Is growth linear, not exponential?(rootsofprogress.org)
rootsofprogress.org
Is growth linear, not exponential?
https://rootsofprogress.org/is-growth-linear-not-exponential
78 comments
It's almost certainly exponential, but the rate of growth depends on a number of dynamic factors. Corrupt elites often shut growth down. It happened in China and Japan several times in the second millennium, and it's happening in the US in the third one; we've backslid since about 1970.
All exponential functions in nature are really S-curves. Slow-downs could absolutely be the result of corruption, but could be natural limits of whatever process enabled the initial growth
Yeah there's always economic miracles or Moore's law but for diodes or some shit, something, they all peter out. All of them. No exceptions. All of them.
Moore's law is in the saggy part of the S-curve, it's been morally dead for fourteen years in some ways, eleven years in others, and for eight years transistors flat out don't shrink at all they just pack more of them in the same space. I saw this coming, I did the math with the size of the transistors and the size of the chips and realized 99.5% of the area on a chip was empty back when 14 nanometer was the smallest. That's what new lithographies go for, they fill up that empty space. It ain't over til it's over, but when it is in fact over, it's over. It's over.
[1] South Korea, China, which is the new one? Is there a new one? A lot of the time it's artificial and carefully presented to market the exponential curve shit, mainly by lying about the beginning of the segment to make it look smaller and the end of the segment to make it look bigger. Shoehorning it. Also defining the segment as unique, like giving it an identity so it's meaningful to isolate it from the bigger picture which is not exponential because it has a past and a future. Moore's Law talks about transistors, why not talk about valves instead and look at the history since the nineteenth century, or since Leibniz for that matter? Like you can make a switch out of wood, people did, they transmitted energy by moving staffs of wood back and forth. Cuckoo clocks. Antikythera mechanism. The abacus. Mental arithmetic.
The big thing there is for usury, usurers have to keep the dream of exponential growth alive to charge exponential compound interest without getting tarred and feathered like so many usurers before them.
Moore's law is in the saggy part of the S-curve, it's been morally dead for fourteen years in some ways, eleven years in others, and for eight years transistors flat out don't shrink at all they just pack more of them in the same space. I saw this coming, I did the math with the size of the transistors and the size of the chips and realized 99.5% of the area on a chip was empty back when 14 nanometer was the smallest. That's what new lithographies go for, they fill up that empty space. It ain't over til it's over, but when it is in fact over, it's over. It's over.
[1] South Korea, China, which is the new one? Is there a new one? A lot of the time it's artificial and carefully presented to market the exponential curve shit, mainly by lying about the beginning of the segment to make it look smaller and the end of the segment to make it look bigger. Shoehorning it. Also defining the segment as unique, like giving it an identity so it's meaningful to isolate it from the bigger picture which is not exponential because it has a past and a future. Moore's Law talks about transistors, why not talk about valves instead and look at the history since the nineteenth century, or since Leibniz for that matter? Like you can make a switch out of wood, people did, they transmitted energy by moving staffs of wood back and forth. Cuckoo clocks. Antikythera mechanism. The abacus. Mental arithmetic.
The big thing there is for usury, usurers have to keep the dream of exponential growth alive to charge exponential compound interest without getting tarred and feathered like so many usurers before them.
Exponential compound interest is just linear simple interest with a fixed term loan with auto renewal of non -paid balance.
Why would anyone ever pay back a loan if it had zero additional interest for extending the term?
Why would anyone ever pay back a loan if it had zero additional interest for extending the term?
Why would anyone ever pay back a loan if it had zero additional interest for extending the term?
I have excellent answers to that. Well first off I would, because I pay loans in order from most benevolent to least benevolent. Second I figure out the mathematical theory that generalizes interest, and includes all types of interest as special cases, and compound interest is the special case number infinity, the most morally wrong case.
I have excellent answers to that. Well first off I would, because I pay loans in order from most benevolent to least benevolent. Second I figure out the mathematical theory that generalizes interest, and includes all types of interest as special cases, and compound interest is the special case number infinity, the most morally wrong case.
That is a great rant. Bravo!
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That first sentence could replace the entire paper. Growth functions are inherently/fundamentally exponential (regardless of whether we're talking about bacterial colonies, populations of rabbits, compound interest or ROI), but the forces that act on the exponent change over time. It really is this simple.
Real exponential growth is fairly rare outside mathematical models, it is almost always attenuated by logistical bottlenecks or resource scarcity after some initial phase.
> Growth functions are inherently/fundamentally exponential but the forces that act on the exponent change over time
By this standard, every function is “fundamentally exponential”.
You can plot literally any (strictly positive, differentiable) function on a log scale and find the slope at every point. BAM! Look, an exponent (changing over time)!
By this standard, every function is “fundamentally exponential”.
You can plot literally any (strictly positive, differentiable) function on a log scale and find the slope at every point. BAM! Look, an exponent (changing over time)!
Don't be facetious. In real life, the forces acting on the exponent correspond to real things that are related to the thing we're observing grow.
If you don't have a causal graph explaining why your exponent is changing, then your plot doesn't make sense.
If you don't have a causal graph explaining why your exponent is changing, then your plot doesn't make sense.
Oh, and what proof do you have to offer that those exponentials are not sigmoids?
That's not really true though. Regardless of whether our growth is exponential or logistic, we haven't hit an inflection point yet. So if the past data is better fit by a linear function, the logistic nature of our growth can't be the reason.
If the rate of growth varies dynamically over time, then it is not exponential growth. Exponential growth only varies in proportion to itself.
It's all epicycles of exponents on exponents. The world is a Fourier series.
You can likewise argue that all the world is a bunch of polynomials or whatever based on Taylor series. It's not particularly informative.
> Corrupt elites often shut growth down.
Yeah so that's oncologists for one, suppose I go to Cuba to a real doctor the likes of which cannot be found in America and get treated for cancer. That's precisely what I want is for that doctor to shut the growth of the tumor down, be a corrupt elite to that tumor, tell it the show's over. It's part of my body like the runaway growth thing is part of the economy, but if either growth is harming its surrounding environment, taking up resources or pushing things out, forget it. Tumor repression, oppression and suppression, all three!
Exponential growth is too unstable, it can't go on forever. If it had taken place since before the Middle Ages, or any other dark age, like it would be too stupid everyone would either own or owe a galaxy. It's not just that's too much, it's also that would be a highly stupid premise for a life to be born into. So it's marketed like "hey we'd all be rich" but come on, for there to be credit there has to be debt, no creditors without debtors. In these fantasies they never talk about who'd be the debtor, but here's a hint, between the speaker talking about we are all going to be too rich for sins to matter, and the listeners, it's not the speaker.
Yeah so that's oncologists for one, suppose I go to Cuba to a real doctor the likes of which cannot be found in America and get treated for cancer. That's precisely what I want is for that doctor to shut the growth of the tumor down, be a corrupt elite to that tumor, tell it the show's over. It's part of my body like the runaway growth thing is part of the economy, but if either growth is harming its surrounding environment, taking up resources or pushing things out, forget it. Tumor repression, oppression and suppression, all three!
Exponential growth is too unstable, it can't go on forever. If it had taken place since before the Middle Ages, or any other dark age, like it would be too stupid everyone would either own or owe a galaxy. It's not just that's too much, it's also that would be a highly stupid premise for a life to be born into. So it's marketed like "hey we'd all be rich" but come on, for there to be credit there has to be debt, no creditors without debtors. In these fantasies they never talk about who'd be the debtor, but here's a hint, between the speaker talking about we are all going to be too rich for sins to matter, and the listeners, it's not the speaker.
Exponential growth can't go on forever (according to our current understanding of physics anyway), but we're a long way away from the cap. Plenty of resources in the solar system, after all. I'm not unconcerned that our resources can only grow as fast as our lightcone, but I don't lose much sleep either.
I used to worry about the heat death of the Universe, before I realized life will annihilate itself first.
Eh, we're a hardy weed. Barring a threat like none we've seen thus far, we'll get there.
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Can’t you model almost anything as a piecewise linear function? I don’t know if this claim is saying much of substance.
Yes, but if your data is actually exponential, the linear segments are not going to be better approximations than an exponential curve. That's what's going on here
I'm not sure that's true in general, nor even frequently. In fact, I'd say it's provably false in general.
The big issue is that you get MANY more curve-fitting parameters to play with if you use a piece-wise linear model vs. an exponential model. (You get to choose HOW MANY breaks to make, what the slope is for each section, and WHERE to make the breaks.)
So... Let's say you created some synthetic data using an underlying exponential plus a normally distributed random number. Obviously, the BEST predictive model is an exponential one. However, for any arbitrary number of observations, I guarantee you there's trivially at least one piece-wise linear model that will have less error than the exponential one. Consider the one that is simply a straight line between EVERY point. Obviously that has zero error compared to the exponential model. Yet, it has very little predictive power compared to the exponential model.
Now, that's not what was done here... but there's actually quite a few parameters in the form of where to make the breaks and how many to make. Doesn't seem like a fair comparison.
The big issue is that you get MANY more curve-fitting parameters to play with if you use a piece-wise linear model vs. an exponential model. (You get to choose HOW MANY breaks to make, what the slope is for each section, and WHERE to make the breaks.)
So... Let's say you created some synthetic data using an underlying exponential plus a normally distributed random number. Obviously, the BEST predictive model is an exponential one. However, for any arbitrary number of observations, I guarantee you there's trivially at least one piece-wise linear model that will have less error than the exponential one. Consider the one that is simply a straight line between EVERY point. Obviously that has zero error compared to the exponential model. Yet, it has very little predictive power compared to the exponential model.
Now, that's not what was done here... but there's actually quite a few parameters in the form of where to make the breaks and how many to make. Doesn't seem like a fair comparison.
Good point.
The paper does cross-validate the models, and I am told that cross-validation properly penalizes overfitting with too many parameters… but I don't understand the statistics well enough here.
The paper does cross-validate the models, and I am told that cross-validation properly penalizes overfitting with too many parameters… but I don't understand the statistics well enough here.
For any sampled data you'll get guaranteed 100% fit by making it pieceways constant with N fragments where N=number of data points. It says nothing about the function you sampled, it's just a way to cheat by overfitting.
The dangerous bit is that an exponential curve will also be a fairly good fit for a logistic function that's not yet fully observed.
Yes, and the one thing you expect on a logistic function is arguments about whether it's linear or exponential.
But with the amount of noise in economical data, I don't think is evidence of anything.
But with the amount of noise in economical data, I don't think is evidence of anything.
Every apparent exponential in the real universe must actually be a logistic or some other bounded curve.
True, fair point.
Yeah like the article mentions, they are basically making an analogy to the idea of “punctuated equilibrium” from evolutionary biology. Here’s a good exploration of how punctuated equilibrium works, vs the alternative which is called gradualism.
https://gvpress.com/journals/IJBSBT/vol3_no4/3.pdf
Yeah like the article mentions, they are basically making an analogy to the idea of “punctuated equilibrium” from evolutionary biology. Here’s a good exploration of how punctuated equilibrium works, vs the alternative which is called gradualism.
https://gvpress.com/journals/IJBSBT/vol3_no4/3.pdf
NB: "Note that both of these charts are on a log scale."
Appears to apply to the two preceding linear-scale charts.
Appears to apply to the two preceding linear-scale charts.
With noise and enough segments, the linear functions can certainly fit better.
If you zoom in far enough every curve looks linear.
> Henri Poincaré famously described them as "monsters" and called Weierstrass' work "an outrage against common sense", while Charles Hermite wrote that they were a "lamentable scourge".
I wonder if there is a really long compound German word for "an achievement whose greatness is best measured by the degree to which it disgusts experts in the field."
I wonder if there is a really long compound German word for "an achievement whose greatness is best measured by the degree to which it disgusts experts in the field."
Heh. Reminds me of how gamers manage to find exploits to cheese speed runs while developers react in dismay.
Not a bad description of the history of analysis. Turns out function spaces are absolutely full of gross things that don't quite fit nicely into your theory.
Thank you :)
Wow that’s super cool, didn’t know about this
I’ve heard this too. How is modeling as a piecewise linear function best done in practice?
You can also model almost anything as a piecewise exponential function.
It takes time for an industry to absorb technologies that require changes in behaviour, organization and values. Even longer if multiple industries are involved, general physical infrastructure, institurions, and consumer culture.
Perhaps even related to a human generation, or even living memory (like physics progresses one funeral at a time). Though the inflection points found lack that duration-scale and periodicity.
Just on curve fitting: you need some penalty for each extra line (noisy data can always be "fitted" better to enough lines). I expect the paper has a large section on this issue of statistical significance, but I can't access it.
I do feel it's kind of hard to say if such a pattern is "really" there, where exactly the breaks are, or if it's just a noisy artifact.
Perhaps even related to a human generation, or even living memory (like physics progresses one funeral at a time). Though the inflection points found lack that duration-scale and periodicity.
Just on curve fitting: you need some penalty for each extra line (noisy data can always be "fitted" better to enough lines). I expect the paper has a large section on this issue of statistical significance, but I can't access it.
I do feel it's kind of hard to say if such a pattern is "really" there, where exactly the breaks are, or if it's just a noisy artifact.
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This idea was originally discussed by Alexey Guzey back in 2021: https://guzey.com/economics/bloom/#bloom-et-al-appear-to-not...
Perhaps there's something I'm missing, but it's weird that he isn't getting any credit for it.
Perhaps there's something I'm missing, but it's weird that he isn't getting any credit for it.
That blog post is cited in the paper the article discusses.
> To demonstrate this, the two models are subjected to various statistical tests on multiple data sets, mostly 20th-century, from the US and about two dozen other countries. In a later section, the models are tested on European data from 1600–1914. The linear model outperforms on pretty much every test
Why not just use minimum description length?
Why not just use minimum description length?
The whole discussion seems to center on the TFP index. I'm not familiar with this, but Wikipedia says "it's also called multi-factor productivity", and it's a ratio of GDP to the "weighted geometric average of labour and capital input". Asking "is growth linear?" is a very different question than "is output/input linear?". Of course it's reasonable for input to grow exponentially as well as output, so yes, the ratio should be linear, which is not inconsistent with just the output being exponential.
And, I can't say without more reading more, what might be hidden by that weighting? Zooming into an exponential enough will make it look linear.
And, I can't say without more reading more, what might be hidden by that weighting? Zooming into an exponential enough will make it look linear.
So sometimes it's quadratic, once in a while cubic. Never exponential. Never, never exponential. Rid that of your mind! If there were infinitely many dimensions, exponential growth would be possible. There are not, so it's impossible.
So cells. Cells do not multiply exponentially. DNA does not move faster than the speed of light! Nothing in the cell does! It may look exponential but actually that's cubic, they're easily confused, along with a shitty excel library, and bad measurement, measuring it on the small side early on. And what else? A bad education, being told exponential growth is real.
So cells. Cells do not multiply exponentially. DNA does not move faster than the speed of light! Nothing in the cell does! It may look exponential but actually that's cubic, they're easily confused, along with a shitty excel library, and bad measurement, measuring it on the small side early on. And what else? A bad education, being told exponential growth is real.
But bunnies in Australia.
Yeah! Bunnies yeah, I had a math book when I was a kid and they showed Fibonacci numbers with bunnies, well actually a pervy old two-year rabbit with a sweet innocent one-year bunny, there has to be an age difference to set up the Fibonacci numbers. And that's still exponential growth, just with a lower base.
I bet if you actually had bunnies you could say fuck it they double every season and deal with them on that basis. You'd be wrong...but by how much, like get real? Plus the bunnies have the INTENTION of doubling, they each WANT to have lots of bunnies per season. I think that's the crux of it, despite the impossibility nature's program is exponential.
I bet if you actually had bunnies you could say fuck it they double every season and deal with them on that basis. You'd be wrong...but by how much, like get real? Plus the bunnies have the INTENTION of doubling, they each WANT to have lots of bunnies per season. I think that's the crux of it, despite the impossibility nature's program is exponential.
Economic expansion debt growth is exponential. However power increase / global warming puts limits on linear power expansion.
These two does not match economy growth vs global warming.
These two does not match economy growth vs global warming.
The last time i did a similar fit with world GDP between 1920ish and now it came out as O(x^a) where a was between 4 and 5. Actually i fitted the ODE dx/dt = \beta*x^\alpha to data, the fitted looked nice. I wish i wrote a blog post about it.
There's an old joke that all growth looks exponential when plotted on log-log paper with a fat marker.
And perhaps there is a new linear section starting with the computer revolution?
Is this new? I remember reading an article a long time ago (maybe 15 years?) discussing how growth in aircraft speed was overall exponential, but linear for each new technology that was introduced.
It would be a change to macro models of economic growth, if accepted. The default is that TFP grows exponentially. There's also a useful comment by Marginal Revolution here (https://marginalrevolution.com/marginalrevolution/2022/04/ad...) where he points out that TFP itself is really just a residual, and perhaps not a very well-defined concept. That is, if you regress
Y = AK^beta L^(1-beta)
with K being the amount of capital and L the amount of labour, A is just the unexplained "scaling factor" and it gets called TFP. But what TFP actually is... is a bit of a whatever-you-like.
Y = AK^beta L^(1-beta)
with K being the amount of capital and L the amount of labour, A is just the unexplained "scaling factor" and it gets called TFP. But what TFP actually is... is a bit of a whatever-you-like.
Why is TFP assumed exponential by default? What's the theory behind that? A time lag in the transfer between exponential input growth (e.g. population) and exponential output growth?
No. The inputs are labour and capital, so that's already taken care of.
One answer might be "TFP reflects technical knowledge; the more knowledge there is, the easier it is to generate new knowledge". But you'd have to get into growth macro for the details. Indeed, Romer is the person.
One answer might be "TFP reflects technical knowledge; the more knowledge there is, the easier it is to generate new knowledge". But you'd have to get into growth macro for the details. Indeed, Romer is the person.
It's not always assumed to be exponential. It was assumed exponential by Paul Romer because long-term economic growth is pretty exponential. Much subsequent work has followed that model, but not all of it.
But TFP is not output. Exponential output growth can come directly from exponential input growth, which would be a trivial obvervation.
> long-term economic growth is pretty exponential
Yeah, that graph with the largest time scale on the article strongly disagrees.
Yeah, that graph with the largest time scale on the article strongly disagrees.
You can get an exponential growth if the time until the next technology decreases linearly with each technology.
For example the time until the next technology is 1/n, then the technologies per unit time increase exponentially.
This is true within a constant which is https://wikipedia.org/wiki/Euler%27s_constant
A justification for maybe why this is the right model is that the previous n technologies all help make the next technology possible in time ~1/n because they (n existing technologies) all contribute to innovating the next technology.
For example the time until the next technology is 1/n, then the technologies per unit time increase exponentially.
This is true within a constant which is https://wikipedia.org/wiki/Euler%27s_constant
A justification for maybe why this is the right model is that the previous n technologies all help make the next technology possible in time ~1/n because they (n existing technologies) all contribute to innovating the next technology.
Aircraft speed maxed out a long time ago. "All the aircraft which went significantly over Mach 3 are now in museums." - Scott Crossfield, former X-15 test pilot.
Well any exponential growth curve in the real world is actually a logistic growth curve in disguise. A civilization would have to go from Kardashev II to Kardashev III in about the same time as it went from I to II to maintain exponential energy growth.
As an aside, the Falcon 9 flew 30 times last year and goes significantly over Mach 3.
As an aside, the Falcon 9 flew 30 times last year and goes significantly over Mach 3.
How is aircraft limited to the speed of sound still?
they're not? it just turns out that the most efficient cruising airspeed for passenger jets is in the upper end of the subsonic range. most people can't afford supersonic travel, and the ones that can overwhelming prefer to spend the extra money on a more comfortable subsonic flight.
interestingly, extremely high speeds turn out not to be that useful for military aircraft either. the fastest production fighter jet of all time, the mig-25, was introduced in 1970. the f-22 has a considerably lower top speed, and it's not because they lacked the funding to make it go faster.
interestingly, extremely high speeds turn out not to be that useful for military aircraft either. the fastest production fighter jet of all time, the mig-25, was introduced in 1970. the f-22 has a considerably lower top speed, and it's not because they lacked the funding to make it go faster.
Also, we don't really build dedicated interceptors anymore; the F-106 is the last US interceptor I can think of.
The F-15 was designed primarily as an air-superiority fighter (and in fact misinterpretation of the Mig-25 design was motivation for increasing its maneuverability; the Mig-25 was a dedicated interceptor).
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In general the Soviets seemed to really like interceptors, the MiG-31 is arguably faster than the Mig-25; altitude and willingness to destroy the engines affect the calculation.
The F-15 was designed primarily as an air-superiority fighter (and in fact misinterpretation of the Mig-25 design was motivation for increasing its maneuverability; the Mig-25 was a dedicated interceptor).
[edit]
In general the Soviets seemed to really like interceptors, the MiG-31 is arguably faster than the Mig-25; altitude and willingness to destroy the engines affect the calculation.
This is largely just military doctrine. Stealth was determined to be a better bet than speed against SAMs and other jets. With improved signal processing techniques and better standoff weapon, speed may be the better bet.
Adding detail to leetcrew's good reply: the energy of an aircraft in flight is proportional to the square of its velocity. Given steady progress in plane technology you'd expect practical top speed to be limited by cost of energy before too long. That happened more than thirty years ago.
Unless we also had progress in energy technology that could overcome this, which unfortunately we haven't.
You can't make something cheap enough to overcome quadratic growth in demand for very long.
Not even CPU cycles.
Not even CPU cycles.
"Growth" isn't necessarily linear nor exponential. It's just a word for when things get bigger -- the rate at which something grows depends on how it grows.
Exponential-growth occurs when each unit of the growing-thing grows at a continuous rate. For example, if Alice invests $100 in a continuously-compounding bond, then keep re-investing the yields into more of the same bonds, then that'ld tend to be an exponential-growth process.
Linear-growth occurs when the growing-thing is produced at a regular rate. For example, if Bob keep making widgets, then the growth-rate of Bob's widget-pile would tend to be linear.
Anyway, apparently [this paper (2020) [PDF]](https://web.stanford.edu/~chadj/IdeaPF.pdf ) had its Equation-(1) basically parse to:
> dA/dt / A = alpha * S
, where "A" would be "ideas" (which seems vaguely defined), "t" is time, "alpha" is a constant-proportionality-factor, and "S" is an amount-of-scientists (who presumably generate the "ideas").
This equation is for an exponential-growth model. For example, if we reduce it to "dA/dt = k * A" (where "k" is a constant for alpha*S, to make this easier on WolframAlpha), then [the solution is an exponential-function](https://www.wolframalpha.com/input?i=dA%2Fdt+%3D+k+*+A ).
By contrast, it'd have been a linear-function if the authors instead assumed
> dA/dt = alpha * S
... this is, no "/ A" on the left-hand-side.
Anyway, a lot of comments on this thread seem to claim that any (first-order continuously-differential) function is approximately linear if we zoom in enough. Which, yup! -- we can look at both the linear-function and exponential-function as linear-functions by zooming in. So let's do that!
Basically, we can compare:
1. dA/dt = alpha * S (the linear-case)
2. dA/dt = alpha * S * A (the exponential-case)
where "dA/dt" is basically the rate at which "ideas" are generated, and then the right-hand-side of both equations is the marginal-rate (or instantaneous-rate), which is basically the slope of the linear-function that we'd see if we zoomed in enough on both functions such that they both appear (at least approximately) linear.
Practically speaking, we can ignore "alpha". It's basically just a fit-constant to be solved for. Then both equations also have "S", which is basically the amount of scientists who're working.
The big difference is that the exponential-case (which the 2020-paper linked above assumed) also includes a factor of "A" -- this is, the ideas. So, does it follow that "ideas" multiply how fast scientists produce more "ideas"? For example, if a scientist is working in a society that has 100 times more "ideas", then would that scientist produce new "ideas" 100 times faster?
If YES, then the exponential-form would seem appropriate. But if NO, then the linear-form would seem appropriate.
---
EDIT: Skimming a few more sources, it looks like various folks may be trying to use the same equations/data/terminology, possibly for different things?
In the above-comment, I was mostly trying to comment on the basic-model that seemed to be presented in [this paper (2020) [PDF]](https://web.stanford.edu/~chadj/IdeaPF.pdf ), which the linked-article seems to be in-response-to.
However, it's unclear if the definitions cited, including of the variable "A", were necessarily representative of their usage elsewhere.
That said, [the linked-article's paper [PDF]](https://pages.stern.nyu.edu/~tphilipp/papers/AddGrowth_macro... ) starts its Section-5, "Conclusion", with:
> TFP growth is not exponential. New ideas add to our stock of knowledge; they do not multiply it.
, which seems to be in-line with the above-comment's interpretation from the other-paper.
Exponential-growth occurs when each unit of the growing-thing grows at a continuous rate. For example, if Alice invests $100 in a continuously-compounding bond, then keep re-investing the yields into more of the same bonds, then that'ld tend to be an exponential-growth process.
Linear-growth occurs when the growing-thing is produced at a regular rate. For example, if Bob keep making widgets, then the growth-rate of Bob's widget-pile would tend to be linear.
Anyway, apparently [this paper (2020) [PDF]](https://web.stanford.edu/~chadj/IdeaPF.pdf ) had its Equation-(1) basically parse to:
> dA/dt / A = alpha * S
, where "A" would be "ideas" (which seems vaguely defined), "t" is time, "alpha" is a constant-proportionality-factor, and "S" is an amount-of-scientists (who presumably generate the "ideas").
This equation is for an exponential-growth model. For example, if we reduce it to "dA/dt = k * A" (where "k" is a constant for alpha*S, to make this easier on WolframAlpha), then [the solution is an exponential-function](https://www.wolframalpha.com/input?i=dA%2Fdt+%3D+k+*+A ).
By contrast, it'd have been a linear-function if the authors instead assumed
> dA/dt = alpha * S
... this is, no "/ A" on the left-hand-side.
Anyway, a lot of comments on this thread seem to claim that any (first-order continuously-differential) function is approximately linear if we zoom in enough. Which, yup! -- we can look at both the linear-function and exponential-function as linear-functions by zooming in. So let's do that!
Basically, we can compare:
1. dA/dt = alpha * S (the linear-case)
2. dA/dt = alpha * S * A (the exponential-case)
where "dA/dt" is basically the rate at which "ideas" are generated, and then the right-hand-side of both equations is the marginal-rate (or instantaneous-rate), which is basically the slope of the linear-function that we'd see if we zoomed in enough on both functions such that they both appear (at least approximately) linear.
Practically speaking, we can ignore "alpha". It's basically just a fit-constant to be solved for. Then both equations also have "S", which is basically the amount of scientists who're working.
The big difference is that the exponential-case (which the 2020-paper linked above assumed) also includes a factor of "A" -- this is, the ideas. So, does it follow that "ideas" multiply how fast scientists produce more "ideas"? For example, if a scientist is working in a society that has 100 times more "ideas", then would that scientist produce new "ideas" 100 times faster?
If YES, then the exponential-form would seem appropriate. But if NO, then the linear-form would seem appropriate.
---
EDIT: Skimming a few more sources, it looks like various folks may be trying to use the same equations/data/terminology, possibly for different things?
In the above-comment, I was mostly trying to comment on the basic-model that seemed to be presented in [this paper (2020) [PDF]](https://web.stanford.edu/~chadj/IdeaPF.pdf ), which the linked-article seems to be in-response-to.
However, it's unclear if the definitions cited, including of the variable "A", were necessarily representative of their usage elsewhere.
That said, [the linked-article's paper [PDF]](https://pages.stern.nyu.edu/~tphilipp/papers/AddGrowth_macro... ) starts its Section-5, "Conclusion", with:
> TFP growth is not exponential. New ideas add to our stock of knowledge; they do not multiply it.
, which seems to be in-line with the above-comment's interpretation from the other-paper.
If you like this do yourself a favor and grab a copy of Vaclav Smil's "Growth - From Microorganisms to Megacities" https://mitpress.mit.edu/books/growth