Convexity of Preferences (Quasi-Concave Utility Functions)

Convex Preferences

Consider an economy with \(N\) commodities. We 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 this vector \(\boldsymbol{x}\) represents the quantity consumed of commodity \(n\). We represent the set of all vectors available to the consumer by the consumption set \[
X\subset\mathbb{R} ^{N}.
\]A consumer facing the consumption set \(X\) compares consumption vectors in \(X\) and chooses the consumption vector that is most desirable to the consumer. We therefore formulate the consumer’s system of preferences as a binary relation
\(\succsim\) on \(X\), called the preference relation. Specifically, for any two consumption vectors \(\boldsymbol{x},\boldsymbol{y}\in X\), we define \(\succsim\) by
\[
\boldsymbol{x}\succsim\boldsymbol{y}\Leftrightarrow\text{the consumer prefers
}\boldsymbol{x}\text{ at least as much as }\boldsymbol{y}.
\]

We have so far formulated basic properties of consumer preferences through rationality, monotonicity, and related assumptions. We now focus on how consumers evaluate combinations of multiple commodities relative to highly unbalanced consumption bundles. Convexity of preferences is used to represent such a tendency to prefer more balanced consumption.

Given a preference relation \(\succsim\) on a consumption set \(X\subset\mathbb{R} ^{N}\), for any consumption vector \(\boldsymbol{x}\in X\), define its upper contour set by \[
U\left( \boldsymbol{x}\right) =\{\boldsymbol{y}\in X\ |\ \boldsymbol{y}\succsim\boldsymbol{x}\}.
\]If \(U\left( \boldsymbol{x}\right) \) is guaranteed to be a convex subset of \(X\) for every \(\boldsymbol{x}\in X\), then \(\succsim\) is said to satisfy convexity. The statement that \(U\left( \boldsymbol{x}\right) \) is convex means that \[
\forall\boldsymbol{y},\boldsymbol{z}\in U\left( \boldsymbol{x}\right)
,\ \forall\alpha\in\left[ 0,1\right] :\alpha\boldsymbol{y}+\left(
1-\alpha\right) \boldsymbol{z}\in U\left( \boldsymbol{x}\right) .
\]By the definition of \(U\left( \boldsymbol{x}\right) \), this is equivalent to requiring, for any \(\boldsymbol{y},\boldsymbol{z}\in X\), \[
\left( \boldsymbol{y}\succsim\boldsymbol{x}\wedge\boldsymbol{z}\succsim\boldsymbol{x}\right) \Rightarrow\forall\alpha\in\left[ 0,1\right] :\alpha\boldsymbol{y}+\left( 1-\alpha\right) \boldsymbol{z}\succsim
\boldsymbol{x}.
\]Thus, \(\succsim\) satisfies convexity if, whenever a consumption vector \(\boldsymbol{x}\) is chosen and two consumption vectors \(\boldsymbol{y},\boldsymbol{z}\) that are at least as desirable as \(\boldsymbol{x}\) are chosen, every convex combination \(\alpha\boldsymbol{y}+\left( 1-\alpha \right) \boldsymbol{z}\) is also at least as desirable as \(\boldsymbol{x}\).
When a preference relation is rational, convexity can also be expressed as follows.

The proposition above shows that a preference relation satisfies convexity if and only if, whenever \(\boldsymbol{x}\) is at least as desirable as \(\boldsymbol{y}\), every convex combination of \(\boldsymbol{x}\) and \(\boldsymbol{y}\) is also at least as desirable as \(\boldsymbol{y}\).

Strictly Convex Preferences

Convexity of a preference relation requires the upper contour set \(U\left( \boldsymbol{x}\right) \) to be convex for every consumption vector \(\boldsymbol{x}\). A stronger requirement is sometimes imposed: whenever two distinct consumption vectors \(\boldsymbol{y},\boldsymbol{z}\) are chosen from \(U\left( \boldsymbol{x}\right) \), every nontrivial convex combination of them is strictly preferred to \(\boldsymbol{x}\). That is, for every \(\boldsymbol{x}\in X\), \[
\forall\boldsymbol{y}\in U\left( \boldsymbol{x}\right) ,\ \forall
\boldsymbol{z}\in U\left( \boldsymbol{x}\right) \backslash\left\{
\boldsymbol{y}\right\} ,\ \forall\alpha\in\left( 0,1\right) :\alpha
\boldsymbol{y}+\left( 1-\alpha\right) \boldsymbol{z}\in U_{s}\left(
\boldsymbol{x}\right) .
\]Here \(U_{s}\left( \boldsymbol{x}\right) \) is the strict upper contour set of the consumption vector \(\boldsymbol{x}\), defined by \[
U_{s}\left( \boldsymbol{x}\right) =\{\boldsymbol{y}\in X\ |\ \boldsymbol{y}\succ\boldsymbol{x}\}.
\]This assumption is called strict convexity.

From the definitions of upper contour sets and strict upper contour sets,
\(\succsim\) satisfies strict convexity if and only if, for every \(\boldsymbol{x}\in X\), \[
\left( \boldsymbol{y}\succsim\boldsymbol{x}\wedge\boldsymbol{z}\succsim\boldsymbol{x}\wedge\boldsymbol{y}\not =\boldsymbol{z}\right)
\Rightarrow\forall\alpha\in\left( 0,1\right) :\alpha\boldsymbol{y}+\left(
1-\alpha\right) \boldsymbol{z}\succ\boldsymbol{x}.
\]Thus, after choosing any consumption vector \(\boldsymbol{x}\) and any two distinct consumption vectors \(\boldsymbol{y},\boldsymbol{z}\) that are at least as desirable as \(\boldsymbol{x}\), every nontrivial convex combination \(\alpha\boldsymbol{y}+\left( 1-\alpha\right) \boldsymbol{z}\) is guaranteed to be strictly more desirable than \(\boldsymbol{x}\).
When a preference relation is rational, strict convexity can also be expressed as follows.

The proposition above shows that a preference relation satisfies strict convexity if and only if, whenever \(\boldsymbol{x}\) and \(\boldsymbol{y}\) are distinct and \(\boldsymbol{x}\) is at least as desirable as \(\boldsymbol{y}\), every nontrivial convex combination of \(\boldsymbol{x}\) and \(\boldsymbol{y}\) is strictly preferred to \(\boldsymbol{y}\).

Relationship Between Convexity and Strict Convexity

Strict convexity of preferences requires the upper contour set associated with each consumption vector to be strictly convex. Since a strictly convex set is also convex, under strictly convex preferences the upper contour set associated with every consumption vector is convex. Thus, strict convexity implies convexity.

The converse of the preceding proposition need not hold. In other words, preferences satisfying convexity need not satisfy strict convexity (See the exercises).

Shape of Indifference Curves Under Convex Preferences

Suppose that a preference relation \(\succsim\) on the consumption set \(\mathbb{R} _{+}^{2}\) satisfies both rationality and convexity. Choose any consumption vector \(\boldsymbol{x}\). Under rational preferences, the consumption set \(\mathbb{R} _{+}^{2}\) is partitioned into the strict upper contour set \(U_{s}\left( \boldsymbol{x}\right) \), the indifference set \(I\left( \boldsymbol{x}\right) \), and the strict lower contour set \(L_{s}\left( \boldsymbol{x}\right) \). Under convexity, the upper contour set \(U\left( \boldsymbol{x}\right) \), which is the union of \(U_{s}\left( \boldsymbol{x}\right) \) and \(I\left( \boldsymbol{x}\right) \), is convex. Therefore, every point on the line segment joining any two points \(\boldsymbol{y},\boldsymbol{z}\) belonging to \(U_{s}\left( \boldsymbol{x}\right) \) or \(I\left( \boldsymbol{x}\right) \), including the endpoints \(\boldsymbol{y},\boldsymbol{z}\), is also guaranteed to belong to either \(U_{s}\left( \boldsymbol{x}\right) \) or \(I\left( \boldsymbol{x}\right) \).
What changes if strict convexity is assumed instead of convexity? Under strict convexity, the upper contour set \(U\left( \boldsymbol{x}\right) \), which is the union of \(U_{s}\left( \boldsymbol{x}\right) \) and \(I\left( \boldsymbol{x}\right) \), is strictly convex. Therefore, every point on the line segment joining any two points \(\boldsymbol{y},\boldsymbol{z}\) belonging to \(U_{s}\left( \boldsymbol{x}\right) \) or \(I\left( \boldsymbol{x}\right) \), excluding the endpoints, is guaranteed to belong to \(U_{s}\left( \boldsymbol{x}\right) \). No claim is made about the endpoints \(\boldsymbol{y}\) and \(\boldsymbol{z}\).

Given the discussion above, suppose an indifference curve is strictly convex toward the origin, as in the figure above. Do the preferences satisfy convexity or strict convexity? Every point on the line segment joining two points \(\boldsymbol{y},\boldsymbol{z}\) belonging to \(I\left( \boldsymbol{x}\right) \) or \(U_{s}\left( \boldsymbol{x}\right) \), including the endpoints \(\boldsymbol{y},\boldsymbol{z}\), belongs to \(U_{s}\left( \boldsymbol{x}\right) \) or \(I\left( \boldsymbol{x}\right) \). Thus, the figure represents preferences satisfying convexity. By the same reasoning, one can also verify that the preferences satisfy strict convexity.

What if the indifference curve is a straight line, as in the figure above? Every point on the line segment joining two points \(\boldsymbol{y},\boldsymbol{z}\) belonging to \(I\left( \boldsymbol{x}\right) \) or \(U_{s}\left( \boldsymbol{x}\right) \), including the endpoints, belongs to \(U_{s}\left( \boldsymbol{x}\right) \) or \(I\left( \boldsymbol{x}\right) \). Thus, the figure represents preferences satisfying convexity. On the other hand, the preferences do not satisfy strict convexity. Indeed, none of the interior points of the line segment joining \(\boldsymbol{y}\) and \(\boldsymbol{z}\) in the figure belong to \(U_{s}\left( \boldsymbol{x}\right) \).

What if the indifference curve is strictly concave toward the origin, as in the figure above? Some points on the line segment joining \(\boldsymbol{y}\) and \(\boldsymbol{z}\) in the figure belong to neither \(U_{s}\left( \boldsymbol{x}\right) \) nor \(I\left( \boldsymbol{x}\right) \). Therefore, the preferences do not satisfy convexity and hence do not satisfy strict convexity.
To summarize, in a two-good model in which monotonicity or similar assumptions guarantee downward-sloping indifference curves, preferences satisfying convexity have indifference curves that are convex toward the origin. If preferences satisfy strict convexity, the indifference curves are strictly convex toward the origin.

Quasi-Concave Utility Functions

A utility function \(u:X\rightarrow\mathbb{R} \) defined on a consumption set \(X\subset\mathbb{R} ^{N}\) is a multivariable function, so we can examine whether it is quasi-concave. Specifically, \(u\) is quasi-concave if its domain, the consumption set \(X\), is a nonempty convex set and \[
\forall\boldsymbol{x},\boldsymbol{y}\in X,\ \forall\alpha\in\left[
0,1\right] :\min\left\{ u\left( \boldsymbol{x}\right) ,u\left(
\boldsymbol{y}\right) \right\} \leq u\left( \alpha\boldsymbol{x}+\left(
1-\alpha\right) \boldsymbol{y}\right) .
\]The domain \(X\) of a quasi-concave function must be a nonempty convex set. The set \(\mathbb{R} _{+}^{N}\) is a typical example.

Given a utility function \(u:\mathbb{R} ^{N}\supset X\rightarrow\mathbb{R} \) and a real number \(v\in\mathbb{R} \), define the set of consumption vectors yielding utility at least \(v\) by \[
U\left( v\right) =\left\{ \boldsymbol{x}\in X\ |\ u\left( \boldsymbol{x}\right) \geq v\right\} ,
\]and call this the upper contour set of \(u\) at \(v\). A utility function \(u\) is quasi-concave if and only if its domain, the consumption set \(X\), is convex and the upper contour set of \(u\) at every real number \(v\) is convex.

Quasi-concavity can also be characterized using partial derivatives. Suppose that the domain \(X\) of a utility function \(u:\mathbb{R} ^{N}\supset X\rightarrow\mathbb{R} \) is a nonempty convex open set and that \(u\) is of class \(C^{1}\). Then \[
\forall\boldsymbol{x},\boldsymbol{y}\in X:\left[ u\left( \boldsymbol{y}\right) \geq u\left( \boldsymbol{x}\right) \Rightarrow\left(
\boldsymbol{y}-\boldsymbol{x}\right) \cdot\nabla u\left( \boldsymbol{x}\right) \geq0\right] \]is a necessary and sufficient condition for \(u\) to be quasi-concave.

Suppose that the domain \(X\) of a utility function \(u:\mathbb{R} ^{N}\supset X\rightarrow\mathbb{R} \) is a nonempty convex open set and that \(u\) is of class \(C^{2}\). If \(u\) is quasi-concave, then at every point \(\boldsymbol{x}\in X\), all odd-order strict principal minors of the bordered Hessian \(D_{u}\left( \boldsymbol{x}\right) \) are nonpositive and all even-order strict principal minors are nonnegative. Conversely, if at every point \(\boldsymbol{x}\in X\), all odd-order strict principal minors of the bordered Hessian \(D_{u}\left( \boldsymbol{x}\right) \) are negative and all even-order strict principal minors are positive, then \(u\) is quasi-concave. Here \[
D_{u}\left( \boldsymbol{x}\right) =\begin{pmatrix}
0 & u_{x_{1}}^{\prime}\left( \boldsymbol{x}\right) & \cdots & u_{x_{N}}^{\prime}\left( \boldsymbol{x}\right) \\
u_{x_{1}}^{\prime}\left( \boldsymbol{x}\right) & u_{x_{1}x_{1}}^{\prime\prime}\left( \boldsymbol{x}\right) & \cdots & u_{x_{1}x_{N}}^{\prime\prime}\left( \boldsymbol{x}\right) \\
\vdots & \vdots & \ddots & \vdots\\
u_{x_{N}}^{\prime}\left( \boldsymbol{x}\right) & u_{x_{N}x_{1}}^{\prime\prime}\left( \boldsymbol{x}\right) & \cdots & u_{x_{N}x_{N}}^{\prime\prime}\left( \boldsymbol{x}\right)
\end{pmatrix}
=\begin{pmatrix}
0 & \frac{\partial u\left( \boldsymbol{x}\right) }{\partial x_{1}} & \cdots
& \frac{\partial u\left( \boldsymbol{x}\right) }{\partial x_{N}}\\
\frac{\partial u\left( \boldsymbol{x}\right) }{\partial x_{1}} &
\frac{\partial^{2}u\left( \boldsymbol{x}\right) }{\partial x_{1}\partial x_{1}} & \cdots & \frac{\partial^{2}u\left( \boldsymbol{x}\right) }{\partial x_{N}\partial x_{1}}\\
\vdots & \vdots & \ddots & \vdots\\
\frac{\partial u\left( \boldsymbol{x}\right) }{\partial x_{N}} &
\frac{\partial^{2}u\left( \boldsymbol{x}\right) }{\partial x_{1}\partial x_{N}} & \cdots & \frac{\partial^{2}u\left( \boldsymbol{x}\right) }{\partial x_{N}\partial x_{N}}\end{pmatrix}
.
\]

The following relationship holds between convexity of preferences and quasi-concavity of utility functions.

Note that this proposition presupposes the existence of a utility function representing the preference relation.

Since a concave function is quasi-concave, if a utility function is concave, the preference relation represented by that utility function satisfies convexity by the preceding proposition.

Strictly Quasi-Concave Utility Functions

A utility function is a multivariable function whose variables are consumption vectors, so we can examine whether it is strictly quasi-concave. Specifically, a utility function \(u:\mathbb{R} ^{N}\supset X\rightarrow\mathbb{R} \) is strictly quasi-concave if its domain, the consumption set \(X\), is a nonempty convex set and \[
\forall\boldsymbol{x}\in X,\ \forall\boldsymbol{y}\in X\backslash\left\{
\boldsymbol{x}\right\} ,\ \forall\alpha\in\left( 0,1\right) :\min\left\{
u\left( \boldsymbol{x}\right) ,u\left( \boldsymbol{y}\right) \right\}
<u\left( \alpha\boldsymbol{x}+\left( 1-\alpha\right) \boldsymbol{y}\right)
.
\]The domain \(X\) of a strictly quasi-concave utility function must be a nonempty convex set. The set \(\mathbb{R} _{+}^{N}\) is a typical convex set.
The following relationship holds between strict convexity of preferences and strict quasi-concavity of utility functions.

Note that this proposition also presupposes the existence of a utility function representing the preference relation.

Since a strictly concave function is strictly quasi-concave, if a utility function is strictly concave, the preference relation represented by that utility function satisfies strict convexity by the preceding proposition.

Exercises