How can I show that the function $u(x_1,x_2)=(x_1x_2)^\alpha$ is quasiconcave, given $\alpha>1,x_i\geq0$?

I managed to find the bordered Hessian, whereby




Nonetheless, that only provides a necessary condition for quasiconcavity.

Is it possible to show quasiconcavity from its definition, i.e., $u(ax_1+(1-a)y_1,ax_2+(1-a)y_2)\geq \textrm{min}\{u(x_1,x_2),u(y_1,y_2)\}$?


1 Answer 1


Is it possible to show quasiconcavity from its definition, i.e., $u(ax_1+(1-a)y_1,ax_2+(1-a)y_2)\geq \textrm{min}\{u(x_1,x_2),u(y_1,y_2)\}$?

Answer: Yes.

A useful trick that can save you some trouble is to perform a monotonic transformation. In preference relation terms you are trying to show

$$ \left( ax_1+(1-a)y_1,ax_2+(1-a)y_2 \right) \succeq \left[ (x_1,x_2) \text{ OR } (y_1,y_2) \right]. $$

If this holds for the utility representation $(x_1x_2)^{\alpha}$, it will also hold for monotonic transformations of this function (the ordering of baskets is unchanged).

Clearly there is a more elegant function to represent the relation, making mathematical calculations easier.

Alternatively you can power through it and make some assumptions w.o.l., like $u(x_1,x_2) \leq u(y_1,y_2)$. The function is clearly strictly monotonic, so that saves you from looking at cases where $x_1 \leq y_1$ AND $x_2 \leq y_2$. All that remains to check (w.o.l.) is the case where $x_1 < y_1$, $x_2 > y_2$.

  • $\begingroup$ Under the second strategy, the condition $u(ax_1+(1−a)y_1,ax_2+(1−a)y_2)=\{[ax_1+(1-a) y_1][ax_2+(1-a) y_2 ]\}^\alpha \geq u(x_1,x_2)=(x_1x_2)^\alpha$ must be fulfilled. Assuming that $x_1<y_1$ and $x_2>y_2$, we have that $ax_1+(1-a) y_1\geq x_1$ and $ax_2+(1-a) y_2\leq x_2$, which does not fulfil the above condition. Am I doing something wrong here? $\endgroup$ Aug 28, 2021 at 5:15
  • 1
    $\begingroup$ @kékszajkók The above is not a complete answer, as the question is a self-study problem. You will also need to use $u(x1,x2) \leq u(y1,y2)$, and still there are some algebraic manupulations to do. $\endgroup$
    – Giskard
    Aug 28, 2021 at 8:05

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