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a) I + 2) 3z = 6 and x + 3y 22 = 6;

b) .r + 2y 3r = 6 and -2x - 4y + hz = IO.

10.39 Find a nonparametric equation of the plane:

o) through the point (I. 2.3) and normal to the vec,or (I, -1, 0),

h) through the point (I, I, I) and perpendicular to the line (x,,xz,xi) =

(4 31, 2 + t, 6 + 3);

C) whose intercepts arc (a. 0, 0), (0. b, II), and (0, 0, c) with a, b, and c all nonzero.

10.40 Find the intersection of the plane x + J + z = 1 and the linex = 3 + f,˜ = 1 - 7t,

I = 3 3t.

10.41 Use Gaussian elimination to find the equation of the line which is lhe intersection

ofthcplanesn+?˜r=4andi+2˜+r=3.

EUCLIDEAN SPACES I1 01

232

10.7 ECONOMIC APPLICATIONS

Budget Sets in Commodity Space

An important application of Euclidean spaces in economic theory is the notion of

a commodity space. In an economy with n commodities, let xi denote the amount

of commodity i. Assume that each commodity is completely divisible so that xi

can be any nonnegative number. The vector

x = (Xj,X2,...,X,)

which assigns a nonnegative quantity to each of the n commodities is called a

commodity bundle. Since we are dealing only with nonnegative quantities, the

set of alI commodity bundles is the positive orthant of RвЂќ

{(x,, , I,,) : x, 2 0,. .,x, 2 O}

and is called a commodity space.

Let pi > 0 denote the price of commodity i. Then, the cost of purchasing

commodity bundle x = (xi,. , x?,) is

/I,вЂњ, + ,I$2 + + p,,x,, = p x.

A consumer with income I can purchase only bundles x such that p x 5 I. This

subset of commodity space is called the consumerвЂ˜s budget set. It is bounded

above by the hyperplane p x = I_ whose normal vector is just the price vector

p. WC have drawn the usual two-dimensional picture for this situation in Figure

10.31.

Figure

10.31

110.71 ECONOMIC APPLICATIONS 233

Input space

Asimilarsituation exists foraproduction process which uses n inputs. Ifx; denotes

input then in

an amount of i, x = (xl,. , x,,) is an input vector input space,

which is also the positive orthant in RвЂќ. If w2 denotes the cost per unit of input

i and w = (IV], , w,,), then the cost of purchasing input bundle x is w x. The

set of all input bundles which have a total cost C, an isocost set, is that part of

Ihe hyperplane w. x = C which lies in the positive orthant. The price vector w is

normal to this hyperplane. If we fix w and let C vary, we obtain isocost hyperplanes

which are parallel to each other.

negative

Depending WI the situation under study, we sometimes write inputs as

numbers. In this case, input space would he the negative orthant in RвЂќ.

Probability Simplex

A hyperplane that arises frequently in applications is the of

space probability

vectors

l},

P,, = {@,I,. ., ˜8,) : pi 2 0 and p, + jar + + p,, =

which WC applications there arc n mutually

call a probability simplex. In these

exclusive states of the world and pi is the prohahility that state i occurs. Since one

of these II states must occur, the 11;вЂ˜s sum to 1. The probability simplex P,, is part

of a hypcrplanc in whose normal vector is = (I, I,. , I); tвЂ™? is pictured in

RвЂќ 1

Figure IO..i?.

One can also consider P,, iis the se, ofbaryccntric coordinates with respect to

the points

e, = (I, 0.. 0). e,, = (0. 0.. 0, I).

Figure

10.32

The Investment Model

The portfolio analysis introduced in Example 5 of Chapter 6 fits naturally into the

geometric framework of this chapter.

Suppose that an investor is choosing the fraction xi of his or her wealth to

invest in asset i. If there are A different investment opportunities, a portfolio is

an A-tuple x = (x1,. , x.4). Since the xiвЂ™s represent fractions of total wealth, they

must sum to I. Therefore the budget constraint is

n, + ,ri f + q = 1

However, since we allow short positions, xi may be negative. In this case, the

budget set is the entire hyperplane

x.1=1

normal to the vector 1 = (1, 1,. _, 1). Figure 10.32 shows the intersection of this

hyperplane with the positive orthant of RвЂќ (for II = 3).

Suppose that there are S possible financial climates or вЂњstates of natureвЂќ in

the coming investment period. Let rri denote the return on asset i if states occurs.

Form the states return vector

6 = (rv,, cc, 2 r;, ).

Then, the return to the investor of portfolio x = (.y,, , x˜) is r,, x. A portfolio

x is riskless if it returns the same return in every state of nature:

r, x = r> x = = r.7 x

IS-LM Analysis

We have discussed a linear Keynesian macroeconomic model and HicksвЂ™ IS-LM

interpretation of it in Chapter 6 and again in Chapter Y. In Exercise 9.18, we

examined a more or less complete version of this model in f&c linear equations

which could he comhincd into two equations as

(l-c,(l-/,)-n,,˜Y+(u+c:)r=˜,,˜(.,˜˜,+I˜˜+G

mY-hr=M,-W.

The first equation represents the production equilibrium and is called the IS

(investment-savings) equation. The second represents the money mark& rqui-

lihrium and is called the LM (liquidity-money) equation. In intermediate macro-

economics courses, one studies this system graphically by dnwing the IS-line and

the LM-line in the plane, as in Figure 10.33. The normal vector to the IS-line is

(1 c,(l - I,) - a,,, Lr + Q)

[lo.71 ECONOMIC APPLICATIONS 235

Figure

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