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 10000 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 10000 cookies tomorrow.

Question

The production function is `q \left( K, L \right) = 3 K^{3} L^{}`. The factory has to produce 62856 cookies in a short period of time with K = 6 ovens.

How many workers does the factory need?

Plugging the values we know gives

$$ \begin{align*} q \left( K, L \right) &= 3 K^{3} L^{} \\ 62856 &= 3 \times 6^{3} L^{} \end{align*} $$

Solving for L:

$$ \begin{align*} L^{} &= \frac{62856}{3 \times 6^{3} } \\ L^{} &= 97 \end{align*} $$

The factory needs `97` grandmas to bake cookies tomorrow.