econ 206 - city university of new...

Prof. Francesc Ortega’s Class

Guest Lecture by Prof. Ryan Edwards

ECON 206Macroeconomic Analysis

October 5, 2011

1

Monday, October 3, 2011

http://www.thehistorybluff.com/?p=1808



http://en.wikipedia.org/wiki/The_Anatomy_Lesson_of_Dr._Nicolaes_Tulp

http://en.wikipedia.org/wiki/The_Anatomy_Lesson_of_Dr._Nicolaes_Tulp

Our objectives today

• Growth:

• Cover the key facts about economic growth that we wish to understand later with models

• Examine the global extent of economic growth

• Understand how growth is a recent phenomenon

• Learn some tools: calculating growth rates and using ratio scales

2


A sketch of the U.S. around 1900

• Life expectancy at birth, the average length of life

starting from birth, was about 50 [today: 77]

• One out of every 10 infants died before his or her first

birthday [today: 7 out of every 1,000]

• 90 percent of households did not have electricity,

refrigerator, telephone, or a car [today most do]

• Fewer than 10 percent of adults had graduated from

high school [today: 85 percent]3


A sobering comparisonUSA 1900

USA today

Kenya today

Life expectancy at birth

50 77 50

Infant mortality 0.1 0.007 0.06

Real GDP per capita (1990 $) $4,100 $28,000 $1,000

4


If growth has been uneven across geographic boundaries, what about across time?

• Growth is a relatively recent phenomenon, only in the past 2 or 3 centuries

• Growth arrived in different countries at different times

• Today, a “Great Divergence”

C.I. Jones — Long-Run Economic Growth, August 1, 2006 56

Figure 3.1: Economic Growth over the Very Long Run

0 500 1000 1500 2000500 2000

5000

10000

15000

20000

25000

30000

Ethiopia

ChinaBrazil

U.K.Japan

U.S.

Year

Per Capita GDP(1990 dollars)

Note: Data from Angus Maddison, The World Economy: Historical Statistics(Paris: OECD Development Center, 2003).

mathematical tools that are extremely useful in studying macroeconomics. Subsequent

chapters in the long-run portion of this book will use these tools to provide economic

theories that help us understand the facts of economic growth.

3.2 Growth over the Very Long Run

One of the most important facts of economic growth is that sustained increases in

standards of living are a remarkably recent phenomenon from the standpoint of world

history. Figure 3.1 makes this point by showing estimates of GDP per capita over

the last 2000 years for a range of countries. For most of history, standards of living

were extremely low, not much different from that in Ethiopia today. The figure shows

this going back for 2000 years, but it is surely true going back even further. Up until

about 12 thousand years ago, humans were hunters and gatherers, living a nomadic

existence. The agricultural revolution then led to the emergence of settlements and


Figure 3.1: Economic Growth over the Very Long Run

0 500 1000 1500 2000500 2000

5000

10000

15000

20000

25000

30000

Ethiopia

ChinaBrazil

U.K.Japan

U.S.

Year


Note: Data from Angus Maddison, The World Economy: Historical Statistics(Paris: OECD Development Center, 2003).

mathematical tools that are extremely useful in studying macroeconomics. Subsequent

chapters in the long-run portion of this book will use these tools to provide economic

theories that help us understand the facts of economic growth.

3.2 Growth over the Very Long Run

One of the most important facts of economic growth is that sustained increases in

standards of living are a remarkably recent phenomenon from the standpoint of world

history. Figure 3.1 makes this point by showing estimates of GDP per capita over

the last 2000 years for a range of countries. For most of history, standards of living

were extremely low, not much different from that in Ethiopia today. The figure shows

this going back for 2000 years, but it is surely true going back even further. Up until

about 12 thousand years ago, humans were hunters and gatherers, living a nomadic

existence. The agricultural revolution then led to the emergence of settlements and

5



Figure 3.2: Per Capita GDP in the United States

1860 1880 1900 1920 1940 1960 1980 2000 2020

2500 5000

10000

15000

20000

25000

30000

35000

Year


Note: Data from 1870 to 1928 from Maddison, cited in Figure 3.1. Data from1929 to 2004 from the Bureau of Economic Analysis.

3.3 Modern Economic Growth

On a scale of thousands of years like that shown in Figure 3.1, the era of modern

economic growth gets so compressed that incomes almost appear to rise as a vertical

line. If we stretch out the time scale and focus on the last 125 years or so, one gets a

better picture of what has been occurring. Figure 3.2 does this for the United States.

Measured in year 2000 prices, per capita GDP in the United States was about $2500

in 1870 and rose to nearly $37,000 by 2004, almost a 15-fold increase. A more mun-

dane way to appreciate the rate of change of per capita GDP is to compare GDP in the

year you were born to GDP in the year your parents were born. In 1985, for example,

per capita income was just over $25,000. Thirty years earlier it was about $13,000.

Speaking loosely and assuming this economic growth continues, the typical U.S. col-

1997, pp. 3–17, as well as Robert E. Lucas, Jr., “Some Macroeconomics for the 21st Century” Journal ofEconomic Perspectives, Volume 14, Winter 2000, pp. 159–68. The phrase “Great Divergence” is borrowedfrom Kenneth Pomeranz, The Great Divergence: China, Europe, and the Making of the Modern WorldEconomy (Princeton University Press, 2000).

• Since 1870, GDP per person in the U.S. has risen 15-fold: from about $2,500 to about $37,000

• Another way to think about this is to compare your lifetime income with your parents’

• You’ll be about twice as rich, because incomes have been doubling every 35 years, or roughly once every generation

6


The math of growth• In Figure 3.2, GDP per capita (yt) is increasing by an

increasing amount

• The increases are roughly proportional to the level at any point, say by some proportion g


lege student will have a lifetime income approximately twice that of his or her parents.

3.3.1 The Definition of Economic Growth

Up to this point, we have been using the phrase “economic growth” generically to refer

to increases in living standards. However, “growth” also has a more precise meaning

related to the exact rate of change of per capita income.

To see this, first notice that the slope of the income series shown in Figure 3.2 has

been rising over time. This means that the annual change in income has been getting

larger over time, at least on average. Our incomes are rising by an ever-increasing

amount. In fact, these income changes are roughly proportional to the level of per

capita income at any point in time.

To see what this statement means, it is helpful to use some algebra. Let y stand for

per capita income. Then, at least as an approximation,

y2005 ! y2004 = g " y2004,

where g is equal to 0.02. That is, the change in per capita income between 2004 and

2005 is roughly proportional to the level of per capita income in 2004, where the factor

of proportionality is 2%.

Dividing both sides of this expression by income in 2004, one discovers another

way of expressing this relationship:

y2005 ! y2004

y2004= g.

Notice that the left-hand side of this equation is the percentage change in per capita

income. This expression says that the percentage change in per capita income is the

constant g, and it is this percentage change that we call a growth rate.

We can look at the growth rate between any two consecutive years, not just 2004

and 2005. Suppose yt is income in some period. Then, we could study the growth rate

between 2003 and 2004, or 1950 and 1951, or more generally between year t and year

t + 1. This leads us to the following general definition for a growth rate:

Definition: A growth rate in some variable y is the percentage change in


lege student will have a lifetime income approximately twice that of his or her parents.

3.3.1 The Definition of Economic Growth

Up to this point, we have been using the phrase “economic growth” generically to refer

to increases in living standards. However, “growth” also has a more precise meaning

related to the exact rate of change of per capita income.

To see this, first notice that the slope of the income series shown in Figure 3.2 has

been rising over time. This means that the annual change in income has been getting

larger over time, at least on average. Our incomes are rising by an ever-increasing

amount. In fact, these income changes are roughly proportional to the level of per

capita income at any point in time.

To see what this statement means, it is helpful to use some algebra. Let y stand for

per capita income. Then, at least as an approximation,

y2005 ! y2004 = g " y2004,

where g is equal to 0.02. That is, the change in per capita income between 2004 and

2005 is roughly proportional to the level of per capita income in 2004, where the factor

of proportionality is 2%.

Dividing both sides of this expression by income in 2004, one discovers another

way of expressing this relationship:

y2005 ! y2004

y2004= g.

Notice that the left-hand side of this equation is the percentage change in per capita

income. This expression says that the percentage change in per capita income is the

constant g, and it is this percentage change that we call a growth rate.

We can look at the growth rate between any two consecutive years, not just 2004

and 2005. Suppose yt is income in some period. Then, we could study the growth rate

between 2003 and 2004, or 1950 and 1951, or more generally between year t and year

t + 1. This leads us to the following general definition for a growth rate:

Definition: A growth rate in some variable y is the percentage change in


that variable. The growth rate between period t and t + 1 is

yt+1 ! yt

yt.

With our definition of a growth rate as a percentage change, we can derive a number

of useful insights. First, notice that if the growth rate of per capita income happens to

equal some number g, then we can express the level of per capita income as

yt+1 = yt(1 + g). (3.1)

This equation is useful because it tells us how to get the value of per capita income

tomorrow if we know the value today and the growth rate.

3.3.2 A Population Growth Example

To see one way in which this equation can be used, consider the following example.

Suppose the population of the world is given by L0. For example, we might suppose L0

is equal to six billion, to reflect roughly the number of people in the world today. Now

consider the possiblity that population growth will be constant over the next century at a

rate given by n. For example, n might equal .02, implying that the world’s population

will grow at 2 percent per year. Under these assumptions, what will the level of the

population be 100 years from now?

First, notice that we can apply the growth equation in (3.1) to this example. Rewrit-

ing that equation with our population notation, we have

Lt+1 = Lt(1 + n). (3.2)

The population next year is equal to the population this year multiplied by 1 plus the

growth rate. Why? Well, the “1” simply reflects the fact that we carry over the people

who were already alive. In addition, for every person at the start, n new people get

added, so that we must add nLt people to the original population Lt.

Let’s apply this equation to our example. We begin at year 0 with L0 people. Then

at year 1 we have

L1 = L0(1 + n). (3.3)

• The left-hand side of the second equation, g, is the percentage change in GDP per capita, the growth rate. A little more math shows us:

7


• With a constant growth rate, level increases are larger and larger over time

• Calculating levels using growth rates over a period of time is mathematically a little complicated

• Example: With 6.5 billion people in the world today and a constant annual growth rate of 2%, how large will world population be in 100 years?

• Hint: It’s not

6.5 billion + 2% x 6.5 billion x 100 years

(which equals 6.5 + 13 billion extra = 19.5 billion)

8



that variable. The growth rate between period t and t + 1 is

yt+1 ! yt

yt.

With our definition of a growth rate as a percentage change, we can derive a number

of useful insights. First, notice that if the growth rate of per capita income happens to

equal some number g, then we can express the level of per capita income as

yt+1 = yt(1 + g). (3.1)

This equation is useful because it tells us how to get the value of per capita income

tomorrow if we know the value today and the growth rate.

3.3.2 A Population Growth Example

To see one way in which this equation can be used, consider the following example.

Suppose the population of the world is given by L0. For example, we might suppose L0

is equal to six billion, to reflect roughly the number of people in the world today. Now

consider the possiblity that population growth will be constant over the next century at a

rate given by n. For example, n might equal .02, implying that the world’s population

will grow at 2 percent per year. Under these assumptions, what will the level of the

population be 100 years from now?

First, notice that we can apply the growth equation in (3.1) to this example. Rewrit-

ing that equation with our population notation, we have

Lt+1 = Lt(1 + n). (3.2)

The population next year is equal to the population this year multiplied by 1 plus the

growth rate. Why? Well, the “1” simply reflects the fact that we carry over the people

who were already alive. In addition, for every person at the start, n new people get

added, so that we must add nLt people to the original population Lt.

Let’s apply this equation to our example. We begin at year 0 with L0 people. Then

at year 1 we have

L1 = L0(1 + n). (3.3)With a constant growth rate n, population after one year is


Similarly, we can get the population in year 2 as

L2 = L1(1 + n).

But we already know the value of L1 from equation (3.3). Substituting from this equa-

tion, we have

L2 = L0(1 + n)(1 + n) = L0(1 + n)2. (3.4)

What about the population in year 3? Well, again we know from our basic growth

equation that

L3 = L2(1 + n).

Now we can use the expression for L2 from equation (3.4), which gives

L3 =!L0(1 + n)2

"(1 + n) = L0(1 + n)3. (3.5)

At this point, you should start to see a pattern. In particular, this reasoning suggests

that the population in any arbitrary year t is

Lt = L0(1 + n)t. (3.6)

This is the key expression we need to answer our original question. Recall that

we are given values for L0 and n and then asked to calculate the size of the world

population 100 years in the future. We can get the answer to this question by evaluating

equation (3.6) at t = 100:

L100 = L0(1 + n)100.

With L0 = 6 billion and n = .02, we find that the population 100 years from now

would equal 43.5 billion people.

More generally, this example illustrates the following important result:

The Constant Growth Rule: If a variable starts at some initial value y0 at

time 0 and grows at a constant rate g, then the value of the variable at some

future time t is given by

yt = y0(1 + g)t. (3.7)

After two years, it is



L2 = L1(1 + n).


tion, we have

L2 = L0(1 + n)(1 + n) = L0(1 + n)2. (3.4)


equation that

L3 = L2(1 + n).


L3 =!L0(1 + n)2

"(1 + n) = L0(1 + n)3. (3.5)



Lt = L0(1 + n)t. (3.6)





L100 = L0(1 + n)100.







yt = y0(1 + g)t. (3.7)

Combining these, we find that



L2 = L1(1 + n).


tion, we have

L2 = L0(1 + n)(1 + n) = L0(1 + n)2. (3.4)


equation that

L3 = L2(1 + n).


L3 =!L0(1 + n)2

"(1 + n) = L0(1 + n)3. (3.5)



Lt = L0(1 + n)t. (3.6)





L100 = L0(1 + n)100.







yt = y0(1 + g)t. (3.7)

So for some year t,

When L0 = 6.5 billion and n = 0.02, L100 = 47.1 billionconsiderably larger than 19.5 billion!

9


We will also need to compute annual growth rates — but how?

• Rule of 70: if something is doubling every t years, you know that the growth rate is 70 ÷ t

(With GDP per person, it’s 70÷35 years = 2%)

• Using the raw data: be a little careful!

• You have to use a calculator or spreadsheet for this

• Note: this will give you 0.02 for 2% growth


us that the growth rate is constant. To get the actual rate, there are two approaches.

One can use the actual underlying data to calculate the growth rate, as described in the

remainder of this section. Alternatively, for a quick estimate of the growth rate, one

can use the Rule of 70. If you look closely at Figure 3.5, you will see that the straight

line is doubling roughly every 35 years. For example, between 1900 and 1935, the

line rises from about $4000 to about $8000. Between 1935 and 1970, the line doubles

again to about $16,000. From our Rule of 70, a process that doubles every 35 years is

growing at 2% per year (70/35=2).

To get a more precise measure of the growth rate, one needs the raw data. If we

have data available every year, we could of course compute the percentage change

across each annual period, and this would be a fine way to measure growth. But what if

instead we are given data only at the start and end of the series? For example, suppose

we know that U.S. per capita GDP was $2500 in 1870 and $37,500 in 2005. What is

the average annual growth rate of income over these 135 years?

The answer can be found by using the Constant Growth Rule in equation (3.7).

Recall that this rule states that for a quantity growing at a constant rate, the level in

year t is given by

yt = y0(1 + g)t.

In our previous use of this equation, we assumed we knew y0 and g and we were asked

to solve for the value of y at some future date t.

Now, though, we are given values of yt and y0 and asked to solve for g. The way

to do this is to rearrange the equation and the take the tth root of the ratio of the two

incomes, as explained in the following rule:

Computing Growth Rates: The average annual growth rate between year

0 and year t is given by

g =

!yt

y0

"1/t

! 1. (3.9)

There are two things worth noting about this rule. First, notice that if there were con-

stant growth between year 0 and year t the growth rate we compute would lead income

to grow from y0 to yt. However, we can apply this formula even to a data series that


us that the growth rate is constant. To get the actual rate, there are two approaches.

One can use the actual underlying data to calculate the growth rate, as described in the

remainder of this section. Alternatively, for a quick estimate of the growth rate, one

can use the Rule of 70. If you look closely at Figure 3.5, you will see that the straight

line is doubling roughly every 35 years. For example, between 1900 and 1935, the

line rises from about $4000 to about $8000. Between 1935 and 1970, the line doubles

again to about $16,000. From our Rule of 70, a process that doubles every 35 years is

growing at 2% per year (70/35=2).

To get a more precise measure of the growth rate, one needs the raw data. If we

have data available every year, we could of course compute the percentage change

across each annual period, and this would be a fine way to measure growth. But what if

instead we are given data only at the start and end of the series? For example, suppose

we know that U.S. per capita GDP was $2500 in 1870 and $37,500 in 2005. What is

the average annual growth rate of income over these 135 years?

The answer can be found by using the Constant Growth Rule in equation (3.7).

Recall that this rule states that for a quantity growing at a constant rate, the level in

year t is given by

yt = y0(1 + g)t.

In our previous use of this equation, we assumed we knew y0 and g and we were asked

to solve for the value of y at some future date t.

Now, though, we are given values of yt and y0 and asked to solve for g. The way

to do this is to rearrange the equation and the take the tth root of the ratio of the two

incomes, as explained in the following rule:

Computing Growth Rates: The average annual growth rate between year

0 and year t is given by

g =

!yt

y0

"1/t

! 1. (3.9)

There are two things worth noting about this rule. First, notice that if there were con-

stant growth between year 0 and year t the growth rate we compute would lead income

to grow from y0 to yt. However, we can apply this formula even to a data series that

10


Growth rate notationThese are all the same:

• The annual growth rate of x

• (xt+1 – xt) / xt = xt+1/xt – 1, or if the data are spaced further apart in time,

• gx or g(x)

All are numbers like 0.02 (which is 2%) per year

gx =

!

xt

x0

"1/t

! 1

11


Properties of growth rates

1. Ratios become differences:

2. Products become sums:

3. Powers become multiples:


was in 1960. Second, the fraction of the world’s population that is impoverished has

fallen dramatically in the last half century. In 1960, two thirds of the world’s population

lived in countries with a per capita GDP less than 5 percent of the 2000 U.S. level. To

put this number in perspective, it corresponds to about $5 per day in today’s prices. By

2000, the fraction living in this kind of poverty had fallen from two thirds to less than

10 percent. In other words, if the distribution were unchanged from its 1960 level, more

than 4 billion people would fall below this poverty threshhold today. Instead, because

of economic growth, only about 600 million people are so impoverished. One of the

major driving forces behind this change has been the rapid economic growth in India

and China, which together contain more than 40 percent of the world’s population.5

3.5 Some Useful Properties of Growth Rates

Before this chapter draws to a close, there is one remaining task to be completed.

This task is to convey a number of simple properties of growth rates that will prove

extremely useful as we develop models of economic growth. These properties can be

summarized as follows:

Growth Rates of Ratios, Products, and Powers: Suppose two variables x

and y have average annual growth rates of gx and gy , respectively. Then

the following rules apply:

If z = x/y, then gz = gx ! gy .

If z = x " y, then gz = gx + gy .

If z = xa, then gz = a " gx.

(where gz is the average annual growth rate of z).

These simple rules explain how to compute the growth rate of the ratios of two

variables, the product of two variables, and a variable that is raised to some power.6 To5For a more sophisticated version of this argument, see Xavier Sala-i-Martin, “The World Distribution of

Income: Falling Poverty and... Convergence, Period,” Quarterly Journal of Economics, May 2006, Volume121, pp. 351–397. In particular, that paper shows that the conclusion is extremely robust to thinking abouthow the income distribution within countries may have changed.

6In terms of the way growth rates are computed in this book, these rules should be thought of as approx-imations that are very good when growth rates are small. With the aid of calculus, these rules can be shown























































(If this looks like logarithms to you, that’s no accident!)

12


Suppose that x grows at rate gx = 0.10 while ygrows at rate gy = 0.03. Then what is gz when...


Table 3.1: Examples of Growth Rate Calculations

Suppose x grows at rate gx = .10 and y grows at rate gy = .03.What is the growth rate of z in the following cases?z = x ! y =" gz = gx + gy = .13z = x/y =" gz = gx # gy = .07z = y/x =" gz = gy # gx = #.07z = x2 =" gz = 2 ! gx = .20

z = y1/2 =" gz = .5 ! gy = .015z = x1/2y!1/4 =" gz = .5 ! gx # .25 ! gy = .0125

Figure 3.9: Population, GDP, and Per Capita GDP for the United States

1860 1880 1900 1920 1940 1960 1980 2000 2020

1.5%

2.0%

3.5%Total GDP

Per Capita GDP

Population

Year

Ratio Scale

Note: Data from Maddison (2003) and the Bureau of Economic Analysis.The average annual growth rate is reported next to each data series.






1860 1880 1900 1920 1940 1960 1980 2000 2020

1.5%

2.0%

3.5%Total GDP

Per Capita GDP

Population

Year

Ratio Scale







1860 1880 1900 1920 1940 1960 1980 2000 2020

1.5%

2.0%

3.5%Total GDP

Per Capita GDP

Population

Year

Ratio Scale







1860 1880 1900 1920 1940 1960 1980 2000 2020

1.5%

2.0%

3.5%Total GDP

Per Capita GDP

Population

Year

Ratio Scale







1860 1880 1900 1920 1940 1960 1980 2000 2020

1.5%

2.0%

3.5%Total GDP

Per Capita GDP

Population

Year

Ratio Scale







1860 1880 1900 1920 1940 1960 1980 2000 2020

1.5%

2.0%

3.5%Total GDP

Per Capita GDP

Population

Year

Ratio Scale


13







1860 1880 1900 1920 1940 1960 1980 2000 2020

1.5%

2.0%

3.5%Total GDP

Per Capita GDP

Population

Year

Ratio Scale







1860 1880 1900 1920 1940 1960 1980 2000 2020

1.5%

2.0%

3.5%Total GDP

Per Capita GDP

Population

Year

Ratio Scale







1860 1880 1900 1920 1940 1960 1980 2000 2020

1.5%

2.0%

3.5%Total GDP

Per Capita GDP

Population

Year

Ratio Scale







1860 1880 1900 1920 1940 1960 1980 2000 2020

1.5%

2.0%

3.5%Total GDP

Per Capita GDP

Population

Year

Ratio Scale


Again, if x grows at rate gx = 0.10 while ygrows at rate gy = 0.03. Then what is gz when...

gz =1

2gx

= 0.05 ! 0.01!

1

3gy

z = x1/2

y!1/3

= 0.0414


A real-world example• Suppose you know the following:

• In 2010, you worked h = 20 hours a week for k = 50 weeks and earned w = $10 per hour

• So your total earnings E were equal to

• With your earnings, you buy donuts priced at P = $2 per donut

• In 2011, times are tough!

• You can only get h’ = 18 hours per week for k’ = 50 weeks, and you didn’t get a raise, so w’ = $10

• But times are also tough for Dunkin’ Donuts, who have to cut donut prices to P’ = $1.80 to stay competitive in a down market

15

E = h k w


Questions you can answer using analytical tools, rather than a calculator

• Q: Has your real wage (wr = w÷P) risen or fallen?

• Risen because w stayed the same while P fell. How much?

• g[wr] =

• Q: Have your real earnings (R = E÷P) risen or fallen?

• g[E÷P] =

16

earnings E = h k w

hours h weeks k wage w price P2010 20 50 $10 $22011 18 50 $10 $1.80

g[w] – g[P] = 0% – (–10%) = 10% because g[P] = –10%

g[E] – g[P] = g[h k w] – g[P] = g[h] + g[k] + g[w] – g[P] =

= –10% + 0% + 0% – (–10%) = 0%

Your real wage rose by the same rate that your hours fell, so your real earnings are unchanged


A key example that will turn up soon:

• Suppose we know that

• What is the growth rate of Yt in terms of the growth rates of At, Kt, and Lt?

• The growth rate of a product is the sum of the growth rates


product of per capita GDP and the population. Therefore the growth rate of GDP is the

sum of the growth rates of GDP per capita and population. This can be seen graphically

in the figure in the different slopes of the three data series. We will use these growth

rules extensively in the chapters that follow, so you should memorize them and be

prepared for their application.

3.5.1 Example: Yt = AtK1/3t L2/3

t

We close this chapter with one final example. This example incorporates one of the key

equations of macroeconomics, so it will be extremely relevant in the coming chapters.

Suppose we have an equation that says a variable Yt is a function of some other

variables At, Kt, and Lt. In particular, this function is

Yt = AtK1/3t L2/3

t .

Question: What is the growth rate of Yt in terms of the growth rates of At, Kt, and

Lt?

To get the answer, we apply our growth rules. Let’s use the notation g(Stuff) to

denote the growth rate of “Stuff.” (This is useful when “Stuff” is a long expression

so that writing gStuff would be a little awkward.) Then our second rule says that the

growth rate of a product is the sum of the growth rates. So

g(Yt) = g(At) + g(K1/3t ) + g(L2/3

t ).

Next, we can use the third rule to compute the growth rates of the last two terms in this

expression. In particular, the growth rate of a variable raised to some power is equal to

that power times the growth rate of the variable. Therefore, we have

g(Yt) = g(At) +1

3! g(Kt) +

2

3! g(Lt).

And that is the answer we were looking for. To anticipate what comes later, this equa-

tion says that the growth rate of output can be decomposed into the growth rate of a

productivity term, A, and the contributions to growth from capital and labor.








t





Yt = AtK1/3t L2/3

t .


Lt?





g(Yt) = g(At) + g(K1/3t ) + g(L2/3

t ).




g(Yt) = g(At) +1

3! g(Kt) +

2

3! g(Lt).




17


• The growth rate of a power is the power times the growth rate

• We will later learn about this function; it tells us that growth in income (Y) comes from

• Growth in productivity (A) plus

• Growth in physical inputs (capital, K; and labor, L)








t





Yt = AtK1/3t L2/3

t .


Lt?





g(Yt) = g(At) + g(K1/3t ) + g(L2/3

t ).




g(Yt) = g(At) +1

3! g(Kt) +

2

3! g(Lt).











t





Yt = AtK1/3t L2/3

t .


Lt?





g(Yt) = g(At) + g(K1/3t ) + g(L2/3

t ).




g(Yt) = g(At) +1

3! g(Kt) +

2

3! g(Lt).



productivity term, A, and the contributions to growth from capital and labor.18


Plotting on a ratio scale (a.k.a. log scale)

• If the y-axis is scaled in terms of ratios or multiples of an amount rather than its levels,

• Then a series that grows at a constant rate...

appears as a straight line on a ratio scale

19



Figure 3.5: Per Capita GDP in the United States: Ratio Scale

1850 1900 1950 2000 2050 2000

4000

8000

16000

32000

2.0% per year

Year

Per Capita GDP(ratio scale, 2000 dollars)

Note: This is the same data shown in Figure 3.2, but plotted using a ratioscale. Notice that the ratios of the equally-spaced labels on the vertical axisare all the same, in this case equal to 2. The dashed line exhibits constantgrowth at a rate of 2.0 percent per year.

On a ratio scale, equal spacings are constant ratios (here, 2:1 or doubling)

GDP per person has grown at a fairly constant rate of 2%

The slopes reveal faster or slower growth

Flatter = slower growth

Steeper = faster growth

20



Figure 3.6: Per Capita GDP since 1870

1860 1880 1900 1920 1940 1960 1980 2000

500

1000

2000

4000

8000

16000

32000

U.K.

U.S.

Ethiopia

China

Brazil

Japan

Germany

Year

Per Capita GDP(ratio scale, 1990 dollars)

Note: Data from Angus Maddison, The World Economy: Historical Statistics(Paris: OECD Development Center, 2003). Observations are presented everydecade after 1950 and less frequently before that as a way of smoothing theseries.

Ratio scales allow us to see and tell stories about shifting growth rates much easier

In 1870, the UK was the richest country

But the U.S. grew more rapidly!

Postwar Germany and Japan caught up

China is growing fast!

21







1860 1880 1900 1920 1940 1960 1980 2000 2020

1.5%

2.0%

3.5%Total GDP

Per Capita GDP

Population

Year

Ratio Scale


Here’s where we see properties of growth rates in action:

The growth rate of GDP/person ...

is equal to the growth rate of GDP minus the growth rate of “person,” a.k.a. population

22


econ 206 - city university of new...

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