Hello. I’m looking forward to model two varieties of money (M_{1,t} and M_{2,t}) in a CIA framework, my question is whether this change to the CIA constraint is correct:

And therefore substitutability between the two varieties is ruled by \phi (e.g. \phi\to\infty would correspond to a regular CIA with two monies and perfect substitutability between them). Since now this seems something more like a “transaction services” production function (with maybe a duality relation to models with transaction costs). Is this mathematically correct? If so does this model still being a CIA? Also, perhaps this has already been developed, I’d be very grateful if you could provide me some references if you know.

Thanks in advance.