P-Set Eight Answers
Long Run Growth
Economics 100b; Spring 1996; Brad DeLong
1. What does it mean for an economy to have constant returns to
scale?
- It means that if you double all the inputs used in production,
output doubles as well.
2. How can you tell whether or not an economy in steady-state growth
is at the "Golden Rule" level of saving and the capital stock?
- Is the marginal product of capital r = aK/Y = n + g + d, the
required gross investment for the economy, the sum of the
depreciation rate, the labor efficiency growth rates, and the
labor force growth rate? If so, then the economy is at the "golden
rule".
3. Why might you think that the "Golden Rule" level of saving and the
capital stock is a good place for an economy to be?
- It is a pareto-optimal distribution of consumption across
points in time. That is, there is no alternative that leaves
consumption higher at every point in time.
4. What do you think were the principal causes of the productivity
slowdown of the 1970s?
- A slowdown in the rate of improvement of the efficiency of
labor (rather than a slowdown in investment). The reduction in the
rate of technological progress had many partial causes, but in
bulk in remains a mystery.
5. Suppose that an economy's production function is Y =
K^{0.5}(EL)^{0.5} ; suppose further that the savings
rate is 40% of GDP, that the depreciation rate d is 4% per year, the
population growth rate n is 0% per year, and the rate of growth g of
the efficiency of the labor force is 2% per year. What is the
steady-state capital-output ratio? What are the steady-state levels
k* of the capital stock per unit of labor power (or per efficiency
unit of labor) and y* of output per unit of labor power?
- steady-state capital-output ratio = 40/6 = 6.67; steady-state
k* = 44.44; steady state y*=6.67
6. Suppose that all variables are the same as in problem 5 save the
production function, which instead is:
K^{0.8}(EL)^{0.2} ; how would your answers be
different? Can you briefly explain why your answers are different?
- the steady state capital-output ratio remains the same: 6.67;
So we have:
y = k/6.67 = k^{0.8} , which implies that:
k^{0.2} = 6.67 , or:
k = 6.67^{5} = 13,168.7243
y = k^{0.8} = 1975.30864
Why are the second results so different, so--enormous? Because the
larger is a, the share of capital in the production function, the
slower do diminishing returns to investment set in because each
marginal unit of capital produces almost as much extra output as
the previous unit of capital. And it is diminishing returns to
capital investment that ultimately lead the capital stock and
output per unit of effective labor to converge to their steady
states.
7. Japan has had a very high savings rate and a high growth rate of
output per worker over the past half century, starting from an
initial post-WWII very low level of capital per worker. What does the
analysis of chapter 4 suggest about Japan's ability to sustain a
higher growth rate than other industrial countries?
- That Japan has grown very fast as it has "converged" to the
steady-state growth path characteristic of industrial economies.
And that in the future Japanese growth will look much more like
that of the typical industrial economy.
8. For what reasons might it make sense for the government to adopt a
"technology policy", and subsidize investment in certain areas?
- Since the bulk of the social benefits from new inventions and
innovations are not the permanent property of the inventor but
diffuse widely through society, the marginal private benefits is
less than the marginal social benefit--hence a competitive market
system will tend to generate "too little' in the way of inventions
and innovations, hence a strong case for government subsidy.
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