Mathematical Modelling for Economists

179 Pages·1988·9.692 MB·English

by  Donald A. R. George (auth.)

Mathematical Modelling
for Economists
Mathematical
Modelling for
Economists
Donald A.R. George
M
MACMILLAN
EDUCATION
© Donald A.R. George 1988
All rights reserved. No reproduction, copy or transmissiOn
of this publication may be made without written permission.
No paragraph of this publication may be reproduced, copied
or transmitted save with written permission or in accordance
with the provisions of the Copyright Act 1956 (as amended),
or under the terms of any licence permitting limited copying
issued by the Copyright Licensing Agency, 33-4 Alfred Place,
London WC1E 7DP.
Any person who does any unauthorised act in relation to
this publication may be liable to criminal prosecution and
civil claims for damages.
First published 1988
Published by
MACMILLAN EDUCATION LTD
Houndmills, Basingstoke, Hampshire RG212XS
and London
Companies and representatives
throughout the world
British Library Cataloguing in Publication Data
George, Donald A.R.
Mathematical modelling for economists.
1. Economics-Mathematical models
I. Title
511' .8'02433 HB135
ISBN 978-0-333-42444-5 ISBN 978-1-349-19238-0 (eBook)
DOI 10.1007/978-1-349-19238-0
To my mother and father
Contents
Preface ix
1 Models, theories and method 1
2 Calculus of several variables 9
Introduction Functions and inverse
functions Continuity and derivatives Functions of
several variables Maxima, minima and
saddlepoints Convexity and concavity Partial and
total derivatives
3 Kuhn-Tucker theory 35
Introduction Constrained maximisation and
mm1m1sation Kuhn-Tucker methods An application
of Kuhn-Tucker methods Multipliers as shadow prices
Modelling the labour-managed firm
4 Linear programming 49
Introduction Primal and dual problems Duality
theorems Another application Numerical methods of
solution
5 Linear algebra 63
Introduction Simultaneous linear equations
Eigenvalues and eigenvectors The open Leontief
model Neo-Ricardian models
6 Differential equations 82
Introduction Single differential equations
Two-dimensional linear dynamical systems
Linear dynamical systems of higher dimension
Linearisation Structural stability and robustness
Rational expectations dynamics
vii
7 Dynamic optimisation 113
Introduction The one-sector neoclassical growth
model A more general maximand Pontryagin's
maximum principle An example from the economics
of resource depletion Rational expectations dynamics
revisited The overtaking criterion
8 Discontinuities and catastrophes 135
Introduction A state-preference model with
non-convexities Thorn's theorem A trade cycle
model Methodological problems
Appendix Brief solutions to selected exercises 159
Bibliography 165
Index 169
viii Contents
Preface
Model building is an activity which students often find difficult and
sometimes rather puzzling. It can be difficult to see just how the
latest 97-equation model with 35 parameters and 53 variables can
have any relevance to the great problems of economics such as
unemployment, poverty, distribution and growth. Students are
frequently tempted to skip the 'nasty mathematical bit' in journal
articles, and yet model building is an indispensable aspect of
modern economics.
The mathematics explained in this book is really quite easy but
the process of model building requires skills other than simply
knowing the appropriate mathematics. It is a process of abstrac
tion involving the construction, manipulation and interpretation of
models and this book places considerable emphasis on these
issues. Modelling is thus treated as a fairly 'down-to-earth' pro
cedure. We eschew the grand questions of the modern scholastics
and the reader wishing to know how many equilibria can dance on
a compact manifold will therefore have to look elsewhere. Having
read the book, students should be able to understand the majority
of models used in the general economics journals and even to
construct and use simple models of their own.
This book is based on courses given at Edinburgh and I am
grateful to generations of Edinburgh students who never fail to
offer comments and criticisms on the material with which they are
presented. The book was revised while I was on leave at the
European University Institute, Florence, to whom I am indebted
for a generous Fellowship and an opportunity to work surrounded
by Tuscan hills and endless supplies of Chianti. I am grateful also
for the constructive and stimulating discussions I have had with
colleagues both in Edinburgh and in Florence. I would mention in
particular Simon Clark (Edinburgh) who read the entire manu
script and provided much useful advice. I am also particularly
grateful to Les Oxley (Edinburgh), who has permitted me to draw
upon research we have conducted jointly, and to Robert Kast
ix
(Florence). Any remaining errors or omissions are, of course, my
own. Chris Barton and Jessica Spataro, who can interpret Ice
landic Runestones with ease, dealt swiftly with my dreadful hand
writing, turning it efficiently into typescript; for this I am most
grateful. Thanks also to Jennifer Pegg and Keith Povey for guiding
the book through the editorial and production processes. Finally,
my greatest debt is to my parents, for their unfailing support and
encouragement in all my endeavours, including those even less
reputable than textbook writing.
Florence DONALD A.R. GEORGE
x Preface
1 Models, theories
and method
Consider an Economics student presented with the following
mathematical relationships:
x =az+~ (1.1)
q=q (1.2)
z =x + y (1.3)
w=x+q (1.4)
w=z (1.5)
and suppose he or she is told that equations (1.1)-(1.5) imply:
dz 1
---- (a of= 1) (1.6)
dq 1-a
He or she is probably not going to be particularly enlightened by
this information. Suppose now that the variables in equations
(1.1)-(1.5) are simply relabelled thus:
C = aY + b (1.1')
I =I (1.2')
Y=C+S (1.3')
E= C+I (1.4')
E=Y (1.5')
The familiar Keynesian-cross model of aggregate income determi
nation is now self-evident and equation (1.6) yields the well
known result relating the investment multiplier dY/dl to the
marginal propensity to consume (a):
dY 1
(1.6')
di 1-a
Equations (1.1)-(1.6) constitute a simple mathematical model of
aggregate income determination. The model consists of a number
1