CW_Concave_functions.pdf - New York University Department of Economics ECON-UA 6 Mathematics for Economists C Wilson Concave Functions of a Single

CW_Concave_functions.pdf - New York University Department...

This preview shows page 1 - 4 out of 9 pages.

New York University Department of Economics ECON-UA 6 C. Wilson Mathematics for Economists February 21, 2012 Concave Functions of a Single Variable If x, y R and λ (0 , 1) , then λy + (1 λ ) x is a convex combination of x and y. Geometrically, a convex combination of x and y is a point somewhere between x and y. A set X R is convex if x, y X implies λy + (1 λ ) x X for all λ [0 , 1] . The de fi nition of a convex set immediately implies that X is convex if and only if X is either empty, a point, or an interval. Throughout this handout, we suppose that X is a convex subset of R . Concave Functions f : X R is concave if for any x, y X , we have, for all λ (0 , 1) , f ( λy + (1 λ ) x ) λf ( y ) + (1 λ ) f ( x ) . f : X R is strictly concave if for any x, y X with x 6 = y , we have, for all λ (0 , 1) , f ( λy + (1 λ ) x ) > λf ( y ) + (1 λ ) f ( x ) . Geometrically, a function f is concave if the cord between any to points on the function lies everywhere on or below the function itself as illustrated in the graph below. A constant function is concave. Why? A linear function is concave. Why? caw1/
Image of page 1

Subscribe to view the full document.

ECON-UA 6: Concave Functions of a Single Variable February 21, 2012 Page 2 Linear Combinations of Concave Functions Consider a list of functions f i : X R for i = 1 , ..., n, and list of numbers α 1 , ..., α n . The function f P n i =1 α i f i is called a linear combination of f 1 , ..., f n . If each of the weights α i 0 , then f is a nonnegative linear combination of f 1 , ..., f n . The next proposition establishes that any nonnegative linear combination of concave functions is also a concave function. Theorem 1: Suppose f 1 , ..., f n are concave functions and ( α 1 , ..., α n ) 0 . Then f P n i =1 α i f i is also a concave function. If at least one f j is also strictly concave and α j > 0 , then f is strictly concave. Proof. Left as an exercise. Since a constant function is concave, Theorem 1 implies If f is (strictly) concave, then any a ne transformation αf + β with α > 0 is also (strictly) concave. The Continuity and Di ff erentiability of Concave Functions A concave function need not be di ff erentiable everywhere. For example f ( x ) = | x | is concave function that is not di ff erentiable at 0. However, we prove in the Appendix that right and left hand derivatives always exist on the interior of the domain and that f ( x ) f + ( x ) . For example, if f ( x ) = | x | , then f (0) = 1 and f + (0) = 1 . Since both righthand and lefthand deriviates exist on the interior, it follows from our earlier results on di ff erentiable functions that f is both right and left continuous and therefore contin- uous. However, concave functions need not be continuous at the boundary as illustrated in the example below. caw1/
Image of page 2
ECON-UA 6: Concave Functions of a Single Variable February 21, 2012 Page 3 A Characterization of Di ff erentiable Concave Functions For di ff erentiable functions, the following theorem provides a simple necessary and su cient con- ditions for concavity.
Image of page 3

Subscribe to view the full document.

Image of page 4
  • Spring '08
  • Todd
  • Derivative, λ, Convex function, function F, linear combination of f1

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Ask Expert Tutors You can ask 0 bonus questions You can ask 0 questions (0 expire soon) You can ask 0 questions (will expire )
Answers in as fast as 15 minutes