In classic models one would set savings equal to investings. Image you alone form the complete population of a small island and you farm coconuts. You produce a total of $10 worth of coconuts a day. You consume $5 worth of coconuts, and you use (invest) the other $5 to plant new coconut trees. You export nothing and there is no government, making your GDP $10 following the expenditure method. The underlying assumption is that everything spend on goods and services within a country should match the value of goods and services produced in this country.
You can compare the expenditure method with summing the asset side of a balance sheet and the income method with the liabilities and equity side of a balance sheet.
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