Springer Texts in Business and Economics Wolfgang Eichhorn Winfried Gleißner Mathematics and Methodology for Economics Applications, Problems and Solutions Springer Texts in Business and Economics More information about this series at http://www.springer.com/series/10099 Wolfgang Eichhorn ã Winfried Gleiòner Mathematics and Methodology for Economics Applications, Problems and Solutions 123 Wolfgang Eichhorn Karlsruhe Institute of Technology (KIT) Karlsruhe, Germany Winfried Gleißner University of Applied Sciences Landshut Landshut, Germany ISSN 2192-4333 ISSN 2192-4341 (electronic) Springer Texts in Business and Economics ISBN 978-3-319-23352-9 ISBN 978-3-319-23353-6 (eBook) DOI 10.1007/978-3-319-23353-6 Library of Congress Control Number: 2016932103 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2016 This work is subject to copyright All rights are reserved 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 Springer International Publishing AG Switzerland (www.springer.com) is part of Springer Science+Business Media Preface This book about mathematics and methodology for economics is the result of the lifelong teaching experience of the authors It is written for university students as well as for students of a university of applied sciences It is completely selfcontained and does not assume any previous knowledge of high school mathematics At the end of all chapters and sections, there are exercises such that the reader can test how familiar she or he is with the material of the preceding stuff After each set of exercises, the answers are given to encourage the reader to tackle the problems The idea to write such a book was born in 1990 during an international meeting on functional equations which took place at the University of Graz, Austria At this meeting a lot of fascinating applications of functional equations to solve mathematically formulated economic problems inspired János Aczél, Distinguished Professor of Mathematics, University of Waterloo, Ontario, Canada: He proposed to one of us (W.E.) to start such an adventure in a form of a textbook for beginners Since then he supported the tentative steps into this direction by a great wealth of brilliant scientific advices Later on he became for both of us the lodestar for our endeavour Dear János, we owe you a great debt of gratitude For a basic course Chaps (sets, vectors, trigonometric functions, complex numbers), (mappings and functions), (vectors, matrices, systems of linear equations), (functions, limits, derivations), (important nonlinear functions), and 10 (integration) are sufficient If a later course will discuss discrete models of economics, Chap 12 (difference equations) should be covered, too For continuous models, Chap 11 (differential equations) is necessary (However, we decided not to go very far into details.) Chapter gives an introduction to linear optimisation and game theory using production systems These ideas are continued in Chaps and 9, which discusses the notion of a Nash Equilibrium Chapter deals with nonlinear optimisation Chapter 13, as the conclusion, reflects methodologically most of all that what we optimistically offered in Chaps 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 and 12 v vi Preface Many thanks go to Thomas Schlink for typing most of the manuscript in LATEX very conscientiously and to Dr Roland Peyrer for his inspiring drawings, which were transformed to PSTricks, an additional package for graphics in Latex Karlsruhe, Germany Landshut, Germany Summer, 2015 Wolfgang Eichhorn Winfried Gleißner Contents Sets, Numbers and Vectors 1.1 Introduction 1.2 Basics 1.2.1 Exercises 1.2.2 Answers 1.3 Subsets, Operations Between Sets 1.3.1 Exercises 1.3.2 Answers 1.4 Cartesian Products of Sets, Rn , Vectors 1.4.1 Exercises 1.4.2 Answers 1.5 Operations for Vectors, Linear Dependence and Independence 1.5.1 Sums, Differences, Linear Combinations of Vectors 1.5.2 Linear Dependence, Independence 1.5.3 Inner Product 1.5.4 Exercises 1.5.5 Answers 1.6 Geometric Interpretations Distance Orthogonal Vectors 1.6.1 Exercises 1.6.2 Answers 1.7 Complex Numbers; the Cosine, Sine, Tangent and Cotangent 1.7.1 Multiplication of Complex Numbers 1.7.2 Trigonometric Form of Complex Numbers; Sine, Cosine 1.7.3 Division of Complex Numbers; Equations 1.7.4 Tangent, Cotangent 1.7.5 Exercises 1.7.6 Answers 1 7 11 12 12 18 19 19 19 21 24 25 26 26 30 30 31 31 34 40 42 43 43 vii viii Contents Production Systems Production Processes, Technologies, Efficiency, Optimisation 2.1 Introduction 2.2 Basics 2.2.1 Exercises 2.2.2 Answers 2.3 Linear Production Models, Linear Optimisation Problems 2.3.1 Exercises 2.3.2 Answers 2.4 Simple Approaches to Linear Optimisation Problems 2.4.1 Exercises 2.4.2 Answers Mappings, Functions 3.1 Introduction 3.2 Basics Domains, Ranges, Images (Codomains) Mappings (Binary Relations), Functions, Injections, Surjections, Bijections Graphs 3.2.1 Exercises 3.2.2 Answers 3.3 Functions of n Variables, n-Dimensional Intervals, Composition of Functions 3.3.1 Exercises 3.3.2 Answers 3.4 Monotonic and Linearly Homogeneous Functions Maxima and Minima 3.4.1 Exercises 3.4.2 Answers 3.5 Convex (Concave) Functions Convex Sets 3.5.1 Exercises 3.5.2 Answers 3.6 Quasi-convex Functions 3.6.1 Exercises 3.6.2 Answers 3.7 Functions in the “Statistical Theory” of Price Indices 3.7.1 Exercises 3.7.2 Answers Affine and Linear Functions and Transformations (Matrices), Linear Economic Models, Systems of Linear Equations and Inequalities 4.1 Introduction 4.2 Proportionality, Linear and Affine Functions Additivity, Linear Homogeneity, Linearity 4.2.1 Exercises 4.2.2 Answers 45 45 46 49 49 49 52 53 53 59 60 61 61 63 72 72 73 77 78 78 84 85 85 92 92 93 99 100 100 103 104 105 105 107 112 113 Contents 4.3 4.4 4.5 4.6 4.7 4.8 ix Additivity, Linear Homogeneity, Linearity of Vector-Vector Functions, Matrices 4.3.1 Exercises 4.3.2 Answers Matrix Algebra 4.4.1 Exercises 4.4.2 Answers Linear Economic Models: Leontief, von Neumann 4.5.1 Exercises 4.5.2 Answers Systems of Linear Equations Solution by Elimination Rank Necessary and Sufficient Conditions 4.6.1 Exercises 4.6.2 Answers Determinant, Cramer’s Rule, Inverse Matrix 4.7.1 Exercises 4.7.2 Answers Applications of Functions of Vector Variables: Aggregation in Economics 4.8.1 Exercises 4.8.2 Answers 113 117 117 118 124 125 126 133 134 135 154 155 156 164 165 165 174 176 Linear Optimisation, Duality: Zero-Sum Games 5.1 Introduction 5.2 Linear Optimisation Problems 5.2.1 Exercises 5.2.2 Answers 5.3 Duality 5.3.1 Exercises 5.3.2 Answers 5.4 Two-Person Zero-Sum Games 5.4.1 Exercises 5.4.2 Answers 177 177 179 192 192 194 200 201 201 207 207 Functions, Their Limits and Their Derivatives 6.1 Introduction 6.2 Limits, Infinity as Limit, Limit at Infinity, Sequences: Trigonometric Functions, Polynomials, Rational Functions 6.2.1 Exercises 6.2.2 Answers 6.3 Continuity, Sectional Continuity, Left and Right Limits 6.3.1 Exercises 6.3.2 Answers 6.4 Derivative, Derivation 6.4.1 Exercises 6.4.2 Answers 209 209 211 220 221 221 226 227 227 233 234 13.5 Theories in the Sciences, in Particular in Economics 615 be In Sect 13.2 we saw that the situation is similar in applied mathematics but that it is, in a way, reversed in pure mathematics and in logic: there a model is a realisation of a theory, a bunch of objects and relations which satisfy the axioms (assumptions) and thus the whole theory Similarly as in Sect 13.3 with graphic representations, also the relation between economic model and systems of assumptions (axioms) or theory can be reversed to conform with the pure mathematical—logical usage Then the system of axioms and thus the whole theory can be considered in an abstract—formal way (as in pure mathematics or logic) and the (say, economic) model is a realisation of the theory As in Sects 13.2 and 13.3 3, again several models (realisations) may be attached to the same theory or system of assumptions For instance, the assumptions in Example (Sect 13.3 4, (13.1), (13.2), (13.3)), may be recognised as pure mathematical equations (with constants A RCC , c 0; 1Œ, d RC ), C D cY C d; I D A; Y D C C I: As model (meaning now “realisation”) we had there national income as Y, expenditure for consumption C/ and investment I/ Another model could be income Y, consumption C and savings I of private household, with the values (13.4) assumed again We state some further requirements for economic theories to be specifically “empirical” (similar specifications apply to theories in other social, behavioural and even natural sciences) In order for a consistent theory to be relevant to economics, its system of assumptions should represent a model of a branch of economics Furthermore, a theory is empirical if it contains statements (“theorems”) which have been checked in practice and are not (yet) falsified It is also desirable that the theory should contain statements about processes of significant extent (“dimension”) in space and time The above shows the importance of the method of deduction also for economics It allows to condense a branch of economics into consistent assumptions from which by logical deduction a theory is built containing statements (“theorems”) which can be confronted with the economic reality and thus justified or falsified Nowadays much of economic theory is created in this way However, there exist useful theories in economics and in other sciences, no assumptions or theorems of which can be exactly verified by experience We mentioned in Sect 13.3 the theory of “ideal gases” in physics and that of “perfect markets” (Sect 13.3, Example 3) in economics Since their basic assumptions are abstractions (idealisations) they cannot be exactly verified by experience Nevertheless, the consequences of these assumptions (the theory built upon them) are approximately correct and help to explain many phenomena of the physical or 616 13 Methodology: Models and Theories in Economics economic reality, respectively In particular, logical consequences of the assumptions describing a perfect market (Sect 13.3, Example 3) and of assumptions about human behaviour, which are also abstractions (such as “maximisation of profit” and “maximisation of utility”) show, among others, that (i) equilibria exist, that is, supply and demand can be balanced and that (ii) such equilibria are efficient, that is, one “economic agent” (e.g., individual enterprise, etc.) can better than in the equilibrium situation only if one or more others worse (see Sect 13.3) The purpose is not so much to find quantitatively (numerically) the points of equilibrium, say The results mentioned are rather important qualitative (structural) results The closer real markets get to fulfilling the perfect market conditions (Sect 13.3, Example 3) as is increasingly the case with stock markets the better the results of this theory approximate the real situation in them 13.6 Why Construct Models and Theories? Types of Models and Theories Depending upon the purpose of constructing models and theories, different types of them evolved In this section we shall touch, by means of examples, on some important types of models and theories, as determined by their purposes, such as description (1), working hypothesis (2), explanation (3), forecasting (4), decision making (5) and political justification (6) Description If one wants to describe a complex situation in the economy one uses (knowingly or intuitively) models For instance, if we wish to describe the flow of goods, services, work and money in the economy of a country, we can use a circular-flow scheme as in Sect 13.3 3, the vertices being the households, enterprises, the state, foreign countries, national savings and investment If we wish to obtain more detailed information we have to disaggregate the national data into those for sectors of the economy, regions of the country, etc In general, whenever we use a system A, that is neither directly nor indirectly interacting with a system B, to obtain information about the system B, we are using A as a model for B Working hypothesis The purpose of some simple models is more modest than explaining processes, they just present “working hypotheses”, the logical consequences of which we can then compare to observations For instance the affine relation C D cY C d (with constant c 0; 1Œ; d RCC ) between consumption and national income in Example (Sect 13.3), which we have quoted repeatedly, can be considered as such a working hypothesis On the one hand, if observations not corroborate it, we can replace it by another working hypothesis, say that C is a function of Y strictly convex from above On the other hand, if observations confirm it (which is the case in the 13.6 Why Construct Models and Theories? Types of Models and Theories 617 long run in most economies) then we can use them to determine or estimate “econometrically” the values of the constants (“parameters”) c and d This gives the model a quantitative character Example We spoke of “econometric” determination or estimation because such relations are rarely “deterministic”, they are more often “stochastic” (depending on chance, on minor or rare accidental oscillations) It is often supposed that two variable quantities (say x and y which may also be vectors, with x uniting all specific influences upon y) are connected by a single-valued deterministic function f , that is, y D f x/, but only “in average” More exactly, this means that there exists a “random variable” u (depending again on chance) so that y D f x/ C u Of course the models differ depending on f and u If f is affine, that is, y D axCbCu, by an abuse of language one often still speaks about a linear (really: affine) model Explanation Models which serve for explanation of observable processes (“explanatory models”) usually contain “laws” (such as “Newton’s law of gravity” in physics), as mentioned in Sect 13.1 For our purposes, a law is a truly universal statement (without temporal and spatial as well as any other, e.g cultural, social etc., limitation of applicability) of interdependencies (i) which stand till now all tests, no matter where or when they were carried out, and (ii) which, based on past experiences, are expected by an overwhelming majority of experts to be valid also in the future Notice that this definition of a law contains not only objective but also subjective criteria (“the overwhelming majority of experts”)—this is the case also with the notation of explanatory model itself, in particular in the social sciences Whether a model contains a law in the above sense or not, it is not enough that one should be able to deduce from it one or a few observed phenomena As already the famous economist VILFREDO PARETO (1848–1923) noted, to every statement or observation B there exists a system of assumptions A1 ; : : : ; An from which B follows by logical deduction The point is that the consequences of the assumptions in an explanatory model should conform with many observations On the other hand there should be enough latitude in the model to explain future or hypothetical situations such as: what would happen if (a) the enterpreneur or the consumers behaved differently (in a certain way), or (b) taxes and/or custom went up or down We have often compared models and theories of economics (or, more generally, the social sciences) to those of physics (or the natural sciences) However, there are differences: The existence of deterministic and stochastic laws in physics permit logical deductive explanations of observed definitive or statistical interdependencies, respectively In economics the “laws” are not 618 13 Methodology: Models and Theories in Economics so general and categorical, and therefore there are proposals to speak only of “quasi laws”, trends like regularities, etc Here the models and theories are based upon suppositions (hypotheses) about human economic behaviour This is in particular true about microeconomic models and theories which deal with the behaviour and plans of individual households and enterprises This behaviour and these plans can change rather abruptly and unpredictably It is doubtful whether laws will ever be found from which these changes can be deduced For statistical reasons macroeconomic models and theories, which deal with aggregate quantities (of national income, consumption, investment, etc.) rather than individual ones, can be based on interdependencies more stable for a longer period An example of such stable macroeconomics interdependency is the affine relation (13.1) in Sect 13.3 C D cY C d/ between national income and consumption The existence of such (relatively) stable macroeconomic laws makes forecasting possible Forecasting While theories in the social sciences which can predict future events with anything even close to the exactness of natural sciences (and even there, predictions are often not very exact: think of weather forecasts), they are still better than lucky (or unlucky) guesses and prophecies: “forecasting models” should contain “laws” as described in and be dynamic, that is, at least one assumption should contain the time (as process, not just one point in time) Models, which are not dynamic, are called static A further classification is into total and partial models The first reflects the entire economy of a country or of a group of countries, while the second is more restricted, for instance, to a sector or to a market We should not forget that in economics (and in other social sciences) the forecast may influence the future That is why one talks about “self-fulfilling” (or “self-destroying”) prophecies A really good model may take also this influence into account But economic forecasting is anyway difficult enough Even shortand medium-term forecasts for the entire economy make the solution of several hundred equations and inequalities necessary, in particular if decision making is to be based on them The solution is nowadays done with computers The following model can be handled without the aid of computers Example Samuelson’s dynamic macroeconomic total model for the “boom and bust” business cycle (Sect 8.7) shows that simple assumptions about the (linear or affine) interdependencies between the national income, consumption and investment may imply, if they take also the time delay into consideration, cyclic oscillations of these quantities around equilibrium values Decision making Mainly in business administration but also in (all of) economics, the importance of models for rational decision making is growing in particular in operations research and game theory Examples are models 13.7 Control, Correction and Applicability of Models and Theories 619 for linear, nonlinear and dynamic optimisation, inventory control, replacement, queueing, games (against nature or other opponents; zero-sum and non-zerosum-games) As in Sect 13.3 3, also for these decision making models, graphs (and graph theory) are often used As to models for (static, deterministic) models for optimisation, the problem is usually to find the maximum or minimum of a function f W RnC ! R under m conditions (restrictions) of the form gj x1 ; : : : ; xn / Ä cj gj I RnC ! RI j D 1; 2; : : : ; m/ (the cj could be immerged into the gj and thus replaced by 0) For instance, the variables could be the quantities of goods produced by an enterprise, the value of the function f , the gain, expected from the production of the goods in these quantities and the restrictions could express bounds of capacity in the enterprise If all the above functions f , g1 , g2 , ,gm are linear then we have a linear optimisation problem (see Sect 4.8), otherwise one of nonlinear optimisation (see Chap 8) If the values of f , g1 , ,gm may in part depend on chance, then this new, different problem belongs to stochastic optimisation Nowadays all these theories are sufficiently advanced to make easier and better decisions possible than “trial and error” More microeconomic models (dealing, say, with individual enterprises) exist and have been used for decision making than macroeconomic ones (dealing with sectors of the national or international economy) but it can be expected that increasingly also the latter will be used in making decisions on questions of national and even global economics Justification of politics In addition to the purposes mentioned till now, construction of models and theories may also be politically or ideologically motivated, and the more so, the less they can be tested by experience The goal of creating a theory conforming to and explaining one’s ideological bias can be done even under semblance of objectivity, for instance by excluding (or including) non-economic influences under the catch phrase “all other things being equal” (compare Sect 13.7 1), by separating production and distribution, or by overidealized assumptions such as unlimited flexibility of all “factors”, complete information or foresight, rigidity of market-evolution, etc 13.7 Control, Correction and Applicability of Models and Theories Of course, models and theories should not remain unchecked, without feedback from (further) observations of the real world This can be done in several ways, considering also what kind of models are involved Control and correction of explanatory models and theories in economics If a theory in economics claims to be able to explain essential phenomena of 620 13 Methodology: Models and Theories in Economics economic reality and thus to assist decision making (compare to and in Sect 13.6) then it should be possible to test (check, control) its assumptions and their consequences empirically (by observation of the real world) But even if they turn out to be correct several times, this does not prove or even guarantee the truth of the theory, since several further observations may disprove (falsify) it Karl Popper (whom we quoted in Sect 13.4 on inductive reasoning) observed that progress in empirical sciences is mostly made not by those who try to justify or “save” a theory, but by those who try to disprove (falsify) it If they succeed, then the theory has to be corrected or a new theory has to be created; if not then the original theory is greatly strengthened (corroborated) This is the basis of “critical rationalism” Figure 13.3, by HANS KARL SCHNEIDER (1920–2011), is itself a graphic representation of a model (compare Sect 13.3 3): it shows how theories in empirical sciences should be created, controlled and corrected It fits natural sciences somewhat better than social sciences, understandable, since the models of social scientists, in particular of economists, have to take into account also a rather difficult and at times irregular subject: human behaviour Unfortunately, even assumptions soundly rejected by experience survive in the social sciences For instance, microeconomic theories are often based on the assumption that entrepreneurs are always moved by the desire to maximise profit While in this generality the assumption has been falsified by counter-examples, it stubbornly keeps reappearing Note that the profit-maximising assumption is legitimate if restricted to carefully outlined occasional activities (see below) It is quite another story that some theories in the social sciences, in particular in economics, cannot be confronted with reality at all This is often achieved by the “all other things being equal” stipulation As mentioned in Sect 13.6 6, in total models (see Sect 13.6 4) this achieves the exclusion of non-economic influences In partial models it can be used to exclude influences from other parts of the economy Whenever observations disprove a tenet of such a theory then its advocates deflect the blame to “not all other things were equal” (they seldom are) For instance, Marxist theory postulated that accumulation of capital causes dramatic increase of poverty, misery, exploitation and degradation of the working masses When this did not materialise then, for instance ROSA LUXEMBURG (1870–1919) explained it by increased exploitation of colonies and other formerly non-capitalistic markets Even after decolonialization it lasted some forty years till the theory collapsed Control and application of models of limited validity When, as it often happens in economics and in other social (and even natural) sciences, no general theories have (yet) been formulated which contain laws explaining certain processes, sometimes “ad hoc models” (models of restricted validity and with less scientific foundation; compare also Sect 13.3 2) can serve well The assumptions in the representation of such models may (a) be of limited validity and/or (b) not be realistic, but their consequences (or some of them), as far as they go, may conform with and be applicable to economic reality Thünen’s “isolated state” 13.7 Control, Correction and Applicability of Models and Theories Subjective realm of science 621 Objective realm of science THEORY intuitive conjecturing of hypotheses deduction assumptions consequenses use for explanation and forecasting statements about the real world application for the purpose of control tests observation result of tests conform with observations or does not conform with observations Finding appropriate tests for theory Fig 13.3 H K Schneider’s graph on creation, control and correction of theories in empirical sciences and the “perfect market” (Examples and 3) in Sect 13.3 are rather of type (b), while the “affine equilibrium” of national income, consumption and investment (Example in Sect 13.3 4) and production functions satisfying E1, E2, E3 in Example (Sect 13.5 1) are rather of type (a) Of course, before one uses such models and their consequences, it has to be checked (empirically, at least “econometrically”, compare Sects 13.5 and 13.6 whether the conditions of their validity are satisfied, at least approximately 622 13 Methodology: Models and Theories in Economics Models of limited validity can be applied to short or medium term (say, one to twelve months) forecasting in the realistic expectation that in time spans of such brevity the economic circumstances not change significantly and that “disturbances” (interdependencies neglected in the model) will not become significant But such changes may make long term forecasts incorrect On the other hand, taking disturbances (more interdependencies) into consideration may unduly complicate the model, making for instance the system of equations and inequalities, which represent it, too cumbersome even for computers Moreover, errors often propagate and increase with the number of steps and calculations In particular, in the case of nonlinear dynamical systems (compare Sect 12.5) small errors in the data (parameters and initial conditions) may lead in a relatively short time to huge deviations from the solutions which would result from correct data 13.8 Concluding Remarks Opinions vary about the role of models, theories and methods in economics and in other social sciences While the reaction of the famous economist ROY FORBES HARROD (1900–1978) to the advocates of methodology was “stop talking and get on with the job”, in the opinion of Vilfredo Pareto (whom we quoted in Sect 13.6 about the role of models) every method is fine, whether using historical analysis or mathematics, as long as it is expedient This points in the direction of interdisciplinary research, for instance by integration of studies in economics, sociology and psychology (such as behavioural analysis), which is indeed gaining in importance nowadays This may lead, as Hans Karl Schneider (also quoted before, in Sect 13.7 1, see Fig 13.3 on theories in sciences) observed, to a general theory connecting social sciences by integrating economic, sociologic, psychologic and other aspects of human behaviour In order to advance towards this goal, no doubt further research in the theory of scientific research is necessary (though probably not sufficient, compare Sect 13.5 1) Such methodological research has gone quite far in the natural sciences but much remains to be done in the social sciences 13.9 Exercises Note Many of the following exercises require longer answers (almost “essays”) than most others in this book Also, there is more than one “correct answer” (the answer we give to some of these exercises at the end of this book is “one” of several correct answers) This situation is more common in economics and even mathematics than one may think at this level Is a model in economics (a) a system of assumptions containing economic notations, (b) a system of equations containing economic variables? 13.10 Answers 623 Give argument for and against models based upon idealistic assumptions (unrealistic but abstracted from reality) What is the role of inductive reasoning in the social sciences? Compare the notions of model in (a) logic and pure mathematics on one hand and in (b) applied mathematics and in the sciences on the other (c) Can they merge? Formulate statements in economics which have (a) all three, (b) exactly two (each couple) of the following properties: realistic, informative, true In what sense can a theory T, which “helps us find our way in the vast and confusing economic reality”, be considered a representation of a model? Show that the assumptions E1, E2, E3 in Example of Sect 13.5 are (a) consistent and (b) for all n 3, independent What is the difference between (a) explanatory models and (b) models based on working hypotheses? Is the model in Example (a) in Sect 13.3 4, (b) in Sect 13.3 1, (c) in Sect 13.3 2, (d) in Sect 13.3 (Samuelson’s model of the business cycle; see Sect 8.7) an affine or nonlinear, deterministic or stochastic, micro- or macroeconomic, total or partial, static or dynamic one? 10 May one see that the practical applicability of a model in economics is better the more realistic its assumptions are? 13.10 Answers (a) A model in economics can be represented by a system of assumptions containing economic notions, but not every system of such assumptions is a model in economics in the sense of a “simplified image of a part of economic reality” The assumptions can, for instance, (i) contradict experience, (ii) be logically inconsistent, (iii) give no information (b) There are models in economics which are not easily representable by a system of equations containing economic variables (see, e.g., Thünen’s model of the “isolated state” in Sect 13.7 2) On the other hand, such a system can be logically inconsistent (i.e., can have no solution) or its solution(s) may contradict experience 624 13 Methodology: Models and Theories in Economics Frequently idealistic assumptions make logical deductions and clear insight possible The logical consequences often can be utilized as useful informations (compare Example in Sect 13.7 and, following there, the “ideal gas” model of physics) That is not necessarily so: Sometimes both the idealistic assumptions and the consequences deduced from them are so idealized that confrontaton with reality makes no sense or is impossible Nevertheless, models based on such assumptions are used for justification of political measures Inductive reasoning supports the finding of hypotheses Since only a finite number of observations or experiments can be made, such hypotheses are preliminary starting points rather than reliable knowledge A model is in (a) logic and pure mathematics any set of objects and relations which satisfy the axioms of an axiom system, (b) in applied mathematics and in the sciences a simplified image of a part of reality (c) The two notions of a model can be merged if for any simplified image A of a part of reality there can be abstractly formulated a system S of assumptions of the following kind There exists a useful covering of the sysmbols in the assumptions with meaning such that S with this realisation of the symbols represents A The following statement is (a) realistic, informative, true: DM 1.- -=US$ 6627 on Tuesday, September 10, 1996, at 11:28 a.m in New York, (b) realistic, informative, wrong: DM 1.- -=US$ 6543 on Tuesday, September 10, 1996, at 11:28 a.m in New York, realistic, not informative, true: if DM 1.- -=US$ 6627 then 1US$=DM 1.5090 (not informative for anybody who knows that 1/.6627=1.5090), not realistic, informative, true: if DM 1.- -had been US$ 7000 on Tuesday, September 10, 1996, at 11.28 a.m in New York then, in view of the transaction costs, all people who had bought DM 1000.- - for less than US$650.- - some time ago had been winners The system of statements, which constitute a theory T, can be considered to be a system of assumptions If this represents a simplified image of (a part of) economic reality, it is a model in economics (a) The assumptions E1, E2, E3 in Example of Sect 12.4 are consistent, since the Cobb-Douglas function F W RnC ! RC given by F.x1 ; x2 ; : : : ; xn / D Cx11 r1 r2 x2 : : : x1n rn (13.8) 13.10 Answers 625 (with a positive constant C; < rj < 1; j D 1; 2; : : : ; n, r1 Cr2 C: : :Crn D n 1) satisfies them all (b) The assumptions E1, E2, E3 are independent, since (13.8) with C > 0, r1 > 1, r2 > 0, , rn > 0, r1 Cr2 C: : :Crn D n n 3/ satisfies E2, E3, but not E1; since (13.8) with C > 0, < rj < 1, r1 C r2 C : : : C rn Ô n satisfies E1,E3, but not E2; since the function F W RnC ! RC given by F.x1 ; : : : ; xn / D x1 C : : : C xn satisfies E1, E2, but not E3 The difference lies in the purpose of the models (a) Explanatory models aim at the logical deduction of events or laws from hypotheses and already known laws (b) Models based on working hypotheses have the purposes of finding laws or (at least) “interdependencies” (a) affine, deterministic, macroeconomic, total, static, (b) nonlinear, deterministic, microecnomic (if F is the production function of an enterprise), macroeconomic (if F is the production function of a country), partial, static, (c) affine or nonlinear (if F is affine or nonlinear), stochastic (further properties depend on the meaning of F), (d) affine, deterministic, macroeconomic, total, dynamic 10 In many cases the answer is no The more details are taken into consideration in the assumptions of a model in economics, the more extrusive becomes the basic structure of the model, that is, in many applications, the system of equations describing this structure The system may be too complicated to be solved But even if such a system of equations or such a structure still can be mastered logically or numerically, practical applicability may be limited because of the following reasons: (i) The inevitable inaccuracy with the determination of data yields propagation of error such that the solutions generally become the more inexact the more extrusive the system of equations is (ii) Propagation of error of a very disappointing kind can also emerge, if one tries to solve initial value problems of certain nonlinear difference equations (which may be not at all complicated or extrusive; see Sect 12.4) Index "-neighbourhood, 278 n-ball, 278 Action plan complete, 495 Addition theorems for sine and cosine, 37 Additive, 114 Additive technology, 49 Affine, 414 Approximately equal, 323 Asset capitalisation value, 528 Asymptotically equal, 323 Bijection, 65 Binary relation, 64 Binomial coefficients, 317 Boundary, 279 Budget equation, 422 Cartesian product, 12 Chain rule, 289 Coefficients, 325 Compact, 390 Competitive equilibrium price vector, 491 Complement, 498 Composite function, 75 Condorcet’s paradox, 487 Cone generated by vectors, 89 Conjugate complex number, 40 Connected, 278 Constant elasticity of substitution (CES), 353 Continuity uniform, 279 Continuous on a set, 279 Contour-line, 75 Convex from below on an interval, 85 Convex function, 85 Convex hull, 88 Cordial, 489 Correspondences, 480, 486, 490 homogeneous of degree r, 481 output, 480 output production, 481 production, 480 Cosine, 35 Cost minimal, 482 Cotangent, 42 Cramer’s rule, 157 Degree, 291, 325 exact, 325 Demand, 485 Demand indifference, 498 Demand set, 489 Derivative, 308 logarithmic, 344 Derivative higher order, partial, 284 Derivative second order, partial, 284 Differentiable, 280 Differential, 279 Differential equation, 291 second order, 351 Discount factor, 323, 526 Distance, 278 Distance of points, 28 Domain, 64, 278 Duopolists, 477 Duopoly, 495 Efficient, 470 Efficient production process, 47 © Springer International Publishing Switzerland 2016 W Eichhorn, W Gleißner, Mathematics and Methodology for Economics, Springer Texts in Business and Economics, DOI 10.1007/978-3-319-23353-6 627 628 Elasticity output, 343 scale, 344, 350 Envelope theorems, 435 Equal desirability, 487 Equilibrium competitive, 479 economic, 485 Equilibrium points, 477, 479 Euclidean norm, 17 Euler Leonhard, 291 Euler’s equation, 291 Exchange equilibrium, 491 Expansion paths linear, 483 Extension, 75 Function Cobb–Douglas, 348 exponential, 305 general representation, 329 homothetic, 337 objective, 422 profit, 436 quadratic, 326 quasi homogeneous, 336 ray-homogeneous, 336 value, 436 Games, 477 constant-sum, 496 extensive form, 494 m-person, 494 normal form, 494 noncooperative, 496 non-zero-sum, 494 zero-sum, 494, 496 two-person, 496 Geometric distance, 473 Geometric mean, 337 Gradient, 283 Growth rate, 132 Homogeneity, 291 linear, 481 Homogeneous, 290, 328 Homogeneous degree, 290 Homogeneous function, 291 Homogeneous function positively, 291 Homogeneous linearly, 292 Index Homogeneous positively, 290 Hyperplanes, 414 Implicite definition, 294 Increasing, 488 strictly, 488 Increasing function, 78 Indifference, 486 Indifference curve, 75 Inefficient production process, 47 Infimum, 392 Inflection points of, 312 Injection, 65 Inner or scalar product, 24 Integer, Integral definite, 511 improper, 527, 530 Interval, 71 Inverse function, 65 Isoquants, 75, 337 Kakutani, Shizuo, 491 Kuhn–Tucker conditions, 465 Lagrange multipliers, 425 Laspeyres’s price index, 100 Leontief matrix, 130 Leontief’s production model, 127 Level set, 337 Limit of a function, 278 Linear, 414 Linear function, 116 Linear optimisation problem, 51 Linear production model, 50 Linear regression, 414 Linear technology, 50 Linear transformation, 116 Linearly dependent, 22 Linearly homogeneous, 83, 106, 114 Linearly homogeneous technology, 50 Linearly independent, 22 Logarithm, 306 natural, 310 Mapping, 64 Match multi-move, 494 Index Matrices Hessian bordered, 427 payoff, 478 Maximal, 470 Maximum, 78, 82 Method of goal priority, 471 Method of goal programming, 472 Method of goal weighting, 471 Method of least squares, 415 Method of steepest ascent, 59, 456 Minimal, 470 Minimal cost combination, 482 Minimum, 78, 82 Monotonic, 81, 488 strictly, 488 Monotonic function, 78 Nash equilibrium points, 478 Natural numbers, Norm of a vector, 28 Norm, Euclidean norm, 17 Objective function, 51 Oligopoly, 479 One-move match, 494 Optimal input vector, 48 Optimal output vector, 48 Optimal production process, 47 Optimal solution, 54 Optimisation backward dynamic, 408 multi objective, 470 vector, 470 Optimisation problem , 47 Order lexicographic, 471 Ordering complete, 486 total, 486 Ordinal, 488 Orthogonal vectors, 29 Pareto-domination, 492 Pareto-optimal, 470, 479 Partially ordered, 13 Path, 278 Payment density, 525 Payoff functions, 478, 494 Payoffs, 477 Players, 477 Points of inflection, 86 629 Polynomials, 327 quotient of, 327 Power set, 480 Preference, 486 strict, 487 Preference relation, 486 Present value, 526 Price index, 62 Price level, 61 Product of a scalar and a matrix, 122 Product of matrices, 118 Product of two complex numbers, 32 Production function microeconomic, 292 Production process, 46 Production surfaces, 338 Production system, 46 Profit, 48 Punctured neighbourhood, 278 Quadratic form, 326 Quadratic optimisation, 462 Quasi-convex , 94 Quasi-convex from above, 94 Quasi-convex from below, 94 Range, 64 Rank of a matrix, 146 Rate of decay, 323 Rational number, Ray, 83 Ray-monotonic, 83 Reflexivity, 486 Region, 278 Return laws of diminishing marginal, 355 Return decreasing, 292 Return increasing, 292 Return to scale, 292, 350 Saddle point, 399 Saturation quantities, 497 Series binomial, 316 Set, Set path-connected, 278 Set valued, 477 Shephard’s axioms, 480 Sine, 35 Singleton, 71 Slater condition, 463 630 Stable, 478 Strategies, 477, 494, 495 Strategy vector, 494 Strictly decreasing function, 78 Strictly increasing function, 78 Strictly monotonic function, 78, 81 Subset, Substitute, 498 Sum of square coefficient, 419 Sum of squares of deviations, 415 Supply, 485 Supremum, 392 Surfaces of inflection, 91 Surjection, 65 System of linear equations, 135 Tangent, 42 Technology, 46 Theory competition, dynamical, 292 Theory of distribution marginal, 292 Theory of games, 477 Theory of zero-sum-games, 477 Theory production, distribution, 292 Total differential, 283 Index Totally ordered , 17 Transitivity, 486 Transpose, 327 Triangle inequality, 28 Unction CES, 353 Unimodal, 79 Unit sphere n-dimensional, 329 Unit vector, 17 Utility function, 422 Variance, 415, 417 Vector, 16 Von Neumann production process, 130 Von Neumann technology, 131 Walras–model, 486 Welfare theorem first, 492 second, 492 ... set For instance you are an element of © Springer International Publishing Switzerland 2016 W Eichhorn, W 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