economics//economic growth//idea production function

An idea production function is an equation that gives the rate at which new technology is produced as a function of research effort and of the technology already accumulated, and it is used to ask whether research is getting harder and whether progress can accelerate. A common form is

An idea production function is an equation that gives the rate at which new technology is produced as a function of research effort and of the technology already accumulated, and it is used to ask whether research is getting harder and whether progress can accelerate. A common form is

dAdt=δ LλA1−β\frac{dA}{dt} = \delta\, L^{\lambda} A^{1-\beta}dtdA​=δLλA1−β

where AAA is the level of technology, LLL the research effort, λ≤1\lambda \le 1λ≤1 captures duplicated work (researchers stepping on each other's toes) and β>0\beta > 0β>0 how fast ideas get harder to find.

The best-known measurement is on chips. Bloom, Jones, Van Reenen and Webb (2020) found that doubling chip density now takes more than eighteen times the researchers it took in the early 1970s; Moore's law held only because the research effort kept growing. They estimate β≈0.2\beta \approx 0.2β≈0.2 for chips against 3.4 for the economy as a whole.

If research is done by software, the exponent decides everything.

With compute fixed, the number of AI researchers grows with AAA itself (L=cAL = cAL=cA), the equation becomes dA/dt=δcλA1+λ−βdA/dt = \delta c^{\lambda} A^{1+\lambda-\beta}dA/dt=δcλA1+λ−β, and progress accelerates when λ>β\lambda > \betaλ>β, equivalently when r=λ/β>1r = \lambda/\beta > 1r=λ/β>1.

rrr is the number of times technology doubles each time cumulative research effort doubles. Published estimates for AI software straddle 1, so the equation maps the possible regimes without saying which one holds. Forecasts of AI progress need it beside scaling laws: these relate a model's loss to the compute spent training it, the idea production function relates the rate of new ideas to research effort.

A fixed number of researchers with β>0\beta > 0β>0 gives growth that slows down, which is the basis of semi-endogenous growth.