Extension to the Simultaneous Equation Case

The implicit-function theorem also comes in a more general and powerful version that deals with the conditions under which a set of simultaneous equations

The generalized version of the theorem states that;

Given the equation system (8.24), if (a) the functions Fl, Fn ail have continuous partial derivatives with respect to all they and* variables, and if (b) at a point {v\q> .. m v„o; xuu____satisfying (8.24). the following Jacobian determinant is nonzero:

will assuredly define a set of implicit functions^

f To view ir another way, what these conditions serve to do is to assure us that the n equations in (8.24) can in principie be solved for the n variables—yi,..., yn—even if we may not be able to obtain the solution (8.25) in an explicit form.

then there exists an w-dimensional neighborhood uf (xio*..., x^o), .V, in which the variables

_V|, yn arc functions of the variables x]} xp, in the form of (8.25). These implicit functions satisfy yw — /'Uio, rWJfl)

They also satisfy (8.24) for e very w-tuple (,vj. ..., xtll ) in the neighborhood N thereby giving (8.24) the status of a set of identities as far as this neighborhood 15 concerned, Moreover, the implicit functions / ],.,., f" are continuous and have continuous partial derivatives with respect to all (he x variables.

As in the single-equation case, it is possible 10 find the partial derivatives of the implicit functions directly from the n equations in (8.24), without having to solve them for the y variables. Taking advantage of the fact that, in the neighborhood M the equations in (8.24) have the status of identities, we can take the total differential of each of these, and write dFJ =0(j = 1,2The result is a set of equations involving the differentials dy\,... ? dyn and dx\ ?... ? dxm ♦ Specifically, after transposing the dx{ terms to the right of Ihe equals signs, we have

—— i/>! + -— dy2 + ■ ■ + ——dyn = - -—dxi + ■ • • + -—dxm 9 vi dy2 ¿yn \ t)jr] dxm /

—dy! + ~dy2 -h ■ -h —dy, = - —dx\ + - ■ +- —dxa,

—dy{ + ——</)2 + ■■■ + ~—dyn ^ - —dxl 4- - ■ ■ +- -—dxm 8>'i dyi dyn \ 0X\ dxm /

Moreover, from (8.25), we can write the differentials of the yj variables as

OX{ 0X2 QXm dy 2 = —dx i 4- --dx 2 + ■ ■ + ~~dxm

dyn dyn Syn dv„ = —dxi + —dx2 H----+ -—dxm dx-2 3xm and these can be used to eliminate the dy} expressions in (8.26). But since the result of substitution would be unmanageably messy, let us simplify matters by considering only what would happen when x\ alone changes while all the other variables ■ ■ remain constant. Letting dx\ i= 0, but setting d.v2 = ■ ■ ■ = dxm = 0 in (8.26) and (8.27), then substituting (8.27) into (8.26) and dividing through by dx\ ^ 0, we obtain the equation system df[ /l)y{ \ i)F] / dvj \ dF] /3v„\ _

(*F* />A | dF" /'dy2\ | | OF" f dyn\ _ BFfl 'dyi \dX] / dyi \i)xj J \3x\ J 9x}

Even this result—for the ease where alone changes—looks Formidably complcx. because it is full of derivatives. But its structure is actually quite easy to comprehend, oncc we learn to distinguish between the two types of derivatives that appear in (8.28). One type, which we have parenthesized for visual distinction, consists of the partial derivatives of the implicit functions with rcspect to that we are seeking. These, therefore, should be viewed as the "variables" to be solved for in (8.28). The other type, on the other hand, consists of the partial derivatives of the F-' functions given in (8,24), Since they would all take specific values when evaluated at the point (yiq, ..., y^: jrl(),..., xmi))—the point around which the implicit functions are defined they appear here not as derivative functions but as derivative values. As such, they can be treated as given constants. This fact makes (8.28) a linear equation system, with a structure similar to (4.1). What is interesting is that such a linear system has arisen during the process of analysis of a problem that is not necessarily linear in itself, since no linearity restrictions have been placcd on the equation system (8.24), Thus we have here an illustration of how linear algebra can comc into play even in nonlinear problems.

Being a linear equation system, (8.28} can be written in matrix notation as

3F2 3F2 3F1

dyi dy2 Syn

Since the determinant of the coefficient matrix in (8.28') is nothing but the particular Jacobian determinant |7| which is known to be nonzero under conditions of the implicit-function theorem, and since the system must be nonhomogeneous (why?), there should be a unique nontrivial solution to (8.28 ). By Cramer's rule, this solution may be expressed analytically as follows:

By a suitable adaptation of this procedure, the partial derivatives of the implicit functions with rcspcct to the other variables, can also be obtained. It is a nice feature of this procedure that, cach time wc allow a particular Xi variable to change, we can obtain in

'dFl ~

WJ

dxi

/>2\

SF2

UJ

ajr,

SF"

WJ

one fell swoop the partial derivatives of all the implicit functions f \ ..., fn with respect to thai particular x-f variable.

Similarly, to the implicit-function rule (8.23) for the single-equation case, the procedure just described calls only for the use of the partial derivatives of the F functions—evaluated at the point { vio? • - - • • , xmo)—in the calculation of the partial derivatives of the implicit functions f[ , /". Thus the matrix equation (8.28') and its analytical solution (8.29) are in effect a statement of the simultaneous-equation version of the implicit-function rule,

Note that the requirement \ rules out a zero denominator in (8.29), just as the requirement Fv ^ 0 did in the implicit-function rule (8.23) and (8.23'). Also, the role played by the condition \J\ / 0 in guaranteeing a unique (albeit implicit) solution (8.23) to the general (possibly nonlinear) system (8.24) is very similar to the role of the nonsingu-larity condition \ A\ ^ 0 in a linear system Ax = d.

Example 5

The following three equations xy - w — 0 F1 = (x, y, w; z) = 0

are satisfied at point P: {x, y, w; z) = 4,1,1), The F functions obviously possess continuous derivatives. Thus, if the Jacobian | /| is nonzero at point P, we can use the implicit-function theorem to find the comparative-static derivative (dxfdz). To do this, we can first take the total differential of the system:

Moving the exogenous differential (and ib coefficients) to the right-hand side and writing in matrix form, we get dz

y x -1

0

0 1 -3w2

dy

3

0 0 (3w2-2z)

dw

2W-i22

where the coefficient matrix on the left-hand side is the Jacobian

y

x -1

1/1 =

=

0

1 -3iv2

Fy

0

0 (3w2 - 2z)

At the point Pf the Jacobian determinant |/ | = 4 (^ 0). Therefore, the implicit-function rule applies and

2w- 1z2

Using Cramer's rule to find an expression for (ax/ttz), we obtain

0 x -1

0

4

-1

3 1 -3 w2

3

1

-3

Iw-lz1 0 {lw2-2z)

-1

0

} -1

} -1

0 1

1 -3

4

4

Let the national-income model (7,1 7) be rewritten in the form

If we take the endogenous variables (Y, Cf T) to be {/i, y2t yi), and take the exogenous variables and parameters (/o, Co, uf yf to be (*i, ..., x6), then the left-side expression in each equation can be regarded as a specific F function, in the form of P(K C, T, iti G0, af p, y, 5). Thus (8.30) is a specific case of (8.24), with n = 3 and m = 6. Since the functions F \ F2t and F3 do have continuous partial derivatives, and since the relevant Jacobian determinant (the one involving only the endogenous variables),

yf1

SF1

af1

dY

3C

37"

3F2

3 F2

dF1

3 Y

3C

37"

af3

3 f3

ay

3C

or

is always nonzero (both (i and ft being restricted to be positive fractions), we can take Y, C and T to be implicit functions of (kf Go, a, yt 8) at and around any point that satisfies (8.30). But a point that satisfies (8.30) would be an equilibrium solution, relating to V*y C* and T*. Hence, what the implicit-function theorem tells us is that we are justified in writing r = f1(/o, Go, a, ft, y, 8)

indicating that the equilibrium values of the endogenous variables are implicit functions of the exogenous variables and the parameters.

The partial derivatives of the implicit functions, such as dY*/dfo and dY*fdGo, are in the nature of comparative-static derivatives. To find these, we need only the partial derivatives of the F functions, evaluated at the equilibrium state of the model. Moreover, since n= 3, three of these can be found in one operation. Suppose we now hold all exogenous variables and parameters fixed except C0. Then, by adapting the result in (8.28'), we may write the equation

1

-1 0"

~ 3 Y*f;)Go~

"l "

1 fl

öCVäCo

0

.-5

0 1

JT'ß C0.

0

from which three comparative-static derivatives (all with respect to Co) can be calculated. The first one, representing the government-expenditure multiplier, will for instance come out to be

9Ym UG0

This is, of course, nothing but the result obtained earlier in (7.19). Note, however, that in the present approach we have worked only with implicit functions, and have completely bypassed the step of solving the system (8.30) explicitly for F, C*, and T*. It is this particular feature of the method that will now enable us to tackle the comparative statics of general-function models which, by their very nature, can yield no explicit solution.

EXERCISE 8.5

1. For each f (x,y) ^ 0, find dy/dx for each of the following; (a) y-

2. For each f(x, y) - 0 use the implicit-function rufe to find dy/dx:

3. For each F(xf y>z) = 0 use the implicit-function rule to find öyßx and 3y/3z:

4( Assuming that the equation F(U, Xi, *2, *n) = 0 implicitly defines a utility function U — f(X], X2____, xn):

(a) Find the expressions for dU/dxi, dU{tixn, tix^l'dxj, and 8x4 (fc) Interpret their respective economic meanings. 5. For each of the given equations F (y, x) = 0, is an implicit function y = f{x) defined around the point (y= 3; x = 1)?

If your answer is affirmative, find dy/dx by the implicit-function rule, and evaluate it at the said point.

6. Given x2 + ^ 2yz-\- y2 - z2 - 11 = 0, is an implicit function z = f(x, y) defined around the point (x = 1, y = 2f z— 0)? If so, find dzjdx and 'dzjdy by the impficit-function rule, and evaluate them at that point.

7. By considering the equation F(y, x) = (x - y)3 = 0 in a neighborhood around the point of origin, prove that the conditions cited in the implicit-function theorem are not in the nature of necessary conditions.

S, If the equation F(x,yt z) = 0 implicitly defines each of the three variables as a function of the other two variables, and if all the derivatives in question exist, find the r l)z Ox dy value of---- .

dx dy dz

9, Justify the assertion in the text that the equation system (8.28') must be rtonhomo-geneous.

10. From the national-income model (8.30), find the nonincome-tax multiplier by the implicit-function rule, Check your results against (7,20).

8.6 Comparative Statics of General-Function Models

When we first considered the problem of comparative-static analysis in Chap. 7, we dealt with the case where the equilibrium values of the endogenous variables of the model are expressible explicitly in terms of the exogenous variables and parameters. There, the technique of simple partial differentiation was all we needed. When a model contains functions expressed in the general form, however, that technique becomcs inapplicable bccause of the unavailability of explicit solutions. Instead, a new technique must be employed that makes use of such concepts as total differentials, total derivatives, as well as the implicit-function theorem and the implicit-function rule. We shall illustrate this lirst with a market model, and then move on to national-income models.

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