Returns to Scale

Returns to scale indicate by how much the production is multiplied when the firm multiplies Capital and Labor by the same factor.

Example

A production function can have decreasing returns to scale, constant returns to scale, or increasing returns to scale.

Let the production function be `q \left( K, L \right) = K^{0.2} L^{0.3}`. Multiplying the capital (K) and labor (L) by the same factor `\lambda > 1` gives $$ \begin{align*} q \left( \lambda K, \lambda L \right) &= \left( \lambda K \right)^{0.2} \left( \lambda L \right)^{0.3} \\ &= \lambda^{0.2} K^{0.2} \lambda^{0.3} L^{0.3} \\ &= \lambda^{0.5} K^{0.2} L^{0.3} \\ &= \lambda^{0.5} q \left( K, L \right) < \lambda q \left( K, L \right) \end{align*} $$

Multiplying by 4 the number of ovens (K) and grandmas (L) in the cookie factory will multiply the number of cookies by `4^{0.5} = 2`.

Let the production function be `q \left( K, L \right) = K^{0.3} L^{0.7}`. Multiplying the capital (K) and labor (L) by the same factor `\lambda > 1` gives $$ \begin{align*} q \left( \lambda K, \lambda L \right) &= \left( \lambda K \right)^{0.3} \left( \lambda L\right)^{0.7} \\ &= \lambda^{0.3} K^{0.3} \lambda^{0.7} L^{0.7} \\ &= \lambda^{1} K^{0.3} L^{0.7} \\ &= \lambda q \left( K, L \right) \end{align*} $$

Multiplying by 2 the number of ovens (K) and grandmas (L) in the cookie factory will multiply the number of cookies by `2`.

Let the production function be `q \left( K, L \right) = K^{3} L`. Multiplying the capital (K) and labor (L) by the same factor `\lambda > 1` gives $$ \begin{align*} q \left( \lambda K, \lambda L \right) &= \left( \lambda K \right)^{3} \left( \lambda L \right)\\ &= \lambda^{3} K^{3} \lambda L \\ &= \lambda^{3} K^{3} L \\ &= \lambda^{3} q \left( K, L \right)> \lambda q \left( K, L \right) \end{align*} $$

Multiplying by 2 the number of ovens (K) and grandmas (L) in the cookie factory will multiply the number of cookies by `2^3=8`.

Question

The cookie factory has the following production function: `q \left( K, L \right) = K^{0.23} L^{0.77}`.

Determine whether the factory has decreasing, constant or increasing returns to scale.

Let `\lambda > 1`. $$ \begin{align*} q \left( \lambda K, \lambda L \right) &= \left( \lambda K \right)^{0.23} \left( \lambda L \right)^{0.77} \\ &= \lambda^{0.23} K^{0.23} \lambda^{0.77} L^{0.77} \\ &= \lambda^{0.23 + 0.77} K^{0.23} L^{0.77} \\ &= \lambda^{1.0} K^{0.23} L^{0.77} \\ &= \lambda^{1.0} q \left( K, L \right) \\ & = \lambda q \left( K, L \right) \end{align*} $$

The production function has constant returns to scale.