What if only the government could create money?
When you dig down, this questions is very nearly a duplicate of this:

Implications of abolishing Fractional Reserve Banking on mortgages and interest rates

but based on the comments, it seems that the connection isn't obvious.

This question asking about what would happen if only a government institution could create money, but equally you say the proposal is not to have a fixed money supply.  They are, in practice, the same thing.

The central bank makes the money base, M0, and then the banks create the money supply, with some multiplier to this.  In a full reserve system, that multiplier is 1.0, and so the money supply is always equal to the money base.

In your question, we consider having a M0, but also have a state run bank that chooses the money multiplier to some number, that may or may not be 1.0.

From that point on things become almost identical - it doesn't really matter from the point of view of the economy, whether you make M0 ten times larger or smaller, if you offset it with changing multiplier.  It should be the total money supply that matters - the money base never leaves the central bank, and so may be thought of as a "construct" onto which the monetary system is hung.  The important point is that the government gets to directly control the money supply.

Before the era of quantitative easing, when a central bank wanted to increase the money supply, they lowered rates and that increased the multiplier.  With quantitative easing, they simply increase M0 and force the money supply up in a 1:1 ratio.  Either works (for a given value of "work").

There is, however, one very important difference:


With full reserve banking borrowing money is very expensive - it is economically the same as capital
With a state run lending bank, borrowing could be done at any rate the government chooses, including at an economic loss.


So, in your system, you could have a state run bank, that effectively subsidises business and/or people's mortgages using money raised from taxes or elsewhere.  This would be like an extreme version of the UK forcing RBS and Lloyds to increase lending.