Two days ago 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 absurdum that 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.

P.S. The Toothpick Problem is also the heart of my response to Robin Hanson’s insistence that growth has to fall to zero:

Our finite universe simply cannot
continue our exponential growth rates for a million years. For
trillions of years thereafter, possibilities will be known and fixed,
and for each person rather limited.

He’s probably right for physical goods.  But why couldn’t the quality of life in virtual reality grow at 4% for ever?  Serious virtual reality wouldn’t be like toothpicks; it would be a vast array of virtual goods and experiences.  And since these goods and experiences would be imaginary, there’s no reason they couldn’t grow forever.  Laugh if you must: Imagination really is infinite!