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Appendix: Isoquants and Isocosts

Isoquants allow us to show all of the various combinations of capital and labor that can be used to produce a level of output. For example, in the table below, 50 units of output can be produced with 1 unit of capital and 8 units of labor, or with 8 units of capital and 1 unit of labor. The two resources can be substituted for each other. We can do a similar analysis for 100 and 150 units of output.
Table 6.A1.gif
Graphing the data, we have the following:
6.A1.gif
While the figure above shows only three isoquants, there are many more (one for each possible level of output). The slope of an isoquant represents the rate at which labor can be substituted for capital (reading down the isoquant from left to right, the amount of capital is decreasing and the amount of labor is increasing). This is shown in the figure below:
6.A2.gif
That is why the slope is called the marginal rate of substitution. It is equal to the ratio of the marginal product of labor to the marginal product of capital (or MPL/MPK) because in order for output to stay the same, the production lost in using less capital must be made up for by the gain in production from using more labor.
But as we have seen, firms need to know what it costs them to produce a level of output. The isocost line allows us to represent the quantities of labor and capital that can be purchased at given input prices, given an amount of total cost.
For example, in the figure below, assuming that the price of labor and the price of capital are both $1 per unit, if the firm bought only labor, its total cost would be $5. If it bought only capital, its total cost would be $5. It can also spend that $5 to buy different combinations of the two inputs. The different isocost lines represent different levels of total cost.
6.A3.gif
The slope of the isocost is the ratio of the two input prices, or PL/PK. This is shown in the figure below:
6.A4.gif
We can bring the isoquants and the isocosts together to determine the least-cost combination of inputs. This is shown in the figure below:
6.A5.gif
In the figure, the cost-minimizing technology is shown by point C, which is the point of tangency of the isoquant and the isocost. At a point of tangency, the slopes of the two curves are equal; therefore, at point C MPL/MPK = PL/PK.
and rearranging terms, this means that at point C: MPL/PL = MPK/PK
which is the condition for producer optimization (marginal product per dollar spent should be equal for all inputs purchased).
Plotting a series of cost-minimizing combinations of inputs, shown as points A, B, and C in the figure below (at left), we can derive the firm’s cost curve (at right) and illustrate the idea that such a cost curve shows the minimum cost of producing any given level of output.
6.A6.gif
6.A7.gif
For more practice in understanding production decision, try the following Active Graph Level Two exercise:


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