Please write your answers on this exam paper.
Name________________
Student ID ________________
Economics 514
Macroeconomic Analysis
Midterm Exam 3
December 14, 2011
Write all answers on the white exam paper. Do not turn in the blue books. 20 points each.
1. A price taking company produces using rented capital. The capital rental rate is
Rt
= .5. The firm must also pay a tax, measured in goods, tw, for each quantity
Pt
of capital it uses so that total tax revenue is tw∙Kt. The firms production function
is of the form Yt  1.5 K t  K t 2 . We can write the optimal level of capital as a
linear function of the tax, tw, Kt *  A  B  tw .
a. Solve for A & B.
FOC : MPKt  1.5  2  Kt  R  tw  .5  tw
P
 K *  .5  .5tw
A = .5; B = -.5
b. Calculate the tax revenue when tw = 0, .5, and 1.
tw∙K* = .5tw-.5tw2
0
0
.5
.125
1
0
c. Solve for the level of the tax that generates the maximum level of revenue.
Max .5tw-.5tw2 FOC: .5 = tw
2. The demand for real balances is given by the Baumol-Tobin theory so that we can
write
Mt
Y
 t
Pt
it
The growth rate of output is always 3%, gY = .03. The real interest rate is always 9%, r=.09.
At time 0, Y0 = M0 = 100. The growth rate of money is set at a permanent level, gM .
Solve for the price level, P0, when gM = .1 and when gM= .03. Explain briefly, in
words, why the answer for gM = .1 is larger than for gM= 0.03.
P0  it
gM π
.1 .07
.03 0
i
.16
.09
P0
.4
.3
When money growth is high, inflation and interest rates are high. This means the cost
of holding cash is high so they are willing to make many trips to the bank. Money
circulates more quickly and prices will be higher.
Please write your answers on this exam paper.
3. Consider an economy in which expenditure is a negative function of the real
interest rate and an exogenous shock:
yt  t  d  rt
Where d = 1 where the demand shock is a completely unpredictable standard
normal variable Et 1 t   0 with a variance Et 1 t 2    2  1 . The central bank
sets the real interest rate as an increasing function of their measure of the inflation
rate rt  b   tFED . The central bank cannot measure current inflation perfectly.
Instead, the measure of inflation equals the true value plus some white noise
 tFED   t  t where Et 1 t   0 with a variance Et 1 t 2    2 .
Workers sign dollar wage contracts to keep their real wages fixed at an
equilibrium level. If prices rise faster than wages, then firms will increase output
above potential output, yt =0
yt  ( t  gtW )
Assume rational expectations, so gtW  Et 1  t  .
a. Assume that b = 1. Write the model consistent expectations of output and
inflation at time t-1.
Et 1  yt   Et 1 ( t  gtW )   Et 1 ( t  Et 1  t )   Et 1  t   Et 1  Et 1  t   0
yt  t  b  ( t  t )
0  Et 1  yt   Et 1 t  b  ( t  t )   Et 1 t   b  Et 1  t   b  Et 1 t   Et 1  t   0
b. Assume that b=1. Write actual output and inflation as a function of t and  t .
yt  ( t  gtW )   t
yt  t  b  ( t  t )  t  b  ( yt  t ) 
(1  b)  yt  t  b  t  yt 
yt 2 
1
 t  b  t    t
1 b
1
 t 2  b 2  t 2  2btt    t2
2
1  2b  b 
1  b 2  E[t 2 ]
1
2
2
2
E[ yt ] 
  E[ ]  b  E[t ] 
1  2b  b2  t
1  2b  b2 
2
c. Solve for the volatility of output under two different monetary policies: Insensitive)
b = 1; and Sensitive) b = 100. Examine the volatility of output E  yt 2  under two
scenarios which depend on how well the central bank measures current inflation. In
the first Clear scenario, the central bank observes inflation perfectly,
Et 1 t 2    2  0 ; in the Unclear scenario, central bank does not observe inflation
easily, Et 1 t 2    2  4 . Which monetary policy generates the lowest output
volatility in each scenario? Explain.
E  yt 2 
Et 1 t      0
Et 1 t 2    2  4
2
2
Insensitive) b = 1
.25
Sensitive) b = 100
1/10,201
5/4 = 1.25
40,001/10,201
1  b 2   2
2


E  yt  =
1  2b  b2 
Under the sensitive monetary policy
Please write your answers on this exam paper.
4. Consider an economy in which expenditure is a negative function of the real
interest rate:
yt  d  rt
Where d = 1. The central bank sets the real interest rate as an increasing function
of the inflation rate rt  b   t .
Workers sign dollar wage contracts to keep their real wages fixed at an
equilibrium level. If prices rise faster than wages, then firms will increase output
above potential output, yt
yt  yt  .5  ( t  gtW )
Assume rational expectations, so gtW  Et 1  t  . Assume that potential output
follows a random walk so yt  yt 1  t where Et 1 t   0
a. Assume that b = 1. Write the model consistent expectations of output and
inflation at time t-1 as a function of yt 1 .
b. Assume that b=1. Write actual output and inflation as a function of yt 1 and  t .
1
1
yt 
yt 1 ) 
b
b
 .5 
 .5 
1    yt  1    yt 1  t 
 b
 b
yt  yt 1  t  .5  (



t
t 
1 
 yt  yt 1 
  t    yt 1 

b 
 .5 
 .5  
1  
1  
 b
 b  

b  1   yt  yt 1 
t
 

,  t    yt 1  t 
1.5
1.5 

c. Solve for the level of b that would always set output equal to potential output, yt .
What would inflation be at this level of b?
If you set b = ∞, then y is potential output and inflation is always zero
Please write your answers on this exam paper.
5. Below is a picture of the Keynesian AS-AD model. Draw what happens to the
equilibrium if there was a shift outward in potential output.
yP
πt
AS
AS2
AD
yt
In the below graph, draw an AD curve that would insure that output was always equal
to potential output even after potential output changes.
yP
πt
AS
AD
AD
yt
What kind of monetary policy might generate such an AD curve?
b = ∞,