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Cobb–Douglas Output, Capital & Labor Elasticities

Evaluate a Cobb–Douglas production scenario or recover an identifiable input, productivity factor or elasticity from the remaining values.

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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Use the result with context

Results are estimates for planning and education. Confirm rates, taxes, fees, and legal requirements with the relevant institution or a qualified professional before making a financial decision.

Formula and method

With productivity 2, labor 100, capital 400 and both elasticities 0.5, output is 2 × √100 × √400 = 400.

Output = productivity × labor^labor elasticity × capital^capital elasticity

Worked example

Enter these known values and leave the other values blank.

Productivity factor calibrated to the input units
2
Labor input
100
Capital input
400 USD
Labor elasticity β
0.5
Capital elasticity α
0.5
Modeled production output
400

Assumptions and limitations

  • Productivity is positive; labor and capital are nonnegative. Each elasticity ranges from zero to one in this model. A zero input requires a strictly positive elasticity to avoid 0^0.
  • The elasticities need not sum to one. Their sum determines returns to simultaneous scaling: below one gives decreasing, one constant, and above one increasing returns.
  • Use consistent calibrated units. Monetary capital is normalized to the captured USD reference; productivity must correspond to that reference and to the chosen labor and output units.
  • Some inverses are not identifiable. An input equal to one provides no information about its exponent, and a zero exponent removes dependence on its input. This scenario does not estimate parameters from economic data.

Common questions

Do the two elasticities have to add up to one?

No. That extra restriction is the constant-returns-to-scale case. The calculator also permits decreasing and increasing returns within its elasticity bounds.

What happens if I double both inputs?

Output is multiplied by 2 raised to the sum of the capital and labor elasticities.

References

Bookify permits zero output when a required input is zero and rejects zero raised to a zero input elasticity.

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