Question

Show that if \(f\left(x_{1}, x_{2}\right)\) is a concave function, then it is also a a quasi-concave function. Do this by comparing Equation 2,100 (defining quasi-concavity) with Equation 2.84 (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

If not, provide a counterexample. Answer: A concave function is also a quasi-concave function because it satisfies the stronger inequality that defines concavity. Intuitively, both concave and quasi-concave functions are "bowl-shaped" or "mountain-shaped" in their domains, but concavity imposes a stricter condition. The converse statement, however, is not true: not all quasi-concave functions are concave. A counterexample to this is the function \(f(x) = x^3\) defined in the domain \([-1,1]\). This function is quasi-concave but not concave.

Step-by-step solution

Recall the definitions of concavity and quasi-concavity for a function

A function \(f\) is said to be concave if it satisfies the following inequality for any two points \(x_1\) and \(x_2\), and for any \(t \in [0, 1]\): (1) \(f(tx_1 + (1-t)x_2) \ge tf(x_1) + (1-t)f(x_2)\) On the other hand, a function \(f\) is said to be quasi-concave if it satisfies the following inequality for any two points \(x_1\) and \(x_2\), and for any \(t \in [0, 1]\): (2) \(f(tx_1 + (1-t)x_2) \ge \min(f(x_1), f(x_2))\)

Compare the defining equations for concavity and quasi-concavity

In order to show that a concave function must also be quasi-concave, we need to compare inequalities (1) and (2). For any given points \(x_1\) and \(x_2\), we need to show that if \(f\) is concave, then inequality (2) holds. From inequality (1), we have: \(f(tx_1 + (1-t)x_2) \ge tf(x_1) + (1-t)f(x_2)\) Since \(f(tx_1 + (1-t)x_2)\) is greater or equal to the convex combination of \(f(x_1)\) and \(f(x_2)\), the value of \(f(tx_1 + (1-t)x_2)\) must be greater or equal to the smaller of \(f(x_1)\) and \(f(x_2)\), i.e., \(\min(f(x_1), f(x_2))\). Therefore, inequality (2) holds and the function is also quasi-concave.

Intuitive explanation

Intuitively, both concave and quasi-concave functions represent "bowl-shaped" or "mountain-shaped" functions in their domains. A concave function is a stricter condition than quasi-concavity as it requires the function to satisfy a stronger inequality (1). Hence, if a function is concave and satisfies this stronger constraint, it must also satisfy the weaker constraint of quasi-concavity.

Is the converse statement true?

Now, we need to verify if quasi-concave functions are necessarily concave. The converse statement would say that if a function is quasi-concave, then it must be concave. To verify this, we can either provide a counterexample or prove it analytically. Let's provide a counterexample: Consider the function \(f(x) = x^3\) defined in the domain \([-1,1]\). This function is not concave since the second derivative, \(f''(x) = 6x\), is positive for \(x > 0\). However, it is quasi-concave because it satisfies the inequality (2). To see this, consider two points \(x_1\) and \(x_2\) in the domain of \(f\). Notice that: - If both \(x_1\) and \(x_2\) are positive, the minimum between \(f(x_1)\) and \(f(x_2)\) is negative, and so is \(f(tx_1+(1-t)x_2)\). - If both \(x_1\) and \(x_2\) are negative, note that \(f\) is strictly increasing in the interval \([-1, 0]\) and so the minimum between \(f(x_1)\) and \(f(x_2)\) will always be smaller than \(f(tx_1+(1-t)x_2)\). - If one of the points is positive, say \(x_2\), then the minimum between \(f(x_1)\) and \(f(x_2)\) is \(f(x_1)<0\), and so the inequality (2) is satisfied. So, we have found a counterexample which shows that the converse statement is not true. Not all quasi-concave functions are necessarily concave.

Key concepts

Microeconomic Theory

Microeconomic theory is a branch of economics that focuses on the behavior of individuals and firms in making decisions regarding the allocation of limited resources. It examines how these decisions and behaviors affect the supply and demand for goods and services, which determines prices and how resources are allocated in markets.

Understanding the nature of functions like concave and quasi-concave functions in microeconomic theory is fundamental because they relate to utility functions, production functions, and cost functions. These functions are used to model preferences, technologies, and costs, respectively. For instance, if a utility function is concave, it implies diminishing marginal utility, which aligns with the observation that, as consumers have more of a good, the less they benefit from an additional unit.

Inequality Constraints in Optimization

Inequality constraints are a critical aspect of optimization problems, where one seeks to maximize or minimize a function subject to certain limitations. In economics, optimization often involves constrained resources, like time or money. When using concavity or quasi-concavity in optimization, particularly in microeconomic theory, these properties can simplify the problem.

Concave functions are particularly important because if an optimization problem entails maximizing a concave utility function or minimizing a convex cost function, then if there exists a solution, it is guaranteed to be a global maximum or minimum, even with inequality constraints present. Quasi-concavity, while not providing the same strong guarantee, still offers useful structure: it ensures that the set of higher-value outputs forms a convex set, facilitating the identification of optimal solutions under constraints.

Concavity in Economics

Concavity in economics is utilized to describe functions where a line segment between any two points on the function's graph will never lie below the graph itself. This implies diminishing returns, a key economic principle, suggesting that as one continues to increase an input, the incremental gains in output will eventually decline.

For a function to be concave, it must satisfy an important inequality known as the concavity inequality, given by formula (1) in our exercise. Such concave functions are often used to represent various economic phenomena, including risk aversion in consumer choice or diminishing marginal productivity in firm production. As per the educational exercise, this property assures that if a function is concave, it is also quasi-concave, which accommodates principles of choice and decision-making under uncertainty.

Counterexamples in Mathematical Economics

Counterexamples in mathematical economics serve as a critical tool to test the limits of economic theories and mathematical models. They can disprove generalizations or demonstrate the existence of exceptions to a proposed rule. The exercise provides a tangible example of how counterexamples are used: showing that quasi-concavity does not imply concavity.

A model or theory might suggest that a certain property should always hold, but a counterexample provides the precise situation where it does not, forcing economists and mathematicians to refine their models. In our case, the function given by the exercise, f(x) = x^3, serves as a counterexample to the incorrect notion that all quasi-concave functions are concave. This clarity is essential for accurate representations of the real-world economic phenomena and theoretical integrity.