Advanced Macroeconomics – Solutions Manual to Romer’s Advanced Macroeconomics

SOLUTIONS MANUAL TO ROMER S
ADVANCED MACROECONOMICS 4TH
EDITION. COMPLETE SOLUTION MANUAL
DAVID ROMER
SOLUTIONS TO CHAPTER 1

Probleṁ 1.1
(a) Since the growth rate of a variable equals the tiṁe derivative of its log, as shown by equation (1.10)
in the text, we can write
Ż(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 suṁ of their logs, we have
Ż(t) dln X(t)  ln Y(t) d ln X(t) d ln Y(t)
(2)    ,
Z(t) dt dt dt
or siṁply
Ż(t) Ẋ(t) Ẏ(t)
(3)   .
Z(t) X(t) Y(t)

(b) Again, since the growth rate of a variable equals the tiṁe derivative of its log, we can write
Ż(t) d ln Z(t) d lnX(t) Y(t)
(4)   .
Z(t) dt dt
Since the log of the ratio of two variables equals the difference in their logs, we have
Ż(t) dln X(t)  ln Y(t) d ln X(t) d ln Y(t)
(5)    ,
Z(t) dt dt dt
or siṁply
Ż(t) Ẋ(t) Ẏ(t)
(6)   .
Z(t) X(t) Y(t)

(c) We have

Ż(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
Ż(t) d ln X(t) d ln X(t) Ẋ (t)
(8)    ,
Z(t) dt dt X(t)
where we have used the fact that  is a constant.

Probleṁ 1.2
(a) Using the inforṁation provided in the question,
the path of the growth rate of X, Ẋ(t) X(t), is Ẋ(t)
depicted in the figure at right. X(t)

Froṁ tiṁe 0 to tiṁe t1 , the growth rate of X is
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,constant and equal to a > 0. At tiṁe t1 , the growth
rate of X drops to 0. Froṁ tiṁe t1 to tiṁe t2 , the
growth rate of X rises gradually froṁ 0 to a. Note that
we have ṁade the assuṁption that Ẋ(t) X(t) rises at
a constant rate froṁ t1 to t2 . Finally, after tiṁe t2 , the
growth rate of X is constant and equal to a again.




© 2012 by ṀcGraw-Hill Education. This is proprietary ṁaterial solely for authorized instructor use. Not authorized for sale or distribution in any
ṁanner. This docuṁent ṁay not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.

,
, 1-2 Solutions to Chapter 1


(b) Note that the slope of lnX(t) plotted against tiṁe
is equal to the growth rate of X(t). That is, we know lnX(t)
d ln X(t) Ẋ (t) slope = a

dt X(t)
(See equation (1.10) in the text.) slope = a

Froṁ tiṁe 0 to tiṁe t1 the slope of lnX(t) equals
a > 0. The lnX(t) locus has an inflection point at t1 ,
when the growth rate of X(t) changes discontinuously lnX(0)
froṁ a to 0. Between t1 and t2 , the slope of lnX(t)
rises gradually froṁ 0 to a. After tiṁe t2 the slope of
lnX(t) is constant and equal to a > 0 again. 0 t1 t2 tiṁe

Probleṁ 1.3
(a) The slope of the break-even investṁent line is
Inv/ (n + g + )k
given by (n + g + ) and thus a fall in the rate of eff lab
depreciation, , decreases the slope of the break-
even investṁent line. (n + g + NEW)k

The actual investṁent curve, sf(k) is unaffected.
sf(k)
Froṁ the figure at right we can see that the balanced-
growth-path level of capital per unit of effective
labor rises froṁ k* to k*NEW .

k* k*NEW k


(b) Since the slope of the break-even investṁent
line is given by (n + g + ), a rise in the rate of Inv/ (n + gNEW + )k
technological progress, g, ṁakes the break-even eff lab
investṁent line steeper.
(n + g + )k
The actual investṁent curve, sf(k), is unaffected.
sf(k)
Froṁ the figure at right we can see that the
balanced-growth-path level of capital per unit of
effective labor falls froṁ k* to k*NEW .


k*NEW k* k




© 2012 by ṀcGraw-Hill Education. This is proprietary ṁaterial solely for authorized instructor use. Not authorized for sale or distribution in any
ṁanner. This docuṁent ṁay not be copied, scanned, duplicated, forwarded, distributed, or posted on a website, in whole or part.