Question

Show that if \(f\left(x_{1}, x_{2}\right)\) is a concave function, it is also a quasi-concave function. Do this by comparing Equation 2.107 (defining quasi- concavity) to Equation 2.88 (defining concavity). Can you give an intuitive reason for this result? Is the converse of the statement true? Are quasiconcave functions necessarily concave?

Short answer

#tag_title# Conclusion #tag_content# In summary, a concave function is also a quasi-concave function, as it satisfies the inequality for quasi-concavity by satisfying the stricter condition of concavity. Intuitively, this is because a concave function's "bowl shape" automatically satisfies the less strict requirement of quasi-concavity. However, the converse statement is not true: not all quasi-concave functions are concave, as evidenced by the example of \(f(x) = x^3\) on \([-1, 1]\).

Step-by-step solution

Concavity Definition

A function \(f(x_1, x_2)\) is concave if, for any \(x, y\) in its domain, and any \(t \in [0, 1]\), we have $$ f(tx + (1-t)y) \geq t f(x) + (1-t) f(y) $$

Quasi-concavity Definition

A function \(f(x_1, x_2)\) is quasi-concave if, for any \(x, y\) in its domain, and any \(t \in [0, 1]\), we have $$ f(tx + (1-t)y) \geq \min \{f(x), f(y)\} $$

Comparing Definitions

Now, let's compare the definitions of concavity and quasi-concavity. Notice that if the inequality for concavity is satisfied, then the inequality for quasi-concavity is satisfied as well. This is because when \(f(tx + (1-t)y) \geq t f(x) + (1-t) f(y)\), it is also greater than or equal to the minimum of \(f(x)\) and \(f(y)\). Therefore, a concave function is also a quasi-concave function.

Intuitive Reason

Intuitively, a concave function represents a "bowl shape," where the function values are always lower or equal to a weighted average of the function values at the endpoints. A quasi-concave function is similar, but the function values only need to be greater than or equal to the minimum of the function values at the endpoints. It's clear that satisfying the stricter condition of concavity also satisfies the less strict condition of quasi-concavity.

Converse Statement

Is the converse of the statement true? That is, are quasi-concave functions necessarily concave? The answer is no. Quasi-concavity is a weaker condition than concavity. Some functions can be quasi-concave but not concave. For example, consider the function \(f(x) = x^3\) on \([-1, 1]\). This function is quasi-concave but not concave.

Key concepts

Concavity

Understanding the concavity of a function is crucial for analyzing the nature of economic problems within microeconomic theory. When we describe a function as concave, we're referring to a unique property that portrays a specific shape or curvature. The formal definition states that a function, say f(x), is concave if, for any two points x and y within its domain, and any t between 0 and 1, the following inequality holds:
\[f(tx + (1-t)y) \geq tf(x) + (1-t)f(y)\]
This mathematical expression assures that the function's value at any point on the line segment connecting (x, f(x)) and (y, f(y)) is not below the line segment itself. This concept is best visualized as a 'bowl' or 'upside-down cap' shape, indicating that the combination of two points on the graph yields a value that's at least as high as their averaged outputs.
In the real world, the concavity of a utility function in microeconomics, for instance, indicates diminishing marginal utility, meaning that as consumption increases, the additional satisfaction gained from consuming an additional unit of a good decreases.
However, when breaking down textbook solutions, it's important to remember that while all concave functions are quasi-concave, they adhere to stricter mathematical requirements, making them a specific subset within the broader category of quasi-concave functions.

Quasi-Concavity

When diving into the realm of quasi-concavity, we are dealing with a less stringent condition compared to concavity. A function is considered quasi-concave if and only if for all x and y in the domain, and any t in the interval [0,1], the function adheres to:
\[f(tx + (1-t)y) \geq \min \{f(x), f(y)\}\]
Quasi-concavity merely requires the function's value at a linear combination of two points to be greater than or equal to the minimum value of the function at those points. This broader condition implies that the function doesn't necessarily have to 'bowl-shaped'; it can, for example, plateau or even have increasing segments while still maintaining quasi-concavity. With regards to comparative statics, a quasi-concave function's local maximum is also a global maximum, which is pivotal in optimization problems often found in microeconomic theory. An intuitive grasp of this concept is essential for students to understand various economic phenomena, such as consumer choice and producer behavior.

Microeconomic Theory

Microeconomic theory serves as the foundation for understanding the complex interactions and behaviors of individual economic participants, including consumers and firms. It examines how entities make decisions about resource allocation and the implications of those decisions on the use and distribution of scarce resources. The theory frequently utilizes mathematical models to represent various scenarios, and this is where concepts like concavity and quasi-concavity come in handy.
Functions that describe utility, production, and cost within these models often exhibit concavity or quasi-concavity due to their implications on diminishing returns and efficiency. For example, a concave utility function aligns with the principle of diminishing marginal utility, a cornerstone of consumer choice theory. Recognizing and understanding these function properties empower students with predictive tools for analyzing changes in market conditions and individual behavior—pivotal aspects of comparative statics within this field of economics.

Comparative Statics

Comparative statics is the study of how a static (unchanging) model's results vary when its input factors change. Essentially, it's a technique used to analyze the comparative positions of equilibrium before and after changes to exogenous variables. In microeconomics, this is often applied to explore how the equilibrium price and quantity of goods respond to shifts in demand and supply conditions.
When employing concave and quasi-concave functions in these analyses, the comparative statics method becomes particularly clear. This is because such functions help us determine how optimal solutions, such as the maximum utility or profit, are affected by changes in external variables. A thorough grasp of these concepts allows students to conduct meaningful comparative statics analysis, which is a fundamental aspect of economic study. It enables the prediction of how policy changes, market shocks, or other alterations outside the model will influence the equilibrium state within the model.