Revealed preference

What we learned before is the utility theory, which builds a framework of consumer's behavior given utility function. However, utility function is never observable in reality. It may not even exist. This topic introduces "revealed preference", which starts from observable data, and try to recover consumer's preference. As data are something that is revealed to the world, we call such preference "revealed preference".

It also suffers from some main problems, such as limited data, so we only get partial knowledge about the preference.

Basic idea: If the consumer buys one bundle instead of another affordable bundle, then the first bundle is considered to be revealed preferred to the second one.

1. WARP

If:

Then($\Rightarrow$$\Rightarrow$):

• Interpretation:

  If $X^0$$X^0$ is revealed preferred to $X^1$$X^1$, then $X^1$$X^1$ can never be revealed preferred to $X^0$$X^0$.

  Notice that chosen consumption bundle is one where superscripts are the same with those of prices. First line (if) means that $X^1$$X^1$ is no more expensive than $X^0$$X^0$ at prices $p^0$$p^0$. Second line (then) means that when $X^1$$X^1$ was chosen it is merely because it was cheaper at prices $p^1$$p^1$ than $X^0$$X^0$. Hence, $X^1$$X^1$ is not revealed preferred to $X^0$$X^0$ merely because it was chosen for the same reason one would not want to infer that Fords are preferred to BMWs.

  

• Some propositions if we have WARP

  Proposition 1: Demand correspondences are homogeneous of degree 0. (under budget balance)

  $$\begin{array}{}\text{(3)}& X_i^M(t{p}_1,\dots ,t{p}_n,tM)\equiv X_i^M(p_1,\dots ,p_n,M),\mathrm{\forall }t>0.\end{array}$$

  Let $X^0(x_1^0,\dots ,x_n^0)$$X^0\left(x_1^0, \ldots, x_n^0\right)$ be chosen at $(p^0,M^0)$$\left(p^0, M^0\right)$ and $X^1(x_1^1,\dots ,x_n^1)$$X^1\left(x_1^1, \ldots, x_n^1\right)$ be chosen at $(p^1,M^1)$$\left(p^1, M^1\right)$. By hypothesis: $p^1=t{p}^0$$p^1=t p^0$ and $M^1=t{M}^0$$M^1=t M^0$.

  Proof:

  Since $t{M}^0=M^1$$t M^0=M^1$, by Walras law $(\underbrace{p^0{X}^0}=M^0;p^1{X}^1=M^1)$$(\underbrace{p^0 X^0}=M^0 ; p^1 X^1=M^1)$, it follows that $t\underbrace{p^0{X}^0}=p^1{X}^1$$t \underbrace{p^0 X^0}=p^1 X^1$ But $p^1=t{p}^0$$p^1=t p^0$, hence $t{p}^0{X}^0=t{p}^0{X}^1$$t p^0 X^0=t p^0 X^1$ or

  $$\begin{array}{}\text{(4)}& p^0{X}^0=p^0{X}^1\end{array}$$

  Equation (1) says that $X^0$$X^0$ is revealed preferred to $X^1$$X^1$ because it was chosen and $X^1$$X^1$ was not and it could have been. Therefore, when $X^1$$X^1$ is chosen, $X^0$$X^0$ must be more expensive:

  $$\begin{array}{}\text{(5)}& p^1{X}^0>p^1{X}^1\end{array}$$

  But $p^1=t{p}^0$$p^1=t p^0$ and (1) can be re-stated as:

  $$\begin{array}{}\text{(6)}& t{p}^0{X}^0=t{p}^0{X}^1\end{array}$$

  and

  $$\begin{array}{}\text{(7)}& p^1{X}^0=p^1{X}^1\end{array}$$

  Equation (6) obviously contradicts (5) and hence $X^1\ne X^0$$X^1 \neq X^0$ is false.

  Proposition 2: Demand correspondences are single-valued, i.e., for any price vector $(\mathbf{p},M)$$(\mathbf{p}, M)$, the consumer chooses a single point of consumption.

  This is a special case of the P.1. for $t=1$$t=1$. Since P.1. holds for any $t>0$$t>0$, it holds for $t=1$$t=1$. Hence, since $p^1=p^0$$p^1=p^0$ and $M^1=M^0$$M^1=M^0$, one and only one consumption bundle is chosen.

  If two different bundles were chosen, each would be revealed preferred to the other, by symmetry, an obvious contradiction.

  Proposition 3: Slutsky matrix $S_{ij}=\frac{\partial X_i^h}{\partial p_j}$$S_{i j}=\frac{\partial X_i^h}{\partial p_j}$ is negative semi-definite under WARP.

  • original income $M^0$$M^0$; after a price drop new income $M^1$$M^1$,

  • $X^1$$X^1$ on $M^1$$M^1$ can be anywhere (no behavioral restrictions),

  • consumer's money income is lowered until he can just buy $X^0$$X^0$ at new price $p^1$$p^1$. Confronted with $M_2$$M_2$ he will buy $x^2$$x^2$.

  

WARP + BB -> NSD + HD0

WARP + BB + 2 goods -> NSD + HD0 + Symmetric -> rational (could be derived from a rational utility function)

2. SARP

One property of the system of consumer demands that WARP does not imply is symmetry. For symmetry we need SARP.

The Strong Axiom of Revealed Preference (SARP) is satisfied if, for every sequence of distinct bundles ${\mathbf{x}}^0,{\mathbf{x}}^1,\dots ,{\mathbf{x}}^k$$\mathbf{x}^0, \mathbf{x}^1, \ldots, \mathbf{x}^k$, where ${\mathbf{x}}^0$$\mathbf{x}^0$ is revealed preferred to ${\mathbf{x}}^1$$\mathbf{x}^1$, and ${\mathbf{x}}^1$$\mathbf{x}^1$ is revealed preferred to ${\mathbf{x}}^2,\dots$$\mathbf{x}^2, \ldots$, and ${\mathbf{x}}^{k-1}$$\mathbf{x}^{k-1}$ is revealed preferred to ${\mathbf{x}}^k$$\mathbf{x}^k$, it is not the case that ${\mathbf{x}}^k$$\mathbf{x}^k$ is revealed preferred to ${\mathbf{x}}^0$$\mathbf{x}^0$. SARP rules out intransitive revealed preferences and therefore can be used to induce a complete and transitive preference relation, $\succsim$$\succsim$, for which there will then exist a utility function that rationalizes the observed behaviour.