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Add reason why transfer payments not counted in GDP.
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The question:

How much would GDP change if during a recession the government raises unemployment benefits by $100 million? can be understood in more than one way. There is the pure accounting question which could be formulated more precisely (albeit in terms of a rather unrealistic scenario) as follows: Suppose, in a certain period, the economic activities and transactions of two countries in recession A and B are exactly the same, except that in A the government pays$100 million more in unemployment benefits than in B. How much does GDP differ between A and B?

The answer to this question, which is presumably what Jones means, is that GDP is the same in both countries, because transfer payments are not a component of GDP. Why not? Because although they are expenditure by the government and income to the recipients, they are not directly associated with any element of production or output.

Then there is the question about economic effects:

Suppose, during a recession, the government raises unemployment benefits by $100 million. What would be the likely economic effects, and how would GDP change as a result? The answer to this question will depend on circumstances, and so it is entirely proper here to raise further questions including how the$100 million is funded and how it will be used. It's a reasonable assumption that the propensity to spend of benefit recipients will be high. If the funding were from increased taxation of people on middle and high incomes who have a somewhat lower propensity to spend, then the net effect would be an increase in aggregate demand, perhaps with a multiplier effect, so GDP would probably increase. (I say 'perhaps' and 'probably' because there could be further complications, eg spending on imports.) If the funding were via government borrowing, then there would potentially be direct effects on both demand and interest rates, and it would be quite challenging (and need further information) to identify all the indirect effects including the effect on GDP.