Cobb-Douglas Production Function

The Cobb-Douglas production function, introduced by Charles Cobb and Paul Douglas in 1928, models how capital and labor combine to produce output. It remains one of the most widely used specifications in macroeconomics and microeconomics.

The function:

Y = A x K^alpha x L^(1-alpha)

where Y is output, A is total factor productivity (TFP), K is capital input, L is labor input, and alpha (typically 0.33 for developed economies) is capital’s share of income.

Marginal products:

MPK = alpha x Y / K (marginal product of capital) MPL = (1-alpha) x Y / L (marginal product of labor)

Both marginal products are positive and diminishing. Doubling capital while holding labor fixed less than doubles output.

Returns to scale. The exponents alpha and (1-alpha) sum to exactly 1, which means constant returns to scale: doubling both K and L exactly doubles output. If the exponents sum to more than 1, there are increasing returns; if less, decreasing returns.

Income shares. A key prediction: capital earns fraction alpha of total output (Y x alpha goes to capital owners) and labor earns fraction (1-alpha). This aligns well with observed income distribution data — capital’s share in the US has been roughly 30-35% historically.

TFP (A) captures everything not explained by capital and labor: technology, institutions, education quality, management practices. Much of the difference in output between rich and poor countries is attributed to TFP differences, not just capital or labor.

A note on the general two-parameter form. Textbooks sometimes write Cobb-Douglas as Y = A · L^α · K^β with α and β as independent exponents. When α + β = 1, that collapses to the constant-returns form this calculator uses. When α + β > 1 the function shows increasing returns to scale (doubling both inputs more than doubles output — typical of industries with strong network effects or scale economies); when α + β < 1 there are decreasing returns. Empirical estimates of the US private sector come out close to α + β ≈ 1, which is why the constant-returns version is the default starting point.