Question

What kind of preferences are represented by a utility function of the form \(u\left(x_{1}, x_{2}\right)=x_{1}+\sqrt{x_{2}} ?\) Is the utility function \(v\left(x_{1}, x_{2}\right)=x_{1}^{2}+2 x_{1} \sqrt{x_{2}}+x_{2}\) a monotonic transformation of \(u\left(x_{1}, x_{2}\right) ?\)

Short answer

Preferences are additive with diminishing returns for good 2. \(v\) is not a monotonic transformation of \(u\).

Step-by-step solution

Understanding the utility function

The utility function \(u(x_1, x_2) = x_1 + \sqrt{x_2}\) suggests that the consumer has additive preferences for the goods, treating them independently. The consumer values good 1 linearly and good 2 with diminishing returns, as indicated by the square root function.

Preferences Indicated by Utility Function

This utility function indicates that the consumer values each additional unit of good 1 equally, but values additional units of good 2 at a decreasing rate. Thus, the preferences represent a mix of linear preference for good 1 and concave preference for good 2.

Checking for Monotonic Transformation

A monotonic transformation of a utility function preserves the order of preferences. That is, if one bundle is preferred to another under \(u\), it should be preferred under \(v\). To check this, \(v(x_1, x_2) = x_1^2 + 2x_1\sqrt{x_2} + x_2\) must be expressed in terms of \(u(x_1, x_2)\), but here it involves quadratic and interaction terms, which do not align with a simple monotonic transformation of the form \(v = f(u)\) where \(f\) is strictly increasing.

Concluding Monotonicity Check

Since the utility function \(v(x_1, x_2)\) includes an \(x_1^2\) term and interaction terms (\(2x_1\sqrt{x_2}\)), it does not simply scale or transform \(u(x_1, x_2)\) with a monotonic function like a logarithm or exponentiation. Hence, \(v\) is not a monotonic transformation of \(u\).

Key concepts

Preferences

When we talk about preferences in the context of utility functions, we are discussing how a consumer values different bundles of goods. In the given utility function \(u(x_1, x_2) = x_1 + \sqrt{x_2}\), the preferences are quite insightful:

• Additive Preferences: The utility function is additive, meaning it treats the goods separately and adds their utilities. Good 1 is valued in a straight linear manner, signifying that each additional unit contributes equally to the utility.
• Concave Preferences: The square root function in \(\sqrt{x_2}\) implies diminishing returns for good 2. This means, as the quantity of good 2 increases, the additional satisfaction or utility gained from each new unit decreases.

These preferences reflect a reality seen often in consumer behavior; while some goods provide constant satisfaction per unit (like good 1), others become less satisfying per additional unit over time (like good 2). This mixed preference structure is crucial in understanding the consumer's choice behavior.

Monotonic Transformation

A monotonic transformation is a way of changing a utility function while preserving the order of preferences among different bundles. In simpler terms, if a consumer prefers bundle A to bundle B using one utility function, they should still prefer A to B after applying a monotonic transformation to that function.To determine if the function \(v(x_1, x_2) = x_1^2 + 2x_1\sqrt{x_2} + x_2\) is a monotonic transformation of \(u(x_1, x_2) = x_1 + \sqrt{x_2}\), these points are examined:

• Preservation of Order: A monotonic transformation should preserve the ranking of preferences defined by the original utility function.
• Strictly Increasing Function: To qualify as a monotonic transformation, the mapping function from \(u\) to \(v\) needs to be strictly increasing.

Upon review, the presence of quadratic and interaction terms in \(v\) suggests that it's not simply a scaled or straightforward transformation of \(u\). These added complexities introduce changes that do not align with a simple increasing function logic, hence \(v\) is not a monotonic transformation of \(u\).

Diminishing Returns

Diminishing returns is a principle where each additional unit of good provides less additional utility compared to the previous unit. In our utility function \(u(x_1, x_2) = x_1 + \sqrt{x_2}\), this is observed in the function \(\sqrt{x_2}\).The square root function in the utility calculation is crucial because:

• Non-linear Increase: As \(x_2\) increases, \(\sqrt{x_2}\) increases at a decreasing rate. Hence, the first few units of good 2 increase utility more significantly compared to later units.
• Utility Context: This models realistic spending habits, where a consumer rapidly achieves satisfaction from the initial amounts of a good, but the desire for additional units wanes.

This concept often aids consumers in deciding optimal consumption levels. Making choices based on diminishing returns ensures that resources provide maximum overall satisfaction, which is central to many economic decisions and analyses.