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(14)

120 bmmumm TO LINMK ALCE˜RA 161

Systems (13) and (14) state that the current price v j of asset,j is equal to a weighted

sum of its returns in each state of nature, with the same weights for each j. The

weight ˜1,˜ for slate .Y is a kind of price f<v wealth in state .s and is often called a

state price. If can price states, then the price of each asset is just the value iit

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the state prices of the rctums in each state. This is the content of the linear hystcm

of equations (13).

Since all the equations in this application are linear, it is not surprising that

techniques of linear algebra can answu questions aboul Ihe existence and chsrac-

tuization of riskless, duplicahlc, and arbitrage portfolios and of insurable states

and state prices.

Lwnz˜l˜˜ 6.1 Suppose that there arc twu assets and three possible slates. If state

R 12 = 3. If state 2 occurs,

I occun, asset I TetвЂќrвЂќs K,, = I and asset 2 returns

Rz, = 2 and Rzl = 2. If state 3 occurs, R II = 3 and Rj2 = I, If both asscfs

have Ihe same current value and if the investor buys n, = 3 shares of asset I

and n1 = I share of asset 2, the corresponding portfolio is ($ i) and the returns

ire

l-3

R,,.;+R,2.z-i in state I,

R?, ; + RI? iI=2 in slate 2.

; + RJ1. I-i

R7, jI IвЂќ state 3.

Purlfolio (4, 4) is a riskfree portfolio since it yields a xlurn of 2 in all three states

(check). The 3.tuplc (A, 4, 4) is a pricing system for this ec˜˜nomy (cheek). As

we will see in Section 7.1. tlxrc xc !no duplicable portfolios and no insumhlc

states.

Thcsc tivc examples illustmtc the important role that linear models play in

wonomics and indeed in all the social sciences. We conclude this chsptcr hy ˜mcn-

ti˜ming three other- instances where economists MC lincx ;dgebra. First. many ot

the elemenwry tcchniqucs ˜Icconometrics. such as (gcneralixd) Icast-squares c\-

firnation. rely heavily on lineusystcms of equations. Second. linear programming.

the optimization of a linear functiw on a xt dcfincd by a syslcm of linear equali-

tics and inequalities. is a fundemcntal economic tccbniquc. As such. a number of

textbook\ arc dcwtcd entirely 10 it and it is the total suh,jcct matter of giaduatc

courses in mathematics and engineering. ar well as economics. Finally. W C will

oi

вЂ˜cl? on linear algrhra techniques when study the gcncraliation the second

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derivative test in C˜IICIIILIS to maximization problems which inwlvc (m˜nlincar)

iunctions of more than one variahlc.

6.2 In Missouri. federal income taxes are deduc,ed from state taxes. Write out and solve

the system of equations which describes the stale and federal taxes and charitable

contribution of the firm in Example I if it were based in Missouri.

6.3 The economy on the island of Bacchus produces only grapes and wine. The production

of I pound of grapes requires 112 pound of grapes, I laborer. and na wine. The

production of I liter of wine requires 112 pound of grapes, I laborer, and 114 liter of

wine. The island has 10 laborers who all together demand I pound of grapes and 3

litcrs of wine for their own consumption. Write out the input-output system for the

economy of this island. Can you solve it?

6.4 Suppose that the produclion of a pound of grapes now requires 7/X liter of wine. If

none of the other input-output coefficients change, write out the new systems for the

OвЂќ,PвЂќ,F.

6.5 Suppose that IO percent of white males of working age and 20 percent of black males

of working age are unemployed in 1966. According to HallвЂ™s model, what will the

corresponding unemployment rates be in 1967вЂ™?

6.6 For the Markw employment model, Hall gives p = ,106 and 4 = ,993 for black

females. and p = .I5 I and <, = ,907 for white females. Write out the Markov systemr

of difference equalions for there two situations. Compute the stationary distributions.

6.1 Consider the IS-LM model of Example 4 with no fiscal policy (G = 0). Suppose

that M, = MвЂќ: that is. the intercept of the LM curve is 0. Suppose that IвЂќ = 1000.

,< = 0.2, h = 1500, u = 2000, and ,n = 0.16. Write out the explicit IS-LM syslem of

equations. Solve rhcm for the equilibrium GNP Y and the interest rater.

6.8 Carry OU, the two checks at the end of Example 5.

NOTES

C H A P T E R 7

Systems of Linear

Equations

As was discussed in the last chapter, systems of linear equations arise in two ways

in economic theory. Some economics models have a natural linear structure, like

the five examples in the last chapter. On the other hand, when the relationships

among the variables under consideration are described by a system of nonlinear

equations, one takes the derivative of these equations to convert them to an approx-

imating linear system. Theorems of calculus tell us that by studying the properties

of this latter linear system, we can learn a lot about the underlying nonlinear

system.

In this chapter we begin the study of systems of linear equations by describing

techniques for solving such systems. The preferred solution technique-Gaussian

elimination-answers the fundamental questions about a given linear system:

does a solution exist, and if so, how many solutions are there?

An implicit system is one in which the equations that describe the economic

relationships under study have the exogenous and endogenous variables mixed in

with each other on the same side of the equal signs. This chapter closes with a

discussion of the Linear Implicit Function Theorem, which tells how to use linear

algebra techniques to quantify the &ect of a change in the exogenous variables

on the endogenous ones in a linear implicit system.

7.1 GAUSSIAN AND GAUSS-JORDAN ELIMINATION

We begin our study of linear phenomena by considering the problem of solving

linear systems of equations, such as

2x, + 3x2 = 7 XI + nz + XJ = 5

or (1)

x,- xz=I XL - x3 = 0.

The general linear system of m equations in n unknowns can be written

(2)

CZ,˜, X˜ + a,,,Zxz + + amx,, = b,,.

h,вЂ˜s are given real numbers; uCj is the coefficient of

In this system, the ujjвЂ™s and

the unknown x i in the ith equation. A of system (2) is an n-tuple of real

solution

numbers x1, x2,. , x,, which satisfies each of the m equations in (2). For example,

xl = 2, x2 = 1 solves the first system in (l), and x, = 5, x1 = 0, xx = 0 solves

the second.

For a linear system such as (2), we are interested in the following three

questions:

(1) Does a solution existвЂ™?

(2) How many solutions arc there?

(3) Is there an efficient algorithm that computes actual solutions?

There are essentially three ways of solving such systems:

(1) substitution,

(2) elimination of variables, and

(3) matrix methods.

Substitution

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