Jacques I. Mathematics for Economics and Business

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Matrices

490

For the money market

= M

= k

Y + k

r + k

, k

> 0, k

< 0)

Show that when the commodity and money markets are both in equilibrium, the income, Y, and

interest rate, r, satisfy the matrix equation

and solve this system for Y and r. Write down the multiplier for r due to changes in M*

and deduce

that interest rates fall as the money supply grows.

13 Find (where possible) the inverse of the matrices

A = B =

Are these matrices singular or non-singular?

14 Find the determinant of the matrix

A =

in terms of a. Deduce that this matrix is non-singular provided a ≠ 1 and find A

−1

in this case.

15 Find the inverse of

Hence find the equilibrium prices of the three-commodity market model given in Practice Problem 13

of Section 1.3.

16 (Maple) Use the Maple instruction det(A)to find the determinant of the 4 × 4 matrix

For what value of

a does this matrix fail to possess an inverse? [Do not forget to load the linear

algebra package before you begin. You do this by typing with(linalg):

]

17 (Maple)

(a) Find the determinant of the general 3 × 3 matrix, A

Deduce that this matrix is singular whenever the second row is a multiple of the first row.

(b) Find the inverse, A

−1

, of the general 3 × 3 matrix given in part (a) and verify that AA

−1

= I.

abc

def

ghi

a 21−1

2341

0 −15 6

124−3

−221

1 −5 −1

2 −1 −6

213

10a

314

145

213

−132

21−1

13 2

−12 1

b + d

− k

1 − a −c

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18 (Maple) The equilibrium levels of income, Y, consumption, C, and taxation, T, satisfy the equations

Y = C + I* + G*

C = a(Y − T ) + b

T = tY

Show that this system can be written in the form Ax = b where

A = x = b =

(a) Find A

−1

and give an interpretation of the element in the first row and second column of A

−1

(b) By using A

−1

, or otherwise, solve this system of equations.

I* + G*

1 −10

−a 1 a

t 0 −1

7.2 • Matrix inversion

491

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section 7.3

Cramer’s rule

In Section 7.2 we described the mechanics of calculating the determinant and inverse of 2 × 2

and 3 × 3 matrices. These concepts can be extended to larger systems in an obvious way,

although the amount of effort needed rises dramatically as the size of the matrix increases. For

example, consider the work involved in solving the system

using the method of matrix inversion. In this case the coefﬁcient matrix has order 4 × 4 and so

has 16 elements. Corresponding to each of these elements there is a cofactor. This is deﬁned to

be the 3 × 3 determinant obtained by deleting the row and column containing the element,

preﬁxed by a ‘+’ or ‘−’ according to the following pattern:

Determinants are found by expanding along any one row or column and inverses are found by

stacking cofactors as before. However, given that there are 16 cofactors to be calculated, even

+−+−

−+−+

+−+−

−+−+

−1

−24

10 23

−15 41

07−36

24 51

Objectives

At the end of this section you should be able to:

 Appreciate the limitations of using inverses to solve systems of linear equations.

 Use Cramer’s rule to solve systems of linear equations.

 Apply Cramer’s rule to analyse static macroeconomic models.

 Apply Cramer’s rule to solve two-country trading models.

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7.3 • Cramer’s rule

493

the most enthusiastic student is likely to view the prospect with some trepidation. To make

matters worse, it frequently happens in economics that only a few of the variables x

are actu-

ally needed. For instance, it could be that the variable x

is the only one of interest. Under these

circumstances it is clearly wasteful expending a large amount of effort calculating the inverse

matrix, particularly since the values of the remaining variables, x

, x

and x

, are not required.

In this section we describe an alternative method that ﬁnds the value of one variable at a

time. This new method requires less effort if only a selection of the variables is required. It is

known as Cramer’s rule and makes use of matrix determinants. Cramer’s rule for solving any

n × n system, Ax = b, states that the ith variable, x

, can be found from

where A

is the n × n matrix found by replacing the ith column of A by the right-hand-side

vector b. To understand this, consider the simple 2 × 2 system

and suppose that we need to ﬁnd the value of the second variable, x

, say. According to

Cramer’s rule, this is given by

where

A = and A

Notice that x

is given by the quotient of two determinants. The one on the bottom is that

of the original coefﬁcient matrix A. The one on the top is that of the matrix found from A

by replacing the second column (since we are trying to ﬁnd the second variable) by the right-

hand-side vector

In this case the determinants are easily worked out to get

det(A

) ==7(12) − (−6)(4) = 108

det(A) ==7(5) − 2(4) = 27

Hence

==4

108

7 −6

412

−6

7 −6

412

4 5

det(A

)

det(A)

−6

4 5

det(A

)

det(A)

Example

Solve the system of equations

using Cramer’s rule to ﬁnd x

−9

123

−416

275



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Matrices

494

Solution

Cramer’s rule gives

where A is the coefﬁcient matrix

and A

is constructed by replacing the ﬁrst column of A by the right-hand-side vector

which gives

If we expand each of these determinants along the top row, we get

det(A

) =

= 9 − 2 + 3

= 9(−37) − 2(−123) + 3(−76)

=−315

and

det(A) =

= 1 − 2 + 3

= 1(−37) − 2(−32) + 3(−30)

=−63

Hence

===5

−315

−63

det(A

)

det(A)

−41

−46

123

−416

275

−91

13 7

−96

13 5

923

−916

1375

923

−916

1375

−9

123

−416

275

det(A

)

det(A)

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7.3 • Cramer’s rule

495

We now illustrate the use of Cramer’s rule to analyse economic models. We begin by con-

sidering the three-sector macroeconomic model involving government activity.

Example

The equilibrium levels of income, Y, disposable income, Y

, and taxation, T, for a three-sector macro-

economic model satisfy the structural equations

Y = C + I* + G*

C = aY

+ b (0 < a < 1, b > 0)

= Y − T

T = tY + T * (0 < t < 1, T* > 0)

Show that this system can be written as Ax = b, where

A = x = b =

Use Cramer’s rule to solve this system for Y.

Solution

In this model the endogenous variables are Y, C, Y

and T, so we begin by manipulating the equations so that

these variables appear on the left-hand sides. Moreover, since the vector of ‘unknowns’, x, is given to be

I* + G*

1 −100

01−a 0

−1011

−t 001

Practice Problem

1 (a) Solve the system of equations

+ 4x

= 16

− 5x

=−9

using Cramer’s rule to find x

(b) Solve the system of equations

+ x

+ 3x

= 8

−2x

+ 5x

+ x

= 4

+ 2x

+ 4x

= 9

using Cramer’s rule to find x

Advice

The incorporation of government expenditure and taxation into the model has already

been considered in Section 5.3, and you might like to compare the working involved in the

two approaches.



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Matrices

496

we need to arrange the equations so that the variables appear in the order Y, C, Y

, T. For example, in the

case of the third equation

= Y − T

we ﬁrst subtract Y and add T to both sides to get

− Y + T = 0

but then reorder the terms to obtain

−Y + Y

+ T = 0

Performing a similar process with the remaining equations gives

Y − C = I* + G*

C − aY

= b

−Y + Y

+ T = 0

−tY + T = T*

so that in matrix form they become

The variable Y is the ﬁrst, so Cramer’s rule gives

Y =

where

and

A =

The calculations are fairly easy to perform, in spite of the fact that both matrices are 4 × 4, because they con-

tain a high proportion of zeros. Expanding A

along the ﬁrst row gives

det(A

) = (I * + G*) − (−1)

b −a 0

011

T*01

1 −a 0

011

001

1 −100

01−a 0

−1011

−t 001

I* + G* −100

b 1 −a 0

0011

T*001

det(A

)

det(A)

I* + G*

1 −100

01−a 0

−1011

−t 001

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7.3 • Cramer’s rule

497

Notice that there is no point in evaluating the last two cofactors in the ﬁrst row, since the corresponding ele-

ments are both zero.

For the ﬁrst of these 3 × 3 determinants we choose to expand down the ﬁrst column, since this column

has only one non-zero element. This gives

= (1) = 1

It is immaterial which row or column we choose for the second 3 × 3 determinant, since they all contain two

non-zero elements. Working along the ﬁrst row gives

= b − (−a) = b − aT*

Hence

det(A

) = (I* + G*)(1) − (−1)(b − aT*) = I* + G* + b − aT*

A similar process can be applied to matrix A. Expanding along the top row gives

det(A) = (1) − (−1)

The ﬁrst of these 3 × 3 determinants has already been found to be 1 in our previous calculations. The sec-

ond 3 × 3 determinant is new and if we expand this along the ﬁrst row we get

=−(−a) = a(−1 + t)

Hence

det(A) = (1)(1) − (−1)a(−1 + t) = 1 − a + at

Finally we use Cramer’s rule to deduce that

Y =

I* + G* + b − aT*

1 − a + at

−11

−t 1

0 −a 0

−111

−t 01

0 −a 0

−111

−t 01

1 −a 0

011

001

T*1

b −a 0

011

T*01

1 −a 0

011

001

Practice Problem

2 Use Cramer’s rule to solve the following system of equations for Y

[Hint: the determinant of the coefficient matrix has already been evaluated in the previous worked

example.]

I* + G*

1 −100

01−a 0

−1011

−t 001

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498

We conclude this section by introducing foreign trade into our model. In all of our previous

macroeconomic models we have implicitly assumed that the behaviour of different countries

has no effect on the national income of the other countries. In reality this is clearly not the case

and we now investigate how the economies of trading nations interact. To simplify the situ-

ation we shall ignore all government activity and suppose that there are just two countries,

labelled 1 and 2, trading with each other but not with any other country. We shall use an obvi-

ous subscript notation so that Y

denotes the national income of country 1, C

denotes the con-

sumption of country 2 and so on. In the absence of government activity the equation deﬁning

equilibrium in country i is

= C

+ I

+ X

− M

where I

is the investment of country i, X

is the exports of country i and M

is the imports

of country i. As usual, we shall assume that I

is determined exogenously and takes a known

value I *

Given that there are only two countries, which trade between themselves, the exports of one

country must be the same as the imports of the other. In symbols we write

= M

and X

= M

We shall assume that imports are a fraction of national income, so that

= m

where the marginal propensity to import, m

, satisﬁes 0 < m

< 1.

Once expressions for C

and M

are given, we can derive a system of two simultaneous

equations for the two unknowns, Y

and Y

, which can be solved either by using Cramer’s rule

or by using matrix inverses.

Example

The equations deﬁning a model of two trading nations are given by

= C

+ I*

+ X

− M

= C

+ I*

+ X

− M

= 0.8Y

+ 200 C

= 0.9Y

+ 100

= 0.2Y

= 0.1Y

Express this system in matrix form and hence write Y

in terms of I*

and I *

Write down the multiplier for Y

due to changes in I*

and hence describe the effect on the national

income of country 1 due to changes in the investment in country 2.

Solution

In this problem there are six equations for six endogenous variables, Y

, C

, M

and Y

, C

, M

. However,

rather than working with a 6 × 6 matrix, we perform some preliminary algebra to reduce it to only two equa-

tions in two unknowns. To do this we substitute the expressions for C

and M

into the ﬁrst equation to get

= 0.8Y

+ 200 + I*

+ X

− 0.2Y

Also, since X

= M

= 0.1Y

, this becomes

= 0.8Y

+ 200 + I*

+ 0.1Y

− 0.2Y

which rearranges as

0.4Y

− 0.1Y

= 200 + I*

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7.3 • Cramer’s rule

499

A similar procedure applied to the second set of equations for country 2 gives

−0.2Y

+ 0.2Y

= 100 + I*

In matrix form this pair of equations can be written as

Cramer’s rule gives

To ﬁnd the multiplier for Y

due to changes in I*

we consider what happens to Y

when I*

changes by an

amount ∆I*

. The new value of Y

is obtained by replacing I*

by I*

+∆I*

to get

so the corresponding change in Y

∆Y

=−

=∆I*

We deduce that any increase in investment in country 2 leads to an increase in the national income in coun-

try 1. Moreover, because

/3 > 1, the increase in national income is greater than the increase in investment.

0.1

0.06

50 + 0.2I*

+ 0.1I*

0.06

50 + 0.2 I*

+ 0.1(I*

+∆I*

)

0.06

50 + 0.2I*

+ 0.1(I*

+∆I*

)

0.06

50 + 0.2I*

+ 0.1I*

0.06

i200 + I*

−0.1i

200 + I*

0.2

0.4 −0.1

−0.2 0.2

200 + I*

100 + I*

0.4 −0.1

−0.2 0.2

Practice Problem

3 The equations defining a model of two trading nations are given by

= C

+ I*

+ X

− M

= C

+ I*

+ X

− M

= 0.7Y

+ 50 C

= 0.8Y

+ 100

= 200 I*

= 300

= 0.3Y

= 0.1Y

Express this system in matrix form and hence ﬁnd the values of Y

and Y

. Calculate the balance of pay-

ments between these countries.

Cramer’s rule A method of solving simultaneous equations, Ax = b, by the use of

determinants. The ith variable x

can be computed using det(A

)/det(A) where A

is the

determinant of the matrix obtained from A by replacing the ith column by b.

Key Terms

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