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3D dP

Hence, if an equilibrium solution (P,Q) exists (as we must assume in order to make it meaningful to talk about comparative statics), the implicit-function theorem tells us that we can write the implicit functions

even though we cannot solve for P and Q explicitly. These functions are known to have continuous derivatives. Moreover, (8.33) will have the status of a pair of identities in some neighborhood of the equilibrium state, so that we may also write

From these, (dP/dY0) and (dQ/dY0) can be found simultaneously.

These two derivatives have as their ingredients the differentials dP, dQ, and dY0. To bring these differential expressions into the picture, we differentiate each identity in (8.36) in turn. The result, upon rearrangement, is a linear system in dP and d&

This system is linear because dP and dQ (the variables) both appear in the first degree, and the coefficient derivatives (all to be evaluated at the initial equilibrium) and dY0 (an arbitrary, nonzero change in the exogenous variable) all represent specific constants. Upon dividing through by dY0 and interpreting the quotient of two differentials as a derivative, we have the matrix equation*

dQ_ dY0

3D dYn

* Without going through the steps of total differentiation and division by dY0, the same matrix equation can be obtained from an adaptation of the implicit-function rule (8.23').

By Cramer's rule, and using (8.34), we then find the solution to be

dP dYn dQ dYn

3D 3Yn

3D 3P dS dP

0 0

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