Short-Run Production

In the short run, to produce a given amount of output, a firm can only adjust its input of labor.

Example

The cookie factory has to produce 10,000 cookies tomorrow using 100 ovens (K). The cookie production function is $$ q \left( K, L \right) = K L^2 $$

How many grandmas (L) does the factory need? $$ \begin{align*} q \left( K, L \right) &= K L^2 \\ 10000 &= 100 L^2 \\ 100 &= L^2 \\ L &= 10 \end{align*} $$

The cookie factory needs 10 grandmas to produces 10,000 cookies tomorrow.

Question

The production function is `q \left( K, L \right) = 7 K^{6} L^{}`. The factory has to produce 12096 cookies in a short period of time with K = 2 ovens.

How many workers does the factory need?

Plugging the values we know gives $$ \begin{align*} q \left( K, L \right) &= 7 K^{6} L^{} \\ 12096 &= 7 \times 2^{6} L^{} \end{align*} $$

Solving for L: $$ \begin{align*} L^{} &= \frac{12096}{7 \times 2^{6} } \\ L^{} &= 27 \end{align*} $$

The factory needs `27` grandmas to bake cookies tomorrow.