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Production in the Long Run 205

MPK18. At this input combination, the MRTS is 28/18 1.55 and the slope

of the isoquant is 1.55 (much steeper).

Suppose the manager sets out to produce an output of 636 units at least

cost. Which combination of inputs along the isoquant will accomplish this

objective? The answer is provided by portraying the firm’s least-cost goal in

graphic terms. Recall that the firm’s total cost of using L and K units of

input is

Using this equation, let’s determine the various combinations of inputs the

firm can obtain at a given level of total cost (i.e., expenditure). To do this, we

rearrange the cost equation to read

To illustrate, suppose the firm faces the input prices of Example 3, PL

$10 and PK$15. If it limits its total expenditures to TC $120, the firm can

use any mix of inputs satisfying K 120/15 (10/15)L or K  8 (2/3)L.

This equation is plotted in Figure 5.2b. This line is called an isocost line

because it shows the combination of inputs the firm can acquire at a given total

cost. We can draw a host of isocost lines corresponding to different levels of

expenditures on inputs. In the figure, the isocost lines corresponding to TC 

$220 and TC $300 are shown. The slope of any of these lines is given by the

ratio of input prices, K/L PL/PK. The higher the price of capital (rela-

tive to labor), the lower the amount of capital that can be substituted for labor

while keeping the firm’s total cost constant.

By superimposing isocost lines on the same graph with the appropriate

isoquant, we can determine the firm’s least-cost mix of inputs. We simply find

the lowest isocost line that still touches the given isoquant. This is shown in

Figure 5.3. For instance, to produce 636 units of output at minimum cost, we

must identify the point along the isoquant that lies on the lowest isocost line.

The figure shows that this is point B, the point at which the isocost line is tan-

gent to the isoquant. Point B confirms Example 3’s original solution: The opti-

mal combination of inputs is 10 units of labor and 8 units of capital. Since

point B lies on the $220 isocost line, we observe that this is the minimum pos-

sible cost of producing the 636 units.

Note that at the point of tangency, the slope of the isoquant and the slope

of the isocost line are the same. The isoquant’s slope is MPL/MPK. In turn,

the isocost line’s slope is PL/PK. Thus, the least-cost combination of inputs is

characterized by the condition

MRTSMPL/MPKPL/PK.

KTC/PK(PL/PK)L.

TCPLLPKK.

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