It has been discussed earlier that in the short run one of the factors of production is variable and the other(s) constant. The marginal product of the variable factor will decrease eventually as more and more of quantities of this factor will be combined with the other constant factor(s). The expansion of output with one factor (at least) constant is described as the Law of Diminishing Marginal Product of the variable factor, which is often referred to as the law of variable proportions.
If the law of diminishing marginal product operates the isoquants will be convex to the origin. A convex isoquant means that the marginal rate of Technical substitution (MRTS) between L and K decreases as L is substituted for K. But the MRTS is equal to the ratio of the marginal productivities of the two factors i.e. MRTS = fL/fK.
In figure, as we move from the point P to the point Q along the isoquant Q0, we employ RQ units more of L and PR units less of K. By moving from P to Q, we increase the labor input by RQ and reduce the capital by PR. That is, we substitute RQ labor for PR. The rate of this substitution is called Marginal Rate of Technical Substitution (MRTSL,K ). For the submission, MRTS = dK/dL=PR/RQ .
What is the MRTSL,K for a movement from Q to S on the isoquant? Here, the increase in labor input is TS, which is equal to RQ. The decrease in capital input in QT. Therefore, MRTSL,K =QT/TS . By comparing MRTSL,K at these two points, it is obvious that MRTSL,K at Q is less than at P. Though RQ = TS, QT < PR. That is, for the same increase in labor input the reduction in capital will be less as we move down the Isoquant. This is because of law of diminishing marginal product. As we shall see later, MRTSL,K at any point on the isoquant is equal to the slope of a tangent at that point. The slope is nothing but the ratio of MP1 and MPk. Therefore, the marginal productivity of L decreases and that of K increases as we move along the isoquant from P to Q., In other words, (f1/fk at Q). Therefore, we can say that MRTSL,K at P > MRTSL,K at Q, i.e. MRTS between L and K decreases as we move from P to Q. Thus it can be said that MRTS between factors follows the diminishing marginal productivities of both the factors of production. Hence the convexity of the isoquant follows directly from the law of diminishing marginal productivity.
As we had discussed in the analysis of indifference curves that two indifferences cannot intersect each other, in the same way it can be proved in the case of isoquants that two isoquants cannot intersects each other.
A set of curves form a family of isoquants as shown in figure. All the inputs combinations, which lie on an isoquant, will result in an output indicated for that curve. The further an isoquant lies from the origin, the greater the output level which it represents: Therefore, Q3 > Q2 >Q1.
The partial derivative of production function ¶Q / ¶L is the marginal productivity of labor (MP1). It is the extra amount of output that can be obtained by employing one additional unit of L, K remaining the same. Similarly, ¶Q / ¶K is the marginal productivity of the capital (MPK). We assume that MP1 or f1 > 0 and MPK or fK>0.