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Why do we tend to use Real GDP Per Capita instead of the real wage? What is the issue of using the real wage instead?

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Thee are several issues including what happens about incomes that are not wages, but the main one is

Not everybody earns a wage

This is an issue for the self-employed (especially in agricultural subsistence economies), but it is also an issue in economies with large numbers of dependants, especially the very young or the very old

It also raises the issue of the employment of women: increasing the proportion of women working will tend to boost total GDP and GDP per capita even though it can in some circumstances reduce average wages if the women entering the job market earn less than men already there

There are also cyclical issues: there have been recessions where unemployment has increased substantially but wages of those still working have been less immediately affected; there have been other recessions where the reverse pattern has predominated. In both cases GDP per capita has fallen and the economy is obviously weaker than it was before the recession

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