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What is a deficit and should I care about party promises related to deficits?
A deficit happens when a government spends more money than it receives in revenue. To pay for a deficit, governments borrow money, which they finance through debt. Many programs should be debt-financed, like long-term infrastructure projects, so debt and deficits in and of themselves are not bad.
Many years of running a deficit is usually a sign of a structural problem, and revenues (like taxes) should be increased. Or, programs should be cut.