Answers

1. Output and capital per capita of a country can be written as
, where Y and K are total output and capital of a country, and L is the size of their
workforce (population).
Let’s denote the growth rates the following:
gK – growth rate of K
gY – growth rate of Y
n – growth rate of Y
a) Kaldor Fact 1 states that k grows at a constant rate, call it . Kaldor Fact 2 states that
K/Y is constant over time, so assume that call it . Show that Kaldor Facts 1 and 2
imply that output per capita, y, grows at the same rate as capital per capita k.
Kaldor Fact 1:
The growth rate of k can be rewritten as
(
)
Kaldor Fact 2:
Let’s analyze what happens to this ratio after t periods:
The only way for the ratio to stay constant is where
the ratio
, when these terms cancel out and
stays constant over time.
The growth rate of y can be rewritten as
(
As the Kaldor Fact 2 implies,
capital per capita k.
)
, therefore, output per capita y grows at the same rate as
b) Now forget about part (a). As in class, suppose Y is produced according to
. Show that the logarithm of output per capita can be written as
Rewriting the production function in per capita terms yields
( )
By taking the logarithm on both sides we obtain the asked expression
c) Suppose both A and k grow at rate g. Show that this implies that the growth rate of y is
also g, and that K/Y is constant over time.
The growth rate of y can be expressed as
[
(
] [
]
)
Let’s analyze what happens to the ratio after t periods:
By multiplying this expression by we obtain
No matter how many periods have passed, the ratio is equal to the ratio of initial values K and Y,
thus the ratio is constant over time.
2. (based on DW, pp78-79, CJ p52) A country has the APF Y =
, and the savings rate is
s = .3. The depreciation rate is = .06 and g = 0.02. Assume there is no population
growth (n = 0) and
= 1. For questions (b) and (c) assume
instead
of ̇
- i.e., assume discrete time rather than continuous time.
(a) What are the steady-state levels of capital per worker, k = K/L, and output per worker,
y = Y/L?
First rewrite the APF in terms of effective unit of worker:
̂
(
) (
̂
)
Remember the Law of Motion is given by:
̇
Taking the logarithm and a derivative with respect to t we obtain
̇
̂
̂
Given the definition of the steady-state level of capital per effective unit of worker derive the
closed-form solution:
̂
(
)
Using the given values for parameters we get the steady-state level of capital per effective
unit of worker
̂
(
)
Hence, level of capital per worker associated with this steady state given level of technology
can be computed as
̂
Plugging in ̂ into the production function we get the steady-state level of output per
effective unit of worker
̂
̂
In the same way we compute the level of output per worker given the level of technology:
̂
(b)-(c) Compute the growth rate of output per worker between years 0 and 1, between years
9 and 10. What is happening to the speed of growth?
The Law of Motion in discrete time is
To obtain the expression in effective terms we divide both sides by
̂
Notice that
and
Substituting this result and ̂
terms:
̂
̂
̂
, so
̂
into the expression above, we obtain the LOM in effective
̂
̂
̂
Create a table that shows how capital and output per effective unit of worker evolves over time
according to our model. The initial capital per effective unit of worker is ̂
. Fill it up to
year 10.
Year
0
1
2
3
4
5
6
7
8
9
10
Capital ̂
1.00
1.22
1.43
1.65
1.87
2.09
2.30
2.51
2.71
2.91
3.10
Output ̂
1.00
1.07
1.13
1.18
1.23
1.28
1.32
1.36
1.39
1.43
1.46
̂
Now, we want to compute the growth rate of output per worker (
), that is
In the table above we have computed the output per effective unit of worker, ̂ . Note that
̂
. Then output per worker can be written as ̂
.
Substituting this in our expression for the growth rate of output per worker
̂
̂
̂
̂
̂
̂
̂
Thus we get that
we get
Note that as we approach the steady state level, not only the growth rate converges to 0.02, but
also the speed of convergence is slowing down.
Now suppose that suddenly, g increases to 0.10.
(a) What is the new steady state output per worker?
Following the same procedure as before we have that
̂
(
)
(
)
̂
̂
̂
̂
(b) How much of the increase in the growth rate of output per worker is due to capital per
worker growth, and how much is due to productivity growth?
Consider the Cobb-Douglas production function in general form:
Taking the logarithm and the derivative with respect to time, we can rewrite the latter expression
in terms of growth rates:
̇
̇
̇
⏟
⏟
⏟
Therefore we can decompose growth in output in capital per worker and productivity growth:
̇
̇
⏟
⏟
̇
⏟
Note that along the balanced growth path
Thus we have that 1/3 of the growth in output per worker is due to capital per worker growth and
2/3 due to productivity growth.
3. (the Golden Rule) Based on the model we learned in class with population growth n and
technology growth g,
(a) Write down an expression for consumption per worker, and find the savings rate, s =s*,
that will maximize consumption per worker on the balanced growth path. This is the socalled “golden rule.”
⏟
⏟
Expressing consumption in per worker terms
Consumption per effective unit of worker is
̂
̂
We can now substitute in for ̂
̂, where ̂ corresponds to the steady state level of
output per effective unit of worker: ̂
̂
̂
(
[
)
. We thus have
]
[
⏟
]
First order conditions with respect to s:
[
]
(b) Suppose that we are at a BGP and that the savings rate is s = s0 < s*. Suddenly, s jumps
to s* — i.e., people start saving more. What happens to consumption per worker in the
short run? What happens in the long run? Explain graphically and mathematically.
For this question, we ignore technology At, as it just shifts the horizontal steady state line to a
positive slope line graphically.
In the short run, consumption per worker will decrease. To explain it, we start with c(t)=(1s)y(t). Let’s take the derivative with respect to time and evaluate it at the point when s0 jumps to
s* by applying the chain rule, which gives
where
Then the function becomes
.
Notice that in a very short period after the rise of savings rate, output won’t change immediately,
which makes y’(t) approximately 0. And
, so
which means in the short run,
consumption per worker will decrease. Intuitively, with an almost fixed production level, in
order to save more, people have to consume less (see Figure 1).
In the long run the economy moves to the new steady state consumption level, which is given by
[
]
. We can obtain how consumption per worker varies with savings rate at the
BGP (Figure 2). Remember the definition of golden rule savings rate, which is defined to be the
maximizer of consumption per worker. So apparently consumption per worker will be higher in
the long run. To make it more clear, higher savings accumulates more capital stock and thus
production in the new steady state is higher. Because the initial steady state is below the golden
rule level, the increase in savings eventually leads to a higher level of consumption.
Figure 1. Transition
Figure 2. Consumption/savings rate at the steady state
(c) Given your answer above, why may the golden rule in fact not be golden unless you
are already saving at s = s*?
There is a direct tradeoff between consumption and saving.
If taking the example stated in part (b), although in the long run, we can have higher production
level and higher consumption level, it happens at the cost of short run's consumption reduction,
which means we are sacrificing current generations’ consumption and utility level to have a
better future result. Whether such tradeoff between a short-run and long-run consumption is
desirable will depend on how the agent weights the short run versus the long run.
If the agents are impatient, so that they are not willing to sacrifice current consumption in order
to have more consumption in the future, a smaller savings rate than sgold would be preferred.
Now let's look at the other case, where an economy with a savings rate higher than the golden
savings rate (sgold), that is with s > sgold. Suppose that starting from the steady state, the savings
rate is reduced permanently to sgold. Then, consumption initially increases by a discrete amount.
Then the level of c falls monotonically during the transition towards its new steady state value,
c*gold. Since c* < c*gold, when s > sgold the economy is oversaving in the sense that per capita
consumption at all points in time could be raised by lowering the saving rate. An economy that
oversaves is said to be economically inefficient, because the path ofper capita consumption lies
below feasible alternatives paths at all points in time. But on the other hand, the reduction of the
savings rate would imply a lower steady state capital and thus a lower output.
4. (DW p182) In the United States, average hourly earnings of production workers in 2007
were $17.45/hour. The national minimum wage was $5.85/hour. If the minimum wage
represents what a worker would receive with no human capital from education, what is
the share of wages that represents payments to human capital and the share that
represents payments to raw labor?
The average wage (w) can be decomposed into two components: payments to raw labour (wmin)
and payment to human capital (wh):
The share that represents payments to raw labour can be computed as
The share that represents payments to human capital can be computed as
5. (based on DW p208) Suppose that a country’s output per capita can be written as
where A is total factor productivity, k is physical capital and h is human capital. Assume that
. Based on the following table, compute the growth rates of productivity and factor
accumulation. In which country does factor accumulation contribute the most to growth? In
which country does productivity contribute the most to growth? Which factor is more important
in each country?
By taking the logarithm and the derivative we can express the production function in terms of
growth rates
̇
̇
̇
̇
⏟
⏟
Append the given table with additional calculations
̇
̇
̇
̇
Production
̇
factors
growth
How to
2
3
4
3+4
calculate
Argentina 0.92 0.46 0.37 0.15 0.25
0.40
Austria
1.10 2.82 0.26 0.94 0.17
1.11
Chile
2.54 0.85 0.73 0.28 0.49
0.77
⏟
Share
̇
Share
⏟
0.43
1.01
0.3
2(3+4)
0.52
-0.01
1.77
0.57
-0.01
0.7
Factor accumulation contributes the most to growth in Austria (101%).
Productivity contributes the most to growth in Chile (70%).
To see how much each factor contributes to production factors growth fill the following table
Country
Share of k in PF
Share of h in PF
More important
growth
growth
factor
Argentina
0.375
0.625
Human capital
Austria
0.85
0.15
Physical capital
Chile
0.36
0.64
Human capital