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tags: micro
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# Preference
## Relation
Given two sets $X, Y$, we can define their product $X \times Y$. If $S$ is a non-empty subset of $X \times Y$, we say $S$ is a relation from $X$ to $Y$. If $S$ is a non-empty subset of $X \times X$, we say $S$ is a relation on $X$.

If $x, y$ are two element in $X$ and $(x, y)$ is in $S$. We can use notation $x \succeq y$ to denote this realtion and use $\succeq$ to refer the realitons based $S$.

### Partial Order
If $X$ is a non-empty set and $\succeq$ is a realtion on $X \times X$. We call $\succeq$ a partial order on $X$ if it satisfied following three criteria:

1. For all $x \in X, x \succeq x$.
2. If $x,y$ in $X$, and both $x \succeq y$, $y \succeq x$, then $x =y$.
3. If $x,y,z$ in $X$, and both $x \succeq y$ $y \succeq z$, then $x \succeq z$.

Sometimes, we call $(X,\succeq)$ as *poset*.

### Total Order
If $X$ is a non-empty set and $\succeq$ is a realtion on $X \times X$. We call $\succeq$ a total order on $X$ if it satisfied following three criteria:

1. For all $x,y \in X$, we have $x \succeq y$ or $x \succeq y$.
2. If $x,y$ in $X$, and both $x \succeq y$, $y \succeq x$, then $x =y$.
3. If $x,y,z$ in $X$, and both $x \succeq y$, $y \succeq z$, then $x \succeq z$.

## Preference Relation
Let $X$ be a non-empty set that consumer might choose, we can define a relation $\succeq$ on $X \times X$ to represent consumer's preference.

We use the notation $x \succ y$ to represent the situation $x \succeq y$ and $y\nsucceq x$; we define $x \sim y$ if $x \succeq y$ and $y\succeq x$.

We usually assume two common properties to the preference. Some scholars even called the preference satisfied following two properties, completeness and transitivity, as rational preference.

### Completeness

The preference relation $\succeq$ is complete if for any $x$ and $y$ in $X$, we have $x \succeq y$, $y \succeq x$, or both.

It means every two items are comparable for the consumer.

### Transitivity

The preference relation $\succeq$ is transitive if for any $x$, $y$, and $z$ in $X$ and $x \succeq y, y \succeq z$, then we have $x \succeq z$.

### Continuity

*Definition 1*
The preference relation $\succeq$ is continuous if for any $x$ in $X$, both the upper contour set $U(x)=\{y \in X| y \succeq x\}$ and lower contour set $L(x)=\{z \in X| x \succeq z\}$ are closed.

*Definition 2*
The preference relation $\succeq$ is continuous if for two sequence $\{x_k\}$ and $\{y_k\}$ in $X$ such that $x_i \succeq y_i$ for all $i$, $\lim_{k \to \infty} x_k = x$ and $\lim_{k \to \infty} y_k = y$ , then $x \succeq y$.

*Note*: Above two definitions are equivalent. This property ensure a smooth preference.

### Convexity
The preference relation $\succeq$ is convex if for any $x \in X$, the upper contour set $U(x)=\{y \in X| y \succeq x\}$ is convex.

The preference relation $\succeq$ is strictly convex if for any $x, y , z$ in $X$ such that $y \succeq x$ $z \succeq x$ and $y \neq z$, we have $ay + (1-a)z \succ x$ for any $a \in (0,1)$.

*Note*: Convexity implies that "average in better than extreme".

### Consumption Set

We usually assume there are $n$ commoties in the world, and the consumer can only condsider non-negative amount of commoties. Hence, we define the consumption set as

$$X = \mathbb{R}^n_+ = \{x \in \mathbb{R}^n | x_i \ge 0, i=1,...,n\}$$

We can define more properties on the preference realtion $\succeq$ on $\mathbb{R}^n_+$.

### Local nonsatiation
The preference relation $\succeq$ is locally nonsatiated if for any $x$ in $\mathbb{R}_+^n$ and every $\varepsilon > 0$, there is $y$ in $\mathbb{R}_+^n$ such that $\|x-y\|$, the distance between $x$ and $y$, are less than $\varepsilon$ and $y \succ x$.

### Monotonic
The preference relation $\succeq$ is monotone if $x$ and $y$ in $\mathbb{R}_+^n$ and $x \gg y$ implies $x \succ y$.

The preference relation $\succeq$ is strongly monotone if $x \ge y$ and $x \neq y$ imply $x \succ y$.

*Note*: Monotone just means "more is better".

### Homothetic
The monotone preference relation $\succeq$ is homothetic if $x$ and $y$ in $\mathbb{R}_+^n$ and $x \sim y$ implies $ax \sim ay$ for $a > 0$.

### Quasilinearity
The preference relation $\succeq$ is quasilinear if for one sepcific commodity (WLOG, we refere to commodity 1) is numeraire commodity such that

1. If $x \sim y$, then $(x + a e_1) \sim (y + a e_1)$, where $e_1 =(1,0,...,0)$ and $a>0$.
2. $(x + a e_1) \succ x$ for all $x$ and $a>0$.

## Utility Representation

A function $u: X \to \mathbb{R}$ is a utility representation of a preference relation $\succeq$ if for all $x$ and $y$ in $X$, we have

$$y(x) > u(y) \iff x \succ y$$

### Monotonic Transformation

We usually refer incresing monotonic transformation as monotonic transformation, which is an operation under a strictly increasing monotone function $h: \mathbb{R} \to \mathbb{R}$ such that

$$h(x) > h(y) \iff x > y$$

One immediately property of utility representation following the definition is that the utility representation of a preference relation $\succeq$ is still the utility representation of the same preference relation after monotonic transformation,

$$u(x) > u(y) \iff x \succ y \\
h(u(x)) > h(u(y)) \iff x \succ y$$

Hence, the utility representation of a preference is not unique. We also called this property as **ordinal**.

### Existence of Utility Representation
**Debreu's theorem:** If $\succeq$ is a complete, transitive, and continuous preference relation on a topologic space, there is a real-valued continuous function that represents the preference relation.

There is a proof restricted on the convex subset of $\mathbb{R}^n$ in Ariel Rubinstein's [Microeconomic Theory](https://arielrubinstein.org/gt/arielDocs/) (P.18). You can also find the math detail in Walter Rudin's [Principles of Mathematical Analysis ](https://www.amazon.com/gp/product/B07BGTTQJB/ref=dbs_a_def_rwt_hsch_vapi_tkin_p1_i2) (Chapter 2).

**Easy version:** If $\succeq$ is a complete, transitive, continuous, and monotonic preference relation on $\mathbb{R}^n_+$, there is a real-valued continuous function that represents the preference relation.

**proof** We can use the set $E=\{(x_1, ...,x_n) \in \mathbb{R}^n_+ | x_1 =x_2=...=x_n \}$ to construct utility representation of $\succeq$. That is, for any $y \in \mathbb{R}^n$

$$u(y)=k \text{ if } (k,...,k) \sim y$$

We need to show that the function is well defined and continuous.

First, we need to show that for any $x \in \mathbb{R}^n_+$, there is at least one element $z$ in $E$ such that $x \sim z$.

1. The continuity of $\succeq$ ensures the upper contour set $U(x)=\{y \in \mathbb{R}_+^n| y \succeq x\}$ and lower contour set $L(x)=\{z \in \mathbb{R}_+^n| x \succeq z\}$ are closed.

3. The compeleness of $\succeq$ ensures that for any element in $E$, it must be in $U(x)$ or $L(x)$ (or both).

3. Suppose that there is not an element in $E$ is equivalent to $x$, then all elements in $E$ are not both in $U(x)$ and $L(x)$.

4. Because $\succeq$ is monotonic, any $x \in \mathbb{R}^n_+$ are bounded between $0$ and some large vector $m \in E$, i.e. $m \succ x \succ 0$ if $x \neq 0$. Hence, both $E \cap U(x)$ and $E \cap L(x)$ are non-empty.

4. Because $E$ is closed, $E \cap U(x)$ and $E \cap L(x)$ are also closed. If all elements in $E$ are not both in $U(x)$ and $L(x)$, then we can seperate the closed set $E$ into two disjoint non-empty closed set, which is impossible.

5. Hence, there is at least one element in $E$ that is both in $U(x)$ and $L(x)$. In other words, that elment is equivalent to $x$.

Second, we need to show that we could not assign two values to the same $y \in \mathbb{R}^n_+$, so $u(\cdot)$ is a function. Since $\succeq$ is monotonic, two different elements in $E$ are not indifferent, i.e.
$$x,y \in E, x \neq y \implies x \succ y \text{ or } y \succ x$$

Third, we need to show that $u$ is continuous. That is, for any $x$ in $\mathbb{R}_+^n$ and $u(x)=k$, given any $\varepsilon >0$, we can find a $\delta >0$ such that $|u(y)-u(x)| < \varepsilon$ if $\|x-y\| < \delta$. Here, $\|\|$ is the Euclidean distance in $\mathbb{R}^n$ space.

We define strict upper contour set $SU(x)=\{y \in \mathbb{R}_+^n| y \succ x\}$ and strict lower contour set $SL(x)=\{z \in \mathbb{R}_+^n| x \succ z\}$. Since $SU(x)$ is the complement of $L(x)$ and $SL(x)$ is the complement of $U(x)$, $SU(x)$ and $SL(x)$ are open sets.

Given $x$ in $\mathbb{R}_+^n$, $u(x)=k$ and $\varepsilon >0$, we choose two points $a=(k+\varepsilon,...,k+\varepsilon)$ and $b=(k-\varepsilon,...,k-\varepsilon)$ on $E$. Based on the construction of $u(\cdot)$, $u(a) = k + \varepsilon$ and $u(b) = k - \varepsilon$. Since $SU(b)$ and $SL(a)$ are open, $S = SU(b) \cap SL(a)$ is also open. By definition, $x$ is in $S$, and there is an open ball $B_{\delta}(x)$ of $x$ is contained in $S$. Hence, any point $y$ in the $B_{\delta}(x)$ will have $|u(y)-u(x)| < \varepsilon$.

### Continuous but not Rational Preference

Here we provide an example of continuous but not rational preference.


For any $(x_1,x_2)$ and $(y_1,y_2)$ in the $\mathbb{R}_+^2$, we define $\succeq$ as

$$(x_1,x_2) \succeq (y_1,y_2) \text{ iff } x_1 \ge x_2 \text{ and } 1\le x_2 \le 2, 1\le y_2 \le 2$$

Hence, if $x_2 \notin [1,2]$ or $y_2 \notin [1,2]$, $(x_1,x_2)$ and $(y_1,y_2)$ are not comparable.

Clearly, $\succeq$ is not complete. We need to show $\succeq$ is continuous.

For any $(z_1,z_2)$ in the $\mathbb{R}_+^2$, if $z_2 \notin [1,2]$, then its upper contour set and lower contour set are both empty and closed.

If $z_2 \in [1,2]$, then its upper contour set is $\{(x_1,x_2)| x_1 \ge z_1, 1\le x_2 \le 2 \}$ and its lower contour set is $\{(x_1,x_2)| x_1 \le z_1, 1\le x_2 \le 2 \}$. Both sets are closed.