Choice-based Demand

现在，假设他们已经做出了选择，现在看消费者的选择有什么样的特征。

Walrasian demand

Definition (Walrasian\Market\Ordinary Demand Correspondence). The consumer's Walrasian (or market, or ordinary) demand correspondence $x(p,w)$ assigns a set of chosen consumption bundles for each price-wealth pair $(p,w)$.

• Definition (demand function). When $x(p,w)$ is single-valued, we refer to it as a demand function.

Assumptions

1. Homogeneous of degree zero. $x(\alpha p,\alpha w)=x(p,w)$ for any $p,w$ and $\alpha >0$​.
2. Walras' law. For every $p\gg 0$​ and $w>0$​, we have $p^T\cdot x=w$​ for all $x\in x(p,w)$​.
3. Demand function. We assume that $x(p,w)$ is always single-valued.
4. When convenient, we also assume $x(p,w)$​ to be continuous and differentiable.

Compensated law of demand

In the consumer demand setting, the weak axiom is essentially equivalent to the compensated law of demand, the postulate that prices and demanded quantities move in opposite directions for price changes that leave wealth unchanged.

$$\mathrm{W}\mathrm{A}\iff \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}\mathrm{l}\mathrm{a}\mathrm{w}\mathrm{o}\mathrm{f}\mathrm{d}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{d}$$

Wealth and Price Effects

Wealth effects

Definition (Consumer's Engel function). For fixed prices $\bar{p}$, the function of wealth $x(\bar{p},w)$​ is called the consumer's Engel function.

• Its image in ${\mathbb{R}}_{+}^L,E_{\bar{p}}=\{x(\bar{p},w):w>0\}$​, is known as the wealth expansion path.​

At any $(p,w)$, the derivative $\partial x_{\ell }(p,w)/\partial w$ is known as the wealth effect for the $\ell$​th good.

• Normal. If $\partial x_{\ell }(p,w)/\partial w\ge 0$​

• Inferior (at $(p,w)$​​). If $\partial x_{\ell }(p,w)/\partial w<0$​​

• Demand is normal. If every commodity is normal at all $(p,w)$​​.

In matrix notation.

$$D_{w}x(p,w)=\left[\begin{array}{c}\frac{\partial x_1(p,w)}{\partial w}\\ \frac{\partial x_2(p,w)}{\partial w}\\ ⋮\\ \frac{\partial x_L(p,w)}{\partial w}\end{array}\right]\in {\mathbb{R}}^L.$$

Price effects

Definition (Offer Curve). Offer curve is the locus of points demanded in ${\mathbb{R}}_{+}^2$ as we range over all possible values of $p_2$.

Definition (Price Effect). The derivative $\partial x_{\ell }(p,w)/\partial p_k$ is known as the price effect of $p_k$ on the demand for good $\ell$.

• Giffen good at $(p,w)$. If $\partial x_{\ell }(p,w)/\partial p_l>0$​.

In matrix form

$$D_{p}x(p,w)=\left[\begin{array}{ccc}\frac{\partial x_1(p,w)}{\partial p_1}& \cdots & \frac{\partial x_1(p,w)}{\partial p_L}\\ & \ddots & \\ \frac{\partial x_L(p,w)}{\partial p_1}& \cdots & \frac{\partial x_L(p,w)}{\partial p_L}\end{array}\right].$$

Implications of assumptions

Implications of homogeneity

Proposition 2.E.1. If the Walrasian demand function $x(p,w)$​ is homogeneous of degree zero, then for all $p$ and $w$:

$$\begin{array}{}\text{(2.E.1)}& \sum _{k=1}^L\frac{\partial x_{\ell }(p,w)}{\partial p_k}{p}_k+\frac{\partial x_{\ell }(p,w)}{\partial w}w=0\text{ for }\ell =1,\cdots ,L.\end{array}$$

In matrix form

$$\begin{array}{}\text{(2.E.2)}& D_{p}x(p,w)p+D_{w}x(p,w)w=0.\end{array}$$

Definition (Elasticities of Demand). The elasticities of demand with respect to prices and wealth.

$${\epsilon }_{\ell k}(p,w)=\frac{\partial x_{\ell }(p,w)}{\partial p_k}\frac{p_k}{x_{\ell }(p,w)}$$$${\epsilon }_{\ell w}(p,w)=\frac{\partial x_{\ell }(p,w)}{\partial w}\frac{w}{x_{\ell }(p,w)}$$

Implications of Walras' law

Definition (Cournot Aggregation). Total expenditure cannot change responding to a change in prices.

Definition (Engel Aggregation). Total expenditure must change by an amount equal to any wealth change.

Implications of the weak axiom

We continue to assume that $x(p,w)$ is single-valued, homogeneous of degree zero, and satisfies Walras' law.

Definition (the WA for the WDF). The Walrasian demand function $x(p,w)$​​​ satisfies the weak axiom of revealed preference (the WA) if the following property holds for any two price-wealth situations $(p,w)$​​​ and $(p^{\prime },w^{\prime })$​​ :

$$\text{if }p\cdot x(p^{\prime },w^{\prime })\le w\text{ and }x(p^{\prime },w^{\prime })\ne x(p,w),\text{ then }{p}^{\prime }\cdot x(p,w)>w^{\prime }.$$

Price changes affect the consumer in two ways.

1. First, they alter the relative cost of different commodities.
2. They also change the consumer's real wealth.

To study the implications of the weak axiom, we need to isolate the first effect.

Proposition 2.F.1

Suppose that the Walrasian demand function $x(p,w)$​ is homogeneous of degree zero and satisfies Walras' law. Then $x(p,w)$​​​ satisfies the weak axiom if and only if the following property holds:

• for any compensated price change from an initial situation $(p,w)$ to a new price-wealth pair $(p^{\prime },w^{\prime })=(p^{\prime },p^{\prime }x(p,w))$​, we have$$\begin{array}{}\text{(2.F.1)}& (p^{\prime }-p)[x(p^{\prime },w^{\prime })-x(p,w)]\le 0,\end{array}$$with strict inequality whenever $x(p,w)\ne x(p^{\prime },w^{\prime })$​.

$(2.F.1)$​​ can be written as $\mathrm{\Delta }p\cdot \mathrm{\Delta }x\le 0$​​.

Proposition 2.F.1 tells us that the law of demand holds for compensated price changes, so we call it the compensated law of demand.

Substitution effects

When consumer demand $x(p,w)$​ is a differentiable function of prices and wealth, Proposition 2.F.1 has a differential implication that is of great importance.

Imagine that we make this a compensated price change by giving the consumer compensation of $dw=x(p,w)\cdot dp$​.

$$S(p,w)=\left[\begin{array}{ccc}{S}_{11}(p,w)& \cdots & S_{1L}(p,w)\\ ⋮& \ddots & ⋮\\ S_{L1}(p,w)& \cdots & S_{LL}(p,w)\end{array}\right],$$

where the $(\mathcal{l},k)$th entry is

$$\begin{array}{}\text{(2.F.10)}& S_{\mathcal{l},k}(p,w)=\frac{\partial x_{\mathcal{l}}(p,w)}{\partial p_k}+\frac{\partial x_{\mathcal{l}}(p,w)}{\partial w}{x}_k(p,w).\end{array}$$

The matrix $S(p,w)$ is known as the substitution/Slutsky matrix, and its elements are known as substitution effects.