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7.21 вЂ˜The following five matrices arc curfficirnt matrices of systems of linear equations.

For each matrix. what can you say ahout the numhcr of solutions of the corrcspondinp

systrm: a) whrn the right-hand sidr is h, = = h, = 0, and h) for gencrat RHS

h,. h1 , вЂќ

150 SYSTEMS OF LINEAR EqlJATloNs [7]

7.5 THE LINEAR IMPLICIT FUNCTION THEOREM

The situation described in Fact 7. IO arises frequently in mathematical models, as

we discussed last section. The hiвЂ™s on the RHS of (2) represent some externally

determined parameters, while the linear equations themselves represent some

equilibrium condition which determines the internal variables x,, , v,,. Ideally,

there should be a unique equilibrium for each choice of the parameters bl, b,,,.

Fact 7. IO tells us exactly when this ideal situation occurs: the number of equations

must equal the number of unknowns and the coefficient matrix mw have maximal

rank.

In this view, consider once again the IS-LM model described in Chapter 6:

SY + a,- = IвЂќ + G

(23)

mY hr = M.7 -MвЂ™.

Choose numerical values for the parameters s. o, 171. h. I*. G, and MвЂќ in system (23).

However, think of M,. the money supply, as a variable policy parameter which a

policymaker can set externally. For each choice of money supply. the economy

reaches an equilibrium in Y and IвЂ™. Since we have two equations in two unknowns,

Fact 7.10 tells us that system (23) will indeed detertnine a unique (Y.r) pair for

each choice of M,, provided the coeflicient matrix

has rank twt,.

In this IS-LM model the variables Y and I are called endugenous variables

becauss their values are determined by the system of equations under consider-

ation. On the other hand. ,W, is called an exogenous variable because its value

is determined outside of system (2.7). If we were to treat .x. o. m. ii. I^. G. and M*

as parameters also. then they too would be exogenous variables. Mathematicians

would call exogenous variables independent variables and endogenous variables

dependent variables.

A genrwl linear model will have 111 equations in II unkmrwns:

U,,,,˜T, T c I,,, 2.12 + + u ,,,,,. I,, = h,,,.

Usually there will be a natural division of the xiвЂ™s into exogenous and endoeenous

variables given by the Imodel. This division will be successful only if. after choosing

values for the exogenous vxiablcs and plugging them into system (24). one can

then unambiguously solve the system for the rest of the variables, the endogenous

ones. Fact 7.10 in the last section tells us the two conditions that must hold in

order for this breakdown into exogenous variables and endogenous variables to

be successful. There must be exactly as many endogenous variables as there are

equations in (24), and the square matrix corresponding to the endogenousvariables

must have maximal rank m. This statement is a version of the Implicit Function

Theorem for linear equations, and is summarized in the following theorem.

Theorem 7.1 Let n,, _, xk and xk+,, .,x. be a partition of the n variables

in (24) into endogenous and exogenous variables, respectively. There is, for

:ach choice of values xF+,, , .rE for the exogenous variables, a unique set of

values xy, _, x: which solves (24) if and only if:

(a) k = m (number of endogenous variables = number of equations) and

(b) the rank of the matrix

corresponding to the endogenous variables, is k

Under the conditions of Theorem 7.1, we can think of system (24) as implicitly

presenting each of the endogenous variables as functions of all the exogenous

variables. Later, we will strengthen this result and use it as motivation for the

Implicit Function Theorem for nonlinear systems of equations-a result which

will be the cornerstone of our treatment of nonlinear equations, especially applied

to comparative St&x in economic models.

EXERCISES

1.25 For each of the following two systems, we want to separate the variables into

exogenous and rndogenous ones so Ihal each choice of values for the exogenous

variables dctcrmines unique YBIUZS for the endogenous variables. For each system,

a) determine hnu many variahlcs can he mdogrnous at any one time, h) determine

a successful separation into exogenous and endagenaus variables, and c) find an

explicit lormula for the endogenous variables in terms of the exogenous ones:

x+*y+ z- w=l

r+2y+z- Iv= 1

9

152 SYSTEMS OF LINEAR EqUATloNs [71

1.26 For Example I in Chapter 6, write out the linear system which corresponds Lo

equation (I) in Chapter 6 but with the $100,000 bcforc-tax profit replaced by a

general before-tax profit P. Solve the resuking system for C, S, and F in tmns of P.

7 . 2 7 For the values of the constants in Exercise 6.7, show that each choice ofM, uniquely

determines an equilibrium (V, I).

7 . 2 8 n) In IS-LM model (23). USC Gaussian elimination to find a general formula in-

valving .s, n, m, and II which, when satisfied. will guarantee that system (23)

determines a unique wlue of Y and I for each choice of I-, MвЂ™. G; and M,.

h) In this case, find an explicit formula for Y and I in terms of all the other variables.

c) Note how changes in each of the exogenous variables affect the values of Y and IвЂ™.

7.29 Consider the system

w - x+3y- r=вЂќ

hвЂ™ + 4.x )вЂ™ + 2r = 3

3w+7x+ y+ z= 6 .

a) Separate the variables into endogenous and exogenous ones so that each choice

of the exogenous variables uniquely dctcrmines values far the sndogcnws ones.

b) For your answer to a, what are the values of the endogenous variables when all

the exogenous variables arc set equal to II?

c) Find a separation into rndogrnous and exogenous variables (aamc number of

each as in part a) that will mat work in the sense of o Find a value of the new

exogenous variables for which thcrc UC infinitely many carrrspondinL: values of

the cndngcnaus variables.

7.30 CвЂ™unsidrr the sy>˜em

IV I - 3x I = 0

IV + 4s .v - I _ 3

3w + 7x - ? - I = 6.

Is there any wcccssful dccompnsition into cndr˜gcnous and erogenous wriahle\вЂ˜.вЂ™

Fwplain.

C H A P T E R 1 0

Euclidean Spaces

As we discussed at the end of Chapter 1, one of the main uses of mathematical anal-

ysis in economic theory is to help construct the appropriate geometric and analytic

generalizations of the two-dimensional geometric models that are the mainstay of

undergraduate economics courses. In this chapter, we begin these constructions by

studying how to generalize notions of points, lines, planes, distances, and angles to

n-dimensional Euclidean spaces. Later, our analyses of n-commodity economies

will make heavy use of these concepts.

The first three sections of this chapter present the basic geometry of coor-

dinates. points and displacements in n-space. If this material is familiar to most

students, it can be left as a background reading assignment.

10.1 POINTS AND VECTORS IN EUCLIDEAN SPACE

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