# PRODUCTION FUNCTION WITH TWO VARIABLE INPUTS

Now we turn to the case of production where two inputs (say capital and labour) are variable. Although, we restrict our analysis to two variable inputs, all of the results hold for more than two also. We are restricting our analysis to two variable inputs because it simply allows us the scope for graphical analysis. When analysing production with more than one variable input, we cannot simply use sets of AP and MP curves like those case of production function with single variable input, because these curves were derived holding the use of all other inputs fixed and letting the use of only one input vary. If we change the level of fixed input, the TP, AP and MP curves would shift. In the case of two variable inputs, changing the use of one input would cause a shift in the MP and AP curves of the other input. For example, an increase in capital would probably result in an increase in the MP of labour over a wide range of labour use.

###### Production Isoquants

In Greek the word ‘iso’ means ‘equal’ or ’same’. A production isoquant (equal output curve) is the locus of all those combinations of two inputs which yields a given level of output. With two variable inputs, capital and labour, the isoquant gives the different combinations of capital and labour, that produces the same level of output. For example, 5 units of output can be produced using either 15 units of capital (K) or 2 units of labour (L) or K=10 and L=3 or K=5 and L=5 or K=3 and L=7. These four combinations of capital and labour are four points on the isoquant associated with 5 units of output as shown in Figure-2. And if we assume that capital and labour are continuously divisible, there would be many more combinations on this isoquant.

Now let us assume that capital, labour, and output are continuously divisible in order to set forth the typically assumed characteristics of isoquants. Figure -3 illustrates three such isoquants. Isoquant I shows all the combinations of capital and labour that will produce 10 units of output. According to this isoquant, it is possible to obtain this output if K0 units of capital and L0 units of  labour inputs are used. Alternately, this output can also be obtained if K1 units of capital and L1 units of labour inputs or K2 units of capital and L2 units of labour are used. Similarly, isoquant II shows the various combinations of capital and labour that can be used to produce 15 units of output. Isoquant III shows all combinations that can produce 20 units of output. Each capital labour combination can be on only one isoquant. That is, isoquants cannot intersect. These isoquants 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-3 isoquant II indicates a higher level of output than isoquant I, and isoquant III indicates a higher level of output than isoquant II.

In general, isoquants are determined in the following way. First, a rate of output, say Q0, is specified. Hence the production function can be written as

Q0 = f (K,L)

Those combinations of K and L that satisfy this equation define the isoquant for output rate Q0.

###### Marginal Rate of Technical Substitution

As we have seen above, generally there are a number of ways (combinations of inputs) that a particular output can be produced. The rate, at which one input can be substituted for another input, if output remains constant, is called the marginal rate of technical substitution (MRTS). It is defined in case of two inputs, capital and labour, as the amount of capital that can be replaced by an extra unit of labour, without affecting total output.

It is customary to define the MRTS as a positive number, since WK/WL, the slope of the isoquant, is negative. Over the relevant range of production the MRTS diminishes. That is, more and more labour is substituted for capital while holding output constant, the absolute value of WK/WL decreases. For example, let us assume that 10 pairs of shoes can be produced using either 8 units of capital and 2 units of labour or 4 units each of capital and 4 UNITS of labour or 2 units of capital and 8 units of labour. From Figure -4 the MRTS of labour for capital between points a and b is equal to WK/WL = (4–8) / (4–2) = –4/2 = –2 or | 2 |. Between points b and c, the MRTS is equal to –2/4 = –½ or | ½ |. The MRTS has decreased because capital and labour are not perfect substitutes for each other. Therefore, as more of labour is added, less of capital can be used (in exchange for another unit of labour) while keeping the output level constant.

There is simple relationship between MRTS of labour for capital and the marginal product MPK and MPL of capital and labour respectively. Since along an isoquant, the level of output remains the same, if WL units of labour are substituted for WK units of capital, the increase in output due to WL units of labour (namely, WL * MPL) should match the decrease in output due to a decrease of WK units of capital (namely, WK * MPK). In other words, along an isoquant,

There are vast differences among inputs in how readily they can be substituted for one another. For example, in some extreme production process, one input can perfectly be substituted for another; whereas in some other extreme production process no substitution is possible. On the other hand, in most of the production processes what we see is imperfect substitution of inputs. These three general shapes that an isoquant might have are shown in Figure -5. In panel I, the isoquants are right angles implying that the two inputs a and b must be used in fixed proportion and they are not at all substitutable. For instance, there is no substitution possible between the tyres and a battery in an automobile production process. The MRTS in all such cases would, therefore, be zero. The other extreme case would be where the inputs a and b are perfect substitutes as shown in panel II. The isoquants in this category will be a straight line with constant slope or MRTS.

A good example of this type would be natural gas and fuel oil, which are close substitutes in energy production. The most common situation is presented in panel III. The inputs are imperfect substitutes in this case and the rate at which input a can be given up in return for one more unit of input b keeping the output constant diminishes as the amount of input b increases.

###### The Economic Region of Production

Isoquants may also have positively sloped segments, or bend back upon themselves, as shown in Figure 6. Above OA and below OB, the slope of the isoquants is positive, which implies that increase in both capital and labour are required to maintain a certain output rate. If this is the case, the MP of one or other input must be negative. Above OA, the MP of capital is negative. Thus output will increase if less capital is used, while the amount of labour is held constant. Below OB, the MP of labour is negative. Thus, output will increase if less labour is used, while the amount of capital is held constant. The lines OA and OB are called ridge lines. And the region bounded by these ridge lines is called economic region of production. This means the region of production beyond the ridge lines is economically inefficient.