Cobb-Douglas production function
Topic: Production Function, Returns and Costs · NCERT: Class 12, Ch 3 "Production and Costs"
Meaning
The Cobb-Douglas production function is q = x₁^α x₂^β, where x₁ and x₂ are inputs such as labour and capital. If both inputs rise t times, output rises t^(α+β) times:
- α + β = 1 means constant returns to scale (CRS);
- α + β > 1 means increasing returns to scale (IRS);
- α + β < 1 means decreasing returns to scale (DRS).
α and β are output elasticities: the % change in output from a 1% change in that input. Under CRS with competitive markets, they equal each input's share of income.
Example
- Q = 5L^½K^½ at L = 100, K = 100 gives 5 × 10 × 10 = 500. This is CRS.
- Q = 2L²K² at (5, 2) gives 200. This is IRS. If L = 0, output is 0.
- Cobb and Douglas (1928) studied US manufacturing for 1899–1922. They found labour ≈ 0.75 and capital ≈ 0.25, which is CRS.