Richland will be overtaken by Poorland sooner or later; you can get a first idea of when by looking at doubling periods through the Rule of 72 Richland's doubling period for income is 72/1 = 72 years; in 72 years time, the income of Richland will be $20,000.
Poorland's doubling period for income is 72/3 = 24 years; its income will be $10,000 in 24 years and $20,000 in 48 years, and $40,000 in 72 years.
At some point between 24 and 48 years, the per capita GDP of Poorland pulled ahead of that of Richland.
To find out exactly when, note that the future value formulas for the incomes of Poorland and Richland are:
| Country | Formula |
|---|---|
| Richland | FV = 10,000 (1 + .01) ^ n |
| Poorland | FV = 5,000 (1 + .03) ^ n |
where FV = future value of income of the country;
and n = number of years into the future.
and ^ indicates exponentiation (raise the (1 + .01) to the nth power)
Some experimentation yields the following numbers:
| Year | Richland | Poorland |
|---|---|---|
| 5 | 10510 | 5796 |
| 10 | 11046 | 6720 |
| 15 | 11610 | 7790 |
| 20 | 12202 | 9030 |
| 25 | 12824 | 10468 |
| 30 | 13478 | 12136 |
| 35 | 14166 | 14069 |
| 40 | 15648 | 16310 |
Poorland will catch up to Richland just after year 35.
Problem 2 -- Growth rates of average labor productivity and growth performance
Note that GDP per worker is what is meant by average labor productibity.
Growth rates from the Penn World Tables are similiar to those reported in the text:
| Year | GDP/L | Pct.change |
|---|---|---|
| 1950 | 20,496 | xxx |
| 1960 | 24,433 | 19.21 |
| 1970 | 30,468 | 24.70 |
| 1980 | 31,698 | 4.04 |
| 1990 | 36,771 | 16.00 |
The given percent changes are over each decade; to find an annual growth rate, divide by 10 (this is not quite correct, since the impact of compounding really should be taken into account, but is good enough for our purposes).
If the U.S. economy had grown at 2.47 percent a year for the entire time span 1950-1990, per worker output would have been:
Instead of 54,392, actual GDP per worker in 1990 was only 36, 771 --
less by $17,621 than it would have been had the top growth rate of the period been maintained.
Problem 3 -- The "graying of America"
With an increasing proportion of the population retired, a smaller share will be employed. Even though labor productivity may increase at the same rate -- nearly doubling over the period 1960-1999 -- "living standards" in the form of GDP per person will not increase as rapidly.
The data of the problem, and the assumed value of labor productivity in 2038, are given in the following table:
| Year | Avg. labor | Share of pop. | GDP per |
|---|---|---|---|
| productivity | employed | capita | |
| 1960 | $35,836 | 0.364 | $13,044 |
| 1999 | $66,381 | 0.489 | $32,460 |
| 2038 | $122,961 | 0.364 | $44,758 |
The computations were:
Values for 1999 and 2038 were found similiarly.
Note that 66,381 / 35,836 = 1.8524
Assuming that it will increase another 1.8524 times by 2038 means that it will increase to:
1.8524 x $66,381 = $122,961.
Given the employment to population ratios in problem 4, and the GDP per person in table 8.1 (page 185), we can find average labor productivity. Since
we need only divide the GDP/Pop figures given in table 8.1 by the L/Pop figures given in the problem. [DIVIDE since the problem gives the L/Pop figure, and we want to multiply by Pop/L]
the result for 1998 is illustrated below:
| Country | GDP/Pop | L/Pop | GDP/L |
|---|---|---|---|
| Canada | $19,406 | 0.46 | $42,187 |
| Germany | $18,723 | 0.33 | $56,736 |
| Japan | $19,379 | 0.51 | $37,98 |