- Báo Cáo Thực Tập
- Luận Văn - Báo Cáo
- Kỹ Năng Mềm
- Mẫu Slide
- Kinh Doanh - Tiếp Thị
- Kinh Tế - Quản Lý
- Tài Chính - Ngân Hàng
- Biểu Mẫu - Văn Bản
- Giáo Dục - Đào Tạo
- Giáo án - Bài giảng
- Công Nghệ Thông Tin
- Kỹ Thuật - Công Nghệ
- Ngoại Ngữ
- Khoa Học Tự Nhiên
- Y Tế - Sức Khỏe
- Văn Hóa - Nghệ Thuật
- Nông - Lâm - Ngư
- Thể loại khác

Tải bản đầy đủ (.pdf) (296 trang)

Bạn đang xem bản rút gọn của tài liệu. Xem và tải ngay bản đầy đủ của tài liệu tại đây (5.38 MB, 296 trang )

Mathematics Textbooks for Science and Engineering

Shapoor Vali

Principles of

Mathematical

Economics II

Solutions Manual, Supplementary

Materials and Supplementary

Exercises

Mathematics Textbooks for Science

and Engineering

Volume 4

More information about this series at http://www.springer.com/series/10785

Shapoor Vali

Principles of Mathematical

Economics II

Solutions Manual, Supplementary Materials

and Supplementary Exercises

Shapoor Vali

Department of Economics

Fordham University

New York, NY

USA

Mathematics Textbooks for Science and Engineering

ISBN 978-94-6239-087-4

ISBN 978-94-6239-088-1

DOI 10.2991/978-94-6239-088-1

(eBook)

Library of Congress Control Number: 2013951796

Published by Atlantis Press, Paris, France www.atlantis-press.com

© Atlantis Press and the authors 2015

This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by any

means, electronic or mechanical, including photocopying, recording or any information storage and

retrieval system known or to be invented, without prior permission from the Publisher.

Printed on acid-free paper

Series Information

Textbooks in the series ‘Mathematics Textbooks for Science and Engineering’ will

be aimed at the broad mathematics, science and engineering undergraduate and

graduate levels, covering all areas of applied and applicable mathematics, interpreted in the broadest sense.

Series editor

Charles K. Chui

Stanford University, Stanford, CA, USA

Atlantis Press

8, square des Bouleaux

75019 Paris, France

For more information on this series and our other book series, please visit our

website www.atlantis-press.com

Editorial

Recent years have witnessed an extraordinarily rapid advance in the direction of

information technology within the scientiﬁc, engineering, and other disciplines, in

which mathematics play a crucial role.

To meet such urgent demands, effective mathematical models as well as innovative mathematical theory, methods, and algorithms must be developed for data

information understanding and visualization.

The revolution of the data information explosion as mentioned above demands

early mathematical training with emphasis on data manipulation at the college level

and beyond. The Atlantis book series, “Mathematics Textbooks for Science and

Engineering (MTSE),” is founded to meet the needs of such mathematics textbooks

that can be used for both classroom teaching and self-study. For the beneﬁt of

students and readers from the interdisciplinary areas of mathematics, computer

science, physical and biological sciences, various engineering specialties, and social

sciences, contributing authors are requested to keep in mind that the writings for the

MTSE book series should be elementary and relatively easy to read, with sufﬁcient

examples and exercises. We welcome submission of such book manuscripts from

all who agree with us on this point of view.

This fourth volume consists of the solution manual and supplementary materials

of the previous volume: “Principles of Mathematical Economics,” by the same

author. It is divided into 13 chapters, covering such topics as: Market equilibrium

model, Rates of change and the derivative, Optimal level of output and long run

price, Nonlinear models, Economics Dynamics, and Mathematics of interest rates

and ﬁnance.

It is an important companion of the author’s previous volume.

Charles K. Chui

vii

Preface

It was part of my original plan in writing my book, Principles of Mathematical

Economics, Volume III of this MTSE book series, to provide answers to some

of the problems in the exercise sections of the book. However, when the text grew

to about 500 pages and the number of problems to over 600, I decided that simply

providing numerical answers, without going through the steps of formulating and

solving the problems, would not be very helpful to students or general readers. My

own experience from teaching quantitative courses convinced me that providing

mere answers was even harmful: in many cases chasing “the answer” leads some

students to set problems up incorrectly, but such that its solution is the same as the

given answer. This, on some occasions, as I am sure many of my academic colleagues have experienced, leads to debate over the “proper” grade for a homework

or exam problem, and periodic sermon, undoubtedly boring, about the impossibility

of consistently reaching correct conclusions through wrong reasoning.

I believe that the old saying “the correct formulation of a problem is 50 % of the

solution” is an expression of the accumulated wisdom of scientiﬁc inquiries. For the

above reasons, I decided to prepare this manual which gives a full-fledged formulation and solution to each of the problems in the text. In some cases I go beyond

setting up and solving a problem and include additional materials relevant to the

subject of the chapter.

This book naturally accompanies Principles of Mathematical Economics, but it

can also be used independently from the text. The book can be treated as a standalone collection of solved problems in different areas of mathematical economics

and as additional sets of exercises, over 500, that can be used to sharpen students’

skill and depth of understandings of many economic topics. Therefore, students can

beneﬁt from this manual even if the course they take in quantitative economics uses

a different textbook.

The manual is organized as follows: exercises from each chapter of the text are

listed, followed by their solutions. Where a problem can be solved using different

methods, sketches of alternative methods are also provided. If the solution references an equation in the text, the equation number is used. But if in the process

of solving a problem a new equation is derived its number is tagged by “SM”

ix

x

Preface

(for “Solution Manual”) to distinguish it from the text equation. For example, the

ﬁrst equation derived in Chap. 4 is labeled (4.1 SM). If a modiﬁed version of a text

formula is introduced, it is tagged “MOD” to indicate the modiﬁcation. Finally,

each set of solutions to chapter exercises is followed by an additional set of

unsolved problems under the heading “Supplementary Exercises”.

In the process of preparing this manual, I discovered a number of typographical

errors in the text. These errors are all mine and escaped me in the process of

proofreading. I have corrected the errors if they appear in the Exercise sections

of the chapters, which are repeated here in the manual. I hope there are no errors in

this manual, but given the sheer volume of numbers and mathematical expressions,

it is still likely that you encounter some errors. In case you do, I would appreciate it

if you would let me know by sending an email to vali@fordham.edu.

As it always the case, writing a book of this nature requires help from many

individuals. In particular, I would like to thank the publishers Dr. Keith Jones and

Dr. Zeger Karssen, as well as the book series editor, Professor Charles Chui, of

Atlantis Press, for their support in the publication of this book. I am grateful to Dr.

Michael Malenbaum, Ellen Fishbein, and Behrang Vali for their help in editing and

proofreading parts of the manuscript. Some of my students also helped me by

checking some of the solutions. I thank them all, especially Tyler Shegerian. Above

all, I am again indebted to my wife Firoozeh for her support and encouragement.

Spring 2014

Shapoor Vali

Contents

1

Household Expenditure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Variables, A Short Taxonomy . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

3

Sets and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

4

Market Equilibrium Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

5

Rates of Change and the Derivative. . . . . . . . . . . . . . . . . . . . . . .

57

6

Optimal Level of Output and Long Run Price . . . . . . . . . . . . . . .

81

7

Nonlinear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123

8

Additional Topics in Perfect and Imperfect Competition. . . . . . . .

151

9

Logarithmic and Exponential Functions. . . . . . . . . . . . . . . . . . . .

177

10

Production Function, Least-Cost Combination of Resources,

and Proﬁt Maximizing Level of Output . . . . . . . . . . . . . . . . . . . .

205

11

Economics Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

227

12

Mathematics of Interest Rates and Finance . . . . . . . . . . . . . . . . .

239

13

Matrices and Their Applications . . . . . . . . . . . . . . . . . . . . . . . . .

263

xi

Figures

Fig.

Fig.

Fig.

Fig.

1.1

1.2

3.1

3.2

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

3.3

3.4

3.5

3.6

4.1

4.2

4.3

4.4

4.5

4.6

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

5.10

5.11

6.1

6.2

Fig. 6.3

Fig. 7.1

Fig. 7.2

Budget Line . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Budget Line and Income-Consumption Path . . . . . .

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

The graph of the positive roots of w ¼ x2 þ 25 . .

wþ2

Graph of f ðwÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

, drawn for À20 w À 6.

w2 À16

wþ2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

p

Graph of f ðwÞ ¼ 2 , drawn for 6 w 15 . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

3

4

25

25

..

w À16

Graph of y ¼ 3x þ 2; y ¼ 10 À 2x; and y ¼ À5 þ 3x .

Graph of f ðxÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Graphs of f ðxÞ and gðxÞ . . . . . . . . . . . . . . . . . . . . . .

Graph of supply and demand functions . . . . . . . . . . .

Graph of supply and demand functions . . . . . . . . . . .

Shift in the demand function. . . . . . . . . . . . . . . . . . .

Supply and demand curves for part (a). . . . . . . . . . . .

Supply and demand curves for part (b) . . . . . . . . . . .

The IS Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

MC curve and Levels of Output where MC ¼ 50 . . . .

Marginal Cost and Average Total Cost curves . . . . . .

Graph of function y ¼ x3 À 9x2 À 48x þ 50 . . . . . . . .

Graph of h ¼ À15t2 þ 96t þ 4 . . . . . . . . . . . . . . . . .

Graph of TR ¼ 1250N À 5N 2 . . . . . . . . . . . . . . . . . .

Graph of y ¼ xð1 À xÞ4 . . . . . . . . . . . . . . . . . . . . . .

Graph of x2 þ y2 À 2y ¼ 2 . . . . . . . . . . . . . . . . . . . .

Graph of f ðx; yÞ ¼ x2 þ y2 À 2y and its Contour Plot .

Graph of z ¼ 3x3 þ 3x2 y À 2y þ 5 . . . . . . . . . . . . . . .

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Graph of z ¼ x2 þ y2 . . . . . . . . . . . . . . . . . . . . . .

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Graph of 49 À x2 À y2 . . . . . . . . . . . . . . . . . . . . .

Graph of A ¼ x þ 2 þ 2x and M ¼ 2x þ 2 . . . . . . . . . .

Graph of A ¼ À2w2 þ 5w þ 10 and

M ¼ À6w2 þ 10w þ 10 . . . . . . . . . . . . . . . . . . . . . .

Graph of ATC ¼ 5Q À 25 þ 240

Q and MC ¼ 10Q À 25 .

3

2

Graph of f ðxÞ ¼ x À 15x À 600 . . . . . . . . . . . . . . .

Graph of 16 Q3 À 2:5Q2 À 100. . . . . . . . . . . . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

25

27

29

29

41

41

42

43

43

50

63

65

67

67

68

70

72

73

75

75

78

108

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

109

110

128

130

xiii

xiv

Figures

Fig. 7.3

Fig. 7.4

Fig. 7.5

Fig. 7.6

Fig.

Fig.

Fig.

Fig.

.............

.............

.............

133

133

134

.............

135

Q41 À

0:6

1800Q21 À 2100Q1 þ 810000

f ðQ1 Þ ¼

700

ð2P À 10Þ À 2Pþ3

þ 10 . . . . . . . . . . . . .

f ðQ1 Þ ¼ 2:43Q41 À 189Q21 þ 10Q1 þ 3295 . .

0:02P

0:02P

3

2

Graph of

Graph of

Graph of

À 10P þ 10Pð3Þ

Graph of f ðPÞ ¼ P ð3Þ

zÀ100P À 1600 . . . . . . . . . . . . . . . . . . . . . . .

Fig. 7.11 Graph of f ðPÞ ¼ 5P1:1 þ 10P0:3 À 110 . . . . . . . .

Fig. 7.12 Graph of the derivative of profit function . . . . . .

Fig. 7.13 Graph of the Derivative of Revenue function . . .

Fig. 8.1 Demand, MR, and MC curves of the monopolist .

P3

.......................

Fig. 9.1 Graph of e0:02Pþ0:1

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

Fig.

7.7

7.8

7.9

7.10

Graph of supply and demand . . . . . . . . . . .

Graph of excess demand . . . . . . . . . . . . . .

700

À 10 À ð2P À 15Þ0:7

Graph of EDðPÞ ¼ 2Pþ3

Graph of ESðPÞ ¼ 100P0:4 À 1000 Pþ25

P2 þ5 . . .

9.2

9.3

9.4

10.1

10.2

10.3

11.1

11.2

11.3

2

P

Graph of e0:02ðPþ0:5Þ

.....................

Graph of MC and MR . . . . . . . . . . . . . . . . .

Graph of TC and TR curves . . . . . . . . . . . . .

Isocost and isoquant . . . . . . . . . . . . . . . . . .

3 Isoquants and expansion path. . . . . . . . . . .

Total, average, and marginal product of labor .

Graph of yt ¼ 6:709ðÀ0:17t Þ À 1:709 . . . . . .

Graph of Kt ¼ Àð0:9Þt þ 6 . . . . . . . . . . . . . .

Â

Ãt

Graph of Pt ¼ 1 þ 0:35ð1:1Þt 100 . . . . . . .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.......

.......

.......

137

138

139

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

143

143

146

148

161

190

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

191

201

201

212

219

221

232

233

235

Tables

Table 1.1

Table 1.2

Table

Table

Table

Table

Table

Table

Table

Table

5.1

9.1

9.2

9.3

11.1

11.2

13.1

13.2

Average annual expenditures and percent changes

by major category of all consumer units, Consumer

Expenditure Survey, 2008–2011 . . . . . . . . . . . . . . .

Percent distribution of total annual expenditures

by major category for all consumer units, Consumer

Expenditure Survey, 2008–2011 . . . . . . . . . . . . . . .

Values of x and y0 . . . . . . . . . . . . . . . . . . . . . . . . .

US GDP from 1978 to 2012, revised. . . . . . . . . . . .

US population from 2000 to 2012. . . . . . . . . . . . . .

National health expenditure 2003–2012 . . . . . . . . . .

Answers to questions number 1 and 2 . . . . . . . . . . .

Actual and estimated values of M2 . . . . . . . . . . . . .

Results of problem # 24 . . . . . . . . . . . . . . . . . . . .

Results of part(d) problem # 25 . . . . . . . . . . . . . . .

......

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

7

9

70

186

200

202

230

236

280

281

xv

Chapter 1

Household Expenditure

Chapter 1, 1.2 Exercises

1. A household allocates its $2,000 monthly income to the purchase of three goods.

Prices of these goods are $30, $40, and $20 per unit.

(a) Write the household monthly budget constraint.

(b) If this household purchases 40 units of good three each month, write its

budget equation and graph it. What is the slope of the budget line?

2. Assume a household with a monthly income of $5,000. This household allocates

its income to the purchasing of food and nonfood products. If the average price

of food products is $20 per unit and nonfood items cost $150 per unit

(a) Write the household’s budget equation.

(b) If this household consumes 100 units of food products, how many units of

nonfood items it can buy?

(c) Assume that the price of nonfood products increases to $160. Write the new

budget equation.

(d) If this household wants to purchase food and nonfood items in the same

proportion as in part (b), what is the household’s new bundle in part (c)?

3. Assume the household’s income in Problem 2 increases by 5 %. Repeat parts

(a)–(d) of problem (2).

4. Assume that due to competition prices of food and nonfood products in Problem

2 decline by 5 % while the household’s income remains the same. Repeat parts

(a) and (b) of problem (2). Compare your result with problem (3).

5. A household splits its $4,000 monthly income between necessity and luxury

goods. The average price of necessities is $30 per unit and that of luxuries is

$100 per unit.

© Atlantis Press and the authors 2015

S. Vali, Principles of Mathematical Economics II, Mathematics Textbooks

for Science and Engineering 4, DOI 10.2991/978-94-6239-088-1_1

1

2

1 Household Expenditure

(a) Write the household budget constraint.

(b) Determine the household equilibrium bundle if its proportion of necessity

and luxury goods purchases is 10 to 1.

(c) What is the equation of the household’s income-consumption path?

(d) Assume the household income declines by 10 %. What is the household’s

new bundle?

(e) Assume no loss of income but an inflation rate of 10 %. What is the household’s new bundle?

(f) Compare your answers in parts (d) and (e).

(g) Assume the original household income increases by 10 % to $4,400. What

is the household’s new bundle?

(h) Assume no change in income but the price level declines by 10 %. What is

the household’s new bundle?

(i) Compare your answers in parts (g) and (h).

(j) Compare your answers to parts (f) and (i). Are you surprised?

Answers to Chapter 1, 1.2 Exercises

#1

(a) Denoting the quantities of goods 1, 2, and 3 by Q 1 , Q 2 , and Q 3 , the household

budget constraint is

30Q 1 + 40Q 2 + 20Q 3 = 2000

(b) If this household purchases 40 units of good 3 each month, then its budget

constraint will be

30Q 1 + 40Q 2 = 2000 − 40 ∗ 20 = 1200

and the budget equation is

40Q 2 = 1200 − 30Q 1

−→

Q 2 = 30 − 0.75Q 1

Figure 1.1 is the graph of the budget equation. The slope of the budget line is

−0.75.

#2

(a) Denoting the quantities of food and nonfood goods by F and NF, we have

20F + 150NF = 5000

and the equation of the budget line is F = 250 − 7.5NF.

1 Household Expenditure

3

Fig. 1.1 Budget Line

(b)

20 ∗ 100 + 150NF = 5000

−→

150NF = 3000

Then we have NF = 3000/150 = 20 units. The household bundle is then

(100F, 20NF).

(c) The new budget constraint is

20F + 160NF = 5000

leading to a new equation for the budget line as

F = 250 − 8NF

(d) From part (b), we have the number of food items that this household purchases as

5 times the number of nonfood items. So the equation of the income-consumption

path is F = 5NF. Therefore, we must solve the following system of two equations with two unknowns for the new bundle.

F = 250 − 8NF

F = 5NF

250 − 8NF = 5NF −→ 13NF = 250 and NF = 19.23

F = 250 − 8 ∗ 19.23 = 96.15. The household new bundle is (96.15F, 19.23NF)

Figure 1.2 shows the budget line and the income-consumption path.

4

1 Household Expenditure

Fig. 1.2 Budget Line and

Income-Consumption Path

#3

(a) After 5 % increase in income, the new budget constraint is

20F + 150NF = 5250 leading to the budget equation

F = 262.5 − 7.5NF

(b) The household bundle is (100F, 21.67NF)

(c) The new equation of the budget line is

F = 262.5 − 8NF

(d) Based on the bundle in part (b), the proportion is now 100 to 21.67, leading to

F = 4.615NF. The household new bundle is then (96.02F, 20.81NF).

#4

The new prices are $19 for food and $142.5 for nonfood, thus

(a) 19F + 142.5NF = 5000 and the equation of the budget line is

F = 263.16 − 7.5NF

Notice no change in the slope of the budget equation.

(b) 19 ∗ 100 + 142.5NF = 5000

−→

142.5NF = 3100 then we have

3100

= 21.75 units. The household bundle is (100F, 21.75NF).

NF =

142.5

Note that this bundle is an improvement over the bundle in part (b) of problem 3.

1 Household Expenditure

5

#5

Let denote quantity of necessities by N and luxuries by L.

(a) The budget constraint is

30N + 100L = 4000

leading to the budget equation

L = 40 − 0.3N

(b) If N = 10L then

L = 40 − 0.3(10L)

−→

L = 10 and N = 100 units

(c) The equation of the income-consumption path is L = 0.1N .

(d) The new budget equation is L = 36 − 0.3N and after substituting 10L for N in

this equation, we find the new bundle as (L = 9, N = 90) units.

(e) With a 10 % inflation rate, the prices of N and L would be $33 and $110,

respectively. The new budget constraint is

33N + 110L = 4000

leading to the budget equation

L = 36.364 − 0.3N

Solving this equation with N = 10L leads to the new bundle (L = 9.091, N =

90.91).

(f) Comparing this bundle with the bundle in part (d) indicates that if the incomeconsumption path is linear, the negative impact of inflation on the real standard

of living of the household is less than that of a comparable loss of income.

(g) Now the budget constraint is

30N + 100L = 4400

−→

L = 44 − 0.3N

Assuming N = 10L, the household optimal bundle is (L = 11, N = 110)

(h) This time the budget constraint is

27N + 90L = 4000

−→

L = 44.444 − 0.3N

Again, with N = 10L the household optimal bundle is (L = 11.11, N = 111.1)

units.

6

1 Household Expenditure

(i) Clearly the bundle in part (h) is better than the bundle in part (g). The implication

is that under the assumption of a linear income-consumption path, the favorable

impact of a p% price decline is greater than that of a p% rise in income.

(j) The results indicate that when the income-consumption path is linear, inflation

is preferable to loss of income and deflation to rise in income. In short, change

in prices is preferable to change in income.

Supplementary Discussion

At the time of this writing (Dec. 2013), the latest available Consumer Expenditure

Survey Report is for year 2011.1 Table 1.1 of the survey is reproduced on the next

page.

As the table shows since the great recession of 2008 the housing expenditure (by

far the largest item on an average household’s budget) among households who own

their homes has declined, and instead, rental expenditure as part of total housing

expenses has grown.

The impact of large scale foreclosure and high unemployment finally reversed

the upward trend of homeownership in the United State. From the peak of 67 % in

2007, the proportion of homeowners dropped to 66 % in 2008 and further to 64 % in

2012. This process naturally resulted in higher demand for rental housing, leading

to higher rents.

According to a report released (December 2013) by the Joint Center for Housing

Studies of Harvard University, “in 1960, about one in four renters paid more than

30 % of income for housing. Today, one in two are cost-burdened”. “Cost-burdened”

refers to the households that pay more than 30 % of their income for housing. Those

who pay more than half of their incomes for housing are “severely cost-burdened”.

The Study reports that “by 2011, 28 % of renters paid more than half of their incomes

for housing, bringing the number with severe cost burdens up by 2.5 million in just

four years, to 11.3 million”.

The next table (Table 1.2) shows the proportion of household income allocated to

14 major categories of expenditure. As this table indicates, over at least a short period

of time, the households allocation of income to the major categories at aggregate

level are relatively stable. This, to some extent, justifies the assumption made in this

chapter that households, on the average, consume goods and services in the same

proportions.

Chapter 1 Supplementary Exercises

1. A household allocates its $3,000 monthly income to the purchase of 4 goods with

prices $25, $50, $20, and $35 per unit.

(a) Write the household monthly budget constraint.

(b) If this household purchases 40 units of good 3 and 20 units of good 1 each

month, write its budget equation and graph it. What is the slope of the budget

line?

1

Tables for 2012 survey are available, but the annual report has not yet been issued.

1 Household Expenditure

7

Table 1.1 Average annual expenditures and percent changes by major category of all consumer

units, Consumer Expenditure Survey, 2008–2011

Item

2008

2009

2010

2011

(%) Change

2008–09 2009–10 2010–11

Number of consumer

120770

units (in thousands)

Consumer unit characteristics

Income before taxes

$63,563

Age of reference person

49.1

Average number in consumer unit

Persons

2.5

Children under 18

0.6

Persons 65 or older

0.3

Earners

1.3

Vehicles

2

Percent homeowner

66

Spending categories

Average annual

$50,486

expenditures

Food

6443

Food at home

3744

Cereals and bakery

507

products

Meats, poultry,

846

fish, and eggs

Dairy products

430

Fruits and vegetables 657

Other food at home

1305

Food away from home 2698

Alcoholic beverages

444

Housing

17109

Shelter

10183

Owned dwellings

6760

Rented dwellings

2724

Other lodging

698

Utilities, fuels, and

3649

public services

Household operations

998

Housekeeping supplies 654

Household furnishings 1624

and equipment

Apparel and services

1801

120847 121107 122287 · · ·

···

···

$62,857 $62,481 $63,685 −1.1

49.4

49.4

49.7

···

−0.6

···

1.9

···

···

···

···

···

···

···

···

···

···

···

···

···

2.5

0.6

0.3

1.3

2

66

2.5

0.6

0.3

1.3

1.9

66

2.5

0.6

0.3

1.3

1.9

65

···

···

···

···

···

···

$49,067 $48,109 $49,705 −2.8

−2.0

3.3

6372

3753

506

6129

3624

502

6458

3838

531

−1.1

0.2

−0.2

−3.8

−3.4

−0.8

5.4

5.9

5.8

841

784

832

−0.6

−6.8

6.1

406

656

1343

2619

435

16895

10075

6543

2860

672

3645

380

679

1278

2505

412

16557

9812

6277

2900

635

3660

407

715

1353

2620

456

16803

9825

6148

3029

648

3727

−5.6

−0.2

2.9

−2.9

−2.0

−1.3

−1.1

−3.2

5.0

−3.7

−0.1

−6.4

3.5

−4.8

−4.4

−5.3

−2.0

−2.6

−4.1

1.4

−5.5

0.4

7.1

5.3

5.9

4.6

10.7

1.5

0.1

−2.1

4.4

2.0

1.8

1011

659

1506

1007

612

1467

1122

615

1514

1.3

0.8

−7.3

−0.4

−7.1

−2.6

11.4

0.5

3.2

1725

1700

1740

−4.2

−1.4

2.4

(continued)

8

1 Household Expenditure

Table 1.1 (continued)

Item

Transportation

Vehicle purchases (net)

outlay)

Gasoline and motor oil

Other vehicle expenses

Public and other transportation

Healthcare

Entertainment

Personal care products and services

Reading

Education

Tobacco products and smoking

supplies

Miscellaneous

Cash contributions

Personal insurance and pensions

Life and other personal insurance

Pensions and social security

2008 2009 2010 2011 (%) Change

2008–09 2009–10 2010–11

8604 7658 7677 8293 −11.0

2755 2657 2588 2669 −3.6

0.2

−2.6

8.0

3.1

2715

2621

513

2976

2835

616

116

1046

317

1986

2536

479

3126

2693

596

110

1068

380

2132

2464

493

3157

2504

582

100

1074

362

2655 −26.9

2454 −3.2

516

−6.6

3313

5.0

2572 −5.0

634

−3.2

115

−5.2

1051

2.1

351

19.9

7.4

−2.8

2.9

1.0

−7.0

−2.3

−9.1

0.6

−4.7

24.5

−0.4

4.7

4.9

2.7

8.9

15

−2.1

−3.0

840

1737

5605

317

5288

816

1723

5471

309

5162

849

1633

5373

318

5054

775

1721

5424

317

5106

−2.9

−0.8

−2.4

−2.5

−2.4

4.0

−5.2

−1.8

2.9

−2.1

−8.7

5.4

0.9

−0.3

1.0

Source U.S, Bureau of Labor Statistics

2. Assume a 5-member household with a monthly income of $6,000. This household

allocates its income to the purchasing of food and nonfood products. If the average

price of food products is $30 per unit and nonfood items cost $120 per unit

(a) Write the households budget equation.

(b) If this household purchases 100 units of food products each month, how

many units of nonfood items it can buy?

(c) Assume that the household’s allocation of its monthly budget to nonfood

products is 5 times that of its budget allocation for food. What is this household optimal bundle? [Note that here the assumption is not that the household

consumption of nonfood products is 5 times of that of food products, but

rather the allocation of household income to nonfood items is 5 times of its

allocation to food products].

(d) Assume the price of food products increases to $35 per unit. Write the household’s new budget equation. If this household wants to purchase food and

nonfood items in the same proportion as in part (c), what is the household’s

new bundle?

(e) Assume the price of food products increases to $35 per unit. If this household

wants to maintain its allocation of budget to purchase of food and nonfood

1 Household Expenditure

9

Table 1.2 Percent distribution of total annual expenditures by major category for all consumer

units, Consumer Expenditure Survey, 2008–2011

Spending category

2008

2009

2010

2011

Average annual expenditures

Food

Food at home

Food away from home

Alcoholic beverages

Housing

Shelter

Utilities, fuels, and public services

Household operations

Housekeeping supplies

Household furnishings and equipment

Apparel and services

Transportation

Vehicle purchases (net outlay)

Gasoline and motor oil

Other vehicle expenses

Public transportation

Healthcare

Entertainment

Personal care products and services

Reading

Education

Tobacco products and smoking supplies

Miscellaneous

Cash contributions

Personal insurance and pensions

Life and other personal insurance

Pensions and social security

100.00

12.8

7.4

5.3

0.9

33.9

20.2

7.2

2.0

1.3

3.2

3.6

17.0

5.5

5.4

5.2

1.0

5.9

5.6

1.2

0.2

2.1

0.6

1.7

3.4

11.1

0.6

10.5

100.0

13.0

7.6

5.3

0.9

34.4

20.5

7.4

2.1

1.3

3.1

3.5

15.6

5.4

4.0

5.2

1.0

6.4

5.5

1.2

0.2

2.2

0.8

1.7

3.5

11.2

0.6

10.5

100.0

12.7

7.5

5.2

0.9

34.4

20.4

7.6

2.1

1.3

3.0

3.5

16.0

5.4

4.4

5.1

1.0

6.6

5.2

1.2

0.2

2.2

0.8

1.8

3.4

11.2

0.7

10.5

100.0

13.0

7.7

5.3

0.9

33.8

19.8

7.5

2.3

1.2

3.0

3.5

16.7

5.4

5.3

4.9

1.0

6.7

5.2

1.3

0.2

2.1

0.7

1.6

3.5

10.9

0.6

10.3

Source U.S Bureau of Labor Statistics

items in the same proportion as in part (c), what is the household’s new

bundle?

3. Assume the household’s income in Problem 2 increases by 5 %. Redo parts (a)–(e)

of problem (2).

4. Assume that due to inflation, prices of food and nonfood products in Problem 2

increase by 5 % while the household’s income remains the same. Repeat parts

(a), (b), and (c) of problem (2). Compare your answer to part (c) of this exercise

to that of exercise 3.

10

1 Household Expenditure

5. A household allocates its $5,000 monthly income to the purchase of necessity

and luxury goods. The average price of necessities is $25 per unit and that of

luxuries is $125 per unit.

(a) Write the household budget constraint.

(b) Determine the household equilibrium bundle if the household’s allocation

of its monthly budget to luxuries is 3 times that of its budget allocation for

necessities.

(c) What is the equation of household income-consumption path?

(d) Assume the household’s income declines by 10 % and the household decides

to reduce its allocation to luxuries by 10 %. What is the household’s new

bundle?

(e) Assume no loss of income but an inflation rate of 10 %. What is the household’s new bundle if the household decides to cut its budget allocation to

luxuries by 10 % ?

(f) Compare your answers in parts (d) and (e).

(g) Assume the original household income increases by 10 % to $5,500 and the

household decides to increase its allocation to luxuries by 10 %. What is the

household’s new bundle?

(h) Assume no change in income but the price level declines by 10 %. What is

the household’s new bundle?

(i) Compare your answers in parts (g) and (h).

6. A household allocates its $3,500 monthly income to food, housing, and medical

care. Assume the household’s budget allocation to housing is twice the sum of

its allocation to food and medical care. Also assume that the household budget

allocation to food is twice its allocation to medical care. If the unit prices of

food, housing, and medical care are $20, $30, and $50 respectively, determine

the household optimal bundle.

Exercises (Appendix A)

1. Write the following sums in sigma notation.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

1 + 2 + 3 + 4 + · · · + 20

a + a2 + a3 + · · · + am

1/2

1/4 +√

1/8 + · · · +√1/64

√ +√

5 + 6 + 7 + · · · + 25

1 + 1/3 + 1/5 + 1/7 + 1/9 + · · · + 1/31

2 + 4 + 6 + 8 + · · · + 100 (the sum of even numbers from 2 to 100)

1 + 4 + 9 + 16 + 25 + · · · + 100

1 Household Expenditure

11

2. Expand and find the sum.

5

5

(a)

( j + 1)2

(b)

i2

j=2

i=1

3

3

(c)

(d)

2 j−1

j=−2

k=0

3k − 2

3k + 2

3. Show that

n

n

(a)

(ai + bi )2 =

i=1

i=1

n

(b)

n

ai2 +

bi2

i=1

n

(xi − yi )2 =

i=1

n

xi2 −

i=1

yi2

i=1

4. Assume n numbers x1 , x2 , ..., xn have mean x¯ and variance s 2 . Show

that if we subtract a constant c from each number, the mean changes to x¯ − c

but the variance stays the same.

5. Assume that in problem 4 instead of adding a constant c to each number, we

multiply the numbers by c. What would be the new mean and variance?

n

6. Show that the sum of the first n positive integers

i is

i=1

n(n + 1)

.

2

[Hint: use the trick that Karl Gauss, a 19th century German mathematician, used

to solve this problem when he was about 9 years old. Write the sum twice, first

in the usual order and second in the reverse order, and then add both sides of the

sums.] Use the result and find

100

(a)

100

(b)

j

j=1

( j + 2)

j=1

n

i = 20100

7. Find the number n such that

i=1

8. True or false?

n

n+2

a j + an+1 + an+2 =

j=1

aj

j=1

12

1 Household Expenditure

9. Evaluate

5

(a)

4

(−1)k k

(b)

( j + 2) j

k=1

j=1

Answers to problems in Chapter 1, Appendix A

#1

20

1 + 2 + 3 + 4 + · · · + 20 =

(a)

k

k=1

m

a + a2 + a3 + · · · + am =

(b)

ai

i=1

32

1/2 + 1/4 + 1/8 + · · · + 1/64 =

(c)

√

(d)

5+

√

1

2i

i=1

25 √

√

√

6 + 7 + · · · + 25 =

k

k=5

15

1 + 1/3 + 1/5 + 1/7 + 1/9 + 1/11 + · · · + 1/31 =

(e)

i=0

1

2i + 1

50

2 + 4 + 6 + 8 + · · · + 100 =

(f)

2i

i=1

10

1 + 4 + 9 + 16 + 25 + · · · + 100 =

(g)

j2

j=1

#2

(a)

5

( j +1)2 = (2+1)2 +(3+1)2 +(4+1)2 +(5+1)2 = 9 + 16 + 25 + 36 = 86

j=2

(b)

5

i 2 = (1)2 + (2)2 + (3)2 + (4)2 + (5)2 = 1 + 4 + 9 + 16 + 25 = 55

i=1