Question

(a) Let \(z=f(x)\) plot as a negatively sloped curve shaped like the right half of a bell in the first quadrant, passing through the points \((0,5),(2,4),(3,2),\) and (5,1) . Let \(z=g(x)\) plot as a positively sloped 45 ' line. Are \(f(x)\) and \(g(x)\) quasiconcave? (b) Now plot the sum \(f(x)+g(x)\). Is the sum function quasiconcave?

Short answer

Both \(f(x)\) and \(g(x)\) are quasiconcave, but \(f(x)+g(x)\) is not necessarily quasiconcave.

Step-by-step solution

Understanding Quasiconcavity

A function is quasiconcave if its upper level sets are convex. This implies that for any points \(x_1\) and \(x_2\) in the domain, the line segment connecting \(f(x_1)\) and \(f(x_2)\) does not cross below the curve. Quasiconcavity allows for downward curves.

Analyzing \(f(x)\)

The function \(f(x)\) describes a curve that is negatively sloped like the right half of a bell. Points include \((0,5)\), \((2,4)\), \((3,2)\), and \((5,1)\). When connecting these points, the curve slopes downwards and passes to the right of each point. This curve resembles a bell due to its diminishing slope.

Determining Quasiconcavity for \(f(x)\)

Since the slope is always decreasing from left to right, for two points \(x_1\) and \(x_2\), \(f(x_1)\) is always greater than or equal to \(f(x_2)\) for \(x_1 < x_2\). Therefore, any line segment between points on the curve will not dip below the function, indicating that \(f(x)\) is quasiconcave.

Analyzing \(g(x)\)

The function \(g(x)\) is a positively sloped 45-degree line which implies that it increases linearly with \(x\). This type of function is linear, implying that any line segment between points is part of the function itself.

Determining Quasiconcavity for \(g(x)\)

Since \(g(x)\) is a linear function creating a straight line, its upper level sets are trivially convex (a straight line forms its own boundary), thereby making \(g(x)\) quasiconcave.

Plotting \(f(x)+g(x)\)

To determine if the sum \(h(x) = f(x) + g(x)\) is quasiconcave, observe how \(f(x)\), which slopes downward, interacts with \(g(x)\), which slopes upward. At each point, add values of \(f(x)\) and \(g(x)\). This will result in a curve that initially rises (due to the linear increase in \(g(x)\) being greater than the decrease in \(f(x)\)), but will eventually resemble the shape of \(f(x)\).

Determining Quasiconcavity for \(h(x)\)

The sum function \(h(x)\) can initially increase before decreasing as the diminishing negative slope of \(f(x)\) influences \(h(x)\). This can potentially lead to some upper level sets which are not convex depending on where the sum changes direction. Therefore, \(f(x) + g(x)\) is not necessarily quasiconcave as changes in curvature and overall shape do not maintain a consistently convex upper set.

Key concepts

Convex sets

To understand quasiconcavity, it's crucial to first grasp the concept of convex sets. Imagine a convex set as a shape where, if you pick any two points within it, the line connecting them stays entirely inside the shape. A simple example is a circle or a rectangle. If you take two points within any of these shapes, the line between them won't leave the border, fulfilling the convexity condition.
Convexity is important because it underlines quasiconcavity. If the upper-level sets of a function are convex, then the function itself is quasiconcave. An upper-level set for a function is the set of points where the function's value is at least a certain level. If you connect any two points inside this upper-level set, the connecting line stays within the set, demonstrating convexity.
For instance, consider a function where the values create such a shape inside the coordinate plane; ensuring quasiconcavity means the midpoints derived by connecting any points at the same level remain above the line.

Quasiconcave functions

Quasiconcave functions are special because they only require the upper-level sets to be convex, not necessarily the function itself. This means that the function can dip and curve as long as its upper sections (the areas where the function value is higher) always maintain a convex form.
Consider the idea of a curve as you might see at the top of a hill. As long as you can draw a line across any two points where the function is the same height or lower, without dropping below the curve elsewhere, the function is quasiconcave.
To illustrate, let's revisit the exercise example. The function \(f(x)\) in the exercise looks like the right side of a bell curve. Since it's decreasing, for any points \(x_1 < x_2\), the value \(f(x_1)\) is greater than \(f(x_2)\). This indicates quasiconcavity because no connecting line between points ever falls below the curve, satisfying the condition of convex upper-level sets.
Linear or even flat functions, such as the line you see with \(g(x)\), also qualify because any straight line can be considered trivially convex, fitting into the definition smoothly.

Mathematical economics

In the realm of mathematical economics, understanding quasiconcavity is vital because many economic models and utility functions depend on this property. It allows economists to model preferences and make predictions about consumer behavior.
Economists frequently assume that utility functions are quasiconcave, as they signify diminishing returns or preferences in more intuitive ways. For example, if an individual's utility from consuming goods behaves like a quasiconcave function, it suggests that while consuming more may lead to higher satisfaction, the incremental benefit decreases as the amount consumed increases.
The mathematical tools of convex sets and quasiconcave functions help in visualizing and solving optimization problems, giving precise answers to economic questions about resource allocations, price setting, and production strategies. For instance, in assessing whether a particular distribution of resources achieves the highest level of utility or satisfaction, ensuring the functions and models employ quasiconcave properties can guide toward more effective economic outcomes.