I've been learning about the IS-LM model recently, and in the long run, a rise in M will cause P to rise such that M/P is unchanged. This will then mean that Y/V(i) is unchanged, since M/P = Y/V(i). Hence, a given level of i will correspond to the same level of Y as it did before the rise in M, so the LM curve will simply return to its original position, leaving Y and i unchanged.
Now, thereal interest rater is decided by the willingness to save and invest - it is decided by real factors, not the level of M. Hence, r is unchanged.
However, inflation has obviously risen, since P rose to cancel out the initial rise in M. Surely this violates the identity i = r + inflation????? If i and r do not change, but inflation does, then the identity cannot hold! What on earth am I doing wrong here??
Thanks so much in advance