Tutorial 2
Chapter 3: The fundamentals of economic growth
Multiple choice questions
Question 1
The figure above represents an economy with a constant population and unchanging
labour force participation, as well as unchanging technology. Assume that output only
has two uses: consumption and investment. At the steady state capital–labour ratio,
which ONE of the following is true?
a. The fraction of output per labour that is saved is equal to the fraction of capital per
labour that depreciates.
Feedback: Incorrect. Note that s does not equal δ generally. See Figure 3.5.
Page reference: 62
*b. Net investment will be zero.
Feedback: Correct. See Figure 3.5. With unchanging technology and unchanging
population, a constant capital–labour ratio requires just enough investment to cover
depreciation.
Page reference: 62
c. Consumption per unit labour will be maximized when the capital–labour ratio is k2.
Feedback: Incorrect. See Figure 3.5.
Page reference: 62
d. Output per unit labour will grow at a steady positive rate because net investment is
positive.
Feedback: Incorrect. See Figure 3.5.
Page reference: 62
Question 2
The figure above represents an economy with a population growing at rate n with an
unchanging labour force participation rate. The rate of technological progress is equal
to a and the rate of depreciation is equal to δ. If the saving rate should fall to a lower
level permanently in the diagram above, which ONE of the following is true?
a. Steady state output per unit of effective labour will fall, although the steady state
capital–effective labour ratio will remain unchanged.
Feedback: Incorrect. See Figure 3.15.
Page reference: 73
*b. The steady state output per unit of effective labour will fall.
Feedback: Correct. See Figure 3.15.
Page reference: 73
c. The capital widening condition will rotate to the left because there is less need to
widen capital with less saving.
Feedback: Incorrect. See Figure 3.15.
Page reference: 73
d. The steady state output per unit of effective labour and the steady state capital–
effective labour ratio will fall in the short run, but return to their long-run levels as
shown in the figure.
Feedback: Incorrect. See Figure 3.15.
Page reference: 73
Question 3
Which ONE of the following is true for effective labour in a Solow model of
economic growth?
a. It grows slower than actual labour input grows.
Feedback: Incorrect. The growth of (AL) is (a + n), which is greater than n if a is
positive, which has been assumed by Solow.
Page reference: 73, 74
b. It is equal to the value of human capital less depreciation.
Feedback: Incorrect. Nothing of the sort was ever written in the text.
Page reference: 73, 74
c. It is equal to the actual labour input less an adjustment for average unemployment
in the economy.
Feedback: Incorrect. Nothing of the sort was ever written in the text.
Page reference: 73, 74
*d. None of the other answers given here are correct.
Feedback: Correct. It grows faster than actual labour input grows by the rate of
technological progress a.
Page reference: 73, 74
Question 4
In the Solow model of economic growth with capital accumulation, positive
population growth, and positive technological progress, which ONE of the following
expressions would grow the fastest in the steady state for a given rate of saving?
*a. K.
Feedback: Correct. Growing at a + n. See Figure 3.16.
Page reference: 74
b. Y/L.
Feedback: Incorrect. Growing at a. See Figure 3.16.
Page reference: 74
c. Y/(AL).
Feedback: Incorrect. Growing at 0 per cent. See Figure 3.16.
Page reference: 74
d. K/(AL).
Feedback: Incorrect. Growing at 0 per cent. See Figure 3.16.
Page reference: 74
Question 5
Suppose that government demographers forecast that our labour force will shrink over
the next sixty years at a rate of 1 per cent per year. The rate of labour-augmenting
technological progress is expected to increase at a rate of 2 per cent per year over the
same period. The saving rate of the economy is 15 per cent and the rate of
depreciation of the capital stock is 10 per cent per year. From this information and in
an economy in a steady state, at what annual rate would we expect to see output per
worker grow?
a. 1 per cent.
Feedback: Incorrect. See Figure 3.16.
Page reference: 74
*b. 2 per cent.
Feedback: Correct. See Figure 3.16.
Page reference: 74
c. 3 per cent.
Feedback: Incorrect. See Figure 3.16.
Page reference: 74
d. 4 per cent.
Feedback: Incorrect. See Figure 3.16.
Page reference: 74
Question 6
Suppose an economy with zero technological progress is in a steady state that does
NOT correspond to the ‘golden rule’ capital–labour ratio and that an increase in
steady-state consumption per unit labour was only possible by decreasing
consumption per unit labour initially from its current steady state value. From this
information, what would you conclude?
a. That the economy is dynamically inefficient because steady-state consumption is
NOT at its maximum value.
Feedback: Incorrect. It is dynamically efficient, because there is no way to increase
both present and future consumption simultaneously.
Page reference: 67
*b. That the economy is dynamically efficient.
Feedback: Correct. There is no free lunch: to increase the capital intensity for larger
future consumption, it will be necessary to save more now, i.e. to consume less now.
Page reference: 67
c. That the economy can achieve the ‘golden rule’ by reducing its capital intensity.
Feedback: Incorrect. It would be going away from the golden rule if it were to do so.
Page reference: 67
d. That all of the answers given here are true.
Feedback: Incorrect.
Page reference: 67
Open question: Exercise 4 and 7 from the book.
4. (a) We know that at the steady state Δk = sy - δk = 0, i.e. sy = δk, so s = δ (k/y) =
δ(K/Y) where y = Y/L and k = K/L. So we need s = 0.05 x 2 = 0.1, i.e. 10%.
(b) We know that at the steady state sy = (δ + n + a)k, so s = (δ + n + a)(k/y) = (δ +
n+ a) (K/Y). We also know that output and capital grow at the rate a + n. So a + n
=0.03 and we need: s = (0.05 + 0.03) x 2 = 0.16, i.e., 16%.
7. The golden rule states the condition under which savings and the implied capital
stock maximize per capita consumption in the steady state. It is important to realize
that the golden rule is a specific/special steady state where consumption per capita is
maximized. The idea is as follows. In the Solow model an economy always reaches a
certain steady state (i.e., long run equilibrium where the capital-(effective) labour
ratio is constant). This does not have to be the golden rule steady state, however (i.e.,
the steady state where consumption per capita is maximized). The idea of the golden
rule shows that given the initial steady state a country finds itself, there might be
another steady state where consumption per capita is maximized (the golden rule
steady state). Theoretically, one way of reaching this golden rule steady state is by
changing the savings rate s in the economy.
While the golden rule can be formulated and studied mathematically, it can also be
read directly from Figure 3.9. Suppose we are to the left of k ' and consider the effects
of raising the capital-labour ratio (equivalently, capital per capita). Because the
production function is steeper than the depreciation line, moving right raises output
per capita more than the required investment to keep us there, so consumption per
capita rises. By increasing consumption per capita, we move in the direction of the
golden rule. Conversely, starting to the right of k ' , the production function is less
steep than the depreciation line, so moving rightwards raises output less than
investment, and per capita consumption declines; we move away from the golden
rule. In fact, reducing capital per capita a little bit from any point strictly greater than
k ' lowers capital by less than the necessary investment (the slope of the depreciation
line). It follows that, starting from k ' , wherever we move, consumption falls. This is
why k ' is the position where consumption is highest – its golden rule level.
Extra questions: Exercises 5 and 6 from the book.
5. In all cases, GDP per capita does not grow in the steady state since there is no
technological progress.
6. (i) If capital and labour inputs double, output is Y = (2K )(2L) = 2 KL = 2Y , so
it doubles too. This implies that we have constant returns to scale.
1
L / K and the marginal
(ii) The marginal productivity of capital is ∂Y/∂K =
2
1
K/L .
productivity of labour is ∂Y/∂L =
2
(iii) The marginal product of each factor decreases in its own use and increases in the
other.