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"Mathematical Economics and Finance Michael Harrison Patrick Waldron December 2, 1998 CONTENTS i Contents List of Tables List of Figures iii v PREFACE vii What Is Economics? . . . . . . . . . . . . . . . . . . . . . . . . . . . vii What Is Mathematics? . . . . . . . . . . . . . . . . . . . . . . . . . . . viii NOTATION ix I MATHEMATICS 1 LINEAR ALGEBRA 1.1 Introduction . . . . . . . . . . . . . . . . 1.2 Systems of Linear Equations and Matrices 1.3 Matrix Operations . . . . . . . . . . . . . 1.4 Matrix Arithmetic . . . . . . . . . . . . . 1.5 Vectors and Vector Spaces . . . . . . . . 1.6 Linear Independence . . . . . . . . . . . 1.7 Bases and Dimension . . . . . . . . . . . 1.8 Rank . . . . . . . . . . . . . . . . . . . . 1.9 Eigenvalues and Eigenvectors . . . . . . . 1.10 Quadratic Forms . . . . . . . . . . . . . 1.11 Symmetric Matrices . . . . . . . . . . . . 1.12 Deﬁnite Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 3 3 7 7 11 12 12 13 14 15 15 15 17 17 17 18 2 VECTOR CALCULUS 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Basic Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Vector-valued Functions and Functions of Several Variables . . . Revised: December 2, 1998 ii 2.4 2.5 2.6 2.7 2.8 2.9 Partial and Total Derivatives . . . . . . . The Chain Rule and Product Rule . . . . The Implicit Function Theorem . . . . . . Directional Derivatives . . . . . . . . . . Taylor’s Theorem: Deterministic Version The Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 21 23 24 25 26 27 27 27 27 29 30 34 39 3 CONVEXITY AND OPTIMISATION 3.1 Introduction . . . . . . . . . . . . . . . . 3.2 Convexity and Concavity . . . . . . . . . 3.2.1 Deﬁnitions . . . . . . . . . . . . 3.2.2 Properties of concave functions . 3.2.3 Convexity and differentiability . . 3.2.4 Variations on the convexity theme 3.3 Unconstrained Optimisation . . . . . . . 3.4 Equality Constrained Optimisation: The Lagrange Multiplier Theorems . . . . 3.5 Inequality Constrained Optimisation: The Kuhn-Tucker Theorems . . . . . . . 3.6 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 . . . . . . . . . . . . . 50 . . . . . . . . . . . . . 58 II APPLICATIONS 4 CHOICE UNDER CERTAINTY 4.1 Introduction . . . . . . . . . . . . . . . . . . . 4.2 Deﬁnitions . . . . . . . . . . . . . . . . . . . . 4.3 Axioms . . . . . . . . . . . . . . . . . . . . . 4.4 Optimal Response Functions: Marshallian and Hicksian Demand . . . . . . . 4.4.1 The consumer’s problem . . . . . . . . 4.4.2 The No Arbitrage Principle . . . . . . . 4.4.3 Other Properties of Marshallian demand 4.4.4 The dual problem . . . . . . . . . . . . 4.4.5 Properties of Hicksian demands . . . . 4.5 Envelope Functions: Indirect Utility and Expenditure . . . . . . . . 4.6 Further Results in Demand Theory . . . . . . . 4.7 General Equilibrium Theory . . . . . . . . . . 4.7.1 Walras’ law . . . . . . . . . . . . . . . 4.7.2 Brouwer’s ﬁxed point theorem . . . . . Revised: December 2, 1998 61 63 . . . . . . . . . . 63 . . . . . . . . . . 63 . . . . . . . . . . 66 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 69 70 71 72 73 73 75 78 78 78 CONTENTS 4.7.3 Existence of equilibrium . . . . . . . . . . . . . . . The Welfare Theorems . . . . . . . . . . . . . . . . . . . . 4.8.1 The Edgeworth box . . . . . . . . . . . . . . . . . . 4.8.2 Pareto efﬁciency . . . . . . . . . . . . . . . . . . . 4.8.3 The First Welfare Theorem . . . . . . . . . . . . . . 4.8.4 The Separating Hyperplane Theorem . . . . . . . . 4.8.5 The Second Welfare Theorem . . . . . . . . . . . . 4.8.6 Complete markets . . . . . . . . . . . . . . . . . . 4.8.7 Other characterizations of Pareto efﬁcient allocations Multi-period General Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii 78 78 78 78 79 80 80 82 82 84 4.8 4.9 5 CHOICE UNDER UNCERTAINTY 85 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2 Review of Basic Probability . . . . . . . . . . . . . . . . . . . . 85 5.3 Taylor’s Theorem: Stochastic Version . . . . . . . . . . . . . . . 88 5.4 Pricing State-Contingent Claims . . . . . . . . . . . . . . . . . . 88 5.4.1 Completion of markets using options . . . . . . . . . . . 90 5.4.2 Restrictions on security values implied by allocational efﬁciency and covariance with aggregate consumption . . . 91 5.4.3 Completing markets with options on aggregate consumption 92 5.4.4 Replicating elementary claims with a butterﬂy spread . . . 93 5.5 The Expected Utility Paradigm . . . . . . . . . . . . . . . . . . . 93 5.5.1 Further axioms . . . . . . . . . . . . . . . . . . . . . . . 93 5.5.2 Existence of expected utility functions . . . . . . . . . . . 95 5.6 Jensen’s Inequality and Siegel’s Paradox . . . . . . . . . . . . . . 97 5.7 Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.8 The Mean-Variance Paradigm . . . . . . . . . . . . . . . . . . . 102 5.9 The Kelly Strategy . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.10 Alternative Non-Expected Utility Approaches . . . . . . . . . . . 104 6 PORTFOLIO THEORY 6.1 Introduction . . . . . . . . . . . . . . . . . . . 6.2 Notation and preliminaries . . . . . . . . . . . 6.2.1 Measuring rates of return . . . . . . . . 6.2.2 Notation . . . . . . . . . . . . . . . . 6.3 The Single-period Portfolio Choice Problem . . 6.3.1 The canonical portfolio problem . . . . 6.3.2 Risk aversion and portfolio composition 6.3.3 Mutual fund separation . . . . . . . . . 6.4 Mathematics of the Portfolio Frontier . . . . . 105 105 105 105 108 110 110 112 114 116 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Revised: December 2, 1998 iv The portfolio frontier in N : risky assets only . . . . . . . . . . . . . . . 6.4.2 The portfolio frontier in mean-variance space: risky assets only . . . . . . . . . . . . . . . 6.4.3 The portfolio frontier in N : riskfree and risky assets . . . . . . . . . . . 6.4.4 The portfolio frontier in mean-variance space: riskfree and risky assets . . . . . . . . . . . Market Equilibrium and the CAPM . . . . . . . . . 6.5.1 Pricing assets and predicting security returns 6.5.2 Properties of the market portfolio . . . . . . 6.5.3 The zero-beta CAPM . . . . . . . . . . . . . 6.5.4 The traditional CAPM . . . . . . . . . . . . 6.4.1 CONTENTS . . . . . . . 116 . . . . . . . 124 . . . . . . . 129 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 130 130 131 131 132 137 137 137 137 137 140 140 6.5 7 INVESTMENT ANALYSIS 7.1 Introduction . . . . . . . . . . . . . . . . . . . . 7.2 Arbitrage and Pricing Derivative Securities . . . 7.2.1 The binomial option pricing model . . . 7.2.2 The Black-Scholes option pricing model . 7.3 Multi-period Investment Problems . . . . . . . . 7.4 Continuous Time Investment Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Revised: December 2, 1998 LIST OF TABLES v List of Tables 3.1 5.1 6.1 6.2 Sign conditions for inequality constrained optimisation . . . . . . 51 Payoffs for Call Options on the Aggregate Consumption . . . . . 92 The effect of an interest rate of 10% per annum at different frequencies of compounding. . . . . . . . . . . . . . . . . . . . . . 106 Notation for portfolio choice problem . . . . . . . . . . . . . . . 108 Revised: December 2, 1998 vi LIST OF TABLES Revised: December 2, 1998 LIST OF FIGURES vii List of Figures Revised: December 2, 1998 viii LIST OF FIGURES Revised: December 2, 1998 PREFACE ix PREFACE This book is based on courses MA381 and EC3080, taught at Trinity College Dublin since 1992. Comments on content and presentation in the present draft are welcome for the beneﬁt of future generations of students. A An electronic version of this book (in LTEX) is available on the World Wide Web at http://pwaldron.bess.tcd.ie/teaching/ma381/notes/ although it may not always be the current version. The book is not intended as a substitute for students’ own lecture notes. In particular, many examples and diagrams are omitted and some material may be presented in a different sequence from year to year. In recent years, mathematics graduates have been increasingly expected to have additional skills in practical subjects such as economics and ﬁnance, while economics graduates have been expected to have an increasingly strong grounding in mathematics. The increasing need for those working in economics and ﬁnance to have a strong grounding in mathematics has been highlighted by such layman’s guides as ?, ?, ? (adapted from ?) and ?. In the light of these trends, the present book is aimed at advanced undergraduate students of either mathematics or economics who wish to branch out into the other subject. The present version lacks supporting materials in Mathematica or Maple, such as are provided with competing works like ?. Before starting to work through this book, mathematics students should think about the nature, subject matter and scientiﬁc methodology of economics while economics students should think about the nature, subject matter and scientiﬁc methodology of mathematics. The following sections brieﬂy address these questions from the perspective of the outsider. What Is Economics? This section will consist of a brief verbal introduction to economics for mathematicians and an outline of the course. Revised: December 2, 1998 x What is economics? PREFACE 1. Basic microeconomics is about the allocation of wealth or expenditure among different physical goods. This gives us relative prices. 2. Basic ﬁnance is about the allocation of expenditure across two or more time periods. This gives us the term structure of interest rates. 3. The next step is the allocation of expenditure across (a ﬁnite number or a continuum of) states of nature. This gives us rates of return on risky assets, which are random variables. Then we can try to combine 2 and 3. Finally we can try to combine 1 and 2 and 3. Thus ﬁnance is just a subset of micoreconomics. What do consumers do? They maximise ‘utility’ given a budget constraint, based on prices and income. What do ﬁrms do? They maximise proﬁts, given technological constraints (and input and output prices). Microeconomics is ultimately the theory of the determination of prices by the interaction of all these decisions: all agents simultaneously maximise their objective functions subject to market clearing conditions. What is Mathematics? This section will have all the stuff about logic and proof and so on moved into it. Revised: December 2, 1998 NOTATION xi NOTATION Throughout the book, x etc. will denote points of n for n > 1 and x etc. will denote points of or of an arbitrary vector or metric space X. X will generally denote a matrix. Readers should be familiar with the symbols ∀ and ∃ and with the expressions ‘such that’ and ‘subject to’ and also with their meaning and use, in particular with the importance of presenting the parts of a deﬁnition in the correct order and with the process of proving a theorem by arguing from the assumptions to the conclusions. Proof by contradiction and proof by contrapositive are also assumed. There is a book on proofs by Solow which should be referred to here.1 N x ∈ N : xi ≥ 0, i = 1, . . . , N is used to denote the non-negative or+ ≡ thant of N , and N ≡ x ∈ N : xi > 0, i = 1, . . . , N used to denote the ++ positive orthant. is the symbol which will be used to denote the transpose of a vector or a matrix. 1 Insert appropriate discussion of all these topics here. Revised: December 2, 1998 xii NOTATION Revised: December 2, 1998 1 Part I MATHEMATICS Revised: December 2, 1998 CHAPTER 1. LINEAR ALGEBRA 3 Chapter 1 LINEAR ALGEBRA 1.1 Introduction [To be written.] 1.2 Systems of Linear Equations and Matrices Why are we interested in solving simultaneous equations? We often have to ﬁnd a point which satisﬁes more than one equation simultaneously, for example when ﬁnding equilibrium price and quantity given supply and demand functions. • To be an equilibrium, the point (Q, P ) must lie on both the supply and demand curves. • Now both supply and demand curves can be plotted on the same diagram and the point(s) of intersection will be the equilibrium (equilibria): • solving for equilibrium price and quantity is just one of many examples of the simultaneous equations problem • The ISLM model is another example which we will soon consider at length. • We will usually have many relationships between many economic variables deﬁning equilibrium. The ﬁrst approach to simultaneous equations is the equation counting approach: Revised: December 2, 1998 4 1.2. SYSTEMS OF LINEAR EQUATIONS AND MATRICES • a rough rule of thumb is that we need the same number of equations as unknowns • this is neither necessary nor sufﬁcient for existence of a unique solution, e.g. – fewer equations than unknowns, unique solution: x2 + y 2 = 0 ⇒ x = 0, y = 0 – same number of equations and unknowns but no solution (dependent equations): x+y = 1 x+y = 2 – more equations than unknowns, unique solution: x = y x+y = 2 x − 2y + 1 = 0 ⇒ x = 1, y=1 Now consider the geometric representation of the simultaneous equation problem, in both the generic and linear cases: • two curves in the coordinate plane can intersect in 0, 1 or more points • two surfaces in 3D coordinate space typically intersect in a curve • three surfaces in 3D coordinate space can intersect in 0, 1 or more points • a more precise theory is needed There are three types of elementary row operations which can be performed on a system of simultaneous equations without changing the solution(s): 1. Add or subtract a multiple of one equation to or from another equation 2. Multiply a particular equation by a non-zero constant 3. Interchange two equations Revised: December 2, 1998 CHAPTER 1. LINEAR ALGEBRA 5 Note that each of these operations is reversible (invertible). Our strategy, roughly equating to Gaussian elimination involves using elementary row operations to perform the following steps: 1. (a) Eliminate the ﬁrst variable from all except the ﬁrst equation (b) Eliminate the second variable from all except the ﬁrst two equations (c) Eliminate the third variable from all except the ﬁrst three equations (d) &c. 2. We end up with only one variable in the last equation, which is easily solved. 3. Then we can substitute this solution in the second last equation and solve for the second last variable, and so on. 4. Check your solution!! Now, let us concentrate on simultaneous linear equations: (2 × 2 EXAMPLE) x+y = 2 2y − x = 7 • Draw a picture • Use the Gaussian elimination method instead of the following • Solve for x in terms of y x = 2−y x = 2y − 7 • Eliminate x 2 − y = 2y − 7 • Find y 3y = 9 y = 3 • Find x from either equation: x = 2 − y = 2 − 3 = −1 x = 2y − 7 = 6 − 7 = −1 Revised: December 2, 1998 (1.2.1) (1.2.2) 6 1.2. SYSTEMS OF LINEAR EQUATIONS AND MATRICES SIMULTANEOUS LINEAR EQUATIONS (3 × 3 EXAMPLE) • Consider the general 3D picture . . . • Example: x + 2y + 3z = 6 4x + 5y + 6z = 15 7x + 8y + 10z = 25 • Solve one equation (1.2.3) for x in terms of y and z: x = 6 − 2y − 3z • Eliminate x from the other two equations: 4 (6 − 2y − 3z) + 5y + 6z = 15 7 (6 − 2y − 3z) + 8y + 10z = 25 • What remains is a 2 × 2 system: −3y − 6z = −9 −6y − 11z = −17 • Solve each equation for y: y = 3 − 2z 17 11 y = − z 6 6 • Eliminate y: 3 − 2z = • Find z: 1 1 = z 6 6 z = 1 • Hence y = 1 and x = 1. Revised: December 2, 1998 (1.2.3) (1.2.4) (1.2.5) 17 11 − z 6 6 CHAPTER 1. LINEAR ALGEBRA 7 1.3 Matrix Operations We motivate the need for matrix algebra by using it as a shorthand for writing systems of linear equations, such as those considered above. • The steps taken to solve simultaneous linear equations involve only the coefﬁcients so we can use the following shorthand to represent the system of equations used in our example: This is called a matrix, i.e.— a rectangular array of numbers. • We use the concept of the elementary matrix to summarise the elementary row operations carried out in solving the original equations: (Go through the whole solution step by step again.) • Now the rules are – Working column by column from left to right, change all the below diagonal elements of the matrix to zeroes – Working row by row from bottom to top, change the right of diagonal elements to 0 and the diagonal elements to 1 – Read off the solution from the last column. • Or we can reorder the steps to give the Gaussian elimination method: column by column everywhere. 1.4 Matrix Arithmetic • Two n × m matrices can be added and subtracted element by element. • There are three notations for the general 3×3 system of simultaneous linear equations: 1. ‘Scalar’ notation: a11 x1 + a12 x2 + a13 x3 = b1 a21 x1 + a22 x2 + a23 x3 = b2 a31 x1 + a32 x2 + a33 x3 = b3 Revised: December 2, 1998 8 1.4. MATRIX ARITHMETIC 2. ‘Vector’ notation without factorisation:  a11 x1 + a12 x2 + a13 x3 b1      a21 x1 + a22 x2 + a23 x3  =  b2  b3 a31 x1 + a32 x2 + a33 x3 3. ‘Vector’ notation with factorisation:     a11 a12 a13 x1 b1       a21 a22 a23   x2  =  b2  a31 a32 a33 x3 b3 It follows that:      a11 a12 a13 x1 a11 x1 + a12 x2 + a13 x3       a21 a22 a23   x2  =  a21 x1 + a22 x2 + a23 x3  a31 a32 a33 x3 a31 x1 + a32 x2 + a33 x3 • From this we can deduce the general multiplication rules: The ijth element of the matrix product AB is the product of the ith row of A and the jth column of B. A row and column can only be multiplied if they are the same ‘length.’ In that case, their product is the sum of the products of corresponding elements. Two matrices can only be multiplied if the number of columns (i.e. the row lengths) in the ﬁrst equals the number of rows (i.e. the column lengths) in the second. • The scalar product of two vectors in n is the matrix product of one written as a row vector (1×n matrix) and the other written as a column vector (n×1 matrix). • This is independent of which is written as a row and which is written as a column. So we have C = AB if and only if cij = k = 1n aik bkj . Note that multiplication is associative but not commutative. Other binary matrix operations are addition and subtraction. Addition is associative and commutative. Subtraction is neither. Matrices can also be multiplied by scalars. Both multiplications are distributive over addition. Revised: December 2, 1998     CHAPTER 1. LINEAR ALGEBRA 9 We now move on to unary operations. i The additive and multiplicative ..."

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