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File: Economics Pdf 125842 | Mathematical Economics Lecture Notes
prof dr frank werner faculty of mathematics institute of mathematical optimization imo http math uni magdeburg de werner math ec new html mathematical economics lecture notes in extracts winter term ...

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       Prof. Dr. Frank Werner
       Faculty of Mathematics
       Institute of Mathematical Optimization (IMO)
       http://math.uni-magdeburg.de/∼werner/math-ec-new.html
               Mathematical Economics
                      Lecture Notes
                         (in extracts)
                       Winter Term 2019/20
                         Annotation:
        1. These lecture notes do not replace your attendance of the lecture. Nu-
          merical examples are only presented during the lecture.
        2. Thesymbol✏pointstoadditional,detailedremarksgiveninthelecture.
        3. I am grateful to Julia Lange for her contribution in editing the lecture
          notes.
                 Contents
                 1 Basic mathematical concepts                                                                              1
                     1.1   Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        1
                     1.2   Convex sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        6
                     1.3   Convex and concave functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .           7
                     1.4   Quasi-convex and quasi-concave functions . . . . . . . . . . . . . . . . . . . . . .             8
                 2 Unconstrained and constrained optimization                                                              11
                     2.1   Extreme points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        11
                           2.1.1    Global extreme points . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        11
                           2.1.2    Local extreme points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       12
                     2.2   Equality constraints     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    14
                     2.3   Inequality constraints     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    16
                     2.4   Non-negativity constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        18
                 3 Sensitivity analysis                                                                                    20
                     3.1   Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       20
                     3.2   Value functions and envelope results . . . . . . . . . . . . . . . . . . . . . . . . .          20
                           3.2.1    Equality constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       20
                           3.2.2    Properties of the value function for inequality constraints        . . . . . . . . .   21
                           3.2.3    Mixed constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      22
                     3.3   Some further microeconomic applications           . . . . . . . . . . . . . . . . . . . . . .   23
                           3.3.1    Cost minimization problem . . . . . . . . . . . . . . . . . . . . . . . . . .          23
                           3.3.2    Profit maximization problem of a competitive firm . . . . . . . . . . . . .              24
                 4 Differential equations                                                                                   25
                     4.1   Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       25
                     4.2   Differential equations of the first order . . . . . . . . . . . . . . . . . . . . . . . .         26
                           4.2.1    Separable equations      . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   26
                           4.2.2    First-order linear differential equations . . . . . . . . . . . . . . . . . . . .       26
                     4.3   Second-order linear differential equations and systems in the plane            . . . . . . . .   28
                 5 Optimal control theory                                                                                  35
                     5.1   Calculus of variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      35
                                                                       i
                 CONTENTS                                                                                                   ii
                     5.2   Control theory     . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    37
                           5.2.1    Basic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      37
                           5.2.2    Standard problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .         38
                           5.2.3    Current value formulations       . . . . . . . . . . . . . . . . . . . . . . . . . .   40
            Chapter 1
            Basic mathematical concepts
            1.1   Preliminaries
            Quadratic forms and their sign
            Definition 1:
            If A = (a ) is a matrix of order n × n and xT = (x ,x ,...,x ) ∈ Rn, then the term
                    ij                              1 2     n
                                        Q(x) = xT ·A·x
            is called a quadratic form.
            Thus:                                    n  n
                               Q(x) = Q(x ,x ,...,x ) = XXa ·x ·x
                                        1  2     n         ij i  j
                                                     i=1 j=1
            Example 1                                                                  ✏
            Definition 2:
            Amatrix A of order n×n and its associated quadratic form Q(x) are said to be
               1. positive definite, if Q(x) = xT · A · x > 0 for all xT = (x ,x ,...,x ) 6= (0,0,...,0);
                                                            1 2     n
               2. positive semi-definite, if Q(x) = xT · A · x ≥ 0 for all x ∈ Rn;
               3. negative definite, if Q(x) = xT ·A·x < 0 for all xT = (x ,x ,...,x ) 6= (0,0,...,0);
                                                            1 2     n
               4. negative semi-definite, if Q(x) = xT · A · x ≤ 0 for all x ∈ Rn;
               5. indefinite, if it is neither positive semi-definite nor negative semi-definite.
            Remark:
            In case 5., there exist vectors x∗ and y∗ such that Q(x∗) > 0 and Q(y∗) < 0.
                                                1
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...Prof dr frank werner faculty of mathematics institute mathematical optimization imo http math uni magdeburg de ec new html economics lecture notes in extracts winter term annotation these do not replace your attendance the nu merical examples are only presented during thesymbolpointstoadditional detailedremarksgiveninthelecture i am grateful to julia lange for her contribution editing contents basic concepts preliminaries convex sets and concave functions quasi unconstrained constrained extreme points global local equality constraints inequality non negativity sensitivity analysis value envelope results properties function mixed some further microeconomic applications cost minimization problem prot maximization a competitive rm dierential equations rst order separable first linear second systems plane optimal control theory calculus variations ii standard current formulations chapter quadratic forms their sign denition if is matrix n xt x rn then ij q called form thus xxa j example ama...

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