Question

Consider the utility function \(u\left(x_{1}, x_{2}\right)=\sqrt{x_{1} x_{2}}\). What kind of preferences does it represent? Is the function \(v\left(x_{1}, x_{2}\right)=x_{1}^{2} x_{2}\) a monotonic transformation of \(u\left(x_{1}, x_{2}\right) ?\) Is the function \(w\left(x_{1}, x_{2}\right)=x_{1}^{2} x_{2}^{2}\) a monotonic transformation of \(u\left(x_{1}, x_{2}\right) ?\)

Short answer

Cobb-Douglas preferences are represented. Both \( v(x_1, x_2) \) and \( w(x_1, x_2) \) are monotonic transformations of \( u(x_1, x_2) \).

Step-by-step solution

Understanding the Utility Function

The given utility function is \( u\left(x_{1}, x_{2}\right) = \sqrt{x_{1} x_{2}} \). This function is often used to represent preferences that exhibit a Cobb-Douglas form, indicating a particular type of consumer preference where goods \(x_1\) and \(x_2\) are consumed in balanced proportions. This utility implies diminishing marginal returns and suggests that the goods are complementary.

Defining Monotonic Transformation

A function \( v(x_1, x_2) \) is a monotonic transformation of \( u(x_1, x_2) \) if there exists a strictly increasing function \( f \) such that \( v(x_1, x_2) = f(u(x_1, x_2)) \). This means that \( f' > 0 \).

Evaluating Function v(x_1, x_2)

Consider \( v(x_{1}, x_{2}) = x_{1}^{2} x_{2} \). Set \( v = (\sqrt{x_{1} x_{2}})^4 \). Then, \( v = (u(x_1, x_2))^4 \). The function \( f(u) = u^4 \) is strictly increasing for \( u > 0 \) because the derivative \( f'(u) = 4u^3 > 0 \) for \( u > 0 \). Hence, \( v(x_{1}, x_{2}) \) is a monotonic transformation of \( u(x_{1}, x_{2}) \).

Evaluating Function w(x_1, x_2)

Consider \( w(x_{1}, x_{2}) = x_{1}^{2} x_{2}^{2} \). Set \( w = (\sqrt{x_{1} x_{2}})^{4} \). \( w = (u(x_1, x_2))^4 \) is also a representative form, and this matches precisely with \( v(x_1, x_2) \) which we have already shown is a monotonic transformation. The expression \( u(x_1, x_2)^4 \) has the same form for \( w \) as it does for \( v \), so \( w(x_1, x_2) \) is a monotonic transformation as well.

Key concepts

Cobb-Douglas Preferences

Cobb-Douglas preferences refer to a specific form of utility function where goods are consumed in a balanced manner, leading to a multiplicative relationship between the quantities of goods. The classic form of a Cobb-Douglas utility function is written as \( u(x_1, x_2) = x_1^{a} \, x_2^{b} \), where \( a \) and \( b \) are positive constants that reflect the consumer's preference for each good. In this context, the utility function \( u\left(x_{1}, x_{2}\right)=\sqrt{x_{1} x_{2}} \) implies that goods \( x_1 \) and \( x_2 \) are consumed such that the product of their square roots represents the utility gained. This indicates that the consumer derives satisfaction by consuming both goods in proportional amounts.

These preferences usually highlight diminishing marginal rates of substitution, meaning that as a consumer increases the amount of one good, they will require greater amounts of the other good to maintain the same level of overall satisfaction. The Cobb-Douglas function is widely used in economics to model scenarios where both goods are essential for achieving a desired level of utility, emphasizing their complementary nature. Its mathematical properties facilitate analyses of consumer behavior and resource allocation under various economic conditions.

Monotonic Transformation

Monotonic transformations play an important role in transforming one utility function into another while preserving the preference ordering of the goods. A transformation is considered monotonic if it can be expressed in the form \( v(x_1, x_2) = f(u(x_1, x_2)) \), where the function \( f \) is strictly increasing.

This means that if \( u \) provides a higher utility to bundle A over bundle B, then \( v \) should also reflect that A is preferred to B. To ensure this, \( f'(u) > 0 \) must hold true, meaning that \( f \) must increase as \( u \) increases.

In the given problems, we explored whether \( v(x_1, x_2) = x_1^2 x_2 \) and \( w(x_1, x_2) = x_1^2 x_2^2 \) are monotonic transformations of the function \( u(x_1, x_2) = \sqrt{x_1 x_2} \). By setting \( v \) and \( w \) equal to a power of \( u \), specifically \( u^4 \), it becomes evident that both are monotonic transformations. The derivative \( f'(u) = 4u^3 \) for \( u^4 \) further confirms the strictly increasing nature, ensuring the preference order is maintained. Thus, such transformations allow us to represent preferences using different yet consistent mathematical expressions.

Complementary Goods

Complementary goods are products that are often consumed together, offering increased satisfaction or utility when used in combination rather than individually. In economics, the concept of complementary goods is essential for understanding how consumer preferences and consumption habits are shaped.

The utility function \( u\left(x_{1}, x_{2}\right)=\sqrt{x_{1} x_{2}} \) indicates that \( x_1 \) and \( x_2 \) are complementary. When these goods are consumed together, they maximize the utility for the consumer due to their inherently linked nature in the Cobb-Douglas context. This interdependency implies that the increased consumption of one good demands the increased consumption of the other to maintain optimal satisfaction.

In practical terms, if you're using two complementary goods – like a smartphone and a mobile app – the utility derived is higher when both are utilized together. The utility function's structure in such cases often demonstrates diminishing marginal returns, where increasing one good without the other leads to less additional utility.

Recognizing the complementary relationship between goods helps in forming strategies for bundling and pricing, as it exploits the increased value that consumers place on combinations of products rather than single items.