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In the past I read something like this:

If basic expenses for family went from 100€ to 200€ but its wage went from 200€ to 400€ it can expend more than before.

Weighting the costs of living in state X, in the median wage in state X, is the best way to compare state X's COL with COLs of other states.

I assume that weighting COL in MW particularly is better than weighting COL in average wage (AW), because, although there are few families with extremely high wages, their wages bias AW extremely upwards, while MW is less effected from such extreme cases.

Is weighting COL in state X in MW in state X the most accurate way to compare that COL to these of other states?

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    $\begingroup$ "Best" way for what purpose? $\endgroup$ – Giskard Nov 21 '19 at 18:57
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    $\begingroup$ You should probably read about the state rather than just look at financial indicators...? I am guessing that relocating expats are not really representative of the state's populations, so that will probably cause a distortion. $\endgroup$ – Giskard Nov 21 '19 at 19:17
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    $\begingroup$ You said expats in your explanation...this is pretty much the problem with your question. Purchasing power has different distributions in states. Any aggregation to an average or a median results in information loss. Without knowing your exact purpose, it is impossible to so say which imperfect aggregation is "better". $\endgroup$ – Giskard Nov 21 '19 at 19:26
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    $\begingroup$ @Giskard I explained what I understand to be the most accurate way to compare COL of state X to that of other states and asked if it is indeed as such; besides that I shared possible reasons and for me this is enough and you fail to recognize these data; in suggestion, spare your time and leave this matter. $\endgroup$ – user24868 Nov 21 '19 at 20:15
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    $\begingroup$ As I have said, different purposes will yield different "best" or "most accurate" answers. Not sure how I can rephrase this. There are trade-offs. There is probably no state/country that is "best" for everyone. $\endgroup$ – Giskard Nov 21 '19 at 22:50
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Quality of life matters too. This sometimes appears in the labor economics literature as "compensating differential". So it's probably not enough to weight cost of living by median or average wage unless you can objectively determine that all other factors are identical (e.g, tax structure, quality of transportation networks, population density, greenspace, etc.).

These factors are in general not likely to be pecuniary, so they are usually treated as a residual - if MW/COL in state A is less than MW/COL in state B, then the difference must be due to improved amenities in state A. This is obviously unsatisfactory, and as a result there's a lot of active literature in this area. And they will, of course, be highly subjective.

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