Preference Relations Satisfying Homotheticity
Consider an economy with \(N\) goods. Each alternative faced by 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 quantity consumed of good \(n\). The set consisting of all consumption 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 in \(X\). We therefore formalize the consumer’s system of preferences as a binary relation \(\succsim\) on \(X\), called a preference relation. Specifically, for any two consumption vectors \(\boldsymbol{x},\boldsymbol{y}\in X\), define \(\succsim\) by \[
\boldsymbol{x}\succsim\boldsymbol{y}\Leftrightarrow\text{the consumer regards
}\boldsymbol{x}\text{ as at least as desirable as }\boldsymbol{y}.
\]
Suppose that the consumption set is \(X=\mathbb{R} _{+}^{N}\). A preference relation \(\succsim\) on \(\mathbb{R} _{+}^{N}\) is said to satisfy homotheticity, or to be homothetic, if
\[
\forall\boldsymbol{x},\boldsymbol{y}\in\mathbb{R} _{+}^{N}:\left( \boldsymbol{x}\sim\boldsymbol{y}\Rightarrow\forall\lambda\in\mathbb{R} _{++}:\lambda\boldsymbol{x}\sim\lambda\boldsymbol{y}\right) .
\]In other words, whenever any two indifferent consumption vectors \(\boldsymbol{x}\) and \(\boldsymbol{y}\) are chosen, the consumption vectors \(\lambda\boldsymbol{x}\) and \(\lambda\boldsymbol{y}\) obtained by changing the quantities of all goods in the same proportion \(\lambda\) are also indifferent.
x\succsim y\Leftrightarrow x\geq y.\quad \cdots (1)\end{equation}Thus, a larger quantity of the good is more desirable. Choose any \(x,y\) satisfying \(x\sim y\). Then \begin{align*}
x\sim y & \Leftrightarrow x\succsim y\wedge y\succsim x\quad\because
\text{definition of }\sim\\
& \Leftrightarrow x\geq y\wedge y\geq x\quad\because\left( 1\right) \\
& \Leftrightarrow x=y.
\end{align*}Since \(x=y\), for any \(\lambda\in\mathbb{R} _{++}\), we have \(\lambda x=\lambda y\). Moreover, \begin{align*}
\lambda x=\lambda y & \Leftrightarrow\lambda x\geq\lambda y\wedge\lambda y\geq\lambda x\\
& \Leftrightarrow\lambda x\succsim\lambda y\wedge\lambda y\succsim\lambda x\quad\because\left( 1\right) \\
& \Leftrightarrow\lambda x\sim\lambda y\quad\because\text{definition of }\sim.
\end{align*}Therefore, \(\succsim\) satisfies homotheticity.
\boldsymbol{x}\succsim\boldsymbol{y}\Leftrightarrow x_{1}x_{2}\geq y_{1}y_{2}.\quad \cdots (1)\end{equation}Thus, a consumption vector is more desirable when the product of the quantities of the two goods is larger. Choose any \(\boldsymbol{x},\boldsymbol{y}\) satisfying \(\boldsymbol{x}\sim\boldsymbol{y}\). Then \begin{align*}
\boldsymbol{x}\sim\boldsymbol{y} & \Leftrightarrow\boldsymbol{x}\succsim\boldsymbol{y}\wedge\boldsymbol{y}\succsim\boldsymbol{x}\quad
\because\text{definition of }\sim\\
& \Leftrightarrow x_{1}x_{2}\geq y_{1}y_{2}\wedge y_{1}y_{2}\geq x_{1}x_{2}\quad\because\left( 1\right) \\
& \Leftrightarrow x_{1}x_{2}=y_{1}y_{2}.
\end{align*}Since \(x_{1}x_{2}=y_{1}y_{2}\), for any \(\lambda\in\mathbb{R} _{++}\), we have \begin{align*}
\lambda^{2}x_{1}x_{2} & =\lambda^{2}y_{1}y_{2},\\
\lambda x_{1}\lambda x_{2} & =\lambda y_{1}\lambda y_{2}.
\end{align*}Hence,
\[
\lambda\boldsymbol{x}\sim\lambda\boldsymbol{y}.
\]Therefore, \(\succsim\) satisfies homotheticity.
A preference relation need not satisfy homotheticity.
\boldsymbol{x}\succsim\boldsymbol{y}\Leftrightarrow x_{1}+\sqrt{x_{2}}\geq y_{1}+\sqrt{y_{2}}.
\]Because the usual ordering of real numbers is complete and transitive, this preference relation is rational. However, it does not satisfy homotheticity. Consider
\begin{align*}
\boldsymbol{x} & =\left( 1,0\right) ,\\
\boldsymbol{y} & =\left( 0,1\right) .
\end{align*}Then
\begin{align*}
x_{1}+\sqrt{x_{2}} & =1,\\
y_{1}+\sqrt{y_{2}} & =1,
\end{align*}so
\[
\boldsymbol{x}\sim\boldsymbol{y}.
\]Now let
\[
\lambda=4.
\]Then
\begin{align*}
4\boldsymbol{x} & =\left( 4,0\right) ,\\
4\boldsymbol{y} & =\left( 0,4\right) ,
\end{align*}and
\begin{align*}
4x_{1}+\sqrt{4x_{2}} & =4,\\
4y_{1}+\sqrt{4y_{2}} & =2.
\end{align*}Therefore,
\[
4\boldsymbol{x}\succ4\boldsymbol{y},
\]so
\[
4\boldsymbol{x}\sim4\boldsymbol{y}\]does not hold. Hence, \(\succsim\) does not satisfy homotheticity.
Indifference Curves under Homothetic Preferences
Suppose that a preference relation \(\succsim\) on \(\mathbb{R} _{+}^{2}\) satisfies rationality, monotonicity, and homotheticity. If \(\succsim\) is rational, the consumption set is partitioned into indifference sets. That is, every consumption vector belongs to some indifference set, and no consumption vector belongs to two distinct indifference sets.
As illustrated above, choose any consumption vectors \(\boldsymbol{x}\) and \(\boldsymbol{y}\) on the same indifference curve \(I\left( \boldsymbol{x}\right) \). If \(\succsim\) satisfies homotheticity, then for every positive real number \(\lambda\), \(\lambda\boldsymbol{x}\) and \(\lambda\boldsymbol{y}\) are indifferent and therefore belong to the same indifference curve. Thus, under homothetic preferences, a homothetic expansion or contraction of an indifference curve about the origin produces another indifference curve.
In general, a complete description of a consumer’s preferences requires determining which of any two consumption vectors in the consumption set is preferred. This entails an enormous amount of information and is practically infeasible. If preferences are assumed to be homothetic, however, once information about an indifference set \(I\left( \boldsymbol{x}\right) \) containing some consumption vector \(\boldsymbol{x}\) is available, other indifference sets can be obtained by homothetic expansion or contraction. For this reason, homotheticity is sometimes assumed when theories are tested empirically.
Homogeneous Utility Functions
Let \(u:\mathbb{R} _{+}^{N}\rightarrow\mathbb{R} \) be a utility function and let \(r\in\mathbb{R} \) be a real number. The function \(u\) is said to be homogeneous of degree \(r\) if \[
\forall\boldsymbol{x}\in\mathbb{R} _{+}^{N},\ \forall\lambda\in\mathbb{R} _{++}:u\left( \lambda\boldsymbol{x}\right) =\lambda^{r}u\left(
\boldsymbol{x}\right) .
\]Thus, starting from any consumption vector\(\boldsymbol{x}\), if the quantities of all goods are changed by the same proportion \(\lambda\), the resulting utility level is multiplied by \(\lambda^{r}\).
In particular, when \(r=1\), \[
u\left( \lambda\boldsymbol{x}\right) = \lambda u\left( \boldsymbol{x}\right) .
\]Such a utility function is called homogeneous of degree 1. Hence, a degree-1 homogeneous function is a special case of a homogeneous function.
\[
u\left( x_{1},x_{2}\right) = x_{1}^{\frac{1}{2}}x_{2}^{\frac{1}{2}}.
\]Then, for any \(\lambda>0\), \begin{align*}
u\left( \lambda x_{1},\lambda x_{2}\right) & = \left( \lambda x_{1}\right)
^{\frac{1}{2}} \left( \lambda x_{2}\right) ^{\frac{1}{2}} \quad
\because\text{definition of }u\\
& = \lambda x_{1}^{\frac{1}{2}}x_{2}^{\frac{1}{2}}\\
& = \lambda u\left( x_{1},x_{2}\right) .
\end{align*}Hence, \(u\) is homogeneous of degree 1.
\[
u\left( x_{1},x_{2}\right) = \min\left\{ x_{1},\frac{x_{2}}{2}\right\} .
\]Then, for any \(\lambda>0\), \begin{align*}
u\left( \lambda x_{1},\lambda x_{2}\right) & = \min\left\{ \lambda x_{1},\frac{\lambda x_{2}}{2}\right\} \\
& = \lambda\min\left\{ x_{1},\frac{x_{2}}{2}\right\} \quad\because
\lambda>0\\
& = \lambda u\left( x_{1},x_{2}\right) .
\end{align*}Hence, \(u\) is homogeneous of degree 1.
\[
u\left( x_{1},x_{2}\right) = x_{1}+2x_{2}.
\]Then, for any \(\lambda>0\), \begin{align*}
u\left( \lambda x_{1},\lambda x_{2}\right) & = \lambda x_{1}+2\lambda x_{2}\\
& = \lambda\left( x_{1}+2x_{2}\right) \\
& = \lambda u\left( x_{1},x_{2}\right) .
\end{align*}Hence, \(u\) is homogeneous of degree 1.
\[
u\left( x_{1},x_{2}\right) = x_{1}x_{2}.
\]Then, for any \(\lambda>0\), \begin{align*}
u\left( \lambda x_{1},\lambda x_{2}\right) & = \left( \lambda x_{1}\right)
\left( \lambda x_{2}\right) \\
& = \lambda^{2}x_{1}x_{2}\\
& = \lambda^{2}u\left( x_{1},x_{2}\right) .
\end{align*}Hence, \(u\) is homogeneous of degree 2.
If a utility function representing a preference relation exists and that utility function is homogeneous, then the preference relation is guaranteed to satisfy homotheticity.
Note that this proposition presupposes the existence of a utility function representing the preference relation.
Marginal Rates of Substitution under Homogeneous Utility Functions
Suppose that \(u:\mathbb{R} _{++}^{N}\rightarrow\mathbb{R} \) is of class \(C^{1}\). At a point \(\boldsymbol{x}\in\mathbb{R} _{++}^{N}\), the marginal rate of substitution of good \(i\) for good \(j\) is defined, provided that \[
\frac{\partial u\left( \boldsymbol{x}\right) }{\partial x_{j}}\neq0,
\]by
\begin{equation}
MRS_{ij}\left( \boldsymbol{x}\right) =\frac{\frac{\partial u\left(
\boldsymbol{x}\right) }{\partial x_{i}}}{\frac{\partial u\left(
\boldsymbol{x}\right) }{\partial x_{j}}}.\quad \cdots (1)\end{equation}
Suppose that \(u\) is homogeneous of degree \(r\). Thus, for any \(\boldsymbol{x}\) and any \(\lambda>0\), \begin{equation}
u\left( \lambda\boldsymbol{x}\right) = \lambda^{r}u\left( \boldsymbol{x}\right) .\quad \cdots (2)\end{equation}
Differentiate both sides of \(\left( 2\right) \) with respect to \(x_{i}\). Since the left-hand side \(u\left( \lambda\boldsymbol{x}\right) \) is the composition of the vector-valued function \(\lambda\boldsymbol{x}\) and the real-valued function \(u\), the chain rule gives \begin{align*}
\frac{\partial u\left( \lambda\boldsymbol{x}\right) }{\partial x_{i}} & =
\left. \nabla u\left( \boldsymbol{y}\right) \right\vert _{\boldsymbol{y}=\lambda\boldsymbol{x}} \cdot\frac{\partial\lambda\boldsymbol{x}}{\partial x_{i}} \quad\because\text{chain rule}\\
& = \left. \left( \frac{\partial u\left( \boldsymbol{y}\right) }{\partial y_{1}}, \cdots, \frac{\partial u\left( \boldsymbol{y}\right) }{\partial y_{i}}, \cdots, \frac{\partial u\left( \boldsymbol{y}\right) }{\partial y_{N}} \right) \right\vert _{\boldsymbol{y}=\lambda\boldsymbol{x}}
\cdot\left( 0,\cdots,\lambda,\cdots,0\right) \\
& = \lambda\left. \frac{\partial u\left( \boldsymbol{y}\right) }{\partial y_{i}} \right\vert _{\boldsymbol{y}=\lambda\boldsymbol{x}}.
\end{align*}For the right-hand side of \(\left( 2\right) \), \[
\frac{\partial\lambda^{r}u\left( \boldsymbol{x}\right) }{\partial x_{i}} =
\lambda^{r} \frac{\partial u\left( \boldsymbol{x}\right) }{\partial x_{i}}.
\]Equating the two expressions yields
\[
\lambda\left. \frac{\partial u\left( \boldsymbol{y}\right) }{\partial y_{i}}
\right\vert _{\boldsymbol{y}=\lambda\boldsymbol{x}} = \lambda^{r}
\frac{\partial u\left( \boldsymbol{x}\right) }{\partial x_{i}}.
\]Since \(\lambda>0\), \begin{equation}
\left. \frac{\partial u\left( \boldsymbol{y}\right) }{\partial y_{i}}
\right\vert _{\boldsymbol{y}=\lambda\boldsymbol{x}} = \lambda^{r-1}
\frac{\partial u\left( \boldsymbol{x}\right) }{\partial x_{i}}.\quad \cdots (3)\end{equation}If we write the partial derivative of \(u\) with respect to its \(i\) th argument as \[
u_{i}\left( \boldsymbol{x}\right) = \frac{\partial u\left( \boldsymbol{x}\right) }{\partial x_{i}},
\]then
\[
u_{i}\left( \lambda\boldsymbol{x}\right) = \left. \frac{\partial u\left(
\boldsymbol{y}\right) }{\partial y_{i}} \right\vert _{\boldsymbol{y}=\lambda\boldsymbol{x}},
\]so \(\left( 3\right) \) may be written as \begin{equation}
u_{i}\left( \lambda\boldsymbol{x}\right) = \lambda^{r-1}u_{i}\left(
\boldsymbol{x}\right) .\quad \cdots (4)\end{equation}Thus, each partial derivative of a degree-\(r\) homogeneous function is homogeneous of degree \(r-1\).
Similarly,
\begin{equation}
u_{j}\left( \lambda\boldsymbol{x}\right) = \lambda^{r-1}u_{j}\left(
\boldsymbol{x}\right) .\quad \cdots (5)\end{equation}Therefore,
\begin{align*}
MRS_{ij}\left( \lambda\boldsymbol{x}\right) & = \frac{u_{i}\left(
\lambda\boldsymbol{x}\right) } {u_{j}\left( \lambda\boldsymbol{x}\right) }
\quad\because\left( 1\right) \\
& = \frac{\lambda^{r-1}u_{i}\left( \boldsymbol{x}\right) } {\lambda
^{r-1}u_{j}\left( \boldsymbol{x}\right) } \quad\because\left( 4\right)
,\left( 5\right) \\
& = MRS_{ij}\left( \boldsymbol{x}\right) \quad\because\left( 1\right) .
\end{align*}Hence,
\[
MRS_{ij}\left( \lambda\boldsymbol{x}\right) = MRS_{ij}\left( \boldsymbol{x}\right) .
\]Thus, under a homogeneous utility function, the marginal rate of substitution \(MRS_{ij}\) is homogeneous of degree 0. Starting from a consumption vector \(\boldsymbol{x}\), changing the quantities of all goods in the same proportion does not change the marginal rate of substitution between goods \(i\) and \(j\).
Exercises
\forall\boldsymbol{x},\boldsymbol{y}\in\mathbb{R} _{+}^{N}:\left( \boldsymbol{x}\succsim\boldsymbol{y}\Rightarrow\forall
\lambda>0:\lambda\boldsymbol{x}\succsim\lambda\boldsymbol{y}\right) .
\]Prove that \(\succsim\) satisfies homotheticity. Also provide a counterexample showing that the converse need not hold.
\[
u\left( x_{1},x_{2}\right) = x_{1}^{\frac{1}{3}}x_{2}^{\frac{2}{3}}.
\]Show that \(u\) is homogeneous of degree 1. Then find the marginal rate of substitution of good 1 for good 2 at each point and discuss what can be inferred from the resulting expression.
\[
u\left( x_{1},x_{2}\right) = 2x_{1}+5x_{2}.
\]Show that \(u\) is homogeneous of degree 1. Then find the marginal rate of substitution of good 1 for good 2 and discuss what can be inferred from it.
\[
u\left( \boldsymbol{x}\right) = kx_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}\cdots x_{N}^{\alpha_{N}},
\]where \(k>0\) and \(\alpha_{n}>0\) for \(n=1,\cdots,N\). This is called a Cobb− −Douglas utility function. If a consumer’s preferences are represented by this utility function, do the preferences satisfy homotheticity? Discuss.
\[
u\left( \boldsymbol{x}\right) = \min\left\{ \alpha_{1}x_{1},\cdots
,\alpha_{N}x_{N} \right\} ,
\]where \(\alpha_{n}>0\) for \(n=1,\cdots,N\). This is called a Leontief utility function. If a consumer’s preferences are represented by this utility function, do the preferences satisfy homotheticity? Discuss.
\[
u\left( \boldsymbol{x}\right) = \alpha_{1}x_{1}+\cdots+\alpha_{N}x_{N},
\]where \(\alpha_{n}>0\) for \(n=1,\cdots,N\). This is called a linear utility function. If a consumer’s preferences are represented by this utility function, do the preferences satisfy homotheticity? Discuss.
=\lambda^{r}u\left( \boldsymbol{x}\right) .
\]Show that\(\succsim\) satisfies homotheticity.
u\left( x_{1},x_{2}\right) =x_{1}+\sqrt{x_{2}}.
\]Show that the preference relation does not satisfy homotheticity.
\[
u\left( x_{1},x_{2}\right) =\left( \sqrt{x_{1}}+\sqrt{x_{2}}\right) ^{2}\]on \(\mathbb{R} _{++}^{2}\).
- Show that \(u\) is homogeneous of degree 1.
- Find \(MRS_{12}\left( x_{1},x_{2}\right) \).
- Show that \(MRS_{12}\) is homogeneous of degree 0.
\[
u\left( x_{1},x_{2}\right) =x_{1}^{\frac{1}{2}}x_{2}^{\frac{1}{2}}\]on \(\mathbb{R} _{++}^{2}\), and define \[
v\left( x_{1},x_{2}\right) =\ln\left( u\left( x_{1},x_{2}\right) \right)
.
\]
- Show that \(u\) is homogeneous of degree 1.
- Show that \(v\) is a strictly increasing transformation of \(u\) and that \(u\) and \(v\) represent the same preference relation.
- Show that \(v\) is not homogeneous of degree 1.
- Explain why the preference relation represented by \(v\) nevertheless satisfies homotheticity.
