Independence of Completeness and Transitivity
Consider an economy with \(N\) commodities. Each alternative available to a consumer is represented by an \(N\)-dimensional vector \[
\boldsymbol{x}=\left( x_{1},\cdots,x_{N}\right) \in\mathbb{R} ^{N}.
\]The \(n\)th component \(x_{n}\) of \(\boldsymbol{x}\) represents the consumption of commodity \(n\). The set of all consumption vectors available to the consumer is represented by the consumption set \[
X\subset\mathbb{R} ^{N}.
\]A consumer facing \(X\) compares the consumption vectors in \(X\) and chooses the one that is most desirable. We formalize the consumer’s system of preferences as a binary relation \(\succsim\) on \(X\), called a preference relation. Specifically, for any two consumption vectors \(\boldsymbol{x},\boldsymbol{y}\in X\), we define \(\succsim\) so that \[
\boldsymbol{x}\succsim\boldsymbol{y}\Leftrightarrow\text{the consumer weakly prefers }\boldsymbol{x}\text{ to }\boldsymbol{y}.
\]
In consumer theory, several conditions are usually imposed on preference relations in order to analyze consumer choice systematically. We have already discussed completeness and transitivity:
\begin{align*}
& \left( a\right) \ \forall\boldsymbol{x},\boldsymbol{y}\in X:(\boldsymbol{x}\succsim\boldsymbol{y}\vee\boldsymbol{y}\succsim
\boldsymbol{x})\\
& \left( b\right) \ \forall\boldsymbol{x},\boldsymbol{y},\boldsymbol{z}\in X:\left[ (\boldsymbol{x}\succsim\boldsymbol{y}\wedge\boldsymbol{y}\succsim\boldsymbol{z})\Rightarrow\boldsymbol{x}\succsim\boldsymbol{z}\right]
.
\end{align*}These two properties are independent. That is, there exist preference relations that satisfy completeness but not transitivity, and conversely, preference relations that satisfy transitivity but not completeness. The following examples illustrate this point.
\boldsymbol{x}\succsim\boldsymbol{y}\Leftrightarrow\left( \boldsymbol{x}\succsim_{1}\boldsymbol{y}\vee\boldsymbol{x}\succsim_{2}\boldsymbol{y}\right)
.
\]Thus, the consumer weakly prefers \(\boldsymbol{x}\) to \(\boldsymbol{y}\) if and only if \(\boldsymbol{x}\) is at least as desirable as \(\boldsymbol{y}\) according to at least one of the two criteria. The relation \(\succsim\) defined in this way satisfies completeness, but need not satisfy transitivity (see the exercises).
\boldsymbol{x}\succsim\boldsymbol{y}\Leftrightarrow\left( \boldsymbol{x}\succsim_{1}\boldsymbol{y}\wedge\boldsymbol{x}\succsim_{2}\boldsymbol{y}\right) .
\]Thus, the consumer weakly prefers \(\boldsymbol{x}\) to \(\boldsymbol{y}\) if and only if \(\boldsymbol{x}\) is at least as desirable as \(\boldsymbol{y}\) according to both criteria. The relation \(\succsim\) defined in this way satisfies transitivity, but need not satisfy completeness (see the exercises).
Rational Preference Relations
The preceding discussion shows that completeness and transitivity play different roles and that neither follows from the other. Completeness requires that any two consumption vectors be comparable, whereas transitivity requires consistency across multiple pairwise comparisons. Therefore, if we want consumer preferences to satisfy a sufficiently strong form of internal consistency, it is natural to impose both conditions simultaneously.
A preference relation \(\succsim\) on a consumption set \(X\subset\mathbb{R} ^{N}\) is called a rational preference relation or a preference ordering if it satisfies both completeness and transitivity, that is, if
\begin{align*}
& \left( a\right) \ \forall\boldsymbol{x},\boldsymbol{y}\in X:(\boldsymbol{x}\succsim\boldsymbol{y}\vee\boldsymbol{y}\succsim
\boldsymbol{x})\\
& \left( b\right) \ \forall\boldsymbol{x},\boldsymbol{y},\boldsymbol{z}\in X:\left[ (\boldsymbol{x}\succsim\boldsymbol{y}\wedge\boldsymbol{y}\succsim\boldsymbol{z})\Rightarrow\boldsymbol{x}\succsim\boldsymbol{z}\right]
\end{align*}both hold.
x\succsim y\Leftrightarrow x\geq y.
\]Here, \(\geq\) on the right-hand side is the usual ordering of real numbers. Thus, \(\succsim\) represents the criterion “more consumption is better.” Since the usual ordering \(\geq\) satisfies both completeness and transitivity, the preference relation \(\succsim\) defined above is rational.
Implications of Rationality
If a preference relation \(\succsim\) on a consumption set \(X\subset\mathbb{R} ^{N}\) satisfies completeness, then for any two consumption vectors \(\boldsymbol{x},\boldsymbol{y}\in X\), exactly one of \[
\boldsymbol{x}\succ\boldsymbol{y},\quad\boldsymbol{x}\sim\boldsymbol{y},\quad\boldsymbol{y}\succ\boldsymbol{x}\]holds. If \(\succsim\) also satisfies transitivity, then the indifference relation \(\sim\) is transitive, \[
\left( a\right) \ \forall\boldsymbol{x},\boldsymbol{y},\boldsymbol{z}\in X:\left[ \left( \boldsymbol{x}\sim\boldsymbol{y}\wedge\boldsymbol{y}\sim\boldsymbol{z}\right) \Rightarrow\boldsymbol{x}\sim\boldsymbol{z}\right]
,
\]and the strict preference relation \(\succ\) is transitive, \[
\left( b\right) \ \forall\boldsymbol{x},\boldsymbol{y},\boldsymbol{z}\in X:\left[ \left( \boldsymbol{x}\succ\boldsymbol{y}\wedge\boldsymbol{y}\succ\boldsymbol{z}\right) \Rightarrow\boldsymbol{x}\succ\boldsymbol{z}\right] .
\]In addition, IP-transitivity and PI-transitivity imply
\begin{align*}
& \left( c\right) \ \forall\boldsymbol{x},\boldsymbol{y},\boldsymbol{z}\in X:\left[ \left( \boldsymbol{x}\succsim\boldsymbol{y}\wedge\boldsymbol{y}\succ\boldsymbol{z}\right) \Rightarrow\boldsymbol{x}\succ\boldsymbol{z}\right] \\
& \left( d\right) \ \forall\boldsymbol{x},\boldsymbol{y},\boldsymbol{z}\in X:\left[ \left( \boldsymbol{x}\sim\boldsymbol{y}\wedge\boldsymbol{y}\succ\boldsymbol{z}\right) \Rightarrow\boldsymbol{x}\succ\boldsymbol{z}\right] \\
& \left( e\right) \ \forall\boldsymbol{x},\boldsymbol{y},\boldsymbol{z}\in X:\left[ \left( \boldsymbol{x}\succ\boldsymbol{y}\wedge\boldsymbol{y}\succsim\boldsymbol{z}\right) \Rightarrow\boldsymbol{x}\succ\boldsymbol{z}\right] \\
& \left( f\right) \ \forall\boldsymbol{x},\boldsymbol{y},\boldsymbol{z}\in X:\left[ \left( \boldsymbol{x}\succ\boldsymbol{y}\wedge\boldsymbol{y}\sim\boldsymbol{z}\right) \Rightarrow\boldsymbol{x}\succ\boldsymbol{z}\right]
.
\end{align*}What can we conclude from these properties?
Choose any two consumption vectors \(\boldsymbol{x},\boldsymbol{x}^{\prime}\in X\). By completeness, they are comparable. Choose a third consumption vector \(\boldsymbol{x}^{\prime\prime}\in X\). Again by completeness, \(\boldsymbol{x}\) and \(\boldsymbol{x}^{\prime\prime}\) are comparable, and so are \(\boldsymbol{x}^{\prime}\) and \(\boldsymbol{x}^{\prime\prime}\). Hence any two of the three vectors are comparable. Moreover, by transitivity of \(\sim\) and \(\succ\), together with IP-transitivity and PI-transitivity, they can be ordered consistently in terms of \(\succ\) and \(\sim\) without generating cycles. The same reasoning applies to any further consumption vectors. Therefore, if \(\succsim\) is rational, any elements of the consumption set \(X\) can be positioned consistently in terms of their relative desirability using \(\succ\) and \(\sim\), without contradiction or cycles.
Utility Functions Representing Rational Preferences
A preference relation \(\succsim\) on a consumption set \(X\) is rational if and only if both \begin{align*}
& \left( a\right) \ \forall\boldsymbol{x},\boldsymbol{y}\in X:\left(
\boldsymbol{x}\succsim\boldsymbol{y}\vee\boldsymbol{y}\succsim\boldsymbol{x}\right) \\
& \left( b\right) \ \forall\boldsymbol{x},\boldsymbol{y},\boldsymbol{z}\in X:\left[ (\boldsymbol{x}\succsim\boldsymbol{y}\wedge\boldsymbol{y}\succsim\boldsymbol{z})\Rightarrow\boldsymbol{x}\succsim\boldsymbol{z}\right]
\end{align*}hold. Suppose there exists a utility function \(u:X\rightarrow\mathbb{R} \) representing \(\succsim\). By the definition of a utility function, conditions (a) and (b) are respectively equivalent to \begin{align*}
& \left( a^{\prime}\right) \ \forall\boldsymbol{x},\boldsymbol{y}\in X:\left[ u\left( \boldsymbol{x}\right) \geq u\left( \boldsymbol{y}\right)
\vee u\left( \boldsymbol{y}\right) \geq u\left( \boldsymbol{x}\right)
\right] \\
& \left( b^{\prime}\right) \ \forall\boldsymbol{x},\boldsymbol{y},\boldsymbol{z}\in X:\left\{ \left[ u\left( \boldsymbol{x}\right) \geq u\left( \boldsymbol{y}\right) \wedge u\left( \boldsymbol{y}\right) \geq u\left( \boldsymbol{z}\right) \right] \Rightarrow u\left( \boldsymbol{x}\right) \geq u\left( \boldsymbol{z}\right) \right\} .
\end{align*}
& \left( a\right) \ \forall\boldsymbol{x},\boldsymbol{y}\in X:\left[
u\left( \boldsymbol{x}\right) \geq u\left( \boldsymbol{y}\right) \vee u\left( \boldsymbol{y}\right) \geq u\left( \boldsymbol{x}\right) \right] \\
& \left( b\right) \ \forall\boldsymbol{x},\boldsymbol{y},\boldsymbol{z}\in X:\left\{ \left[ u\left( \boldsymbol{x}\right) \geq u\left(
\boldsymbol{y}\right) \wedge u\left( \boldsymbol{y}\right) \geq u\left(
\boldsymbol{z}\right) \right] \Rightarrow u\left( \boldsymbol{x}\right)
\geq u\left( \boldsymbol{z}\right) \right\}
\end{align*}both hold.
The usual ordering \(\geq\) on the real numbers satisfies completeness and transitivity: \begin{align*}
& \left( a\right) \ \forall a,b\in\mathbb{R} :\left( a\geq b\vee b\geq a\right) \\
& \left( b\right) \ \forall a,b,c\in\mathbb{R} :\left[ \left( a\geq b\wedge b\geq c\right) \Rightarrow a\geq c\right] .
\end{align*}Therefore, any real-valued function \(u:X\rightarrow\mathbb{R}\) satisfies \begin{align*}
& \left( a\right) \ \forall\boldsymbol{x},\boldsymbol{y}\in X:\left[
u\left( \boldsymbol{x}\right) \geq u\left( \boldsymbol{y}\right) \vee u\left( \boldsymbol{y}\right) \geq u\left( \boldsymbol{x}\right) \right]
\\
& \left( b\right) \ \forall\boldsymbol{x},\boldsymbol{y},\boldsymbol{z}\in X:\left\{ \left[ u\left( \boldsymbol{x}\right) \geq u\left(
\boldsymbol{y}\right) \wedge u\left( \boldsymbol{y}\right) \geq u\left(
\boldsymbol{z}\right) \right] \Rightarrow u\left( \boldsymbol{x}\right)
\geq u\left( \boldsymbol{z}\right) \right\} .
\end{align*}Hence, if a utility function representing \(\succsim\) exists, then the preceding conditions automatically hold, and the proposition above implies that \(\succsim\) is rational. In other words, a utility function representing a preference relation need not exist in general, but whenever one does exist, the represented preference relation is guaranteed to be rational.
The converse need not hold. Even if a preference relation \(\succsim\) is rational, there need not exist a utility function representing it. Conditions for the existence of a utility representation will be discussed in detail elsewhere.
Exercises
\boldsymbol{x}\succsim\boldsymbol{y}\Leftrightarrow\left( \boldsymbol{x}\succsim_{1}\boldsymbol{y}\vee\boldsymbol{x}\succsim_{2}\boldsymbol{y}\right)
.
\]Show that \(\succsim\) satisfies completeness but need not satisfy transitivity.
\boldsymbol{x}\succsim\boldsymbol{y}\Leftrightarrow\left( \boldsymbol{x}\succsim_{1}\boldsymbol{y}\wedge\boldsymbol{x}\succsim_{2}\boldsymbol{y}\right) .
\]Show that \(\succsim\) satisfies transitivity but need not satisfy completeness.
- Any two consumption vectors \(\left( x_{1},x_{2}\right) \) and \(\left( y_{1},y_{2}\right) \) can always be compared under \(\succsim\).
- For any three consumption vectors \(\left( x_{1},x_{2}\right) \), \(\left( y_{1},y_{2}\right) \), and \(\left( z_{1},z_{2}\right) \), if \[
\left( x_{1},x_{2}\right) \succ\left( y_{1},y_{2}\right)
\]and
\[
\left( x_{1},x_{2}\right) \succ\left( z_{1},z_{2}\right)
\]both hold, then
\[
\left( y_{1},y_{2}\right) \succ\left( z_{1},z_{2}\right)
\]must also hold. - For any two quantities \(x_{1}\) and \(x_{2}\), the option of consuming \(x_{1}\) units of commodity \(1\) and the option of consuming \(x_{2}\) units of commodity \(2\) can always be compared under \(\succsim\).
\left( x_{1},x_{2}\right) \succsim\left( y_{1},y_{2}\right)
\Leftrightarrow x_{1}+2x_{2}\geq y_{1}+2y_{2}.
\]Prove that this preference relation is rational.
