The formal study of preferences uses a general vector notation. A bundle of
\(n\) commoditics 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 \(\mathbf{x}^{1} \approx \mathbf{x}^{2}\)
The preference relation is "complete" if for any two bundles the individual is
able to state cither \(\mathbf{x}^{1}>\mathbf{x}^{2},
\mathbf{x}^{2}>\mathbf{x}^{1}\), or \(\mathbf{x}^{1} \approx\)
\(\mathbf{x}^{2}\). 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 \(x\). 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, \(\mathbf{x}^{1}>\mathbf{x}^{2}\) if and only
if \(\sum_{i=1}^{n} x_{i}^{1}>\sum_{i=1}^{n} x_{i}^{2} .\) If \(\sum_{i=1}^{n}
x_{i}^{1}=\sum_{i=1}^{n} x_{i}^{2}, \mathbf{x}^{1} \approx \mathbf{x}^{2}\)
b. Lexicographic preferences: In this case the preference relation is
organized as a dictionary: If \(x_{1}^{1}>x_{1}^{2}, x^{1} \succ x^{2}\)
(regardless of the amounts of the other \(n-1\) goods). If \(x_{1}^{1}=x_{1}^{2}\)
and \(x_{2}^{1}>x_{2}^{2}, x^{1}>x^{2}\) (regardless of the amounts of the other
\(n-2\) 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 (x") that provides complete "bliss." The ranking of
all other bundles is determined by how close they are to \(\mathbf{x}^{*}\),
That is, \(\mathbf{x}^{1}>\mathbf{x}^{2}\) if and only if
\\[
\left|\mathbf{x}^{1}-\mathbf{x}^{*}\right|<\left|\mathbf{x}^{2}-\mathbf{x}^{\prime}\right|
\text { where
}\left|\mathbf{x}^{l}-\mathbf{x}^{*}\right|=\sqrt{\left(x_{1}^{\prime}-x_{1}^{*}\right)^{2}+\left(x_{2}^{\prime}-x_{x}^{*}\right)^{2}+\ldots+\left(x_{n}^{\prime}-x_{n}^{*}\right)^{2}}
\\]
