An important tool of analysis when two inputs are variable is the *production isoquant* or simply *isoquant*. An **isoquant** is a curve showing all possible combinations of the inputs physically capable of producing a given (fixed) level of output. Each point on an isoquant is technically efficient; that is, for each combination on the isoquant, the maximum possible output is that associated with the given isoquant. The concept of an isoquant implies that it is possible to substitute some amount of one input for some of the other, say, labor for capital, while keeping output constant. Therefore, if the two inputs are continuously divisible, as we will assume, there are an infinite number of input combinations capable of producing each level of output.

To understand the concept of an isoquant, return for a moment to Table 8.1 in the preceding chapter. This table shows the maximum output that can be produced by combining different levels of labor and capital. Now note that several levels of output in this table can be produced in two ways. For example, 108 units of output can be produced using either 6 units of capital and 1 worker or 1 unit of capital and 4 workers. Thus these two combinations of labor and capital are two points on the isoquant associated with 108 units of output. And if we assumed that labor and capital were continuously divisible, there would be many more combinations on this isoquant.

Other input combinations in Table 8.1 that can produce the same level of output are:

Q = 258: using K = 2, L = 5 or K = 8, L = 2

Q = 400: using K = 9, L = 3 or K = 4, L = 4

Q = 453: using K = 5, L = 4 or K = 3, L = 7

Q = 708: using K = 6, L = 7 or K = 5, L = 9

Q = 753: using K = 10, L = 6 or K = 6, L = 8

Each pair of combinations of K and L is two of the many combinations associated with each specific level of output. Each demonstrates that it is possible to increase capital and decrease labor (or increase labor and decrease capital) while keeping the level of output constant. For example, if the firm is producing 400 units of output with 9 units of capital and 3 units of labor, it can increase labor by 1, decrease capital by 5, and keep output at 400. Or if it is producing 453 units of output with K = 3 and L = 7, it can increase K by 2, decrease L by 3, and keep output at 453. Thus an isoquant shows how one input can be substituted for another while keeping the level of output constant.

**Characteristics of Isoquants **

We now set forth the typically assumed characteristics of isoquants when labor, capital, and output are continuously divisible. Figure 9.1 illustrates three such isoquants. Isoquant Q_{1} shows all the combinations of capital and labor that yield 100 units of output. As shown, the firm can produce 100 units of output by using 10 units of capital and 75 of labor, or 50 units of capital and 15 of labor, or any other combination of capital and labor on isoquant Q_{1}. Similarly, isoquant Q_{2}shows the various combinations of capital and labor that can be used to produce 200 units of output. And isoquant Q_{3} shows all combinations that can produce 300 units of output. Each capital–labor combination can be on only one isoquant. That is, isoquants cannot intersect.

Isoquants Q_{1}, Q_{2} and Q_{3} are only three of an infinite number of isoquants that could be drawn. A group of isoquants is called an isoquant map. In an **isoquant map**, all isoquants lying above and to the right of a given isoquant indicate higher levels of output. Thus in Figure 9.1 isoquant Q_{2}indicates a higher level of output than isoquant Q_{1} and Q_{3}, indicates a higher level than Q_{2}.

**Marginal Rate of Technical Substitution**

As depicted in Figure 9.1, isoquants slope downward over the relevant range of production. This negative slope indicates that if the firm decreases the amount of capital employed, more labor must be added in order to keep the rate of output constant. Or if labor use is decreased, capital usage must be increased to keep output constant. Thus the two inputs can be substituted for one another to maintain a constant level of output. The rate at which one input is substituted for another along an isoquant is called the **marginal rate of technical substitution (MRTS)** and is defined as

MRTS = -∆K/∆L

The minus sign is added to make MRTS a positive number, since ∆K/∆L, the slope of the isoquant, is negative.

Over the relevant range of production, the marginal rate of technical substitution diminishes. As more and more labor is substituted for capital while holding output constant, the absolute value of ∆K/∆L decreases. This can be seen in Figure 9.1. If capital is reduced from 50 to 40 (a decrease of 10 units), labor must be increased by 5 units (from 15 to 20) in order to keep the level of output at 100 units. That is, when capital is plentiful relative to labor, the firm can discharge 10 units of capital but must substitute only 5 units of labor in order to keep output at 100. The marginal rate of technical substitution in this case is ∆K/∆L = -(-10)5 = 2, meaning that for every unit of labor added, 2 units of capital can be discharged in order to keep the level of output constant. However, consider a combination where capital is more scarce and labor more plentiful. For example, if capital is decreased from 20 to 10 (again a decrease of 10 units), labor must be increased by 35 units (from 40 to 75) to keep output at 100 units. In this case the MRTS is 10/35, indicating that for each unit of labor added, capital can be reduced by slightly more than one-quarter of a unit.

As capital decreases and labor increases along an isoquant, the amount of capital that can be discharged for each unit of labor added declines. This relation is seen in Figure 9.1. As the change in labor and the change in capital become extremely small around a point on an isoquant, the absolute value of the slope of a tangent to the isoquant at that point is the MRTS (-∆K/∆L) in the neighborhood of that point. In Figure 9.1, the absolute value of the slope of tangent T to isoquant Q_{1} at point A shows the marginal rate of technical substitution at that point. Thus the slope of the isoquant reflects the rate at which labor can be substituted for capital. As you can see, the isoquant becomes less and less steep with movements downward along the isoquant, and thus MRTS declines along an isoquant.

**Relation of MRTS to Marginal Products**

For very small movements along an isoquant, the marginal rate of technical sub- stitution equals the ratio of the marginal products of the two inputs. We will now demonstrate why this comes about.

The level of output, Q, depends on the use of the two inputs, L and K. Because Q is constant along an isoquant, ∆Q must equal zero for any change in L and K that would remain on a given isoquant. Suppose that, at a point on the isoquant, the marginal product of capital (MP_{K}) is 3 and the marginal product of labor (MP_{L}) is 6.If we add 1 unit of labor, output would increase by 6 units. To keep Q at the original level, capital must decrease just enough to offset the 6-unit increase in output generated by the increase in labor. Because the marginal product of capital is 3, 2 units of capital must be discharged in order to reduce output by 6 units. In this case the MRTS = -∆K/∆L = -(-2)/1 = 2, which is exactly equal to MP_{L}/MP_{K} = 6/3 = 2.

In more general terms, we can say that when L and K are allowed to vary slightly, the change in Q resulting from the change in the two inputs is the marginal product of L times the amount of change in L plus the marginal product of K times its change. Put in equation form,

∆Q = (MP_{L})(∆L) + (MP_{K})(∆K)

In order to remain on a given isoquant, it is necessary to set ∆Q equal to 0. Then, solving for the marginal rate of technical substitution yields

MRTS = -∆K/∆L = MP_{L}/MP_{K}

Using this relation, the reason for diminishing MRTS is easily explained. As additional units of labor are substituted for capital, the marginal product of labor diminishes. Two forces are working to diminish labor’s marginal product: (1) Less capital causes a downward shift of the marginal product of labor curve, and (2) more units of the variable input (labor) cause a downward movement along the marginal product curve. Thus, as labor is substituted for capital, the marginal product of labor must decline. For analogous reasons the marginal product of capital increases as less capital and more labor are used. The same two forces are present in this case: a movement along a marginal product curve and a shift in the location of the curve. In this situation, however, both forces work to increase the marginal product of capital. Thus, as labor is substituted for capital, the marginal product of capital increases. Combining these two conditions, as labor is substituted for capital, MP_{L} decreases and MP_{K} increases, so MP_{L}/MP_{K} will decrease.