Suppose an single bank in a competitive banking system wanted to expand its lending, perhaps by holding a larger portfolio of bonds. It would need to obtain funding. In perfect competition, it would pay the going interest rate on deposits. It would expand its balance sheet until the marginal cost of providing intermediation services equals to interest margin.
However, I never think in terms of perfect competition. With imperfect competition, an individual bank expanding its balance sheet would both charge less on loans and pay more on deposits. The interest rate margin is driven down as the marginal cost of providing intermediation services rises. When marginal revenue equals marginal cost, profits are maximized.
To me, Williamson's account of quantitative easing is similar. A central bank wanting to expand its portfolio of long term bonds, perhaps because it wants to drive down the yield on them, must provide a higher yield on its liabilities. Of course, a single bank is mostly seeking to gain market share at the expense of other banks offering very similar liabilities. A central bank in a closed economy is competing with other sorts of assets, and so the liquidity premium on its liabilities are central.
What about deflation?
Consider Hayek's model of banks issuing private currency. Unlike a conventional banking system where each bank accepts the other banks' monetary liabilities for deposit at par and settle net clearing balances, Hayek's system had each bank's monetary liabilities float. He also emphasized a scenario where all the banks seek to stabilize the purchasing power of their monetary liabilities. In other words, the price level in terms of each bank's money would be stable. Perhaps because the private and competitive issue of deposits is conventional, Hayek at least seemed to emphasize the issue of banknotes--hand-to-hand currency. Still, given this picture of the monetary order, any bank could expand it lending--undertake quantitative easing--by paying a higher nominal interest rate on deposits, just as would occur in a more conventional banking system.
Interestingly, Benjamin Klein described the same institutional order as Hayek, but he emphasized the return each bank would provide on hand-to-hand currency by generating inflation or deflation. From that perspective, a bank wanting to expand its asset portfolio would need to provide a increased yield on its currency. A bank would need to lower the inflation rate. In other words, a bank wanting to expand its asset portfolio would generate a higher deflation rate for the currency it issues.
How could a bank actually implement the policy? Obviously, it would announce and advertise its new policy, generate added demand for its liabilities, and purchase the assets it wants. Of course, it would need to adjust its policy to generate the promised rate of deflation.
Consider a bank paying an increased nominal interest rate on deposits. We could imagine the bank just adding funds to deposit accounts. It would hope that its customers would notice this occurrence, and come to expect a higher nominal interest rate. But what bank would do that? They tell their customers that they are paying a higher interest rate and would advertise what they are doing.
Consider then a central bank. If it wanted to expand the size of its balance sheet, and it didn't want to pay a higher nominal interest rate on its reserve deposits, it could instead promise a lower inflation rate. And how would it do that? Step one would be to promise lower inflation or deflation.
Now, imagine a single bank in a competitive banking system that simply purchases the assets it wants and does nothing to provide funding. The result would be adverse net clearings. It would be compelled to take action to pay off its claims at the clearinghouse. With a Hayek/Klein regime, the result would not be an automatic appreciation of the currency. On the contrary, the result would be a depreciation of the currency on the interbank exchange.
And suppose a central bank creates money and buys assets, keeps the interest rate on reserve balances unchanged or even lowers it. What happens? Inflation, not deflation.
Finally, suppose a central bank wants to expand its balance sheet, and it promises lower inflation (even to the point of deflation) to raise money demand, and expands its balance sheet beyond that greater demand to hold money and generates inflation. Suppose it breaks its word? This, of course, is exactly the problem that concerned Klein. Why not undertake an inflationary expropriation?
Larry White insists that this is why banks need to issue redeemable banknotes and deposits. Inflationary default is prohibited by contract. And for a central bank, James Buchanan always insisted that this is why constitutional restrictions of the issue of money are necessary.