I read the following statement.
“ A utility maximizer with strictly convex and strongly monotonic preferences satisfies weak axiom of revealed preferences.”
How can I prove or show this? I cannot realize this statement on my mind. Please show me it’s proof in order to understand its statement.
As far as I can see this comes just from definitions:
As given in MWG definition 1.C.1:
The choice structure $(\mathscr{B},C(\cdot))$ satisfies the weak axiom of revealed preference if the following property holds:
If for some $B \in \mathscr{B}$ with $x,y \in B$ we have $x\in C(B)$, then for any $B'\in \mathscr{B}$ with $x,y\in B'$ and $y\in C(B')$, we must also have $x\in C(B')$.
This basically states that under the weak axiom of revealed preference (WARP) if there is any choice set which contains both $x$ and $y$ and $x$ is preferred then there can be no set containing both $x$ and $y$ where $y$ would be chosen over $x$. Or we can say that "if x is revealed at least as good as y, then y cannot be revealed preferred to x."
Next according to MWG definition 3.B.2:
The preference relation $\succeq$ on $X$ is monotone if $x\in X$ and $y >> x$ implies $y \succ x$. It is strongly monotone if $y \geq x$ and $y \neq x$ imply that $y\succ x$.
In addition, according to MWG definition 3.B.4:
The preference relation $\succeq $ on $X$ is convex if for every $x\in X$, the upper countour set $\{y \in X: y \succeq x\}$ is convex; that is, if $y \succeq x$ and $z \succeq x$, then $\alpha y + (1-\alpha) z \succeq x$ for any $\alpha \in [0,1]$.
Lastly, utility maximization is often used to imply rationality (Simon 2001) and preference relation can be rational only according to MWG definition 1.B.1:
The preference relation $\succeq$ is rational if it possesses the following two properties:
(i) Completeness: for all $x,y \in X$ we have that $x\succeq y$ or $y \succeq x$ or both.
(ii) Transitivity: For all $x,y,z \in X$ if $x \succeq y$ and $y \succeq z$ then $x \succeq z$.
Given the definitions above if they are satisfied then if $y$ is revealed to be at least as good as $x$, $x$ cannot later be revealed as better. So it follows from the definitions of those properties. For example, WARP would be violated if the following choices are made $C(\{x,y\}) = y$ and in $C(\{x,y,z\}) = {x}$, by a person holding their preferences constant - such situation would clearly violate several of the above definitions.
