# Cobb-Douglas Production Function

Cobb-Douglas production function is a model that tells us about the relationship between total product, total factor productivity, quantities of labor and capital and their output elasticities.

The Cobb-Douglas production function is the most widely used production function because it allows different combination of labor and capital. Other versions of the production functions such as the linear production function and fixed-proportion (Leontief) production function represent extreme case-scenarios i.e. perfect substitution between labor and capital and zero substitution respectively.

The Cobb-Douglas production function was developed by Paul Douglas, an economist, and Charles Cobb, a mathematician.

## Formula

The mathematical equation for Cobb-Douglas function is as follows:

$$ \text{Q}=\text{A}\times \text{K}^\alpha\times \text{L}^\beta $$

Where Q is the total product, K represents the units of capital, L stands for units of labor, A is the total factor productivity, and α and β are the output elasticities of capital and labor respectively.

Historical data on total production, labor hours and real value of capital is used to determine A, α and β using the least squares method. Total factor productivity (A) is a coefficient that represents the effect of factors other than labor and capital on the total product. The output elasticities of labor (a) and capital (b) measure the responsiveness of output to changes in the labor and capital respectively i.e. a and b tell us the percentage change in total output that corresponds to a 1% increase in labor and capital respectively.

The Cobb-Douglas function represents the typical convex isoquant.

## Marginal Product of Labor/Capital

When the Cobb-Douglas production function is partially differentiated with reference to L and K, we get the marginal product of labor (MPL) and marginal product of capital (MPK) respectively:

$$ {\text{MP}} _ \text{L}=\beta\times \text{A}\times \text{K}^\alpha\times \text{L}^{\beta-\text{1}} $$

$$ {\text{MP}} _ \text{K}=\alpha\times \text{A}\times \text{K}^{\alpha-\text{1}}\times \text{L}^\beta $$

The Cobb-Douglas production function also tells about the returns to scale. The following table summarizes what the sum of α and β tells us about returns to scale:

This historical basis of Cobb-Douglas function is its most significant weakness. Besides its statistical elegance, it has no theoretical basis and the assumption that the output elasticities of labor and capital and total factor productivity in future will be the same as in past is not very sound.

Written by Obaidullah Jan, ACA, CFA and last modified on