Skip to main content

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):



  • Then, solve for L:


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

  • 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:

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

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:


Let's apply this formula to our problem:

  • Derive the MPL and MPK:


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

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


       



Comments

Popular posts from this blog

Macroeconomics: multiplier and crowding out effects

Multiplier effect: whenever   any of the components of AD increases, the increase in GDP will be greater than the initial increase in expenditures. The impact on GDP of a particular increase in spending depends on the proportion of the new income that is taken out of the system to the proportion that continues to circulate in the economy. The multiplier effect tells us the impact a particular change in one the components of AD will have on the total income (GDP).  Let k denote the spending multiplier, which is a function of MPC and MPS. The larger the marginal propensity to consume, the larger the spending multiplier. Notice that the larger the MPC, the greater the impact a particular change in the spending variables will have on the nation's GDP. The crowding out effect: If government spending increases without an increase in taxes, the government must borrow funds from the private sector to finance its deficit, thereby increasing the interest rate. This increase in interest ...

Microeconomics: Factor Markets

Definition: Factor markets: markets for the factors of production (example: labor and capital). Markets are formed whenever consumers and producers meet to exchange goods or services. Deriving factor demand: the demand for goods or services in the product markets creates demand for the factors of production.  An increase (decrease) in demand for good X leads the suppliers to increase their production thereby increasing (decreasing) the demand for the factors of production.   Marginal revenue product: The demand for the factor of production is formed by multiplying a firm's marginal revenue by its marginal product.  Remember that by taking the derivative of the TR function with respect to Q we are able to find the MR. Marginal product on the other hand is found by taking the derivative of the production function with respect to a factor of production (L or K for example). Marginal revenue product (MRP): the change in total revenue when one more input is employed. It decrea...

A short interlude...

Hello all, it's been quite some time since I have made any major announcements since the creation of this blog about a year and a half ago. I am currently in graduate school and will soon take a portion of my comprehensive examination (i.e exams in Micro, Macro, and Metrics that will allow me to continue my studies in my graduate program), wish me luck! Thus, I will not be able to post consistently until at least mid-June. The roadmap is as such, if I pass, I will start introductory lessons in Econometrics and Statistics (if not, then I'll have to study until I can retake the comps in August). I hope that once I am done, I will be able to add more advance materials to the blog, such as general equilibrium, indirect utility functions, and game theory/mechanism design for Micro. The Solow, Ramsey, RBC, New Keynesian models, permanent income hypothesis (PIH) and more for Macro. By the way I think I still need to add notes on the IS-LM curves, so I will do that before jumping to t...