STUDY NOTES & REFERENCE GUIDE
SOLUTIONS MANUAL TO ROMER'S
··
ADVANCED MACROECONOMICS 4TH
··
EDITION. COMPLETE 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 ln X(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
d lnX(t) Y(t)
(4) Z(t) = d ln Z(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
Z(t) d ln Z(t) d ln[X(t)
(7) = = ]
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)
(8) = = = ,
Z(t) dt dt 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 X(t)
depicted in the figure at right. X(t)
From time 0 to time t1 , the growth rate of X is
constant and equal to a > 0. At timeCourseroomsity
t1 , the growth Stuvia | Page 1 of 323
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website,
in whole or part.
, STUDY NOTES & REFERENCE GUIDE
© 2012 by McGraw-Hill Education. This is proprietary material solely for authorizedainstructor use. Not authorized for sale or
distribution in any
© 2012 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or
Courseroomsity Stuvia | Page 2 of 323
distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted
on a website, in whole or part.
, STUDY NOTES & REFERENCE GUIDE
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.
Courseroomsity Stuvia | Page 3 of 323
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website,
in whole or part.
, STUDY NOTES & REFERENCE GUIDE
Courseroomsity Stuvia | Page 4 of 323
SOLUTIONS MANUAL TO ROMER'S
··
ADVANCED MACROECONOMICS 4TH
··
EDITION. COMPLETE 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 ln X(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
d lnX(t) Y(t)
(4) Z(t) = d ln Z(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
Z(t) d ln Z(t) d ln[X(t)
(7) = = ]
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)
(8) = = = ,
Z(t) dt dt 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 X(t)
depicted in the figure at right. X(t)
From time 0 to time t1 , the growth rate of X is
constant and equal to a > 0. At timeCourseroomsity
t1 , the growth Stuvia | Page 1 of 323
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website,
in whole or part.
, STUDY NOTES & REFERENCE GUIDE
© 2012 by McGraw-Hill Education. This is proprietary material solely for authorizedainstructor use. Not authorized for sale or
distribution in any
© 2012 by McGraw-Hill Education. This is proprietary material solely for authorized instructor use. Not authorized for sale or
Courseroomsity Stuvia | Page 2 of 323
distribution in any manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted
on a website, in whole or part.
, STUDY NOTES & REFERENCE GUIDE
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.
Courseroomsity Stuvia | Page 3 of 323
manner. This document may not be copied, scanned, duplicated, forwarded, distributed, or posted on a website,
in whole or part.
, STUDY NOTES & REFERENCE GUIDE
Courseroomsity Stuvia | Page 4 of 323
