Eco 610 Mathematical Economics I Notes M. Jerison 8/24/14

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1 Eco 610 Mathematical Economics I Notes M. Jerison 8/24/14 1. Introduction Economists analyze qualitative relations (e.g., who is whose boss, or which ownership pattern is more efficient) and quantitative relations (e.g., prices, which are ratios of amounts of goods exchanged). Mathematics is a language for describing qualitative and quantitative relations, so it is a natural language for economic analysis. Economists study effects of policies and events using mathematical models to formulate counterfactual questions such as What would the euro/dollar exchange rate be now if the U.S. had defaulted on part of its debt in August 2011? or, more generally, What would have happened if a different policy had been adopted or a different event had occurred? The models are collections of sets along with relations among the elements of the sets. Counterfactuals have the form of logical implications: If A, then B (e.g., If consumers had more income, then their total saving would be higher. ). An important part of economic analysis consists in discovering which sets of hypotheses imply which interesting conclusions. This involves finding and proving theorems. That is why this course emphasizes proofs along with other mathematical tools. The main tools we work with are constrained optimization and systems of equations. The main theorems in these subject areas are proved using methods of real analysis, which will also be a topic of the course. We begin with some concepts and notation in logic and set theory that are prerequisites for the course. The symbol will be used to mean is defined to be. The name of an object being defined will be written in boldface. Logic Items in a mathematical model are often labeled by letters. The letters in economic models usually represent variables that take numerical values, but they can represent other things. For example, a statement such as Every student in Eco 610 last year got the grade A might be labeled A. Statements can be combined to form other statements: A or B, or A and B. We are mainly concerned with whether statements are true or not. The statement not A, also written A, is an abbreviation for the statement statement A is false, which means statement A is not true. The statement A and B is true if and only if A is true and B is true. The mathematical term or is inclusive. The statement A or B is true if and only if any one of the following holds: A is true and B is false; A is false and B is true; A is true and B is true. If we want to exclude the last possibility, we must say either A or B, not both. Implication: A ñ B means [A is true and B is true] or [A is false] and can be written B ð A. Other ways of saying the same thing: If A then B; B if A; A only if B; B is necessary for A; A is sufficient for B; [(not Bq ñ (not A)]. This last implication is called the contrapositive of ra ñ Bs. It is sometimes easier to prove an implication by proving its contrapositive. (Keep this in mind when you try to solve the exercises in these notes.) The converse of A ñ B is the statement B ñ A. Note that A ñ B may be true and its converse false, or A ñ B may be false and its converse true. (Prove these claims.) If A ñ B and its converse, B ñ A, are both true, we say that A is equivalent to B or that A and B are equivalent. Then we write A ô B, which means [A is true and B is true] or [A is false and B is false]. There are many other ways to say or write the same thing: A if and only if B; A iff B; A is necessary and sufficient for B. 1

2 2 If the statement A ñ B is true, it is called a theorem. One of the main problems of economic analysis is to determine which implications are theorems when A and B contain interesting statements about economic relations. The main way of convincing people that a theorem is a theorem is to break it into a sequence of statements A i ñ A i 1, i 1, 2,..., n that are each accepted as true, with A A 1 and B A n 1. The intermediate statements A i may be theorems and axioms from mathematics or from an economic model. The sequence is called a proof. Examples are given in these notes, in SB pp , and in the books by Solow and Velleman in the syllabus. In ordinary English there are statements that are neither true nor false, e.g., This statement is false. Standard logical systems in mathematics avoid this by allowing statements to refer only to a fixed universal set of objects and to sets constructed from subsets of that set through mathematical grammar. In that case, the sentence in quotes, above, is not a statement. On the other hand, in mathematical logic systems that include basic arithmetic, some statements are undecidable, i.e., impossible to prove or disprove (Gödel (1931)). The most important examples of these statements deal with infinite sets. So far in economics, this issue has only come up in a narrow part of game theory. Some Sets and Notation A collection of objects is called a set. The objects are called elements or members of the set. If x is an element or member of a set S, we write x P S and say that x is in S. A set can be defined by enumeration (listing of its elements): talbany, Schenectady, Troyu, or by partial enumeration, when the pattern is clear: N t1, 2, 3,... u, the set of natural numbers, P t2, 3, 5, 7, 11, 13, 17,... u, the set of prime numbers, Z t0, 1, 1, 2, 2, 3, 3,... u, the set of integers. Q the set of rational numbers or rationals, R the set of real numbers or reals. H the empty set (with no elements). Alternatively, sets can be defined by properties that their elements satisfy, e.g., the set of even integers tn P Z : n 2k for some k P Zu t2k : k P Zu, where : means such that. Sets can be defined from operations involving other sets: X S Y T union of S and T ; Y S X T intersection of S and T ; SzT tx : x is in S and not in T u complement of T in S. It is denoted zt or T c if the set S is understood. S T means that every element of the set S is also an element of T. If it is true, we say that S is a subset of T or is contained in T, and we also write T S. Two sets S and T are equal (and we write S T ) if they have exactly the same elements, in other words, if every element of S is also an element of T and vice versa (so S T and T S). S is a proper subset of T if S T, but S T. The set of subsets of a set S is called the power set of S. The set C tcu with the single element c has two different subsets. (What are they? What is the power set of C?) The power set of a set S is denoted 2 S. (Why?) X Y is the set of ordered pairs px, yq with x P X and y P Y. We use the term ordered pair to refer to the fact that px, yq is treated as different from py, xq if x y.

3 Let S α be a set for each α in a set A. The set A is called an index set and an element of A is an index. The set of sets S α is called a collection of sets. Y αpa S α tx : x P S α for some α P Au is the union of the sets S α. We write Y n α1s α If A t1, 2,..., nu. X αpa S α tx : x P S α for every α P Au is the intersection of the sets S α. The symbol α used for an index can be replaced by any other symbol: Y αpa S α Y βpa S β. A collection of sets is disjoint if every pair of sets in the collection has empty intersection. A partition of a set S is a disjoint collection of nonempty subsets of S with union S. Important sets: ra, bs R, the closed interval tx P R : a x bu, a b. pa, bq R, the open interval tx P R : a x bu. Note: a could be 8; b could be 8. ra, bq tx P R : a x bu, pa, bs tx P R : a x bu are half open intervals, neither open nor closed. An interval that is not open need not be closed. An interval that is not closed need not be open. R tx P R : x 0u r0, 8q, the nonnegative real numbers R tx P R : x 0u p 8, 0s, the nonpositive real numbers R tx P R : x 0u, the positive real numbers R tx P R : x 0u, the negative real numbers Identities: A X pb Y Cq pa X Bq Y pa X Cq; A Y pb X Cq pa Y Bq X pa Y Cq; DeMorgan s Laws: zpa Y Cq pzaq X pzcq; zpa X Cq pzaq Y means for each or for all. D means there exists. P R, Dx P R : x a. D 1 means there exists exactly one. P N, D 1 m P N : m n 1. A slash through a symbol means not, as in, R,, E. P N, Em P N : n m n 1., addition symbol: 3 i1 a i a 1 a 2 a 3 ; n i2 a i a 2 a 3 a 4 a n. Π, product symbol: Π n i1a i a 1 a 2 a 3 a n, where each a i P R. a n a a a a to the power n (a multiplied by itself n times, where a P R, n P N). n! n pn 1q 2 1 n factorial for n P N. Also 0! 1. Π n i1s i S 1 S 2 S 3 S n tpx 1, x 2,..., x n q : x i P S i, i 1,..., nu is the product of the sets S i, with n P Z or n 8. An element of Π n i1s i is a list or an n-tuple. S n Π n i1s i, where S i S, i 1,..., n p P N is prime if p 1 and p is divisible only by p and 1 in N (p mn for m, n P N implies m 1 or n 1.) Important property of the natural numbers: Every m P N (m 1) has a unique prime factorization: There is a unique list of prime numbers p i and natural numbers n i, i 1,..., I, such that p i p i 1 for i 1,..., I 1, and m Π I i1p n i sign a 1 if a 0. sign a 0 if a 0. sign a 1 if a 0. a psign aqa absolute value of (magnitude of) a P R; S # elements in set S. The statement n i1 i npn 1q{2 depends on the index n P N. We can label it Spnq. 3 i.

4 4 Mathematical Induction: Every statement P pnq for n P N, n m, is true if the following conditions are satisfied: (a) P pmq is true; (b) for every k P N with k m, P pk 1q is true if P pkq is true. The assumption that P pkq is true in (b) is called the induction hypothesis. The statement Spnq above is proved by induction for all n P N in SB pp Exercises X1. Prove that if m balls are put in n boxes (n m), then at least one box contains more than one ball. X2. Prove that Π n i1 p1 a iq 1 n i1 a i if a i P X3. The following argument claims to prove that all people are of the same sex. Find the first sentence that contains an incorrect step in the argument and explain clearly the mistake(s) in it. Proof : Call a set of people uniform if all the people are of the same sex. We want to prove that every set of k people is uniform for each k P N. The conclusion is correct for k 1. Next, suppose that every set of k people is uniform. (This is the induction hypothesis.) We must prove that every set of k 1 people is uniform. Start with k 1 people and remove one person. The remaining set of people is uniform by the induction hypothesis. Now bring the removed person back and remove someone else. Again, the set of remaining people is uniform. So the set of k 1 people is uniform and the claim is proved. X4. Consider a set of n people and assume that no person is her or his own friend and that whenever person A is a friend of person B, B is a friend of A. Is it possible that no two people in the set have the same number of friends in the set? Hint: Try to answer this without using induction. Consider the number of friends different people in the set must have if they all have different numbers of friends. Relations, Functions and Correspondences This section develops terminology and notation for studying economic relations. We might want to ask, for example, which people are members of the boards of directors of which firms, or ask if one nation is wealthier than another, or if a consumer prefers one bundle of goods to another. These are all examples of relations. To specify a relation formally, we need to specify which elements x of some set X are related to which elements y of a set Y. The pairs px, yq that are related in this way are identified as being elements of some set R X Y. The relation is identified with the set R. When px, yq P R, we say that x is related to y under R and write xry. Example 1. A relation between board members and firms can be specified by listing potential members, labeled A, B, C and D, and firms, labeled 1, 2 and 3. The relation tpa, 1q, pa, 2q, pb, 2q, pc, 1q, pc, 3q, pd, 1q, pd, 3qu is interpreted as specifying that firm 1 has A, C and D on its board, firm 2 has A and B, and firm 3 has C and D. Example 2. A (weak) preference relation on a set X is a subset of X X that is reflexive px P Xq and transitive px y and y z ñ x y, z P Xq. Example 3. An equivalence relation E on a set X is relation that is reflexive, transitive, and symmetric pxey ñ y P Xq. In the theory of the consumer, indifference (between bundles of goods) is an equivalence relation. Every partition ts i u ipi of a set X determines an equivalence relation E in which xey iff x and y are in the same S i X. (Prove this last claim.) An equivalence class of an equivalence relation E on X is a set S X such that y P S.

5 Example 4. A rational number is a ratio of integers m and n. But what is a ratio? We can define the rationals formally by associating m{n with the integer pair pm, nq where n 0. But then the same rational number is also associated with pkm, knq for any integer k 0 (since km{pknq m{n. Thus, formally, each rational number m{n must be defined as an equivalence class of pairs of integers for an equivalence relation such that pm, nq pp, qq whenever mq np for integers n 0, q 0, m, and p. One of the most important relations is the functional relation. Many textbooks call a function f a rule that assigns to each element of a set X a unique element of a set Y. This is expressed by writing f : X Ñ Y or x P X ÞÑ fpxq P Y. The set X is called the domain or source. The set Y is called the range or target of the function. But what is a rule? Can we define what it means for something to be a function, using only terms that have a clear meaning? The answer is yes. The rule is specified by the graph of the function, the set of pairs px, yq P X Y that the function associates with each other. But the function itself is more than its graph. Example: Each hand in the classroom belongs to a unique person. This defines a function with domain the set of hands in the room and range the set of people in the room. The same relation can be represented by other functions simply by making the range a larger set. For example, the domain could be the set of hands in the room and the range could be the set of people in the world. It is important to distinguish between these functions. The function associates hands in the room to people in the room is onto (every person in the range of the function has a hand in the domain), whereas the function that associates hands in the room to people in the world is not onto. X5. Give a formal definition of the term function. Define a function to be a list of sets satisfying some restriction(s). Your definition must include everything that determines a function and nothing extra. That way, two lists of sets satisfying your restrictions represent the same function if and only if they are the same list. Your definition should not include any undefined terms except for the names of the sets in the list. An element gpxq of Y, the unique element of Y assigned by the function g : X Ñ Y to some x in X is called a value of g. The function is called real-valued if Y R. An element of the domain is called an argument of the function. The set of values assigned to arguments in S X is gpsq ty P Y : px, yq P G for some x P Su, called the image of g on S. It is simply called the image of g if S X, and is denoted Image g or Im g. The preimage or inverse image of V Y under g is g 1 pv q tx P X : gpxq P V u, the set of arguments assigned values in V. The letters used to label the domain, range and graph of a function do not matter. What matters are the sets themselves. The restriction of f : X Ñ Y to S X (also called f on S) is the function f S from S to Y, defined by f S pxq P S. A function f : X Ñ Y is injective (or one-to-one) if fpxq fpyq implies x y; different arguments are assigned different values by the function. A function f : X Ñ Y is surjective or onto if Image f Y, i.e., if every y P Y equals fpxq for some x P X. A function that is injective and surjective is bijective (a one-to-one correspondence). A bijective f : X Ñ Y has a unique inverse f 1 : Y Ñ X with f 1 pfpxqq P X. A set X is finite if there is a bijective f : X Ñ t1, 2,..., nu for some n P N. Then this n is the number of elements in X and is called the cardinality of X and is denoted X or 5

6 6 #X. A set that is not finite is infinite. Sets X and Y (finite or infinite) have the same cardinality if there is a bijection between them. Y has higher cardinality than X if there is an injection from X to Y but not from Y to X. A set is countable if there is a bijection between it and N. The set Q of rationals is countable. The set R of reals is not; it has higher cardinality than N and Q. This follows from two facts: (1) R has the cardinality of 2 N ; (2) every set has lower cardinality than its power set. X6. Prove that cardinality of a finite set is well-defined (defined without ambiguity) by proving that there cannot be two one-to-one correspondences f : X Ñ t1, 2,..., nu and g : X Ñ t1, 2,..., mu with m n. X7. Prove that a set is infinite iff it has the same cardinality as one of its proper subsets. A permutation of a finite set can be thought of as a listing of the elements of the set in a particular order. For example, one permutation of the set t1, 2, 3u is p1, 2, 3q and another is p3, 2, 1q. (Are there others? If so, what are they?) Formally, a permutation of the set S with #S n is a bijection π : t1, 2,..., nu Ñ S. The function π determines the list, with πp1q P S coming first, followed by πp2q, etc. X8. Prove that every set of n P N elements has n! different permutations. A k-combination from a set S is a set of k distinct elements of S. For 1 k n, n there are k n!{rk!pn kq!s different k-combinations from a set of n elements. The n definition 0! 1 makes the same formula hold when k n and k 0. The number k is referred to as n choose k or n take k. To see why the formula for it is correct, note that every permutation of a set S of n elements is a listing of k distinct elements of S followed by a listing of the n k remaining elements. For each k-combination from S and each permutation of these k elements, followed by any permutation of the n k remaining elements, we get a distinct permutation of S. Therefore the number of permutations of S n is k k!pn kq! n!, which yields the formula above. Binomial Theorem. px hq n n n k0 k xk h n k for x, h P R, n P N. n X9. (a) Prove that n k k 1 n 1 k for n P N and k 1,..., n. (b) Use induction to prove the binomial theorem. A sequence in a set S is a function f : M Ñ S, where M is an infinite subset of N. Alternative notation for such a sequence: tx i u ipm or tx i u 8 i1 if M N, or tx i u i, where x i fpiq P S for i P M. The composition or composite of f : X Ñ Y and g : W Ñ X 1 with Im g X is the function f g with domain W and range Y, defined by f gpxq P W. Example: g : R Ñ R, gpxq 3x 1. Note that this function could have been defined just as well by g : R Ñ R, gp q 3 1, or by gp3x 1q 3p3x 1q 1 9x 2. It is the rule (determined by the graph of g) that matters, not the symbol used for the argument. The formula gpxq 3x 1 alone is not quite a complete definition of a function since it does not specify the domain (the set of x s) or the range (set of y s). By convention, we will assume that the domain is the largest set for which the formula is defined (unless specified otherwise). The function g in this example is affine, i.e., of the form: Gpxq ax b for x P R. The graph of an affine function on R is a straight line. x P R ÞÑ fpxq sin x does not completely specify a function since the range is not determined. The function is onto only if the range is r 1, 1s.

7 Y apa S a tx : x P S a for some a P Au. Π apa S a tpx a q apa : x a P S P Au. X apa S a tx : x P S P Au. The Real Numbers To obtain conditions under which solutions to optimization problems exist, we need to describe formally the properties of the set of reals R. We can define R as a set satisfying the following five axioms. a1. R contains the set of rational numbers Q. a2. The properties of addition, multiplication and order for Q also apply to R. For all x, y, z P R, addition and multiplication are associative: px yq z x py zq and pxyqz xpyzq; commutative: x y y x and xy yx; distributive: xpy zq xy xz; with x 0 x, x 0 0, x p xq 0, and for x 0, x p1{xq 1. a3. If x y then x z y z. If, in addition, z 0 then xz yz. a4. If x y then there is a rational number q with x q y. An upper bound [respectively, lower bound] for a set S R is a number b P R such that b x [resp. b xs for every x P S. A least upper bound for S is an upper bound b for S such that no upper bound for S is less than b. A least upper bound for S is called the supremum of S and denoted sup S. A greatest lower bound for S is a lower bound b for S such that no lower bound for S is greater than b. A greatest lower bound for S is called the infimum of S and denoted inf S. Example: For a, b P R with a b, supra, bq b. So sup S is not necessarily an element of S. If S has no upper bound, then sup S 8. If S H then sup S 8. a5. Every nonempty set of reals with an upper bound has a real least upper bound. Axiom a5 ensures that the set of real numbers has no holes. This distinguishes the reals from the rationals. There are numbers such as? 2 that can be approximated arbitrarily closely by rationals yet are not rational. Axiom a5 does not hold if the set of reals is replaced by the set of rationals. The set of rationals less than? 2 has an upper bound, but no least upper bound that is rational. Exercises: X10. Prove that for each p P P,? p is not rational (i.e., it is not the ratio of two integers). X11. Use the axioms above to prove that x y implies x y for x, y P R. X12. Use the axioms above to prove that every nonempty set of reals with a lower bound has a unique infimum Basic Calculus Topics: Limits, derivatives, monotone functions, monotonicity gap, exponents, logs, elasticity, chain rule, critical points, maximizers, minimizers, inflection points, first and second order conditions, convex sets, concave and convex functions, antiderivatives, integrals. SB chs. 2 5, 21 pp , Appendix A4.

8 8 A sequence tx i u ipn in R has a limit x P R if for each open interval B containing x there is a number N such that i N ñ x i P B. Then x i can be made as close to x as we wish by making i big enough. In that case we say that the sequence approaches or converges to x and we write x i Ñ x and lim iñ8 x i x or lim x i x. Examples of Sequences: a. x i 1{i converges to 0 for i P N. This sequence can be written as 1, 1{2, 1{3,..., representing the values x 1, x 2, x 3,.... This sequence converges monotonically since the function i ÞÑ x i 1{i is strictly decreasing. b. x i p 1q i {i Ñ 0. The sequence, 1, 1{2, 1{3, 1{4, 1{5,..., cycles above and below its limit, 0, so it is not monotonic. But the distance x i between the value of the sequence and the limit is strictly decreasing. The sequence steadily approaches its limit from one side or the other. c. x i r2 p 1q i s{i Ñ 0. The sequence is 1, 3{2, 1{3, 3{4, 1{5, 3{6,.... The distance between x i and the limit, 0, repeatedly falls, then rises, then falls again, etc. If 0 P p a, bq, no matter how small a and b are, there is some N such that x i P p a, N. Since 1{i x i 3{i, we have x i P p a, bq if 3{i b or i N 3{b. d. x i i has no limit in R. Some authors write lim x i 8 or x i Ñ 8. d. x i p 1q i, the sequence 1, 1, 1, 1,..., is bounded since there is some r P R with x i (We can take r 2.) But the sequence has no limit. To prove this, note that for any x, the interval I p x.5, x.5q does not contain either 1 or 1. So there is no N with p 1q i P N. Limit of a function: We say that f : S Ñ R approaches c or has a limit c at x if fpx i q Ñ c for every tx i u in S converging to x. In that case, we write fpxq Ñ c as x Ñ x or lim xñ x fpxq c. If the limit c equals fp xq, then we call f continuous at x. Note that f is continuous at x P S if x is isolated from the other elements of S (i.e., if some open interval contains x and no other element of S). (Explain why.) A function is called continuous if it is continuous at every element of its domain. It is easy to show that the identity function fpxq x and all constant functions are continuous. Then it can be shown that sums, multiples, quotients, powers, roots, and compositions of continuous functions are continuous wherever they are defined. Many continuous functions can be constructed by combining other simpler continuous functions in these ways. The derivative of a function f : S R Ñ T R at x P S approximates the slope of a segment joining the points p x, fp xqq and px, fpxqq in the graph of f for x near x. Formally, the function f is differentiable at x there is a function F, called the slope function for f at x, such that fpxq fp xq px xqf P S, such that F is continuous at x. Then the derivative of f at x is f 1 p xq F p xq lim xñ x rfpxq fp xqs{px xq. We call f differentiable if it is differentiable at every element of its domain. In that case f 1 (called the derivative of f) is a function from S to R. If f 1 is differentiable, its derivative is denoted f 2 or f p2q, and if f pkq is differentiable its derivative is denoted f pk 1q for k 2, 3,.... When f pkq exists, it is the kth order derivative of f and f is called differentiable of order k. If f pkq is continuous, then f is called C k or continuously differentiable of order k and we write f P C k. If f P C k for every k P N, we write f P C 8 and call f infinitely differentiable. We call f continuously differentiable if f P C 1. Sums, multiples, quotients, powers, roots, and compositions of functions differentiable of order k are differentiable of order k wherever they are defined. Nearly all the functions used in economics are C 1 at least piecewise (i.e., on subsets forming partitions of their domains). The functions may have (rare) breaks in their graphs

9 or may have kinks. A kink is a point in the graph where the function is continuous, but not differentiable. For the most commonly used functions in economics, derivatives can be defined without using limits. Let such a function f be differentiable at x. For h near 0, there is a continuous function g satisfying hgpx, hq fpx hq fpxq. Then f 1 pxq gpx, 0q. For example, let fpxq x 2. Then fpx hq fpxq px hq 2 x 2 2xh h 2 hgpx, hq, where gpx, hq 2x h. So f 1 pxq gpx, 0q 2x. A real-valued function f is nondecreasing on a set S R if for x, y P S, x y implies fpxq fpyq. The function is nonincreasing on S if for x, y P S, x y implies fpxq fpyq. The function is strictly increasing, [respectively, strictly decreasing] on S if for x, y P S, x y implies fpxq fpyq [resp., fpxq fpyq]. We omit reference to S (and say that f is nondecreasing, strictly increasing, etc.) if S is the entire domain of f. A function is called weakly monotone if it is nondecreasing or nonincreasing. It is called monotone if it is strictly increasing or strictly decreasing. Some authors use the term monotone to mean strictly increasing alone. A differentiable function f : S Ñ R with S R is nondecreasing [respectively, nonincreasing] if and only if f 1 0 [f 1 0]. (Note that f 1 0 means that f 1 pxq 0 for every x in the domain of f.) Thus the sign of the derivative can tell us if the function is weakly monotone. If f 1 0 then f is strictly increasing. If f 1 0 then f is strictly decreasing. However the converses of these two statements are false. In particular, f 1 0 might not be true when f is strictly increasing. (Prove this.) I call this fact the monotonicity gap referring to the gap between the condition of having a positive derivative and the weaker condition of being strictly increasing. If f is nondecreasing, then it is differentiable except at rare points (to be defined below). If f is strictly increasing, f 1 pxq may be 0, but only at rare x. (Certainly not on any open interval, because then f is constant on that interval.) Convexity plays an central role in economic theory as a way of representing decreasing returns from changes in production or consumption. A set in R n is convex if it contains every segment that has endpoints in the set. A function f : X R n Ñ R is convex if the set of points px, yq with y fpxq, i.e., the set of points lying on or above its graph is convex. (Note that these are not quite complete definitions since we have not yet defined a segment.) The function f is concave if the set of points px, yq with y fpxq (the set of points on or below the graph of f) is convex. Note that we do not speak of concave sets. These definitions are geometric. We will give algebraic versions of them and generalize them to functions of several variables below A differentiable function f : R Ñ R is convex rrespectively, concaves if and only if f 1 is nondecreasing rrespectively nonincreasings. If f is twice differentiable, then it is convex rrespectively, concaves if and only if f 2 0 rf 2 0s. These characterizations of convex and concave functions using derivatives apply only to functions of a single real variable. Versions for functions of several variables will be discussed below. Note that convex and concave functions need not be differentiable. Their graphs can have kinks. But they are differentiable except at rare points. We will state this more precisely later. A function f : X R n Ñ R is strictly convex [respectively, strictly concave] if for every segment with distinct endpoints in the graph of f, the other points of the segment are above [respectively, below] the graph of f.

10 10 2. A differentiable function f : R Ñ R is strictly convex rrespectively, strictly concaves if f 1 is increasing rrespectively decreasings. If f is twice differentiable, then it is strictly convex rrespectively, strictly concaves if f 2 0 rf 2 0s. The converses of these statements are false. For example, a twice differentiable, strictly concave function f : R Ñ R might not satisfy f 2 0. It is possible that f 2 pxq 0 at some value(s) of x. (X Find an example of such a function.) But if f is twice differentiable and strictly concave, we cannot have f 2 pxq 0 for all x in an open interval. In section 12 we define an exponential function fpxq b x with base b 0, for x P R. We prove that it has the following important properties: E1. If x P N, then fpxq is b multiplied by itself x times. E2. f is positive and has a positive derivative of every order. E3. fpx yq fpxqfpyq. E4. fpxyq b xy pb x q y pb y q x. The limit lim nñ8 r1 p1{nqs n 8 exists and equals n0p1{n!q, and is denoted e If fpxq e x, then f 1 f. fpxq b x has an inverse function, the logarithmic function log b x, with these properties: L1. log b fpxq x and fplog b xq x. L2. log b xy log b x log b y. L3. log b x y y log b x. The natural log function is ln x log e x. L4. If gpxq log b x, then g 1 pxq 1{px ln bq. These functions are important in economics because the growth rates of so many economic variables do not change drastically over time. The growth rate of a differentiable function f at x is f 1 pxq{fpxq. If f is the exponential function b x above, then its growth rate at x is f 1 pxq{fpxq ln b for all x. Exponential functions have constant growth rates. X13. Use properties E1, E2, E3, E4, L1, L2, L3 to prove that fpxq b x satisfies fp0q 1, fp yq 1{fpyq and fpxq{fpyq fpx yq, and that log b px{yq log b x log b y. A differentiable function f : S Ñ R with S R has elasticity xf 1 pxq{fpxq at x where fpxq 0. The elasticity measures the response of the function to variations in its argument. It is especially useful for economic applications since it is unaffected by changes in the units in which the variables in the domain and range of the function are measured. To see why this is true, first consider a change of units for elements of the range of the function. This has the effect of multiplying the value of the function by a constant, say k. The elasticity at x is then xkf 1 pxq{rkfpxqs xf 1 pxq{fpxq. Suppose instead that the units in which the argument of the function change. Let the unit size be divided by k. Then x units become kx units. The function with these new units is F, where F pkxq fpxq. Since kf 1 pkxq f 1 pxq, we see that xf 1 pxq{fpxq kxf 1 pkxq{f pkxq. The elasticity of F at the correct argument kx, which represents the same quantity in the new units as x in the old, is the same the elasticity of f at x. The Darboux-Stieltjes Integral: This integral provides a general notation for expected values of discrete or continuous random variables. Consider a nondecreasing function F : ra, bs Ñ R, where a b P R and F paq F pbq. Define F pt q suptf pxq : x tu for t a and F pt q inftf pxq : x tu for t b, and let F pa q F paq and F pb q F pbq. If F is continuous at t, then F pt q F ptq F pt q. Otherwise, F pt q F pt q is called the jump of F at t. Let f : ra, bs Ñ R be bounded. For S ra, bs, define Mpf, Sq suptfpxq : x P Su

11 and mpf, Sq inftfpxq : x P Su. Call P pt i q n i0, with a t 0 t 1 t n 1 t n b, a division of ra, bs. (It is sometimes called a partition, but it is not a set of subsets of n ra, bs.) Define J F pf, Pq k1 fpt kq rf pt k q F pt k qs, and the upper sum: Upf, Pq J F pf, Pq lower sum: Lpf, Pq J F pf, Pq ņ k1 ņ k1 Mpf, rt k 1, t k sq rf pt k q F pt k 1qs and mpf, rt k 1, t k sq rf pt k q F pt k 1 qs of f for F on P. Define U F pfq inftupf, Pq : P a division of ra, bsu and L F pfq suptlpf, Pq : P a division of ra, bsu. It can be shown that U F pfq ³ L F pfq. If ³ these terms b are equal, their value is called the F -integral of f and is denoted fdf or b fpxqdf pxq; a then ³ the function f is called F -integrable on ra, bs. The subscript a and superscript b on are called limits of integration and may be omitted when it does not cause confusion. A function f is called piecewise continuous on a subset of R if it is continuous at all but finitely many points of its domain. Theorem 1. If a function is piecewise continuous or if it is bounded and either nondecreasing or nonincreasing then it is F -integrable. More general F -integrable functions can be constructed by piecing together functions that are nondecreasing or nonincreasing on adjacent intervals. Functions that are not F -integrable are unbounded or have too many discontinuities. Theorem ³ 2. Let f and g be F -integrable and G-integrable functions on ra, bs. For c P R, (a) ³ ³ ³ pcf gqdf cp fdf q gdf, (b) ³ ³ f df f df ³, ³ (c) fdpf Gq f df fdg, (d) f g ñ ³ f df ³ g df, and (e) if g is continuous and F is C 1, then ³ g df ³ b a gpxqf 1 pxqdx. The next result generalizes integration by parts and the Fundamental Theorem of Calculus. Theorem 3. Integration by Parts: Then Let F and G be nondecreasing on ra, bs, and define F ptq rf pt q F pt qs{2 and G ptq rgpt q Gpt P ra, bs. ³ b ³ F b dg a a G df F pbqgpbq F paqgpaq. ³ b X14. Use theorems 2(e) and 3 to prove F 1 pxqdx F pbq F paq for F P C 1 with F 1 0 a on ra, bs. 11 ³ b ³ f df and lim 0 0 añ 8 f df a ³ b If f df exists for every interval ra, a bs ³ R and the limits lim bñ8 both exist, then the improper integral f df is defined to be the sum of those two limits. R Integrals on unions of disjoint intervals are defined as sums of the integrals on the intervals. If S T, then an integral on T zs tx P T : x R Su equals ³ the integral on T minus the integral on S. Applying these definitions we can define f df for any set S constructed S by countable unions and finite intersections of intervals in R. Given g : R Ñ R and a real valued random variable X with distribution function F (where F pxq is the probability that X takes a value no greater than x), the expected value ³ of gpxq is g df, denoted EgpXq. The expected value of gpxq conditional on X taking a R ³ value in a set S is ErgpXq X in Ss p g df q{p³ df pxqq, also denoted ErgpXq Ss. S S

12 12 3. Vectors, Functions of Several Variables and Their Derivatives We want to study functions that depend on several variables, or more generally study models in which the values of some variables are determined by values of others. It is convenient to treat each set of variables as a single object called a vector. A vector can be thought of as an abstract object, an element of a vector space, but we will start out with the most important examples of vectors, elements of Euclidean space, R n, represented as lists of n real numbers px 1, x 2,..., x n q (also written px i q n i1 or px i q). The list is called an n-vector and a number x i in the list is called a component or entry or element of the vector. We interpret each n-vector as a displacement, i.e., a direction and a length. A displacement can be represented geometrically as an arrow with a base point and a tip. Arrows with the same direction and length but different base points represent the same vector. Example A. We may treat the quantities of goods bought by a group of consumers as depending on the consumers incomes and the prices of all the goods. The lists of quantities of goods, of prices and of consumer incomes are all vectors. Vectors can be added can be added to each other and can be multiplied by scalars (real numbers). The sum of the vectors x px 1, x 2,..., x n q and y py 1, y 2,..., y n q is x y px 1 y 1, x 2 y 2,..., x n y n q. Multiplying t P R by y yields ty pty 1, ty 2,..., ty n q. There is a unique vector 0 such that 0 x x for every vector x, and for each vector y, there is a unique vector y such that y y 0. Vector addition is commutative px y y xq and associative px yq w x py wq. Scalar multiplication satisfies the distributive law, tpx yq tx ty, and is associative, tpuyq ptuqy, for t, u P R. Be sure you understand the geometric interpretation for these operations. (See SB ch. 10.) For example, the negative of a vector has the same length and points in the opposite direction. Two vectors are called collinear if one is a scalar multiple of the other. Draw pictures of examples of collinear vectors and convince yourself that they point in the same or opposite direction (from 0). If two vectors x and y are not collinear, then they determine a plane (the plane determined by the three points x, y and 0). Each point in this plane can be written as αx βy, for some α and β in R. Such a vector is called a linear combination of x and y. More generally, a linear combination of a set of K vectors tv k u K k1 is a K vector k1 α kv k, where each α k is real. The set of linear combinations of the vectors v k is called their span and denoted span tv k u. The dot product (or inner or scalar product) of the vectors x px 1, x 2,..., x n q and v pv 1, v 2,..., v n q in R n n is x v i1 x iv i, also written xv. The length or Euclidean norm of x is }x}? x x. The set R n with this norm is called n-dimensional Euclidean space. (Other norms can be defined and used for various purposes, e.g., the sup norm, }x} 8 maxt x 1, x 2,..., x n u. The term space refers to a set with additional structure, in this case, sums, products and norms.) The dot product and norm satisfy the following identities for all vectors x, v and w in R n and scalars t, τ P R: x v v x, ptxq v x ptvq tpx vq, x pv wq px wq px wq, }x} 0, r}x} 0 ñ x 0s, }tx} t }x} and }x v} }x} }v}, the triangle inequality. A function f : R n Ñ R m is called linear if fpx yq fpxq fpyq and fptxq tfpxq for all x, y P R n and t P R. A linear function f : R n Ñ R can be written as fpxq a x for some vector a. This class of functions generalizes the class of functions from R to R that have straight line graphs passing through the origin.

13 The distance between points x and v is the length of the vector x v (why?). It is also the length of the vector v x (why?). For subsets X and Y of R n and a P R, we define the sets: ax tax : x P Xu, X Y tx y : x P X, y P Y u and X Y tx y : x P X, y P Y u. (3.1) Two collinear vectors that are both nonzero point in the same or exact opposite direction. To make this precise, we define a direction to be a vector of length 1. Such a vector is also called a unit vector. The direction of a vector v 0 is the vector p1{}v}qv, which we also write v{}v}. It has length 1 (prove this) and points in the same direction as v. Useful facts about dot products: The dot product of COLLINEAR vectors is 1 or 1 times the product of their lengths, depending on whether the vectors point in the same or opposite directions. The dot product of orthogonal (perpendicular) vectors is 0. The dot product of vectors u and v is the dot product of the orthogonal (perpendicular) projection of u on v. To prove these results we must define orthogonality and orthogonal projection. The (orthogonal) projection of u on v is the vector that is closest to u among the vectors collinear to v. In SB Figure 10.20, p. 217, R is the projection of u on v. Algebraically, tv is the (orthogonal) projection of u on v if and only if }u τv} is minimized with respect to τ at τ t. Vectors u and v are orthogonal (perpendicular to each other) if and only if the projection of u on v is the 0 vector. As in SB Figure 10.20, if R tv is the orthogonal projection of u on v, then u tv and v are orthogonal. 3. (a) If u tv, then u v psign tq}u}}v}. (Prove this directly.) (b) If u and v are orthogonal (perpendicular), then u v 0. (c) u v w v psign tq}w} }v}, where w tv is the projection of u on v. (d) If θ is the angle between vectors u 0 and v 0 based at 0, then u v }u}}v} cos θ. Proof of (d): Let w t v be the orthogonal projection of u on v 0. By definition of the projection, t minimizes the quadratic polynomial fptq pu tvq pu tvq u u 2tu v t 2 v v. So 0 f 1 pt q 2u v 2t v v and t pu vq{}v} 2. If u 0 and θ is the angle between u and v, then cos θ psign t q}w}{}u} psign t q t }v}{}u} t }v}{}u} pu vq{r}v}}u}s. Parts (a), (b) and (c) follow from (d). A real valued function of several variables is a function f : S Ñ R, where S R n. Such function can have derivatives. Roughly speaking, f is differentiable (or has a derivative) at x if there is a vector g such that fpxq is well approximated by fp xq g px xq (a linear function plus a constant) for x near x. (The precise definition is given in section 8, below.) The vector g is called the gradient of f at x, and is denoted Bfp xq. Its ith component is the ith partial derivative of f evaluated at x, denoted f i p xq or Bfp xq{bx i. (It is the derivative of f treated as a function of its ith argument, x i, with every other argument x j fixed at the value x j ). The set of points at which f takes the constant value k is tx : fpxq ku, the k level set of f. Suppose we allow the argument of f to be a function x depending on an argument of its own: xptq px 1 ptq,..., x n ptqq for t P S, where S is an interval in R. The function x is called a curve in R n. If each x i ptq is differentiable, then the vector x 1 ptq px 1 1 ptq,..., x1 n ptqq is called the tangent vector to x at xptq. This terminology is justified by the fact that for small δ 0, the vector p1{δqrxpt δq xptqs is well approximated by x 1 ptq. The chain rule (proved in section 8 below) states that if f is differentiable at xpτq and hptq fpxptqq for t near τ, then h 1 pτq Bfpxpτqq x 1 pτq. It follows that if the curve x lies in a level set of f, then 13

14 14 the tangent vector x 1 ptq is orthogonal to the gradient of f at xptq, i.e., x 1 ptq Bfpxptqq 0. The gradient is perpendicular to the level set. An important special case of a curve is xptq x tv, where v is a unit vector in R n. The image of this function x is a line passing through x. For h f x, h 1 p0q Bfp xq v is the directional derivative of f at x in the direction v. It measures the rate of change in the value of f per unit distance that the argument x travels in the direction v starting at x. The directional derivative is highest when v is the direction of the gradient Bf p xq. (Why?) Therefore, the gradient vector is the direction of steepest ascent on the graph of f (the direction in which the value of f rises fastest). A function f : S Ñ R is homogeneous of degree k at x P S if there is an open interval T containing 1 such that fptxq t k P T ; and f is homogeneous of degree k if it is homogeneous of degree k at every element of its domain. A function is called linear homogeneous if it is homogeneous of degree 1, and is called homogeneous if it is homogeneous of some degree. If fpxq is the maximum output producible with the vector of inputs x we call f a production function. Then we say that f exhibits constant returns to scale if it is homogeneous of degree 1. In that case, multiplying all the inputs by t multiplies the output level by t. If f is differentiable and homogeneous of degree k, then Bfptxq{Bx i f i ptxqt t k f i pxq, where f i pxq Bfpxq{Bx i is the partial derivative of f with respect to its ith argument evaluated at x. It follows that the partial derivative f i is homogeneous of degree k 1. (Why?) Also, the derivative of hptq fptxq t k fpxq at t 1 is kfpxq Bfpxq x. So if f is homogeneous of degree 0, then its gradient Bfpxq is orthogonal to its argument x. For differentiable f with fpxq 0, we define the scale elasticity of f at x to be the elasticity of hptq f ptxq at t 1. The scale elasticity equals k if f is homogeneous of degree k at x. (Prove this.) The scale elasticity of a production function might vary depending on the input vector x. In standard models of competitive firms with U-shaped long run average cost functions, the scale elasticity is above 1 (exhibiting locally increasing returns to scale) at low input levels and below 1 (exhibiting locally decreasing returns) at high input levels. 4. Introduction to Optimization in Several Variables The goal of this section is to develop intuition about the Lagrange method for solving problems such as: (P1) Maximize fpxq subject to constraints g i pxq b i, i 1,..., k, where f and all g i are functions from R n to R (functions of several variables if n 1). In this maximization problem, f is called the objective function; each g i is a constraint function and b i is a constraint variable. Components of x, the argument of f, are called choice (or control) variables. The constraint set is tx P R n : g i pxq b the set of x satisfying all the constraints. We say that g i and the ith constraint bind at x if g i pxq b i. To characterize solutions to the maximization problem, we use derivatives to study how the value of the objective function changes when its argument changes. Suppose that x solves the maximization problem (P1). This means that x is in the constraint set and that fp xq fpxq for every x in the constraint set. Suppose we vary xptq x tv by varying t. Suppose that, for every binding i at x, g i is differentiable at x and v is a direction satisfying Bg i p xq v 0. This means that v is a direction pointing into the constraint set from x. Then the directional derivative of each binding g i in direction v is negative. Therefore, xptq stays in the constraint set when t is raised slightly above

15 0. It follows that the value of fpxptqq cannot increase with t, so its derivative at t 0 (the directional derivative Bf p xq v) is nonpositive. To summarize, we have Bf p xq v 0 whenever v satisfies Bg i p xq v 0 for every constraint i that binds at x. If at least one such v exists, then, by the Theorem of the Alternative (to be stated and proved below), there are scalars λ i 0 such that Bfp xq λ i Bg i p xq, where the sum is over constraints i that bind at x. The geometric intuition for this result will be discussed in class. The Theorem of the Alternative is simply a statement about vectors, not involving differentiation. This conclusion can be expressed in an equivalent way. Define the Lagrange function Lpx, λq fpxq k i1 λ irg i pxq b i s, where λ pλ 1,..., λ k q. 4. If x solves the maximization problem (P1) at the beginning of this section, and if there is some v such that Bg i p xq v 0 for every binding constraint i at x, then there exist λ i 0, i 1,..., k, such that BLp x, λq{bx i Bfp xq{bx i i λ ibg i p xq{bx i 0, i 1,..., n, and λ i pg i p xq b i q 0 for i 1,..., k. The number λ i is the Lagrange multiplier of constraint i. The equation λ i pg i p xq b i q 0 is called a complementary slackness condition. It says that if constraint i is slack, with g i p xq b i, then its multiplier equals 0, and if the multiplier is positive the constraint binds. Note that a binding constraint can have a Lagrange multiplier equal to 0. Result 4 gives conditions that are necessary but not generally sufficient for x to be a solution to the maximization problem. If problem (P1) has a solution and only one point x satisfies the necessary conditions in result 4, then x is the unique solution. (See the sample problem below.) The condition that there is some v such that Bg i p xq v 0 for every binding constraint i at x is called a constraint qualification at x. Note that it depends only on the constraint functions, not on the objective function. In section 10, we will prove Result 4 and also the following somewhat different Lagrange theorem. Consider functions g i : R n Ñ R for i 0, 1, 2,..., k and the problem (P2) min g 0 pxq subject to g i pxq b 1,..., k. 5. If x solves (P2) and g i is differentiable at x for i 0, 1,..., k, then there exists λ pλ 0, λ 1,..., λ k q 0 with λ i rg i p xq b i s 1,..., k, such that the function Lpxq k i0 λ ig i pxq satisfies BLp xq 0. It is convenient to work with minimization problems because the second order conditions necessary for a solution are easier to state. But in economic theory it is more common for optimization to be formulated as a maximization problem. Fortunately, theorems for minimization apply to maximization since the solutions to the problem max f are the same as the solutions to the problem minp fq. Thus in any maximization problem with objective function f we can apply Result 5 letting g 0 f. If we transform problem (P1) into a minimization problem and apply Result 5, we obtain λ 0 Bfp xq k i1 λ ibg i p xq. (Show why.) In Result 4, this λ 0 equals 1. The stronger conclusion holds because of the constraint qualification. A constraint qualification is a condition ensuring that λ in Result 5 satisfies λ 0 0. Here is an example of a constraint qualification different from the one in Result 4. Every element of the constraint set satisfies this constraint qualification if each g i is affine (linear plus a constant) for i 1,..., k and g i pxq b 1,..., k for some x. Note that Result 5 itself does not require any constraint qualification. Result 4 provides an interpretation for the Lagrange multipliers λ i. Define the value function V of (P1) with V pbq suptfpxq : g i pxq b 1,..., ku for b pb 1,..., b k q. 15