Subscripting Helps!

The accounting identities equating aggregate expenditures to production and of both to incomes at market prices are inescapable, no matter which variety of Keynesian or classical economics you espouse. I tell students that respect for identities is the first piece of wisdom that distinguishes economists from others who expiate on economics. The second? … Identities say nothing about causation.

– James Tobin, 1997, p. 300, ‘Comment’, in B.D. Bernheim and J.B. Shoven (eds), National Saving and Economic Performance, Chicago: University of Chicago Press.

This is a continuation of my post Stock-Flow Inconsistent? which was a reply to Jason Smith’s blog post More like stock-flow inconsistent on his blog Information Transfer Economics. If you had checked my post before around noon UTC yesterday, you might want to check the updated version.

Jason Smith also has updated his post and proposes a new equation:

ΔH = Γ·(G – T)

(incorrect equation)

Now, that’s quite wrong because it violates rules of accounting.

Morever, Jason Smith insists that it is a behavioral equation.

A lot of clarity can be achieved if one uses subscripts, so that things are clearer.

So we have two equations:

ΔH = GT

dH/dt = GT

Although these two are related, they are not exactly the same: the former is in a difference equation form and the latter in the differential equation form. The in the former has no time dimensions and the in the latter has time dimension equal to –1. The in the former is total expenditure in a period, the in the latter is a rate.

Since stock-flow consistent models are written typically in difference equations, rather than differential equations, let us avoid subscripts for difference equations for the former and use it for latter.

So it is better to write the equations as:

ΔH = G – T

dHcontinuous/dt = Gcontinuous – Tcontinuous

Each time step in the formalism of difference equations is Δt and hence

G = Gcontinuous ·Δt

T = Tcontinuous ·Δt

Hcontinuous = H

So,

ΔHt = Gcontinuous – Tcontinuous

(approximately)

Or,

ΔH/ Δt = Gt – Tt

Or,

ΔH = G – T

So instead of reaching the correct equation which is:

ΔH= Δt · (Gcontinuous – Tcontinuous)

Jason reaches the equation:

ΔH = Γ·(G – T)

(incorrect equation)

But

ΔH = G – T

as it is an accounting identity in the model!