Financial
Cobb-Douglas production function (Y = A * K^alpha * L^beta), MPK & MPL
Calculate economic output Y, Marginal Product of Capital (MPK), Marginal Product of Labor (MPL), and Returns to Scale from TFP, Capital, and Labor.
Model output elasticities, marginal productivity, and returns to scale
This hypothetical Cobb-Douglas model holds productivity A fixed and uses consistently calibrated units for capital K, labor L and output Y. It is a model of supplied elasticities, not evidence that changing a real input causes the predicted output change. Doubling K and L multiplies output by 2^(alpha+beta).
MPK=alpha*Y/K and MPL=beta*Y/L are derivatives: local changes per unit of input, not exact finite one-unit output increments. This example requires positive A, K and L and nonnegative elasticities.
| Cobb-Douglas Metric (A=10, K=81, L=16, alpha=0.25, beta=0.75) | Formula | Result (rounded where shown) |
|---|---|---|
| Total Output (Y) | 10 * (81 ^ 0.25) * (16 ^ 0.75) = 10 * 3 * 8 | 240.0 |
| Marginal Product of Capital (MPK) | alpha * (Y / K) = 0.25 * (240 / 81) | 0.7407 units / K |
| Marginal Product of Labor (MPL) | beta * (Y / L) = 0.75 * (240 / 16) | 11.25 units / L |
| Returns to Scale Check (alpha + beta) | 0.25 + 0.75 = 1.00 | "Constant Returns" |
Cobb-Douglas total output Y
A2=TFP (10), B2=Capital K (81), C2=Labor L (16), D2=alpha (0.25), E2=beta (0.75).
=IF(OR(COUNT(A2:E2)<>5,MIN(A2:C2)<=0,MIN(D2:E2)<0),"Invalid inputs",ROUND(A2 * (B2 ^ D2) * (C2 ^ E2), 2))Marginal product of labor (MPL)
The formula calculates beta*Y/L. Separately, alpha+beta equal to 1 gives constant returns to scale; above 1 gives increasing returns and below 1 gives decreasing returns, with A held fixed.
=IF(OR(COUNT(A2:E2)<>5,MIN(A2:C2)<=0,MIN(D2:E2)<0),"Invalid inputs",LET(y, A2 * (B2 ^ D2) * (C2 ^ E2), ROUND(E2 * (y / C2), 4)))Try the example
Use the original values or change them to test your understanding.