concavity_proof

concavity_proof - Spring 2004 Economics 425 John Rust...

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John Rust Economics 425 University of Maryland Properties of Concave and Convex Functions Defnition 1: A function f : R n R n is concave if and only if for every x , y R n and for every θ [ 0 , 1 ] f (( 1 - θ ) x + θ y ) ( 1 - θ ) f ( x )+ θ f ( y ) (1) Defnition 2: A function f : R n R n is convex if and only if for every x , y R n and for every θ [ 0 , 1 ] f (( 1 - θ ) x + θ y ) ( 1 - θ ) f ( x )+ θ f ( y ) (2) Comment 0: Note that the ( 1 - θ ) multiplying the x and f ( x ) and the θ multiplying the y and f ( y ) in the deFnitions above is just an arbitrary convention. We can switch this, and get an equivalent deFnition of concavity: f is concave if and only if for each x and y and every θ [ 0 , 1 ] we have f ( θ x + ( 1 - θ ) y ) θ f ( x )+ ( 1 - θ ) f ( y ) (3) Lemma 0: A function f is concave if and only if its negative, - f , is convex. ProoF: If f is concave then we have from DeFnition 1, f (( 1 - θ ) x + θ y ) ( 1 - θ ) f ( x )+ θ f ( y ) (4) Multiplying both sides of this inequality by - 1 we get - f (( 1 - θ ) x + θ y ) ( 1 - θ )[ - f ( x )]+ θ [ - f ( y )] (5) (Notice that multiplying an inequality by - 1 ±ips the direction of the inequality). But this latter in- equality, (5), satisFes the second deFnition for convexity, i.e. - f is a convex function. Comment 1: The simple geometric intuition is that a concave function f has the curvature of an over- turned bowl. But taking the negative, - f , is just like ±ipping the bowl up, so an upturned bowl is convex. ²igure 1, below, illustrates the deFnition of concavity for a particular concave function, log ( x ) (where log denotes the natural logarithm, i.e. the inverse of the exponential function, log ( exp { x } ) = exp { log ( x ) } = x ). The Fgure shows two points x and y and a point ( 1 - θ ) x + θ y lying on the line seg- ment (on the x-axis) between x and y . According to the deFnition of concavity, log ( x ) is concave if and only if for any x and y and any θ [ 0 , 1 ] we have log (( 1 - θ ) x + θ y ) ( 1 - θ ) log ( x ) + θ log ( y ) . This latter point, ( 1 - θ ) log ( x ) + θ log ( y ) , is just a point on the line joining log ( x ) and log ( y ) (to see this, note that the function l ( θ ) ( 1 - θ ) log ( x )+ θ log ( y ) = log ( x )+ θ [ log ( y ) - log ( x )] is linear with slope log ( y ) - log ( x ) and goes through the two points l ( 0 ) = log ( x ) and l ( 1 ) = log ( y ) ). Thus, the geometric intuition behind concavity is that a function is concave if the line seqment joining the function at any two points x and y lies below the function. In the case of the function f ( x ) = log ( x ) we can see visually that this is the case, i.e. ( 1 - θ ) log ( x )+ θ log ( y ) log (( 1 - θ ) x + θ y ) . 1
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This note was uploaded on 10/25/2011 for the course ECON 306 taught by Professor Cramton during the Spring '06 term at Maryland.

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concavity_proof - Spring 2004 Economics 425 John Rust...

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