Problems book to accompany mathematics for economists

625 Pages·2016·89.57 MB·English

by  Todorova T.

PROBLEMS BOOK TO ACCOMPANY

Tamara Todorova
Problems Book to Accompany

Mathematics for Economists

Tamara Todorova

®

JOHN WILEY &' SONS, INC.
ACQUISITIONS EDITOR Lacey Vitetta

MARKETING MANAGER Diane Mars
EDIORIAL ASSISTANT Emily Mes
SENIOR DESIGNER Jim 2'Shea
PRODUCTION MANAGER Micheline Frederick
PRODUCTION FOITCR ‘Amy Weintraub

This book was printed and bound by Courier Westford. The cover was printed by Ca.rier Westford
This bookis printed on aid free paper. «

Copyrign: ©) 201 john wiley & Sons, Inc. All rignts reserved. No part ofthis publication may be
reprocuced, stared ina retrieval system cr ranemitted in any farm ar by ary means, electronic
mechanical, photocopying, recarding, scanning or otherw se, except as permitted under Sections 107 or
108 ofthe 1976 Uni:ed States Copyright Act, without ether the prior written permission of the
Publisher, or authorization thraugh payment of the appraprte per~zapy tee te the Copyright Clearance
Center, Ine. 222 Rosewood Drive, Danvers, MA01923, website warwicopyrignt.com, Recvests tothe
ublisher for permission should be addressed to the Permissions Department, Jahn Wiley & Sans; Ine,
111 River Slee, Hoboken, 4 07030-5774, [201)788-6011, fx [phone], website
ripi/iuwn.iley.com/zo/permissions

Evaluation copies are provided to qualified academics and profeslenals for review purposes only, for
se in their courses during tre next academic yea". These copies are licensed and may not be sold or
transferred toa third par-y. Upon completion of the review period, please return the evaluation coxy to
Wiley. Retura instructors and ‘ree ct charge return shioping label are ava lable at

ww suley.com/gofretumlabel, Outside of the United States, pleose contact your lozal representative

Lbrary of Congress Cataloging in Publication Data:

ISBN-13 978-0-470-59181-9

Printed in the Jnited States of Americe

wessesssza
Table of Contents

Matrix Algebra

2, Comparative Staies I: Parti Derivatives
3. Comparative Staties I: Differentials

ne Variable

4, Optimization of Functions of
§. Fxponential and Logarithm Functions
6. Functions of More than One Variable

Constrained Optimization

9, Simple Diffecential Equations

10, Simple Difference Equations

1, Advanced Differential and Difference Equations

electronic website:

12, Inroduetion to Dynemie Optimizatior
The Caleulus of Variations
electronic website:

www wiley comical

61s
com‘collegeitodorova

70

rodorova
Detailed Table of Contents

Chapter 1: Matrix Ak ~ ransposes und inverse, nansingulariy ua mais,
ddterminas, mats inversen, Cram’s rule, Cass method, market equlirium analysis, Seyesian
rational-income model, Leoatiesiapu-ouput model, problems.

Jura near mo

Chapter 2: Comparative Staties I Partial Derivatives derivative, rules of ferentiation,
iteremiation of functions of more thar one vavahle, inverss-function rule, partial derivatives
Jacobian detereinans, problems

Chapter 3: Comparative Staties Il: Differentials point elsticity.pstal and tral
A ferentials, es af dilfereils, total derivatives af composite functions, implieit-antion rae
inns function thea, aeneral-Funeion mocels ~ market equilisrcm: model, IS-L model,
protlems

Chapter 4: Optimization of Functions of One Variabl
eriative test, polit mission, eubie cost function, eubie prot Funston, ah derivative test fr
relative extremum, problems

Chapter S: Exponential and Logarithmic Functions exponcntal factions, the ntanbore.
meres compounding, ineanuarcous rae cf growth, daunting, logaitbns ules of logatthes,

ithe “unctons, base conceiscn,de-isatives oF expaneatial sad logaithm funetions,eptimal
timiag ate of geewth af composite functions, port slastcity revisited, prablems

Chapter 6: Functions of More than One Vari
three and mono variables,

ch

fire, inp

functions of
‘arable quadratie forms, esnitiems For optimization withoul consti,

ferstic-ool test fr sign deiniteress, tious and nlinarket fe, inp
ins and

onparasive satis asposts of optimizatioe, problems

Chapter 7: 19 Lagrange-maliplior msthod,»-varuble and
rmukiconsraint case, conditions for optimization with consists, utility maximization, Shusky
‘iin inl constsints, lotapgencs nets, Cold-Dowglas peraduct on Fusetion,
Slasiigy of substation, CES prouction funtion, problems

Chapter 8: Integration —indefinice integral rules of integration, dete ineptsls, isproper
integrals investment and capital formation, consusnet and areducer surplus, probability anc detinite
inearls, present value of a cashflow, Dantar grower mod, problems
ter 9: Simple Differential Equations —frst-onler linear differential equations with
constant coefficient and constant term, dynamie model of market pice the case
‘coefficient and variable ter

act diferamtal equations, integrating factors, separation of variables,
Bemoulli equations, phase ciggroms, Solow growih model, problems

‘Chapter 10: Simple Difference Equations ~frst-onler difference
solving, dynamic stability of equilibrium, the Cobweb model, pro

Chapter 11: Advanced Differential and Difference Equations ~ second
Aiferenval equations the Arve-Pratt measure of risk aversion, rearket equilibrium wit price
expeciatons, the relationship between inflation acd unctaploymient, higher-order differental
‘equations, secand-ordereifference equations, muliplie-accelratar model, inflation and
uemployerent in discrete time, higher-order diference equations, problems

Chapter 12: Introduction to Dynamic Optimization: The Calculus of Va
introduction to dynamic optimization, the eae

‘cost-minimizing firm, constrained dynami opti
problems

tions
tion, Ramsey growth
ization, a glimpse of optimal contol theory
Preface

his aupplement i itended to complement any couse in quanialive msthuds in exoromice, mathematical
‘eonamies for undergraduate student or inlrductory malhomatical economics for grate stents, The Key

snd srengthen the theory they have leat from standard exthooks ia mathematical evonomics. The
book supalements sry ofthe mest videly used texts, such as Mathomatise for Economists by Cal Simon ed

[Caurence Bhime end Fundamental Methods of Shthematcal Exonamics by Alpha Chi

‘While these books are predominantly theoretical xs, the problens book i a practice text iustrating a a
umber ef matiematial tools ob

‘applied in econcmics. Al problems are economic applications, and no
problem s purely mathematical without elevance to evo. In fc, nan probes eas be sta a itaple
conomic models that challeige student hinking an knowles, While non aerate textbooks i he

ara sess the math theory and Musa it with sone economic aplication, dis buck iss sch an

economics bok es tsa math book, For completeness, sh: Ihereticalsuaraaries uf Us relevant Urey ae
fdesiptions of econ ecpleaions ae pid in (ve hen cee chapter i eine ste
to he essentials ofthe subject, However, ee sutetivalsumamaes ave only prefory aera, seoralary Wy
the asin purpse of te book, ith elinw uriginaity uy put except in we sections, The thew
‘nly meant orient the renders before they plunge in he rublems, which I have tried to mae ws wide
ging as possible, I ear the problems and solutions asthe essence othe Bock ad the justification fort

The book is designed for sucenis af economics who are taking mathematical economics as par of thee
‘disaton, About 70 persent ofthe esomomics conten, herein, is drawn fom microeconomics, and the est
from macroeconomics. Thus, courses in intemede micto-and macroeconomics are sable prerequisites for
this book. fist course in calelus is sirable as well,

An essential aim ofthe book isto popularize the subject of maematical economics sad make it more
accessible to beginning eonom ss, While a large numberof students around the world ae ntigved by
‘sonomies nd finance, particularly amid glabal nancial and economic ers, ewer ae eager lo pursue
scmamies beens of 1 math "hey WoLId ive fo daa ecenmmies majors, Hanl-core mattematies bordering
fon the graduate evel scares a great man students and deters them from slyng economics. This book
redices the intimidation with math und helps make economics # mote interesting ard upproachable subj

While the problens book may be use by heyinn ng graduate students of ecoromics i: isnot perfsedy suited
for them, Rather, at his acvanced level, students might use the s00k as reference gue to reltesh some ost
knowledge. Therefore, he book is appropriate forthe masters level and peshaps

arious quantitative fools wo ther feds of study may also use the Book. Yet, tis oka perfect substitute for a
ook in applied athena for business 0: Finacial mathematics, Nester i La sibs Fora suse i
intermediate microeconomics, although many of te robles i this book are drawn om sich a course

fom: Wook economists was writen a a response oan insuticency inthe
etd ef matnematical economics 0 appropriate dy (aos and exercise bonks, Students af economies ac
robust hemework-ype problems with which o fest ther knosledge. Asa professor Ihave found att bea
Serous handicap my leeching. My students have hac no substan! learring sds or tools, and hence have
shed foc a book hat gives exercises tthe level of exams in the course, This resulted inthe design ofa
rember of problems, the prodost of years of wark, The proms ars eae for sudents to work with as they
Closely follow the tacory covered inthe beginning of seh shaper. Many problems expand onthe ceoncmic
tnadee and apliestions proviced inthe thsoretical pat, Problane are aranged thematically tts wit) che

jr oyes ard moving tothe move challenging ones, Sues can ty to solve a yusblem without Inoking

is slution ~ this ie wha call acive selving. Only aftorvard should they Took up the slution and compare it
With their own results. Otherwise, they would be tempted to fallow the solution pasvely and without mus

| Understanding

| Interns of eentent,the problems book cavers the most essential applications of mthematss in eecnomiss

‘matrix algebra, exponential functions, logarithmic functions, unconstrained and constrsined,
optimization, end integral calettus The last part of the book diac1scedifersntial an difference equations,
string with irs-order eqtions

nd moving t higher-order anes, Thes last topics are more chal enging for
1rdergras, and there acl tine to dacust them ina ne-semester couse. Vet they ar included for those
studi taking rigorous math eourscs or double-msjoring in mathematics and economies, a well To some

beginning graduate students who want to brush spon thei enowledge of differen ealeuius. Tot also

salle with
‘optimal contol theory, These lst opis are discuss: in bom chapters loested a the Wiley website

[thank my editor, George Lobel, who believed inthis project an gave mca helping band. is ski
sidance and editing ofthe book helped me overcome numercus problems and my inexaerience as an author, 1
am also thankful to Constance Acler, Emily MsCie, Lacey Vite, and Amy Weiatrau from Wiley
Blackwsl for ecordiaating the proces of book aepatation an! assisting my work. Secial thanks go to Amy
‘Weintraub who was oy reduction Elior ud whe fllewed br dealine sty. an ndcbied to Joan
Diesel for pootieading + manscrpt fl f extn and yatsbles, Lalo tank te illo reviewers
w rose help eae ented tpesernnt ne Lok: Steven Aner (Roxseve iivesiy) Mak
Arvin (Trent Universiy, Cad), Manica Costa Dit (University College Lamon), Bests Gross (University
‘of California, Berkeley), Karee Isl Joins Hopkins University, Mins Batre Lute [University
‘NonlFlorca), Wing Suen (University of Heng Kong), Rosia Wooster (Porn State Universi), nd
Wendy Wu (Willi Laurier University) Baral Ramachandran, who i my sp
re mich inspiration in writing the bok. He was always there io advise me ané encour
‘omens of isess und disappointment. Lar also truly grateful to three work-study’ stents of mine
‘Yordansa Kestova snd Moin Miley meticulously solv! and eorested problems for me, the fs forthe etry
| version ofthe bok ard the second forthe ae. [express my Thank to A exandru Androni, one of myst
‘work-say students, who gave me he ides a writing My problems electronically. Alex Was the Fist 9 write
| those in Mitossft Equation, This ulate brought me tothe Kea

ing the book, I shoul also mention
‘hat all my’stadens raking Quaniative Methods in Econoenics with me athe American University in Bulgaria
reeoive my appreciation, The students fom te ls fue Years helped re the mast asthe book was ested
‘hem numerous ies. Finally, 1 vish to tank my Family, my mom and ia, who had patience the etn
writing ofthe book. dedicate this book to my daughter Tayana, who grew aang with ihe book and was
livered several moras Before the manuscript ws complete

Tamara Todorova
>on
I

Suggested Readings

1, Arvo, Kenneth J, Hollie 8, Chenery, Ragichs S. Minhas, un Robert. M, Solow. “Capital Labor
bsttton and Eeonontic Etexney.” Review of Economie i Sansfioe Avast 1961}: 225 280,
2, Baldun, ete, Jemes Beaded and Rober Tumor Akathersorial Ezonomice aston, MAS
Thomson Lasring, 20
Blanchard, Olivier Macroecomoe
1. Chiang, Alpha C, Fuerst
Til I, 198.

ed. Enplowood Clits, Nl Pearce Hall Inert ons, 2000,
Methods of Muthematcel Economics. 3 Now Yorks Mere

S. dee Fount, Angel Mahemotia! Movin und Macels for Boowonits Catsidge: Cambridg
Universily Press 200
6 sey, “Capial Expansion, Rate if Growth, sd Fnplytnent,” Eeamomet i, Api 1946
75 reprinted im Domar Ess inthe Phe ef Beaman Groth. Oxted iver Pas

Dixit, Avinash, Optimization bx Eoonaaie Paory. 2% of, Ostoré Lniveity Pres, 1990.

8. Bowing, Edvard T. ntodheton vo Mathrsncc!Lcowomis, ix Schur’ Outline Serie
New Yale MeGiaw-Hil, 192,

9. Inoue, fa, Mathematics for Bonnie ane Busines.

od. Englewood Clits, NI: Prete Hal,

206,
10, Keyes, John M. The Goneral Thay af Engparment Stare ad Mone, Aachen, NV: Prometheus
11, Leona. Danie. and Ngo Yan Lang, Optima! Canin Ther ed State Optician in Sconce

Cambridge: Carbide Universiy Press, 1998

Leontil, Wassily W. tnpnt-Onfpus coments, 2°, New Vers: Orb University Pros (986,

15, MaeColell Andrew, Micisel D. Whinetn, an Cory, Gen, Micnaenonomie Teor: Ost
University Press, 198

14 Nichotsoa, Walle
Pres, 382

15. Penfieon, Malcolm, ard Nicho'as Rau. Mathias for Beonomies Mi
Univesity Pen, 2001

snore Theory: Base Principles and Extonons, Yort Worth, TX: Dryden

6, Pili, Alla “The Relationship betwesn Unemployoien: aud the Ret of Change of Me

Wage Rakes in Usa! Kingbnn HGT=1957" Faveenacs Neve 1958) 283-299

17 Rainville, Bal Davi, Philp E, Bolics, ad Rigid Baloo, Flementary Dievental Equations
2h ed. Lpper Sie iver, i Prentice Hal, 1996

18, Suruekon, Pal. Foundations of Economic Anais, Caubeidye, MA; Harsad Univesity Press,
sr,

19, Silberberg, Eugene, and Wing Suen, The Sinem af Boronic" A Masons Arie. Now
"York: MeCi-iiIrvn, 200
Simon, Carl und Lawrence Bume. Mfariomaries jor eanomsts. New York: W. W. Nort 19,

Solow, Rober M.A Contribiton 29 she Theary of Eeomosis Grth.” Qrarter'y Jaw
Foote (Ferry, 956) 65-9
22, Solow, Rebert M, “Technical Change and the Agaregate Production Function.” The Review
Economics end Statistics, 3: 90.3 (August 1957) i

23. Siilta, Joseph E.,and Car E. Walsh. Economics 3 ea. New York: W..W. Norton and Compan

24, Sydsueer, Knut, and Peter Hammond, Esientval Mariersanes for Beononte A “od
Englewond Cis, NJ: Petes Hal, 2906

5, Varian, Hal, ieraeconomi nays, New Vonks W.W., Norton, L984