Question

(a) Verify that a cubic function \(z=a x^{3}+b x^{2}+c x+d\) is in general neither quasiconcave nor quasiconvex. (b) is it possible to impose restrictions on the parameters such that the function becomes both quasiconcave and quasiconvex for \(x \geq 0 ?\)

Short answer

The general cubic function is neither quasiconcave nor quasiconvex. It can be both if \(a = 0\) and \(b = 0\), making it linear for \(x \geq 0\).

Step-by-step solution

Understand Quasiconcavity and Quasiconvexity

A function is quasiconcave if its upper level sets are convex, and quasiconvex if its lower level sets are convex. This often requires analyzing the sign changes and the restrictions on the parameters of the function.

Analyze General Cubic Function

Consider the cubic function \(z = ax^3 + bx^2 + cx + d\). The function is neither always increasing nor decreasing, indicating it is neither fully quasiconcave nor quasiconvex over its entire domain without restrictions. This is because it has the potential to change inflection, depending on the roots of its derivative.

Differentiate the Cubic Function

Find the first derivative: \(z' = 3ax^2 + 2bx + c\). This will help identify where the function increases or decreases, important for understanding concavity and convexity properties.

Analyze the Derivative for Sign Changes

The sign of \(z' = 3ax^2 + 2bx + c\) changes at roots \(x_1, x_2\), which are determined from solving \(3ax^2 + 2bx + c = 0\). The function is neither quasiconcave nor quasiconvex where this derivative changes sign (indicating changes in increasing or decreasing behavior).

Consider Restrictions for Non-negative Domain

For the function to be both quasiconcave and quasiconvex for \(x \geq 0\), it must be linear, i.e., \(a = 0\) and \(b = 0\), since a linear function has no inflection points and does not change concavity.

Conclusion on Parameter Restrictions

Setting \(a = 0\) and \(b = 0\) makes \(z = cx + d\), a linear function over \(x \geq 0\). This ensures the function is both quasiconcave and quasiconvex on its domain as lines are both.

Key concepts

Understanding Quasiconcavity

Quasiconcavity is an intriguing concept primarily used in economic models and mathematical analysis. A function is considered quasiconcave if, for any two points on its graph, all points along the line segment connecting these two are above the function. This means that the function's upper level sets, regions where output is at least a certain value, are convex. Unlike strict concavity, quasiconcavity does not necessarily imply the function is curved entirely upwards or downwards. It's more about the arrangement of the function's values.
To determine quasiconcavity, visualizing the function graphically or analyzing its derivative can be useful. Particularly, checking where the first derivative changes sign is insightful, as it helps understand how the function's slope behaves.
For cubic functions like \(z = ax^3 + bx^2 + cx + d\), without restrictions on the parameters, they often change in curvature due to their nature. This makes a cubic function not inherently quasiconcave. Thus, parameter restrictions are essential to manipulate its form and maintain the required convexity in its upper level sets.

Unveiling Quasiconvexity

Quasiconvexity is closely related to quasiconcavity, but it flips its conditions. A function is quasiconvex if its lower level sets are convex. Imagine looking at the lower part of a graph and all points on a line between two points below the graph lying beneath the graph at those points. Quasiconvexity ensures that, no matter how you choose those two points, they follow this rule.
For cubic functions, analyzing their direction changes and where they turn upwards or downwards is key. In the function \(z = ax^3 + bx^2 + cx + d\), this is determined by its first derivative. Cubic functions can exhibit patterns that are neither entirely increasing nor decreasing without the proper restrictions, making them not inherently quasiconvex.
Thus, understanding where the derivative changes sign helps us identify segments where the function loses its quasiconvexity, similar to how we analyze quasiconcavity.

Differentiating Cubic Functions

To analyze the behavior of cubic functions, examining their derivatives plays a vital role. The first derivative tells us how the function's slope changes across different points. For our general cubic function \(z = ax^3 + bx^2 + cx + d\), the derivative is \(z' = 3ax^2 + 2bx + c\).
By setting \(z' = 0\), we can find critical points which indicate where the function could change from increasing to decreasing or vice versa. These critical points are located at the solutions, or roots, of the equation \(3ax^2 + 2bx + c = 0\).
Identifying these roots and understanding the sign of the derivative on either side helps us ascertain the presence of any extrema and intervals of increase or decrease. Such an analysis is fundamental in determining the intervals over which the cubic function may exhibit quasiconcavity or quasiconvexity traits, although, without restrictions, it cannot fully achieve either across all its domain.

Exploring Parameter Restrictions

For a cubic function to exhibit both quasiconcavity and quasiconvexity, certain parameter restrictions must be implemented. By observing that linear functions inherently lack inflection points and consistently maintain concavity or convexity, turning the cubic function into a linear form is key.
Here, this means setting the coefficients of higher-degree terms to zero. Specifically, making \(a = 0\) and \(b = 0\) transforms the cubic function \(z = ax^3 + bx^2 + cx + d\) into the linear function \(z = cx + d\). This transformation results in a function that is both quasiconcave and quasiconvex for \(x \geq 0\).
Linear functions don't have sign changes in their first derivative as they have no curvature change, allowing them to maintain the convexity of both their upper and lower level sets continuously, satisfying both conditions of quasiconcavity and quasiconvexity for non-negative domains.