Even though the number of product categories is relevant to reconstructing GDP from deconstructed GDP, the number of product categories available is not Quantity in this equation, it's n.
That's why you're not going to get "annual sales of Schmartphones would 60x over the time period, while maintaining constant GDP share" when the rest of the economy is growing at 0%. And why "The growth experienced by consumers however would be linear by any common sense understanding of the situation" is not true: the growth in the number of categories might be linear, but consumers would be buying exponentially more product value (quantity * price).
On the product categories vs quantity: I agree, I intended Schmartphones to be a new category in the post, apologies if that wasn't clear[1].
That's why you're not going to get "annual sales of Schmartphones would 60x over the time period, while maintaining constant GDP share" when the rest of the economy is growing at 0%
I believe this actually is possible, and works in my example, though I accept that it's not very realistic and the path would look more like the "Trajectory of a newly invented good" graph.
The point is that the GDP shares of the other categories can change while still growing at 0%, because growth is just based on the quantity of goods delivered. Going through the example again with more explicit numbers (and changing growth to 100% YoY to simplify it):
t = 0: There are n=9 types of goods, 1000 units are sold of each, And they all have the same price of $1, so each gets 1/9th GDP share. New good has been invented but has 0% GDP share so far.t = 1: The new type of good sells 2 units, at $500 each, so it goes straight to GDP share parity with all the other goods, and each now have 1/(9+1) = 1/10 GDP share. But each of the other categories still sells 1000 units, so they are all at 0% growth, and per thet = 2: The new type of code sells 4 units, at $250. It maintains its 1/10 GDP share, grows by 100%, and adds 10% to overall GDP growth. The other types of goods still sell 1000 units each and contribute nothing to GDP growth.^perhaps an important point glossed over in the post is that the price of the new good has to go down by the inverse of the growth rate. And as mentioned briefly in the footnote, I think the simplest way to imagine this type of economy is that people work more hours to provide the new type of good, while otherwise changing nothing about what they do for the other goods.
Sidenote: This setup would keep nominal GDP constant, and the economy overall would be judged as being in steep deflation because you can buy more and more "product value" in the form of the new good, for the same nominal dollars.
the growth in the number of categories might be linear, but consumers would be buying exponentially more product value (quantity * price).
I think this gets to the crux of it, and is the main point I'm trying to make: Using either fixed base year accounting or chain weighting, the value of a new product essentially gets baked into GDP at the [price when it is introduced] x [quantity when it is mature]. If the price at the time of introduction is proportional to GDP at that time (say, people always spend 1% of their income on the new gizmo, or 1% of people have a ton of disposable income and are price insensitive), then a linear flow of new goods will translate into an exponential increase in GDP.
Standard economics interprets this as an exponential increase in product value over time. This is reasonable, and I think this assumption is well understood by economists going through this calculation.
The main reason I think this is an interesting observation is that there is some laundering that happens between the calculation and the practical conception of GDP. People (including economists) often conflate exponential growth in GDP with exponential growth in ability to have impact on the world. For instance "if the economy keeps doubling every 30 years, we need to have tiled the light cone or hard plateaued within a few hundred years". But the calculation supports this continued exponential doubling with only a linear stream of new goods, and these goods may not be of the type that let you tile the light cone.
Whether this mechanism explains a large or small fraction of measured GDP growth over the last ~100 years is an empirical question. I'm not claiming that the exponential growth comes from linear introduction of new goods, just that the accounting + a plausible trajectory for new goods permits this.
Also: I am aware that the way categories are counted practically means that a new good would probably fit into an existing category (e.g. the 71 the US uses) rather than creating a whole new one. I think this still works because within categories the same logic applies
In an interview, Philip Trammell presented a model of the economy where growth is dominated by “mid-life” technologies, which are mature enough to have a meaningful share of GDP, but are still growing rapidly:
— Philip Trammell, Epoch After Hours, around 00:56:00
For context, GDP growth is calculated like this:
This is a sum over types of good[1] (e.g. motor vehicles & parts, chemical products).
The idea is that for new goods, growth is high but GDP share is low. For mature goods, GDP share may be high, but growth is flat because demand is satiated[2]. As a result, the growth number is dominated by these mid-life goods. Some examples of goods that fit each of these stages from recent years:
As stated this is just a toy model that bears some resemblance to the real economy. As far as I can find there has been no quantitative study evaluating this. The point of this post is to show that if you work this through, it can easily produce exponential GDP growth from linear growth in the number of goods available, without these goods being particularly exceptional. For this and other reasons I’ve become convinced that the way GDP (growth) is calculated makes it quite far from the common sense conception of it as “amount of stuff you can buy” or “amount of impact you can have on the world”.
Example: A linear increase in number of goods produces exponential GDP growth
A scenario that is a plausible description of the real economy generates this result. At
t = 0, suppose the GDP share is:In this economy growth is 0% overall. Suppose now that a new good type is invented, the Schmartphone, which follows this trajectory:
t = 0: It is invented, has a high price and fast growth among early adopters, but still ~0% GDP sharet = 1tot = 10: Mid-life phase: It is adopted by more people as they learn about it. Plus, further development means the price comes down and/or the product quality increases, driving demand from people previously on the fence. During this phase it has moderate GDP share (say 10%[3]), and still high growth (say 50% YoY)t = 10onwards: Saturation and commodification: Eventually everyone who wants one has one and replacements are sold at a constant rate, so growth falls to 0% YoY. You can also suppose that continued innovation + amortising the initial investment drives the price down until Schmartphones are <1% of GDPFrom
t = 1tot = 10this economy had 5% growth overall. (50% growth) * (10% GDP share) for Schmartphones, (0% growth) * (90% GDP share) for everything else. Over the 10 years, the addition of this new good would ~1.6x overall GDP. 50% YoY growth for 10 years means that annual sales of Schmartphones would 60x over the time period, while maintaining constant GDP share. This is not crazy, but obviously for the real economy you’d need slightly lower numbers to be realistic (and the real peak GDP share of smartphones was more like 1%).Now suppose at
t = 10another new good type is invented (the Schmair Fryer), and goes through the same cycle: Early adoption, mid-life high growth high GDP share phase, saturation and merging into the mass of “other mature goods”. Again fromt = 11tot = 20the economy would have 5% growth, 1.6x-ing again byt = 20.You can repeat this indefinitely, and after 5 new goods GDP would be 11.5x its original size. After 10 it would be 132x, and so on.
The growth experienced by consumers however would be linear by any common sense understanding of the situation. At
t = 0, you work all day to spend 60% of your income on necessities, and 40% on several of say 100 “other mature goods”. Att = 100you have 110 to choose from, but GDP has apparently increased by 132x. You do buy and enjoy these new goods (they must be good to have accounted for 10% of GDP during their mid-life phase), but it’s not the same as being able to buy 132x what you could att = 0, and your real economic constraint of buying necessities is unaffected.Relationship to the real economy
As mentioned above, it seems that there’s no existing research that can definitively say whether this is a description of the real economy or not, so I’m keeping my cached view as “individual economic power grows exponentially over time” for now.
James Bessen’s ”AI and Jobs: The Role of Demand” documents this hump-shaped pattern empirically for textiles, steel, and automotive over 200 years of data. Each shows a rise in employment and expenditure share during its growth phase, followed by a decline as the industry matures.
I tried to do a similar exercise with recent technologies. Smartphones fit the pattern nicely: explosive growth from 2007-2015, peak GDP share around 1-1.5%, now essentially flat. Cloud computing looks mid-hump right now at ~0.6% of GDP and 20% growth. EVs similarly at ~0.5% of GDP and 20% growth. Personal computers are clearly post-hump, having peaked around 2010. These are very rough so I think it’s not particularly worth including the sources. To make the case strongly, you would need to categorise all sources of GDP growth and fit them into this framework.
Furthermore there’s a question of whether this exponential-from-linear effect can apply to other methods for calculating GDP, as the “chain weighting” method above was introduced fairly recently (1996 in the US). I’m not sure about this. From spending a few minutes thinking about it, I think a similar argument applies to “fixed base year” GDP calculations. In that case a new good is counted going forward at whatever price it entered the market at. Two things would need to be true to produce a result that is similar to the one above, with a similar newly-invented good:
Bonus example: No increase in number of goods produces exponential GDP growth
Here’s a more extreme version that doesn’t even require new goods to be invented. I don’t believe this looks that much like the real economy, but it’s nice to illustrate the point that GDP growth can become divorced from common sense. Suppose you have one good that goes in and out of fashion, in an economy where everything else is static (0% growth, though GDP shares can move around to make room for this fashionable good):
Over the full cycle, the good’s quantity and GDP share are back where they started, but GDP has grown by ~9.5%. The expansion happened at high share (10%) and the contraction at low share (1%), so they don’t cancel out. You can repeat this indefinitely for persistent GDP growth from a fixed number of goods with no net change in quantities. I think this maps onto what is called “Chain drift” in the economics literature.
Actually I lied, only 30% of the GDP calculation in the US is from countable goods. Another 50% is from services where there is an agreed price index, the logic of this post goes through for that case as well (as long as new services are invented, even if they fit within an existing category). The other 20% is services with no agreed price index, in which case some kind of input-based accounting is used. I’m not sure if the post’s logic works there.
Additionally, further innovation or amortisation of earlier investments will tend to just drive the price down, which will also lower GDP share. Though in practice we see some mature goods (housing) that have persistently high GDP share and some which have it frittered away over time (televisions).
This requires the GDP share of our “necessities” + “mature innovated goods” to shrink to 90% during this period. In a more continuous case you can imagine that there is a gradual price reduction among “mature innovated goods” which makes room for this (but no growth in the number of these goods due to saturated demand, so still 0% GDP growth from this). It could also be due to people temporarily working more hours to create the new type of good.