Ok, this is an elementary question that has been bothering me for a while. I have a function like
$z(x,y)=x^{0.6}y^{0.4}$.
Defined for $x,y\in R^{++}$
Plainly spoken, this function is concave but not strictly concave. On the other hand another function like $z(x,y)=x^{0.6}y^{0.3}$ is indeed strictly concave.
I am interested to visualize their difference. Now, I know that from the definition of strict concavity, that every line segment of the subset U of $R^n$ is a concave function of one variable. Ok! I draw the above function $z(x,y)=x^{0.6}y^{0.4}$ with some software and get a plot that looks frankly to me very strictly concave. No matter the levels sets of all possible combinations, I aways obtain strictly convex sets. In fact I don't see a visual difference between both functions above. How do I visualize my non strict convexity?
Best Answer
It contains line segment (on the surface of the 3d graph), namely the line $x=y=z$, and applying the concaveness condition for any of these points we will get it with equality.