Problem Set 4
FE411 Spring 2008
Rahman
Some Answers
1) An Alternative Production Function with Human Capital
Here’s a production function with a slight twist from the book - consider a Cobb-Douglas
production function with three inputs. K is capital (the number of machines), L is labor
(the number of workers), and H is human capital (the number of college degrees among
the workers). The production function is:
Y  K 1/ 3 L1/ 3 H 1/ 3
a. Derive an expression for the marginal product of labor. How does an increase in
the amount of human capital affect the marginal product of labor?
MPL = 1/3*K1/3H1/3 L-2/3. Increase in human capital raises MPL.
b. Derive an expression for the marginal product of human capital. How does an
increase in the amount of human capital affect the marginal product of human
capital?
MPH = 1/3*K1/3H-2/3 L1/3. Increase in human capital lowers MPH.
c. What is the income share paid to labor? What is the income share paid to human
capital? In the national income accounts of this economy, what share of total
income do you think all workers would appear to receive?
1/3, 1/3, and 2/3, respectively.
d. Say an unskilled worker earns the marginal product of labor, whereas a skilled
worker earns the marginal product of labor plus the marginal product of human
capital. Using your answers to (a) and (b), find the ratio of the skilled wage to the
unskilled wage. How dies an increase in the amount of human capital affect this
ratio? Explain.
Wskilled/Wunskilled = (MPL + MPH) / MPL = 1 + (L/H). When H increases,
this ratio falls because the diminishing returns to human capital lower its
return, while at the same time increasing the marginal product of
unskilled workers.
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Problem Set 4
FE411 Spring 2008
Rahman
e. Some people advocate government funding of college scholarships as a way of
creating a more egalitarian society. Others argue that scholarships help only those
who are able to go to college. Do your answers to the preceding questions shed
light on this debate?
If more college scholarships increase H, then it does lead to a more
egalitarian society. The policy lowers the returns to education,
decreasing the gap between the wages of more and less educated workers.
More importantly, the policy even raises the absolute wage of unskilled
workers because their marginal product rises when the number of skilled
workers rises.
2) The Solow Growth Model with Growing Efficiency of Labor
Assume that production is a function of capital and effective labor, and that the rate of
savings, depreciation, population growth, and labor-augmenting technological progress
are all constant, as described in Chapter 8’s version of the Solow Model. Further, assume
that the production per effective worker can be described by the function:
y  f (k )  k
(1 )
2
where k is capital per effective worker.
a. Using the functional form for y, write the functions for both consumption and
investment in the Solow Model.
i = sk0.5
c = (1-s)k0.5
b. Write the Law of Motion of Capital (using the functional form for y). What
condition for the Law of Motion of Capital must hold for the economy to be in a
steady state? Show the steady state condition on a graph with the investment function
and the depreciation of capital per effective worker.
Δk = sk0.5 – (δ+n+g)k
For steady state, this expression must equal zero. [Graph not shown, see book &
notes].
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Problem Set 4
FE411 Spring 2008
Rahman
c. Solve for the steady-state value of y as a function of s, n, g, and δ.
sk*0.5 = (δ+n+g)k*. Solving for k* gives us k* = (s/( δ+n+g))2. Plugging this
into our production function gives us:
y* = s/(n+δ+g)
d. A developed country has a saving rate of 28% and a population growth rate of 1%
per year. A less-developed country has a saving rate of 10% and a population growth
rate of 4% per year. In both countries, g = 0.02 and δ = 0.04. Find the steady-state
value of y for each country.
Plug into equation above.
Developed country: y* = 4
Developing country: y* = 1
e. What policies might the less-developed country pursue to raise its level of income?
Induce savings: tax incentives, balancing budget, etc.
Induce reduction in population growth: explicit laws, tax incentives, etc.
f. Explain why the savings rate in an economy is so important for the steady state. If
the less-developed country increases its savings rate, show on the graph from b what
happens to the investment function and the steady state level of capital. What is the
cost today of increasing the savings rate? What are the benefits of doing so?
Savings translates into permanent increases of living standards, through raising
capital stocks per person. On the graph [not shown], you would draw a new
investment line, and illustrate a higher k* and y*.
The cost of raising s is a lower consumption per person today. The benefits come
in the form of higher output per person. Whether or not this also translates into
higher long-run consumption per person is uncertain.
g. Write the Golden Rule for maximizing consumption. Use the Golden Rule to
solve for the Golden level of steady state k, y, c, and i.
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Problem Set 4
FE411 Spring 2008
Rahman
The Golden Rule in this problem is where MPK = n + δ + g. Using our
functional form, this means that MPK = 0.5k*-0.5 = n + δ + g. Solving for this capital
per person level, we find that:
k*(g.r.) = (0.5/(n+δ+g))2.
This means that y*(g.r.) = (0.5/(n+δ+g)). c*(g.r.) = (1-s)(0.5/(n+δ+g)).
i*(g.r.) = s(0.5/(n+δ+g)).
Notice that the golden rule income level is (0.5/(n+δ+g)), while our GENERAL
production function in steady state is s/(n+δ+g) (from part c). So how far is each
country from the golden rule level of output production?
For the developed country with s = 0.28, it is producing only 0.28/0.5 = 56% as
much as it should be if it were attempting to maximize long-run consumption.
For the developing country with only s = 0.1, it is even worse; it is producing 0.1/0.5
= 20% as much as it should be if it were attempting to maximize long-run
consumption.
The bottom line is it appears that each country is short-changing the future! Getting
to the golden rule level would clearly require saving 50% of income, and that’s a lot.
3) Predictions of the Real Economy from Solow
Prove each of the following statements about the steady state with population growth and
technological progress. (Note: You need to understand section 8-1 fairly thoroughly for
this one. Note also that these are not very rigorous proofs – you just need a bit of
deductive logic).
a. The capital-output ratio is constant.
In steady state, sy = (δ+n+g)k. This means that k/y = s/(δ+n+g), which is a
constant.
b. Capital and labor each earn a constant share of an economy’s income [Hint:
Recall the definition of MPK].
Capital’s share of income is MPK*(K/Y). From part a we know that K/Y is
constant. Further, given that k is constant in steady state, its marginal
productivity must be constant as well.
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Problem Set 4
FE411 Spring 2008
Rahman
c. The real rental price of capital is constant, and the real wage grows at the rate of
technological progress g. [Hint: The real rental price of capital equals total
capital income divided by the capital stock, and the real wage equals total labor
income divided by the labor force.]
The real rental price of capital R = Total Capital Income/Capital Stock
= (MPK*K)/K = MPK.
This we know is constant, so the real rental price of capital is constant.
The real wage is w = Total Labor Income/Labor Force
This means that w*L = Total Labor Income (TFI)
Converting to rates of change: Δw/w +ΔL/L = ΔTLI/TLI
We know that the labor force grows at rate n, and that total labor income grows
at rate n+g. Thus, real wages grow at rate g.
Cool, huh? Well, I think so anyway…
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