Simultaneous Equation Approach

The analysis of model (8.32) was carried out on the basis of a single equation, namely, (8.35). Since only one endogenous variable can fruitfully be incorporated into one equation, the inclusion of P* means the exclusion of Q*. As a result, we were compelled to find (dP*fdYo) first and then to infer (dQ*fdYo) in a subsequent step. Now we shall show how P* and Of can be studied simultaneously. As there are two endogenous variables, we shall

accordingly set up a two-equation system. First, letting Q = Qd — Q, in (8.32) and rearranging, we can express our market model as

which is in the form of (8.24), with n— 2 and m = 1. It becomes of interest, once again, to check the conditions of the implicit-function theorem. First, since the demand and supply functions are both assumed to possess continuous derivatives, so must the functions F1 and F1. Second, the endogenous-variable Jacobian (the one involving P and Q) indeed turns out to be nonzero, regardless of where it is evaluated, because

3F]

9F>

im

ÖP

HQ

BP

dF2

3F2

dS dP

dP

»Q

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, (8.38) 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*jdY^) and {dQ* jdY$) can be found simultaneously by using the implicit-function rule (8.28').

)n the present context, with F1 and F2 as defined in (8.41), and widi two endogenous variables P* and 0* and a single exogenous variable Yfj, the implicit-function rule takes tlie specific form

ÖF1

aF

dQ"

,) F2

■SF2

ip*

Note that the comparative-stain; derivatives are wriüen here with the symbol drather than 3, because there is only one exogenous variable in the present problem. More specifically, the last equation can be expressed as dP* dS

W0 0

By Cramer's rule, and using (8.39), we then find the solution to be dD

w"

byq o

\dYoJ

dD

dD

BP*

9Y,n

dS

0

dS 3D

dP*

dP* a/«

W\

where all the derivatives of the demand and supply functions (including those appearing in the Jacohian) are to be evaluated at the initial equilibrium. You can check that the results just obtained are identical with those obtained earlier in (8.36) and (8.37), by means of the single-equation approach,

Instead of directly applying the implicit-function rule, we can also reach the same result by first differentiating totally each identity in (8,41) in turn, to get a linear system of equations in the variables dP* and dQ*:

dYo dY*

and then dividing through by dYn ^ 0, and interpreting each quotient of two differentials as a derivative.

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