Mathematics textbooks for science and engineering

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 />

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
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 . . . .

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

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25
27
29
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41
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43
43

50
63
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78
108

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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 . . . . . . .

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137

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143
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161
190

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

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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