Student’s Manual
Essential Mathematics for
Economic Analysis
5th edition
Knut Sydsæter
Peter Hammond
Andrés Carvajal
Arne Strøm
For further supporting resources, please visit:
www.mymathlab.com/global
,Preface
This Student’s solutions manual accompanies Essential Mathematics for Economic Analysis, 5th
Edition, Pearson, 2016. Its main purpose is to provide more detailed solutions to the problems
marked ⊂ S⊃M in the book. The answers provided in this manual should be used in combination with
any shorter answers provided in the main test. There are a few cases where only part of the answer
is set out in detail, because the rest follows the same pattern.
We would appreciate suggestions for improvements from our readers, as well as help in weeding
out inaccuracies and errors.
Coventry, Davis, and Oslo, April 2016
Peter Hammond ()
Andrés Carvajal ()
Arne Strøm ()
Contents
1 Essentials of Logic and Set Theory 1
Review exercises ..................................................................................................................................... 1
2 Algebra 2
Rules of algebra ...................................................................................................................... 2
Fractions .................................................................................................................................. 2
Fractional powers .......................................................................................................................... 3
Inequalities..................................................................................................................................... 4
2.8 Summation .............................................................................................................................. 5
2.11 Double sums .................................................................................................................................. 6
Review exercises ..................................................................................................................................... 7
3 Solving Equations 8
Solving equations .......................................................................................................................... 8
Equations and their parameters............................................................................................. 9
Quadratic equations ...................................................................................................................... 9
Nonlinear equations .................................................................................................................... 10
Using implication arrows ...................................................................................................... 11
Two linear equations in two unknowns .............................................................................. 12
Review exercises ................................................................................................................................... 12
4 Functions of One Variable 12
4.2 Basic definitions .................................................................................................................... 12
4.4 Linear functions ..................................................................................................................... 13
Quadratic functions ............................................................................................................... 13
Polynomials ........................................................................................................................... 13
Power functions........................................................................................................................... 15
i
, 4.10 Logarithmic functions ........................................................................................................... 15
Review exercises ................................................................................................................................... 16
5 Properties of Functions 17
Inverse functions ................................................................................................................... 17
Graphs of equations .................................................................................................................... 17
Distance in the plane ............................................................................................................ 18
General functions .................................................................................................................. 18
Review exercises ................................................................................................................................... 18
6 Differentiation 18
6.2 Tangents and derivatives ...................................................................................................... 18
6.5 A dash of limits ..................................................................................................................... 19
Sums, products and quotients.............................................................................................. 20
The chain rule ....................................................................................................................... 21
Exponential functions ........................................................................................................... 21
Logarithmic functions............................................................................................................ 21
Review exercises ................................................................................................................................... 23
7 Derivatives in Use 23
Implicit differentiation .......................................................................................................... 23
Economic examples ..................................................................................................................... 23
Differentiating the inverse .................................................................................................... 24
Linear approximations .......................................................................................................... 24
Polynomial approximations .................................................................................................. 24
Taylor’s formula .................................................................................................................... 24
Elasticities .............................................................................................................................. 25
Continuity ........................................................................................................................ 25
More on limits ............................................................................................................................. 25
The intermediate value theorem and Newton’s method ................................................... 26
7.12 L’Hôpital’s rule .......................................................................................................................... 27
Review exercises ................................................................................................................................... 27
8 Single-variable Optimization 29
Simple tests for extreme points ................................................................................................. 29
Economic examples ..................................................................................................................... 29
The extreme value theorem ....................................................................................................... 29
Further economic examples........................................................................................................ 30
Local extreme points ............................................................................................................ 30
Inflection points .................................................................................................................... 32
Review exercises ................................................................................................................................... 32
9 Integration 34
Indefinite integrals ................................................................................................................ 34
Area and definite integrals ................................................................................................... 35
ii
, Properties of indefinite integrals ......................................................................................... 35
Economic applications .......................................................................................................... 36
Integration by parts .............................................................................................................. 36
Integration by substitution ................................................................................................... 38
Infinite intervals of integration ............................................................................................ 39
A glimpse at differential equations ............................................................................... 41
Separable and linear differential equations ........................................................................ 41
Review exercises ................................................................................................................................... 42
10 Topics in Financial Mathematics 43
10.2 Continuous compounding .................................................................................................... 43
10.4 Geometric series.......................................................................................................................... 44
10.7 Internal rate of return .......................................................................................................... 44
Review exercises ................................................................................................................................... 45
11 Functions of Many Variables 45
Partial derivatives with two variables .................................................................................. 45
Geometric representation ........................................................................................................... 45
Functions of more variables ....................................................................................................... 45
Partial derivatives with more variables ............................................................................... 46
Economic applications .......................................................................................................... 47
Partial elasticities .................................................................................................................. 47
Review exercises ................................................................................................................................... 47
12 Tools for Comparative Statics 47
A simple chain rule ......................................................................................................... 47
Chain rules for many variables ............................................................................................. 48
Implicit differentiation along a level curve .......................................................................... 49
More general cases ..................................................................................................................... 49
Elasticity of substitution ....................................................................................................... 50
Homogeneous functions of two variables ................................................................................. 50
Homogeneous and homothetic functions .................................................................................. 51
Linear approximations .......................................................................................................... 52
Differentials.................................................................................................................................. 52
12.11Differentiating systems of equations ......................................................................................... 53
Review exercises ................................................................................................................................... 53
13 Multivariable Optimization 54
Two variables: sufficient conditions .................................................................................... 54
Local extreme points ............................................................................................................ 55
Linear models with quadratic objectives ............................................................................. 56
The Extreme Value Theorem ................................................................................................ 57
The general case.......................................................................................................................... 60
Comparative statics and the Envelope Theorem ................................................................ 60
Review exercises ................................................................................................................................... 61
iii
Essential Mathematics for
Economic Analysis
5th edition
Knut Sydsæter
Peter Hammond
Andrés Carvajal
Arne Strøm
For further supporting resources, please visit:
www.mymathlab.com/global
,Preface
This Student’s solutions manual accompanies Essential Mathematics for Economic Analysis, 5th
Edition, Pearson, 2016. Its main purpose is to provide more detailed solutions to the problems
marked ⊂ S⊃M in the book. The answers provided in this manual should be used in combination with
any shorter answers provided in the main test. There are a few cases where only part of the answer
is set out in detail, because the rest follows the same pattern.
We would appreciate suggestions for improvements from our readers, as well as help in weeding
out inaccuracies and errors.
Coventry, Davis, and Oslo, April 2016
Peter Hammond ()
Andrés Carvajal ()
Arne Strøm ()
Contents
1 Essentials of Logic and Set Theory 1
Review exercises ..................................................................................................................................... 1
2 Algebra 2
Rules of algebra ...................................................................................................................... 2
Fractions .................................................................................................................................. 2
Fractional powers .......................................................................................................................... 3
Inequalities..................................................................................................................................... 4
2.8 Summation .............................................................................................................................. 5
2.11 Double sums .................................................................................................................................. 6
Review exercises ..................................................................................................................................... 7
3 Solving Equations 8
Solving equations .......................................................................................................................... 8
Equations and their parameters............................................................................................. 9
Quadratic equations ...................................................................................................................... 9
Nonlinear equations .................................................................................................................... 10
Using implication arrows ...................................................................................................... 11
Two linear equations in two unknowns .............................................................................. 12
Review exercises ................................................................................................................................... 12
4 Functions of One Variable 12
4.2 Basic definitions .................................................................................................................... 12
4.4 Linear functions ..................................................................................................................... 13
Quadratic functions ............................................................................................................... 13
Polynomials ........................................................................................................................... 13
Power functions........................................................................................................................... 15
i
, 4.10 Logarithmic functions ........................................................................................................... 15
Review exercises ................................................................................................................................... 16
5 Properties of Functions 17
Inverse functions ................................................................................................................... 17
Graphs of equations .................................................................................................................... 17
Distance in the plane ............................................................................................................ 18
General functions .................................................................................................................. 18
Review exercises ................................................................................................................................... 18
6 Differentiation 18
6.2 Tangents and derivatives ...................................................................................................... 18
6.5 A dash of limits ..................................................................................................................... 19
Sums, products and quotients.............................................................................................. 20
The chain rule ....................................................................................................................... 21
Exponential functions ........................................................................................................... 21
Logarithmic functions............................................................................................................ 21
Review exercises ................................................................................................................................... 23
7 Derivatives in Use 23
Implicit differentiation .......................................................................................................... 23
Economic examples ..................................................................................................................... 23
Differentiating the inverse .................................................................................................... 24
Linear approximations .......................................................................................................... 24
Polynomial approximations .................................................................................................. 24
Taylor’s formula .................................................................................................................... 24
Elasticities .............................................................................................................................. 25
Continuity ........................................................................................................................ 25
More on limits ............................................................................................................................. 25
The intermediate value theorem and Newton’s method ................................................... 26
7.12 L’Hôpital’s rule .......................................................................................................................... 27
Review exercises ................................................................................................................................... 27
8 Single-variable Optimization 29
Simple tests for extreme points ................................................................................................. 29
Economic examples ..................................................................................................................... 29
The extreme value theorem ....................................................................................................... 29
Further economic examples........................................................................................................ 30
Local extreme points ............................................................................................................ 30
Inflection points .................................................................................................................... 32
Review exercises ................................................................................................................................... 32
9 Integration 34
Indefinite integrals ................................................................................................................ 34
Area and definite integrals ................................................................................................... 35
ii
, Properties of indefinite integrals ......................................................................................... 35
Economic applications .......................................................................................................... 36
Integration by parts .............................................................................................................. 36
Integration by substitution ................................................................................................... 38
Infinite intervals of integration ............................................................................................ 39
A glimpse at differential equations ............................................................................... 41
Separable and linear differential equations ........................................................................ 41
Review exercises ................................................................................................................................... 42
10 Topics in Financial Mathematics 43
10.2 Continuous compounding .................................................................................................... 43
10.4 Geometric series.......................................................................................................................... 44
10.7 Internal rate of return .......................................................................................................... 44
Review exercises ................................................................................................................................... 45
11 Functions of Many Variables 45
Partial derivatives with two variables .................................................................................. 45
Geometric representation ........................................................................................................... 45
Functions of more variables ....................................................................................................... 45
Partial derivatives with more variables ............................................................................... 46
Economic applications .......................................................................................................... 47
Partial elasticities .................................................................................................................. 47
Review exercises ................................................................................................................................... 47
12 Tools for Comparative Statics 47
A simple chain rule ......................................................................................................... 47
Chain rules for many variables ............................................................................................. 48
Implicit differentiation along a level curve .......................................................................... 49
More general cases ..................................................................................................................... 49
Elasticity of substitution ....................................................................................................... 50
Homogeneous functions of two variables ................................................................................. 50
Homogeneous and homothetic functions .................................................................................. 51
Linear approximations .......................................................................................................... 52
Differentials.................................................................................................................................. 52
12.11Differentiating systems of equations ......................................................................................... 53
Review exercises ................................................................................................................................... 53
13 Multivariable Optimization 54
Two variables: sufficient conditions .................................................................................... 54
Local extreme points ............................................................................................................ 55
Linear models with quadratic objectives ............................................................................. 56
The Extreme Value Theorem ................................................................................................ 57
The general case.......................................................................................................................... 60
Comparative statics and the Envelope Theorem ................................................................ 60
Review exercises ................................................................................................................................... 61
iii
