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 \(\left|\mathbf{x}^{1}-\mathbf{x}^{*}\right|=\)
\(\sqrt{\left(x_{1}^{i}-x_{1}^{*}\right)^{2}+\left(x_{2}^{i}-x_{2}^{*}\right)^{2}+\cdots+\left(x_{n}^{i}-x_{n}^{*}\right)^{2}}\)
