The best technical explanation of Keynes' famous multiplier remains that of Murray Rothbard here.
Thus, income = independent expenditures (private investment + government deficit) + passive consumption expenditures. Using our illustrative consumption function, income = independent expenditures + 90 percent of income. Now, by simple arithmetic, income equals ten times independent expenditures. For every increase in independent expenditures, there will be a ten-fold increase in income. Similarly, a decrease in independent expenditures will lead to a ten-fold drop in income. This "multiplier" effect on income will be achieved by any type of independent expenditure — whether private investment or government deficit. Thus, in the Keynesian model, government deficits and private investment have the same economic effect.
Please note that:
1) Rothbard is EXPLAINING the multiplier, not defending or agreeing with it;
2) Rothbard selected the 90% figure to make the math easy. That figure is a VARIABLE which is, according to Keynes, based on research. It is derived from the Keynesian consumption function which, like marginal utility, is mathematically derived from aggregate economic data.
3) Please note the last two sentences in boldface. Keynesian theory says the multiplier applies whether the expenditure is public or private.
My contention is that the entire concept of government deficit spending creating multiple increases in real income is baloney. Deficit spending merely increases economic activity in that it increases the total of money in circulation. Any increases in money income are a matter of prospective and MUST be accompanied by increases in expenses and vice versa. From the prospective of the government, deficit spending must be paid for by a corresponding amount of income, either obtained from taxes or debt financing. From the prospective of the "receivers" of government money, their new income must be accompanied by expenditures of their own before that money can become anyone else's income and so on and so on. Any difference in income and expenditures from the prospective of each individual involved becomes profit or loss in terms of money. Profits realized from this process are what benefits the economy, not an increase in the aggregate income of all concerned. However, whether these profits are "real" is also a question we must consider and this depends upon the amounts of government deficit spending and its effect on inflation. If inflation is significant, profits may appear on individual income statements, but these profits may not appear for the individual in real terms.
In sum, if any kind of "multiplier" exists in an exchange economy it is real profit.
Money can be created in our current fiat monetary system by the government and the Fed by means of deficit spending and public debt financing, or money can be created by the private fractional reserve banking system making loans to private individuals.
So, the larger question is: Which of the above two options is in the best interests of individual Americans?
Keep in mind, from the Keynesian prospective any fictional "multiplier" works equally well whether government deficit spending or private investment, so that part of the equation shouldn't come into play. The question really boils down to which political/economic system do you think best satisfies the desires (felt needs) of the American consumer: government spending directed by 535 federal politicians (and thousands of unelected bureaucrats) or the unfettered free market?