How this calculator works
The Cobb-Douglas function describes how capital and labour combine to produce output, with each input raised to an elasticity and scaled by total factor productivity. The elasticity alpha is the percentage change in output for a one percent change in capital, holding labour fixed; beta does the same for labour. When the elasticities sum to one the function has constant returns to scale: doubling both inputs doubles output. Sums above one imply increasing returns and sums below one imply decreasing returns. Total factor productivity captures everything else, from technology and management to institutions, and is measured as a residual rather than observed directly.
Formula
Y = A × K^alpha × L^beta. Rearranged, A = Y / (K^alpha × L^beta), K = (Y / (A × L^beta))^(1/alpha) and L = (Y / (A × K^alpha))^(1/beta). Y is output, A total factor productivity, K capital, L labour, and alpha and beta the output elasticities.
Worked example
With A = 1, K = 1000, L = 100, alpha = 0.3 and beta = 0.7, output is 1 × 1000^0.3 × 100^0.7 = 199.5262. Doubling both capital and labour to 2000 and 200 doubles output to 399.0524 because the elasticities sum to one.
Advertisement
Frequently asked questions
What do alpha and beta mean in practice?
They are output elasticities and, under competitive markets, also approximate the shares of income paid to capital and labour. A common estimate for advanced economies puts capital at roughly 0.3 and labour at roughly 0.7.
Why can the tool not solve for alpha or beta?
Those elasticities enter a nonlinear function and are usually estimated from many observations rather than computed from a single point. A single output, capital and labour figure does not identify them uniquely.
What happens if alpha plus beta is not one?
Returns to scale are increasing when the sum exceeds one and decreasing when it is below one. The tool reports the sum so you can see which case your parameters describe.