Advanced Macroeconomics 5th Edition By David Romer © 2019 Solution Manual

[email protected]

for the Facebook page click here 

for more books  for  ( Test Bank and Solution Manual) click here

sample free

SOLUTIONS TO CHAPTER 1
Problem 1.1
(a) Since the growth rate of a variable equals the time derivative of its log, as shown by equation (1.10)
in the text, we can write
(1)
 ( )  
( )
Z t ln ( ) ln ( ) ( )
Z t
d Z t
dt
d X t Y t
dt
  .
Since the log of the product of two variables equals the sum of their logs, we have
(2)
 ( )  
( )
Z t ln ( ) ln ( ) ln ( ) ln ( )
Z t
d X t Y t
dt
d X t
dt
d Y t
dt


  ,
or simply
(3)
 ( )
( )
 ( )
( )
 ( )
( )
Z t
Z t
X t
X t
Y t
Y t
  .
(b) Again, since the growth rate of a variable equals the time derivative of its log, we can write
(4)
 ( )  
( )
Z t ln ( ) ln ( ) ( )
Z t
d Z t
dt
d X t Y t
dt
  .
Since the log of the ratio of two variables equals the difference in their logs, we have
(5)
 ( )  
( )
Z t ln ( ) ln ( ) ln ( ) ln ( )
Z t
d X t Y t
dt
d X t
dt
d Y t
dt


  ,
or simply
(6)
 ( )
( )
 ( )
( )
 ( )
( )
Z t
Z t
X t
X t
Y t
Y t
  .
(c) We have
(7)
 ( )
( )
Z t ln ( ) ln[ ( ) ]
Z t
d Z t
dt
d X t
dt
 

.
Using the fact that ln[X(t) ] = lnX(t), we have
(8)
 ( )  
( )
ln ( ) ln ( )  ( )
( )
Z t
Z t
d X t
dt
d X t
dt
X t
X t
  

  ,
where we have used the fact that  is a constant.
Problem 1.2
(a) Using the information provided in the question,
the path of the growth rate of X, X (t) X(t), is
depicted in the figure at right.
From time 0 to time t1 , the growth rate of X is
constant and equal to a > 0. At time t1 , the growth
rate of X drops to 0. From time t1 to time t2 , the
growth rate of X rises gradually from 0 to a. Note that
we have made the assumption that X (t) X(t) rises at
a constant rate from t1 to t2 . Finally, after time t2 , the
growth rate of X is constant and equal to a again.
 ( )
( )
X t
X t
a
0 t1 t2 time