02 Facts Practice

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Chapter 1The Facts to Be Explained

Problems and Solutions3. Suppose that, in a given country, per-capita Gdp was 1000 $ in 1900 and 4000

$ in 1948. Using the rule of 72 (NOT a calculator), calculate the growth rate ofper-capita Gdp in that country in 1900-48.

ANSWER Using the rule of 72, we know that GDP per capita will double every72/gyears, wheregis the annual growth rate of GDP per capita. Working

backwards, if we start in the year 1900 with a GDP per capita of $1,000, toreach $4,000 by the year 1948, GDP per capita must have doubled twice.To see this, note that after doubling once, GDP per capita would be $2,000 insome year, and doubling again, GDP per capita would be $4,000, exactly theGDP per capita in year 1948. Using the fact that GDP doubled twice within 48years and assuming a constant annual growth rate, we conclude that GDP per

capita doubles every 24 years. Solving for the equation, 72/g 24, we getg, theannual growth rate, to be three percent per year.

4. Suppose that the entire world population consists of four (4) people, dividedinto two countries of two people each. The following table shows data on theirincome and nationality.

Person Nationality Income

Alfred Country A 1000

Bob Country B 2000

Carol Country B 3000

Doris Country A 4000

Based on this table, which is the most importance source of income inequality inour simplified world: between-country inequality or within country inequality?


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2 Weil Economic Growth, Second Edition

ANSWER Between-country inequality is the inequality associated with averageincomes of different countries. Country As average income is given by addingAlfreds Income and Doriss Income and then dividing by 2. This yields an

average income of 2,500 for Country A. Similar calculations reveal that

Country Bs average income is 2,500. Because the average income for CountryA is equal to that of Country B, there is no between-country inequality in thisworld.

Within-country inequality is the inequality associated with incomes of people inthe same country. In Country A, Alfred earns 1,000 while Doris earns 4,000,making it an income disparity of 3,000. In Country B, the income disparity is1,000. Therefore, we see within-country income inequality in both Country Aand Country B. Because there is no between-country inequality, worldinequality can be entirely attributed to within-country inequality.

The picture below (Figure 1.8 in the Weil textbook) shows that most of the increasein total world inequality since 1820 (when capitalist development started) has

been the result of increased inequality BETWEEN countries. Contrary topopular wisdom, world inequality has, instead, declined. This decline has beenthe net effect of increased inequality WITHIN countries and reduced inequalityBETWEEN countries. Essentially, the rapid growth of China, India andIndonesiaall populous countrieshas accelerated the overall process ofconvergence in the world economy and driven down world inequality.


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Chapter 1 The Facts to Be Explained 3


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4 Weil Economic Growth, Second Edition

5. In 1900, per-capita Gdp in Japan was $1433. By the year 2000 it hasgone up to $ 23971. Calculate the growth rate of the Japanese Gdp during this

period. Then, suppose that Japan keeps growing at the same rate as in 1900-2000 for the next 100 years. What would be the level of per-capita Gdp then in

2100?

ANSWER (use a calculator or excel, this time). We can solve for the averageannual growth rate,g, by substituting the appropriate values into the equation:

(Y1900) (1 g)100Y2000.

Letting Y1900 $1,433, Y2000 $23,971, and rearranging to solve forg, we get:

g ($23,971/$1,433)(1/100)1,

g0.0286.

Convertingginto a percent, we conclude that the growth rate of income per capitain Japan over this period was approximately 2.86 percent per year.

To find the income per capita of Japan 100 years from now, in 2100, we solve

(Y2000) (1 g)100Y2100.

Letting Y2000 $23,971 andg 0.0286,

($23,971) (1 0.0286)100Y2100,

Y2100 $402,103.76.

That is, if Japan grew at the average growth rate of 2.86 percent per year, we wouldfind the income per capita of Japan in 2100 to be about $402,103.76.

7. Between 1970 and 2005, Chinas per capita Gdp grew at an average rate of7.3% per year. The US Gdp grew instead at the much lower rate of 2.2%. As aresult, in 2005, per-capita Gdp was $ 36,806 in the US and just $ 5955 in

China. Assuming that both countries will continue to grow at this rate, in whatyear will China overtake the US in terms of per-capita Gdp?

ANSWER In order to calculate the year in which income per capita in China willovertake the income per capita in the United States, we first need to find t, thenumber of years it will take for the income per capita in both countries to beequal. That is,

(YU.S., 2005) (1 .022)t (YChina, 2005) (1 .075)

t.


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Chapter 1 The Facts to Be Explained 5

Since YU.S., 2005$36,806, YChina, 2005 $5,955, we then substitute in these values andsolve fort.

(1 0.075/1 + .022)t ($36,806/$5,955).

We can solve fortby trying out different values on a calculator. Alternatively,taking the Natural Log of both sides, and noting that ln(xy) y ln(x), we get

tln(1.05) ln($36,806/$5,955)

t 37.33.

That is, in 37.33 years, assuming they grow at the current growth rates, theincome per capita of China will surpass that of the United States. This year will

roughly be 2005 t, i.e., the year 2042.

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02 Facts Practice