Bryan Caplan:

The Toothpick Problem: I posed the following hypothetical:

Suppose half of the sectors of the economy grow forever at 4%, while the rest completely stagnate. I'm strongly tempted to say that this economy's growth rate equals 2% forever. Anyone tempted to disagree? If so, why?

Answers in the comments spanned the entire range. Some said that the growth rate would asymptote to 4%. Phil:

The growth rate will increase over time to approach 4% asymptotically as the 4% sector grows relative to the 0% sector.

Tyler, in contrast, said that the growth rate would asymptote to 0%:

Eventually the growth rate converges to zero or near-zero...the growing sectors become quite small in gdp terms and their future gains cease to matter very much.

My claim: Both the 4% and the 0% answers are vulnerable to a

reductio ad absurdumthat I call the Toothpick Problem. Imagine an economy with a million goods. One of them is toothpicks.Reductio ad absurdum for the 4%-ers:

Suppose toothpick production grows at 4% forever, and the rest of the economy stagnates. Doesn't your position imply that "economic growth" asymptotes to 4% despite near-total stagnation?

Reductio ad absurdum for the 0%-ers:

Suppose toothpick production stagnates, but production of every other good grows at 4% forever. Doesn't your position imply that "economic growth" asymptotes to 0% despite near-universal progress?

The absurdity of both extreme positions is what draws me so strongly to the 2% answer in my original hypothetical. Weighing output using initial shares isn't perfect, but it's reasonable relative to the alternatives…

I read things like this, and I am reminded of something I once saw on Twitter: "My shoes give me a migraine. Every time I listen to [redacted], I take them off and bang myself on the head with them."

It's not a question that has "an" answer. To pose it as a question that does is to fail to grasp the issue. To propose that "the" answer is a split-the-difference 2%/year

isn't perfect, but it's reasonable relative to the alternatives

is an epic fail. For the only sane answer is: "it depends".

Look: the growth in real GDP from one year to the next is how much the measured flow of production would grow if all prices in the economy stayed the same from the one year to the next.

On what does that growth depend?

If the relative prices of the goods and services in the two sectors stay the same on average, as one grows by 4%/year and the other grows by 0%/years, then as time passes a larger and larger share of total expenditure is spent on the products of the fast-growing sector. The growth rate asymptotes to 4%/year.

If the relative prices of the goods and services in the fast-growing sector fall relative to the stagnant sector by 4%/year on average, then as time passes total expenditure remains divided 50-50 between the two sectors. The growth rate is always 2%/year.

If the relative prices of the goods and services in the fast-growing sector fall relative to the stagnant sector by more than 4%/year on average, then as time passes the fast-growing sector becomes a smaller and smaller part of total expenditure. The growth rate asymptotes to 0%/year.

"The" answer is not one number. "The" answer depends on what the elasticities of demand are. That in turn depends on who gets the income, and on what the people who get the income value as technological change proceeds.