Finding the Total Derivative

To carry on the discussion tn a general framework, let us consider any function y = f(xy w) where x — g(w)

FIGURE 8.4

FIGURE 8.4

Example 1

The two functions f and g can also be combined into a composite function y = f[g(n%w] (8.11')

The three variables v, v, and vv are related to one another as shown in fig. 8.4. In this figure, which we shall refer to as a channel map, it is clearly seen that w—the ultimate source of change—can affect y through two separate channels: (1) indirectly, via the function g and then / (the straight arrows), and (2) directly via the function / (the curved arrow). The direct effect can simply be represented by the partial derivative fw. But the indirect effect dx 9 v dx can only be expressed by a product of two derivatives, fx —. or —, by the chain rule

' aw' dx aw for a composite function. Adding up the two effects gives us the desired total derivative of y with respect to w.

aw aw

This total derivative can also be obtained by an alternative method: We may first differentiate the function v — f(x, w) totally, to get the total differential dy - fx dx + fw dw and then divide through by dw. The result is identical with (8,12). Either way, the process of finding the total derivative dy/dw is referred to as the total differentiation of y with respect to vt:

It is extremely important to distinguish between the two look-alike symbols dy/dw and dy/dw in (8.12). The former is a total derivative, and the latter, a partial derivative. The latter is in fact merely a component of the former.

Find the total derivative dy/dw, given the function y— f(x, w) = 3x - w2 where x = g(w) = 2w2 + w + 4 By virtue of (8.12), the total derivative should be

As a check, we may substitute the function g into the function f, to get y = 3(Iw1 -f- w -f A) - w2 = 5 w2 + 3w+ 12

which is now a function of w alone. The derivative dy/dw is then easily found to be IOw-i- 3, the identical answer.

Chapter 8 Comparative-Static Analysix ofGenerut-Fuwtion Models 191

FIGURE 8.5

Chapter 8 Comparative-Static Analysix ofGenerut-Fuwtion Models 191

FIGURE 8.5

Example 2

If we have a utility function U = L/(c, s), where cis the amount of coffee consumed and s is the amount of sugar consumed; and another function s = indicating the complementarity between these two goods, then we can simply write the composite function from which it follows that

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