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If the probable reason is because it must reflect the government's ability to pay back the deficit, wouldn't government income be a better indicator, rather than GDP?

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In short: yes, it very well might be and this has been argued by some (see, for example: http://www.forbes.com/sites/jeffreydorfman/2014/07/12/forget-debt-as-a-percent-of-gdp-its-really-much-worse/#29b8530f6e0c).

The use of Debt/GDP is the current international norm, but Debt/Tax revenue might very well be a better, more concise option. Using Debt/GDP might be because it has been common to compare countries to each other with GDP or GDP/capita and having GDP as a factor relates the debt to these measures.

It can also be argued that in the end the whole population of a country is responsible for paying their country's debts, so in an extreme situation it's not just the government's income which might be used to pay off debts.

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