University of California, Santa Cruz - ECON 100B

CHAPTER 3: An Overview of Long-Run Economic Growth

MULTIPLE CHOICE

1)Economic growth can be useful in describing why Europeans are:

1. 1. better off in 2015 than they were in 1915 and why Europeans are better off than Thais.
   2. better off in 2015 than they were in 1915 and why Europeans are worse off than Thais.
   3. worse off in 2015 than they were in 1915.
   4. worse off than Thais.
   5. worse off in 2015 than they were in 1915 and why Europeans are better off than Thais.

                                

 

2. What country or countries do the following characteristics possibly describe?

• Life expectancy at birth is under 50 years.
• More than 90 percent of households do not have electricity.
• Fewer than 10 percent of young adults have graduated from high school.
  1. Kenya                                                      d. Russia
  2. the United States in the late 1800s           e.   All of these answers are correct.
  3. Bangladesh

                                

 

3. According to historical data, the wages in ancient Greece and Rome were          in fifteenth- century Britain or seventeenth-century France.
   1. somewhat lower than                               d. about the same as
   2. a lot higher than                                       e.   None of these answers is correct.
   3. a lot lower than

 

ANS:

D

 

4. If the 130,000-year period since anatomically modern humans made their first appearance were compressed into a single day, economic growth would have begun in the last:
   1. three hours.                                              d. half hour.
   2. hour.                                                        e.   two hours.
   3. three minutes.

                                

 

5. English philosopher Thomas Hobbes is notable for observing that life was          for thousands of years.
   1. nasty, brutish, and short
   2. just like the legend of King Arthur
   3. great, if you were a king
   4. halcyon, fulsome, but short
   5. comfortable and prosperous

                                

6. The era of modern economic growth began about:
   1. the time of the Egyptian pharaohs.          d. the time of the Renaissance.
   2. 500 years ago.                                         e.   300 years ago.
   3. the time of Caesar.

                                

 

7. The birthplace of modern economic growth was in            during the                century.
   1. Japan; mid-twentieth
   2. the United States; mid-nineteenth
   3. the United Kingdom; mid-eighteenth
   4. China; late twentieth
   5. Germany; early nineteenth

                                

 

8. Developed countries’ average incomes rose from about            in 1700 to about               today. a. $1,000; $30,000                                           d. $500; $45,000

b.   $2,500; $50,000                                      e.   $500; $100,000

c.                                $500; $70,000

                                

 

9. Until about 12,000 years ago, humans were           ; at that point               led to the first towns and true economic development.
   1. farmers; the Bronze Age
   2. hunters and gatherers; the Iron Age
   3. hunters and gatherers; agriculture
   4. hunters and gatherers; universities
   5. farmers; industry

                                

 

10. In 2009 prices, U.S. per capita GDP was about            in 1870, and by 2015 it had grown to about

               .

a.   $500; $37,000                                         d. $10,000; $37,000

b.   $3,200; $50,800                                      e.   $100; $100,000

c.                                $500; $17,000

                                

 

11. Per capita GDP in Japan and the UK is about             United States per capita GDP, while in Argentina it is about               U.S. per capita GDP, and in Ghana only            .
    1. equal to; one-third; one-tenth
    2. one and a quarter; one-half ; one-third
    3. three-fourths; one-third; one-fifteenth
    4. four-fifths; one-half; one-quarter
    5. one-half; one-half; one-quarter

                                

 

12. Assuming the current rate of economic growth continues, the average college student will have a lifetime income               that of his or her parents.
    1. five times                                                 d. about twice
    2. 10 times                                                   e.   half
    3. about the same as

                                

 

13. In 2000, average per capita income was about           , while in 1970, average per capita income was                     .
    1. $75,000; one-fifth that amount
    2. $25,000; twice that amount
    3. $36,000; one-third that amount
    4. $45,000; about half that amount
    5. $32,000; more or less the same

                                

 

14. Assuming the current rate of economic growth continues, the average parent of a college student will have a lifetime income              that of his or her son or daughter.
    1. 10 times                                                   d. five times
    2. half                                                          e.   about twice
    3. about the same as

                                

 

15. Economic growth is defined as:
    1. the percent change in per capita income, or GDP.
    2. the percent change in prices, or GDP.
    3. the decline in the unemployment rate.
    4. the difference between the nominal and real GDP.
    5. changes in technology.

                                

 

16. The study of economic growth concentrates on understanding the determinants of the:
    1. rate of price changes.
    2. short-term change in per capita GDP.
    3. rate of population growth.
    4. change in per capita GDP over time.
    5. None of these answers is correct.

                                

17. Defining per capita GDP in 2016 as y₂₀₁₆ and per capita GDP in 2015 as y₂₀₁₅, the growth rate of per capita GDP, , from 2015 to 2016 is given by:

a.

b.

.

c.

.

d.

.

e.

.

                                

 

18. If per capita GDP in 2015 was $1,000 and in 2016 was $1,200, the growth rate of per capita GDP was:
    1. 1.2 percent.
    2. about 17 percent.
    3. 20 percent.
    4. 120 percent.
    5. Not enough information is given.

                                

 

19. If per capita GDP in 2014 was $900, in 2015 was $1,000, and in 2016 was $1,200, the growth rate of per capita GDP between 2014 and 2016 was:
    1. 25 percent.
    2. about 11 percent.
    3. 20 percent.
    4. about 133 percent.
    5. about 33 percent.

                                

 

20. The growth rate of any variable y between periods t and t + 1 is the           and is given by the term

               .

1. 1. percentage of that variable;
   2. percentage change in that variable;
   3. change in that variable; d.

 

percent of that variable;

e.   percent change in that variable;

                                

21. According to the constant growth rate rule, if a variable starts at some initial value y₀ at t = 0 and grows at                                 at a constant rate   then the value of the variable in three periods is given by:
    1. .

1. 2. .
   3. .
   4. .
   5. .

 

22. The rule of 70 states that if y_t grows at a rate of g percent per year, then the number of years it takes y_t

to:

1. 1. double is approximately equal to 70/g.
   2. double is exactly equal to 70/g.
   3. double is approximately equal to g/70.
   4. triple is approximately equal to 70/g.
   5. double is approximately equal to 70/(1 + g).

                                

 

23. According to the rule of 70, if an economy averages a 4 percent growth rate, it will take about

                years to double in size.

a.   2.8                                                           d. 5.7

b.   1,750                                                       e.   17.5

c.                                0.06

                                

24. If instead of labeling the vertical axis in the usual “1, 2, 3, 4, . . .” fashion we label it as “1, 2, 4, 8, ”

so that equal intervals represent a doubling; we call this the           scale.

 

 

 

 

 

 

25.  

quadratic logarithmic c.     exponential ratio geometric         If we compress the vertical axis at “key doubling points,” we call this the

logarithmic ratio c.     exponential quadratic geometric         The compression of the vertical axis at “key doubling points” is called the

scale.

 

 

 

 

 

 

26. scale.

quadratic logarithmic c.	exponential	d. e.	ratio geometric

27. Over the past 50 years, Brazil’s population growth rate has averaged about 2.3 percent. According to the rule of 70, Brazil’s population will double in about           years.

3 30 c.

33

d. 161 e.   1.6

 

28. Between 1970 and 1976, Israel’s average inflation rate was about 65 percent per year. With that rate of inflation, prices would double about every            years, using the rule of 70.

a.   93                                                            d.   1.1

b.   107.7                                                       e.   9.3

c.                                0.95

                                

 

29. If the population of Romania was about 20.3 million in 1970 and the average population growth rate is

0.2 percent, then Romania’s population would have been about          million in 2010. a.      20.3                                                         d. 17.4

b.   22.0                                                         e.   23.6

c.                                40.6

                                

 

30. If the population of Romania was about 22 million in 2010 and the average population growth rate is

0.2 percent, then Romania’s “initial” population was about           million in 1970. a.      23.9                                                         d. 22.0

b.   18.9                                                         e.   23.8

c.                                20.3

                                

 

31. Suppose there are L₀ people in the world today. If the population growth rate equals , then in 50 years, the world population will be:
    1. .
    2. .

c.

.

4. .
5. .

 

 

32. Suppose population growth is given by                    , where L₀ is the population today,  is the population in t periods, and is the population constant growth rate. If we do not know what value

takes but do know the values of L₀, , and t, we can calculate  by punching               into our calculators.

a.

 

 

b.

 

 

c.

 

 

d.

 

 

e.

 

                                

 

33. The president of the World Bank has asked you to calculate the average population growth rate of Hungary from 1970 to 2010. You know the population was about 10.4 million in 1970 and about 9.5 million in 2010. The average growth rate is about            percent.

a.   0.2                                                           d. 2.7

b. ?0.2                                                         e.   ?200

c.                                98.4

                                

34. The president of the World Bank is on his way to a meeting with the president of Uruguay. He bumps into you in the hallway and wants to know how long it will take for Uruguayan per capita GDP to double. All he knows is that the average growth rate has been about 1 percent. You quickly tell him it will take about               years because you know             .

60; the rule of 60 7; the rule of 70 c.

700; percent change

70; the rule of 70 1; exponential growth

 

35. The president of the World Bank has asked you to calculate the average per capita GDP growth rate of Rwanda from 1980 to 2010. In 1980, per capita GDP was about $728 and in 2010 was about $1,025. You tell him the average growth rate of per capita GDP is about           percent.

a.   1.1                                                           d. ?2.3

b. ?1.1                                                         e.   20.2

c.                                0.0

                                

 

36. The president of the World Bank has asked you to calculate the average population growth rate of Nigeria from 1960 to 2010. You know the population in 1960 was about 42 million and in 2010 was about 152 million. The average growth rate is about            percent.

a.   16.4                                                         d. ?4.1

b.   0.9                                                           e.   2.6

c.                                0.0

 

 

37. In the late nineteenth century,             was the richest country in the world, but it now lags behind the United States because of                   .
    1. China; a lower rate of inflation
    2. the United Kingdom; a lower economic growth rate
    3. Germany; a higher economic growth rate
    4. Japan; consistently being at war
    5. China; a higher economic growth rate

                                

 

38. Suppose that in 1965 Japan had an initial per capita GDP of $12,000 per year and China had a per capita GDP of $5,000. But China is growing at 5 percent per year and Japan is growing at 3 percent per year.                     would have been richer in 2015 with a per capita GDP of approximately         .

a.   Japan; $5,000 b. Japan; $31,500 c.	China; $7,500	d. China; $60,023 e.   Not enough information is given.

 

39. Suppose that in 1955 Japan had an initial per capita GDP of $15,000 per year and China had a per capita GDP of $2,500. But China is growing at 7 percent per year and Japan is growing at 2 percent per year. In 2015,              would have been the lower-income country, with a per capita GDP of approximately              .

a.   China; $2,500                                          d. Japan; $46,000

b.   Japan; $15,000                                        e.   Japan; $105,000

c.                                China; $50,000

                                

 

40. Since approximately 1980,              has been one of the fastest growing economies.
    1. Germany                                                  d. the United States
    2. the United Kingdom                                e.   Mexico
    3. China

                                

 

41. Of the following countries in 2014,             had the highest per capita GDP, and in 1950,           

had the lowest per capita GDP.

1. 1. Japan; the United Kingdom
   2. Luxembourg; China
   3. Germany; Germany
   4. the United States; Germany
   5. the United Kingdom; the United States

                                

 

42. Over the past two decades, China’s per capita GDP grew an average of about         percent.
    1. 8                                                              d.   4

b.   10                                                            e.   2

c.                                5

43. Between 1986 and 2014, India’s per capita GDP growth rate averaged about         percent, which was slower than in China. As a result, by 2014 China’s per capita GDP was about                of the

U.S. level, whereas in India it was about            .

1. 4.4; one-half; one-fifth
2. 8; three-fourths; one-half
3. 5.8; one-fourth; one-tenth
4. 10; one-half; one-fourth
5. 2; one-third; one-fourth

                                

 

44. When a lower-income economy’s GDP is able to “catch up” with a higher-income economy’s GDP, this behavior is related to an important concept in the study of economic:
    1. growth.                                                    d. fluctuations.
    2. divergence.                                              e.   asset markets.
    3. convergence.

                                

 

45. If France’s per capita GDP is $5,000 in 1950 and Portugal’s is $2,500, but Portugal is growing faster, the expectation that sometime in the future Portugal’s per capita GDP will equal that of France is called economic:
    1. growth.                                                    d. dynamics.
    2. divergence.                                              e.   justice.
    3. convergence.

                                

 

46. In 2014, which of the following had a per capita GDP higher than the United States?
    1. the United Kingdom                                d. Japan
    2. Germany                                                  e.   Greece
    3. Singapore

 

 

47. Which of the following had a negative growth rate over the past 50 years?
    1. Tanzania
    2. Ethiopia
    3. Burundi
    4. the Democratic Republic of Congo
    5. All of these answers are correct.

                                

 

48. Between 1960 and 2014, which of the following countries was among the slowest growing?

South Korea Ireland c.	Thailand	Hong Kong Uganda

 

49. Between 1960 and 2014, which of the following countries was among the fastest growing?
    1. Niger
    2. Nicaragua
    3. Madagascar
    4. Venezuela
    5. None of these answers is correct.

                                

 

50. In 1960, approximately              of the world’s population lived on less than $7 per day; by 2004, it

                to                of the population.

1. 1. 10 percent; rose; over one-third
   2. 10 percent; rose; over two-thirds
   3. two-thirds; fell; less than 10 percent
   4. one-third; fell; less than 1 percent
   5. one-third; rose; over two-thirds

                                

 

51. In 1960, approximately              of the world’s population lived on less than $7 per day, and the rate has                steadily since.
    1. 10 percent; fallen                                     d.   one-third; fallen
    2. two-thirds; fallen                                     e.   two-thirds; risen
    3. 10 percent; risen

                                

 

52. In 2014, approximately              of the world’s population lived on $7 a day or less.
    1. one-twelfth                                              d. one-third
    2. two-thirds                                                e.   12 percent
    3. 10 percent

                                

 

53. Between 1960 and 2014, the fraction of the world’s population that lived on $7 a day or less

               . This can be attributed to, in large part,            .

1. 1. fell; a shrinking global population
   2. stayed about the same; global warming
   3. fell; economic growth in China and India
   4. rose; rising oil prices
   5. rose; the financial crisis

                                

54. Suppose k, l, and m grow at constant rates given by       and      What is the growth rate of a.

b.

 

c.

 

d.

 

e.

 

 

                                

 

55. Suppose k, l, and m grow at constant rates given by       and      What is the growth rate of y if

 

a.

b.

 

c.  

d.

 

 

e.

 

 

                                

 

56. Suppose k, l, and A grow at constant rates given by       and       What is the growth rate of y if

 

a.

b.

c.  

d.

e.

 

                                

 

57. 
    Suppose k and l grow at constant rates given by and     What is the growth rate of y if

a.  

b.

c.

d.

e.

 

                                

 

 

58. Suppose K and L grow at constant rates given by  and      What is the growth rate of y if a.

b.

c.

 

 

d.

e.  

                                

59. Suppose k, l, and A grow at constant rates given by        and       What is the growth rate of y if a.

b.

c.

d.

e.

 

                                

60. Suppose k, l, and A grow at constant rates given by       and     . What is the growth rate of y if a.

b.

c.  

d.

e.  

                                

 

61. Suppose x grows at a rate of        percent and y grows at a rate of          percent. If z = y ? x, then z

grows at                percent; if z = x/y, z grows at              percent.

a.   7; ?17		d. 60; 42
b. ?17; 7		e.   60; ?42
c.	17; ?7

 

                                

 

62. Suppose x grows at a rate of        percent and y grows at a rate of         percent. If z = x ? y, then z

grows at                percent; if z = x/y, z grows at              percent. a.     0; 6                            d. ?6; 6

b.   6; 0                                                          e.   1; 0

c.                                6; 12

                                

 

63. Suppose k grows at a rate of         percent and l grows at a rate of          percent. If                  then

y grows at                percent.

1. 1. 5                                                              d.   4

b.   15                                                            e.   3

c.                                9

                                

 

 

64. Suppose k grows at a rate of         percent and l grows at a rate of        percent. If                 then y

grows at                percent.

a.   ?6                                                            d. ?10

b.   5                                                              e.   12

c.                                6

                                

 

65. Suppose k grows at a rate of         percent and l grows at a rate of        percent. If                  then

y grows at                percent.

6 5 c.

7

?6 12

 

66. Suppose k grows at a rate of         percent and l grows at a rate of        percent. If                  then

y grows at                percent.

a.   3                                                              d.   12

b.   4                                                              e.   ?4

c.                                ?3

 

67. Assume that both Japan’s and the United States’ average annual per capita GDP growth rates are 2 percent per year, and both countries began with an initial per capita GDP of $1,000. However, the United States has been growing since 1915 and Japan only since 1965. In 2015, the United States would have been               than Japan.
    1. 0.37 times poorer                                     d. 4,555 times richer
    2. 99 times richer                                         e.   0.269 times poorer
    3. 2.69 times richer

                                

 

68. Assume that both Mexico’s and Argentina’s average annual per capita GDP growth rates are 3 percent per year, and both countries began with an initial per capita GDP of $1,000. However, Argentina has been growing since 1940 and Mexico only since 1965. In 2015, Mexico’s per capita GDP would have been about              , while Argentina’s would have been about           .

a.   $19.42; $12.94                                        d. $48,544; $72,816

b.   $9,179; $4,384                                        e.   $4,384; $9,179

c.                                $51,500; $77,250

                                

 

69. Assume that Mexico’s average annual per capita GDP growth rate is 3 percent per year, while Argentina’s is 2.5 percent. Next, assume that both countries began with an initial per capita GDP of

$1,000 in 1965. By 2015, per capita GDP would have been          in Mexico and               in Argentina.

a.   $228; $291                                              d. $4,384; $3,437

b.   $3,437; $4,384                                        e.   Not enough information is given.

c.                                $4,515; $3,523

                                

 

70. Which of the following is/are the benefit(s) of economic growth?
    1. increases in life expectancy
    2. reductions in infant mortality
    3. higher incomes
    4. an expansion in the range of goods and services available
    5. All of these answers are correct.

                                

 

71. Which of the following is/are the benefit(s) of economic growth?
    1. an expansion in the range of goods and services available
    2. shorter expected life spans
    3. more pollution
    4. lower government deficits
    5. lower productivity

                                

 

72. The costs of economic growth include which of the following?

1. 1. pollution and the depletion of natural resources
   2. global warming
   3. increased income inequality
   4. technological advances, which may lead to the loss of specific jobs and industries
   5. All of these answers are correct.

                                

 

73. The costs of economic growth include which of the following?
    1. pollution and the depletion of natural resources
    2. global cooling
    3. the gradual release of individuals from physical labor
    4. increased productivity
    5. longer life spans

                                

 

74. Despite the costs associated with economic growth, most believe:
    1. they will only get higher.
    2. there are not too many benefits.
    3. they are higher than the benefits.
    4. the benefits far outweigh the costs.
    5. the benefits are negative.

                                

 

75. The relationship between pollution and per capita GDP is documented as:
    1. U-shaped.                                                d. Nonexistent.
    2. an inverse U.                                           e.   negatively related.
    3. positively related.

                                

 

76. In dynamic economies, it is true that economic growth may contribute to         , but it also leads to

               .

1. 1. unemployment; inflation
   2. job creation; job destruction
   3. job destruction; job creation
   4. crime; unemployment
   5. pollution; price instability

                                

 

TRUE/FALSE

 

1. According to historical data, wages in ancient Greece and Rome were about the same as wages in fifteenth-century Britain.

 

                                

 

2. The “birthplace” of modern economic growth was the mid-nineteenth-century United States.

 

 

 

3. Economic growth is defined as the percent change in per capita income or GDP.

 

                                

 

4. If the 130,000-year period since anatomically modern humans made their first appearance were compressed into a single day, economic growth would have begun in the last three minutes.

 

                                

 

5. One of the nice properties of the rule of 70 is that it simply approximates how long it takes for a variable to double independent of the level of the variable.

 

                                

 

6. Suppose there are L₀ people in the world today. If the population growth rate equals then in 50 years, the world population will be                                     .

 

 

7. One of the nice properties of the rule of 70 is that it simply approximates how long it will take for a variable to double, but it is dependent on the level of the variable.

 

 

 

8. Defining per capita GDP in 2014 as       and per capita GDP in 2015 as        the growth rate of per capita GDP,              from 2014 to 2015 is given by                                   .

 

 

9. The growth rate of any variable y between periods t and t + 1 is the percentage change in that variable, given by                      .

                                

 

10. When a lower-income economy’s GDP is able to “catch up” with a higher-income economy’s GDP, this behavior is related to an important concept called economic convergence.

 

                                

 

11. If x grows at 3 percent and y grows at ?2 percent and w = x/y, then w grows at 5 percent.

 

                                

 

12. If x grows at 3 percent and y grows at ?2 percent and w = x/y, then w grows at 1 percent.

 

 

 

13. Define                  . If               and               then

 

 

 

14. Define                    . If              ,              , and if             , then

 

                                

 

 

15. Let                      If                               and if               then

 

 

16. 
    Suppose k, l, and A grow at constant rates given by       and       The growth rate of y if               is

 

                                

 

17. Suppose k, l, and A grow at constant rates given by       and      The growth rate of y if is

                                

 

18. Suppose k and l grow at constant rates given by and     and A is a constant. If                   the growth rate of y is

 

 

19. A benefit of economic growth is the expansion in the variety of goods and services available to individuals.

 

                                

 

20. A benefit of economic growth is increased resource depletion.

 

 

 

21. Each of the following is a benefit of economic growth:
    1. an expansion in the range of goods and services available for purchase,
    2. reductions in infant mortality, and
    3. increased income inequality.

 

 

 

22. Each of the following is a benefit of economic growth:
    1. loss of jobs in some sectors,
    2. pollution, and
    3. increased income inequality.

 

 

 

23. In 2010, one country to have a higher per capita GDP than the United States was Sweden.

 

 

 

24. Between 1960 and 2010, Madagascar was one of the fastest-growing countries in the world.

 

 

 

25. Assuming the current rate of economic growth continues, the average parent of a college student will have a lifetime income twice that of his or her son or daughter.

 

 

 

 

SHORT ANSWER

 

1. In your day-to-day life, how have you been impacted by economic growth, and how has economic growth changed your quality of life, as compared to your grandparents or great grandparents? Give specific examples.

 

 

2. According to the text, in which countries did economic growth begin? Which countries lag behind?

 

 

 

3. In 1955 per capita real GDP in Argentina was $5,200 per year and it grew at a constant growth rate of

1.5 percent per year. In Mexico, 1955 per capita real GDP was $3,600, but it grew at a constant rate of 2 percent (both in 2005 prices).What would per capita real GDP be in each country in 61 years (2015)? What about in 91 years (2045)? Will per capita real GDP in Mexico ever surpass that in Argentina? If so, when, approximately? What inference do we make about growth rates from this example?

 

4. The following table contains per capita real GDP for 11 countries for the years 1950 and 2014. Comment on each country’s GDP in each year as compared to Argentina. Are there any “unusual” observations you can make about the data? Based on your knowledge of the countries, discuss some possible explanations of why some countries do better than others. Next, calculate the average real GDP growth rate for each country. Comment.

 

Argentina 2,890 20,222 Colombia 3,179 12,599 Costa Rica 3,223 14,186 Cyprus 2,784 28,602 Japan 2,616 35,358 Mauritius 4,665 17,942 Nicaragua 3,404 4,453 Portugal 2,727 28,476 Turkey 3,054 19,236 Uruguay 6,259 20,396

Table 3.1: Real Per Capita GDP in $US Country                1950          2014    

 

 

 

 

 

 

 

 

 

 

Venezuela            5,862        14,134   (Source: Penn World Tables 9.0)

 

To get the growth rate we use the following equation:

 

 

.

 

 

 

 

 

Table 3.1: Real Per Capita GDP in $US

 

Country                1950          2014        Growth (%)

Argentina	2,890	20,222	3.04
Colombia	3,179	12,599	2.14
Costa Rica	3,223	14,186	2.31
Cyprus	2,784	28,602	3.65
Japan	2,616	35,358	4.09

Mauritius	4,665	17,942	2.09
Nicaragua	3,404	4,453	0.41
Portugal	2,727	28,476	3.68
Turkey	3,054	19,236	2.87
Uruguay	6,259	20,396	1.83

Venezuela            5,862        14,134            1.36       

 

Once we calculate the average growth rates, we can see considerable differences. The highest and lowest growth rates are Japan (4.1 percent) and Nicaragua (0.4 percent), respectively.

 

 

 

5. Suppose y₀ = 100 and assume k grows at a constant rate      percent per year. If            what is the approximate value of y in 50 years?

 

 

6. What are the “stylized facts” of Japan’s growth experience over the past 150 years or so?

 

 

 

7. Name three of the fastest-growing countries from 1960 to 2010. Name three of the slowest over the same period. Are there any similarities?

 

 

 

 

8. Express the following expressions in terms of growth rates: (a)