Use 'exchequer' to refer specifically to financial governmental departments, not all government parts. The national exchequer is responsible for tax collection.
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Specific Usage
Most common in the UK, it's rarely used to describe government finance departments in other countries. The UK's exchequer was referenced several times during the debate.
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Budget Relation
'Exchequer' often appears in discussions about government budgets and spending. The recent exchequer statement outlined the new fiscal policies.