Chapter 4
Facts About Economic Growth
In this chapter we set the table for the growth model to come. Before jumping into the
economic model, we start by describing some basic facts of economic growth. First, we look
at the time series growth in the United States which is more or less representative of the
average high-income country. Next, we look at economic growth over the world.
4.1
Economic Growth over Time: The Kaldor Facts
In an influential 1957 article Nicholas Kaldor listed a set of stylized facts characterizing
the then relatively recent economic growth across countries (Kaldor (1957)).1 “Stylized”
means that these facts are roughly true over sufficiently large periods of time – they do not
exactly hold, especially over short time frequencies. The “Kaldor Facts” continue to provide
a reasonably accurate description of economic growth across developed countries including
the United States.
1. Output per worker grows at a sustained, roughly constant, rate over long periods of
time.
How rich are Americans today relative to several generations ago? To make such a
comparison requires a standard unit of account. As we discussed in Chapter 1, it is
common to use price indexes to distinguish changes in prices from changes in quantities.
Hence, we focus on real, rather than nominal, GDP.
A natural measure of the productive capacity of an economy is real GDP per worker.
GDP can go up either because there are more people working in an economy or because
the people working in an economy are producing more. For thinking about an economy’s
productive capacity, and for making comparisons across time, we want a measure of
GDP that controls for the number of people working in an economy.
The log of real GDP per worker in the US is shown in Figure 4.1. Why do we plot this
relationship in logs? GDP grows exponentially over time which implies the slope gets
1
Also see the Wikipedia entry on this.
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steeper as time goes by. The log of an exponential function is a linear function which
is much easier to interpret. The slope of a plot in the log is approximately just the
growth rate of the series.
Figure 4.1: Real GDP per Worker in the US 1950-2011
11.6
Actual Series
Trend Series
11.4
log(RGDP per Worker)
11.2
11
10.8
10.6
10.4
10.2
1950
1960
1970
1980
1990
2000
2010
2020
Year
The figure also plots a linear time trend, which is depicted with the dotted straight
line. While the actual series is occasionally below or above the trend line, it is clear
that GDP per worker grows at a sustained and reasonably constant rate. The average
growth rate over this period is about 1.7 percent annually. How does a 1.7 percent
annual growth rate translate into absolute differences in income over time? A helpful
rule of thumb is called the “Rule of 70.” The Rule of 70 (or sometimes rule of 72) is
a way to calculate the approximate number of years it takes a variable to double. To
calculate this, divide 70 by the average growth rate of this series. This gives you the
approximate number of years it takes the variable to double. To see why it is called the
“Rule of 70” consider the following example. Let Y0 be a country’s initial level of income
per person and suppose the annual rate of growth is g percent per year. We can find
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how long it takes income per person to double by solving the following equation for t:
2Y0 = (1 + g)t Y0
⇔ 2 = (1 + g)t
⇔ ln 2 = t ln(1 + g).
Provided g is sufficient small, ln(1 + g) ≈ g. ln 2 is approximately equal to 0.7. Hence,
70 divided by the annual percent rate of growth equals the required time for a country’s
income per person to double. In the U.S. case, t = 70/1.7 ≈ 41. This means that, at
this rate, GDP per worker in the US will double twice every 80 years or so. Measured
in current dollar terms, US GDP per capita in 1948 was about $32,000. In 2016, it is
$93,000. In other words, over this 60 year period GDP per work in the US has doubled
about 1.5 times.
Small differences in average growth rates can amount to large differences in standards
of living over long periods of time. Suppose that US real GDP per capita were to
continue to grow at 1.7 percent for the next one hundred years. Using the rule of 70,
this means that real GDP per capita would double approximately 2.5 times over the
next century. Suppose instead that the growth rate were to increase by a full percentage
point to 2.7 percent per year. This would imply that real GDP per capita would double
approximately 4 times over the next century, which is a substantial difference relative
to double 2.5 times.
2. Capital per worker grows at a sustained, approximately constant, rate over long periods
of time.
Figure 4.2 shows the time series of the log of capital per worker over the period 1950-2011
along with a linear time trend. Capital constitutes the plant, machinery, and equipment
that is used to produce output. Similar to output per worker, the upward trend is
unmistakable. Over the period under consideration, on average capital per worker grew
about 1.5 percent per year, which is only slightly lower than the growth rate in output
per capita.
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Figure 4.2: Capital per worker in the US 1950-2011.
12.6
Actual Series
Trend Series
12.4
log(Capital per Worker)
12.2
12
11.8
11.6
11.4
1950
1960
1970
1980
1990
2000
2010
2020
Year
The fact that capital and output grow at similar rates leads to the third of Kaldor’s
facts.
3. The capital to output ratio is roughly constant over long periods of time.
If capital and output grew at identical rates from 1950 onwards, the capital to output
ratio would be a constant. However, year to year the exact growth rates differ and,
on average, capital grew a little slower than did output. Figure 4.3 plots the capital
to output ratio over time. The capital to output ratio fluctuated around a roughly
constant mean from 1950 to 1990, but then declined substantially during the 1990s.
The capital output ratio picked up during the 2000s. Nevertheless, it is not a bad first
approximation to conclude that the capital to output ratio is roughly constant over
long stretches of time.
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Figure 4.3: Capital to Output Ratio in the U.S. 1950–2011
3.5
3.4
K/Y
3.3
3.2
3.1
3
2.9
1950
1960
1970
1980
1990
2000
2010
2020
Year
Over the entire time period, the ratio moved from 3.2 to 3.1 with a minimum value a
little less than 3 and a maximum level of about 3.5. However, the ratio moves enough
to say that the ratio is only approximately constant over fairly long periods of time.
4. Labor’s share of income is roughly constant over long periods of time..
Who (or what) earns income? This answer to this question depends on how broadly
(or narrowly) we define the factors of production. For instance, should we make a
distinction between those who collect rent from leasing apartments to those who earn
dividends from owning a share of Facebook’s stock? Throughout most of this book,
we take the broadest possible classification and group income into “labor income” and
“capital income.” Clearly, when people earn wages from their jobs, that goes into labor
income and when a tractor owner rents his tractor to a farmer, that goes into capital
income. Classifying every type of income beyond these two stark cases is sometimes not
as straightforward. For instance, a tech entrepreneur may own his computers (capital),
but also supply his labor to make some type of software. The revenue earned by the
entrepreneur might reasonably be called capital income or labor income. In practice,
countries have developed ways to deal with the assigning income problem in their
National Income and Product Accounts (also known by the acronym, NIPA). For now,
assume that everything is neatly categorized as wage income or capital income.
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Labor’s share of income at time t equals total wage income divided by output, or:
LABSHt =
wt Nt
.
Yt
Here wt is the (real) wage, Nt total labor input, and Yt output. Output must equal
income, and since everything is classified as wage or capital income, capital’s share is
CAP SHt = 1 − LABSHt .
Clearly, these shares are bounded below by 0 and above by 1. Figure 4.4 shows the
evolution of the labor share over time.
Figure 4.4: Labor Share in the US 1950-2011.
0.68
0.67
Labors Share
0.66
0.65
0.64
0.63
0.62
1950
1960
1970
1980
1990
2000
2010
2020
Year
The labor share is always between a 0.62 and 0.7 with an average of 0.65. Despite a
downward trend over the last decade, labor’s share has been relatively stable. This also
implies capital’s share has been stable with a mean of about 0.35. The recent trends
in factor shares have attracted attention from economists and we return to this later
in the book, but for now take note that the labor share is relatively stable over long
periods of time.
5. The rate of return on capital is relatively constant.
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The return to capital is simply the value the owner gets from “renting” capital to
someone else. For example, if I own tractors and lease them period-by-period to a
farmer for $10 per tractor, capital income is simply $10 times the number of tractors.
If a producer owns his or her own capital this “rent” is implicit. We will lose Rt to
denote the rental rate on capital and Kt the total stock of capital. We can infer the
rate of return on capital from the information we have already seen. Start with the
formula for capital’s share of income:
CAP SHt = 1 − LABSHt
Rt Kt
=
Yt
⇒
Rt = (1 − LABSHt )
Yt
Kt
Rt is the rate of return on capital. Since we already have information on labor’s share
and the capital to output ratio, we can easily solve for Rt . It’s also straightforward to
see why, given previously documented facts, Rt ought to be approximately constant
Yt
– LABSHt and K
are both roughly constant, so the product of the two series ought
t
also to be close to constant. The implied time series for Rt is displayed in Figure 4.5.
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0.125
0.12
Return on Capital
0.115
0.11
0.105
0.1
0.095
1950
1960
1970
1980
1990
2000
2010
2020
Year
Figure 4.5: Return on capital in the US 1950-2011.
The rate of return on capital varies between 0.095 and 0.125 with an upwards trend
since the mid 1980s. Therefore, the upward trend in capital’s share since 2000 can be
attributed more to the rise in the real return of capital rather than an increase in the
capital to output ratio. The rate of return on capital is closely related to the real interest
rate, as we will see in Part III. In particular, in a standard competitive framework the
return on capital equals the real interest rate plus the depreciation rate on capital. If
the depreciation rate on capital is roughly 0.1 per year, these numbers suggest that real
interest rates are quite low on average. An interesting fact not necessarily relevant for
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growth is that the return on capital seems to be high very recently in spite of extremely
low real interest rates.
6. Real wages grow at a sustained, approximately constant, rate.
Finally, we turn our attention to the time series evolution of wages. By now, you should
be able to guess what such a time path looks like. If the labor share is relatively stable
and output per worker rises at a sustained rate, then wages must also be rising at a
sustained rate. To see this, go back to the equation for the labor share. If Yt /Nt is
going up then Nt /Yt must be going down. But the only way for the left hand side to be
approximately constant is for the wage to increase over time. Figure 4.6 plots the log
of wages against time and shows exactly that.
3.6
Actual Series
Trend Series
3.4
log(Wages)
3.2
3
2.8
2.6
2.4
2.2
1950
1960
1970
1980
1990
2000
2010
2020
Year
Figure 4.6: Wages in the US 1950-2011.
Annual wage growth averaged approximately 1.8 percent over the entire time period.
Resembling output and capital per worker, there is a clear sustained increase. Moreover,
wages grow at approximately the same rate as output per worker and capital per worker.
The Kaldor facts can be summarized as follows. Wages, output per worker, and capital
per worker grow at approximately the same sustained rate and the return on capital
is approximately constant. All the other facts are corollaries to these. A perhaps
surprising implication of these facts is that economic growth seems to benefit labor
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(real wages rise over time) and not capital (the return on capital is roughly constant).
Over the next two chapters we show that our benchmark model of economic growth is
potentially consistent with all these facts.
4.2
Cross Country Facts
Kaldor’s facts pertain to the economic progress of rich countries over long periods of time.
However, there is immense variation in income across countries at any given point in time.
Some of these countries have failed to grow at all, essentially remaining as poor today as
they were forty years ago. On the other hand, some countries have become spectacularly
wealthy over the last several decades. In this section we discuss the variation in economic
performance across a subset of countries. We measure economic performance in terms of
output per person. Because not everyone works, this is more indicative of average welfare
across people than output per worker. This does not mean output per capita is necessarily
an ideal way to measure economic well-being. Output per capita does not capture the value
of leisure, nor does it capture many things which might impact both the quality and length
of life. For example, output per capita does not necessarily capture the adverse consequences
of crime or pollution. In spite of these difficulties, we will use output per capita as our chief
measure of an economy’s overall standard of living.
When comparing GDP across countries, a natural complication is that different countries
have different currencies, and hence different units of GDP. In the analysis which follows,
we measure GDP across country in terms of US dollars using a world-wide price index that
accounts for cross-country differences in the purchasing power of different currencies.
1. There are enormous variations in income across countries.
Table 4.1 shows the differences in the level of output per person in 2011 for a selected
subset of countries.
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Table 4.1: GDP Per Capita for Selected Countries
GDP per Person
High income countries
Canada
Germany
Japan
Singapore
United Kingdom
United States
$35,180
$34,383
$30,232
$59,149
$32,116
$42,426
China
Dominican Republic
Mexico
South Africa
Thailand
Uruguay
$8,640
$8,694
$12,648
$10,831
$9,567
$13,388
Cambodia
Chad
India
Kenya
Mali
Nepal
$2,607
$2,350
$3,719
$1636
$1,157
$1,281
Middle income countries
Low income countries
Notes: This data comes from the Penn World Tables, version 8.1. The real GDP
is in terms of chain-weighted PPPs.
In purchasing power parity terms, the average person in the United States was 36.67
times ($42,426/$1,157) richer than the average person in Mali. This is an enormous
difference. In 2011, 29 countries had an income per capita of five percent or less of that
in the U.S. Even among relatively rich countries, there are still important differences in
output per capita. For example, in the US real GDP per capita is about 30 percent
larger than it is in Great Britain and about 25 percent larger than in Germany.
2. There are growth miracles and growth disasters.
Over the last four decades, some countries have become spectacularly wealthy. The
people of Botswana, for instance, subsisted on less than two dollars a day in 1970, but
their income increased nearly 20 fold over the last forty years.
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Table 4.2: Growth Miracles and Growth Disasters
South Korea
Taiwan
China
Botswana
Growth Miracles
1970 Income
2011 Income
$1918
$27,870
$4,484
$33,187
$1,107
$8,851
$721
$14,787
% change
1353
640
700
1951
Growth Disasters
1970 Income
2011 Income
Madagascar
$1,321
$937
Niger
$1,304
$651
Burundi
$712
$612
Central African Republic
$1,148
$762
% change
-29
-50
-14
-34
Notes: This data comes from the Penn World Tables, version 8.1. The real GDP is in terms of chain-weighted PPPs.
As Table 4.2 shows, the countries of East Asia are well represented in the accounting of
growth miracles. On the other hand, much of continental Africa has remained mired in
poverty. Some countries, like those shown under the growth disasters column actually
saw GDP per person decline over the last forty years. Needless to say, a profound task
facing leaders in developing countries is figuring out how to get to the left side of this
table and not fall on the right side.
3. There is a strong, positive correlation between income per capita and human capital.
Human capital refers to the stock of knowledge, social attributes, and habits possessed
by individuals or groups of individuals. It is capital in the sense that human capital
must itself by accumulated over time (i.e. it is not something with which an individual
is simply endowed), is useful in producing other goods, and does not get completely used
up in the process of producing other goods. Unlike physical capital, it is intangible and
possessed by individuals or groups of individuals. As such, measuring what economists
call “human capital” is rather difficult. A natural proxy for human capital is years of
education. On one hand, calculating the average number of years citizens spend in
school seems like a reasonable proxy, but what if teachers do not show up to school?
What if there are no books or computers? Clearly, the quality of education matters as
much as the quantity of education. While imperfect, economists have devised measures
to deal with cross country heterogeneity in quality. With that caveat in mind, Figure
4.7 shows how the level of human capital varies with income per person.
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11
log(RGDP per Person)
8
9
10
7
6
1
1.5
2
2.5
Index of Human Capital
3
3.5
Figure 4.7: Relationship Between Human Capital and Income per
Person
As the level of human capital per person increases, income per person also increases.
This does not mean that more education causes an increase in income. Indeed, the
arrow of causation could run the other direction. If education is a normal good, then
people in richer countries will demand more education. However, it is reasonable to
think that people who know how to read, write, and operate a computer are more
productive than those who do not. Understanding the direction of causation is difficult,
but carries very important policy implications.
4.3
Summary
• In this chapter we covered a number of cross country and within country growth facts.
• The main time series facts are that output, capital, and wages grow at a sustained rate
and that the capital to output ratio and real interest rate do not have sustained growth.
• From a cross country perspective there are enormous variations in living standards.
• Rich countries tend to have more educated populations.
Questions for Review
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1. Write down and briefly discuss the six Kaldor stylized facts about economic
growth in the time series dimension.
2. Write down and discuss the three stylized facts about economic growth in
the cross-sectional dimension.
3. It has been widely reported that income inequality within the US and other
industrialized countries is growing. Yet one of the stylized facts is that
capital does not seem to benefit from economic growth (as evidenced by the
approximate constancy of the return to capital across time). If this is the
case, what do you think must be driving income inequality?
Exercises
1. [Excel Problem] Download quarterly data on output per worker in the
nonfarm business sector for the US for the period 1947 through 2016. You
can do so here.
(a) Take natural logs of the data (which appears as an index) and then
compute first differences of the natural logs (i.e. compute the difference
between the natural log of the index in 1947q2 with the value in 1947q1,
and so on). What is the average value of the first difference over the
entire sample? To put this into annualized percentage units, you may
multiply by 400.
(b) Compute the average growth rate by decade for the 1950s, 1960s, 1970s,
1980s, 1990s, 2000s, and 2010s (even though this decade isn’t complete).
Does the average growth rate look to be constant by decade here? What
pattern do you observe?
2. [Excel Problem] In this problem we investigate the relationship between
the size of government and growth in GDP per person between 1960-2010
for the following countries: Australia, Canada, Germany, Japan, Spain, and
the United States.
(a) Go to this Saint Louis FRED page, and download the “Share of Government Consumption at Current Purchasing Power Parities” for the
relevant subset of countries. Plot the trends of government’s share over
time for the countries. Comment on the trends. Do they seem to be
moving in the same direction for all the countries?
(b) Next, we have to construct real GDP per capita. First, go to this, page,
and download ”‘Expenditure-Side Real GDP at Chained Purchasing
71