Download Macroeconomics I, UPF Professor Antonio Ciccone SOLUTIONS

Macroeconomics I, UPF
Professor Antonio Ciccone
SOLUTIONS PROBLEM SET 2
2.1 Show that income growth depends on distance from steady state;
show also that the rate of convergence to steady state depends on
the elasticity of output with respect to capital. Derive the rate
of convergence to the balanced growth path of the Solow model,
assuming that economies are close to their balanced-growth-path
level of income per e¢ ciency worker. You can derive this convergence rate in 2 steps:
(a) Assume a Cobb-Douglas production function and showing that
the growth of capital per e¢ ciency worker at any point in time
can be written as a function of LN-income-per-e¢ ciency worker,
ln yt , and (exogenous) production function parameters (LN refers
to the natural logarithm).
Let k
Y
AL ;
K
AL ; y
k_
and k^ denote percent change in k, i.e k^ =
@k
@t
Assume F (K; L) = K (AL)1
k_
k
) f (k) = k
As seen in class,
k_ = sf (k)
(n + + a)k
) k^ = sk
1
(n + + a)
We know y = k ) ln y =
ln k and ln k =
1
ln y
Then we can rewrite the growth rate of k as
k^ = se(
1) ln k
(n + + a) = se
1
ln y
(n + + a)
(b) Take a linear approximation (…rst-order Taylor approximation)
of the growth of capital per e¢ ciency worker around ln y (LNincome-per-e¢ ciency-worker in the balanced growth path).
First-order Taylor approximation:
i
h
h
1
0
ln y
(n + + a) (ln y ln y ) + se
k^ = se
|
{z
}
= 0 at the steady state
h
i
1
k^ = 1 1 s (y )
(ln y ln y )
1
1
ln y
1
i
(ln y
ln y )
Show that the rate of convergence of economies to their balanced
growth path is greater the lower the elasticity of output with respect
to capital (the share of income that goes to capital). Why?
We see here that if output per e¢ ciency worker is over its steady state level,
capital and output per worker shrink (k^ < 0). This is so because, in the last
expression, the term in square brackets is negative, since < 1. Note also that,
1
at the steady state, s (k )
=n+ +a
) k^ =
1
1 (n + + a) (ln y
|
{z
}
speed of convergence
ln y )
where we have factored (-1) out in order to focus exclusively on the magnitude of the speed of convergence. Finally, note that in the case of a CobbDouglas production function, the elasticity of output with respect to capital is
equal to:
@ ln y
@ ln k
=
=
MP K
AvP K
=
rK
Y
From the last expression, we see that the speed of convergence depends
negatively on the elasticity of output with respect to capital. Note that a higher
value of implies a higher value of k in the steady state:
1
k^ = 0 implies k =
s
n+ +a
1
and
@k
@
>0
Now compare two economies with di¤erent values of and, hence, of k .
The rate of convergence indicates how fast an economy converges to its BGP,
given the distance that separates its current output per capita from its steadystate output per capita. Since higher values of imply higher values of k , when
both economies are located at the same distance from their respective steady
states, the economy with higher will also have higher values of k. But average
product is a decreasing function of k; hence, average product of capital is lower
in the economy with higher elasticity. Finally, lower average product of capital
is translated into slower growth of total output which, in turn, is translated into
slower accumulation of capital and a slower rate of convergence (recall that in
the model accumulation of total capital is a linear function of total output).
2.2 (TBG) Assuming that capital does not move internationally (as
in the Solow model), international di¤erences in savings rate
translate into high returns to capital in low savings economies.
The Solow model assumes that economies are closed to international …nancial markets. In reality, however, there would be
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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 di¤erent
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;
and sH = 2sL
H
=
L
= ; aH = aL = a
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):
1
k
K
AL
=
s
n+ +a
1
The real interest rate is the marginal product of capital minus depreciation:
1
r = MPK
s
n+ +a
1
= (k)
s
n+ +a
. Then in the BGP, r =
1
=
1
Comparing the returns in both countries, we see that
rL
rh =
sL
n+ +a
1
2sL
n+ +a
1
=
2
sL
n+ +a
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 di¤erence is always
positive.
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2.3 Productivity di¤erences with international capital mobility. Consider two Solow economies with identical Cobb-Douglas production functions. Suppose that there is no technological change.
Suppose also that the rate of depreciation of capital and that
the level of labor e¢ ciency is the same in both economies.
Since the statement does not say anything in particular about the saving
rates, we let them be di¤erent: s2 > s1 : Let all other parameters be equal
across economies, as in the previous question.
(a) (TGB) 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 di¤erent levels of
output per e¢ ciency worker (not necessarily the steady-state values).
We also assume that capital chases productivity di¤erentials 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 ‡ow .
0
0
M P K1 = M P K2 ) f (k1 + ) = f (k2
)
Using the assumption that the technology is Cobb-Douglas, we get
(k1 + )
1
=
(k2
)
1
)
=
k2 k 1
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 k_ 2 = k_ 1 =
_ otherwise, it would be possible for the values of k to be di¤erent accross
k,
economies. Thus, the modi…ed laws of motion can be written as:
k_ 1 = s1 f (k1 )
(n + )k1 + F and k_ 2 = s2 f (k2 )
(n + )k2
F
where F is a strictly non-negative capital ‡ow, since economy 2 always saves
more and would tend to move faster towards the steady state equilibrium. The
magnitude of the ‡ow can be calculated using the previous equations:
F =
s2 s1
2 f (k)
We can then …nd the steady state to which both economies move jointly:
4
k_ 1 = 0 ) s1 k1
) k1 =
(n + )k1 +
1 =2
)
( s2 +s
2
s2 s1
2 k1
=0
1
1
= k2
n+
Plugging this in the production function, we …nd that production per worker
is:
Ay = A
1 =2
( s2 +s
)
2
1
n+
Hence, both economies converge to the same output per worker
(b) Will the result in (a) continue to hold if countries have di¤erent
labor e¢ ciency levels?
From (a) we see that both economies always converge to the same level
of capital per e¢ ciency worker if they di¤er in the parameter s only. This
equilibrium is not a¤ected by the economies having a di¤erent level of worker
e¢ ciency, A.
Output per worker, nevertheless, will be di¤erent and the di¤erence in A
(constant for each economy in this question) rescales output per worker.
yi = Ai
( s2 2 s1 )=2
n+
1
for i = 1; 2
2.4 (TBG) Rapid growth (convergence) could be due to rapid technological progress or rapid capital accumulation. After World
War II, Germany, France, and Italy grew faster in terms of income per capita than the United States. This happened although
the savings rate and the population growth rates in these countries were similar. Could this be explained by the destruction of
physical capital during WWII? How would you use data to see
whether your argument is right or wrong?
We rely on the growth accounting framework to answer the question:
Y = F (A; K; L) = AF (K; L) (Assume technology enters linearly for simplicity)
Y_
Y
=
_ (K;L)
_
_
AF
+ AFYK K + AFYL L
Y
=
_ K
_ L
AFK K
AFL L
A_
A+
Y
K+ Y
L
5
= Td
FP+
^
^
K K + LL
where i represents the share of income that goes to factor i, T F P represents
Total Factor Productivity and x
^ denotes the growth rate of variable x.
Finally, using the fact that with constant returns to scale techonology K +
L = 1;
Y^
^ = Td
L
FP +
K
^
K
^
L
This expression can be interpreted as saying that, in this framework, growth
in income per capita has two possible sources: (i) growth of total factor productivity (improvements in the techniques used to aggregate inputs) and (ii) capital
deepening (the excess of growth of capital stock over that of labor supply or,
equivalently, growth of capital per capita).
To focus initially on the e¤ect of physical capital only, we carry out the
analysis under the following assumptions:
(i) The same production function describes technology both in the US and
in Europe
(ii) Pre-war levels of e¢ ciency and output per capita were the same in both
regions
As seen in Figure 4.1, the economic e¤ect of the war can be understood
as a destruction of physical capital in per capita terms, without any alteration
in the fundamentals of the economy. It can be seen that, in this scenario,
growth in Europe needed to be greater than that of the US, strictly due to
capital accumulation. That is, the destruction of capital triggers always some
additional growth compared to steady state growth, given the parameters of the
economy.
But from the growth accounting equation, we see that growth of income
per capita has two possible sources, only one of which is capital accumulation.
Hence, we must also see whether TFP was increasing signi…cantly during this
period and, if this were the case, which of the two sources was behind rapid
European growth. Data should be used to assess this issue. In particular, we
could replicate Young’s estimations, but using the European region as a whole,
and thus identify the relative importance of each source.
(i) First, we could take the average values for the whole period 1945-1990
(as seen in class, Europe stops converging, to a certain extent, around 1990).
This will give a …rst indication of whether the nature of this growth was more
"transitional" (i.e., due to capital accumulation) or "long-run" (i.e., due to
technological progress).
(ii) It would also be interesting to divide the sample, as Young did, in …veor ten-year periods. This will allow us to see how the importance of each
source varies in time. If the hypothesis that capital accumulation was the main
factor is correct, we would observe a decrease in the importance of capital per
capita accumulation as Europe approaches its steady state. TFP’s behavior,
on the other hand, would be di¢ cult to predict. We would expect its relative
6
importance to increase as we move to the latter years of the sample, because it
is unlikely that Europe were able to develop technology too rapidly right after
WWII. In fact, if our assumption of equal pre-war e¢ ciency levels were correct,
it would be unlikely to observe that Td
F P EU R > Td
F P U S throughout the whole
sample, as this would imply that Europe’s overall level of e¢ ciency surpassed
that of the US.
In conclusion, if the Solow model describes adequately this economies, physical capital destruction should induce faster growth in Europe. In fact, if it is
true that European growth is mostly due to capital accumulation, we should
observe that a signi…cant part of it can be accounted for by capital deepening (most likely, for the earlier observations of the sample). However, if TFP
growth also proved to be a source of growth, we could not argue that growth
was uniquely due to capital deepening. As explained in the next question, both
factors seem to have been important.
2.5 Do you know how to use total factor productivity growth (the
Solow residual) to answer economic questions? Some argue that
the fact that Germany, France, and Italy grew faster than the
US in terms of income per capita after WWII has to do with
these countries catching up to modern technologies (developed
in the US and unavailable during WWII). How would you use
data to see whether this argument is right or wrong?
From the previous question we conclude that it is not unlikely that the
destruction of capital per capita would push Europe to start growing faster
than before, because this destruction moves the economy below its BGP.
Note, however, that data tells us that output per capita in Europe was
nowhere near that of the US (actually, the latter was roughly twice as large that
of some European economies in the interwar period). Thus, it is unlikely that
both economies could be charaterized by the same level of production e¢ ciency
(given that population growth and savings are assumed to be the same). Then,
since the gap was closed signi…cantly in the period under analysis, both capital
deepening and TFP growth may be needed as explanations.
What we would need to do in this case is an analysis similar to the one
proposed in the previous question, but with data disaggregated by country for
Europe. If the statement is true, we should observe TFP growing faster in
Germany, France and Italy than it did in the US, as these countries were catching
up faster than the US was developing new technologies. In fact, previous growth
accounting exercises show that growth in Europe came roughly equally from
both sources, while that of the USA relied much less on TFP growth. As
explained before, this must also imply that the US level of e¢ ciency was higher
before the war, which gives room for thinking that Europe was doing some
technological catch-up too (maybe even induced by the destruction of capital).
The last point is also related to the possibility of technological transfers
from the US to Europe as part of the reconstruction process after the war (in
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fact, this could help explain the di¤erence in TFP growth in both regions).
In this sense, perhaps a good proxy for this sort of transfer would be data on
the importance of capital-goods imports from the US in overall investment in
Europe (for the data and the estimations mentioned in this question, see Barro
and Sala-i-Martin, 1999, Economic Growth, Ch. 10)
2.6 (TBG) Changes in output per worker and output (income) per
capita due to changes in the participation rate (the fraction of
the population that is working). Consider a Solow economy
without technological progress that is on its balanced growth
path. Suppose that a fraction 0 < < 1 of the population works,
i.e. L = P , where P is the size of the population of the country
and L is the number of workers.
Assumption: a = 0 ) A
(a) Suppose now that the participation rate increases. What happens to output per worker and income per capita in the short
and in the long run? Why?
In the short run, output per worker drops because there are more workers,
each equipped with less capital than workers had before the shock. L is larger
K
) is smaller, resulting in lower output per worker (i.e., lower
but k (i.e., AL
f (k)). Once the increase in occurs, k begins to climb quickly as saving excedes
e¤ective depreciation (see Figure 6.1). As usual, k increases more slowly as it
approaches its steady state value and stops upon reaching it. Thus, in the
long run, output per worker is the exact same as it was before the increase in
. Output per capita (not output per worker), on the other hand, increases
discretely at the time of the shock as more people out of the total population
are producing. However, each of the workers is less productive than before, so
output per worker decreases discretely. Output per person continues to grow,
though at a slowing rate, until settling at a constant level in the long run. This
long run level is higher than the initial level, unlike that of output per worker
which reaches its initial level (see Figures 6.2-6.5 for the evolution of the relevant
variables).
(b) And what happens when the participation rate
decreases?
What, then, does the Solow model predict will happen to output
per worker in a society where the participation rate falls because
of an increase in the share of older people?
8
A drop in causes exactly the reverse e¤ect of the increase in . At the
time of the shock, capital per worker rises because the same stock of capital is
used by less workers (see Figure 6.6). Thus, output per worker rises as well.
However, e¤ective depreciation is greater than new investment and, right after
the shock, the capital stock starts declining until it slows and comes to rest at
its steady state level. Thus, output per worker ends also at its original level.
On the other hand, output per capita falls discretely at the time of the shock
and declines rapidly before slowing and reaching a constant long-term level that
is below its initial level (see Figures 6.7-6.10)
Thus, in the absence of technological progress, economies experiencing a
demographic shift towards many retirees per worker can expect lower per capita
income in the future. However, output per worker will be unchanged in the long
run.
2.7 In the Solow model, low savings rates are a cause for underdevelopment (remember that the savings rate is assumed to be
exogenous in the model); a low savings rate could, however, be
partly a response to low incomes. Consider a Solow economy
without technological progress and without population growth.
Suppose, however, that the savings rate is not constant. Instead,
the savings rate is a function of income per capita. The shape
of this function is shown in Figure 7.1. Show that now it is no
longer the case that two economies which are identical in everything except the initial level of capital per worker converge to
the same balanced growth path. Explain.
Assume the usual Solow economy with the following variations:
s=
s1 if k < k
¯
s2 if k > k
As seen in Figure 7.2, this situation determines two possible steady state
equilibria. Moreover, it is clear that, depending on the starting point, the economy will converge to one of those equilibria. The equilibria are stable in that,
once the conomy is located above or below the threshold, it always converges to
the corresponding equilibrium. Speci…cally, the equilibria are de…ned by:
k =
k1 : s1 f (k1 ) = k1
k2 : s2 f (k2 ) = k2
This can be thought of in terms of subsistence levels of income. We can
imagine that individuals in poorer countries devote a large share of their income
to consumption and other basic needs that, in general, do not acumulate capital.
Furthermore, it is safe to assume that consumption will not increase signi…cantly
9
in per capita terms past a certain threshold, since individuals start accumulating
relatively more capital. If y > f (k) individuals can devote a higher share of their
¯
income to buying capital goods or put their savings in the bank, making those
resources available for others to invest. In consequence, a low-income economy
is, in this model, doomed to a low-capital steady-state equilibrium, because it
will never be able to save enough to invest its way beyond k.
¯
2.8 Much of international productivity di¤erences must be explained
by international di¤erences in e¢ ciency. Suppose the following
economies can be represented by a Solow model with technological progress and population growth:
y(i)/y(USA)
1
0.864
0.453
0.233
0.048
USA
Canada
Argentina
Thailand
Cameroon
s(i)
0.204
0.246
0.144
0.213
0.102
n(i)
0.0096
0.0122
0.0141
0.0153
0.028
A(i)/A(US)
1
0.972
0.517
0.468
0.264
The …rst column gives the level of GDP per capita relative to the
US in 1997 (data are from the Penn World Tables). Assume that the
rate of technological progress a and the rate of capital depreciation
add up to 0.075. A(i)=A(US) is the level of e¢ ciency relative to the
US. Suppose also that the production function is Y = K (AL)1
with
equal to 1/3.
Derive the expression for income per capita in the balanced growth
path as a function of the savings rate, the population growth rate,
and the level of technology. Use the values in the Table above to
compute long run income per capita of all countries relative to the
US under two possible scenarios:
(a) The level of technology in these countries relative to the US
stays constant at its current value
Expression for income per capita in the BGP:
Production function: Y = K (AL)1
Law of motion of capital: K_ = sF (K; L)
De…ne k =
K
AL
10
K
Then, k_ =
=
sF (K;L)
AL
_
_
_
K:AL
K(AL+
LA)
(AL)2
K A_
A2 L
K
_
KL
AL2
=
K A_
A2 L
_
K
AL
= sf (k)
_
KL
AL2
(n + + a)k
In the long run, k_ = 0. Using the assumption of a Cobb-Douglas technology
and the assumption of a + = 0:075, we have:
1
k=
1
s
0:075+n
) income per capita equals yi = Ai :
si
0:075+ni
1=2
Assuming Ai =AU SA is constant at its current value:
1. CANADA:
yCAN
yU SA
=
ACAN
AU SA
h
= 0:972
:
h
sCAN =sU SA
(0:075+nCAN )=(0:075+nU SA )
0:246=0:204
(0:075+0:0122)=(0:075+0:0096)
i1=2
i1=2
= 1:05
In the same fashion, for the other countries:
2. ARGENTINA:
h
i1=2
sARG =sU SA
yARG
AARG
=
:
yU SA
AU SA
(0:075+nARG )=(0:075+nU SA )
i1=2
h
0:144=0:204
= 0:423
= 0:517
(0:075+0:0141)=(0:075+0:0096)
3. THAILAND
yT HA
yU SA
h
= 0:468
4. CAMEROON
yCM R
yU SA
h
= 0:264
i1=2
= 0:463
i1=2
= 0:1692
0:213=0:204
(0:075+0:0153)=(0:075+0:0096)
0:102=0:204
(0:075+0:028)=(0:075+0:0096)
(b) All countries have the same level of technology as the US in the
long run.
Assuming that, in the long run, Ai =AU S = 1,
1. CANADA:
yCAN
yU SA
=1
=
h
ACAN
AU SA
:
h
sCAN =sU SA
(0:075+nCAN )=(0:075+nU SA )
0:246=0:204
(0:075+0:0122)=(0:075+0:0096)
i1=2
11
i1=2
= 1: 081 6
2. ARGENTINA
h
i1=2
0:144=0:204
yARG
=
1
= 0:818 68
yU SA
(0:075+0:0141)=(0:075+0:0096)
3. THAILAND
h
i1=2
0:213=0:204
yT HA
= 0:989 0
yU SA = 1
(0:075+0:0153)=(0:075+0:0096)
4. CAMEROON
h
i1=2
0:102=0:204
yCM R
=
1
= 0:640 84
yU SA
(0:075+0:028)=(0:075+0:0096)
The result is expected. The US has a higher level of e¢ ciency (A) and since,
when calculating output per worker, A enters linearly, it pushes output per
capita in the US upwards, relative to that of the other countries. Therefore, if
we assume the e¢ ciency level to be equal in the long run equilibrium, then all
the ratios should be scaled up.
12
Figure 4.1
Figure 6.1
(n+d)k
(n+d)k
f(k)
sf(k)
sf(k)
k
k
k
kEUROPE
kshock
kus
Figure 6.2
kss
Figure 6.3
Ln L
Ln K/AL
n
n
tshock
t
tshock
t
Figure 6.4
Figure 6.5
Ln π Y/AL
= Ln Y/AP
Ln Y/AL
Y = F(K,AL)
L
Y
t
tshock
t
tshock
Figure 6.6
Figure 6.7
(n+d)k
Ln L
k
sf(k)
n
n
k
kss
kshock
tshock
t
Figure 6.8
Figure 6.9
Ln Y/AL
Ln K/AL
t
tshock
t
tshock
Figure 6.10
Figure 7.1
Ln π Y/AL
= Ln Y/AP
s(y)
tshock
t
y
y
Figure 7.2
(n+d)k
f(k)
sf(k)
k
k*1
k
k*2