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.

Comments

Microeconomics: Firms and cost in the short run

In Economics, the term short run refers to a time period where at least one variable of interest does not change . In our case, the short run for a firm is when at least one input (labor, land, capital) stays fixed. Usually land and capital are considered fixed in the short run. If an input is fixed during a period time, no matter how much the total product a firm produces, its cost stays the same. This cost is commonly known as fixed cost (FC). Examples of fixed costs: rent, property taxes, loan payments. Labor is often considered to be a part of the variable cost (VC) . Variable cost can be defined as the cost a firm has control over during the short run. Unlike fixed cost, variable cost increases (decreases) as a firm's total product increases (decreases). Examples of variable costs include: utility bills, wages, raw materials A firm's total cost (TC) is the sum of its variable and fixed costs. As you can see, the fixed cost...

Macroeconomics: equilibrium in the AD/AS model

Real output and price level in the AD/AS model: The short run equilibrium in the AD/AS model, is when the intersection between the short run aggregate supply curve and the aggregate demand curve determines the level of output and price level. This short equilibrium output can be compared to the full-employment (i.e long run equilibrium) output by adding the long run aggregate supply curve in the graph. If the difference between the short run equilibrium output and full employment output is negative than we have a recessionary gap. This could be caused by a decrease in households' consumption, a fall in private investments, or a decrease in government spending. If the difference is positive, we have an inflationary gap, where the short run equilibrium output is bigger than its full employment output. This can be caused by an increase in households expenditures, expansionary fiscal policy, etc... If the gap is zero, this means that t...

Microeconomics: Shift in Supply

We recently talked about the factors that cause a shift in the demand. It is now time to discuss the factors that shift the supply curve which cause the equilibrium price and demand to change. Determinants of Supply: changes in supply are caused by changes in the price of inputs, the number of firms in the market, technology, changes in the price of related goods and services, and changes in expected prices. An increase (decrease) in the price of an input results in less (more) supply as per unit production costs rise (fall). More competition increases supply, and less competition leads to less supply. If a new firm enters the market for good X, then the supply of good X increases. Improvement in technology can result in an increase in the ability of producers to supply their products. For example, the invention of the printing press increased the supply of books. An increase (decrease) in the price of a related good leads to an increase (decrease) in the supply of the other...

Economics and Statistics

Search This Blog