Solution Key: Homework 3
Economics 101
Chapters 7 & 8
1)
a) Y/L = (K/L)1/2.
b) Y = 20,000; labor productivity (and not marginal productivity) is defined as the
ratio of output to labor (Y/L), labor productivity from per capita production
function is Y/L= (40,000/10,000)1/2 = 2, to find labor productivity from the
original production function first calculate
Y = (40,000)1/2 (10,000)1/2 = (200)(100) = 20000,
so Y/L = 20000/10000 = 2; yes
c) From sf(K/L*) = dK/L* we have saving rate s = 0.2.
d) Consumption is C = (1-s)Y and consumption per worker is C/L = (1-s)Y/L = 1.6.
2) a) Country A will have the higher level of output per worker
(n + d)K/L
f(K/L)
(Y/L)A
(Y/L) B
sAf(K/L)
sBf(K/L)
(K/L*) B
(K/L*) A
K/L
b) In the steady state the growth rate of output per worker will be zero in both country
A and B
3)
a) According to the convergence hypothesis, both the U.S. and China should
converge to the same steady state level of output per person and then both stop
growing. So both disparity in levels and growth rates should disappear in time.
b) If China has a higher saving rate than the U.S., this would tend to make China’s
steady state levels of capital and output per worker higher than the U.S. levels.
But per capita growth rates still will eventually reach zero in both countries.
c) If China has a higher population growth rate, it would tend to make China’s
steady state levels of capital and output per worker lower than the U.S. levels. Per
capita growth rates still go to zero.
d) In this case, the levels of output per person in the two countries will converge to
each other, and the rates of growth in per-person income both will converge to the
rate of technological progress.
e) With this endogenous growth model, the growth rates will remain constant at their
current levels. So China’s level of output per person would eventually surpass that
of the U.S.
4)
a. The destruction of some of the country’s capital stock in a war would have no
effect on the steady state, because there has been no change in s, f, n, or d.
Instead, k is reduced temporarily, but equilibrium forces eventually drive k to the
same steady-state value as before.
b. Immigration raises n from n1 to n2 in the figure below. The rise in n lowers
steady-state k, leading to a lower steady-state output per worker.
c. The rise in energy prices reduces the productivity of capital per worker. This
causes sf(k) to shift down from sf1(k) to sf2(k) in figure below. The result is a
decline in steady-state k. Steady-state output per worker falls for two reasons: (1)
each unit of capital has a lower productivity, and (2) steady-state k is reduced
y
(n2 + d)k
(n1+ d)k
sf(k)
(n + d)k
sf1 (k)
sf2(k)
k
k
d. A temporary rise in s has no effect on the steady-state equilibrium.
e. Let’s make a distinction between population N and labor force L. The increase in
the size of the labor force L does not affect the growth rate of the labor force, so
there is no impact on the steady-state capital-labor ratio K/L or on the output per
worker Y/L. However, because larger labor force implies higher total out put Y.
Therefore with an increase in the fraction of working people in a population,
national income and income per capita Y/N increases.
5a) In a solow growth model a rise in the rate of capital depreciation shifts up the (n +
d)k line from (n + d1)k to (n + d2)k. The equilibrium steady-state capital-labor ratio,
output per worker ratio is lower, so consumption per worker is lower There is no
effect on the long-run growth rate of the total capital stock, because in the long run
the capital stock must grow at the same rate (n) as the labor force grows, so that the
capital-labor ratio is constant.
b) In an endogenous growth model, the growth rate of output is Y/Y = sA – d, so the
rise in the depreciation rate reduces the economy’s growth rate. Similarly, the growth
rate of capital equals /K = sA – d, which also declines when the depreciation rate
rises. Since consumption is constant fraction of output, its growth rate declines as
well. So the increase in the depreciation rate reduces the long-run growth rate of the
capital stock, as well as long-run capital, output, and consumption per worker.
6) a) False. Standard of living improves if income per capita grows.
b) False. Since 1950 only OECD countries had converging GDP per capita.
c) False. Increase in saving rate increases capital accumulation and hence the growth
rate. But as capital labor ratio reaches the new steady state per capita growth rate
converges to the rate of technological change.
e) True.
f) True.
g) False. Evidence shows that there is convergence among OECD and some Asian
countries. There is evidence of convergence among in states in the U.S. But is
little evidence of convergence among other countries (i.e. countries in Africa, the
Middle East and most countries in Latin America).
h) Convergence among countries is mainly the result of technological progress.
7) Some poor countries grow more rapidly than affluent ones and some less. There is no
general rule; the same for country size.
8) Any policy that leads to an improvement in efficiency can increase economic growth.
Less developed countries can increase their growth by capital accumulation and
acquisition of the best technology from advanced economies. The former can be achieved
by high investment (and thus high saving rate) and the latter with high investment and
high exports of manufactured goods.
9) Need to know what government expenditure is cut following the tax cut. Depending
of the type and the timing of government the cut could hurt or help economic growth rate.
10) Increased depreciation of capital resulting from low-quality investment in the past
reduces efficiency and thus the economic growth. Both type of depreciations lead to
lower steady state level of k and y and lower transitory growth rate. In endogenous model
the effect of lower efficiency on growth depends on saving rates. The higher the saving
rate (and thus the investment), the higher the cost of low quality investment