SciencesPo
Master E&B
Coeurdacier Nicolas
Understanding the world economy
Sample quantitative exercise
Steady-state income per capita and the role of savings
More difficult for those willing to practice more quantitative models. Parts 1 and 2 are still fairly
close to the course. Part 3 is more difficult.
Output (Y ) is produced in an economy using physical capital (K) and labour input (L) according
to the following Cobb-Douglas production function1 :
Y = AK 0.4 L0.6
where A is total factor productivity (TFP). For simplicity, we consider variables per capita; then,
Y denotes output per capita, K denotes physical capital per capita and L = 1. We also assume to
simplify calculations that A = 1. The output per capita is then:
Y = K 0.4
Assume that saving which equals investment is equal to s% of output (Y ). The depreciation rate
of capital is 5% per period.
1) Explain why this economy converges towards a steady-state without growth.
Show that in the steady-state, we must have:
sY ∗ = 0.05K ∗
where (*) denotes the variable at the steady-state value.
2) Deduce from 1) that the steady-state capital stock and output per capita must verify:
K∗ = (
1
s
) 0.6
0.05
;
Y∗ =(
s 2
)3
0.05
3) Use excel to calculate Y ∗ for different values of s [choose at least 5 values between 0 and 1
(excluding 0)]. Plot on a graph the logarithm of Y ∗ (log(Y ∗ )) as a function of the logarithm of s
(log(s)) [x-axis: log(s); y-axis: log(Y ∗ )]. Comment. What is the slope of the straight line you obtain?
Part 2: A simple test of the Solow model
We propose a simple test of the theory by looking at the relationship between the log of income
per capita and the log of savings across countries.
We plot on a graph the log of income per capita as a function of the log of savings [x-axis:log(s);
y-axis: log(Y )] (see figure below).
1
This production function is consistent with a share of capital income in value added of 40%.
1
ln Y
12
y = 1.5628x + 4.6346
R2 = 0.4974
10
8
6
4
2
0
0.5
1
1.5
2
2.5
3
3.5
4
ln s
s in %
4) What is the slope obtained in the data? Compare the value of the slope obtained with real data
and the one predicted by the model (question 3). How would you explain the difference?
Part 3: A Solow model augmented with Human capital
We want to improve the theory by considering human capital (H) per capita as an input of
production (H denotes the education and the skills of a worker...). Output (Y ) is produced using
physical capital (K) and human capital (H) according to the following production function:
Y = AK 0.4 H 0.6
Like in the first part, we consider variables per capita; Y denotes output per capita, K physical
capital per capita and H human capital per capita.
We assume that when physical in the economy is higher, so is the level of human capital such that:
H = λK b
where λ and b are positive numbers. For instance, this happens because every time money is spent
on acquiring new machines it enhances the skills of the workers that use them.
2
5) Explain why with b = 1, allowing for the interaction between physical and human capital yields
a constant marginal product of capital schedule. Does the economy converge towards a steady-state
when b = 1?
We now assume b = 1/3. To simplify algebra, we also assume A = λ = 1.
6) Show that this implies the following relationship: Y = K 0.6 . Does the economy converge towards
a steady-state?
7) Following the same steps as in questions 1), 2) and 3), compute the steady-state capital stock
(K ∗ ) and the steady-state output per capita (Y ∗ ) as a function of the savings rate and use excel to
calculate Y ∗ for different values of s. Plot on a graph the logarithm of Y ∗ (log(Y ∗ )) as a function of
the logarithm of s (log(s)) [x-axis: log(s); y-axis: log(Y ∗ )]. What is the slope of the straight line you
obtain? Compare with the data. Comment briefly.
3
Correction
Part 1: The standard Solow model
Output (Y ) is produced in an economy using physical capital (K) and labour input (L) according
to the following Cobb-Douglas production function2 :
Y = AK 0.4 L0.6
where A is total factor productivity (TFP). For simplicity, we consider variables per capita; then,
Y denotes output per capita, K denotes physical capital per capita and L = 1. We also assume to
simplify calculations that A = 1. The output per capita is then:
Y = K 0.4
Assume that saving which equals investment is equal to s% of output (Y ). The depreciation rate
of capital is 5% per period.
1) Explain why this economy converges towards a steady-state without growth.
Show that in the steady-state, we must have:
sY ∗ = 0.05K ∗
where (*) denotes the variable at the steady-state value.
The model converges towards a steady-state because of diminishing marginal productivity of capital
dY
(MPK): dK
= 0.4K −0.6 is decreasing with the stock of K.
The capital accumulation equation between t and t + 1 tells us:
Kt+1 = Kt + It − 0.05Kt
In the steady-state K ∗ = Kt+1 = Kt , then:
I ∗ = 0.05K ∗
where (*) denotes the variable at the steady-state value.
Aggregate investment must equal aggregate savings, so I ∗ = sY ∗
sY ∗ = 0.05K ∗
2) Deduce from 1) that the steady-state capital stock and output per capita must verify:
K∗ = (
1
s
) 0.6
0.05
;
Y∗ =(
s 2
)3
0.05
sY ∗ = 0.05K ∗ gives:
s (K ∗ )0.4 = 0.05K ∗
s
(K ∗ )0.6 =
0.05
1
s
∗
K = (
) 0.6
0.05
2
This production function is consistent with a share of capital income in value added of 40%.
4
Using the production function gives Y ∗ .
Y ∗ = (K ∗ )0.4 = (
s 2
)3
0.05
3) Use excel to calculate Y ∗ for different values of s [choose at least 5 values between 0 and 1
(excluding 0)]. Plot on a graph the logarithm of Y ∗ (log(Y ∗ )) as a function of the logarithm of s
(log(s)) [x-axis: log(s); y-axis: log(Y ∗ )]. Comment. What is the slope of the straight line you obtain?
ln Y
2.5
2
y = 0.6667x - 1.073
1.5
1
0.5
0
ln s
1.5
2
2.5
3
3.5
4
4.5
5 s in %
Obviously taking log of the previous expression gives:
ln(Y ∗ ) =
2
2
ln(s) − ln(0.05)
3
3
So the slope on the graph must be 2/3. In the steady-state, the only difference in income per
capita is driven by differences in savings: higher savings increases the steady-state capital stock,
which increases income per capita.
Part 2: A simple test of the Solow model
We propose a simple test of the theory by looking at the relationship between the log of income
per capita and the log of savings across countries.
We plot on a graph the log of income per capita as a function of the log of savings [x-axis:log(s);
y-axis: log(Y )] (see figure below).
5
ln Y
12
y = 1.5628x + 4.6346
R2 = 0.4974
10
8
6
4
2
0
0.5
1
1.5
2
2.5
3
3.5
4
ln s
s in %
4) What is the slope obtained in the data? Compare the value of the slope obtained with real data
and the one predicted by the model (question 3). How would you explain the difference?
Slope = 1.56. The slope is higher than the one predicted by the model.
GDP per capita and savings are positively related as predicted by the model. Obviously the
correlation is not perfect like in the model because many other factors than the savings rate affect
income per capita (although differences in savings still explain a large part of the variability in income
per capita, R2 close to 50%).
The slope is too high compared to what the simple Solow model predicts (1.5 versus 2/3). In other
words in the data the impact of savings on income per capita is larger than what is predicted by the
model.
This simple model does not account for many variables that are both correlated with
the savings rate and income per capita. For instance, in countries where TFP is high, the savings
rate might be higher (because of higher return on capital) and so is the income per capita. Another
potential missing variable is human capital: when savings are high, so is capital. Higher capital stock
raises the return on education and increases human capital. As a consequence income per capita
is even higher and the impact of savings is reinforced. Development of financial markets can be an
another obvious candidate (increases both savings and income per capita).
6
Part 3: A Solow model augmented with Human capital
We want to improve the theory by considering human capital (H) per capita as an input of
production (H denotes the education and the skills of a worker...). Output (Y ) is produced using
physical capital (K) and human capital (H) according to the following production function:
Y = AK 0.4 H 0.6
Like in the first part, we consider variables per capita; Y denotes output per capita, K physical
capital per capita and H human capital per capita.
We assume that when physical in the economy is higher, so is the level of human capital such that:
H = λK b
where λ and b are positive numbers. For instance, this happens because every time money is spent
on acquiring new machines it enhances the skills of the workers that use them.
5) Explain why with b = 1, allowing for the interaction between physical and human capital yields
a constant marginal product of capital schedule. Does the economy converge towards a steady-state
when b = 1?
With b = 1, we have:
Y = AK 0.4 H 0.6 = AK 0.4 (λK)0.6 = Aλ0.6 K
dY
Then : dK
= Aλ0.6 = cst. We have a constant marginal productivity of capital. The interaction
beween physical and human capital generates constant return to capital (even if each factor exhibits
decreasing MPK). This prevents convergence towards a steady-state. To see this, rewrite the capital
accumulation equation:
Kt+1 − Kt = It − 0.05Kt = sYt − 0.05Kt = sAλ0.6 Kt − 0.05Kt
or rewriting in terms of the growth rate of K (=gK ):
gK =
Kt+1 − Kt
= sAλ0.6 − 0.05
Kt
Assuming sAλ0.6 −0.05 > 0, the capital stock grows without bound at a constant (positive) growth
rate: a case of endogenous growth.
We now assume b = 1/3. To simplify algebra, we also assume A = λ = 1.
6) Show that this implies the following relationship: Y = K 0.6 . Does the economy converge towards
a steady-state?
b = 1/3 and A = λ = 1.implies:
Y = AK 0.4 H 0.6 = K 0.4 K 1/3
0.6
= K 0.6
Yes convergence towards a steady-state because decreasing MPK!
7
Saving which equals investment is still equal to s% of output. The depreciation rate of capital is
5% per period.
7) Following the same steps as in questions 1), 2) and 3), compute the steady-state capital stock
and the steady-state output per capita (Y ∗ ) as a function of the savings rate and use excel to
calculate Y ∗ for different values of s. Plot on a graph the logarithm of Y ∗ (log(Y ∗ )) as a function of
the logarithm of s (log(s)) [x-axis: log(s); y-axis: log(Y ∗ )]. What is the slope of the straight line you
obtain? Compare with the data. Comment briefly.
(K ∗ )
In the steady-state:
sY ∗ = 0.05K ∗ and s (K ∗ )0.6 = 0.05K ∗
1
s
s
or: K ∗ = (
) 0.4
(K ∗ )0.4 =
0.05
0.05
And:
Y∗ =(
s 0.6
s 1.5
) 0.4 = (
)
0.05
0.05
Then taking log-, we have:
ln(Y ∗ ) = 1.5 [ln(s) − ln(0.05)]
The slope now must be equal to 1.5! Allowing for a broader definition of capital (human and
physical capital) makes the model more consistent with the data (in the data we found 1.56). The
reason is the following: the interaction between human and physical capital reinforces the
impact of savings on income per capita; higher savings, implies higher steady-state physical
capital stock which increases income per capita (standard effect of the Solow model). But on the top
of that, higher capital stock raises human capital which in turns raises even more income per capita.
8
ln Y
9
8
y = 1.5x + 1.0397
7
6
5
4
3
2
1
0
1.5
2
2.5
3
3.5
9
4
4.5
5
ln s
s in %