I saw an article by Robert Shiller in the NYT in July: "How to make the case for a new stimulus" and was reminded of this by another article by Shiller that I saw yesterday.
The mechanism relies on expenditures all being on consumption or investment, but matched taxes coming from a mixture of consumption and savings. The higher the proportion that comes from savings, the higher the boost to national income, and that's before multiplier effects.
Go back to my previous post but now add in private savings (firms still own all capital and produce all the output, so that gross profits, P = Y - W, and it's firms that make investments in capital stock K) so:
P = Sf + d = Y - W = C + G + I - W = W + d - Sa - T + G + I - W
i.e. Y = d + W + I + G - T - Sa
(the only difference now is that savings by firms are labelled Sf and savings by agents are labelled Sa). Therefore, assuming wages, dividends, investment all constant in short run, and assuming that raising T by 1 lowers Sa by 0.5 (i.e. 50% of taxes raised come from income that would otherwise have been saved), then raising government expenditure and taxes by same amount (i.e. balance budget expansion in government expenditure) would raise national income by 50% of the increase in the government budget. This would then be subject to multiplier effects since wages, dividends and investment are not constant in the slightly longer run.
Obviously fully deficit financed expansion in government expenditure would be even more expansionary (100% of the increase rather than 50% as above). Ideally you'd set tax rates and expenditure such the goverment was running a small surplus at full employment, so that it could afford to engage in this counter-cyclical policy during downturns. Also, just need debt levels not to grow as a percentage of GDP over the cycle - you don't need to balance the budget in pounds and pence over the cycle if there's economic growth.
However, maybe there won't be economic growth in future?