Preference Relations Satisfying Non-Satiation
Consider an economy with \(N\) goods, and represent each alternative available to 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 of all vectors available to the consumer is represented by the consumption set\[ X\subset\mathbb{R} ^{N}\]The consumer facing the consumption set \(X\) chooses the most desirable consumption vector by comparing the elements of \(X\). We formulate the consumer\’s system of preferences as a binary relation \(\succsim\) on \(X\), called a preference relation. Specifically, for any two consumption vectors \(\boldsymbol{x},\boldsymbol{y}\in X\),\[ \boldsymbol{x}\succsim\boldsymbol{y}\Leftrightarrow\text{the consumer prefers
}\boldsymbol{x}\text{ at least as much as }\boldsymbol{y}\]and define \(\succsim\) accordingly.
If a preference relation \(\succsim\) on the consumption set \(X\subset\mathbb{R} ^{N}\) satisfies the condition\[ \forall\boldsymbol{x}\in X,\ \exists\boldsymbol{y}\in X:\boldsymbol{y}\succ\boldsymbol{x}\]that is, if for every feasible consumption vector \(\boldsymbol{x}\) there exists another feasible consumption vector \(\boldsymbol{y}\) that is strictly preferred to it, then \(\succsim\) is said to satisfy non-satiation.
\]If the preference relation \(\succsim\) on \(\mathbb{R} _{+}\) is defined in this way, it satisfies \(\left( 1\right) \) and hence satisfies local non-satiation. Indeed, since \(\mathbb{R} _{+}\) is not bounded above, for any consumption vector \(x\in\mathbb{R} _{+}\) there exists a consumption vector \(y\in\mathbb{R} _{+}\) such that \(y>x\). By the definition of \(\succsim\), this implies \(y\succ x\). Thus, \(\left( 1\right) \) holds.
\Leftrightarrow\left( y_{1}\geq x_{1}\wedge y_{2}\geq x_{2}\right)
\]If the preference relation \(\succsim\) on \(\mathbb{R} _{+}^{2}\) is defined in this way, it satisfies \(\left( 1\right) \) and hence satisfies local non-satiation (see the exercises).
If the preference relation \(\succsim\) does not satisfy non-satiation, then\[ \exists\boldsymbol{x}\in X,\ \forall\boldsymbol{y}\in X:\lnot\left(
\boldsymbol{y}\succ\boldsymbol{x}\right)
\]holds. In particular, if \(\succsim\) satisfies completeness, then exclusivity also holds, and the preceding statement is equivalent to\[ \exists\boldsymbol{x}\in X,\ \forall\boldsymbol{y}\in X:\boldsymbol{x}\succsim\boldsymbol{y}\]Thus, if the preference relation \(\succsim\) fails to satisfy non-satiation, there exists a consumption vector \(\boldsymbol{x}\) that is at least as desirable as every consumption vector \(\boldsymbol{y}\). Such a consumption vector \(\boldsymbol{x}\) is called a satiation point or bliss point. Conversely, the assumption of non-satiation guarantees that there is no satiation point in the consumption set.
Preference Relations Satisfying Local Non-Satiation
If a preference relation \(\succsim\) on the consumption set \(X\subset\mathbb{R} ^{N}\) satisfies the condition\[ \forall\boldsymbol{x}\in X,\ \forall\varepsilon>0,\ \exists\boldsymbol{y}\in X\cap N_{\varepsilon}\left( \boldsymbol{x}\right) :\boldsymbol{y}\succ\boldsymbol{x}\]then \(\succsim\) is said to satisfy local non-satiation. Here, \(N_{\varepsilon}\left( \boldsymbol{x}\right) \) denotes the neighborhood of radius \(\varepsilon\) centered at \(\boldsymbol{x}\), and is defined by\begin{align*} N_{\varepsilon}\left( \boldsymbol{x}\right) & =\left\{ \boldsymbol{z}\in\mathbb{R} ^{N}\ |\ \left\Vert \boldsymbol{z}-\boldsymbol{x}\right\Vert <\varepsilon
\right\} \\
& =\left\{ \boldsymbol{z}\in\mathbb{R} ^{N}\ |\ \sqrt{\sum_{n=1}^{N}\left( z_{n}-x_{n}\right) ^{2}}<\varepsilon
\right\}
\end{align*}Thus, \(N_{\varepsilon}\left( \boldsymbol{x}\right) \) is the set of points in \(\mathbb{R} ^{N}\) whose distance from \(\boldsymbol{x}\) is less than \(\varepsilon\). Since \(\varepsilon\) is an arbitrary positive real number in the preceding definition, it can be taken arbitrarily small. Accordingly, local non-satiation means that for any feasible consumption vector \(\boldsymbol{x}\), there exists another feasible consumption vector \(\boldsymbol{y}\) arbitrarily close to \(\boldsymbol{x}\) that is strictly preferred to \(\boldsymbol{x}\).
N_{\varepsilon}\left( x\right) =\left( x-\varepsilon,x+\varepsilon\right)
\]For any \(x,y\in\mathbb{R} _{+}\), suppose the preference relation is defined by\[ y\succsim x\Leftrightarrow y\geq x
\]If the preference relation \(\succsim\) on \(\mathbb{R} _{+}\) is defined in this way, it satisfies \(\left( 1\right) \) and therefore satisfies local non-satiation. Indeed, choose any consumption vector \(x\in\mathbb{R} _{+}\) and any radius \(\varepsilon>0\). Then \[
x+\frac{\varepsilon}{2}\in\mathbb{R} _{+}\cap\left( x-\varepsilon,x+\varepsilon\right)
\]holds, and
\[
x+\frac{\varepsilon}{2}>x
\]also holds. By the definition of \(\succsim\), this implies \(x+\frac {\varepsilon}{2}\succ x\). Therefore, \(\left( 1\right) \) holds.
N_{\varepsilon}\left( x_{1},x_{2}\right) =\left\{ \left( z_{1},z_{2}\right) \in\mathbb{R} ^{2}\ |\ \sqrt{\left( z_{1}-x_{1}\right) ^{2}+\left( z_{2}-x_{2}\right)
^{2}}<\varepsilon\right\}
\]For any \(\left( x_{1},x_{2}\right) ,\left( y_{1},y_{2}\right) \in\mathbb{R} _{+}^{2}\), suppose the preference relation is defined by\[ \left( y_{1},y_{2}\right) \succsim\left( x_{1},x_{2}\right)
\Leftrightarrow\left( y_{1}\geq x_{1}\wedge y_{2}\geq x_{2}\right)
\]If the preference relation \(\succsim\) on \(\mathbb{R} _{+}^{2}\) is defined in this way, it satisfies \(\left( 1\right) \) and therefore satisfies local non-satiation (see the exercises).
Relationship Between Non-Satiation and Local Non-Satiation
Suppose the preference relation \(\succsim\) satisfies local non-satiation. For any consumption vector \(x\), local non-satiation guarantees the existence of another consumption vector \(y\) arbitrarily close to \(x\) that is strictly preferred to \(x\). Thus, \(x\) is not a satiation point. Since the same argument applies to every consumption vector, there is no satiation point in the consumption set. Therefore, the following proposition holds.
The converse of the preceding proposition does not necessarily hold. That is, a preference relation satisfying non-satiation need not satisfy local non-satiation.
\]where \(\left\lfloor \cdot\right\rfloor \) is the floor function, and \(\left\lfloor x\right\rfloor \) is the greatest integer less than or equal to \(x\). This preference relation \(\succsim\) satisfies non-satiation but does not satisfy local non-satiation.
Relationship Between Local Non-Satiation and Monotonicity
When the consumption set is \(\mathbb{R} _{+}^{N}\), if a preference relation \(\succsim\) on \(\mathbb{R} _{+}^{N}\) satisfies strict monotonicity, then \(\succsim\) is guaranteed to satisfy local non-satiation.
The converse of the preceding proposition does not necessarily hold. That is, preferences satisfying local non-satiation need not satisfy strict monotonicity, as the following example shows.
Because the indifference curve \(I\left( \boldsymbol{x}\right) \) contains an upward-sloping segment, this preference relation \(\succsim\) does not satisfy strict monotonicity. Indeed, if \(\succsim\) satisfied strict monotonicity, then for a point \(\boldsymbol{y}\) such that \(\boldsymbol{y}>\boldsymbol{x}\) as in the figure above, we would have \(\boldsymbol{y}\succ\boldsymbol{x}\). However, since both \(\boldsymbol{x}\) and \(\boldsymbol{y}\) lie on \(I\left( \boldsymbol{x}\right) \), we also have \(\boldsymbol{x}\sim\boldsymbol{y}\), a contradiction. On the other hand, if any point on \(I\left( \boldsymbol{x}\right) \) and any open neighborhood of that point are chosen, the neighborhood intersects \(U_{s}\left( \boldsymbol{x}\right) \). Thus, for every point on \(I\left( \boldsymbol{x}\right) \), there are consumption vectors arbitrarily close to it that are strictly preferred to it. Hence this preference relation \(\succsim\) is consistent with local non-satiation.
Monotonicity is the property that a consumption vector becomes more desirable when the amount of at least one good is increased without reducing the consumption of any other good. Thus, monotonicity is a strong assumption that specifies the direction “more is better” for every good in the consumption vector.
On the other hand, monotonicity generally fails when the consumption vector contains an object for which a larger quantity is undesirable to the consumer. Even in such a case, however, there may be consumption vectors arbitrarily close to any given consumption vector that are strictly preferred to it. This property is captured by local non-satiation. Local non-satiation does not specify the direction in which a good must be increased or decreased to raise the consumer\'{s} satisfaction; it requires only that every neighborhood of every consumption vector contain a more desirable consumption vector. The following is an example.
Utility Functions Satisfying Non-Satiation
Let the consumption set be \(X\subset\mathbb{R} ^{N}\), and let \(u:X\rightarrow\mathbb{R} \) be a utility function defined on it satisfying the condition\[ \forall\boldsymbol{x}\in X,\ \exists\boldsymbol{y}\in X:u\left(
\boldsymbol{y}\right) >u\left( \boldsymbol{x}\right)
\]If this condition holds, that is, if for every feasible consumption vector
\(\boldsymbol{x}\) there exists another feasible consumption vector \(\boldsymbol{y}\) that yields strictly greater utility, then \(u\) is said to satisfy non-satiation.
\]This utility function \(u\) is non-satiated. Indeed, choose any consumption vector \(x\in\mathbb{R} _{+}\). For example, consider the following consumption vector:\[ x+1\in\mathbb{R} _{+}\]Then\begin{align*}
u\left( x+1\right) & =x+1\quad\because\text{by the definition of }u\\
& >x\\
& =u\left( x\right) \quad\because\text{by the definition of }u
\end{align*}holds.
u\left( x_{1}+1,x_{2}+1\right) & =\left( x_{1}+1\right) +\left(
x_{2}+1\right) \quad\because\text{by the definition of }u\\
& >x_{1}+x_{2}\\
& =u\left( x_{1},x_{2}\right) \quad\because\text{by the definition of }u
\end{align*}holds.
The following relationship holds between non-satiation of preferences and non-satiation of utility functions.
Note that this proposition presupposes the existence of a utility function representing the preference relation.
Locally Non-Satiated Utility Functions
Let the consumption set be \(X\subset\mathbb{R} ^{N}\), and let \(u:X\rightarrow\mathbb{R} \) be a utility function defined on it satisfying the condition\[ \forall\boldsymbol{x}\in X,\ \forall\varepsilon>0,\ \exists\boldsymbol{y}\in X\cap N_{\varepsilon}\left( \boldsymbol{x}\right) :u\left( \boldsymbol{y}\right) >u\left( \boldsymbol{x}\right)
\]If this condition holds, then \(u\) is said to satisfy local non-satiation. Here, \(N_{\varepsilon}\left( \boldsymbol{x}\right) \) is the neighborhood of radius \(\varepsilon\) centered at \(\boldsymbol{x}\). In other words, local non-satiation means that for every feasible consumption vector \(\boldsymbol{x}\), there exists another feasible consumption vector \(\boldsymbol{y}\) arbitrarily close to \(\boldsymbol{x}\) that yields a greater payoff.
\]This utility function \(u\) satisfies local non-satiation. Indeed, choose any consumption vector \(x\in\mathbb{R} _{+}\) and any radius \(\varepsilon>0\). Then \[
x+\varepsilon\in\mathbb{R} _{+}\cap\left( x-\varepsilon,x+\varepsilon\right)
\]holds, and
\[
x+\varepsilon>x
\]also holds. By the definition of \(u\), it follows that\[ u\left( x+\varepsilon\right) >u\left( x\right)
\]holds.
The following relationship holds between local non-satiation of preferences and local non-satiation of utility functions.
Note that this proposition also presupposes the existence of a utility function representing the preference relation.
Exercises
\Leftrightarrow\left( y_{1}\geq x_{1}\wedge y_{2}\geq x_{2}\right)
\]Prove that \(\succsim\) satisfies both non-satiation and local non-satiation.
& \left( b\right) \ \forall x,y\in\mathbb{R} _{+}\backslash\left\{ 1\right\} :x\sim y
\end{align*}Show that this preference relation \(\succsim\) satisfies neither non-satiation nor local non-satiation.
\]Show that this preference relation \(\succsim\) satisfies non-satiation but does not satisfy local non-satiation.
u\left( x_{1},x_{2}\right) =x_{1}-x_{2}\]Prove that this utility function satisfies local non-satiation but is not monotonically increasing.
u\left( x_{1},x_{2}\right) =\min\left\{ x_{1},x_{2}\right\}
\]Show that this utility function is monotonically increasing but not strictly monotonically increasing. Also prove that it satisfies local non-satiation.
u\left( x_{1},x_{2}\right) =x_{1}+x_{2}\]Show that this utility function is strictly monotonically increasing on \(X\) but does not satisfy local non-satiation. Explain why in terms of the shape of the consumption set \(X\).
