for the constant
level of inflation when  t =1.05.
Using  t = 1.05, we can write the inflation equation function as a dynamic equation.
8
 t   t 1 . The output and real interest rate functions are written as equations in
9
1.125  .25 t 1
.1125  .5 t 1
2
4
 t 1 . as qt 
 1   t 1 and rt 
 .1   t 1 .
9
9
1.125
1.125
I use Excel to solve the dynamic behavior of these equations over time.
π
t-1
t
t+1
t+2
t+3
t+4
t+5
t+6
t+7
t+8
t+9
t+10
t+∞
0.2 t-1
0.177778 t
0.158025 t+1
0.140466 t+2
0.124859 t+3
0.110986 t+4
0.098654 t+5
0.087692 t+6
0.077949 t+7
0.069288 t+8
0.061589 t+9
0.054746 t+10
0 t+∞
q
r
1 t-1
0.95555556 t
0.96049383 t+1
0.9648834 t+2
0.96878525 t+3
0.97225355 t+4
0.97533649 t+5
0.97807688 t+6
0.98051278 t+7
0.98267803 t+8
0.98460269 t+9
0.9863135 t+10
1 t+∞
0.2
0.188889
0.179012
0.170233
0.16243
0.155493
0.149327
0.143846
0.138974
0.134644
0.130795
0.127373
0.1
Those equations will reach their steady state at infinity with π = 0, q = q = 1, r = .1.
e. Re solve 3a and 3b for qt-1 and qt under the assumption that the central bank sets
an aggressive response to inflation, d = 2. How does an aggressive response to
inflation affect the response of output to a demand shock? Explain
Fill in the new numbers. qt 
 t  .45   t 1
1.5
, rt 
 t  .9  2 t 1
1.5
and
.5   t  .525   t 1
. When  t 1 = 1.10, and  t 1   t 2 then
1.5
.5 1.1  .525   t 1
 t 1 
 .05 and qt-1 = q =1 and
1.5
  .45   t 1 1.05  .45  .05 1.45
qt  t


 0.966666667 . Output doesn’t declince
1.5
1.5
1.5
as much and the output gap is not as large. When demand shifts down, inflation goes
down. When the inflation rate goes down, under the second monetary policy the
t 
central bank reduces interest rates sharply which offsets some of the drop in demand.
Weak
Stance
d=2
π
SRAS
Strong
Stance
d=2
Q
q
4. (20 points) A government starts time 0 with outstanding financial wealth equal to
(1+r) B-1. In the long run, government debt (i.e. negative financial wealth) must
equal the present value of primary surpluses, as the government must collect more
in future taxes to repay the debt.
(1  r ) B1  TAX 0  G0 
(TAX1  G1 ) (TAX 2  G2 ) (TAX 3  G3 )


 ....
2
3
1  r 
1  r 
1  r 
The ratio of outstanding debt to GDP is 90%. If the real interest rate is 21% (r =
.21) and the GDP growth rate is 10%, go = .1, solve for a permanent level of
primary surpluses to GDP, psy, which will keep the debt to GDP ratio constant at
90%.
Hint:

(1  r ) B1
1 g
 .9  psy [1 
Q0
1  r 
Q
1 g   1 g 

2
Q
1  r 
2
3
Q
1  r 
3
 ....