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Which is a better measure of the health of an economy between Debt-to-GDP and Debt-to-Revenue? Why?

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You may want to consider this as the difference between the potential to pay and the actual means to pay on hand. Since government revenues will ultimately come from GDP (excepting foreign aid etc.), you will be certain that debt to revenue will be higher (smaller denominator).

How useful is this as a measure? I suppose, as with all things, it depends. Debt to GDP is a well understood and widely used statistic, and so it lends itself to comparison between other nations. Ultimately GDP is a pool from which the government, in principle, could raise revenues from.

In contrast, revenues are your means to pay right now. Debts generally don't all need to be paid at once, and so your means of paying right now may not matter as much as the fact you are able to pay them when they're due (after all, the money you have right now may be better put to use in some productive activity that will enhance your means of paying the debt in the future). If the bill collectors are at the door, suddenly your revenues matter quite a bit!

I suppose the closest I can get to a direct answer to your question might be that if you doubted the ability of a government to tap into these potential revenues (ie. people won't pay taxes), then debt to revenue would likely be more appropriate given that it is an accurate portrayal of the means to pay.

I'll close with a numerical example. Suppose there were two governments that were identical across all respects, only tax to cover their expenses and have \$10 billion in debt with two different payment plans. Country A must pay back \$1 billion a year over 10 years and Country B must pay back \$2 billion a year over 5 years.

Country A's debt to revenue would be 10, while Country B's debt to revenue would be 5. The numbers are different, but we haven't really learned much about the health of these economies.

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