May 31, 2007

Thou Shalt Not Extrapolate Exponential Growth Rates

If you think housing in the DC area is expensive now, just wait 50 years:

The findings, discussed at a forum sponsored by the Metropolitan Washington Council of Governments, show that the price of the average home in the area will grow to $14 million in 2057 from today’s average of $477,000.

Should we take this seriously? After all, I saw it in a free newspaper. But it quotes a real study, and its author, a professor at George Mason University. Let's read a little more:

Researchers predicted, however, the average household income would only climb to $1.3 million from today’s $137,000 during the same period.

“We already have an affordability problem,” said George Mason University professor Stephen Fuller, who calculated the estimates. “But this is really scary. It is going to take 11 times the average household income to afford the average-priced house.”

Wait a minute. Who will buy those houses? At 6% annual interest rate, a 30-year mortgage on an average home would require payments exceeding three-quarters of an average household's income. And that's just mortgage payments - no property tax, insurance, maintenance, or utilities. Even at the entirely unrealistic zero interest, mortgage payments would equal 36% of income. Prices can be astronomical in a limited area, where only the region's elite can afford to live, but the study refers to the entire Washington Metropolitan Area, which, according to the article, will then span from Baltimore to Richmond and have a population of 9.9 million. If only the top 5-10% can afford housing in the area, where will the other 9 million live? There is a violation of the law of supply and demand in the study's results.

That study is yet another example of the dangers of extrapolating exponential growth rates over long periods. If you calculate the annual growth rates implicit in the study's results, they look reasonable: housing prices increasing 7% per year, and incomes 4.6% per year. Assuming 3% inflation, the real (net of inflation) rates are 4% and 1.6%. I might use those numbers myself if I had to predict prices and earnings 2-3 years from now.

The problem is that the difference in growth rates of 2.4% compounds to a factor of more than 3 (in other words, a difference of over 200%) over 50 years. What makes sense over short periods doesn't necessarily make sense in the long run.

For a related example, consider health care costs. Their share of GDP grew from 7.2% in 1965 to 16% in 2005, at an average annual rate of 2.7%. If that trend continued for another 50 years, health care would account for 60% of GDP in 2055. Keep in mind that health care is itself a sector of the economy, so its product is counted in the GDP, and if it makes up 60% of GDP, it means that it is one-and-a-half times as big as the rest of the economy combined. If that doesn't make the absurdity of the result obvious, try to ponder what would happen just 8 years later, when health care expenditures would exceed 100% of GDP.

Similarly, if you looked at the rate of growth of market capitalization of Nortel or Enron in the years before their respective stocks crashed, and extrapolated those rates into the future, you would conclude that those companies would own everything in North America at a time not too far into the future.

Or take population groups. Population growth in Muslim countries has been higher than the world's average, so Islam has been increasing its share, which currently stands at about 19% of the world's population. According to some estimates, the number of Muslims has been growing approximately 0.6 percentage points faster, annually, than the world's population. At that rate, by the end of 24th century, everyone in the world will be Muslim. The only problem is, Pentecostalism, with about 2% of the world's population as its adherents, is growing about 6% faster than the world's population, so by 2075, everybody in the world will be Pentecostals.

A more cheerful extrapolation shows that in just 10 years, every computer in the US will be a Mac.

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