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Until now I've learned that the Fed controls the interest rates and that affects the money supply. For instance,

The Fed increases Interest Rate ----> makes firms/households reluctant to borrow and Banks loan less ---> less money supply.

However, what is the economic intuition for the opposite sequence? So

"Fed increase money supply ----> interest rate decrease" "Fed decrease money supply ----> interest rate increase"

I have never thought of this sequence - I have always thought of the interest rate as the causal factor and the money supply change as the result. I would appreciate it if somebody could explain how the opposite sequence is also possible. So to put it simply,

if Fed increases/decreases the money supply, why does interest rate decrease/increase? not in economic theory, but in real-life explanation please.

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The answer to this question is complicated because the way the Fed operates has changed over time. In particular, Quantitative Easing (QE) changed things a lot.

Firstly, the Fed largely controls the Fed Funds rate, which is an overnight inter-bank rate. Other interest rates are set as a spread to that rate, but are not directly controlled by the Fed. Instead, those rates are generally set by what market participants expect the Fed to do.

Secondly, we need to be careful about what “money supply” you are talking about. The Fed directly operates on the monetary base, since that roughly corresponds to the size of its balance sheet. Other monetary aggregates include other instruments that the Fed cannot directly control.

In the pre-QE era, the Fed could control interest rates by creating/destroying bank reserves - a major component of the monetary base. Banks lend reserves to eachother in the Fed Funds market. If there is a shortage of reserves, banks will bid up the cost of reserves - the Fed Funds rate. The Fed creates reserves by buying bonds (or lending against them),and withdraws reserves by selling them. They need to conduct operations to keep interest rates near target.

However, the Fed is effectively forced to supply whatever reserves are needed by the banking system, otherwise they will fail their reserves requirements. Central bankers started listening to Monetarists in the 1970s, and tried setting their interest rate target so that money growth would hit a target rate. However, since they did not publicly acknowledge that they had a target interest rate, it appeared that they were targeting the money supply.

In the post-QE era, there is an excess of reserves in the system, so there is no issue of banks being short. Interest rates are instead determined by the level of interest paid on reserves, which can evolve completely independently of monetary aggregates.

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It’s just supply and demand. The interest rate is the price at which you can borrow money. For a fixed demand curve, if there’s less money available (the supply curve shifts left), the market will clear at a higher price (higher interest rate). Similarly, if there’s more money available for lending, the equilibrium interest rate will be lower.

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In the real world, the Fed actually just sets a target for Fed Funds rate - which is the rate at which US Banks lend to each other for overnight funds (1 day). It uses Open Market Operations to manage the Fed Funds Rate at its desired target level.

Each FOMC meeting sets a target for the fed funds rate. It can't force the banks to use its targeted rate. Instead, it uses open market operations to push the fed funds rate to its target. 

If the FOMC wants the rate lower, the Fed purchases securities from its member banks. It deposits credit onto the banks' balance sheets, giving them more reserves than they need. It forces the banks to lower the fed funds rate so they can lend out the extra funds to each other. That's how the Fed lowers interest rates.

When the Fed wants rates higher, it does the opposite. It sells its securities to banks and consequently removes funds from their balance sheet. This gives banks fewer reserves which allow them to raise rates. Since 2015,  the Fed has been raising interest rates. It does this to control inflation.

Source

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So first, let's understand that the Fed Funds Rate which the Fed had dropped 3% back in early 2008 is the rate charged to banks, not consumers. While there is a correlation between the Fed lowering rates and the cost of credit to various financial players, it is virtually nonexistent in terms of consumer credit.

A factor that hardly ever gets asked in this forum when someone poses the question of interest rates is the role that supply and demand for debt plays. I can understand why, as an individual you are thinking, who the heck wants to buy debt?

Well, it happens everyday, institutions are buying and selling debt all day long.

So if we are in a period, I am haven't checked recently, where the supply of available credit has shrunk. Then it doesn't matter if the Fed raises or lowers the coupon rate. The cost of credit would be rising regardless.

Now, why are they talking about raising the coupon rate now? Here is the real-world multiple objectives of the Fed in raising the coupon rate:

  1. It's attempting to preserve its credibility, which is threatened by a zero-interest-rate forever policy.
  2. It's attempting to maintain its political capital, which is eroding as even the mainstream media has accepted the reality that Fed policy has enriched the already-wealthy at the expense of everyone else.
  3. It's attempting to "normalize" interest rates without killing its favorite child, the stock market.
  4. It's attempting to raise rates without upsetting the fragile global currency/debt shopping cart.
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