Choice and Preference

两种决策建模方法

基于选择的方法：将个人的选择行为作为本源特征，直接对行为作出假设，假设决策者的行为满足一致性（显示偏好弱公理）。

1. 在理论上为更一般的个人行为留足了空间。
2. 能对直接观察到的东西作出假设。
3. 说明个人决策理论未必需要建立在自省过程之上，可以全部建立在行为基础之上。

基于偏好的方法：将决策者的爱好 (taste) 作为个人的本源特征，假设决策者在他的可能选择集上有满足某些理性公理的偏好关系，然后分析偏好对他的选择行为的影响结果。

• Utility represents an individual's choices.

  • Individual choices are primitive data that economists can observe.
  • Choices are taken to reveal individual's preferences.
  • Utility is a convenient mathematical construction for modeling choices and preferences.

Choice

不使用抽象概念，根据观察到的行为，考虑是否满足一致性的要求。

Commodity Constraints

任何个人问题的起点都是一个可能的被选物集合 (set of possible alternatives) ，这些备选物是互斥的 (mutually exclusicommoditiesve) ，个人必须从这些备选物中进行选择。

Commodity

Definition (Commodity). We call the goods and services available for purchase in the market commodities.

Assumptions

For simplicity, we assume that the number of commodities is finite and equal to $L$​​ (indexed by $\ell =1,\cdots ,L$​​​​).

• Definition (Commodity Bundle). A commodity vector (or commodity bundle) is a list of amounts of the different commodities

  $$x=\left[\begin{array}{c}{x}_1\\ ⋮\\ x_L\end{array}\right]$$

  and can be viewed as a point in ${\mathbb{R}}^L$, the commodity space.

  • The $\ell$​​th entry of the commodity vector stands for the amount of commodity $\ell$​​ consumed.

Consumption Set

Definition (Consumption Set). The consumption set is a subset of the commodity space ${\mathbb{R}}^L$, denoted by $X\subset {\mathbb{R}}^L$, whose elements are the consumption bundles that the individual can conceivably consume given the physical constraints imposed by his environment.

Assumptions

Basically, we assume that $X={\mathbb{R}}_{+}^L=\{x\in {\mathbb{R}}^L:x_{\ell }\ge 0\text{ for }\ell =1,\cdots ,L\}$​​​​.

One special feature of the set ${\mathbb{R}}_{+}^L$​​ is that it is convex. Much of the theory to be developed applies for general convex consumption sets as well as for ${\mathbb{R}}_{+}^L$.

Definition (Convex Set). If any two consumption bundles $x$​ and $x^{\prime }$​ are both elements of a set $S$​, and the bundle $x^{″}=\alpha x+(1-\alpha )x^{\prime }$​ is also an element of $S$​ for any $\alpha \in [0,1]$​​.

Competitive Budgets

The consumer's choice is limited to those commodity bundles that he can afford.

Assumptions

The Principle of Completeness/Universality of Markets. We suppose that the $L$​​ commodities are all traded in the market at monetary prices that are publicly quoted.

Formally, these prices are represented by the price vector

$$p=\left[\begin{array}{c}{p}_1\\ ⋮\\ p_L\end{array}\right]\in {\mathbb{R}}^L.$$

Assumptions

All Prices Positive Assumption. Assume that $p\gg 0$; that is, $p_{\ell }>0$ for every $\ell$.

Price-taking Assumption. We assume that these prices are beyond the influence of the consumer.

Positive Assumption. To focus on the case in which the consumer has a nondegenerate choice problem, we always assume $w>0$.

The affordability of a consumption bundle depends on two things:

1. the market prices $p^T=(p_1,\cdots ,p_L)$, and
2. the consumer's wealth level (in monetary) $w$​​​.

The consumption bundle is affordable if

$$p^T\cdot x=p_1{x}_1+\cdots +p_L{x}_L\le w.$$

Definition (Walrasian/Competitive Budget Set). The Walrasian/competitive budget set $B_{p,w}=\{x\in {\mathbb{R}}_{+}^L:p^T\cdot x\le w\}$​ is the set of all feasible consumption bundles for the consumer who faces market prices $p$​ and has wealth $w$​.

• Definition(Budget Hyperplane). The set $\{x\in {\mathbb{R}}_{+}^L:p\cdot x=w\}$​​ is called budget hyperplane.

Proposition (Convex Budget Set). If the consumption set X is a convex set, the Warlrasian budget set $B_{p,w}$​​​​ is convex too.

Choice Structure

Definition (Choice Structure). Formally, choice behavior is represented by means of a choice structure. A choice structure $(\mathcal{B},C(\cdot ))$ consists of two ingredients:

1. $\mathcal{B}$​ is a family (a set) of nonempty subsets of $X$​. That is, every element of $\mathcal{B}$​ is a set $B\subset X$​​. The budget sets in $\mathcal{B}$ should be thought of as an exhaustive listing of all the choice experiments that the institutionally, physically, or otherwise restricted social situation can conceivably pose to the decision maker.
2. $C(\cdot )$ is a choice rule that assigns a nonempty set of chosen elements $C(B)\subset B$​ for every budget set $B\in \mathcal{B}$​. The element/elements of $C(B)$​ is/are the alternative/alternatives that the decision maker will/might choose from $B$.

Assumptions

Definition (Weak Axiom of Revealed Preference). The choice structure $(\mathcal{B},C(\cdot ))$​​ satisfies the weak axiom of revealed preference if the following property holds:

• If for some $B\in \mathcal{B}$​ with $x,y\in B$​ we have $x\in C(B)$​, then for any $B^{\prime }\in \mathcal{B}$​ with $x,y\in B^{\prime }$​ and $y\in C(B^{\prime })$​, we must also have $x\in C(B^{\prime })$​​​.

A simpler statement of the weak axiom can be obtained by defining a revealed preference relation from the observed choice behavior in $C(\cdot )$​.

Definition (Revealed Preference Relation). Given a choice structure $(\mathcal{B},C(\cdot ))$​​​ the revealed preference relation ${\succsim }^{\ast }$​​​ is defined by

• $x{\succsim }^{\ast }y\iff \text{ there is some }B\in \mathcal{B}\text{ such that }x,y\in B\text{ and }x\in C(B)$​.​​​

We read $x{\succsim }^{\ast }y$​ as "$x$​ is revealed at least as good as $y$​."

Note that the revealed preference relation need not be either complete or transitive.

Definition (Weak Axiom of Revealed Preference). If $x$​​​​ is revealed at least as good as $y$​​, then $y$​​ cannot be revealed preferred to $x$​​​​​.

Preference

Preference relate observable choice data to preferences over X.

Preference Relation

Definition (Preference Relation). $\succsim$​​​ 是在备选物集$X$​​​上的二元关系，任何一对备选物$x,y\in X$​​​都可以进行比较，我们将$\succsim$​​​读为“$x$​​​至少和$y$​​​一样好”。我们可从$\succsim$​​​​​推出另外两种关系：

1. 严格偏好关系 $\succ$: $x\succ y\iff x\succ y\text{ but not }y\succ x$​，读为“$x$比$y$好”。
2. 无差异关系$\sim$: $x\sim y\iff x\succsim y\text{ and }y\succsim x$​​, 读为“ $x$ 与 $y$ 无差异”。

在大部分经济理论中，都假设个人偏好是理性的。

Assumptions

Definition (Rational Preference Relation). 若偏好关系$\succsim$​具有下列两个性质，则它是理性的：

1. 完备性 (completeness). 对于所有 $x,y\in X$​​​，都有 $x\succsim y$​​​ 或 $y\succsim x$​​​（或两者都成立）。
2. 传递性 (transitivity). 对于所有 $x,y,z\in X$​​​，若 $x\succsim y$​​​ 且 $y\succsim z$​​​，则 $x\succsim z$​​​。

Proposition 1.B.1. 如果$\succsim$​是理性的，则：

1. $\succ$为非反身的 (irreflexive) 和传递的。
2. $\sim$为反身的、传递的和对称的 (symmetric)。
3. 若$x\succ y\succsim z$，则$x\succ z$​。

Relationship: Preference & Choice

Proposition 1.D.1. Suppose that $\succsim$​ is a rational preference relation. Then the choice structure generated by $\succsim ,(\mathcal{B},C^{\ast }(\cdot ,\succsim ))$​, satisfies the weak axiom.

Definition (Rationalize). Given a choice structure $(\mathcal{B},C(\cdot ))$​, we say that the rational preference relation $\succsim$​ rationalizes $C(\cdot )$​ relative to $\mathcal{B}$​ if

$$C(B)=C^{\ast }(B,\succsim )$$

for all $B\in \mathcal{B}$​, that is, if $\succsim$​ generates the choice structure $(\mathcal{B},C(\cdot ))$​.

• An alternative notion of a rationalizing preference that appears in the literature requires only that $C(B)\subset C^{\ast }(B,\succsim )$ for every budget $B\in \mathcal{B}$​.

Proposition 1.D.2: If $(\mathcal{B},C(\cdot ))$​ is a choice structure such that

1. the weak axiom is satisfied,
2. $\mathcal{B}$ includes all subsets of $X$ of up to three elements,

then there is a rational preference relation $\succsim$​ that rationalizes $C(\cdot )$​ relative to $\mathcal{B}$​​. Furthermore, this rational preference relation is the only preference relation that does so.

Preference Relations: Basic Properties

Rationality

Assumptions

Rational. Preference Relation is complete and transitive.

Desirability

Desirability assumptions. Larger amounts of commodities are preferred to smaller ones.

Assumptions

Definition (monotonicity). The preference relation $\succsim$ on $X$ is monotone if $x\in X$ and $y\gg x$ implies $y\succ x$. It is strongly monotone if $y\ge x$ and $y\ne x$ imply that $y\succ x$.

Under the assumption of monotonicity, a consumer will always choose a bundle on the boundary of the budget set.

For much of the theory, a weaker desirability assumption than monotonicity, known as local nonsatiation, actually suffices.

Assumptions

Definition (Local Nonsatiation). The preference relation $\succsim$ on $X$ is locally nonsatiated if for every $x\in X$ and every $\epsilon >0$, there is $y\in X$ such that $\parallel y-x\parallel \le \epsilon$ and $y\succ x$​.

The assumption of monotonicity or local nonsatiation ensures that $I_x=\{y\in X|y\sim x\}$​ are thin.

Given $\succsim$ and a consumption bundle $x$, we can define three sets of consumption bundles.

1. The indifference set: $\{y\in X:y\sim x\}$
2. The upper contour set: $U(x)=\{y\in X:y\succsim x\}$​​
3. The lower contour set: $\{y\in X:x\succsim y\}$​

Convexity

Convexity/Diversity assumptions. Concerns the trade-offs that the consumer is willing to make among different goods.

Assumptions

Definition (Convex Preference Relation). The preference relation $\succsim$​ on $X$​ is convex if for every $x\in X$​, the upper contour set $\{y\in X:y\succsim x\}$​ is convex; that is, if $y\succsim x$​ and $z\succsim x$​, then $\alpha y+(1-\alpha )z\succsim x$​ for any $\alpha \in [0,1]$​.

Definition (Strictly Convex Preference Relation): The preference relation $\succsim$ on $X$ is strictly convex if for every $x$, we have that $y\succsim x$, $z\succsim x$, and $y\ne z$ implies $\alpha y+(1-\alpha )z\succ x$ for all $\alpha \in (0,1)$.

Convexity is a strong but central hypothesis in economics.

• Choice: It can be interpreted in terms of diminishing marginal rates of substitution: That is, with convex preferences, from any initial consumption situation $x$​, and for any two commodities, it takes increasingly larger amounts of one commodity to compensate for successive unit losses of the other.
• Preference: Convexity can also be viewed as the formal expression of a basic inclination of economic agents for diversification. A taste for diversification is a realistic trait of economic life. Economic theory would be in serious difficulty if this postulated propensity for diversification did not have significant descriptive content.
• Choice situations violated: you may like both milk and orange juice but get less pleasure from a mixture of the two.

The convexity assumption can hold only if X is convex.

1. First, a good number (although not all) of the results of this chapter extend without modification to the nonconvex case.
2. Second, as we show in Appendix A of Chapter 4 and in Section 17.1, nonconvexities can often be incorporated into the theory by exploiting regularizing aggregation effects across consumers.

Deducible

Deducible: In applications (particularly those of an econometric nature), it is common to focus on preferences for which it is possible to deduce the consumer’s entire preference relation from a single indifference set. Two examples are the classes of homothetic and quasilinear preferences.

Assumptions

Definition (homothetic preference/位似偏好). A monotone preference relation $\succsim$​ on $X={\mathbb{R}}_{+}^L$​​ is homothetic if all indifference sets are related by proportional expansion along rays; that is, if $x\sim y$​, then $\alpha x\sim \alpha y$​ for any $\alpha \ge 0$​​​.

Definition (quasilinear preference/拟线性偏好). The preference relation $\succsim$​​ on $X=(-\mathrm{\infty },\mathrm{\infty })\times {\mathbb{R}}_{+}^{L-1}$​​ is quasilinear with respect to (w.r.t.) commodity 1 (called, in this case, the numeraire commodity) if

1. All the indifference sets are parallel displacements of each other along the axis of commodity 1. That is, if $x\sim y$, then $(x+\alpha e_1\sim (y+\alpha e_1)$ for $e_1=(1,0,\cdots ,0)$ and any $\alpha \in \mathbb{R}$.
2. Good 1 is desirable; that is, $x+\alpha e_1\succ x$ for all $x$ and $\alpha >0$​.

Even if we know many things on preferences, it is not very useful to analyze consumer behavior.

• The trick is to find a way to use mathematical formulas consistent with preferences : the tool we use is a utility function.
• So by studying a utility function and its properties, one can analyze the consumer's preferences.