The impact of quantitative easing

There is a theory that I read (I don't remember where), that quantitative easing has no impact at all, because it is only a change on the Fed's balance sheet.  I disagree.

So let me set the stage.  There are two types of money, type A and type B.  (There are actually more but I want to keep this simple).  Type A is central bank money.  Cash in your hand is type A.  Bank reserves parked at the Fed are also Type A.  Type B is bank liabilities.  Your checking account is type B.  They are interchangeable.

Now consider two different scenarios.  First, a bank, call it TBTF Bank, owns a 30 year treasury bond for $1 million that they agree to sell to the Fed.  The Fed writes them a check for $1 million.  (It would actually be a wire transfer, but bear with me).  The check is type A.  TBTFB would deposit is in its account with the FRB, and this would be excess reserves.  In this scenario, it is true that QE would have no impact.  The excess bank reserves serve the same purpose as the bond did and the only difference is the reserves pay less interest.

Second scenario.  A hedge fund owns a bond that they sell to the Fed for $1 million, and they get a check.  The hedge fund doesn't have a bank account with the Fed.  The check is type A.  The hedge fund deposits it in their account with TBTF bank.  So what happens here?  The bank in turn deposits the $1 million in their account at FRB.  So the bank's assets (the FRB deposit) and their liabilities (the hedge fund deposit), both go up by $1 million.  Type A money goes up by $1 million, but so does type B money.  So the Type B money goes up dollar for dollar with the type A money.  The hedge fund is now free to buy stocks or whatever else they want with the money.

Conclusion:  Quantitative easing (the expansion of the central bank's balance sheet) increases the money supply dollar-for-dollar, unless it results in excess reserves being stored with the central bank.