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If government expenditure increases without rise in current tax rate, consumer expects future tax to rise...

If government expenditure increases without rise in current tax rate, consumer expects future tax to rise from which makes expected future income fall, in turn, private consumption decreases and private savings goes up, however, National savings decreases because Public savings decreases more than increases in private savings, that is why national savings decreases. This is how savings decreases when government expenditure increases. My question is How do we know that Public savings decreases more than increases in private savings ? please provide detail explanation

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Decrease in the public saving is represented by of full value of the increase in the government spending denoted by G. However private savings are increased by a lower amount because a given change in private savings is MPS times a given change in income, where MPS is the marginal propensity to save. therefore we see that while the government spending is increased by a full amount and therefore public saving is decreased by a full amount, private saving is in only by MPS fraction of income. Due to this reason decline in public savings is greater than increase in private savings.

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