Cobb-Douglas Production Function
Q = A × L^α × K^β.
The Cobb-Douglas production function models output from labor and capital inputs.
Widely used in economics and econometrics.
The Formula
The Cobb-Douglas production function models how inputs of labor (L) and capital (K) combine to produce output (Q). The parameter A is total factor productivity (technological efficiency), α (alpha) is the output elasticity of labor, and β (beta) is the output elasticity of capital.
Output elasticity measures the percentage change in output for a 1% change in an input. If α = 0.7, a 1% increase in labor → 0.7% increase in output.
Returns to scale depend on α + β: If α + β = 1: constant returns to scale (doubling inputs doubles output). If α + β > 1: increasing returns (economies of scale). If α + β < 1: decreasing returns (diminishing returns at scale).
Empirical estimates for the US economy: α ≈ 0.7, β ≈ 0.3. This means labor accounts for about 70% of national income — consistent with observed wage share of GDP.
The Cobb-Douglas function has been fitted to data for many industries and countries. It forms the basis of the Solow growth model for long-run economic growth. Limitations: it assumes perfect substitutability between inputs, which may not hold in practice.
Variables
| Symbol | Meaning | Unit |
|---|---|---|
| Q | Total output | units or $ |
| A | Total factor productivity | Dimensionless |
| L | Labor input | worker-hours |
| K | Capital input | machine-hours or $ |
| α | Output elasticity of labor | Dimensionless |
| β | Output elasticity of capital | Dimensionless |
Example 1
A = 2, α = 0.6, β = 0.4, L = 100 workers, K = 50 machines.
Q = 2 × 100^0.6 × 50^0.4 = 2 × 15.85 × 8.70
Q ≈ 275.8 units of output
Example 2
If both L and K double (200 workers, 100 machines) with α + β = 1 (constant returns):
Q = 2 × 200^0.6 × 100^0.4 = 2 × (2^0.6 × 15.85) × (2^0.4 × 8.70)
Q ≈ 551.6 units (exactly doubled — confirming constant returns to scale)
When to Use It
- Estimating returns to scale in industry or national economies
- Growth accounting (separating productivity from input growth)
- Econometric estimation of production relationships
- Macroeconomic modeling and the Solow growth model