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Currently competition authorities usually decide whether to approve a merger or not based on the total revenues or capacities of the merging firms and some rule of thumbs. A reason for this is that they do not always have a stronger mandate because of the legal background. Merger simulation models promise a more analytical policy approach. For a good survey see Budsinki & Ruhmer (2010).

It is my understanding that the kind of data needed to estimate the demand function of a market, or usually demand functions in the case of differentiated products, is not publicly available. Yet rival companies frequently have similar data which they use to estimate their market share.

How would one go about legally acquiring such data?
Are there any ways with which to estimate cross elasticities without said data?

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In your case, you might try to reach the said antitrust authority and ask for the dataset. You also could reach the parties themselves and ask for the shared data. Authorities acquire these datasets through parties BI data and/or third parties that might have some connection to the merger (companies concerned with the operation, clients etc). I believe it is highly improbable that the data is made publicly available anywhere.

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