Solution Manual to
Romer's Advanced
Macroeconomics 5th
Edition.
, Solutions Manual to Romer's Advanced Macroeconomics 5th
Edition. Solution Manual David Romer
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
˙Z (t) d ln Z(t) d lnX(t)Y(t)
(1) .
Z(t) dt dt
Since the log of the product of two variables equals the sum of their logs, we have
˙Z (t) dln X(t) ln Y(t) d ln X(t) d ln Y(t)
(2) ,
Z(t) dt dt dt
or simply
˙Z (t) ˙X (t) ˙Y (t)
(3) .
Z(t) X(t) Y(t)
(b) Again, since the growth rate of a variable equals the time derivative of its log, we can write
ln Z(t) (4) ˙Z (t) d d lnX(t) Y(t).
Z(t) dt dt
Since the log of the ratio of two variables equals the difference in their logs, we have
˙Z (t) dln X(t) ln Y(t) d ln X(t) d ln Y(t)
(5) ,
Z(t) dt dt dt
or simply
˙Z (t) ˙X (t) ˙Y (t)
(6) .
Z(t) X(t) Y(t)
(c) We have
(7) ˙Z (t) d ln Z(t) d ln[X(t) ] .
Z(t) dt dt
Using the fact that ln[X(t) ] = lnX(t), we have
˙Z (t) d ln X(t) d ln X(t) ˙X (t)
, lOMoARcPSD|58847208
(8)
dt X(t) , Z(t) dt where we have used the fact that is a
constant.
Problem 1.2
(a) Using the information provided in the question, the
Ẋ(t)
path of the growth rate of X, ˙X (t) X(t), is depicted in X(t)
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 a
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.
Romer's Advanced
Macroeconomics 5th
Edition.
, Solutions Manual to Romer's Advanced Macroeconomics 5th
Edition. Solution Manual David Romer
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
˙Z (t) d ln Z(t) d lnX(t)Y(t)
(1) .
Z(t) dt dt
Since the log of the product of two variables equals the sum of their logs, we have
˙Z (t) dln X(t) ln Y(t) d ln X(t) d ln Y(t)
(2) ,
Z(t) dt dt dt
or simply
˙Z (t) ˙X (t) ˙Y (t)
(3) .
Z(t) X(t) Y(t)
(b) Again, since the growth rate of a variable equals the time derivative of its log, we can write
ln Z(t) (4) ˙Z (t) d d lnX(t) Y(t).
Z(t) dt dt
Since the log of the ratio of two variables equals the difference in their logs, we have
˙Z (t) dln X(t) ln Y(t) d ln X(t) d ln Y(t)
(5) ,
Z(t) dt dt dt
or simply
˙Z (t) ˙X (t) ˙Y (t)
(6) .
Z(t) X(t) Y(t)
(c) We have
(7) ˙Z (t) d ln Z(t) d ln[X(t) ] .
Z(t) dt dt
Using the fact that ln[X(t) ] = lnX(t), we have
˙Z (t) d ln X(t) d ln X(t) ˙X (t)
, lOMoARcPSD|58847208
(8)
dt X(t) , Z(t) dt where we have used the fact that is a
constant.
Problem 1.2
(a) Using the information provided in the question, the
Ẋ(t)
path of the growth rate of X, ˙X (t) X(t), is depicted in X(t)
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 a
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.
