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

Cubic functions are typically neither quasiconcave nor quasiconvex, but constraints like constant slope on \(x \geq 0\) can unify these properties for some functions.

Step-by-step solution

Definitions of Quasiconcavity and Quasiconvexity

A function is quasiconcave if for any two points in its domain, the function value at any point on the line segment connecting them is not less than the minimum of the function values at these two points. A function is quasiconvex if for any two points, the function value at any point on the line segment is not greater than the maximum of the values at these points.

Analyze the Cubic Function

Consider the cubic function \( z = ax^3 + bx^2 + cx + d \). The nature of cubic functions typically means that they can have inflection points and changes in concavity, which generally prevents uniform quasiconcavity or quasiconvexity across their domains.

Demonstrate Non-Uniform Nature

To verify a lack of quasiconcavity or quasiconvexity generally, examine the second derivative \( z'' = 6ax + 2b \). This varies with \( x \), indicating change of concavity. At different intervals in \( x \), \( z \) can switch between concave and convex regions, precluding broader quasiconcavity or quasiconvexity.

Conditions for Both Quasiconcave and Quasiconvex

Simultaneous quasiconcavity and quasiconvexity implies a constant function. However, here we seek such on \( x \geq 0 \). A function that is of single-layer inflection, like a monotonic expression (e.g., \( z = a(x+1)^3 - a \)) restricted so the derivative doesn't change sign on \( x \geq 0 \) can be both. Setting \( a \geq 0 \), \( b \leq 0 \), and ensuring \( z'(x) = 3ax^2 + 2bx + c \) maintains monoticity can unify both properties.

Conclusion

Generally, cubic functions exhibit changes in slope and curvature preventing common quasiconcavity or quasiconvexity unless imposing strict constraints that prevent inflection or hierarchy shifts.

Key concepts

Quasiconcavity

In mathematical economics, the term "quasiconcavity" refers to a special property of certain functions. A function is deemed quasiconcave if, for any two points in the function's domain, the value of the function at any point on the line segment connecting these two points is at least the smaller of the values at these two points.
This can be particularly useful in optimization, as quasiconcave functions often allow us to find optimal solutions easily.

• A quasiconcave function does not need to be perfectly concave; it only requires that if you are "moving" between two points on the graph, the "path" of the function stays on or above a straight line drawn between these two points.
• This property becomes crucial when dealing with various economic models and utility functions, where maximizing or minimizing a function is key.

Quasiconvexity

Quasiconvexity is the counterpart to quasiconcavity. Here, a function is considered quasiconvex if for any two points, the value of the function at any point on the line segment connecting these two points is not greater than the larger of the values at these two points.
This property is essential in ensuring that the function can uphold certain types of economies of scale or returns to scale in economic theories.

• For example, if you were to consider a production function in economics, quasiconvexity could imply that combining resources more effectively (along the line segment between two different points) will not produce less output than the smaller value at these points.
• It provides useful insight and flexibility in settings where strict convexity is too limiting.

Cubic Function Analysis

Analyzing cubic functions—all third-degree polynomials like \(z = ax^3 + bx^2 + cx + d\)—involves understanding their behavior through their inflection points and changes in concavity.
Cubic functions are known for their complex behavior, which includes possibly displaying multiple turns and changing curvature.

• The inflection point of a cubic function is where the function changes from being concave (curving down) to convex (curving up), or vice versa.
• Studying such points, especially via their first and second derivatives, can help determine where these changes occur.

These characteristics generally prevent a cubic function from being uniformly quasiconcave or quasiconvex without imposing specific constraints on its coefficients and domain.

Derivative Analysis

Derivatives are powerful tools in mathematical analysis, particularly for assessing the behavior of functions. When we apply derivative analysis to a cubic function, we focus on the first and second derivatives.
The first derivative \(z'(x)\), expressed as \(3ax^2 + 2bx + c\), indicates where the function increases or decreases. A zero or change in sign of \(z'(x)\) can reveal critical points where the function changes direction.
The second derivative \(z''(x) = 6ax + 2b\) provides information about the concavity of the function.

• If \(z''(x) > 0\) for a certain \(x\), the function is convex at that point; if \(z''(x) < 0\), it is concave.
• By examining where the second derivative is positive or negative, it helps identify inflection points—critical in studying changes in concavity.

Derivative analysis thus serves as a fundamental approach to understanding and interpreting the complex shapes of cubic functions, and acts as a window to grasp their behavioral complexities.