Is growth linear, not exponential?

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.

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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.

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.