## Saturday, August 29, 2015

### Scott Sumner's information equilibrium model

Earlier this month, Scott Sumner started a blog post off with an identity using the price level and the money supply:

$$P = \frac{M}{M/P}$$

and identified the numerator with the money supply and the denominator with real money demand allowing some conclusions to be teased out.

This is an information equilibrium model. We have the price of money being $1/P$ (a higher price level means money is less valuable -- see Sumner here), so this is inverted from our usual formulation:

$$\frac{dD}{dS} \equiv p = k \; \frac{D}{S}$$

In this format, we'd write (with $k = 1$):

$$\frac{d(M/P)}{dM} \equiv \frac{1}{P} = \frac{M/P}{M}$$

Note that the differential equation is the correct marginal thinking in traditional economics -- the price of money should be the marginal exchange rate of a unit of demand (i.e. a unit of utility) for a unit of supply. This equation appears in Irving Fisher's 1892 thesis, and the information equilibrium model is only a minor generalization of it (adding the $k$).

This is also the simplest differential equation consistent with long run neutrality of money (homogeneity of degree zero in supply and demand), Bennett McCallum's definition of the quantity theory of money [1].

Now let's follow this trolley all the way to the end of the line.

First, let's replace $M/P$ with money demand $D$ and solve the differential equation (in general equilibrium where $D$ and $M$ vary):

$$\log \left( \frac{D}{D_{ref}}\right) = \log \left( \frac{M}{M_{ref}}\right)$$

$$\frac{D}{D_{ref}} = \frac{M}{M_{ref}}$$

where the $ref$ identifies constants introduced in integrating the differential equation. If we substitute Sumner's form for $D = M/P$ (or just by substituting the above equation in the definition of the price), we can show

$$\frac{M}{P D_{ref}} = \frac{M}{M_{ref}}$$

$$\frac{1}{P} = \frac{D_{ref}}{M_{ref}}$$

$$P = \frac{M_{ref}}{D_{ref}}$$

i.e. the price level is constant in general equilibrium and there is no inflation. That's what we'd expect from $k = 1$.

[Update 8/31/2015: There was a sign error in the equations and incorrect discussion in these last paragraphs which should have referred to movement of the supply and demand curves ($X_{0} \rightarrow X_{0} + \delta X$), not movement along them (changes in $\Delta X = X - X_{ref}$). I marked the two changed sentences with an initial *. H/T Tom Brown in comments below.]

Next, let's check out partial equilibrium. We can solve the differential equation (constraining $D$ or $M$ alternately to be slowly varying around $D_{0}$ and $M_{0}$, respectively) to arrive at:

$$P = \frac{M_{ref}}{D_{0}} \exp \left(+ \frac{D - D_{ref}}{D_{0}}\right)$$

$$P = \frac{M_{0}}{D_{ref}} \exp \left(- \frac{M - M_{ref}}{M_{0}}\right)$$

which are supply and demand curves (this is essentially the same as the AD-AS model in the information equilibrium framework). *In partial equilibrium, an increase in demand for money (a shift in the demand curve, $D_{0} \rightarrow D_{0} + \delta D$) leads to a rise in the price of money ($1/P$) and a fall in the price level ($P$). *An increase in supply of money (a shift in the supply curve $M_{0} \rightarrow M_{0} + \delta M$) leads to a fall in the price of money ($1/P$) and a rise in the price level ($P$). In a sense, we only get inflation (or deflation) from changes in money demand and money supply. However, since $k = 1$, these should go away in the long run and inflation should be constant -- given the general equilibrium solution.

In Sumner's post, he says that $D$ is actually real GDP and "other stuff", thus we come to the conclusion that growth is deflationary.

So what about that "other stuff"? Well, in the information equilibrium model, we put the other stuff in two places -- (exogenous) nominal shocks and a changing $k$. It's the latter that becomes Sumner's $V$ (velocity in the quantity theory of money). And the best place to see that model is in my draft paper available here.

Footnotes:

[1] Long-Run Monetary Neutrality and Contemporary Policy Analysis Bennett T. McCallum (2004)

1. Jason, remind me: what's the difference between p and P?

1. P is price level
p is a generic price

2. I'm good down to here:

"Next, let's check out partial equilibrium. We can solve the differential equation (constraining D or M alternately to be slowly varying around D0 and M0, respectively) to arrive at:"

So the 1st equation under there, D is constrained to slowly vary around D0. What about M? Is M held constant? Why is "slowly vary" significant?

There's only one differential equation I see there involving P: the one right after this:

"In this format, we'd write (with k=1):"

so that must be the one. You're also folding in P = Mref/Dref I guess. This is where I've been lost before I think.

1. This is where I can just refer you to the paper. It's also the same partial equilibrium solution that's on the second or third post on this blog.

dD/D = dM/M

Integrate

dD/D0 = dM/M

Or

dD/D = dM/M0

2. Makes sense, thanks!

3. Following the development of eqs. 13 and 14 in your paper:

P = M/D0

dD/D0 = dM/M

(1/D0)*{integral from Dref to D of dD'} = integral from Mref to M of dM'/M'

(1/D0)*(D - Dref) = log(M) - log(Mref) = log(M/Mref)

exp((D - Dref)/D0) = M/Mref

(Mref/D0)*exp((D - Dref)/D0) = M/D0 = P

I'm missing a minus sign on the argument to exp(). Where did I go wrong? Likewise (I think) I end up with a sign reversal on your other equation:

P = (M0/Dref)*exp(-(M-Mref)/M0)

4. I read about the fall process of a body in a gravitational field in here:
http://arxiv.org/pdf/0905.0610v4.pdf
Eq. 34 defines R_Planet. So the delta time and delta length reference variables (with the subscript "ref") are thus not arbitrary, but represent some solution up to that point: the object was observed to have fallen delta l meters in delta t amount of time. I think I asked you elsewhere if the "ref" variables were arbitrary. Clearly not then (if your problems here are analogous).

I'm still having trouble with |delta y| const in eq. 20 and definition 4. Is there a way to relate the constant-restriction-part (giving rise to the solution with the exponential) to the simple gravitational example? The part I always find troubling is that |delta y| const = a constant, yet |delta y| seems to be written as if it's varying right there in the differential equation. Also your Taylor series expansion near a Yref, ... but what about Y0?... which are we closer to? I have a hard time thinking of what these mean in a concrete way, but the gravitational example is simple and I can see what's going on there.

5. I've gone back to re-read your April 2013 posts. You answered some questions then. I see you used Qref = Q0 = 1 in your example plots in

"Supply and demand from information transfer"

6. Hi Tom,

I think you are right; in the process of thinking Sumner's discussion was correct I confused myself between e. g. shifts of a demand curve and shifts along a demand curve. I will fix this and the discussion. Actually this is related to your questions about D0 and Dref.

Also when F&B say floating and constant restriction, you should think general and partial equilibrium.

7. (That is you are right about the sign error.)

3. Jason, sorry for my abundance of stream of consciousness comments. Erase any that you like. (I left a few back on some of your April 2013 posts as well, especially the supply and demand post). BTW, you should check out Cochrane's latest: he get's all "mathy"

1. That's fine Tom. Actually, a lot of the earlier posts never got any "peer review" from comments, so it could definitely help clear things up in some cases.

The info eq model can't have a permanent one-time increase in the interest rate with a finite RGDP, but in general the effects are somewhat similar under certain conditions (i.e. kappa dependent):

http://informationtransfereconomics.blogspot.com/2014/09/the-liquidity-effect-and.html