Consider two function f(x) and g(x) defined on [0,1] and f(0)=g(0)=0 and f(1)=g(1)=1. f(x) is concave and weakly increasing, while g(x) is convex and weakly increasing. Define the semi-elasticity of a function h(x) as h′(x)/h(x). Can you prove that the semi-elasticity of g is greater than the semi-elasticity of f for all x in (0,1)?

Alexander Lewis

Alexander Lewis

Answered question

2022-10-31

Consider two function f ( x ) and g ( x ) defined on [ 0 , 1 ] and f ( 0 ) = g ( 0 ) = 0 and f ( 1 ) = g ( 1 ) = 1.
Prove that the semi-elasticity of g is greater than the semi-elasticity of f for all x ( 0 , 1 )?

Answer & Explanation

Davion Fletcher

Davion Fletcher

Beginner2022-11-01Added 9 answers

Yes, it is true. Let x ( 0 , 1 ) and let ϕ ( t ) be the linear function such that ϕ ( 0 ) = 0 and ϕ ( x ) = f ( x ). The graph of f will lie above the graph of ϕ on ( 0 , x ), since f is concave, but then a drawing shows that necessarily f ( x ) ϕ ( x ) = f ( x ) x . In other words, f ( x ) f ( x ) 1 x .
Similarly, we get g ( x ) g ( x ) x , and hence g ( x ) g ( x ) 1 x , by comparing the graph of the convex function g with the straight line through ( 0 , 0 ) and ( x , g ( x ) ).
Note that the assumption f ( 1 ) = g ( 1 ) = 1 was not needed. This can also be seen by noting that h ( x ) h ( x ) is unchanged under vertical scaling, as h ( x ) and h ( x ) will be multiplied by the same constant.

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