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Lecture notes for the Bachelor degree programmesIB/IMC/IMA/ITM/IEVM/ACM/IEM/IMMat Karlshochschule International UniversityModule0.1.1 IMQM: Introduction to Management and its Quantitative Methods H ENK VAN E LST
August 30, 2015Fakult¨at I: Betriebswirtschaft und ManagementKarlshochschuleInternational UniversityKarlstraße 36–3876133 KarlsruheGermanyE–mail: [email protected]
E–Print: arXiv:1509.04333v2 [q-fin.GN]c (cid:13) bstract
These lecture notes provide a self-contained introduction to the mathematical methods required in a Bachelordegree programme in Business, Economics, or Management. In particular, the topics covered comprise real-valued vector and matrix algebra, systems of linear algebraic equations, Leontief’s stationary input–outputmatrix model, linear programming, elementary financial mathematics, as well as differential and integralcalculus of real-valued functions of one real variable. A special focus is set on applications in quantitativeeconomical modelling.
Cite as: arXiv:1509.04333v2 [q-fin.GN]These lecture notes were typeset in L A TEX 2 ε . ontents AbstractQualification objectives of the module (excerpt) 1Introduction 21 Vector algebra in Euclidian space R n R n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Euclidian scalar product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 ( n × n ) -matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 213.6 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 n . . . . . . . . . . . . . . . . . . . . . . . . . . . 527.1.2 Rational functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537.1.3 Power-law functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537.1.4 Exponential functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537.1.5 Logarithmic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547.1.6 Concatenations of real-valued functions . . . . . . . . . . . . . . . . . . . 547.2 Derivation of differentiable real-valued functions . . . . . . . . . . . . . . . . . . 557.3 Common functions in economic theory . . . . . . . . . . . . . . . . . . . . . . . . 577.4 Curve sketching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587.5 Analytic investigations of economic functions . . . . . . . . . . . . . . . . . . . . 597.5.1 Total cost functions according to Turgot and von Th¨unen . . . . . . . . . . 597.5.2 Profit functions in the diminishing returns picture . . . . . . . . . . . . . . 627.5.3 Extremal values of rational economic functions . . . . . . . . . . . . . . . 647.6 Elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 A Glossary of technical terms (GB – D) 75Bibliography 81 ualification objectives of the module(excerpt)
The qualification objectives shall be reached by an integrative approach.A broad instructive range is aspired. The students shall acquire a 360 degree orientation concerningthe task- and personnel-related tasks and roles of a manager and supporting tools and methods andbe able to describe the coherence in an integrative way. The knowledge concerning the tasks andthe understanding of methods and tools shall be strengthened by a constructivist approach basedon case studies and exercises.Students who have successfully participated in this module will be able to • . . . , • solve problems in Linear Algebra and Analysis and apply such mathematical methods toquantitative problems in management. • apply and challenge the knowledge critically on current issues and selected case studies.1 CONTENTS ntroduction These lecture notes contain the entire material of the quantitative methods part of the first semestermodule at Karl-shochschule International University. The aim is to provide a selection of tried-and-tested math-ematical tools that proved efficient in actual practical problems of
Economics and
Management .These tools constitute the foundation for a systematic treatment of the typical kinds of quantita-tive problems one is confronted with in a Bachelor degree programme. Nevertheless, they providea sufficient amount of points of contact with a quantitatively oriented subsequent Master degreeprogramme in
Economics , Management , or the
Social Sciences .The prerequisites for a proper understanding of these lecture notes are modest, as they do not gomuch beyond the basic A-levels standards in
Mathematics . Besides the four fundamental arith-metical operations of addition, subtraction, multiplication and division of real numbers, you shouldbe familiar, e.g., with manipulating fractions, dealing with powers of real numbers, the binomialformulae, determining the point of intersection for two straight lines in the Euclidian plane, solv-ing a quadratic algebraic equation, and the rules of differentiation of real-valued functions of onevariable.It might be useful for the reader to have available a modern graphic display calculator (GDC) fordealing with some of the calculations that necessarily arise along the way, when confronted withspecific quantitative problems. Some current models used in public schools and in undergraduatestudies are, amongst others, • Texas Instruments
TI–84 plus , • Casio
CFX–9850GB PLUS .However, the reader is strongly encouraged to think about resorting, as an alternative, to a spread-sheet programme such as EXCEL or OpenOffice to handle the calculations one encounters inone’s quantitative work.The central theme of these lecture notes is the acquisition and application of a number of effec-tive mathematical methods in a business oriented environment. In particular, we hereby focus on quantitative processes of the sort INPUT → OUTPUT , for which different kinds of functional relationships between some numerical INPUT quantities and some numerical
OUTPUT quantities are being considered. Of special interest in this context3 CONTENTSwill be ratios of the structure OUTPUTINPUT . In this respect, it is a general objective in
Economics to look for ways to optimise the value ofsuch ratios (in favour of some economic agent ), either by seeking to increase the OUTPUT whenthe INPUT is confined to be fixed, or by seeking to decrease the INPUT when the OUTPUT isconfined to be fixed. Consequently, most of the subsequent considerations in these lecture noteswill therefore deal with issues of optimisation of given functional relationships between some variables , which manifest themselves either in minimisation or in maximisation procedures.The structure of these lecture notes is the following. Part I presents selected mathematical methodsfrom
Linear Algebra , which are discussed in Chs. 1 to 5. Applications of these methods focus onthe quantitative aspects of flows of goods in simple economic models, as well as on problems inlinear programming. In Part II, which is limited to Ch. 6, we turn to discuss elementary aspectsof
Financial Mathematics . Fundamental principles of
Analysis , comprising differential and in-tegral calculus for real-valued functions of one real variable, and their application to quantitativeeconomic problems, are reviewed in Part III; this extends across Chs. 7 and 8.We emphasise the fact that there are no explicit examples nor exercises included in these lecturenotes. These are reserved exclusively for the lectures given throughout term time.Recommended textbooks accompanying the lectures are the works by Asano (2013) [2], Dowl-ing (2009) [11], Dowling (1990) [10], Bauer et al (2008) [3], Bosch (2003) [6], and H¨ulsmann et al (2005) [16]. Some standard references of Applied Mathematics are, e.g., Bronstein et al (2005) [7] and Arens et al (2008) [1]. Should the reader feel inspired by the aesthetics, beauty,ellegance and efficiency of the mathematical methods presented, and, hence, would like to knowmore about their background and relevance, as well as being introduced to further mathematicaltechniques of interest, she/he is recommended to take a look at the brilliant books by Penrose(2004) [21], Singh (1997) [23], Gleick(1987) [13] and Smith (2007) [24]. Note that most of thetextbooks and monographs mentioned in this Introduction are available from the library at Karl-shochschule International University.Finally, we draw the reader’s attention to the fact that the *.pdf version of these lecture notes con-tains interactive features such as fully hyperlinked references to original publications at the web-sites dx.doi.org and jstor.org , as well as active links to biographical information on sci-entists that have been influential in the historical development of
Mathematics , hosted by the web-sites The MacTutor History of Mathematics archive ( )and en.wikipedia.org . hapter 1Vector algebra in Euclidian space R n Let us begin our elementary considerations of vector algebra with the introduction of a specialclass of mathematical objects. These will be useful at a later stage, when we turn to formulatecertain problems of a quantitative nature in a compact and elegant way. Besides introducing thesemathematical objects, we also need to define which kinds of mathematical operations they can besubjected to, and what computational rules we have to take care of.
Given be a set V of mathematical objects a which, for now, we want to consider merely as acollection of n arbitrary real numbers a , . . . , a i , . . . , a n . In explicit terms, V = a = a ... a i ... a n | a i ∈ R , i = 1 , . . . , n . (1.1)Formally the n real numbers considered can either be assembled in an ordered pattern as a columnor a row. We define Def.:
Real-valued column vector with n components a := a ... a i ... a n , a i ∈ R , i = 1 , . . . , n , (1.2)Notation: a ∈ R n × ,and 5 CHAPTER 1. VECTORALGEBRA IN EUCLIDIAN SPACE R N Def.:
Real-valued row vector with n components a T := ( a , . . . , a i , . . . , a n ) , a i ∈ R , i = 1 , . . . , n , (1.3)Notation: a T ∈ R × n .Correspondingly, we define the n -component objects := ... ... and T := (0 , . . . , , . . . , (1.4)to constitute related zero vectors .Next we define for like objects in the set V , i.e., either for n -component column vectors or for n -component row vectors, two simple computational operations. These are Def.: Addition of vectors a + b := a ... a i ... a n + b ... b i ... b n = a + b ... a i + b i ... a n + b n , a i , b i ∈ R , (1.5)and Def.: Rescaling of vectors λ a := λa ... λa i ... λa n , λ, a i ∈ R . (1.6)The rescaling of a vector a with an arbitrary non-zero real number λ has the following effects: • | λ | > — stretching of the length of a • < | λ | < — shrinking of the length of a • λ < — directional reversal of a ..2. DIMENSION AND BASISOF R N a will be made precise shortly.The addition and the rescaling of n -component vectors satisfy the following addition and multipli-cation laws: Computational rules for addition and rescaling of vectors
For vectors a , b , c ∈ R n :1. a + b = b + a ( commutative addition )2. a + ( b + c ) = ( a + b ) + c ( associative addition )3. a + = a ( addition identity element )4. For every a , b ∈ R n , there exists exactly one x ∈ R n such that a + x = b ( invertibility of addition )5. ( λµ ) a = λ ( µ a ) with λ ∈ R ( associative rescaling )6. a = a ( rescaling identity element )7. λ ( a + b ) = λ a + λ b ; ( λ + µ ) a = λ a + µ a with λ, µ ∈ R ( distributive rescaling ).In conclusion of this section, we remark that every set of mathematical objects V constructed inline with Eq. (1.1), with an addition and a rescaling defined according to Eqs. (1.5) and (1.6), andsatisfying the laws stated above, constitutes a linear vector space over Euclidian space R n . R n Let there be given m n -component vectors a , . . . , a i , . . . , a m ∈ R n , as well as m real numbers λ , . . . , λ i , . . . , λ m ∈ R . The new n -component vector b resulting from the addition of arbitrarilyrescaled versions of these m vectors according to b = λ a + . . . + λ i a i + . . . + λ m a m =: m X i =1 λ i a i ∈ R n (1.7)is referred to as a linear combination of the m vectors a i , i = 1 , . . . , m . Def.:
A set of m vectors a , . . . , a i , . . . , a m ∈ R n is called linearly independent when the condi-tion ! = λ a + . . . + λ i a i + . . . + λ m a m = m X i =1 λ i a i , (1.8) This is named after the ancient greek mathematician Euclid of Alexandria (about 325 BC–265 BC). A slightly s horter notation for n -component column vectors a ∈ R n × is given by a ∈ R n ; likewise a T ∈ R n for n -component row vectors a T ∈ R × n . CHAPTER 1. VECTORALGEBRA IN EUCLIDIAN SPACE R N i.e., the problem of forming the zero vector ∈ R n from a linear combination of the m vectors a , . . . , a i , . . . , a m ∈ R n , can only be solved trivially, namely by λ = . . . = λ i = . . . = λ m .When, however, this condition can be solved non-trivially, with some λ i = 0 , then the set of m vectors a , . . . , a i , . . . , a m ∈ R n is called linearly dependent .In Euclidian space R n , there is a maximum number n (!) of vectors which can be linearly indepen-dent. This maximum number is referred to as the dimension of Euclidian space R n . Every setof n linearly independent vectors in Euclidian space R n constitutes a possible basis of Euclidianspace R n . If the set { a , . . . , a i , . . . , a n } constitutes a basis of R n , then every other vector b ∈ R n can be expressed in terms of these basis vectors by b = β a + . . . + β i a i + . . . + β n a n = n X i =1 β i a i . (1.9)The rescaling factors β i ∈ R of the a i ∈ R n are called the components of vector b with respectto the basis { a , . . . , a i , . . . , a n } . Remark:
The n unit vectors e := ... , e := ... , . . . , e n := ... , (1.10)constitute the so-called canonical basis of Euclidian space R n . With respect to this basis, allvectors b ∈ R n can be represented as a linear combinationen b = b b ... b n = b e + b e + · · · + b n e n = n X i =1 b i e i . (1.11) Finally, to conclude this section, we introduce a third mathematical operation defined for vectorson R n . Def.:
For an n -component row vector a T ∈ R × n and an n -component column vector b ∈ R n × ,the Euclidian scalar product a T · b := ( a , . . . , a i , . . . a n ) b ... b i ... b n = a b + . . . + a i b i . . . + a n b n =: n X i =1 a i b i (1.12).3. EUCLIDIAN SCALAR PRODUCT 9defines a mapping f : R × n × R n × → R from the product set of n -component row and columnvectors to the set of real numbers. Note that, in contrast to the addition and the rescaling of n -component vectors, the outcome of forming a Euclidian scalar product between two n -componentvectors is a single real number .In the context of the Euclidian scalar product, two non-zero vectors a , b ∈ R n (wit a = = b )are referred to as mutually orthogonal when they exhibit the property that a T · b = b T · a . Computational rules for Euclidian scalar product of vectors
For vectors a , b , c ∈ R n :1. ( a + b ) T · c = a T · c + b T · c ( distributive scalar product )2. a T · b = b T · a ( commutative scalar product )3. ( λ a T ) · b = λ ( a T · b ) with λ ∈ R ( homogeneous scalar product )4. a T · a > for all a = ( positive definite scalar product ).Now we turn to introduce the notion of the length of an n -component vector. Def.:
The length of a vector a ∈ R n is defined via the Euclidian scalar product as | a | := √ a T · a = q a + . . . + a i + . . . + a n =: vuut n X i =1 a i . (1.13)Technically one refers to the non-negative real number | a | as the absolute value or the Euclidiannorm of the vector a ∈ R n . The length of a ∈ R n has the following properties: • | a | ≥ , and | a | = 0 ⇔ a = ; • | λ a | = | λ || a | for λ ∈ R ; • | a + b | ≤ | a | + | b | ( triangle inequality ).Every non-zero vector a ∈ R n , i.e., | a | > , can be rescaled by the reciprocal of its length. Thisprocedure defines the Def.: Normalisation of a vector a ∈ R n ; ˆ a := a | a | ⇒ | ˆ a | = 1 . (1.14)By this method one generates a vector of length , i.e., a unit vector ˆ a . To denote unit vectors wewill employ the “hat” symbol.Lastly, also by means of the Euclidian scalar product, we introduce the angle enclosed betweentwo non-zero vectors.0 CHAPTER 1. VECTORALGEBRA IN EUCLIDIAN SPACE R N Def.: Angle enclosed between a , b = ∈ R n cos[ ϕ ( a , b )] = a T | a | · b | b | = ˆ a T · ˆ b ⇒ ϕ ( a , b ) = cos − (ˆ a T · ˆ b ) . (1.15) Remark:
The inverse cosine function cos − ( . . . ) is available on every standard GDC or spread-sheet. The notion of on inverse function will be discussed later in Ch. 7. hapter 2Matrices
In this chapter, we introduce a second class of mathematical objects that are more general thanvectors. For these objects, we will also define certain mathematical operations, and a set of com-putational rules that apply in this context.
Consider given a collection of m × n arbitrary real numbers a , a . . . , a ij , . . . , a mn , which wearrange systematically in a particular kind of array. Def.:
A real-valued ( m × n ) -matrix is formally defined to constitute an array of real numbersaccording to A := a a . . . a j . . . a n a a . . . a j . . . a n ... ... . . . ... . . . ... a i a i . . . a ij . . . a in ... ... . . . ... . . . ... a m a m . . . a mj . . . a mn , (2.1)where a ij ∈ R , i = 1 , . . . , m ; j = 1 , . . . , n .Notation: A ∈ R m × n .Characteristic features of this array of real numbers are: • m denotes the number of rows of A , n the number of columns of A . • a ij represents the elements of A ; a ij is located at the point of intersection of the i th row andthe j th column of A . • elements of the i th row constitute the row vector ( a i , a i , . . . , a ij , . . . , a in ) , elements of the112 CHAPTER2. MATRICES j th column the column vector a j a j ... a ij ... a mj .Formally column vectors need to be viewed as ( n × -matrices, row vectors as (1 × n ) -matrices.An ( m × n ) -zero matrix , denoted by , has all its elements equal to zero, i.e., := . . .
00 0 . . . ... ... . . . ... . . . . (2.2)Matrices which have an equal number of rows and columns, i.e. m = n , are referred to as quadratic matrices . In particular, the ( n × n ) -unit matrix (or identity matrix) := . . . . . .
00 1 . . . . . . ... ... . . . ... . . . ... . . . . . . ... ... . . . ... . . . ... . . . . . . (2.3)holds a special status in the family of ( n × n ) -matrices.Now we make explicit in what sense we will comprehend ( m × n ) -matrices as mathematicalobjects. Def.:
A real-valued matrix A ∈ R m × n defines by the computational operation A x := a a . . . a j . . . a n a a . . . a j . . . a n ... ... . . . ... . . . ... a i a i . . . a ij . . . a in ... ... . . . ... . . . ... a m a m . . . a mj . . . a mn x x ... x j ... x n := a x + a x + . . . + a j x j + . . . + a n x n a x + a x + . . . + a j x j + . . . + a n x n ... a i x + a i x + . . . + a ij x j + . . . + a in x n ... a m x + a m x + . . . + a mj x j + . . . + a mn x n =: y y ... y i ... y m = y (2.4).2. BASIC CONCEPTS 13a mapping A : R n × → R m × , i.e. a mapping from the set of real-valued n -component columnvectors (here: x ) to the set of real-valued m -component column vectors (here: y ).In loose analogy to the photographic process, x can be viewed as representing an “object,” A a“camera,” and y the resultant “image.”Since for real-valued vectors x , x ∈ R n × and real numbers λ ∈ R , mappings defined by real-valued matrices A ∈ R m × n exhibit the two special properties A ( x + x ) = ( A x ) + ( A x ) A ( λ x ) = λ ( A x ) , (2.5)they constitute linear mappings . We now turn to discuss the most important mathematical operations defined for ( m × n ) -matrices,as well as the computational rules that obtain. Def.: Transpose of a matrixFor A ∈ R m × n , we define the process of transposing A by A T : a Tij := a ji , (2.6)where i = 1 , . . . , m und j = 1 , . . . , n . Note that it holds that A T ∈ R n × m .When transposing an ( m × n ) -matrix, one simply has to exchange the matrix’ rows with its columns(and vice versa): the elements of the first row become the elements of the first column, etc. Itfollows that, in particular, ( A T ) T = A (2.7)applies.Two special cases may occur for quadratic matrices (where m = n ): • When A T = A , one refers to A as a symmetric matrix . • When A T = − A , one refers to A as an antisymmetric matrix . Def.: Addition of matricesFor A , B ∈ R m × n , the sum is given by A + B =: C : a ij + b ij =: c ij , (2.8)with i = 1 , . . . , m and j = 1 , . . . , n .Note that an addition can be performed meaningfully only for matrices of the same format . It is important to note at this point that many advanced mathematical models designed to describe quantitativeaspects of some natural and economic phenomena do not satisfy the conditions (2.5), as they employ non-linearmappings for this purpose. However, in such contexts, linear mappings often provide useful first approximations.
Def.: Rescaling of matricesFor A ∈ R m × n and λ ∈ R \{ } , let λ A =: C : λa ij =: c ij , (2.9)where i = 1 , . . . , m and j = 1 , . . . , n .When rescaling a matrix, all its elements simply have to be multiplied by the same non-zero realnumber λ . Computational rules for addition and rescaling of matrices
For matrices A , B , C ∈ R m × n :1. A + B = B + A ( commutative addition )2. A + ( B + C ) = ( A + B ) + C ( associative addition )3. A + = A ( addition identity element )4. For every A and B , there exists exactly one Z such that A + Z = B .( invertibility of addition )5. ( λµ ) A = λ ( µ A ) with λ, µ ∈ R \{ } ( associative rescaling )6. A = A ( rescaling identity element )7. λ ( A + B ) = λ A + λ B ; ( λ + µ ) A = λ A + µ A with λ, µ ∈ R \{ } ( distributive rescaling )8. ( A + B ) T = A T + B T ( transposition rule 1 )9. ( λ A ) T = λ A T with λ ∈ R \{ } . ( transposition rule 2 )Next we introduce a particularly useful mathematical operation for matrices. Def.:
For a real-valued ( m × n ) -matrix A and a real-valued ( n × r ) -matrix B , a matrix multipli-cation is defined by AB =: C a i b j + . . . + a ik b kj + . . . + a in b nj =: P nk =1 a ik b kj =: c ij , (2.10)with i = 1 , . . . , m and j = 1 , . . . , r , thus yielding as an outcome a real-valued ( m × r ) -matrix C ..3. MATRIX MULTIPLICATION 15The element of C at the intersection of the i th row and the j th column is determined by thecomputational rule c ij = Euclidian scalar product of i th row vector of A and j th column vector of B . (2.11)It is important to realise that the definition of a matrix multiplication just provided depends inan essential way on the fact that matrix A on the left in the product needs to have as many (!)columns as matrix B on the right rows . Otherwise, a matrix multiplication cannot be defined in ameaningful way. GDC:
For matrices [ A ] and [ B ] edited beforehand, of matching formats, their matrix multiplicationcan be evaluated in mode MATRIX → NAMES by [ A ] ∗ [ B ] . Computational rules for matrix multiplication
For A , B , C real-valued matrices of correspondingly matching formats we have:1. AB = is possible with A = , B = . ( zero divisor )2. A ( BC ) = ( AB ) C ( associative matrix multiplication )3. A 1 |{z} ∈ R n × n = |{z} ∈ R m × m A = A ( multiplicative identity element )4. ( A + B ) C = AC + BCC ( A + B ) = CA + CB ( distributive matrix multiplication )5. A ( λ B ) = ( λ A ) B = λ ( AB ) with λ ∈ R ( homogeneous matrix multiplication )6. ( AB ) T = B T A T ( transposition rule ).6 CHAPTER2. MATRICES hapter 3Systems of linear algebraic equations In this chapter, we turn to address a particular field of application of the notions of matrices andvectors, or of linear mappings in general.
Let us begin with a system of m ∈ N linear algebraic equations, wherein every single equation canbe understood to constitute a constraint on the range of values of n ∈ N variables x , . . . , x n ∈ R .The objective is to determine all possible values of x , . . . , x n ∈ R which satisfy these constraintssimultaneously. Problems of this kind, namely systems of linear algebraic equations , are oftenrepresented in the form • Representation 1: a x + . . . + a j x j + . . . + a n x n = b ... a i x + . . . + a ij x j + . . . + a in x n = b i (3.1)... a m x + . . . + a mj x j + . . . + a mn x n = b m . Depending on how the natural numbers m and n relate to one another, systems of linear algebraicequations can be classified as follows: • m < n : fewer equations than variables; the linear system is under-determined , • m = n : same number of equations as variables; the linear system is well-determined , • m > n : more equations than variables; the linear system is over-determined .A more compact representation of a linear system of format ( m × n ) is given by178 CHAPTER 3. SYSTEMSOF LINEARALGEBRAIC EQUATIONS • Representation 2: A x = a . . . a j . . . a n ... . . . ... . . . ... a i . . . a ij . . . a in ... . . . ... . . . ... a m . . . a mj . . . a mn x ... x j ... x n = b ... b i ... b m = b . (3.2)The mathematical objects employed in this variant of a linear system are as follows: A takes thecentral role of the coefficient matrix of the linear system, of format ( m × n ) , x is its variablevector , of format ( n × , and, lastly, b is its image vector , of format ( m × .When dealing with systems of linear algebraic equations in the form of Representation 2, i.e. A x = b , the main question to be answered is: Question:
For given coefficient matrix A and image vector b , can we find a variable vector x that A maps onto b ?In a sense this describes the inversion of the photographic process we had previously referred to:we have given the camera and we already know the image, but we have yet to find a matchingobject. Remarkably, to address this issue, we can fall back on a simple algorithmic method due tothe German mathematician and astronomer Carl Friedrich Gauß (1777–1855). The algorithmic solution technique developed by Gauß is based on the insight that the solution setof a linear system of m algebraic equations for n real-valued variables, i.e. A x = b , (3.3)remains unchanged under the following algebraic equivalence transformations of the linear sys-tem:1. changing the order amongst the equations,2. multiplication of any equation by a non-zero real number c = 0 ,3. addition of a multiple of one equation to another equation,4. changing the order amongst the equations.Specifically, this implies that we may manipulate a given linear system by means of these fourdifferent kinds of equivalence transformations without ever changing its identity. In concrete cases,however, one should not apply these equivalence transformations at random but rather follow atarget oriented strategy. This is what Gaußian elimination can provide..3. RANK OF AMATRIX 19 Target:
To cast the augmented coefficient matrix ( A | b ) , i.e., the array a . . . a j . . . a n b ... . . . ... . . . ... ... a i . . . a ij . . . a in b i ... . . . ... . . . ... ... a m . . . a mj . . . a mn b m , (3.4)when possible, into upper triangular form . . . ˜ a j . . . ˜ a n ˜ b ... . . . ... . . . ... ... . . . ˜ a ij . . . ˜ a in ˜ b i ... . . . ... . . . ... ... . . . . . . ˜ a mn ˜ b m , (3.5)by means of the four kinds of equivalence transformations such that the resultant simpler finallinear system may easily be solved using backward substitution .Three exclusive cases of possible solution content for a given system of linear algebraic equationsdo exist. The linear system may possess either1. no solution at all, or2. a unique solution , or3. multiple solutions . Remark:
Linear systems that are under-determined, i.e., when m < n , can never be solveduniquely due to the fact that in such a case there not exist enough equations to constrain the valuesof all of the n variables. GDC:
For a stored augmented coefficient matrix [ A ] of format ( m × n + 1) , associated with a given ( m × n ) linear system, select mode MATRIX → MATH and then call the function rref ([ A ]) . It ispossible that backward substitution needs to be employed to obtain the final solution.For completeness, we want to turn briefly to the issue of solvability of a system of linear algebraicequations. To this end, we need to introduce the notion of the rank of a matrix. Def.:
A real-valued matrix A ∈ R m × n possesses the rank rank ( A ) = r , r ≤ min { m, n } (3.6)if and only if r is the maximum number of row resp. column vectors of A which are linearlyindependent. Clearly, r can only be as large as the smaller of the numbers m and n that determinethe format of A .0 CHAPTER 3. SYSTEMSOF LINEARALGEBRAIC EQUATIONSFor quadratic matrices A ∈ R n × n , there is available a more elegant measure to determine itsrank. This (in the present case real-valued) measure is referred to as the determinant of matrix A , det( A ) , and is defined as follows. Def.: (i) When A ∈ R × , its determinant is given by det( A ) := (cid:12)(cid:12)(cid:12)(cid:12) a a a a (cid:12)(cid:12)(cid:12)(cid:12) := a a − a a , (3.7)i.e. the difference between the products of A ’s on-diagonal elements and A ’s off-diagonalelements.(ii) When A ∈ R × , the definition of A ’s determinant is more complex. In that case it isgiven by det( A ) := (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a a a a a a a a a (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) := a ( a a − a a ) + a ( a a − a a ) + a ( a a − a a ) . (3.8)Observe, term by term, the cyclic permutation of the first index of the elements a ij accordingto the rule → → → .(iii) Finally, for the (slightly involved) definition of the determinant of a higher-dimensionalmatrix A ∈ R n × n , please refer to the literature; e.g. Bronstein et al (2005) [7, p 267].To determine the rank of a given quadtratic matrix A ∈ R n × n , one now installs the followingcriteria: rank ( A ) = r = n , if det( A ) = 0 , and rank ( A ) = r < n , if det( A ) = 0 . In the first case, A is referred to as regular , in the second as singular . For quadratic matrices A that are singular,rank ( A ) = r (with r < n ) is given by the number r of rows (or columns) of the largest possiblenon-zero subdeterminant of A . GDC:
For a stored quadratic matrix [ A ] , select mode MATRIX → MATH and obtain its determinantby calling the function det ([ A ]) . Making use of the concept of the rank of a real-valued matrix A ∈ R m × n , we can now summarisethe solution content of a specific system of linear algebraic equations of format ( m × n ) in a table.For given linear system A x = b , with coefficient matrix A ∈ R m × n , variable vector x ∈ R n × and image vector b ∈ R m × , thereexist(s).5. INVERSE OFA REGULAR ( N × N ) -MATRIX 21 b = b =
1. rank ( A ) = rank ( A | b ) no solution ——–2. rank ( A ) = rank ( A | b ) = r (a) r = n a unique x = solution(b) r < n multiple multiplesolutions: solutions: n − r free n − r freeparameters parameters ( A | b ) here denotes the augmented coefficient matrix.Next we discuss a particularly useful property of regular quadratic matrices. ( n × n ) -matrix Def.:
Let a real-valued quadratic matrix A ∈ R n × n be regular , i.e., det( A ) ∈ R \{ } . Then thereexists an inverse matrix A − to A defined by the characterising properties A − A = AA − = . (3.9)Here denotes the ( n × n ) -unit matrix [cf. Eq. (2.3)].When a computational device is not at hand, the inverse matrix A − of a regular quadratic matrix A can be obtained by solving the matrix-valued linear system AX ! = (3.10)for the unknown matrix X by means of simultaneous Gaußian elimination . GDC:
For a stored quadratic matrix [ A ] , its inverse matrix can be simply obtained as [ A ] − , wherethe x − function key needs to be used.2 CHAPTER 3. SYSTEMSOF LINEARALGEBRAIC EQUATIONS Computational rules for the inverse operation
For A , B ∈ R n × n , with det( A ) = 0 = det( B ) , it holds that1. ( A − ) − = A ( AB ) − = B − A − ( A T ) − = ( A − ) T ( λ A ) − = 1 λ A − .The special interest in applications in the concept of inverse matrices arises for the followingreason. Consider given a well-determined linear system A x = b , with regular quadratic coefficient matrix A ∈ R n × n , i.e., det( A ) = 0 . Then, for A , there exists aninverse matrix A − . Matrix-multiplying both sides of the equation above from the left (!) by theinverse A − , results in A − ( A x ) = ( A − A ) x = x = x | {z } left-hand side = A − b | {z } right-hand side . (3.11)In this case, the unique solution (!) x = A − b of the linear system arises simply from matrixmultiplication of the image vector b by the inverse matrix of A . (Of course, it might actuallyrequire a bit of computational work to determine A − .) There are a number of exciting advanced topics in
Linear Algebra . Amongst them one findsthe concept of the characteristic eigenvalues and associated eigenvectors of quadratic matrices ,which has particularly high relevance in practical applications. The question to be answered hereis the following: for given real-valued quadratic matrix A ∈ R n × n , do there exist real numbers λ n ∈ R and real-valued vectors v n ∈ R n × which satisfy the condition A v n ! = λ n v n ? (3.12)Put differently: for which vectors v n ∈ R n × does their mapping by a quadratic matrix A ∈ R n × n amount to simple rescalings by real numbers λ n ∈ R ?By re-arranging, Eq. (3.12) can be recast into the form ! = ( A − λ n ) v n , (3.13).6. OUTLOOK 23with an ( n × n ) -unit matrix [cf. Eq. (2.3)] and an n -component zero vector. This conditioncorresponds to a homogeneous system of linear algebraic equations of format ( n × n ) . Non-trivialsolutions v n = to this system exist provided that the so-called characteristic equation ! = det ( A − λ n ) , (3.14)a polynomial of degree n (cf. Sec. 7.1.1), allows for real-valued roots λ n ∈ R . Note that symmetric quadratic matrices (cf. Sec. 2.2) possess exclusively real-valued eigenvalues λ n . When theseeigenvalues turn out to be all different , then the associated eigenvectors v n prove to be mutuallyorthogonal.Knowledge of the spectrum of eigenvalues λ n ∈ R and associated eigenvectors v n ∈ R n × of a real-valued matrix A ∈ R n × n provides the basis of a transformation of A to its diagonalform A λ n , thus yielding a diagonal matrix which features the eigenvalues λ n as its on-diagonalelements; cf. Leon (2009) [19].Amongst other examples, the concept of eigenvalues and eigenvectors of quadratic real-valuedmatrices plays a special role in Statistics , in the context of exploratory principal componentanalyses of multivariate data sets, where the objective is to identify dominant intrinsic structures;cf. Hair et al (2010) [14, Ch. 3] and Ref. [12, App. A].4 CHAPTER 3. SYSTEMSOF LINEARALGEBRAIC EQUATIONS hapter 4Leontief ’s stationary input–output matrixmodel
We now turn to discuss some specific applications of
Linear Algebra in economic theory. Tobegin with, let us consider quantitative aspects of the exchange of goods between a certain numberof economic agents . We here aim at a simplified abstract description of real economic processes.
The quantitative model to be described in the following is due to the Russianeconomist Wassily Wassilyovich Leontief (1905–1999), cf. Leontief (1936) [20],for which, besides other important contributions, he was awarded the 1973Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel.Suppose given an economic system consisting of n ∈ N interdependent economic agents ex-changing between them the goods they produce. For simplicity we want to assume that every oneof these economic agents represents the production of a single good only. Presently we intendto monitor the flow of goods in this simple economic system during a specified reference periodof time . The total numbers of the n goods leaving the production sector of this model constitutethe OUTPUT quantities . The
INPUT quantities to the production sector are twofold. On theone hand, there are exogenous
INPUT quantities which we take to be given by m ∈ N differentkinds of external resources needed in differing proportions to produce the n goods. On the otherhand, due to their mutual interdependence, some of the economic agents require goods made bytheir neighbours to be able to produce their own goods; these constitute the endogenous INPUTquantities of the system. Likewise, the production sector’s total OUTPUT during the chosen ref-erence period of the n goods can be viewed to flow through one of two separate channels: (i) the exogenous channel linking the production sector to external consumers representing an openmarket, and (ii) the endogenous channel linking the economic agents to their neighbours (thusrespresenting their interdependencies). It is supposed that momentum is injected into the economicsystem, triggering the flow of goods between the different actors, by the prospect of increasing
256 CHAPTER 4. LEONTIEF’SINPUT–OUTPUT MATRIXMODEL the value of the INPUT quantities, in line with the notion of the economic value chain .Leontief’s model is based on the following three elementary
Assumptions :1. For all goods involved the functional relationship between INPUT and OUTPUT quantitiesbe of a linear nature [cf. Eq. (2.5)].2. The proportions of “INPUT quantities to OUTPUT quantities” be constant over the refer-ence period of time considered; the flows of goods are thus considered to be stationary .3.
Economic equilibrium obtains during the reference period of time: the numbers of goodsthen supplied equal the numbers of goods then demanded.The mathematical formulation of Leontief’s quantitative model employs the following
Vector- and matrix-valued quantities :1. q — total output vector ∈ R n × , components q i ≥ units (dim: units)2. y — final demand vector ∈ R n × , components y i ≥ units (dim: units)3. P — input–output matrix ∈ R n × n , components P ij ≥ (dim: 1)4. ( − P ) — technology matrix ∈ R n × n , regular, hence, invertible (dim: 1)5. ( − P ) − — total demand matrix ∈ R n × n (dim: 1)6. v — resource vector ∈ R m × , components v i ≥ units (dim: units)7. R — resource consumption matrix ∈ R m × n , components R ij ≥ , (dim: 1)where denotes the ( n × n ) -unit matrix [cf. Eq. (2.3)]. Note that the components of all thevectors involved, as well as of the input–output matrix and of the resource consumption matrix,can assume non-negative values (!) only. We now turn to provide the definition of the two central matrix-valued quantities in Leontief’smodel. We will also highlight their main characteristic features.
Suppose the reference period of time has ended for the economic system in question, i.e. thestationary flows of goods have stopped eventually. We now want to take stock of the numbers ofgoods that have been delivered by each of the n economic agents in the system. Say that duringthe reference period considered, agent delivered of their good the number n to themselves, thenumber n to agent , the number n to agent , and so on, and, lastly, the number n n to agent n ..2. INPUT–OUTPUT MATRIX ANDRESOURCE CONSUMPTION MATRIX 27The number delivered by agent to external consumers shall be denoted by y . Since in this modela good produced cannot all of a sudden disappear again, and since by Assumption 3 above thenumber of goods supplied must be equal to the number of goods demanded, we find that for thetotal output of agent it holds that q := n + . . . + n j + . . . + n n + y . Analogous relationshold for the total output q , q , . . . , q n of each of the remaining n − agents. We thus obtain theintermediate result q = n + . . . + n j + . . . + n n + y > (4.1)... q i = n i + . . . + n ij + . . . + n in + y i > (4.2)... q n = n n + . . . + n nj + . . . + n nn + y n > . (4.3)This simple system of balance equations can be summarised in terms of a standard input–outputtable as follows:[Values in units] agent · · · agent j · · · agent n external consumers Σ : total outputagent n . . . n j . . . n n y q ... ... . . . ... . . . ... ... ...agent i n i . . . n ij . . . n in y i q i ... ... . . . ... . . . ... ... ...agent n n n . . . n nj . . . n nn y n q n The first column of this table lists all the n different sources of flows of goods (or suppliers ofgoods), while its first row shows the n + 1 different sinks of flows of goods (or consumers ofgoods). The last column contains the total output of each of the n agents in the reference periodof time .Next we compute for each of the n agents the respective values of the non-negative ratios P ij := INPUT (in units) of agent i for agent j (during reference period)OUTPUT (in units) of agent j (during reference period) , (4.4)or, employing a compact and economical index notation, P ij := n ij q j , (4.5) Note that the normalisation quantities in these ratios P ij are given by the total output q j of the receiving agent j and not by the total output q i of the supplying agent i . In the latter case the P ij would represent percentages of thetotal output q i . i, j = 1 , . . . , n . These n × n = n different ratios may be naturally viewed as the elements ofa quadratic matrix P of format ( n × n ) . In general, this matrix is given by P = n n + ... + n j + ... + n n + y . . . n j n j + ... + n jj + ... + n jn + y j . . . n n n n + ... + n nj + ... + n nn + y n ... . . . ... . . . ... n i n + ... + n j + ... + n n + y . . . n ij n j + ... + n jj + ... + n jn + y j . . . n in n n + ... + n nj + ... + n nn + y n ... . . . ... . . . ... n n n + ... + n j + ... + n n + y . . . n nj n j + ... + n jj + ... + n jn + y j . . . n nn n n + ... + n nj + ... + n nn + y n , (4.6)and is referred to as Leontief’s input–output matrix of the stationary economic system underinvestigation.For the very simple case with just n = 3 producing agents, the input–output matrix reduces to P = n n + n + n + y n n + n + n + y n n + n + n + y n n + n + n + y n n + n + n + y n n + n + n + y n n + n + n + y n n + n + n + y n n + n + n + y . It is important to realise that for an actual economic system the input–output matrix P can bedetermined only once the reference period of time chosen has come to an end .The utility of Leontief’s stationary input–output matrix model is in its application for the purposeof forecasting . This is done on the basis of an extrapolation , namely by assuming that an input–output matrix P reference period obtained from data taken during a specific reference period also is valid(to an acceptable degree of accuracy) during a subsequent period, i.e., P subsequent period ≈ P reference period , (4.7)or, in component form, P ij | subsequent period = n ij q j (cid:12)(cid:12)(cid:12)(cid:12) subsequent period ≈ P ij | reference period = n ij q j (cid:12)(cid:12)(cid:12)(cid:12) reference period . (4.8)In this way it becomes possible to compute for a given (idealised) economic system approximatenumbers of INPUT quantities required during a near future production period from the knownnumbers of
OUTPUT quantities of the most recent production period. Long-term empirical ex-perience has shown that this method generally leads to useful results to a reasonable approximation.All of these calculations are grounded on linear relationships describing the quantitative aspects ofstationary flows of goods, as we will soon elucidate.
The second matrix-valued quantity central to Leontief’s stationary model is the resource con-sumption matrix R . This may be interpreted as providing a recipe for the amounts of the m dif-ferent kinds of external resources (the exogenous INPUT quantities ) that are needed in the pro-duction of the n goods (the OUTPUT quantities ). Its elements are defined as the ratios R ij := amounts (in units) required of resource i for the production of one unit of good j , (4.9).3. STATIONARYLINEARFLOWSOF GOODS 29with i = 1 , . . . , m und j = 1 , . . . , n . The rows of matrix R thus contain information concerning the m resources, the columns information concerning the n goods. Note that in general the ( m × n ) resource consumption matrix R is not (!) a quadratic matrix and, therefore, in general not invertible. We now turn to a quantitative description of the stationary flows of goods that are associatedwith the total output q during a specific period of time considered. According to Leontief’sAssumption 1, there exists a linear functional relationship between the endogenous vector-valued INPUT quantity q − y and the vector-valued OUTPUT quantity q . This may be represented interms of a matrix-valued relationship as q − y = P q ⇔ q i − y i = n X j =1 P ij q j , (4.10)with i = 1 , . . . , n , in which the input–output matrix P takes the role of mediating a mappingbetween either of these vector-valued quantities. According to Assumption 2, the elements of the input–output matrix P remain constant for the period of time considered, i.e. the correspondingflows of goods are assumed to be stationary .Relation (4.10) may also be motivated from an alternative perspective that takes the physical sci-ences as a guidline. Namely, the total numbers q of the n goods produced during the period oftime considered which, by Assumption 3, are equal to the numbers supplied of the n goods satisfya conservation law : “whatever has been produced of the n goods during the period of time con-sidered cannot get lost in this period.” In quantitative terms this simple relationship may be castinto the form q |{z} total output = y |{z} final demand (exogenous) + P q |{z} deliveries to production sector (endogenous) . For computational purposes this central stationary flow of goods relation (4.10) may be rearrangedas is convenient. In this context it is helpful to make use of the matrix identity q = q , where denotes the ( n × n ) -unit matrix [cf. Eq. (2.3)]. Examples: (i) given/known: P , q Then it applies that y = ( − P ) q ⇔ y i = n X j =1 ( δ ij − P ij ) q j , (4.11)with i = 1 , . . . , n ; ( − P ) represents the invertible technology matrix of the economicsystem regarded.0 CHAPTER 4. LEONTIEF’SINPUT–OUTPUT MATRIXMODEL(ii) given/known: P , y Then it holds that q = ( − P ) − y ⇔ q i = n X j =1 ( δ ij − P ij ) − y j , (4.12)with i = 1 , . . . , n ; ( − P ) − here denotes the total demand matrix , i.e., the inverse of thetechnology matrix. Likewise, by Assumption 1, a linear functional relationship is supposed to exist between the ex-ogenous vector-valued
INPUT quantity v and the vector-valued OUTPUT quantity q . In matrixlanguage this can be expressed by v = R q ⇔ v i = n X j =1 R ij q j , (4.13)with i = 1 , . . . , m . By Assumption 2, the elements of the resource consumption matrix R remain constant during the period of time considered, i.e., the corresponding resource flows are supposedto be stationary .By combination of Eqs. (4.13) and (4.12), it is possible to compute the numbers v of resourcesrequired (during the period of time considered) for the production of the n goods for given finaldemand y . It applies that v = R q = R ( − P ) − y ⇔ v i = n X j =1 n X k =1 R ij ( δ jk − P jk ) − y k , (4.14)with i = 1 , . . . , m . GDC:
For problems with n ≤ , and known matrices P and R , Eqs. (4.11), (4.12) and (4.14) canbe immediately used to calculate the quantities q from given quantities y , or vice versa. Leontief’s input–output matrix model may be extended in a straightforward fashion to includemore advanced considerations of economic theory . Supposing a closed though not necessarilystationary economic system G comprising n interdependent economic agents producing n differ-ent goods, one may assign monetary values to the INPUT quantity v as well as to the OUTPUTquantities q and y of the system. Besides the numbers of goods produced and the associatedflows of goods one may monitor with respect to G for a given period of time, one can in additionanalyse in time and space the amount of money coupled to the different goods, and the corre-sponding flows of money . However, contrary to the number of goods, in general there does not .4. OUTLOOK 31exist a conservation law for the amount of money with respect to G . This may render the analysisof flows of money more difficult, because, in the sense of an increase in value , money can eitherbe generated inside G during the period of time considered or it can likewise be annihilated ; it is not just limited to either flowing into respectively flowing out of G . Central to considerations ofthis kind is a balance equation for the amount of money contained in G during a given period oftime, which is an additive quantity. Such balance equations constitute familiar tools in Physics (cf.Herrmann (2003) [15, p 7ff]). Its structure in the present case is given by rate of change in time of the amount of money in G [in CU/TU] = (cid:18) flux of money into G [in CU/TU] (cid:19) + (cid:18) rate of generation of money in G [in CU/TU] (cid:19) . Note that, with respect to G , both fluxes of money and rates of generation of money can in prin-ciple possess either sign, positive or negative. To deal with these quantitative issues properly,one requires the technical tools of the differential and integral calculus which we will discussat an elementary level in Chs. 7 and 8. We make contact here with the interdisciplinary scienceof Econophysics (cf., e.g., Bouchaud and Potters (2003) [5]), a very interesting and challengingsubject which, however, is beyond the scope of these lecture notes.Leontief’s input–output matrix model, and its possible extension as outlined here, provide thequantitative basis for considerations of economical ratios of the kindOUTPUT [in units]INPUT [in units] , as mentioned in the Introduction. In addition, dimensionless (scale-invariant) ratios of the formREVENUE [in CU]COSTS [in CU] , referred to as economic efficiency , can be computed for and compared between different economicsystems and their underlying production sectors. In Ch. 7 we will briefly reconsider this issue. Here the symbols CU and TU denote “currency units” and “time units,” respectively. hapter 5Linear programming
On the backdrop of the economic principle , we discuss in this chapter a special class of quanti-tative problems that frequently arise in specific practical applications in
Business and
Manage-ment . Generally one distinguishes between two variants of the economic principle : either (i) todraw maximum utility from limited resources, or (ii) to reach a specific target with minimum ef-fort (costs). With regard to the ratio ( OUTPUT ) / ( INPUT ) put into focus in the Introduction, theissue is to find an optimal value for this ratio under given boundary conditions . This aim can berealised either (i) by increasing the (positive) value of the numerator for fixed (positive) value ofthe denominator, or (ii) by decreasing the (positive) value of the denominator for fixed (positive)value of the numerator. The class of quantitative problems to be looked at in some detail in thischapter typically relate to boundary conditions according to case (i). To be maximised is a (non-negative) real-valued quantity z , which depends in a linear functionalfashion on a fixed number of n (non-negative) real-valued variables x , . . . , x n . We suppose thatthe n variables x , . . . , x n in turn are constrained by a fixed number m of algebraic conditions,which also are assumed to depend on x , . . . , x n in a linear fashion . These m constraints, orrestrictions, shall have the character of imposing upper limits on m different kinds of resources. Def.:
Consider a matrix A ∈ R m × n , a vector b ∈ R m × , two vectors c , x ∈ R n × , and a constant d ∈ R . A quantitative problem of the formmax (cid:8) z = c T · x + d | A x ≤ b , x ≥ (cid:9) , (5.1)334 CHAPTER 5. LINEARPROGRAMMINGor, expressed in terms of a component notation,max z ( x , . . . , x n ) = c x + . . . + c n x n + d (5.2) a x + . . . + a n x n ≤ b (5.3)... a m x + . . . + a mn x n ≤ b m (5.4) x ≥ (5.5)... x n ≥ , (5.6)is referred to as a standard maximum problem of linear programming with n real-valued vari-ables. The different quantities and relations appearing in this definition are called • z ( x , . . . , x n ) — linear objective function , the dependent variable, • x , . . . , x n — n independent variables , • A x ≤ b — m restrictions , • x ≥ — n non-negativity constraints . Remark:
In an analogous fashion one may also formulate a standard minimum problem oflinear programming , which can be cast into the formmin (cid:8) z = c T · x + d | A x ≥ b , x ≥ (cid:9) . In this case, the components of the vector b need to be interpreted as lower limits on certaincapacities.For given linear objective function z ( x , . . . , x n ) , the set of points x = ( x , . . . , x n ) T satisfyingthe condition z ( x , . . . , x n ) = C = constant ∈ R , (5.7)for fixed value of C , is referred to as an isoquant of z . Isoquants of linear objective functionsof n = 2 independent variables constitute straight lines, of n = 3 independent variables Euclid-ian planes, of n = 4 independent variables Euclidian 3-spaces (or hyperplanes), and of n ≥ independent variables Euclidian ( n − -spaces (or hyperplanes).In the simplest cases of linear programming , the linear objective function z depends on n = 2 variables x and x only. An illustrative and efficient method of solving problems of this kind willbe looked at in the following section..2. GRAPHICAL SOLUTION METHOD 35 The systematic graphical solution method of standard maximum problems of linear programming with n = 2 independent variables comprises the following steps:1. Derivation of the linear objective function z ( x , x ) = c x + c x + d in dependence on the variables x and x .2. Identification in the x , x –plane of the feasible region D of z which is determined by the m restrictions imposed on x and x . Specifically, D constitutes the domain of z (cf. Ch. 7).3. Plotting in the x , x –plane of the projection of the isoquant of the linear objective function z which intersects the origin ( x = x ). When c = 0 , this projection is described by theequation x = − ( c /c ) x .
4. Erecting in the origin of the x , x –plane the direction of optimisation for z which is deter-mined by the constant z -gradient ( ∇ z ) T = (cid:18) ∂z∂x ∂z∂x (cid:19) = (cid:18) c c (cid:19) . Parallel displacement in the x , x –plane of the projection of the (0 , -isoquant of z alongthe direction of optimisation ( ∇ z ) T across the feasible region D out to a distance where theprojected isoquant just about touches D .6. Determination of the optimal solution ( x , x ) as the point resp. set of points of inter-section between the displaced projection of the (0 , -isoquant of z and the far boundaryof D .7. Computation of the optimal value of the linear objective function z O = z ( x , x ) fromthe optimal solution ( x , x ) .8. Specification of potential remaining resources by substitution of the optimal solution ( x , x ) into the m restrictions.In general one finds that for a linear objective function z with n = 2 independent variables x and x , the feasible region D , when non-empty and bounded , constitutes an area in the x , x –plane with straight edges and a certain number of vertices. In these cases, the optimal values ofthe linear objective function z are always to be found either at the vertices or on the edges of thefeasible region D . When D is an empty set, then there exists no solution to the correspondinglinear programming problem. When D is unbounded, again there may not exist a solution to thelinear programming problem, but this then depends on the specific circumstances that apply.6 CHAPTER 5. LINEARPROGRAMMING Remark:
To solve a standard minimum problem of linear programming with n = 2 indepen-dent variables by means of the graphical method, one needs to parallelly displace in the x , x –plane the projection of the (0 , -isoquant of z along the direction of optimisation ( ∇ z ) T untilcontact is made with the feasible region D for the first time. The optimal solution is then given bythe point resp. set of points of intersection between the displaced projection of the (0 , -isoquantof z and the near boundary of D . The main disadvantage of the graphical solution method is its limitation to problems withonly n = 2 independent variables. In actual practice, however, one is often concerned with linear programming problems that depend on more than two independent variables . Todeal with these more complex problems in a systematic fashion, the US-American mathemati-cian George Bernard Dantzig (1914–2005) has devised during the 1940ies an efficient algorithmwhich can be programmed on a computer in a fairly straightforward fashion; cf. Dantzig(1949,1955) [8, 9].In mathematics, simplex is an alternative name used to refer to a convex polyhedron, i.e., a body offinite (hyper-)volume in two or more dimensions bounded by linear (hyper-)surfaces which inter-sect in linear edges and vertices. In general the feasible regions of linear programming problemsconstitute such simplexes. Since the optimal solutions for the independent variables of linearprogramming problems , when they exist, are always to be found at a vertex or along an edge ofsimplex feasible regions, Dantzig developed his so-called simplex algorithm such that it system-atically scans the edges and vertices of a feasible region to identify the optimal solution (when itexists) in as few steps as possible.The starting point be a standard maximum problem of linear programming with n indepen-dent variables in the form of relations (5.2)–(5.6). First, by introducing m non-negative slackvariables s , . . . , s m , one transforms the m linear restrictions (inequalities) into an equivalent setof m linear equations. In this way, potential differences between the left-hand and the right-handsides of the m inequalities are represented by the slack variables. In combination with the definingequation of the linear objective function z , one thus is confronted with a system of m linearalgebraic equations for the n + m variables z, x , . . . , x n , s , . . . , s m , given by Maximum problem of linear programming in canonical form z − c x − c x − . . . − c n x n = d (5.8) a x + a x + . . . + a n x n + s = b (5.9) a x + a x + . . . + a n x n + s = b (5.10)... a m x + a m x + . . . + a mn x n + s m = b m . (5.11)As discussed previously in Ch. 3, a system of linear algebraic equations of format (1 + m ) × (1 + n + m ) is under-determined and so, at most, allows for multiple solutions . The general.3. DANTZIG’SSIMPLEX ALGORITHM 37 (1 + n + m ) -dimensional solution vector x L = ( z L , x ,L , . . . , x n,L , s ,L , . . . , s m,L ) T (5.12)thus contains n variables the values of which can be chosen arbitrarily . It is very important to beaware of this fact. It implies that, given the linear system is solvable in the first place, one has a choice amongst different solutions, and so one can pick the solution which proves optimal for thegiven problem at hand. Dantzig’s simplex algorithm constitues a tool for determining such an optimal solution in a systematic way.Let us begin by transferring the coefficients and right-hand sides (RHS) of the under-determinedlinear system introduced above into a particular kind of simplex tableau . Initial simplex tableau z x x . . . x n s s . . . s m RHS − c − c . . . − c n . . . d a a . . . a n . . . b a a . . . a n . . . b ... ... ... . . . ... ... ... . . . ... ... a m a m . . . a mn . . . b m (5.13)In such a simplex tableau one distinguishes so-called basis variables from non-basis variables .Basis variables are those that contain in their respective columns in the number tableau a (1 + m ) -component canonical unit vector [cf. Eq. (1.10)]; in total the simplex tableau contains m ofthese. Non-basis variables are the remaining ones that do not contain a canonical basis vectorin their respective columns; there exist n of this kind. The complete basis can thus be perceivedas spanning a (1 + m ) -dimensional Euclidian space R m . Initially, always z and the m slackvariables s , . . . , s m constitute the basis variables, while the n independent variables x , . . . , x n classify as non-basis variables [cf. the initial tableau (5.13)]. The corresponding so-called (first) basis solution has the general appearance x B = ( z B , x ,B , . . . , x n,B , s ,B , . . . , s m,B ) T = ( d, , . . . , , b , . . . , b m ) T , since, for simplicity, each of the n arbitrarily specifiable non-basis variables may be assigned thespecial value zero. In this respect basis solutions will always be special solutions (as opposedto general ones) of the under-determined system (5.8)–(5.11) — the maximum problem of linearprogramming in canonical form.Central aim of the simplex algorithm is to bring as many of the n independent vari-ables x , . . . , x n as possible into the (1 + m ) -dimensional basis, at the expense of one of the m slack variables s , . . . , s m , one at a time, in order to construct successively more favourablespecial vector-valued solutions to the optimisation problem at hand. Ultimately, the simplex algo-rithm needs to be viewed as a special variant of Gaußian elimination as discussed in Ch. 3, with aset of systematic instructions concerning allowable equivalence transformations of the underlyingunder-determined linear system (5.8)–(5.11), resp. the initial simplex tableau (5.13). This set ofsystematic algebraic simplex operations can be summarised as follows: Simplex operations − c j ≥ for all j ∈ { , . . . , n } ? If so, then thecorresponding basis solution is optimal . END . Otherwise goto S2.S2: Choose a pivot column index j ∗ ∈ { , . . . , n } such that − c j ∗ := min {− c j | j ∈{ , . . . , n }} < .S3: Is there a row index i ∗ ∈ { , . . . , m } such that a i ∗ j ∗ > ? If not, the objective function z isunbounded from above. END . Otherwise goto S4.S4: Choose a pivot row index i ∗ such that a i ∗ j ∗ > and b i ∗ /a i ∗ j ∗ := min { b i /a i ∗ j ∗ | a i ∗ j ∗ > , i ∈{ , . . . , m }} . Perform a pivot operation with the pivot element a i ∗ j ∗ . Goto S1.When the final simplex tableau has been arrived at, one again assigns the non-basis variables thevalue zero. The values of the final basis variables corresponding to the optimal solution of thegiven linear programming problem are then to be determined from the final simplex tableau bybackward substitution, beginning at the bottom row. Note that slack variables with positive valuesbelonging to the basis variables in the optimal solution provide immediate information on existingremaining capacities in the problem at hand. hapter 6Elementary financial mathematics In this chapter we want to provide a brief introduction into some basic concepts of financial math-ematics . As we will try to emphasise, many applications of these concepts (that have immediatepractical relevance) are founded on only two simple and easily accessible mathematical structures:the so-called arithmetical and geometrical real-valued sequences and their associated finite series. An arithmetical sequence of n ∈ N real numbers a n ∈ R , ( a n ) n ∈ N , is defined by the property that the difference d between neighbouring elements in the sequence be constant , i.e., for n > a n − a n − =: d = constant = 0 , (6.1)with a n , a n − , d ∈ R . Given this recursive formation rule, one may infer the explicit representa-tion of an arithmetical sequence as a n = a + ( n − d with n ∈ N . (6.2)Note that any arithmetical sequence is uniquely determined by the two free parameters a and d ,the starting value of the sequence and the constant difference between neighbours in the sequence,respectively. Equation (6.2) shows that the elements a n in a non-trivial arithmetical sequence exhibit either linear growth or linear decay with n .When one calculates for an arithmetical sequence of n + 1 real numbers the arithmetical mean of the immediate neighbours of any particular element a n (with n ≥ ), one finds that
12 ( a n − + a n +1 ) = 12 ( a + ( n − d + a + nd ) = a + ( n − d = a n . (6.3)390 CHAPTER 6. ELEMENTARYFINANCIAL MATHEMATICSSummation of the first n elements of an arbitrary arithmetical sequence of real numbers leads toa finite arithmetical series , S n := a + a + . . . + a n = n X k =1 a k = n X k =1 [ a + ( k − d ] = na + d n − n . (6.4)In the last algebraic step use was made of the Gaußian identity (cf., e.g., Bosch (2003) [6, p 21]) n − X k =1 k ≡
12 ( n − n . (6.5) A geometrical sequence of n ∈ N real numbers a n ∈ R , ( a n ) n ∈ N , is defined by the property that the quotient q between neighbouring elements in the sequence be constant , i.e., for n > a n a n − =: q = constant = 0 , (6.6)with a n , a n − ∈ R and q ∈ R \{ , } . Given this recursive formation rule, one may infer the explicit representation of a geometrical sequence as a n = a q n − with n ∈ N . (6.7)Note that any geometrical sequence is uniquely determined by the two free parameters a and q ,the starting value of the sequence and the constant quotient between neighbours in the sequence,respectively. Equation (6.7) shows that the elements a n in a non-trivial geometrical sequence exhibit either exponential growth or exponential decay with n (cf. Sec. 7.1.4).When one calculates for a geometrical sequence of n + 1 real numbers the geometrical mean ofthe immediate neighbours of any particular element a n (with n ≥ ), one finds that √ a n − · a n +1 = p a q n − · a q n = a q n − = a n . (6.8)Summation of the first n elements of an arbitrary geometrical sequence of real numbers leads toa finite geometrical series , S n := a + a + . . . + a n = n X k =1 a k = n X k =1 (cid:2) a q k − (cid:3) = a n − X k =0 q k = a q n − q − . (6.9)In the last algebraic step use was made of the identity (cf., e.g., Bosch (2003) [6, p 27]) n − X k =0 q k ≡ q n − q − for q ∈ R \{ , } . (6.10) Analogously, the modified Gaußian identity n X k =1 (2 k − ≡ n applies. .2. INTEREST ANDCOMPOUND INTEREST 41 Let us consider a first rather simple interest model. Suppose given an initial capital of positivevalue K > paid into a bank account at some initial instant, and a time interval consistingof n ∈ N periods of equal lengths. At the end of each period, the money in this bank accountshall earn a service fee corresponding to an interest rate of p > percent. Introducing thedimensionless interest factor q := 1 + p > , (6.11)one finds that by the end of the first interest period a total capital of value (in CU) K = K + K · p
100 = K (cid:16) p (cid:17) = K q will have accumulated. When the entire time interval of n interest periods has ended, a finalcapital worth of (in CU) recursively: K n = K n − q , n ∈ N , (6.12)will have accumulated, where K n − denotes the capital (in CU) accumulated by the end of n − interest periods. This recursive representation of the growth of the initial capital K due to a totalof n interest payments and the effect of compound interest makes explicit the direct link with themathematical structure of a geometrical sequence of real numbers (6.6).It is a straightforward exercise to show that in this simple interest model the final capital K n isrelated to the initial capital K byexplicitly: K n = K q n , n ∈ N . (6.13)Note that this equation links the four non-negative quantities K n , K , q and n to one another.Hence, knowing the values of three of these quantities, one may solve Eq. (6.13) to obtain thevalue of the fourth. For example, solving Eq. (6.13) for K yields K = K n q n =: B . (6.14)In this particular variant, K is referred to as the present value B of the final capital K n ; this isobtained from K n by an n -fold division with the interest factor q .Further possibilities of re-arranging Eq. (6.13) are:(i) Solving for the interest factor q : q = n r K n K , (6.15)(ii) Solving for the contract period n : n = ln ( K n /K )ln( q ) . (6.16) Inverting this defining relation for q leads to p = 100 · ( q − . n ∈ N shall denote the number of full years that have passed in a specific interestmodel.Now we turn to discuss a second, more refined interest model. Let us suppose that an initialcapital K > earns interest during one full year m ∈ N times at the m th part of a nominalannual interest rate p nom > . At the end of the first out of m periods of equal length /m , theinitial capital K will thus have increased to an amount K /m = K + K · p nom m ·
100 = K (cid:16) p nom m · (cid:17) . By the end of the k th ( k ≤ m ) out of m periods the account balance will have become K k/m = K (cid:16) p nom m · (cid:17) k ; the interest factor (cid:16) p nom m · (cid:17) will then have been applied k times to K . At the end of the fullyear, K in this interest model will have increased to K = K m/m = K (cid:16) p nom m · (cid:17) m , m ∈ N . This relation defines an effective interest factor q eff := (cid:16) p nom m · (cid:17) m , (6.17)with associated effective annual interest rate p eff = 100 · h(cid:16) p nom m · (cid:17) m − i , m ∈ N , (6.18)obtained from re-arranging q eff = 1 + p eff .When, ultimately, n ∈ N full years will have passed in the second interest model, the initialcapital K will have been transformed into a final capital of value K n = K (cid:16) p nom m · (cid:17) n · m = K q n eff , n, m ∈ N . (6.19)The present value B of K n is thus given by B = K n q n eff = K . (6.20)Finally, as a third interest model relevant to applications in Finance , we turn to consider the con-cept of installment savings . For simplicity, let us restrict our discussion to the case when n ∈ N equal installments of constant value E > are paid into an account that earns p > percentannual interest (i.e., q > ) at the beginning of each of n full years. The initial account balance.3. REDEMPTION PAYMENTSIN CONSTANT ANNUITIES 43be K = 0 CU . At the end of a first full year in this interest model, the account balance will haveincreased to K = E + E · p
100 = E (cid:16) p (cid:17) = Eq .
At the end of two full years one finds, substituting for K , K = ( K + E ) q = ( Eq + E ) q = E ( q + q ) = Eq ( q + 1) . At the end of n full years we have, recursively substituting for K n − , K n − , etc., K n = ( K n − + E ) q = · · · = E ( q n + . . . + q + q ) = Eq ( q n − + . . . + q + 1) = Eq n − X k =0 q k . Using the identity (6.10), since presently q > , the account balance at the end of n full years canbe reduced to the expression K n = Eq q n − q − , q ∈ R > , n ∈ N . (6.21)The present value B associated with K n is obtained by n -fold division of K n with the interestfactor q : B := K n q n Eq. . z}|{ = = E ( q n − q n − ( q − . (6.22)This gives the value of an initial capital B which will grow to the same final value K n after n annual interest periods with constant interest factor q > .Lastly, re-arranging Eq. (6.21) to solve for the contract period n yields. n = ln [1 + ( q − K n /Eq )]ln( q ) . (6.23) The starting point of the next discussion be a mortgage loan of amount R > that an economic agent borrowed from a bank at the obligation of annual service payments of p > percent (i.e., q > ) on the remaining debt . We suppose that the contract between the agent andthe bank fixes the following conditions:(i) the first redemption payment T amount to t > percent of the mortgage R ,(ii) the remaining debt shall be paid back to the bank in constant annuities of value A > at the end of each full year that has passed.The annuity A is defined as the sum of the variable n th interest payment Z n > and thevariable n th redemption payment T n > . In the present model we impose on the annuitythe condition that it be constant across full years, A = Z n + T n ! = constant . (6.24)4 CHAPTER 6. ELEMENTARYFINANCIAL MATHEMATICSFor n = 1 , for example, we thus obtain A = Z + T = R · p
100 + R · t
100 = R (cid:18) p + t (cid:19) = R (cid:20) ( q −
1) + t (cid:21) ! = constant . (6.25)For the first full year of a running mortgage contract, the interest payment, the redemption payment,and, following the payment of a first annuity, the remaining debt take the values Z = R · p
100 = R ( q − T = A − Z R = R + Z − A substitute for Z z}|{ = R + R · p − A = R q − A .
By the end of a second full year, these become Z = R ( q − T = A − Z R = R + Z − A substitute for Z z}|{ = R q − A substitute for R z}|{ = R q − A ( q + 1) . At this stage, it has become clear according to which patterns the different quantities involved inthe redemption payment model need to be formed. The interest payment for the n th full year in amortgage contract of constant anuities amounts to (recursively) Z n = R n − ( q − , n ∈ N , (6.26)where R n − denotes the remaining debt at the end of the previous full year. The redemptionpayment for full year n is then given by (recursively) T n = A − Z n , n ∈ N . (6.27)The remaining debt at the end of the n th full year then is (in CU)recursively: R n = R n − + Z n − A = R n − q − A , n ∈ N . (6.28)By successive backward substitution for R n − , R n − , etc., R n can be re-expressed as R n = R q n − A ( q n − + . . . + q + 1) = R q n − A n − X k =0 q k . Now employing the identity (6.10), we finally obtain (since q > )explicitly: R n = R q n − A q n − q − , n ∈ N . (6.29)All the formulae we have now derived for computing the values of the quantities { n, Z n , T n , R n } form the basis of a formal redemption payment plan , given by.3. REDEMPTION PAYMENTSIN CONSTANT ANNUITIES 45 n Z n [CU] T n [CU] R n [CU] – – R Z T R Z T R ... ... ... ... ,a standard scheme that banks must make available to their mortgage customers for the purpose offinancial orientation. Remark:
For known values of the free parameters R > , q > and A > , the simplerecursive formulae (6.26), (6.27) and (6.28) can be used to implement a redemption payment planin a modern spreadsheet programme such as EXCEL or OpenOffice.We emphasise the following observation concerning Eq. (6.29): since the constant annuity A con-tains implicitly a factor ( q − [cf. Eq. (6.25)], the two competing terms in this relation each growexponentially with n . For the redemption payments to eventually terminate, it is thus essential tofix the free parameter t (for known p > ⇔ q > ) in such a way that the second term on theright-hand side of Eq. (6.29) is given the possibility to catch up with the first as n progresses (thelatter of which has a head start of R > at n = 0 ). The necessary condition following fromthe requirement that R n ! ≤ R n − is thus t > .Equation (6.29) links the five non-negative quantities R n , R , q , n and A to one another. Givenone knows the values of four of these, one can solve for the fifth. For example:(i) Calculation of the contract period n of a mortgage contract, knowing the mortgage R , theinterest factor q and the annuity A . Solving the condition R n ! = 0 imposed on R n for n yields (after a few algebraic steps) n = ln (cid:0) pt (cid:1) ln( q ) ; (6.30)the contract period is thus independent of the value of the mortgage loan, R .(ii) Evaluation of the annuity A , knowing the contract period n , the mortgage loan R , and theinterest factor q . Solving the condition R n ! = 0 imposed on R n for A immediately yields A = q n ( q − q n − R . (6.31)Now equating the two expressions (6.31) and (6.25) for the annuity A , one finds in additionthat t
100 = q − q n − . (6.32)6 CHAPTER 6. ELEMENTARYFINANCIAL MATHEMATICS Quantitative models for pension calculations assume given an initial capital K > thatwas paid into a bank account at a particular moment in time. The issue is to monitor the subse-quent evolution in discrete time n of the account balance K n (in CU), which is subjected to twocompeting influences: on the one-hand side, the bank account earns interest at an annual interestrate of p > percent (i.e., q > ), on the other, it is supposed that throughout one full year a totalof m ∈ N pension payments of the constant amount a are made from this bank account, alwaysat the beginning of each of m intervals of equal duration per year.Let us begin by evaluating the amount of interest earned per year by the bank account. An impor-tant point in this respect is the fact that throughout one full year there is a total of m deductions ofvalue a from the bank account, i.e., in general the account balance does not stay constant through-out that year but rather decreases in discrete steps. For this reason, the account is credited by thebank with interest only at the m th part of p > percent for each interval (out of the total of m )that has passed, with no compound interest effect. Hence, at the end of the first out of m intervalsper year the bank account has earned interest worth of (in CU) Z /m = ( K − a ) · pm ·
100 = ( K − a ) ( q − m . The interest earned for the k th interval (out of m ; k ≤ m ) is then given by Z k/m = ( K − ka ) ( q − m . Summation over the contributions of each of the m intervals to the interest earned then yields forthe entire interest earned during the first full year (in CU) Z = m X k =1 Z k/m = m X k =1 ( K − ka ) ( q − m = ( q − m " mK − a m X k =1 k . By means of substitution from the identity (6.5), this result can be recast into the equivalent form Z = (cid:20) K −
12 ( m + 1) a (cid:21) ( q − . (6.33)Note that this quantity decreases linearly with the number of deductions m made per year resp. withthe pension payment amount a .One now finds that the account balance at the end of the first full year that has passed is given by K = K − ma + Z Eq. ( . ) z}|{ = K q − (cid:20) m + 12 ( m + 1)( q − (cid:21) a . At the end of a second full year of the pension payment contract the interest earned is Z = (cid:20) K −
12 ( m + 1) a (cid:21) ( q − , .4. PENSION CALCULATIONS 47while the account balance amounts to K = K − ma + Z substitute for K and Z z}|{ = K q − (cid:20) m + 12 ( m + 1)( q − (cid:21) a ( q + 1) . At this stage, certain fairly simple patterns for the interest earned during full year n , and the account balance after n full years, reveal themselves. For Z n we have Z n = (cid:20) R n − −
12 ( m + 1) a (cid:21) ( q − , (6.34)and for K n one obtains K n = K n − − ma + Z n substitute for K n − and Z n z}|{ = K q n − (cid:20) m + 12 ( m + 1)( q − (cid:21) a n − X k =0 q k . The latter result can be re-expressed upon substitution from the identity (6.10). Thus, K n canfinally be given byexplicitly: K n = K q n − (cid:20) m + 12 ( m + 1)( q − (cid:21) a q n − q − , n, m ∈ N . (6.35)In a fashion practically identical to our discussion of the redemption payment model in Sec. 6.3,the two competing terms on the right-hand side of Eq. (6.35) likewise exhibit exponential growthwith the number n of full years passed. Specifically, it depends on the values of the parameters K > , q > , a > , as well as m ≥ , whether the second term eventually manages tocatch up with the first as n progresses (the latter of which, in this model, is given a head start ofvalue K > at n = 0 ).We remark that Eq. (6.35), again, may be algebraically re-arranged at one’s convenience (as longas division by zero is avoided). For example:(i) The duration n (in full years) of a particular pension contract is obtained from solving thecondition K n ! = 0 accordingly. Given that [ . . . ] a − K ( q − > , one thus finds n = ln (cid:16) [ ... ] a [ ... ] a − K ( q − (cid:17) ln( q ) . (6.36)(ii) The present value B of a pension scheme results from the following consideration: forfixed interest factor q > , which initial capital K > must be paid into a bank accountsuch that for a duration of n full years one can receive payments of constant amount a at thebeginning of each of m intervals (of equal length) per year? The value of B = K is againobtained from imposing on Eq. (6.35) the condition K n ! = 0 and solving for K . This yields B = K = (cid:20) m + 12 ( m + 1)( q − (cid:21) a q n − q n ( q − . (6.37) To avoid notational overload, the brackets [ . . . ] here represent the term (cid:2) m + ( m + 1)( q − (cid:3) . everlasting pension payments of amount a ever > is based on thestrategy to consume only the annual interest earned by an initial capital K > residingin a bank account with interest factor q > . Imposing now on Eq. (6.35) the condition K n ! = K to hold for all values of n , and then solving for a , yields the result a ever = q − m + ( m + 1)( q + 1) K ; (6.38)Note that, naturally, a ever is directly proportional to the initial capital K ! Attempts at the quantitative description of the process of declining material value of industrialgoods, properties or other assets, of initial value K > , are referred to as depreciation . In-ternational tax laws generally provide investors with a choice between two particular mathematicalmethods of calculating depreciation . We will discuss these options in turn. When the initial value K > is supposed to decline to in the space of N full years byequal annual amounts, the remaining value R n (in CU) at the end of n full years is described by R n = K − n (cid:18) K N (cid:19) , n = 1 , . . . , N . (6.39)Note that for the difference of remaining values for years adjacent one obtains R n − R n − = − (cid:18) K N (cid:19) =: d < . The underlying mathematical structure of the straight line depreciationmethod is thus an arithmetical sequence of real numbers, with constant negative difference d between neighbouring elements (cf. Sec. 6.1.1). The foundation of the second depreciation method to be described here, for an industrial good of initial value K > , is the idea that per year the value declines by a certain percentage rate p > of the value of the good during the previous year. Introducing a dimensionless depreciationfactor by q := 1 − p < , (6.40)the remaining value R n (in CU) after n full years amounts torecursively: R n = R n − q , R ≡ K , n ∈ N . (6.41)The underlying mathematical structure of the declining balance depreciation method is thus a geometrical sequence of real numbers, with constant ratio < q < between neighbouring.6. SUMMARISING FORMULA 49elements (cf. Sec. 6.1.2). With increasing n the values of these elements become ever smaller. Bymeans of successive backward substitution expression (6.41) can be transformed toexplicitly: R n = K q n , < q < , n ∈ N . (6.42)From Eq. (6.42), one may derive results concerning the following questions of a quantitative na-ture:(i) Suppose given a depreciation factor q and a projected remaining value R n for some industrialgood. After which depreciation period n will this value be attained? One finds n = ln ( R n /K )ln( q ) . (6.43)(ii) Knowing a projected depreciation period n and corresponding remaining value R n , at which percentage rate p > must the depreciation method be operated? This yields q = n r R n K ⇒ p = 100 · − n r R n K ! . (6.44) To conclude this chapter, let us summarise the results on elementary financial mathematics thatwe derived along the way. Remarkably, these can be condensed in a single formula which containsthe different concepts discussed as special cases. This formula, in which n represents the numberof full years that have passed, is given by (cf. Zeh–Marschke (2010) [26]): K n = K q n + R q n − q − , q ∈ R > \{ } , n ∈ N . (6.45)The different special cases contained therein are:(i) Compound interest for an initial capital K > : with R = 0 and q > , Eq. (6.45)reduces to Eq. (6.13).(ii) Installment savings with constant installments
E > : with K = 0 CU , q > and R = Eq , Eq. (6.45) reduces to Eq. (6.19).(iii) Redemption payments in constant annuities : with K = − R < , q > and R = A > , Eq. (6.45) reduces to the negative (!) of Eq. (6.29). In this dual formulation,remaining debt K n = − R n is (meaningfully) expressed as a negative account balance.(iv) Pension payments of constant amount a > : with q > and R = − (cid:20) m + 12 ( m + 1)( q − (cid:21) a , Eq. (6.45) transforms to Eq. (6.35).(v) Declining balance depreciation of an asset of initial value K > : with R = 0 and < q < , Eq. (6.45) converts to Eq. (6.42) for the remaining value K n = R n .0 CHAPTER 6. ELEMENTARYFINANCIAL MATHEMATICS hapter 7Differential calculus of real-valuedfunctions of one real variable In Chs. 1 to 5 of these lecture notes, we confined our considerations to functional relationshipsbetween
INPUT quantities and
OUTPUT quantities of a linear nature. In this chapter now, weturn to discuss characteristic properties of truly non-linear functional relationships between one
INPUT quantity and one
OUTPUT quantity . Let us begin by introducing the concept of a real-valued function of one real variable . Thisconstitutes a special kind of a mapping that needs to satisfy the following simple but very strictrule:a mapping f that assigns to every element x from a subset D of the real numbers R (i.e., D ⊆ R ) one and only one element y from a second subset W of the real numbers R (i.e., W ⊆ R ). Def.: A unique mapping f of a subset D ⊆ R of the real numbers onto a subset W ⊆ R of thereal numbers, f : D → W , x y = f ( x ) (7.1)is referred to as a real-valued function of one real variable .We now fix some terminology concerning the concept of a real-valued function of one real variable: • D : domain of f , • W : target space of f , • x ∈ D : independent variable of f , also referred to as the argument of f , • y ∈ W : dependent variable of f , Cf. our introduction in Ch. 2 of matrices as a particular class of mathematical objects.
512 CHAPTER 7. DIFFERENTIAL CALCULUS OFREAL-VALUED FUNCTIONS • f ( x ) : mapping prescription , • graph of f : the set of pairs of values G = { ( x, f ( x )) | x ∈ D } ⊆ R .For later analysis of the mathematical properties of real-valued functions of one real variable, weneed to address a few more technical issues. Def.:
Given a mapping f that is one-to-one and onto , with domain D ( f ) ⊆ R and target space W ( f ) ⊆ R , not only is every x ∈ D ( f ) assigned to one and only one y ∈ W ( f ) , but also every y ∈ W ( f ) is assigned to one and only one x ∈ D ( f ) . In this case, there exists an associatedmapping f − , with D ( f − ) = W ( f ) and W ( f − ) = D ( f ) , which is referred to as the inversefunction of f . Def.:
A real-valued function f of one real variable x is continuous at some value x ∈ D ( f ) whenfor ∆ x ∈ R > the condition lim ∆ x → f ( x − ∆ x ) = lim ∆ x → f ( x + ∆ x ) = f ( x ) (7.2)obtains, i.e., when at x the left and right limits of the function f coincide and are equal to the value f ( x ) . A real-valued function f as such is continuous when f is continuous for all x ∈ D ( f ) . Def.:
When a real-valued function f of one real variable x satisfies the condition f ( a ) < f ( b ) for all a, b ∈ D ( f ) with a < b , (7.3)then f is called strictly monotonously increasing . When, however, f satisfies the condition f ( a ) > f ( b ) for all a, b ∈ D ( f ) with a < b , (7.4)then f is called strictly monotonously decreasing .Note, in particular, that real-valued functions of one real variable that are strictly monotonous andcontinous are always one-to-one and onto and, therefore, are invertible.In the following, we briefly review five elementary classes of real-valued functions of one real vari-able that find frequent application in the modelling of quantitative problems in economic theory . n Polynomials of degree n are real-valued functions of one real variable of the form y = f ( x ) = a n x n + a n − x n − + . . . + a i x i + . . . + a x + a x + a with a i ∈ R , i = 1 , . . . , n, n ∈ N , a n = 0 . (7.5)Their domain comprises the entire set of real numbers, i.e., D ( f ) = R . The extent of their targetspace depends specifically on the values of the real constant coefficients a i ∈ R . Functions in thisclass possess a maximum of n real roots ..1. REAL-VALUEDFUNCTIONS 53 Rational functions are constructed by forming the ratio of two polynomials of degrees m resp. n ,i.e., y = f ( x ) = p m ( x ) q n ( x ) = a m x m + . . . + a x + a b n x n + . . . + b x + b with a i , b j ∈ R , i = 1 , . . . , m, j = 1 , . . . , n, m, n ∈ N , a m , b n = 0 . (7.6)Their domain is given by D ( f ) = R \{ x | q n ( x ) = 0 } . When for the degrees m and n of thepolynomials p m ( x ) and q n ( x ) we have(i) m < n , then f is referred to as a proper rational function , or(ii) m ≥ n , then f is referred to as an improper rational function .In the latter case, application of polynomial division leads to a separation of f into a purely poly-nomial part and a proper rational part. The roots of f always correspond to those roots of thenumerator polynomial p m ( x ) for which simultaneously q n ( x ) = 0 applies. The roots of the de-nominator polynomial q n ( x ) constitute poles of f . Proper rational functions always tend for verysmall (i.e., x → −∞ ) and for very large (i.e., x → + ∞ ) values of their argument to zero. Power-law functions exhibit the specific structure given by y = f ( x ) = x α with α ∈ R . (7.7)We here confine ourselves to cases with domains D ( f ) = R > , such that for the corespondingtarget spaces we have W ( f ) = R > . Under these conditions, power-law functions are strictlymonotonously increasing when α > , and strictly monotonously decreasing when α < . Hence,they are inverted by y = α √ x = x /α . There do not exist any roots under the conditions stated here. Exponential functions have the general form y = f ( x ) = a x with a ∈ R > \{ } . (7.8)Their domain is D ( f ) = R , while their target space is W ( f ) = R > . They exhibit strictmonotonous increase for a > , and strict monotonous decrease for < a < . Hence, theytoo are invertible. Their y -intercept is generally located at y = 1 . For a > , exponential functionsare also known as growth functions .Special case: When the constant (!) base number is chosen to be a = e , where e denotes theirrational Euler’s number (according to the Swiss mathematician Leonhard Euler, 1707–1783)defined by the infinite series e := ∞ X k =0 k ! = 10! + 11! + 12! + 13! + . . . , natural exponential function y = f ( x ) = e x =: exp( x ) . (7.9)In analogy to the definition of e , the relation e x = exp( x ) = ∞ X k =0 x k k ! = x
0! + x
1! + x
2! + x
3! + . . . applies.
Logarithmic functions, denoted by y = f ( x ) = log a ( x ) with a ∈ R > \{ } , (7.10)are defined as inverse functions of the strictly monotonous exponential functions y = f ( x ) = a x — and vice versa. Correspondingly, D ( f ) = R > and W ( f ) = R apply. Strictly monotonouslyincreasing behaviour is given when a > , strictly monotonously decreasing behaviour when < a < . In general, the x -intercept is located at x = 1 .Special case: The natural logarithmic function (lat.: logarithmus naturalis) obtains when theconstant basis number is set to a = e . This yields y = f ( x ) = log e ( x ) := ln( x ) . (7.11) Real-valued functions from all five categories considered in the previous sections may be combinedarbitrarily (respecting relevant computational rules), either via the four fundamental arithmeticaloperations , or via concatenations . Theorem:
Let real-valued functions f and g be continuous on domains D ( f ) resp. D ( g ) . Thenthe combined real-valued functions1. sum/difference f ± g , where ( f ± g )( x ) := f ( x ) ± g ( x ) with D ( f ) ∩ D ( g ) ,2. product f · g , where ( f · g )( x ) := f ( x ) g ( x ) with D ( f ) ∩ D ( g ) ,3. quotient fg , where (cid:18) fg (cid:19) ( x ) := f ( x ) g ( x ) with g ( x ) = 0 and D ( f ) ∩ D ( g ) \{ x | g ( x ) = 0 } ,4. concatenation f ◦ g , where ( f ◦ g )( x ) := f ( g ( x )) mit { x ∈ D ( g ) | g ( x ) ∈ D ( f ) } ,are also continuous on the respective domains..2. DERIVATION OFDIFFERENTIABLE REAL-VALUEDFUNCTIONS 55 The central theme of this chapter is the mathematical description of the local variability of con-tinuous real-valued function of one real variable, f : D ⊆ R → W ⊆ R . To this end, letus consider the effect on f of a small change of its argument x . Supposing we affect a change x → x + ∆ x , with ∆ x ∈ R , what are the resultant consequences for f ? We immediately find that y → y + ∆ y = f ( x + ∆ x ) , with ∆ y ∈ R , obtains. Hence, a prescribed change of the argument x by a (small) value ∆ x triggers in f a change by the amount ∆ y = f ( x + ∆ x ) − f ( x ) . It is ofgeneral quantitative interest to compare the sizes of these two changes. This is accomplished byforming the respective difference quotient ∆ y ∆ x = f ( x + ∆ x ) − f ( x )∆ x . It is then natural, for given f , to investigate the limit behaviour of this difference quotient as thechange ∆ x of the argument of f is made successively smaller. Def.:
A continuous real-valued function f of one real variable is called differentiable at x ∈ D ( f ) , when for arbitrary ∆ x ∈ R the limit f ′ ( x ) := lim ∆ x → ∆ y ∆ x = lim ∆ x → f ( x + ∆ x ) − f ( x )∆ x (7.12)exists and is unique. When f is differentiable for all x ∈ D ( f ) , then f as such is referred to asbeing differentiable .The existence of this limit in a point ( x, f ( x )) for a real-valued function f requires that the latterexhibits neither “jumps” nor “kinks,” i.e., that at ( x, f ( x )) the function is sufficiently “smooth.”The quantity f ′ ( x ) is referred to as the first derivative of the (differentiable) function f at posi-tion x . It provides a quantitative measure for the local rate of change of the function f in thepoint ( x, f ( x )) . In general one interprets the first derivative f ′ ( x ) as follows: an increase of theargument x of a differentiable real-valued function f by (one) unit leads to a change in the valueof f by approximately f ′ ( x ) · units.Alternative notation for the first derivative of f : f ′ ( x ) ≡ d f ( x )d x . The differential calculus was developed in parallel with the integral calculus (see Ch. 7) during thesecond half of the th Century, independent of one another by the English physicist, mathemat-iccian, astronomer and philosopher Sir Isaac Newton (1643–1727) and the German philosopher,mathematician and physicist Gottfried Wilhelm Leibniz (1646–1716).Via the first derivative of a differentiable function f at an argument x ∈ D ( f ) , i.e., f ′ ( x ) , onedefines the so-called linearisation of f in a neighbourhood of x . The equation describing theassociated tangent to f in the point ( x , f ( x )) is given by y = f ( x ) + f ′ ( x )( x − x ) . (7.13)6 CHAPTER 7. DIFFERENTIAL CALCULUS OFREAL-VALUED FUNCTIONS GDC:
Local values f ′ ( x ) of first derivatives can be computed for given function f in mode CALC using the interactive routine dy/dx .The following rules of differentiation apply for the five families of elementary real-valued functionsdiscussed in Sec. 7.1, as well as concatenations thereof:
Rules of differentiation ( c ) ′ = 0 for c = constant ∈ R ( constants )2. ( x ) ′ = 1 ( linear function )3. ( x n ) ′ = nx n − for n ∈ N ( natural power-law functions )4. ( x α ) ′ = αx α − for α ∈ R and x ∈ R > ( general power-law functions )5. ( a x ) ′ = ln( a ) a x for a ∈ R > \{ } ( exponential functions )6. ( e ax ) ′ = ae ax for a ∈ R ( natural exponential functions )7. (log a ( x )) ′ = 1 x ln( a ) for a ∈ R > \{ } and x ∈ R > ( logarithmic functions )8. (ln( x )) ′ = 1 x for x ∈ R > ( natural logarithmic function ).For differentiable real-valued functions f and g it holds that:1. ( cf ( x )) ′ = cf ′ ( x ) for c = constant ∈ R ( f ( x ) ± g ( x )) ′ = f ′ ( x ) ± g ′ ( x ) ( summation rule )3. ( f ( x ) g ( x )) ′ = f ′ ( x ) g ( x ) + f ( x ) g ′ ( x ) ( product rule )4. (cid:18) f ( x ) g ( x ) (cid:19) ′ = f ′ ( x ) g ( x ) − f ( x ) g ′ ( x )( g ( x )) ( quotient rule )5. (( f ◦ g )( x )) ′ = f ′ ( g ) | g = g ( x ) · g ′ ( x ) ( chain rule )6. (ln( f ( x ))) ′ = f ′ ( x ) f ( x ) for f ( x ) > ( logarithmic differentiation )7. ( f − ( x )) ′ = 1 f ′ ( y ) (cid:12)(cid:12)(cid:12)(cid:12) y = f − ( x ) , if f is one-to-one and onto.( differentiation of inverse functions ).The methods of differential calculus just introduced shall now be employed to describe the localchange behaviour of a few simple examples of functions in economic theory , and also to deter-mine their local extremal values. The following section provides an overview of such frequentlyoccurring economic functions ..3. COMMON FUNCTIONSIN ECONOMIC THEORY 57 total cost function K ( x ) ≥ (dim: CU)argument: level of physical output x ≥ (dim: units)2. marginal cost function K ′ ( x ) > (dim: CU/unit)argument: level of physical output x ≥ (dim: units)3. average cost function K ( x ) /x > (dim: CU/unit)argument: level of physical output x > (dim: units)4. unit price function p ( x ) ≥ (dim: CU/unit)argument: level of physical output x > (dim: units)5. total revenue function E ( x ) := xp ( x ) ≥ (dim: CU)argument: level of physical output x > (dim: units)6. marginal revenue function E ′ ( x ) = xp ′ ( x ) + p ( x ) (dim: CU/unit)argument: level of physical output x > (dim: units)7. profit function G ( x ) := E ( x ) − K ( x ) (dim: CU)argument: level of physical output x > (dim: units)8. marginal profit function G ′ ( x ) := E ′ ( x ) − K ′ ( x ) = xp ′ ( x ) + p ( x ) − K ′ ( x ) (dim:CU/unit)argument: level of physical output x > (dim: units)9. utility function U ( x ) (dim: case dependent)argument: material wealth, opportunity, action x (dim: case dependent)The fundamental concept of a utility function as a means to capture in quantitative termsthe psychological value (happiness) assigned by an economic agent to a certain amountof money, or to owning a specific good, was introduced to economic theory in 1738 bythe Swiss mathematician and physicist Daniel Bernoulli FRS (1700–1782); cf. Bernoulli(1738) [4]. The utility function is part of the folklore of the theory, and often taken to bea piecewise differentiable, right-handedly curved (concave) function, i.e., U ′′ ( x ) < , onthe grounds of the assumption of diminishing marginal utility (happiness) with increasingmaterial wealth.10. economic efficiency W ( x ) := E ( x ) /K ( x ) ≥ (dim: 1)argument: level of physical output x > (dim: units)11. demand function N ( p ) ≥ , monotonously decreasing (dim: units)argument: unit price p , ( ≤ p ≤ p max ) (dim: CU/unit)12. supply function A ( p ) ≥ , monotonously increasing (dim: units)argument: unit price p , ( p min ≤ p ) (dim: CU/units).8 CHAPTER 7. DIFFERENTIAL CALCULUS OFREAL-VALUED FUNCTIONSA particularly prominent example of a real-valued economic function of one real variableconstitutes the psychological value function , devised by the Israeli–US-American experi-mental psychologists Daniel Kahneman and Amos Tversky (1937–1996) in the context oftheir Prospect Theory (a pillar of
Behavioural Economics ), which was later awarded aSveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel in 2002 (cf. Kahne-man and Tversky (1979) [18, p 279], and Kahneman (2011) [17, p 282f]). A possible representa-tion of this function is given by the piecewise description v ( x ) = a log (1 + x ) for x ∈ R ≥ − a log (1 − x ) for x ∈ R < , (7.14)with parameter a ∈ R > . Overcoming a conceptual problem of Bernoulli’s utility function, here,in contrast, the argument x quantifies a change in wealth (or welfare) with respect to some givenreference point (rather than a specific value of wealth itself). Before we turn to discuss applications of differential calculus to simple quantitative problemsin economic theory , we briefly summarize the main steps of curve sketching for a real-valuedfunction of one real variable, also referred to as analysis of the properties of differentiability of areal-valued function.1. domain : D ( f ) = { x ∈ R | f ( x ) is regular } symmetries : for all x ∈ D ( f ) , is(i) f ( − x ) = f ( x ) , i.e., is f even , or(ii) f ( − x ) = − f ( x ) , i.e., is f odd , or(iii) f ( − x ) = f ( x ) = − f ( x ) , i.e., f exhibits no symmetries ?3. roots : identify all x N ∈ D ( f ) that satisfy the condition f ( x ) ! = 0 .4. local extremal values :(i) local minima of f exist at all x E ∈ D ( f ) , for which thenecessary condition f ′ ( x ) ! = 0 , and thesufficient condition f ′′ ( x ) ! > are satisfied simultaneously.(ii) local maxima of f exist at all x E ∈ D ( f ) , for which thenecessary condition f ′ ( x ) ! = 0 , and thesufficient condition f ′′ ( x ) ! < are satisfied simultaneously..5. ANALYTICINVESTIGATIONSOFECONOMIC FUNCTIONS 595. points of inflection : find all x W ∈ D ( f ) , for which thenecessary condition f ′′ ( x ) ! = 0 , and thesufficient condition f ′′′ ( x ) ! = 0 are satisfied simultaneously.6. monotonous behaviour :(i) f is monotonously increasing for all x ∈ D ( f ) with f ′ ( x ) > (ii) f is monotonously decreasing for all x ∈ D ( f ) with f ′ ( x ) < local curvature :(i) f behaves left-handedly curved for x ∈ D ( f ) with f ′′ ( x ) > (ii) f behaves right-handedly curved for x ∈ D ( f ) with f ′′ ( x ) < asymptotic behaviour :asymptotes to f are constituted by(i) straight lines y = ax + b with the property lim x → + ∞ [ f ( x ) − ax − b ] = 0 or lim x →−∞ [ f ( x ) − ax − b ] = 0 (ii) straight lines x = x at poles x / ∈ D ( f ) range : W ( f ) = { y ∈ R | y = f ( x ) } . According to the law of diminishing returns , which was introduced to economic theory by theFrench economist and statesman Anne Robert Jacques Turgot (1727–1781) and also by the Ger-man economist Johann Heinrich von Th¨unen (1783–1850), it is meaningful to model non-negative total cost functions K ( x ) (in CU) relating to typical production processes, with argument levelof physical output x ≥ units, as a mathematical mapping in terms of a special polynomial ofdegree 3 [cf. Eq. (7.5)], which is given by K ( x ) = a x + a x + a x | {z } = K v ( x ) + a |{z} = K f with a , a > , a < , a ≥ , a − a a < . (7.15)The model thus contains a total of four free parameters. It is the outcome of a systematic regres-sion analysis of agricultural quantitative–empirical data with the aim to describe an inherently non-linear functional relationship between a few economic variables. As such, the functionalrelationship for K ( x ) expressed in Eq. (7.15) was derived from a practical consideration. It is areflection of the following observed features:0 CHAPTER 7. DIFFERENTIAL CALCULUS OFREAL-VALUED FUNCTIONS(i) for levels of physical output x ≥ units, the total costs relating to typical productionprocesses exhibit strictly monotonously increasing behaviour; thus(ii) for the total costs there do not exist neither roots nor local extremal values; however,(iii) the total costs display exactly one point of inflection.The continuous curve for K ( x ) resulting from these considerations exhibits the characteristic shapeof an inverted capital letter “S”: beginning at a positive value corresponding to fixed costs, the totalcosts first increase degressively up to a point of inflection, whereafter they continue to increase,but in a progressive fashion.In broad terms, the functional expression given in Eq. (7.15) to model totals costs in dependenceof the level of physical output is the sum of two contributions, the variable costs K v ( x ) and the fixed costs K f = a , viz. K ( x ) = K v ( x ) + K f . (7.16)In economic theory , it is commonplace to partition total cost functions in the diminishing returnspicture into four phases , the boundaries of which are designated by special values of the level ofphysical output of a production process: • phase I (interval units ≤ x ≤ x W ):the total costs K ( x ) possess at a level of physical output x W = − a / (3 a ) > units a point of inflection . For values of x smaller than x W , one obtains K ′′ ( x ) < CU / unit , i.e., K ( x ) increases in a degressive fashion. For values of x larger than x W , the opposite applies, K ′′ ( x ) > CU / unit , i.e., K ( x ) increases in a progressive fashion. The marginal costs ,given by K ′ ( x ) = 3 a x + 2 a x + a > CU / unit for all x ≥ units , (7.17)attain a minimum at the same level of physical output, x W = − a / (3 a ) . • phase II (interval x W < x ≤ x g ):the variable average costs K v ( x ) x = a x + a x + a , x > units (7.18)become minimal at a level of physical output x g = − a / (2 a ) > units. At this valueof x , equality of variable average costs and marginal costs applies, i.e., K v ( x ) x = K ′ ( x ) , (7.19) The last condition in Eq. (7.15) ensures a first derivative of K ( x ) that does not possess any roots; cf. the case ofa quadratic algebraic equation ! = ax + bx + c , with discriminant b − ac < . .5. ANALYTICINVESTIGATIONSOFECONOMIC FUNCTIONS 61which follows by the quotient rule of differentiation from the necessary condition for anextremum of the variable average costs, ! = (cid:18) K v ( x ) x (cid:19) ′ = ( K ( x ) − K f ) ′ · x − K v ( x ) · x , and the fact that K ′ f = 0 CU/unit. Taking care of the equality (7.19), one finds for the tangentto K ( x ) in the point ( x g , K ( x g )) the equation [cf. Eq. (7.13)] T ( x ) = K ( x g )+ K ′ ( x g )( x − x g ) = K v ( x g )+ K f + K v ( x g ) x g ( x − x g ) = K f + K v ( x g ) x g x . Its intercept with the K -axis is at K f . • phase III (interval x g < x ≤ x g ):The average costs K ( x ) x = a x + a x + a + a x , x > units (7.20)attain a minimum at a level of physical output x g > units, the defining equation of whichis given by CU ! = 2 a x g + a x g − a . At this value of x , equality of average costs andmarginal costs obtains, viz. K ( x ) x = K ′ ( x ) , (7.21)which follows by the quotient rule of differentiation from the necessary condition for anextremum of the average costs, ! = (cid:18) K ( x ) x (cid:19) ′ = K ′ ( x ) · x − K ( x ) · x . Since a quotient can be zero only when its numerator vanishes (and its denominator remainsnon-zero), one finds from re-arranging the numerator expression equated to zero the property K ′ ( x ) K ( x ) /x = x K ′ ( x ) K ( x ) = 1 for x = x g . (7.22)The corresponding extremal value pair ( x g , K ( x g )) is referred to in economic theory as the minimum efficient scale (MES) . From a business economics perspective, at a level of phys-ical output x = x g the (compared to our remarks in the Introduction inverted) ratio “INPUTover OUTPUT,” i.e., K ( x ) x , becomes most favourable. By respecting the property (7.21),the equation for the tangent to K ( x ) in this point [cf. Eq. (7.13)] becomes T ( x ) = K ( x g ) + K ′ ( x g )( x − x g ) = K ( x g ) + K ( x g ) x g ( x − x g ) = K ( x g ) x g x . Its intercept with the K -axis is thus at CU.2 CHAPTER 7. DIFFERENTIAL CALCULUS OFREAL-VALUED FUNCTIONS • phase IV (half-interval x > x g ):In this phase K ′ ( x ) /K ( x ) /x > obtains; the costs associated with the production of oneadditional unit of a good, approximately the marginal costs K ′ ( x ) , now exceed the aver-age costs, K ( x ) /x . This situation is considered unfavourable from a business economicsperspective. In this section, we confine our considerations, for reasons of simplicity , to a market sitution withonly a single supplier of a good in demand. The price policy that this single supplier may thusinact defines a state of monopoly . Moreover, in addition we want to assume that for the marketsituation considered economic equilibrium obtains. This manifests itself in equality of supply and demand , viz. x ( p ) = N ( p ) , (7.23)wherein x denotes a non-negative supply function (in units) (which is synonymous with the sup-plier’s level of physical output) and N a non-negative demand function (in units), both of whichare taken to depend on the positive unit price p (in CU / unit) of the good in question. The supplyfunction , and with it the unit price , can, of course, be prescribed by the monopolistic supplier inan arbitrary fashion. In a specific quantitative economic model, for instance, the demand func-tion x ( p ) (recall that by Eq. (7.23) x ( p ) = N ( p ) obtains) could be assumed to be either a linear ora quadratic function of p . In any case, in order for x ( p ) to realistically describe an actual demand–unit price relationship, it should be chosen as a strictly monotonously decreasing function, and assuch it is invertible . The non-negative demand function x ( p ) features two characteristic points,signified by its intercepts with the x - and the p -axes. The prohibitive price p proh is to be deter-mined from the condition x ( p proh ) ! = 0 units; therefore, it constitutes a root of x ( p ) . The saturationquantity x sat , on the other hand, is defined by x sat := x (0 CU/unit ) .The inverse function associated with the strictly monotonously decreasing non-negative demandfunction x ( p ) , the unit price function p ( x ) (in CU / unit), is likewise strictly monotonouslydecreasing. Via p ( x ) , one calculates, in dependence on a known amount x of units sup-plied/demanded (i.e., sold), the total revenue (in CU) made by a monopolist according to (cf.Sec. 7.3) E ( x ) = xp ( x ) . (7.24)Under the assumption that the non-negative total costs K ( x ) (in CU) underlying the productionprocess of the good in demand can be modelled according to the diminishing returns picture ofTurgot and von Th¨unen, the profit function (in CU) of the monopolist in dependence on the levelof physical output takes the form G ( x ) = E ( x ) − K ( x ) = x unit price z}|{ p ( x ) | {z } total revenue − (cid:2) a x + a x + a x + a (cid:3)| {z } total costs . (7.25).5. ANALYTICINVESTIGATIONSOFECONOMIC FUNCTIONS 63The first two derivatives of G ( x ) with respect to its argument x are given by G ′ ( x ) = E ′ ( x ) − K ′ ( x ) = xp ′ ( x ) + p ( x ) − (cid:2) a x + 2 a x + a (cid:3) (7.26) G ′′ ( x ) = E ′′ ( x ) − K ′′ ( x ) = xp ′′ ( x ) + 2 p ′ ( x ) − [6 a x + 2 a ] . (7.27)Employing the principles of curve sketching set out in Sec. 7.4, the following characteristic valuesof G ( x ) can thus be identified: • break-even point x S > units, as the unique solution to the conditions G ( x ) ! = 0 CU (necessary condition) (7.28)and G ′ ( x ) ! > CU / unit (sufficient condition) , (7.29) • end of the profitable zone x G > units, as the unique solution to the conditions G ( x ) ! = 0 CU (necessary condition) (7.30)and G ′ ( x ) ! < CU / unit (sufficient condition) , (7.31) • maximum profit x M > CU, as the unique solution to the conditions G ′ ( x ) ! = 0 CU / unit (necessary condition) (7.32)and G ′′ ( x ) ! < CU / unit (sufficient condition) . (7.33)At this point, we like to draw the reader’s attention to a special geometric property of the quan-titative model for profit that we just have outlined: at maximum profit, the total revenue func-tion E ( x ) and the total cost function K ( x ) always possess parallel tangents . This is due to thefact that by the necessary condition for an extremum to exist, one finds that CU / unit ! = G ′ ( x ) = E ′ ( x ) − K ′ ( x ) ⇔ E ′ ( x ) ! = K ′ ( x ) . (7.34) GDC:
Roots and local maxima resp. minima can be easily determined for a given stored functionin mode
CALC by employing the interactive routines zero and maximum resp. minimum .To conclude these considerations, we briefly turn to elucidate the technical term
Cournot’s point ,which frequently arises in quantitative discussions in economic theory ; this is named after theFrench mathematician and economist Antoine–Augustin Cournot (1801–1877).
Cournot’s point ( x M , p ( x M )) , for the unit price function p ( x ) of a good in a monopolistic market situation.Note that for this specific combination of optimal values the Amoroso–Robinson formula applies,which was developed by the Italian mathematician and economist Luigi Amoroso (1886–1965)and the British economist Joan Violet Robinson (1903–1983). This states that p ( x M ) = K ′ ( x M )1 + ε p ( x M ) , (7.35)with K ′ ( x M ) the value of the marginal costs at x M , and ε p ( x M ) the value of the elasticity of theunit price function at x M (see the following Sec. 7.6). Starting from the defining equation of the total revenue E ( x ) = xp ( x ) , the Amoroso–Robinson formula is derived by evaluating the firstderivative of E ( x ) at x M , so E ′ ( x M ) = p ( x M ) + x M p ′ ( x M ) = p ( x M ) (cid:20) x M p ′ ( x M ) p ( x M ) (cid:21) Sec. 7.6 z}|{ = = p ( x M ) [1 + ε p ( x M )] , and then re-arranging to solve for p ( x M ) , using the fact that E ′ ( x M ) = K ′ ( x M ) . Remark:
In a market situation where perfect competition applies, one assumes that the unitprice function has settled to a constant value p ( x ) = p = constant > CU/unit (and, hence, p ′ ( x ) = 0 CU/unit obtains). Now we want to address the determination of extremal values of economic functions that constituteratios in the sense of the construction OUTPUTINPUT , a topic raised in the Introduction.Let us consider two examples for determining local maxima of ratios of this kind.(i) We begin with the average profit in dependence on the level of physical output x ≥ units, G ( x ) x . (7.36)The conditions that determine a local maximum are [ G ( x ) /x ] ′ ! = 0 CU / unit and [ G ( x ) /x ] ′′ ! < CU / unit . Respecting the quotient rule of differentiation (cf. Sec. 7.2),the first condition yields G ′ ( x ) x − G ( x ) x = 0 GE / ME . (7.37)Since a quotient can only be zero when its numerator vanishes while its denominator remainsnon-zero, it immediately follows that G ′ ( x ) x − G ( x ) = 0 CU ⇒ x G ′ ( x ) G ( x ) = 1 . (7.38).6. ELASTICITIES 65The task at hand now is to find a (unique) value of the level of physical output x whichsatisfies this last condition, and for which the second derivative of the average profit becomesnegative.(ii) To compare the performance of two companies over a given period of time in a meaningfulway, it is recommended to adhere only to measures that are dimensionless ratios , and soindependent of scale . An example of such a dimensionless ratio is the measure referred toas economic efficiency , W ( x ) = E ( x ) K ( x ) , (7.39)which expresses the total revenue (in CU) of a company for a given period as a multiple ofthe total costs (in CU) it had to endure during this period, both as functions of the level ofphysical output . In analogy to our discussion in (i), the conditions for the existence of alocal maximum amount to [ E ( x ) /K ( x )] ′ ! = 0 × / unit and [ E ( x ) /K ( x )] ′′ ! < × / unit .By the quotient rule of differentiation (see Sec. 7.2), the first condition leads to E ′ ( x ) K ( x ) − E ( x ) K ′ ( x ) K ( x ) = 0 × / unit , (7.40)i.e., for K ( x ) > CU, E ′ ( x ) K ( x ) − E ( x ) K ′ ( x ) = 0 CU / unit . (7.41)By re-arranging and multiplication with x > unit, this can be cast into the particular form x E ′ ( x ) E ( x ) = x K ′ ( x ) K ( x ) . (7.42)The reason for this special kind of representation of the necessary condition for a localmaximum to exist [and also for Eq. (7.38)] will be clarified in the subsequent section. Again,a value of the level of physical output which satisfies Eq. (7.42) must in addition lead toa negative second derivative of the economic efficiency in order to satisfy the sufficientcondition for a local maximum to exist. Finally, we pick up once more the discussion on quantifying the local variability of differentiablereal-valued functions of one real variable, f : D ⊆ R → W ⊆ R , though from a slightly differentperspective. For reasons to be elucidated shortly, we confine ourselves to considerations of regimesof f with positive values of the argument x and also positive values y = f ( x ) > of the functionitself.As before in Sec. 7.2, we want to assume a small change of the value of the argument x andevaluate its resultant effect on the value y = f ( x ) . This yields x ∆ x ∈ R −→ x + ∆ x = ⇒ y = f ( x ) ∆ y ∈ R −→ y + ∆ y = f ( x + ∆ x ) . (7.43)6 CHAPTER 7. DIFFERENTIAL CALCULUS OFREAL-VALUED FUNCTIONSWe remark in passing that relative changes of non-negative quantities are defined by the quotientnew value − old valueold valueunder the prerequisite that “old value > ” applies. It follows from this specific construction thatthe minimum value a relative change can possibly attain amounts to “ − ” (corresponding to adecrease of the “old value” by 100%).Related to this consideration we identify the following terms: • a prescribed absolute change of the independent variable x : ∆ x , • the resultant absolute change of the function f : ∆ y = f ( x + ∆ x ) − f ( x ) , • the associated relative change of the independent variable x : ∆ xx , • the associated resultant relative change of the function f : ∆ yy = f ( x + ∆ x ) − f ( x ) f ( x ) .Now let us compare the order-of-magnitudes of the two relative changes just envisaged, ∆ xx and ∆ yy . This is realised by considering the value of their quotient, “resultant relative change of f divided by the prescribed relative change of x ”’: ∆ yy ∆ xx = f ( x + ∆ x ) − f ( x ) f ( x )∆ xx . Since we assumed f to be differentiable, it is possible to investigate the behaviour of this quotientof relative changes in the limit of increasingly smaller prescribed relative changes ∆ xx → ⇒ ∆ x → near some x > . One thus defines: Def.:
For a differentiable real-valued function f of one real variable x , the dimensionless (i.e.,units-independent) quantity ε f ( x ) := lim ∆ x → ∆ yy ∆ xx = lim ∆ x → f ( x + ∆ x ) − f ( x ) f ( x )∆ xx = x f ′ ( x ) f ( x ) (7.44)is referred to as the elasticity of the function f at position x .The elasticity of f quantifies its resultant relative change in response to a prescribed infinitesimallysmall relative change of its argument x , starting from some positive initial value x > . As such itconstitutes a measure for the relative local rate of change of a function f in a point ( x, f ( x )) . In.6. ELASTICITIES 67 economic theory , in particular, one adheres to the following interpretation of the elasticity ε f ( x ) :when the postive argument x of some positive differentiable real-valued function f is increased by , then in consequence f will change approximately by ε f ( x ) × .In the scientific literature one often finds the elasticity of a positive differentiable function f of apositive argument x expressed in terms of logarithmic differentiation. That is, ε f ( x ) := d ln[ f ( x )]d ln( x ) for x > and f ( x ) > , since by the chain rule of differentiation it holds that d ln[ f ( x )]d ln( x ) = d f ( x ) f ( x )d xx = x d f ( x )d xf ( x ) = x f ′ ( x ) f ( x ) . The logarithmic representation of the elasticity of a differentiable function f immediately explainswhy, at the beginning, we confined our considerations to positive differentiable functions of posi-tive arguments only. A brief look at the list of standard economic functions provided in Sec. 7.3reveals that most of these (though not all) are positive functions of non-negative arguments.For the elementary classes of real-valued functions of one real variable discussed in Sec. 7.1 onefinds:
Standard elasticities f ( x ) = x n for n ∈ N and x ∈ R > ⇒ ε f ( x ) = n ( natural power-law functions )2. f ( x ) = x α for α ∈ R and x ∈ R > ⇒ ε f ( x ) = α ( general power-law functions )3. f ( x ) = a x for a ∈ R > \{ } and x ∈ R > ⇒ ε f ( x ) = ln( a ) x ( exponential functions )4. f ( x ) = e ax for a ∈ R and x ∈ R > ⇒ ε f ( x ) = ax ( natural exponential functions )5. f ( x ) = log a ( x ) for a ∈ R > \{ } and x ∈ R > ⇒ ε f ( x ) = 1ln( a ) log a ( x ) ( logarithmic functions )6. f ( x ) = ln( x ) for x ∈ R > ⇒ ε f ( x ) = 1ln( x ) ( natural logarithmic function ).In view of these results, we would like to emphasise the fact that for the entire family of gen-eral power-law functions the elasticity ε f ( x ) has a constant value , independent of the value ofthe argument x . It is this very property which classifies general power-law functions as scale-invariant . When scale-invariance obtains, dimensionless ratios, i.e., quotients of variables of the To extend the regime of applicability of the measure ε f , one may consider working in terms of absolute values | x | and | f ( x ) | . Then one has to distinguish between four cases, which need to be looked at separately: (i) x > , f ( x ) > ,(ii) x < , f ( x ) > , (iii) x < , f ( x ) < and (iv) x > , f ( x ) < . constants . In this context, we would like to remark that scale-invariant (fractal) power-law functions of the form f ( x ) = Kx α , with K > and α ∈ R < \{− } ,are frequently employed in Economics and the
Social Sciences for modelling uncertainty of eco-nomic agents in decision-making processes , or for describing probability distributions of rareevent phenomena ; see, e.g., Taleb (2007) [25, p 326ff] or Gleick (1987) [13, Chs. 5 and 6]. Thisis due, in part, to the curious property that for certain values of the exponent α general power-lawprobability distributions attain unbounded variance; cf. Ref. [12, Sec. 8.9].Practical applications in economic theory of the concept of an elasticty as a measure of relativechange of a differentiable real-valued function f of one real variable x are generally based on thefollowing linear (!) approximation: beginning at x > , for small prescribed percentage changesof the argument x in the interval < ∆ xx ≤ , the resultant percentage changes of f amountapproximately to ( percentage change of f ) ≈ ( elasticity of f at x ) × ( percentage change of x ) , (7.45)or, in terms of a mathematical formula, to f ( x + ∆ x ) − f ( x ) f ( x ) ≈ ε f ( x ) ∆ xx . (7.46)We now draw the reader’s attention to a special kind of terminology developed in economic theory to describe the relative local change behaviour of economic functions in qualitative terms. For x ∈ D ( f ) , the relative local change behaviour of a function f is called • inelastic , whenever | ε f ( x ) | < , • unit elastic , when | ε f ( x ) | = 1 , and • elastic , whenever | ε f ( x ) | > .For example, a total cost function K ( x ) in the diminishing returns picture exhibits unit elasticbehaviour at the minimum efficient scale x = x g where, by Eq. (7.22), ε K ( x g ) = 1 . Also, atthe local maximum of an average profit function G ( x ) /x , the property ε G ( x ) = 1 applies; cf.Eq. (7.38).Next, we review the computational rules one needs to adhere to when calculating elasticities forcombinations of two real-valued functions of one real variable in the sense of Sec. 7.1.6: Computational rules for elasticities If f and g are differentiable real-valued functions of one real variable, with elasticities ε f and ε g ,it holds that:1. product f · g : ε f · g ( x ) = ε f ( x ) + ε g ( x ) ,2. quotient fg , g = 0 : ε f/g ( x ) = ε f ( x ) − ε g ( x ) ,3. concatenation f ◦ g : ε f ◦ g ( x ) = ε f ( g ( x )) · ε g ( x ) ,.6. ELASTICITIES 694. inverse function f − : ε f − ( x ) = 1 ε f ( y ) (cid:12)(cid:12)(cid:12)(cid:12) y = f − ( x ) .To end this chapter, we remark that for a positive differentiable real-valued function f of onepositive real variable x , a second elasticity may be defined according to ε f [ ε f ( x )] := x dd x (cid:20) xf ( x ) d f ( x )d x (cid:21) . (7.47)Of course, by analogy this procedure may be generalised to higher derivatives of f still.0 CHAPTER 7. DIFFERENTIAL CALCULUS OFREAL-VALUED FUNCTIONS hapter 8Integral calculus of real-valued functions ofone real variable In the final chapter of these lecture notes we give a brief overview of the main definitions andlaws of the integral calculus of real-valued functions of one variable. Subsequently we considera simple application of this tool in economic theory . Def.:
Let f be a continuous real-valued function of one real variable and F a differentiable real-valued function of the same real variable, with D ( f ) = D ( F ) . Given that f and F are relatedaccording to F ′ ( x ) = f ( x ) for all x ∈ D ( f ) , (8.1)then F is referred to as a primitive of f . Remark:
The primitive of a given continuous real-valued function f cannot be unique. By therules of differentiation discussed in Sec. 7.2, besides F also F + c , with c ∈ R a real-valuedconstant, constitutes a primitive of f since ( c ) ′ = 0 . Def.: If F is a primitive of a continuous real-valued function f of one real variable, then Z f ( x ) d x = F ( x ) + c , c = constant ∈ R , with F ′ ( x ) = f ( x ) (8.2)defines the indefinite integral of the function f . The following names are used to refer to thedifferent ingredients in this expression: • x — the integration variable , • f ( x ) — the integrand , • d x — the differential , and, lastly, 712 CHAPTER 8. INTEGRALCALCULUS OFREAL-VALUED FUNCTIONS • c — the constant of integration .For the elementary, continuous real-valued functions of one variable introduced in Sec. 7.1, thefollowing rules of indefinite integration apply: Rules of indefinite integration R α d x = αx + c with α = constant ∈ R ( constants )2. Z x d x = x c ( linear functions )3. Z x n d x = x n +1 n + 1 + c for n ∈ N ( natural power-law functions )4. Z x α d x = x α +1 α + 1 + c for α ∈ R \{− } and x ∈ R > ( general power-law functions )5. Z a x d x = a x ln( a ) + c for a ∈ R > \{ } ( exponential functions )6. Z e ax d x = e ax a + c for a ∈ R \{ } ( natural exponential functions )7. R x − d x = ln | x | + c for x ∈ R \{ } .Special methods of integration need to be employed when the integrand consists of a concatanationof elementary real-valued functions. Here we provide a list with the main tools for this purpose.For differentiable real-valued functions f and g , it holds that1. R ( αf ( x ) ± βg ( x )) d x = α R f ( x ) d x ± β R g ( x ) d x with α, β = constant ∈ R ( summation rule )2. R f ( x ) g ′ ( x ) d x = f ( x ) g ( x ) − R f ′ ( x ) g ( x ) d x ( integration by parts )3. R f ( g ( x )) g ′ ( x ) d x u = g ( x ) and d u = g ′ ( x )d x z}|{ = R f ( u ) d u = F ( g ( x )) + c ( substitution method )4. Z f ′ ( x ) f ( x ) d x = ln | f ( x ) | + c for f ( x ) = 0 ( logarithmic integration ). Def.:
Let f be a real-valued function of one variable which is continuous on an interval [ a, b ] ⊂ D ( f ) , and let F be a primitive of f . Then the expression Z ba f ( x ) d x = F ( x ) | x = bx = a = F ( b ) − F ( a ) (8.3)defines the definite integral of f in the limits of integration a and b .For definite integrals the following general rules apply:.3. APPLICATIONS IN ECONOMICTHEORY 731. Z aa f ( x ) d x = 0 ( identical limits of integration )2. Z ab f ( x ) d x = − Z ba f ( x ) d x ( interchange of limits of integration )3. Z ba f ( x ) d x = Z ca f ( x ) d x + Z bc f ( x ) d x for c ∈ [ a, b ] ( split of integration interval ). Remark:
The main qualitative difference between an (i) indefinite integral and a (ii) definite in-tegral of a continuous real-valued function of one variable reveals intself in the different kinds ofoutcome: while (i) yields as a result a real-valued (primitive) function , (ii) simply yields a singlereal number . GDC:
For a stored real-valued function, the evaluation of a definite integral can be performed inmode
CALC with the pre-programmed function R f(x)dx . The corresponding limits of integrationneed to be specified interactively.As indicated in Sec. 7.6, the scale-invariant power-law functions f ( x ) = x α for α ∈ R and x ∈ R > play a special role in practical applications. For x ∈ [ a, b ] ⊂ R > and α = − it holdsthat Z ba x α d x = x α +1 α + 1 (cid:12)(cid:12)(cid:12)(cid:12) x = bx = a = 1 α + 1 (cid:0) b α +1 − a α +1 (cid:1) . (8.4)Problematic in this context can be considerations of taking limits of the form a → resp. b → ∞ ,since for either of the two cases(i) case α < − : lim a → Z ba x α d x → ∞ , (8.5)(ii) case α > − : lim b →∞ Z ba x α d x → ∞ , (8.6)one ends up with divergent mathematical expressions. The starting point shall be a simple market situation for a single product. For this product, on theone-hand side, there be a demand function N ( p ) (in units) which is monotonously decreasing onthe price interval [ p u , p o ] ; the limit values p u and p o denote the minimum and maximum prices perunit (in CU/u) acceptable for the product. On the other hand, the market situation be described bya supply function A ( p ) (in units) which is monotonously increasing on [ p u , p o ] .The equilibrium unit price p M (in CU / unit) for this product is defined by assuming a state of economic equilibrium of the market, quantitatively expressed by the condition A ( p M ) = N ( p M ) . (8.7)4 CHAPTER 8. INTEGRALCALCULUS OFREAL-VALUED FUNCTIONSGeometrically, this condition defines common points of intersection for the functions A ( p ) and N ( p ) (when they exist). GDC:
Common points of intersection for stored functions f and g can be easily determined inter-actively in mode CALC employing the routine intersect .In a drastically simplified fashion, we now turn to compute the revenue made on the market by thesuppliers of a new product for either of three possible strategies of market entry .1.
Strategy 1:
The revenue obtained by the suppliers when the new product is being soldstraight at the equilibrium unit price p M , in an amount N ( p M ) , is simply given by U = U ( p M ) = p M N ( p M ) ( in CU ) . (8.8)2. Strategy 2:
Some consumers would be willing to purchase the product intially also at a unitprice which is higher than p M . If, hence, the suppliers decide to offer the product initiallyat a unit price p o > p M , and then, in order to generate further demand, to continuously (!) reduce the unit price to the lower p M , the revenue obtained yields the larger value U = U ( p M ) + Z p o p M N ( p ) d p . (8.9)Since the amount of money K := Z p o p M N ( p ) d p ( in CU ) (8.10)is (theoretically) safed by the consumers when the product is introduced to the market ac-cording to strategy 1, this amount is referred to in the economic literature as consumersurplus .3. Strategy 3:
Some suppliers would be willing to introduce the product to the market initiallyat a unit price which is lower than p M . If, hence, the suppliers decide to offer the productinitially at a unit price p u < p M , and then to continuously (!) raise it to the higher p M , therevenue obtaines amounts to the smaller value U = U ( p M ) − Z p M p u A ( p ) d p . (8.11)Since the suppliers (theoretically) earn the extra amount P := Z p M p u A ( p ) d p ( in CU ) (8.12)when the product is introduced to the market according to strategy 1, this amount is referredto in the economic literature as producer surplus . This is a strong mathematical assumption aimed at facilitating the actual calculation to follow. See previous footnote. ppendix AGlossary of technical terms (GB – D) A absolute change: absolute ¨Anderungabsolute value: Betragaccount balance: Kontostandaddition: Additionanalysis: Analysis, Untersuchung auf Differenzierbarkeitseigenschaftenarithmetical mean: arithmetischer Mittelwertarithmetical sequence: arithmetische Zahlenfolgearithmetical series: arithmetische Reiheaugmented coefficient matrix: erweiterte Koeffizientenmatrixaverage costs: St¨uckkostenaverage profit: Durchschnittsgewinn, Gewinn pro St¨uck B backward substitution: r¨uckwertige Substitutionbalance equation: Bilanzgleichungbasis: Basisbasis solution: Basisl¨osungbasis variable: BasisvariableBehavioural Economics: Verhaltens¨okonomikboundary condition: Randbedingungbreak-even point: Gewinnschwelle C chain rule: Kettenregelcharacteristic equation: charakteristische Gleichungcoefficient matrix: Koeffizientenmatrixcolumn: Spaltecolumn vector: Spaltenvektorcomponent: Komponentecompound interest: Zinseszinsconcatenation: Verschachtelung, Verkn¨upfungconservation law: Erhaltungssatz 756 APPENDIX A. GLOSSARY OFTECHNICAL TERMS(GB – D)constant of integration: Integrationskonstanteconstraint: Zwangsbedingungcost function: Kostenfunktionconsumer surplus: Konsumentenrentecontinuity: Stetigkeitcontract period: LaufzeitCournot’s point: Cournotscher Punktcurve sketching: Kurvendiskussion D decision-making: Entscheidungsfindungdeclining-balance depreciation method: geometrisch–degressive Abschreibungdefinite integral: bestimmtes Integraldemand function: Nachfragefunktiondependent variable: abh¨angige Variabledepreciation: Abschreibungdepreciation factor: Abschreibungsfaktorderivative: Ableitungdeterminant: Determinantedifference: Differenzdifference quotient: Differenzenquotientdifferentiable: differenzierbardifferential: Integrationsdifferenzialdifferential calculus: Differenzialrechnungdimension: Dimensiondirection of optimisation: Optimierungsrichtungdivergent: divergent, unbeschr”anktdomain: Definitionsbereich E economic agent: Wirtschaftstreibende(r) (meistens ein homo oeconomicus )economic efficiency: Wirtschaftlichkeiteconomic equilibrium: ¨okonomisches Gleichgewichteconomic principle: ¨okonomisches Prinzipeconomic theory: WirtschaftstheorieEconophysics: ¨Okonophysikeigenvalue: Eigenwerteigenvector: Eigenvektorelastic: elastischelasticity: Elastizit¨atelement: Elementend of profitable zone: Gewinngrenzeendogenous: endogenequilibrium price: Marktpreisequivalence transformation: ¨Aquivalenztransformation7exogenous: exogenexponential function: Exponentialfunktionextrapolation: Extrapolation, ¨uber bekannten G¨utigkeitsbereich hinaus verallgemeinern F feasible region: zul¨assiger Bereichfinal capital: Endkapitalfixed costs: Fixkostenforecasting: Vorhersagen erstellenfunction: Funktion G Gaußian elimination: Gauß’scher AlgorithmusGDC: GTR, grafikf¨ahiger Taschenrechnergeometrical mean: geometrischer Mittelwertgeometrical sequence: geometrische Zahlenfolgegeometrical series: geometrische Reihegrowth function: Wachstumsfunktion HI identity: Identit¨atimage vector: Absolutgliedvektorindefinite integral: unbestimmtes Integralindependent variable: unabh¨angige Variableinelastic: unelastischinitial capital: Anfangskapitalinstallment: Ratenzahlunginstallment savings: Ratensparenintegral calculus: Integralrechnungintegrand: Integrandintegration variable: Integrationsvariableinterest factor: Aufzinsfaktorinterest rate: Zinsfußinverse function: Inversfunktion, Umkehrfunktioninverse matrix: inverse Matrix, Umkehrmatrixisoquant: Isoquante JKL law of diminishing returns: Ertragsgesetz8 APPENDIX A. GLOSSARY OFTECHNICAL TERMS(GB – D)length: L¨angelevel of physical output: Ausbringungsmengelimits of integration: Integrationsgrenzenlinear combination: Linearkombinationlinearisation: Linearisierunglinear programming: lineare Optimierunglocal rate of change: lokale ¨Anderungsratelogarithmic function: Logarithmusfunktion M mapping: Abbildungmarginal costs: Grenzkostenmaximisation: Maximierungminimisation: Minimierungminimum efficient scale: Betriebsoptimummonetary value: Geldwertmonopoly: Monopolmonotonicity: Monotoniemortgage loan: Darlehen N non-basis variable: Nichtbasisvariablenon-negativity constraints: Nichtnegativit¨atsbedingungennon-linear functional relationship: nichtlineare Funktionalbeziehung O objective function: Zielfunktionone-to-one and onto: eineindeutigoptimal solution: optimalen L¨osungoptimal value: optimaler Wertoptimisation: Optimierungorder-of-magnitude: Gr¨oßenordnungorthogonal: orthogonalover-determined: ¨uberbestimmt P parallel displacement: Parallelverschiebungpension calculations: Rentenrechnungpercentage rate: Prozentsatzperfect competition: totale Konkurrenzperiod: Periodepivot column index: Pivotspaltenindexpivot element: Pivotelementpivot operation: Pivotschrittpivot row index: Pivotzeilenindexpole: Polstelle, Singularit¨atpolynomial division: Polynomdivision9polynomial of degree n : Polynom vom Grad n power-law function: Potenzfunktionpresent value: Barwertprimitive: Stammfunktionprincipal component analysis: Hauptkomponentenanalyseproducer surplus: Produzentenrenteproduct rule: Produktregelprofit function: Gewinnfunktionprohibitive price: PohibitivpreisProspect Theory: Neue Erwartungstheoriepsychological value function: psychologische Wertfunktion Q quadratic matrix: quadratische Matrixquotient: Quotientquotient rule: Quotientenregel R range: Wertespektrumrank: Rangrare event: seltenes Ereignisrational function: gebrochen rationale Funktionreal-valued function: reellwertige Funktionreference period: Referenzzeitraumregression analysis: Regressionsanalyseregular: regul¨arrelative change: relative ¨Anderungremaining debt: Restschuldremaining resources: Restkapazit¨atenremaining value: Restwertrescaling: Skalierungresources: Rohstofferesource consumption matrix: Rohstoffverbrauchsmatrixrestrictions: Restriktionenroot: Nullstellerow: Reiherow vector: Zeilenvektor S saturation quantity: S¨attigungsmengescale: Skala, Gr¨oßenordnungscale-invariant: skaleninvariantsimplex: Simplex, konvexer Polyedersimplex tableau: Simplextabellesingular: singul¨arsink: Senke0 APPENDIX A. 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