"<i>Growth</i>" 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.<p>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.<p>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.<p>Anyway, apparently [this paper (2020) [PDF]](<a href="https://web.stanford.edu/~chadj/IdeaPF.pdf" rel="nofollow">https://web.stanford.edu/~chadj/IdeaPF.pdf</a> ) had its Equation-(1) basically parse to:<p>> dA/dt / A = alpha * S<p>, where "<i>A</i>" would be "<i>ideas</i>" (which seems vaguely defined), "<i>t</i>" is time, "<i>alpha</i>" is a constant-proportionality-factor, and "<i>S</i>" is an amount-of-scientists (who presumably generate the "<i>ideas</i>").<p>This equation is for an exponential-growth model. For example, if we reduce it to "<i>dA/dt = k * A</i>" (where "<i>k</i>" is a constant for alpha*S, to make this easier on WolframAlpha), then [the solution is an exponential-function](<a href="https://www.wolframalpha.com/input?i=dA%2Fdt+%3D+k+*+A" rel="nofollow">https://www.wolframalpha.com/input?i=dA%2Fdt+%3D+k+*+A</a> ).<p>By contrast, it'd have been a linear-function if the authors instead assumed<p>> dA/dt = alpha * S<p>... this is, no "/ A" on the left-hand-side.<p>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!<p>Basically, we can compare:<p>1. dA/dt = alpha * S (the linear-case)<p>2. dA/dt = alpha * S * A (the exponential-case)<p>where "<i>dA/dt</i>" is basically the rate at which "<i>ideas</i>" 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.<p>Practically speaking, we can ignore "<i>alpha</i>". It's basically just a fit-constant to be solved for. Then both equations also have "<i>S</i>", which is basically the amount of scientists who're working.<p>The big difference is that the exponential-case (which the 2020-paper linked above assumed) also includes a factor of "<i>A</i>" -- this is, the ideas. So, does it follow that "<i>ideas</i>" multiply how fast scientists produce more "<i>ideas</i>"? For example, if a scientist is working in a society that has 100 times more "<i>ideas</i>", then would that scientist produce new "<i>ideas</i>" 100 times faster?<p>If <i>YES</i>, then the exponential-form would seem appropriate. But if <i>NO</i>, then the linear-form would seem appropriate.<p>---<p>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?<p>In the above-comment, I was mostly trying to comment on the basic-model that seemed to be presented in [this paper (2020) [PDF]](<a href="https://web.stanford.edu/~chadj/IdeaPF.pdf" rel="nofollow">https://web.stanford.edu/~chadj/IdeaPF.pdf</a> ), which the linked-article seems to be in-response-to.<p>However, it's unclear if the definitions cited, including of the variable "<i>A</i>", were necessarily representative of their usage elsewhere.<p>That said, [the linked-article's paper [PDF]](<a href="https://pages.stern.nyu.edu/~tphilipp/papers/AddGrowth_macro.pdf" rel="nofollow">https://pages.stern.nyu.edu/~tphilipp/papers/AddGrowth_macro...</a> ) starts its Section-5, "<i>Conclusion</i>", with:<p>> TFP growth is not exponential. New ideas add to our stock of knowledge; they do not multiply it.<p>, which seems to be in-line with the above-comment's interpretation from the other-paper.