Preference Relations Satisfying Monotonicity
Consider an economy with \(N\) goods, and represent each alternative facing the consumer 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 the vector \(\boldsymbol{x}\) represents the consumption of good \(n\). The set of all vectors available to the consumer is represented by the consumption set \[
X\subset\mathbb{R} ^{N}.
\]A consumer facing the consumption set \(X\) chooses the most desirable consumption vector by comparing the consumption vectors that are elements of \(X\). We therefore formulate the consumer’s system of tastes as a binary relation \(\succsim\) on \(X\) and call it a preference relation. Specifically, for any two consumption vectors \(\boldsymbol{x},\boldsymbol{y}\in X\), we define \(\succsim\) so that \[
\boldsymbol{x}\succsim\boldsymbol{y}\Leftrightarrow\text{the consumer prefers
}\boldsymbol{x}\text{ at least as much as }\boldsymbol{y}.
\]
Given vectors \(\boldsymbol{x},\boldsymbol{y}\in\mathbb{R} ^{N}\) in Euclidean space \(\mathbb{R} ^{N}\), define the binary relation \(\geq\) on \(\mathbb{R} ^{N}\) by the condition \[
\boldsymbol{y}\geq\boldsymbol{x}\Leftrightarrow\forall n\in\left\{
1,\cdots,N\right\} :y_{n}\geq x_{n},
\]and call this the standard order. Thus, after comparing the corresponding components of two vectors \(\boldsymbol{x}\) and \(\boldsymbol{y}\), \(\boldsymbol{y}\geq\boldsymbol{x}\) holds if and only if every component of \(\boldsymbol{y}\) is at least as large as the corresponding component of \(\boldsymbol{x}\). When \(\boldsymbol{y}\geq\boldsymbol{x}\) holds, we say that \(\boldsymbol{y}\) is greater than or equal to \(\boldsymbol{x}\).
With this in mind, a preference relation \(\succsim\) on the consumption set \(X\subset\mathbb{R} ^{N}\) is said to satisfy monotonicity, or to be nondecreasing, if it satisfies
\[
\forall\boldsymbol{x},\boldsymbol{y}\in X:\left( \boldsymbol{y}\geq\boldsymbol{x}\Rightarrow\boldsymbol{y}\succsim\boldsymbol{x}\right) .
\]A preference relation \(\succsim\) satisfies monotonicity if, whenever a consumption vector \(\boldsymbol{x}\) is chosen and none of the quantities of the goods are reduced from \(\boldsymbol{x}\), the consumer’s satisfaction does not decrease.
For any consumption vector \(\boldsymbol{x}\in X\), its upper contour set is defined as \[
U\left( \boldsymbol{x}\right) =\left\{ \boldsymbol{y}\in X\ |\ \boldsymbol{y}\succsim\boldsymbol{x}\right\} .
\]Therefore, the preference relation \(\succsim\) satisfies monotonicity if and only if \[
\forall\boldsymbol{x},\boldsymbol{y}\in X:\left[ \boldsymbol{y}\geq\boldsymbol{x}\Rightarrow\boldsymbol{y}\in U\left( \boldsymbol{x}\right)
\right] .
\]In other words, if, for every consumption vector \(\boldsymbol{x}\), every consumption vector greater than or equal to \(\boldsymbol{x}\) belongs to the upper contour set \(U\left( \boldsymbol{x}\right) \), then \(\succsim\) satisfies monotonicity.
\(\mathbb{R} _{+}\), a preference relation \(\succsim\) satisfies monotonicity if, for all \(x,y\in\mathbb{R} _{+}\), \[
y\geq x\Rightarrow y\succsim x.
\]Thus, for any consumption vector \(x\in\mathbb{R} _{+}\), its upper contour set is \begin{align*}
U\left( x\right) & =\left\{ y\in\mathbb{R} _{+}\ |\ y\geq x\right\} \\
& =[x,+\infty),
\end{align*}which means that every consumption vector belonging to the bounded half-open interval \([x,+\infty)\) is at least as desirable as \(x\).
\(\mathbb{R} _{+}^{2}\), a preference relation \(\succsim\) satisfies monotonicity if, for all \(\left( x_{1},x_{2}\right) ,\left( y_{1},y_{2}\right) \in\mathbb{R} _{+}^{2}\), \[
\left( y_{1}\geq x_{1}\wedge y_{2}\geq x_{2}\right) \Rightarrow\left(
y_{1},y_{2}\right) \succsim\left( x_{1},x_{2}\right) .
\]This preference relation \(\succsim\) is illustrated below.
Thus, as shown in the figure above, if a consumption vector \(\boldsymbol{x}=\left( x_{1},x_{2}\right) \) is chosen arbitrarily, the shaded region, including its boundary, is the upper contour set \(U\left( \boldsymbol{x}\right) \) of \(\boldsymbol{x}\), that is, the set of consumption vectors that are at least as desirable as \(\boldsymbol{x}\).
\left( y_{1}\geq x_{1}\wedge y_{2}\geq x_{2}\right) \Rightarrow\left(
y_{1},y_{2}\right) \succsim\left( x_{1},x_{2}\right) .
\]Thus, the household’s preference relation satisfies monotonicity.
X=\{\left( h,x\right) \in\mathbb{R} _{+}^{2}\ |\ 10\leq h\leq24\} .
\]If the worker regards more leisure and more consumption of goods as better, then, for all consumption vectors \(\left( h_{1},x_{1}\right) ,\left( h_{2},x_{2}\right) \in X\), the worker’s preference relation \(\succsim\) satisfies
\[
\left( h_{2}\geq h_{1}\wedge x_{2}\geq x_{1}\right) \Rightarrow\left(
h_{2},x_{2}\right) \succsim\left( h_{1},x_{1}\right) .
\]Thus, the worker’s preference relation satisfies monotonicity.
Preference Relations Satisfying Strict Monotonicity
Suppose that the standard order \(\geq\) on Euclidean space \(\mathbb{R} ^{N}\) is given. That is, for any vectors \(\boldsymbol{x},\boldsymbol{y}\in\mathbb{R} ^{N}\), \[
\boldsymbol{y}\geq\boldsymbol{x}\Leftrightarrow\forall n\in\left\{
1,\cdots,N\right\} :y_{n}>x_{n}.
\]Then define the binary relation \(>\) on \(\mathbb{R} ^{N}\) by requiring that, for any vectors \(\boldsymbol{x},\boldsymbol{y}\in\mathbb{R} ^{N}\), \[
\boldsymbol{y}>\boldsymbol{x}\Leftrightarrow\left( \boldsymbol{y}\geq\boldsymbol{x}\wedge\boldsymbol{y}\not =\boldsymbol{x}\right) ,
\]and call it the standard strict order. By definition, \(\boldsymbol{y}>\boldsymbol{x}\) means that both \begin{align*}
\forall n & \in\left\{ 1,\cdots,N\right\} :y_{n}\geq x_{n}\\
\exists n & \in\left\{ 1,\cdots,N\right\} :y_{n}>x_{n}\end{align*}hold. When \(\boldsymbol{y}>\boldsymbol{x}\) holds, we say that \(\boldsymbol{y}\) is greater than \(\boldsymbol{x}\).
With this in mind, a preference relation \(\succsim\) on the consumption set \(X\subset\mathbb{R} ^{N}\) is said to satisfy strict monotonicity, or to be strictly increasing, if it satisfies
\[
\forall\boldsymbol{x},\boldsymbol{y}\in X:\left( \boldsymbol{y}>\boldsymbol{x}\Rightarrow\boldsymbol{y}\succ\boldsymbol{x}\right) .
\]A preference relation \(\succsim\) satisfies strict monotonicity if, whenever a consumption vector \(\boldsymbol{x}\) is chosen, the consumer’s satisfaction increases when none of the quantities of the goods are reduced and the quantity of at least one good is increased.
For any consumption vector \(\boldsymbol{x}\in X\), its strict upper contour set is defined as \[
U_{s}\left( \boldsymbol{x}\right) =\left\{ \boldsymbol{y}\in X\ |\ \boldsymbol{y}\succ\boldsymbol{x}\right\} .
\]Therefore, the preference relation \(\succsim\) satisfies strict monotonicity if and only if \[
\forall\boldsymbol{x},\boldsymbol{y}\in X:\left[ \boldsymbol{y}>\boldsymbol{x}\Rightarrow\boldsymbol{y}\in U_{s}\left( \boldsymbol{x}\right) \right] .
\]In other words, if, for every consumption vector \(\boldsymbol{x}\), every consumption vector greater than \(\boldsymbol{x}\) belongs to the strict upper contour set \(U_{s}\left( \boldsymbol{x}\right) \), then \(\succsim\) satisfies strict monotonicity.
x>y\Rightarrow x\succ y.
\]
\left( x_{1}\geq y_{1}\wedge x_{2}\geq y_{2}\right) \wedge\left(
x_{1}>y_{1}\vee x_{2}>y_{2}\right) \Rightarrow\left( x_{1},x_{2}\right)
\succ\left( y_{1},y_{2}\right) .
\]This preference relation \(\succsim\) is illustrated below.
Thus, as shown in the figure above, if a consumption vector \(x=\left( x_{1},x_{2}\right) \) is chosen arbitrarily, the shaded region, including its boundary but excluding the point \(x\) itself, is the strict upper contour set \(U_{s}\left( x\right) \) of \(x\), that is, the set of consumption vectors more desirable than \(x\).
Relationship between Monotonicity and Strict Monotonicity
A preference relation that satisfies strict monotonicity also satisfies monotonicity.
The converse of the proposition above does not necessarily hold. That is, a preference relation satisfying monotonicity does not necessarily satisfy strict monotonicity, as the following example shows.
\boldsymbol{x}\sim\boldsymbol{y}\]for all \(\boldsymbol{x},\boldsymbol{y}\in X\). Choose any consumption vectors \(\boldsymbol{x},\boldsymbol{y}\in X\) satisfying \(\boldsymbol{y}\geq \boldsymbol{x}\). By the definition of \(\succsim\), \(\boldsymbol{y}\sim\boldsymbol{x}\) holds. Then \begin{align*}
\boldsymbol{y}\sim\boldsymbol{x} & \Leftrightarrow\boldsymbol{y}\succsim\boldsymbol{x}\wedge\boldsymbol{x}\succsim\boldsymbol{y}\quad
\because\text{definition of }\sim\\
& \Rightarrow\boldsymbol{y}\succsim\boldsymbol{x}.
\end{align*}Thus, \(\boldsymbol{y}\succsim\boldsymbol{x}\), so \(\succsim\) satisfies monotonicity. On the other hand, choose any consumption vectors \(\boldsymbol{x},\boldsymbol{y}\in X\) satisfying \(\boldsymbol{y}>\boldsymbol{x}\). By the definition of \(\succsim\), \(\boldsymbol{y}\sim\boldsymbol{x}\) holds. Hence \(\boldsymbol{y}\succ\boldsymbol{x}\) does not hold, so \(\succsim\) does not satisfy strict monotonicity.
Shape of Indifference Sets under Monotonic Preferences
Suppose that a preference relation \(\succsim\) on the consumption set \(\mathbb{R} _{+}^{2}\) satisfies both rationality and strict monotonicity. That is, suppose that all of the following conditions hold: \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]
\\
& \left( c\right) \ \forall\boldsymbol{x},\boldsymbol{y}\in X:\left(
\boldsymbol{y}>\boldsymbol{x}\Rightarrow\boldsymbol{y}\succ\boldsymbol{x}\right) .
\end{align*}Choose any consumption vector \(\boldsymbol{x}\in\mathbb{R} _{+}^{2}\), and let its indifference set be \[
I\left( \boldsymbol{x}\right) =\left\{ \boldsymbol{y}\in\mathbb{R} _{+}^{2}\ |\ \boldsymbol{y}\sim\boldsymbol{x}\right\} .
\]
Suppose that \(I\left( \boldsymbol{x}\right) \) has two-dimensional thickness, as shown in the figure above. Then there exists an interior point \(\boldsymbol{y}\) of \(I\left( \boldsymbol{x}\right) \), and by the definition of an interior point, for sufficiently small \(\varepsilon>0\), both \begin{align*}
\boldsymbol{y} & \in I\left( \boldsymbol{x}\right) \\
\boldsymbol{y}+\left( \varepsilon,0\right) & \in I\left( \boldsymbol{x}\right)
\end{align*}hold. By the definition of the indifference set, it follows that
\[
\boldsymbol{y}+\left( \varepsilon,0\right) \sim\boldsymbol{y}.
\]On the other hand,
\[
\boldsymbol{y}+\left( \varepsilon,0\right) >\boldsymbol{y}.
\]Therefore, strict monotonicity of \(\succsim\) implies \[
\boldsymbol{y}+\left( \varepsilon,0\right) \succ\boldsymbol{y},
\]which is a contradiction. Hence, \(I\left( \boldsymbol{x}\right) \) cannot have thickness.
Next, choose any consumption vector \(\boldsymbol{x}\in\mathbb{R} _{+}^{2}\) and divide the consumption set \(\mathbb{R} _{+}^{2}\) into the four regions \(A,B,C,D\) shown in the figure above by two lines passing through \(\boldsymbol{x}\). Choose any consumption vector \(\boldsymbol{y}\) in region \(B\) or on its boundary, excluding the point \(\boldsymbol{x}\). Since \(\boldsymbol{y}>\boldsymbol{x}\), \(\boldsymbol{y}\) is not an element of the indifference set \(I\left( \boldsymbol{x}\right) \). Therefore, \(I\left( \boldsymbol{x}\right) \) does not intersect region \(B\) or its boundary. Similarly, choose any consumption vector \(\boldsymbol{y}\) in region \(D\) or on its boundary, excluding the point \(\boldsymbol{x}\). Since \(\boldsymbol{x}>\boldsymbol{y}\), \(\boldsymbol{y}\) is not an element of the indifference set \(I\left( \boldsymbol{x}\right) \). Therefore, \(I\left( \boldsymbol{x}\right) \) does not intersect region \(D\) or its boundary. It follows that \(I\left( \boldsymbol{x}\right) \) can pass only through the interior of region \(A\) and the interior of region \(C\). Since this argument applies to every consumption vector in \(\mathbb{R} _{+}^{2}\), \(I\left( \boldsymbol{x}\right) \) is a set of points arranged in a downward-sloping direction.
Next, choose any two distinct indifference sets \(I\left( \boldsymbol{x}\right) ,I\left( \boldsymbol{y}\right) \). In this case, \(\boldsymbol{x}\sim\boldsymbol{y}\) does not hold. Suppose that \[
I\left( \boldsymbol{x}\right) \cap I\left( \boldsymbol{y}\right)
\not =\phi.
\]Then there exists a consumption vector \(\boldsymbol{z}\) satisfying \[
\boldsymbol{z}\in I\left( \boldsymbol{x}\right) \cap I\left( \boldsymbol{y}\right) ,
\]and both
\begin{align*}
\boldsymbol{z} & \sim\boldsymbol{x}\\
\boldsymbol{z} & \sim\boldsymbol{y}\end{align*}hold. By transitivity of \(\sim\), \[
\boldsymbol{x}\sim\boldsymbol{y}\]then holds, which is a contradiction. Therefore, distinct indifference sets do not intersect.
Next, choose any consumption vectors \(x,y\in\mathbb{R} _{+}^{2}\) satisfying \(\boldsymbol{y}>\boldsymbol{x}\), as shown in the figure above. Strict monotonicity of \(\succsim\) then implies \(\boldsymbol{y}\succ\boldsymbol{x}\). Therefore, every consumption vector in \(I\left( \boldsymbol{y}\right) \) is more desirable to the consumer than every consumption vector in \(I\left( \boldsymbol{x}\right) \). In this sense, \(I\left( \boldsymbol{y}\right) \) corresponds to a higher preference level than \(I\left( \boldsymbol{x}\right) \) and, geometrically, lies farther to the upper right.
To summarize, if a preference relation \(\succsim\) on the consumption set \(\mathbb{R} _{+}^{2}\) satisfies rationality and strict monotonicity, its indifference sets are not regions with positive area or thickness but are sets of points arranged in a downward-sloping direction. Distinct indifference sets do not intersect, and a consumption set corresponding to a more desirable consumption vector lies farther to the upper right. However, these properties alone do not guarantee that an indifference set can be represented as a continuous curve. To treat indifference sets as continuous indifference curves in the usual sense, additional conditions such as continuity of preferences are required.
Validity of the Monotonicity Assumption
The monotonicity assumption on preferences requires that, if the quantity of one good is increased without reducing the quantities of the other goods, the resulting consumption vector be at least as desirable to the consumer as the original one. Under strict monotonicity, if the quantity of at least one good is increased, the resulting consumption vector must be strictly more desirable than the original one. Thus, monotonicity assumes that, for every good constituting the consumption vector, “more is no worse.”
This assumption is useful in consumer theory involving many goods, but it need not be valid in every situation. For example, once the consumption of a certain good exceeds some level, further consumption may fail to increase satisfaction or may even reduce it. If such a satiation point lies in the interior of the consumption set, monotonicity does not hold on the entire consumption set.
However, because the consumption set is specified as the set of consumption vectors available to the consumer, monotonicity can be assumed on a restricted range if no satiation occurs within the range under analysis. Thus, the validity of monotonicity depends not only on the nature of the goods being considered but also on how the consumption set is specified.
Monotonicity also generally fails when an object whose greater quantity is undesirable to the consumer is used directly as a component of the consumption vector. For example, other things being equal, many people prefer fewer hours of labor. Hence, if labor hours are included as a component of the consumption vector, this is inconsistent with the monotonicity assumption that more is better.
Even in such cases, however, the problem can sometimes be formulated in a way that satisfies monotonicity by changing the choice of variables. For example, instead of using labor hours themselves as a variable, one can use leisure time, defined as available time minus labor time. Then, other things being equal, preferences can be expressed so that more leisure is more desirable.
It follows that monotonicity is not a property that holds unconditionally for every consumer or every object. Nevertheless, by appropriately specifying the goods and the consumption set under analysis, it can serve as a useful and reasonable assumption in many consumer problems.
Monotonically Increasing Utility Functions
A utility function \(u:X\rightarrow\mathbb{R} \) defined on the consumption set \(X\subset\mathbb{R} ^{N}\) is a monotonically increasing function or a monotonically non-decreasing function if
\[
\forall\boldsymbol{x},\boldsymbol{y}\in X:\left[ \boldsymbol{y}\geq\boldsymbol{x}\Rightarrow u\left( \boldsymbol{y}\right) \geq u\left(
\boldsymbol{x}\right) \right] .
\]That is, a utility function \(\succsim\) is monotonically increasing if, whenever a consumption vector \(\boldsymbol{x}\) is chosen, utility does not decrease when none of the quantities of the goods are reduced from \(\boldsymbol{x}\).
u\left( x\right) =x.
\]This utility function \(u\) is monotonically increasing. Indeed, for any \(x,y\in\mathbb{R} _{+}\) satisfying \(y\geq x\), \begin{align*}
u\left( y\right) & =y\quad\because\text{definition of }u\\
& \geq x\quad\because y\geq x\\
& =u\left( x\right) \quad\because\text{definition of }u.
\end{align*}
u\left( x_{1},x_{2}\right) =x_{1}+x_{2}.
\]This utility function \(u\) is monotonically increasing. Indeed, choose any \(\left( x_{1},x_{2}\right) ,\left( y_{1},y_{2}\right) \in\mathbb{R} _{+}^{2}\) satisfying \(\left( y_{1},y_{2}\right) \geq\left( x_{1},x_{2}\right) \), that is, \(y_{1}\geq x_{1}\) and \(y_{2}\geq x_{2}\). Then \begin{align*}
u\left( y_{1},y_{2}\right) & =y_{1}+y_{2}\quad\because\text{definition of
}u\\
& \geq x_{1}+x_{2}\quad\because\left( y_{1},y_{2}\right) \geq\left(
x_{1},x_{2}\right) \\
& =u\left( x_{1},x_{2}\right) \quad\because\text{definition of }u.
\end{align*}
The following relationship holds between monotonicity of preferences and monotonicity of a utility function.
Note that this proposition presupposes the existence of a utility function representing the preference relation. Can anything be inferred about the existence of a utility function merely from monotonicity of a preference relation? If a preference relation satisfies monotonicity, does there exist a utility function representing it? Conditions for the existence of a utility function will be discussed in detail elsewhere.
Strictly Monotonically Increasing Utility Functions
A utility function \(u:X\rightarrow\mathbb{R} \) defined on the consumption set \(X\subset\mathbb{R} ^{N}\) is strictly monotonically increasing if \[
\forall\boldsymbol{x},\boldsymbol{y}\in X:\left[ \boldsymbol{y}>\boldsymbol{x}\Rightarrow u\left( \boldsymbol{y}\right) >u\left(
\boldsymbol{x}\right) \right] .
\]That is, a utility function \(u\) is strictly monotonically increasing if, whenever a consumption vector \(\boldsymbol{x}\) is chosen, utility increases when none of the quantities of the goods are reduced and the quantity of at least one good is increased.
u\left( x\right) =x.
\]This utility function \(u\) is monotonically increasing. Indeed, choose any \(x,y\in\mathbb{R} _{+}\) satisfying \(y>x\). Then \begin{align*}
u\left( y\right) & =y\quad\because\text{definition of }u\\
& >x\quad\because y>x\\
& =u\left( x\right) \quad\because\text{definition of }u.
\end{align*}
u\left( x_{1},x_{2}\right) =x_{1}+x_{2}.
\]This utility function \(u\) is monotonically increasing. Indeed, choose any \(\left( x_{1},x_{2}\right) ,\left( y_{1},y_{2}\right) \in\mathbb{R} _{+}^{2}\) satisfying \(\left( y_{1},y_{2}\right) >\left( x_{1},x_{2}\right) \), that is, \(y_{1}\geq x_{1}\) and \(y_{2}\geq x_{2}\), with at least one of \(y_{1}>x_{1}\) and \(y_{2}>x_{2}\) holding. Then \begin{align*}
u\left( y_{1},y_{2}\right) & =y_{1}+y_{2}\quad\because\text{definition of
}u\\
& >x_{1}+x_{2}\quad\because\left( y_{1},y_{2}\right) >\left( x_{1},x_{2}\right) \\
& =u\left( x_{1},x_{2}\right) \quad\because\text{definition of }u.
\end{align*}
The following relationship holds between strict monotonicity of preferences and strict monotonicity of a utility function.
Note that this proposition also presupposes the existence of a utility function representing the preference relation.
Exercises
\left( x_{1}>y_{1}\wedge x_{2}=y_{2}\right) \Rightarrow\left( y_{1},y_{2}\right) \succ\left( x_{1},x_{2}\right) .
\]Does this preference relation satisfy monotonicity or strict monotonicity? Discuss.
\left( x_{1},x_{2}\right) \succsim\left( y_{1},y_{2}\right)
\Leftrightarrow x_{1}+x_{2}\geq y_{1}+y_{2}.
\]Does this preference relation satisfy monotonicity or strict monotonicity? If it does, prove it. If it does not, provide a counterexample.
\left( x_{1},x_{2}\right) \succsim\left( y_{1},y_{2}\right)
\Leftrightarrow\left[ x_{1}>y_{1}\vee\left( x_{1}=y_{1}\wedge x_{2}\geq y_{2}\right) \right] .
\]Such preferences are called lexicographic preferences. Do lexicographic preferences satisfy monotonicity or strict monotonicity? If they do, prove it. If they do not, provide a counterexample.
u\left( x_{1},x_{2}\right) =x_{1}+2x_{2}.
\]Prove that this preference relation satisfies monotonicity and strict monotonicity.
u\left( x_{1},x_{2}\right) =\max\left\{ x_{1},x_{2}\right\} .
\]Show that this preference relation satisfies monotonicity but not strict monotonicity.
\boldsymbol{y}>\boldsymbol{x}.
\]Prove that, for every \(\boldsymbol{x}^{\prime}\in I\left( \boldsymbol{x}\right) \) and every \(\boldsymbol{y}^{\prime}\in I\left( \boldsymbol{y}\right) \), \[
\boldsymbol{y}^{\prime}\succ\boldsymbol{x}^{\prime}.
\]
u\left( c,h\right) =c-h,
\]where \(T>0\).
- If \(\left( c,h\right) \) is regarded as the consumption vector, is this utility function monotonically increasing?
- Define leisure time by \(l=T-h\) and regard \(\left( c,l\right) \) as the consumption vector. Express the utility function as a function of \(c\) and \(l\), and show that this utility function is strictly monotonically increasing.
\left( x_{1},x_{2}\right) \succsim\left( y_{1},y_{2}\right)
\Leftrightarrow\min\left\{ x_{1},x_{2}\right\} \geq\min\left\{ y_{1},y_{2}\right\} .
\]Does this preference relation satisfy monotonicity or strict monotonicity? If it does, prove it. If it does not, provide a counterexample.
u\left( \boldsymbol{x}\right) =kx_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}\cdots x_{N}^{\alpha_{N}},
\]where \(k>0\) and \(\alpha_{n}>0\ \left( n=1,\cdots,N\right) \). Such a function \(u\) is called a Cobb− −Douglas utility function. If a consumer’s preferences are represented by a Cobb− −Douglas utility function, does the preference relation satisfy monotonicity or strict monotonicity? Discuss.
