# 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^{0.07} L^{0.25}.

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)^{0.07} \left( \lambda L \right)^{0.25} \\ &= \lambda^{0.07} K^{0.07} \lambda^{0.25} L^{0.25} \\ &= \lambda^{0.07 + 0.25} K^{0.07} L^{0.25} \\ &= \lambda^{0.32} K^{0.07} L^{0.25} \\ &= \lambda^{0.32} q \left( K, L \right) \\ & < \lambda q \left( K, L \right) \end{align*} The production function has decreasing returns to scale.