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Tutorial 8: The Costs of Production (cont.)

 

In this part of Tutorial 8 we turn our focus from the short run to the long run. We faced this same choice in Tutorial 7 on production: how do we model production in the long run, when all inputs are variable, in a two-dimensional graph? The answer there was isoquants -- a modeling showing all the combinations of two inputs, labor and capital, that could produce a given level of output. [We conveniently removed Resources from our list of inputs to make this modeling easier -- no results change when we remove it from consideration.] So now we must develop a model of the cost of production when the firm has two variable inputs.

 

Cost in the Long Run Top of page.

When all inputs are variable, the firm can choose its inputs such that costs are minimized for a given level of output. In developing this cost model we make two simplifying assumptions:

  1. There are only two variable inputs, L (hours/yr) and K (machine hours/yr); and
  2. Both can be hired in competitive markets, L @ $w/yr, and K @ $r/yr, so the firm takes the prices as given.

With these assumptions we can build a model that looks at the cost of production when the firm hires two variable inputs.

 

The Isocost Line Top of page.

We can construct a graphical model that shows all possible combinations of L and K the firm can hire for a given total cost. We start by writing out a total cost equation similar to what we used in part a, only this time both inputs are variable. It simply states that the total cost of production, C, is the sum of expenditures on labor, w · L, and expenditures on capital, r · K;

C = w·L + r·K. 

This three variable equation can be written in a form that will plot out as a straight line in a two-dimensional graph by solving for K in terms of C and L (you'll see why in just a minute):

r · K = C - w · L
K(C, L) = (C/r) - (w/r) · L.

If we fix the amount of money the firm has to spend on a given level of output, C1, then we have a single equation with a single unknown [remember, w and r are known to us]:

K(L | C1) = (C1/r) - (w/r) · L.

This function reveals that the amount of capital a firm may hire at a given total cost, C1, is equal to C1/r, if the firm hires no labor, and that amount is reduced by (w/r) for every additional worker hired. Figure 5 shows this graphically.

 

Figure 5


Graph of the isocost line.

 

Let C1 = $20, w = $2 per labor hour, and r = $2 per capital hour. Look at the vertical intercept. If the firm only hires capital, then, at $4 per hour, they will only be able to afford 10 machine hours per day with the $20 per day they have to spend. Thus the vertical intercept shows the amount of K the firm can afford to hire if it hires no L.

For each additional labor hour hired, the firm must give up $2/$2 = 1 machine hour of capital. (Remember the slope of the line is -w/r.) Thus the slope tells the firm how many units of K they must trade to free up enough money to buy 1 more unit of L.

Now look at the horizontal intercept. It tells a similar story. If the firm only hires labor, then, at $4 per hour, they will only be able to afford 10 labor hours per day with the $20 per day they have to spend. Thus the horizontal intercept shows the amount of L the firm can afford to hire if it hires no K.

In between the intercepts lie all the combinations of labor and capital for which the following cost equation holds:

$20 = $2 · L + $2 · K.

 

Suppose the amount of money available to hire inputs rises. That means that the amount of K the firm could hire, with L = 0, increases, raising the vertical intercept. It also means that the amount of L the firm could hire, with K = 0, increases, raising the horizontal intercept. The isocost line shifts outward parallel to the original. Figure 6 shows this for C = $30.

 

Figure 6


Graph of isocost line showing an increase in isocosts.

 

Why doesn't the slope change? The slope is equal to -w/r. Because the wage rate and the rental rate on capital have not changed, the slope of the isocost line does not change. That means the firm must still give up 1 unit of K in order to free up enough funds to buy one more unit of L.

 

Now it's time to "do the thing".

Click on the following link to download the Minimizing Production Costs Workbook. Work through Tutorial 8 Question 1. This will help to improve your understanding of how the isocost line is used to model production costs when all inputs are variable (in the long run).

Return here when you have finished.

Need help downloading the Excel file?

 

 

This second part of Tutorial 8 presents a discussion of the properties of the production costs faced by firms in the long run, when no inputs are fixed. In it we learned that a linear equation, called an isocost line can be used to model the total cost of hiring two variable inputs.

The next step is to combine this isocost model together with the isoquant model developed in Tutorial 7 and use it to see how a firm can minimize the cost of producing a given level of output using two variable inputs.