Topic 1: Preference and Utility

1. Preference relations

• Preference, At least as good as: $\succsim$$\succsim$

• Strict preference: $\succ$$\succ$ : $x\succsim y,y\succsim ̸x$$x \succsim y, y \not \succsim x$.

• Rational: The preference relation $\succsim$$\succsim$ is rational if it possesses the following two properties:

  • Completeness: for all $x,y\in \mathrm{X}$$x, y \in \mathrm{X}$, we have that $x\succsim y$$x \succsim y$ or $y\succsim x$$y \succsim x$ (or both).

  • Transitivity: For all $x,y,z\in X$$x, y, z \in X$, if $x\succsim y$$x \succsim y$ and $y\succsim z$$y \succsim z$, then $x\succsim z$$x \succsim z$.

• If $\succsim$$\succsim$ is rational, then

  • $\succ$$\succ$ is both irreflexive ( $x\succ x$$x\succ x$ never holds), and transitive ($x\succ y,y\succ z\Rightarrow x\succ z$$x \succ y, y\succ z \Rightarrow x \succ z$)

  • $\sim$$\sim$ is reflexive, and transitive, and symmetric

  • if $x\succ y\succsim z$$x \succ y \succsim z$, then $x\succsim z$$x \succsim z$.

Additional assumptions on preference.

• Continuity : For every converging sequence of pairs of consumption bundles ${\left\{({\mathbf{x}}^n;{\mathbf{y}}^n)\right\}}_{n=1}^{+\mathrm{\infty }}$$\left\{\left(\mathbf{x}^n ; \mathbf{y}^n\right)\right\}_{n=1}^{+\infty}$ and $\underset{n\to \mathrm{\infty }}{lim}{\mathbf{x}}^n=\mathbf{x},\underset{n\to \mathrm{\infty }}{lim}{\mathbf{y}}^n=\mathbf{y}$$\lim _{n \rightarrow \infty} \mathbf{x}^n=\mathbf{x}, \lim _{n \rightarrow \infty} \mathbf{y}^n=\mathbf{y}$. If ${\mathbf{x}}^n\succsim {\mathbf{y}}^n$$\mathbf{x}^n \succsim \mathbf{y}^n$ for $\mathrm{\forall }n$$\forall n$, then $\mathbf{x}\succsim \mathbf{y}$$\mathbf{x} \succsim \mathbf{y}$.

  • A violation: Lexicographic preference（字典preference）

• Local Non-satiation: For $\mathrm{\forall }\mathbf{x}\in {\mathbb{R}}_{\mathbb{+}}^{\mathbb{L}}$$\forall \mathbf{x} \in \mathbb{R^L_+}$, and $\mathrm{\forall }ϵ>0$$\forall \epsilon >0$, $\mathrm{\exists }\mathbf{x}$$\exist \mathbf{x}$ such that $d({\mathbf{x}}^{\mathbf{0}},\mathbf{x})\le ϵ$$d(\mathbf{x^0},\mathbf{x})\leq \epsilon$, $\mathbf{x}\succ {\mathbf{x}}^{\mathbf{0}}$$\mathbf{x} \succ \mathbf{x^0}$.

• Monotone:

  • (MWG) The preference relation $\succsim$$\succsim$ is monotone if ${\mathbf{x}}^1\gg {\mathbf{x}}^2$$\mathbf{x}^1 \gg \mathbf{x}^2$ implies ${\mathbf{x}}^1\succ {\mathbf{x}}^2$$\mathbf{x}^1 \succ \mathbf{x}^2$. It is strongly monotone if ${\mathbf{x}}^1>{\mathbf{x}}^2$$\mathbf{x}^1>\mathbf{x}^2$ implies that ${\mathbf{x}}^1\succ {\mathbf{x}}^2$$\mathbf{x}^1 \succ \mathbf{x}^2$.

  • (JR) The preference relation $\succsim$$\succsim$ is strictly monotone if ${\mathbf{x}}^1\ge {\mathbf{x}}^2$$\mathbf{x}^1 \geq \mathbf{x}^2$ implies ${\mathbf{x}}^1\succsim {\mathbf{x}}^2$$\mathbf{x}^1 \succsim \mathbf{x}^2$ AND ${\mathbf{x}}^1\gg {\mathbf{x}}^2$$\mathbf{x}^1 \gg \mathbf{x}^2$ implies ${\mathbf{x}}^1\succ {\mathbf{x}}^2$$\mathbf{x}^1 \succ \mathbf{x}^2$. Votation

  • ${\mathbf{x}}^1\ge {\mathbf{x}}^2$$\mathbf{x}^1 \geq \mathbf{x}^2$ if and only if $\mathrm{\forall }i,x_i^1\ge x_i^2$$\forall i, x_i^1 \geq x_i^2$.

  • ${\mathbf{x}}^1>{\mathbf{x}}^2$$\mathbf{x}^1>\mathbf{x}^2$ if and only if ${\mathbf{x}}^1\ge {\mathbf{x}}^2$$\mathbf{x}^1 \geq \mathbf{x}^2$ and ${\mathbf{x}}^1\ne {\mathbf{x}}^2$$\mathbf{x}^1 \neq \mathbf{x}^2$

  • ${\mathbf{x}}^1\gg {\mathbf{x}}^2$$\mathbf{x}^1 \gg \mathbf{x}^2$ if and only if $\mathrm{\forall }i,x_i^1>x_i^2$$\forall i, x_i^1>x_i^2$

• Convexity: for $\mathrm{\forall }\mathbf{x}\in \mathbf{X}$$\forall \mathbf{x} \in \mathbf{X}$, the upper level set is convex

  • Strict convexity: ${\mathbf{x}}^2\succsim {\mathbf{x}}^1$$\mathbf{x}^2 \succsim \mathbf{x}^1$ and ${\mathbf{x}}^3\succsim {\mathbf{x}}^1$$\mathbf{x}^3 \succsim \mathbf{x}^1$, then $t{\mathbf{x}}^2+(1-t){\mathbf{x}}^3\succ {\mathbf{x}}^1$$t \mathbf{x}^2+(1-t) \mathbf{x}^3 \succ \mathbf{x}^1$ for $\mathrm{\forall }t\in (0,1)$$\forall t \in(0,1)$

  • $\Rightarrow {\mathbf{x}}^2\sim {\mathbf{x}}^1$$\Rightarrow \mathbf{x}^2 \sim \mathbf{x}^1$ and ${\mathbf{x}}^3\sim {\mathbf{x}}^1$$\mathbf{x}^3 \sim \mathbf{x}^1$ then $t{\mathbf{x}}^2+(1-t){\mathbf{x}}^3\succ {\mathbf{x}}^1$$t \mathbf{x}^2+(1-t) \mathbf{x}^3 \succ \mathbf{x}^1$ indifference

  • Indifference curves demonstrate diminishing marginal rate of substitution

  • preference is convex, but utility function is quasi-concave.

2. Utility function

Binary preference relation $\Rightarrow$$\Rightarrow$ utility function

Representation theorem:

• Function $U:\mathbf{X}\to \mathbb{R}$$U: \mathbf{X} \rightarrow \R$ represents preference relation $\succsim$$\succsim$ if for $\mathrm{\forall }({\mathbf{x}}^1,{\mathbf{x}}^2)\in \mathbf{X}\times \mathbf{X},U\left({\mathbf{x}}^1\right)\ge U\left({\mathbf{x}}^2\right)\iff {\mathbf{x}}^1\succsim {\mathbf{x}}^2$$\forall\left(\mathbf{x}^1, \mathbf{x}^2\right) \in \mathbf{X} \times \mathbf{X}, U\left(\mathbf{x}^1\right) \geq U\left(\mathbf{x}^2\right) \Leftrightarrow \mathbf{x}^1 \succsim \mathbf{x}^2$

Rational & continuous preference $\iff$$\Leftrightarrow$ continuous utility function

• " $\Leftarrow$$\Leftarrow$ " is obvious. If a preference $\succsim$$\succsim$ could be represented by a utility function, then it has to be rational and continuous.

• “ $\Rightarrow$$\Rightarrow$ " we also need continuous for this property. If we have continuous, complete and transitive preference, then we are able to obtain a continuous utility function.

• Proof

We want to show that $\mathrm{\exists }U()$$\exist U()$ such that $U(x)\ge U(x^{\prime })$$U(x)\geq U(x^\prime)$ if and only if $x\succsim x^{\prime }$$x \succsim x^\prime$.

Let $U_{x}e\sim x$$U_x e \sim x$, where $U_x\in \mathbb{R}$$U_x\in \R$, and

$$\begin{array}{}\text{(1)}& \begin{array}{c}e={\left[\begin{array}{c}1\\ 1\\ ...\\ 1\end{array}\right]}_{L\times 1}\in {\mathbb{R}}^L\end{array}\end{array}$$

If for $\mathrm{\forall }x$$\forall \ x$, we have $U_x$$U_x$ exists and is unique, then $U(x)=U_x$$U(x) = U_x$ is the utility function that we are looking for.

Check for existence:

Define $A=\{t\ge 0\mid te\succsim x\}$$A = \left\{t\geq 0\mid te\succsim x \right\}$ and $B=\{t\le 0\mid te\precsim x\}$$B = \left\{t \leq 0 \mid te \precsim x \right\}$, and $A$$A$ and $B$$B$ are both closed.

By completeness, $A\cap B\ne \varnothing$$A\cap B \neq \varnothing$, there exist $t$$t$ such that $U(x)=t$$U(x) = t$.

Suppose $A\cap B=\varnothing$$A\cap B = \varnothing$, then it implies that two preference cannot be simultaneously $x\succsim y$$x\succsim y$ and $x\precsim y$$x\precsim y$, which violates completeness of preference. Therefore, we know that $A\cap B\ne \varnothing$$A\cap B\neq \varnothing$.

Check for uniqueness:

We can denote $A=[\underset{―}{t},\mathrm{\infty }]$$A=[\underline{t}, \infty]$, and $B=[0,\overline{t}]$$B = \left[0,\bar t\right]$.

if $\underset{―}{t}\ne \overline{t}$$\underline{t} \neq \bar{t}$, then, there must be some $t_1$$t_1$ and $t_2$$t_2$ , say $t_1<t_2$$t_1<t_2$, and $t_1,t_2\in [\underset{―}{t},\overline{t}]$$t_1, t_2 \in[\underline{t}, \bar{t}]$ such that

$$\begin{array}{}\text{(2)}& t_{1}e\sim x\sim t_{2}e\end{array}$$

, which is impossible!

Therefore, $t$$t$ must be unique.

• Example - Lexicographic preference

  This preference is rational, but not continuous. Hence, we cannot express this preference with a utility function.

  Let $x^n=(1+\frac{1}{n},1)\to x=(1,1)$$x^n=\left(1+\frac{1}{n}, 1\right) \rightarrow x=(1,1)$, and $y^n=(1,2)\to y=(1,2)$$y^n=(1,2) \rightarrow y=(1,2)$

  $x^n\succsim y^n\mathrm{\forall }n$$x^n \succsim y^n \quad \forall n$.

  but $y>x$$y>x$.

Convexity of Utility/Preference/Indifference curve:

The following statements are equivalent:

• Utility function is quasi-concave

• Preference is convex

• Indifference curve is convex

• The upper contour set is convex

3. Summary

For every rational (complete and transitive), continuous, strongly monotonic and (strictly) convex preference, we can find a continuous, strictly increasing and (strictly) quasi-concave utility function to represent it.

Utility functions are ordinal, and positive monotonic transform of a utility function can represent the same preference.