ECA5102 Macroeconomics
Dr. Huang Kui, Angela
National University of Singapore
Assignment 3 Solution
Intellectual Property: You are not allowed to reproduce or distribute any teaching mate-
rials without the permission of the lecturer.
1. This problem considers a variation on the Solow model. Suppose that instead of the
population growing at a constant rate, that people have fewer children as they become
wealthier. In particular, as always suppose that output is produced via a Cobb-Douglas
production function Y = KαL1−α, where technology is A = 1 and thus its growth rate
is g = 0 for simplicity. Consumers save a constant fraction s of their income, and the
capital stock depreciates at rate δ. The capital stock evolves according to the following
equation
dK
dt
≡ K˙ = sY − δK
Define the per-capita variables as k = K/L, y = Y/L, and c = C/L. However, instead
of being a constant, now the population growth rate is proportional to the marginal
product of capital (MPK):
L˙
L
= n ·MPK
where 0 < n < s/α. Since the MPK falls as capital increases, this captures the
declining population growth rates of wealthier nations.
(a) Derive the equation of motion for per-capita quantities of capital k˙ ≡ dk
dt
= d(K/L)
dt
.
Solution: Notice that
y =
Y
L
= kα
L˙
L
= n ·MPK = nαKα−1L1−α = nα
(
K
L
)α−1
= nαkα−1
Capital stock evolves according to the following equation
K˙ = sY − δK.
Divide by L in the capital accumulation equation:
K˙
L
= sy − δk = skα − δk.
Then remember that k = K/L, so:
k˙ =
K˙L− L˙K
L2
=
K˙
L
− K
L
L˙
L
= skα − δk − knαkα−1 = (s− nα)kα − δk.
1
(b) Determine the steady state per-capita quantities of capital, output, consumption,
and population growth.
Solution: The steady state is at k˙ = 0.
Solve for steady state:
0 = (s− nα) (k∗)α − δk∗ ⇒ k∗ =
(
s− nα
δ
) 1
1−α
Then
y∗ = (k∗)α =
(
s− nα
δ
) α
1−α
c∗ = (1− s) y∗ = (1− s)
(
s− nα
δ
) α
1−α
(
L˙
L
)
steady state
= nα (k∗)α−1 =
δ
s− nαnα =
δα
s/n− α
(c) What are the growth rates of aggregate capital stock K, aggregate output Y , and
aggregate consumption C in the steady state?
Solution: Because K = kL, and the steady state per-capita quantities of capital
k∗ is a constant, taking logs and differentiating with respect to time t, we have(
K˙
K
)
steady state
=
(
L˙
L
)
steady state
=
δα
s/n− α.
Similarly, (
Y˙
Y
)
steady state
=
(
L˙
L
)
steady state
=
δα
s/n− α,
and (
C˙
C
)
steady state
=
(
L˙
L
)
steady state
=
δα
s/n− α.
(d) What are the short run (transitional dynamics) and long run (steady state) effects
of an increase in n on the per-capita quantities of capital, output, consumption,
and the population growth rate?
Solution:
Long-run effect:
Clearly, steady state (long run) capital, output and consumption are decreasing in
n, but steady state population growth is increasing.
Short-run effect:
Recall that the equation for capital evolution is:
k˙t = (s− nα)kαt − δkt
2
(s− nα)kαt shifts downwards as n increases and δkt doesn’t shift (Figure 1 below),
thus kt falls in the short run and converges downwards to the new steady state. yt
and ct have the same tendency. However, population growth has a different one.
Notice that L˙
L
= nαkα−1, where α − 1 < 0. The population growth will jump
upwards because of an increase in n initially, and will continue to grow up to its
new steady state level as k converges to the new k∗′ (Figure 2 below).
Figure 1: Shifts of the Curves
Figure 2: Population Growth Rate
2. In 1861 many southern US states seceded and formed the Confederacy. Treat the North
and South as two separate countries, and analyze the effects of the secession and the
Civil War using the Solow model. Suppose that the rate of population growth n is the
same in the North and South, but the North had a higher rate of productivity growth
g, a higher savings rate s and a higher productivity level A.
(a) What does the model predict about the long run (Balanced Growth Path) compar-
ative economic performance (both growth rates and levels of per capita output)
3
of the North and South?
Solution: The North has a higher g so will grow faster than the South in the
long run in terms of per capita output (since g is higher). Note that the ranking
of the long run (balanced growth path) capital per effective labor k˜ = K/(AL)
is ambiguous since the North has a higher s (which would increase k˜) but also a
higher g (which would reduce k˜). Thus, the ranking of the long run (balanced
growth path) output per effective labor y˜ = f(k˜) is ambiguous, and of output per
capita y = Ay˜ is ambiguous.
(b) During the Civil War, much of the capital stock in the South was destroyed, and
after the war the country was reunited. Suppose that after war the whole US had
the same (high) TFP growth rate g, savings rates s and TFP level A, that were
predominant in the North. What does the model predict about the comparative
economic performance (both growth rates and levels of per capita output) of
the North and South after re-unification? Consider both the long run (Balanced
Growth Path) and short run (transitional dynamics).
Solution: We assume that the North is on its balanced growth path, so that
k˜ = K/(AL) is constant prior to re-unification. Under our assumptions here, re-
unification has no effect on the North, so that it stays on the balanced growth path,
growing at g in per capita terms. The South however now faces an increase in s and
g, and so its balanced growth path shifts to equal that of the North. In the short
run, after the war, the capital stock and output in the South is very low, and it is
far from not only its pre-war balanced growth path, but also from its new post-war
one. Thus the South will grow (much) faster along the transition. In the long run
there is convergence, as the South converges to the same balanced growth path as
the North. They will have the same growth rates and levels of per capita output.
3. (Romer 1.7) Find the elasticity of output per unit of effective labor on the balanced
growth path, y˜∗, with respect to the rate of population growth, n, where we define the
elasticity as ηy˜n = ∂y˜
∗/y˜∗
∂n/n
. If αK(k˜∗) = f ′(k˜∗)k˜∗/f(k˜∗) = 1/3 , g = 2%, and δ = 3%, by
about how much does a fall in n from 2 percent to 1 percent raise y˜∗? (Hint: check the
relevant part in the textbook: Romer Section 1.5 Quantitative Implications)
Solution: Define the elasticity of y˜∗ with respect to n as
ηy˜n =
∂y˜∗/y˜∗
∂n/n
=
∂y˜∗
∂n
n
y˜∗
Because y˜∗ = f(k˜∗),
∂y˜∗
∂n
= f ′
[
k˜∗ (s, n, g, δ)
] ∂k˜∗ (s, n, g, δ)
∂n
(1)
k˜∗ is defined by ˙˜k = 0 ⇒ sf
[
k˜∗ (s, n, g, δ)
]
=(n+ g + δ)k˜∗ (s, n, g, δ)
Differentiate the last equation with respect to n:
sf ′
[
k˜∗ (s, n, g, δ)
] ∂k˜∗ (s, n, g, δ)
∂n
=(n+g+δ)
∂k˜∗ (s, n, g, δ)
∂n
+ k˜∗ (s, n, g, δ)
4
and solve it for ∂k˜∗ (s, n, g, δ) /∂n:
∂k˜∗ (s, n, g, δ)
∂n
=
k˜∗ (s, n, g, δ)
sf ′
[
k˜∗ (s, n, g, δ)
]
− (n+ g + δ)
(2)
Substitute (2) into (1) and then (1) into the equation for ηy˜n:
ηy˜n = f
′(k˜∗)
k˜∗
sf ′
(
k˜∗
)
− (n+ g + δ)
n
f(k˜∗)
Recall
s =
(n+ g + δ)k˜∗
f(k˜∗)
Substitute the last equation in the equation for ηy˜n:
ηy˜n =
f ′(k˜∗)k˜∗
(n+ g + δ)k˜∗f ′(k˜∗)/f(k˜∗)− (n+ g + δ)
n
f(k˜∗)
=
f ′(k˜∗)k˜∗
(n+ g + δ)
[
f ′(k˜∗)k˜∗/f(k˜∗)− 1
] n
f(k˜∗)
Divide both the numerator and the denominator by f(k˜∗) and simplify:
ηy˜n =
f ′(k˜∗)k˜∗/f(k˜∗)
(n+ g + δ)
[
f ′(k˜∗)k˜∗/f(k˜∗)− 1
] n
f(k˜∗)/f(k˜∗)
Let αK(k˜∗) = f ′(k˜∗)k˜∗/f(k˜∗). The last equation above reduces to:
ηy˜n =
nαK(k˜∗)
(n+ g + δ)
[
αK(k˜∗)− 1
]
We are given δ = 0.03, αK(k˜∗) = 1/3, g = 0.02 and n changes from 0.02 to 0.01.
Substitute these values in the last equation:
ηy˜n =
0.015× 1
3
(0.015 + 0.02 + 0.03)
(
1
3
− 1) = −0.12
Note that I have used n = 0.015, which is the average of 0.02 and 0.01. The reason is
that the elasticity is defined for a small change in n but we have a substantial change
from 0.02 to 0.01. So we approximate the elasticity by taking the average of the old
and new values of n.
Based on ηy˜n = −0.12, when n decreases by 50% (from 0.02 to 0.01), y˜∗ is expected to
increase by 6% (−0.12× 50%).
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4. (Romer 1.12) Embodied technological progress. (This follows Solow, 1960, and
Sato, 1966.) One view of technological progress is that the productivity of capital
goods built at t depends on the state of technology at t and is unaffected by subsequent
technological progress. This is known as embodied technological progress (technological
progress must be “embodied” in new capital before it can raise output). This problem
asks you to investigate its effects.
As a preliminary, Let us modify the basic Solow model to make technological progress
capital-augmenting rather than labor-augmenting. So that a balanced growth path ex-
ists, assume that the production function is Cobb–Douglas: Y (t) = [A(t)K(t)]αL(t)1−α.
Assume that A grows at rate µ: ˙A(t) = µA(t).
Show that the economy converges to a balanced growth path, and find the growth rates
of Y and K on the balanced growth path.
(Hint: Show that we can write Y/(A
α
1−αL) as a function of K/(A
α
1−αL). Then analyze
the dynamics of K/(A
α
1−αL).)
Solution: The production function is:
Y (t) = [A(t)K(t)]αL(t)1−α.
The growth rate of technology is given by:
˙A(t)
A(t)
= µ.
Because the production function is Cobb-Douglas, we can rewrite it in such a way that
the technology becomes labor augmenting:
Y (t) = K(t)α
[
A(t)
α
1−αL(t)
]1−α
.
Divide both sides by A(t)
α
1−αL(t):
Y (t)
A(t)
α
1−αL(t)
=
[
K(t)
A(t)
α
1−αL(t)
]α
.
Let y(t) ≡ Y (t)
A(t)
a
1−αL(t)
and k(t) ≡ K(t)
A(t)
a
1−αL(t)
. The last equation becomes:
y(t) = k(t)α.
6
Find k˙(t):
k˙(t) =
dk(t)
dt
=
d
(
K(t)
A(t)
α
1−αL(t)
)
dt
=
[
A(t)
α
1−αL(t)
]
K˙(t)−K(t)
[
A(t)
α
1−α L˙(t) + α
1−αA(t)
α
1−α−1A˙(t)L(t)
]
[
A(t)
α
1−αL(t)
]2
=
K˙(t)
A(t)
α
1−αL(t)
− K(t)
A(t)
α
1−αL(t)
(
L˙(t)
L(t)
+
α
1− α
A˙(t)
A(t)
)
=
sY (t)− δK(t)
A(t)
α
1−αL(t)
− k(t)
(
n+
α
1− αµ
)
=
sY (t)
A(t)
α
1−αL(t)
− k(t)
(
δ + n+
α
1− αµ
)
= sy(t)− k(t)
(
δ + n+
α
1− αµ
)
We have
k˙(t) = sk(t)α −
(
δ + n+
α
1− αµ
)
k(t) (3)
Equation (1) is very similar to the fundamental equation governing the dynamics of
the Solow model with labor-augmenting technological progress. Here, however, we are
measuring in units of effective labor A(t)
α
1−αL(t) rather than in units of effective labor
A(t)L(t). Using the same graphical technique as with the Solow model we learned in
class, we can graph both components of k˙(t). See the figure below.
When actual investment per unit of effective labor A(t)
α
1−αL(t), sk(t)α, exceeds break-
even investment per unit of effective labor A(t)
α
1−αL(t), given by
(
δ + n+ α
1−αµ
)
k(t),
7
k will rise toward k∗. When actual investment per unit of effective labor A(t)
α
1−αL(t)
falls short of break-even investment per unit of effective labor A(t)
α
1−αL(t), k will fall
toward k∗.
On the balanced growth path k˙(t) = 0. This implies:
sk(t)α =
(
δ + n+
α
1− αµ
)
k(t).
We can find k∗ from the last equation and show that the system converges to k∗ from
any k > 0. Since y = kα, y will also be constant when the economy converges to k∗. On
the balanced growth path, because Y (t) = y∗A(t)
a
1−αL(t), and K(t) = k∗A(t)
a
1−αL(t),
we have
gY = gK = gL +
α
1− αgA = n+
α
1− αµ.
5. (Final Exam AY2019/20) This problem considers a variant of the Solow growth model
which includes human capital that we learned in class. The production function is given
by
Y (t) = [K (t)]α [H (t)]β [A (t)L (t)]1−α−β ,
where Y (t) is output, K(t) is physical capital, H(t) is human capital, A(t) is the level of
technology, and L(t) is population. The dynamic equation for physical capital is given
by
dK (t)
dt
≡ K˙ (t) = sKY (t) ,
where sK ∈ (0,1) is the fraction of output that is saved and invested in physical capital,
and the depreciation rate δK is zero. The dynamic equation for human capital is given
by
dH (t)
dt
≡ H˙ (t) = sHY (t) ,
where sH ∈ (0,1) is the fraction of output that is saved and invested in human capital,
and the depreciation rate δH is zero. Technology and population grow at fixed rates of
A˙(t)
A(t)
= g > 0 and L˙(t)
L(t)
= n > 0, respectively.
(a) Write the production function in intensive form, i.e., in terms of y˜ (t) = Y (t)
A(t)L(t)
,
k˜ (t) = K(t)
A(t)L(t)
, and h˜ (t) = H(t)
A(t)L(t)
.
Solution: Divide the production function by A (t)L (t) to get
Y (t)
A (t)L (t)
=
[
K (t)
A (t)L (t)
]α [
H (t)
A (t)L (t)
]β
,
which in intensive form is
y˜ (t) =
[
k˜ (t)
]α [
h˜ (t)
]β
.
8
(b) Derive the dynamic equations of physical and human capital per unit of effective
labor, i.e., ˙˜k ≡ dk˜
dt
and ˙˜h ≡ dh˜
dt
. Show your working clearly.
Solution: To derive the dynamic equation for physical capital per unit of effective
labor, recall that
˙˜k (t) ≡ dk˜ (t)
dt
=
d
(
K(t)
A(t)L(t)
)
dt
.
Taking the derivative, we get
˙˜k (t) =
A (t)L (t) K˙ (t)−K (t)
[
A (t) L˙ (t) + A˙ (t)L (t)
]
[A (t)L (t)]2
,
which simplifies to
˙˜k (t) =
K˙ (t)
A (t)L (t)
− (n+ g) k˜ (t) .
Substituting for K˙ (t) = sKY (t) gives
˙˜k (t) = sK y˜ (t)− (n+ g) k˜ (t) .
Finally, substituting for y˜(t) we get the following dynamic equation for physical
capital per unit of effective labor,
˙˜k (t) = sK
[
k˜ (t)
]α [
h˜ (t)
]β
− (n+ g) k˜ (t) .
Following similar steps, to derive the dynamic equation for human capital per unit
of effective labor, recall that
˙˜h (t) ≡ dh˜ (t)
dt
=
d
(
H(t)
A(t)L(t)
)
dt
.
Taking the derivative, we get
˙˜h (t) =
A (t)L (t) H˙ (t)−H (t)
[
A (t) L˙ (t) + A˙ (t)L (t)
]
[A (t)L (t)]2
,
which simplifies to
˙˜h (t) =
H˙ (t)
A (t)L (t)
− (n+ g) h˜ (t) .
Substituting for H˙ (t) = sHY (t) gives
˙˜h (t) = sH y˜ (t)− (n+ g) h˜ (t) .
Finally, substituting for y˜(t) we get the following dynamic equation for human
capital per unit of effective labor,
˙˜h (t) = sH
[
k˜ (t)
]α [
h˜ (t)
]β
− (n+ g) h˜ (t) .
9
(c) Solve for the Balanced-Growth-Path expressions for k˜(t), h˜(t), and y˜(t). Denote
these expressions as k˜∗, h˜∗, and y˜∗, respectively.
Solution: Set ˙˜k (t) = 0 and ˙˜h (t) = 0, in the last two equations and solve them
simultaneously for k˜∗ and h˜∗ to get
k˜∗ =
(
sK
n+ g
) 1−β
1−α−β
(
sH
n+ g
) β
1−α−β
,
h˜∗ =
(
sK
n+ g
) α
1−α−β
(
sH
n+ g
) 1−α
1−α−β
,
Thus
y˜∗ =
(
sK
n+ g
) α
1−α−β
(
sH
n+ g
) β
1−α−β
.
(d) Consider two countries. One is rich and the other is poor. The two countries have
the same values for parameters α, β, g and n. Denote the savings rates in the poor
country by sPK and sPH . Denote the savings rates in the rich country by sRK and
sRH . Denote the Balanced-Growth-Path output per unit of effective labor in the two
countries by y˜P and y˜R. Derive an expression for y˜R/y˜P .
Solution: The steady-state output in the rich country is given by
y˜R =
(
sRK
n+ g
) α
1−α−β
(
sRH
n+ g
) β
1−α−β
,
and in the poor country by
y˜P =
(
sPK
n+ g
) α
1−α−β
(
sPH
n+ g
) β
1−α−β
.
Taking the ratio of the two and simplifying, we get
y˜R
y˜P
=
(
sRK
sPK
) α
1−α−β
(
sRH
sPH
) β
1−α−β
.
(e) You are given the following parameter values,
Parameter: α β
Value: 1/3 0
Assume that sPH = sRH . Using the expression in (d), solve for the numerical values
of sRK/sPK that can generate a Balanced-Growth-Path income ratio of 5 : 1 (i.e.,
y˜R/y˜P = 5) between the two countries.
Solution: Because sPH = sRH , the last equation becomes
y˜R
y˜P
=
(
sRK
sPK
) α
1−α−β
.
10
Thus
sRK
sPK
=
(
y˜R
y˜P
) 1−α−β
α
.
Substitute the given values to get
sRK
sPK
= 52 = 25.
(f) Now you are given the following parameter values,
Parameter: α β
Value: 1/3 1/3
Assume that sPH = sRH . Using the expression in (d), solve for the numerical values
of sRK/sPK that can generate a Balanced-Growth-Path income ratio of 5 : 1 (i.e.,
y˜R/y˜P = 5) between the two countries.
Solution: Similarly, we have
sRK
sPK
=
(
y˜R
y˜P
) 1−α−β
α
.
Substitute the given values to get
sRK
sPK
= 51 = 5.
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