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 )?

Question

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  )?

Davion Fletcher · Accepted Answer

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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