economics//production function

A production function is an equation that gives the output a firm or an economy can produce from given quantities of inputs, such as labour \(L\) and capital \(K\), and it is used to reason about how output responds when one input grows and the others do not. The simplest common form is Cobb-Douglas,

A production function is an equation that gives the output a firm or an economy can produce from given quantities of inputs, such as labour LLL and capital KKK, and it is used to reason about how output responds when one input grows and the others do not. The simplest common form is Cobb-Douglas,

Y=A KαL1−αY = A\,K^{\alpha} L^{1-\alpha}Y=AKαL1−α

in which each input has diminishing returns but neither can ever cap output on its own.

How easily one input stands in for another is measured by the elasticity of substitution, which Cobb-Douglas fixes at 1.

The CES function lets that elasticity take any value, and with it the question of whether running short of one input caps output.

In AI research the inputs are researchers' work and the compute their experiments need, and the shape of the function decides whether more researchers can make up for missing chips (compute bottleneck).