Question

The formal study of preferences uses a general vector notation. A bundle of \(n\) commodities is denoted by the vector \(\mathbf{x}=\left(x_{1}, x_{2}, \ldots, x_{n}\right),\) and a preference relation \((>)\) is defined over all potential bundles. The statement \(\mathbf{x}^{1}>\mathbf{x}^{2}\) means that bundle \(\mathbf{x}^{1}\) is preferred to bundle \(\mathbf{x}^{2}\). Indifference between two such bundles is denoted by \(\mathbf{x}^{1}=\mathbf{x}^{2}\) The preference relation is "complete" if for any two bundles the individual is able to state either \(\mathbf{x}^{1}>\mathbf{x}^{2}, \mathbf{x}^{2}>\mathbf{x}^{1},\) or \(\mathbf{x}^{1}=\mathbf{x}^{2} .\) The relation is "transitive" if \(\mathbf{x}^{1}>\mathbf{x}^{2}\) and \(\mathbf{x}^{2}>\mathbf{x}^{3}\) implies that \(\mathbf{x}^{1}>\mathbf{x}^{3}\). Finally, a preference relation is "continuous" if for any bundle \(y\) such that \(y>x,\) any bundle suitably close to y will also be preferred to \(\mathbf{x}\). Using these definitions, discuss whether each of the following preference relations is complete, transitive, and continuous. a. Summation preferences: This preference relation assumes one can indeed add apples and oranges. Specifically, $$\begin{array}{l} \mathbf{x}^{1}>\mathbf{x}^{2} \text { if and only if } \sum_{i=1}^{n} x_{i}^{1}>\sum_{i=1}^{n} x_{i}^{2} \text { . If } \sum_{i=1}^{n} x_{i}^{1}=\sum_{i=1}^{n} x_{i}^{2} \\ \mathbf{x}^{1} \approx \mathbf{x}^{2} \end{array}$$ b. Lexicographic preferences: In this case the preference relation is organized as a dictionary: If \(x_{1}^{1}>x_{1}^{2}, \mathbf{x}^{1}>\mathbf{x}^{2}\) (regardless of the amounts of the other \(n-1\) goods). If \(x_{1}^{1}=x_{1}^{2}\) and \(x_{2}^{1}>x_{2}^{2}, \mathbf{x}^{1}>\mathbf{x}^{2}\) (regardless of the amounts of the other \(n-2 \text { goods }) .\) The lexicographic preference relation then continues in this way throughout the entire list of goods. c. Preferences with satiation: For this preference relation there is assumed to be a consumption bundle ( \(\mathbf{x}^{*}\) ) that provides complete "bliss." The ranking of all other bundles is determined by how close they are to \(\mathbf{x}^{*}\). That is, \(\mathbf{x}^{1}>\mathbf{x}^{\mathbf{2}}\) if and only if \(\left|\mathbf{x}^{1}-\mathbf{x}^{*}\right|<\left|\mathbf{x}^{2}-\mathbf{x}^{*}\right|\) where \(\left|\mathbf{x}^{1}-\mathbf{x}^{*}\right|=\) \(\sqrt{\left(x_{1}^{i}-x_{1}^{*}\right)^{2}+\left(x_{2}^{i}-x_{2}^{*}\right)^{2}+\cdots+\left(x_{n}^{i}-x_{n}^{*}\right)^{2}}\)

Short answer

Question: Analyze the following three preferences for completeness, transitivity, and continuity: a. Summation Preferences, b. Lexicographic Preferences, c. Preferences with Satiation. Answer: a. Summation Preferences: complete, transitive, and continuous; b. Lexicographic Preferences: complete, transitive, but not continuous; and c. Preferences with Satiation: complete, transitive, and continuous.

Step-by-step solution

Completeness

For any two given bundles, \(\mathbf{x}^{1}\) and \(\mathbf{x}^{2}\), the summation will result in either greater, equal or smaller number depending on the summation of each bundle. Therefore, the summation preference relation is complete.

Transitivity

If \(\mathbf{x}^{1}>\mathbf{x}^{2}\) and \(\mathbf{x}^{2}>\mathbf{x}^{3}\), it implies that \(\sum_{i=1}^{n} x_{i}^{1}>\sum_{i=1}^{n} x_{i}^{2}\) and \(\sum_{i=1}^{n} x_{i}^{2}>\sum_{i=1}^{n} x_{i}^{3}\). Considering that the sums can be directly compared, we can conclude that \(\mathbf{x}^{1}>\mathbf{x}^{3}\). Therefore, the summation preference relation is transitive.

Continuity

If y > x, any bundle close to y, let's say z, will also result in z > x since the sum of the small differences in each component of the bundles remains small. So, the summation preference relation is continuous. b. Lexicographic Preferences

Completeness

For any given bundles, there will always be a decision made based on the first element, then the second element if needed, and so on. Therefore, the lexicographic preference relation is complete.

Transitivity

If \(\mathbf{x}^{1}>\mathbf{x}^{2}\) and \(\mathbf{x}^{2}>\mathbf{x}^{3}\), lexicographically comparing each element in the bundles, we can conclude that \(\mathbf{x}^{1}>\mathbf{x}^{3}\). Therefore, the lexicographic preference relation is transitive.

Continuity

If y > x, it does not guarantee that a bundle close enough to y will be preferred to x because there could be a small change in a higher order element that reverses the relationship. Thus, this preference relation is not continuous. c. Preferences with Satiation

Completeness

Given any two bundles, this preference relation ranks them based on the distance to the bliss point \(\mathbf{x}^{*}\). So, either one will be closer, farther, or at the same distance from bliss. Therefore, this preference relation is complete.

Transitivity

If \(\mathbf{x}^{1}>\mathbf{x}^{2}\) and \(\mathbf{x}^{2}>\mathbf{x}^{3}\), that means \(\mathbf{x}^{1}\) is closer to the bliss point than \(\mathbf{x}^{2}\), and \(\mathbf{x}^{2}\) is closer to the bliss point than \(\mathbf{x}^{3}\). It follows that, \(\mathbf{x}^{1}\) must be closer to the bliss point than \(\mathbf{x}^{3}\). Thus, this preference relation is transitive.

Continuity

If y > x, and z is a bundle close enough to y, then the distance of z to the bliss point will be very similar to the distance of y to the bliss point. Therefore, we can also say that z > x. Thus, this preference relation is continuous.

Key concepts

Preference Relations

In microeconomic theory, preference relations are fundamental in understanding how consumers choose between different sets of goods, or 'bundles'. They represent the attitudes of a consumer towards the desirability of one bundle over another. A typical preference statement looks like this: \( \mathbf{x}^{1} > \mathbf{x}^{2} \) means that the consumer prefers bundle \( \mathbf{x}^{1} \) to bundle \( \mathbf{x}^{2} \). Alternatively, the expression \( \mathbf{x}^{1} = \mathbf{x}^{2} \) indicates indifference between the two bundles.

Preference relations play a crucial role in the analysis of consumer behavior and are used to derive the demand curve for goods. Preferences are assumed to have certain properties such as completeness, transitivity, and continuity, which are necessary for a well-defined preference ordering.

Completeness

The completeness property of preference relations ensures that an individual can always compare any two bundles and state a preference for one bundle over the other or indicate that they are indifferent between them. This is the basis for the consumer being able to make decisions between all possible options. In mathematical terms, for any given bundles \( \mathbf{x}^{1} \) and \( \mathbf{x}^{2} \), completeness implies that one of the following is true: \( \mathbf{x}^{1} > \mathbf{x}^{2} \) or \( \mathbf{x}^{2} > \mathbf{x}^{1} \) or \( \mathbf{x}^{1} = \mathbf{x}^{2} \). Without this property, there would be no way to define a coherent preference ordering, which is essential for any further economic analysis of consumer choice.

Transitivity

Transitivity is a property that, if held, allows for consistent decision-making. If a consumer prefers bundle A over bundle B, and bundle B over bundle C, then transitivity guarantees that the consumer will also prefer bundle A over bundle C. Formally, if \( \mathbf{x}^{1} > \mathbf{x}^{2} \) and \( \mathbf{x}^{2} > \mathbf{x}^{3} \), then it must follow that \( \mathbf{x}^{1} > \mathbf{x}^{3} \).

When the preference relation is transitive, it avoids circular preferences and paradoxical situations where a consumer might prefer a bundle A to B, B to C, but then C to A, leading to an indefinable 'best' choice. Therefore, transitivity is crucial for rational decision making within the framework of microeconomics.

Continuity

The continuity property of preference relations asserts that small changes in a bundle shouldn't lead to a sudden reversal in preference. Continuity ensures that if one bundle is preferred to another, slightly modifying one of the bundles doesn't abruptly change this preference ordering. For instance, if a consumer prefers bundle y to bundle x represented as \( y > x \), and we have another bundle z that is very similar to y, then continuity dictates that \( z > x \) as well.

This property allows for the creation of indifference curves in graphical representations of consumer preferences. These curves depict points along which a consumer is indifferent between different combinations of two goods. Without the continuity property, indifference curves could have gaps or jumped segments, which would complicate the analysis of consumer choice.