Is the utility function u ( x ) = x 1 x 2 + γ x 2 concave or quasi-concave?

Question

Is the utility function    u  (  x  )  =      x    1        x    2    +  γ      x    2    concave or quasi-concave?

Sean Sutton · Accepted Answer

Consider       A    α    =  {  (      x    1    ,      x    2    )  ∈            R        +    2    :  (      x    1    +  γ  )      x    2    ≥  α  }.If   α  ≤  0, then       A    α    =            R        +    2   which is convex but not strictly convex.If   α  &amp;gt;  0, then       A          α        =  {  (      x    1    ,      x    2    )  ∈            R        +    2    :      x    2    ≥      α                  x        1            +      γ        } which is convex.Hence   u is quasi-concave.It is easay to check that it is not concave by examining the Hessian.

Mark Rosales · Accepted Answer

The hessian matrix is  H  =      [                            0                          1                                      1                          0                      ]  Therefore      x    T    H  x  =            [                                                  x              1                                                                          x              2                                          ]        T        [                            0                          1                                      1                          0                      ]        [                                        x            1                                                            x            2                                ]    =  2      x    1        x    2    ≥  0

Is the utility function u(x)=x_1x_2+gamma x_2 concave or quasi-concave?

Answered question

Answer & Explanation

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