Applied Microeconomics: The Strong Axiom of Revealed Preference,Aggregation, and Rational Preferences

Professionally most of my focus has been empirical and to a great extent, has been agnostic when it comes to micro theory. Take data mining and machine learning, social network analysis,  text mining or issues related to big data for instance. Or many of the other issues I have taken up at EconometricSense. A lot of what I have worked on has been more about data, algorithms, and experimental design than the nuts and bolts of microeconomic theory.

However, there are some theoretical issues in microeconomics that I either have forgotten, or never really understood that well.

Particularly these issues have to do with the strong axiom of revealed preference, the market aggregated demand function, and welfare analysis as discussed in one of my graduate texts (Microeconomic Theory by Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green).  From that text  (MWG) I basically get the following:

• The strong axiom (SA) of revealed preference is equivalent to the symmetry and negative semi-definiteness of the slutsky substitution matrix
• Violations of the SA mean cycling of choices or violations of transitivity
• If observed demand follows the SA, then preferences that rationalize demand can always be recovered
• It is impossible to find preferences that rationalize a demand function when the slutsky matrix is not symmetric

That means, for an individuals' observed demand function, if the slutsky matrix is not symmetric, you can't make welfare statements based on the area beneath the demand curve.

What happens when we aggregate individual demand to get a market demand function? It seems to me that the data of interest in most applied work is going to be related to an aggregate market demand curve. Based on Green et al:

• the chances of the SA "being satisfied by a real economy are essentially zero"
• If we allow individuals in an economy to have different preference relations/utility functions, when we aggregate to get a market demand function, the negative semi-definiteness of the slutsky matrix (equivalent to the weak axiom of revealed preference) might hold, but "symmetry will almost certainly not." 
• While positive effects of an equilibrium might hold, without symmetry the SA does not and we therefore cannot make statements about consumer welfare based on the area beneath an observed empirical market demand function

This last conclusion leaves me with a lot of questions to ponder:

1. What does that imply with regard to empirical work? It seems to not matter for positive effects (for instance a conclusion that a wage set above equilibrium causes a surplus of labor). 
2. But, how much does it matter that I can't use an empirical demand function to calculate changes in consumer surplus for a change in prices? Maybe it only matters if I am interested in calculating some amount? 
3. For any individual, if the SA might holds (which is possible), we certainly know a price increase would reduce consumer surplus, put them on a lower indifference curve and make them worse off. Regardless of the conclusions above, wouldn't that hold for all consumers represented by the aggregate market demand curve? Can't we make a normative statement  (in  terms of a qualitative directional sense even if we can't calculate total surplus) about all consumers even if the SA fails in the aggregate but holds for each individual?

Now,  the MWG text mentioned above does go on in later chapters to discuss the notions of a positive and normative representative consumer as well as a social welfare function and wealth distribution mechanisms and implications for welfare analysis. But I'd really like to know about #3. Can we make directional statements about changes in welfare as long as we know that any attempt at quantification or calculation of surplus would be invalid due to violations of the SA?

Is this a case where one should just take the example of Milton Freidman's pool players who behave as if they know physics? Maybe all of the assumptions (like the SA) fail to hold for a market demand function, but we still feel confident making directional or qualitative welfare statements about price changes because everything else about the model predicts so well?

Any thoughts from readers?

I found it interesting, that the issues in the bulleted statements related to the MWG text were never addressed that I can tell in any of my undergraduate principles or intermediate micro texts, nor even in Nicholson's more advanced graduate text. It just seems like these texts jump from individual demand to market demand as a horizontal summation of individual's demand curves and go straight to welfare analysis and discussions about consumer surplus without discussing these issues related to the SA.

Note: I definitely spent some time with the issue of consumer surplus calculations based on compensated vs uncompensated demand curves. I don't think that is the issue here at all.

****updated modified on October 29, 2015