Exercise: How to derive an average long run total cost function

Let's have an exercise on deriving a long run average cost curve in order to fully understand how this process works:

Assume that you have the following production function, wage (w) , and rent (r):

$Q=\frac{\sqrt L \cdot K^3}{10}$

$r= 30 \textit{ and } w=15$

Then, solve for L:

$L= \frac{100 \cdot Q^2}{K^6}$

Create a short run total cost function where the whole function is expressed in terms of Q and K:

$\\ SRTC= w \cdot L + r \cdot K\\ \\ SRTC= 15 \cdot \frac{100\cdot Q^2}{K^6}+30 \cdot K$

Find the level of capital (K) that minimizes the total cost function. Simply take the derivative of the short run total cost function with respect to K and set it equal to zero. Solve for K:

$\\ 1). \frac{dSRTC}{dK}= 15 \cdot (-6 \cdot (\frac{100 \cdot Q^2}{K^7}))+ 30=0 \\ \\ 2). K= (300 \cdot Q^2)^{\frac{1}{7}}$

We have enough information to derive the long run cost function. Simply replace L and K by equivalent functions that are expressed in terms of Q only:

$LRTC= 15 \cdot \frac{100 \cdot Q^2}{(300 \cdot Q^2)^{\frac{6}{7}}} + 30 \cdot (300 \cdot Q^2)^{\frac{1}{7}}$

The last step needed to obtain the average long run cost function is to divide the long term cost function by Q:

$ALRTC= \frac{1500}{(300)^{\frac{6}{7}} \cdot Q^{\frac{3}{7}}} + \frac{30 \cdot (300^\frac{1}{7})}{Q^\frac{5}{7}}$

We can see that this production function is experiencing economies of scale, as the quantity produced increases, the long term average cost diminishes.

Let's graph this:

Minimize short run average cost:

Remember the formula to optimize the cost of an average cost function:

$\frac{MPL}{w}= \frac{MPK}{r}$

Let's apply this formula to our problem:

Derive the MPL and MPK:

$\\ MPL= \frac{\partial Q}{\partial L}= \frac{K^3}{20 \sqrt L} \\ \\ MPK=\frac{\partial Q}{\partial K}= \frac{3 \sqrt L \cdot K^2 }{10}$

Find the relationship between K and L by following the formula:

$\\1). \ \frac{K^3}{300 \sqrt L}= \frac{3 \sqrt L \cdot K^2}{300} \\ \\ \\ 2). \ K^3= 3K^2L\rightarrow K= 3L$

We can see that as long as the firm uses three times more machines as labor, it minimizes its cost.

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