economics//production function//elasticity of substitution

The elasticity of substitution \(\sigma\) measures how easily one input of production can replace another while output stays the same, and it decides whether running short of one input caps output or can be made up with more of the other. Formally it is the percentage change in the ratio of the two inputs for a 1% change in their relative marginal products.

The elasticity of substitution σ\sigmaσ measures how easily one input of production can replace another while output stays the same, and it decides whether running short of one input caps output or can be made up with more of the other. Formally it is the percentage change in the ratio of the two inputs for a 1% change in their relative marginal products.

With σ>1\sigma > 1σ>1 the inputs are substitutes: more of one makes up for less of the other without limit.

With σ<1\sigma < 1σ<1 they are complements: if one input is fixed, output approaches a ceiling however much of the other is added, as with drivers and trucks.

With σ=1\sigma = 1σ=1 (Cobb-Douglas) the case sits between the two.

Estimates for US manufacturing, labour against capital, sit near 0.7, so factories behave as complements. For AI research the inputs are researchers' work and experiment compute, and the only direct estimate (Whitfill and Wu, 2025, on data from four labs) gives 2.58 in one specification and about 0 in another that accounts for the size of frontier experiments: the value that decides whether thinking can replace experiments has estimates on both sides of 1, so the compute bottleneck, the claim that experiments limit AI research, is still undecided.

Elasticity of substitution can be read as the quality of a cheap test.

Where a small experiment predicts what a large one would show, thought substitutes for compute and σ\sigmaσ is high; where behaviour only appears at scale, the only valid check is the large run and σ\sigmaσ falls towards zero.