### Dynamic Time Warping -or- The Problem of Monetary Economics

8/10/2012 01:34:00 PM
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I stumbled across this post at one of my new favorite blogs (it is about math and programming). It reminded me of a long standing problem in modern models of money.

There are two basic (non-trivial) ways to include money in a macroeconomic model. The first way is to impose a cash-in-advance (CIA) constraint, which says basically that people's nominal spending this period must be less than or equal to the nominal stock of money they deposited in their checking accounts at the end of last period. This attempts to capture the notion that people can only buy things with money, instead of, say, lugging your couch to the grocery store to exchange for food. But if we simply impose a cash-on-hand constraint then people will always maintain a zero balance in their checking accounts--they will only acquire cash at the moment they need to spend it, which is clearly not what happens in the real world. So to get around this we constrain the model so that the money has to be acquired at least one period in advance. But this too conflicts what we see in the data: it implies that the velocity of money is constant over time, because money only changes hands once per period.

The alternative to the CIA model is a money-in-utility (MIU) model, which removes the cash in advance constraint, and instead pushes real money balances into the utility function. A handy interpretation of this is to say that people value having a positive real money balances (or more precisely that they hate, hate, hate not having any cash on hand, since a necessary condition for the model to work is that the utility from zero real balances is negative infinity), although this is not a necessary interpretation--the utility function is a description of people's behaviors, so all that putting money balances in the utility function says is that people do hold real money balances, regardless of whether that means they wanted or prefer to. The MIA model gives us a result in which the velocity of money is variable, and can be calibrated to approximately match what we find in the data.

Now, I'm going to assert that these two models are actually equivalent to each other, even though we have constant velocity of money in one, but variable velocity in the other. The reason I'm willing to claim this is that it all comes down to how we, the economists, define the length of one period in the model. In the data that we test our models against, one period is measured in terms of units of real-world time (a month or quarter, for example), but this need not be so--instead we could measure periods in units of, say, money velocities. This is, in my opinion, exactly what the CIA model does: one period in a CIA model includes everything that happens in the time it takes for a dollar to be spend exactly once. That is, CIA-time is real-world time but warped, much like how time works in Einstein's relativity. In this case, CIA-time slows down when the MIA velocity of money is high, and speeds up when the MIA velocity is low.

I've been thinking lately on what it would take to prove my hypothesis that the two models are identical (ok, technically, I mean that a generalized form of the CIA model is equivalent under certain restrictions to the a specific class of MIA models). However, if I'm reading the implications of that post over at Math ∩ Programming correctly, that could be way harder than I thought (Einstein, you had it easy!).

Addendum: Why do we care? Analytically, the MIA model is much better at explaining the data than the CIA model. In my view this is because we measure the economy in MIA-time and not in CIA-time. But, even though the MIA model is far more accurate, it is deeply unsatisfying because it is very hard to microfound the assumption that money is in the utility function. Indeed, while the MIA model accurately describes how much money households have in their checking accounts at any point in time, money is never actually spent in the MIA economy. That's really weird. More over, there is no particular reason for people to care about money balances per se--they care about consumption, not money, so we would like to be able to derive this utility value from real money balances through some type of microfoundation, such as a cash-in-advance constraint. Therefore, proving that there is aversion of the CIA model that is analytically equivalent to the MIA model would provide some much needed microfoundations for monetary macro. Yeah, there is currently a paper showing that you can motivate the MIA assumption with "shopping time," but the fact is that that suffers all the same micro problems as MIA--even in the shopping model money is never actually spent.