Mathematical Methods for Microeconomics Andreas M. Hefti 15. August 2014 Draft Introductory Maths course composed of three sequences: Real analysis and maths for micro se- 1 quence: Andreas Hefti Dynamic programming sequence (maths 2 for macro): Michelle Rendall Probability theory sequence (maths for 3 econometrics): Marc Sommer See http://www.econ.uzh.ch/dpe/courses.html for course material, dates and venues Draft Micro sequence consists of a lecture and an exercise session Lecture: 18.08: 08:15 - 12:00, 14:00 - 15:30 @ KOL-E-21 19.08, 21.08, 22.08: 08:15 - 12:00 @ KOL-E-21 Exercises (Jean-Michel Benkert) 25.08 - 26.08: 08:30 - 12:00 @ KOL-E-21 27.08 - 28.08: 08:30 - 12:00 @ PLD-E-04 Draft Exam: The exam (22.09. 10:00 - 11:30 @ KOL-H-312) is 90 minutes { 30 minutes for each the micro, macro and empirics sequence The exam is closed book For the micro sequence we expect you to have command over all definitions (≈ 30) as well as over the theorems (≈ 40), but not the propositions, lemmata, corollaries (of course, you still need to understand them) While linear algebra is a course prerequisite, there will not be any specific questions in the exam Draft In this dense (crash) course you 1 study the main mathematical building blocks required by modern microeconomic and game theory 2 are trained in their usage (proofs!) Focus is on deriving the mathematical apparatus required to understand (develop) economic theory. =) We concentrate more on (abstract) tools and less on their applications (they will come in the core courses). Draft Overview (Micro sequence): 1 Preliminaries: Mathematical statements; implication; proof techniques; sets; functions; relations 2 Topology: Metric spaces; norms; sequences; convergence; open, closed, compact, convex sets 3 Continuity (Lipschitz and uniform); uniform convergence 4 Differentiability; directional and partial derivatives 5 Implicit function theorem 6 Homogeneous functions; (quasi) concavity /convexity Draft 7 Constrained and unconstrained optimization; Lagrange's method; Constraint qualification; Envelope theorem 8 Uni- and multivariate (Riemann) Integration; fundamental theorem; convergence theorem; techniques; Leibnitz rule; Fubini's theorem 9 Correspondences: closed graph; upper/lower hemi-continuity 10 Fixpoint theory: Brouwer; Kakutani; Tarski; Banach 11 Convex analysis: Hyperplanes; supporting and separating hyperplane theorems 12 (Course prerequisite): Linear Algebra Draft 1. Preliminaries Draft 1.1. Mathematical statements A statement is a sentence\ that can either be true or " false, but never both (\tertium non datur") Examples: 4 > 2 What about: I am lying now\ " What about I always lie\ " Let A be a statement. Then :A ( not A\) also is a " statement. :A is true (false) if A is false (true). Example: A =\Every student loves micro\. Hence :A = No student " loves micro\. Correct? Draft Suppose A and B are statements. Then a new statement, usually called the implication, can be defined: A B A ^ B A _ B A ) B T T T T T F T F T T F F F F T T F F T F If A ) B is true, then A is sufficient for (the truth of) B and B is necessary for (the truth of) A. A ) B :,:A _ B A =\Madonna is a man\, B =\Snow is black\, A ) B true? The truth of the implication depends ONLY on the truth of the involved partial statements! If A is both necessary and sufficient for B, then A , B ( A is equivalent " to B\). A = X is yellow\. B = X is a lemon\. A and B equivalent? " " Draft There are three ways to prove the truth of A ) B. 1 Direct proof Uses ((A ) C) ^ (C ) B)) ) (A ) B). Show that A ) C and C ) B are true. 2 Proof by contraposition Prove :B ):A. 3 Indirect Proof (by contradiction) Assume that B is false but A is true. Show that A and :B together imply C, but it is already known that C is false. Hence :B must be false if A true. Thus A implies B. Draft 1.2. Sets (Georg Cantor) A set is an unordered collection of distinct items, called elements. For a 6= b: fa; bg = fb; ag = fa; b; ag N = f1; 2; 3; :::g fn 2 N :n divides 15g = f1; 3; 5g ; ≡ fx 2 X : x 6= xg empty set (of X) (Bertrand Russell) Is the following recursive definition a set? Let M be the set of all sets that do not contain themselves as elements. Note: Neither sets\ nor elements\ are actually " " well-defined objects in mathematics. Only allowed operations ( calculus") are relevant! " Draft Definition 1.1 (Quantors) Let M denote a set and E is a property, which any x 2 M either satisfies or not. 1 8 (all quantor) 8x 2 M : E(x) , fx 2 M : E(x)g = M 2 9 (existence quantor) 9x 2 M : E(x) , fx 2 M : E(x)g= 6 ; Proposition 1.1 1 :(8x 2 M : E(x)) , (9x 2 M : :E(x)) 2 :(9x 2 M : E(x)) , (8x 2 M : :E(x)) Negation: revert\ quantors and junctors (keep the order)! " (8x9y : E(x; y)) and (9y8x : E(x; y)) are different statements! Example: E(x; y) = Proposition y is trivial to student x\ " Draft X ⊂ Y ≡ 8x 2 X : x 2 Y X = Y , (X ⊂ Y ) ^ (Y ⊂ X) Let A; B ⊂ X Intersection: A \ B ≡ fx 2 X :(x 2 A) ^ (x 2 B)g Union: A [ B ≡ fx 2 X :(x 2 A) _ (x 2 B)g Difference: AnB ≡ fx 2 X :(x 2 A) ^ (x2 = B)g Complement: Ac ≡ XnA Power set: P(X) = fY : Y ⊂ Xg ;X ≡ fx 2 X : x 6= xg Use Venn-diagram for illustrations (but not for proofs) Proposition 1.2 Let X; Y be two sets. 1 The elements of ;X have ANY property 2 ;X = ;Y ≡ ; (empty set is unique) 3 ; ⊂ X; Y Is P(;) = P(f;g)? Draft Proposition 1.3 (DeMorgan Laws) Let fAi : i 2 Ig be a family of subsets of a set. Then: 1 T c S c ( i Ai) = i Ai 2 S c T c ( i Ai) = i Ai Draft Two objects a; b together form a new object (a; b), the (ordered) pair If X; Y are sets, then the (cartesian) product X × Y is a set, the set of all ordered pair (x; y), x 2 X and y 2 Y . n Q Similarly X1 × ::: × Xn ≡ Xj is a new set, the product j=1 set. n Q x 2 Xj is written as (x1; :::; xn). j=1 xj is the j-th component of x, sometimes xj = prj(x) (j-th projection of x) Draft 1.3. Functions Definition 1.2 (Function) X; Y are sets. A function (mapping) f from X to Y is a rule that assigns to each element of X exactly one element of Y . A function is specified by its domain (X), its codomain (Y ) and its mapping rule f, notation: f : X ! Y; x 7! f(x) Two functions f; g are equal (f = g) if the have the same domain, codomain and f(x) = g(x), x 2 X Image of f: im(f) = f(X) = fy 2 Y : 9x 2 X : y = f(x)g Draft Let f : X ! Y be a function. f is injective (or one-to-one) if x 6= y ) f(x) 6= f(y) f is surjective (or onto) if im(f) = Y f is bijective if f is both one-to-one and onto Inverse image: f −1(C) = fx 2 X : f(x) 2 Cg, C ⊂ Y Composition: If f : X ! U and g : U ! Y then g ◦ f : X ! Y ; x 7! g(f(x)) Examples of functions: Identity: idX : X ! X ,x 7! x x x ≥ 0 Absolute value: |·| : ! [0; 1) ,jxj ≡ R −x x < 0 (Infinite) Sequence: ' : N ! X is a sequence (in X), often just denoted by (xn), '(n) = xn Draft Proposition 1.4 For C ⊂ Y we have f(f −1(C)) ⊂ C. Moreover, f is bijective if and only if (iff) 9 a mapping g : Y ! X with g ◦ f = idX and f ◦ g = idY . Then g is uniquely determined. If f is bijective then the inverse function f −1 is the uniquely −1 −1 determined mapping f : Y ! X with f(f ) = idY and −1 f (f) = idX Proposition 1.5 Let f : X ! Y , A; B ⊂ X and C; D ⊂ Y . 1 f −1(C [ D) = f −1(C) [ f −1(D) 2 f −1(C \ D) = f −1(C) \ f −1(D) 3 f(A [ B) = f(A) [ f(B) 4 f(A \ B) ⊂ f(A) \ f(B) Draft 1.4. Relations Frequently, we care about how the elements of a set X relate to each other. For example: order relation >, or preference relation . A (binary) relation on X is any subset R ⊂ X × X. Notation: (x; y) 2 R denoted as xRy. Further R is reflexive if xRx, x 2 X transitive if (xRy) ^ (yRz) ) (xRz) symmetric if xRy ) yRx A relation R on X is called an equivalence relation if R is reflexive, transitive and symmetric. Frequently, equivalence relations are denoted by ∼ For any x 2 X the set [x] ≡ fy 2 X : y ∼ xg is the equivalence class of x. Example: X is population of Zurich. Let x ∼ y if x and y have the same parents. Draft A relation ≤ is called order on X if ≤ is reflexive, transitive and antisymmetric, i.e. (x ≤ y) ^ (y ≤ x) ) (y = x) If ≤ is an order on X, then the pair (X; ≤) is a (partially) ordered set.