These two first-year textbook models -- the simple money supply multiplier model; and the simple keynesian income-expenditure multiplier model -- are formally identical. Translated into math, or game theory, you can't tell the difference between them. They contain exactly the same important insight: that what is true for the individual bank/household is not true for all banks/households. And they both contain the same flaw: they ignore interest rates.
This is really just a rhetorical device, designed for people who understand and appreciate the keynesian multiplier but not the money supply multiplier. Or vice versa.
Assume a closed economy, with no government. For "bank" read "household" throughout. The other translations are obvious (S=savings, I=investment, Y=income, C=consumption).
Let S be desired reserves. Let Y be deposits. Assume desired reserves are a constant fraction s of deposits: S=sY. Let I be the supply of reserves. In equilibrium the supply of reserves must equal the demand for reserves (desired reserves equal actual reserves), so S=I. In equilibrium therefore, deposits Y must adjust so that sY=I, which means we get the multiplier Y=(1/s)I. Just to complete the model, recognise that accounting reminds us that loans C plus reserves S must equal deposits Y, so that C+S=Y=C+I.
One important insight of this model is that the supply of reserves I determines desired reserves S, not the other way round. Deposits Y adjust to ensure that desired reserves S equals the supply of reserves I.
A second important insight is the difference between the individual bank and the banking system as a whole. An individual bank, one that is small relative to the system as a whole, will take deposits Y as given. The individual bank that increases loans C will suffer a dollar for dollar reduction in reserves S. But for the banking system as a whole, an increase in loans C will cause an equal increase in deposits Y, with no change in reserves S.
If banks are at less than "full employment", they could all be better off if they jointly increased loans C and deposits Y. They don't need extra reserves S to do this. And aggregate reserves S would not decrease if they did this. But this expansion of loans C and deposits Y can never happen, without an increase in the supply of reserves I. That's because any individual bank that increased its own loans C would see no corresponding increase in its own deposits Y, merely a reduction in its reserves S. All the benefits of increased loans C would be an increased level of deposits Y at other banks.
People who miss this important insight, who miss the distinction between desired reserves S and actual reserves I, end up with Say's Law of Banking, which states that there can never be a general shortage of reserves I, so banks must always be at full employment.
Now it is true that one bank's deposits Y are not strictly speaking exogenous to that bank. By offering better terms to its customers, it can increase its deposits Y at the expense of other banks' deposits Y. But that does nothing to increase aggregate deposits Y to the banking system as a whole, and get all banks to full employment. It simply redistributes a fixed amount of deposits Y among the different banks. One bank gets to full employment, but only by reducing employment for other banks. The only way to get to full employment of all banks, without an increase in the supply of reserves I is if all banks decrease their desired reserve ratio s. But no individual bank has any incentive to do that.
So the policy message is clear. The only way to get the total level of deposits Y to that required for full employment of banks is to increase the total supply of reserves I. And then hope that banks don't just increase their desired reserves S by the same amount. Ricardian Equivalence Theory, based on the permanent deposits Y theory of loans C, argues that any transitory increase in the supply of reserves I will indeed be mostly hoarded with a corresponding increase in desired reserves S. So lets make the increased supply of reserves I permanent.