Study Skills in Mathematics. Edited by D. Burkhardt and D. Rutherford. Nottingham: Shell Centre for Mathematical Education (Revised edn 1981).

Transcription

1 Study Skills in Mathematics. Edited by D. Burkhardt and D. Rutherford. Nottingham: Shell Centre for Mathematical Education (Revised edn 1981). (Copies are available from the Shell Centre for Mathematical Education, University of Nottingham, Nottingham NG7 2RI; 0.75.) Weber, J. E. Mathematical Analysis (4th edn, 1982). (New York: Harper-Row). Glass, J. C. An Introduction to Mathematical Methods in Economics (1980). (New York: McGraw-Hill). Kennedy, G. Mathematics for Innumerate Economists (1982). (Duckworth). Chiang, A. C. Fundamental Methods of Mathematical Economics (2nd edn, 1974). (New York: McGraw-Hill). Black, J. and Bradley, J. F. Essential Mathematics for Economists (2nd edn, 1980). (Chichester: Wiley). Dowling, E. T. Mathematics for Economists (1980). (New York: McGraw-Hill). Holden, K. and Pearson, A. W. Introductory Mathematics for Economists (1983). (Basingstoke: Macmillan). Yamane, T. Mathematics for Economists (2nd edn, 1968). (Englewood Cliffs, N.J.: Prentice-Hall).

2 Index Abscissa, 16 Absolute sign, 10, 237 Additional earnings, 230, 233 Additional cost, 217, 233 Additional revenue, 230 Adjoint matrix, 240, 241 Algebraic manipulation, 12 Annuities, 25 Area under a curve, ,218, 224 Arithmetic progression, 25 Array, 235 Associative, 12,236,237, 242 Asymptote, 21 Augmented matrix, 241, 243, 248 Autonomous investment, 33, 35, 262 Average product, 11 0 Average profit, 58 Average revenue, 58 function, 77, 217 Average total cost, 21, 58, 76, 77, 86, 90, 113, 114, 220, 229 and relation to marginal cost, 77, 113, 114 Average variable cost, 114 and relation to marginal cost, 114 Balance of trade, 256 Base, 11 of exponential function, 19 of logarithm, 12, 20, 55 Behavioural equation, 7 Bordered Hessian, 177, 178 Boundary condition, 209 Brand switching, 25 2 Breakeven point, 42 Budget constraint, 18 2, 184, 194 Calculus, 5 1, 223 Cancellation, 23 7 Capital Rental Bill, 130 Capital stock, 225 Chain rule, 54, 55, 123,210 Closed economic system, 37, 132 Cobb-Douglas production function, 129, 136 Coefficient, 7 Coefficient matrix, 240 Cofactor, 240 Cofactor matrix, 240, 241 Column vector, 235 Commodity market model, 22,32 Common logarithm, 12 Commutative, 12, 236, 23 7 Comparative-static derivatives, 140 Competitive, 140, 142 Completing the square, 23 Complex numbers, 6, 24 Complimentary relationship, 129 Conformability conditions, 235, 237 Consistent equation, 22, 244, 245, 248 Constant, 7, 121, 122 Constant elasticity of substitution function, 144 Constant function, 17 Constant-function rule, 52, 121 Constant of integration, 105, 209, 216, 217, 224 Constant returns to scale, 137, 141 Constrained extrema, 176 Constrained optimisation, 1 76 Constraint: budget, 182, 184, 194 production, 179,192 Consumer's expenditure, 231 Consumer's surplus, 208, 220, 222, 229, 231 Consumption function, 7, 32, 58, 74, 132, 229, 231 Continuous function, 52, 209, 217 Cost, minimisation of, 179, 197, 202 Cramer's rule, 242, 254, 255, 259, 263 Cross-multiplication, 8 Cross-partial elasticities, 1 24 Cross-price elasticity of demand, 127, 128 and complements, 1 28 and substitutes, 128 Cubic function, 18 Definite integral, 208, 211, 212, 215 as area under a curve, 211 Degree of homogeneity, 13 7 Degree of polynomial, 13, 18 Demand function, 26-43, 58, 61, 67, 72, 77, 86, 90, 96, , , 127, 138, for capital, , 153, , 192, 202,208, 215,220,229,247,249,254, 262 derivative of, 62, 73, 79, 82, 90, 105, 140, 156 for labour, 113 for money, 33, 37, 132 sketching of, Dependence: among equations, 22 linear, 239 among rows, 239, 248 Dependent variable, 6, 120 Derivative, 51 as function, 51 higher-order, 57 partial, 120, 145 as rate of change, 51 second, 57 and slope of curve, 5 1 total, 120 Determinant, 23 7 Laplace expansion of,

3 Determinant Cont'd, order of, 23 7, 238 properties of, 238 Diagonal matrix, 235, 240 Differentiability, 208, 209 Differential equations, 208 Differentiation, 51 partial, 120, 145 total, 1 25, 1 26 Differentiation rules: chain rule, 54 constant-function rule, 52 exponential-function rule, 56 function-of-a-function rule, 54 inverse-function rule, 56 linear-function rule, 52 log-function rule, 55 power-function rule, 52 product rule, 53 quotient rule, 54 sum-difference rule, 53 Dimension, 235 Discrimination, see Price Disposable income, 3 7, 13 2 Distributive, 12,237,255 Domain, 16 e, the number, 20 Economic life, 230 Elasticity, 58 cross price, 12 7 income, 127 and logarithms, 128 partial, 140 point, 62 price of demand, 58, 73, 77, 86, 103, 127 price of supply, 58, 69, 73 of substitution, 137, 144 Element, 15, 235, 238 Elimination of variables, 22, 29, 31 Ellipse, 190 Ellipsis, 5 Endogenous variable, 6, 259 Equalities, 7 Equation, 6 behavioural, 7 equilibrium, 7 identity, 7 Equilibrium, 22, 26, 29, 33 general, 25 linear, 22, 26, 33 non-linear, 23, 29 Equilibrium condition, 43, I 08 Equilibrium income level, 33 Equilibrium interest rate, 33 Equilibrium point 220, 222 Equilibrium price, 22, 26, 28, 31, 36, 42, 43, 68, 73,138,222,254,262 Equilibrium quantity, 22, 26, 28, 31, 36, 42, 43, 68, 73, 138, 222, 262 Equilibrium values, 38, 132, 259,262 Euler's Theorem, 13 7 Excess demand, 22, 26 Exogenous variable, 6, 133, 134, 259 Exponent, II Exponential function, 19, 25 derivative of, 56 and growth, 19 integration of, 209 natural, 20 Extrema: constrained, 1 76 of a function of one variable, 84 of a function of two variables, 145 global, 84 local, 84 relative, 84 unconstrained, 84, 145 Extreme point, 145, I 70 Extremum, 84 Factoring, 13, 23, 43, 44, 166 difference of two squares, 14 perfect square, 14 sum and product method, 14 Final demand, 249 First-order condition, 84, 176 First derivative, 57 Fraction, 6 Function, 16 constant, I 7 continuous, 52 decay, 20 exponential, 18, 20 growth, 20 identity, 17 implicit, 29 linear, 17 logarithmic, 20 multivariate, 16 polynomial, 18 power, 19 quadratic, 18 rectangular, 21 Function-of-a-function rule, 54, 59 Fundamental Theorem of integral calculus, 212 Gaussian method, 241, , 257, 261, 262, 264 General equilibrium, 235 General system of linear equations, 244 Geometrical progression, 25 Government expenditure, 37, 141 Government-expenditure multiplier, 133, 135 Graph of a function, 15, 16, 26, 29, 33, 37, 43,45,48, 226 Higher-order derivatives, 57 Homogeneous function, 13 7 Hyperbola, 41 Idempotent matrix, 235 Identity function, 17 Identity matrix, 235, 237, 241, 242 Imaginary numbers, 6, 23 Implicit function, 29, 126, 135 Implicit-function rule, 126 Income, 58 elastic, 128 inelastic, I 28 inferior, 128 level, 33 superior, 128 Income elasticity of demand, 127, 128 Income-tax multiplier, 133 Inconsistent equations, 22, 244,

4 Indefinite integral, 208, 209 Independece, 245, 248 Independent variable, 6, 120, 121, 145 Indeterminate, 244 Index, 11 Index of summation, 10 Indices, 8, 9, 11 Indifference curve, 196 Inequalities, 8 Inflection points, 85,146,148,171,172 criteria for, 85,146 non-stationary, 85 stationary, 85 Initial condition, 209, 216, 217 Input-output analysis, 249, 256, 263 Integers, 5 Integral, 208 definite, 208,211,212,215 improper, 215 indefinite, 105, 208, 209, 212 methods, , 213, 216, 218, 225, 230 sign, 209 Integrand, 209 Integration, 223 constant of, 105, 216, 217, 224 of exponential function, 209 limits of, 213, 218 rules of, 209 Intercept, 18, 19, 52, 139 Interest rate, 33, 37, 132, 134, 141 Intersect, 69 Inverse-function rule, 56 Inverse matrix, 240, 242 calculation of, , 265 Inverse relationship, 16 Investment, 32, 58, 189,202,225 Investment function, 132 Irrational numbers, 6, 23 IS curve, 33, 34, 37, 39, 134,259 Isocost, 197 Isoquant, 136, 199 Labour, 113 Lagrange multiplier, 176 interpretation of, 192, 194 Lagrange-multiplier method, 176, 1 79 Lagrangian function, , 182, 185, 187, 190, 192, 195, 198, Laplace expansion, 239 Leading diagonal, 235, 239, 241 Limit, 120 Linear dependence, 22, 244, 245 Linear equations, 22, 26ff., 244ff., 262 Linear function, 1 7 Linear-function rule, 52 Linear independence, 245, 248 Linearly homogeneous function, 13 7 LM curve, 34, 37-9, 134, 259 Logarithm, 12, 55 common, 12 conversion formulae, 12 and elasticity, 128 natural, 12, 25, 55 properties, 12 Logarithmic function, 20 derivative of, 55 Logarithmic-function rule, 55 Maintenance cost, 217, 226, 230 Mapping, 16 Marginal change, 131 Marginal cost, 57, 64, 76, 77, 85, 86, 113, 140,155,208,215,223,230 and relation to average total cost, 77, 113 of production, 181, 229 Marginal productivity, 13 5 of capital, 121, 129, 130, 140, 141 diminishing, 7 7 ofinput, 149,151 oflabour, 58, 77, 121, 129, 140, 141 ofprofit,57,58,64 and relation to average product, 110 value of the, 110 Marginal propensity to consume, 58, 74, 229 Marginal propensity to save, 58, 74, 229 Marginal rate of substitution, 195, 196 Marginal rate of technical substitution, 135, 136, 141, 197, 199 Marginal revenue, 57, 58, 64, 77, 113, 155, 208,215,223,229,230 and elasticity of demand, 77 Marginal utility, 125 ofmoney, 195,196 Market models, 26, 28, 30, 35, 42, 67,72 Matrix algebra, 235 addition, 236 adjoint, 240, 241 augmented, 241, 243, 248 coefficient, 240 cofactor, 240, 241 conformability, 235, 237 determinant, 237 diagonal, 235, 240 dimension, 235 equality, 236 idempotent, 235 identity, 235, 237, 241, 242 inverse, 240, 242, 265 inversion of, 240, 241 manipulation, 236 minor, 238 multiplication, 236, 246 non-singular, 240, 242, 244 null,235 order of, 235 representation, 239,246-48,251,254, 256,264,266 scalar, 236 scalar multiplication of, 236 singular, 240 square, 235, 240, 242 stochastic, 253 subtraction, 236 symmetric, 235 technological coefficients, 249 transpose of, 235 triangular, 23 9 unit, 235 zero, 235 Markov chain, 254, 260 Maxima, 84, 85, 145,147,171,176,178 Maxima output, 110 Mean rate of return, 189 Minima, 85,145,147,171,176,178 Minor, 238 Money, 225 flow of, 225 marginal utility of, 195,

5 Money demand, 32,37 Money market, 32, 37, 134 Money supply, 32, 37, 141 Money supply multiplier, 135 Multiple solutions, 244, 245 Multiplier: government expenditure, 133, 135 income-tax, 133 Lagrange, 176 money supply, 135 Multi-product firm, 15 2 Multi-variable demand function, 120, 124 National-income models, 22, 32, 37, 132, 141,262 Natural exponential functions, 5, 123 N atura11ogarithms, 12, 25, 55 Natural numbers, 5 Necessary condition, 84, 145 Negative areas, 213 Negative integers, 5 Net earnings, 234 Net investment, 225 Net present value, 25 Net savings, 226 Non-singular matrix, 240, 242, 244 Non-stationary point, 85, 110 nth derivative test, 85 Null matrix, 235 Number system, 5 Numerical constants, 7, 235 One-to-one correspondence, 16 Operator symbol, 5 1, 120 Optimisation: constrained, 1 76 unconstrained, 84, 145 Optimum production, 149, 171, 182, 197, 202 Order, 235 Ordered pairs, 15, 16 Ordered set, 15 Ordinate, 16 Origin, 5 Parabola, 18, 51 Parameter, 7, 134, 138, 235 Partial derivative, 120, 176 cross, 124 first-order, 120, 140, 176 as function, 124 higher-order, 124 mixed, 124 second-order, 124, 140, 176 Partial differentiation: and comparative-static analysis, 140 techniques of, Partial elasticities, 140 Partial equilibrium, 22 Perfect square, 14 Pi (ll), 10 Point elasticity of demand, 62 Pointofinflection, 85,110,146,148,171, 172 Polynomial functions, 13 degree, 13 Positive numbers, 5 Post-multiply, 237, 242, 247 Power-function rule, 52, 121,209,215 Pre-multiply, 237, 242 Price discrimination, 96, 113, 114, 153, 164, 170 Price elasticity of demand, 58, 61, 73, 77, 86, 103, 127, 153, 155 and marginal revenue, 77, 103 Price elasticity of supply, 58, 69, 73 Principal diagonal, see Leading diagonal Probability, 252, 262 laws, 25 2, 265 transition, 252, 253, 260, 262 Producer's surplus, 208, 220, 223, 229, 232 Product, 21 0, 211 Production constraint, 179, 192 Production function, 16, 43, 71, 77, 110, 135, 141, 148, 170, 171, 182, 197, 202, 229 Cobb-Douglas, 129, 136 and homogeneity, 136, 13 7 and returns to scale, 136 Production theory, 110, 13 7 Product matrix, 236 Product rule, 53, 55, 56,59-61,74, 121 Product transformation curve, 43 Profit, 6, 7, 18, 42, 57, 64, 71, 113,215, 220,223,224,246 Profit maximisation, 44, 86, 91, 100, 103, 113,145,186, 19~ 202,223,230 and input decisions, 145,149,151,170, 171 and multi-markets, 96, 113, 145, 153, 164, 167,170,202 and multi-plant firm, 158 and multi-product firm, 145, 161 and price discrimination, 96, 113, 114, 153,164,170 and transfer prices, 106 Quadratic equation, 23, 29, 30, 44, 66, 89 roots of, 23, 29, 30, 66 Quadratric formula, 24 Quadratic function, 18, 23, 42 Quotient, 15, 210, 211 Quotient rule, 54, 60, 63, 73, 75, 76, 116, 122, 187 Range, 16 Rate of capital formation, 58 Rate of change, 51, 57 instantaneous, 51, 120, 121 Rate of increase, 227 Rate of interest, 33, 37, 132, 134, 141 Rate of manufacture, 229 Rational numbers, 6, 23 Real number line, 6 Real numbers, 5, 6, 23, 24 Reciprocal, 11, 240 Rectangular system, 243 Regression analysis, 16 2 Relative extrema, 84 criteria for, maximum, 84, 85, 145 minimum, 85, 145 Relation, 16 Repair costs, 217,219,230 Returns risk, 189, 202 Returns to scale: constant, 137, 141 and homogeneous function, 137 Roots, 18, 23, 29, 44, 46,48 Row manipulation, 241,

6 Row vector, 235 Saddle point, 145, 148, 167, 171 Sales function, 130, 184, 186 Sales revenue maximisation, 103, 220 Saving function, 75, 226 Scalar, 236 Scalar multiplication, 236 Scientific notation, 11 Scrap value, 230, 234 Second derivative, 57 Second-order condition, 84, 176 Sequences, 25 Series: arithmetic, 25 geometric, 25 Set, 15 Sigma (k), 10 Simultaneous-equation system, 22, 42, 183, 185, 187, 190, 244 Singular matrix, 240 Sinking funds, 25 Slope, 18,51,52,139 Solution of equations: completing the square, 23 factorisation, 23 square-root formula, 24, 30, 66, 73, 89, 166, 180 system, 235, 237, 239, 242, 262 Square matrix, 235, 240, 242 Square root, 11 Standard integrals, 208 Stationary point, 84, 85, 87, 146, 176 Steady state, 253 Stochastic matrix, 253, 254 Subdeterminant, 238 Subsidy, 43 Substitution method, 22, 210 Substitute good, 129, 192 Substitute relationship, 129 Sufficient condition, 84, 145, 176 Sum-difference rule, 53, 121 Summation index, 10 Summation sign, 10 Supply function, 26-43,67, 69, 72, 108, 114,138,208,220,229,254,262 derivative of, 70, 74 sketching of, Supply of money, 32, 37, 132 Surplus production, 250 Symmetric matrix, 235 Tax, 26, 36, 37, 43, 67, 77, 100, 108, 114,132 revenue, 43,67,69, 10~ 108,114 Technological coefficients, 249, 256, 263 Theory of the Firm, 64 Theory of production, 110, 13 7 Total absolute area, 214 Total area, 213 Total cost, 7, 57, 91, 208, 247 Total cost function, 18, 21, 42, 64, 77, 90, 96, 100, 103, 106, 113, 140, 150, 151, 153, 154, 158, 161, 162, 164,167, 179,192,202, 215, 221,223, 229, 263 Total derivative, 125, 197 Total differential, 125 Total differentiation, 120 Total factor payments, 129 Total fixed cost, 90, 112, 113, 170, 215, 224, 230 Total net earnings, 230 Total net savings, 226, 227 Total production, 249 Total revenue, 7, 57, 59, 154, 158, 208 Total revenue function, 64, 77, 84, 86, 90, 103, 113, 150, 152, 221, 223, 229 Total variable cost, 90, 112, 113, 170 Transformation, 16 Transfer price, 106 Transaction-precautionary demand for money, 33,37 Transition probabilities, 252, 253, 260, 262 Transpose matrix, 235 Triangular matrix, 239 Two-sector economy, 32, 249, 250, 258 Unconstrained extrema, 84, 145, 177 Undertermined Multiplier, 176 Unique solution, 237, 248, 259 Unit matrix, 235 Utility, function, 125,140,165,170,189,194 marginal, 125, 142, 166, 174 maximisation of, 165, 176, 190, 194 Variables, 6, 235 continuous, 6 dependent and independent, 6, 120, 121, 125,145 discrete 6 endogenous and exogenous, 6, 133, 134 Variance of the rate of return, 189 Vector, 235, 236, 247 Vertical intercept, 18 Wage Bill, 130 Wages Curve, 113 Wealth, 58 Young's Theorem, 124 Zero-factor property, 23 Zero matrix, 235 Zero, the number, 5 275