Download Chapter 15: Neoclassical Growth Theory

Chapter 15: Neoclassical Growth Theory
Instructor’s Manual
Chapter 15: Neoclassical Growth Theory
Problem 1
The three facts neoclassical growth theory was designed to explain are:
i.
GDP per capita in the U.S. grew at an average rate of 1.64% per year over the past
century.
ii.
Consumption and saving were constant fractions of GDP during the same period.
iii.
Labor's share of domestic income was constant during the same period.
Problem 2
Constant returns to scale exists when the doubling of all inputs doubles output. Diminishing
marginal product refers to the fact that the marginal product of a single input decreases as the
level of that input increases, holding all other inputs constant. Yes, many production functions
are consistent with both of these properties, including the Cobb-Douglas production function.
Problem 3
No, the production function does not exhibit constant returns to scale. Consider increasing both K
and L by the multiple m. Then
(mK )( mL)1/ 2  m 3 / 2 ( KL1/ 2 )  m 3 / 2Y  mY .
Therefore, output increases by more than m.
For this production function, the marginal product of capital is MPK = L1/2, while the marginal
product of labor is MPL = (1/2)KL-1/2. Thus notice that MPK is independent of K (i.e., the marginal
product of capital is not decreasing but constant) while MPL is decreasing in L.
wL (1 / 2) KL1 / 2 L 1
Labor's share of GDP will be

 .
Y
Y
2
Capital's share of GDP is
rK KL1 / 2

 1.
Y
Y
The assumption of competitive factor markets breaks down in this economy since it implies
that competitive firms would pay factors more than 100% of the total product available.
Problem 4
See equation 15.3 in the text. The three factors that increase growth in the neoclassical model are
(i) increasing the capital per person ratio, (ii) increasing the labor per person ratio, and (iii)
increasing total factor productivity. Over 1890-1995, roughly 56% of growth is explained by
increases in total factor productivity with 34% and 10% explained by increases in capital per
person and labor per person, respectively. However, looking at Figure 15.3 you can see that the
growth rate of total factor productivity has slowed since 1970.
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Chapter 15: Neoclassical Growth Theory
Instructor’s Manual
Problem 5
Total factor productivity is the slope of the production function. Total factor productivity is
measured as the portion of growth that cannot be explained by increases in capital per person and
labor per person. As a result, it is often referred to as the Solow residual, meaning that it is the
part of growth that it left over after input growth has been accounted for. Total factor
productivity in large part measures changes in productivity, but it may also capture other factors
in the economy that cannot be easily separated from changes in productivity. As a result, it may
not be a very accurate measure of true productivity.
Problem 6
The steady states of the difference equation
1/2
xt+1 = 2xt - 3xt + 2,
are obtained by solving the equation
1/2
x = 2x - 3x + 2 or, y2 - 3y + 2 = 0,
where we have defined a new variable y = x1/2 to simplify the algebra.
The solutions for the equation are y = 1 and 2. Hence the corresponding solutions for the
steady state values of x are 1 and 4.
Figure 2 below shows the graph of the difference equation.
xt+1
450 line
xt 1  2 xt  3 xt1 / 2  2
1
4
xt
Figure 2
Since the curve is steeper than the 45o line at the steady-state value 1, it is unstable. On the other
hand, the curve is flatter than the 45o line at the steady state value 4, so it is stable. This is seen
quite easily in Figure 2 above, if you trace out the path starting from two initial values of x, one
close to 1 and another close to 4; the arrows trace out two such paths.
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Chapter 15: Neoclassical Growth Theory
Instructor’s Manual
Problem 7
Since  = 1, N = 1, the neoclassical growth equation becomes Kt+1 = (1 -  + s) Kt, which is a
straight line through the origin with slope (1-  + s). We assume that this slope exceeds 1.0 (in
other words, we assume that  < s). Figure 3 below illustrates this case. There is only one steady
state, at zero. Moreover, since the graph is a straight line with slope greater than that of the 45o
line, the zero steady state is unstable.
Note that, if we had assumed that (1 -  + s) < 1 (i.e., s < , then the slope of the line would
have been less than 1.0, and the steady state at zero would have been stable.
Given  = 0.1 and s = 0.2, 1-  + s = 1.1 > 1. So in this case the neoclassical growth equation
is Kt+1 = 1.1 Kt, which implies that the ratio Kt+1/Kt is 1.1. In other words, the capital stock always
grows at the rate 10% per year. Since Yt = Kt, GDP also grows by 10% per year forever. This
economy is therefore capable of growth.
This example differs from the model in Chapter 13 in a fundamental way. If  = 1, the
production function is Y = K K. The marginal product of capital for this production
function is always equal to 1, and never falls. Hence this production function does not exhibit the
key feature that the production function in Chapter 13 exhibits: diminishing returns to capital.
Labor's share in GDP will gradually go to zero in this economy, since all contribution to
output growth comes from capital.
Kt+1
Kt+1 = (1-+s)Kt
Kt+1 = Kt
45
K0
Kt
Figure 3
Problem 8
An increase in the depreciation rate shifts the capital accumulation difference equation
downward, which reduces the steady state level of capital in country A relative to country B. As
a result, country B will have a higher steady state level of per capita GDP in the long run. In the
short run, if A and B currently have the same level of capital, then B will grow faster than A as
they each approach their own steady states. However, in the long run their growth rates will be
the same and will be equal to the growth rate of productivity.
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Chapter 15: Neoclassical Growth Theory
Instructor’s Manual
Problem 9
The steady state level of the capital stock is the level of capital at which changes in the capital per
person ratio are equal to zero, i.e kt+1 = kt = k . Thus, k = (1 - ) k + As k . Setting  = .1,
s = .16, and  = .5, the steady state level of capital is k = 2.56.
Problem 10
With population growth, the capital accumulation difference equation is
k t 1  Ask   (1    n)k t . Setting k t 1  k t  k you can solve for k = 9.
Problem 11
a.
In the short run and in the long run, country B will have a higher steady state capital
stock than country A and as a result will have a higher steady state level of GDP.
b.
In the short run and long run, country B will grow at the rate of productivity growth, 2%,
because it is in the steady state. In the short run country A is above its steady state and as
a result will grow at less than 2% until it reaches its new lower steady state at which point
it will grow at 2% once again.
c.
Growth in country A and B is equal to 2% at time zero. Growth in country B stays
constant at 2%. Growth in country A falls immediately below 2% and gradually rises
until it equals 2% once again.
Problem 12
An increase in population growth shifts the capital accumulation difference equation downward
and the steady state level of capital is lower.
a.
Per capita GDP is lower in the short run in the long run.
b.
The level of GDP will actually be larger in the short run and long run when population
increases, even though per capita GDP falls.
c.
The growth rate of per capita GDP is lower than the rate of productivity growth in the
short run but in the long run when the economy returns to its new steady state the growth
rate will equal the productivity growth rate.
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Chapter 15: Neoclassical Growth Theory
Instructor’s Manual
Problem 13
According to the neoclassical model, a country’s saving and investment rate is positively
correlated with the level of GDP in the long run. However, there is no correlation between
savings rates and a country’s long run rate of output growth, which is solely determined by a
country’s productivity growth rate. This seems to be inconsistent with the real world, as you
would expect countries that save more would be both richer and grow faster. This is one reason
for the development of endogenous growth models, which will be in more detail in Chapter 16.
Problem 14
In the long run, if two countries have the same steady states then they should be converging
towards the same level of per capita GDP. However, the obvious condition to this statement is
that these two countries must have the same steady state, meaning the same levels of population
growth, savings, technology, and depreciation. If two countries do not have the same steady
states, then the neoclassical model does not predict convergence.
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