Bryan Caplan  

Tell Me the Growth Rate

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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?

Before you answer: Would it matter if the 4%-growth sectors were all in "virtual reality" - and the stagnant sectors were for actual food, shelter, cars, etc.?  If it matters, why does it matter?

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COMMENTS (44 to date)
Chris writes:

Doesn't the size of the sectors matter? Even if they started off exactly the same then, over time, total growth would exceed 2% as the productive sector would overwhelm the non-productive.

I don't think there's enough information here to give an answer.

I'd want to know how the population was changing over time. If the population was growing but production of food, shelter and cars was static, then real consumption per capita falls regardless of how fast the virtual sector is growing and misery follows.

If population remains static (maybe because we're spending all our time in VR), then the answer depends on why the physical world sectors aren't growing. If it's because people choose to invest only in VR sectors, then growth is 2%. If it's because something is preventing them from investing in more physical goods, even though they'd prefer physical to virtual goods, then grow is less than 2%

Arthur_500 writes:

You point out that Growth Rate is an aggregate of all the sectors. Sectors such as Food, Shelter and Cars are based on population. With the current population decreasing, it comes as no suprise that these sectors will be stagnant until there is necessary demand.

However, there are other sectors that will be growth industries and may pull along the entire growth rate. Who could have foretold of the demise of those growing hay and the outrageous growth of petroleum products only 100 years ago? Until petroleum was developed we couldn't develop styrofoam, various plastics, etc.

Why does it matter? Because growth allows for excess which enables us to live more comfortably. As long as our growth matches our population then we are stagnant. As it exceeds then we have demand for Lazy-boy chairs (comfort) and yachts (pleasure).

A few years ago the Internet caused a great growth industry. Possibly the first industry in history based on nothing (outside of Ponzi). I believe it is the unknown industries that will bring along a growth rate even in the face of declining populations.

Steve writes:

The growth rate would start out at 2%, but assuming this pattern held it would reach 3.9% in 95 years.

joeftansey writes:

The math is wrong. Matlab code


for i = 1:50



So its actually "better" if you have a small part of the economy growing faster. But who cares about GDP right?

Phil writes:

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

But that's not what you're getting at, I don't think.

Steve writes:

I guess I was supposed to say why. If production in the economy is 200 at time 0 then half of this, 100, grows to 104. The other half remains at 100. The new total, 204, is 2% greater than 200 thus a 2% growth rate in year 1. However, in year 2 the 104 grows to 108.16, while the 100 remains. The new total, 208.16, is 2.04% greater than the old total of 204. The growth will keep increasing in such a way and will form an asymptote at 4%.

gappy writes:

The long-term growth rate would be 4%, and it would not matter if the sector were virtual reality.

The thrust of the post is obvious.

Gary Rogers writes:

I also agree. Even though the two halves of the economy start out the same size, an exponentially increasing segment will soon dwarf the stagnent segment. The total growth will start out at 2% but soon aproach 4%.

Nickolaus writes:

C'mon Brian. Learn2exponentiate. The answers suggesting it approaches 4% asymptotically are correct.

coyote writes:

Growth rate begins at 2% but as the growth sector increases as a percent of economy, the total growth rate approaches 4 in the limit. Agriculture is a good example. It has been greatly outgrown over the last 150 years by other sectors so it has declined as a percentage of the economy. As it declines, its slow growth is less of a drag on the total.

As economy matures, won't food, clothing, shelter undergrow the total, as, per Mazlov, people satisfy these needs first, then move on to other stuff.

I think the exponential answers are missing the focus of the question: Economically, is there a difference between the physical and the virtual world? Is growth and innovation in the virtual world a perfect substitute for growth and innovation in the physical world?

If that wasn't the point of the exercise, Dr. Caplan could have just said sector A and sector B. But he didn't do that. He very specifically called out the difference between real reality and virtual reality and asked you to explain why that difference does/doesn't matter.

Plus, he's posting from GMU where Robin Hanson likes to talk about the "Singularity" and unbounded growth in the coming virtual economy.

I'll stand by my answer.

matt writes:

Coyote is correct.

If the population is constant and everyone has enough to eat should we expect growth in the food/agriculture industry. We might change what gets produced as taste changes but there should be no expectation of growth in that sector.

One can easily imagine a situation where we have enough energy generation from renewable sources, the population is near constant, and most things are produced by 3d-printers at home, printed out of materials that can't be depleted. At some point we should only expect growth in virtual goods. How does investment work in this type of economy?

mike shupp writes:

Let's assume an island, which we might call Hispanoila, which has been settled and maintained in a basically stable state for some time -- centuries.

Basically VERY stable, The population, several million people, never varies more than a few hundred souls, sometimes more, sometimes less, but overall not changing. A few houses fall down each year; a few new ones are built, but basically the housing stock does not change. Most Hispanoilans are uneducated; this has been true for centures, and no changes are anticipated. The farms grow the same crops year after after, producing roughly the same quantities of rice and goats milk and wheat and ganja. The diet is monotonous. Incomes and income distribution is pretty well fixed (the level of technology is very low and, of course, unchanging).

But we need not feel too sorry for the Hispanoilans. The low living costs, beautiful beaches and gorgeous sunsets and the many attractive young ladies have brought a steady flow of artists (and photographers and cable television production units) to the island for several centuries now. Many of them have donated some of their work to to the island's inhabitants -- more precisely, to a score of local museums. Some of these artists have become very famous indeed, and the value of their donated art approaches astronomical amounts. Reputatable authorities claim the worth of the art is at least equal to the purchase price of every house and church on the island, plus the sales price of a year's harvest, plus the worth of all farm implements, etc.

Best of all, the museums are administered by a beneficent government for the benefit of all. The museums are open for all natives and all vistors, for at least 12 hours a day, and there is no admission charge. Truly it can be said that every citizen of Hispanoile shares in the extraordinary riches of local culture!

So. Are the local people becoming richer?

I'd say NO. Imagine again you are a local resident, with a sick child. You wish to keep the kid alive. This takes medicine you don't have. Medicine you can't buy, because the stocks of every commodity are fixed and being poor you have nothing to trade for medicine or the money to buy medicine. If perhaps a few of those artworks could be sold... what are millions to gringos, after all? ... then perhaps Hispaniola could afford some medical clinics, or import a few trained nurses. But the artwork can't be sold, and even as your financial value on paper goes up 2% this year and the next year and the year after that... your child dies. Suck it up, dude. If you can't change the relative frequencies of available goods, your economy is stagnant.

WHY THIS MATTERS: Second Life and World of Warcraft are NOT making all of us happier and richer and better off and are NOT substituting for ongoing lower and middle class wage stagnation.

Brian writes:

To say that a sector of an economy grows by 4% is an inchoate feeling or a matter of opinion dressing itself up as mathematical precision. The goods produced in year 2 will not be the same in nature or quality as those from year 1, the location of delivery and desires of the customers will be different. You can claim to measure the difference in dollars but the demand of the market for different fashions of goods will have changed. Furthermore, the medium of exchange is different; there is no canonical exchange rate between year 1 dollars and year 2 dollars.

In practice, we have experienced experts track long term trends in every industry and add up their established opinions on the relative qualities and dressing of goods to estimate the relative value and volume from year to year. Then we have another set of experts measure the relative values of our media of exchange in different markets and produce a measure of inflation or deflation. Actually, they produce many different measures because there is no such thing as inflation -- there is only prices and every possible combination of price changes would produce a different definition of inflation useful only in its own context. There is no such thing as a general, common, or economy-wide inflation.

So we have an assembly of expert opinion masquerading as a precise measure of growth. If your opinion is that the sum of 4% growth and 0% growth is 2% growth or an asymptotic trend toward 4% then you can combine and average the opinions of your experts to get that result. Or you can get whichever other result you like.

But we must remember that growth rates are a matter of opinion, not facts of reality. You choose informed and reasonable opinions and combine them in a way that makes sense to you for your purposes. If your purpose is to privilege material comforts, you may arrange to hit 2%. If your dream is to celebrate creative mental exercises and entertainments, you can arrive at 4%. Both are equally valid opinions.

Treating this as a math problem, on the other hand, is dead wrong.

Tyler Cowen writes:

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.

OneEyedMan writes:

Can we assume that the prices are good? That is, are they prices that clear the markets without externalities? If so, then under the assumptions growth asymptotes to 4% over time as many others describe. The comment about VR raises the concern that the goods produced may not be worth their prices in the same way that goods in the real economy are. But if the prices are good, that shouldn't be possible. Preferences and production functions would determine the quantity and price of goods produces, just like for physical goods.

Nick Rowe writes:

Tyler assumes that the growing sectors will see a decline in the price of output relative to the static sectors. But whether this happens depends on the income elasticity of demand, not just on productivity growth.

To avoid Tyler's and Nickolaus's critiques, you need to specify that the half that is growing is continuously updated to reflect half the value of GDP. It's not just those sectors that are half of GDP today.

Given that interpretation of your question, I would say it's 2%. Can't see why anyone would disagree. Whether it's growth in stuff you can kick, or can't kick, doesn't matter. Suppose GDP consists of singing songs to each other. Every year the songs we sing are 2% more beautiful than the year before. That's 2% real growth.

Kevin Harris writes:

Wasn't a lot of the growth leading up the bust in the 'virtual' sector?

mbka writes:

Well. I would have answered as Nickolaus did, assuming that growth of the productive sector simply carried on from time zero with the same assumptions. Tyler's answer makes sense from a different perspective but isn't starting from that standard assumption. And then Nick gives a third plausible way to define the problem as growth, always updated to then-current conditions.

My conclusion: this should be a textbook example of how ill-posed problems pervade and confuse all of economics.

A more cynical interpretation would be that it shows how little the word "growth" actually means.

Duane McMullen writes:

The mathematical answer of growth asymptoting to 4% remains valid.

This is not a theoretical question as it has played out numerous times already.

Agriculture used to be the majority of global output, and employment. Now, agricultural production is vastly greater than ever before while also a much smaller proportion of the global economy.

'Not real' sectors of the economy are nothing new, the pioneer being 'services'. Services were once a very small part of the global economy. Now they are a significant and growing part of our economy, and employment.

I don't see how virtual reality as a sector would be substantially different from services.

Imagine a world where meeting our physical needs has become trivial and the majority of us have all voluntarily moved to the Matrix. When I look at the time I spend on the internet, I think that in many ways we have already started to do that.

At the point that most of us are in the Matrix ('Cloud 2.0'), except for the small minority of throwbacks who insist on remaining in the real world, the economy that matters to all of us would be the virtual one within the matrix.

In this world, our physical are met as easily and trivially as is our present need for air to breathe. The fundamental elements of scarcity then would be software, bandwidth, accessible memory and computation cycles. Agriculture production may be vastly greater than at present, yet also far less a proportion of our economy as measured by whatever is used as units of exchange - perhaps something tangible like computation cycles, bandwidth or accessible memory.

The 4% growth of the economy in the virtual world would be, for all intents and purposes, actual economic growth despite the stagnation in the 'real economy', of trivial size reletive to the virtual economy.

Martin writes:

Isn't this 'simply' Baumol's cost disease dressed up in a different way?

Becky Hargrove writes:

The only reason the meaning of growth has not been penned down is that production differences in the single unit of money have not been measured or calculated: hence the problem of matching nations to one another at 'same' rates and the attempt to match service economies to manufacturing and production economies at same rates. Were those differences in monetary efficiencies actually documented it would be easier for budgets of all kinds to happen.

Even so there is more than one way to do that. If it is only done by the monetary unit of measurement, as Tyler suggested, the growth rate might eventually converge to zero as service economies continue to be forced to accede to (the distilled) parameters of production.

However, should any portion of future services come to rely upon internal efficiency, or infinitely variable (person to person) measurements, real growth is possible that is only limited by human time. The songs that we sing to one another are more easily measured because we do not have to rely on a single 'world' efficiency rate in order for them to be included as growth.

The Sheep Nazi writes:

For those of us out there who are interested but unlettered, can someone walk through Tyler Cowen's argument. Thanks in advance...

Chad Reese writes:

I'm offended nobody mentioned this sooner:

"Never tell me the growth rate." -Young Han Solo in his macro class

Becky Hargrove writes:

There is another aspect to growth I'd really like to see some discussion about. On Mankiw's blog recently he posted "The Monetary System of the Future?" in which he asked his students to imagine money that was not linked to debt-based fiat currencies or fractional reserve and compound interest banking. It would be interesting to know how the discussion went, in that present growth does seem to be limited in a credit-based sense.

Ken writes:
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?

This statement is loaded with implicit assumptions that have to be there for this to work. Assuming X is the combined 50% of the economy that stagnates at time 0 and Y is the 50% of the economy that has a 4% growth rate at time 0. Assume that X and Y make up 1 unit each of the economy (so the economy is 2 units at time 0), then at time k, the economy will be 1 + 1*(1.04)^k, meaning that asymptotically the economic growth rate will be 4%, not 2%. For example in 100 years the economy will be 1+1.04^100 = 51.5 and in 101 years the economy will be 1+1.04^101 = 53.5. Since 53.5/51.5 = 1.038, the economy grows at 3.8% in the 101st year.

For the situation you to occur is that some industries in Y become stagnant.


jb writes:

I don't understand Tyler's argument either - if a sector is growing at 4%, how can it become a smaller and smaller part of the overall economy relative to other sectors?

Also, I respectfully decline to agree with Mr "Hispanolia-it-only-counts-as-growth-if-I-*say*-it-counts-as-growth". Growth in the digital economy creates specific demands against other sectors (power, computer hardware, software, networking, artists, etc) and will have all sorts of follow-on benefits.

Nick Rowe writes:

Tyler's argument, as I understand it:

The economy initially produces 50 apples and 50 bananas, each selling for $1 each. Productivity of apple orchards grows at 4% per year, while productivity of banana plantations stays the same. As the supply of apples grows, relative to bananas, the price of apples drops relative to bananas. If the demand curve is inelastic, the price of apples (relative to bananas) will fall faster than 4%, so the value of apples (price x quantity) falls relative to bananas. Eventually the value of apples, as a percentage of GDP, falls towards zero.

Bill writes:

So if I have this, Tyler is talking about quantity output, while most here are thinking in terms of real value (dollars)?

In dollar terms, in Tyler,s example, if the price of apples are falling, quantity output would increase at a greater that 4% rate to keep a 4% dollar growth rate. But this is only for commodity pricing - there would surely be product improvement and intra-sectoral product development to maintain the 4% value output growth.

Bill writes:

No. What I posted is wrong. I'm just confused at this point.

mark writes:

Thank you Nick Rowe for that explanation.

As posed by Bryan Caplan, the question asked us to assume a 4% growth rate "forever." As Nick Rowe explains Tyler's comment, I perceive that comment to essentially challenge the "forever" in the assumption, and say it is not plausible. So the question blows up.

As to whether it is or not plausible, I defer. But, as the question was asked, the only correct answer is the 4% asymptote.

Noah Yetter writes:

Extrapolating to "forever" is a fool's errand.

Nathan Smith writes:


If the products of the growing sectors are perfect substitutes for the goods in the stagnant sectors, the growth rate will asymptotically approach 4%.

More likely, the growing sectors' products will yield diminishing marginal utility relative to those of the stagnant sectors as their supply increases. In terms of national income accounts, this will be reflected in prices, and then something like what Tyler predicts will happen, where the growing sectors' products become virtually free and GDP growth asymptotically approaches 0%.

But we can't really say anything without making specific assumptions about the functional form of utility. It all depends on the utility function.

Nick Rowe writes:

Nathan: Yep. Nathan and I are saying the same thing in different ways. If we knew the utility function we would know the price and income elasticities of demand.

Put quantity of bananas on the horizontal axis, and quantity of apples on the vertical. Draw an indifference map. If the quantity of apples grows at 4%, and the quantity of bananas stays constant, the growth path of the economy follows up a vertical line. Draw a tangent to the indifference curve from a point on that line, and follow that tangent down to the horizontal axis. That measures GDP, in units of bananas as the numeraire. Follow that tangent up to the vertical axis, and it measures GDP, in units of apples as the numeraire.

Now, how the hell do we measure GDP using a proper chain-linked GDP deflator as numeraire, on that diagram?

PrometheeFeu writes:

It does not matter because if it did, it would not happen. We are not going to starve because everyone starts playing video-games for the simple reason that starving is an unpleasant business and so if we have a lot of resources we will dedicate some to making food.

I think the math is not really true because the relative size of the sectors change over time which means that even if the growing and stagnating sectors start at the same size, after a year, one is bigger than the other pushing the rate of growth up. But also, because national accounting is kind of a silly exercise that can be useful but is too often confused for an observation of reality.

PrometheeFeu writes:

@Nathan Smith and Nick Rowe:
Why do we keep growing one sector over the other if the marginal utility of the stagnant sector is greater than that of the growing sector? If we accept Bryan's premise, we have to conclude that the stagnant sector has lower marginal utility.

The Sheep Nazi writes:

@Nathan Smith and Nick Rowe: thank you for your patience. My unschooled view was something more like this. I've got a barn with a thousand crates of apples in it, and another barn with a thousand crates of bananas in it, and I am worth their sum. If I don't lose any bananas, and every year I get forty more crates of apples, compounded, then it looks to me as though I should eventually gain four percent a year forever -- Steve's arithmetic looks right. It had not crossed my mind that I have to go to market (where apples are sold for fertilizer and bananas are a delicacy) to find out what I am really worth.

mbka writes:

But Nick, Nathan, Sheep: conventionally a phrase like "growth forever" would likely mean "nominal output growth as measured in usual accounting units". So it's GDP growth already alright. The apples aren't devaluing. The apple output is already measured in unit of account, not in apples. While Tyler's point is smart if one redefines it as quantity growth with decreasing nominal value (as in someone's above comment, similar to Baumol's cost disease), the problem as posed appears to be defined as nominal growth. It is already the nominal output that grows.

And something else: while we might get to 0% growth under the quantity interpretation and to 4% using the conventional interpretation, I don't see how one might get to the 2% of Bryan's post.

Nick Rowe writes:

mbka: But when Bryan says "growth rate" I assume he means *real* growth rate, i.e. adjusted for inflation. And there is a correct (at least conventional) way to adjust GDP changes for inflation. You value this year's output at last years prices, and see how much the "real" value has gone up. Then you value next year's output at this year's prices, and so on, in a chain.

And if I could do math (I can't) I could figure out how the growth rate of GDP would be related to the rate of change of the relative price of apples to bananas. The answer to Bryan's question is probably something like: growth rate = 4% minus rate of change of price of apples in terms of bananas. (That can't be exactly right)

Any mathematically competent person could figure this out easily. I can't. Never was good at math.

Nick Rowe writes:

OK. I just remembered L'Hopital's Rule. Perhaps I can do math after all :-)

Here's the answer to Bryan's question, "what does the growth rate converge to, as time goes to infinity?":

1. If the price of apples in terms of bananas falls at greater than 4% per year: 0%

2. If the price of apples in terms of bananas falls at less than 4% per year: 4%

3. And I can't figure out how to apply L'Hopital's Rule if the price of bananas falls at exactly 4% per year, but I think the answer is: 2%

Nick Rowe writes:


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%.

mbka writes:

Nick, I shouldn't have used the word 'nominal' since it allows to introduce inflation which wasn't anywhere in the problem as posed. My point was, whether real or nominal, whichever growth rate you use, the initial problem was posed as "growth" either 0 or 4% (stagnant vs. growing sectors). Both should reasonably be considered as output measured by a shared unit of account - and not as output once measured as growing apples and once as stagnant oranges. One is still stagnant, the other still grows at 4% p.a. Even more to the point if you assume the growth is in real GDP terms then the growing apple output should already be corrected for any price changes in apples.

Cameron Murray writes:

4% growth in virtual reality? What does that mean? Does it become 4% more virtual each year?

Virtual reality is a game. It looks a little foolish if you ask the question with a slight change -

"Before you answer: Would it matter if the 4%-growth sectors were all in the boardgame "monopoly" - and the stagnant sectors were for actual food, shelter, cars, etc.? If it matters, why does it matter?"

Do I need to go further?

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