Chiang Solution

144 Pages·2005·1.03 MB·English

Instructor’s Manual

to accompany

Fundamental Methods
of
Mathematical Economics

Fourth Edition

Alpha C. Chiang
University of Connecticut

Kevin Wainwright
British Columbia Institute of Technology

www.mhhe.com

Title of Supplement to accompany
FUNDAMENTAL METHODS OF MATHEMATICAL ECONOMICS
Alpha C. Chiang, Kevin Wainwright

Published by McGraw-Hill, an imprint of The McGraw-Hill Companies, Inc., 1221 Avenue of the Americas,
New York, NY 10020. Copyright  2005, 1984, 1974, 1967 by The McGraw-Hill Companies, Inc.
All rights reserved.

The contents, or parts thereof, may be reproduced in print form solely for classroom use with
FUNDAMENTAL METHODS OF MATHEMATICAL ECONOMICS
provided such reproductions bear copyright notice, but may not be reproduced in any other form or for any
other purpose without the prior written consent of The McGraw-Hill Companies, Inc., including, but not limited
to, in any network or other electronic storage or transmission, or broadcast for distance learning.

ISBN 0-07-286591-1 (CD-ROM)

Contents

CONTENTS
1

CHAPTER 2
6

Exercise 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6

Exercise 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6

Exercise 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7

CHAPTER 3
9

Exercise 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9

Exercise 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9

Exercise 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10

Exercise 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11

CHAPTER 4
13

Exercise 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13

Exercise 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14

Exercise 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15

Exercise 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17

Exercise 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19

Exercise 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20

Exercise 4.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21

CHAPTER 5
22

Exercise 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22

Exercise 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23

Exercise 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24

Exercise 5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25

Exercise 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26

Exercise 5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27

Exercise 5.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29

CHAPTER 6
32

1

Exercise 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32

Exercise 6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32

Exercise 6.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33

Exercise 6.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33

Exercise 6.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34

CHAPTER 7
35

Exercise 7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35

Exercise 7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35

Exercise 7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37

Exercise 7.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37

Exercise 7.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38

Exercise 7.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39

CHAPTER 8
40

Exercise 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40

Exercise 8.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41

Exercise 8.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43

Exercise 8.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44

Exercise 8.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45

Exercise 8.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47

CHAPTER 9
51

Exercise 9.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51

Exercise 9.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51

Exercise 9.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52

Exercise 9.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54

Exercise 9.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55

CHAPTER 10
56

Exercise 10.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56

Exercise 10.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56

Exercise 10.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57

Exercise 10.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58

2

Exercise 10.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59

Exercise 10.6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60

Exercise 10.7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61

CHAPTER 11
63

Exercise 11.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63

Exercise 11.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64

Exercise 11.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66

Exercise 11.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69

Exercise 11.6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71

Exercise 11.7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73

CHAPTER 12
76

Exercise 12.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76

Exercise 12.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77

Exercise 12.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78

Exercise 12.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81

Exercise 12.6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83

Exercise 12.7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85

CHAPTER 13
87

Exercise 13.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87

Exercise 13.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88

Exercise 13.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91

CHAPTER 14
92

Exercise 14.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92

Exercise 14.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93

Exercise 14.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95

Exercise 14.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96

Exercise 14.6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
97

CHAPTER 15
98

Exercise 15.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98

3

Exercise 15.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99

Exercise 15.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
100

Exercise 15.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
101

Exercise 15.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
102

Exercise 15.6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
103

Exercise 15.7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
104

CHAPTER 16
106

Exercise 16.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
106

Exercise 16.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
107

Exercise 16.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
109

Exercise 16.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
110

Exercise 16.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
112

Exercise 16.6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
114

Exercise 16.7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
115

CHAPTER 17
117

Exercise 17.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
117

Exercise 17.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
118

Exercise 17.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
118

Exercise 17.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
120

Exercise 17.6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
121

CHAPTER 18
123

Exercise 18.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
123

Exercise 18.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
124

Exercise 18.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
125

Exercise 18.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
126

CHAPTER 19
129

Exercise 19.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
129

Exercise 19.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
131

Exercise 19.4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
133

Exercise 19.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
135

4

Exercise 19.6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
138

CHAPTER 20
141

Exercise 20.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
141

5

CHAPTER 2

Exercise 2.3

1.
(a) {x | x > 34}
(b) {x | 8 < x < 65}

2. True statements: (a), (d), (f), (g), and (h)

3.
(a) {2,4,6,7}
(b) {2,4,6}
(c) {2,6}

(d) {2}
(e) {2}
(f) {2,4,6}

4. All are valid.

5. First part: A∪(B∩C) = {4, 5, 6}∪{3, 6} = {3, 4, 5, 6} ; and (A∪B)∩(A∪C) = {3, 4, 5, 6, 7}∩

{2, 3, 4, 5, 6} = {3, 4, 5, 6} too.

Second part: A ∩ (B ∪ C) = {4, 5, 6} ∩ {2, 3, 4, 6, 7} = {4, 6} ; and (A ∩ B) ∪ (A ∩ C) =

{4, 6} ∪ {6} = {4, 6} too.

6. N/A

7. ∅, {5}, {6}, {7}, {5, 6}, {5, 7}, {6, 7}, {5, 6, 7}

8. There are 24 = 16 subsets: ∅, {a}, {b}, {c}, {d}, {a,b}, {a,c}, {a,d}, {b,c}, {b,d}, {c,d},

{a,b,c}, {a,b,d}, {a,c,d}, {b,c,d}, and {a,b,c,d}.

9. The complement of U is ˜U = {x | x /∈ U}. Here the notation of ”not in U” is expressed via the

/∈ symbol which relates an element (x) to a set (U). In contrast, when we say ”∅ is a subset

of U,” the notion of ”in U” is expressed via the ⊂ symbol which relates a subset(∅) to a set

(U). Hence, we have two different contexts, and there exists no paradox at all.

Exercise 2.4

1.
(a) {(3,a), (3,b), (6,a), (6,b) (9,a), (9,b)}

(b) {(a,m), (a,n), (b,m), (b,n)}

(c) { (m,3), (m,6), (m,9), (n,3), (n,6), (n,9)}

2. {(3,a,m), (3,a,n), (3,b,m), (3,b,n), (6,a,m), (6,a,n), (6,b,m), (6,b,n), (9,a,m), (9,a,n), (9,b,m),

(9,b,n),}

6

Chiang/Wainwright: Fundamental Methods of Mathematical Economics
Instructor’s Manual

3. No. When S1 = S2.

4. Only (d) represents a function.

5. Range = {y | 8 ≤ y ≤ 32}

6. The range is the set of all nonpositive numbers.

7. (a) No.
(b) Yes.

8. For each level of output, we should discard all the inefficient cost figures, and take the lowest

cost figure
as the total cost for that output level. This would establish the uniqueness as

required by the definition of a function.

Exercise 2.5

1. N/a

2. Eqs. (a) and (b) differ in the sign of the coefficient of x; a positive (negative) sign means an

upward (downward) slope.

Eqs. (a) and (c) differ in the constant terms; a larger constant means a higher vertical intercept.

3. A negative coefficient (say, -1) for the x2 term is associated with a hill. as the value of x is

steadily increased or reduced, the −x2 term will exert a more dominant influence in determining

the value of y. Being negative, this term serves to pull down the y values at the two extreme

ends of the curve.

4. If negative values can occur there will appear in quadrant III a curve which is the mirror image

of the one in quadrant I.

5.
(a) x19
(b) xa+b+c
(c) (xyz)3

6.
(a) x6
(b) x1/6

7. By Rules VI and V, we can successively write xm/n = (xm)1/n =
n√xm; by the same two rules,

we also have xm/n = (x1/n)m = ( n√x)m

8. Rule VI:

(xm)n = xm × xm × ... × xm
|
{z
}
n terms
= x × x × ... × x
|
{z
} =

mn terms
xmn

7

Chiang/Wainwright: Fundamental Methods of Mathematical Economics
Instructor’s Manual

Rule VII:

xm × ym
=
x × x × ... × x
|
{z
}
m terms
× y × y . . . × y
|
{z
}
m terms
=
(xy) × (xy) × . . . × (xy)
|
{z
}
m terms

= (xy)m

8

Chiang/Wainwright: Fundamental Methods of Mathematical Economics
Instructor’s Manual

CHAPTER 3

Exercise 3.2

1.
(a) By substitution, we get 21 − 3P = −4 + 8P or 11P = 25. Thus P ∗ = 2 3

11. Substituting

P ∗ into the second equation or the third equation, we find Q∗ = 14 2

11.

(b) With a = 21, b = 3, c = 4, d = 8, the formula yields

P ∗ = 25

11 = 2 3

11
Q∗ = 156

11 = 14 2

11

2.

(a)

P ∗ = 61

9 = 6 7

9
Q∗ = 276

9 = 30 2

3

(b)

P ∗ = 36

7 = 5 1

7
Q∗ = 138

7 = 19 5

7

3. N/A

4. If b+d = 0 then P ∗ and Q∗ in (3.4) and (3.5) would involve division by zero, which is undefined.

5. If b + d = 0 then d = −b and the demand and supply curves would have the same slope

(though different vertical intercepts). The two curves would be parallel, with no equilibrium

intersection point in Fig. 3.1

Exercise 3.3

1.
(a) x∗
1 = 5;
x∗
2 = 3
(b) x∗
1 = 4;
x∗
2 = −2

2.
(a) x∗
1 = 5;
x∗
2 = 3
(b) x∗
1 = 4;
x∗
2 = −2

3.

(a) (x − 6)(x + 1)(x − 3) = 0, or x3 − 8x2 + 9x + 18 = 0

(b) (x − 1)(x − 2)(x − 3)(x − 5) = 0, or x4 − 11x3 + 41x2 − 61x + 30 = 0

4. By Theorem III, we find:

(a) Yes.
(b) No.
(c) Yes.

9

Chiang/Wainwright: Fundamental Methods of Mathematical Economics
Instructor’s Manual

5.

(a) By Theorem I, any integer root must be a divisor of 6; thus there are six candidates: ±1,

±2, and ±3. Among these, −1, 1

2, and − 1

4

(b) By Theorem II, any rational root r/s must have r being a divisor of −1 and s being a

divisor of 8. The r set is {1, −1}, and the s set is {1, −1, 2, −2, 4, −4, 8, −8}; these give

us eight root candidates: ±1, ± 1

2, ± 1

4, and ± 1

8. Among these, −1, 2, and 3 satisfy the

equation, and they constitute the three roots.

(c) To get rid of the fractional coefficients, we multiply every term by 8.
The resulting

equation is the same as the one in (b) above.

(d) To get rid of the fractional coefficients, we multiply every term by 4 to obtain

4x4 − 24x3 + 31x2 − 6x − 8 = 0

By Theorem II, any rational root r/s must have r being a divisor of −8 and s being a

divisor of 4. The r set is {±1, ±2, ±4, ±8}, and the s set is {±1, ±2, ±4}; these give us

the root candidates ±1, ± 1

2, ± 1

4, ±2, ±4, ±8. Among these, 1

2, − 1

2, 2, and 4 constitute the

four roots.

6.

(a) The model reduces to P 2 + 6P − 7 = 0. By the quadratic formula, we have P ∗
1 = 1 and

P ∗
2 = −7, but only the first root is acceptable. Substituting that root into the second or

the third equation, we find Q∗ = 2.

(b) The model reduces to 2P 2−10 = 0 or P 2 = 5 with the two roots P ∗
1 =
√

5 and P ∗
2 = −
√

5.

Only the first root is admissible, and it yields Q∗ = 3.

7. Equation (3.7) is the equilibrium stated in the form of ”the excess supply be zero.”

Exercise 3.4

1. N/A

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Chiang/Wainwright: Fundamental Methods of Mathematical Economics
Instructor’s Manual

2.

P ∗
1
=
(a2 − b2)(α0 − β0) − (a0 − b0)(α2 − β2)
(a1 − b1)(α2 − β2) − (a2 − b2)(α1 − β1)

P ∗
2
=
(a0 − b0)(α1 − β1) − (a1 − b1)(α0 − β0)
(a1 − b1)(α2 − β2) − (a2 − b2)(α1 − β1)

3. Since we have

c0 = 18 + 2 = 20
c1 = −3 − 4 = −7
c2 = 1

γ0 = 12 + 2 = 14
γ1 = 1
γ2 = −2 − 3 = −5
it follows that

P ∗
1 = 14+100

35−1 = 57

17 = 3 6

17
and
P ∗
2 = 20+98

35−1 = 59

17 = 3 8

17

Substitution into the given demand or supply function yields

Q∗
1 = 194

17 = 11 7

17
and
Q∗
2 = 143

17 = 8 7

17

Exercise 3.5

1.

(a) Three variables are endogenous: Y, C, and T.

(b) By substituting the third equation into the second and then the second into the first, we

obtain

Y = a − bd + b(1 − t)Y + I0 + G0

or

[1 − b(1 − t)]Y = a − bd + I0 + G0

Thus

Y ∗ = a − bd + I0 + G0

1 − b(1 − t)

Then it follows that the equilibrium values of the other two endogenous variables are

T ∗ = d + tY ∗ = d(1 − b) + t(a + I0 + G0)

1 − b(1 − t)

and

C∗ = Y ∗ − I0 − G0 = a − bd + b(1 − t)(I0 + G0)

1 − b(1 − t)

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Chiang/Wainwright: Fundamental Methods of Mathematical Economics
Instructor’s Manual

2.

(a) The endogenous variables are Y, C, and G.

(b) g = G/Y = proportion of national income spent as government expenditure.

(c) Substituting the last two equations into the first, we get

Y = a + b(Y − T0) + I0 + gY

Thus

Y ∗ = a − bT0 + I0

1 − b − g

(d) The restriction b + g 6= 1 is needed to avoid division by zero.

3. Upon substitution, the first equation can be reduced to the form

Y − 6Y 1/2 − 55 = 0

or

w2 − 6w − 55 = 0
(where w = Y 1/2)

The latter is a quadratic equation, with roots

w∗
1, w∗
2 =
∙1

26 ± (36 + 220)1/2
¸
= 11, −5

From the first root, we can get

Y ∗ = w∗2
1 = 121
and
C∗ = 25 + 6(11) = 91

On the other hand, the second root is inadmissible because it leads to a negative value for C:

C∗ = 25 + 6(−5) = −5

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Chiang/Wainwright: Fundamental Methods of Mathematical Economics
Instructor’s Manual

CHAPTER 4

Exercise 4.1

1.

Qd
−Qs
= 0

Qd
+bP
= a

Qs
−dP
= −c

Coefficient Matrix:
⎡

⎢⎢⎢⎣

1
−1
0

1
0
b

0
1
−d

⎤

⎥⎥⎥⎦

Vector of Constants:
⎡

⎢⎢⎢⎣

0

a

−c

⎤

⎥⎥⎥⎦

2.
Qd1
−Qs1
= 0

Qd1
−a1P1
−a2P2
= a0

Qs1
−b1P1
−b2P2
= b0

Qd2
−Qs2
= 0

Qd2
−α1P1
−α2P2
= α0

Qs2
−β1P1
−β2P2
= β0

Coefficient matrix:
⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1
−1
0
0
0
0

1
0
0
0
−a1
−a2

0
1
0
0
−b1
−b2

0
0
1
−1
0
0

0
0
1
0
−α1
−α2

0
0
0
1
−β1
−β2

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Variable vector:
⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Qd1

Qs1

Qd2

Qs2

P1

P2

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Constant vector:
⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0

a0

b0

0

α0

β0

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

3. No, because the equation system is nonlinear

4.

Y − C
=
I0 + G0

−bY + C
=
a

The coefficient matrix and constant vector are
⎡

⎣
1
−1

−b
1

⎤

⎦

⎡

⎣ I0 + G0

a

⎤

⎦

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Chiang/Wainwright: Fundamental Methods of Mathematical Economics
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