Math Prep Notes1 Joel Sobel and Joel Watson 2008, under revision 1 c 2006, 2008 by Joel Sobel and Joel Watson. For use only by Econ 205 students at UCSD in 2008. These notes are not to be distributed elsewhere. ii Preface These notes are the starting point for a math-preparation book, primarily for use by UCSD students enrolled in Econ 205 (potentially for use by folks outside UCSD as well). The first draft consists of a transcription of Joel Watson’s handwritten notes, as well as extra material added by Philip Neary, who worked on the tran- scription in 2006. Joel Sobel and Joel Watson have revised parts of these notes and added material, but the document is still rough and disorganized. Surely there are many mistakes. These material here is incomplete and contain many mistakes. If you find an error, a notational inconsistency, or other deficiency, please let one of the Joels know. Contents 1 Sets, Functions, and the Real Line 3 1.1 Sets . .3 1.2 Functions . .5 1.3 The Real Line . .7 1.4 Methods of Proof . 13 1.5 Some helpful notes . 16 2 Sequences 25 2.1 Introduction . 25 2.2 Sequences . 25 3 Functions and Limits of Functions 37 4 Differentiation 47 5 Taylor’s Theorem 59 6 Univariate Optimization 63 7 Integration 69 7.1 Introduction . 69 7.2 Fundamental Theorems of Calculus . 73 7.3 Properties of Integrals . 75 7.4 Computing Integrals . 76 8 Basic Linear Algebra 79 8.1 Preliminaries . 79 8.2 Matrices . 81 8.2.1 Matrix Algebra . 83 8.2.2 Inner Product and Distance . 87 8.3 Systems of Linear Equations . 90 iii iv CONTENTS 8.4 Linear Algebra: Main Theory . 93 8.5 Eigenvectors and Eigenvalues . 95 8.6 Quadratic Forms . 98 9 Multivariable Calculus 101 9.1 Linear Structures . 101 9.2 Linear Functions . 104 9.3 Representing Functions . 105 9.4 Limits and Continuity . 108 9.5 Sequences . 110 9.6 Partial Derivatives and Directional Derivatives . 111 9.7 Differentiability . 112 9.8 Properties of the Derivative . 116 9.9 Gradients and Level Sets . 120 9.10 Homogeneous Functions . 123 9.11 Higher-Order Derivatives . 124 9.12 Taylor Approximations . 125 10 Convexity 127 10.1 Preliminary: Topological Concepts . 127 10.2 Convex Sets . 128 10.3 Quasi-Concave and Quasi-Convex Functions . 130 10.3.1 How to check if a function f is quasiconcave or not . 131 10.3.2 Relationship between Concavity and Quasiconcavity . 132 10.3.3 Ordinal “vs” Cardinal . 133 11 Unconstrained Extrema of Real-Valued Functions 137 11.1 Definitions . 137 11.2 First-Order Conditions . 138 11.3 Second Order Conditions . 139 11.3.1 S.O. Sufficient Conditions . 140 11.3.2 S.O. Necessary Conditions . 140 12 Invertibility and Implicit Function Theorem 143 12.1 Inverse Functions . 143 12.2 Implicit Functions . 146 12.3 Examples . 151 12.4 Envelope Theorem for Unconstrained Optimization . 153 CONTENTS v 13 Constrained Optimization 155 13.1 Equality Constraints . 155 13.2 The Kuhn-Tucker Theorem . 159 13.3 Saddle Point Theorems . 164 13.4 Second-Order Conditions . 173 13.5 Examples . 174 14 Monotone Comparative Statics 179 vi CONTENTS CONTENTS 1 Notes on notation and items to be corrected: 1. Sets are typically denoted by italic (math type) upper case letters; elements are lower case. 2. PN used script letters to denote sets and there may be remnants of this throughout (such as in some figures, all which must be redrawn anyway). 3. " (varepsilon) is the preferred epsilon symbol. 4. Standard numerical sets (reals, positive integers, etc.) are written using the mathbb symbols: R, P, and so on. 2 CONTENTS Chapter 1 Sets, Functions, and the Real Line This chapter reviews some basic definitions regarding sets and functions, and it contains a brief overview of the construction of the real line. 1.1 Sets The most basic of mathematical concepts is a set, which is simply a collection of objects. For example, the “days of the week” is a set comprising the following objects: Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, and Sunday. The set of Scandinavian countries consists of: Sweden, Norway, Denmark, Fin- land and Iceland. Sets can be composed of any objects whatsoever. We often use capital italic letters to denote sets. For instance, we might let D denote the set of days of the week and S denote the set of Scandinavian countries. We will use lowercase italic letters to denote individual objects (called elements or points) in sets. Using the symbol “2,” which means “is an element of,” we thus write x 2 X to indicate that x is an element of set X. By the way, the symbol “62” means “is not an element of,” so we would write x 62 X to mean that X does not contain x. To define a set, it is sometimes convenient to list its elements. Formally, we do this by enclosing the list in curly brackets and separating the elements with commas. For instance, the set of Scandinavian countries is S = fSweden, Norway, Denmark, Finland, Icelandg and the set of days of the week is D = fMonday, Tuesday, Wednesday, . , Sundayg: 3 4 CHAPTER 1. SETS, FUNCTIONS, AND THE REAL LINE When a set contains many elements, it is useful to define it by making reference to a generic element x, using the “such as” symbol “j”, and including qualifying statements (properties that x is required to have). For instance, the set of numbers between 2 and 5 (including the endpoints) is fx j 2 ≤ x ≤ 5g: The set that contains no elements is called the empty set and is denoted ;. If a set has at least one element, it is called nonempty. Here are other common definitions that we will use frequently: Definition 1. A set Y is called a subset of set X, written Y ⊂ X, if every element of Y is also an element of X (that is, x 2 Y implies x 2 X). If, in addition, Y 6= X, then we say that Y is a proper subset of X. Note that the set of “weekend days,” fSaturday, Sundayg, is a proper subset of the set of days of the week. Observe that if sets X and Y are both subsets of each other, then they must be equal. Using the symbol ), which means “implies,” this conclusion can be expressed as: X ⊂ Y and Y ⊂ X ) X = Y . Sometimes when we want to prove that two sets are equal, we perform the two steps of showing that each is contained in the other. Definition 2. The union of sets X and Y , written X [ Y , is the set formed by combining the distinct elements of the two sets. That is, X [ Y ≡ fx j x 2 X or x 2 Y (or both)g: The symbol “≡”, used first in the definition above, means “is defined as” or “is equivalent to.” Definition 3. The intersection of sets X and Y , written X \ Y , is the set formed by collecting only the points that are common to both sets. That is, X \ Y ≡ fx j x 2 X and x 2 Y g: Definition 4. The difference between sets X and set Y , written X n Y , is the set formed by removing from X all points in the intersection of X and Y . That is, X n Y ≡ fx j x 2 X and x 62 Y g: When we are analyzing various sets, we typically have in mind a grand set that contains all objects of interest, such that every set we study is a subset of the grand set. This grand set is called the space, or universe, and is sometimes denoted U. The next definition makes reference to the space. 1.2. FUNCTIONS 5 Figure 1.1: Proper subset. A ⊂ B Figure 1.2: Union of two sets. A[B Definition 5. The complement of the set X, denoted X0 or Xc, is defined as U n X. Example 1. Suppose that the space is all of the lowercase letters: U ≡ fa; b; c;:::; zg: Consider X = fa, b, c, dg and Y = fd, e, f, gg. Then we see that X [ Y = fa, b, c, d, e, f, gg, X \ Y = fdg, X n Y = fa, b, cg, and Xc = fe,f,g,...,zg. For visual thinkers, Venn diagrams can be useful to represent the relations between sets. Figure 1.1 represents that set A (the points inside the smaller circle) is a proper subset of set B (the points inside the larger circle). The shaded region of Figure 1.2 shows the union of sets A and B, whereas the shaded region in Figure 1.3 is the intersection. 1.2 Functions Often we are interested in representing ways in which elements of one set might be related to, or associated with, elements of another set. For example, for sets X Figure 1.3: Intersection of two sets A\B 6 CHAPTER 1. SETS, FUNCTIONS, AND THE REAL LINE Figure 1.4: A function α.. and Y , we might say that every x 2 X points to, or “maps to,” a point y 2 Y . The concept of a “function” represents such a mapping. Definition 6. A function f from a set X to a set Y is a specification (a mapping) that assigns to each element of X exactly one element of Y . Typically we express that f is such a mapping by writing f :X ! Y . The set X (the points one “plugs into” the function) is called the domain of the function, and the set Y (the items that one can get out of the function) is called the function’s codomain.