Frank Cowell Microeconomics June 2006 Convexity MICROECONOMICS Principles
![Frank Cowell: Microeconomics June 2006 Convexity MICROECONOMICS Principles and Analysis Frank Cowell Frank Cowell: Microeconomics June 2006 Convexity MICROECONOMICS Principles and Analysis Frank Cowell](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-1.jpg)
![Convex sets Frank Cowell: Microeconomics Ideas of convexity used throughout microeconomics n Restrict attention Convex sets Frank Cowell: Microeconomics Ideas of convexity used throughout microeconomics n Restrict attention](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-2.jpg)
![Overview. . . Convexity Frank Cowell: Microeconomics Sets Basic definitions Functions Separation Overview. . . Convexity Frank Cowell: Microeconomics Sets Basic definitions Functions Separation](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-3.jpg)
![Convexity in 2 R Frank Cowell: Microeconomics § A set A in R 2 Convexity in 2 R Frank Cowell: Microeconomics § A set A in R 2](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-4.jpg)
![Strict Convexity in 2 R Frank Cowell: Microeconomics § A set A in R Strict Convexity in 2 R Frank Cowell: Microeconomics § A set A in R](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-5.jpg)
![The simplex Frank Cowell: Microeconomics x 1 + x 2 + x 3 = The simplex Frank Cowell: Microeconomics x 1 + x 2 + x 3 =](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-6.jpg)
![The ball Frank Cowell: Microeconomics x 1 Si [xi– ai]2 § A ball centred The ball Frank Cowell: Microeconomics x 1 Si [xi– ai]2 § A ball centred](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-7.jpg)
![Overview. . . Convexity Frank Cowell: Microeconomics Sets For scalars and vectors Functions Separation Overview. . . Convexity Frank Cowell: Microeconomics Sets For scalars and vectors Functions Separation](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-8.jpg)
![Convex functions Frank Cowell: Microeconomics §A function f: R R § Draw A, the Convex functions Frank Cowell: Microeconomics §A function f: R R § Draw A, the](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-9.jpg)
![Concave functions (1) Frank Cowell: Microeconomics §A function f: R R y §Draw the Concave functions (1) Frank Cowell: Microeconomics §A function f: R R y §Draw the](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-10.jpg)
![Concave functions (2) Frank Cowell: Microeconomics y §A function f: R 2 R y Concave functions (2) Frank Cowell: Microeconomics y §A function f: R 2 R y](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-11.jpg)
![Convex and concave function Frank Cowell: Microeconomics y y = f(x) §An affine function Convex and concave function Frank Cowell: Microeconomics y y = f(x) §An affine function](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-12.jpg)
![Quasiconcavity Frank Cowell: Microeconomics x 2 §Draw contours of a function f: R 2 Quasiconcavity Frank Cowell: Microeconomics x 2 §Draw contours of a function f: R 2](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-13.jpg)
![Overview. . . Convexity Frank Cowell: Microeconomics Sets Fundamental relations Functions Separation Overview. . . Convexity Frank Cowell: Microeconomics Sets Fundamental relations Functions Separation](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-14.jpg)
![Convexity and separation Frank Cowell: Microeconomics convex §Two convex sets in R 2 § Convexity and separation Frank Cowell: Microeconomics convex §Two convex sets in R 2 §](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-15.jpg)
![A hyperplane in 2 R §Hyperplane in R 2 is a straight line {x: A hyperplane in 2 R §Hyperplane in R 2 is a straight line {x:](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-16.jpg)
![A hyperplane separating A and y Frank Cowell: Microeconomics x 2 §A convex set A hyperplane separating A and y Frank Cowell: Microeconomics x 2 §A convex set](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-17.jpg)
![A hyperplane separating two sets Frank Cowell: Microeconomics §Convex sets A and B. §A A hyperplane separating two sets Frank Cowell: Microeconomics §Convex sets A and B. §A](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-18.jpg)
![Supporting hyperplane Frank Cowell: Microeconomics §Convex sets A and B. §A and B only Supporting hyperplane Frank Cowell: Microeconomics §Convex sets A and B. §A and B only](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-19.jpg)
- Slides: 19
![Frank Cowell Microeconomics June 2006 Convexity MICROECONOMICS Principles and Analysis Frank Cowell Frank Cowell: Microeconomics June 2006 Convexity MICROECONOMICS Principles and Analysis Frank Cowell](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-1.jpg)
Frank Cowell: Microeconomics June 2006 Convexity MICROECONOMICS Principles and Analysis Frank Cowell
![Convex sets Frank Cowell Microeconomics Ideas of convexity used throughout microeconomics n Restrict attention Convex sets Frank Cowell: Microeconomics Ideas of convexity used throughout microeconomics n Restrict attention](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-2.jpg)
Convex sets Frank Cowell: Microeconomics Ideas of convexity used throughout microeconomics n Restrict attention to real space Rn ( 1, x 2, . . . , xn) n I. e. sets of vectors (x n Use the concept of convexity to define n Convex functions u Concave functions u Quasiconcave functions u
![Overview Convexity Frank Cowell Microeconomics Sets Basic definitions Functions Separation Overview. . . Convexity Frank Cowell: Microeconomics Sets Basic definitions Functions Separation](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-3.jpg)
Overview. . . Convexity Frank Cowell: Microeconomics Sets Basic definitions Functions Separation
![Convexity in 2 R Frank Cowell Microeconomics A set A in R 2 Convexity in 2 R Frank Cowell: Microeconomics § A set A in R 2](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-4.jpg)
Convexity in 2 R Frank Cowell: Microeconomics § A set A in R 2 § Draw a line between any two points in A x 2 l Any point on this line also belongs to A. . . l. . . so A is convex x 1
![Strict Convexity in 2 R Frank Cowell Microeconomics A set A in R Strict Convexity in 2 R Frank Cowell: Microeconomics § A set A in R](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-5.jpg)
Strict Convexity in 2 R Frank Cowell: Microeconomics § A set A in R 2 § Draw a line between any two boundary points of A x 2 l Any intermediate point on this line is in interior of A. . . l. . . so A is strictly convex x 1 Examples of convex sets in R 3
![The simplex Frank Cowell Microeconomics x 1 x 2 x 3 The simplex Frank Cowell: Microeconomics x 1 + x 2 + x 3 =](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-6.jpg)
The simplex Frank Cowell: Microeconomics x 1 + x 2 + x 3 = const x 2 0 x 3 § The simplex is convex, but not strictly convex
![The ball Frank Cowell Microeconomics x 1 Si xi ai2 A ball centred The ball Frank Cowell: Microeconomics x 1 Si [xi– ai]2 § A ball centred](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-7.jpg)
The ball Frank Cowell: Microeconomics x 1 Si [xi– ai]2 § A ball centred on the point (a 1, a 2, a 3) >0 = const x 2 0 x 3 §It is strictly convex
![Overview Convexity Frank Cowell Microeconomics Sets For scalars and vectors Functions Separation Overview. . . Convexity Frank Cowell: Microeconomics Sets For scalars and vectors Functions Separation](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-8.jpg)
Overview. . . Convexity Frank Cowell: Microeconomics Sets For scalars and vectors Functions Separation
![Convex functions Frank Cowell Microeconomics A function f R R Draw A the Convex functions Frank Cowell: Microeconomics §A function f: R R § Draw A, the](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-9.jpg)
Convex functions Frank Cowell: Microeconomics §A function f: R R § Draw A, the set "above" the function f y A : = {(x, y): y f(x)} § If A is convex, f is a convex function y = f(x) § If A is strictly convex, f is a strictly convex function x
![Concave functions 1 Frank Cowell Microeconomics A function f R R y Draw the Concave functions (1) Frank Cowell: Microeconomics §A function f: R R y §Draw the](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-10.jpg)
Concave functions (1) Frank Cowell: Microeconomics §A function f: R R y §Draw the function –f § Draw A, the set "above" the function –f y = f(x) x § If –f is a convex function, f is a concave function § Equivalently, if the set "below" f is convex, f is a concave function §If –f is a strictly convex function, f is a strictly concave function
![Concave functions 2 Frank Cowell Microeconomics y A function f R 2 R y Concave functions (2) Frank Cowell: Microeconomics y §A function f: R 2 R y](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-11.jpg)
Concave functions (2) Frank Cowell: Microeconomics y §A function f: R 2 R y = f(x) § Draw the set "below" the function f x 2 0 x 1 § Set "below" f is strictly convex, so f is a strictly concave function
![Convex and concave function Frank Cowell Microeconomics y y fx An affine function Convex and concave function Frank Cowell: Microeconomics y y = f(x) §An affine function](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-12.jpg)
Convex and concave function Frank Cowell: Microeconomics y y = f(x) §An affine function f: R R § Draw the set "above" the function f § Draw the set "below" the function f § The graph in R 2 is a straight line. x § Both "above" and “below" sets are convex. §So f is both concave and convex. § Corresponding graph in R 3 would be a plane. §The graph in Rn would be a hyperplane.
![Quasiconcavity Frank Cowell Microeconomics x 2 Draw contours of a function f R 2 Quasiconcavity Frank Cowell: Microeconomics x 2 §Draw contours of a function f: R 2](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-13.jpg)
Quasiconcavity Frank Cowell: Microeconomics x 2 §Draw contours of a function f: R 2 R §Pick the contour for some specific y-value y 0. § Draw the "better-than" set for y 0. § If the "better-than" set B(y 0) is convex, f is a concave-contoured function B(y 0) § An equivalent term is a "quasiconcave" function y 0 = f(x) §If B(y 0) is strictly convex, f is a strictly quasiconcave" x 1 function
![Overview Convexity Frank Cowell Microeconomics Sets Fundamental relations Functions Separation Overview. . . Convexity Frank Cowell: Microeconomics Sets Fundamental relations Functions Separation](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-14.jpg)
Overview. . . Convexity Frank Cowell: Microeconomics Sets Fundamental relations Functions Separation
![Convexity and separation Frank Cowell Microeconomics convex Two convex sets in R 2 Convexity and separation Frank Cowell: Microeconomics convex §Two convex sets in R 2 §](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-15.jpg)
Convexity and separation Frank Cowell: Microeconomics convex §Two convex sets in R 2 § Convex and nonconvex sets § Convex sets can be separated by a hyperplane. . . §. . . but nonconvex sets sometimes can't be separated convex non-convex
![A hyperplane in 2 R Hyperplane in R 2 is a straight line x A hyperplane in 2 R §Hyperplane in R 2 is a straight line {x:](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-16.jpg)
A hyperplane in 2 R §Hyperplane in R 2 is a straight line {x: Si pixi c} §Parameters p and c determine the slope and position §Draw in points "above" H H( §Draw in points "below" H : = c) p, : S {x i p xi i =c } Frank Cowell: Microeconomics x 2 {x: Si pixi c} x 1
![A hyperplane separating A and y Frank Cowell Microeconomics x 2 A convex set A hyperplane separating A and y Frank Cowell: Microeconomics x 2 §A convex set](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-17.jpg)
A hyperplane separating A and y Frank Cowell: Microeconomics x 2 §A convex set A H §A point y "outside" A l §The point x* in A that is closest to y y §The separating hyperplane A l x* § y A. § y lies in the "above-H" set § x* lies in the "below-H" set x 1
![A hyperplane separating two sets Frank Cowell Microeconomics Convex sets A and B A A hyperplane separating two sets Frank Cowell: Microeconomics §Convex sets A and B. §A](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-18.jpg)
A hyperplane separating two sets Frank Cowell: Microeconomics §Convex sets A and B. §A and B only have no points in common. H §The separating hyperplane. A B § All points of A lie strictly above H § All points of B lie strictly below H
![Supporting hyperplane Frank Cowell Microeconomics Convex sets A and B A and B only Supporting hyperplane Frank Cowell: Microeconomics §Convex sets A and B. §A and B only](https://slidetodoc.com/presentation_image_h/a003e829f7b669cddf3fa98f86e93efc/image-19.jpg)
Supporting hyperplane Frank Cowell: Microeconomics §Convex sets A and B. §A and B only have boundary points in common. H §The supporting hyperplane. A l B § Interior points of A lie strictly above H § Interior points of B lie strictly below H
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