Intermediate macro

Solutions #9

Tutorial 9: week starting October 12th

Solutions

Solutions to pre-tutorial problems

1. You should be able to tell whether the following statements are true, false or uncertain and also
be able to justify your answer.

(i) Empirical evidence shows that countries with better institutions have higher real GDP per
capita. Therefore, better institutions lead to higher productivity.
(ii) The capital income share has declined in recent decades in major advanced countries.
(iii) With competitive firms and the constant return to scale Cobb-Douglas production function,
a profit maximizing firm would demand for capital if the amount of labor input decreases.

Solutions:

(i) Uncertain. It is true that countries with better institutions have higher real GDP per
capita, but it is less clear whether better insititutions lead to higher GDP per capita or
higher GDP per capita promotes better institutions.
(ii) True.
(iii) False. A profit maximizing firm chooses the amount of capital and labor inputs to maximize
its profit. The demand for capital is increasing in the amount of labor input. This is
because more labor input raises the marginal product of capital, which leads to a higher
demand for capital. Capital and labor are complements in the production function.

2. Figure 1 depicts the path of Brazil’s real GDP per person in the period 1950–2003.
The graph reveals that growth slowed significantly over this period. Your task is to use growth
accounting to understand why. Here are some data that you will find useful:

Year GDP Y Capital K Population P Employment N Education H

1960 190 380 72 25 2.9

1975 610 1,260 110 40 3.0
2003 1,310 3,190 180 85 5.1

GDP and capital are measured in billions of 2000 US dollars (PPP basis). Population and em-
ployment (labor) are in millions of people. Education is measured as average years of schooling.
Intermediate macro: Solutions #9 2

real GDP per person (in 2000 US dollars, PPP basis)

Figure 1: Slowdown in Brazilian economic growth.

(i) Compute the average annual growth rates of output per person Y /P and output per worker
Y /N over the period 1960–2003. Are these growth rates different? Why or why not?
(ii) Suppose that Brazil has a production function

Y = AK α (HN )1−α

with α = 1/3. Use the growth accounting methodology to allocate growth in output per
worker Y /N over the period 1960–2003 to growth in total factor productivity A, capital
K, employment N and education H. What factors are most important?
(iii) Consider the sub-periods 1960–1975 and 1975–2003 in isolation. What are the main dif-
ferences between the two? What may have been the causes of such differences?

Solutions:

(i) Average annual growth in output, population and employment over the 43 years 1960–2003
are given by, respectively

ln(Y2003 /Y1960 ) ln(1310/190) 1.93

gY = = = = 0.0449
43 43 43
ln(P2003 /P1960 ) ln(180/72) 0.92
gP = = = = 0.0213
43 43 43
ln(N2003 /N1960 ) ln(85/25) 1.22
gN = = = = 0.0285
43 43 43
Hence average annual growth in output per person was

gY − gP = 0.0449 − 0.0213 = 0.0236

Intermediate macro: Solutions #9 3

or 2.36% annual. Similarly, average annual growth in output per worker was

gY − gN = 0.0449 − 0.0285 = 0.0164

or 1.64% annual. Growth in output per person was quite a bit faster than growth in
output per worker. Why? Because population growth was slower than labor force growth,
probably because of demographics.
(ii) From the given production function

gY = gA + αgK + (1 − α)(gH + gN )

Following the same approach as in (i) above

ln(K2003 /K1960 ) ln(3190/380) 2.13

gK = = = = 0.0495
43 43 43
ln(H2003 /H1960 ) ln(5.1/2.9) 0.56
gH = = = = 0.0131
43 43 43
Which implies productivity growth was

gA = gY − αgK − (1 − α)(gH + gN )

1 2
= 0.0449 − (0.0495) − (0.0285 + 0.0131)
3 3

= 0.0007

Growth in output per worker is

gY − gN = gA + α(gK − gN ) + (1 − α)gH

Using our various calculations gives the decomposition

1 2
gY − gN = 0.0164 = 0.0007 + (0.0495 − 0.0285) + (0.0131)
3 3
or
gY − gN = 0.0164 = 0.0007 + 0.0070 + 0.0088
The main contribution comes from human capital accumulation, namely 0.0088/0.0164 =
0.54 or just over half of the growth in output per worker is accounted for by human capital
accumulation. After that, physical capital accumulation is the next most important factor.
The contribution from TFP is practically zero.
(iii) Computing the various annual growth rates we have that for 1960–1975, output per worker
grew at 4.64%, physical capital per worker at 4.86%, human capital at 0.23% and TFP
at 2.87%. By contrast, for 1975–2003, output per worker grew at 0.04%, physical capital
per worker at 0.63%, human capital at 1.90% and TFP at −1.43%. So there was a
significant slowdown mostly because physical capital growth slew down and TFP growth
went negative. This was partly offset by faster human capital accumulation.
Intermediate macro: Solutions #9 4

3. Consider output per person in France and the US. In France, output per capita is lower, but
since a smaller fraction of the population works and each worker works fewer hours, output per
hour worked is not much different.
Your goal is to explain these differences using a levels decomposition and the following data:

Country GDP Y Capital K Employment N Hours h Population P

France 1,350 3,850 28 1500 59

US 9,170 19,600 142 1830 285

GDP and capital are reported in billions of 2000 US dollars (PPP basis) while employment
and population are in millions. Hours is the average number of hours per year worked by an
employed person. Total hours worked can be computed as the product of employment (the
number of people working) and hours (the number of hours per worker).

(i) Compute output per person, output per worker, and output per hour worked. How do
they differ?
(ii) Suppose both countries have a production function of the form

Y = AK α (N h)1−α

where N h is total hours worked and where α = 1/3. Use a levels decomposition to
determine the primary sources of the difference in output per person. Why do you think
hours per worker and the ratio of employment to population are lower in France than in
the US?

Solutions:

(i) GDP per person, per worker and per hour worked are simply the ratios Y /P , Y /N and
Y /(N h), respectively. The numbers work out to be:

GDP per person Y /P per worker Y /N per hour Y /(N h)

France 22,881 48,214 32.14

US 32,175 64,577 35.29
Ratio 0.711 0.747 0.911

Note that the ratios (the bottom line) increase as we move to the right: French output per
hour is much closer to US output per hour than is French output per worker.
(ii) We are asked to account for differences in output per person. This is

Y NY
=
P PN
Intermediate macro: Solutions #9 5

where N/P is the fraction of the population that is employed and where output per worker
is, from the the production function,
 α
Y K
=A h1−α
N N
In short, we attribute differences in output per person first to differences in the employment
participation and differences in output per worker and then further break down differences
in output per worker into differences in capital per worker, hours per worker, and TFP. In
particular, output per person can be decomposed as

Y N Y
ln
= ln + ln
P P N
and output per worker can be decomposed as
Y K
ln = ln A + α ln + (1 − α) ln h
N N
The numbers work out to be:

Y /P N/P Y /N K/N h A

France 22,881 0.474 48,214 137,500 1500 7.13

US 32,175 0.498 64,577 138,028 1830 8.35

Ratio 0.711 0.952 0.747 0.996 0.820 0.854

Log difference −0.341 −0.049 −0.292 −0.00383 −0.199 −0.158
Contribution −0.341 −0.049 −0.0013 −0.133 −0.158
| these add{zto −0.292 }

To summarise, output per person is 34.1% below that of the US. Of this, about 4.9% comes
France’s lower employment participation and the other 29.2% comes from lower output per
worker. Of that 29.2%, about 15.8% comes from lower TFP and 13.3% from lower hours
with differences in capital intensity being negligible. In short, TFP is the most important
source of difference in output per person, accounting for about 15.8/34.1 = 0.46 of the
difference followed by hours, accounting for 13.3/34.1 = 0.39 of the difference, followed by
employment participation, accounting for 4.9/34.1 = 0.14 of the difference, with capital
intensity accounting for the (tiny) remainder.

Solutions to in-tutorial problems

1. Make sure that you understand the problems set in the pre-tutorial sheet for this week’s tutorial.
Ask others in your group if you are still unsure about any of the pre-tutorial sheet problems.

2. Consider a Solow growth model with Cobb-Douglas production function

Y = K α (AN )1−α
Intermediate macro: Solutions #9 6

with constant savings rate s, depreciation rate δ and no growth in productivity or labor (gA =
gN = 0).
(a) Suppose A = 1, α = 1/3, s = 0.2 and δ = 0.1 (annual). Calculate the steady state capital
per worker and steady state output per worker.
(b) Suppose that the real wage w and real return to capital r are equal to the marginal products
of labor and capital respectively. Calculate the steady state wage rate and return to capital.
(c) Now suppose the saving rate increases to s = 0.25. What happens to the steady state w
and r? Do they rise or fall? Give intuition for your results.
(d) What if we still have s = 0.2 but productivity increases by 10% from A = 1 to A = 1.1.
How does this change the wage rate and return in the short run? What about the long
run? Again, give intuition for your results.
(e) Now suppose that gA = gN = 0.05 and we still have s = 0.2. Calculate the steady state
capital per effective worker and steady state output per effective worker. What is the long
run growth rate of the wage rate? What is the long run growth rate of the return to
capital? Explain.
Solutions:
Let k = K/AN and y = Y /AN denote steady state capital and output per effective worker.
Using the Cobb-Douglas production function, these are related by
y = k α ≡ f (k)
The steady state condition is
sy ∗ = (δ + gA + gN )k ∗
In terms of the capital per effective worker, the steady state condition is
sk α = (δ + gA + gN )k
Solving this gives
 1
 1−α
∗ s
k = (1)
δ + gA + gN
And so steady state output/worker is
 α
 1−α
∗ ∗ ∗ α s
y = f (k ) = (k ) = (2)
δ + gA + gN
The marginal productivity conditions can be written in per effective worker terms as
w = (1 − α)AK α (AN )−α
and it follows that
w = (1 − α)Ak ∗α = (1 − α)Ay ∗
For the return to capital
y∗
r = αK α−1 (AN )1−α = αk ∗α−1 = α
k∗
Once k and y are in the steady state, the growth rate of w is the same as the growth rate of A,
which is gA and r is a constant.
Intermediate macro: Solutions #9 7

(a) With gA = gN = 0, A = 1, α = 1/3, s = 0.20 and δ = 0.10, from (1) steady state capital
per worker is
  32
0.2
k∗ = = 2.828
0.1
From (2) steady state output/worker is
1
y ∗ = 2 2 = 1.414

(b) With α = 1/3, s = 0.20 and δ = 0.10, the return on capital is

r = (1/3)(1/2) = 0.167

or about 16.7% annual. Similarly, the wage rate is

w = (1 − α)Ay ∗ = (2/3)(1)(1.414) = 0.943

(c) Since
(δ + gA + gN )
r=α
s
an increase in the savings rate s lowers the return to capital (as the supply of capital
increases). With the given numbers, the return falls from r = 0.167 (when s = 0.2) to
(1/10)
r = (1/3) = 0.133
(1/4)
or about 13.3% annual when s = 0.25. The increase in capital delivered by the higher
saving rate also increases the demand for labor and the wage rate increases. In particular,
with the given numbers, the wage rate increases from 0.943 (when s = 0.2) to
1
w = (2/3)(1)(2.5) 2 = 1.054

(d) As above, the return to capital and wage rate are, in steady state,
(δ + gA + gN )
r=α
s
and  α
 1−α
s
w = (1 − α)A
δ + gA + gN
The increase in productivity has no effect on the return to capital in steady state (in the
long run). While the increase in productivity increases the demand for capital, the increase
in productivity also increases capital accumulation so that in the long run the supply of
capital rises too. In fact, the long run supply curve for capital is perfectly horizontal
(completely elastic). The wage rate rises as the increase in productivity and the resulting
increase in capital both increase the demand for labor (capital and labor are complements
in production). With the given numbers, the new wage rate is
1
w = (2/3)(1.1)(2) 2 = 1.037

which is indeed higher than the w = 0.943 when productivity is only A = 1.

Intermediate macro: Solutions #9 8

(e) Using the steady state equation for k, we can find

  32
∗ 0.2
k = =1
0.1 + 0.05 + 0.05

and
1
y∗ = 1 2 = 1
Notice that the long run growth rate of w is given by gA , so the growth rate of w is 0.05.
The return to capital is constant in the long run, so the changes in gA and gN do not affect
the long run growth rate of r.