Advanced Microeconomics


 Myra Marshall
 1 years ago
 Views:
Transcription
1 Advanced Microeconomics Ordinal preference theory Harald Wiese University of Leipzig Harald Wiese (University of Leipzig) Advanced Microeconomics 1 / 68
2 Part A. Basic decision and preference theory 1 Decisions in strategic (static) form 2 Decisions in extensive (dynamic) form 3 Ordinal preference theory 4 Decisions under risk Harald Wiese (University of Leipzig) Advanced Microeconomics 2 / 68
3 Ordinal preference theory Overview 1 The vector space of goods and its topology 2 Preference relations 3 Axioms: convexity, monotonicity, and continuity 4 Utility functions 5 Quasiconcave utility functions and convex preferences 6 Marginal rate of substitution Harald Wiese (University of Leipzig) Advanced Microeconomics 3 / 68
4 The vector space of goods Assumptions households are the only decision makers; nite number ` of goods de ned by place time contingencies example: an apple of a certain weight and class to be delivered in Leipzig on April 1st, 2015, if it does not rain the day before Harald Wiese (University of Leipzig) Advanced Microeconomics 4 / 68
5 The vector space of goods Exercise: Linear combination of vectors Problem Consider the vectors x = (x 1, x 2 ) = (2, 4) and y = (y 1, y 2 ) = (8, 12). Find x + y, 2x and 1 4 x y! Harald Wiese (University of Leipzig) Advanced Microeconomics 5 / 68
6 The vector space of goods Notation For ` 2 N: R` := f(x 1,..., x`) : x g 2 R, g = 1,..., `g. 0 2 R` null vector (0, 0,..., 0); vectors are called points (in R`). Positive amounts of goods only: n o x 2 R` : x 0 Vector comparisons: R`+ := x y :, x g y g for all g from f1, 2,..., `g ; x > y :, x y and x 6= y; x y :, x g > y g for all g from f1, 2,..., `g. Harald Wiese (University of Leipzig) Advanced Microeconomics 6 / 68
7 Distance between x and y y euclidian distance: c c b city block distance: a+b x a infinity distance: max(a, b) Harald Wiese (University of Leipzig) Advanced Microeconomics 7 / 68
8 Distance De nition In R`: q euclidian (or 2) norm: kx yk := kx yk 2 := `g =1 (x g y g ) 2 in nity norm: kx yk := max g =1,...,` jx g y g j cityblock norm: kx yk 1 := ` g =1 jx g y g j Harald Wiese (University of Leipzig) Advanced Microeconomics 8 / 68
9 Distance Exercise Problem What is the distance (in R 2 ) between (2, 5) and (7, 1), measured by the 2norm kk 2 and by the ini nity norm kk? Harald Wiese (University of Leipzig) Advanced Microeconomics 9 / 68
10 Distance and balls Ball De nition Let x 2 R` and ε > 0. ) n o K = x 2 R` : kx x k < ε (open) εball with center x. x * x ε kx x k = ε holds for all x on the circular line; K all the points within Problem Assuming the goods space R 2 +, sketch three 1balls with centers (2, 2), (0, 0) and (2, 0), respectively. Harald Wiese (University of Leipzig) Advanced Microeconomics 10 / 68
11 Distance and balls Boundedness De nition A set M is bounded if there exists an εball K such that M K. Example The set [0, ) = fx 2 R : x 0g is not bounded. Harald Wiese (University of Leipzig) Advanced Microeconomics 11 / 68
12 Sequences and convergence Sequence De nition A sequence x j j2n in R` is a function N! R`. Examples In R: 1, 2, 3, 4,... or x j := j; In R 2 : (1, 2), (2, 3), (3, 4),... 1, 2 1, 1, 1 3, 1, 1 4, 1, 1 5,.... Harald Wiese (University of Leipzig) Advanced Microeconomics 12 / 68
13 Sequences and convergence Convergence De nition x j j2n in R` converges towards x 2 R` if for every ε > 0 there is an N 2 N such that x j x < ε for all j > N holds. Convergent a sequence that converges towards some x 2 R`. Examples 1, 2, 3, 4,... is not convergent towards any x 2 R; 1, 1, 1, 1,... converges towards 1; 1, 1 2, 1 3,... converges towards zero. Harald Wiese (University of Leipzig) Advanced Microeconomics 13 / 68
14 Sequences and convergence Lemma 1 Lemma Let x j j2n be a sequence in R`. If x j converges towards x and y ) x = y. j2n x j x j2n = j 1,..., x j` converges towards (x 1,..., x`) i xg j j2n converges towards x g for every g = 1,..., `. Problem Convergent? or (1, 2), (1, 3), (1, 4),... 1, 1, 1, 1, 1, 1, 1, 1, Harald Wiese (University of Leipzig) Advanced Microeconomics 14 / 68
15 Boundary point De nition A point x 2 R` is a boundary point of M R` i there is a sequence of points in M and another sequence of points in R`nM so that both converge towards x. M * x R 2 \ M Harald Wiese (University of Leipzig) Advanced Microeconomics 15 / 68
16 Interior point De nition A point in M that is not a boundary point is called an interior point of M. * x M R 2 \ M Note: Instead of R`, we can consider alternative sets, for example R`+. Harald Wiese (University of Leipzig) Advanced Microeconomics 16 / 68
17 Closed set De nition A set M R` is closed i every converging sequence in M with convergence point x 2 R` ful lls x 2 M. Problem Are the sets closed? f0g [ (1, 2), K = x 2 R` : kx x k < ε, K = x 2 R` : kx x k ε Harald Wiese (University of Leipzig) Advanced Microeconomics 17 / 68
18 Preference relations Overview 1 The vector space of goods and its topology 2 Preference relations 3 Axioms: convexity, monotonicity, and continuity 4 Utility functions 5 Quasiconcave utility functions and convex preferences 6 Marginal rate of substitution Harald Wiese (University of Leipzig) Advanced Microeconomics 18 / 68
19 Preference relations and indi erence curves Preference relation De nition (weak) preference relation  a relation on R`+ that is complete (x  y or y  x for all x, y) transitive (x  y and y  z implies x  z for all x, y, z) and re exive (x  x for all x); indi erence relation: strict preference relation: x y :, x  y and y  x; x y :, x  y and not y  x. Harald Wiese (University of Leipzig) Advanced Microeconomics 19 / 68
20 Relations and equivalence classes Exercise Problem For any two inhabitants from Leipzig, we ask whether one is the father of the other. This relation ful lls the following properties (?). Fill in "yes" or "no": property re exive transitive complete is the father of Harald Wiese (University of Leipzig) Advanced Microeconomics 20 / 68
21 Preference relations and indi erence curves Exercise Problem Fill in: property indi erence strict preference re exive transitive complete Harald Wiese (University of Leipzig) Advanced Microeconomics 21 / 68
22 Preference relations and indi erence curves transitivity of strict preference We want to show x y ^ y z ) x z Proof: x y implies x  y, y z implies y  z. Therefore, x y ^ y z implies x  z. Assume z  x. Together with x y, transitivity implies z  y, contradicting y z. Therefore, we do not have z  x, but x z. Harald Wiese (University of Leipzig) Advanced Microeconomics 22 / 68
23 Preference relations and indi erence curves Better and indi erence set De nition Let % be a preference relation on R`+. ) B y := x 2 R`+ : x % y better set B y of y; W y := x 2 R`+ : x  y worse set W y of y; I y := B y \ W y = x 2 R`+ : x y y s indi erence set; indi erence curve the geometric locus of an indi erence set. Harald Wiese (University of Leipzig) Advanced Microeconomics 23 / 68
24 Preference relations and indi erence curves Indi erence curve numbers to indicate preferences: x x x 1 x 1 Harald Wiese (University of Leipzig) Advanced Microeconomics 24 / 68
25 Indi erence curves must not intersect x 2 C Two di erent indi erence curves, Thus C B A B But C A ^ A B ) C B Contradiction! x 1 Harald Wiese (University of Leipzig) Advanced Microeconomics 25 / 68
26 Preference relations and indi erence curves Exercise Problem Sketch indi erence curves for a goods space with just 2 goods and, alternatively, good 2 is a bad, good 1 represents red matches and good 2 blue matches, good 1 stands for right shoes and good 2 for left shoes. Harald Wiese (University of Leipzig) Advanced Microeconomics 26 / 68
27 Preference relations and indi erence curves Lexicographic preferences In twogood case: x  lex y :, x 1 < y 1 or (x 1 = y 1 and x 2 y 2 ). Problem What do the indi erence curves for lexicographic preferences look like? Harald Wiese (University of Leipzig) Advanced Microeconomics 27 / 68
28 Axioms: convexity, monotonicity, and continuity Overview 1 The vector space of goods and its topology 2 Preference relations 3 Axioms: convexity, monotonicity, and continuity 4 Utility functions 5 Quasiconcave utility functions and convex preferences 6 Marginal rate of substitution Harald Wiese (University of Leipzig) Advanced Microeconomics 28 / 68
29 Convex preferences Convex combination De nition Let x and y be elements of R`. ) kx + (1 k) y, k 2 [0, 1] the convex combination of x and y. Harald Wiese (University of Leipzig) Advanced Microeconomics 29 / 68
30 Convex preferences Convex and strictly convex sets M not convex De nition c x M y boundary point convex, but not strictly convex x interior point y M strictly convex A set M R` is convex if for any two points x and y from M, their convex combination is also contained in M. A set M is strictly convex if for any two points x and y from M, x 6= y, kx + (1 k) y, k 2 (0, 1) is an interior point of M for any k 2 (0, 1). Harald Wiese (University of Leipzig) Advanced Microeconomics 30 / 68
31 Convex preferences Convex and concave preference relation De nition A preference relation % is convex if all its better sets B y are convex, strictly convex if all its better sets B y are strictly convex, concave if all its worse sets W y are convex, strictly concave if all its worse sets W y are strictly convex. Preferences are de ned on R`+ (!): x 2 x 2 better set interior point in R 2 + worse set interior point in R 2 + Harald Wiese (University of Leipzig) x 1 Advanced Microeconomics x 1 31 / 68
32 Convex preferences Exercise Problem Are these preferences convex or strictly convex? x 2 x 2 better set better set (a) x 1 (b) x 1 x 2 x 2 better set better set (c) x 1 (d) x 1 Harald Wiese (University of Leipzig) Advanced Microeconomics 32 / 68
33 Monotonicity of preferences Monotonicity De nition A preference relation % obeys: weak monotonicity if x > y implies x % y; strict monotonicity if x > y implies x y; local nonsatiation at y if in every εball with center y a bundle x exists with x y. Problem Sketch the better set of y = (y 1, y 2 ) in case of weak monotonicity! Harald Wiese (University of Leipzig) Advanced Microeconomics 33 / 68
34 Exercise: Monotonicity and convexity Which of the properties (strict) monotonicity and/or (strict) convexity do the preferences depicted by the indi erence curves in the graphs below satisfy? x 2 6 x 2 [ 1] [2] x 1 x 1 x [3] x [4] x 1 Harald Wiese (University of Leipzig) Advanced Microeconomics 34 / 68 x 1
35 Monotonicity of preferences Bliss point x x 1 Harald Wiese (University of Leipzig) Advanced Microeconomics 35 / 68
36 Continuous preferences De nition A preference relation  is continuous if for all y 2 R`+ the sets n o W y = x 2 R`+ : x  y and B y = n o x 2 R`+ : y  x are closed. Harald Wiese (University of Leipzig) Advanced Microeconomics 36 / 68
37 Lexicographic preferences are not continuous Consider the sequence x j j2n = j, 2! (2, 2) All its members belong to the better set of (2, 4). But (2, 2) does not. x x 1 Harald Wiese (University of Leipzig) Advanced Microeconomics 37 / 68
38 Utility functions Overview 1 The vector space of goods and its topology 2 Preference relations 3 Axioms: convexity, monotonicity, and continuity 4 Utility functions 5 Quasiconcave utility functions and convex preferences 6 Marginal rate of substitution Harald Wiese (University of Leipzig) Advanced Microeconomics 38 / 68
39 Utility functions De nition De nition For an agent i 2 N with preference relation % i, U i : R`+ 7! R utility function if U i (x) U i (y), x % i y, x, y 2 R`+ holds. U i represents the preferences % i ; ordinal utility theory. Harald Wiese (University of Leipzig) Advanced Microeconomics 39 / 68
40 Utility functions Examples of utility functions Examples CobbDouglas utility functions (weakly monotonic): U (x 1, x 2 ) = x1 a x2 1 a with 0 < a < 1; perfect substitutes: U (x 1, x 2 ) = ax 1 + bx 2 with a > 0 and b > 0; perfect complements: U (x 1, x 2 ) = min (ax 1, bx 2 ) with a > 0 and b > 0. Harald Wiese (University of Leipzig) Advanced Microeconomics 40 / 68
41 Utility functions DixitStiglitz preferences for love of variety U (x 1,..., x`) = where X := ` j=1 x j implies ` j=1! ε x ε ε 1 ε 1 j with ε > 1 U X X,..., ` ` = ` X j=1 ` = ` X ε ε 1 ε 1! ε ε 1 ε 1 ` = ε 1! ε ε 1 ε ` j=1 X ε ε 1 ε 1! ε ε 1 1 ε ` = ` ε ε 1 X 1` = ` ε ε 1 1 X = ` ε 1 1 X and hence, by U ` > 0, a love of variety. Harald Wiese (University of Leipzig) Advanced Microeconomics 41 / 68
42 Utility functions Exercises Problem Draw the indi erence curve for perfect substitutes with a = 1, b = 4 and the utility level 5! Problem Draw the indi erence curve for perfect complements with a = 1, b = 4 (a car with four wheels and one engine) and the utility level for 5 cars! Does x 1 denote the number of wheels or the number of engines? Harald Wiese (University of Leipzig) Advanced Microeconomics 42 / 68
43 Equivalent utility functions De nition (equivalent utility functions) Two utility functions U and V are called equivalent if they represent the same preferences. Lemma (equivalent utility functions) Two utility functions U and V are equivalent i there is a strictly increasing function τ : R! R such that V = τ U. Problem Which of the following utility functions represent the same preferences? a) U 1 (x 1, x 2, x 3 ) = (x 1 + 1) (x 2 + 1) (x 3 + 1) b) U 2 (x 1, x 2, x 3 ) = ln (x 1 + 1) + ln (x 2 + 1) + ln (x 3 + 1) c) U 3 (x 1, x 2, x 3 ) = (x 1 + 1) (x 2 + 1) (x 3 + 1) d) U 4 (x 1, x 2, x 3 ) = [(x 1 + 1) (x 2 + 1) (x 3 + 1)] 1 e) U 5 (x 1, x 2, x 3 ) = x 1 x 2 x 3 Harald Wiese (University of Leipzig) Advanced Microeconomics 43 / 68
44 Existence Existence is not guaranteed Assume a utility function U for lexicographic preferences: U (A 0 ) < U (B 0 ) < U (A 00 ) < U (B 00 ) < U (A 000 ) < U (B 000 ) ; x 2 within (U (A 0 ), U (B 0 )) at least one rational number (q 0 ) etc. B' B '' B'' ' q 0 < q 00 < q 000 ; injective function f : [r 0, r 000 ]! Q; not enough rational numbers; r ' A' r '' A '' r'' ' A'' ' x 1 contradiction > no utility function for lexicographic preferences Harald Wiese (University of Leipzig) Advanced Microeconomics 44 / 68
45 Existence Existence of a utility function for continuous preferences Theorem If the preference relation  i of an agent i is continuous, there is a continuous utility function U i that represents  i. Wait a second for the de nition of a continuous function, please! x 2 ( x,x) ( x 1,x 2 ) 45 U ( x) : = x x 1 Harald Wiese (University of Leipzig) Advanced Microeconomics 45 / 68
46 Existence Exercise Problem Assume a utility function U that represents the preference relation . Can you express weak monotonicity, strict monotonicity and local nonsatiation of  through U rather than ? Harald Wiese (University of Leipzig) Advanced Microeconomics 46 / 68
47 Continuous functions De nition De nition f : R`! R is continuous at x 2 R` i for every x j in R` j2n that converges towards x, f x j j2n converges towards f (x). f( x) f( x) Fix x 2 R`. Consider sequences in the domain x j that converge towards x. j2n Convergence of sequences in the range f x j towards f (x) > f j2n continuous at x No convergence or convergence towards y 6= f (x) > no continuity Harald Wiese (University of Leipzig) Advanced Microeconomics 47 / 68 x x
48 Continuous functions Counterexample f( x) f( x) excluded x x Speci c sequence in the domain x j that converges towards x j2n but sequence in the range f x j does not converge towards f (x) j2n Harald Wiese (University of Leipzig) Advanced Microeconomics 48 / 68
49 Quasiconcave utility functions and convex preferences Overview 1 The vector space of goods and its topology 2 Preference relations 3 Axioms: convexity, monotonicity, and continuity 4 Utility functions 5 Quasiconcave utility functions and convex preferences 6 Marginal rate of substitution Harald Wiese (University of Leipzig) Advanced Microeconomics 49 / 68
50 Quasiconcavity De nition f : R`! R is quasiconcave if f (kx + (1 k) y) min (f (x), f (y)) holds for all x, y 2 R` and all k 2 [0, 1]. f is strictly quasiconcave if f (kx + (1 k) y) > min (f (x), f (y)) holds for all x, y 2 R` with x 6= y and all k 2 (0, 1). Note: quasiconcave functions need not be concave (to be introduced later). Harald Wiese (University of Leipzig) Advanced Microeconomics 50 / 68
51 Quasiconcavity Examples f( x) Also quasi concave: f ( kx+ ( 1 k) y) f( y) x kx+ ( 1 k)y y x Example Every monotonically increasing or decreasing function f : R! R is quasiconcave. Harald Wiese (University of Leipzig) Advanced Microeconomics 51 / 68
52 Quasiconcavity A counter example f( x) f f( y) ( kx+ ( 1 k) y) x kx+ ( 1 k)y y x Harald Wiese (University of Leipzig) Advanced Microeconomics 52 / 68
53 Better and worse sets, indi erence sets De nition Let U be a utility function on R`+. B U (y ) := B y = x 2 R`+ : U (x) U (y) the better set B y of y; W U (y ) := W y = x 2 R`+ : U (x) U (y) the worse set W y of y; I U (y ) := I y = B y \ W y = x 2 R`+ : U (x) = U (y) y s indi erence set (indi erence curve) I y. Harald Wiese (University of Leipzig) Advanced Microeconomics 53 / 68
54 Concave indi erence curve De nition Let U be a utility function on R`+. I y is concave if U (x) = U (y) implies U (kx + (1 k) y) U (x) for all x, y 2 R`+ and all k 2 [0, 1]. I y is strictly concave if U (x) = U (y) implies U (kx + (1 k) y) > U (x) for all x, y 2 R`+ with x 6= y and all k 2 (0, 1). Harald Wiese (University of Leipzig) Advanced Microeconomics 54 / 68
55 Concave indi erence curve Examples and counter examples x kx+ ( 1 k)y x y y (a) (b) x x y y (c) (d) Harald Wiese (University of Leipzig) Advanced Microeconomics 55 / 68
56 Lemma Lemma Let U be a continuous utility function on R`+. A preference relation % is convex i : all the indi erence curves are concave, or U is quasiconcave. Us better sets strictly convex U strictly quasi concave c U quasiconcave c Us better sets convex Us better sets strictly convex and local nonsatiation Us indifference curves strictly concave U s indifference curves concave Harald Wiese (University of Leipzig) Advanced Microeconomics 56 / 68
57 Marginal rate of substitution Overview 1 The vector space of goods and its topology 2 Preference relations 3 Axioms: convexity, monotonicity, and continuity 4 Utility functions 5 Quasiconcave utility functions and convex preferences 6 Marginal rate of substitution Harald Wiese (University of Leipzig) Advanced Microeconomics 57 / 68
58 Marginal rate of substitution Mathematics: Di erentiable functions De nition Let f : M! R be a realvalued function with open domain M R`. f is di erentiable if all the partial derivatives f i (x) := f x i (i = 1,..., `) exist and are continuous. f 0 (x) := 0 f 1 (x) f 2 (x)... f` (x) 1 C A f s derivative at x. Harald Wiese (University of Leipzig) Advanced Microeconomics 58 / 68
59 Marginal rate of substitution Mathematics: Adding rule Theorem Let f : R`! R be a di erentiable function and let g 1,..., g` be di erentiable functions R! R. Let F : R! R be de ned by F (x) = f (g 1 (x),..., g` (x)). ) ` df dx = f dg i i=1 g i dx. Harald Wiese (University of Leipzig) Advanced Microeconomics 59 / 68
60 Marginal rate of substitution Economics x 2 x 2 y 2 ( ) y 1,y 2 x 1 y 1 x 1 I y = (x 1, x 2 ) 2 R 2 + : (x 1, x 2 ) (y 1, y 2 ). I y : x 1 7! x 2. Harald Wiese (University of Leipzig) Advanced Microeconomics 60 / 68
61 Marginal rate of substitution De nition and exercises De nition If the function I y is di erentiable and if preferences are monotonic, di y (x 1 ) dx 1 the MRS between good 1 and good 2 (or of good 2 for good 1). Problem What happens if good 2 is a bad? Harald Wiese (University of Leipzig) Advanced Microeconomics 61 / 68
62 Marginal rate of substitution Perfect substitutes Problem Calculate the MRS for perfect substitutes (U (x 1, x 2 ) = ax 1 + bx 2 with a > 0 and b > 0.)! Solve ax 1 + bx 2 = k for x 2! Form the derivative of x 2 with respect to x 1! Take the absolute value! Harald Wiese (University of Leipzig) Advanced Microeconomics 62 / 68
63 Marginal rate of substitution Lemma 1 Lemma Let % be a preference relation on R`+ and let U be the corresponding utility function. If U is di erentiable, the MRS is de ned by: U x 1, U x 2 marginal utility. U MRS (x 1 ) = di y (x 1 ) dx 1 = x 1 U x 2. Harald Wiese (University of Leipzig) Advanced Microeconomics 63 / 68
64 Marginal rate of substitution Lemma: proof Proof. U (x 1, I y (x 1 )) constant along indi erence curve; di erentiating U (x 1, I y (x 1 )) with respect to x 1 (adding rule): di y (x 1 ) dx 1 0 = U x 1 + U x 2 di y (x 1 ) dx 1. can be found even if Iy were not given explicitly (implicitfunction theorem). Problem Again: What is the MRS in case of perfect substitutes? Harald Wiese (University of Leipzig) Advanced Microeconomics 64 / 68
65 Marginal rate of substitution Lemma 2 Lemma Let U be a di erentiable utility function and I y an indi erence curve of U. I y is concave i the MRS is a decreasing function in x 1. x 2 x 1 < y 1 ; MRS (x 1 ) > MRS (y 1 ). x 1 y 1 x 1 Harald Wiese (University of Leipzig) Advanced Microeconomics 65 / 68
66 Marginal rate of substitution CobbDouglas utility function MRS = U (x 1, x 2 ) = x1 a x2 1 a, 0 < a < 1 U x 1 U a 1 1 x2 1 a = ax (1 x 2 a) x1 ax 2 a = a x 2. 1 a x 1 CobbDouglas preferences are monotonic so that an increase of x 1 is associated with a decrease of x 2 along an indi erence curve. Therefore, CobbDouglas preferences are convex (CobbDouglas utility functions are quasiconcave). Harald Wiese (University of Leipzig) Advanced Microeconomics 66 / 68
67 Further exercises Problem 1 De ne strict antimonotonicity. Sketch indi erence curves for each of the four cases: strict convexity strict concavity strict monotonicity strict antimonotonicity Problem 2 (Strictly) monotonic, (strictly) convex or continuous? (a) U(x 1, x 2 ) = x 1 x 2, (b) U(x 1, x 2 ) = min fa x 1, b x 2 g where a, b > 0 holds, (c) U(x 1, x 2 ) = a x 1 + b x 2 where a, b > 0 holds, (d) lexicographic preferences Harald Wiese (University of Leipzig) Advanced Microeconomics 67 / 68
68 Further exercises Problem 3 (di cult) Let U be a continuous utility function representing the preference relation  on R l +. Show that  is continuous as well. Also, give an example for a continuous preference relation that is represented by a discontinuous utility function. Hint: De ne a function U 0 that di ers from U for x = 0, only Harald Wiese (University of Leipzig) Advanced Microeconomics 68 / 68
c 2008 Je rey A. Miron We have described the constraints that a consumer faces, i.e., discussed the budget constraint.
Lecture 2b: Utility c 2008 Je rey A. Miron Outline: 1. Introduction 2. Utility: A De nition 3. Monotonic Transformations 4. Cardinal Utility 5. Constructing a Utility Function 6. Examples of Utility Functions
More informationMathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions
Natalia Lazzati Mathematics for Economics (Part I) Note 5: Convex Sets and Concave Functions Note 5 is based on Madden (1986, Ch. 1, 2, 4 and 7) and Simon and Blume (1994, Ch. 13 and 21). Concave functions
More informationEconomic Principles Solutions to Problem Set 1
Economic Principles Solutions to Problem Set 1 Question 1. Let < be represented b u : R n +! R. Prove that u (x) is strictl quasiconcave if and onl if < is strictl convex. If part: ( strict convexit of
More informationPrice Elasticity of Supply; Consumer Preferences
1 Price Elasticity of Supply 1 14.01 Principles of Microeconomics, Fall 2007 ChiaHui Chen September 12, 2007 Lecture 4 Price Elasticity of Supply; Consumer Preferences Outline 1. Chap 2: Elasticity 
More informationRevealed Preference. Ichiro Obara. October 8, 2012 UCLA. Obara (UCLA) Revealed Preference October 8, / 17
Ichiro Obara UCLA October 8, 2012 Obara (UCLA) October 8, 2012 1 / 17 Obara (UCLA) October 8, 2012 2 / 17 Suppose that we obtained data of price and consumption pairs D = { (p t, x t ) R L ++ R L +, t
More informationFirst Welfare Theorem
First Welfare Theorem Econ 2100 Fall 2015 Lecture 17, November 2 Outline 1 First Welfare Theorem 2 Preliminaries to Second Welfare Theorem Last Class Definitions A feasible allocation (x, y) is Pareto
More informationWalrasian Demand. u(x) where B(p, w) = {x R n + : p x w}.
Walrasian Demand Econ 2100 Fall 2015 Lecture 5, September 16 Outline 1 Walrasian Demand 2 Properties of Walrasian Demand 3 An Optimization Recipe 4 First and Second Order Conditions Definition Walrasian
More informationUCLA. Department of Economics Ph. D. Preliminary Exam MicroEconomic Theory
UCLA Department of Economics Ph. D. Preliminary Exam MicroEconomic Theory (SPRING 2011) Instructions: You have 4 hours for the exam Answer any 5 out of the 6 questions. All questions are weighted equally.
More informationEconomics 326 (Utility, Marginal Utility, MRS, Substitutes and Complements ) Ethan Kaplan
Economics 326 (Utility, Marginal Utility, MRS, Substitutes and Complements ) Ethan Kaplan September 10, 2012 1 Utility From last lecture: a utility function U (x; y) is said to represent preferences if
More informationAdvanced Microeconomics
Advanced Microeconomics General equilibrium theory I: the main results Harald Wiese University of Leipzig Harald Wiese (University of Leipzig) Advanced Microeconomics 1 / 52 Part F. Perfect competition
More informationEconomics 121b: Intermediate Microeconomics Problem Set 2 1/20/10
Dirk Bergemann Department of Economics Yale University s by Olga Timoshenko Economics 121b: Intermediate Microeconomics Problem Set 2 1/20/10 This problem set is due on Wednesday, 1/27/10. Preliminary
More informationIndifference Curves and the Marginal Rate of Substitution
Introduction Introduction to Microeconomics Indifference Curves and the Marginal Rate of Substitution In microeconomics we study the decisions and allocative outcomes of firms, consumers, households and
More informationConsumer Theory. The consumer s problem
Consumer Theory The consumer s problem 1 The Marginal Rate of Substitution (MRS) We define the MRS(x,y) as the absolute value of the slope of the line tangent to the indifference curve at point point (x,y).
More informationPreferences. M. Utku Ünver Micro Theory. Boston College. M. Utku Ünver Micro Theory (BC) Preferences 1 / 20
Preferences M. Utku Ünver Micro Theory Boston College M. Utku Ünver Micro Theory (BC) Preferences 1 / 20 Preference Relations Given any two consumption bundles x = (x 1, x 2 ) and y = (y 1, y 2 ), the
More informationFrom preferences to numbers Cardinal v ordinal Examples MRS. Utility. Intermediate Micro. Lecture 4. Chapter 4 of Varian
Utility Intermediate Micro Lecture 4 Chapter 4 of Varian Preferences and decisionmaking 1. Last lecture: Ranking consumption bundles by preference/indifference 2. Today: Assigning values (numbers) to
More informationRepresentation of functions as power series
Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions
More informationEconomics 326: Duality and the Slutsky Decomposition. Ethan Kaplan
Economics 326: Duality and the Slutsky Decomposition Ethan Kaplan September 19, 2011 Outline 1. Convexity and Declining MRS 2. Duality and Hicksian Demand 3. Slutsky Decomposition 4. Net and Gross Substitutes
More informationEconomics 401. Sample questions 1
Economics 40 Sample questions. Do each of the following choice structures satisfy WARP? (a) X = {a, b, c},b = {B, B 2, B 3 }, B = {a, b}, B 2 = {b, c}, B 3 = {c, a}, C (B ) = {a}, C (B 2 ) = {b}, C (B
More informationMTH4100 Calculus I. Lecture notes for Week 8. Thomas Calculus, Sections 4.1 to 4.4. Rainer Klages
MTH4100 Calculus I Lecture notes for Week 8 Thomas Calculus, Sections 4.1 to 4.4 Rainer Klages School of Mathematical Sciences Queen Mary University of London Autumn 2009 Theorem 1 (First Derivative Theorem
More informationFollow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu
COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part
More informationLecture 4: Equality Constrained Optimization. Tianxi Wang
Lecture 4: Equality Constrained Optimization Tianxi Wang wangt@essex.ac.uk 2.1 Lagrange Multiplier Technique (a) Classical Programming max f(x 1, x 2,..., x n ) objective function where x 1, x 2,..., x
More informationNo: 10 04. Bilkent University. Monotonic Extension. Farhad Husseinov. Discussion Papers. Department of Economics
No: 10 04 Bilkent University Monotonic Extension Farhad Husseinov Discussion Papers Department of Economics The Discussion Papers of the Department of Economics are intended to make the initial results
More informationMore is Better. an investigation of monotonicity assumption in economics. Joohyun Shon. August 2008
More is Better an investigation of monotonicity assumption in economics Joohyun Shon August 2008 Abstract Monotonicity of preference is assumed in the conventional economic theory to arrive at important
More informationDefinition and Properties of the Production Function: Lecture
Definition and Properties of the Production Function: Lecture II August 25, 2011 Definition and : Lecture A Brief Brush with Duality CobbDouglas Cost Minimization Lagrangian for the CobbDouglas Solution
More informationFixed Point Theorems
Fixed Point Theorems Definition: Let X be a set and let T : X X be a function that maps X into itself. (Such a function is often called an operator, a transformation, or a transform on X, and the notation
More information36 CHAPTER 1. LIMITS AND CONTINUITY. Figure 1.17: At which points is f not continuous?
36 CHAPTER 1. LIMITS AND CONTINUITY 1.3 Continuity Before Calculus became clearly de ned, continuity meant that one could draw the graph of a function without having to lift the pen and pencil. While this
More information4.6 Null Space, Column Space, Row Space
NULL SPACE, COLUMN SPACE, ROW SPACE Null Space, Column Space, Row Space In applications of linear algebra, subspaces of R n typically arise in one of two situations: ) as the set of solutions of a linear
More informationFollow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu
COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part
More informationChoice under Uncertainty
Choice under Uncertainty Part 1: Expected Utility Function, Attitudes towards Risk, Demand for Insurance Slide 1 Choice under Uncertainty We ll analyze the underlying assumptions of expected utility theory
More informationNotes V General Equilibrium: Positive Theory. 1 Walrasian Equilibrium and Excess Demand
Notes V General Equilibrium: Positive Theory In this lecture we go on considering a general equilibrium model of a private ownership economy. In contrast to the Notes IV, we focus on positive issues such
More informationCourse 221: Analysis Academic year , First Semester
Course 221: Analysis Academic year 200708, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................
More informationIntroduction. Agents have preferences over the two goods which are determined by a utility function. Speci cally, type 1 agents utility is given by
Introduction General equilibrium analysis looks at how multiple markets come into equilibrium simultaneously. With many markets, equilibrium analysis must take explicit account of the fact that changes
More informationLecture Notes on Elasticity of Substitution
Lecture Notes on Elasticity of Substitution Ted Bergstrom, UCSB Economics 20A October 26, 205 Today s featured guest is the elasticity of substitution. Elasticity of a function of a single variable Before
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationFunctions of 2 Variables
Functions of 2 Variables Functions and Graphs In the last chapter, we extended di erential calculus to vectorvalued functions. In this chapter, we extend calculus primarily to functions of two variables,
More informationEconomics 2020a / HBS 4010 / HKS API111 FALL 2010 Solutions to Practice Problems for Lectures 1 to 4
Economics 00a / HBS 4010 / HKS API111 FALL 010 Solutions to Practice Problems for Lectures 1 to 4 1.1. Quantity Discounts and the Budget Constraint (a) The only distinction between the budget line with
More information2.3 Convex Constrained Optimization Problems
42 CHAPTER 2. FUNDAMENTAL CONCEPTS IN CONVEX OPTIMIZATION Theorem 15 Let f : R n R and h : R R. Consider g(x) = h(f(x)) for all x R n. The function g is convex if either of the following two conditions
More information1. Let X and Y be normed spaces and let T B(X, Y ).
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: NVP, Frist. 20050314 Skrivtid: 9 11.30 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
More informationLecture 2: Consumer Theory
Lecture 2: Consumer Theory Preferences and Utility Utility Maximization (the primal problem) Expenditure Minimization (the dual) First we explore how consumers preferences give rise to a utility fct which
More informationPOWER SETS AND RELATIONS
POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty
More informationCost Minimization and the Cost Function
Cost Minimization and the Cost Function Juan Manuel Puerta October 5, 2009 So far we focused on profit maximization, we could look at a different problem, that is the cost minimization problem. This is
More informationUtility. M. Utku Ünver Micro Theory. M. Utku Ünver Micro Theory Utility 1 / 15
Utility M. Utku Ünver Micro Theory M. Utku Ünver Micro Theory Utility 1 / 15 Utility Function The preferences are the fundamental description useful for analyzing choice and utility is simply a way of
More informationLecture Notes on Elasticity of Substitution
Lecture Notes on Elasticity of Substitution Ted Bergstrom, UCSB Economics 210A March 3, 2011 Today s featured guest is the elasticity of substitution. Elasticity of a function of a single variable Before
More informationName. Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions.
Name Final Exam, Economics 210A, December 2011 Here are some remarks to help you with answering the questions. Question 1. A firm has a production function F (x 1, x 2 ) = ( x 1 + x 2 ) 2. It is a price
More informationTastes and Indifference Curves
C H A P T E R 4 Tastes and Indifference Curves This chapter begins a chapter treatment of tastes or what we also call preferences. In the first of these chapters, we simply investigate the basic logic
More informationMetric Spaces. Chapter 1
Chapter 1 Metric Spaces Many of the arguments you have seen in several variable calculus are almost identical to the corresponding arguments in one variable calculus, especially arguments concerning convergence
More informationAnalysis MA131. University of Warwick. Term
Analysis MA131 University of Warwick Term 1 01 13 September 8, 01 Contents 1 Inequalities 5 1.1 What are Inequalities?........................ 5 1. Using Graphs............................. 6 1.3 Case
More informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More informationA Simpli ed Axiomatic Approach to Ambiguity Aversion
A Simpli ed Axiomatic Approach to Ambiguity Aversion William S. Neilson Department of Economics University of Tennessee Knoxville, TN 379960550 wneilson@utk.edu March 2009 Abstract This paper takes the
More informationCritical points of once continuously differentiable functions are important because they are the only points that can be local maxima or minima.
Lecture 0: Convexity and Optimization We say that if f is a once continuously differentiable function on an interval I, and x is a point in the interior of I that x is a critical point of f if f (x) =
More information1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
More informationMeasuring Rationality with the Minimum Cost of Revealed Preference Violations. Mark Dean and Daniel Martin. Online Appendices  Not for Publication
Measuring Rationality with the Minimum Cost of Revealed Preference Violations Mark Dean and Daniel Martin Online Appendices  Not for Publication 1 1 Algorithm for Solving the MASP In this online appendix
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationI. Pointwise convergence
MATH 40  NOTES Sequences of functions Pointwise and Uniform Convergence Fall 2005 Previously, we have studied sequences of real numbers. Now we discuss the topic of sequences of real valued functions.
More informationNotes on metric spaces
Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.
More informationWeak topologies. David Lecomte. May 23, 2006
Weak topologies David Lecomte May 3, 006 1 Preliminaries from general topology In this section, we are given a set X, a collection of topological spaces (Y i ) i I and a collection of maps (f i ) i I such
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationOn Lexicographic (Dictionary) Preference
MICROECONOMICS LECTURE SUPPLEMENTS Hajime Miyazaki File Name: lexico95.usc/lexico99.dok DEPARTMENT OF ECONOMICS OHIO STATE UNIVERSITY Fall 993/994/995 Miyazaki.@osu.edu On Lexicographic (Dictionary) Preference
More informationLECTURE NOTES IN MEASURE THEORY. Christer Borell Matematik Chalmers och Göteborgs universitet 412 96 Göteborg (Version: January 12)
1 LECTURE NOTES IN MEASURE THEORY Christer Borell Matematik Chalmers och Göteborgs universitet 412 96 Göteborg (Version: January 12) 2 PREFACE These are lecture notes on integration theory for a eightweek
More informationReadings. D Chapter 1. Lecture 2: Constrained Optimization. Cecilia Fieler. Example: Input Demand Functions. Consumer Problem
Economics 245 January 17, 2012 : Example Readings D Chapter 1 : Example The FOCs are max p ( x 1 + x 2 ) w 1 x 1 w 2 x 2. x 1,x 2 0 p 2 x i w i = 0 for i = 1, 2. These are two equations in two unknowns,
More informationLecture 8: Market Equilibria
Computational Aspects of Game Theory Bertinoro Spring School 2011 Lecturer: Bruno Codenotti Lecture 8: Market Equilibria The market setting transcends the scenario of games. The decentralizing effect of
More informationGeneral Equilibrium Theory: Examples
General Equilibrium Theory: Examples 3 examples of GE: pure exchange (Edgeworth box) 1 producer  1 consumer several producers and an example illustrating the limits of the partial equilibrium approach
More informationk=1 k2, and therefore f(m + 1) = f(m) + (m + 1) 2 =
Math 104: Introduction to Analysis SOLUTIONS Alexander Givental HOMEWORK 1 1.1. Prove that 1 2 +2 2 + +n 2 = 1 n(n+1)(2n+1) for all n N. 6 Put f(n) = n(n + 1)(2n + 1)/6. Then f(1) = 1, i.e the theorem
More informationDuality of linear conic problems
Duality of linear conic problems Alexander Shapiro and Arkadi Nemirovski Abstract It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least
More informationRi and. i=1. S i N. and. R R i
The subset R of R n is a closed rectangle if there are n nonempty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an
More informationDiscrete Mathematics
Discrete Mathematics ChihWei Yi Dept. of Computer Science National Chiao Tung University March 16, 2009 2.1 Sets 2.1 Sets 2.1 Sets Basic Notations for Sets For sets, we ll use variables S, T, U,. We can
More informationMarket Power, Forward Trading and Supply Function. Competition
Market Power, Forward Trading and Supply Function Competition Matías Herrera Dappe University of Maryland May, 2008 Abstract When rms can produce any level of output, strategic forward trading can enhance
More informationPRACTICE FINAL. Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 10cm.
PRACTICE FINAL Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 1cm. Solution. Let x be the distance between the center of the circle
More informationMath 5311 Gateaux differentials and Frechet derivatives
Math 5311 Gateaux differentials and Frechet derivatives Kevin Long January 26, 2009 1 Differentiation in vector spaces Thus far, we ve developed the theory of minimization without reference to derivatives.
More informationAdvanced Microeconomics
Advanced Microeconomics Static Bayesian games Harald Wiese University of Leipzig Harald Wiese (University of Leipzig) Advanced Microeconomics / 7 Part E. Bayesian games and mechanism design Static Bayesian
More informationMultivariable Calculus and Optimization
Multivariable Calculus and Optimization Dudley Cooke Trinity College Dublin Dudley Cooke (Trinity College Dublin) Multivariable Calculus and Optimization 1 / 51 EC2040 Topic 3  Multivariable Calculus
More informationn k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...
6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence
More informationSequences and Convergence in Metric Spaces
Sequences and Convergence in Metric Spaces Definition: A sequence in a set X (a sequence of elements of X) is a function s : N X. We usually denote s(n) by s n, called the nth term of s, and write {s
More informationMathematical Economics: Lecture 15
Mathematical Economics: Lecture 15 Yu Ren WISE, Xiamen University November 19, 2012 Outline 1 Chapter 20: Homogeneous and Homothetic Functions New Section Chapter 20: Homogeneous and Homothetic Functions
More informationAnalysis via Uniform Error Bounds Hermann Karcher, Bonn. Version May 2001
Analysis via Uniform Error Bounds Hermann Karcher, Bonn. Version May 2001 The standard analysis route proceeds as follows: 1.) Convergent sequences and series, completeness of the reals, 2.) Continuity,
More informationON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction
ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZPÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that
More informationTHE BANACH CONTRACTION PRINCIPLE. Contents
THE BANACH CONTRACTION PRINCIPLE ALEX PONIECKI Abstract. This paper will study contractions of metric spaces. To do this, we will mainly use tools from topology. We will give some examples of contractions,
More informationANALYTICAL MATHEMATICS FOR APPLICATIONS 2016 LECTURE NOTES Series
ANALYTICAL MATHEMATICS FOR APPLICATIONS 206 LECTURE NOTES 8 ISSUED 24 APRIL 206 A series is a formal sum. Series a + a 2 + a 3 + + + where { } is a sequence of real numbers. Here formal means that we don
More informationTastes and Indifference Curves
C H A P T E R 4 Tastes and Indifference Curves This chapter begins a 2chapter treatment of tastes or what we also call preferences. In the first of these chapters, we simply investigate the basic logic
More informationConstrained Optimisation
CHAPTER 9 Constrained Optimisation Rational economic agents are assumed to make choices that maximise their utility or profit But their choices are usually constrained for example the consumer s choice
More information1. Periodic Fourier series. The Fourier expansion of a 2πperiodic function f is:
CONVERGENCE OF FOURIER SERIES 1. Periodic Fourier series. The Fourier expansion of a 2πperiodic function f is: with coefficients given by: a n = 1 π f(x) a 0 2 + a n cos(nx) + b n sin(nx), n 1 f(x) cos(nx)dx
More information9 More on differentiation
Tel Aviv University, 2013 Measure and category 75 9 More on differentiation 9a Finite Taylor expansion............... 75 9b Continuous and nowhere differentiable..... 78 9c Differentiable and nowhere monotone......
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationChapter 7. Continuity
Chapter 7 Continuity There are many processes and eects that depends on certain set of variables in such a way that a small change in these variables acts as small change in the process. Changes of this
More informationPART II THEORY OF CONSUMER BEHAVIOR AND DEMAND
1 PART II THEORY OF CONSUMER BEHAVIOR AND DEMAND 2 CHAPTER 5 MARSHALL S ANALYSIS OF DEMAND Initially Alfred Marshall initially worked with objective demand curves. However by working backwards, he developed
More informationCITY AND REGIONAL PLANNING 7230. Consumer Behavior. Philip A. Viton. March 4, 2015. 1 Introduction 2
CITY AND REGIONAL PLANNING 7230 Consumer Behavior Philip A. Viton March 4, 2015 Contents 1 Introduction 2 2 Foundations 2 2.1 Consumption bundles........................ 2 2.2 Preference relations.........................
More informationDifferentiability and some of its consequences Definition: A function f : (a, b) R is differentiable at a point x 0 (a, b) if
Differentiability and some of its consequences Definition: A function f : (a, b) R is differentiable at a point x 0 (a, b) if f(x)) f(x 0 ) x x 0 exists. If the it exists for all x 0 (a, b) then f is said
More informationTowards an evolutionary cooperative game theory
Towards an evolutionary cooperative game theory Andre Casajus and Harald Wiese March 2010 Andre Casajus and Harald Wiese () Towards an evolutionary cooperative game theory March 2010 1 / 32 Two game theories
More informationExact Nonparametric Tests for Comparing Means  A Personal Summary
Exact Nonparametric Tests for Comparing Means  A Personal Summary Karl H. Schlag European University Institute 1 December 14, 2006 1 Economics Department, European University Institute. Via della Piazzuola
More information1. Prove that the empty set is a subset of every set.
1. Prove that the empty set is a subset of every set. Basic Topology Written by MenGen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since
More informationCAPM, Arbitrage, and Linear Factor Models
CAPM, Arbitrage, and Linear Factor Models CAPM, Arbitrage, Linear Factor Models 1/ 41 Introduction We now assume all investors actually choose meanvariance e cient portfolios. By equating these investors
More informationMicroeconomic Theory: Basic Math Concepts
Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts
More informationPractice with Proofs
Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using
More information160 CHAPTER 4. VECTOR SPACES
160 CHAPTER 4. VECTOR SPACES 4. Rank and Nullity In this section, we look at relationships between the row space, column space, null space of a matrix and its transpose. We will derive fundamental results
More informationPROBLEM SET. Practice Problems for Exam #2. Math 2350, Fall Nov. 7, 2004 Corrected Nov. 10 ANSWERS
PROBLEM SET Practice Problems for Exam #2 Math 2350, Fall 2004 Nov. 7, 2004 Corrected Nov. 10 ANSWERS i Problem 1. Consider the function f(x, y) = xy 2 sin(x 2 y). Find the partial derivatives f x, f y,
More informationMathematics Notes for Class 12 chapter 12. Linear Programming
1 P a g e Mathematics Notes for Class 12 chapter 12. Linear Programming Linear Programming It is an important optimization (maximization or minimization) technique used in decision making is business and
More informationLecture 4: Random Variables
Lecture 4: Random Variables 1. Definition of Random variables 1.1 Measurable functions and random variables 1.2 Reduction of the measurability condition 1.3 Transformation of random variables 1.4 σalgebra
More informationIntroduction to Algebraic Geometry. Bézout s Theorem and Inflection Points
Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a
More informationFunctions and Equations
Centre for Education in Mathematics and Computing Euclid eworkshop # Functions and Equations c 014 UNIVERSITY OF WATERLOO Euclid eworkshop # TOOLKIT Parabolas The quadratic f(x) = ax + bx + c (with a,b,c
More informationRANDOM INTERVAL HOMEOMORPHISMS. MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis
RANDOM INTERVAL HOMEOMORPHISMS MICHA L MISIUREWICZ Indiana University Purdue University Indianapolis This is a joint work with Lluís Alsedà Motivation: A talk by Yulij Ilyashenko. Two interval maps, applied
More information