Mathematics textbooks for science and engineering

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.. d If this household wants to purchase food and nonfoo

Principles of

Mathematical Economics II

Solutions Manual, Supplementary Materials and Supplementary

Exercises

Shapoor Vali

and Engineering

Volume 4

Principles of Mathematical Economics II

Solutions Manual, Supplementary Materials and Supplementary Exercises

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 (eBook)

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

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

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

be aimed at the broad mathematics, science and engineering undergraduate andgraduate levels, covering all areas of applied and applicable mathematics, inter-preted in the broadest sense

Recent years have witnessed an extraordinarily rapid advance in the direction ofinformation technology within the scientific, engineering, and other disciplines, inwhich mathematics play a crucial role.

To meet such urgent demands, effective mathematical models as well as vative mathematical theory, methods, and algorithms must be developed for datainformation understanding and visualization

inno-The revolution of the data information explosion as mentioned above demandsearly mathematical training with emphasis on data manipulation at the college leveland beyond The Atlantis book series, “Mathematics Textbooks for Science andEngineering (MTSE),” is founded to meet the needs of such mathematics textbooksthat can be used for both classroom teaching and self-study For the benefit ofstudents and readers from the interdisciplinary areas of mathematics, computerscience, physical and biological sciences, various engineering specialties, and socialsciences, contributing authors are requested to keep in mind that the writings for theMTSE book series should be elementary and relatively easy to read, with sufficientexamples and exercises We welcome submission of such book manuscripts fromall 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 sameauthor It is divided into 13 chapters, covering such topics as: Market equilibriummodel, Rates of change and the derivative, Optimal level of output and long runprice, Nonlinear models, Economics Dynamics, and Mathematics of interest ratesandfinance

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

Charles K Chui

vii

It was part of my original plan in writing my book, Principles of MathematicalEconomics, 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 simplyproviding numerical answers, without going through the steps of formulating andsolving the problems, would not be very helpful to students or general readers Myown experience from teaching quantitative courses convinced me that providingmere answers was even harmful: in many cases chasing“the answer” leads somestudents to set problems up incorrectly, but such that its solution is the same as thegiven answer This, on some occasions, as I am sure many of my academic col-leagues 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 thesolution” is an expression of the accumulated wisdom of scientific inquiries For theabove reasons, I decided to prepare this manual which gives a full-fledged for-mulation and solution to each of the problems in the text In some cases I go beyondsetting up and solving a problem and include additional materials relevant to thesubject of the chapter

This book naturally accompanies Principles of Mathematical Economics, but itcan also be used independently from the text The book can be treated as a stand-alone collection of solved problems in different areas of mathematical economicsand 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 canbenefit 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 arelisted, followed by their solutions Where a problem can be solved using differentmethods, sketches of alternative methods are also provided If the solution refer-ences 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

(for“Solution Manual”) to distinguish it from the text equation For example, thefirst equation derived in Chap.4is labeled (4.1 SM) If a modified version of a textformula is introduced, it is tagged “MOD” to indicate the modification Finally,each set of solutions to chapter exercises is followed by an additional set ofunsolved problems under the heading“Supplementary Exercises”.

In the process of preparing this manual, I discovered a number of typographicalerrors in the text These errors are all mine and escaped me in the process ofproofreading 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 inthis 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 manyindividuals 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, ofAtlantis 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 andproofreading parts of the manuscript Some of my students also helped me bychecking some of the solutions I thank them all, especially Tyler Shegerian Aboveall, I am again indebted to my wife Firoozeh for her support and encouragement

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

Fig 1.1 Budget Line 3

Fig 1.2 Budget Line and Income-Consumption Path 4

Fig 3.1 The graph of the positive roots of w¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2þ 25 25

Fig 3.2 Graph of fðwÞ ¼ wffiffiffiffiffiffiffiffiffiffiþ2 w 2 16 p , drawn for20  w   6 25

Fig 3.3 Graph of fðwÞ ¼ wffiffiffiffiffiffiffiffiffiffiþ2 w 2 16 p , drawn for 6 w  15 25

Fig 3.4 Graph of y¼ 3x þ 2; y ¼ 10  2x; and y ¼ 5 þ 3x 27

Fig 3.5 Graph of fðxÞ 29

Fig 3.6 Graphs of fðxÞ and gðxÞ 29

Fig 4.1 Graph of supply and demand functions 41

Fig 4.2 Graph of supply and demand functions 41

Fig 4.3 Shift in the demand function 42

Fig 4.4 Supply and demand curves for part (a) 43

Fig 4.5 Supply and demand curves for part (b) 43

Fig 4.6 The IS Curve 50

Fig 5.1 MC curve and Levels of Output where MC¼ 50 63

Fig 5.2 Marginal Cost and Average Total Cost curves 65

Fig 5.3 Graph of function y¼ x3 9x2 48x þ 50 67

Fig 5.4 Graph of h¼ 15t2þ 96t þ 4 67

Fig 5.5 Graph of TR¼ 1250N  5N2 68

Fig 5.6 Graph of y¼ xð1  xÞ4 70

Fig 5.7 Graph of x2þ y2 2y ¼ 2 72

Fig 5.8 Graph of fðx; yÞ ¼ x2þ y2 2y and its Contour Plot 73

Fig 5.9 Graph of z¼ 3x3þ 3x2y 2yþ 5 75

Fig 5.10 Graph of z¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffix2þ y2 75

Fig 5.11 Graph of ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 49 x2 y2 p 78

Fig 6.1 Graph of A¼ x þ 2 þ2 xand M¼ 2x þ 2 108

Fig 6.2 Graph of A¼ 2w2þ 5w þ 10 and M¼ 6w2þ 10w þ 10 109

Fig 6.3 Graph of ATC¼ 5Q  25 þ240 Q and MC¼ 10Q  25 110

Fig 7.1 Graph of fðxÞ ¼ x3 15x2 600 128

Fig 7.2 Graph of 16Q3 2:5Q2 100 130

xiii

Fig 7.3 Graph of supply and demand 133

Fig 7.4 Graph of excess demand 133

Fig 7.5 Graph of EDðPÞ ¼ 700 2P þ3 10  ð2P  15Þ0 :7 . 134

Fig 7.6 Graph of ESðPÞ ¼ 100P0 :4 1000 P þ25 P 2 þ5   135

Fig 7.7 Graph of fðQ1Þ ¼ Q4 1 1800Q2 1 2100Q1þ 810000 137

Fig 7.8 Graph of ð2P  10Þ0 :6 700 2P þ3þ 10 138

Fig 7.9 Graph of fðQ1Þ ¼ 2:43Q4 1 189Q2 1þ 10Q1þ 3295 139

Fig 7.10 Graph of fðPÞ ¼ P3ð3Þ0 :02P 10P2þ 10Pð3Þ0 :02P z100P  1600 143

Fig 7.11 Graph of fðPÞ ¼ 5P1 :1þ 10P0 :3 110 143

Fig 7.12 Graph of the derivative of profit function 146

Fig 7.13 Graph of the Derivative of Revenue function 148

Fig 8.1 Demand, MR, and MC curves of the monopolist 161

Fig 9.1 Graph of P 3 e 0:02Pþ0:1 190

Fig 9.2 Graph of e0:02ðPþ0:5ÞP2 191

Fig 9.3 Graph of MC and MR 201

Fig 9.4 Graph of TC and TR curves 201

Fig 10.1 Isocost and isoquant 212

Fig 10.2 3 Isoquants and expansion path 219

Fig 10.3 Total, average, and marginal product of labor 221

Fig 11.1 Graph of yt¼ 6:709ð0:17tÞ  1:709 232

Fig 11.2 Graph of Kt¼ ð0:9Þtþ 6 233

Fig 11.3 Graph of Pt¼ 1 þ 0:35ð1:1Þ tt 100 235

Table 1.1 Average annual expenditures and percent changes

by major category of all consumer units, Consumer

Expenditure Survey, 2008–2011 7

Table 1.2 Percent distribution of total annual expenditures by major category for all consumer units, Consumer Expenditure Survey, 2008–2011 9

Table 5.1 Values of x and y0 . 70

Table 9.1 US GDP from 1978 to 2012, revised 186

Table 9.2 US population from 2000 to 2012 200

Table 9.3 National health expenditure 2003–2012 202

Table 11.1 Answers to questions number 1 and 2 230

Table 11.2 Actual and estimated values of M2 236

Table 13.1 Results of problem # 24 280

Table 13.2 Results of part(d) problem # 25 281

xv

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 itsbudget 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 allocatesits 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 ofnonfood items it can buy?

(c) Assume that the price of nonfood products increases to $160 Write the newbudget equation

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

(a) Write the household budget constraint.

(b) Determine the household equilibrium bundle if its proportion of necessityand 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’snew bundle?

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

house-(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 isthe 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

Fig 1.1 Budget Line

(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

equa-tions 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)

Figure1.2shows the budget line and the income-consumption path

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

NF= 3100

142.5 = 21.75 units The household bundle is (100F, 21.75NF).

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

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

(a) The budget constraint is

(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

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

(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 ExpenditureSurvey Report is for year 2011.1Table1.1of the survey is reproduced on the nextpage

As the table shows since the great recession of 2008 the housing expenditure (byfar the largest item on an average household’s budget) among households who owntheir homes has declined, and instead, rental expenditure as part of total housingexpenses has grown

The impact of large scale foreclosure and high unemployment finally reversedthe 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

The next table (Table1.2) 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 aggregatelevel are relatively stable This, to some extent, justifies the assumption made in thischapter that households, on the average, consume goods and services in the sameproportions

Chapter 1 Supplementary Exercises

1 A household allocates its $3,000 monthly income to the purchase of 4 goods withprices $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 eachmonth, write its budget equation and graph it What is the slope of the budgetline?

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

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

units, Consumer Expenditure Survey, 2008–2011

2008–09 2009–10 2010–11 Number of consumer

units (in thousands)

120770 120847 121107 122287 · · · ·

Consumer unit characteristics Income before taxes $63,563 $62,857 $62,481 $63,685 −1.1 −0.6 1.9 Age of reference person 49.1 49.4 49.4 49.7 · · · ·

Average number in consumer unit Persons 2.5 2.5 2.5 2.5 · · · ·

Children under 18 0.6 0.6 0.6 0.6 · · · ·

Persons 65 or older 0.3 0.3 0.3 0.3 · · · ·

Earners 1.3 1.3 1.3 1.3 · · · ·

Vehicles 2 2 1.9 1.9 · · · ·

Percent homeowner 66 66 66 65 · · · ·

Spending categories

Average annual

expenditures

$50,486 $49,067 $48,109 $49,705 −2.8 −2.0 3.3

products

fish, and eggs

public services

and equipment

(continued)

Public and other transportation 513 479 493 516 −6.6 2.9 4.7

Pensions and social security 5288 5162 5054 5106 −2.4 −2.1 1.0

Source U.S, Bureau of Labor Statistics

2 Assume a 5-member household with a monthly income of $6,000 This householdallocates its income to the purchasing of food and nonfood products If the averageprice 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, howmany units of nonfood items it can buy?

(c) Assume that the household’s allocation of its monthly budget to nonfoodproducts is 5 times that of its budget allocation for food What is this house-

hold 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 hold’s new budget equation If this household wants to purchase food andnonfood items in the same proportion as in part (c), what is the household’snew bundle?

house-(e) Assume the price of food products increases to $35 per unit If this householdwants to maintain its allocation of budget to purchase of food and nonfood

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

units, Consumer Expenditure Survey, 2008–2011

Source U.S Bureau of Labor Statistics

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

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 2increase 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

5 A household allocates its $5,000 monthly income to the purchase of necessityand luxury goods The average price of necessities is $25 per unit and that ofluxuries 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 fornecessities

(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 newbundle?

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

house-(f) Compare your answers in parts (d) and (e)

(g) Assume the original household income increases by 10 % to $5,500 and thehousehold decides to increase its allocation to luxuries by 10 % What is thehousehold’s new bundle?

(h) Assume no change in income but the price level declines by 10 % What isthe 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 medicalcare Assume the household’s budget allocation to housing is twice the sum ofits allocation to food and medical care Also assume that the household budgetallocation to food is twice its allocation to medical care If the unit prices offood, housing, and medical care are $20, $30, and $50 respectively, determinethe household optimal bundle

2 Expand and find the sum.

4 Assume n numbers x1, x2, , x n have mean ¯x and variance s2 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?

6 Show that the sum of the first n positive integers

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 thesums.] Use the result and find



i=1

i2= (1)2+ (2)2+ (3)2+ (4)2+ (5)2= 1 + 4 + 9 + 16 + 25 = 55

#4 After subtracting c from each observation, the new set is x1− c, x2− c, ,

x n − c The mean of this new set, ¯x ne w , is

The new variance is

#5 After multiplying each observation by a constant c, the new set is cx1, cx2, ,

cx n The mean of this new set, ¯x ne w , is

The new variance is c2times the original variance

#6 Let’s denote the sum of first n integers n

Using this result, we have

4



j=1

( j + 2) j = (1 + 2)1+ (2 + 2)2+ (3 + 2)3+ (4 + 2)4= 3 + 16 + 125 + 1296 = 1440

Chapter 1, Appendix A : Supplementary Exercises

1 Write the sum in sigma notation

2 +2

3 + · · · + n

n+ 1

2 Find the value of following sums:

[Series (e), (f), (g), (h), (i), (j), and (k) are called telescoping series, since the

terms of the series, other than the first and last terms, cancel out in pairs.]

Variables, A Short Taxonomy

(h) US government budget deficit since 2002

(i) US trade deficit with China since 2002

(j) Time takes to run 100 meters

(k) Record of the top 100 runners in 2010 New York Marathon

(l) Consumer credit outstanding

(m) Last quarter corporate profits

(n) Closing price of stocks listed on NYSE on January 28, 2008

(o) Population of major cities in the US based on the 2000 census

(p) Unemployment rate of 27 members of the European Union in the last

10 years

(q) Unemployment rate of 27 members of European Union in 2007

(r) Closing price of stocks listed on NYSE from Jan 2, 1990 to Jan 2, 2008(s) Record of the top 100 runners in New York Marathon in the last 5 years

2 Classify the qualitative variables in Exercise 1 as dichotomous or multinomial

3 Classify the quantitative variables in Exercise 1 as discrete or continuous

4 Classify the economic variables in Exercise 1 as time-series, cross-section, ormixed

5 Classify k and s in Problem 1 as time-series, cross-section, or mixed

© 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_2

17

6 Classify the economic variables in Exercise 1 as stock or flow.

7 Determine the size of the table—the number of rows, columns, and cells—youwould need in order to present each of the variables that you identified as mixed

in Exercise 1 (for (r) assume 3500 stocks listed on NYSE and 252 trading days)

8 How would you present the unemployment rate of the white and non-whitepopulations in the last 20 years for 50 states of the United States?

Answers to Chapter 2, Exercises 2.1

#1 Variables in (b), (e), (f), and (g) are qualitative The rest are quantitative.

#2 All qualitative variables in exercise #1 are multinomial.

#3 The quantitative variables in (c) and (o) are discrete The rest are continuous.

#4 Variables in (h), (i), and (l) are time series Variables in (n), (o), and (q) are

cross section Variables in (p), and (r) are mixed Depending how one interprets thequestion, the variable “corporate profit” in (m) could be cross section or time series

#5 The variable in (k) is cross section and in (s) is mixed.

#6 Variables in (h), (i), and (m) are flows Variables in (l), (n), (o), (p), (q), and (r)

are stocks

#7 For (p) we need a table of 27 by 10: 27 rows for 27 members of the EU and 10

columns for 10 years This table has 27 * 10=270 cells If we designate the variable

EU Unemployment rate as EUU, it can be expressed as

EUU i j i = 1, 2, , 27; j = 1, 2, , 10

For (r) we need a table with 3500 rows for 3500 corporations listed on NYSE and

252 * 18= 4,536 columns for 252 trading days over an 18-year period This tablehas 3500 * 4536=15,876,000 cells The variable Closing Price of Stock CPS can

be expressed as

CPS i j i = 1, 2, , 3500; j = 1, 2, , 4536

For part (s) we need a table of 100 rows and 5 columns This table has 500 cells

#8 Here the unemployment rate can be expressed as

Urate i j k i = 1, 2, , 50; j = 1, 2, , 20; k = 1, 2

where i subscript refers to “state”, j to “time”, and k to “race”, classified as White and

non-White To present this data, we need a 3-dimensional table Those familiar with ahigh-level computer programming language, such as FORTRAN and C, know how touse a 3-dimensional “array” to store the data in a computer For a paper presentation,however, we need two separate tables, one for White and one for non-White Eachtable has 50 rows for States and 20 columns for years, a total of 1,000 cells

Chapter 2 Supplementary Exercises

1 Classify the following as quantitative or qualitative variables

(a) Gender

(b) Brand of soft drink

(c) Race (classified as white or non-white)

(d) US trade deficit with China

(e) Price index

(f) Price of Gold

(g) Exchange rate between $ ande

(h) Military rank

(i) The US money supply

(j) Energy use pere1 of GDP of EU members

(k) Farm productivity in developing countries

(l) Average manufacturing unit labor costs of G7 countries

(m) Federal grant to 50 states in 2012

(n) Real private fixed investment

(o) Number of spectators in a 90,000 seats stadium

(p) Population of NAFTA countries in the last 20 years

(q) Money supply of OECD countries in 2012

(r) Inflation rate of Latin American countries for the last 10 years

(s) Closing prices of stocks of companies in S&P 500 index

(t) Employment status of a person in the work force

(u) Religion (classified as Christian or non-Christian)

(v) Ethnicity (classified as Italian or non-Italian)

(w) Inflation rate of US major metropolitan areas in 2012

2 Classify the qualitative variables in Exercise 1 as dichotomous or multinomial

3 Classify the quantitative variables in Exercise 1 as discrete or continuous

4 Classify the economic variables in Exercise 1 as time-series, cross-section, ormixed

5 Classify the economic variables in Exercise 1 as stock or flow

6 Determine the size of the table—number of rows, columns, and cells—you wouldneed in order to present each of the variables that you identified as mixed inExercise 1

7 How would you present the age distribution of the white and non-white tions in the last 20 years for 50 states of the United States?

popula-Sets and Functions

2 Write the complement of A from Exercise 1 using specification.

3 Write the set of all subsets of C in Exercise 1

4 Given A = {a, b, c, d}, B = {c, d}, and C = {a, b, e, f, c, d}, verify

whether the following statements are valid:

© 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_3

21

6 In general, A × B = B × A Under what condition they are equal?

7 Find the Cartesian product of X = {3, 4, 9} and Y = {9, 16, 81} Write the function that maps X to Y

8 Assume A is the set of names of 100 faculty members at a small college and B

is the set of academic ranks a faculty member could have:

B = {Professor, Associate Professor, Assistant Professor, Lecturer}

Is a mapping from A to B a function?

9 Determine the range of the following functions for the given domain:

(a) f (3) (b) g(3) (c) f (−3) (d) g(−3)

(e) f (x + 2) (f) g(x + 2) (g) f (√x) (h) g(x2+ 1)

(i) f [ f (x)] (j) g[g(x)] (k) f [g(x)] (l) g[ f (x)]

15 Write the following sets using specification:

(a) Set of integers less than 10

(b) Set of positive integer numbers

(c) Set of real numbers between−15 and 20

(d) Set of real numbers square of which is greater than 50

(e) Set of real numbers with square roots less than 200

(f) Set of natural numbers with square roots less than 10

16 Graph the functions

(a) y = 3x + 2 (b) y = 10 − 2x (c) y = −5 + 3x

Find the coordinate of point of intersection between (a) and (b) Why can’t we

do the same for (a) and (c)?

Answers to Chapter 3, 3.7 Exercises

#2 ¯A = {a ∈ R | a /∈ A}

#3 Denoting the power set of C by P, the set of all possible subsets of C is

P = {∅, {−4}, {1}, {15}, {19}, {−4, 1}, {−4, 15}, {−4, 19}, {1, 15}, {1, 19}, {15, 19}, {−4, 1, 15}, {−4, 1, 19}, {−4, 15, 19}, {1, 15, 19}, {−4, 1, 15, 19}}

#4 Parts (d) and (h) are false, and the rest are true.

#5 Only (c) y =√x2+ 1 is not a function; for each value of x there are two values for y If, for example, x = 3 then y = ±3.162.

#6 The Cartesian product of two sets A and B is equal to the Cartesian product of B

and A if and only if A and B are equal.

#7 The Cartesian product of X and Y is

X × Y = {3, 4, 9} × {9, 16, 81}

= {(3, 9), (3, 16), (3, 81), (4, 9), (4, 16), (4, 81), (9, 9), (9, 16), (9, 81)}

A subset of X × Y is {(3, 9), (4, 16), (9, 81)} and a function that maps X to Y is

y = x2.

#8 A mapping from A to B is a function.

#9 To avoid confusion between R as range and R as the set of real numbers, I will use

R A to denote range

(a) R A = {y ∈ R | y = 4}

(b) R A = {y ∈ R | 45 ≤ y ≤ 135}

(c) R A = {y ∈ R | 5 ≤ y ≤ 105}

(d) R A = {w ∈ R | 5 ≤ w ≤ 6.403} considering only positive roots

Figure3.1is the graph of the function w =√x2+ 25 in part (d)

(e) R A = { f (z) ∈ R | f (z) >√2} considering only positive roots

Considering only the positive square roots of real numbers, we have the following rangefor part (f)

(f) R A = { f (w) ∈ R | − 1 ≤ f (w) ≤ −0.866 or 1 ≤ f (w) ≤ 1.7889}

Figures3.2and3.3depict the behavior of the function for values of−20 ≤ w ≤ −6

and 6≤ w ≤ 15.

Fig 3.1 The graph of the

To determine the domain of function in part (f), we first solve the quadratic equation

x2− 9x + 20 = 0 This equation has two roots, namely x = 4 and x = 5 The values

of x2− 9x + 20 for x between 4 and 5 are negative and the function f (x) has no real value Therefore, the domain of f (x) must be expressed as

+ 13



2x+1 3x−5

(a) X = {1, 2, 3, 4} (b) A = {10, 11, 12, 13, 14} (c) Y = {2, 3, 4, 5, 6, 7, 8} (d) X = {−4, −3, −2, −1, 0, 1, 2, 3, 4} (e) X = {1, 4, 9, 16, 25, 36, 49, 64, 81}

#14

(a) Counting numbers greater than 1 and less than 9

(b) Real numbers greater than or equal to−4 and less than 9

(c) Real numbers greater than 0, or positive real numbers

(d) Real numbers greater than−10 and less than 10

Chapter 3 Supplementary Exercises

1 Consider the following subsets of the lower-case English alphabet

A = {a, b, d} B = {c, d, e, h} C = {d, h, v, w, z}

Find the following:

(a) A ∪ B (b) A ∩ B (c) A ∪ C (d) C ∪ B (e) A ∪ B ∪ C (f) A ∩ B ∩ C (g) (A ∪ B) ∩ C (h) A ∩ C

2 Find the cardinality of the sets in Exercise 1

3 Use A , B and C in Exercise 1 and find the following Cartesian products:

5 Write the power set of B in Exercise 1 What is its cardinality?

6 Express the following set using enumeration

(a) Set of positive even numbers less than 20

(b) Set of prime numbers between 10 and 50

(c) Set of companies on the Dow Jones Utility Index (Google the Dow Jones Index)(d) Set of US branches of government

7 Express the following sets using specification:

(a) Set of real numbers between and including−20 and 100

(b) Set of real negative numbers with absolute values greater than 200

(c) Set of integer numbers whose squares are greater than 100 and less than 5000.(d) Set of real numbers with inverse greater than 10

(e) Set of real numbers with cubic roots less than 250

8 Describe the following sets:

(a) {a ∈ R | − 5 < a < 20}

(b) {z ∈ R | 5 ≤ z < 10}

(c) {w ∈ Z | w > 0}