Advanced Microeconomics

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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 Quasi-concave 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 city-block 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 2-norm 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) ε-ball 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 1-balls 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 ε-ball 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 Quasi-concave 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 two-good 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 Quasi-concave 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 non-satiation at y if in every ε-ball 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 Quasi-concave 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 Cobb-Douglas 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 Dixit-Stiglitz 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 non-satiation 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 Quasi-concave 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 Quasi-concave utility functions and convex preferences 6 Marginal rate of substitution Harald Wiese (University of Leipzig) Advanced Microeconomics 49 / 68

50 Quasi-concavity De nition f : R`! R is quasi-concave 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 quasi-concave 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: quasi-concave functions need not be concave (to be introduced later). Harald Wiese (University of Leipzig) Advanced Microeconomics 50 / 68

51 Quasi-concavity 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 quasi-concave. Harald Wiese (University of Leipzig) Advanced Microeconomics 51 / 68

52 Quasi-concavity 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 quasi-concave. 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 Quasi-concave 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 real-valued 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 (implicit-function 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 Cobb-Douglas 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 Cobb-Douglas preferences are monotonic so that an increase of x 1 is associated with a decrease of x 2 along an indi erence curve. Therefore, Cobb-Douglas preferences are convex (Cobb-Douglas utility functions are quasi-concave). Harald Wiese (University of Leipzig) Advanced Microeconomics 66 / 68

67 Further exercises Problem 1 De ne strict anti-monotonicity. Sketch indi erence curves for each of the four cases: strict convexity strict concavity strict monotonicity strict anti-monotonicity 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

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