economics//production function//CES function

A CES (constant elasticity of substitution) function is a production function that combines inputs with one parameter fixing how substitutable they are, and it is the standard form for asking whether a scarce input caps output. For labour \(L\) and capital \(K\),

A CES (constant elasticity of substitution) function is a production function that combines inputs with one parameter fixing how substitutable they are, and it is the standard form for asking whether a scarce input caps output. For labour LLL and capital KKK,

Y=[αLρ+(1−α)Kρ]1/ρ,σ=11−ρY = \left[\alpha L^{\rho} + (1-\alpha) K^{\rho}\right]^{1/\rho}, \qquad \sigma = \frac{1}{1-\rho}Y=[αLρ+(1−α)Kρ]1/ρ,σ=1−ρ1​

where σ\sigmaσ is the elasticity of substitution. When ρ→0\rho \to 0ρ→0 it becomes Cobb-Douglas, and when ρ→−∞\rho \to -\inftyρ→−∞ the inputs become perfect complements and output is set by whichever is scarcer.

With ρ<0\rho < 0ρ<0 (σ<1\sigma < 1σ<1) and capital fixed, adding labour without limit sends LρL^{\rho}Lρ to zero, and output approaches (1−α)1/ρK(1-\alpha)^{1/\rho} K(1−α)1/ρK: a hard ceiling. Applied to AI research with factory values of σ\sigmaσ, this ceiling caps the speed of software progress at a few times today's, which is the core of the compute-bottleneck argument.

The form is fitted on data where the ratio of labour to capital varies little, about twofold in factories, so applying it to an automated lab with orders of magnitude more research labour per chip is an extrapolation far outside the data.