What this calculator does
The Cobb-Douglas function is the standard way economists relate output to labour and capital. Each input is raised to an exponent representing how responsive output is to it, and the two are multiplied together with a productivity term.
The exponents carry most of the meaning. When they sum to exactly 1 the function has constant returns to scale, meaning doubling both inputs doubles output. A sum above 1 gives increasing returns and below 1 gives decreasing returns.
The formula
Output equals total factor productivity multiplied by labour raised to its elasticity and capital raised to its elasticity. The sum of the two elasticities determines the returns to scale.
| Term | Meaning |
|---|---|
| Total factor productivity (A) | Everything affecting output other than the quantity of labour and capital: technology, organisation, institutions. |
| Output elasticity | The percentage change in output from a one per cent change in that input. |
| Returns to scale | The sum of the elasticities, indicating what happens when all inputs scale together. |
The inputs explained
| Field | What to enter |
|---|---|
| Total factor productivity (A) | Total factor productivity, capturing technology and efficiency. |
| Labour input (L) | Labour input, usually in hours or worker-count. |
| Capital input (K) | Capital input, usually the value of the capital stock. |
| Output elasticity of labour (β) | Output elasticity of labour. Empirically around 0.6 to 0.7 for many economies. |
| Output elasticity of capital (α) | Output elasticity of capital. Empirically around 0.3 to 0.4. |
When to use it
Modelling growth
The function underlies most growth accounting, separating input growth from productivity growth.
Assessing returns to scale
The elasticities say directly whether expanding proportionally raises output proportionally.
Estimating the effect of investment
Raising capital alone shows the diminishing returns that follow from an exponent below 1.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What does adding capital alone achieve?
The same labour force with three levels of capital.
Questions
Why do the exponents usually sum to 1?
Because constant returns to scale is both theoretically convenient and empirically close to what is observed. It also means the elasticities can be read as the share of output going to each factor, which matches income share data reasonably well.
What does total factor productivity actually represent?
Everything that raises output without raising measured inputs: better technology, better management, better institutions. It is calculated as a residual, which is why it is sometimes called the measure of our ignorance.
Why does adding capital alone give diminishing returns?
Because the capital exponent is below 1. Each additional machine has fewer workers to use it, so output rises but by progressively less. This is the core insight behind the convergence prediction in growth theory.
What are the model's limitations?
It assumes a fixed substitution relationship between labour and capital, constant elasticities and smooth substitutability. Real production processes often have fixed proportions over some ranges, which the function cannot represent.
For how fixed costs amplify output changes in a firm, see the degree of operating leverage calculator. For market structure, see the HHI calculator.