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Homework 2
1. Consider the Solow model without technical progress. In class we derived
the steady-state values for capital per worker (k ) and output per worker
(y ) as a function of saving rate (s), population rate (n), and depreciation
rate (d):
(a) Suppose that s = 20%; n = 1% and d = 10%: Calculate k ; y and
consumption per worker (c ).
You need to assume a value for : Basically, using the information
we have about the real economy, you should assume either = 0:3
or = 1=3: I am assuming = 0:3:
k
=
0:2
0:1 + 0:01
y
=
0:2
0:1 + 0:01
c
=
1
1
1
0:3
= 2:349
0:3
0:3
= 1:292
0:2
0:1 + 0:01
0:8
1
0:3
0:3
= 1:034
(b) Repeat the exercise for s = 20%; n = 4% and d = 10%. What do
you observe as compared to your answer in part (a)? Comment.
k
=
0:2
0:1 + 0:04
y
=
0:2
0:1 + 0:04
c
=
1
1
1
0:3
= 1:665
0:3
0:3
= 1:165
0:2
0:1 + 0:04
0:8
1
0:3
0:3
= 0:932
In the steady state, with higher population growth, capital per worker
is less. Thus, output per worker is less and so is consumption: standards of livings are worse in this case.
(c) Now suppose that saving rate raises permanently from 20% to 30%.
How does this increase in the saving rate a¤ect k ; y and c ? (Again,
compare to your answer in part a.)
k
y
c
1
=
0:3
0:1 + 0:01
1
=
0:3
0:1 + 0:01
=
1
0:3
= 4:192
0:3
0:3
0:3
0:1 + 0:01
0:7
4
= 1:537
1
0:3
0:3
= 1:076
In the steady state, with a higher saving rate, capital per worker
is more. Thus, output per worker is more and, in this case, so is
consumption: standards of livings are better. Now, this saving rate
corresponds to the golden rule (with my assumptions, at least). So
if saving rate grows any higher, capital and output per worker are
more but not so consumption. So standards of living actually worsen
if saving rate passes this 30% benchmark. (You can try with a 40%
saving rate, for example).
2. Output per worker (labour productivity) is de…ned as y = Y =L: Show
that the growth rate in labour productivity depends on growth in total
factor productivity and growth in the capital-labour ratio (k = K=L). In
particular, show that
B_
k_
y_
=
+
:
y
B
k
If the production function is Y = BK L1 ; labour productivity
y = Bk : To calculate the growth rate, …rst take natural logarithms
ln y = ln B +
ln k;
and then di¤erentiate with respect to time to yield
y_
B_
k_
=
+
:
y
B
k
3. Suppose an economy described by the Solow model is in a steady state with
population growth n of 1.1 percent per year and technological progress g
of 1.5 percent per year. Total capital and total output grow at 2.6 percent
per year. Suppose further that the capital share of output is 0.4. If you
used the growth accounting equation to divide output growth into three
sources— capital, labour, and total factor productivity, how much would
you attribute to each source?
If the production function is Y = BK L1 ; by calculating the natural
log
ln Y = ln B + ln K + (1
) ln L;
and then di¤erentiating with respect to time, we derive the growth equation
Y_
B_
K_
L_
=
+
+ (1
) :
(1)
Y
B
K
L
Capital contributed
K_
= 0:4 2:6 = 1:04%:
K
5
Labour contributed
(1
)
L_
= 0:6 1:1% = 0:66%:
L
Then, if total output grows at 3%,
2:6 =
B_
+ 1:04 + 0:66;
B
_
total factor productivity growth B=B
= 0:9%:
(Note: the rest is optional or an alternative way to derive the same answer.)
Now remember the relation between Total Factor Productivity and the
e¢ ciency index: B = A1 : Thus, the relation between their growth rates
is
B_
B
B_
B
A_
A
=
(1
)
=
0:6
1:5% = 0:9%;
i.e., technical progress contributed 0:9%.
4. In the economy of Solovia,...
(a) The men of Solovia stay at home performing household chores, while
the women work in factories. If some of the men decided to start
working outside of the home so that the labour force increased by 4
percent, what would happen to the measured output of the economy?
(Hint: use the growth accounting equation).
Y_
B_
K_
L_
=
+
+ (1
) :
Y
B
K
L
Other things (capital and TFP) being constant, output would grow
by
Y_
L_
= (1
) = 0:3 4% = 1:2%:
Y
L
Let us say that the economy was growing at the rate g: This means
that the measured output of the economy would grow at (g + 1:2): It
says the measured output of the economy because, presumably, the
men in Solovia were previously engaged in household production, not
accounted in the GDP.
(b) In year 1, the capital stock was 6, the labour input was 3, and output
was 12. In year 2, the capital stock was 7, the labour input was 4, and
output was 14. What happened to total factor productivity growth
between the two years?
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In year 1
Y = BK L1
12 = B1 60:7
B1 = 2:46
30:3
In year 2
14 = B2 70:7
B2 = 2:37
40:3
So TFP actually decreased between these two years by
2:37 2:46
2:46
100 =
3:9%:
(Alternatively, you can calculate the TFP growth rate by using the
growth equation— equation 1. The answer will not be the same because the log approach is an approximation that works better for the
long run. So the method in this answer is somewhat superior but
you will not be penalized for using the other method.)
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