I'm extremely skeptical that this is a meaningful question. There are so many modeling choices in the division into sectors, and in what single value is measured for growth. It's simply not useful to project more than a few orders of magnitude without having to rethink your whole system.
If it _WERE_ sane to ask, then of course it converges to 4%, because after a few centuries, that sector (or product, for toothpicks) is the only relevant thing in the economy. This should show the absurdity of the question, not the incorrectness of the answer.
I don't understand...first you're asking us to split the economy in half, then we're only talking about toothpicks (or presumable any single other good)?
Also, Virtual Life isn't free, it costs energy. I understand the point that there's no limits to what a virtual economy might provide, but it's all rooted in physical goods somewhere along the line, no?
If the toothpick sector grows at 4% for ever, the growth rate asymptotically approach 4%. The maths is just obvious.
If the point is that it can not, in practice, grow at 4% for ever, well, yes, of course, but the question was really weirdly formulated if that was the point.
This is an example of why unrealistic thought experiments can be misleading. If toothpick production is growing 4% per year, after a while nearly the whole economy is the toothpick economy. Huge amounts of toothpicks are being produced. Almost everyone works in the toothpick industry.
And business in the toothpick industry is GOOD: it is still growing at 4% per year! How? No idea, it’s an unrealistic example. But that’s what is happening. And so since nearly the whole economy is the toothpick economy, the whole economy is growing at nearly 4% per year. That’s just how things work in the thought experiment as you specified it.
There is nothing subjective about the answer here. It’s 4%. For it not to be, the growth in the toothpick economy has to drop below 4%.
What is growing? Sheer output? Productivity? Something else?
If output is growing, I think I agree with Tyler. At some point they hit the point where everyone has more toothpicks than they want, the price goes to near zero, and further growth in that market becomes relatively insignificant. Maybe aggregate growth does not go quite to zero, but no one cares about the difference. It will not be the case that everyone gets a job in the toothpick industry or has a mountain of toothpicks.
Can one industry grow so consistently without causing growth in the industries that produce its physical capital and raw inputs, or in industries that use its products, or in industries affected by the wealth or substitution effects of its growth on consumers? Does the toothpick industry generate its own power and grow its own trees and tree-substitutes? There is some static thinking dominating this supposedly dynamic thought experiment. The thought experiment is too extreme, it is incoherent. It isn’t possible for only one industry to grow indefinitely. Either its growth will spark growth in other industries, or the stagnation of other industries will stifle its growth.
Could the other industries just ignore the effects of external growth? Doesn’t the thought experiment assume some degree of competition? Suppliers of inputs for the growing industry just raise their prices, but do not increase output? Maybe the growing industry never reduce their prices? The supply and demand curves must look pretty odd.
Is there a good for which even a cornucopia isn’t enough? Maybe we can always use more computing power, to make things faster or more detailed or more whatever we still think we lack. More (clean, side-effectless) energy can always be applied to a number of things. But these would tend to create growth in other sectors, wouldn't they?
Suppose the toothpicks get produced with zero costly inputs. There is still a limit to how many toothpicks the world wants, even if we can get a million for a penny, and ten for the price of one every weekend. At some point they switch from being an economic good to a form of pollution. More is better, until it's not.
But what does this illustrate? Maybe that the numbers don’t mean what we would like them to mean. Certainly that this thought experiment, though interesting, is difficult to apply to anything real.
As others have commented, it seems like this is just math. Because I have too much time on my hands, I created the following Jupyter notebook that illustrates why the growth rate of the entire economy asymptotically approaches the growth rate of the non-stagnate sector.
Cowen's pessimistic estimate is only applicable if we assume the demand increases as fast as income for the output of the sector where productivity doesn't improve. To a first approximation this is the actual state of affairs in healthcare.
> But why couldn’t the quality of life in virtual reality grow at 4% for ever?
People who are good at math should understand that exponentials are never sustainable -- they grow too fast. Anything that grows at 4% will outnumber the number of molecules in the universe in a mere 5000 years. Reducing the rate to 1% doesn't help much -- you still run out of matter in universe in under 20000 years. Exponential growth is ALWAYS temporary.
The answer depends on substitutability and satiation.
If everything in the zero growth sector has a perfect substitute in the 4% sector, then the economy eventually grows at 4%. The substitute goods in the 4% sector eventually become cheap enough relative to their 0% counterparts that people switch to them, and the 0% growth sectors disappear.
If all goods in the 4% sector correspond to preferences that can be absolutely satiated, then the economy eventually produces the amount required to satiate the preferences (or get asymptotically close) using a smaller share of land/labour/capital over time. In the toothpick example, toothpicks are eventually produced by one guy who enjoys the smell the toothpick replicator makes. Then, the economy doesn't grow anymore.
If all goods in the 0% sector correspond to preferences that can be absolutely satiated, and the current economy could satiate those preferences, with X fraction of the economy, then eventually the economy converges to a situation where X of peoples budgets are spent on the 0% sector, and the rest of the economy grows at 4%. If X=0.75, then the 4% sector accounts for 25% of the economy, and so living standards rise by 1%. This is the toothpicks grow at 0% situation, with X=0.00001, so growth is almost 4%.
If it would take over 100% of the current economy to satiate preferences for the 0% sector, I think this is qualitatively equivalent to having goods in the 0% sector that can't be satiated, so I'll handle them together. In this case, the 4% goods eventually become so abundant that their marginal utility is very low compared to the 0% sectors. Consumers, therefore, substitute towards 0% goods and the 4% goods eventually make up a smaller portion of the economy, similar to the absolute satiation situation. Eventually, growth in living standards approaches 0.
This is mostly a math problem. But a practical consideration comes to mind. Large things generally quite growing at the rate that got them large. As the US economy continues to grow, it can be expected that the rate of growth will eventually decline. Decline to what? If I knew that, I'd be rich; I'm not rich.
Do you think your last point on the growth of virtual reality has any bearing on a moral dilemma such as the experience machine one? Should we choose the virtual "fake" world if it's the only one that can keep improving forever?
These problems can, at least to some, be fun and some economists like to do this a lot so they can say,..ha, gotcha! I think some of them really wanted to be magicians, pulling rabbits out of a hat. But many times the questions/problems are vaguely presented. Steven Landsburg is another one who likes to play these games.
"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? " You should have expressed this as "At the BEGINNNING of the growth period, half the sector grows at...etc" otherwise seems to make no sense. Period 0: Z sector 1, Y sector 1. Period 1: Z sector 1.04, Y sector 1 (so Z sector now makes up 1.04/2.04 > 0.50 of the economy). Period 2: Z sector 1.04*1.04 = 1.082, Y sector 1 (Z sector is now 1.082/2.082 even further above 0.5 and so.). 100 years from now Z: 1.04^100= 50.1, Y is 1 (Z sector share 50.1/51.1). At period 0: gdp is 2, 100 years later it is 51.1, so 2 (1+x)^100 = 51.1--> 3.3% growth. Keeping going on, approaches 4%.
But that is impossible for supply and demand reasons. At some point, the goods produced by the growing sectors approach satiation of demand, and the price goes to near zero. No one wants more, even at a lower (positive) price, and output growth must stop.
Maybe if t-shirts end up selling for one cent each we will start using them for something else, or throw them away instead of washing them, but I don’t want more than 365 of them per year. Well, maybe a few more if I change clothes during the day. And that ignores the cost of disposing of them. Anyhow, the point is, for ordinary goods, there is a point where output growth doesn’t matter, and beyond that it becomes a problem to be dealt with.
Maybe productivity growth can continue instead of output growth?
Maybe there are goods like computer power or energy that can create infinite demand, so we always want more without sending the price to zero?
I made a comment below that goes into this further.
You can always come up a demand curve that never hits the horizontal axis (speaking of course in theory). Picture a non-linear demand curve asymptotically approaching horizontal axis. So price can infinitely fall without hitting zero.
Or more directly, we can draw that curve, but is that what we see in real life? Most demand curves eventually cross into negative territory. As noted, energy, computation, maybe a few other things might be exceptions.
Or again seeking realism, when a good gets below a certain price, it becomes something you trade by the carload in the Chicago commodities exchange. If the price goes too low, even that becomes infeasible. It would require some alternative enabling technology that can underprice and outperform the CCE and railroads.
That works as a formalism, but what would it mean in practice? Very one would be living in a pile of toothpicks. Everyone wants 100 classic cars? We don’t have space for them. There is a point for most ordinary goods where there are just too many of them, and no one can sell any. This is probably not true of (clean) energy and maybe a few other things. Maybe if the scenario is that half the economy is growing, these constraints don’t bite. I got distracted by the toothpicks.
Yes, it is theoretical, definitely not meant in literal or practical way. I was responding to Bryan's hypothetical. But to respond to your examples: what if "consumption" of toothpicks is to use them and then burn them to ashes. The ashes then shot into space which as far as we know is infinite. What if consumption of classic cars is viewing them for a year, getting bored, then launching them into space.
Launch them into space, and then buy 100 more? How do you get 4% growth if it's 100 per year every year? I guess you need to buy 104 next year. But even though they only cost one cent each, do you really want to spend four whole cents to get four more cars, when you have 100 every year? If you buy and launch one every day, what makes you want to increase to buying a launching 1.04 per day next year? At some point, it becomes tedious. You have other things to do. And even if you are a 24-hour-a-day car fanatic, that just means you are not consuming more pure-bred dogs and elegant furniture. Please don’t launch those dogs into space.
You can only eat so much in a day (unless you're a Roman, I suppose). It can always be a bit more delicious and a bit more healthy, but as some point you are ready to stop. The demand curve goes negative, temporarily.
In 100 years you’ll have .5 * 1.04^100 + .5. The .5 that didn’t grow is insignificant. If only toothpicks production grows eventually the rest will be insignificant as well, but it’ll take a lot longer.
Isn't this a context-dependent issue of substitutability, rather than a question with a single "answer"? If the horse industry stagnates, but automobiles grow at 4%, then the transport sector (eventually) grows at 4%. But if healthcare stagnates, and the music industry grows at 4%, then healthcare will simply take an ever-larger share of GDP compared to music (i.e. Baumol).
I'm extremely skeptical that this is a meaningful question. There are so many modeling choices in the division into sectors, and in what single value is measured for growth. It's simply not useful to project more than a few orders of magnitude without having to rethink your whole system.
If it _WERE_ sane to ask, then of course it converges to 4%, because after a few centuries, that sector (or product, for toothpicks) is the only relevant thing in the economy. This should show the absurdity of the question, not the incorrectness of the answer.
I don't understand...first you're asking us to split the economy in half, then we're only talking about toothpicks (or presumable any single other good)?
Also, Virtual Life isn't free, it costs energy. I understand the point that there's no limits to what a virtual economy might provide, but it's all rooted in physical goods somewhere along the line, no?
If the toothpick sector grows at 4% for ever, the growth rate asymptotically approach 4%. The maths is just obvious.
If the point is that it can not, in practice, grow at 4% for ever, well, yes, of course, but the question was really weirdly formulated if that was the point.
This is an example of why unrealistic thought experiments can be misleading. If toothpick production is growing 4% per year, after a while nearly the whole economy is the toothpick economy. Huge amounts of toothpicks are being produced. Almost everyone works in the toothpick industry.
And business in the toothpick industry is GOOD: it is still growing at 4% per year! How? No idea, it’s an unrealistic example. But that’s what is happening. And so since nearly the whole economy is the toothpick economy, the whole economy is growing at nearly 4% per year. That’s just how things work in the thought experiment as you specified it.
There is nothing subjective about the answer here. It’s 4%. For it not to be, the growth in the toothpick economy has to drop below 4%.
What is growing? Sheer output? Productivity? Something else?
If output is growing, I think I agree with Tyler. At some point they hit the point where everyone has more toothpicks than they want, the price goes to near zero, and further growth in that market becomes relatively insignificant. Maybe aggregate growth does not go quite to zero, but no one cares about the difference. It will not be the case that everyone gets a job in the toothpick industry or has a mountain of toothpicks.
Can one industry grow so consistently without causing growth in the industries that produce its physical capital and raw inputs, or in industries that use its products, or in industries affected by the wealth or substitution effects of its growth on consumers? Does the toothpick industry generate its own power and grow its own trees and tree-substitutes? There is some static thinking dominating this supposedly dynamic thought experiment. The thought experiment is too extreme, it is incoherent. It isn’t possible for only one industry to grow indefinitely. Either its growth will spark growth in other industries, or the stagnation of other industries will stifle its growth.
Could the other industries just ignore the effects of external growth? Doesn’t the thought experiment assume some degree of competition? Suppliers of inputs for the growing industry just raise their prices, but do not increase output? Maybe the growing industry never reduce their prices? The supply and demand curves must look pretty odd.
Is there a good for which even a cornucopia isn’t enough? Maybe we can always use more computing power, to make things faster or more detailed or more whatever we still think we lack. More (clean, side-effectless) energy can always be applied to a number of things. But these would tend to create growth in other sectors, wouldn't they?
Suppose the toothpicks get produced with zero costly inputs. There is still a limit to how many toothpicks the world wants, even if we can get a million for a penny, and ten for the price of one every weekend. At some point they switch from being an economic good to a form of pollution. More is better, until it's not.
But what does this illustrate? Maybe that the numbers don’t mean what we would like them to mean. Certainly that this thought experiment, though interesting, is difficult to apply to anything real.
As others have commented, it seems like this is just math. Because I have too much time on my hands, I created the following Jupyter notebook that illustrates why the growth rate of the entire economy asymptotically approaches the growth rate of the non-stagnate sector.
https://github.com/jstark997/sim-partial-stagnate-econ/blob/master/SimPartialStagnateEconomy.ipynb
I am a software engineer, not an economist, so maybe I am missing something? If so, please let me know. Thanks.
Cowen's pessimistic estimate is only applicable if we assume the demand increases as fast as income for the output of the sector where productivity doesn't improve. To a first approximation this is the actual state of affairs in healthcare.
> But why couldn’t the quality of life in virtual reality grow at 4% for ever?
People who are good at math should understand that exponentials are never sustainable -- they grow too fast. Anything that grows at 4% will outnumber the number of molecules in the universe in a mere 5000 years. Reducing the rate to 1% doesn't help much -- you still run out of matter in universe in under 20000 years. Exponential growth is ALWAYS temporary.
The answer depends on substitutability and satiation.
If everything in the zero growth sector has a perfect substitute in the 4% sector, then the economy eventually grows at 4%. The substitute goods in the 4% sector eventually become cheap enough relative to their 0% counterparts that people switch to them, and the 0% growth sectors disappear.
If all goods in the 4% sector correspond to preferences that can be absolutely satiated, then the economy eventually produces the amount required to satiate the preferences (or get asymptotically close) using a smaller share of land/labour/capital over time. In the toothpick example, toothpicks are eventually produced by one guy who enjoys the smell the toothpick replicator makes. Then, the economy doesn't grow anymore.
If all goods in the 0% sector correspond to preferences that can be absolutely satiated, and the current economy could satiate those preferences, with X fraction of the economy, then eventually the economy converges to a situation where X of peoples budgets are spent on the 0% sector, and the rest of the economy grows at 4%. If X=0.75, then the 4% sector accounts for 25% of the economy, and so living standards rise by 1%. This is the toothpicks grow at 0% situation, with X=0.00001, so growth is almost 4%.
If it would take over 100% of the current economy to satiate preferences for the 0% sector, I think this is qualitatively equivalent to having goods in the 0% sector that can't be satiated, so I'll handle them together. In this case, the 4% goods eventually become so abundant that their marginal utility is very low compared to the 0% sectors. Consumers, therefore, substitute towards 0% goods and the 4% goods eventually make up a smaller portion of the economy, similar to the absolute satiation situation. Eventually, growth in living standards approaches 0.
This is mostly a math problem. But a practical consideration comes to mind. Large things generally quite growing at the rate that got them large. As the US economy continues to grow, it can be expected that the rate of growth will eventually decline. Decline to what? If I knew that, I'd be rich; I'm not rich.
I think it depends a very great deal on how you define "half of the sectors of the economy". And on why the bifurcation occurs.
Do you think your last point on the growth of virtual reality has any bearing on a moral dilemma such as the experience machine one? Should we choose the virtual "fake" world if it's the only one that can keep improving forever?
These problems can, at least to some, be fun and some economists like to do this a lot so they can say,..ha, gotcha! I think some of them really wanted to be magicians, pulling rabbits out of a hat. But many times the questions/problems are vaguely presented. Steven Landsburg is another one who likes to play these games.
"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? " You should have expressed this as "At the BEGINNNING of the growth period, half the sector grows at...etc" otherwise seems to make no sense. Period 0: Z sector 1, Y sector 1. Period 1: Z sector 1.04, Y sector 1 (so Z sector now makes up 1.04/2.04 > 0.50 of the economy). Period 2: Z sector 1.04*1.04 = 1.082, Y sector 1 (Z sector is now 1.082/2.082 even further above 0.5 and so.). 100 years from now Z: 1.04^100= 50.1, Y is 1 (Z sector share 50.1/51.1). At period 0: gdp is 2, 100 years later it is 51.1, so 2 (1+x)^100 = 51.1--> 3.3% growth. Keeping going on, approaches 4%.
But that is impossible for supply and demand reasons. At some point, the goods produced by the growing sectors approach satiation of demand, and the price goes to near zero. No one wants more, even at a lower (positive) price, and output growth must stop.
Maybe if t-shirts end up selling for one cent each we will start using them for something else, or throw them away instead of washing them, but I don’t want more than 365 of them per year. Well, maybe a few more if I change clothes during the day. And that ignores the cost of disposing of them. Anyhow, the point is, for ordinary goods, there is a point where output growth doesn’t matter, and beyond that it becomes a problem to be dealt with.
Maybe productivity growth can continue instead of output growth?
Maybe there are goods like computer power or energy that can create infinite demand, so we always want more without sending the price to zero?
I made a comment below that goes into this further.
You can always come up a demand curve that never hits the horizontal axis (speaking of course in theory). Picture a non-linear demand curve asymptotically approaching horizontal axis. So price can infinitely fall without hitting zero.
Or more directly, we can draw that curve, but is that what we see in real life? Most demand curves eventually cross into negative territory. As noted, energy, computation, maybe a few other things might be exceptions.
Or again seeking realism, when a good gets below a certain price, it becomes something you trade by the carload in the Chicago commodities exchange. If the price goes too low, even that becomes infeasible. It would require some alternative enabling technology that can underprice and outperform the CCE and railroads.
That works as a formalism, but what would it mean in practice? Very one would be living in a pile of toothpicks. Everyone wants 100 classic cars? We don’t have space for them. There is a point for most ordinary goods where there are just too many of them, and no one can sell any. This is probably not true of (clean) energy and maybe a few other things. Maybe if the scenario is that half the economy is growing, these constraints don’t bite. I got distracted by the toothpicks.
Yes, it is theoretical, definitely not meant in literal or practical way. I was responding to Bryan's hypothetical. But to respond to your examples: what if "consumption" of toothpicks is to use them and then burn them to ashes. The ashes then shot into space which as far as we know is infinite. What if consumption of classic cars is viewing them for a year, getting bored, then launching them into space.
Launch them into space, and then buy 100 more? How do you get 4% growth if it's 100 per year every year? I guess you need to buy 104 next year. But even though they only cost one cent each, do you really want to spend four whole cents to get four more cars, when you have 100 every year? If you buy and launch one every day, what makes you want to increase to buying a launching 1.04 per day next year? At some point, it becomes tedious. You have other things to do. And even if you are a 24-hour-a-day car fanatic, that just means you are not consuming more pure-bred dogs and elegant furniture. Please don’t launch those dogs into space.
You can only eat so much in a day (unless you're a Roman, I suppose). It can always be a bit more delicious and a bit more healthy, but as some point you are ready to stop. The demand curve goes negative, temporarily.
In 100 years you’ll have .5 * 1.04^100 + .5. The .5 that didn’t grow is insignificant. If only toothpicks production grows eventually the rest will be insignificant as well, but it’ll take a lot longer.
Isn't this a context-dependent issue of substitutability, rather than a question with a single "answer"? If the horse industry stagnates, but automobiles grow at 4%, then the transport sector (eventually) grows at 4%. But if healthcare stagnates, and the music industry grows at 4%, then healthcare will simply take an ever-larger share of GDP compared to music (i.e. Baumol).