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Chapter 4
The Theory of Economic Growth

The Solow growth model allows us to analyze the determinants of the long-run
trend value of the standard of living and its growth rate.

When the economy is in its balanced-growth equilibrium, the growth rates of
output per worker, the capital-labor ratio, and labor efficiency will all be the
same. The balanced-growth equilibrium occurs when the capital-output ratio is
constant.

In balanced-growth equilibrium and assuming a Cobb-Douglas production
function, the level of output per worker (Y/L) depends upon the saving rate (s),
the rate of growth of the labor force (n), the rate of growth of labor efficiency (g),
the rate of depreciation of capital (), the parameter of the production function (),
and the current level of labor efficiency (E). In balanced-growth equilibrium, the
growth rate of labor efficiency alone determines how fast output per worker
grows.
Learning Guide
You may find this chapter challenging. There is only one model in this chapter ­ the
Solow growth model ­ but it is abstract. Once you see the key point however, the Solow
model becomes suddenly simple.
Some professors believe the material covered in Chapter 4 is the most important material
of the course. You will especially want to be able to answer the Section C and Section D
questions in this chapter and in Chapter 5.
You will need some math skills in this chapter. Math skills covered in Chapter B are
indicated with
Short on time?
If you are short on time, you are in a bind. It is almost
impossible to study this chapter quickly. At a minimum,
learn the equations for
really: do not try to save time on Chapter 4.
K/Y and Y/L. But
You must understand the Cobb-Douglas production function. The determinants of longrun economic growth and the determinants of balanced-growth equilibrium are very
important. Distinguishing between events that change only the level of output per worker
and those that also change the growth rate of output per worker is key. You need to
acquire both a technical mastery of the topics and a conceptual one.
A. BASIC DEFINITIONS
Before you apply knowledge, you need a basic grasp of the fundamentals. In other words, there are some
things you just have to know. Knowing the material in this section won't guarantee a good grade in the
course, but not knowing it will guarantee a poor or failing grade.
USE THE WORDS OR PHRASES FROM THE LIST BELOW TO
COMPLETE THE SENTENCES. SOME ARE USED MORE THAN ONCE;
SOME ARE NOT USED AT ALL.
balanced-growth equilibrium
capital stock
capital-labor ratio
capital-output ratio
Cobb-Douglas
Depreciation
efficiency of labor
growth rate
investment
Keynesian
labor force
output per worker
saving rate
Solow
1. ____________________ occurs when the various forces determining economic growth
are balanced so that output per worker is increasing from period to period at the same rate
as the capital-labor ratio.
2. The ____________________ production function is a particular algebraic form of the
general abstract function
.
3.
The ____________________ has increased due to improvements in technology
and in business organization.
4.
The ____________________ production function states
.
5.
Total saving equals ____________________ spending.
6.
____________________ reduces the capital stock due to wear-and-tear and
obsolescence.
7.
Output per worker is a function of two inputs: the ____________________ and
the ____________________.
8.
The two major factors generating differences between economies' productive
potential are the ____________________ and the ____________________.
9.
When the economy is in ____________________, the growth rate of output per
worker equals the growth rate of labor efficiency.
10.
The ____________________ is the ratio of the sum of household saving,
government saving, and foreign saving to total output.
11.
The growth model is also called the ____________________ model, named after
the economist who received the Nobel Prize for developing the model.
12.
The best proxy we have for the material standard of living is
____________________.
13.
The ____________________ is the total amount of business equipment,
machinery and buildings available for producing goods and services.
14.
When graphing the production function and the balanced-growth equilibrium line,
the ____________________ goes on the horizontal axis and ____________________
goes on the vertical axis.
15.
____________________ refers to construction of buildings and purchases of
business equipment and machinery.
16.
The ____________________ is the average amount of physical capital available
to workers for production.
17.
The ____________________ is the share of real output (GDP) that is saved.
18.
Real GDP per worker converges to its ____________________ path as the
capital-output ratio converges to its equilibrium value.
HINT: In Chapters 4 and 5 we are looking at the long-run economy. In the long
run, the economy is always at full employment, always on the production possibilities
frontier, always on the aggregate production function. Output and income (real GDP, Y)
always equal their potential.
CIRCLE THE CORRECT WORD OR PHRASE IN EACH OF THE
FOLLOWING SENTENCES
19.
In the very long run, a higher saving rate will / will not increase output per worker
and will / will not permanently increase the growth rate of output per worker.
20.
Increased obsolescence of capital will increase / decrease the depreciation rate.
21.
The economy is / is not in balanced-growth equilibrium when %Δ(Y/L) =
%Δ(K/L) = %(E).
22.
The economy is / is not in balanced-growth equilibrium when K/Y is constant.
23.
The balanced-growth equilibrium line is a straight line / nonlinear curve
emanating from the origin.
24.
Output per worker increases / decreases when capital per worker increases.
25.
Output per worker increases / decreases when technology improves.
26.
Output per worker increases / decreases when efficiency of labor increases.
27.
The growth rate of the labor force is endogenous / exogenous to the growth
model.
28.
The growth rate of labor efficiency is endogenous / exogenous to the growth
model.
29.
The growth rate of output is endogenous / exogenous to the growth model.
30.
The growth rate of the capital stock is endogenous / exogenous to the growth
model.
31.
An increase in government spending, all else constant, causes a(n) increase /
decrease in government saving and a(n) increase / decrease in the saving rate.
32.
An increase in consumption spending, all else constant, causes a(n) increase /
decrease in household saving and a(n) increase / decrease in the saving rate.
33.
An increase in imports, all else constant, causes a(n) increase / decrease in foreign
saving and a(n) increase / decrease in the saving rate.
_______________________________
SELECT THE ONE BEST ANSWER FOR EACH MULTIPLE-CHOICE
QUESTION.
34. In balanced-growth equilibrium,
A. the amount of capital per worker is constant.
B. investment per worker is constant.
C. the amount of output per worker is constant.
D. the capital-to-output ratio is constant.
35. The aggregate production function tells us
A. how a firm combines its inputs to produce its output.
B. how the economy's total labor force, capital, and technology can be used to
produce output.
C. the balanced-growth equilibrium.
D. the rate of diminishing returns.
36. To determine the balanced-growth equilibrium value of K/Y, the Solow growth
model requires information about all of the following variables EXCEPT
A. the rate of growth of the labor force.
B. the size of the labor force.
C. the saving rate.
D. the rate of depreciation of capital.
37. Over the last 200 years, the U.S. standard of living has been
A. smoothly increasing from year to year.
B. increasing from decade to decade but not necessarily from year to year.
C. constant.
D. sometimes increasing and sometimes decreasing with no clear trend over time.
TO THE CHALKBOARD
Explaining Figure 4.8
Textbook Figure 4.8 shows how equilibrium output per worker (Y/L) and efficiency of labor
(E) grow over time when the economy is in balanced-growth equilibrium. Remember that in
that equilibrium, Y/L and K/L will grow at the same rate as labor efficiency: g. A
constant rate of growth provides smooth growth but is not graphed as a straight line. A
straight line would depict increases that were the same amount each period (such as,
$5,000 per month) but would then be a declining rate of growth (percentage change) each
period. In equilibrium, the rate of growth (percentage change) is constant from period to
period, which means the amount of growth is increasing from period to period. The
graphical result is above: a smooth curve whose slope continually increases.
If we were to use a logarithmic scale to depict how equilibrium output per worker (Y/L)
and efficiency of labor (E) grow over time when the economy is in balanced-growth
equilibrium, then we would have a straight line graph. A constant percentage change over
time produces a constant slope when log(Y/L) is shown. The graphical result is at the right:
a curve with a constant slope. In balanced growth equilibrium, the slopes of the two curves
will be equal.
38. The efficiency of labor can increase when
A. workers acquire new and better skills.
B. employers reorganize the work place to increase sales with fewer workers.
C. scientific discoveries make machines more productive.
D. all of the above.
39. Ultimately, the most important factor determining the growth of output per worker
over time is the
A. saving rate.
B. level of output per worker.
C. growth rate of the labor force.
D. growth rate of labor efficiency.
40. A decrease in the growth rate of the labor force, all else constant, will permanently
A. increase the level of Y/L and its growth rate.
B. increase the level of Y/L but have no effect on its growth rate.
C. change neither the level of Y/L nor its growth rate.
D. decrease the level of Y/L but have no effect on its growth rate.
_______________________________
TO THE CHALKBOARD
The Key Equations
There are a number of equations in this chapter. Here is a list of the key equations, with a brief
description of each. Be sure you learn especially equations [4] and [5].
[1]
[2]
[3]
[4]
[5]
The general production function. Output per worker depends upon the capitallabor ratio (also known as capital per worker) and the efficiency of labor.
The Cobb-Douglas production function, which specifies the functional
form of the relationship between output per worker, capital per worker,
and labor efficiency.
How capital stock changes from one period to the next. Capital stock at
the beginning of period t + 1 equals capital stock at the beginning of the
previous period plus the saving rate (s) times last period's output (Yt)
minus the depreciation rate () times capital stock at the beginning of
period t.
The balanced-growth equilibrium value of the capital-output ratio equals
the saving rate (s) divided by the sum of the labor force growth rate (n),
the labor efficiency growth rate (g), and the depreciation rate ().
The balanced-growth value of output per worker. When the economy is in
equilibrium, output per worker is a constant proportion of labor efficiency
E: Y/L equals the saving rate (s) divided by the sum of the labor force
growth rate (n), the labor efficiency growth rate (g), and the depreciation
rate (), all raised to the power ( divided by one minus ), and then multiplied
by the value of labor efficiency, E.
B. MANUPULATION OF CONCEPTS AND MODELS
Most instructors expect you to be able to do basic manipulation of the concepts. Being able to do so often means you
can earn a C in a course. But if you want a better grade, you'll need to be able to complete this section easily and move
on to sections C and D.
NOTE: The distinguishing feature of a Cobb-Douglas production function is
that the exponents sum to one (1). For instance, Y = AKβL(1-β) is also Cobb-Douglas
because + β (1 - β) = 1. The Cobb-Douglas production function is convenient to use
because it has some very nice mathematical properties. A Cobb-Douglas production
function exhibits constant returns to scale: if you double all of the inputs, output will
also double. Mathematically, the function exhibits constant returns to scale because the
exponents sum to 1.
1. The Cobb-Douglas production function is
.
A. Suppose labor efficiency, E, is 10,000. Graph the Cobb-Douglas production
function when = 0.2. (You might find it easier to do the graph using a spreadsheet
package such as Quattro Pro or Excel, or using a graphing calculator.)
B. Suppose labor efficiency, E, is 10,000. Graph the Cobb-Douglas production
function when = 0.8.
C. When does a change in the capital to labor ratio generate the larger change in
output per worker, when = 0.2 or when = 0.8? Explain your answer, using the
economic concept of diminishing returns.
Many students are tempted to ask, "Is the Cobb-Douglas production function
realistic?" That's the wrong question. The right question is, "Is the CobbDouglas production function a good enough approximation of reality to allow us
to do reasonable analysis and come up with useful conclusions?" Yes.
2. A. Suppose E = 10,000 and = 0.4. Assume
production can be described by the Cobb-Douglas
production function. Compute the value of output
per worker at each level of capital per worker
shown at the right.
B. When K/L doubles, does Y/L double? Why or why not?
In Questions 3, 4, and 5, you will work with the definitions of labor force, labor
efficiency, and capital stock, and see how their values change over time.
3. Compute the average value of n, the growth rate of the labor force, by decade, 19502000. Consult the Economic Report of the President to locate labor force data.
Labor Force
(Thousands age 16 & over)
1950
1960
1970
1980
1990
Average Annual Growth
Rate
2000
The "efficiency of labor" is sometimes thought of as simply "technology" but
includes business organization too. In general, E is telling us how much output
each worker is able to produce with capital. If someone invents a faster or
more reliable computer (a piece of capital), a worker will probably be able to
produce more output with the same number of computers. Is the worker really
more "efficient" in the usual sense of the word? Does he spend less time on the
phone? Does she organize her work space so it is more productive? No. The
"efficiency of labor" rises in this case not because of any "efficient" action by
the worker, but because a technological advance that affected capital increased
the worker's productivity.
4. Suppose that the labor force is initially 50,000,000 and labor efficiency is initially
10,000. Suppose g = 0.025 and n = 0.03. Complete the following table.
Period
1
2
3
4
5
Et
Lt (in thousands)
10,000
5. Use the relationship describing changes in the capital stock, Kt+1 = Kt + sYt Kt, to
answer the following questions.
A. Suppose Kt = 2,500, δ = 4 percent, the saving rate is 15 percent, and Yt = 1,000.
What is the value of the capital stock at the end of year t + 1?
B. Suppose Kt+1 = 8,000, Kt = 7,000, s = 10 percent, and δ = 0.03. What was the value
of output in year t?
C. Suppose Yt = 300, s = 0.20, Kt+1= 1,000, and the depreciation rate is 4 percent.
What was the value of Kt?
6. Suppose
s = 16%
n = 2%
g = 2%
δ=4
A. What is the equilibrium level of the capital-to-output ratio, K/Y?
B. For each of the following levels of K/L, calculate the balanced-growth equilibrium
value of output per worker:
K/L
1,000
2,000
3,000
4,000
equilibrium
Y/L
C. At the right, graph these four
equilibrium combinations of K/L
and Y/L. Connect the four points
with a straight line.
D. What is the intercept of the line you drew? What is its slope?
TO THE CHALKBOARD
Finding Equilibrium K/L and Y/L: The Graphical Approach
Figures 4.5 and 4.9 show how to find the longrun equilibrium level of output per worker. Let's
review. The production function is a behavioral
relationship. We use the Cobb-Douglas
production function:
This
function can be presented in a two-dimensional
graph with , the dependent variable, on the
vertical axis and an independent variable, on the
horizontal axis. In general, the graph will look as
shown at the right.
The equilibrium condition
be transformed to
can
,
or.
This condition can also be
presented in a two-dimensional graph, again with
on the vertical axis and
on the horizontal
axis. In general, the graph will look as shown at
the right: a straight line with an intercept of 0
and a slope of
.
Combining these two curves onto one graph gives
us the graph at the right. The point where the
two curves cross is the one combination of
output per worker
and capital per worker
that satisfies both the behavioral relationship
and the equilibrium condition. It is the only
combination of
and
that is both on the
production function ­ where the economy always
is in the long run ­ and on the balanced-growth
equilibrium line.
7. Suppose
s = 10%
n = 2%
A. What is the equation for the production
function? Graph the production
function at the right.
B. What is the equilibrium capital-output
ratio? At the right, graph the balanced
growth equilibrium line, Y/K.
C. From your graph, what is the balancedgrowth equilibrium level of output per
worker?
g = 1%
δ=3
α=.5
E=500
What if the economy is not at balanced-growth equilibrium? Suppose the
economy is initially at a combination of K/L and Y/L that is on the production
function ­ where the economy always is in the long run ­ but above the balancedgrowth equilibrium line. Then the economy is producing more output than is
needed for balanced growth, so there is more saving and thus more investment
and thus faster growth of the capital stock than we would have at the balanced
growth equilibrium. And when the capital stock grows, output per worker
increases. So if the economy is initially at a point above the balanced-growth
equilibrium line, K/L and Y/L increase until the economy converges to
balanced-growth equilibrium.
8. Suppose
s = 24%
n = 2%
g = 0%
δ= 4%
A. At right, graph the balanced-growth
equilibrium line, Y/K.
B. Suppose the capital-labor ratio is 2,000.
What is the approximate value of output
per worker? Label that point A.
C. Answer this question simply by looking
at your graph: Is point A a balancedgrowth equilibrium combination of K/L
and Y/L? How do you know?
D. When the capital-labor ratio is 2,000, what is the balanced-growth equilibrium level
of output per worker? Is the actual level of output per worker greater than or less
than the balanced-growth equilibrium level?
E. From one period to the next, the level of output per worker determines the amount
of saving per worker which determines investment per worker which determines the
next periods capital-labor ratio. In the next period, will K/L still equal 2,000? Why?
9. Suppose
s = 24%
n = 2%
A. At right, graph the balanced-growth
equilibrium line, Y/K.
B. Suppose the capital-labor ratio is
8,000. What is the approximate value
of output per worker? Label that point
B.
g = 0%
δ= 4%
C. Answer this question simply by
looking at your graph: Is point B a
balanced-growth equilibrium
combination of K/L and Y/L? How do
you know?
D. When the capital-labor ratio is 8,000,
what is the balanced-growth
equilibrium level of output per
worker? Is the actual level of
output per worker greater than or less
than the balanced-growth equilibrium
level?
E. In the next period, will K/L still equal
8,000? Why?
NOTE: In Questions 10, 11, and 12, you will see how capital, labor, and
output change over time as the economy moves toward a balanced-growth
equilibrium. In Question 10, we begin with the capital-output ratio below its
equilibrium level. In Question 11, we begin with the capital-output ratio
above its equilibrium level. In Question 12, we begin with the capital-output
ratio at its equilibrium level.
10. Suppose n = 0.015, g = 0.008, = 0.03, s = 0.25, E1 = 100, K1 = 75,000, L1 = 15, and
the production function is Cobb-Douglas, with = 2/3. Complete the following table.
Use a separate sheet of paper for your calculations.
Period
1
2
3
4
K
75,000
L
15
E
100
Y/L
Y
K/Y
Is the economy at its balanced-growth equilibrium? Why or why not? Answer the
question without computing the equilibrium value of the capital-output ratio.
11. Suppose n = 0.015, g = 0.008, = 0.03, s = 0.25, E1 = 100, K1 = 75,000, L1 = 5, and
the production function is Cobb-Douglas, with = 2/3. Complete the following table. Use a
separate sheet of paper for your calculations.
Period
1
2
3
4
K
75,000
L
5
E
100
Y/L
Y
K/Y
Is the economy at its balanced-growth equilibrium? Why or why not? Answer the
question without computing the equilibrium value of the capital-output ratio.
12. Suppose n = 0.015, g = 0.008, = 0.03, s = 0.25, E1 = 100, K1 = 75,000, L1 = 7.14, and
the production function is Cobb-Douglas, with = 2/3. Complete the following table. Use a
separate sheet of paper for your calculations.
Period
1
2
3
4
K
75,000
L
7.4
E
100
Y/L
Y
K/Y
Is the economy at its balanced-growth equilibrium? Why or why not? Answer the
question without computing the equilibrium value of the capital-output ratio.
NOTE: So have you seen the powerful simplicity of the model yet? The
standard of living ­ output per worker ­ depends upon a bunch of things: the
saving rate, the rate at which the labor force is growing, the depreciation rate,
the value of labor efficiency. But one thing and only one thing determines
whether the standard of living increases from generation to generation: the rate
of growth of labor efficiency. If efficiency isn't growing ­ if the workforce isn't
acquiring new and better skills, if kids aren't learning in school and becoming
smarter than their parents, if businesses aren't finding ways to reorganize to
improve worker efficiency, if scientists aren't making new discoveries ­ then
there will be no improvement in the standard of living over time.
TO THE CHALKBOARD:
Deriving Equilibrium Y/L=(s/(n + g + δ))(α/α-1)-Et
K/Y is the capital-output ratio. Given values for s, n, g, and , its balanced-growth equilibrium value is a
number.
Y/L is output per worker. Given values for s, n, g, δ, and α, the balanced-growth equilibrium value of
Y/L is a constant proportion of labor efficiency, E. Given a value for E as well, Y/L is a number.
To derive the expression for Y/L requires that we remember two things: [1] manipulate K/Y until you
have an expression with K/L in it and [2] use the Cobb-Douglas production function.
Now we will substitute this last expression into the production function.
13. This question asks you to work with the equations for the balanced-growth
equilibrium values of the capital-output ratio and output per worker. Use the formula
for the balanced-growth equilibrium value of the capital-output ratio,
.
A. Suppose the saving rate is 20 percent, the labor force is increasing 3 percent
annually, labor efficiency is increasing by 2 percent each year, and capital
depreciates 3.5 percent annually. What is the balanced-growth value of the
capital-output ratio? Suppose = 0.6 and E = 100. What is the equilibrium value of
output per worker?
B. Suppose the saving rate is 25 percent, the labor force is increasing 3 percent
annually, labor efficiency is increasing by 2 percent each year, and capital
depreciates 3.5 percent annually. (That is, use the values from part A, but change
the saving rate to 25 percent.) Now what is the balanced-growth value of the
capital-output ratio? Suppose α= 0.6 and E = 100. What is the equilibrium value
of output per worker?
C. Suppose the saving rate is 20 percent, the labor force is increasing 4 percent
annually, labor efficiency is increasing by 2 percent each year, and capital
depreciates 3.5 percent annually. (That is, use the values from part A, but change
the growth rate of the labor force to 4 percent.) Now what is the balanced-growth
value of the capital-output ratio? Suppose α = 0.6 and E = 100. What is the
equilibrium value of output per worker?
D. Suppose the saving rate is 20 percent, the labor force is increasing 3 percent
annually, labor efficiency is increasing by 2.5 percent each year, and capital
depreciates 3.5 percent annually. (That is, use the values from part A, but change
the growth rate of labor efficiency to 2.5 percent.) Now what is the balancedgrowth value of the capital-output ratio? Suppose α = 0.6 and E = 100. What is
the equilibrium value of output per worker?
E. Suppose the saving rate is 20 percent, the labor force is increasing 3 percent
annually, labor efficiency is increasing by 2 percent each year, and capital
depreciates 5.5 percent annually. (That is, use the values from part A, but change
the depreciation rate to 5.5 percent.) Now what is the balanced-growth value of
the capital-output ratio? Suppose α = 0.6 and E = 100. What is the equilibrium
value of output per worker?
NOTE: Why does investment spending ­ not income ­ change when
saving changes? We are looking at the economy in the long run when
output is always equal to its potential; the economy is always on the
production function. So a change in spending and saving by one sector of
the economy ­ say households ­ doesn't . . . indeed can't . . . change total
output. The economy remains on the production function. For reasons
we'll explore in Chapter 7, a change in spending and saving by
households, government agencies, or the rest of the world causes a change
in investment spending by businesses.
14. Suppose
C = $8,000 billion
I = $1,000 billion
G = $1,000 billion
T = $1,000 billion
GX = $2,000 billion
IM = $2,000 billion
Y = $10,000 billion
A. What is the annual value of household saving SH? Of government saving SG? Of
foreign saving SF? Of total saving? What is the saving rate s?
B. Suppose consumption spending increases to $8,500 billion per year. What is the
new annual value of household saving SH? Of total saving? What is the saving
rate s? What is the new annual value of investment spending?
15. Suppose that each year
C = $7,000 billion
I = $1,500 billion
G = $2,000 billion
T = $1,500 billion
GX = $3,000 billion
IM = $3,500 billion
Y = $10,000 billion
A. What is the annual value of household saving SH? Of government saving SG? Of
foreign saving SF? Of total saving? What is the saving rate s?
b. Suppose government spending increases to $2,500 billion per year and taxes are
cut to $1,000 billion per year. What is the new annual value of household saving
SH? Of government saving SG? Of total saving? What is the saving rate s? What is
the new annual value of investment spending?
16. Suppose
s = 15%
n = 2%
g = 0%
δ== 4%
α== 2/3
E = 500
A. What is the equation for the production
function? Graph the production
function at the right.
B. What is the equilibrium capital-output
ratio? At the right, graph the balanced
growth equilibrium line, Y/K.
C. What is the balanced-growth
equilibrium level of output per worker?
Use the equation
Confirm your answer by finding the
balanced-growth equilibrium point on
your graph.
D. Suppose the saving rate increases to 19.5%. What is the new equilibrium capitaloutput ratio? Graph the new balanced-growth equilibrium line above.
E. Answer this question simply by looking at your graph: Is the balanced-growth
equilibrium combination of K/L and Y/L that you found in part C still the
equilibrium? How do you know?
F. What is the new balanced-growth equilibrium level of output per worker? What is the
new equilibrium capital-labor ratio? Explain how the economy moves from the initial
balanced-growth equilibrium level to this new level of output per worker.
G. Once the economy has moved to the new balanced-growth equilibrium level of output
per worker, will there be any further changes in K/L and Y/L? Why?
17. This question asks you to compute the change in the equilibrium level of output per
worker as a result of changes in the parameters. Use the equations
and
A. Suppose the saving rate is 20 percent, the depreciation rate is 2.7 percent, the
labor force is growing by 1.9 percent annually, and labor efficiency is growing by
2 percent a year. Suppose the production function is Cobb-Douglas with α= 0.7.
Suppose E currently equals 4,000. What is the equilibrium value of output per
worker?
B. Suppose the saving rate is 21 percent, and the remainder of the parameters are
the same as in part A. What is the new equilibrium value of output per worker?
What is the percentage change in the equilibrium value of output per worker
when the saving rate increases by one percentage point?
C. Suppose the depreciation rate is 3.7 percent, and the remainder of the parameters
are the same as in part A. What is the new equilibrium value of output per
worker? What is the percentage change in the equilibrium value of output per
worker when the depreciation rate increases by one percentage point?
D. Suppose the labor force is growing by only 0.9 percent annually and the
remainder of the parameters are the same as in part A. What is the new
equilibrium value of output per worker? What is the percentage change in the
equilibrium value of output per worker when the labor force growth rate
decreases by one percentage point?
TO THE CHALKBOARD:
When Labor Efficiency Grows
How do the graphs look when labor efficiency is increasing at a constant rate g over time?
The balanced-growth equilibrium line is
unaffected. The capital-output ratio in
equilibrium equals s / (n + g + δ). Increases in E
over time simply mean that g is positive, not
that g is increasing. (Review Section B,
Question 4.)
But the production function is affected. When
E increases, the production function shifts up.
(Review Chapter 3, Section B, Question 9E.)
So when E is increasing, K/L and Y/L increase
along the economy's balanced-growth
equilibrium line as shown here.
18. Suppose
s = 15%
n = 2%
g = 5%
δ=3%
α=2/3
A. What is the balanced-growth equilibrium value of K/Y?
B. When E = 10,000, what is the balanced-growth equilibrium value of Y/L?
C. Fill in the table at right, showing
how changes in E (remember:
g = 5%) lead to changes in the
balanced-growth equilibrium
value of Y/L.
D. Between periods 1 and 2, what is the growth rate of Y/L? Of E?
C. APPLYING OF CONCEPTS AND MODELS
Now we're getting to the good stuff. Being able to apply a specific concept or model to a real world situation -- where
you are told which model to apply but you have to figure out how to apply it -- is often what you need to earn a B in a
course. This is where macroeconomics starts to become interesting and the world starts to make more sense.
1. The personal saving rate in the United States averaged 8.3 percent in the 1960s but
only 5.9 percent in the 1990s. According to the Solow growth model, what is the longrun effect on output per worker? On the long-run growth rate of output per worker?
2. Total saving includes saving by government agencies. When the government runs a
budget surplus ­ when government revenues exceed government outlays ­ government
is increasing the nation's total saving. When the government runs a budget deficit,
government is decreasing the nation's total saving. During the Clinton Administration
(1993 - 2000), the federal government budget changed from a deficit of $290 billion
(about 4.7 percent of GDP) to a surplus of $230 billion (about 2.4 percent of GDP). If
the surplus had not been subsequently eliminated, what would have been the long-run
effect on standards of living? On the long-run growth rate of output per worker?
3. During the George W. Bush Administration (Bush 43, the younger George Bush,
2001 - 2008), the federal government's budget balance shifted from a surplus of $230
billion (about 2.4 percent of GDP) to a deficit of over $400 billion (about 3.5 percent
of GDP). If there is no reversal of these fiscal policies, what is the long-run effect on
the standard of living of the movement from budget surplus to deficit? On the longrun growth rate of output per worker?
4. If engineers were to develop a way to make machines last much longer, that would
dramatically lower depreciation rates. According to the Solow growth model, what is
the long-run effect on output per worker? On the long-run growth rate of output per
worker?
5. Competition has led scores of U.S. corporations to reorganize, cutting costs without
cutting output. What is the effect on labor efficiency? On output per worker? On the
long-run growth rate of labor efficiency? On the long-run growth rate of output per
worker?
6. Politicians concerned over the burgeoning trade deficit ­ the excess of imports into the
United States over exports from the United States ­ have called upon Americans to
stop buying foreign goods and "Buy American." If the people do as the politicians
urge and permanently reduce imports, what is the long-run effect on foreign saving?
On investment? On output per worker? On the long-run growth rate of output per
worker?
D. EXPLAINING THE REAL WORLD
Most instructors are delighted when you are able to figure out which concept or model to apply to a real world
situation. Being able to do so means you thoroughly understand the material and is often what you need to do to earn
an A in a course. This is where you experience the power of macroeconomic theory.
1. What is the long-run effect of the increase in women's labor force participation on
output per worker? On output per capita? On the long-run growth rate of output per
2. The United Nations sponsors several programs aimed at helping women in poor
nations gain reproductive control. If fertility is lowered, what will be the long-run
effect on standards of living in those nations? On the long-run growth rate of the
standard of living? worker?
3. In the mid-1800s, steam power replaced water power in New England manufacturing.
What should have been the long-run impact of this development on standards of
living? On the long-run growth rate of the standard of living? Why?
4. In the late 1990s, everyone worried that the "Y2K Bug" would lead to collapse of
computer-driven systems on January 1, 2000. As a result, most businesses replaced
their computer equipment. What effect would the computer purchases have had on
standards of living in the long run? On the long-run growth rate of the standard of
living?
5. The government is choosing between spending $100 billion on funding scientific
research and lowering personal taxes by $100 billion. Which action will increase longrun standards of living? Why?
6. A $1.3 trillion tax cut was approved by Congress in 2001. Consider three possible
uses of the tax cut: [1] the recipients spend all of the tax cut, purchasing consumer
goods and services; [2] the recipients save the entire tax cut, placing the funds into
various financial assets; [3] the recipients save half of the tax cut and use the other
half to pay off their credit card debt and other loans. In each case, what is the long-run
effect of the tax cut? Are there long-run economic benefits of the tax cut?
7. Someone says to you, "There is no good reason to come up with policies that would
raise the economy's saving rate. The growth rate of output per worker just winds up
where it was initially." The statement is both right and wrong. Identify one thing about
the statement that is right and explain why it is right. Identify one thing about the
statement that is wrong and explain why it is wrong.
E. POSSIBILITIES TO PONDER
The more you learn, the more you realize you have more to learn. These questions go beyond the material in the text.
They are the sort of questions that distinguish A+ or A work from A- work. Some of them may even serve as decent
starting points for junior or senior year research papers.
1. If America and India share knowledge regarding production methods, technology, and
organization of the workplace, will Indian standards of living equal American
standards of living in the long run?
2. In the 1940s, many economic historians thought that a country had to have a railroad
sector to experience economic growth. After all, the United States economy had
boomed at the same time that the railroad was being constructed throughout and
across America, replacing dirt roads and waterways. According to the Solow model,
what aspects of railroad development might have been key to increasing economic
growth? Would a railroad be necessary today for an industrializing nation to
experience economic growth?
3. If you are advising the government on how it spends money and you want to increase
long-run standards of living, what advice do you give regarding which projects the
government should fund? What advice do you give regarding tax cuts?
4. Should a government have "increase the rate of economic growth" as one of its policy
goals? Should this be a government's only policy goal?
5. The AIDS epidemic is killing thousands of people each day in sub-Saharan Africa.
(http://www.avert.org/subaadults.htm) If the standard by which government decides
whether to address a problem is "Does this government action contribute to economic
growth?," will the government establish programs to end or at least reduce the number
of deaths by AIDS?
SOLITIONS
SOLUTIONS
SOLUTIONS
A. Basic Definitions
* indicates there are notes below related to this question.
SOLUTIONS
1. Balanced-growth
equilibrium
2. Cobb-Douglas
3. efficiency of labor
4. Cobb-Douglas
5. investment
6. Depreciation
7. Capital-labor ratio;
efficiency of labor
8. Capital-output ratio;
efficiency of labor
9. Balanced-growth
equilibrium
10. Saving rate
11. Solow
12. Output per worker
13. Capital stock
14. Capital-labor ratio; output per
worker
15. Investment*
16. Capital-labor ratio
17. Saving rate
18. Balanced-growth equilibrium
*15. Remember: In economics, investment is not purchasing stocks and bonds.
Investment always refers to purchases by business that add to the capital stock.
19. Will; will not
24. Increases
29. Endogenous
20. Increase
25. Increases*
30. Endogenous*
21. Is
26. Increases
31. Decrease; decrease
22. Is*
27. Exogenous*
32. Decrease; decrease
23. Straight line*
28. Exogenous
33. Increase; increase
*22. The two definitions of balanced-growth equilibrium in questions 21 and 22 are
equivalent.
*23. The production function is the nonlinear curve in the graph that helps you determine
the equilibrium values of K/L and Y/L.
*25. Technology is synonymous with labor efficiency in the Solow model as developed
in Textbook Chapter 4.
*27. The growth rate of the labor force, n, is not determined by any of the factors that are
part of the Solow growth model. Therefore, n is exogenous to the model.
*30. The growth rate of the capital stock depends upon two exogenous factors ­ s and ­
and upon one endogenous factor ­ output. Therefore the growth rate of K is itself
endogenous.
34. D. The definition of "balanced-growth equilibrium" is that this is the point where the
capital-output ratio is constant, which is equivalent to stating that capital per worker and
output per worker are growing at the same rate.
35. B. The aggregate production function is usually expressed as
. It is a
statement about production of total output for the entire economy, not about production
within one firm.
36. B. The balanced-growth equilibrium value is
. We need information
about the saving rate (s), the labor force growth rate (n), the growth rate of labor
efficiency (g), and the depreciation rate () only.
37. B. Check out Textbook Figure 4.1. Real GDP per capita ­ our proxy for the standard
of living ­ has increased a great deal since 1800; the value for 2000 is much higher than
the value for 1800. But real GDP per capita does not increase smoothly. The Great
Depression of the 1930s jumps out from that figure. But even ignoring the Great
Depression, you can see that there are years when real GDP per capita falls. In Chapter 4,
we are not concerned with the year-to-year movements in the standard of living, but in
the determinants of that long-run decades-long trend.
38. D. The efficiency of labor refers only to how much output per worker is produced
with a given amount of capital per worker. "Efficiency" can be thought of in its usual
sense, as an action that someone takes to accomplish a goal with fewer resources. But in
the context of the production function, anything that increases output given capital is said
to increase "efficiency" of labor. So improving worker skills increases "efficiency of
labor", as do reorganizing business to improve worker efficiency and technological
improvement in physical capital.
39. D. When the economy is in equilibrium, it is the growth rate of labor efficiency ­ and
only the growth rate of labor efficiency ­ that determines the growth rate of output per
worker.
40. B. A decrease in the growth rate of the labor force will raise the level of output per
worker (Y/L). While the economy is transitioning to its new balanced-growth path, the
growth rate of Y/L will be temporarily higher than the growth rate of labor efficiency.
("Temporarily" may take many years ­ this is a long-run analysis.) But the decrease in the
growth rate of the labor force will have no permanent effect on the growth rate of Y/L.
Only the growth rate of labor efficiency determines the permanent equilibrium growth
rate of output per worker.
B. Manipulation of Concepts and Models
A. Y/L = (K/L)0.2E0.8 appears as shown at
the right. Notice that at low levels of K/L,
the returns to additional capital per worker
are quite large; the curve is nearly vertical
between K/L = 0 and K/L = 1000. But as
capital per worker rises, the returns become
smaller and smaller.
B. Y/L = (K/L)0.8E0.2 appears as shown at
the right. Notice that the returns to
additional capital per worker do not change
much as capital per worker changes. That
is, the slope of the production function is
nearly constant.
C. At very low levels of K/L, a change in the capital to labor ratio generates a larger
change in output per worker when α= 0.2; but at moderate and high levels of K/L, a
change in the capital to labor ratio generates a larger change in output per worker when α
= 0.8. Regardless of the value of α, increases in the amount of capital per worker generate
additional output per worker (returns are positive), but the amount of additional output
per worker gets smaller with each increase in the amount of capital per worker (returns
are diminishing as K/L rises). When α = 0.2, diminishing returns to inputs set in quickly.
When α = 0.8, diminishing returns are slow to appear. So over most of the range of K/L,
increases in capital per worker generates a larger change in output per worker when α =
0.8.
2. A. Y/L = (5,000)0.4(10,000)0.6 = 7,578.58
Y/L = (10,000)0.4(10,000) 0.6 =10,000.00
Y/L = (20,000) 0.4(10,000) 0.6 = 13,195.08
B. When K/L doubles, Y/L does not double.
With the Cobb-Douglas production
function, additions to capital per worker
do increase output per worker, but the
size of increases in output per worker is
not a constant proportion of additions to
capital per worker. When K/L doubles,
Y/L increases by 32 percent. If both E
and K/L doubled, then Y/L would
double.
3. In the Economic Report of the President 2005, the data for Labor Force are found in
Table B35. The average annual growth rate is found by taking the 10th root of the ratio
of, for instance, the 1960 to 1950 value, and subtracting 1 from that value. That is,
(69,628/62,208)0.1 - 1 = 0.0113 = 1.13%.
1950
1960
1970
Labor Force Thousands age
16 & over
62,208
69,628
82,771
Average Annual
Growth Rate
1.13%
1.74%
106,940
125,840
142,583
1980
1990
2000
2.60%
1.64%
1.26%
4.
Period
1
2
3
4
5
Lt (in thousands)
50,000.00
51,500.00
53,045.00
54,636.35
56,275.44
Et
10,000.00
10,000.00
10,506.25
10,768.91
11,038.13
5. A. Kt+1 = 2,550
Kt+1 = Kt + sYt- δKt
K t+1 = 2,500 + 0.15(1,000) 0.04(2,500)
K t+1 = 2,500 + 150 100 = 2,550
B. Yt = 12,100
K t+1 = Kt + sYt -δKt
=Yt = 12,100
C. Kt = 979.17
K t+1 = Kt + sYt –δKt
K t+1 - sYt = Kt – δKt
K t+1 - sYt = (1-δ)Kt
= Kt
= Kt = 979.17
6. A.
B. In equilibrium, K/Y = 2. So multiplying
both sides by Y/2 and then dividing by
L we see that in equilibrium,
C. The graph is at the right.
D. The intercept is 0. The slope is the rise
Δ(Y/L) over run Δ(K/L) which is
, equal to 1 over the
balanced-growth equilibrium capitaloutput ratio. This is always true. The
slope of the line of equilibrium
combinations of K/L and Y/L will always
equal 1 over the balanced-growth
equilibrium capital-output ratio.
7. A. The production function is
Y/L = (K/L)0.5(900) 0.5= 30(K/L) 0.5. The
graph is at the right. The balancedgrowth equilibrium line has a slope of
1/1.67 = 0.6.
B. K/Y = s / (n + g + δ) = 0.10 / (0.02 + 0.01
+ 0.03) = 0.10 / 0.06 = 1.67. The graph is
at the right.
C. In balanced-growth equilibrium, Y/L =
1,500.
8. A. In balanced growth equilibrium, K/Y = 4.
Because we graph K/L on the horizontal (not
vertical) axis and Y/L on the vertical (not
horizontal) axis, the balanced-growth
equilibrium line is a straight line from the
origin with slope equal to 1 over the
balanced-growth equilibrium capital-output
ratio. So the balanced-growth equilibrium
line has a slope of 1/4. As seen at the right,
the balanced-growth equilibrium line begins
at (0,0) and includes the point (4000, 1000).
B. We read the value of output per worker off of the production function. When K/L is
2,000, Y/L is about 900.
C. Point A is not a balanced-growth equilibrium combination of K/L and Y/L. We
know this because point A is not on the balanced-growth equilibrium line.
D. When K/L is 2,000, the balanced-growth equilibrium level of output per worker is
500 = 2000/4. So at point A the actual level of output per worker is greater than the
balanced-growth equilibrium level of output per worker.
E. Were Y/L equal to its balanced-growth equilibrium level of 500, the capital-labor
ratio would not change from period to period. [Important: A constant value of K/L
occurs in equilibrium in this question because we have assumed that labor efficiency
is constant (g = 0).] Because Y/L is greater than the balanced-growth equilibrium
level (900 > 500), then there will be more saving and thus more investment than is
needed to maintain K/L at its existing level of 2000. Additional investment will add
to the capital stock. So K/L will increase from one period to the next.
9. A. These values are the same as in
Question 8. In balanced growth
equilibrium, K/Y = 4. So the
balanced-growth equilibrium line
has a slope of 1/4.
B. We read the value of output per
worker off of the production
function. When K/L is 8,000, Y/L
is about 1,500.
C. Point B is not a balanced-growth
equilibrium combination of K/L
and Y/L. We know this because
point B is not on the balancedgrowth equilibrium line.
D. When K/L is 8,000, the balanced-growth equilibrium level of output per worker is
2,000. So at point B the actual level of output per worker is less than the balancedgrowth equilibrium level of output per worker (1500 < 2000).
E. Were Y/L equal to its balanced-growth equilibrium level of 2,000, the capital-labor
ratio would not change from period to period. [Important: A constant value of K/L
occurs in equilibrium in this question because we have assumed that labor
efficiency is constant (g = 0).] Because Y/L is less than the balanced-growth
equilibrium level (1,500 < 2,000), then there will be less saving and thus less
investment than is needed to maintain K/L at its existing level of 2000. A shortage
of investment will reduce the capital stock. So K/L will decrease from one period
to the next.
10.
Period
1
2
3
4
K
75,000.00
77,839.52
80,761.45
83,767.66
L
15.00
15.22
15.45
15.69
E
100.00
100.80
101.61
102.42
Y/L
1,357.21
1,381.18
1,405.27
1,429.50
Y
20,358.13
21,028.39
21,716.22
22,422.01
Sample calculations are for period 2.
K2 = K1 + sY1 - δK1 = 75,000 + 0.25(20,358.13) - 0.03(75,000) = 77,839.52
L2 = L1(1 + n) = 15.00(1 + 0.015) = 15.22
E2 = E1 (1 + g) = 100.00(1 + 0.008) = 100.8
=
= 1,381.18
K/Y
3.68
3.70
3.72
3.74
Y2 = (Y2 / L2) L2 = 1,381.18(15.22) = 21,028.39
(K/Y) 2 = K2 / Y2 = 77,839.53 / 21,028.39 = 3.70
The economy is not at its balanced-growth equilibrium; K/Y is increasing. If the
economy was at its balanced growth equilibrium, K/Y would be constant. Because K/Y is
increasing, K/L and Y/L must be increasing, so we are initially at a level of output per
worker that is below equilibrium.
12.
Period
1
2
3
4
K
75,000.00
76,723.87
78,487.54
80,291.93
L
7.14
7.25
7.36
7.47
E
100.00
100.80
101.61
102.42
Y/L
2,226.26
2,243.87
2,261.63
2,279.52
Y
15,895.48
16,261.55
16,636.08
17,019.26
K/Y
4.72
4.72
4.72
4.72
Sample calculations are for period 2.
K2 = K1 + sY1 - δK1 = 75,000 + 0.25(15,895.48) - 0.03(75,000) = 76,723.87
L2 = L1 (1 + n) = 7.14(1 + 0.015) = 7.25
E2 = E1 (1 + g) = 100.00(1 + 0.008) = 100.8
=
= 2,243.87
Y2 = (Y2 / L2) L2 = 2,243.87(7.25) = 16,261.55
(K/Y) 2 = K2 / Y2 = 76,723.87 / 16,261.55 = 4.72
The economy is at its balanced-growth equilibrium; K/Y is constant. Notice that at the
balanced-growth equilibrium, capital, labor, labor efficiency, the capital per worker ratio,
and the output per worker ratio are all increasing. Further calculations would show you
that output per worker and capital per worker are both increasing at a constant rate of 0.8
percent per period.
13. A. K/Y = 0.20 / (0.03 + 0.02 + 0.035) = 2.35
Y/L = (2.35)(0.6/0.4)(100) = 360.92
B. K/Y = 2.94 and Y/L = 504.41
C. K/Y = 2.11 and Y/L = 305.46
D. K/Y = 2.22 and Y/L = 331.27
E. K/Y = 1.90 and Y/L = 262.88
14. A. SH = Y - T - C = 10,000 - 1,000 - 8,000 = $1,000 billion per year
SG = T - G = 1,000 - 1,000 = $0 billion per year
SF = IM - GX = 2,000 - 2,000 = $0 billion per year
Total saving = SH + SG + SF = 1,000 + 0 + 0 = $1,000 billion per year
s = Total saving / Y = 1,000 / 10,000 = 0.10 = 10%
B. SH = Y - T - C = 10,000 - 1,000 - 8,500 = $500 billion per year
Total saving = SH + SG + SF = 500 + 0 + 0 = $500 billion per year
s = Total saving / Y = 500 / 10,000 = 0.05 = 5%
I = sY = Total saving = $500 billion per year. Investment decreases when
household saving falls.
15. A. SH = Y - T - C = 10,000 - 1,500 - 7,000 = $1,500 billion per year
SG = T - G = 1,500 - 2,000 = $500 billion per year
SF = IM - GX = 3,500 - 3,000 = $500 billion per year
Total saving = SH + SG + SF = 1,500 + (500) + 500 = $1,500 billion per year
s = Total saving / Y = 1,500 / 10,000 = 0.15 = 15%
B. SH = Y - T - C = 10,000 - 1,000 - 7,000 = $2,000 billion per year
SG = T- G = 1,000 2,500 = $1,500 billion per year
Total saving = SH + SG + SF = 2,000 + (1,500) + 500 = $1,000 billion per year
s = Total saving / Y = 1,000 / 10,000 = 0.10 = 10%
I = sY = Total saving = $1,000 billion per year. Investment falls when government
spending rises.
16. A. The production function is
.The production
function is graphed at the right.
B. The balanced-growth equilibrium
capital-output ratio is
=
0.15 / 0.06 = 2.5. It is graphed as a
straight line with a slope of 1/2.5 = 0.4.
C. Y/L = 3,125.
= (2.5)2(500) = 6.25(500) = 3,125. From the
graph, it appears that the balanced-growth equilibrium level of Y/L ­ the point
where the balanced-growth equilibrium line and the production function cross ­
is indeed 3,125. K/L equals about 8,000.
D. The new balanced-growth
equilibrium K/Y ratio is (0.195)/(0.02 +
0 + 0.04) = 3.25. The initial and new
balanced-growth equilibrium lines are
shown at the right.
E. No, the old combination of K/L and
Y/L (about 8,000 and 3,125) is no
longer the balanced-growth equilibrium
combination. When K/L equals about
8,000, output per worker ­ which we
read from the production function ­ is
greater than the balanced-growth
equilibrium level of output per worker
which we read from the second
equilibrium line.
F. The new balanced-growth equilibrium level of output per worker is Y/L =
(3.25)2(500) = 5,281.25. To find the new equilibrium level of K/L we use the
production function:
. We know the values of everything but K/L.
, so
.
The new higher saving level generates additional resources and funds for 3
investment, increasing the capital stock and thus the capital-labor ratio. As the
capital-labor ratio increases, output per worker increases along the production
function. Because of diminishing returns to investment ( is less than 1), the
increases in Y/L are progressively smaller for each increase in K/L. Gradually the
economy therefore converges to the new balanced-growth equilibrium
combination of K/L = about 17,000 and Y/L = 5,281.25.
G. Once the economy has moved to its new balanced-growth equilibrium level of
output per worker, there will be no further changes in K/L or Y/L because we
assumed g = 0. The production function is stationary from period to period when
the efficiency of labor is constant (when g=0). And so once the economy grows
to its new higher level of output per worker, there will be no further economic
growth. Barring further changes in the saving rate, depreciation rate, labor force
growth rate, or efficiency of labor, the economy will stagnate.
17. A. Equilibrium output per worker is 53,153.0.
K/Y = 0.2 / (0.019 + 0.02 + 0.027) = 3.03
Y/L = (3.03)(0.7/0.3)(4,000) = 53,152.96
B. Equilibrium output per worker is now 56,562, an increase of 12.1 percent.
K/Y = 0.21 / (0.019 + 0.02 + 0.027) = 3.18
Y/L = (3.18) (0.7/0.3) (4,000) = 56,561.98.
%Δ(Y/L) = 56,561.98 / 53,152.96 - 1 = 0.121 = 12.1%
C. Equilibrium output per worker is now 38,244, a decrease of 28.0 percent.
K/Y = 0.20 / (0.019 + 0.02 + 0.037) = 2.63
Y/L = (2.63)(0.7/0.3)(4,000) = 38,244.14.
%Δ(Y/L) = 38,244.14 / 53,152.96 - 1 = -0.280 = -28.0%
D. Equilibrium output per worker is now 77,987, an increase of 46.7 percent.
K/Y = 0.2 / (0.009 + 0.02 + 0.027) = 3.57
Y/L = (3.57) (0.7/0.3) (4,000) = 77,987.43.
%Δ(Y/L) = 77,987.43 / 53,152.96 - 1 = 0.467 = 46.7%
18. A. The balanced-growth equilibrium capital-output ratio is
B. Y/L = 22,500.
C. See table.
D. Between periods 1 and 2, Y/L increases by
5 percent (23,625/22,500 = 1.05). That is
the same rate at which E is increasing: 5
percent. This is no surprise! When the
economy moves from one balancedgrowth equilibrium to another, K/L, Y/L,
and E all grow at exactly the same rate,
here 5 percent.
C. Applying Concepts and Models
1. A lower saving rate lowers K/Y in balanced-growth equilibrium, lowering Y/L for
any given value of E. But in balanced-growth equilibrium, Y/L and K/L will grow
at the rate g which is unchanged. So the value of Y/L for any value of E will be
lower, but in equilibrium, standards of living will continue to grow at the rate g.
2. The increase in government saving will increase the saving rate and thus increase
the balanced-growth value of the capital-output ratio. For a given value of labor
efficiency, E, output per worker (the standard of living) will also increase. The
equilibrium rate of growth of the standard of living will not change, however,
unless there is a change in the growth rate of labor efficiency.
3. The decrease in government saving has the opposite effect of what we found in
#2. The drop in government saving lowers the saving rate, decreasing in the long
run the funds available for investment. The balanced-growth equilibrium capitaloutput ratio falls, lowering output per worker for a given value of labor efficiency.
The equilibrium rate of growth of the standard of living will not change, however,
unless there is a change in the growth rate of labor efficiency.
4. A fall in the depreciation rate increases the balanced-growth equilibrium capitaloutput ratio. Given the level of labor efficiency, E, equilibrium levels of output
per worker will rise. The equilibrium rate of growth of output per worker will not
change, however, unless there is a change in the growth rate of labor efficiency.
5. Changes in business organization can increase labor efficiency, which will
increase output per worker. But unless there are ongoing year-after-year
improvements in organization which generate annual increases in labor efficiency,
there is no permanent long-run change in the growth rate of labor efficiency and
thus no permanent change in the growth rate of output per worker.
6. If Americans reduce their purchases of imported goods and services, substituting
domestically-produced goods and services in their place, foreign saving will fall.
The resulting drop in total saving will reduce investment spending, and thus lower
output per worker for a given level of labor efficiency as the balanced-growth
equilibrium capital-output ratio declines. The equilibrium rate of growth of output
per worker will not change, however, unless there is a change in the growth rate
of labor efficiency.
D. Explaining the Real World
1. The question is asking about the effect of a change in the labor force growth rate
on equilibrium output per worker.
The increase in women's labor force participation provides a period when the
labor force is growing at a faster rate. For example, if women's labor force
participation began at 20 percent and grew to 70 percent over a 50-year period,
but thereafter remained steady at 70 percent, then during those 50 years the labor
force will be growing at a higher rate than before women's participation began to
grow. After the 50-year period, labor force growth would fall back to the rate of
growth of the population. During the 50-year period, faster labor force growth
lowers K/Y in balanced-growth equilibrium, lowering Y/L given E at equilibrium.
Increased labor force participation increases the worker-population ratio ­ the
number of workers per capita. So even though output per worker will be lower
(given E), output per capita will be higher. The equilibrium rate of growth of
output per worker will not change, however, unless there is a change in the
growth rate of labor efficiency.
2. The question is asking about the impact of changes in population growth and thus
in labor force growth on balanced-growth equilibrium output per worker.
A decrease in population growth rates should also lower labor force growth rates,
with a lag of about 15 years. A decrease in labor force growth rates will increase
the balanced-growth equilibrium capital-output ratio, increasing the equilibrium
level of output per worker given efficiency. The equilibrium rate of growth of the
standard of living will not change, however, unless there is a change in the growth
rate of labor efficiency.
3. The question is asking about the effect of a change in labor efficiency on
equilibrium output per worker.
All else constant, the development of steam power should increase standards of
living in the long run because it would increase labor efficiency. Whether or not
the growth rate of output per worker increased permanently would depend upon
the continuation of technological developments that subsequently increased labor
efficiency further.
4. The question is asking about the effect of a change in depreciation on equilibrium
output per worker.
All else constant, replacing computer equipment because of the Y2K Bug did not
increase the usable capital stock. The Y2K Bug essentially increased the rate of
depreciation of computer equipment, rendering existing computer systems
obsolete. That is, it increased the depreciation rate. An increase in the
depreciation rate would lower the balanced-growth equilibrium capital-output
ratio and, given a level of E, would lower output per worker. However if labor
efficiency continued to grow at the same rate, g, then in the long run, equilibrium
output per worker and capital per worker would also continue to grow at rate g.
On the other hand, if the new computer equipment not only corrected the Y2K
Bug but also enabled workers to produce more output with the same quantity of
capital, it would have increased labor efficiency. The increase in labor efficiency
would increase output per worker in equilibrium.
5. The question is asking about the different impacts of increasing labor efficiency
and decreasing saving.
Funding scientific research has the better chance of increasing standards of living
in the long run if the research increases labor efficiency. Lowering taxes will
lower government saving. All else constant, lower government saving decreases
the equilibrium capital-output ratio and, given labor efficiency, lowers
equilibrium output per worker. If consumers save the entire tax cut, then personal
saving will rise to offset the drop in government saving, leaving total saving
unchanged. In this case, however, there is still no long-run increase in the
standard of living.
6. The question is asking about the long-run effect of changes in saving.
In case [1], total saving declines, so the equilibrium capital-output ratio declines,
as does the balanced-growth equilibrium level of output per worker. In case [2],
total saving remains the same, so the balanced-growth equilibrium capital-output
ratio and the equilibrium level of output per worker also remain the same. Case
[3] is the same as case [2]! When consumers pay off debt, they are saving. Saving
is simply "not spending on currently produced goods and services" so whether
consumers put funds into their savings account or pay off their credit card bills,
they are saving. In case [1], there are no long-run economic benefits. In cases [2]
and [3], there are long-run economic benefits of the tax cut only if you assume
that the government will lower its spending commensurate with the tax cut and
that investment spending by the private sector is economically better than
spending by the government. For instance, if investment spending by the private
sector has a lower depreciation rate than government spending on infrastructure or
other investment, then transferring funds from the government to the private
sector can raise balanced-growth equilibrium levels of the capital-output ratio and
output per worker. It all depends upon the types of spending cuts imposed by
Congress and the type of new spending undertaken by the private sector.
7. The question is asking about the difference between changes in the level of output
per worker and in the rate of growth of output per worker.
It is true that an increase in the saving rate has no permanent effect on the growth
rate of output per worker. In balanced-growth equilibrium, the growth rate of
output per worker equals the growth rate of labor efficiency. A change in the
saving rate has no effect on g, the growth rate of labor efficiency. But the
statement is incorrect in saying "there is no good reason" to encourage saving.
Granted, the growth rate of output per worker will return to its initial rate, but
during the transition period from the initial balanced growth equilibrium to the
new balanced-growth equilibrium, output per worker will be growing faster than
labor efficiency. And it can take decades for the economy to adjust to a new
balanced-growth equilibrium! Few people would truly scoff at this opportunity to
enjoy 30, 40, or 50 years of faster growth in standards of living.
E. Possibilities to Ponder
No solutions are given to these questions. The questions are designed to be somewhat
open ended. Each question draws on your understanding of the concepts covered in this
chapter.