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:

P_tC_t \le \left( a_1M_{1,t-1}^{\frac{\phi-1}{\phi}}+a_1M_{2,t-1}^{\frac{\phi-1}{\phi}}\right)^{\frac{\phi}{\phi-1}}

a_1,a_2>0\\
\phi\in\mathbb{R}

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.

You are combining two types of money using a CES-function, so the intuition for \psi\to\infty is correct and it also seems mathematically correct (except for the a_1 appearing twice; fixing those shares is a bit involved). Strictly speaking, I don’t think this qualifies as cash in advance where you stipulate that you need to have all means of payment in the full amount in advance. Here, you instead seem to have some “payment technology in advance”. But that distinction may not be overly important.

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Thanks for the answer! You’re right actually I meant the second one to be a_2.