Continuity of a Preference Relation (Continuous Utility Function)

Continuity of a Preference Relation

Consider an economy with \(N\) goods. Represent each alternative faced by the consumer by an \(N\)-dimensional vector\[ \boldsymbol{x}=\left( x_{1},\cdots,x_{N}\right) \in\mathbb{R} ^{N}\]where the \(n\)th component \(x_{n}\) of the vector \(\boldsymbol{x}\) denotes the consumption of good \(n\). The set consisting of all vectors available to the consumer is represented as the consumption set\[ X\subset\mathbb{R} ^{N}\]A consumer facing the consumption set \(X\) chooses the most desirable consumption vector by comparing consumption vectors that are elements of \(X\). We formulate the consumer’s system of tastes as a binary relation \(\succsim\) on \(X\), called a preference relation. More specifically, choose any two consumption vectors \(\boldsymbol{x},\boldsymbol{y}\in X\). Then\[ \boldsymbol{x}\succsim\boldsymbol{y}\Leftrightarrow\text{the consumer regards
}\boldsymbol{x}\text{ as at least as desirable as }\boldsymbol{y}\text{ }\]defines \(\succsim\).
The consumption set \(X\subset\mathbb{R} ^{N}\) is a subset of the Euclidean space \(\mathbb{R} ^{N}\) and therefore inherits a topology. Given a preference relation \(\succsim\) on \(X\), if, for every consumption vector \(\boldsymbol{x}\in X\), the strict upper contour set and strict lower contour set\begin{align*} U_{s}\left( \boldsymbol{x}\right) & =\left\{ \boldsymbol{y}\in X\ |\ \boldsymbol{y}\succ\boldsymbol{x}\right\} \\
L_{s}\left( \boldsymbol{x}\right) & =\left\{ \boldsymbol{y}\in X\ |\ \boldsymbol{x}\succ\boldsymbol{y}\right\}
\end{align*}are both open sets in \(X\), then \(\succsim\) is said to satisfy continuity.

We review the meaning of continuity of \(\succsim\) while recalling the definition of an open set. Given a consumption vector \(\boldsymbol{a}\in X\) and a positive real number \(\varepsilon>0\), the neighborhood of radius \(\varepsilon\) centered at \(\boldsymbol{a}\) is the set of points in \(X\) whose distance from \(\boldsymbol{a}\) is less than \(\varepsilon\), and is denoted by\begin{align*} N_{\varepsilon}\left( \boldsymbol{a}\right) & =\left\{ \boldsymbol{x}\in X\ |\ d\left( \boldsymbol{x},\boldsymbol{a}\right) <\varepsilon\right\} \\
& =\left\{ \boldsymbol{x}\in X\ |\ \left\Vert \boldsymbol{x}-\boldsymbol{a}\right\Vert <\varepsilon\right\} \\
& =\left\{ \boldsymbol{x}\in X\ |\ \sqrt{\sum_{i=1}^{N}\left( x_{i}-a_{i}\right) ^{2}}<\varepsilon\right\}
\end{align*}where \(d:X\times X\rightarrow\mathbb{R} \) is the Euclidean distance function.
Choose any consumption vector \(\boldsymbol{x}\in X\) and consider its strict upper contour set\[ U_{s}\left( \boldsymbol{x}\right) =\left\{ \boldsymbol{y}\in X\ |\ \boldsymbol{y}\succ\boldsymbol{x}\right\}
\]The set \(U_{s}\left( \boldsymbol{x}\right) \) is open in \(X\) if, for every point \(\boldsymbol{a}\in U_{s}\left( \boldsymbol{x}\right) \), there exists a neighborhood \(N_{\varepsilon}\left( \boldsymbol{a}\right) \) centered at \(\boldsymbol{a}\) that is a subset of \(U_{s}\left( \boldsymbol{x}\right) \); that is,\[ \forall\boldsymbol{a}\in U_{s}\left( \boldsymbol{x}\right) ,\ \exists
\varepsilon>0:N_{\varepsilon}\left( \boldsymbol{a}\right) \subset U_{s}\left( \boldsymbol{x}\right)
\]holds.

To understand continuity more deeply, consider what may occur when preferences satisfy rationality but fail to satisfy continuity. In this case, the following statement, which is the negation of the preceding statement, holds:\[
\exists\boldsymbol{a}\in U_{s}\left( \boldsymbol{x}\right) ,\ \forall
\varepsilon>0:N_{\varepsilon}\left( \boldsymbol{a}\right) \cap X\backslash U_{s}\left( \boldsymbol{x}\right) \not =\phi
\]If \(\succsim\) satisfies completeness, however, exclusivity holds. Hence, for every consumption vector \(\boldsymbol{y}\in X\),\begin{align*} \boldsymbol{y}\in X\backslash U_{s}\left( \boldsymbol{x}\right) &
\Leftrightarrow\lnot\left( \boldsymbol{y}\succ\boldsymbol{x}\right)
\quad\because\text{definition of }U_{s}\\
& \Leftrightarrow\boldsymbol{x}\succsim\boldsymbol{y}\quad\because
\succsim\text{ satisfies exclusivity}\\
& \Leftrightarrow\boldsymbol{y}\in L\left( \boldsymbol{x}\right)
\quad\because\text{definition of }L_{s}\end{align*}holds. Therefore, the preceding statement is equivalent to\[
\exists\boldsymbol{a}\in U_{s}\left( \boldsymbol{x}\right) ,\ \forall
\varepsilon>0:N_{\varepsilon}\left( \boldsymbol{a}\right) \cap L\left(
\boldsymbol{x}\right) \not =\phi
\]What does this mean? First, \(\boldsymbol{a}\in U_{s}\left( \boldsymbol{x}\right) \) means \(\boldsymbol{a}\succ\boldsymbol{x}\), so \(\boldsymbol{a}\) is a consumption vector that is more desirable than \(\boldsymbol{x}\). Because \(\varepsilon>0\) is arbitrary in the preceding statement, \(N_{\varepsilon }\left( \boldsymbol{a}\right) \) and \(L\left( \boldsymbol{x}\right) \) intersect even for arbitrarily small \(\varepsilon\). Thus, among points arbitrarily close to \(\boldsymbol{a}\) (elements of \(N_{\varepsilon}\left( \boldsymbol{a}\right) \)), there are points that are no more desirable than \(\boldsymbol{x}\) (elements of \(L\left( \boldsymbol{x}\right) \)). In summary, if \(\succsim\) fails continuity, then for some consumption vector \(\boldsymbol{x}\), there may be consumption vectors that are no more desirable than \(\boldsymbol{x}\) arbitrarily close to a consumption vector that is more desirable than \(\boldsymbol{x}\). The consumer’s tastes do not vary continuously: even a slight change in a consumption vector can cause a large change in the consumer’s preference over consumption vectors. Conversely, the continuity assumption guarantees that the consumer’s tastes vary continuously.
The same argument applies to the strict lower contour set \(L_{s}\left( \boldsymbol{x}\right) \).

Example (A Preference Relation Satisfying Continuity)

In a two-good model, suppose the consumption set is \(\mathbb{R} _{+}^{2}\) and, under the given preference relation \(\succsim\), consider a consumption vector \(\boldsymbol{x}\in\mathbb{R} _{+}^{2}\) whose indifference curve \(I\left( \boldsymbol{x}\right) \) is represented by the dotted curve in the figure below.

Suppose the strict upper contour set \(U_{s}\left( \boldsymbol{x}\right) \) is the gray region. Points on the indifference curve \(I\left( \boldsymbol{x}\right) \), which forms the boundary, are not included in \(U_{s}\left( \boldsymbol{x}\right) \). If we choose any point in \(U_{s}\left( \boldsymbol{x}\right) \), such as the point \(\boldsymbol{a}\) in the figure, there is a neighborhood \(N_{\varepsilon}\left( b\right) \) centered at that point that is a subset of \(U_{s}\left( \boldsymbol{x}\right) \) (the region inside the dotted circle in the figure). Because the same argument applies to every point \(\boldsymbol{a}\) of \(U_{s}\left( \boldsymbol{x}\right) \), \(U_{s}\left( \boldsymbol{x}\right) \) is an open set in \(\mathbb{R} _{+}^{2}\). The same is true for the strict lower contour set \(L_{s}\left( \boldsymbol{x}\right) \). Therefore, this preference relation \(\succsim\) is consistent with the continuity assumption.

Example (A Preference Relation That Does Not Satisfy Continuity)

In a two-good model, suppose the consumption set is \(\mathbb{R} _{+}^{2}\) and the preference relation \(\succsim\) on it is defined for any consumption vectors \(\left( x_{1},x_{2}\right) ,\left( y_{1},y_{2}\right) \in\mathbb{R} _{+}^{2}\) by\[ \left( x_{1},x_{2}\right) \succ\left( y_{1},y_{2}\right) \Leftrightarrow x_{1}>y_{1}\vee\left( x_{1}=y_{1}\wedge x_{2}>y_{2}\right)
\]In this case, for the consumption vector \(\boldsymbol{x}\in X\) shown below, the strict upper contour set \(U_{s}\left( \boldsymbol{x}\right) \) is the gray region. The solid-line portion is included in \(U_{s}\left( \boldsymbol{x}\right) \), whereas the dotted-line portion is not included in \(U_{s}\left( \boldsymbol{x}\right) \).

If we choose a point \(\boldsymbol{a}\in U_{s}\left( \boldsymbol{x}\right) \) as in the figure above, every neighborhood \(N_{\varepsilon}\left( \boldsymbol{a}\right) \) intersects the lower contour set \(L\left( \boldsymbol{x}\right) \). Thus, although \(\boldsymbol{a}\) is more desirable than \(\boldsymbol{x}\) because it is an element of \(U_{s}\left( \boldsymbol{x}\right) \), arbitrarily close to it there are consumption vectors that are elements of \(L\left( \boldsymbol{x}\right) \), that is, consumption vectors that are no more desirable than \(\boldsymbol{x}\). Therefore, this preference relation \(\succsim\) does not satisfy continuity.

Definition of Continuous Preferences Using Closed Sets

Using the definition of a closed set, continuity of a preference relation \(\succsim\) can also be expressed as follows.

A closed set can also be characterized using sequences, so we reconsider the meaning of the preceding proposition in terms of sequences. Choose any consumption vector \(\boldsymbol{x}\in X\) and consider its upper contour set \(U\left( \boldsymbol{x}\right) \). Then choose any convergent sequence \(\{\boldsymbol{x}_{v}\}\) whose terms are points of \(U\left( \boldsymbol{x}\right) \). That is, choose any sequence \(\{\boldsymbol{x}_{v}\}\) satisfying both\begin{align*} & \left( a\right) \ \forall v\in\mathbb{N} :\boldsymbol{x}_{v}\in U\left( \boldsymbol{x}\right) \\
& \left( b\right) \ \lim_{v\rightarrow+\infty}\boldsymbol{x}_{v}\in X
\end{align*}In general, the limit of such a convergent sequence need not be a point of \(U\left( \boldsymbol{x}\right) \). The guarantee that the limit of every such convergent sequence is a point of \(U\left( \boldsymbol{x}\right) \), that is, that\[ \left( c\right) \ \lim_{v\rightarrow+\infty}\boldsymbol{x}_{v}\in U\left(
\boldsymbol{x}\right)
\]holds, is necessary and sufficient for \(U\left( \boldsymbol{x}\right) \) to be closed in \(X\). The definition that the lower contour set \(L\left( \boldsymbol{x}\right) \) is closed in \(X\) is analogous.
To understand the meaning of continuity more deeply, consider what may occur for preferences that satisfy rationality but fail continuity. In this case, there must exist a sequence \(\{\boldsymbol{x}_{v}\}\) that satisfies \(\left( a\right) ,\left( b\right) \) but does not satisfy \(\left( c\right) \). By the definition of the upper contour set \(U\left( \boldsymbol{x}\right) \), the fact that the sequence \(\{\boldsymbol{x}_{v}\}\) satisfies \(\left( a\right) \) means that every element of \(\{\boldsymbol{x}_{v}\}\) is a consumption vector at least as desirable as \(\boldsymbol{x}\). By \(\left( b\right) \), \(\{\boldsymbol{x}_{v}\}\) converges, so there are infinitely many points of \(\{\boldsymbol{x}_{v}\}\) arbitrarily close to the consumption vector corresponding to its limit \(\lim\limits_{v\rightarrow+\infty}\boldsymbol{x}_{v}\). In other words, arbitrarily close to the consumption vector \(\lim\limits_{v\rightarrow+\infty}\boldsymbol{x}_{v}\), there are infinitely many consumption vectors at least as desirable as \(\boldsymbol{x}\). On the other hand, the failure of \(\left( c\right) \) means that the consumption vector \(\lim\limits_{v\rightarrow+\infty}\boldsymbol{x}_{v}\) is not at least as desirable as \(\boldsymbol{x}\), that is, \(\lnot\left( \lim\limits_{v\rightarrow+\infty}\boldsymbol{x}_{v}\succsim \boldsymbol{x} \right) \) holds. Since the rationality of \(\succsim\) implies completeness, this is equivalent to \(\boldsymbol{x} \succ \lim\limits_{v\rightarrow+\infty}\boldsymbol{x}_{v}.\) Thus, \(\lim\limits_{v\rightarrow+\infty}\boldsymbol{x}_{v}\) is a consumption vector that is less desirable than \(\boldsymbol{x}\). In summary, if preferences fail to satisfy continuity, then, when a consumption vector \(\boldsymbol{x}\) is chosen, there may exist infinitely many consumption vectors that are at least as desirable as \(\boldsymbol{x}\) and arbitrarily close to \(\lim\limits_{v\rightarrow+\infty}\boldsymbol{x}_{v}\), even though this limiting consumption vector itself is less desirable than \(\boldsymbol{x}\). The consumer’s tastes do not vary continuously, and even a slight change in a consumption vector can cause a large change in the consumer’s preference over consumption vectors. Conversely, the continuity assumption guarantees that the consumer’s tastes vary continuously.
The same argument applies to the lower contour set \(L\left( \boldsymbol{x}\right) \).
 

Definition of Continuous Preferences Using Sequences

Continuity of a preference relation can also be expressed using sequences as follows.

If the preference relation \(\succsim\) does not satisfy continuity, then by the preceding proposition there exist convergent sequences \(\{\boldsymbol{x}_{v}\},\{\boldsymbol{y}_{v}\}\) in \(X\) satisfying\begin{equation} \forall v\in\mathbb{N} :\boldsymbol{x}_{v}\succsim\boldsymbol{y}_{v}\quad \cdots (1)\end{equation}such that\begin{equation}
\lim\limits_{v\rightarrow+\infty}\boldsymbol{y}_{v}\succ\lim
\limits_{v\rightarrow+\infty}\boldsymbol{x}_{v}\quad \cdots (2)\end{equation}holds. The consumption vector \(\lim\limits_{v\rightarrow+\infty}\boldsymbol{x}_{v}\in X\) is the limit of \(\{\boldsymbol{x}_{v}\}\), and the consumption vector \(\lim\limits_{v\rightarrow+\infty}\boldsymbol{y}_{v}\in X\) is the limit of \(\{\boldsymbol{y}_{v}\}\). Hence, for sufficiently large \(v\), \(\boldsymbol{x}_{v}\) is arbitrarily close to \(\lim\limits_{v\rightarrow +\infty}\boldsymbol{x}_{v}\) and \(\boldsymbol{y}_{v}\) is arbitrarily close to \(\lim\limits_{v\rightarrow+\infty}\boldsymbol{y}_{v}\). Thus, for sufficiently large \(v\), although \(\boldsymbol{x}_{v}\) is at least as desirable as \(\boldsymbol{y}_{v}\) by \(\left( 1\right) \), by \(\left( 2\right) \), the preference between the arbitrarily close \(\lim\limits_{v\rightarrow+\infty }\boldsymbol{x}_{v}\) and \(\lim\limits_{v\rightarrow+\infty}\boldsymbol{y}_{v}\) is reversed, so that \(\lim\limits_{v\rightarrow+\infty}\boldsymbol{y}_{v}\) is more desirable than \(\lim\limits_{v\rightarrow+\infty}\boldsymbol{x}_{v}\). In other words, even a slight change in a consumption vector can cause a large change in the preference over consumption vectors. Conversely, the continuity assumption guarantees that the consumer’s tastes vary continuously.

Continuous Utility Functions

Let the consumption set be \(X\subset\mathbb{R} ^{N}\). A utility function \(u:X\rightarrow\mathbb{R} \) defined on it is a multivariable function, so its continuity can be considered. Specifically, a utility function \(u\) is continuous at a point \(\boldsymbol{a}\in X\) in its domain if, as \(\boldsymbol{x}\rightarrow \boldsymbol{a}\), \(u\left( \boldsymbol{x}\right) \) converges to a finite real number and the limit equals \(u\left( \boldsymbol{a}\right) \); that is, if both\begin{align*} & \left( a\right) \ \lim_{\boldsymbol{x}\rightarrow\boldsymbol{a}}u\left(
\boldsymbol{x}\right) \in\mathbb{R} \\
& \left( b\right) \ \lim_{\boldsymbol{x}\rightarrow\boldsymbol{a}}u\left(
\boldsymbol{x}\right) =u\left( \boldsymbol{a}\right)
\end{align*}hold.

Using the epsilon-delta definition, continuity of a utility function \(u:X\rightarrow\mathbb{R} \) at a point \(\boldsymbol{a}\in X\) can be expressed as\[ \forall\varepsilon>0,\ \exists\delta>0,\ \forall\boldsymbol{x}\in X:\left[
d\left( \boldsymbol{x},\boldsymbol{a}\right) <\delta\Rightarrow\left\vert u\left( \boldsymbol{x}\right) -u\left( \boldsymbol{a}\right) \right\vert
<\varepsilon\right] \]where \(d\left( \boldsymbol{x},\boldsymbol{a}\right) \) is the Euclidean distance between the two points \(\boldsymbol{x}\) and \(\boldsymbol{a}\), given by\begin{align*} d\left( \boldsymbol{x},\boldsymbol{a}\right) & =\left\Vert \boldsymbol{x}-\boldsymbol{a}\right\Vert \\
& =\sqrt{\sum_{i=1}^{N}\left( x_{i}-a_{i}\right) ^{2}}.
\end{align*}

Continuity of a utility function \(u:X\rightarrow\mathbb{R} \) can also be expressed using sequences. Namely, for every sequence \(\left\{ \boldsymbol{x}_{v}\right\} \) in \(X\) converging to a point \(\boldsymbol{a}\in X\), the sequence of real numbers \(\left\{ u\left( \boldsymbol{x}_{v}\right) \right\} \) defined by that sequence and the utility function \(u\) converges to \(u\left( \boldsymbol{a}\right) \in\mathbb{R} \) if and only if \(u\) is continuous at \(\boldsymbol{a}\).
Let the consumption set be \(X\subset\mathbb{R} ^{N}\). If a utility function \(u:X\rightarrow\mathbb{R} \) is continuous at every point in its domain \(X\), then \(u\) is said to be continuous.

Relationship Between Continuous Preferences and Continuous Utility Functions

What relationship holds between continuity of preferences and continuity of a utility function? First, if there exists a utility function \(u\) representing a preference relation \(\succsim\) and \(u\) is continuous, then the preference relation \(\succsim\) represented by \(u\) is guaranteed to satisfy continuity.

We have established that if there exists a utility function \(u\) representing a preference relation \(\succsim\) and \(u\) is continuous, then \(\succsim\) necessarily satisfies continuity. In other words, a preference relation that fails continuity cannot be represented by a continuous utility function.

On the other hand, a utility function representing a preference relation that satisfies continuity need not itself be continuous. Thus, a continuous preference relation may be represented by a discontinuous utility function. Indeed, if a preference relation \(\succsim\) satisfying continuity is represented by a continuous utility function \(u\), then any positive monotonic transformation of that utility function also represents the same preference relation \(\succsim\). Consequently, even if we apply a positive monotonic transformation that turns \(u\) into a discontinuous function \(u^{\prime}\), the resulting function \(u^{\prime}\) still represents \(\succsim\).

What can be said about a preference relation that fails continuity? Such a preference relation \(\succsim\) may still be represented by some utility function \(u\). In that case, however, the utility function \(u\) cannot be continuous. If it were continuous, the preference relation it represents would satisfy continuity, contradicting the assumption that \(\succsim\) is not continuous. Thus, any utility function representing a preference relation that fails continuity is discontinuous.

The preceding discussion assumes the existence of a utility function representing the preference relation. Can anything be concluded about the existence of a utility function from continuity of the preference relation itself? If a preference relation satisfies continuity, does there exist a utility function representing it? Conditions for the existence of a utility function will be discussed in detail separately.

Exercises