## Thursday, April 18, 2013

### On the Weak and General Axiom of Revealed Preference

"I have two related questions on Choice.
I know that we can satisfy WARP but have nevertheless a violation of GARP. My question is if we can have a situation where WARP is violated, but GARP is satisfied?
Secondly, from the definition of GARP it is always spoken of a bundle being revealed preferred to another bundle through a chain or ""sequence"" of revealed preferences. My question is, if this defined ""sequence"" can consist of only two observations, so that we have actually a direct revealed preference after all?  In other words, does ""revealed preferred"" include the case of ""direct revealed preferred""?"

My response:

The quick answers are, to the first question, no, and to the second question, yes.  In other words, GARP implies WARP and the chain can have only two elements, which is WARP directly.

A longer response would include making a comparison between revealed preference and the usual assumptions made about preference.  These are about a preference relation, R.  xRy is then read as, x is preferred to or indifferent to y.   So from R one can also define P and I by:

• xPy if xRy and not yRx.
• xIy if xRy and yRx.

There are three "logical" assumption about preference orderings.

R is complete.  For every x and y in the Consumption set (the set of possible consumption bundles) either xRy or yRx.  This means comparisons can be made between any two consumption bundles.  Note that neither P nor I are complete.

R is reflexive.  For every x in the Consumption set xRx.

R is transitive.  For every x, y, and z in the Consumption set, if xRy and yRz then xRz.

These properties allow one to define a choice, provided the choice set is closed.  In this case if the choice set is C then x in C is a choice (a maximal element under R) if xRy for all y in C.  Note that there is no greatest number less than 100.  If you posit it is 99, then 99.9 is greater and you can always add an additional 9 to the right.  So for a choice to exist, the choice set must be closed.  100 is the greatest number less than or equal to 100.  The choice set being closed means it includes its boundary.

To these logical assumptions, one usually adds an economic assumption - monotonicity or more is preferred to less.  This assumption rules about satiation points as well as "thick indifference curves,"  The upshot of this assumption is that when the choice problem is given by a budget set, the choice will always be on the budget line, never inside the line.

A further assumption that is frequently made is that preferences are Convex, which gives indifference curves their usual shape.  This is done do when the Budget environment changes in a small way, the choice also changes at most in a small way.  Or, if you prefer, the demands are continuous function of the budget environment.

Now, with all this machinery, what does WARP get you?  In this way of thinking, WARP is equivalent to completeness, reflexivity, and monotonicity.  You need GARP to bring in transitivity.  There is also something called SARP, the strong axiom.  It is WARP plus the assumption that preferences are strictly convex, so the choice is always unique.

I hope that helps.

1. Dear Professor,

I am not sure if we apply the same definitions for WARP and GARP, as it appears there are subtle differences around in the literature. If we, for example, check in Varian p.133, it states:

"GARP: If x_t is revealed preferred to x_s, then x_s cannot be strictly directly revealed preferred to x_t"

"WARP: If x_t is directly revealed preferred to x_s, then x_s cannot be directly revealed preferred to x_t"

To be clear about revealed preferences:

direct revealed preference (RD) means p_t*x_t>=p_t*x.

strictly direct revealed preference (PD) means: p_t*x_t>p_t*x.

revealed preference (R) means: A chain of direct revealed preferences (x_t(RD)x_s, x_s(RD)x_n...=>x_t(R)x_n)

Now suppose we take an example:

p_1=(1,1,2); x_1=(1,0,0)
p_2=(1,1,1); x_2=(0,1,0)
p_3=(1,3,1); x_3=(0,0,2)

It is easy to see that WARP is violated between observation 1 and 2. However, GARP is not violated in all possible cases. We have here a situation where the relationship is transitive and complete, but not antisymmetric as required by WARP.

2. Thanks for the comment.

The comparison between the first two bundles and price regimes is interesting and helps to clarify definitions, as you suggest. Expenditure on either bundles is 1, in either regime. Indeed, the regimes are the same except with respect to the price of good 3, while the consumption of good 3 is zero in both cases. So, in a pathological sense WARP is violated. In a different sense, the choices in the two regimes are indifferent and varying the price of the third good is a clever way to make the price regimes not the same, but still allow this indifference in the comparison.

I'm not sure what the third regime shows. Expenditure on that bungle in either of the first two regimes is greater than 1, so it is unaffordable in those regimes.

I'd also say this is somewhat beyond intermediate micro, at least as I teach it. I don't do consumption sets with higher dimension than two.