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concavity_proof

# concavity_proof - Spring 2004 Economics 425 John Rust...

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Spring 2004 John Rust Economics 425 University of Maryland Properties of Concave and Convex Functions Definition 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) Definition 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 definitions above is just an arbitrary convention. We can switch this, and get an equivalent definition 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 Definition 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 flips the direction of the inequality). But this latter in- equality, (5), satisfies the second definition 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 flipping the bowl up, so an upturned bowl is convex. Figure 1, below, illustrates the definition 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 figure 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 definition 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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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 7 6 5 4 3 2 1 0 1 Example of Concavity Definition: log(x) x y (1 θ )x+ θ y log(y) log(x) log((1 θ )x+ θ y) (1 θ )log(x)+ θ log(y) However it seems that the general definition of concavity is a hard one to verify. How can we determine in general whether for any particular function f whether for any two points x and y the function lies above the line segment joining f ( x ) and f ( y ) ? For example take the function f ( x ) = x .
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concavity_proof - Spring 2004 Economics 425 John Rust...

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