Costs

Cobb-Douglas Production Function

A widely used special form of the production function is the Cobb-Douglas production function:

$q = x_1^{\alpha}\, x_2^{\beta}$

where $x_1$ and $x_2$ are the quantities of the two factors, $q$ is output, and $\alpha$ and $\beta$ are positive constants. Its convenience is that its returns to scale can be read straight off the exponents.

Starting from $q_0 = \bar{x}_1^{\alpha}\,\bar{x}_2^{\beta}$ and scaling both inputs by $t > 1$:

$q_1 = (t\bar{x}_1)^{\alpha}(t\bar{x}_2)^{\beta} = t^{\alpha+\beta}\,\bar{x}_1^{\alpha}\,\bar{x}_2^{\beta} = t^{\alpha+\beta}\,q_0$

So output is multiplied by $t^{\alpha+\beta}$, and the sum of the exponents settles the returns to scale:

• If $\alpha + \beta = 1$: output rises exactly in proportion — CRS. …