Table of contents for Maths for economics / Geoff Renshaw.

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Brief contents
Detailed contents	ix
About the author	xiii
About the book	xv
How to use the book	xvii
Chapter map	xviii
Guided tour of the textbook features	xx
Guided tour of the Online Resource Centre	xxii
Acknowledgements	xxiii
	Part One Foundations
	1	Arithmetic	3
	2	Algebra	43
	3	Linear equations	63
	4	Quadratic equations	109
	5	Some further equations and techniques	134
	Part Two Optimization with one independent variable
	6	Derivatives and differentiation	165
	7	Derivatives in action	184
	8	Economic applications of functions and derivatives	213
	9	Elasticity	256
	Part Three Mathematics of finance and growth
	10	Compound growth and present discounted value	297
	11	The exponential function and logarithms	328
	12	Continuous growth and the natural exponential function	342
	13	Derivatives of exponential and logarithmic functions and their applications	368
	Part Four Optimization with two or more independent variables
	14	Functions of two or more independent variables	389
	15	Maximum and minimum values, the total differential and applications	441
	16	Constrained maximum and minimum values	479
	17	Returns to scale and homogeneous functions; partial elasticities; logarithmic scales; growth accounting	519
	Part Five Some further topics
	18	Integration	551
	19	Matrix algebra	577
	20	Difference and differential equations	597
	W21	Extensions and future directions (on the Online Resource Centre)	000
Appendix: Answers to chapter 1 self-test	623
Glossary	624
Index	632
Detailed contents
About the author	xiii
About the book	xv
How to use the book	xvii
Chapter map	xviii
Guided tour of the textbook features	xx
Guided tour of the Online Resource Centre	xxii
Acknowledgements	xxiii
	Part 1 Foundations
	1	Arithmetic	3
	1.1	Introduction	3
	1.2	Addition and subtraction with positive and negative numbers	4
	1.3	Multiplication and division with positive and negative numbers	7
	1.4	Brackets and when we need them	10
	1.5	Factorization	13
	1.6	Fractions	14
	1.7	Addition and subtraction of fractions	16
	1.8	Multiplication and division of fractions	20
	1.9	Decimal numbers	24
	1.10	Adding, subtracting, multiplying, and dividing decimal numbers	26
	1.11	Fractions, proportions, and ratios	27
	1.12	Percentages	28
	1.13	Index numbers	33
	1.14	Powers and roots	35
	1.15	Standard index form	40
	1.16	Some additional symbols	41
		Self-test exercises	42
	2	Algebra	43
	2.1	Introduction	43
	2.2	Rules of algebra	44
	2.3	Addition and subtraction of algebraic expressions	44
	2.4	Multiplication and division of algebraic expressions	45
	2.5	Brackets and when we need them	47
	2.6	Fractions	49
	2.7	Addition and subtraction of fractions	50
	2.8	Multiplication and division of fractions	52
	2.9	Powers and roots	55
	2.10	Extending the idea of powers	56
	2.11	Negative and fractional powers	57
	2.12	The sign of an	59
	2.13	Necessary and sufficient conditions	60
		Appendix: The Greek alphabet	62
	3	Linear equations	63
	3.1	Introduction	63
	3.2	How we can manipulate equations	64
	3.3	Variables and parameters	69
	3.4	Linear and non-linear equations	69
	3.5	Linear functions	72
	3.6	Graphs of linear functions	73
	3.7	The slope and intercept of a linear function	75
	3.8	Graphical solution of linear equations	80
	3.9	Simultaneous linear equations	81
	3.10	Graphical solution of simultaneous linear equations	84
	3.11	Existence of a solution to a pair of linear simultaneous equations	87
	3.12	Three linear equations with three unknowns	90
	3.13	Economic applications	91
	3.14	Demand and supply for a good	91
	3.15	The inverse demand and supply functions	94
	3.16	Comparative statics	97
	3.17	Macroeconomic equilibrium	102
	4	Quadratic equations	109
	4.1	Introduction	109
	4.2	Quadratic expressions	110
	4.3	Factorizing quadratic expressions	112
	4.4	Quadratic equations	114
	4.5	The formula for solving any quadratic equation	116
	4.6	Cases where a quadratic expression cannot be factorized	117
	4.7	The case of the perfect square	118
	4.8	Quadratic functions	120
	4.9	The inverse quadratic function	122
	4.10	Graphical solution of quadratic equations	123
	4.11	Simultaneous quadratic equations	126
	4.12	Graphical solution of simultaneous quadratic equations	127
	4.13	Economic application 1: supply and demand	128
	4.14	Economic application 2: costs and revenue	131
	5	Some further equations and techniques	134
	5.1	Introduction	134
	5.2	The cubic function	135
	5.3	Graphical solution of cubic equations	138
	5.4	Application of the cubic function in economics	141
	5.5	The rectangular hyperbola	142
	5.6	Limits and continuity	143
	5.7	Application of the rectangular hyperbola in economics	146
	5.8	The circle and the ellipse	149
	5.9	Application of circle and ellipse in economics	151
	5.10	Inequalities	152
	5.11	Examples of inequality problems	156
	5.12	Applications of inequalities in economics	159
	Part 2 Optimization with one 	independent variable
	6	Derivatives and differentiation	165
	6.1	Introduction	165
	6.2	The difference quotient	166
	6.3	Calculating the difference quotient	167
	6.4	The slope of a curved line	168
	6.5	Finding the slope of the tangent	170
	6.6	Generalization to any function of x	172
	6.7	Rules for evaluating the derivative of a function	173
	6.8	Summary of rules of differentiation	182
	7	Derivatives in action	184
	7.1	Introduction	184
	7.2	Increasing and decreasing functions	185
	7.3	Optimization: finding maximum and minimum values	187
	7.4	A maximum value of a function	187
	7.5	The derivative as a function of x	188
	7.6	A minimum value of a function	189
	7.7	The second derivative	191
	7.8	A rule for maximum and minimum values	191
	7.9	Worked examples of maximum and minimum values	192
	7.10	Points of inflection	195
	7.11	A rule for points of inflection	198
	7.12	More about points of inflection	199
	7.13	Convex and concave functions	206
	7.14	An alternative notation for derivatives	209
	7.15	The differential and linear approximation	210
	8	Economic applications of functions and derivatives	213
	8.1	Introduction	213
	8.2	The firm's total cost function	214
	8.3	The firm's average cost function	216
	8.4	Marginal cost	218
	8.5	The relationship between marginal and average cost	220
	8.6	Worked examples of cost functions	222
	8.7	Demand, total revenue, and marginal revenue	229
	8.8	The market demand function	229
	8.9	Total revenue with monopoly	231
	8.10	Marginal revenue with monopoly	232
	8.11	Demand, total and marginal revenue functions with monopoly	234
	8.12	Demand, total and marginal revenue with perfect competition	235
	8.13	Worked examples on demand, marginal and total revenue	236
	8.14	Profit maximization	239
	8.15	Profit maximization with monopoly	240
	8.16	Profit maximization using marginal cost and marginal revenue	242
	8.17	Profit maximization with perfect competition	244
	8.18	Comparing the equilibria under monopoly and perfect competition	246
	8.19	Two common fallacies concerning profit maximization	248
	8.20	The second order condition for profit maximization	248
		Appendix 8.1: The relationship between total cost, average cost, and marginal cost	253
		Appendix 8.2: The relationship between price, total revenue, and marginal revenue	254
	9	Elasticity	256
	9.1	Introduction	256
	9.2	Absolute, proportionate, and percentage changes	257
	9.3	The arc elasticity of supply	259
	9.4	Elastic and inelastic supply	260
	9.5	Elasticity as a rate of proportionate change	260
	9.6	Diagrammatic treatment	261
	9.7	Shortcomings of arc elasticity	263
	9.8	The point elasticity of supply	263
	9.9	Reconciling the arc and point supply elasticities	265
	9.10	Worked examples on supply elasticity	265
	9.11	The arc elasticity of demand	268
	9.12	Elastic and inelastic demand	270
	9.13	An alternative definition of demand elasticity	272
	9.14	The point elasticity of demand	273
	9.15	Reconciling the arc and point demand elasticities	274
	9.16	Worked examples on demand elasticity	275
	9.17	Marginal revenue and the elasticity of demand	279
	9.18	The elasticity of demand under perfect competition	282
	9.19	Worked examples on demand elasticity and marginal revenue	284
	9.20	Other elasticities in economics	288
	9.21	The firm's total cost function	288
	9.22	The aggregate consumption function	290
	9.23	Generalizing the concept of elasticity	292
	Part 3 Mathematics of finance 	and growth
	10	Compound growth and present discounted value	297
	10.1	Introduction	297
	10.2	Arithmetic and geometric series	298
	10.3	An economic application	300
	10.4	Simple and compound interest	304
	10.5	Applications of the compound growth formula	307
	10.6	Discrete versus continuous growth	309
	10.7	When interest is added more than once per year	309
	10.8	Present discounted value	314
	10.9	Present value and economic behaviour	316
	10.10	Present value of a series of future receipts	316
	10.11	Present value of an infinite series	319
	10.12	Market value of a perpetual bond	320
	10.13	Calculating loan repayments	322
	11	The exponential function and logarithms	328
	11.1	Introduction	328
	11.2	The exponential function y = 10x	330
	11.3	The function inverse to y = 10x	331
	11.4	Properties of logarithms	333
	11.5	Using your calculator to find common logarithms	333
	11.6	The graph of y = log10x	334
	11.7	Rules for manipulating logs	335
	11.8	Using logs to solve problems	337
	11.9	Some more exponential functions	338
	12	Continuous growth and the natural exponential function	342
	12.1	Introduction	342
	12.2	Limitations of discrete compound growth	343
	12.3	Continuous growth: the simplest case	343
	12.4	Continuous growth: the general case	346
	12.5	The graph of y = aerx	347
	12.6	Natural logarithms	349
	12.7	Rules for manipulating natural logs	351
	12.8	Natural exponentials and logs on your calculator	351
	12.9	Continuous growth applications	353
	12.10	Continuous discounting and present value	358
	12.11	Graphs with semi-log scale	361
	13	Derivatives of exponential and logarithmic functions and their applications	368
	13.1	Introduction	368
	13.2	The derivative of the natural exponential function	369
	13.3	The derivative of the natural logarithmic function	370
	13.4	The rate of proportionate change, or rate of growth	371
	13.5	Discrete growth	371
	13.6	Continuous growth	374
	13.7	Instantaneous and nominal growth rates compared	377
	13.8	Semi-log graphs and the growth rate again	378
	13.9	An important special case	379
	13.10	Logarithmic scales and elasticity	381
	Part 4 Optimization with two or 	more independent variables
	14	Functions of two or more independent variables	389
	14.1	Introduction	389
	14.2	Functions with two independent variables	390
	14.3	Examples of functions with two independent variables	393
	14.4	Partial derivatives	398
	14.5	Evaluation of first order partial derivatives	401
	14.6	Second order partial derivatives	403
	14.7	Economic applications 1: the production function	411
	14.8	The shape of the production function	411
	14.9	The Cobb-Douglas production function	420
	14.10	Alternatives to the Cobb-Douglas form	425
	14.11	Economic applications 2: the utility function	428
	14.12	The shape of the utility function	429
	14.13	The Cobb-Douglas utility function	434
		Appendix 14.1: A variant of the partial derivatives of the Cobb-Douglas function	439
	15	Maximum and minimum values, the total differential, and applications	441
	15.1	Introduction	441
	15.2	Maximum and minimum values	442
	15.3	Saddle points	448
	15.4	The total differential of z = f(x, y)	452
	15.5	Differentiating a function of a function	457
	15.6	Marginal revenue as a total derivative	458
	15.7	Differentiating an implicit function	460
	15.8	Finding the slope of an iso-z section	463
	15.9	A shift from one iso-z section to another	463
	15.10	Economic applications 1: the production function	465
	15.11	Isoquants of the Cobb-Douglas production function	468
	15.12	Economic applications 2: the utility function	470
	15.13	The Cobb-Douglas utility function	472
	15.14	Economic application 3: macroeconomic equilibrium	473
	15.15	The Keynesian multiplier	473
	15.16	The IS curve and its slope	474
	15.17	Comparative statics: shifts in the IS curve	475
	16	Constrained maximum and minimum values	479
	16.1	Introduction	479
	16.2	The problem, with a graphical solution	480
	16.3	Solution by implicit differentiation	482
	16.4	Solution by direct substitution	485
	16.5	The Lagrange multiplier method	486
	16.6	Economic applications 1: cost minimization by the firm	490
	16.7	Economic applications 2: profit maximization	496
	16.8	A worked example	501
	16.9	Some problems with profit maximization	502
	16.10	Profit maximization by a monopolist	508
	16.11	Economic applications 3: utility maximization by the consumer	510
	16.12	Deriving the consumer's demand functions	512
	17	Returns to scale and homogeneous functions; partial elasticities; growth accounting; logarithmic scales	519
	17.1	Introduction	519
	17.2	The production function and returns to scale	520
	17.3	Homogeneous functions	522
	17.4	Properties of homogeneous functions	525
	17.5	Partial elasticities	531
	17.6	Partial elasticities of demand	532
	17.7	The proportionate differential of a function	534
	17.8	Growth accounting	537
	17.9	Elasticity and logs	539
	17.10	Partial elasticities and logarithmic scales	540
	17.11	The proportionate differential and logs	542
	17.12	Log linearity with several variables	544
	Part 5 Some further topics
	18	Integration	551
	18.1	Introduction	551
	18.2	The definite integral	552
	18.3	The indefinite integral	554
	18.4	Rules for finding the indefinite integral	555
	18.5	Finding a definite integral	562
	18.6	Economic applications 1: deriving the total cost function from the marginal cost function	565
	18.7	Economic applications 2: deriving total revenue from the marginal revenue function	567
	18.8	Economic applications 3: consumers' surplus	569
	18.9	Economic applications 4: producers' surplus	570
	18.10	Economic applications 5: present value of a continuous stream of income	572
	19	Matrix algebra	577
	19.1	Introduction	577
	19.2	Definitions and notation	578
	19.3	Transpose of a matrix	579
	19.4	Addition/subtraction of two matrices	579
	19.5	Multiplication of two matrices	580
	19.6	Vector multiplication	582
	19.7	Scalar multiplication	583
	19.8	Matrix algebra as a compact notation	583
	19.9	The determinant of a square matrix	584
	19.10	The inverse of a square matrix	587
	19.11	Using matrix inversion to solve linear simultaneous equations	589
	19.12	Cramer's rule	590
	19.13	A macroeconomic application	592
	19.14	Conclusions	594
	20	Difference and differential equations	597
	20.1	Introduction	597
	20.2	Difference equations	598
	20.3	Qualitative analysis	601
	20.4	The cobweb model of supply and demand	605
	20.5	Conclusions on the cobweb model	610
	20.6	Differential equations	612
	20.7	Qualitative analysis	615
	20.8	Dynamic stability of a market	616
	20.9	Conclusions on market stability	620
	W21	Extensions and future directions (on the Online Resource Centre)	000
Appendix: Answers to chapter 1 self-test	623
Glossary	624
Index	632

Library of Congress Subject Headings for this publication:

Economics, Mathematical.