Cobb-Douglas Production

The Cobb-Douglas production function is the workhorse of production theory. Invented by mathematician Charles Cobb and economist Paul Douglas in 1928, it fits real-world data remarkably well and has elegant mathematical properties that make it ideal for economic modeling.

Key Term

Cobb-Douglas Production Function

$q = A L^{α} K^{β}$

where:

$A$ = total factor productivity (technology level)
$α$ = output elasticity of labor (% change in q per % change in L)
$β$ = output elasticity of capital
$α + β$ determines returns to scale

The Cobb-Douglas production function — the most widely used functional form in economics

The Cobb-Douglas function packs a lot of economics into a simple formula:

MPL = $α A L^{α - 1} K^{β} = α q / L$ . Since $α < 1$ , MPL is diminishing.
Returns to scale = $α + β$ . If $α + β = 1$ , constant returns to scale.
MRTS = $(α / β) (K / L)$ , which diminishes as L increases and K decreases.
Factor shares: with competitive markets, labor’s share of income = $α$ , capital’s share = $β$ (when $α + β = 1$ ).

Interactive — Cobb-Douglas Returns to Scale

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Cobb-Douglas isoquants with adjustable exponents.

Worked Example

A factory has the production function $q = 10 L^{0.7} K^{0.3}$ . Currently L = 100 and K = 50. (a) What is output? (b) What are the returns to scale? (c) What is the MPL?

Check yourself · no marks

In the Cobb-Douglas function $q = A L^{α} K^{β}$ with $α + β = 1$ , if labor receives wage $w = M P_{L}$ and capital receives rental rate $r = M P_{K}$ , what fraction of total revenue goes to labor?

Commit to one first. Guessing and being wrong beats reading the answer cold.

Practice · 1 / 4

No penalty for a wrong answer

Test your understanding of Cobb-Douglas production:

In the Cobb-Douglas function $q = A L^{α} K^{β}$ , the parameter $A$ represents:

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