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

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 ...

Econometrics: Bivariate population model

Hello I'm finally back from my extended break. I thought that we should start studying Econometrics. Let's begin by analyzing a simple bivariate regression. Assume that this equation describes the relationship between two variables X and Y. We say that Y is the dependent variable, whereas X is the independent variable. In other words, we assume that Y (the output) depends on X (the input). Epsilon is the error term, it represents other factors that affect Y. The error term must be uncorrelated with the variable X so that we do not need to include them in our regression, and thus the coefficient of X (beta) should not change, even though Epsilon also determines Y. beta-0 is the constant term. It tells us what would be Y if X=0. beta-1 is the effect on Y if X changes by one unit. To see this, assume X=education is a continuous function, let's take the derivative of Y=wage with respect to X: Thus, if education goes up by one unit, we should expect, on average, wage to go up ...

Exercise: short and long run effects of a fiscal policy

You are in charge of the government budget for the year 2021. You are told that schools and public roads need to be updated, and you estimate an appropriate budget in order to carry out the task. Describe what will happen in the short run to the economy, and what type of inflation do we see? Describe the supply side effect from this policy, explain what happens in the long run. Assume that there is no crowding out effect. First, note that updating public infractures, assuming taxes stay the same, will require an increase in government spending, which will also have an impact on the investment and consumption level in the economy (remember the spending multiplier?). Therefore, the aggregate demand curve will shift to the right. This will cause inflation (i.e positive change in the price level) and a higher output in the short-run. Note that if we were to assume a large crowding out effect, the short run aggregate supply curve would shift to the left (increase in production c...

Economics and Statistics

Search This Blog