If I may be so bold I think the debate over IS-LM boils down to simple miscommunication.
However, if you prefer IS-MP to IS-LM then in my words you just think IS-LM is inconvenient because it’s the same model. To my knowledge there is nothing we could describe in IS-MP that we can’t describe in IS-LM you just have to move the curves more and they represent implicit actions rather than explicit ones.
If you have an MP curve the you can say: And the Fed decides to lower interest rates and its really clear what happens. But, you could also say that the Fed conducts open market operation which increases the supply of reserves and lowers the overnight rate.
As a note, I think it is easiest to call “the interest rate” the overnight rate and for that to be what is depicted on both graphs. The long term rates and “real rates” descend from the overnight rate.
However, in all cases banks stand at the center of the transmission apparatus between Investment, Savings and Monetary Policy and banks look at the overnight rate.
This is also helpful because your investment decisions over the long term must match up against what you think the long term evolution of the overnight rate is. Inflation be damned.
If you know exactly what the overnight rate is going to be over the next several years then you know exactly what your net return on lending needs to be. What the rate of inflation is, is irrelevant.
If you beat the overnight rate you win. If you don’t beat the overnight rate you lose. Absolutely nothing else matters as long as the overnight market is completely liquid. Now, it might be that the path overnight rate is consistent with everyone taking home a wad of cash. Or the overnight rate is consistent with everyone going out of business.
However, that’s really neither here nor there. There is nothing you can do about that. The only thing you can do is beat the overnight rate. Its literally the opportunity cost of not having that cash.