The first rule of Post-Keynesian Economics is: You do not talk make accounting mistakes. The second rule of Post-Keynesian Economics is: You do not talk make accounting mistakes.

– Anonymous.

Jason Smith—who is a physicist—but writes a blog in Macroeconomics, wonders how equations in the simplest stock-flow consistent model given in the textbook Monetary Economics written by Wynne Godley and Marc Lavoie make any sense from a dimensional analysis viewpoint.

He says he

seem[s] to have found a major flaw.

He sees the equation:

ΔH = G – T

and wonders where the time dimensions are. For, H is the stock of money and hence has no time dimension, whereas the right hand side has flows and has time dimensions of inverse of time. For example if the US government spends $4 tn in one year, G is $4 tn/year.

In continuous time, the above equation is:

dH/dt = G – T

So how are these two equations the same?

Perhaps, Jason is not familiar with difference equations. He instead seems to prefer:

τ·ΔH = G – T

Well that’s just wrong if τ is anything different from 1, as a matter of accounting.

Now moving on to time scales, it is true that in difference equations some time scale is implicit. But it doesn’t mean the methodology itself is wrong. Many physicists for example set all constants to 1 and then talk of numbers which are dimensionless.

So if a relativist sets “c=1”, i.e, the speed of light to 1, all velocities are in relation to the speed of light. So if somebody says the speed is 0.004, he/she means the speed is 0.004 times the speed of light.

But Jason Smith says:

Where does this time scale come from over which the adjustment happens? There is some decay constant (half life). It’s never specified (more on scales here and here). If you think this unspecified time scale doesn’t matter, then we can take Δt →l_p and the adjustment happens instantaneously. Every model would achieve its steady state in the Planck time.

That’s not true. String theorists for example set the parameter α’ = 1. But nobody ever claims that macroscopic adjustments happen at Planckian length scales or time scales.

Coming back to economics, there’s nothing wrong in

ΔH = G – T

There’s an implicit time scale yes, such as a day, or a month, or a year, or even an infinitesimal. But parameters change accordingly. So in G&L models we have the consumption function

 C = α₁ ·YD + α₂ ·W

where C is household consumption, YD, the disposable income and W, the household wealth.

Let’s say I start with a time period of 1 year for simplicity. α₂ might be 0.4. But if I choose a time period of 1 quarter, α₂ will correspondingly change to 0.1. In English: if households consume of 4/10^th  of their wealth in one year, they consume in 1/10^th one quarter.

So if we were to model using a time scale of a quarter instead of a year, α₂ will change accordingly.

But the equation

ΔH = G – T

won’t change because it is an accounting identity!

It’s the difference equation version of the differential equation:

dH/dt = G – T

Physicists can pontificate on economic matters. I myself know string theory well. But boy, they shouldn’t make mathematical errors and embarrass themselves!

In other words, accounting identities can be written as accounting identities in difference equations. What changes is values of parameters when one chooses a time scale for difference equations.

Wynne Godley’s model is touched by genius. In fact according to one of the reviewers of Monetary Economics, Lance Taylor says that it is out of choice that Wynne Godley chose a difference equation framework. They can be changed to differential equations and we’ll obtain the same underlying dynamics.

Here’s Lance Taylor in A foxy hedgehog: Wynne Godley and macroeconomic modelling

Godley has always preferred to work in discrete time, responding to the way the data are presented.

Question: is the equation ΔH = G – T consistent with dimensional analysis?

Answer: Yes. H is the stock of money at the end of previous period. ΔH is the change in stock of money in a period. G and T are the government expenditure and tax revenues in that period. So H, ΔH, G and T have no times dimensions in difference equations. All are in the unit of account. Such as $10tn, $400bn, $4 tn, $3.6tn. Time dynamics is captured by model parameters.

In G&L’s book Monetary Economics, in Appendix 3 of Chapter 3, there’s a mean-lag theorem, which tells you the mean lag between two equilibrium (defined as a state where stock/flow ratios have stabilized):

it is:

[(1 − α₁)/α₂ ]· [(1 – θ)/θ]

where θ is the tax rate.

So, in the model, assuming a value of 0.6 for α₁, 0.4 for α₂, and 0.2 for θ we have the mean-lag equal to 4.

Let’s assume that time period is yearly. This means the mean lag is 4 years.

If instead, we were to use quarterly time periods, α₂ would be 0.1 and the mean lag evaluates to 16, i.e., sixteen quarters, which is 4 years, same as before.

So there is really no inconsistency in stock-flow consistent models.

tl;dr summary: In difference equations, there’s nothing wrong with equations such as ΔH = G – T. It is an accounting identity. By a choice of a time scale, one implicity chooses a time scale for parameter values. What’s wrong? Jason Smith would obtain the same results as the simplest Godley/Lavoie model if he were to work in continuous time and write equations such as dH/dt = G – T. I will leave it to him as an exercise!