Ch05Revision_Long-Run Economic Growth3.29.10.docx

8/11/2019 Ch05Revision_Long-Run Economic Growth3.29.10.docx

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Chapter 5: Long-Run Economic Growth

After reading this chapter, you should be able to:

5.1 Discuss the connection between labor productivity and the standard of living (pages xx)

5.2 Use the Solow Growth Model to Explain the Effect of Capital Accumulation on Labor

Productivity (pages xx)

5.3 Explain how total factor productivity affects labor productivity (pages xx)

5.4 Explain the balanced growth path and convergence and the long-run equilibrium (pages x

x)

Chapter Opener: Labor Productivity and the Standard of Living in China and the United

States

In 2008 China had a population of 1.3 billion people and its economy employed 772.8 million

workers while the United States has a population of just 301.3 million people and the U.S.

economy employs 156.3 million workers. Given these differences in population and growth

rates of real GDP, you might conclude that China had a much larger economy and a higher

standard of living than the United States. However, the opposite is true; the standard of living

was 5.5 times higher in the United States than China. How was this possible? The average

worker in the United States produced 6.4 times as many goods and services than the average

worker in China. Productivity is the key to understanding the standard of living. Understanding


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why these productivity differences exist is the key to understanding why the standard of living

varies across countries.

Even though the standard of living is higher in the United States, the Chinese economy has

been growing much more rapidly than the U.S. economy. The growth rate of real GDP in China

has averaged 8.5 percent per year from 1980 to 2008 while the growth rate of real GDP in the

United States has averaged just 2.9 percent over the same time period. Furthermore, the Chinese

Premier Wen Jiabao on March 5, 2010 reaffirmed his countrys commitment to achieve 8 percent

real GDP growth for 2010 and China is expected to aim for similarly high growth rates in the

future. The rapid Chinese growth rates mean that the Chinese economy is growing larger

relative to the U.S. economy. However, as we just saw the standard of living is still higher in the

United States than in China.

SOURCE: Penn World Tables. Poon, Terrence; Back, Aaron; and Wu, J.R. China Economy

Still Needs Support, Wall Street Journal, March 5, 2010.

An Inside Look [or An Inside Look at Policy] on page xx explores.

Big Question for This Chapter:

In Chapter 1, we introduced 10 big questions in macroeconomics. Here are the big questions we

return to in this chapter:

Big Question 1: Why has the standard of living increased over the last two hundred years?

Big Question 2: Why have some countries failed to achieve sustained economic growth?


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We start by explaining the connection between labor productivity and the standard of living.

As you read this chapter, see if you can answer these questions. You can check your answers

against those we provide at the end of the chapter.

Continued on page xx

[Transition statement]

In Chapter 4, we learned that capital per worker hour and Total Factor Productivity (TFP) are the

key determinants of potential real GDP per worker hour. In this chapter, we explain the

determinants of capital per worker hour and total factor productivity. Real GDP per worker hour

is important because both potential real GDP and potential real GDP per worker hour are closely

tied to labor productivity.

[End transition statement]

5.1 Discuss the Connection between Labor Productivity and the Standard of Living

The circular flow diagram in Figure 2-1 showed us that real GDP equals the income

generated in an economy. So, real GDP per person is a measure of both the income available to

the average person in the country and how many goods and services the average person can

purchase. Economists assume that people are rational and purchase goods and services that

make themselves better off. For example, if you are hungry then you buy food. If you are cold,

then you buy shelter. As your income increases, you use the extra income to purchase goods and

services that make you better off. For this reason, economists often use real GDP per person as a

measure of the standard of living.


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Figure 5-1 shows real GDP per person from 1820 to 2008 for several countries and

Figure 5-1 Real GDP per Person, 1820-2008

SOURCE: Angus Maddison. Data available at:http://www.ggdc.net/maddison/ .

Caption:The levels of real GDP per person varied significantly in 1820 with Africa at $420 per

person and the United Kingdom at $1,706. However, the relative rankings of countries by real

GDP per person have changed because the growth rates of real GDP per person have varied. For

example, the growth rate of real GDP per person averaged just 0.8 percent per year in Africa.

This is an extremely low growth rate and in 2008 real GDP per person was just $1,780 in Africa.

Some African countries such as Malawi, Niger, and Zambia still have real GDP per persons of
http://www.ggdc.net/maddison/http://www.ggdc.net/maddison/http://www.ggdc.net/maddison/http://www.ggdc.net/maddison/


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$744, $514, and $845. For all intents and purposes, growth has not come to many countries in

Africa. In contrast, Japans growth rate averaged 1.9 percent per year so real GDP per person in

Japan rose from $669 in 1820 to $22,816 in 2008. Real GDP per person in China was stagnant

until the 1970s and then accelerated rapidly from essentially zero percent to over eight percent

per year. As a result, of these differences in growth rates, the relative rankings of regions by real

GDP per person have also changed. In 1820, the United Kingdom had the highest real GDP per

person, but now the United States has a higher and Japan has the same real GDP per person.

End Caption

regions around the world. The levels of real GDP per person varied significantly in 1820 with

Africa at $420 per person and the United Kingdom at $1,706. However, the relative rankings of

countries by real GDP per person have changed because the growth rates of real GDP per person

have varied. For example, the growth rate of real GDP per person averaged just 0.8 percent per

year in Africa. This is an extremely low growth rate and in 2008 real GDP per person was just

$1,780 in Africa. Some African countries such as Malawi, Niger, and Zambia still have real

GDP per persons of $744, $514, and $845. For all intents and purposes, growth has not come to

many countries in Africa. In contrast, Japans growth rate averaged 1.9 percent per year so real

GDP per person in Japan rose from $669 in 1820 to $22,816 in 2008. Real GDP per person in

China was stagnant until the 1970s and then accelerated rapidly from essentially zero percent to

over eight percent per year. As a result, of these differences in growth rates, the relative rankings

of regions by real GDP per person have also changed. In 1820, the United Kingdom had the

highest real GDP per person, but now the United States has a higher and Japan has the same real

GDP per person.


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The standard of living is related to labor productivity through the following equation:

(5.1)

,

so the standard of living equals labor productivity, , times labor input,

.

Both labor productivity and labor inputs influence the standard of living, but the most important

determinant of the standard of living is labor productivity. Even if every man, woman, and child

in a country worked 24 hours a day 365 days a year, each person can work no more than 8,760

hours per year because people need sleep and eat, children go to school, and older people retire.

Although there is a clear limit to how much the labor input can increase real GDP per person,

there is no limit to labor productivityas long as productivity increases, real GDP per person

can also increase. Once we have explained labor productivity, we have explained most of real

GDP per person and the standard of living.

Labor productivity rose from $13.47 per worker hour in 1949 to $47.26 per worker hour

in 2008, but labor inputs actually decreased somewhat from 837.3 hours per person in 1949 to

830.0 hours per person in 2008 for reasons that we will discuss in Chapter 7. The decrease in

labor inputs should reduce the potential real GDP per person. So, the entire increase in potential

real GDP per person is due to increased productivity. We can use equation (5.1) to calculate

what potential real GDP per person would have been if labor productivity had remained at 1949

levels and if labor inputs had remained at 1949 levels. Figure 5-2 shows these two series along

with the actual level of potential real GDP

Figure 5-2 Influence of Labor Productivity and Labor Inputs on Real GDP per Person,

1949 - 2008


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SOURCE: Bureau of Economic Analysis, Bureau of the Census, and Congressional Budget

Office.

Caption:The blue line represents potential real GDP per person, the green line represents real

GDP per person if hours per person remained at 1949 levels, and the red line represents potential

real GDP per person if labor productivity remained at 1949 levels. If labor productivity remains

constant at the 1949 level, then potential real GDP per person barely changes and is just $11,177

by 2007. However, if labor inputs remain constant at the 1949 level, then potential real GDP per

person rises all the way to $39,574 by 2007. Clearly, labor productivity is the main determinant

of the increase in potential real GDP per person.


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End Caption

per person. The blue line represents potential real GDP per person, the green line represents real

GDP per person if hours per person remained at 1949 levels, and the red line represents potential

real GDP per person if labor productivity remained at 1949 levels. If labor productivity remains

constant at the 1949 level, then potential real GDP per person barely changes and is just $11,177

by 2007. However, if labor inputs remain constant at the 1949 level, then potential real GDP per

person rises all the way to $39,574 by 2007. Clearly, labor productivity is the main determinant

of the increase in potential real GDP per person.

Problems with Real GDP per person as a Measure of the Standard of Living

Real GDP per person is not a perfect measure of the standard of living, but it is likely the

best measure that we have. As long as people use their income to purchase goods and services

that make them better off, then the standard of living should increase with real GDP per person.

Nevertheless, there are several objections to using real GDP per person as a measure of the

standard of living that we consider in more detail:

Distribution of income

Value of leisure time

Happiness

Life Expectancy

Distribution of Income


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Real GDP per person is just an averageit tells you what the average person in the

economy can consume. However, an average can be misleading because it does not tell you

about the distributionof income. Table 5-2 illustrates this problem using an example of two

people in two countries.

Table 5-2 Income Distribution and Real GDP per Person

Country 1 Country 2

Person 1 $50,000 $99,000

Person 2 $50,000 $1,000

GDP per Person $50,000 $50,000

In country 1, each person earns exactly $50,000 so GDP per person is $50,000 and tells you how

many goods and services the typical person can consume. In this case, real GDP per person is a

very good measure of the standard of living for the typical person. However, this approximation

is not very good for country 2. In country 2, person 1 has an income of $99,000, and person 2

has an income of just $1,000. GDP per person is still $50,000, but person 1 can consume much

more than that amount and person 2 can consume much less. Person 1 has a higher standard of

living and person 2 has a lower standard of living than GDP per person indicates.

Uneven distribution of income is important to keep in mind when using GDP per person

as a measure of the standard of living for the typical person. However, also keep in mind that

two analyses suggest that increases in real GDP per person make the poor better off. First, as

real GDP per person for the world rose by 1.8 percent per year from 1981 to 2005, the number of


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people living on less than $1.25 per day fell from 1.9 billion in 1981 to 1.4 billion in 2005.1 That

represents a 26 percent decrease in the number of extremely poor individuals in just 24 years.

Even if economic growth did not cause the decrease in poverty, economic growth and substantial

decreases in poverty are compatible. Second, if all the gains from economic growth went to only

those individuals at the top of the income distribution, then increases in GDP per person would

have little benefit for those at the bottom of the income distribution. However, there is some

evidence that the poor do benefit from economic growth just as much as the rest of society.

David Dollar and Aart Kraay, economists with the World Bank, found that as real GDP per

person increased by about one percent, the income of the individuals in the bottom 20 percent of

the income distribution also increased by one percent.2 As an economy grows, there is a

tendency for the incomes of the individuals in the bottom of the income distribution to rise.

Therefore, the rich do not get richer at the expense of the poor. Figure 5-3 shows the relationship

between the

Figure 5-3 Relationship between Real GDP per Person and Average Income of Individuals

in the Lower 20 Percent of the Income Distribution

1Data come from the World Banks PovcalNet and are available at

http://web.worldbank.org/WBSITE/EXTERNAL/EXTDEC/EXTRESEARCH/EXTPROG

RAMS/EXTPOVRES/EXTPOVCALNET/0,,contentMDK:21867101~pagePK:64168427~pi

PK:64168435~theSitePK:5280443,00.html . Data downloaded on May 6, 2009.

2Growth is Good for the Poor,Journal of Economic Growthvol.7, no.3, (2002) pp.195-225.
http://web.worldbank.org/WBSITE/EXTERNAL/EXTDEC/EXTRESEARCH/EXTPROGRAMS/EXTPOVRES/EXTPOVCALNET/0,,contentMDK:21867101~pagePK:64168427~piPK:64168435~theSitePK:5280443,00.htmlhttp://web.worldbank.org/WBSITE/EXTERNAL/EXTDEC/EXTRESEARCH/EXTPROGRAMS/EXTPOVRES/EXTPOVCALNET/0,,contentMDK:21867101~pagePK:64168427~piPK:64168435~theSitePK:5280443,00.htmlhttp://web.worldbank.org/WBSITE/EXTERNAL/EXTDEC/EXTRESEARCH/EXTPROGRAMS/EXTPOVRES/EXTPOVCALNET/0,,contentMDK:21867101~pagePK:64168427~piPK:64168435~theSitePK:5280443,00.htmlhttp://web.worldbank.org/WBSITE/EXTERNAL/EXTDEC/EXTRESEARCH/EXTPROGRAMS/EXTPOVRES/EXTPOVCALNET/0,,contentMDK:21867101~pagePK:64168427~piPK:64168435~theSitePK:5280443,00.htmlhttp://web.worldbank.org/WBSITE/EXTERNAL/EXTDEC/EXTRESEARCH/EXTPROGRAMS/EXTPOVRES/EXTPOVCALNET/0,,contentMDK:21867101~pagePK:64168427~piPK:64168435~theSitePK:5280443,00.htmlhttp://web.worldbank.org/WBSITE/EXTERNAL/EXTDEC/EXTRESEARCH/EXTPROGRAMS/EXTPOVRES/EXTPOVCALNET/0,,contentMDK:21867101~pagePK:64168427~piPK:64168435~theSitePK:5280443,00.htmlhttp://web.worldbank.org/WBSITE/EXTERNAL/EXTDEC/EXTRESEARCH/EXTPROGRAMS/EXTPOVRES/EXTPOVCALNET/0,,contentMDK:21867101~pagePK:64168427~piPK:64168435~theSitePK:5280443,00.html


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SOURCE: Dollar and Kraay (2002). Data is available on the web at:http://econ.worldbank.org/WBSITE/EXTERNAL/EXTDEC/0,,contentMDK:20311740~pagePK:

64165401~piPK:64165026~theSitePK:469372,00.html#Growth__inequality_and_poverty

Caption:As real GDP per person increases, the average income for individuals in the bottom 20

percent of the income distribution also increases. On average, when real GDP per person

increases by one percentage point, the average income for individuals in the bottom 20 percent of

the income distribution also increases by about one percentage point. Therefore, there is no

tendency for inequality to increase as real GDP per person increases. We see a similar pattern

when we look at individual countries and regions such as Africa, China, Japan, the United

Kingdom and the United States.
http://econ.worldbank.org/WBSITE/EXTERNAL/EXTDEC/0,,contentMDK:20311740~pagePK:64165401~piPK:64165026~theSitePK:469372,00.html#Growth__inequality_and_povertyhttp://econ.worldbank.org/WBSITE/EXTERNAL/EXTDEC/0,,contentMDK:20311740~pagePK:64165401~piPK:64165026~theSitePK:469372,00.html#Growth__inequality_and_povertyhttp://econ.worldbank.org/WBSITE/EXTERNAL/EXTDEC/0,,contentMDK:20311740~pagePK:64165401~piPK:64165026~theSitePK:469372,00.html#Growth__inequality_and_povertyhttp://econ.worldbank.org/WBSITE/EXTERNAL/EXTDEC/0,,contentMDK:20311740~pagePK:64165401~piPK:64165026~theSitePK:469372,00.html#Growth__inequality_and_povertyhttp://econ.worldbank.org/WBSITE/EXTERNAL/EXTDEC/0,,contentMDK:20311740~pagePK:64165401~piPK:64165026~theSitePK:469372,00.html#Growth__inequality_and_poverty


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End Caption

real GDP per person and average income for the bottom 20 percent of the income distribution.

As real GDP per person increases, the average income for individuals in the bottom 20 percent of

the income distribution also increases. On average, when real GDP per person increases by one

percentage point, the average income for individuals in the bottom 20 percent of the income

distribution also increases by about one percentage point. Therefore, there is no tendency for

inequality to increase as real GDP per person increases. We see a similar pattern when we look

at individual countries and regions such as Africa, China, Japan, the United Kingdom and the

United States.

Value of Leisure Time

Because real GDP per person measures the income of the average person in a country, it

tells us how many goods and services the average person can consume. But people care about

more than the goods and services they can purchase. For example, people want time to spend

with their friends, their spouses, or their children. In other words, individuals also care about

leisure time. If the large increases in the standard of living shown in Figure 5-1 came solely

from increases in labor inputs, then the increase in real GDP per person would have come at the

expense of less leisure time. Whether individuals are better off would therefore depend on the

value of the lost leisure time relative to the value of the goods and services that individuals

gained.


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As we saw earlier, all of the increase in potential real GDP per person has come from

increased productivity and average annual hours worked per person has remained essentially

constant for the United States. Figure 5-4 shows average annual

Figure 5-4 Annual Average Hours per worker hour in Western Countries, 1870 - 2000

SOURCE: Huberman and Minns (2007).

Caption:As real GDP per person has increased, annual average hours per worker hour tends to

decrease. So, leisure time tends to rise as real GDP per person increases. Hours per year for the

average worker in the United States decreased from 3,096 hours per year in 1870 to 1,878 hours

per year in 2000. That is, a 39.3 percent decrease in the amount of time working and a

significant amount of extra time available for leisure activities such as being with friends and


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family. All western nations experienced a similar decrease in average hours per worker hour.

For example, hours per year for the average worker in France decreased from 3,168 in 1870 to

1,443 in 2000. The decrease in hours worked was larger outside of the United States, but

average annual hours worked did decrease in the United States for reasons that we will explain in

Chapter 7.

End Caption

hours of work per worker hour for several western countries: France, Germany, Italy, the United

Kingdom, and the United States. Hours per year for the average worker in the United States

decreased from 3,096 hours per year in 1870 to 1,878 hours per year in 2000. That is, a 39.3

percent decrease in the amount of time working and a significant amount of extra time available

for leisure activities such as being with friends and family. All western nations experienced a

similar decrease in average hours per worker hour. For example, hours per year for the average

worker in France decreased from 3,168 in 1870 to 1,443 in 2000. The decrease in hours worked

was larger outside of the United States, but average annual hours worked did decrease in the

United States for reasons that we will explain in Chapter 7.

Happiness

Richard Easterlin of the University of Pennsylvania looks at results from nineteen

different countries and concludes that there is little relationship between self-reported happiness

in surveys and real GDP per person across countries or within a country.3 This lack of

3Does Economic Growth Improve the Human Lot? Some Empirical Evidence, (1974) In Nations and Households

in Economic Growth: Essays in Honor of Moses Abramowitz edited by Paul David and Melvin Reder. Academic

Press.


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correlation is called theEasterlin Paradoxbecause happiness does not necessarily increase with

incomes. Why the lack of relationship? Easterlin argues that individuals judge themselves

relative to their peer groups. Therefore, if your income increases by 10 percent, but the income

of all your peers also increases by 10 percent then you will not report yourself as any happier

because you have not improved relative to your peers. However, if your income increases by 10

percent while everyone elses income remains constant then you have improved relative to your

peers so you will report yourself as happier.4 Because the increased real GDP per person does

not lead to increased happiness, Easterlin argues that economists and policymakers place too

much emphasis on economic growth. Richard Layard of the London School of Economics notes

that at low levels of real GDP per person an increase in income does lead to increased

happiness.5 However, he also argues that there is little relationship between average income and

happiness once real GDP per person exceeds $15,000 per year. At low levels of income,

economic growth produces vital goods and services like food, shelter, and clothing. However,

once these basic necessities are met economic growth tends to produce luxuries that do not

necessarily make individuals better off. For example, investment bankers and lawyers often

work long hours and neglect their personal lives. They have little time to interact with spouses,

children, or friends. In this example, the extra goods and services may come at the cost of less

fulfilling personal lives. Once individuals reach a certain level of income, they begin to make

judgments about their happiness based on their income relative to their peers rather than their

absolute level of income. Therefore, an increase in income will not make a person happier if

4Will Raising the Incomes of All Increase the Happiness of All?Journal of Economic Behavior and Organization

(1995) pp.35-47.

5Happiness: Has Social Science a Clue? Lionel Robbins Memorial Lecture 2002/2003, London School of

Economics. Available at:http://cep.lse.ac.uk/events/lectures/layard/RL030303.pdf.
http://cep.lse.ac.uk/events/lectures/layard/RL030303.pdfhttp://cep.lse.ac.uk/events/lectures/layard/RL030303.pdfhttp://cep.lse.ac.uk/events/lectures/layard/RL030303.pdfhttp://cep.lse.ac.uk/events/lectures/layard/RL030303.pdf


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everyone elses income increases by just as much. Absolute income is no longer important, but

relative income is important.

The Easterlin Paradox shows that money cannot buy happiness as the saying goes. Or

does it? Justin Wolfers and Betsey Stevenson of the Wharton School at the University of

Pennsylvania reexamined this issue using a wider range of countries.6 They find a robust

positive relationship between self-reported happiness and GDP per person across 131 countries.

In addition, Wolfers and Stevenson also find a positive relationship between economic growth

and happiness within a country. Their work suggests that economic growth and higher absolute

levels of income do make individuals happier.

Life Expectancy

Modern economic growth generates pollution. Consumers create air pollution by burning

gasoline to power their cars and natural gas to heat their homes. Firms create air pollution when

they produce electricity, pesticides, or plastics. This pollution affects the air we breathe, the

water we drink, and the food we eat. Sometimes pollution is a minor irritant that spoils scenic

views, but pollution can also contribute to dangerous diseases such as cancer. In addition, busy

workers have less leisure time, which can create stress and negative health consequences,

potentially shortening life spans. Therefore, the costs to higher real GDP per person might offset

the benefits of the increase in the amount of goods and services that individuals can purchase.

Does life expectancy decrease with real GDP per person? No. In fact, as real GDP per person

6Economic Growth and Subjective Well-Being: Reassessing the Easterlin Paradox in Brookings Paper on Economic

Activity (Spring 2008), 1-102.


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increases, life expectancy at birth increases suggesting that health outcomes actually improve as

real GDP per person increases. Figure 5-5 shows the relationship

Figure 5-5 Relationship between Real GDP per Person and Life Expectancy at

Birth, 2007

SOURCE: World Banks World Development Indicators

Caption:There is a clear tendency for life expectancy to increase as GDP per person increases.

When real GDP per person is $5,000 the life expectancy is 66.9 years and this rises to 74.1 years

as real GDP per person increases to $15,000. However, if real GDP per person increases by

another ten thousand dollars to $25,000 then life expectancy only increases to 77.4. Therefore,

life expectancy increases at a decreasing rate with real GDP per person. Despite the pollution


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costs associated with economic growth, the higher real GDP per person the longer the average

person can expect to live.

End Caption

between life expectancy at birth and GDP per person for 185 countries in 2007. There is a clear

tendency for life expectancy to increase as GDP per person increases. When real GDP per

person is $5,000 the life expectancy is 66.9 years and this rises to 74.1 years as real GDP per

person increases to $15,000. However, if real GDP per person increases by another ten

thousand dollars to $25,000 then life expectancy only increases to 77.4. Therefore, life

expectancy increases at a decreasing rate with real GDP per person. Despite the pollution costs

associated with economic growth, the higher real GDP per person the longer the average person

can expect to live.

Part of the reason that pollution costs do not shorten life spans as real GDP per person

increases is that a clean environment is a normal good, which means that individuals purchase

more of it as their incomes increase. This idea lies behind the environmental Kuznets curve. The

relationship depicted by the curve states that as an economy expands, pollution initially

increases, reaches a maximum, and then begins to decline as individuals choose to spend more of

their income on a cleaner environment. Therefore, rather than destroying the environment and

reducing life expectancy, economic growth may improvethe environment and increase life

expectancy. As real GDP per person increases, individuals can also purchase more health care

which should also increase life expectancy.


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5.2 Use the Solow Growth Model to Explain the Effect of Capital Accumulation onLabor

Productivity

Labor productivity is the key determinant of real GDP per person and, therefore, the standard of

living. In Chapter 4, we learned that capital accumulation and total factor productivity (TFP) are

the primary determinants of the labor productivity. Therefore, if we want to understand why the

standard of living increases over time, we need to understand capital accumulation and the

determinants of total factor productivity. In this section, we discuss capital accumulation and

assume that total factor productivity remains constant. Economists use the Solow growth model

to explain how capital accumulation influences labor productivity. The model is named after

Nobel Laureate Robert Solow of the Massachusetts Institute of Technology who developed the

model during the 1950s.7 His work and this model has become the foundation for how

economists think about economic growth. We work with the intensive form of the aggregate

production function that we described in Chapter 4, in whichyis potential real GDP per worker

hour and kis capital per worker hour. Economists also call kthe capital-labor ratio. As we saw

in Chapter 4, potential real GDP per worker hour and the capital-labor ratio are related through

the aggregate production function:

(5.2)

where is capitals share of income andAis total factor productivity. We learned in Chapter 4

that for the United States capitals share ofincome equals 0.32 and that total factor productivity

in 2007 was 9.4. Figure 5-6 shows:

7Solow, Robert. A Contribution to theTheory of Economic Growth, Quarterly Journal of Economics, 70, (February

1956) pp.65-94.


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Figure 5-6 The Aggregate Production Function for the United States, 2007

Caption:The capital-labor ratio for the United States was $144.94 per hour in 2007, and

potential real GDP per worker hour was $46.36 per hour. The production function shows us

what happens to labor productivity as the capital-labor ratio increases while keeping total factor

productivity constant. If the capital-labor ratio increases to $174.94 per hour, then labor

productivity increases to $49.24. The marginal product of the extra capital is $2.88 per hour. If

the capital-labor ratio increases again by $30 to $204.94 per hour, then labor productivity also

increases to $51.79 per hour. The marginal product of the extra capital is still positive but

decreases to $2.56 per hour. Why does the marginal product of capital decrease? The marginal

product of capital decreases due to diminishing marginal returns when total factor productivity is

constant, so the contribution of capital accumulation to labor productivity growth eventually

becomes zero. The fact that capital experiences diminishing marginal returns means that the


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sustained increase in labor productivity and the standard of living that the United States and

other countries have experienced must be due to factors other than capital accumulation such as

total factor productivity growth. We will return to this issue later in the chapter.

End Caption

the aggregate production function for the United States. The capital-labor ratio for the United

States was $144.94 per hour in 2007, and potential real GDP per worker hour was $46.36 per

hour. The production function shows us what happens to labor productivity as the capital-labor

ratio increases while keeping total factor productivity constant. If the capital-labor ratio

increases to $174.94 per hour, then labor productivity increases to $49.24. The marginal

product of the extra capital is $2.88 per hour. If the capital-labor ratio increases again by $30 to

$204.94 per hour, then labor productivity also increases to $51.79 per hour. The marginal

product of the extra capital is still positive but decreases to $2.56 per hour. Why does the

marginal product of capital decrease? The marginal product of capital decreases due to

diminishing marginal returns when total factor productivity is constant, so the contribution of

capital accumulation to labor productivity growth eventually becomes zero. The fact that capital

experiences diminishing marginal returns means that the sustained increase in labor productivity

and the standard of living that the United States and other countries have experienced must be

due to factors other than capital accumulation such as total factor productivity growth. We will

return to this issue later in the chapter.

Capital Accumulation and the Bath Tub Analogy


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Labor productivity is the critical determinant of the standard of living. We learned in

Chapter 4 that 41.7 percent of the labor productivity growth is due to capital accumulation. The

Solow growth model helps explain capital accumulation. If you understand a bath tub, then you

understand the essential elements of this model. Figure 5-7 shows a bath tub with water flowing

into the tub through the faucet

Figure 5-7 Capital Accumulation and the Bath Tub Analogy

Caption:Understanding the basics of capital accumulation are as simple as understanding why

the level of water rises and falls in a bath tub. Investment per worker hour is the water flowing

into the bath tub and the amount of investment necessary to keep the capital-labor ratio constant

is water flowing out of the bath tub. The level of water in the bath tub is the capital-labor ratio.

End Caption

and water flowing out of the bath tub through the drain. The level of water is a stock variable

because we measure it at a point in time, while the water flowing into and flowing out of the tub

Water flowing into the bath tub

Water flowing out of the bath tu

Level of water in the bath tub


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are flow variables that are measuredper time period. When is the level of water in the bath tub

constant? The answer is simple. The level of the water in the tub is constant when the water

flowing into the bath tub is exactly equal to the water flowing out of the bath tub. How does the

level of water in the tub change? The level of water in the tub increases when the water flowing

into the tub is greater than the water flowing out of the tub, and the level of water in the tub

decreases when the water flowing out of the tub is greater than the water flowing into the tub.

To apply the bath tub analogy to labor productivity, we just need to identify the stock and

flow variables in the Solow growth model. The capital-labor ratio is the stock variable because

we measure it at a point in time as the amount of capital goods per worker. As a reminder, the

capital-labor ratio is defined as:

,

whereKis the stock of capital goods andLis the labor force. The capital-labor ratio can change

for one two reasons: either the capital stock changes or labor force changes.

Investment and Water Flowing into the Bath Tub

The level of water in the bath tub increases when water flows into the tub, but what

causes the capital-labor ratio to increase? The capital-labor ratio increases when the stock of

machines, tools, buildings, and roads increases. In other words, when households, firms, or the

government purchase investment goods. For simplicity, we assume that the investment rate of

for the economy,s, is a constant ratio between zero and one. Investment per worker hour, i,

equals:

(5.3) .


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Figure 5-8 shows the investment rate for the United States from 1949 to 2008. During this

Figure 5-8 The Investment Rate for the United States, 1949 - 2008

SOURCE: Bureau of Economic Analysis and Congressional Budget Office

Caption:the private sector conducts the vast majority of investment activity in the United

States. From 1949 to 2008, private sector investment averaged 16.0 percent of potential GDP

while government investment averaged just 4.1 percent of potential real GDP. You should also

notice a distinct decrease in the rate of government investment after the 1960s, from 6.5 percent

in 1953 to 3.4 percent in 2008.

End Caption

period, the investment rate averaged 0.201 of potential GDP, so:


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i= 0.201y

for the United States. Figure 5-9 shows that the private sector conducts the vast majority of

investment activity in the United States. From 1949 to 2008, private sector investment averaged

16.0 percent of potential GDP while government investment averaged just 4.1 percent of

potential real GDP. You should also notice a distinct decrease in the rate of government

investment after the 1960s, from 6.5 percent in 1953 to 3.4 percent in 2008. Remember that

, so:

, and

for the United States,

.

Figure 5-9 shows how investment per worker hour changes as the capital-labor ratio increases.

Figure 5-9 Investment per worker hour, Real GDP per worker hour, and the Capital-

Labor Ratio


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Caption:Notice that the investment function has the same general shape as the production

function from Figure 4.6 (a) in Chapter 4. This similarity occurs because we have assumed a

constant investment rate for the economy. Therefore, as the capital-labor ratio increases real

GDP per worker hour also increases and that causes investment per worker hour to increase.

However, because of diminishing marginal returns, the increase in investment per worker hour

gets smaller and smaller as the capital-labor ratio increases.

End Caption

Notice that the investment function has the same general shape as the production function from

Figure 4.6 (a) in Chapter 4. This similarity occurs because we have assumed a constant

investment rate for the economy. Therefore, as the capital-labor ratio increases real GDP per


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worker hour also increases and that causes investment per worker hour to increase. However,

because of diminishing marginal returns, the increase in investment per worker hour gets smaller

and smaller as the capital-labor ratio increases.

The Role of the Financial Sector in Capital Accumulation

Capital accumulation plays an important role in explaining labor productivity. But how

do households, firms, and the government finance the purchase of new capital goods? This

question highlights the importance of the financial sector because investment is often financed

by funds obtained in financial markets. If Ford wants to build a new factory in the United States,

then it is likely to obtain the funds in financial market by either borrowing or issuing new stock.

Where do financial markets obtain the funds to finance Fords investment? Other individuals in

the household, government, and foreign sectors have decided to save. When you save by putting

some of your income in a savings account, this saving allows the bank to lend those funds toa

household or a firm, like Ford, that wants to invest. Therefore, a well functioning financial

market is essential for allowing households, firms, and the government to finance the investment

expenditures that lead to higher labor productivity.

Investment is the purchase of capital goods by households and firms, , and thegovernment, . Therefore, the total amount of capital goods purchased in aneconomy,I, is:

.


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The funds for purchasing these investment goods come from private savings, ,government savings, , and foreign sector savings, . Therefore, nationalsavings, S, is:

.

Private saving equals household disposable income minus consumption expenditures. Therefore,

a tax increase, a decrease in income, or an increase in consumption all lead to less private

savings. If government saving is positive then the government runs a budget surplus, but if

government saving is negative then the government runs a budget deficit. A budget surplus, like

the United States had in the late 1990s, leads to higher national saving, all else equal. In

contrast, the large budget deficits that the government expects to run from 2010 to 2019 means

that national saving is lower. Foreign sector saving is the sum of saving by foreign households

and foreign governments. When individuals in these sectors save, then there is a larger pool of

funds available to finance investment expenditures.

When we discussed the loanable funds model in Chapter 3, we learned that saving equals

investment, so:

,

and

.

The results here are very similar to the results with the loanable funds model in Chapter 3. A

budget deficit means government savings becomes negative. Unless private or foreign savings


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increases to compensate, investment expenditures must decrease. Whether the budget deficit

causes private or government investment expenditures to decrease depends on the circumstances.

Similarly, if the household sector saves more and government and foreign savings remain

constant, then investment expenditures must increase. Again, whether private or government

investment expenditures increase depends on the circumstances.

Break-Even Investment and Water Flowing out of the Bath Tub

Break-even investmentis the level of investment necessary to keep the capital-labor ratio

constant. When investment is greater than the break-even level, the capital-labor ratio increases

and when investment is less than the break-even level, the capital-labor ratio decreases. Two

factors determine the break-even level of investment. First, the capital stock depreciates over

time. We assume that the depreciation rate, d, is a constant fraction of the capital-labor ratio and

that the depreciation rate is between zero and one, so:

.

Second, the capital-labor ratio can decrease when the capital stock is constant and the number of

workers increases. The capital-labor ratio decreases because the existing capital stock is spread

across a larger labor force. You can think of this as the dilution of the existing capital stock. We

use nto represent the growth rate of the labor force and ntakes a value between zero and one, so:

.

Therefore, we can think of break-even investment as:

(5.4)


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.

Notice that break-even investment is just a constant fraction of the capital-labor ratio, so at

higher levels of the capital-labor ratio the break-even investment is higher. For the United

States, the growth rate of potential labor hours has averaged 1.2 percent or 0.012 per year from

1949 - 2008. The depreciation rate depends on the type of capital good. For example, buildings

can last for decades while computers may only be useful for a few years so computers depreciate

much more quickly than buildings. However, a depreciation rate of 10 percent or 0.10 is a

common value to use. Hence, break-even investment for the United States is:

Break-Even Investment= (0.10 + 0.012)k= 0.112k.

When we graph the break-even investment line in Figure 5-10, we see that it is a straight line

with

Figure 5-10 Break-Even Investment and the Capital-Labor Ratio


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Caption: The break-even investment line has a slope equal to the sum of the depreciation rate

and the labor force growth rate. At higher levels of the capital-labor ratio, more investment is

required to keep the capital-labor ratio constant, so the break-even level of investment is also

higher. An increase in the depreciation or labor force growth rates leads to a steeper break-even

investment line, while a decrease in the depreciation rate or labor force growth rates leads to a

flatter break-even investment line.

End Caption

a positive slope equal to (d+ n); at higher levels of the capital-labor ratio, more investment is

required to keep the capital-labor ratio constant, so the break-even level of investment is also

higher. An increase in the depreciation or labor force growth rates leads to a steeper break-even

investment line, while a decrease in the depreciation rate or labor force growth rates leads to a

flatter break-even investment line.


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Equilibrium and the Steady State

The change in the level of water in the bath tub equals the water flowing into the bath tub

minus the water flowing out of the bath tub. For the level of water in the bath tub to remain

constant, the water flowing into the tub must equal the water flowing out of the tub. Think of

this as the equilibrium for the bath tub. In terms of the Solow growth model, the level of water is

the capital-labor ratio so equilibrium occurs when the capital-labor ratio is constant. Economists

call this equilibrium asteady state. A steady stateis an equilibrium in which the capital-labor

ratio and output per worker hour are constant, but capital, labor, and output are growing. The

steady-state is the long-run equilibrium so if an economy is not at the steady-state then the

economy will gradually move toward the steady state. To find the steady state, we first need to

find an equation for the change in the capital-labor ratio and the change in the capital-labor ratio

equals investment minus break-even investment:

.

We can express this relationship as,

.

If we plug in the relationship for investment per worker hour in equation (5.4) we get

(5.5) .

Equation (5.14) is the key equation for the Solow growth model because it tells us how the

capital-labor ratio evolves over time and allows us to determine the equilibrium. We know that


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in the steady state. We can plug this fact into equation (5.10) to solve for the steady-state capital-labor ratio k* as:

(5.6) * +

.

Using the aggregate production function, the steady state real GDP per worker hour is:

(5.7) * +

.

For the United States, the formula for the steady state values is

* +

and

.

(MD: Steady State An equilibrium in which the capital-labor ratio and output per worker hour

are constant, but capital, labor, and output are growing. The steady-state is the long-run

equilibrium so if an economy is not at the steady-state then the economy will gradually move

toward the steady state.)

Figure 5-11 shows both the investment and break-even

Figure 5-11 Equilibrium in the Solow Growth Model


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Caption:The investment curve and the break-even investment line intersect at pointA. In the

steady state, the capital-labor ratio is constant, so that the change in the capital-labor ratio is zero.

This point occurs where the investment line intersects the break-even investment line at pointA

in Figure 5-12. At point A, the capital-labor ratio is $63.8 per worker hour and the levels of

investment and break-even investment are $7.1 per worker hour. Suppose that the initial capital-

labor ratio is $22 per worker hour which we label as in Figure 5-12. At that capital-laborratio, the level of investment, , is $5.1 per worker hour and is greater than break-eveninvestment, , of $2.5 per worker hour. According to equation (5.8), and thecapital-labor ratio increases toward the steady state capital-labor ratio, . Now suppose theinitial capital-labor ratio is $125 per worker hour which we label as in Figure 5-12. At thatcapital-labor ratio, the level of investment, , is $8.9 per worker hour and is greater than break-


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even investment, , of $14.0 per worker hour. According to equation (5.8), and the capital-labor ratio decreases toward the steady-state capital-labor ratio, .

End Caption

investment lines. In the steady state, the capital-labor ratio is constant, so that the change in the

capital-labor ratio is zero. This point occurs where the investment line intersects the break-even

investment line at pointAin Figure 5-12. At point A, the capital-labor ratio is $63.8 per worker

hour and the levels of investment and break-even investment are $7.1 per worker hour.

The steady state at pointAis stable because there is a built-in tendency for the economy

to move toward the equilibrium. For example, suppose that the initial capital-labor ratio is $22

per worker hour which we label as in Figure 5-12. At that capital-labor ratio, the level ofinvestment, , is $5.1 per worker hour and is greater than break-even investment, , of$2.5 per worker hour. According to equation (5.8), and the capital-labor ratio increasestoward the steady state capital-labor ratio, . The increase in the capital-labor ratio is thevertical distance between the investment and break-even investment lines. Notice that this

vertical distance decreases as the capital-labor ratio increases. Why does this happen? As the

economy accumulates more capital goods per worker hour, capital goods become less productive

because of diminishing marginal returns. As a result, the extra output and investment that the

economy receives from additional capital decreases as the economy accumulates more capital.

The increase in the capital-labor ratio continues until and that does not occur until thecapital-labor ratio equals .


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Now suppose the initial capital-labor ratio is $125 per worker hour which we label as in Figure 5-12. At that capital-labor ratio, the level of investment, , is $8.9 per worker hourand is greater than break-even investment, , of $14.0 per worker hour. According toequation (5.8), and the capital-labor ratio decreases toward the steady-state capital-laborratio, . The decrease in the capital-labor ratio is the vertical distance between the investmentand break-even investment lines. Notice that this vertical distance decreases as the capital-labor

ratio decreases due to diminishing marginal returns. Why does this happen? Once again it is due

to diminishing marginal returns. As the economy reduces capital goods per worker hour, capital

goods become more productive. As a result, the extra output and investment that the economy

receives from additional capital increases as the economy reduces the capital-labor ratio. The

decrease in the capital-labor ratio continues until and that does not occur until thecapital-labor ratio equals . So, the steady-state is the equilibrium for the economy. We willcome back to this point when we discuss the balanced growth path in the next section.

The Investment Rate and Real GDP Per Worker Hour

Now that we have an equilibrium model for the capital-labor ratio and real GDP per

worker hour, we can ask what causes the equilibrium to change. Figure 5-12 shows what

happens when the investment rate increases from 0.201, , to 0.302, . First,

Figure 5-12 An Increase in the Investment Rate


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Caption:An increase in the investment rate from 0.201 to 0.302 shifts the investment curve

shifts upward from to , so the economy is now producing more investment goods for anygiven level of the capital-labor ratio. Next, the capital-labor ratio increases. The level of

investment is now greater than the level necessary to replace depreciation and provide the new

workers with just as much capital as the existing workers. As a result, the capital-labor ratio

begins to rise from the original steady-state value of $63.8, , to the new steady-state value of$115.8, . Because the capital-labor ratio is one of the inputs of the production function, thehigher capital-labor ratio increases real GDP per worker-hour. So, the Solow growth model

predicts that a higher investment rate will increase labor productivity from $35.5 dollars per

worker hour, , to $43.0 dollars per worker hour, , and a higher standard of living. Noticethat the increase in real GDP per worker hour eventually decreases to zero due to diminishing

marginal returns to capital goods.


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End Caption

the investment curve shifts upward from to , so the economy is now producing moreinvestment goods for any given level of the capital-labor ratio. Next, the capital-labor ratio

increases. The level of investment is now greater than the level necessary to replace depreciation

and provide the new workers with just as much capital as the existing workers. As a result, the

capital-labor ratio begins to rise from the original steady-state value of $63.8, , to the newsteady-state value of $115.8,

. Because the capital-labor ratio is one of the inputs of the

production function, the higher capital-labor ratio increases real GDP per worker-hour. So, the

Solow growth model predicts that a higher investment rate will increase labor productivity from

$35.5 dollars per worker hour, , to $43.0 dollars per worker hour, , and a higher standard ofliving. Notice that the increase in real GDP per worker hour eventually decreases to zero due to

diminishing marginal returns to capital goods.

[Box Begins]

Macro Data: The American Reinvestment and Recovery Act and Real GDP Per Person

Congress passed and President Obama signed into law the American Reinvestment and Recovery

Act in February 2009. The Act is designed to stimulate the economy in the short run, but it may

have a negative long-run effect on labor productivity and the standard of living. The

Congressional Budget Office estimates that the Act will reducereal GDP by between 0.0 percent


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and 0.2 percent in 2019.8 Why is that? The Act will increase the budget deficit by $787 billion

dollars, which the government is likely to pay for by borrowing. In other words, the government

will run a budget deficit and government savings will decrease. Because investment

expenditures equal saving, the investment rate should fall and the decrease in the investment rate

should lead to a lower capital-labor ratio and a lower level of potential real GDP per worker

hour. Figure 5-13 shows the effect of the decrease in the investment due to the borrowing

Figure 5-13 The Long-Run Effect of the American Reinvestment and Recovery Act

8Estimated Macroeconomic Impacts of the American Recovery and Reinvestment Act of 2009. Letter from

Douglas Elmendorf Director of the Congressional Budget Office to Senator Charles E. Grassley (Iowa). March 2,

2009. Available at:http://www.cbo.gov/ftpdocs/100xx/doc10008/03-02-Macro_Effects_of_ARRA.pdf.
http://www.cbo.gov/ftpdocs/100xx/doc10008/03-02-Macro_Effects_of_ARRA.pdfhttp://www.cbo.gov/ftpdocs/100xx/doc10008/03-02-Macro_Effects_of_ARRA.pdfhttp://www.cbo.gov/ftpdocs/100xx/doc10008/03-02-Macro_Effects_of_ARRA.pdfhttp://www.cbo.gov/ftpdocs/100xx/doc10008/03-02-Macro_Effects_of_ARRA.pdf


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Caption:The budget deficits will reduce government savings which will reduce the funds

available to the private sector to invest. As a result the investment rate will decrease froms1tos2

and the investment curve will shift downward froms1ytos2y. As a result, the steady-state

capital-labor ratio will decrease from to and output per worker hour will decrease from to . Because workers will be less productive, potential real GDP will also decrease.

End Caption

associated with the American Reinvestment and Recovery Act. The budget deficits will reduce

government savings which will reduce the funds available to the private sector to invest. As a

result the investment rate will decrease froms1tos2and the investment curve will shift

downward froms1ytos2y. As a result, the steady-state capital-labor ratio will decrease from to and output per worker hour will decrease from to . Because workers will be lessproductive, potential real GDP will also decrease. However, part of the funds from the Act will

go to infrastructure and other investment projects for which the government is responsible. If

the Act adds government investment dollars to balance off falling private investment dollars, the

overall investment rate in the economy, and the investment curve, will shift down only slightly.

The end result will be a relatively small decrease in potential real GDP per worker hour and

potential real GDP.However, if the Act finances government consumption or investment

projects with little or no value, then the investment rate for the economy will decrease and so

will labor productivity and potential real GDP. The key point is that a deficit may reduce

potential real GDP depending upon what the government does with the funds it borrows.

See related problem XXX on page XXX.


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[End Box]

Depreciation, Labor Force Growth Rate, and Real GDP per worker hour

Figure 5-14 shows what happens when the depreciation rate decreases. First, the break-

Figure 5-14 A Decrease in the Depreciation Rate

Caption:A decrease in the depreciation rate flattens the break-even investment line. Next, the

capital-labor ratio increases for the same reason as we found when the investment rate increases;

the level of investment is now greater than the level necessary to replace depreciation and

provide the new workers with just as much capital as the existing workers. As a result, the

capital-labor ratio begins to rise from the original steady-state value of to the new steady-statevalue of . Because the capital-labor ratio is one of the inputs of the production function, thehigher capital-labor ratio increases potential real GDP per worker hour. Therefore, the Solow


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growth model predicts that a lower depreciation rate will lead to higher productivity and higher

standard of living.

End Caption

even investment line flattens. Next, the capital-labor ratio increases for the same reason as we

found when the investment rate increases; the level of investment is now greater than the level

necessary to replace depreciation and provide the new workers with just as much capital as the

existing workers. As a result, the capital-labor ratio begins to rise from the original steady-state

value of to the new steady-state value of . Because the capital-labor ratio is one of theinputs of the production function, the higher capital-labor ratio increases potential real GDP per

worker hour. Therefore, the Solow growth model predicts that a lower depreciation rate will

lead to higher productivity and higher standard of living.

What about the growth rate of the labor force? Notice that the growth rate of the labor

force and the depreciation rate both influence the slope of the break-even investment line in the

same way. Therefore, a decrease in the labor force growth rate will have exactly the same effect

on the standard of living as a decrease in the depreciation rate. Therefore, the Solow growth

model predicts that a lower labor force growth rate will lead to higher productivity and higher

standard of living.

Solved Problem 5.1: A Decrease in the Labor Force Growth Rate and Real GDP per

worker hour. According to the United Nations Population Division, the population growth rate


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averaged 1.7 percent per year between 1950 and 2005. The following table shows the

Population Division forecasts that the population growth rates for different regions in the world.

Population Growth Rates across the World

Period Africa Asia Europe North

America

South

America

World

1950 to 2005 2.6 1.9 0.5 2.2 1.2 1.7

2005 to 2050 1.7 0.6 -0.1 0.6 0.6 0.8

Change in thePopulationGrowth Rate -0.9 -1.3 -0.6 -1.6 -0.6 -0.9

SOURCE: United Nations Population Division. The calculations are based upon the Median

Variant forecast.

The slower population growth rate should also reduce the growth rate of the labor force. What

effect will this reduction have on labor productivity and the standard of living in the world?

Solving the Problem:

Step 1: Review the chapter material. The problem asks you to determine the effect of a

decrease in the labor force growth rate on labor productivity and the standard of living, so you

may want to review the section Depreciation, Labor Force Growth Rate, and Real GDP per

worker hour, which begins on page x.


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Step 2: Use a graph to determine how a decrease in labor force growth rate influences the

Solow growth model.

The Solow growth model consists of three curves: the aggregate production function, the

investment curve, and the break-even investment line. To determine the effect of a decrease in

the labor force growth rate, we must determine which, if any, of these curves the labor force

growth rate influences. Earlier we learned that the break-even investment line is so theslope of the break-even investment depends on the depreciation rate and the labor force growth

rate. When the labor force growth rate decreases, the slope of the break-even investment line

will decrease and the line will flatten or pivot downwards. The equation for the aggregate

production function is and equation the equation for the investment curve is then youwill see so that the labor force growth rate does not influence either of these curves. Therefore,

the labor force growth rate influences only break-even investment. Your graph showing the

effect of the decrease in the labor force growth rate should look like this:


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Step 3:Determine the effect on the capital-labor ratio. The break-even investment line shifts

downwards. At the initial capital-labor ratio,

, the level of investment,

is greater than the

new level of break-even investment, . Equation (5.14) tells us how the capital-laborratio evolves over time. Using that equation and what we know about the current levels of

investment and break-even investment:

.

As a result, the capital-labor ratio begins to increase towards the new steady-state capital-labor

ratio, . The change in the capital-labor ratio is the vertical distance between the investmentcurve and the break-even investment line. The vertical distance gets smaller as the capital-labor

ratio increases due to diminishing marginal returns, so the increase in the capital-labor ratio gets


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smaller and smaller as the world approaches the new steady state. Growth stops when the

economy reaches the new steady state at pointB. The steady state level of labor productivity has

increased from to .

Step 4: Determine the effect of the capital-labor ratio on the standard of living. Economists

use real GDP per person to measure the standard of living, and equation (5.1) tells us that real

GDP per person equals labor productivity multiplied by labor input. We measure labor input as

average annual hours worked per person. If labor input remains constant, then the increase in

labor productivity from to will increase the standard of living. The ultimate effect of thedecrease in the population growth rate is to increase the standard of living for the average person

in the world.

The United Nations predicts that the population growth rate will decrease during the 2005

to 2050 period for all regions of the world. However, the table at the beginning of this Solved

Problems shows that the decrease in the population growth rate will vary across regions of the

world. For example, Africa is expected to have a population growth rate of 1.7 percent per year.

Asia, North America, and South America are expected to have population growth rates of about

0.6 percent per year, but Europe is expected to have a negativepopulation growth rate of -0.1

percent per year. The largest decrease in the population growth rate will occur in North America

so, all else equal, you should expect that labor productivity and the standard of living increases

from slower labor force growth rates will be highest in North America.


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Your Turn:See related problem XXX on page XXX.

**********************************************************************

Table 5-3 summarizes how changes in the Solow growth model change the steady-state

Table 5-3 Summary of Changes in the Steady-State

An increase in will leading to and

the investment rate shift the investmentcurve up

an increase in thecapital-labor ratio

increase potential realGDP per worker hour.

the level of total

factor productivity

shift the investment

curve up

an increase in the

capital-labor ratio

increase potential real

GDP per worker hour.

the depreciation

rate

shift the break-eveninvestment live up

a decrease in thecapital-labor ratio

a decrease in potentialreal GDP per worker

hour.

the labor force

growth rate

shift the break-eveninvestment live up

a decrease in thecapital-labor ratio

a decrease in potentialreal GDP per worker

hour.

potential real GDP per worker hour. Increases in the investment rate and total factor productivity

lead to higher real GDP per worker hour in the steady-state while increases in the depreciation

rate and the growth rate of the labor force lead to lower real GDP per worker hour in the steady

state.

5.3 Explain how Total Factor Productivity Affects Labor Productivity


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Total factor productivity measures the overall efficiency of the economy in transforming capital

and labor into final goods and services that households can consume. Total factor productivity

growth, along with capital accumulation are the two sources for increases in labor productivity.

We just learned that increases in labor productivity from capital accumulation eventually

decrease to zero due to diminishing marginal returns. As a consequence, total factor productivity

is the ultimate source of labor productivity growth and, hence, the increase in the standard of

living.

Total Factor Productivity and Real GDP per worker hour

Figure 5-15 shows the effect of an increase in total factor productivity for the United

Figure 5-15 An Increase in Total Factor Productivity


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Caption:For the United States in 2007, total factor productivity is initially 9.4 and the capital-

labor ratio is $144.9 per hour so potential real GDP per worker hour is $46.4 and the economy is

at pointAin Figure 5-16. If the capital-labor ratio remains constant and total factor productivity

increases by one point to 10.4 then the production function shifts up and the economy is now at

pointB, so that at any given capital-labor ratio, real GDP per worker hour increases to $51.3 per

hour. In this example, an increase in total factor productivity has a similar effect as an increase

in the investment rate. But there is an important difference. The marginal product of capital

decreases as the economy accumulates more capital holding all else constant, but the extra output

from increasing total factor productivity does not. If total factor productivity increases to by

another point to 11.4 then the economy moves to point Cand potential real GDP per worker hour

increases to $56.2. Therefore, each time total factor productivity increases by one point,

potential real GDP per worker hour increases by $4.9. In contrast to capital goods, there are no

diminishing marginal returns for total factor productivity. Therefore, there is no limit to growth

from increases in total factor productivity and total factor productivity growth must be the

explanation for increases in labor productivity and the standard of living.

End Caption

States in 2007 assuming that the capital-labor ratio equals $144.9 per hour. Total factor

productivity is initially 9.4 so potential real GDP per worker hour is $46.4 and the economy is at

pointAin Figure 5-16. If the capital-labor ratio remains constant and total factor productivity

increases by one point to 10.4 then the production function shifts up and the economy is now at

pointB, so that at any given capital-labor ratio, real GDP per worker hour increases to $51.3 per

hour. In this example, an increase in total factor productivity has a similar effect as an increase


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in the investment rate. But there is an important difference. The marginal product of capital

decreases as the economy accumulates more capital holding all else constant, but the extra output

from increasing total factor productivity does not. If total factor productivity increases to by

another point to 11.4 then the economy moves to point Cand potential real GDP per worker hour

increases to $56.2. Therefore, each time total factor productivity increases by one point,

potential real GDP per worker hour increases by $4.9. In contrast to capital goods, there are no

diminishing marginal returns for total factor productivity. Therefore, there is no limit to growth

from increases in total factor productivity and total factor productivity growth must be the

explanation for increases in labor productivity and the standard of living.

New technology is one source of higher total factor productivity. If a new technology is

discovered that increases the processing power of computers, then total factor productivity would

increase fromA1toA2. The production function would shift upward and real GDP per worker

hour would increase. Similarly, if a second technology is discovered that reduces congestion on

the internet, then the productivity of all existing computers would increase so total factor

productivity would increase fromA2toA3. The production function would again pivot upward

and real GDP per worker hour would increase again.

What effect does an increase in total factor productivity have on the steady-state values of

the capital-labor ratio and potential real GDP per worker hour? Figure 5-16 shows the effect of a

one-time

Figure 5-16 A One-Time Increase in Total Factor Productivity


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Caption:The one-time increase in total factor productivity shifts the production function from

to. At the initial capital-labor ratio, investment,, is now greater thanbreak-even investment. As a result, the capital-labor ratio increases and this causes potential real

GDP per worker hour to increase and the steady state to move from pointAto pointB.

End Caption

increase in total factor productivity. The one-time increase in total factor productivity shifts the

production function from

to

. At the initial capital-labor ratio, investment,

, is now greater than break-even investment. As a result, the capital-labor ratioincreases and this causes potential real GDP per worker hour to increase and the steady state to

move from pointAto pointB.


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In fact, the growth rate of total factor productivity,gA, is the key determinant for labor

productivity growth and the growth rate of the standard of living when labor inputs per person

are constant. Table 5-4

Table 5-4 Steady-State Growth Rates

Variable Symbol Steady-State Growth Rate

Capital-labor ratio (

)

Potential real GDP per worker hour (

)

Capital Stock ( ) Potential real GDP ( )

shows the steady-state growth rates for the capital-labor ratio and potential real GDP per worker

hour when the hours per person (labor input) is constant. The appendix at the end of the chapter

shows the how to derive the steady-state growth rates. For the United States, capitals share of

income, , is 0.32. TFP growth has averaged 0.0125 or 1.25 percent per year and the growth rate

of potential labor hours, n, averaged 0.0121 or 1.21 percent per year. So, for the United States:

or 1.84 percent per year,

and

or 3.05 percent per year.


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Labor productivity increases by about 1.84 percent per year, and if labor inputs per person are

constant, then potential real GDP per person also grows at about 1.84 percent per year.

Similarly, when labor inputs per person are constant, potential labor hours grow at about 1.21

percent per year so potential real GDP grows at about 3.05 percent per year.

What Explains Differences in TFP?

Now that we know that TFP growth is the key factor behind labor productivity growth

and the growth rate of the standard of living, we need to determine why TFP increases over time.

Economists have spent a tremendous amount of time trying to answer this question. Although no

single theory has emerged, economists have identified several important factors, which we

discuss next.

Research and Development and the Level of Technology

As we mentioned in Chapter 4, total factor productivity measures the overall efficiency of

the economy in transforming inputs into real GDP. One of the most important factors

influencing total factor productivity is the stock of knowledge that the world possess and the

associated level of technology. The invention of computers made workers more productive by

giving them new and better types of capital goods to work with. For example, word processors

allow one administrative assistant today to do work that would have taken a team of

administrative assistants in 1949. Assembly line workers in automobile plants now operate and

oversee robots rather than doing the manual labor themselves. As a result, one worker today can

produce many more automobiles than a team of workers could in 1949. In both examples, the

new capital goods has made labor more productive.


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New capital goods do not just appear out of thin air. Private firms and the government

devote a significant amount of resources to research and development (R&D) activities to come

up with ideas for new capital goods or new goods and services for consumption. A considerable

amount of the economys resources are devoted to discovering and testing these new ideas. The

Commerce Department calculates that the United States conducted $317 billion worth of R&D in

2004, with the private sector responsible for $209 billion of R&D and the government

responsible for the remaining $108 billion of R&D.

[Begin Box]

Research and Development Expenditures and Labor Productivity Differences between

China and the United States

On page xx we saw that labor productivity in the United States was 6.3 times higher than in

China. Because of the higher labor productivity, the United States is able to maintain a higher

standard of living. Part of the reason the United States has a higher level of labor productivity is

that the United States devotes more resources to develop new technology and accumulate human

capital. For example, in 2004 (the most recent year data is available for both countries) the

United States devoted 2.6 percent of GDP to research and development while China devoted just

1.2 percent and as recently as 1996 China devoted just 0.6 percent of GDP research and

development. The higher level of investment in new technology helps increase knowledge and

total factor productivity so U.S. workers remain more productive.

SOURCE: World Development Indicators, the World Bank.

[End Box]


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H3: Quality of Labor

The quality of labor, like the quality of capital goods, can change over time. Workers

become more skilled as they acquire human capital. Human capitalis the accumulated

knowledge and skills that workers acquire from education and training or from life experiences.

There are two basic ways for workers to acquire human capital.

(MD: Human Capital Theaccumulated knowledge and skills that workers acquire from

education and training or from life experiences.)

First, a worker can go to school for formal training to learn basic skills that are useful in

the workplace. Students go to school to learn science, math, and other subjects that make them

better workers. And as students learn and acquire new skills, their human capital increases, and

they become more productive workers. Through this learning, education transforms low-skilled

high school graduates into high-skilled engineers and scientists.

Second, as Nobel Laureate Kenneth Arrow of Stanford University points out, workers

can accumulate skills through learning by doing.9 Arrow argued that the more workers perform

a task, the more they learn about how to do the task quicklythereby improving their

productivity in doing the task. Arrow cited evidence from engineering studies showing that the

amount of time it takes to build an airplane decreases as workers build more airplanes. This

relationship emerges because the workers have acquired knowledge and skills through building

the previous airplanes, making them more productive.

9Economic Implications of Learning by Doing, The Review of Economic Studies(June 1962), pp.155-173.


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H3: Government and Social Institutions

Nobel Laureate Douglass North of Stanford University and economist Robert Thomas of

the University of Washington have emphasized the importance of government and social

institutions in explaining differences in labor productivity and the standard of living across

countries.10 North and Thomas, and many other economists, believe that markets and property

rights are important institutions that lead to economic growth. Individuals and firms are unlikely

to risk their own funds, and investors are unlikely to lend their funds to individuals and firms,

unless they can keep the profits from risky investment projects. In other words, property rights

must be secure to encourage investment and capital accumulation. In some countries, property

rights are not secure due to government corruption because it is impossible for individuals and

firms to start or expand businesses without paying bribes to at least one government official.

Historical evidence suggests that government institutions and property rights do matter.

After Germanys defeat in World War II and the onset of the Cold War between the United

States and the Soviet Union, Germany was divided into West Germany which was a

parliamentary democracy with a market economy and secure property rights and East Germany

which was a communist dictatorship without a strong market or secure property rights. Both

East and West Germany were devastated by World War II. The two countries were reunified

into a single country in 1990 and at the time of reunification real GDP per person in then West

Germany was 2.6 times the level of real GDP per person in East Germany. The case of North

and South Korea provides a second example of the importance of government institutions. Japan

had occupied the Korean peninsula during World War II, but after Japans surrender Soviet

10The Rise of the Western World: A New Economic History. Cambridge University Press (1973).


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troops occupied what would become North Korea while U.S. troops occupied what would

become South Korea. Just like East Germany, North Korea was a communist dictatorship

without strong markets or secure property rights while South Korea had strong markets and

secure property rights. While economic data for North Korea are unreliable because the North

Korean government does not make official data available, the CIAFact Bookestimates that real

GDP per person for South Korea was 14.4 times the level in North Korea in 2008.

Economists Daron Acemoglu of MIT, Simon Johnson of MIT, and James Robinson of

the University of California at Berkeley tried to find systematic evidence showing the effect of

government and social institutions on real GDP per person.11 European countries colonized large

regions of the world between the 1600s and the 1800s. In countries such as the United States,

Australia and New Zealand, Europeans came as settlers and established institutions that enforced

the rule of law. These favorable institutions encouraged investment which led to faster economic

growth and higher real GDP per person. However, in Africa and other areas European countries

did not come as settlers. Instead, the Europeans came to extract natural resources and so did not

establish government institutions that favored investment. The areas of the world in which the

Europeans established strong property rights are generally rich today while the regions in which

Europeans did not establish strong property rights are generally not rich.

The experiences of the Germany, Korea, and the former European colonies have

convinced many economists that government institutions play a critical role in encouraging

economic growth.

11The Colonial Origins of Comparative Development: An Empirical Investigation,American Economic Review

(December 2001), pp. 1369-1401.


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Geography

Not all economists today agree with the primary importance of institutions for explaining

the standard of living. In fact, as long ago as Adam Smiths 1776 bookAn Inquiry into the

Nature and Causes of the Wealth of Nations, economists have pointed out that geography

influences a nations natural resources. For example, access to navigable rivers and coast line

makes trade easier and should increase labor productivity and the standard of living. For

example, the United States has a large coast line and extensive navigable rivers while countries

such as Bolivia and Tibet are landlocked and mountainous so transportation is difficult.

However, Jeffrey Sachs of Columbia University argues that geography plays an important role in

economic growth for another reason. Sachs, along with economists Andrew Mellinger and John

Gallup, argues that tropical climates experience higher rates of infectious disease such as

malaria.12 The countries that are poor today are often the countries with high rates of infe

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