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Summary Quantitative Modelling 1 Department of Decision Sciences 2023 This is the only study guide for DSC1520
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Quantitative Modelling 1 Department of Decision Sciences 2023 This is the only study guide for DSC1520

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  • September 1, 2023
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DSC1520/001/4/2023




SG001/4/2023


Quantitative Modelling 1
DSC1520

Semesters 1 and 2


Department of Decision Sciences


This is the only study guide for DSC1520.




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,Contents


1 Introduction 1

1.1 The module ............................................................................................................................................ 1

1.1.1 Knowledge assumed to be in place .......................................................................................... 1

1.1.2 Notation ...................................................................................................................................... 2

1.1.3 Software ...................................................................................................................................... 2

1.1.4 Activities.....................................................................................................................................2

1.2 Modelling ................................................................................................................................................ 3

1.2.1 Development of a mathematical model ................................................................................... 3



I Linear functions 5

2 Applications of linear functions 6

2.1 Revenue, cost and profit ....................................................................................................................... 6

2.1.1 Cost ............................................................................................................................................. 6

2.1.2 Revenue....................................................................................................................................... 7

2.1.3 Profit ........................................................................................................................................... 7

2.2 Demand and supply ............................................................................................................................... 9

2.2.1 Demand ..................................................................................................................................... 10

2.2.2 Supply .......................................................................................................................................11

2.3 Price elasticity of demand and supply ................................................................................................ 13

2.3.1 Price elasticity of demand ....................................................................................................... 14

2.3.2 Price elasticity of supply .......................................................................................................... 16

ii


3 Applications of simultaneous functions 17

, 3.1 Equilibrium ..........................................................................................................................................17
3.1.1 Equilibrium in the goods market ............................................................................................ 17
3.1.2 Equilibrium in the labour market............................................................................................ 18
3.1.3 Price controls and government intervention in markets ....................................................... 19
3.1.4 Complementary and substitute goods .................................................................................... 22
3.1.5 Taxes and subsidies ................................................................................................................. 24
3.2 Break-even analysis.............................................................................................................................. 27
3.3 Consumer and producer surplus ......................................................................................................... 28
3.3.1 Consumer surplus .................................................................................................................... 28
3.3.2 Producer surplus ...................................................................................................................... 29
3.3.3 Total surplus ............................................................................................................................. 30
3.4 Linear programming ............................................................................................................................. 31



II Non-linear functions 39

4 Applications of non-linear functions 40
4.1 Non-linear functions in economics ..................................................................................................... 40
4.2 Quadratic functions ............................................................................................................................. 41
4.2.1 Optimising quadratic revenue, profit and cost functions ...................................................... 42
4.2.2 Market equilibrium .................................................................................................................. 45
4.2.3 Break-even analysis.................................................................................................................. 46
4.3 Cubic functions ..................................................................................................................................... 48
4.4 Exponential and logarithmic functions ............................................................................................... 49
4.4.1 Unlimited growth ..................................................................................................................... 50
4.4.2 Limited growth ......................................................................................................................... 51
4.4.3 Logistic growth ......................................................................................................................... 51
4.5 Hyperbolic functions ............................................................................................................................ 53



III Differentiation 56

5 Differentiation theory 57
5.1 Slope of a curve and differentiation ................................................................................................... 57
5.2 Differentiation rules ............................................................................................................................. 59
5.2.1 The power rule for differentiation .......................................................................................... 60


5.2.2 Exponential and natural logarithmic functions ...................................................................63
5.2.3 The chain rule .......................................................................................................................... 63

, 5.2.4 The product rule ...................................................................................................................... 65
5.2.5 The quotient rule ..................................................................................................................... 67
5.3 Higher derivatives ................................................................................................................................ 69
5.4 Optimisation of functions in one variable .......................................................................................... 70
5.4.1 The nature of stationary points .............................................................................................. 72
5.4.2 Intervals along which a function is increasing or decreasing ................................................ 73
5.4.3 Sketching functions using differentiation ............................................................................... 75

6 Applications of differentiation 77
6.1 Marginal and average functions ......................................................................................................... 77
6.1.1 Marginal revenue ...................................................................................................................... 78
6.1.2 Marginal cost............................................................................................................................ 78
6.1.3 Average revenue and cost functions ...................................................................................... 79
6.1.4 Production functions ................................................................................................................ 82
6.2 Optimisation of economic functions ..................................................................................................... 83
6.3 Elasticity of demand – non-linear demand functions ........................................................................ 89



IV Integration 92

7 Integration theory 93
7.1 Integration as the reverse of differentiation ...................................................................................... 93
7.2 Integration rules ................................................................................................................................... 94
7.2.1 The power rule for integration ............................................................................................... 94
7.2.2 Integration by substitution ...................................................................................................... 96
7.3 The definite integral and the area under a curve............................................................................ 100

8 Applications of integration 105
8.1 From marginal to total functions ...................................................................................................... 105
8.1.1 Total cost from marginal cost ............................................................................................... 105
8.1.2 Total revenue from marginal revenue .................................................................................. 106
8.1.3 Rate of change to total quantity........................................................................................... 106
8.2 Consumer and producer surplus ....................................................................................................... 109
8.2.1 Consumer surplus .................................................................................................................. 109


8.2.2 Producer surplus .................................................................................................................... 111

8.2.3 Total surplus ........................................................................................................................... 112

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