Question

Show that if \(f\left(x_{1}, x_{2}\right)\) is a concave function then it is also a quasi-concave function. Do this by comparing Equation 2.114 (defining quasi-concavity) with Equation 2.98 (defining concavity). Can you give an intuitive reason for this result? Is the converse of the statement true? Are quasi-concave functions necessarily concave? If not, give a counterexample.

Short answer

Question: Prove that a concave function is also a quasi-concave function. Answer: To prove that a concave function is also a quasi-concave function, we need to show that if function \(f\) is concave, it also satisfies the definition of a quasi-concave function. Since, for any two points \(x\) and \(y\) and for any scalar \(\alpha \in [0,1]\), a concave function \(f\) satisfies: $$ f(\alpha x + (1-\alpha) y) \ge \alpha f(x) + (1-\alpha) f(y) $$ We can see that for any \(x, y\) and \(\alpha \in [0,1]\), if \(f(x) \le f(y)\), then $$ f(\alpha x + (1-\alpha) y) \ge f(x) $$ and if \(f(x) > f(y)\), then $$ f(\alpha x + (1-\alpha) y) \ge f(y) $$ In both cases, we have that $$ f(\alpha x + (1-\alpha) y) \ge \min\{f(x), f(y)\} $$ meaning that \(f\) is also quasi-concave.

Step-by-step solution

Definition of Concave Function

A function \(f\left(x_{1}, x_{2}\right)\) is said to be concave if, for any two points \(x\) and \(y\) and for any scalar \(\alpha \in [0,1]\): $$ f(\alpha x + (1-\alpha) y) \ge \alpha f(x) + (1-\alpha) f(y) $$

Definition of Quasi-Concave Function

A function \(f\left(x_{1}, x_{2}\right)\) is said to be quasi-concave if, for any two points \(x\) and \(y\) and for any scalar \(\alpha \in [0,1]\): $$ f(\alpha x + (1-\alpha) y) \ge \min\{f(x), f(y)\} $$ Now, let's prove the given statement.

Comparing Concave with Quasi-Concave Definition

Suppose \(f\) is a concave function, then for any \(x, y\) and \(\alpha \in [0,1]\), $$ f(\alpha x + (1-\alpha) y) \ge \alpha f(x) + (1-\alpha) f(y) $$ Now, we want to show that \(f(\alpha x + (1-\alpha) y)\) is at least as big as the minimum of \(f(x)\) and \(f(y)\). Consider the two cases below: 1. If \(f(x) \le f(y)\), then $$ \alpha f(x) + (1-\alpha) f(y) \ge \alpha f(x) + (1-\alpha) f(x) = f(x) \Rightarrow f(\alpha x + (1-\alpha) y) \ge f(x) $$ 2. If \(f(x) > f(y)\), then $$ \alpha f(x) + (1-\alpha) f(y) \ge \alpha f(y) + (1-\alpha) f(y) = f(y) \Rightarrow f(\alpha x + (1-\alpha) y) \ge f(y) $$ In both cases, we have that \(f(\alpha x + (1-\alpha) y) \ge \min\{f(x), f(y)\}\), meaning that \(f\) is also quasi-concave.

Intuitive Reason

Intuitively, this result suggests that concave functions are a special case of quasi-concave functions where not only do they fulfill the quasi-concavity requirement, but are also "more concave" due to following the stricter condition posed by concavity.

Converse Statement

The converse is not always true. Quasi-concave functions are not necessarily concave. A counterexample is the function \(f(x) = x^3\), which we illustrate in the following steps: 1. \(f(x) = x^3\) is quasi-concave because for any \(x, y\) and \(\alpha \in [0,1]\): $$ f(\alpha x + (1-\alpha) y) = (\alpha x + (1-\alpha) y)^3 \ge \min\{x^3, y^3\} = \min\{f(x), f(y)\} $$ 2. However, \(f(x) = x^3\) is not concave because its second derivative \(f''(x) = 6x\) is positive for \(x>0\), which violates the condition for concavity (a concave function should have a non-positive second derivative). In conclusion, a concave function is always quasi-concave, but a quasi-concave function is not necessarily concave.

Key concepts

Concave Functions

Concave functions are an important concept in microeconomic theory. They help to describe how outputs change with respect to inputs in various economic models. A function is said to be concave if, for any two points within the function's domain, the line segment connecting these points lies entirely below or on the graph of the function. Mathematically, this is defined as:

• For function \(f(x_1, x_2)\), and any points \(x\) and \(y\), with scalar \(\alpha\) such that \(0 \leq \alpha \leq 1\), the inequality \(f(\alpha x + (1-\alpha) y) \geq \alpha f(x) + (1-\alpha) f(y)\) holds true.

This property indicates that the average of the function is greater than or equal to the function at the average point, showcasing diminishing increases or even declines in value as inputs increase. In business and economics, this can model scenarios where returns or benefits decrease after a certain level of input is reached. A classic example is the utility function in consumer theory, where satisfaction from consumption experiences diminishing returns.

Quasi-Concave Functions

Quasi-concave functions are a broader category than concave functions. They are particularly useful in economic optimization problems where we deal with preferences, production, or utility functions. A function is quasi-concave if

• For any points \(x\) and \(y\), and any value of \(\alpha\) where \(0 \leq \alpha \leq 1\), \(f(\alpha x + (1-\alpha) y) \geq \min\{f(x), f(y)\}\).

This condition implies that the function retains at least the minimal value of the two points it's evaluated at, ensuring the level sets of the function are convex sets. Quasi-concavity is often related to preferences in economics which are not strictly diminishing like concave functions. For instance, while concave utility functions always show risk-averse behavior, quasi-concave utility functions can model more nuanced preferences where the consumer could have varying levels of risk tolerance.

Mathematical Proof

In mathematical terms, proving properties like these involves rigorously showing that one condition implies another. To prove that a concave function is also quasi-concave, you'd compare their respective defining inequalities. Given:

• Concavity: \(f(\alpha x + (1-\alpha) y) \geq \alpha f(x) + (1-\alpha) f(y)\)
• Quasi-concavity: \(f(\alpha x + (1-\alpha) y) \geq \min\{f(x), f(y)\}\)

You start by assuming the stronger condition, concavity, and show it satisfies the weaker condition, quasi-concavity, through logical steps. Intuitively, once the average in concave functions is greater than individual values, it stretches to cover minimums in quasi-concave functions. A counterexample like \(f(x) = x^3\) shows that the reverse is not true; it demonstrates that a function can be quasi-concave but not concave, because its second derivative changes sign, which violates the necessary conditions for concavity. This highlights the nuanced differences in these mathematical constructs critical for economic theory.