Quasi-concavity of a function of two variables such as z = ( x a + y b ) 2 with a and b both greater than one... is it enough to show that it is not quasiconcave by showing that the second derivatives are not negative?

Question

Quasi-concavity of a function of two variables such as   z  =  (      x    a    +      y    b        )    2   with a and b both greater than one... is it enough to show that it is not quasiconcave by showing that the second derivatives are not negative?

Medwsa1c · Accepted Answer

Step 1The quasi- part matters. A quasiconcave function need not have negative second derivative. It seems that the definition of quasiconcavity that you are using is  f  (  t  a  +  (  1  −  t  )  b  )  ≥  min  (  f  (  a  )  ,  f  (  b  )  )  ,    0  ≤  t  ≤  1It is a bit more convenient to work with an equivalent form of the above: for every   λ the set   {  x  :  f  (  x  )  ≥  λ  } is convex (on a line this means that the set is an interval). For example, every increasing function on   [  0  ,  ∞  ) is quasiconcave, and       x    2   is an example of that.Concerning   f  (  x  ,  y  )  =  (      x    a    +      y    b        )    2   : note that the upper level set   {  (  x  ,  y  )  :  f  (  x  ,  y  )  ≥  λ  } can be written as   {  (  x  ,  y  )  :      x    a    +      y    b    ≥      λ          1              /            2        } . To show this set is not always convex, I would try   λ  =  1 and the points   (  1  ,  0  ) and   (  0  ,  1  ) . The midpoint       (          1      2        ,               1      2        )   is not in the upper level set.

Quasi-concavity of a function of two variables such as z=(x^a+y^b)^2 with a and b both greater than one... is it enough to show that it is not quasiconcave by showing that the second derivatives are not negative?

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