The Cobb-Douglas function is the workhorse of growth economics. With α ≈ 0.3–0.4 for capital, about two-thirds of national income goes to labor, which roughly matches the data for the U.S.
How to use the Cobb-Douglas Production Function Calculator
- Total factor productivity (A)
- Capital (K)
- Labor (L)
- Output elasticity of capital (α)
- Output elasticity of labor (β)
Results update as you type, and every field accepts the units shown in its menu.
How it works
Enter total factor productivity, capital, labor and the output elasticities α and β.
Tips
- Use real (inflation-adjusted) figures to compare across years.
- Check whether data are seasonally adjusted.
- Use the same currency and base year throughout.
- Economic models simplify reality – treat results as illustrations, not forecasts.
Common mistakes to avoid
- Mixing nominal and real values.
- Comparing percentage changes with percentage-point changes.
- Treating correlations as causes.
Inputs explained
| Field | What to enter |
|---|---|
| Total factor productivity (A) | A number |
| Capital (K) | A number |
| Labor (L) | A number |
| Output elasticity of capital (α) | A number |
| Output elasticity of labor (β) | A number |
What each result means
- Output (Q) = [Total factor productivity (A)]×[Capital (K)]^[Output elasticity of capital (α)]×[Labor (L)]^[Output elasticity of labor (β)]
- Marginal product of capital = [Output elasticity of capital (α)]×[Output (Q)]÷[Capital (K)]
- Marginal product of labor = [Output elasticity of labor (β)]×[Output (Q)]÷[Labor (L)]
Example values
How the main result changes when Total factor productivity changes and everything else stays at the example values above:
| Total factor productivity (A) | Output (Q) |
|---|---|
| 0.75 | 461.68 |
| 1.13 | 692.52 |
| 1.5 | 923.36 |
| 1.88 | 1,154.2 |
| 2.25 | 1,385.04 |
| 3 | 1,846.72 |
Formula
Q = A × K^α × L^βMPK = α × Q ÷ K; MPL = β × Q ÷ Lα + β = 1 constant, > 1 increasing, < 1 decreasing returns to scaleWorked example
Inputs — Total factor productivity (A): 1.5; Capital (K): 1,000; Labor (L): 500; Output elasticity of capital (α): 0.3; Output elasticity of labor (β): 0.7.
Results — Output (Q) = 923.36; Marginal product of capital = 0.277; Marginal product of labor = 1.2927; Returns to scale = Constant returns to scale.
Frequently asked questions
What is the Cobb-Douglas production function?
Q = A K^α L^β, relating output to capital and labor.
What do α and β mean?
The share of output growth from a 1% increase in each input.
What is total factor productivity?
Output growth not explained by capital and labor – technology and efficiency.
What formula does the Cobb-Douglas Production Function Calculator use?
Q = A × K^α × L^β.