Bryan Caplan  

The Toothpick Problem

Democracy Without Government... Quiggin the Pacifist?...
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!

Comments and Sharing

COMMENTS (32 to date)
Tom writes:
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?

I don't understand the issue? Yes- if toothpick production grows at 4% forever growth rates for the entire economy would approach 4%. Is there a problem with this logically?

You can't say there is near-total stagnation if people REALLY like toothpicks enough to continue driving the economy more and more heavily toward toothpick production/conssumption, instead you are forced to say that the economy is growing in the most important consumor good area.

Finch writes:

> But why couldn't the quality of life in virtual
> reality grow at 4% for ever?

Hanson assumes there's some lowest level at which we can organize things and a discreteness to how those things can be organized: atoms, quarks, Planck-length fluctuations, whatever, positioned in some sort of lattice. So there's a finite number of ways a given amount of matter and volume can be arranged, and therefore a finite amount of information that can be encoded. So your virtual reality is finite too. It might be a large set of possibilities, but eventually we'll approach the best one, and then progress will only occur as we expand the volume we're working with, at, presumably, the speed of light. But obviously that only helps beings that live on the edge, which would be an ever declining fraction of all beings. Therefore most of us would be living in stagnation.

This (information density is necessarily finite) is an assumption I completely don't understand. It seems speculative at best, and likely wrong at worst. If it's wrong, and the speed of light is still a hard limit, we'll grow forever by making increasingly rich and dense worlds, as you describe.

alec resnick writes:

I might be misunderstanding something, but isn't this completely analogous to having two bank accounts, one with a 4% interest rate and one with a 0% interest rate where we're interested in the total amount of money we have? If so, I think this plot should answer the question with an asymptotic 4%, given two 'accounts' starting with $100. The economy with 'a million goods' would be represented by a 999,998 more bank accounts with 0% interest rates--it would take longer to converge, but it would converge. You should be able to verify this yourself by throwing in more "100+" in the second, sum term.

I think the piece that you're missing is basically a way to weight the sectors of the economy by something other than their size. e.g. in the reductio ad absurdum for the 4%'ers, you say that you'd end up with 4% growth despite "near total stagnation." It's not clear what 'near total' means, here, if at some point the toothpick industry is larger than everything else in the economy combined. If you set a limit on the relative stagnation you'd get a very different answer.

Nick Rowe writes:

I still think my answer is right. It depends on what happens to the relative price of toothpicks. Which depends on preferences.

(But yes, your general point is correct: there is no physical limit on growth.)

From my comment on your last post:

If the quantity of apples grows at 4% per year, but the price of apples in terms of bananas falls at exactly 4% per year, the two sectors each remain at 50% of total GDP, in value terms. So the growth rate of real GDP is 2%. Which is Bryan's answer.

If the price of apples in terms of bananas falls at less than 4% per year, the value of apple production eventually converges to 100% of GDP. So growth of GDP converges to 4%.

If the price of apples in terms of bananas falls at more than 4% per year, the value of apple production eventually converges to 0% of GDP. So growth of GDP converges to 0%.

Finch writes:

> there is no physical limit on growth

There is if you assume a finite speed of light that cannot be exceeded, which seems reasonable, and a finite density of information that cannot be exceeded, which does not seem reasonable.

Hanson makes both assumptions. Which lead you to conclude we are at best going to be a maximally dense sphere of maximal utility expanding at the speed of light.

If you're on the leading edge of that sphere, things can get better for you, but if you're in the middle, and in the limit everyone will be in the middle, you're stuck with zero growth. This is the core of Hanson's argument. The question of distribution of wealth, ems, and all that stuff, is just detail.

Brian, the answer is 4% if there are no changes in relative prices. The 4% sector comes to dominate GDP (prices times quantities), as many commenters have pointed out.

In your example, if people really like toothpicks so much that they are willing to spend most of their money on them, they must be really great toothpicks. That's not stagnation. That's an economy producing what people are demanding, even if it seems like an absurd set of preferences.

If people did not want more toothpicks, their prices would fall and their share of GDP would go to zero as productivity improved, which is what Tyler seems to be assuming. However, your question posited 4% growth, which I interpret as meaning that output and demand was increasing in the 4% sector, not shrinking.

You can get your 2% answer if you assume that prices fall in the 4% sector by 4% per year even as real output increases by 4%, yielding constant shares of GDP for both sectors.

JohnE writes:

I think the confusion lies with what you mean by 'growth' of a sector. The original question seems to imply that you mean growth in terms of a sector's contribution to GDP. In which case the answer is the economy asymptotes to 4% growth. To see this, note that if x+y=z and x grows at 4% while y stays constant, then the growth of z will approach 4%.

However now it seems that you interpret growth of a sector to mean a growth in quantity, which is something completely different. In that case then the answer is that the economy's growth rate is anything over 0%. 4% is not even a bound in this case. To see why suppose the economy consists of one person (me) and there are two goods, toothpicks and popcorn. Suppose my marginal rate of substitution between toothpicks and popcorn is increasing in toothpicks. Then if the quantity of toothpicks increase at 4%, GDP (measured in popcorn) will grow at more than 4%.

Admitedly, this requires strange assumptions on preferences. The point however is that if you are talking about growth in quantities, then there will be price effects, which makes the economy's growth indeterminate. Tyler's answer is correct given normal assumptions on people's preferences. Your critque of Tyler's answer is wrong because it doesn't take into account price effects.

Phil writes:

You quoted me and challenged us 4%ers with a reductio ad absurdum. I was going to respond, but my answer would be the same as Tom's (first comment) so I'll just go with his.

Chris T writes:

As originally framed, the 4% asymptote is correct and is still correct even when formulated with toothpicks. Any other answer relies on information outside of the parameters of the question.

The growth rates of the sectors are invariant and relative price, personal preferences, and any other considerations do not matter. The problem is you're looking for an economics answer when you've created a purely mathematical problem.

A.B. writes:

+1 to Tom

@Finch: our current understanding of physics is that information encoding has a maximum density. It may be that new physics challenge that view, but there is no reason to expect it. The most parsimonious theories of physics that successfully explain the experimental data available simply entail a maximum finite density of information.

Khoth writes:

I didn't answer before, but I'm in the 4% camp.

For your reductio, it does seem absurd, but not for the reason you want. An economy consisting almost entirely of toothpicks is absurd whether you call the growth rate 0, 2% or 4%.

Or are you disputing that toothpicks would dominate the economy, even in the long run?

mark writes:

Agree with Chris T. Unless you label it a trick question, and say you really wanted people to focus on the premises, the only answer to the question as asked is the asymptote at 4%. Your original question contained the phrase "grow forever". This post says, the "forever" part is impossible, silly readers! Maybe so, but then the question was wrongly phrased.

Finch writes:


Thank you.

Do you have a citation or review you can point to? My understanding was that the bounds that are hypothesized to exist (the Bousso bound?) were tied up in theories about quantum gravity that were basically guesses at this point. But my understanding is shallow, so I could easily be wrong.

If that's not the case and this is solid, then Hanson is right and it's 0% for everybody, eventually.

jb writes:

I still don't understand Tyler's argument.

In any case, even virtual worlds and goods take up some miniscule amount of physical matter (8 gigabytes of ram weighs about 4 ounces). So eventually even transistors and gates (assuming they cannot get smaller than the quantum level) will have a minimum size, and again we reach the 'no more growth'

But I think that size is mindbogglingly large by current standards - on the order of Sextillions of people, if not Septillions per galaxy.

PrometheeFeu writes:

'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?'

Yes. This shows one of two things:

1) National accounting is meaningless.

2) That hypothetical example describes a situation in which toothpicks are highly valued items (Why else would resources flow towards toothpick production?) and so if an economy is producing more and more of highly valued items, it is not stagnating.

PrometheeFeu writes:

I think Bryan's plan concerns growth over a really long time and is not really about the laws of physics implying a limitation on growth when you start running out of atoms or exceed the speed of light.

Becky Hargrove writes:

Bryan, had I known you were making the argument against Robin's I definitely would have insisted upon the infinity of imagination, laughs or not.

Nick, my slow brain has finally figured out your explanation, I had to look at the curve in a giant dictionary by my desk to do it. In my argument I was trying to measure what happens between production/manufacturing and services. I wish it were as simple as the production ratios of apples and oranges.

GMU writes:

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Julien Couvreur writes:

From a mechanical perspective, it should converge towards 4%. Basically, the growing sector takes over the economy (99% of everything that's produced is toothpicks, and that continues growing at 4%).

From an economic perspective, the problem is unrealistic and stupid. Sectors are not independent (we'd have a shortage of trucks to bring all that wood to the toothpick factory), and there are constraints on resources (there can't be that much wood) and demand (we don't need that many toothpicks).

Generally, I am quite suspect of people talking of economic growth. I think a much more relevant and useful concept is economic development. Unfortunately, development is qualitative and hard to measure, as opposed to GDP growth (which I suspect is why econometrists and the media latch onto it).

Jon Leonard writes:

This seems like more an exercise in mathematical modeling than either economics or pure math, and an object lesson in choosing models.

Saying that one sector is static and another grows at 4% pretty clearly implies a specific model: Total = a + b(1.04)^t, which pretty clearly has asymptotic growth of 4%, assuming b is not 0. (High-school calculus level proof, easiest with L'Hopital's rule).

But that begs the question of whether that's an appropriate model, and for what context. Extrapolation is harder than interpolation, and extrapolation far from the supporting data is even harder. So the absurdity of permanent 4% growth merely tells us that the question isn't posed particularly well. A real economy has shifts in growth rates; perhaps a different question might be more interesting. (Or maybe not, if the point was to have a trick question.)

lemmy caution writes:

"This (information density is necessarily finite) is an assumption I completely don't understand."

I am pretty sure that there is a physical reason for that assumption. Here is a hard to understand Wikipedia page:

Cameron Murray writes:

I will reiterate my comments from the previous post.

Virtual reality is a game. I could grow my quality of life in a monopoly board game at 4% forever, but no one would seriously suggest that we can aggregate the 'virtual' gains being made from monopoly players into the real economy.

Don Boudreaux writes:

Julian Simon would - rightly - applaud this post, Bryan.

joecushing writes:

Nobody seems to be getting it. Economies are always going to have high and low growth sectors. Generally; it's new sectors that grow fast and they level off as they age. By the time the high growth sector becomes large enough to pull the overall growth rate up--it's not the high growth sector anymore. The economy has to find a new growth sector. This is why the best answer is today's average growth rate.

Finch writes:

Thank lemmy caution.

The Bekenstein bound and the Bousso bound are the same thing, or very similar things, I believe.

They both are derived from heavy thinking about general relativity, quantum mechanics, thermodynamics, and how they all play together. My understanding is that they are not regarded as fringe, but that they are a long way from being accepted scientific fact.

A.B., you made it sound like you were a physicist. If you're still listening, I'd appreciate a comment - I'm happy to correct my understanding. If this is too far off topic, I yield.

If the original question is one of economics, then at the end of the day physics will give the answer. If it's one of accounting, the answer is either "Four percent" or "Hey?! I've discovered there's something wrong with my system of accounting..."

Finch writes:

Finch: I didn't mean to sound like I'm a physicist, I'm not.

I was thinking indeed of the Bekenstein bound. It draws on general relativity, quantum mechanics and thermodynamics, both of which we have strong empirical evidence for.

Finch writes:

Finch at 10:17 is presumably A.B. It's not me.

Thank you; it seems like we're not far apart.

We have strong evidence for each of those theories, but very little idea of how they tie together. And this is necessarily a question of how they tie together. If a physicist told me "Yeah, but in this particular case we think that virtually any theory that ties these theories together would have to have a bound like this as a consequence," I'd update my beliefs. But as it is, it seems speculative. And if I remember correctly seeing this in class a long time ago, it was taught as if it was speculative.

Michael Wengler writes:

If growth means ECONOMIC VALUE growth, then clearly the only possible answer is that the far future economy is completely dominated by the 50% of technologies with a 4%/year growth rate and so has an overall 4%/year growth rate. That is just math and logic.

If growth means "number of toothpicks" for the toothpick industry, "number of widgets" for the widget industry, irregardless of whether toothpicks or widgets are rising or falling in value at these production rates, then the answer is STILL 4%. Here's why. In order to compare numbers of widgets and numbers of toothpicks we need to pick some ratio to equate them. If the ratios we pick are static in time, then in the long run we will STILL find the 4%/year growers dominating the future sums of all goods.

I can't imagine how anybody could justify 0%, and 2% just seems like an innumerate answer to me.

JohnE writes:

@Michael Wengler

If the ratios we pick are static in time

This is where your reasoning leads you astray. Almost surely this ratio (which is usually called a price) will not be static over time. If consumers have decreasing marginal rates of substitution, then as people consume more and more toothpicks, the price they are willing to pay for toothpicks (in terms of other goods) will lower. This means the economy will grow at a rate between 0 and 4%. As I pointed out above it is theoretically possible for the growth rate to exceed 4% if consumers have increasing marginal rates of substitution.

Nick Rowe writes:

JohnE: right.

"As I pointed out above it is theoretically possible for the growth rate to exceed 4% if consumers have increasing marginal rates of substitution."

Yep. I didn't see that possibility. Weird preferences though. The growing good would be Giffen, I think?

jamesoswald writes:

4% growth in productivity or in size of the economy? If a sector grows as a part of the economy, it will eventually become to 100% of the economy. If a sector grows in productivity, it will run into decreasing returns on the consumption side and wind up like agriculture - efficient but small.

Chris writes:

I don't get why there is discussion on this? It's a freshman level Calculus problem. As the problem is defined, this is a simple limit problem and the answer, as several others have stated, is asymptotic growth to 4%. For giggles, I ran it through excel and found that in year 100, the growth rate is about 3.9% and the stagnant sectors of the economy are roughly 1/25th of the overall economy (after starting as 1/2 of the economy in year 0).

I'm a little disheartened that three PhDs (including Kling and Cowen) are getting so wrapped around the axle on such a trivial problem.

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