JyTI2 Geometric Interpretation of Partial Derivatives

As a special type of derivative, a partial derivative is a measure of the instantaneous rates of change of some variable, and in that capacity it again has a geometric counterpart in the slope of a particular curve.

Let us consider a production function Q = Q{K, L), where Q, K, and L denote output, capital input, and labor input, respectively. This function is a particular two-variable version of (7,12). with n = 2. We can therefore define two partial derivatives 3 Qjd K {or QK ) and 3 Qj'dL (or Qf ). The partial derivative relates to the rates of change of output with respect to infinitesimal changes in capital, while labor input is held constant. Thus Qk symbolizes the marginal-physical-product-of-capital (MPP^) function. Similarly, the partial derivative QL is the mathematical representation of the MPPL function.

Geometrically, the production function Q = Q(K, L) can be depicted by a production surface in a 3-space, such as is shown in Fig. 7.4. The variable Q is plotted vertically, so that for any point (K, L) in the base plane (KL plane)* the height of the surface will indicate the output Q. The domain of the function should consist of the entire nonnegative quadrant of the base plane, but for our purposes it is sufficient to consider a subset of it, the

FIGURE 7,4

rectangle QK^BL^ As a consequence, only a small portion of the production surface is shown in the figure.

Let us now hold capital fixed at the level and consider only variations in the input L. By setting K = Kq, all points in our (curtailed) domain become irrelevant except those on the line segment K$B. By the same token, only the curve K^CDA (a cross section of the production surface) is germane to the present discussion. This curve represents a total-physical-produet-of-labor (TPPA) curve for a fixed amount of capital K — Ko; thus we may read from its slope the rate of change of Q with respect to changes in L while K is held constant. It is clear, therefore, that the slope of a curve such as K0CDA represents the geometric counterpart of the partial derivative Ql . Once again, we note that the slope of a total (TPPi) curve is its corresponding marginal (MPPA = Qi) curve.

As mentioned earlier, a partial derivative is a function of all the independent variables of the primitive function. That Qi, is a function of L is immediately obvious from the KQCDA curve itself When L = L \, the value of QL is equal to the slope of the curve at point CT; but when L = the relevant slope is the one at point A Why is Qi also afunction of AT? The answer is that K can be fixed at various levels, and for each fixed level of K, there results a different TPPi curve (a different cross section of the production surface), with inevitable repercussions on the derivative Qi. Hence Qi is also a function of K,

An analogous interpretation can be given to the partial derivative Qk . If the labor input is held constant instead of K (say, at the level of ¿0)* the line segment L0B will be the relevant subset of the domain, and the curve LqA will indicate the relevant subset of the production surface. The partial derivative QK can then be interpreted as the slope of the curve LoA—bearing in mind that the K axis extends from southeast to northwest in Tig. 7.4. It should be noted that Qk is again a function of both the variables L and K.

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