# A Simple Lesson About Money and Models

Imagine you are in your high school algebra class and you are presented with the following two equations:

$x + y = 20$
$2x + 10y = 100$

Two linear equations with 2 unknowns. This is a simple problem to solve.

Now suppose that your teacher gives you the following three equations:

$x + y = 20$
$2x + 10y = 100$
$x + z = 5$

Note that this is still a simple problem to solve. The first two equations are identical to the previous example. You can use those first two equations to solve for x and y. Then, knowing x, you can solve for z. The central point is that the third equation is not important for determining the value of x. The first two equations are sufficient to solve for x and y.

So why am I bringing this up?

This is precisely how the benchmark New Keynesian model deals with money. The baseline New Keynesian model does not include money. The model is complete and a solution exists. Subsequently, to examine whether money would be important in the model, a money demand function is added to this system of equations. There is a solution to the model that exists. Money is then shown to be irrelevant in the determination of the other variables. But, then again, so was z.

UPDATE: I have updated the post to read “benchmark New Keynesian model” to reflect the fact that some have attempted to integrate money into the NK model in other ways, specifically through non-separable utility. This is, in fact, where I am going to take this argument in the future. Nonetheless, for now, see the excellent comment by Jonathan Benchimol below with some links to his related research.

### 8 responses to “A Simple Lesson About Money and Models”

1. Dear Josh,
you forgot to mention that this post, and its conclusion, is true for a NK model derived with a separable household’s utility function.

However, it does not hold when non-separability between money and consumption is assumed in the household’s utility function of the NK model: https://doi.org/10.1016/j.jmacro.2011.10.003

Although other methodologies are available (for instance, cash in advance) for NK models, they all lead to the same result: a fully interdependent 3 equations NK model. In fact it is a 4 equations NK model (inflation, output, interest rate and money equations… then asking the question about introducing money in the central bank reaction function).

Doing so, money influences other variables during crisis but less before and after. Introducing money in a non separable utility function also improve forecasting performance during crisis: https://doi.org/10.1017/S1365100515000644

This is also true for Israel: https://doi.org/10.1016/j.jpolmod.2015.12.007

Another way is to introduce money held by households and firms through a separable household’s utility function. It then allows for a fully interdependent system of equations with a role for money: http://dx.doi.org/10.4284/0038-4038-2011.197

Well, a very interesting field indeed 😉

Best,
Jonathan

• You are correct, but you are getting a bit ahead of me. I was going to discuss the implications of this in a subsequent post and discuss the implications of and the (sometimes questionable) microeconomic foundations of non-separability. You will be happy to know that David Beckworth and I cite your work in a project we are working on along these lines.

Nonetheless, you are correct that I should have said “benchmark New Keynesian model” in the post. I will fix it.

And thanks, it is always an honor to be cited by people like you and David.
Best,
Jonathan

3. Nick Rowe

Josh: Yes.
We need a name for this. Where M is caused, but does not cause.
“Endogenous” (of course) is not the right name (though I suspect that when some people talk about “endogenous money” this might be what they mean.
“Epiphenomenon” might be a better name, but it’s hardly helpful for most people’s understanding.

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5. mulp

Money is irrelevant to the economy as long as workers have a means to produce and consume goods, services, capital aka durable goods.

Money is merely a proxy for past or future labor.

E.g., imagine robots can replace humans in every aspect including producing new robots. Given robots are not consumers, but capital that merely produces goods, services, and capital, the money price to humans must be zero. Unless the robots pay money to consumers so consumers can buy what robots produce.

If consumers need to work for money, the only ones they will work for are other consumers, which will not be robots which produce everything without labor. If money is defined as what you pay robots who don’t pay humans, then people will create an economy without money, only trading labor for labor in the form of goods and services.

Economies are zero sum. Costs must equal incomes, so costs minus income equal zero. If just one person in an economy consumes less than they earn in terms of dollars, they will end up with all the money, unless money is created from nothing, ie with zero value. Ie, the money is paid for not producing just to balance consumption and production.