Cobb-Douglas Production Function Calculator: Output, Marginal Products and Returns to Scale
Calculate output Y = A·K^α·L^β, or solve for A, K or L. Shows marginal and average products, MRTS, returns to scale and the cost-minimizing mix of capital and labor.
How it works
Y = A × K^α × L^β · MPL = ∂Y/∂L = β × Y ÷ L · MPK = ∂Y/∂K = α × Y ÷ K · MRTS = MPL ÷ MPK = (β/α) × (K/L) · returns to scale: α + β (= 1 constant, > 1 increasing, < 1 decreasing) · solving: A = Y ÷ (K^α L^β), K = (Y ÷ (A L^β))^(1/α), L = (Y ÷ (A K^α))^(1/β) · cost-minimizing K/L = (α/β) × (w/r)
The Cobb-Douglas production function models output as total factor productivity A times capital K and labor L, each raised to a power. The exponents α and β are output elasticities: a 1% increase in capital raises output by about α%, and a 1% increase in labor by about β%. Their sum gives returns to scale. Doubling both inputs multiplies output by 2^(α+β), so α + β = 1 means output doubles exactly (constant returns). Enter A, K, L and the two elasticities to get output, or switch the Solve for buttons to back out productivity, or the capital or labor needed for a target output. Results include the marginal products (the extra output from one more unit of each input, which falls as you add more), average products, and the marginal rate of technical substitution. Two tables show scaling both inputs and adding labor alone (diminishing returns). The isoquant chart traces every K–L mix that gives the same output. Add a wage and a capital rental rate under More options to get total cost and the cost-minimizing input mix. Some textbooks swap the letters and put α on labor; here α always belongs to capital and β to labor. Inputs can be in any units as long as you use them consistently. A has no meaning on its own; it depends on how Y, K and L are measured. The model assumes a constant elasticity of substitution of 1 between capital and labor. No other calculator on the site covers it. It is built for economics courses and simple productivity analysis.
Worked example
A plant has productivity A = 1.5, uses K = 200 units of capital and L = 100 units of labor, with α = 0.3 and β = 0.7. Output Y = 1.5 × 200^0.3 × 100^0.7 = 1.5 × 4.901 × 25.119 = 184.67 units. The marginal product of labor is 0.7 × 184.67 ÷ 100 = 1.293, and of capital 0.3 × 184.67 ÷ 200 = 0.277. MRTS = 1.293 ÷ 0.277 = 4.667 units of capital per unit of labor. Since α + β = 1, returns to scale are constant: doubling both inputs doubles output to 369.34.
Sources: Omni Calculator: Cobb-Douglas production function (glass-ball examples, returns to scale) · Wikipedia: Cobb–Douglas production function (elasticities, returns to scale, factor shares)
Frequently asked questions
What do α and β mean in the Cobb-Douglas function?
They are the output elasticities of capital and labor. With α = 0.3, a 1% rise in capital raises output by about 0.3%. Under constant returns and competitive markets they also equal capital’s and labor’s shares of income.
How do I tell the returns to scale?
Add the exponents. α + β = 1 gives constant returns (double the inputs, double the output), above 1 gives increasing returns and below 1 gives decreasing returns.
How do I calculate the marginal product of labor?
Take the partial derivative: MPL = β × A × K^α × L^(β−1), which simplifies to β × Y ÷ L. With β = 0.7, Y = 184.67 and L = 100, MPL = 1.293 extra units of output per extra unit of labor.
Why does the marginal product fall as I add labor?
With β below 1, each extra worker has less capital to work with, so output rises by less each time. This is the law of diminishing marginal returns, shown in the “Adding labor with capital fixed” table.
What is total factor productivity (A)?
The part of output not explained by measured capital and labor: technology, management, skills and efficiency. Growth in A, also called the Solow residual, raises output without any more inputs.
What is the cheapest combination of capital and labor?
Costs are lowest when the output per unit of money spent is equal for both inputs, MPL/w = MPK/r. For Cobb-Douglas this gives K/L = (α/β) × (w/r). Enter a wage and a rental rate under More options to get the exact mix and cost.
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This calculator is for educational purposes. Double-check important results before acting on them.