How do you prove from definition (no Hessians) that $U(x_1,x_2)=x_1^2 x_2$ is quasi-concave?

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    $\begingroup$ What is the definition of Quasi-Concavity? $\endgroup$ – Brennan Nov 12 '19 at 18:40

Take $(1,1)$ and $(-1,1)$: we have that $U(1,1)=U(-1,1)=1$. However, $U(\frac12(1,1) + \frac12(-1,1)) = U(0,1) - 0 < 1 = \min\{U(1,1),U(-1,1)\}$. Hence the function, at least defined globally over $\mathbb R^2$ is not quasi-concave.

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Just to build up on @201p answer:

You probably wanted to prove that the utility function $u: \mathbb{R^2_{++}} \to \mathbb{R}$ given by: $u(x_1,x_2) = x^2_1 x_2$ is quasiconcave. As @201p pointed - allowing the function to be defined globally over $\mathbb{R^2}$ gives you trouble.

To see the intuition behind it, let's start with a definition. The function $u$ of many variables defined on a convex set $S$ is quasiconcave if every upper level set of function $u$ is convex.

Draw a level set of aforementioned utility function allowing it to be defined globally over $\mathbb{R^2}$:

enter image description here

Shaded area is the upper level set (at some level of utility). You can clearly see that it is not convex, therefore utility function defined that way is not quasiconcave.

However, if you allow utility function to be $u: \mathbb{R^2_{++}} \to \mathbb{R}$ given by: $u(x_1,x_2) = x^2_1 x_2$. The level set would consist only with the part on the right, clearly being convex.

Let $u: \mathbb{R^2_{++}} \to \mathbb{R}$ given by: $u(x_1,x_2) = x^2_1 x_2$.

Another way of thinking about it could be:

Proposition. The function $f$ of many variables defined on a convex set $S$ is quasiconcave if and only if for all $x \in S$ and $x' \in S$ such that $f(x) ≥ f(x')$ we have $f((1−λ)x + λx') ≥ f(x')$ for all $λ ∈ [0, 1]$.

But all this actually means is that a function is quasiconcave if and only if the line segment joining the points lying on two level curves lies nowhere below the level curve corresponding to the lower value of the function. Higher level curve represents higher utility. Let point $A$ represent higher utility and $B$ lower utility. If the segment joining $A$ and $B$ lies on or above the indifference curve corresponding to the smaller value of the utility function (speaking about point $B$), then the function $u$ of many variables is quasiconcave.

enter image description here

For more interesting examples you should check out: https://mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/qcc/t

Finally, Cobb-Douglas function $f: \mathbb{R^2_{++}} \to \mathbb{R}$ given by: $u(x_1,x_2) = x^a_1 x^b_2$ for positive $a,b>0$ is:

  • strictly concave if $a + b < 1$
  • concave if $a + b = 1$
  • neither concave nor convex if $a + b > 1$
  • quasiconcave for all $a,b > 0$

For proofs you could check out this!

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