This note tries to provide a graphical explanation to the quasiconcavity of real valued functions. Please refer to Daniel Wilhem's lecture note and Pemberton & Rau.
Here we first remind ourselves of the relevant definitions. After that, we illustrate each of them.
A set \( V \subset \mathbb{R}^n \) is called convex if the line segment between any two points in \( V \) is contained in the set itself, i.e. if \[ \alpha x + (1-\alpha)y \in V \text{ for all }\alpha \in [0,1], x,y \in V \]
Here we quickly remind ourselves of upper and lower contour, as well as level sets. We will graphically illustrate those below.
The upper contour sets of a function \( f:A \mapsto \mathbb{R} \), defined on the convex set \( A \subset \mathbb{R}^n \), are defined as the sets of values \( x \in A \) for which the function attains at least value \( c\in\mathbb{R} \): \[ \{ x \in \mathbb{R}^n: f(x) \geq c \} \]
The lower contour sets of a function \( f:A \mapsto \mathbb{R} \), defined on the convex set \( A \subset \mathbb{R}^n \), are defined as the sets of values \( x \in A \) for which the function attains at most value \( c\in\mathbb{R} \): \[ \{ x \in \mathbb{R}^n: f(x) \leq c \} \]
The level set (there is just one for each level) of a function \( f:A \mapsto \mathbb{R} \), defined on the convex set \( A \subset \mathbb{R}^n \), is defined as the set of values \( x \in A \) for which the function attains at most value \( c\in\mathbb{R} \): \[ \{ x \in \mathbb{R}^n: f(x) \leq c \} \]
To reiterate, in general there are many upper and lower contour sets (unless the value \( c \) is the global max or min of \( f \)), but only one level set at level \( c \).
\( f \) is a concave function if all the points on or below its graph form a convex set, i.e. if the set \[ \{(x,y) \in \mathbb{R}^n \times \mathbb{R}: y\leq f(x)\} \] is convex.
\( f \) is a quasi-concave function if for any number \( c\in\mathbb{R} \), the following sets are convex: \[ \{ x \in \mathbb{R}^n: f(x) \geq c \} \] Of course those are the upper contour sets.
\( f \) is a quasi-convex function if for any number \( c\in\mathbb{R} \), the following sets are convex: \[ \{ x \in \mathbb{R}^n: f(x) \leq c \} \] Those are the lower contour sets.
We plot the function \( f(x,y) = x^{0.25} y^{0.25},x,y \geq 0 \). You might recognize this as the canonical “Cobb Douglas” functional form. Here goes:
Now, why is this a concave function? We have to check all the points on or below it's graph, i.e. all the points on the shape and in the shape's interior. Line segments connecting any two points must lie inside the shape. Here I draw two lines, both of them are in the interior.
Let's plot a quasi-concave function now. It's the same function as before, but with different powers: \( f(x,y) = x^2 y^2,x,y \geq 0 \)
The definition of a quasi-concave function basically talks about the upper contour sets of \( f \), i.e. \( \{ x \in \mathbb{R}^n: f(x) \geq c \} \), saying that each of those sets should be convex. To graphically illustrated this, let's draw a couple of levels into our quasi-concave function and then look at the corresponding contour plot (i.e. a plot that looks “from above” on to the function and traces the level sets. See below.). First we introduce some level sets into \( f \):
Notice here that the upper contour sets are the collection above any one of the blue planes you could draw in this graph above a certain level \( c \).
Next, here's the contour plot. This is like looking from above at the previous figure and drawing the lines where the original (red) shape and the inserted (blue) level sets intersect. Notice that the figure labels each curve with it's associated level (i.e. the value of \( f \) at that level)
The important thing to note here is that each of the level sets is indeed a convex set: pick any two points and connect them, you'll draw a line inside the corresponding level set.
That is easy to see if we draw the level sets of our concave function from above:
The level sets are convex, therefore this is quasiconcave.
Remember that concavity requires that connecting any two points on or below the graph of \( f \) must produce a line below the graph. Let's try that with our quasi-concave function:
You can see that the line lies outside the shape, so this does not qualify as a concave function.
You can see that whenever \( f \) is quasi-concave, level sets are convex. We want to find a unique point of tangency between the level sets of the objective \( f \) and the constraint set. Convex level sets guarantee this.
You can download a program called R and run a demo file, where you can actually move around the 3D shapes from above. It's for free. To try this out, follow those steps:
1. go to r-project to download R
2. go to github-gist and download the demo file
3. launch R
4. click on File > Source File and choose the file “rgl.r” from the location you downloaded to in step 2
5. the program will produce many plotting windows. you can place them on your screen, resize them, and grab the 3d objects with your mouse to move them around.