Mathematical Analysis For Economists [Hardcover] R.G.D. Allen

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R G D Allen | Publisher: Brunton Press (November 4, 2008) | 564 | English | DJVU


MATHEMATICAL ANALYSIS FOR ECONOMISTS by R. G. D. ALLEN. Originally published in 1937. FOREWORD; THIS book, which is based on a series of lectures given at the London School of Economics annually since 1931, aims at providing a course of pure mathematics developed in the directions most useful to students of economics. At each stage the mathematical methods described are used in the elucidation of problems of economic theory. Illustrative examples are added to all chapters and it is hoped that the reader, in solving them, will become familiar with the mathematical tools and with their applications to concrete economic problems. The method of treatment rules out any attempt at a systematic development of mathematical economic theory but the essentials of such a theory are to be found either in the text or in the examples. I hope that the book will be useful to readers of different types. The earlier chapters are intended primarily for the student with no mathematical equipment other than that obtained, possibly many years ago, from a matriculation course. Such a student may need to accustom himself to the application of the elementary methods before proceeding to the more powerful processes described in the later chapters. The more advanced reader may use the early sections for purposes of revision and pass on quickly to the later work. The experienced mathematical economist may find the book as a whole of service for reference and discover new points in some of the chapters. I have received helpful advice and criticism from many mathe maticians and economists. I am particularly indebted to Professor A. L. Bowley and to Dr. J. Marschak and the book includes numerous modifications made as a result of their suggestions on reading the original manuscript. I am also indebted to Mr. G. J. Nash who has read the proofs and has detected a number of slips in my construction of the examples.

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CONTENTS
CHAP. РАДИ
Foreword ν
A Short Bibliography xiv
The Use op Greek Letters in Mathematical
Analysis - xvi
I. Numbers and Variables ------- l
1.1 Introduction --------- 1
1.2 Numbers of various types 3
1.3 The real number system ------- 6
1.4 Continuous and discontinuous variables - 7
1.5 Quantities and their measurement ----- 9
1.6 Units of measurement 13
1.7 Derived quantities -------- 14
1.8 The location of points in space - - - - - 16
1.9 Variable points and their co-ordinates 20
Examples I—The measurement of quantities ; graphical
methods --------- 23
II. Functions and their Diagrammatic Representation 28
2.1 Definition and examples of functions - - - - 28
2.2 The graphs of functions 32
2.3 Functions and curves - - - - - - - 30
2.4 Classification of functions - - - - - - 38
2.5 Function types - - 41
2.6 The symbolic representation of functions of any form - 45
2.7 The diagrammatic method 48
2.8 The solution of equations in one variable 50
2.9 Simultaneous equations in two variables 54
Examples II—Functions and graphs ; the solution of
equations 67
III. Elementary Analytical Geometry - - - - 61
3.1 Introduction ----.---- 61
3.2 The gradient of a straight line ----- 63
3.3 The equation of a straight line ----- 66

CHAP. PAG В
3.4 The parabola 69
3.6 The rectangular hyperbola 72
3.6 The circle 75
3.7 Curve classes and curve systems - - - - - 76
3.8 An economic problem in analytical geometry 80
Examples III—The straight line ; curves and curve systems 82

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关键词：Mathematical mathematica Economists Mathematic Hardcover 2011 developed providing published economic

IV. Limits and Continuity of Functions 85
4.1 The fundamental notion of a limit 85
4.2 Examples of the limit of a function ----- 88
4.3 Deiinition of the limit of a single-valued function - - 91
4.4 Limiting and approximate values - - - - - 95
4.5 Some properties of limits 97
4.6 The continuity of functions 98
4.7 Illustrations of continuity and discontinuity of functions - 100
4.8 Multi-valued functions 102
Examples IV—Limits of functions ; continuity of functions - 103
V. Functions and Diagrams in Economic Theory - 107
6.1 Introduction - - - - - - - - -107
5.2 Demand functions and curves - - - - 108
5.3 Particular demand functions and curves - - - 112
6.4 Total revenue functions and curves - - - - 116
6.6 Cost functions and curves 117
6.6 Other functions and curves in economic theory - - 121
6.7 Indifference curves for consumers' goods - - 124
5.8 Indifference curves for the flow of income over time - - 127
Examples V—Economic functions and curves - - 129
VI. Derivatives and their Interpretation - - 134
6.1 Introduction - - - - - - - - 134
6.2 The definition of a derivative 137
6.3 Examples of the evaluation of derivatives - 140
6.4 Derivatives and approximate values - - - 142
6.5 Derivatives and tangents to curves 143
6.6 Second and higher order derivatives - - - 148
6.7 The application of derivatives in the natural sciences - 149
6.8 The application of derivatives in economic theory - - 152
Examples VI—Evaluation and interpretation of derivatives - 167
VII. The Technique oe Derivation 160
7.1 Introduction 160
7.2 The power function and its derivative - - - - 161
7.3 Rules for the evaluation of derivatives - - - 163
PAGH
7.4 Examples of the evaluation of derivatives · · -166
7.5 The function of a function rule - - - - - 168
7.6 The inverse function rule 171
7.7 The evaluation of second and higher order derivatives - 172
Examples VII—Practical derivation - - - - 175
Applications op Derivatives 179
8.1 The sign and magnitude of the derivative - - - 179
8.2 Maximum and minimum values - - - - 181
8.3 Applications of the second derivative - - - 184
8.4 Practical methods of finding maximum and minimum
values -..-.---- 186
8.6 A general problem of average and marginal values - 190
3.6 Points of inflexion ...-..- 191
8.7 Monopoly problems in economic theory - - · 196
8.8 Problems of duopoly 200
8.9 A note on necessary and sufficient conditions - - 204
Examples VIII—General applications of derivatives ;
economic applications of derivatives - 205
Exponential and Logarithmic Functions - - 211
9.1 Exponential functions - - - - - - -211
9.2 Logarithms and their properties 213
9.3 Logarithmic functions - - - - - - -217
9.4 Logarithmic scales and graphs - - - - - 219
9.5 Examples of logarithmic plotting ----- 223
9.6 Compound interest .-.---. 228
9.7 Present values and capital values - - - 232
9.8 Natural exponential and logarithmic functions - - 234
Examples IX—Exponential and logarithmic functions;
compound interest problems - - - - - 238
Logarithmic Derivation 242
10.1 Derivatives of exponential and logarithmic functions - 242
10.2 Logarithmic derivation - - - - - 246
10.3 A problem of capital and interest .... 248
10.4 The elasticity of a function 251
10.5 The evaluation of elasticities - - - - 252
10.6 The elasticity of demand 254
10.7 Normal conditions of demand 267
10.8 Cost elasticity and normal cost conditions - - 260
Examples X—Exponential and logarithmic derivatives ;
elasticities and their applications - 264
CHAP. PAGB
XI. Functions of Two or More Variables - · - 268
11.1 Functions of two variables - - - - - -268
11.2 Diagrammatic representation of functions of two
variables 270
11.3 Plane sections of a surface - 272
11.4 Functions of more than two variables - - - 275
11.5 Non-measurable variables 276
11.6 Systems of equations ------- 278
11.7 Functions of several variables in economic theory - 281
11.8 The production function and constant product curves - 284
11.9 The utility function and indifference curves - - 289
Examples XI—Functions of two or more variables ;
economic functions and surfaces ----- 292
XII.)Partial Derivatives and their Applications - 296
12.1 Partial derivatives of functions of two variables - 296
12.2 Partial derivatives of the second and higher orders - 300
12.3 The signs of partial derivatives 303
12.4 The tangent plane to a surface ----- 305
12.5 Partial derivatives of functions of more than two
variables 309
12.6 Economic applications of partial derivatives - - 310
12.7 Homogeneous functions 315
12.8 Euler's Theorem and other properties of homogeneous
functions - - - - - - - -317
12.9 The linear homogeneous production function - - 320
Examples XII—Partial derivatives ; homogeneous
functions ; economic applications of partial
derivatives and homogeneous functions - - - 322
XIILi Differentials and Differentiation - 326
13.1 The variation of a function of two variables - - 326
13.2 The differential of a function of two variables - - 328
13.3 The technique of differentiation 330
13.4 Differentiation of functions of functions - - - 332
13.5 Differentiation of implicit functions - - - 334
13.6 The differential of a function of more than two
variables 339
13.7 The substitution of factors in production - - - 340
13.8 Substitution in other economic problems - - - 344
13.9 Further consideration of duopoly problems - - - 345
Examples XIII—Differentiation ; economic applications
of differentials 347
XIV. Problems op Maximum and Minimum Values - 351
14.1 Partial stationary values - - · - - -351
14.2 Maximum and minimum values of a function of two
or more variables - - - - - - «362
14.3 Examples of maximum and mininium values - - 366
14.4 Monopoly and joint production - - - - -369
14.5 Production, capital and interest ----- 362
14.6 Relative maximum and minimum values - - - 364
14.7 Examples of relative maximum and minimum values 367
14.8 The demand for factors of production - 369
14.9 The demand for consumers' goods and for loans - - 374
Examples XIV—General maximum and minimum
problems ; economic maximum and minimum problems 378
XV. Integrals of Functions of One Variable - - 384
16.1 The definition of a definite integral - 384
16.2 Definite integrals as areas - - - - - -387
16.3 Indefinite integrals and inverse differentiation - - 390
15.4 The technique of integration 393
16.5 Definite integrals and approximate integration - - 396
15.6 The relation between averago and marginal concepts - 400
15.7 Capital valuos -------- 401
16.8 A problem of durable capital goods - 404
15.9 Average and dispersion of a frequency distribution - 406
Examples XV—Integration ; integrals in economic
problems 408
XVI. Differential Equations 412
16.1 The nature of the problem - 412
16.2 Linear differential equations and their integration - 417
16.3 The general integral of a linear differential equation - 422
16.4 Simultaneous linear differential equations - - - 425
16.5 Orthogonal curve and surface systems - - - 429
16.6 Other differential equations 430
16.7 Dynamic forms of demand and supply functions - 434
16.8 The general theory of consumers' choice - - - 438
Examples XVI—Differential equations ; economic
applications of differential equations - - - - 442
XVII. Expansions, Taylor's Series and Higher Order
Differentials -------- 446
17.1 Limits and infinite series - - 446
17.2 The expansion of a function of one variable (Taylor's
series) 449
17.3 Examples of the expansion of functions - - - 454
17.4 The expansion of a function of two or more variables 456
17.5 A complete criterion for maximum and minimum
valuos - - 459
17.6 Second and higher order differentials - - 461
17.7 Differentials of a function of two independent
variables -------- 463
17.8 Differentials of a function of two dependent variables 465
Examples XVII—Infinite series ; expansions ; highor
order differentials - - - - - -469
XVIII. Determinants, Linear Equations and Quadratic
Forms 472
18.1 The general notion of a determinant - - 472
18.2 The definition of dotorminants of various orders - 473
18.3 Properties of determinants - - - - 477
18.4 Minors and co-factors of determinants - - - 478
18.5 Linear and homogeneous functions of several
variables 481
18.6 The solution of linear equations .... 482
18.7 Quadratic forms in two and three variables - - 485
18.8 Examples of quadratic forms ----- 489
18.9 Two general results for quadratic forms - - - 491
Examples XVIII—Determinants ; linear equations :
quadratic forms 492
XIX^) Farther Problems of Maximum and Minimum
Values - 495
19.1 Maximum and minimum values of a function of
several variables ------- 496
19.2 Relative maximum and minimum values - 498
19.3 Examples of maximum and minimum values - - 500
19.4 The stability of demand for factors of production - 502
19.5 Partial elasticities of substitution - - - - 503
19.6 Variation of demand for factors of production - - 505
19.7 The demand for consumers' goods (integrability
case) --------- 509
19.8 Demands for three consumers' goods (general case) - 613
Examples XIX—General maximum and minimum
problems ; economic maximum and minimum problems 617
XX. Some Problems in the Calculus of Variations - 521
20Л The general theory of functionals - 521
20.2 The calculus of variations - - - - - 523
20.3 The method of the calculus of variations - - - 524
20.4 Solution of the simplest problem - - - - 526
20.5 Special cases of Euler's equation - - - 529
20.6 Examples of solution by Euler's equation - - 530
20.7 A dynamic problem of monopoly - - - 533
20.8 Other problems in the calculus of variations - - 536
Examples XX—Problems in the calculus of variations - 540
Index :
Mathematical Methods ...... 543
Economic Applications --.... 646
Authors -..... 648

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