Μια εισαγωγή στα Μαθηματικά για Οικονομολόγους

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Letitia Stafford
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1 Μια εισαγωγή στα Μαθηματικά για Οικονομολόγους

2 Κυρτά σύνολα (covex sets) στον S είναι κυρτό σύνολο εάν για κάθε 1 x S και 2 x S 1 2 tx t x S t [ ] + (1 ), 0,1

3 Μια συνάρτηση f : A B λέγεται 1-1 ή αμφιμονοσήμαντη, όταν αντιστοιχίζει κάθε όρισμα σε αποκλειστικά δική του τιμή ή αλλιώς, όταν διαφορετικά ορίσματα απεικονίζονται σε διαφορετικές τιμές: αν a a τότε f ( a) f( a ) Μια συνάρτηση f : A B λέγεται επί (oto), όταν δεν υπάρχει στοιχείο στο Β που να μην είναι η εικόνα κάποιου στοιχείου του Α: Μια συνάρτηση είναι αντιστρέψιμη ( b B, a A: b= f( a) f 1 ) αν είναι 1-1 και επί: 1 1 f B A f b = a b= f a : ( ) ( )

4 Λίγη τοπολογία Μετρική είναι απλά ένα μέγεθος μέτρησης της απόστασης Μετρικός χώρος είναι ένα σύνολο στο οποίο έχει οριστεί μια έννοια απόστασης (μια μετρική). Μετρικοί χώροι Μετρική {( x, x )\ x, x } 2 = {( x, x,.. x )\ x, i 1,2... } 1 2 i = = dx (, x) = x x dx (, x) = ( x x) + ( x x) dx (, x) = ( x x) ( x x)

5 Defiitio: Ope ad Closed ε-balls 1. The ope ε-ball with ceter 0 x ad radius ε > 0 (a real umber) is the subset of poits i : { ε} B x x d x x < 0 0 ε ( ) \ (, ) 2. The closed ε-ball with ceter 0 x ad radius ε > 0 (a real umber) is the subset of poits i : { ε} 0 0 Bε x x d x x ( ) \ (, )

6 Defiitio: Ope sets i S is a ope set if, for all x S, there exists some ε > 0 such that B ( x) S ε Theorem: O Ope Sets i 1. is a ope set 2. is a ope set 3. The uio of ope sets is a ope set 4. The itersectio of ay fiite collectio of ope sets is a ope set

7 Defiitio: Closed sets i S is a closed set if, its complemet c S, is a ope set Theorem: O Closed Sets i 1. is a closed set 2. is a closed set 3. The uio of ay fiite collectio of closed sets is a closed set 4. The itersectio of closed sets is a closed set

8 Defiitio: Bouded sets i A set S is called bouded if it is etirely cotaied withi some ε-ball (either ope or closed). That is S is bouded if there exists some ε > 0 such that S B ( x) ε for some x.

9 Defiitio (Heie Borel): Compact sets i A set S is called compact if it is closed ad bouded

10 Defiitio: Caushy cotiuity Let m D ad let f : if for every ε > 0, there is a δ > 0 such that: D. The fuctio f is cotiuous at the poit 0 x D 0 0 ( δ( ) ) ε ( ( )) f B x D B f x If f is cotiuous at every poit x D, the it is called a cotiuous fuctio.

11 Theorem: Cotiuity ad iverse images Let m D. The followig coditios are equivalet: 1. f : D is cotiuous 2. For every ope ball B i, f 1 ( B) is ope i D 3. For every ope S, f 1 ( S) is ope i D Theorem: The Cotiuous Image of a Compact Set is a Compact Set Let m D ad let f : D be a cotiuous fuctio. If S D (closed ad bouded) the its image f ( S) is a compact set. is compact

12 Theorem (Weierstrass): Existece of Extreme Values Let f : S be a cotiuous real-valued fuctio, where S is a oempty compact subset of that:. The there exists a vector x S ad a vector x S such f ( x ) f( x) f( x), x S

13 Defiitio: Real Valued Fuctios f : D R is a real-valued fuctio if D is ay set ad R Defiitio: Icreasig Real Valued Fuctios Let f : D, where D. The f is icreasig if f x f x wheever 0 1 ( ) ( ) x x 0 1 f is strictly icreasig if f x > f x wheever 0 1 ( ) ( ) x >> x 0 1 f is strogly icreasig if f x > f x wheever 0 1 ( ) ( ) x x ad 0 1 x x 0 1

14 Defiitio: Decreasig Real Valued Fuctios Let f : D, where D. The f is decreasig if f x f x wheever 0 1 ( ) ( ) x x 0 1 f is strictly decreasig if f x < f x wheever 0 1 ( ) ( ) x >> x 0 1 f is strogly decreasig if f x < f x wheever 0 1 ( ) ( ) x x ad 0 1 x x 0 1

15 Defiitio: Level Sets 0 L( y ) is a level set of the real-valued fuctio f : D R if-f { } 0 0 L( y ) = x\ x D, f ( x) = y, where 0 y R Defiitio: Superior ad Iferior Sets 1. S ( y 0 { x \ x D, f ( x ) y 0 } 2. I ( y 0 { x \ x D, f ( x ) y 0 } 3. S ( y 0 { x \ x D, f ( x ) y 0 } 4. I ( y 0 { x \ x D, f ( x ) y 0 } is called the superior set for level is called the iferior set for level > is called the strictly superior set for level < is called the strictly iferior set for level 0 y 0 y 0 y 0 y

16 Cocave Fuctios Defiitio: Cocave Fuctios f : D R is a cocave fuctio if for all 1 2 x, x D f x tf x + t f x t 1 2 ( ) ( ) (1 ) ( ) [0,1] where 1 2 = + (1 ) deotes the covex combiatio of x tx t x x, x 1 2

17 Cocave Fuctios Theorem: Poits o ad below the Graph of a Cocave fuctio always form a Covex Set Let A {( x, y)\ x D, f ( x) y} be the set of poits o ad below the graph of f : D R, where D is a covex set ad R. The f is cocave fuctio A is a covex set

18 Defiitio: Strictly Cocave Fuctios Cocave Fuctios f : D R is a strictly cocave fuctio if-f for all x x i D: f ( tx + (1 t) x ) > tf( x ) + (1 t) f ( x ) t (0,1) 1 2

19 Defiitio: Quasi-Cocave Fuctio Quasi-Cocave Fuctios f : D R is quasi-cocave if-f for all 1 2 x, x D: ( ) + (1 ) mi ( ), ( ) [0,1] f tx t x f x f x t Theorem: Quasi-Cocavity ad the Superior Sets f : D is a quasi-cocave fuctio if-f S( y ) is a covex set for all y

20 Quasi-Cocave Fuctios Defiitio: Strictly Quasi-Cocave Fuctio f : D R is strictly quasi-cocave if-f for all 1 2 x x D: ( ) + (1 ) > mi ( ), ( ) (0,1) f tx t x f x f x t Theorem: Strictly Quasi-Cocavity ad the Superior Sets f : D is a strictly quasi-cocave fuctio if-f S( y ) is a strictly covex set for all y

21 Theorem: Cocavity implies Quasi-cocavity A (strictly) cocave fuctio is always (strictly) quasi-cocave Theorem: Cobb-Douglas Fuctio a b Every Cobb-Douglas fuctio f ( x1, x2) = Ax1x2 with A, ab>0, is quasi-cocave.

22 Covex ad Quasi-Covex Fuctios Defiitio: Covex ad Strictly Covex Fuctios 1. f : D R is a covex fuctio if for all 1 2 x, x D ( ) f tx t x tf x t f x t + (1 ) ( ) + (1 ) ( ) [0,1] 2. f : D R is a strictly covex fuctio if for all 1 2 x x D ( ) f tx t x tf x t f x t + (1 ) < ( ) + (1 ) ( ) (0,1)

23 Covex ad Quasi-Covex Fuctios Theorem: Poits o ad Above the Graph of a Cocave fuctio always form a Covex Set Let A {( x, y)\ x D, f ( x) y} be the set of poits o ad above the graph of f : D R, where D is a covex set ad R. The f is covex fuctio A is a covex set

24 Covex ad Quasi-Covex Fuctios Defiitio: Quasi-Covex ad Strictly q-covex Fuctio 1. f : D R is quasi-covex if-f for all 1 2 x, x D: ( ) + (1 ) max ( ), ( ) [0,1] f tx t x f x f x t 2. f : D R is strictly quasi-covex if-f for all 1 2 x x D: ( ) + (1 ) < max ( ), ( ) (0,1) f tx t x f x f x t

25 Covex ad Quasi-Covex Fuctios Theorem: (Strictly) Quasi-Covex ad the Iferior Sets f : D is a (strictly) quasi-covex fuctio if-f I ( y ) is a (strictly) covex set for all y Theorem: (Strictly) Quasi-Covex ad (Strictly) Quasi-Cocave fuctios f ( x ) is a (strictly) quasi-cocave fuctio if-f f ( x) is a (strictly) quasi-covex fuctio

26 Calculus ad Optimizatio Fuctios of a sigle variable

27 Calculus ad Optimizatio Fuctios of a sigle variable Theorem: Suppose f : D R, D, R is twice cotiuously differetiable 1. f is cocave f ( x) 0, x D 2. f is covex f ( x) 0, x D Moreover, 1. if f ( x) < 0, x D the f is strictly cocave 2. if f ( x) > 0, x D the f is strictly covex

28 Calculus ad Optimizatio Fuctios of several variables

29 Calculus ad Optimizatio Fuctios of several variables Theorem: Youg s Theorem For ay twice cotiuously differetiable fuctio f ( x), x : 2 2 f ( x) f ( x) =, i, j x x x x i j j i

30 (Leadig) Pricipal Miors of a Matrix Defiitio: Let A be a matrix. A k k submatrix of A formed by deletig k colums ad the same k rows from A is called a k th order pricipal submatrix of A. The determiat of that pricipal submatrix is called a k th order pricipal mior of A. Defiitio: Let A be a A obtaied by deletig the last matrix. The k th order pricipal submatrix of k rows ad the last k colums from A is called the k th order leadig pricipal submatrix of A. Its determiat is called the k th order leadig pricipal mior of A.

31 Defiiteess of a Matrix Theorem: Defiiteess of a matrix Let A be a symmetric matrix (a) A is positive defiite if-f all its leadig pricipal miors are strictly positive (b) A is egative defiite if-f its leadig pricipal miors alterate i sig as follows: A < 0, A > 0, A <

32 Semi-Defiiteess of a Matrix Theorem: Semi-Defiiteess of a matrix Let A be a symmetric matrix (a) A is positive semi-defiite if-f every pricipal mior is 0 (b) A is egative semi-defiite if-f every pricipal mior of odd order 0 ad every pricipal mior of eve order 0

33 Border Matrices H 0 f1 f2... f f1 f11 f12... f 1 = f2 f21 f22... f 2... f f 1 f2... f

34 Border Matrices Theorem: Defiiteess of a bordered matrix Let H be a symmetric bordered matrix (a) H is positive defiite if-f all its bordered pricipal miors are strictly egative i.e 0 f f 0 f H = < 0 H = f f f < 0... H < f1 f11 f2 f21 f22 (b) H is egative defiite if-f its bordered pricipal miors alterate i sig as follows: 0 f f 0 f H = < 0 H = f f f > 0 H < f1 f11 f2 f21 f22

35 Border Matrices Theorem: Semi-Defiiteess of a bordered matrix Let H be a symmetric bordered matrix (a) H is positive semi-defiite if-f every bordered pricipal mior is 0 (b) H is egative semi-defiite if-f every bordered pricipal mior of odd order 0 ad every bordered pricipal mior of eve order 0

36 (Border) Matrices ad (Quasi) Cocavity/Covexity Theorem: Cocavity Covexity i May Variables Let D be a covex subset of o which f is twice cotiuously differetiable f is cocave (covex) H ( x ) is egative (positive) semi-defiite, x D Moreover If H ( x ) is egative (positive) defiite x D the f is strictly cocave (covex)

37 Theorem: Cocavity Covexity ad Secod-Order Ow Partial Derivatives Let f : D R be a twice cotiuously differetiable fuctio 1. If f cocave fii ( x) 0 i = 1, 2,... x 2. If f covex fii ( x) 0 i = 1, 2,... x

38 (Border) Matrices ad (Quasi) Cocavity/Covexity Theorem: Quasi-Cocavity (Covexity) i may variables Let D be a covex subset of o which f is twice cotiuously differetiable f is quasi cocave (covex) H ( x ) is egative (positive) semi-defiite, x D Moreover If ( ) H x is egative (positive) defiite x D the f is strictly quasi -cocave (quasi-covex)

39 Homogeeous Fuctios Defiitio: Homogeeous Fuctios A real-valued fuctio f ( x ) is called homogeeous of degree k, if f ( tx) = t k f ( x) t > 0

40 Homogeeous Fuctios Theorem: Partial Derivatives of Homogeeous Fuctios If f ( x ) is h.o.d. k, its partial derivatives are h.o.d. k-1 Theorem: Euler s Theorem Let f ( x ) be a cotiuously differetiable homogeeous fuctio of degree k o The for all x + f ( x) f ( x) f ( x) x + x x = kf ( x) x x x

41 Optimizatio Maxima ad miima for sigle-variable fuctios Cosider the fuctio of a sigle-variable f ( x) = y ad assume it is differetiable whe we say the fuctio achieves a local maximum at x, we mea that f x f x x B x ( ) ( ), ε ( ) whe we say the fuctio achieves a global maximum at x, we mea that f ( x ) f ( x), x D uique local maximum at x if f ( x ) > f ( x), x x B ( x ) ε uique global maximum at x if f ( x ) > f ( x), x x D

42 Optimizatio Maxima ad miima for sigle-variable fuctios Theorem: (a) If f ( x0) = 0 ad f ( x0) < 0 the x 0 is local max of f (b) If f ( x0) = 0 ad f ( x0) > 0 the x 0 is local mi of f (c) If f ( x0) = 0 ad f ( x0) = 0 the x 0 ca be max, mi, or either

43 Optimizatio Maxima ad miima for sigle-variable fuctios If f ( x ) is a twice cotiuously differetiable fuctio whose domai is a iterval I, the (a) If f ( x0 ) = 0 ad f ( x) < 0, x I the x 0 is a global max of f (b) If f ( x0 ) = 0 ad f ( x) > 0, x I the x 0 is a global mi of f

44 Optimizatio Real-valued fuctios of -variables Defiitio: Let f : D, D (1) A poit x D is a global max if f ( x ) f ( x), x D (1) A poit x D is a uique global max if f ( x ) > f ( x), x D ad x x (2) A poit x D is a local max if f ( x ) f ( x), x Bε ( x ) D (2) A poit x D is a uique local max if f ( x ) > f ( x), x Bε ( x ) D ad x x

45 Optimizatio Real-valued fuctios of -variables Theorem: Let f : D, D be a twice cotiuously differetiable fuctio If x D is a local max or mi of f ad if x is a iterior poit of D, the x solves the system f x x... ( ) 1 ( ) f x x 2 ( ) f x x = = = 0 0 0

46 Optimizatio Real-valued fuctios of -variables SECOND ORDER CONDITIONS Theorem: Sufficiet Coditios Let f : D, D be a twice cotiuously differetiable fuctio Suppose that x satisfies f ( x ) = 0, i = 1, 2,... x ad that the leadig pricipal i miors of H ( x ) alterate i sig at x. The H < 0, H > 0, H < x is a uique local max of f

47 Optimizatio Real-valued fuctios of -variables SECOND ORDER CONDITIONS Theorem: Necessary Coditios Let f : D, D be a twice cotiuously differetiable fuctio (1) If f ( x ) reaches a local iterior maximum at x the f x x ( ) i = 0, i = 1, 2,... ad H ( x ) is egative semi-defiite (2) If f ( x ) reaches a local iterior miimum at x the ad H ( x ) is positive semi-defiite f ( x ) x i = 0, i = 1, 2,...

48 Optimizatio Real-valued fuctios of -variables Theorem: Global Theorem Let f : D, D be a twice cotiuously differetiable fuctio which is [strictly] CONCAVE (covex) o D. The followig statemets are equivalet, where x is a iterior poit of D : (1) f x x ( ) i = 0, for i = 1, 2,... (2) f achieves a [uique] GLOBAL MAXIMUM (global miimum) at x

49 Costraied Optimizatio Equality Costraits max f ( x, x ) s.t. g( x, x ) = 0 x, x

50 Costraied Optimizatio Equality Costraits max f ( x, x ) s.t. g( x, x ) = 0 x, x Solve: 1. By substitutio

51 Costraied Optimizatio Equality Costraits max f ( x, x ) s.t. g( x, x ) = 0 x, x Solve: 1. By substitutio 2. Lagrage s Method

52 Costraied Optimizatio Equality Costraits Theorem: Sufficiet Coditios for Local Optima with Equality Costraits Let the objective fuctio be f ( x ) ad the m costraits be j g ( x) = 0, j = 1, 2,... m Let ( x, Λ ) solve the F.O.C. The 1. x is a local maximum of f ( x ) subject to the costraits, if the bordered pricipal miors, evaluated at ( x, Λ ), alterate i sig begiig with egative 2. x is a local miimum of f ( x ) subject to the costraits, if the bordered pricipal miors, evaluated at ( x, Λ ), are all egative

53 Costraied Optimizatio Iequality Costraits Theorem: Necessary Coditios for Optima of Real-valued fuctios s.t. Noegative Costraits Let the objective fuctio f ( x ) be cotiuously differetiable 1. If x maximizes f ( x ) s.t. x 0, the x satisfies: (i) f ( x) x i 0, i = 1, 2... (ii) (iii) f( x ) xi = 0, i = 1, 2... xi x 0 i = 1, 2,... i

54 Costraied Optimizatio Iequality Costraits KUHN-TUCKER CONDITIONS

55 Value Fuctios M ( a) f ( xa ( ), a) Theorem: Theorem of the Maximum If the objective fuctio ad the costrait are cotiuous i the parameters, ad if the domai is a compact set, the M ( a ) ad x( a ) are cotiuous i a

56 THE ENVELOPE THEOREM Cosider the problem max f ( x ; a ) s.t. g( x; a) = 0 ad x 0 x ad suppose the objective fuctio ad costrait are cotiuously differetiable i a. For each a, let xa ( ) >> 0 uiquely solve the problem ad assume that it is also cotiuously differetiable i the parameters a. Let Lxaλ (,, ) be the problem s associated Lagragia fuctio ad let ( x( a), λ ( a)) solve the Kuh-Tucker coditios. Fially, let M ( a ) be the problem s associated maximum-value fuctio. The the Evelope Theorem states that M( a) L = j = 1, 2,... m a a j j x( a) λ ( a)

Brief Review of Functions of Several Variables

Brief Review of Functions of Several Variables Brief Review of Fuctios of Several Variables Differetiatio Differetiatio Recall, a fuctio f : R R is differetiable at x R if ( ) ( ) lim f x f x 0 exists df ( x) Whe this limit exists we call it or f(