Long-Run Cost Minimization

A firm chooses levels capital and labor to minimize costs when:
  1. MRTS = MRTT
  2. `q\left( K, L \right) = q`

Example

Ziggy hires grandmas to bake premium quality cookies. He knows that over the course of the year, the company has to produce 320000 cookies. He wants to plan the number of grandmas (L) and ovens (K) that minimizes costs.

Ziggy pays a grandma (L) $100000 a year, the cost of a oven (K) for the year is $10000 and the production function is `q \left( L, K \right) = KL^2`.

Step 1: MRTS = MRTT

$$ \begin{align*} -\frac{MP_L}{MP_K} &= -\frac{w}{r} \\ \frac{2KL}{L^2} &= \frac{100000}{10000} \\ \frac{2K}{L} &= 10 \\ K &= 5L \end{align*} $$

Step 2: Plug back in the production function

The factory produces q = 320000 cookies, so

$$ \begin{align*} KL^2 &= q \\ 5L \times L^2 &= 320000 \\ L^3 &= 64000 \\ L &= 40 \end{align*} $$

The factory needs `L = 40` grandmas and therefore `K = 5L = 200` ovens.

Question

The production function is now `q \left( K, L \right) = 5 K^3 L^2`. The cookie factory has to bake `945` cookies. The rent of capital `r = 3` and the cost of labor is `w = 6`.

What levels of capital (K) and labor (L) minimize the factory's costs?

The factory needs `L = 7^{\frac{1}{5}}` grandmas and `K = 3 \times 7^{\frac{1}{5}}` ovens..

Step 1: MRTS = MRTT

$$ \begin{align*} -\frac{MP_L}{MP_K} &= -\frac{w}{r} \\ \frac{5 \times 2 K^{3} L^{1}}{5 \times 3 K^{2} L^{2}} &= \frac{6}{3} \\ \frac{K}{L} &= \frac{3 \times 6}{2 \times 3} \\ K &= 3 L \end{align*} $$

Step 2: Plug into the production function

$$ \begin{align*} 5 K^3 L^2 &= 945 \\ 5 \left( 3 L \right)^{3} L^{2} &= 945 \\ L^{3 + 2} &= \frac{945}{5 \times 3^{3}} \\ L^{5} &= 7 \\ L &= 7^{\frac{1}{5}} \end{align*} $$

So `L = 7^{\frac{1}{5}}` and `K = 3 \times 7^{\frac{1}{5}}`.