Micro Lab 4

Revealed Preference

Revealed Preference

A binary relationship based on observable choices

If $\mathbf{x}$$\mathbf{x}$ is chosen when ${\mathbf{x}}^{\prime }$$\mathbf{x}^{\prime}$ is affordable, i.e, $\mathbf{p}{\mathbf{x}}^{\prime }\le (<)w$$\mathbf{p} \mathbf{x}^{\prime} \leq(<) w$, then $\mathbf{x}$$\mathbf{x}$ is (strictly)revealed preferred to $x^{\prime }$$x^{\prime}$

WARP:

Mathematically,

If ${\mathbf{x}}^0$$\mathbf{x}^0$ is (weakly) revealed preferred to ${\mathbf{x}}^1$$\mathbf{x}^1$ and they are different consumption bundles, ${\mathbf{x}}^1$$\mathbf{x}^1$ can not be (weakly) revealed preferred to ${\mathrm{x}}^0$$\mathrm{x}^0$ . In other word, we only choose ${\mathbf{x}}^1$$\bold x^1$ when ${\mathbf{x}}^0$$\bold x^0$ is unaffordable under new price ${\mathbf{p}}^1$$\bold p^1$.

WARP + Walras' Law $\Rightarrow$$\Rightarrow$

• Choice function is homogeneous of degree 0 in $\mathbf{p}$$\mathbf{p}$ and $w$$w$

  Proof

  $$\begin{array}{}\text{(3)}& \begin{array}{rll}{\mathbf{p}}^0,w^0& \to & {\mathbf{x}}^0\\ {\mathbf{p}}^1=t{\mathbf{p}}^0,w^1=t{w}^0,t>0& \to & {\mathbf{x}}^1\end{array}\end{array}$$

  If the choice function is not homogeneous of degree 0 in $\mathbf{p}$$\mathbf{p}$ and $w,{\mathbf{x}}^0$$w, \mathbf{x}^0$ and ${\mathbf{x}}^1$$\mathbf{x}^1$ would have to be different

  $$\begin{array}{}\text{(4)}& \begin{array}{rl}\underset{―}{{\mathbf{p}}^1{\mathbf{x}}^1}=w^1& =t{w}^0=t\underset{―}{{\mathbf{p}}^0{\mathbf{x}}^0}\\ {\mathbf{p}}^0{\mathbf{x}}^1& ={\mathbf{p}}^0{\mathbf{x}}^0\\ {\mathbf{p}}^1{\mathbf{x}}^1& ={\mathbf{p}}^1{\mathbf{x}}^0\end{array}\end{array}$$

  Contradict with WARP, so ${\mathbf{x}}^1={\mathbf{x}}^0$$\mathbf{x}^1=\mathbf{x}^0$.

• negative semi-definite substitution matrix.

  If choice function $\mathbf{x}(\mathbf{p},w)$$\mathbf{x}(\mathbf{p}, w)$ is differentiable

  $$\begin{array}{}\text{(5)}& \begin{array}{rl}{\mathbf{x}}^c& =\mathbf{x}({\mathbf{p}}^2,{\mathbf{p}}^2{\mathbf{x}}^1)\\ d{x}_i^c& =(\frac{\partial x_i}{\partial p_j}+\frac{\partial x_i}{\partial w}{x}_j^1)d{p}_j\end{array}\end{array}$$

  Assume the prices of more than one goods have changed,

  $$\begin{array}{}\text{(6)}& \begin{array}{rl}d{x}_i^c& =\sum _{j=1}^L(\frac{\partial x_i}{\partial p_j}+\frac{\partial x_i}{\partial w}{x}_j^1)d{p}_j={\mathbf{s}}_{i}d\mathbf{p}\\ d{\mathbf{x}}^c& =Sd\mathbf{p}\\ d\mathbf{p}d{\mathbf{x}}^c& \le 0\Rightarrow d\mathbf{p}Sd\mathbf{p}\le 0\end{array}\end{array}$$

  Note
  

  • With revealed preference approach, Substitution Matrix is NSD.

  • With utility function approach (Last chapter), Substitution Matrix is NSD and symmetric, because

  $$\begin{array}{}\text{(7)}& s_{ij}=\frac{\partial h_i(p,u)}{\partial p_j}=\frac{{\partial }^{2}e(p,u)}{\partial p_i\partial p_j}\end{array}$$

SARP

Strong Axiom of Revealed Preference: A demand $\mathbf{x}(\mathbf{p},w)$$\mathbf{x}(\mathbf{p}, w)$ satisfies SARP if, for every sequence of distinct bundles ${\mathbf{x}}^1,\dots ,{\mathbf{x}}^N$$\mathbf{x}^1, \ldots, \mathbf{x}^N$, where ${\mathbf{x}}^1$$\mathbf{x}^1$ is revealed preferred to ${\mathbf{x}}^2$$\mathbf{x}^2$, and ${\mathbf{x}}^2$$\mathbf{x}^2$ is revealed preferred to ${\mathbf{x}}^3,\dots$$\mathbf{x}^3, \ldots$, and ${\mathbf{x}}^{N-1}$$\mathbf{x}^{N-1}$ is revealed preferred to ${\mathbf{x}}^N$$\mathbf{x}^N$, it is not the case that ${\mathbf{x}}^N$$\mathbf{x}^N$ is revealed preferred to ${\mathbf{x}}^1$$\mathbf{x}^1$.

• SARP=WARP+ transitivity; SARP rules out intransitive revealed preference.

• SARP implies WARP for sure. *WARP implies SARP if $L=2$$L=2$.

Questions

JR 2.9

Suppose there are only two goods and that a consumer's choice function $\mathbf{x}(\mathbf{p},y)$$\mathbf{x}(\mathbf{p}, y)$ satisfies budget balancedness, $\mathbf{p}\cdot \mathbf{x}(\mathbf{p},y)=y\mathrm{\forall }(\mathbf{p},y)$$\mathbf{p} \cdot \mathbf{x}(\mathbf{p}, y)=y \forall(\mathbf{p}, y)$. Show the following: (a) If $\mathbf{x}(\mathbf{p},y)$$\mathbf{x}(\mathbf{p}, y)$ is homogeneous of degree zero in ( $\mathbf{p},y$$\mathbf{p}, y$ ), then the Slutsky matrix associated with $\mathbf{x}(\mathbf{p},y)$$\mathbf{x}(\mathbf{p}, y)$ is symmetric. (b) If $\mathbf{x}(\mathbf{p},y)$$\mathbf{x}(\mathbf{p}, y)$ satisfies WARP, then the 'revealed preferred to' relation, $R$$R$, has no intransitive cycles. (By definition, $x^{1}R{x}^2$$x^1 R x^2$ if and only if $x^1$$x^1$ is revealed preferred to $x^2$$x^2$.)

JR 2.10

Hicks (1956) offered the following example to demonstrate how WARP can fail to result in transitive revealed preferences when there are more than two goods. The consumer chooses bundle $x^i$$x^i$ at prices ${\mathbf{p}}^i,i=0,1,2$$\mathbf{p}^i, i=0,1,2$, where

$$\begin{array}{}\text{(8)}& \begin{array}{rl}& p^0=\left(\begin{array}{c}1\\ 1\\ 2\end{array}\right)x^0=\left(\begin{array}{c}5\\ 19\\ 9\end{array}\right)\\ & p^1=\left(\begin{array}{c}1\\ 1\\ 1\end{array}\right)x^1=\left(\begin{array}{c}12\\ 12\\ 12\end{array}\right)\\ & p^2=\left(\begin{array}{c}1\\ 2\\ 1\end{array}\right)x^2=\left(\begin{array}{c}27\\ 11\\ 1\end{array}\right)\end{array}\end{array}$$

(a) Show that these data satisfy WARP. Do it by considering all possible pairwise comparisons of the bundles and showing that in each case, one bundle in the pair is revealed preferred to the other. (b) Find the intransitivity in the revealed preferences.

2022 Midterm 1 Q4

The weak axiom of reveled preference (WARP) implies that the demand relations $x_i=$$x_i=$ $x_i^M\left(p_1 \ldots, p_n, M\right)$ are single valued, i.e., for any price-income vector $(\mathbf{P},M)$$(\mathbf{P}, M)$ the consumer chooses a single point of consumption. Prove this result.