THE GLOBAL BUSINESS ENVIRONMENT:
INTERNATIONAL MACROECONOMICS AND FINANCE
Professor Diamond
ECONOMIC GROWTH:
FACTS AND THEORY
Class Notes: 3
GROWTH RATES AROUND THE WORLD: WHY SOME NATIONS GROW
AND OTHERS DO NOT
The economic well-being of a country’s population depends on three factors: the
nation’s past and current rate of economic growth; its ability to control the business
cycle; and the degree of equality in the distribution of income. Clearly economic growth
is the most important because it is the fundamental determinant of a nation’s standard of
living.
As noted earlier, we always measure economic growth in the long run in terms of
real per capita income. Table 4.1 shows real income per capita in 1997 for the world’s
most populous nations. The United States enjoyed the highest standard of living with per
capita income of $28,740, followed fairly closely by $23,4000 for Japan and $21,300 for
Germany. Thereafter there is a significant drop in per capita income beginning with
Mexico at $1,120 and declining to Nigeria with only $880 per person. Unfortunately the
majority of the world’s population lives in countries that have experienced low rates of
economic growth.
These differences in living standards are also apparent in the per capita incomes
of OECD countries, which we examined in Class Notes 1. Again the United States
registered the highest income per capita at $29,326 (based on purchasing power parity
exchange rates) while Turkey has the lowest living standard at $6,463 per capita.
It also should be noted that economic growth rates may vary significantly for a
given country at different time periods. In the case of the U.S. there is a marked
difference in the growth rates of the immediate decades following World War II vs. the
decades at the end of the 20th century. The 1950’s and 1960’s were periods of rapid
economic growth with per capita income increasing at an annual rate of 2.2 per cent from
1948-1972. In contrast the annual growth rate slowed to 1.5 per cent (a decline of 32 per
cent) in the 1972-1995 period.
The 1960’s are regarded as the gold standard of the rising levels of living for the
U.S. in the last half of the 20th century. In that decade real median family income rose by
almost 40 per cent. In contrast median family income rose by only 11% in the 1970’s,
4% in the 1980’s and 3.9% in the 1990’s (through 1998). In terms of those members of
the population living below the poverty line there was a 10.3 point reduction in the
1960’s vs. a decline of only 0.4 points in the 1970’s, a 1.1 point increase in the 1980’s
and a 0.1 reduction in the 1990’s (through 1998).
It is pertinent to note that the substantial gains in median family income in the
1960’s (and in the 1950’s) was achieved typically with only one wage earner in a family
and with a modest work week of 40 hours or less for most members of the labor force. In
contrast the lesser gains of the 1970’s, 1980’s and 1990’s required two wage earners in
the great majority of households and a significantly longer work week. Juliet Schor in
her 1992 book entitled The Overworked American estimates that the typical U.S. worker
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of the 1980’s and early 1990’s worked one full month longer per year than their
counterparts in the 1950’s and 1960’s.
These variances in the rates of growth between nations as well as for a given
country at different time periods raises the questions as to why some nations grow and
others do not and what factors account for differential growth rates for the same
country at different time periods? Our task in this part of the course is to try and
answer these questions.
THE PRODUCTION FUNCTION
We start with the simple observation that to produce goods and services requires
the input of the factors of production – capital K and Labor L being the most important.
The quantity of output that capital and labor can produce at any given time depends on
the available technology. The interaction of capital, labor and technology is expressed by
a production function – the mathematical relationship showing how the quantities of the
factors of production determine the quantity of goods and services produced. It can be
expressed as:
Y = F(K,L)
The equation states that output is a function of the amount of capital and labor and the
current state of technology. Figure 4-1 depicts a production function where additional
quantities of capital are combined with a fixed amount of labor. As the input of capital
increases the production function levels off due to the law of diminishing marginal
productivity.
THE SOLOW NEOCLASSICAL GROWTH MODEL
The principal analytical tool to study economic growth is the Solow model.
Robert Solow of MIT used neoclassical theory to examine how an economy evolves over
time. His paradigm shows how changes in capital stock, population (labor) and
technological progress affect the growth of output over time. The model also enables us
to identify some of the reasons why growth rates in a particular country vary from time to
time and why nations vary so markedly in their standards of living. Solow’s work
became the basic foundation for subsequent research on economic growth. He was the
Nobel laureate in economics in 1987.
CAPITAL ACCUMULATION
The model begins with an examination of the growth in the capital stock. Initially
it makes the following assumptions:
1. The supply of labor and technology are held constant.
2. Y – the supply of goods is based on the production function: Y = F(K,L)
2
3. There are constant returns to scale – a proportionate increase in all factors
of production leads to an increase in output of the same proportion.
In any given period the volume of physical capital is determined by the level of
investment in previous years less depreciation. As long as the level of investment
exceeds depreciation the capital stock will increase. Capital is composed of both private
and government capital – roads, bridges, airports, communication facilities, buildings etc.
Investment is financed out of saving – the amount of national income not utilized for
consumption. Domestic investment can exceed domestic saving in any given year due to
foreign investments.
Step 1: The Supply of Goods. Given that the production function reflects
constant returns to scale then:
zY = F(zK,zL)
where z is any positive number. By multiplying capital and labor by z we increase output
by z.
To simplify the analysis we express all quantities relative to the size of the labor
force. Our assumption of constant returns to scale enables us to do so since output per
worker depends only on the amount of capital per worker. To show this we set z = 1/L in
the above equation to obtain:
Y/L = F(K/L,1)
which states that output per worker Y/L is a function of capital per worker K/L.
To further simplify the analysis we use lower case letters to denote quantities per
worker. Thus y = Y/L, k = K/L, and y = f(k,1). In addition we define f(k) as being equal
to F(k,1) so:
y = f(k)
By graphing the production function we can see how the amount of capital per
worker k determines the quantity of output per worker y = f(k) – see Figure 4-1.
Step 2: The Demand for Goods. The demand for goods comes from
consumption and investment needs. Thus, output per worker is divided between
consumption per worker c and investment per worker i. Thus:
y=c+i
(a closed economy with no government spending or taxes).
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Consumption per worker c – the consumption function is proportional to income:
c = cy
Since income per worker can be divided between consumption and saving, we can
substitute l-s for c. Then:
c = (l-s)y
If we substitute (l-s)y for c in the national income identity:
y = (l-s)y+i
and rearranging terms:
i = sy
Thus, investment per worker like consumption per worker is proportional to
income. Moreover, since saving must equal investment the rate of saving s is the fraction
of output devoted to investment.
Step 3: The Capital Stock and the Steady State. The steady state is a condition
where the key variables are stable i.e. not changing. It is the long term equilibrium of the
economy.
We are now in a position to utilize the two major components in the Solow model
– the production function y = f (k) and the investment function i = sy to see how
increases in the capital stock over time result in economic growth. There are two
principal factors that cause the capital stock to change: investment and depreciation.
To determine the amount of investment per worker we substitute the production
function y = f (k) in the investment function i = sy thereby expressing investment per
worker as a function of the capital per worker:
i = sf (k)
The higher the level of k the greater the level of output f (k) and investment i = sf (k).
This equation thus relates the existing stock of capital k to accumulation of new capital i.
Figure 4-2 graphs this relationship as well as how the saving rate determines the
allocation of output between consumption and investment for every value of value of
capital per worker k.
To incorporate depreciation into the model we assume a certain fraction of the
capital stock wears out each year,  is the depreciation rate. The amount of the capital
stock that depreciates each year is k. We express the impact of investment and
depreciation on the capital stock as follows:
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k = i - k
where change in the capital stock from one year to the next is the difference between
investment and the amount of depreciated capital.
Since investment equals saving we can express the change in the capital stock as:
k = sf (k) - k
Figure 4-4 graphs the investment and depreciation functions. It shows investment i and
depreciation k for different levels of the capital stock. The higher the capital stock the
greater the output and the greater the depreciation. There is a single capital stock where
the amount of investment equals the amount of depreciation – i = k and k = 0. This is
called the steady state level of capital k*. The capital stock per worker will not change
over time since the two factors that cause it to change balance one another.
The steady state depicts the long run equilibrium of the economy. The economy
ends up with this level of capital regardless of the level of capital with which it began. If
it is less than the steady state, the level of investment exceeds the amount of depreciation
and over time the capital stock (and the level of output) will grow until it reaches the
steady state. If the economy starts with more than the steady state level of capital,
investment is less than depreciation and over time the capital stock will fall again
approaching the steady state level – see Figure 4-4. Once the capital stock reaches the
steady state the quantity of new capital (investment) equals the amount wearing out
(depreciation) and the capital stock neither increases nor decreases.
To see how an economy approaches the steady state we assume the production
function is in the form of the famous Cobb-Douglas production function. Paul Douglas
of the University of Chicago, who later served as U.S. Senator from Illinois working with
mathematician Charles Cobb determined a production function as follows:
YK
1
2
L
1
2
which assuming constant returns to scale implies that workers and owners of capital
would share equally in the division of national income as the output of the economy grew
over time. This conclusion is borne out by the historical record, which indicates that the
ration of labor income to total income has remained constant at about 0.7 for long periods
of U.S. history. This fact is not only borne out by the data observed by Cobb and
Douglas but by more recent analysis of national income accounts figures – see Figure
3-14.
By appropriate algebraic manipulation of the Cobb-Douglas production function
we find:
y k
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that is output per worker is the square root of capital per worker. Utilizing this
relationship we can develop a table illustrating how an economy approaches the steady
state year by year. Alternatively we can utilize the equation that determines changes in
the capital stock - k  sf (k )  k . By appropriate substitutions and algebraic steps we
can conclude that the steady state of the capital stock is reached when k * / k  s /  .
Thus, if an economy’s saving rate s is 0.3 and its capital stock deprecation rate is 0.1,
then by squaring both sides of the equation the steady state capital stock is 9 units per
worker. If the saving rate increases to 0.35 the steady state capital stock per worker
increases to 12.25.
Thus the saving rate is a key variable in determining the steady state capital stock
and thus the rate of economic growth since the larger the capital stock the higher the level
of output. Figure 4-5 illustrates how an increase in the saving rate shifts the investment
function upward (s = i) thereby permitting a higher steady state of capital stock. The
Solow model demonstrates a nation’s saving rate is a key determinant of economic
growth. If the saving rate is high the economy will have a large capital stock and a
high level of output. If the saving rate is low it will have small capital stock and low
level of output.
Thus a nation that raises its saving rate will experience faster economic growth,
but only in the short run. An increase in the saving rate raises growth but only until the
economy reaches the new steady state. If the high saving rate is maintained the economy
will experience a large capital stock and a high level of output but it will not maintain a
high rate of growth forever.
Step 4: The Golden Rule of Capital. Assuming policy makers can set the
saving rate at any level they wish and thus the capital stock steady state, which should
they choose, what is the optimal?
The goal of economic growth is to maximize the well being of individuals in the
nation. People are not concerned with the size of the capital stock or even the latest GDP
number. They focus on the amount of goods and services they can consume.
Accordingly they would choose the steady state level with the highest level of
consumption per worker. This is called the Golden Rule of Capital Accumulation k*
gold. How can we tell when the economy is at the Golden Rule level? To do so we must
first determine the steady state consumption per worker. We begin with the national
income identity on a per worker basis:
y = c + i and
c=y-i
Consumption is output minus investment.
To determine the steady state consumption level we substitute steady state values
for output y and investment i as follows:
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-- Steady state output per worker is f(k*).
-- k* is steady state capital stock per worker
-- At the steady state, investment is equal to depreciation i =  k*.
-- Substituting f(k*) for y and  k for I
c* = f(k*) -  k*
Thus steady state consumption is the difference between steady state output and steady
state depreciation. It shows that increased capital has two effects on steady state
consumption. It causes output to rise but more output must be used to replace
depreciating capital.
Figure 4-7 graphs these relationships. It shows that there is one steady state level
of the capital stock that maximizes steady state consumption – the Golden Rule of Capital
Accumulation k* gold. This point occurs when the output f(k*) and depreciation  k
curves have the same slope as shown by the tangent drawn on the output curve.
Consumption is maximized c* gold. If the capital stock is below the Golden Rule level,
an increase in the capital stock raises output more than depreciation and consumption
rises. Conversely if the capital stock is above the Golden Rule level an increase in the
capital a stock causes an increase in output smaller than the rise in depreciation and
consumption is reduced.
Another way to determine the Golden Rule level of capital is to find the point
where the marginal product of MPK equals the depreciation rate MPK = . If the steady
state capital is below the Golden Rule level, increases in capital cause consumption to
rise because the marginal product of capital (addition to output) is greater than the
depreciation rate. If the steady state capital stock exceeds the Golden Rule level
increases in capital cause consumption to fall because the marginal product of capital is
less then the depreciation rate.
It is important to note that an economy does not automatically gravitate towards
the Golden Rule steady state. Policy makers must choose a particular saving rate that
produces the Golden Rule level of capital accumulation. Figure 4-8 shows there is one
saving rate s gold [recall that i = sf(k)] which will yield the Golden Rule steady state
capital level. A change in the saving rate would shift the investment curve and cause the
economy to move to a steady state with a lower level of consumption.
The Transition to the Golden Rule Steady State. In virtually all instances
policy makers are faced with an economy that either has more capital or less capital than
in the Golden Rule steady state. The most common situation is when there is less capital.
This requires an increase in saving which immediately reduces consumption. This
sacrifice is necessary in order to increase investment, output and eventually consumption
in the future. The decision to increase saving/lower consumption to achieve the Golden
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Rule steady state may be politically difficult because the current consuming population
bears the burden while a future generation will reap the rewards, see Figure
4-10.
When an economy’s capital stock level is above the Golden Rule steady state the
process of returning to the optimum level is far less demanding. Reducing saving results
in an immediate rise in the level of consumption accompanied by a drop in investment
and output. As the level of the capital stock falls, output, investment and consumption
decline to the new steady state. When the economy has achieved the Golden Rule steady
state both output and investment are lower than when the process began. However,
consumption is higher, see Figure 4-9.
POPULATION GROWTH
Capital by itself cannot produce output nor can it totally explain economic
growth. We know that capital must combine with other factors of production,
particularly labor for goods and services to result. Thus we add population to the model.
We assume that the population and the labor force grow at a constant rate n. How does
the growth of the labor force influence the accumulation of capital per worker in the
steady state? Like depreciation it causes capital per worker to fall. Thus the change in
the capital stock per worker is:
k = i – ( + n)k
New investment increases k and depreciation and population growth decrease k.
We can think of ( + n)k as break-even investment – the amount of investment
necessary to keep the capital stock per worker constant. Depreciation reduces k by
wearing out the capital stock while population growth reduces k by spreading the capital
stock more thinly over a larger work force.
To facilitate the analysis we substitute sf(k) for i. Thus:
k = sf(k) – ( + n)k
To determine the steady state level of capital per worker we extend our earlier graphic
analysis – see Figure 4-11. ( + n)k is the amount of investment to keep constant the
capital stock per worker. For the economy to be in a steady state, investment sf(k) must
offset the effects of depreciation and population growth . This occurs when the two
curves intersect. Here k is unchanging and thus is at the steady state which we designate
as k*.
The Effects of Population Growth. Population growth alters the Solow model
in three ways:
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1.
It brings us closer to explaining sustained economic growth. In the steady
state with population growth, capital per worker k and output per worker y
are unchanging because the number of workers is growing at the same rate
as total capital and total output. Thus population growth cannot explain
sustained growth in the standard of living because output per worker is
constant at the steady state. But population growth can explain sustained
growth in total output.
2.
Population growth gives us another explanation for why some countries
are rich and others poor. Figure 4-12 shows that an increase in the rate of
population growth not matched by an increase in investment causes the
steady state level of capital per worker to fall. Thus k* is lower and since
y* = f(k*) the level of output per worker y* is also lower. Thus the model
predicts that countries with higher rates of population growth will have
lower levels of income per person. Figure 4-13, which provides a scatter
plot of income per person and the population growth rate for 113
countries, confirms this conclusion.
3.
Population growth affects the conditions for determining the Golden Rule
level of capital accumulation. We again begin with the simplifies nation
income identity:
y=c+i
c=y–i
Given that steady state output per worker y = f(k*) and steady state investment is equal to
( + n)k* we can express steady state consumption as:
c* = f(k*) - ( + n)k*
Using the same logic as we did previously the level of k* that maximizes consumption
per worker is the one where:
MPK =  + n
MPK -  = n
Thus in the Golden Rule steady state the marginal productivity of capital net of
depreciation equals the rate of population growth.
TECHNOLOGICAL PROGRESS
We can now introduce the third factor in the Solow Growth model that can
influence economic growth – technological progress. Solow does this by developing the
concept of the efficiency of labor. The efficiency of labor reflects society’s knowledge
about production methods, and the health, education and skill of the labor force.
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To do this we return to our basic production function:
Y = F(K,L)
We add the efficiency of labor E:
Y = F(K,LxE)
LxE is the labor force measured in efficiency units. Thus the augmented production
function states that total output Y depends on the number of units of capital K and the
number of efficiency units of labor, LxE.
We assume that technological progress causes the efficiency of labor to grow at
some constant rate g. Assume g = 0.02, then each unit of labor become 2% more
efficient each year. Output increases as if the labor force had increased by an additional
2% This is called labor augmenting technological progress. Since the labor force L is
growing at rate n, and the efficiency of each unit of labor is growing at rate g, the number
of efficiency units of labor LxE is growing at rate n+g.
Steady State with Technological Progress. Stating technological progress as
labor augmenting enables us to express its impact in terms of the capital stock as we did
in the case of population growth.
We now define k in terms of capital per efficiency unit of labor:
k = K/(LxE)
And output per efficiency unit of labor is:
y = Y/(LxE)
Using these definitions we can state:
y = f(k)
We can now utilize the same process as we did with population growth to see how
the economy evolves over time to reach the steady state:
k = sf(k) – ( + n +g)k
The new term g, the rate of technological change, is included in the equation because k is
now defined as the quantity of capital per efficiency unit of labor. If g is high then the
number of efficiency units is growing rapidly. This causes the amount of capital per
efficiency unit of labor to fall. However, keep in mind that at the same time it causes
output per worker Y/L (economic growth) to increase.
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As with our previous analysis there is one level of k denoted k* where capital per
efficiency unit of labor and output per efficiency unit of labor are constant – namely
where the quantity of new investment offsets the reduction in k due to depreciation,
population growth and technological progress – see Figure 5-1.
Table 5-1 shows how four key variables behave in the steady state with
technological progress. Capital k and output y per efficiency unit of labor are constant.
The number of efficiency units per worker is growing at rate g. Thus output per worker
Y/L (often called productivity) is also growing at rate g. Total output (GDP) is growing
at rate n + g.
The Effects of Technological Progress. With the inclusion of technological
progress we can now explain why some countries experience increases in the standard of
living and others do not. The model demonstrates that technological progress can lead to
sustained growth in output per worker. Unlike a high rate of national saving, which
causes a high rate of growth only until the steady state is reached, technological progress
can result in continuous improvements in output per worker. Once the economy is in
steady state the rate of growth of output per worker depends only on the rate of
technological progress. Thus only technological progress can explain persistently
rising living standards.
The Golden Rule with Technological Progress.
The introduction of
technological progress also modifies the conditions for the Golden Rule steady state.
Building on what we have learned earlier, steady state consumption per efficiency unit of
labor occurs when:
c* = f(k*) – ( + n +g)k*
Steady state consumption is maximized when:
MPK =  + n + g
MPK -  = n + g
The net marginal product of capital equals the rate of growth of total output n + g. This is
where the Golden Rule of capital and consumption are achieved.
We can utilize these relationships to determine whether economies have more or
less capital than at the Golden Rule steady state.
STEADY STATE GROWTH IN THE UNITED STATES
How well does the Solow model fit the United States experience? The model
predicts in the steady state with technological progress how the economy’s key variables
will behave. Both output per worker and the capital stock per worker grow at the rate of
technological progress g. Data over the past 40 years for the United States indicates that
both capital and output per worker grew approximately at the same rate of 2% per year.
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Technological progress also affects factor prices. In the steady state real wages grow at
the rate of technological progress (2%) and the rental price of capital is constant. These
predictions also hold true for the United States.
ACCELERATING UNITED STATES ECONOMIC GROWTH: SAVING,
GROWTH AND ECONOMIC POLICY
Let us now apply the Solow model to assist in developing policy to accelerate
United States economic growth.
The U.S. Golden Rule Steady State. The Solow model tells us that one
particular saving rate produces the Golden Rule steady state where consumption per
worker is maximized. If the marginal product of capital net of depreciation is greater
than the growth rate of total output the economy is operating with less capital than the
Golden Rule steady state. Increasing the rate of saving and the capital stock will
eventually lead to a steady state with higher consumption per worker. If MPK -  is less
than n + g the economy has too much capital and reducing the rate of saving and the
capital stock will in time have a beneficial effect on consumption.
To assess the United States situation we need to compare the economy’s growth
rate and the net return to capital. Real GDP grows at an average rate of 3% per year, so
n + g = 0.03. To determine the net marginal product of capital we utilize the following
data:
1. U.S. capital stock is about 2.5 times one year’s GDP, KxY = 2.5 or k = 2.5y.
2. Depreciation of the capital stock is about 10% of GDP, k = 0.1y.
3. Capital’s share in output is about 30%. Thus:
 = (k) / k
= (0.1y) / (2.5y)
= 0.04
About 4% of the capital stock depreciates each year. To obtain the marginal product of
capital we recall that capital is paid its marginal product. Thus:
Capital’s share = MPKxK/Y
0.30 = MPKx2.5
MPK = 0.30 / 2.5
MPK = 0.12
The marginal product of capital is about 12%. The net marginal product of capital is
about 8% (0.12-0.4), well in excess of the economy’s growth rate of 3% per year.
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The high return to capital indicates that the U.S. capital stock is well below
the Golden Rule steady state level. This finding suggests that the American
economy would incur substantial benefits in its growth rate if it increased its rate of
saving and investment.
Changing the Rate of Saving. Policy makers can influence the saving rate in
two ways:
1. directly through government (public) saving
2. incentives to private saving (households and business).
Government saving is the difference between receipts (revenues) and spending. If
the government sector operates with a budgetary deficit there is negative saving. In
addition the borrowing required to finance the deficit crowds out private investment. If
the government budget is balanced there is zero saving. A surplus yields an increase in
national saving.
Private saving depends primarily on the division of the national income between
consumption and saving and on the real rate of return to saving and investment. We
know that the level of national consumption is a function of national income. However,
it can be influenced by a number of other factors such as taxes on consumption i.e. value
added tax, national sales tax and disincentives to utilize credit for consumer purchases.
The real rate of return on saving is adversely affected by high tax rates on income to
capital. Conversely exempting saving income from taxes such as IRA’s encourages
household saving. Similarly rapid depreciation allowances, thereby reducing corporate
income taxes, encourages business saving and investment in capital goods. We will
discuss other policies, which could increase the national saving rate throughout the
course.
Allocating the Economy’s Investment. The Solow model makes the simplified
assumption that there is only on type of capital. In actuality there are three major
categories: 1. physical (traditional) private business capital – plant and equipment; 2.
public capital – infra-structure such as roads, bridges, dams, transit systems etc.; 3.
human capital – the increased knowledge, skills, expertise and health that workers
acquire through education, technical training, on-the-job training and quality medical
care. Recent research on economic growth has revealed that human capital is at least as
important as physical capital in explaining international differences in standards of living.
There is little doubt that physical, public and human capital all contribute to
economic growth. The question as to how to allocate an economy’s available investment
resources both within and among these three types of capital is much more difficult.
Again, we will talk about his throughout the semester.
Encouraging Technical Progress. The Solow model shows that technological
progress is the source of sustained growth in output per worker and thus in living
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standards. However, the model treats technological progress as exogenous to the
economy and does not explain it. Moreover, the determinants of technological programs
are not well understood. Nevertheless there are some programs in place which are
regarded as beneficial to the process. They include the patent system, tax breaks for
research and development expenditures, and government agencies such as the National
Science Foundation.
THE POST 1972 GRWOTH/PRODUCTIVITY SLOWDOWN
One of the major limitations of the Solow model, which it shares with other
growth paradigms, is its inability to explain the significant slowdown in the growth rate
of all of the world’s major economies in the period after 1972. Table 5-2 shows the
changes in the annual rate of growth in output per person for the United States and other
developed nations for 1948-1972 and 1972-1995. For the United States the growth rate
fell from 2.2.to 1.5, a decline of 32 percent. The other countries experienced similar or
even steeper reductions.
A number of possible explanations are put forth:
1.
The sharp increase in the world price of oil following OPEC’s (the oil
cartel) unwillingness to sell its petroleum to western nations in 1973.
Higher energy prices forced companies to discontinue the use of capital
goods that required large amounts of fuel. In addition, funds that
ordinarily would have been directed to capital formation and research
and development, were diverted to cover the high cost of energy.
2.
A substantial increase in the number of teenagers (baby boomers)
entering the labor force beginning in the 1970s lowered the experience
level of the labor force resulting in a decline in labor productivity.
3.
A substantial increase in government health, environmental and safety
regulations reduced the rate of productivity growth.
4.
There was a slow down in the introduction of new technologies to
improve the efficiency of the production process. The decades
immediately following World War II were exceptional in this regard
due to a catch-up from the depression decade of the 1930s and technical
advances made during World War II.
5.
An acceleration in the shift away from high productivity manufacturing
to the low productivity service sector.
BEYOND THE SOLOW GROWTH MODEL
Solow received his Nobel Prize in Economics in 1987 for his pioneering work on
economic growth models, which he began in the 1950’s. Despite the invaluable insights
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and policy implications of the Solow model, economists for some time have noted a
major weakness of the model with regard to its treatment of technological change. As
noted earlier the Solow model assumes that technology is exogenous – that is, it is
outside of the model and thus not explained. This is, in the minds of many, a major flaw
given Solow’s conclusion that the only source of persistent gains in a nation’s growth rate
is technological progress. Nations seeking to increase their rate of economic growth via
technological change have little or no direction as to how to achieve their goal.
Accordingly a major thrust of economic growth theory, particularly since the late
1980’s by economists such as Paul Romer of Stanford University, has focused on the
development of endogenous growth theory. Unlike Solow it includes technological
change as an integral part of the economic growth model and thus attempts to explain the
rate of technological progress.
Utilizing a simple endogenous growth model we can gain some insight into this
new approach to explaining economic growth. Assume the following production
function:
Y = AK
where Y = national output
A = a constant
K = capital stock
The equation states that output is equal to the amount of the capital stock times a
constant, A, which indicates the amount of output produced for each unit of capital stock.
This production function does not exhibit the law of diminishing returns. An extra unit of
capital stock produces A units of output regardless of the number of units of capital stock.
We can now use this production function to explain economic growth. As we have done
previously we assume that income will be divided between consumption and saving. The
saving will be utilized for investment. Thus, as in the Solow model the amount of
additional capital stock in any given time period is:
K = sY - K
combining this equation with the endogenous production function Y = AK, after some
algebraic manipulation it yields:
Y / Y = K / K = sA - 
This equation shows that the rate of economic growth will increase as long as the saving
rate times the constant A exceed the depreciation rate – sA-. A country’s output can
grow forever, even without the assumption of technological progress. In the endogenous
growth model where there is no diminishing return as the amount of capital stock is
increased, saving and investment can lead to continuous growth.
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Is this assumption of the absence of diminishing returns creditable? According to
endogenous growth theorists it is, if one rejects the traditional view that K, the capital
stock, only includes an economy’s stock of plant and equipment. They postulate that K
includes knowledge as well. Knowledge, particularly the development of new ideas,
leads to production of new goods and services. It also results in improving the efficiency
of producing older products. Unlike the production of physical capital stock there is no
evidence to suggest that the production of knowledge exhibits diminishing returns.
Indeed, some would argue that not only is the assumption of constant returns feasible, but
in addition there is historical evidence to support the view that knowledge exhibits
increasing returns.
Endogenous growth theory has progressed substantially beyond the above simple
production function.
In particular two sector models and the application of
microeconomic analysis to research and development decision making in the private
sector are two important avenues of research. The former approach divides the
economy’s production function into two sectors – manufacturing and research
universities. The latter focuses on the motivations which lead firms to invest in the
research and development of new products and how other firms build on these
innovations to produce the next generation of new products and techniques.
GROWTH ACCOUNTING
Another approach to explaining economic growth is growth accounting. Growth
accounting measures empirically the relative importance of three major sources of output
growth – increases in capital, increases in labor and advances in technology. It is the
production function expressed in growth rate terms.
We begin by examining how the factors of production contribute to increases in
output. Similar to the Solow model we initially assume there is no technological change.
Thus the production function is:
Y = F(K,L)
The quantity of capital and labor determines the level of output. Since there is no
technological progress the production function does not change over time.
Increases in Capital. First we increase the amount of capital and hold labor
constant. To determine how much an increase in the capital stock, K, will contribute to
raising output, we recall that this is a function of the marginal physical product of capital,
MPK. This can be expressed as follows:
MPK = F(K+1, L) – F (K, L)
The output resulting from an increase in capital by one unit minus the previous output is
the marginal physical product of capital, MPK. Thus, assuming the change in K is
relatively small and therefore the marginal product is relatively constant, an increase in
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K units causes output to increase by approximately MPK X K. This enables us to
convert changes in capital into changes in output.
Increases in Labor. Next let us increase the amount of labor while holding the
capital stock constant. Again, the marginal physical product of labor, MPL, indicates
how much an additional unit of labor contributes to a rise in output. Thus:
MPL = F(K,L+1) – F(K,L)
The total output resulting from an addition of one unit of labor minus the previous output
is the marginal physical product of labor, MPL. Thus as with changes in K, when the
amount of labor increases by L units, output increases by approximately MPL X L.
Increases in Capital and Labor. Now let us consider what occurs when both
capital and labor increase. Combining their respective contribution to total output we
arrive at the following:
Y = (MPK X K) + (MPL X L)
This equation shows how to attribute the growth in output to each factor of production.
To do this we need to convert the above equation to an expression of economic
growth. First by algebraic manipulation it can be restated as follows:
Y / Y = (MPK X K/Y) (K/K) + (MPL X L/Y) (L/L)
This form of the equation indicates how the growth of capital, K/K and the growth of
labor, L/L contribute to the growth rate of output Y/Y.
Next we need to be able to measure the terms in the parenthesis in the above
equation. We recall that the marginal product of capital equals its real rental price. Thus,
(MPK X K) is the total return to capital and (MPK X K)/Y is capital’s share of output.
Similarly the marginal product of labor equals its real wage. Thus (MPL X L) is the total
compensation labor receives and (MPL X L)/Y is labor’s share of output. Assuming the
production function has constant returns to scale we can utilize Euler’s theorem to
determine that capital’s and labor’s share sum to one. Euler’s theorem is a mathematical
result that economists use to show that economic profit must be zero if the production
function has constant returns to scale and factors of production are paid their marginal
products. Thus:
Y / Y =  (K/K) + (1 - ) / (L/L)
where  = capital’s share
1 -  = labor’s share
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The above equation provides a formula to show how changes in capital and labor
cause changes in output. Now we must weight the respective factors growth rates by
their historical shares of total income. As we have observed earlier capital’s share of
U.S. income is about 30 per cent,  = 0.30 and labor’s share is about 70 per cent, 1 -  =
0.70. Thus a 10 per cent increase in the capital stock (K/K = 0.10), leads to a 3 per cent
increase in output (Y/Y = 0.03). And, a 10 per cent increase in labor ( L.L = 0.10)
leads to a 7 per cent increase in output (Y/Y = 0.07).
Technological Progress. We can now drop the unrealistic assumption that there
is no technological profess. We include technology in our production function as
follows:
Y = A F (K,L)
where A is a measure of the current level of technology called total factor productivity.
It is a measure of level of technology that is closely related to the efficiency of labor in
the Solow model. Output can now increase not only because of increases in capital and
labor, but also due to increases in total factor productivity. If the quantity of capital and
labor is unchanged and total factor productivity increases by 1 per cent output increases
by 1 per cent.
Adding technological change to our growth accounting equation:
Y / Y =  (K/K) + (1 - ) / (L/L) + A/A
The growth in capital, labor, and total factor productivity equals the growth in
output. This is the key equation in growth accounting. It identifies and allows us to
measure the three sources of economic growth: changes in the amount of capital, labor
and total factor productivity.
Because total factor productivity is not measurable directly, it is derived
indirectly. Data is available on the growth in output, capital and labor and on capital and
labor’s share of output. From this information and the growth accounting equation we
can compute the growth in total factor productivity as follows:
A/A = Y / Y -  (K/K) - (1 - ) (L/L)
A/A is the change in output that cannot be explained by changes in the inputs of capital
and labor.
We have already examined some of the reasons why total factor productivity may
change, particularly the impact of increased knowledge. Other changes that may
influence total factor productivity include improvements in the health and education of
labor and government regulations. As a residual, total factor productivity captures
anything that changes the relation between measured inputs and measured output.
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Sources of Growth in the United States. Growth accounting enables us to
measure the sources of economic growth. Table 5-3 identifies the contributions of the
three major sources of growth in the U.S. from 1950 to 1996. The Table shows that
economic growth grew at an average annual rate of 3.2 per cent: 0.9 per cent is attributed
to increases in the capital stock; 1.2 per cent to increases in the labor input; and 1.1 per
cent to increases in total factor productivity. Thus over this recent 46 year period capital,
labor and productivity contributed almost equally to U.S. economic growth. Note also
that there is a significant decline in the annual growth rate of total factor productivity
after 1970, and that it continued to grow at a substandard rate for the balance of the 19701996 period vis-à-vis the 1950’s and 1960’s. This data helps to further explain the
sources of the slowdown in U.S. economic growth in the post-1972 period.
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