On the topic of what is the monetary base...that is pretty easy. It is the supply of money that government directly created. Coins, dollar bills, electronic dollars (confusingly conflated with terms such as reserves or federal reserve credit) are all included in MB. It is absolutely a measure of the money supply, albeit a narrow one because it doesn't include non-government sources of money. We exchange gold for services, bank deposits, savings deposits and more for what we want... That is why higher monetary aggregates that include deposits are more accurate in determining the supply of money to the economy and therefore its affect on inflation.
Now the supply of MB is still very important because the banks can only use government money to meet reserve requirements and withdrawals which makes them somewhat unique. The supply of MB helps determine how much banks can in turn leverage that to multiply their own bank deposits (non-government money). A high ratio between M2 and MB may mean we're on the verge of a collapse. A low ratio may mean we're on the verge of hyper-inflation as banks will surely multiple MB to the max once possible.
On the subject of the money multiplier...this is a very important idea that needs to be understood to knowing how banks work, create money and create inflation/bankruns/bailouts. However in a modern economy, things are done in a slightly different fashion (although still fraudulent and they still multiply the money supply).
A modern bank (and they've done studies on this) is not really constrained with lending by its reserves...nor it is overly in danger by withdrawals. This is because banks have what is called the money market which is an effective cartel. If I want to loan you 100 million and I think you'll pay it back...I make the loan as the loan officer. The asset manager will then borrow that 100 from money center banks. But wait...what if the money center banks don't have that money? They always will! That is because the money center banks borrow from the Primary Dealers and the Primary dealers 'borrow' (or sell assets) to the Fed to get WHATEVER reserves they need. In fact the whole system is on auto-pilot so banks in effect determine the supply of MB...the tail wags the dog.
Perhaps a modern multiplier example would look like this.
Bank A incorporates with $1,000 equity and $1,000 cash.
Customer A wants and gets a $10,000 deposit in exchange for a $10,000 loan.
Customer A writes a check for $10,000 to Vendor X who deposits at Bank B
Bank A borrows $9,000 from Bank B through the money market to meet its withdrawal
Bank A ends up with 10k in loan assets and 9k in loan liabilities and 1k in equity
Bank B ends up with +10k in deposits and +1k reserves and +9k in loan assets
In this fashion you can clearly see how 1k of money create 10X without the steps illustrated by most econ 101 texts with the money multiplier. But wait...how did Bank A's withdrawal clear if they didn't have they money. Well ultimately Bank B financed it (or any other indirect banking agent thought the money market). But more immediately this apparent contradiction was made possible by Fed float and lax clearing rules. But this is not significant. Even if the Fed played had very strict requirements, the banks could use a small percentage of equity to grease the distribution of reserves and loans to each other. That or the bank temporarily converts the deposits to near deposits and then back again. Either way they multiply the money supply with impunity and efficiency.
So if the banks max on the reserve ratios...they can still create deposits...they just borrow the money from the fed (mostly indirectly) to get the reserves they need.
To understand the banking system you have to imagine it has one entity...that takes in all MB and creates say 5X of M2. The money market is the circulatory system that distributes the MB throughout the body to the various organs. And the Fed is the intravenous drip. When the banks stop lending to each other because of credit concerns...then body gets quite sick...and the Fed tries to inject more fluid into the body.
One last note...while the conventional money multiplier considers really demand deposits (M1) and base money (MB) this is not an accurate picture of the economy because many of the deposits held by businesses and the wealthy are 'near-deposits'. They yield a small percentage of interest and/or have slight time restrictions on their withdrawals. Banks also play games by shuffling what should be demand deposit into near-deposits to skirt reserve requirements...truly one has to look at the higher aggregate to understand the banking system as a whole.
