In modern banking the act of depositing is completely severed from the act of lending. Banks lend based on capital ratios, availability of borrowers, and the Fed's interest rates. Deposits and reserves no longer matter.
Effectively, when you deposit money, you deposit in a 100% reserve account with the Fed. The Fed as a completely separate action lends money to the banks.
Thus last year when people panicked and decided to hoard more, the result was deflation, (not lower interest rates on business loans and mortgages as your model would predict).
For further reading: http://www.winterspeak.com/2009/09/loans-create-deposits-how...
http://www.moslereconomics.com/mandatory-readings/soft-curre...
http://reservedplace.blogspot.com/2009/04/easing-understandi...
And the comment thread between RebelEconomist and JKH here: http://blogs.cfr.org/setser/2007/05/11/rising-deficit-in-the...
These folks actually work in banking, and make their living understanding how the system works. They are very much worth reading.
The problem with the hoarding scenario is that if you know that a greater than X percent of people are hoarding, then you better too because the economy will do worse for it, and you better have a hedge when that happens. This is called a deflationary spiral.
With fiat currency, the central banks can attempt to discourage hoarding by printing money. However, the amount of money destroyed by the recent financial crash has nearly outpaced the Fed's ability to print money.
"“Saving,” in short, in the modem world, is only another form of spending. The usual difference is that the money is turned over to someone else to spend on means to increase production. So far as giving employment is concerned, Benjamin’s “saving” and spending combined give as much as Alvin’s spending alone, and put as much money in circulation. The chief difference is that the employment provided by Alvin’s spending can be seen by anyone with one eye; but it is necessary to look a little more carefully, and to think a moment, to recognize that every dollar of Benjamin’s saving gives as much employment as every dollar that Alvin throws around."
Spending increases the velocity of money ( http://en.wikipedia.org/wiki/Velocity_of_money ) which can, in the short term, increase employment.
Hate to comment on something submitted so many times, but the above use of "hoarding" no longer applies. There was a time when people actually took physical money or gold and hoarded it...hid it away. With the gold standard, this amounted to a drop in the money supply. Such deflation has well-understood problems. It isn't possible now. When you "hoard" money in your BAC account, BAC loans it out at anywhere from 5 to 25 percent interest and life goes on. This isn't the 1930s.