Jacques I. Mathematics for Economics and Business
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microeconomics
comparative statics 380–3
difference equations 561–6
differential equations 579–83
matrices 475–6
non-linear equations 125–7
see also supply and demand analysis
minima 299, 316, 388, 398, 669
minimization see optimization
modelling 49, 63, 669
money
demand and supply 104–5, 110, 233–4, 669
precautionary demand for 105, 111, 670
speculative demand for 105, 111, 671
transactions demand for 104–5, 111, 672
money market equilibrium 104–7
monopolies 265–6, 273, 322, 669
constrained optimization 414–16
unconstrained optimization 393–4
MPC see marginal propensity to consume
MPS see marginal propensity to save
MRCS see marginal rate of commodity substitution
MRTS see marginal rate of technical substitution
multiplication
algebraic fractions 76–7, 80
brackets 71–6
cross-multiplication 83
fractions 76–7, 80
matrices see matrices
negative numbers 17
scalar 459–60
by scale factors 180, 181
of successive scale factors 182
multipliers 669
autonomous consumption 375, 383, 478
autonomous export 380
autonomous taxation 378
balanced budget 378–9, 383, 663
government expenditure 378
in input–output analysis 511–12
investment 375, 376–7, 383, 478
Lagrange 411–20, 668
marginal propensity to consume 375, 383
marginal propensity to import 380
national economy models 99–112, 502
national income 96, 110, 271, 374, 669
determination
difference equations 559–61
differential equations 577–9
linear equations 96–112
natural logarithms 168–9, 172, 669
continuous compounding of interest 200–1
derivatives 331–40
differentiation of 331–40, 426
integration and 426
negative indices/powers 142–3, 146, 158, 163, 221, 247,
570
negative numbers
addition of 18–19
division of 17
multiplication of 17
square roots 119
subtraction of 18–19
net investment 445, 669
net present value (NPV) 222–3, 226–8, 234, 669
nominal data 189, 191, 669
non-linear equations 113–74
difference equations 564–6
quadratic equations 115–28
revenue, cost and profit 129–40
simultaneous 390–1
non-negativity constraints 524, 527, 530, 532, 537, 541,
669
non-renewable resources 215–17
non-singular matrices 473, 474, 489, 669
NPV see net present value
number line 67, 84, 669
numbers
index 184–8, 191, 667
negative see negative numbers
tables of, extraction of formulae from 169–72
numerators 76, 84, 669
objective functions 401, 410, 669
constrained optimization of 401–2, 404–9, 410,
415–16
linear programming 524–8, 532, 539–41, 542–5
operations see mathematical operations
optimization
average costs 310–12
average product of labour 306–7, 320, 327–8, 338–9
constrained see constrained optimization
economic functions 298–319
meaning of term 304, 316, 669
production functions 304–7
profits see profit(s)
tax revenue 312–14, 330
total revenue 307–8
unconstrained 386–99, 672
orders of matrices 454, 468, 669
origin of graph 15, 33, 669
oscillatory time paths 559, 563
output see production
output constraint 408–9
own price elasticity of demand 357, 359
Index
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Paasche indices 188, 191, 193, 669
parabolas 121–7, 669
parameters 49, 63, 669
partial derivatives 346–50, 354, 392, 395, 414, 512,
594, 669
partial differentiation 341–420
comparative statics 374–85
constrained optimization 400–10
elasticity of demand 357–9
functions of several variables 343–55
Lagrange multipliers 411–20
production functions 365–71
unconstrained optimization 386–99
utility functions 359–65
particular solutions (PS)
difference equations 555, 556, 557, 558, 560, 562,
566, 669
differential equations 573, 574, 575, 580, 583, 670
percentages 177–93
index numbers 184–8
inflation 188–90, 191, 663
interest 203–4, 206, 663
scale factors 180–3
perfect competition 50, 266, 273, 392, 443–4, 670
point elasticity 285–6, 670
points of inflection 299, 316, 670
power functions
differentiation of 245, 587–8
integration of 426, 427
power(s) 142–5, 158, 159, 670
fractional 144–5, 158, 247
negative 142–3, 146, 158, 163, 221, 247, 570
rules of 145–53, 158
of zero 142, 143, 158
precautionary demand for money 105, 111, 670
present value 221–30, 234, 447–8, 670
see also net present value (NPV)
price discrimination 322–6, 394–5
price elasticity of demand 284–90, 296, 357, 358, 371,
670
marginal revenue and, relationship between 292–6,
329
price elasticity of supply 291–2, 296, 670
price equilibrium 47, 380–3, 475–6, 486
primitives see integrals
principal (simple or compound interest) 196, 206,
670
see also present value
problem formulation 535–8
producer’s surplus 443–4, 670
product rule of differentiation 278–80, 335, 592,
593
production, factors of 53, 96, 110, 149, 159, 666
production functions 149–52, 159, 268–73, 670
Cobb–Douglas see Cobb–Douglas production
functions
constrained optimization of 401–2, 406–9, 417–19
differentiation of 268–73
exponential functions 338–9
homogeneous 151–2
input–output analysis 503
isoquants 366, 369, 370
natural logarithms 339
optimization of 304–7
partial differentiation of 365–71
productivity, labour 306–7, 316
profit
definition 129, 139, 670
maximum 136, 138
optimization of 308–9, 320, 321–6, 339
linear programming 536–8
unconstrained 387, 391–6
PS see particular solutions
pure competition see perfect competition
quadratic equations 115–28, 302, 304–6
demand elasticity 290–1
quadratic functions 115, 670
demand functions 125–7
factorization of 119–20
graphs representing 121–7, 298–9, 302
supply functions 125–7
quantity equilibrium 47
quotient rule of differentiation 281–2, 327, 335
real data 189, 191, 670
reciprocals 146, 221, 570
differentiation of natural logs 335, 339, 426, 429
integration of 426, 429
rectangular hyperbolas 132–3, 139, 670
recurrence relations see difference equations
reduced form (macroeconomic model) 375, 383, 477,
670
relative maxima and minima 299, 316
revenue
average 266, 273, 663
continuous streams 448
differentiation of 262–6
marginal see marginal revenue
total see total revenue
reverse flow charts 90–1, 94, 670
roots
fractional powers as 144–5, 158, 247
see also square roots
row vectors in matrices 456–7, 462, 468, 670
rows, matrix 454
Index
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rules
Cramer’s rule 492–501, 664
differentiation see differentiation
indices 145–53, 158
integration 427–8
logarithms 155–8, 159, 168, 336–7
powers 145–53, 158
saddle points 388–9, 398, 670
saturation levels 166–7
savings 96
autonomous 99, 110, 663
differentiation and 271–3
marginal propensity see marginal propensity to save
savings function 97, 99
integration and 432
savings plans, geometric series 211–13
scalar multiplication of matrices 459–60
scalar quantities, distinguished from vector quantities
457
scale factors 180–3, 191, 196, 670
multiplication by 180, 181
multiplication of successive 182
second-order derivatives 256–8, 300–1, 347–50, 392,
395, 414, 594, 671
second principal minors in matrices 594, 595, 597, 671
simple interest 195, 206, 671
compared with compound interest 195
simplex algorithm 546
simultaneous linear equations 26, 33, 671
algebraic solutions 35–46
graphical solutions 24–34
infinitely many solutions 40
matrix-based solutions 467–8, 475– 8
no solution 39
with three unknowns 41–5
simultaneous linear inequalities 521–32
simultaneous non-linear equations 390–1
singular matrices 473, 474, 489, 671
sinking fund 211, 217, 671
slopes
of curves 241–5, 249
slopes of 256
of straight lines 26–7, 33, 239–41, 242, 249,
671
small increments formula 351–3, 354, 378, 671
speculative demand for money 105, 111, 671
interest rates and, relationship between 233–4
spreadsheets
difference equations 564–6
Laspeyre indices 187
see also Excel
square function (x
2
)
curve 121, 244, 587
differentiation of 245, 587–8
values 121, 243
square matrices 472, 489, 671
square roots 116, 127, 671
of negative numbers 119
stable models
difference equations 559, 566, 671
differential equations 576, 578, 579, 583, 671
statics 375, 383, 671
stationary points 298–304, 316, 671
constrained optimization 404–9
Hessian matrices 595
of inflection 299, 316
profits 321
unconstrained optimization 387–91
stock holding problems 329
straight lines
slopes 26–7, 33, 239–41, 242, 671
see also graphs
structural equations 375, 383, 477, 495, 671
substitutable goods 51, 60, 63, 671
substitution, method of 404–9, 410, 669
subtraction
fractions 78–9, 80
matrices 458
negative numbers 18–19
sum rule of differentiation 252–3, 347
sum rule of integration 427–8
superior good(s) 53, 63, 671
supply
analysis see supply and demand analysis
money 104, 110, 233–4, 669
price elasticity of 291–2, 296
supply and demand analysis
difference equations 561–6
differential equations 579–83
linear equations 47–65
two-commodity market model 60–1
supply functions 53, 63, 671
producer’s surplus 443–4
quadratic 125–7
surplus
consumer’s 441–3, 664
producer’s 443–4, 670
symmetric functions 354, 671
tables of numbers
extraction of formulae from 169–72
sketching curves from 122
tangents 241–5, 249, 264, 671
Index
682
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taxation 101–2, 111, 377–9, 495–7, 671
autonomous taxation multiplier 378
optimization of 312–14, 330
supply and demand analysis affected by 56
technical coefficients, matrices of 503–11, 513, 668
technical substitution, marginal rate of 366–71, 402,
668
technology matrices 503–11, 513, 671
three-dimensional graphs 345–6, 370, 397
three-sector model 377–9
time, in economic models see difference equations;
differential equations
time paths
exploding 557
oscillatory 559, 563
uniformly converging 558, 559
uniformly diverging 557, 559
time series 184, 191, 671
total costs 131, 139, 671
graphs 133, 134–5
integration of marginal costs 430–4
non-linear equations 131–3, 145
optimization of economic functions 323–6
total demand (for money) 105
total output 504–11
total revenue 129, 139, 284, 672
differentiation of 262–7, 273, 339
graphs 129–31, 134–5
integration of marginal revenue 430 –4
non-linear equations 129–31, 134–5
optimization of 307–8
trading nations 498–9
transactions demand for money 104–5, 111, 672
transposition
of formulae 87–95, 672
of matrices 455–7, 468, 672
turning points see stationary points
two-commodity market model, supply and demand
analysis 60–1
U-shaped curves 121–7, 672
sketching from function formula 123–4, 243–4
unbounded feasible regions 531–2, 672
unconstrained optimization 386–99, 672
uniformly converging time paths 558, 559, 566, 672
uniformly diverging time paths 557, 559, 566, 672
unit elasticity of demand 284, 285, 296, 672
unstable models
difference equations 559, 566, 671, 672
differential equations 576–7, 671, 672
utility 360, 371, 672
utility functions
constrained optimization of 386–7, 403–4
partial differentiation of 359–65
unconstrained optimization of 396–8
variable costs 131, 139, 672
variables
dependent 49, 62, 344, 354, 665
endogenous 52, 62, 665
exogenous 52, 53, 62, 665
functions of several 343–4, 354
partial differentiation of 343–55
functions of two variables 368, 666
independent 49, 62, 344, 354, 667
vectors
distinguished from scalar quantities 457
see also column vectors; row vectors
wages optimization, linear programming 542–5
x axis of graph 15, 20, 26, 33, 672
y axis of graph 15, 20, 26, 33, 672
zero
division by 22, 426
power of 142, 143, 158
zero matrices 458–9, 468, 672
Index
683
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