The Interest Rate & Money Supply
You may find the relationship between the interest rate and money supply somewhat confusing if you have studied the standard economics text books, and I suspect the reason for that may be due to a misplaced loyalty to the standard IS-LM model that underpins much of Keynesian economics.
Even Gregory Mankiw (one of the world's most renowned experts on New-Keynesian economics) has argued, in previous years, that control of both the interest rate and money supply is possible. Well, he is correct in a sense, but as I'll show below this gives a very misleading impression of how monetary policy works.
These two variables are directly related, meaning that independent targets for both are not possible. To achieve a desired level for either one necessarily requires that the other will move to whatever level the money market dictates. In the proof below you'll need an understanding of how the the IS-LM model works, so click the link if needed.
The Interest Rate and Money Supply Relationship
In the diagram above lets assume that we live in the Keynesian world where there is spare capacity in the economy to expand income/output without causing any upward pressure on prices. The government in this scenario is able to issue more bonds (raising national debt) and spend more money in order to boost the economy and stimulate growth.
In terms of the IS-LM analysis, the government boost causes more spending in the goods market independently of the existing interest rate, and this is represented by a rightward shift in the IS curve from IS1 to IS2.
Now, with the existing money supply there is a problem. It is impossible to maintain the interest rate at R1 given the government boost to the economy. Why is that? Because with the higher income level of Y2 resulting from the government boost, people will want to increase their bank balances in order to increase their purchasing power in the economy.
As with supply and demand for other products/assets, the increase in demand for money (with a fixed money supply) will put upward pressure on the price of money. And what is the price of money? It's the interest rate that must be paid for borrowing it (or foregone for holding it instead of investing it).
So, even in the Keynesian world, if the government wants to boost the economy then it will need to compromise on one or both of its interest rate and money supply targets. If the government settles for accepting an increased interest rate as a consequence of its fiscal expansion, then it should at least be able to maintain a constant money supply.
If, on the other hand, the government is determined to maintain the original interest rate, then the government could instruct the central bank to increase the money supply in order to satisfy the extra demand for money at the existing interest rate. However, not only would this compromise any claim of central bank independence, it would be a highly controversial move in its own right. It would constitute something called 'monetizing the debt' whereby the central bank purchases the extra bonds that the government has issued.
The controversy here surrounds what most economists would regard as a highly inflationary policy (although the MMT mob that is currently gaining influence in the corridors of power promote exactly this sort of policy).
In terms of the diagram above, an increase in the money supply would be represented by a rightward shift of the LM curve, as illustrated in the right side of the diagram. National income and output in this idealized Keynesian world-view would then grow way beyond the government boosted level of Y2 and go all the way to Y3.
What could possibly go wrong with this you might wonder? The answer is that we don't live in this sort of Keynesian world, we live in a world where inflation will result from such expansionary fiscal and monetary policy, and I turn to that in a later section. First though, we need a few words to explain the difficulty of estimating the interest elasticity of demand for money.