# Proof on weak axiom of revealed preferences

“ 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.

• In my own opinion - it is must more instructive to think of WARP in terms of transitivity (consistency) of preferences under constrained decision-making. Deep down, WARP is about consistency, and more to the point: if consistency cannot be assumed in economic decision-making then it is hard to arrive at welfare conclusions from any impact on available choices. Any glance at an advanced micro text will give you the proof, but understanding why it is important is more informative. Nov 10, 2020 at 22:50
• Hmm thanks for comment. Which book? I look at MGW. But I cannot see @EB3112 Nov 10, 2020 at 22:55
• Tried Varian?... Nov 10, 2020 at 23:02

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.

• very explanatory! really many thanks! Nov 11, 2020 at 17:35