## Tuesday, January 26, 2016

### Paul Romer on Compounding

From Paul Romer's blog:

12 Oct, 2015

### Compounding

In an update on an old story, an investment banker asks the client to pay by placing one penny on the first square of a chessboard, two pennies on the second square, four on the third, doubling the number on each square that follows. If the banker had asked for this on only the white squares, the initial penny would double thirty-one times to \$21,474,836 on the last square. Using both the black and the white squares, the sum on the last square is \$92,233,720,368,547,758.

People are reasonably good at estimating how things add up, but for compounding, which involved repeated multiplication, we fail to appreciate how quickly things grow. As a result, we often lose sight of how important even small changes in the average rate of growth can be. For an investment banker, the choice between a payment that doubles with every square on the chessboard and one that doubles with every other square is more important than any other part of the contract. Who cares whether the payment is in pennies, pounds, or pesos? For a nation, the choices that determine whether income doubles in one generation or two dwarf all other economic policy concerns.

Growth in Income per Capita
A rough guide to the doubling time for any rate of growth is to divide it into 70. For example, if something grows at 7% per year, you can infer that it doubles every 10 years because 70 / 7 = 10. If it grows at 3.5% per year, it takes 20 years to double. Taking twice as long to double may not sound so bad, but remember the difference between using just the white squares or all the squares on the chessboard. Or consider what happens over the course of a century. A doubling time of 20 years means doubling 5 times in a century, which produces an increase by a factor of 32. Doubling 10 times produces an increase by a factor of 1024.
Figure 1
Is growth at 7% per year even possible? For a country that starts at a low level of income, we now know that growth this fast can be sustained for decades. Figure 1 shows GDP per capita in Shenzhen, a new city that the national government in China started as a reform zone where it could pilot controversial new policies such as letting foreign firms enter and hire Chinese workers. When a graph has a ratio scale on the vertical axis, as this figure does, the slope of a plotted line is equal to the growth rate. The steeper slope from 1980 to 1985 shows that growth was higher then. In this exceptional initial period, it grew at an exponential rate of 23% per year. In the subsequent interval from 1985 to 2011, it grew at more than 7% per year. Growth at this pace lifted GDP per capita (measured in the purchasing power of a dollar in 2005) from about \$2500 in 1985 to about \$17,000 in 2011.
Figure 2
Figure 2 shows that Shenzhen’s population also grew rapidly, at an average at the rate of 11% per year, increasing from about 300,000 in 1980 to more than 10 million in 2011....MORE
HT: I am guessing it was at The Browser who seem to have gotten around to the  "Crusade against multiple regression analysis" story at the Edge as today's feature.