Utility

Know how infer preferences from choice, next is representing utility with a utility function.

Utility Existence

A function is a utility function representing preference relation, it is not unique.

Definition (Utility Function). A function $u:X\to \mathbb{R}$ is a utility function representing preference relation $\succsim$ if, for all $x,y\in X$,

$$x\succsim y\iff u(x)\ge u(y).$$

• Property: ordinal property.
  • For any strictly increasing function $f:\mathbb{R}\to \mathbb{R},u(x)=f(u(x))$ is a newly utility function representing the same preferences as u.

Proposition 1.B.2. A preference relation $\succsim$​​ can be represented by a utility function only if it is rational.

Question: Can any rational preference relation be described by utility function?

No.

• One case in which we can always represent a rational preference relation with a utility function arises when $X$ is finite.

• With infinitely many alternatives, some problems emerge.

  • Example: The Lexicographic Preference Relation.
    • Definition (Lexicographic Preference Relation): For simplicity, assume that $X={\mathbb{R}}_{+}^2$​​. Define $x\succ y$​​ if either “$x_1>y_1$​​” or “$x_1=y_1$​​ and $x_2>y_2$​​.” This is known as the lexicographic preference relation.

The assumption that is needed to ensure the existence of a utility function is that the preference relation be continuous.

Utility properties

Continuity

Assumptions

Definition (Continuous). The preference relation $\succsim$​ on $X$​ is continuous if it is preserved under limits. That is, for any sequence of pairs $\{(x^n,y^n){\}}_{n=1}^{\mathrm{\infty }}$​ with $x^n\succsim y^n$​ for all $n,x=\underset{n\to \mathrm{\infty }}{lim}{x}^n$​, and $y=\underset{n\to \mathrm{\infty }}{lim}{y}^n$​, we have $x\succsim y$​​.

The assumption of continuity says that preferences cannot exhibit a sudden reversal (or jumps).

Proposition 3.C.1: Suppose that the rational preference relation $\succsim$​​ on $X$​​ is continuous. Then there is a continuous utility function $u(x)$​​ that represents $\succsim$​​​​​.

Differentiable

Assumptions

For analytical purposes, it is also convenient if $u(\cdot )$​​ can be assumed to be differentiable.

Whenever convenient in the discussion that follows, we nevertheless assume utility functions to be twice continuously differentiable.

Intuitively, what is required is that indifference sets be smooth surfaces that fit together nicely so that the rates at which commodities substitute for each other depend differentiably on the consumption levels.

Transition Properties

Restrictions on preferences translate into restrictions on the form of utility functions.

The property of monotonicity, for example, implies that the utility function is increasing: $u(x)>u(y)$ if $x\gg y$​.

The property of convexity of preferences, on the other hand, implies that $u(\cdot )$​ is quasiconcave [and, similarly, strict convexity of preferences implies strict quasiconcavity of $u(\cdot )$].

• Definition (quasiconcave): The utility function $u(\cdot )$ is quasiconcave if the set $\{y\in {\mathbb{R}}_{+}^L:u(y)\ge u(x)\}$ is convex for all $x$ or, equivalently, if $u(\alpha x+(1-\alpha )y)\ge min\{u(x),u(y)\}$ for any $x$, $y$ and all $\alpha \in [0,1]$. If the inequality is strict for all $x\ne y$ and $\alpha \in (0,1)$, then $u(\cdot )$ is strictly quasiconcave;

Deducible.

1. A continuous $\succsim$ on $X={\mathbb{R}}_{+}^L$​ is homothetic if and only if it admits a utility function $u(x)$​ that is homogeneous of degree one [i.e., such that $u(\alpha x)=\alpha u(x)$ for all $\alpha >0$​].
2. A continuous $\succsim$​ on $(-\mathrm{\infty },\mathrm{\infty })\times {\mathbb{R}}_{+}^L$​​ is quasilinear with respect to the first commodity if and only if it admits a utility function $u(x)$​​ of the form $u(x)=x_1+\varphi (x_2,\cdots ,x_L)$.