10 ms·
Is growth linear, not exponential?
- apolloartemis 4y agoCan’t you model almost anything as a piecewise linear function? I don’t know if this claim is saying much of substance.
- jasoncrawford 4y agoYes, 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
- apolloartemis 4y agoTrue, 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 https://gvpress.com/journals/IJBSBT/vol3_no4/3.pdf
- adamsmith143 4y agoIf you zoom in far enough every curve looks linear.
- mathieutd 4y agoNot quite! https://en.wikipedia.org/wiki/Weierstrass_function https://en.wikipedia.org/wiki/Weierstrass_function
- apolloartemis 4y agoWow that’s super cool, didn’t know about this
- foobarian 4y agoHeh. Reminds me of how gamers manage to find exploits to cheese speed runs while developers react in dismay.
- wardedVibe 4y agoNot 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.
- AlchemistCamp 4y agoThank you :)
- bee_rider 4y ago> 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."
- dredmorbius 4y agoNB: "Note that both of these charts are on a log scale." Appears to apply to the two preceding linear-scale charts.
- Fomite 4y agoThe dangerous bit is that an exponential curve will also be a fairly good fit for a logistic function that's not yet fully observed.
- LeegleechN 4y agoEvery apparent exponential in the real universe must actually be a logistic or some other bounded curve.
- marcosdumay 4y agoYes, 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.
- Enginerrrd 4y agoI'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.
- jasoncrawford 4y agoGood 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.
- ajuc 4y agoFor 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.
- marcosdumay 4y agoWith noise and enough segments, the linear functions can certainly fit better.
- melling 4y agoI’ve heard this too. How is modeling as a piecewise linear function best done in practice?
- adrianN 4y agoYou can also model almost anything as a piecewise exponential function.
- aidenn0 4y agoIs 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.
- dash2 4y agoIt 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/additive-growth.html 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.
- pishpash 4y agoWhy 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?
- jasoncrawford 4y agoIt'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.
- marcosdumay 4y ago> long-term economic growth is pretty exponential Yeah, that graph with the largest time scale on the article strongly disagrees.
- pishpash 4y ago
- georgewsinger 4y agoThis idea was originally discussed by Alexey Guzey back in 2021: https://guzey.com/economics/bloom/#bloom-et-al-appear-to-not-realize-that-most-of-the-data-they-analyze-in-the-paper-including-the-us-tfp-does-not-exhibit-exponential-growth 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.
- spekcular 4y agoThat blog post is cited in the paper the article discusses.
- carapace 4y agoIf 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 https://mitpress.mit.edu/books/growth
- robocat 4y agoAnd perhaps there is a new linear section starting with the computer revolution?
- 21723 4y agoIt'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.
- epgui 4y agoThat 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.
- jasonwatkinspdx 4y agoOh, and what proof do you have to offer that those exponentials are not sigmoids?
- marginalia_nu 4y agoReal exponential growth is fairly rare outside mathematical models, it is almost always attenuated by logistical bottlenecks or resource scarcity after some initial phase.
- ravi-delia 4y agoThat'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.
- deleted 4y ago[deleted]
- Xcelerate 4y ago> 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?
- hyperpallium2 4y agoIt 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.
- daniel-cussen 4y agoSo 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.
- gkop 4y agoBut bunnies in Australia.
- daniel-cussen 4y agoYeah! 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.
- alanbernstein 4y agoThe 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.
- _Nat_ 4y ago"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 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 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 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.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.
- acd 4y agoEconomic 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.
- freemint 4y agoThe 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.
- brandmeyer 4y agoThere's an old joke that all growth looks exponential when plotted on log-log paper with a fat marker.