Macroeconomics I, UPF
Professor Antonio Ciccone
SOLUTIONS PROBLEM SET 2
1. (from Romer ”Advanced Macroeconomics” Chapter 1) Describe how,
if at all, each of the following developments affects the break-even and actual
investment lines in our basic diagram of the Solow model:
(a) The rate of depreciation falls.
-see Figures at the end of the solutions-
The slope of the line representing Break-Even-Investment (BEI) decreases,
i.e. BEI becomes less steep. This means that for any k̃t less investment is
required to sustain the initial value. Thus, a decrease in the rate of depreciation
increases the BGP-value of k̃t .
(b) The rate of technological progress rises.
The slope of BEI increases, i.e. the line becomes steeper. Hence, an increase
in a leads to an acceleration of A lowering k̃t and thus at any initial level more
investment is required to keep k̃t constant. k̃BGP decreases.
(c) The production function is Cobb-Douglas, f (k) = k α , and capital’s share,
α, rises
The line representing actual
investment (AI) moves “upward” i.e. it asigns
a greater value of s · f k̃t to any value k̃t . If the capital share rises more
investment is needed at any value of capital per effective worker. k̃BGP increases.
(d) Workers exert more effort, so that output per unit of effective labor for a
given value of capital per unit of effective labor is higher than before.
Same effect as (c). AI moves upward because for any k̃t , s · f k̃t is greater.
As this does not result in a constant increase of labor effectivity, BEI remains
unchanged
2. Can an increase in the savings rate end up increasing long-run
income but decreasing long-run consumption per worker? Consider
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a Solow economy without technological progress that is on its balanced growth
path. Now suppose there is a permanent increase in the savings rate. Show the
evolution of consumption per worker over time (from the time of the increase
in the savings rate to the new balanced growth path).
In a Solow Economy without technological progress, we have a condition for
the BGP:
s · f (k) = (n + δ) · k,
k=
K
L
Furthermore
C
L
=
Y
L
−s·
Y
L
= y − s · y = f (k) − s · f (k) = f (k) − (n + δ) · k
This means, that in the basic diagramm of the Solow model, consumption
per worker is displayed as the difference between the production function and
the Break-Even-Line (if the economy is on its BGP). Maximizing consumption
per worker, we get: f 0 (k) = n + δ which gives the so-called golden-rule value
of k that goes along with the maximum consumption level. If an increase of
the savings rate might lead to a decrease of long-run consumption per worker
depends on the question if the economy is initially endowed with a level of
capital per worker that is lower, equal to, or higher than the golden rule level.
If initially k is smaller than kGR a marginal increase in the savings rate increases
both, income and consumption, if k is equal to kGR a marginal increase in s
leaves c unaltered. While if k kGR the change in s increases y but decreases
c (see illustration).
-see Figures-see Figures-
The intuitive explanation says, that for k kGR the increase in f (k) caused
by the increase in s does not suffice to maintain k at the higher BGP level,
such that consumption needs to be reduced in order to be able to keep k at the
higher level.
3. Solow model with government. Consider a Solow economy with
government. The government taxes households and consumes all of the tax
revenue. Households consume a constant fraction c of their disposable income
Y −T , where T is taxes paid to the government. Show that national savings Y −
C −G (savings by households and the government), where G denotes government
consumption, is equal to (1 − c) (Y − T ). How does, therefore, an increase in
taxes affect output per worker in the short and in the long run (assume that
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there is no technological progress and that the economy is on a balanced growth
path when taxes are increased)?
Y − C − G = Y − c (Y − T ) − T = (Y − T ) − c (Y − T ) = (1 − c) (Y − T )
Effect on output graphically: -see Figures-
The introduction of taxes immediately decreases actual investment which is a
flow variable. This leads to a situation where Break-Even-Investment is greater
than actual investment. Hence, k decreases until it reaches its new Balanced
Growth Path value. The stock variable k adjusts slower than investment. Output per worker does not alter in the short run because the share of investment
that now goes to taxes is used entirely for government consumption. However,
as k decreases in the long-run the “missing” investment leads to a decrease in
output per worker. The new stable equilibrium is in E new .
Formally:
Balanced Growth Path:
s · [f (k) − T ] = (n + δ) · k
⇒ s · f 0 (k) ·
⇔
∂k
∂T
=
∂k
∂T
− s = (n + δ) ·
s
s·f 0 (k)−(n+δ)
=
∂k
∂T
s
s
s·f 0 (k)− k
·[f (k)−T ]
≺0
This is true, because disposable income is always greater than the share of
output that goes to capital.
Moreover, because:
∂y
∂k
0, we have:
∂y
∂T
≺0
4. (from Romer ”Advanced Macroeconomics” Chapter 1) Consider an economy with technological progress but without popultion growth that is on its balanced growth path. Now suppose there is a one-time jump in the number of
workers.
(a) At the time of the jump, does output per unit of effective labor rise, fall,
or stay the same? Why?
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At the time of the jump output per unit of effective labor instantly falls, as
well as capital per unit of effective labor does. While the capital stock is unable
to change instantly and technological progress does not alter other than before
Y
K
must decrease. The same argument applies for ỹ = AL
. Intuitively,
k̃ = AL
it is less capital per effective worker available and output has to be shared by
more ”effective workers”.
Formally:
∂ k̃
∂L
K
= − AL
2 ≺ 0
∂ ỹ
∂L
Y
= − AL
2 ≺ 0
(b) After the initial change (if any) in output per effective labor when the new
workers appear, is there any further change in output per unit of effective
labor? If so, does it rise or fall? Why?
The one-time jump of L may be intepreted as a new “initial” situation for
the economy that does not matter in the long run. The decrease in k̃ leads to a
situation where actual investment exceeds Break-Even-investment. Intuitively
a decrease in k̃ increases M P K and thus leads to M P K R. Firms therefore
need to pay less for their capital than the capital yields for them which makes
investment lucrative. Hence, the capital stock and thereby k̃ and ỹ increase
until the BGP-value of k̃ that was valid before the increase in L is obtained
again. k̃ and ỹ end up at the initial situation again.
(c) Once the economy has again reached a balanced growth path, is output per
unit of effective labor higher, lower, or the same as it was before the new
workers appeared? Why?
Due to the argumentation in (b) the economy ends up in its initial situation.
This result refers to the fact that a jump in L does not alter any growth rate,
savings rate or production function but simply effects an initial condition. As
one of the basic results of the Solow-Swan Analysis is that initial conditions of
K, A and L do not matter, it is obvious that the economy moves to it’s balanced
growth path again. A comparable situation might be the destruction of capital
(e.g. in a war) because this would also alter k̃ without affecting growth rates of
K, A and L.
Graphically:-see Figures5.(from Romer ”Advanced Macroeconomics” Chapter 1) Consider a Solow
economy that is on its balanced growth path. Assume for simplicity that there is
no technological progress. Now suppose that the rate of population growth falls.
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(a) What happens to the balanced growth path values of capital per worker,
output per worker and consumption per worker? Sketch the paths of this
variables as the economy moves to its new balanced growth path.
-see FiguresBGP value of k :
s · f (k) = (n + δ) · k ⇒ s · f 0 (k) ·
⇒
∂k
∂n
=
k
s·f 0 (k)−(n+δ)
=
∂k
∂n
= k+n·
k2
s·(k·f 0 (k)−f (k))
∂k
∂n
+δ·
∂k
∂n
≺0
BGP value of y :
∂f
∂k
∂n
0 ∂f
∂n
≺ 0 ∂k⇒
=
∂y
∂n
≺0
BGP value of c :
cBGP = f (k) − s · f (k) = (1 − s) · f (k)
∂k
∂c
= (1 − s) · f 0 (k) · ∂n
≺0
⇒ ∂n
Graphically:-see Figures-
(b) Describe the effect of the fall in population growth on the path of output
(that is, total output, not output per worker).
We know that in the steady state, i.e. if the economy is on its BGP, in an
Y
economy where there is no technological progress k = K
L and therefore y = L
are constant. That means, that in this case total output has to grow at the same
rate as the equilibrium population. Hence, total output grows at the growth rate
n1 in the initial equilibrium and at the - decreased - growth rate n2 in the new
steady state. Furthermore, we know that y = YL increases if population growth
decreases (part (a)). This results from the fact that the population instantly
grows slower as the growth rate of the population drops. However, adjustment
of the growth rate of total output is gradually as compared to the one time drop
of the growth rate of the population, because as y = YL increases total output
has to grow at a rate n2 ≺ n ≺ n1 during the adjustment process.
6. Assuming that capital does not move internationally (as in the Solow
model), international differences in savings rate translate into high returns to
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capital in low savings economies. The Solow model assumes that economies are
closed to international financial markets. In reality, however, there would be
a tendency for countries with high marginal products of capital (and therefore
high real interest rates) to attract the savings of countries with low marginal
product of capital (and therefore low real interest rates). To see how strong this
tendency may be, assume that there are two countries that function as described
in the Solow model. Assume that these countries are identical in everything,
except their savings rates. In particular, country H has a savings rate that is
twice that of country L. How different will the marginal products of capital and
therefore the real interest rates of the two countries be once they have reached
their balanced growth paths?
Assume that at the initial position:
LH = LL = L, AH = AL = A, nH = nL = n, δH = δL = δ, aH = aL = a
and sH = 2sL
Assume also, for simplicity, that technology in both countries is described
by identical Cobb-Douglas production functions. Then it is true that for each
country, in the BGP (see Question 1):
k∗ ≡
K
AL
=
s
n+δ+a
1
1−α
The real interest rate is the marginal product of capital minus depreciation:
r = M P K − δ = α(k)α−1 − δ. Then in the BGP, r = α
−1
s
α n+δ+a
−δ
s
n+δ+a
sL
n+δ+a
α−1
1−α
−δ =
Comparing the returns in both countries, we see that
rL − rh = α
sL
n+δ+a
−1
−1
2sL
− δ − α n+δ+a
−δ =
α
2
−1
>0
This is the expected result that MPK will be higher in the country with
lower savings. This is because higher savings mean higher BGP capital and,
consequently (due to decreasing returns to this factor), lower MPK. Logically,
this is translated into an inequality in real interest rates. The rate of return
in the high-saving country is not half that of the low-saving country because
MPK in the BGP is not linear in saving rates. However, the difference is always
positive.
7.Productivity differences with international capital mobility. Consider two
Solow economies with identical Cobb-Douglas production functions. Suppose
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that there is no technological change. Suppose also that the rate of depreciation
of capital and that the level of labor efficiency is the same in both economies.
Since the statement does not say anything in particular about the saving
rates, we let them be different: s2 > s1 . Let all other parameters be equal
across economies, as in the previous question.
(a) Show that if capital moves to the country with the higher return
(higher marginal product), both countries will have the same
output per worker in the steady state.
Assume initially that the economies are closed and have different levels of
output per efficiency worker (not necessarily the steady-state values).
We also assume that capital chases productivity differentials and that the
capital markets clear instantaneously. Therefore, we assume that right after
the capital markets open, marginal productivity of capital is equalized in both
economies after an initial capital flow φ.
0
0
M P K1 = M P K2 ⇒ f (k1 + φ) = f (k2 − φ)
Using the assumption that the technology is Cobb-Douglas, we get
α−1
α (k1 + φ)
α−1
= α (k2 − φ)
⇒φ=
k2 −k1
2
Because we characterized the capital markets to clear immediatly M P K1 =
M P K2 holds at all times. Hence, from then on (and on the road to the steady
state), k2 = k1 = k also holds. Note that this also implies that k̇2 = k̇1 =
k̇, otherwise, it would be possible for the values of k to be different accross
economies. Thus, the modified laws of motion can be written as:
k̇1 = s1 f (k1 ) − (n + δ)k1 + F and k̇2 = s2 f (k2 ) − (n + δ)k2 − F
where F is a strictly non-negative capital flow, since economy 2 always saves
more and would tend to move faster towards the steady state equilibrium. The
magnitude of the flow can be calculated using the previous equations:
F =
s2 −s1
2 f (k)
We can then find the steady state to which both economies move jointly:
k̇1 = 0 ⇒ s1 k1α − (n + δ)k1 +
s2 −s1 α
2 k1
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=0
⇒ k1∗ =
1 /2
( s2 +s
)
2
1
1−α
n+δ
= k2∗
Plugging this in the production function, we find that production per worker
is:
Ay = A
1 /2
( s2 +s
)
2
α
1−α
n+δ
Hence, both economies converge to the same output per worker
(b) Will the result in (a) continue to hold if countries have different
labor efficiency levels?
From (a) we see that both economies always converge to the same level
of capital per efficiency worker if they differ in the parameter s only. This
equilibrium is not affected by the economies having a different level of worker
efficiency, A.
Output per worker, nevertheless, will be different and the difference in A
(constant for each economy in this question) rescales output per worker.
y i = Ai
1 /2
)
( s2 −s
2
n+δ
α
1−α
for i = 1, 2
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Figure 1:
Figure 2
( n + a + δ ) ⋅ kt
BEI
s ⋅ f kt
( )
AI
kt
kBGP
Figure 3
Output per
worker
increases
f kt
( )
( n + a + δ ) ⋅ kt
s2 ⋅ f kt
( )
s1 ⋅ f kt
( )
Length of the dotted
line, i.e. steady-state
consumption per
worker decreases
kGR
<
9
k1
k2
kt
Figure 2:
Figure 1
y% =
( a + δ ) ⋅ k%
Y
AL
( )
f k%
( )
s ⋅ f k%
K
k% =
AL
k%2
k%BGP1
Figure 2
( n1 + δ ) ⋅ k
( n2 + δ ) ⋅ k
Y
y=
L
f (k )
c2
c1
s ⋅ f (k )
k=
k BGP1
10
k BGP2
K
L
Figure 3:
Figure 3
k BGP
t
yBGP
t
cBGP
t
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