Question

What kind of preferences are represented by a utility function of the form \(u\left(x_{1}, x_{2}\right)=\sqrt{x_{1}+x_{2}} ?\) What about the utility function \(v\left(x_{1}, x_{2}\right)=\) \(13 x_{1}+13 x_{2} ?\)

Short answer

The utility \(u(x_{1}, x_{2}) = \sqrt{x_{1}+x_{2}}\) represents substitutable goods with diminishing returns. The utility \(v(x_{1}, x_{2}) = 13x_{1}+13x_{2}\) represents perfect substitutes.

Step-by-step solution

Analyze First Utility Function

The utility function \(u(x_{1}, x_{2}) = \sqrt{x_{1} + x_{2}}\) is increasing and concave. This means that it represents preferences that are monotonic and exhibit diminishing marginal utility. As both \(x_1\) and \(x_2\) increase, the utility increases, but at a decreasing rate. This reflects perfect substitutes without constant substitution between the goods, as well as diminishing returns when adding more quantities of either good.

Interpret First Set of Preferences

Considering the concave nature of \(u(x_{1}, x_{2}) = \sqrt{x_{1}+x_{2}}\), it represents preferences for goods that are substitutable, but not perfect substitutes. Consumers may replace one good with the other to some extent, but modifying quantities may affect overall satisfaction due to diminishing marginal returns.

Analyze Second Utility Function

The utility function \(v(x_{1}, x_{2}) = 13x_{1} + 13x_{2}\) is linear. This linear form implies that the marginal rate of substitution is constant because the slope of the indifference curve remains unchanged even with increased consumption. Therefore, this utility function models perfect substitutes, as consumers are willing to substitute one good for the other at a constant rate, retaining the same level of utility.

Interpret Second Set of Preferences

For the linear utility function \(v(x_{1}, x_{2}) = 13x_{1} + 13x_{2}\), the preferences represent perfect substitutability between the two goods, where each additional unit of either good provides the same utility increment.

Key concepts

monotonic preferences

Monotonic preferences are a key concept in understanding utility functions. This idea revolves around the simple notion that more is always better. If you have more of a good, your utility, or satisfaction, increases. This principle holds true as long as the goods are desirable. In economics, utility functions like \( u(x_{1}, x_{2}) = \sqrt{x_{1} + x_{2}} \) display monotonic preferences.

As you increase quantities of either \( x_1 \) or \( x_2 \), the utility strictly rises. Importantly, this happens without hitting a maximum utility threshold—utility keeps climbing as long as you have more. The important note here is that the utility increases but never decreases with more consumption.

For instance, consider having two different snacks; more of either means more satisfaction. Monotonicity reflects this reality, significant in consumer choice theory, because it ensures that individuals always prefer larger bundles.

diminishing marginal utility

Diminishing marginal utility relates to how the additional satisfaction or utility from consuming extra units of a good decreases as you consume more of it. Think of it like eating chocolate bars: the first bar might be delightful, the second satisfying, but by the third or fourth, each bar adds less joy than the previous one.

The utility function \( u(x_{1}, x_{2}) = \sqrt{x_{1} + x_{2}} \) demonstrates diminishing marginal utility. Here, as you add more of \( x_1 \) or \( x_2 \), the increase in utility lessens. This happens because the square root function grows at a decreasing rate; hence, each additional good contributes a smaller increment of utility.

This concept is vital because it explains why individuals diversify their consumption. When the added satisfaction decreases, people might opt to consume more varied goods instead of just increasing the quantity of one.

perfect substitutes

Perfect substitutes occur when one good can entirely replace another in consumption without affecting the consumer's overall satisfaction. This happens when goods provide the same utility swap for swap.

The utility function \( v(x_{1}, x_{2}) = 13x_{1} + 13x_{2} \) exemplifies perfect substitutes. Because this function is linear, the increase in one good gives the same rise in utility as the other. The slope of the indifference curve here is constant, meaning the marginal rate of substitution remains the same.

In practical terms, consider tea and coffee being perfect substitutes for a caffeine lover. If the consumer is indifferent between having one cup of tea or one cup of coffee, then they are perfect substitutes.

• Linearity indicates this direct exchangeability of goods.
  
• Utility equals increments for both goods.
• Preferences don't change with consumption changes between these goods.

This scenario differs starkly from other preferences where substitution might key into different utility increments.