Use of Total Derivatives

In both the single-equation and the simultaneous-equation approaches illustrated above, we have taken the total differentials of both sides of an equilibrium identity and then equated the two results to arrive at the implicit-function rule. Instead of taking the total differentials, however, it is possible to take, and equate, the total derivatives of the two sides of the equilibrium identity with respect to a particular exogenous variable or parameter. In the single-equation approach, for instance, the equilibrium identity is

Taking the total derivative of the equilibrium identity with respect to Yq—which takes into account the indirect as well as the direct effects of a change in Yq—will therefore give us the equation i)D_ _ dS /dP*\

I indirect effect \ / direct effect \ / indirect effect \ ^ of Yn on 1 \ of ^u on D ) ^ of Y» on 5 }

When this is solved for (dP*/dY^)? the result is identical with the one in (8.36).

FIGURE 8.7

FIGURE 8.7

In the simultaneous-equation approach, on the other hand, there is a pair of equilibrium identities:

The various effects of are now harder to keep track of, but with the help of the channel map in Fig. 8,7, the pattern should become clear. This channel map tells us, for instance, that when differentiating the D function with respect to Jo, we must allow for the indirect effect of Y{} upon D through P*f as well as the direct effect of ic (curved arrow), in differentiating the S function with respect to on the other hand, there is only the indirect effect (through P*) to be taken into account. Thus the result of totally differentiating the two identities with respect to is, upon rearrangement, the following pair of equations:

These are, of course, identical with the equations obtained by the total-differential method, and they lead again to the comparative-static derivatives in (8,42).

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