# Chapter 3

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```					Chapter 03 – Growth and Accumulation

3           GROWTH AND ACCUMULATION

FOCUS OF THE CHAPTER

• In this chapter we study how potential outputthe output that would be produced if all
factors were fully employedgrows over time.

• To better accomplish this, we learn growth accounting and the fundamentals of neoclassical
growth theory. Together, they tell us that output growth results both from improvements in
technology and from increases in one or more of the inputs to the production processcapital,
labor, and natural resources. Neoclassical growth theory also tells us that in the long run,
growth in potential output results entirely from technological improvement.

* Note: The authors in this chapter and the next use the term “long run” in a way that is
inconsistent with the rest of the textbook. They should be saying “very long run.”

SECTION SUMMARIES

1.    Growth Accounting
Output grows because of increases in factors of production like capital and labor, and because of
improvements in technology. The production function provides a link between the level of
technology (A), the amount of capital (K), labor (N), and other inputs used, and the amount of
output (Y) created. The generic formula for the production function is:

Y = AF(K,N)

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The Cobb-Douglas production function, a more specific formula, is frequently used as well, as it
provides a good approximation of production in the actual economy. The formula for the Cobb-
Douglas production function is:

Y = AKN 1  

 pronounced “theta”, represents capital’s share of incometotal payments to capital, as a
fraction of output, or (iK)/Y. (1  ) is labor’s share of income, given by (wN)/Y. To derive these
results algebraically, you need one more fact: When the markets for capital and labor are in
equilibrium (i.e., when the supply of capital equals the demand for capital, and the supply of
labor equals the demand for labor), capital and labor are each paid their marginal product.
For the Cobb-Douglas function, the marginal product of capital (MPK) is AKN.
The marginal product of labor (MPL) is (1  )AKN 1  

We can express our production function in terms of growth rates rather than levels:

Y/Y = [(1  )x N/N] + [x K/K] + A/A

The symbol  pronounced “delta” means “change in”. The term Y/Y, then, should be
interpreted as the growth rate of output. The terms N/N and K/K should be interpreted as the
growth rates of labor and capital, respectively. The last term, A/A is the rate of improvement of
technology, often called the growth rate of total factor productivity (TFP). It is the amount that
output increases as a result of technological progress alone (plug in N/N = K/K = 0, and you’ll
see why).

Because growth in GDP per capita (output per person) tells us more about increases in the
standard of living, it is useful to subtract the rate of population growth (N/N) from both sides,
and write the above equation in per capita terms:

y/y = (x k/k) + A/A

The terms y and k represent output and capital per person: y = Y/N, k = K/N. (There is an
implicit assumption here that the fraction of the population in the labor force is constant. This is
why we can get away with using the terms “population” and “labor supply” interchangeably.)
The terms (y/y) and (k/k) are the growth rates of output and capital per person: y/y = Y/Y 
N/N, and k/k = K/K  N/N. The term K/N is often called the capital-labor ratio.

2.    Empirical Estimates of Growth
Since 1929, U.S. economic growth has averaged about 2.9 percent a year. Of this, estimates
suggest that about 1.09 percent has been due to increases in the labor supply, 0.32 percent has
been due to capital accumulation, and 1.49 percent has been the result of technological progress.

Physical capital and labor are not the only inputs to production. Two other important factors of
production are natural resources and human capitalthe skills and talents of workers. The

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shares of income (also called factor shares) of physical capital, human capital, and raw labor are
estimated to be roughly 1/3 each.

3.    Growth Theory: The Neoclassical Model
Neoclassical growth theory studies the way that growth in the capital stock per worker affects the
long-run level of per-capita potential output. A key result is that, while the rate of saving has a
significant impact on the level of per capita potential output in the long run, the rate of
improvement in technology entirely determines its growth rate.

y

y = f(k)

In the constant returns to scale
production function, capital & labor
have diminishing marginal returns.

k

Figure 31
THE PER-CAPITA PRODUCTION FUNCTION

To build our model, we begin with a few simplifying assumptions: (1) the level of technology is
fixed, so that there is no growth in total factor productivity; (2) the production function has
constant returns to scale (see Review of Technique 4), so that increasing the amount every input
used in production will increase output by the same amount.

A consequence of this second assumption is that all factors of production must have diminishing
marginal productsas more of one input is added, and the others are held constant, each unit
contributes less to output than did the previous one. (Buying more tractors for your construction
company without hiring any more workers to drive them will not help increase your output
much.)

We also need to write our variables in per capita form; as before, y = Y/N and k = K/N. We write
the per capita production function:
y = f(k)

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(n + d)k
We then consider the flows into and out kthe
(flows out of k)
stock of capital per worker.

Investment increases the total stock of capital (K),                                              sf(k)
(flow into k)
which increases k. It can be thought of as a flow
into each worker’s pool of capital.

Both depreciation and population growth decrease
kdepreciation because it decreases the stock of
functional capital, and population growth because                            steady– state k    k
it increases the number of workers sharing this
Figure 32
capital. Both depreciation and population growth
can be visualized as flows out of each worker’s                STEADY-STATE IN THE NEOCLASSICAL MODEL
pool of capital. When the flow into this pool is
greater than the flows out, k grows. When the
flows out of this pool are greater than the flow in, k shrinks. And when the flows in and out
exactly balance, the level of capital per worker will remain fixed.

We call this last case the steady-state, because it is the point at which the level of capital in each
worker’s pool remains steady, or stable. It is the point of equilibrium in our model; we will find
that the capital stock per worker grows or shrinks toward this point, and that once it gets there it
stays. At least until some shock forces it to move.

It is not difficult to express the dynamics described above as an equation. We know that saving
must equal investment. If we assume that people save a constant fraction (s) of their incomes, we
can write the flow into k as (s x y), or, using our per capita production function, as (s x f(k)). The
standard assumption about depreciation is that a constant fraction (d) of the capital stock
becomes obsolete each period. Using this assumption, we can express the flow out of each
worker’s pool of capital that result from depreciation as (d x k). Similarly, when population
grows at a constant rate (n), we can express the flow out of this pool resulting from population
growth as (n x k). Putting all of these terms together, we get the following equation:

k = sf(k)  (n + d)k

We find an expression for the steady-state by simply plugging in the requirement k = 0:

sf(k*) = (n + d)k*

k* represents the steady-state value of k. The steady-state value of y is y* = f(k*).

The growth process can be studied graphically as well. Figure 32 graphs the flows into and out
of k against the level of k. The outflows are graphed as a straight line, with slope (n + d). This is
often called the investment requirement line, as it shows the amount that must be invested, if the
capital stock is to remain constant. The slope of the line that represents the flow into k shrinks as

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k increases, because we have assumed it to have diminishing marginal returns. Where these two
lines intersect, the flows into and out of k balance, and k is at its steady-state. Whether the
savings line lies above the investment requirement line, so that k is increasing, or the investment
requirement line lies above the savings line, so that k is decreasing, k always moves toward the

Using this graph, we can examine the consequences of changes in s, n, or d. An increase in the
savings rate (s) will shift the savings line, sf(k), upward, increasing the steady-state capital-labor
ratio, and hence the steady-state level of per capita potential output. It will also, temporarily,
increase the growth rate of both y and k (remember, the growth rate of k is zero at the steady-
state, and without improvements in technology per capita, output has no other reason to grow).
An increase in either the rate of depreciation or the rate of population growth will increase the
slope of the investment requirement line, decreasing the steady-state levels of k and y, and
causing both to “grow”, temporarily, at a negative rate.

(n + d)k

A reduction in the savings rate
s0 f(k)
(flow into k at higher s)   will decrease steady-state

levels of capital and output .
s1 f(k)
(flow into k at low er s)

lower s                                   k
lower s          higher
higher s s

Figure 33
A DECREASE IN THE SAVINGS RATE REDUCES THE

When the level of technology is permitted to change, we add in another term: “g”, or the growth
rate of technology. Technological improvement is represented on our graph as an upward shift
in the savings line, now written s x Af(k). Notice that it causes the steady-state levels of k and y
to rise.

Notice also, though, that with the addition of technological growth our production function no
longer has constant returns to scaledoubling capital and labor will more than double output.
In order to fix this problem, growth theorists often assume that technology has a very particular
characteristic: It is assumed to be labor augmenting, so that technological improvements increase
the productivity specifically of labor.

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The production function, under this assumption, is written as follows:

Y = F(K, AN)
or, in its Cobb-Douglas form,

Y = K (AN) 1  

Now when we divide through by N we get something a little different:

y = f(k,A)

(trust me on this one), or

y = k A 1

Now, when technology increases at a constant rate (the savings line shifts continually upward),
the steady-state levels of k and y will grow at that same rate:

%k* = %y* = %A

This is not true when technology takes a more general form (the one where it affects the
productivity of both capital and labor identically, for example).

(n + d)k

A technological improvement
SA 1 f(k)
(im p roved technology)
levels of capital and output.
SA 0 f(k)
(original technology )

lower A                                   k
higher A

Figure 34
A TECHNOLOGICAL IMPROVEMENT INCREASES THE

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Appendix
The appendix discusses, in some detail, how to derive the fundamental growth equation Y/Y =
[(1  )x N/N] + [x K/K] + A/A. There is one assumption needed to do this that markets are
perfectly competitive, so that capital and labor are each paid their marginal products.

Note: The fundamental growth equation takes a slightly different form when labor-augmenting
technology is assumed,

Y/Y = [(1  )x N/N] + [x K/K] + [(1  )x A/A]

or, in per-capita terms,
y/y = [x k/k] + [(1  )x A/A].
Just in case you were curious.

KEY TERMS

growth accounting
growth theory
production function
Cobb-Douglas production function
marginal product of labor (MPN)
marginal product of capital (MPK)
total factor productivity
GDP per capita
capital-labor ratio
diminishing marginal returns
convergence
Solow residual
human capital
neoclassical growth theory

GRAPH IT 3

When two economies converge, their growth rates, and, in some cases, the levels of their output
eventually become equal. This graph asks you to plot some historical growth rates for the U.S.
and Japan between 1950 and 1970, to see if convergence was occurring. Table 3–1 provides these
growth rates. All that you need to do is plot each country’s growth rate every year, and connect
the dots. You will get two lines. You will notice that the rate of growth in per capita output is
noticeably higher for Japan than for the U.S. during much of this period. It is this high rate of
growth that has brought the Japanese standard of living into line with the standards of living
enjoyed by others in the industrialized world. Convergence, however, cannot be said to have

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occurred unless Japan’s growth eventually slows, approaching the average growth of other
nations. Can you see this happening in Chart 31? Can these data help you to argue that Japan’s
high growth during this period was a transitory phenomenon?

TABLE 3–1

Year                  Percentage growth in GDP        Percentage growth in GDP
(Japan)                          (U.S.)

1951                            23.6                             13.3

1952                            11.8                               3.2

1953                             6.8                               3.6

1954                             5.5                             1.7

1955                             7.9                               7.6

1956                            10.4                               3.5

1957                            11.0                               3.4

1958                             7.5                             0.2

1959                             9.8                               6.3

1960                            14.5                               2.4

1961                            14.3                               1.6

1962                             7.7                               6.1

1963                            10.9                               3.9

1964                            13.4                               5.3

1965                             5.1                               6.9

1966                            13.2                               8.2

1967                            14.0                               4.5

1968                            16.8                               7.9

1969                            15.1                               6.7

1970                            16.8                               5.7

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25

20

15

10

5

0

1952     1954    1956    1958     1960    1962     1964    1966     1968    1970

Chart 31
PERCENTAGE CHANGE IN GDP FOR JAPAN AND THE U.S.: 19511970

THE LANGUAGE OF ECONOMICS 3

Stocks and Flows, or “About Your Bathtub”

It is always important to know whether a given variable is a stock or a flow. It is easiest to
understand the difference between stocks and flows in the context of your bathtub. The level of
the water in your bathtub is a stock variableit rises and falls depending on the amount of water
entering through the faucet, and, if the drain is unplugged, leaving through the drain. The
amounts of water flowing into and out of the tub are, not surprisingly, flow variables.

Let’s think through some examples of stock and flow variables. We already know that capital is a
stock variable. We also know that investment is a flow into the capital stock, and that depreciation
is a flow out of it. Population is a stock variable. Its level is affected by the birth rate (births are a
flow into the pool of living, breathing people) and the death rate (deaths are a flow out of that
pool). The unemployment rate, despite its name, is yet another stock variable. In Chapter 7 we
will see that those losing their jobs and those entering the labor force are a flow into this pool, and
that those who find jobs or leave the labor force are a flow out of it.

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REVIEW OF TECHNIQUE 3

Balancing Flows into and out of the Tub: Finding a Steady-State

When more water flows into a bathtub than flows out of it, the level of water in the bathtub will
rise. When more water flows out than in, that level will fall. Most importantly, when the flow of
water in and out of this bathtub is exactly the same, the level of water will remain constant. This is
all that a steady-state isa point at which the flows into and out of some “pool” balance.

In the neoclassical growth model, each worker has an identical pool of capital. Investment is a
flow into these pools. Depreciation is a flow out. Population growth, as it increases the number of
pools without adding any more “water”, decreases amount available to each individual and can
also be viewed as a flow out. Thus, at the steady-state, per capita savings = the flow out of each
worker’s pool of capital caused by depreciation + the flow out of that pool caused by population
growth.

If we wanted to find the steady-state level of a population, we would find the point where the
number of people being born was exactly equal to the number of people dying. There can, in some
cases, be more than one steady-state, but that’s an issue for another day.

CROSSWORD

ACROSS

1 MPK assumed in neoclassical model

6 Type of variable, the growth rate of technology           DOWN
in this chapter is an example
2 Type of capital, includes knowledge
9 Kind growth theory studied in this chapter
3 _____ returns to scale
10 Growth rate of steady-state, per-capita output
4 If the growth rates of two countries become
when TFP is fixed
equal over time, they ________
11 Type of output, its growth is modeled in this
5 Type of variable, capital is an example
chapter
7 Affects the steady-state level of output, but not
13 Type of function, provides link between inputs
its growth rate
and outputs
8 Type of variable, investment is an example
14 Type of growth, reduces the steady-state
12 At the steady-state, flows into and out of the
capital labor ratio
capital stock are _________

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1                         2

3

4         5            6                              7

8

9

10

11           12

13

14

FILL-IN QUESTIONS

1.   ______________ helps us to determine how much of the growth in total output is the result of
growth in different factors of production.

2.   The ______________ provides a quantitative link between the inputs to production, and the
output produced.

3.   We call the amount by which output increases when 1 more unit of labor is used the
____________________. It ______________ as more labor is used in production.

4.   When more output can be generated without using more inputs, it must be the case that
____________________ has increased.

5.   Output per person (total output divided by population) is called ____________________.

6.   We call the stock of machines and buildings used in production ______________ capital.

7.   The stock of knowledge and skills is called ______________ capital.

8.   The capital/labor ratio and output per person are constant at the ___________________.

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9.   If two countries have the same rate of population growth, and access to the same
technologies, the rate at which their potential output grows will ______________ over time,
and be identical at the steady state.

10.   When they also have the same rate of savings, the ______________ of their potential output
will be identical at this steady state as well.

TRUE-FALSE QUESTIONS

T    F     1.    An increase in the rate of population growth will change the rate at which per
capita potential output grows in the steady state.

T    F     2.    An increase in the rate of population growth will change the level of per capita
potential output at the steady state.

T    F     3.    An increase in the rate of population growth will change the rate at which total
potential output grows in the steady state.

T    F     4.    An increase in the savings rate will change the rate at which total potential
output grows in the steady state.

T    F     5.    An increase in the savings rate will immediately change the growth rate of total
potential output.

T    F     6.    After this, the growth rate will remain constant.

T    F     7.    An increase in the rate of depreciation will change the levels of the capital/labor
ratio and of per capita potential output at the steady state.

T    F     8.    When we allow productivity to grow over time, the rate at which per capita
potential output grows in the steady state is both positive and exogenous.

T    F     9.    Two countries with the same rate of population growth, and with access to the
same technologies will have the same level of output (and therefore income) at

T    F     10.   The growth rate of their potential output, at the steady state, will (also) be the
same.

MULTIPLE-CHOICE QUESTIONS

1. The rate of growth of potential output is the same as the
b. growth rate of the economy

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c. trend path of GDP                                    d. output gap

2. Which of the following can affect the growth rate of per capita potential output in the steady-
state?
c. rate of depreciation
a. savings rate
d. rate of productivity growth
b. rate of population growth

3. Which of the following cannot affect the growth rate of total potential output in the steady-
state?
a. savings rate                                  c. rate of depreciation
b. rate of population growth                     d. (a) & (c)

4. Which of the following need to be equal, if two countries are to achieve the same growth rate
of per-capita potential output at the steady state?
c. rate of depreciation
a. savings rate
b. rate of population growth

5. Which of the following is a realistic estimate of capital’s share of income (the fraction of total
output that is used to pay capital’s “wage”) in the U.S.?
a. 0.25                                              c. 0.75
b. 0.5                                               d. 1

6. Which of the following would not increase labor’s productivity (measured as Y/N)?
a. technological progress                            c. more natural resources
b. an increase in the capital/labor ratio            d. an increase in the population growth rate

7. Which of the following would not increase capital’s productivity (measured as Y/K)?
a. technological progress                            c. more natural resources
b. an increase in the capital/labor ratio            d. an increase in the population growth rate

8. With the production function y = k¼, a 1% increase in the capital/labor ratio should increase
per capita potential output by
a. 0.25%                                             c. 0.75%
b. 0.5%                                              d. 1%

9. Which of the following would not increase the stock of human capital?
a. education                                         c. more natural resources
b. on the job training                               d. vaccinations

10. Human capital’s share of income in industrialized countries is roughly
a. 0.1                                           c. 0.5
b. 0.33                                          d. 0.75

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CONCEPTUAL PROBLEMS

1. Can human capital depreciate? Does it have diminishing marginal returns?

2. Do natural resources depreciate? Do they have diminishing marginal returns?

3. Name two important assumptions at the foundation of the neoclassical model of growth.

4. Which of the following are stock variables? Which are flow variables?

a) capital, b) per-capita GDP, c) depreciation, d) investment

TECHNICAL PROBLEMS

1.   Consider the following production function: Y = K1/4N3/4

a) What is capital’s share of income?

b) Find an equation for the productivity of capital (Y/K).

c) What is labor’s share of income?

d) Find an equation for the productivity of labor (Y/N).

e) Does this production function have constant returns to scale?
(Translation: Do the exponents add to 1?)

f) Write this production function in per capita terms.
(Translation: Divide both sides by N.)

2.   If in a fixed population the number of people in the labor force doubles, what will happen to

a) the steady-state level of per capita potential output?

b) the steady-state growth rate of per capita potential output?

c) labor’s share of income?

d) labor productivity?

3.   If, instead, the number of people in the population doubles (you may assume that the number
of people in the labor force doubles as well) what will happen to

a) the steady-state level of per capita potential output?

b) the steady-state growth rate of per capita potential output?

c) labor’s share of income?

d) labor productivity?

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Chapter 03 – Growth and Accumulation

4.   Now suppose that, in an economy initially at steady-state, there is an exogenous increase in
the savings rate. Show how per capita output changes over time.

5.   Use the growth accounting equation to answer the following question:
If capital’s share of income is 25% and labor’s share of income is 75%, the stocks of both
capital and labor increase by 50% (K/K = N/N = 0.5), and there is no technology growth, at
what rate will potential output grow? Will the capital-labor ratio increase at all?

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