My current understanding of the banking money multiplication process goes as follows: Alice comes along and deposits 100 cash into Bank A. Bank A gains 100 in vault cash (reserves) as an asset and 100 in liabilities in the form of Alice's checkable deposit. Bank A then loans out 90 (they cannot loan more since RR=0.1) and so the banks asset are 10 reserves and 90 loans and everything balances. The individual who gets the loan from Bank A takes the 90 and deposits in say Bank B and the multiplying process continues until total nation-wide deposits=10*reserves=$1000.
But now my textbook (Mishkin) states that
Because a single bank can create deposits equal only to the amount of its excess reserves, it cannot by itself generate multiple deposit expansion.
He then goes on to say that
A single bank cannot make loans greater in amount than its excess reserves
as well as
the banking system as a whole can generate a multiple expansion of deposits, because when a bank loses its excess reserves, these reserves do not leave the banking system, even though they are lost to the individual bank
But this doesn't make sense to me at all. I see no reason why Bank A in my earlier example is not able to multiply deposits all on its own. Say that in my example, the original 90 loan goes out to Bob to finance one of his business endevours. Bob does not want 90 in cash so instead he opens up a checkable deposit account at bank A and so bank A's reserves go back to 100 with 90 in loans and now they have deposits =190 . Clearly they have excess reserves. So they now make another loan to Mary equal to to 81 who then also does not wish to hold cash and so she stores the money in a deposit account at bank A. This process then repeats as Bank A makes loans and each borrower gets a deposit at bank A until bank A has 100 in reserves, 900 in loans and 1000 in deposits.
Thus Bank A, a single bank, has generated multiple deposit expansion. But Minkish seems to say this is only possible if the borowers deposit their money in other banks instead of the original lending bank. So is he wrong or am I wrong and missing something here? Any help on this would be most appreciated!