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THE THEORY OF
INTERNATIONAL
BUSINESS
Economic Models
and Methods

Mark Casson


The Theory of International Business



Mark Casson

The Theory of
International Business
Economic Models and Methods


Mark Casson
Department of Economics
University of Reading
Reading, United Kingdom

ISBN 978-3-319-32296-4
ISBN 978-3-319-32297-1
DOI 10.1007/978-3-319-32297-1

(eBook)


Library of Congress Control Number: 2016946226
© The Editor(s) (if applicable) and the Author(s) 2016
This work is subject to copyright. All rights are solely and exclusively licensed by the
Publisher, whether the whole or part of the material is concerned, specifically the rights of
translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on
microfilms or in any other physical way, and transmission or information storage and retrieval,
electronic adaptation, computer software, or by similar or dissimilar methodology now
known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publication does not imply, even in the absence of a specific statement, that such names are
exempt from the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information
in this book are believed to be true and accurate at the date of publication. Neither the
publisher nor the authors or the editors give a warranty, express or implied, with respect to
the material contained herein or for any errors or omissions that may have been made.
Printed on acid-free paper
This Palgrave Macmillan imprint is published by Springer Nature
The registered company is Springer International Publishing AG Switzerland


In memory of Alan Rugman: scholar, friend, colleague



PREFACE

AND

ACKNOWLEDGEMENTS


The ideas presented in this book have been gestating for a long time. The
premature and unexpected death of my good friend Alan Rugman has
stimulated me to reflect on how far theories of international business (IB)
have changed in the forty years since we first met. Discussions with Peter
Buckley have led me to ask whether these changes have been for the better or not. I have therefore resolved to set down what, in my opinion, the
theory of IB might look like now if the direction originally charted had
been followed up in a systematic way.
IB scholars seem to have lost the initiative in theory development. In
the 1970s and early 1980s, IB exported key ideas to economics, but now
IB is largely an importer of ideas from other fields. IB scholars seem to
think that concepts derived from strategic management and the resourcebased theory of the firm provide all the economics that is required to
understand IB. This is a big mistake. A major objective of this book is to
help to rectify this error.
This book is based on a series of ‘Masterclass’ lectures delivered at the
Henley Business School, University of Reading, in November 2015. The
class was attended by an international group of scholars and students. I
am grateful to the participants for giving up their valuable time and for
providing feedback which helped me to turn my lectures into book.
The Masterclasses are funded by Helen Rugman to perpetuate the
intellectual legacy of Alan Rugman. I hope that these notes will advance
the agenda pursued by Alan in his years at Reading, which was to restore
the theory of the firm to its proper place at the heart of IB studies. I am

vii


viii

PREFACE AND ACKNOWLEDGEMENTS


grateful to James Walker for inviting me to present this Masterclass, and
for providing the stimulus to prepare the original notes.
The book draws on recent work that I have been undertaking in collaboration with colleagues and, in particular, in joint research with Nigel
Wadeson (University of Reading) and Lynda Porter (University of Bath).
I have also benefited from discussions with Rajneesh Narula, Quyen
Nguyen and Maggie Cooper. Davide Castellani and Janet Casson provided valuable comments on the final draft. I should also like to thank
Daria Radwan for her cheerful and meticulous administrative support.
The book assumes little or no prior knowledge of economics. It does,
however, address some popular misconceptions in IB regarding economics which need to be dispelled. Central to prevailing misconceptions is
the failure to realise why economists develop models and to appreciate
why such models are essential for analysing business behaviour in a global
economy.


CONTENTS

1

2

The Relationship Between Economics and International
Business Studies
Introduction to Modelling Techniques

1
21

3 Introduction to Monopoly

41


4 Introduction to Location

53

5

71

Division of Labour and Modularisation

6 Analysis of Ownership
7

Modelling Contractual Arrangements

93
111

8 Global Rivalry

129

9 Extensions of the Models

147

ix



x

CONTENTS

10

The Management of the Firm

157

11

Conclusions: A Model-Building Agenda

165

Index

167


LIST

Fig. 2.1
Fig. 2.2
Fig. 2.3
Fig. 2.4
Fig. 2.5
Fig. 3.1
Fig. 3.2

Fig. 4.1
Fig. 4.2
Fig. 4.3
Fig. 4.4
Fig. 5.1
Fig. 5.2
Fig. 5.3

Fig. 5.4
Fig. 6.1

OF

FIGURES

Utility expressed as a quadratic function of two variables
Constrained maximisation of utility using the method of
substitution: first-order condition
Determination of optimal consumption mix
Comparison of solving a constrained maximum problem by
substitution and by the Lagrange multiplier
Demand analysis of optimal consumption strategy
Monopoly equilibrium with a one-part tariff
Monopoly equilibrium with a two-part tariff
The Edgeworth Box: international consumption patterns
under complete specialisation of production
Determination of consumption under complete specialisation
in production: demand analysis
Supply and demand analysis of trade with no international
price discrimination

Monopoly equilibrium with international price discrimination
Monopoly equilibrium with international transport costs
and local marketing costs
Monopoly equilibrium with local marketing costs: the case of
international price discrimination
International supply chains: a schematic diagram of product
development, production and marketing in a two-country
world
Decision tree for the optimal location of production and
development
Contractual arrangements for supply chain coordination

26
34
35
37
39
50
51
59
61
64
69
75
79

84
86
101


xi


xii

LIST OF FIGURES

Fig. 6.2
Fig. 6.3
Fig. 7.1
Fig. 7.2
Fig. 8.1
Fig. 9.1

Schematic representation of internalisation, subcontracting,
franchising, putting out and licensing
The role of headquarters as a coordination hub
Decision tree for supply chain optimisation: the interaction of
contractual arrangements and location
Interaction of location and contractual arrangements in the
optimisation of supply chain strategy for a foreign market
Decision tree for solution of a sequential innovation game
with negative competitive profits
Decision tree for government–business interaction with
economies of scale in production

102
104
113
120

136
155


LIST

Table 1.1
Table 1.2
Table 2.1
Table 3.1
Table 4.1
Table 4.2
Table 4.3
Table 5.1
Table 5.2
Table 5.3
Table 5.4
Table 5.5

Table 5.6
Table 6.1
Table 7.1
Table 7.2

Table 7.3
Table 7.4

OF

TABLES


The sequence of models
Key thinkers, listed in logical sequence of concepts
Comparative statics of Model 1
Comparative statics of Model 2
Comparative statics of price for Model 3.1
Comparative statics for Model 4.1
Comparative statics for Model 4.2
Comparative statics for Model 5.1
Unit costs of market supply with transport costs
Comparative statics for Model 5.2
Unit costs of knowledge transfer
Unit costs of supply to each market conditional on
the location of production and the location of
development
Comparative statics for Model 6
Coordination costs incurred by alternative contractual
arrangements for a supply chain serving a given market
Three-way classification of costs in Model 7
Supplying a foreign market: coordination costs for licensing
and full internalisation in a two-country world where
headquarters is located in country 1 and the market
in country 2
Supplying the home market: coordination costs where both
headquarters and the market are located in country 1
Total unit costs of four strategies for supplying a foreign
market

14
19

32
49
58
67
68
74
77
78
84

87
89
109
114

117
118
120

xiii


xiv

LIST OF TABLES

Table 7.5

Table 7.6
Table 8.1

Table 8.2
Table 8.3
Table 8.4
Table 8.5

Supplying a foreign market: coordination costs for five
contractual arrangements where headquarters is located
in country 1 and the market in country 2
Supplying the home market: coordination costs where
both headquarters and the market are located in country 1
General form of profits in a 2 × 2 game of rivalry
Profits generated in four cases where monopoly profits
are positive
Profits in a two-rival game with choice of locations for
development and headquarters
Probabilistic outcomes for alternative innovation decision
processes
Expected profit outcomes with innovation rivalry under
uncertainty

124
125
131
134
140
144
146


CHAPTER 1


The Relationship Between Economics
and International Business Studies

Abstract The literature on international business (IB) studies relies heavily on concepts from business strategy and makes relatively little use of
concepts from economics. This is a mistake. This chapter introduces the
concepts used by economists to analyse IB issues. It describes ‘how economists think’ and what they do and explains why their approach is so useful
in IB studies.
Keywords Model • Mathematics • Economics • Rationality •
Equilibrium

THE ROLE OF ECONOMICS IN INTERNATIONAL
BUSINESS STUDIES
This book concerns the application of economic theory to international
business (IB). There is a particular focus on the multinational enterprise
(MNE). A major stimulus to the development of modern IB theory was
the need to explain the dramatic expansion of US MNEs into Europe (and
elsewhere) in the 1950s and 1960s. The economic theory of IB developed
as a branch of applied economics. Most of the early writers had trained
as economists and worked in either business economics or economic
development.
© The Editor(s) (if applicable) and the Author(s) 2016
M. Casson, The Theory of International Business,
DOI 10.1007/978-3-319-32297-1_1

1


2


M. CASSON

Mainstream models of international trade and investment could not
analyse MNEs satisfactorily as they could not explain why a firm would
own and control assets in a foreign country. It is interesting that, with
the exception of Stephen Magee, trade theorists did not play an important part in developing early IB theory—probably because they were too
strongly attached to the standard ‘factor-proportions’ approach to trade.
Some IB scholars have drawn the wrong conclusion from this. They
argue that this story shows that economic methods do not work in IB. In
fact the opposite is true. It shows that economic methods were successful
in developing a creative alternative to standard trade theory.
It is widely acknowledged that IB is essentially an inter-disciplinary
subject. Full understanding of IB behaviour requires insights from a
range of disciplines. Ideally these should be synthesised. But synthesis
is difficult because different branches of the social sciences are based on
different assumptions. These differences extend to fundamental issues
concerning human nature. Economists’ assumptions on this subject
often appear to be an outlier. In particular, mainstream economists assert
human rationality and every other social science discipline seems to deny
it. This has led, in practice, to the notion that IB theory should include
every relevant discipline except economics. The gap left by economics
should be filled by newly developed subjects such as the theory of IB
strategy or the resource-based theory of the firm. These are supposed to
encapsulate relevant notions from economics whilst leaving the objectionable material out.
This is a mistake on three counts.

The Concept of Rationality in Economics Is Widely Misunderstood
Rationality has a specific meaning in economics which differs from
its connotation in everyday use. Critics often ignore this. Economic
rationality asserts that each person possesses a coherent set of preferences that allow them to place alternative courses of action in a consistent order according to the desirability of the expected outcome.

When a person chooses one course of action instead of another it
is because the expected outcome of the chosen course of action is
better.


THE RELATIONSHIP BETWEEN ECONOMICS AND INTERNATIONAL BUSINESS...

3

For example, suppose that an individual faces a series of choice involving three options A, B and C. To start with they face a choice between
A and B because C is not available. They choose A. The rational interpretation of their action is that they prefer A to B. Next they are forced
to choose between B and C and they choose B. Finally, they are asked to
choose between A and C. Rationality predicts that they will choose A.
This is because rationality implies that preferences are transitive: if A is
preferred to B and B is preferred to C then A must be preferred to C. If C
is chosen then the individual is either irrational, or their preferences have
changed during the process.
Rationality is often confused with perfect information and perfect foresight, which are different things altogether. Rationality combined with
perfect information implies that people never make mistakes; rationality
alone does not imply that, however, because mistakes may be due entirely
to missing information.

The IB System Is Complex
Complexity makes it easy for IB scholars to make mistakes. A purely
verbal argument is fraught with risks. The same word often has different connotations in different contexts. Its meaning can therefore
change as the context changes and as the argument develops. To standardise the meaning of words it is it is important to make explicit the
context in which they are used. Stringing together a set of plausible
sounding statements without making clear the context may result in
misleading conclusions. To achieve intellectual rigour it helps to have
a formal model. The model creates a virtual world in which abstract

concepts are carefully defined within a context that is clear.

The more complex the argument, the greater the risk of error and the
more important it is to have a model. It is important to keep a model
simple, however. This is where rationality comes in. Rationality does not
reflect some doctrinaire view of human nature favoured by economists.
It is simply an instrumental assumption made to simplify a potentially
complex model. It is necessary to assume rationality because researchers
themselves are not fully rational. If they were fully rational they could


4

M. CASSON

weave arguments of incredible complexity without falling into error. In
practice they cannot do this. It is because of our limited intellectual powers that when analysing complex systems it is useful to assume rationality.

Substitutes for Economic Theory Are Inadequate
Neither IB strategy nor the resource-based theory is normally
articulated in mathematical terms. Both involved errors in their
original formulations.

The resource-based theory assumed that the competencies of employees constituted capabilities of the firm from which the firm could earn
exceptional profit. The theory ignored the way that the labour market
works. The labour market allows employees to profit from their own competencies. As the resource-based theory included no formal model of the
labour market this point was overlooked. Firms employ teams of workers
and compete to hire the members of these teams. For example, football
clubs in the English Premier League combine highly talented players into
teams. It is the players and not the clubs that appropriate the gains from

teamwork. Very high salaries are paid to attract and retain the best team
players. This is because their alternative earnings reflect what they would
be worth to a rival team and not what they would be worth if they played
on their own. If these salaries do not exhaust the profits from the team
then the manager’s salary will normally absorb whatever remains.
The flaw in IB strategy is that every strategy has a competitive response.
If the word strategy is taken literally then it implies degree of rivalry, yet
many so-called theories of strategy (in IB and elsewhere) do not analyse
rivalry in any detail. A firm can often neutralise a rival’s strategy simply by
matching it with a similar strategy. By ignoring rivals’ responses, the profits of ‘winning strategies’ are over-stated.
The flaws in these two theories have a common failing: they do not analyse
competition properly. Resource-based theory fails to analyse labour market
competition and business strategy fails to analyse product market competition. Economists have devoted a lot of effort to analysing competition
and it seems foolish to ignore the product of those efforts. Modern economists usually analyse strategic rivalry using non-cooperative game theory.
As demonstrated below, game theory can clarify quite complex situations
because it relies on the rational action principle to simplify the analysis.


THE RELATIONSHIP BETWEEN ECONOMICS AND INTERNATIONAL BUSINESS...

5

THE NATURE OF ECONOMIC MODELLING
A Model as an Abstraction
Modelling is often regarded as a purely technical exercise. Both modellers and their critics often take this view. Modellers take pride in the
intellectual ingenuity used to construct their models. While all models are abstract, good models capture the salient points of reality too.
Bad models address unreal situations and their practical irrelevance
gives modelling a bad name.

The real world is complex, and this complexity makes it messy. A good

model abstracts from the messy stuff and concentrates just on the things
that really matter for the problem in hand. The messy world is what you
get when you take a photograph; a busy background diminishes the force
of the subject matter in the foreground. A good economic model is like a
work of art. A figurative artist will blur the background and sharpen up the
foreground to give it added prominence. They may even edit out the background altogether to produce a pure abstraction. In fact, good models are
often described in artistic terms—as elegant, or even beautiful. This is more
than just hyperbole. Models are valuable not only for their practical utility
in clarifying problems; they can be appreciated on aesthetic grounds as well.
Good models are based on explicit definitions. Variables are carefully
defined and then related to each other. These relationships are typically
deduced from a small set of basic assumptions, which are also made explicit.
The idea is that the assumptions are relatively weak and the conclusions
are relatively strong, that is, the assumptions appear perfectly reasonable
whilst the conclusions are quite striking. The conclusions are not just a
trivial re-statement of the assumptions in a different form. The result is a
powerful model in which the logic of the analysis has an important role.
Good models have real-world implications. The relationships deduced
from the model can be translated into relationships (such as correlations)
between observable variables. These observable variables may be either
quantitative, for example, sales, employment, profits, patents, advertising
expenditure and R&D expenditure, or qualitative, for example, whether
a firm innovates and if so where it locates its R&D. A good model is based
on plausible assumptions and leads to conclusions that can be tested (and
preferably corroborated) through quantitative or qualitative research.


6

M. CASSON


Since models are based on explicit definitions, it may be useful at this
stage to offer a definition of a model. It is quite surprising that, while economic literature is full of definitions and full of models, it is hard to find
a definition of an economic model. The definition given below should be
interpreted in the light of the discussion above.
An economic model is a symbolic representation of an abstract world.
It comprises a set of relationships, deduced from a parsimonious set of
explicit assumptions.
These relationships translate into observable relationships between realworld economic variables.

This definition is quite broad and many different types of model fall
within its scope. Models may be classified in various ways.
• Level of analysis. Many economic models are formulated at the level
of the national economy, but the models in this book refer to the
global economy.
• Types of variable. Macroeconomic models involve aggregate quantities, such as gross domestic product or the consumer price index,
which relate to bundles of commodities, while microeconomic models typically involve individual products and their prices. IB studies
involve both types of model; macroeconomic models are used to
explain aggregate flows of foreign direct investment (FDI), as measured in national accounts, whilst microeconomic models are used to
analyse the behaviour of individual industries and firms. This book
focuses on microeconomic modelling.
• Degree of homogeneity. Most microeconomic models assume that
products are divisible into identical units which are perfect substitutes for each other, that is, customers are indifferent as to which
unit they consume. If customers perceive different units as identical they will always buy the cheapest and, with perfect information
on price, this implies that in equilibrium there will be just a single
price for each product. The demands of individual consumers for
a homogeneous product can therefore be aggregated to generate
the total demand at a given price. Likewise, if all producers supply
identical products then their supplies can be aggregated to generate
the total supply forthcoming at any given price. This is the basis

for conventional demand and supply analysis of a product market.
These notes follow this general approach by assuming that, while
products may be available in different varieties, units of any given


THE RELATIONSHIP BETWEEN ECONOMICS AND INTERNATIONAL BUSINESS...

7

variety are homogeneous. Knowledge is not homogeneous, however; different firms exploit different knowledge and this means
that the market for knowledge is very different from the markets
for ordinary products.
• Spatial heterogeneity. Many economic models make no reference
to space. In IB space is, of course, central. Space in IB consists of
the two-dimension surface of the earth, which is itself embedded
in a three-dimensional space. Geographical distances on this surface are not strictly Euclidean: someone who set off along a great
circle would finish up back where they started after 25,000 miles.
Many spatial models constrain the spatial options, by assuming the
economic activity is confined to points on a circle or points on a
straight line. In IB, however, realism demands that every location is
treated as fundamentally different from every other, and this is the
approach adopted in these notes.
• Number of different types of decision-maker. In rational action modelling
the decision-maker plays a central role. Most economic models involve
multiple decisions-makers, and a key aspect of the ‘economic problem’
is to reconcile the different decisions that they make. The model must
specify who has the right to make what decisions. The right to make a
decision is generally conferred by ownership of property, and consists
of the right to decide how this property is used (in particular the right
to consume it and to sell it). Decisions may be delegated to institutions,

however, including governments (empowered by the collective delegation of citizens) and firms (empowered by the collective delegation of
shareholders). These institutions then delegate their own decisions to
specific individuals who occupy specific roles (e.g. government ministers, chief executives). The determination of decision-making powers is
therefore quite a subtle process. Simple models involve just a few decision-makers. Where there are many decision-makers, the complexity of
the model can be reduced by classifying decision-makers into types and
assuming that all decision-makers of the same type are identical. This
homogeneity assumption is often applied to workers and consumers,
and sometimes to firms as well. The models below invoke homogeneity regarding workers and consumers at each location. Institutions are
analysed individually, however.
• Contractibility. In economic models the decisions of different
decision-makers are very often coordinated by contracts between
them. Contracting refers to the entire process of searching out a
partner, negotiating terms and enforcing compliance. There is an


8

M. CASSON

important distinction between models where contracting is costly
and models where it is free. In the IB literature contracting between
firms is always regarded as costly, although contracting between firms
and households is often regarded as free. There is another aspect of
contractibility, however, whose significance is sometimes overlooked.
This arises where the costs of certain types of contract are prohibitive, so that those types of contract are never used. The fact that a
particular type of contract is never used means that the possibility
of using it is often overlooked. But in fact many of the results that
are obtained from economic models are most readily explained by
the nature of the contracts that they exclude rather than the nature
of the contracts they include. The models below include a wider

range of contracts than those included in mainstream models and,
in particular, a range of contractual options for knowledge transfer.
However, they follow mainstream literature in excluding collusion
between firms (e.g. cartels) and agreements between firms and consumer cooperatives. These exclusions may be justified on grounds of
realism. The IB system functions very differently from the way that
it would if such contracts were in regular use.
Models are constructed according to basic rules. These are the rules of
the model-maker’s craft. Some approaches work well and others usually
fail. The experienced modeller knows the rules but the novice typically
does not. Where rational action models are concerned, the concept of
equilibrium is key. A complex system like an economy can exist in many
states. A rational action model identifies equilibrium states. The modeller
then predicts that it is the equilibrium states that are observed.

The Concept of Equilibrium
An equilibrium is usually understood as a balance of forces, for
example, market equilibrium balances the opposing ‘forces’ of supply and demand. A balance of forces is, however, a physical concept
rather than an economic one. In economic models an equilibrium is
usually understood as a state in which individual plans are mutually
compatible and no individual has any reason to modify their plans.
This means that when the plans are implemented everything turns
out exactly as expected and everyone is satisfied with the outcome.


THE RELATIONSHIP BETWEEN ECONOMICS AND INTERNATIONAL BUSINESS...

9

It is convenient to focus on market equilibrium at this stage; nonmarket equilibrium is discussed in the context of rivalry in Chap. 8. The
definition of market equilibrium is basically in two parts.

• Consistency. When individual plans are aggregated, the total commitment of resources is equal to the total quantity available.
• Social efficiency. No one can be better off, given the options available to them. Efficiency implies that there is no available type of
contract that makes someone better off and is acceptable to others.
The only way of making someone better off is to make others worse
off and, being rational, they would never agree to it. When further
contracting is impossible, it means that each person’s choices are best
responses to other people’s choices.
Models are often classified by their equilibrium properties
• Multiplicity: How many equilibria are there? There could be no
equilibrium. It is often assumed that there is just one equilibrium,
but this is not always the case. The model discussed in Chap. 8 has
multiple equilibria.
• Stability: If the economy is close to an equilibrium will it converge
on it or move away from it? If it is disturbed from equilibrium will
‘negative feed back’ return it to equilibrium? The models presented
here are stable.
• Efficiency: Not all equilibria are efficient. In particular, monopolistic
equilibrium is usually inefficient (see Chap. 3), and the outcome of
rivalry in innovation may be inefficient too.
• Fairness or justice: Ethical judgements can be made on equilibrium outcomes. Different criteria lead to different judgements.
Some economists claim that ethical judgements should be avoided
because they are subjective and ‘unscientific’; economics is only
concerned with efficiency, they claim. This is over-stated. The point
is that economists have no particular expertise in making ethical
judgements. Their role is to usually clarify the issues and leave
policy-makers to make the final judgements. Ethical judgements
are often related to the distribution of income. The models presented below provide predictions about the distribution of income;
they provide sufficient information for third parties to make ethical
judgements if desired.



10

M. CASSON

The concept of equilibrium attracts a lot of criticism. It is hard to accept
it as a literal description of reality. But it is not meant to be that; it is simply
a method of deriving predictions from an abstract model by assuming a
tendency for a system to converge. The value of an equilibrium model lies
not in the literal truth of its assumptions but in the practical utility of its
conclusions.
It is often said that equilibrium analysis is inherently static but that is
not quite correct. It is true in one sense but not in another: an equilibrium outcome is the product of instantaneous communication, and in that
sense timeless, but outcomes can change over time.
In mainstream economic models competitive market equilibrium is
achieved by instantaneous adjustments to traders’ plans. This involves
a trial and error process which is completed before any actual trades
take place. An example is the fictional auction process described by the
nineteenth-century French economist Leon Walras. A more realistic
account involves sellers posting prices, buyers searching sellers for the best
price, and sellers adjusting their prices to match buyers’ demands to their
own supplies. This approach is protracted, but for days or weeks rather
than years. Auction markets are often cited as real-world examples of equilibrium price-setting; prices may vary from minute to minute, but the
quantity on offer at any given moment is usually fixed. Multi-lateral faceto-face negotiation can also come close to the theoretical ideal.
An equilibrium outcome is determined by prevailing economic conditions, and as these conditions change equilibrium outcomes change as well.
Comparative static analysis, as described in Chap. 2, analyses the effect
on equilibrium of a persistent one-off change in conditions. Equilibrium
models exhibit adaptive behaviour; if conditions change continually then
the equilibrium changes continuously too.
The Role of Mathematics in Modelling

Many IB scholars find economic models hard to understand. This is usually blamed on the difficulty of the mathematics and, to some extent, on
the abstract nature of the model. In fact the mathematics in IB models is
relatively simple. To understand the models discussed below, the ability to
add, subtract, multiply and divide is essential. It is also useful to know how
to handle a mathematical function by calculating the value of the function from the values of the variables. Finally, basic calculus is helpful. A
knowledge of calculus is not assumed in these notes; however some of the


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