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Tutorial 7: Production (cont.)

 

Production with Two Variable Inputs: Isoquants

In the first part of this Tutorial you viewed an Excel workbook containing a 3-D Surface chart illustrating the following production function:

Q(L, K) = 1 * L0.5 * K0.5

If you haven't done so already, download the workbook now. [NOTE: The size of this file is larger (~285kb) than all of the others, with the exception of the Comparative Statics Quiz, so allow for a longer than normal download time.]

Click on the Chart tab. When you look at the chart you notice a line running around this "hill" at various intervals, representing higher levels of output (Q) = 20 units/t (at the top of the blue range), 40 units/t (top of the purple range), and 60 units/t (top of the yellow range). Each of these lines shows that it is possible to get a given level of output, say Q = 20 units/t, with various combinations of L and K. These lines are referred to as isoquants because they show a constant ('iso') level of quantity ('quant') resulting from various combinations of L and K.

If you click on the Data tab and look at the table used to generate the chart, you will see much the same thing. Various combinations of L and K have been highlighted that would generate an output level Q = 10 units/t (see Table 5 below).

Table 5

Output Q = 10 units/t

Labor (L/t)

Capital (K/t)

4

25

5

20

10

10

20

5

25

4

 

This reveals a fact of fundamental importance in the theory of production. It is possible, using the production method modeled by the Cobb-Douglas production function, to substitute less of one input for more of another and still keep output constant. In the case illustrated by the Data, we can increase the amount of labor used in the production process from 10 to 20 units/t, and simultaneously drop the amount of capital used from 10 to 5 units/t with no change in output. There is more than one way to produce a given level of output!

Diminishing Returns Again… Top of page.

So what have we got? The Chart (and Data) help us see that, for a given level of output, when L is increased, K can be decreased. These lines of constant output are referred to as isoquants. You can easily modify the chart type in Excel to view these isoquants in two dimensions. In the Production.xls workbook, click once on the Chart tab then, from the Chart menu, choose Chart type. Select the chart subtype Contour (second row first chart) and click OK.

The slope of the isoquant reveals how much K may be given up when one more L is used to produce a given level of Q. The absolute value of the rate at which L may be substituted for K is referred to as the Marginal Rate of Technical Substitution between L and K (MRTSLK). Let's work with this mathemagically and see what we find...

MRTS is measured as the negative of the slope of an isoquant (which, of course, is negative). So

MRTS = —DK/DL > 0

for a given level of Q, where DK is the "rise" and DL is the "run". If you look back at the Data in the Production.xls workbook, you will see that as the amount of labor used in production increases from 4 to 5 units/t, the amount of capital that may be "retired", keeping output constant at 10 units/t, is 5 (from 25 to 20 units/t). So MRTS is 5 here. But when the amount of labor used in production increases from 10 to 20 units/t, the amount of capital that may be "retired", keeping output constant at 10 units/t, is also 5 (from 10 to 5 units/t). So MRTS is 0.5 here. So MRTS diminishes as the quantity of labor hired increases. But why?

Let's see. Normally, when a firm increases the amount of one input used, keeping the other constant, output increases. In this example, an increase in L will cause an increase in Q = MPL*DL. That is, the additional amount of labor hired, DL, will raise output by an amount equal to the additional output that extra labor makes (i.e., its marginal product, MPL).

Similarly, when the firm decreases the amount of one input used, keeping the other constant, output decreases by an amount equal to the additional output that input would have produced. In this example, a decrease in K will cause a decrease in Q = MPK*DK. That is, the amount of K laid off, DK, will decrease output by an amount equal to the marginal product of that capital, MPK.

If we substitute more labor for less capital along an isoquant, then the change in output would be zero -- the additional output created by the new labor (MPL*DL) would be exactly offset by the loss of output resulting from the decrease in capital (MPK*DK). Thus:

DQ = 0; MPL*DL + MPK*DK = 0

Rearranging terms:

MPL*DL = —MPK*DK

DK/DL = MPL/MPK

\ MRTS = MPL/MPK

So we have found that the rate at which L may be substituted for K is equal to the ratio of their respective marginal products. Earlier we found that the marginal rate of technical substitution declines as we move along the isoquant. How does knowing how to calculate MRTS answer the question of why MRTS declines as more L is substituted for K?

Well, in the beginning of this Tutorial we learned that the marginal product of a single variable input eventually declines as more of that input is used with all other inputs held constant. This "law of diminishing returns" was due to the fact that that variable input, say labor, had to share the fixed amount of capital that was available. It is that sharing that promotes diminishing returns. Turning this around, as less labor is used marginal product rises because less sharing of capital is required.

So as you move left to right along our isoquant, L increases, causing MPL to decline, and K increases, causing MPK to increase. If MRTS = MPL/MPK, then as you move left to right along an isoquant the numerator decreases (e.g., 5, 4, 3, 2, 1) while the denominator increases (e.g., 1, 2, 3, 4, 5), resulting in a smaller value for the ratio (e.g., 5, 2, 1, 0.5, 0.2).

 

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

Click on the following link to download the Minimizing Production Costs Workbook. Work through Tutorial 7 Questions 2 - 4. This will help to improve your understanding of how we model the production of output using two variable inputs.

Return here when you have finished.

Need help downloading the Excel file?

 

Now we see that we can model production with two variable inputs by using a graphical model of isoquants. Isoquants show various combinations of two inputs that produce a given level of output. As it turns out, isoquants are convex to the origin because their slope, called the marginal rate of technical substitution (MRTS), diminishes as more labor is used with less capital. The reason the slope diminishes is because it is equal to the ratio of the marginal products of the inputs, MPL/MPK, and as more L is used, MPL declines (law of diminishing returns), as less K is used, MPK increases (law of diminishing returns in reverse!).

Next we take a look at some other properties of an isoquant model and introduce the concept of production in the long run.