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Aggregated consumers as a biased concept (in case of cross-price elasticity)?

I try to approach aggregated consumption data as if it was a new consumer (similarly to approaching average data as if it was representative consumer). Is this approach valid? Or do you know some literature solving this?

EDIT: This is more concise version of what I try to ask based on @Giskard comments. We have consumers with some utility functions $U_i(\boldsymbol{x})$. We know that we could (potentially) derive analythically the cross-price elasticity for each consumer and each pair of goods just from utility functions. This would give us the TRUE/REAL cross-price relation between goods. Thus, we could also find some average cross-price relation between goods. BUT, here is the thing... Does aggregation of consumers per day approximate this REAL relation or it is biased? I would say the bias is there because cross-price relations emerge even where they (by analythic solution) could not be... But what about other relations that were there? Would they be biased as well?

Assume that we have data about quantity purchased and relevant prices as such:

ID Good1 Good2 Good3 Good4 Price1 Price2 Price3 Price4 date
ID1 3 5 0 0 4 2 3 2 day1
ID2 0 7 8 3 4 2 3 2 day1
ID3 0 8 8 5 3 5 4 2 day2
ID4 5 7 2 0 3 5 4 2 day2

As we can see, during first day two consumers came into the shop while another two different consumers came the next day. Here is the thing: Consumers do not buy all the goods... Therefore, some purchased amounts remain zero for some consumers.

I would like to find the cross-price elasticity between all pairs of goods by regression, thus identifying what would be the cross-price elasticity for some average/representative consumer. However, those zeros are a little bit problematic, because I have to use logarithms.

Nevertheless, I had an idea to aggregate by days getting the following:

ID Good1 Good2 Good3 Good4 Price1 Price2 Price3 Price4 date
ID1 + ID2 3 + 0 5 + 7 0 + 8 0 + 3 4 2 3 2 day1
ID3 + ID4 0 + 5 8 + 7 8 + 2 5 + 0 3 5 4 2 day2

Although, right now I am overthinking what does this tell me... Before, there was no substitution between Good 4 and Good 1 but now it emerges... However, we have not seen a single consumer to perform some substitution action between these goods. From simulations I know that if there were no zeros, I could just estimate the cross-price elasticity by classic approach and obtain "as if" the result of some average preferences (solved in this question). This leads me to believe that this aggregation is actually harmful.

The questions are following:

  • Is it correct to think that applying cross-price elasticity on aggregated consumers provides biased results?
  • What would the application of cross-price elasticity in this case imply (or tell)?
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  • $\begingroup$ Hi! In "biased results", what exactly do you mean by "biased"? Do you have reason to believe that the OLS regression estimate of the cross-price elasticity of the aggregated demand is not the same as the expected value of the cross-price elasticity of the aggregated demand? $\endgroup$
    – Giskard
    Feb 20 at 11:07
  • $\begingroup$ Hi! I mean the following: If I think about consumers with certain preferences $U_1(x_1;x_2;x_3)$ ; $U_2(x_2;x_3;x_4)$, their actions on behalf of these preferences create some consumption patterns... I see only those patterns but would like to describe the preferences behind as accurately as possible. If I was searching for some average cross-price elasticity between products that corresponds to those hidden preferences, the spontaneous formation of some cross-price elasticity between pair of goods none of the consumers bought together would pose a problem, imho. Maybe this reasoning is wrong. $\endgroup$
    – Athaeneus
    Feb 20 at 14:23
  • $\begingroup$ "accurately as possible" is unfortunately not well defined; neither is "pose a problem". $\endgroup$
    – Giskard
    Feb 20 at 17:14
  • $\begingroup$ Sorry, so I would define the problem in the following way: There is some true/real cross-price elasticity between products. This makes sense because cross-price elasticity is a function of elasticity of substitution, which in turn is a function of consumer's preferences. We know that we can find some virtual representative consumer who has as if average preferences that would produce the pattern we can observe (see linked question). However, with aggregation I am not sure if it can provide these "as if" average preferences, if the output reflects the preferences of some average customer... $\endgroup$
    – Athaeneus
    Feb 20 at 17:30

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