(quasi) convex (quasi) concave function

[goldiecr12]

In my advanced microeconomic class, i have the following question

determine if the fonction ? = min {?1 , ?2} -> ℝ2 is (strictly) concave/convex, (strictly) quasiconcave/quasiconvexe, on its domain.

I have no idea how to approch the problem

 

[goldiecr12]

 

[Steven G]

I am not sure how the function y = min{x,y} maps onto R^2.

After all y =min{3,8} =3 and is not in R^2. Am I missing something here?

 

[goldiecr12]

Thank for your response !

But its not min(x,y) its min (x1, x2) so x is a vector x=(x1,x2)

 

[Steven G]

Generally vectors are written as <x1, x2>.

 

[Dr.Peterson]

I think you need to give us the entire context of the question, and define the notation. It appears that your notation is not what we would consider standard.

If this means the minimum of all vectors in some set, I don't even know what that means, since vectors can't be compared - unless some such comparison has been defined for you.

 

[goldiecr12]

oh i'm sorry for the bad notation. It's my first post on this site. If i change the question for :

For each of the next functions, determine if it is (strictly) concave/convex, (strictly) quasiconcave/quasiconvexe, on its domain.



I know how to answers 1 to 8 but i don't know 9 and 10. Maybe it's not a vector but i know the fonction is in 3D and have 3 variables because the domain is R2+. I'm not really good in english so it's tough for me.

 

[Dr.Peterson]

Thanks. Although this doesn't show the definition of the notation, seeing more problems for comparison somehow helps me see what this is about.

The notation means just what we initially thought it means: the minimum (that is, smaller) of the two variables x₁ and x₂.

But we are to think of [MATH](x_1, x_2)[/MATH] as an element of the set [MATH]\mathbb{R}_+^2[/MATH]. So the value of the function is [MATH]x_1[/MATH] on the region where [MATH]x_1 < x_2[/MATH], and [MATH]x_2[/MATH] on the region where [MATH]x_1 \ge x_2[/MATH].

Does that help you get started? If so, and if you need more help, please show us how the definitions of convex, etc., apply to this function.