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Since unemployment benefit is a form of government spending, which is a component of aggregate expenditure, then shouldn't unemployment benefits be part of national income measurement?

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It is included. It is a transfer payment, so the trick is to ensure that the income flow is only counted one time in the national accounting.

Statistics Canada, National Gross Domestic Product by Income and by Expenditure Accounts link to description

The Income and Expenditure Accounts record the distribution and use of income by the six main institutional sectors: households, non-profit institutions serving households, general governments, non-financial corporations, financial corporations and non-residents. These accounts articulate revenues to the sector (including current transfers from other sectors, such as employment insurance received) and current expenditures of the sector (including transfers to other sectors, such as income taxes paid to government).

("Employment insurance" is the Canadian equivalent of unemployment insurance.)

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