Advanced Microeconomics

Advanced Microeconomics Ordinal preference theory Harald Wiese University of Leipzig Harald Wiese (University of Leipzig) Advanced Microeconomics 1 / 68

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

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

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

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 + 3 4 y! Harald Wiese (University of Leipzig) Advanced Microeconomics 5 / 68

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

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

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

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

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

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

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

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

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,.... 2 3 4 5 Harald Wiese (University of Leipzig) Advanced Microeconomics 14 / 68

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

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

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

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

Relations and equivalence classes De nition De nition Relations R (write xry) might be complete (xry or yrx for all x, y) transitive (xry and yrz implies xrz for all x, y, z) re exive (xrx for all x); Problem For any two inhabitants from Leipzig, we ask whether one is the father of the other. Fill in "yes" or "no": property re exive transitive complete is the father of Harald Wiese (University of Leipzig) Advanced Microeconomics 19 / 68

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 20 / 68

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

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

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

Preference relations and indi erence curves Indi erence curve numbers to indicate preferences: x 2 3 6 x 2 6 3 x 1 x 1 Harald Wiese (University of Leipzig) Advanced Microeconomics 24 / 68

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

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

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

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

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

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

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

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

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

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] 6 3 3 x 1 x 1 x 2 6 3 [3] x 2 3 6 [4] x 1 Harald Wiese (University of Leipzig) Advanced Microeconomics 34 / 68 x 1

Monotonicity of preferences Bliss point x 2 9 7 6 3 x 1 Harald Wiese (University of Leipzig) Advanced Microeconomics 35 / 68

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

Lexicographic preferences are not continuous Consider the sequence x j j2n = 2 + 1 j, 2! (2, 2) All its members belong to the better set of (2, 4). But (2, 2) does not. x 2 4 2 2 x 1 Harald Wiese (University of Leipzig) Advanced Microeconomics 37 / 68

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

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

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

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

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

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

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

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 1, x 2 ) ( x 1,x 2 ) U ( x) : = x1 x 1 Harald Wiese (University of Leipzig) Advanced Microeconomics 45 / 68

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

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

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

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

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

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

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

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

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

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

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 Us indifference curves concave Harald Wiese (University of Leipzig) Advanced Microeconomics 56 / 68

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

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 B @ 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

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

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

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

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

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

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

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

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

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

Further exercises Problem 3 (di cult) Let U be a continuous utility function representing the preference relation - on R`+. 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