# Information equilibrium as an economic principle

Jason Smith Associate Technical Fellow, The Boeing Company. P. O. Box 3707,
Seattle, Washington 98124. Email<EMAIL_ADDRESS>
###### Abstract

A general information equilibrium model in the case of ideal information
transfer is defined and then used to derive the relationship between supply
(information destination) and demand (information source) with the price as
the detector of information exchange between demand and supply. We recover the
properties of the traditional economic supply-demand diagram. Information
equilibrium is then applied to macroeconomic problems, recovering some common
macroeconomic models in particular limits like the AD-AS model, IS-LM model
(in a low inflation limit), the quantity theory of money (in a high inflation
limit) and the Solow-Swan growth model. Information equilibrium results in
empirically accurate models of inflation and interest rates, and can be used
to motivate a “statistical economics”, analogous to statistical mechanics for
thermodynamics.

_Keywords:_ Information theory, macroeconomics, microeconomics

_Journal of Economic Literature Classification:_ C00, E10, E30, E40 _._

##  1 Introduction

In the natural sciences, complex non-linear systems composed of large numbers
of smaller subunits provide an opportunity to apply the tools of statistical
mechanics and information theory. From this intuition Lee Smolin, (2009)
suggested a new discipline of statistical economics to study the collective
behavior of economies composed of large numbers of economic agents.

A serious impasse to this approach is the lack of well-defined or even
definable constraints enabling the use of Lagrange multipliers, partition
functions and the machinery of statistical mechanics for systems away from
equilibrium or for non-physical systems. The latter – in particular economic
systems – lack e.g. fundamental conservation laws like the conservation of
energy to form the basis of these constraints. In order to address this
impasse, Fielitz and Borchardt, (2014) introduced the concept of natural
information equilibrium. They produced a framework based on information
equilibrium and showed it was applicable to several physical systems. The
present paper seeks to apply that framework to economic systems.

The idea of applying mathematical frameworks used in the physical sciences to
economic systems is an old one; even the idea of applying principles from
thermodynamics is an old one. Willard Gibbs – who coined the term "statistical
mechanics" – supervised Irving Fisher’s thesis [Fisher, (1892)] in which he
applied a rigorous approach to economic equilibrium. Samuelson later codified
the Lagrange multiplier approach to utility maximization commonly used in
economics today.

The specific thrust of Fielitz and Borchardt, (2014) is that it looks at how
far you can go with the maximum entropy or information theoretic arguments
without having to specify constraints. This refers to partition function
constraints optimized with the use of Lagrange multipliers. In thermodynamics
language it’s a little more intuitive: basically the information transfer
model allows you to look at thermodynamic systems without having defined a
temperature (Lagrange multiplier) and without having the related constraint
(that the system observables have some fixed value, i.e. equilibrium).

A word of caution before proceeding; the term "information" is somewhat
overloaded across various technical fields. Our use of the word information
differs from its more typical usage in economics, such as in “information
economics” or “perfect information” in game theory. Instead of focusing on a
board position in chess, we are assuming all possible board positions (even
potentially some impossible ones such as those including three kings). The
definition of information we use is the definition required when specifying a
random chess board out of all possible chess positions, and it comes from
Hartley and Shannon. It is a quantity measured in bits (or nats), and has a
direct connection to probability. As stated in Shannon, (1949), “information
must not be confused with meaning”.

This is in contrast to Akerlof information asymmetry, for example, where
knowledge (meaningful information) of the quality of a vehicle is better known
to the seller than the buyer. We can see that this is a different use of the
term information – how many bits this quality score requires to store (and
hence how many available ‘quality states’ there are) is irrelevant to
Akerlof’s argument. The perfect information in a chess board $C$ represents
$I(C)<64\log_{2}13\simeq 237$ bits; this quantity is irrelevant in an analysis
of chess strategies in game theory (except as a practical limit to computation
of all possible chess moves).

We propose the idea that information equilibrium should be used as a guiding
principle in economics and organize this paper as follows. We will begin in
Section 2 by introducing and deriving the primary equations of the information
equilibrium framework, and proceed to show how the information equilibrium
framework can be understood in terms of the general market forces of supply
and demand. This framework will also provide a definition of the regime where
market forces fail to reach equilibrium through information loss.

Since the framework itself is agnostic about the goods and services sold or
the behaviors of the relevant economic agents, the generalization from widgets
in a single market to an economy composed of a large number of markets is
straightforward. We will describe macroeconomics in Section 3, and demonstrate
the effectiveness of the principle of information equilibrium both empirically
an in derivations of standard macroeconomic models. In particular we will
address the price level and the labor market where we show that information
equilibrium leads to well-known stylized facts in economics. The quantity
theory of money will be shown to be an approximation to information
equilibrium when inflation is high, and Okun’s law will be shown to follow
from information equilibrium. Lastly, we establish in Section 4 an economic
partition function, define a concept of economic entropy and discuss how
nominal rigidity and the so-called _liquidity trap_ in Krugman, (1998) may be
best understood as entropic forces for which there are no microfoundations.

##  2 Information equilibrium

We will describe the economic laws of supply and demand as the result of an
information transfer model. Much of the description of the information
transfer model follows Fielitz and Borchardt, (2014). Following Shannon,
(1948) we have a system that transfers information111This follows the notation
of one of the earlier versions of Fielitz and Borchardt, (2014). The German
word for source is quelle. We did not want to create confusion by using $S$
and $D$ for source and destination and then for supply and demand, since they
appear in reverse order in the equations. $I_{q}$ from a source $q$ to a
destination $u$ (see Figure 1). Any process can at best transfer complete
information, so we know that $I_{u}\leq I_{q}$.

Figure 1: Information transfer from source to destination.

We will follow Fielitz and Borchardt, (2014) and use the Hartley
definition222The Hartley definition is equivalent to the Shannon definition
for states with equal probabilities. As this definition enters into the
information transfer index which is later taken as a free parameter, this
distinction is not critical. of information $I=K(s)n$ where $K(s)=K_{0}\log s$
where $s$ is the number of symbols and $K_{0}$ defines the unit of information
(e.g. $1/\log 2$ for bits). If we take a measuring stick of length $|q|$
(process source) and subdivide it in to segments $\delta|q|$ (process source
signal) then $n_{q}=|q|/\delta|q|$. In that case, the information transfer
relationship $I_{u}\leq I_{q}$ becomes

$\frac{|u|}{\delta|u|}K_{u}(s_{u})\leq\frac{|q|}{\delta|q|}K_{q}(s_{q})$ (2.1)

Let us define $k\equiv K_{q}(s_{q})/K_{u}(s_{u})$ which we will call the
_information transfer index_ and rearrange so that

$\frac{|u|}{\delta|u|}\leq k\frac{|q|}{\delta|q|}$ (2.2)

Compared to Fielitz and Borchardt, (2014), we have changed some of the
notation, e.g. $|\Delta q|$ becomes $|q|$. We have set up the condition
required by information theory for a signal $\delta|q|$ measured by the stick
of length $|q|$ to be received as a signal $\delta|u|$ and measured by a stick
of length $|u|$. These signals will contain the same amount of information if
$I_{u}=I_{q}$.

Now we define a process signal detector that relates the process source signal
$\delta|q|$ emitted from the process source $q$ to a process destination
signal $\delta|u|$ that is detected at the process destination $u$ and
delivers an output value:

$p\equiv\left(\frac{\delta|q|}{\delta|u|}\right)_{\mbox{\footnotesize
detector}}$

If our source and destination are large compared to our signals
($n_{q},n_{u}\gg 1$) we can take $\delta|q|\rightarrow d|q|$, we can re-
arrange the information transfer condition:

$p=\frac{d|q|}{d|u|}\leq k\frac{|q|}{|u|}$ (2.3)

In the following, we will use the notation333We can consider an information
transfer model to be an ‘information preserving’ morphism in category theory.
The morphism itself is defined by the differential equation (2.3), but we will
label it with the detector $p$. $p:q\rightarrow u$ to designate an information
transfer model with source $q$, destination $u$ and detector $p$ for the
general case where $I_{u}\leq I_{q}$, and use the notation
$p:q\rightleftarrows u$ to designate an information equilibrium relationship
where $I_{u}=I_{q}$. I will also occasionally use the notations $q\rightarrow
u$ and $q\rightleftarrows u$ to designate an information transfer (information
equilibrium) model without specifying the detector. Next, we derive supply and
demand using this model.

###  2.1 Supply and demand

At this point we will take our information transfer process and apply it to
the generic economic problem of supply and demand. We will drop the absolute
values and use positive quantities. In that case, we will identify the
information transfer process source as the demand $D$, the information
transfer process destination as the supply $S$, and the process signal
detector as the price $P$. The price detector relates the demand signal
$\delta D$ emitted from the demand $D$ to a supply signal $\delta S$ that is
detected at the supply $S$ and delivers a price $P$. We translate Condition 1
in Fielitz and Borchardt, (2014) for the applicability of our information
theoretical description into the language of supply and demand:

> _Condition 1_ : The considered economic process can be sufficiently
> described by only two independent process variables (supply and demand:
> $D,S$) and is able to transfer information.

We are now going to solve the differential equation 2.3. But first we assume
ideal information transfer $I_{S}=I_{D}$ such that:

$P=k\frac{D}{S}$ (2.4) $\frac{dD}{dS}=k\frac{D}{S}$ (2.5)

Note that Eq. (2.4) represents movement of the supply and demand curves where
$D$ is a “floating-restriction” information source in the language of Fielitz
and Borchardt, (2014), as opposed to movement along the supply and demand
curves where $D=D_{0}$ is a “constant-restriction information source”, again
in the language of Fielitz and Borchardt, (2014). The differential equation
(2.5) can be solved by integration

$\displaystyle\int_{D_{ref}}^{D}\frac{dD^{\prime}}{D^{\prime}}$
$\displaystyle=$ $\displaystyle
k\int_{S_{ref}}^{S}\frac{dS^{\prime}}{S^{\prime}}$ (2.6) $\displaystyle\log
D-\log D_{ref}$ $\displaystyle=$ $\displaystyle k\left(\log S-\log
S_{ref}\right)$ (2.7) $\displaystyle\frac{D}{D_{ref}}$ $\displaystyle=$
$\displaystyle\left(\frac{S}{S_{ref}}\right)^{k}$ (2.8)

and we can then solve for the price using Eq. (2.4)

$\displaystyle P$ $\displaystyle=$ $\displaystyle k\frac{D}{S}$ (2.9)
$\displaystyle=$ $\displaystyle
k\frac{1}{S}D_{ref}\left(\frac{S}{S_{ref}}\right)^{k}$ (2.10) $\displaystyle=$
$\displaystyle
k\frac{1}{S}D_{ref}\frac{S}{S_{ref}}\left(\frac{S}{S_{ref}}\right)^{k-1}$
(2.11) $\displaystyle=$ $\displaystyle
k\frac{D_{ref}}{S_{ref}}\left(\frac{S}{S_{ref}}\right)^{k-1}$ (2.12)

These equations represent the general equilibrium solution where $D$ and $S$
change in response to each other.

If we hold the information source or destination effectively constant,
responding only slowly to changes in the other variable, we can describe
‘partial equilibrium’ solutions that will lead to supply and demand diagrams.
We will take $D=D_{0}$ to be a constant-restriction information source in the
language of Fielitz and Borchardt, (2014) and integrate the differential
equation Eq. (2.5)

$\frac{1}{D_{0}}\int_{D_{ref}}^{D}dD^{\prime}=k\int_{S_{ref}}^{S}\frac{1}{S}\,dS^{\prime}$

We find

$\Delta D=D-D_{ref}=kD_{0}\log\left(\frac{S}{S_{ref}}\right)$ (2.13)

Equation (2.13) represents movement along the demand curve, and the
equilibrium price $P$ moves according to Eq. (2.4) based on the expected value
of the supply and our constant demand source:

$\displaystyle P$ $\displaystyle=$ $\displaystyle k\frac{D_{0}}{S}$ (2.14)
$\displaystyle\Delta D$ $\displaystyle=$ $\displaystyle
kD_{0}\log\left(\frac{S}{S_{ref}}\right)$ (2.15)

Equations (2.14,2.15) define a demand curve. A family of demand curves can be
generated by taking different values for $D_{0}$ assuming a constant
information transfer index $k$.

Analogously, we can define a supply curve by using a constant information
destination $S_{0}$ and follow the above procedure to find:

$\displaystyle P$ $\displaystyle=$ $\displaystyle k\frac{D}{S_{0}}$ (2.16)
$\displaystyle\Delta S$ $\displaystyle=$
$\displaystyle\frac{S_{0}}{k}\log\left(\frac{D}{D_{ref}}\right)$ (2.17)

So that equations (2.16, 2.17) define a supply curve. Again, a family of
supply curves can be generated by taking different values for $S_{0}$.

Note that equations (2.14,2.15) and (2.16, 2.17) linearize (Taylor series
around $D=D_{ref}$ and $S=S_{ref}$)

$\displaystyle D$ $\displaystyle\simeq$ $\displaystyle
D_{ref}+kD_{0}-S_{ref}P$ (2.18) $\displaystyle S$ $\displaystyle\simeq$
$\displaystyle S_{ref}-\frac{S_{0}}{k}+\frac{S_{0}{}^{2}}{k^{2}D_{ref}}P$
(2.19)

plus terms of order $P^{2}$ such that

$D\simeq\alpha-\beta P$ $S\simeq\gamma+\delta P$

where $\alpha=D_{ref}+kD_{0}$, $\beta=S_{ref}$ ,$\gamma=S_{ref}-S_{0}/k$ and
$\delta=S_{0}{}^{2}/(k^{2}D_{ref})$. This recovers a simple linear model of
supply and demand (where you could add a time dependence to the price e.g.
$\frac{dP}{dt}\propto S-D$ to produce a simple dynamic model).

We can explicitly show the supply and demand curves using equations
(2.14,2.15) and (2.16, 2.17) and plotting price $P$ vs change in quantity
$\Delta Q=\Delta S$ or $\Delta D$ in Figure 2. In the figure we also show a
shift in the supply curve (red) to the right. The new (lower) equilibrium
price is the intersection of the new displaced supply curve and the unchanged
demand curve.

(a) Supply and demand curves (b) Shift of the supply curve

Figure 2: Left: Supply and demand curves Right: Shift of the supply curve
resulting in a new lower equilibrium price.

If we use the linearized version of the supply and demand relationships Eqs.
(2.18, 2.19) near the equilibrium price, we can find the (short run) price
elasticities of demand and supply

$e^{d}=\frac{dD/D}{dP/P}=\frac{D-D_{0}/k-D_{ref}}{D}$

Expanding around

$\Delta D=D-D_{ref}$ $e^{d}\simeq-\frac{kD_{0}}{D_{ref}}+O(\Delta D)$

And analogously

$e^{s}\simeq\frac{S_{0}}{kS_{ref}}+O(\Delta S)$

from which we could measure the information transfer index $k$.

There is a third way to solve Eq. (2.5) where both supply and demand are
considered to vary slowly (i.e. be approximately constant). In that case the
integral becomes

$\frac{1}{D_{0}}\int_{D_{ref}}^{D}dD^{\prime}=\frac{k}{S_{0}}\int_{S_{ref}}^{S}dS^{\prime}$

If we define

$\displaystyle\Delta D$ $\displaystyle=$ $\displaystyle D-D_{ref}$
$\displaystyle\Delta S$ $\displaystyle=$ $\displaystyle S-S_{ref}$

solving the integral shows us that the price is also constant

$P=\frac{\Delta D}{\Delta S}=k\frac{D_{0}}{S_{0}}$ (2.20)

###  2.2 Physical analogy with ideal gases

In the original paper, Fielitz and Borchardt, (2014) use the information
transfer model to build the ideal gas law. This specific application gives us
some analogies that are useful. In the model we have

$p=\frac{2}{f}\frac{E}{V}$

the pressure $p$ is the price $P$, volume $V$ is the supply $S$ and the
energy444Substituting the energy in the formula you get $pV=Nk_{B}T$
$E=(f/2)Nk_{B}T$ is the demand $D$. The information transfer index contains
the number of degrees of freedom $f$ in the ideal gas as well as the factor of
$1/2$ that comes from the integral of a normal distribution in the derivation
from statistical mechanics. In Fielitz and Borchardt, (2014), the general
equilibrium solution corresponds to an isentropic process (and more generally,
a polytropic process), while the partial equilibrium solution for the demand
curve correspond to an isothermal process.

###  2.3 Alternative motivation

We would like to provide an alternative and more macro- and micro-economic
motivation of Eq. (2.5) rooted in two economic principles: homogeneity of
degree zero and marginalism. For example, according to Bennett McCallum,
(2004), the quantity theory of money (QTM) is the macroeconomic observation
that the economy obeys long run neutrality of money which is captured in the
assumption of homogeneity constraints. In particular, supply and demand
functions will be homogeneous of degree zero, i.e. ratios of $D$ to $S$ such
that if $D\rightarrow\alpha D$ and $S\rightarrow\alpha S$ then
$g(D,S)\rightarrow\alpha^{0}g(D,S)=g(D,S)$. The simplest differential
equation555We might consider this the most important term in an effective
theory of supply and demand, analogous to effective field theory in physics
where a full expansion would look like
$\frac{dD}{dS}=c_{0}+c_{1}\frac{D}{S}+c_{2,0}\frac{D^{2}}{S^{2}}+\cdots+c_{2,1}\frac{D}{S}\frac{dD}{dS}+c_{2,2}D\frac{d^{2}D}{dS^{2}}+\cdots$
consistent with this observation is

$\frac{dD}{dS}=k\frac{D}{S}$ (2.21)

Fisher, (1892) looks at the exchange of some number of gallons of $A$ for some
number of bushels of $B$ and states: "the last increment $dB$ is exchanged at
the same rate for $dA$ as $A$ was exchanged for $B$". Fisher writes this as an
equation on page 5:

$\frac{A}{B}=\frac{dA}{dB}$ (2.22)

Fisher notes that this marginalist argument was introduced by both Jevons and
Marshall. Of course it is generally false. Many goods exhibit economies of
scale, fixed costs or other effects so that either the last increments of $dA$
and $dB$ are cheaper (e.g. software) or more expensive (e.g. oil) than the
first increments. The simplest way to account for this is by multiplying one
side of Eq. (2.22) by a constant. Thus we can say using information
equilibrium as an economic principle enforces a generalized marginal thinking.
The information equilibrium approach can also be interpreted as an application
of information theory to Irving Fisher’s measuring stick.

##  3 Macroeconomics

Since the information equilibrium framework depends on a large number of
states for the information source and destination, it ostensibly would be
better applied to the macroeconomic problem. Below we make a connection to
some classic macroeconomic toy models and a macroeconomic relationship: AD-AS
model, Okun’s law, the IS-LM model, the Solow growth model, and the quantity
theory of money. A summary of the models described in Section 3 appears in
Appendix A. The details of the _Mathematica_ codes used to fit the parameters
are provides in Appendix B.

###  3.1 AD-AS model

The AD-AS model uses the price level $P$ as the detector, aggregate demand $N$
(NGDP) as the information source and aggregate supply $S$ as the destination,
or $P:N\rightleftarrows S$, which immediately allows us to write down the
aggregate demand and (short run) aggregate supply (SRAS) curves for the case
of partial equilibrium.

$P=\frac{N_{0}}{k_{A}S_{ref}}\exp\left(-k_{A}\frac{\Delta N}{N_{0}}\right)$
$P=\frac{N_{ref}}{k_{A}S_{0}}\exp\left(+\frac{\Delta S}{k_{A}S_{0}}\right)$

Positive shifts in the aggregate demand curve raise the price level along with
negative shifts in the supply curve. Traveling along the aggregate demand
curve lowers the price level (more aggregate supply at constant demand).

(a) AD-AS model (b) Shift of the aggregate supply curve

Figure 3: Left: AD-AS model with AD curve in blue, SRAS curve in red and LRAS
curve as dashed red. Right: Shift of the aggregate supply curve. The values
$N=\langle N\rangle$ and $S=\langle S\rangle$ parameterize the supply and
demand curves, respectively.

The long run aggregate supply (LRAS) curve would be vertical in Figure 3
representing the general equilibrium solution

$\frac{N}{N_{ref}}=\left(\frac{S}{S_{ref}}\right)^{k_{A}}$

with price $P\sim S^{k_{A}-1}$.

Another interesting result in this model is that it can be used to illuminate
the role of money in macroeconomics as a tool of information mediation. If we
start with the AD-AS model information equilibrium condition

$P\equiv\frac{dN}{dS}=k\;\frac{N}{S}$

we can in general make the following transformation using a new variable $M$
(i.e. money):

$P=\frac{dN}{dM}\frac{dM}{dS}=k\;\frac{N}{M}\frac{M}{S}$ (3.1)

If we take $N$ to be in information equilibrium with the intermediate quantity
$M$, which is in information equilibrium with $S$, i.e.

$P:N\rightleftarrows M\rightleftarrows S$

then we can use the information equilibrium condition

$\frac{dM}{dS}=k_{s}\;\frac{M}{S}$

to show that equation (3.1) can be re-written

$\displaystyle P$ $\displaystyle=$
$\displaystyle\frac{dN}{dM}\frac{dM}{dS}=k\;\frac{N}{M}\frac{M}{S}$ (3.2)
$\displaystyle=$
$\displaystyle\frac{dN}{dM}\left(k_{s}\;\frac{M}{S}\right)=k\;\frac{N}{M}\frac{M}{S}$
(3.3) $\displaystyle=$
$\displaystyle\frac{dN}{dM}=\frac{k}{k_{s}}\;\frac{N}{M}$ (3.4) $\displaystyle
P$ $\displaystyle=$ $\displaystyle\frac{dN}{dM}=k_{n}\;\frac{N}{M}$ (3.5)

where we have defined $k_{n}\equiv k/k_{s}$. The solution to the differential
equation (3.5) defines a quantity theory of money where the price level goes
as

$\log P\sim(k_{n}-1)\log M$

We will discuss this more in Section 3.4 on the price level and inflation.

###  3.2 Labor market and Okun’s law

The description of the labor market uses the price level $P$ as the detector,
aggregate demand $N$ as the information source and total hours worked666You
can also use the total employed $L$ as an information destination. $H$ as the
destination. We define the market $P:N\rightleftarrows H$ so that we can say:

$P=\frac{1}{k_{H}}\;\frac{N}{H}$

Re-arranging and taking the logarithmic derivative of both sides:

$\displaystyle H$ $\displaystyle=$ $\displaystyle\frac{1}{k_{H}}\;\frac{N}{P}$
(3.6) $\displaystyle\frac{d}{dt}\log H$ $\displaystyle=$
$\displaystyle\frac{d}{dt}\log\frac{N}{P}-\frac{d}{dt}\log k_{H}$ (3.7)
$\displaystyle\frac{d}{dt}\log H$ $\displaystyle=$
$\displaystyle\frac{d}{dt}\log\frac{N}{P}-0=\frac{d}{dt}\log R$ (3.8)

where $R$ is RGDP. The total hours worked $H$ (or total employed $L$)
fluctuates with the change in RGDP growth. This is one form of Okun’s law,
from Okun, (1962). The model is shown in Figure 4. The model parameters are
listed in Appendix A.

Figure 4: The model of US inflation using N = NGDP and total hours worked H is
shown in blue. Inflation data (CPI all items) is in green.

###  3.3 IS-LM and interest rates

The classical Hicksian Investment-Savings Liquidity-Money Supply (IS-LM) model
uses two markets along with an information equilibrium relationship. Let $p$
be the price of money in the money market (LM market) $p:N\rightleftarrows M$
where $N$ is aggregate demand and $M$ is the money supply. We have:

$p=\frac{1}{k_{p}}\;\frac{N}{M}$

We assume that the interest rate $i$ is in information equilibrium with the
price of money $p$, so that we have the information equilibrium relationship
$i\rightleftarrows p$ (no need to define a detector at this point). Therefore
the differential equation is:

$\frac{di}{dp}=\frac{1}{k_{i}}\;\frac{i}{p}$

with solution (we will not need the additional constants $p_{ref}$ or
$i_{ref}$):

$i^{k_{i}}=p$

And we can write:

$i^{k_{i}}=\frac{1}{k_{p}}\;\frac{N}{M}$

Already this is fairly empirically accurate as we can see in Figure 5.

Figure 5: The model of US long term interest rate using N = NGDP and the
monetary base minus reserves is shown in blue. The green dotted line is the
long term interest rate data is from FRED, (2015); the data shown is the
10-year constant maturity rate, series GS10.

We can now rewrite the money (LM) market and add the goods (IS) market as
coupled markets with the same information source (aggregate demand) and same
detector (interest rate, directly related to – i.e. in information equilibrium
with – the price of money):

$\displaystyle i^{k_{i}}$ $\displaystyle:$ $\displaystyle N\rightleftarrows M$
(3.9) $\displaystyle i^{k_{i}}$ $\displaystyle:$ $\displaystyle
N\rightleftarrows S$ (3.10)

(a) IS market (b) LM market

Figure 6: Left: IS market with IS curve in blue. Right: LM market with money
demand curves in green. The three points represent the equilibrium solution as
well as the equilibrium solutions after shifts of $\pm\Delta N$. The values
$M=\langle M\rangle$ and $S=\langle S\rangle$ parameterize the money demand
and IS curves, respectively.

where $S$ is the aggregate supply. Changes in the LM market manifest as
increases in the money supply $M$ as well as shifts in the information source
$N_{0}\rightarrow N_{0}+\Delta N$, so we write the LM curve as a demand curve
Eqs. (2.14, 2.14) with shifts:

$i^{k_{i}}=\frac{N_{0}+\Delta N}{k_{p}M_{ref}}\exp\left(-k_{p}\frac{\Delta
M}{N_{0}+\Delta N}\right)$

The IS curve can be straight-forwardly be written as the demand curve in the
IS market:

$i^{k_{i}}=\frac{N_{0}}{k_{S}S_{ref}}\exp\left(-k_{S}\frac{\Delta
N}{N_{0}}\right)$

This model assumes that $N$ does not move strongly with $M$, so only applies
to a low inflation scenario. For high inflation, $N$ acquires a strong
dependence on $M$ and the quantity theory of money in Section 3.4 becomes a
more accurate description.

Figure 7: The IS-LM model. The IS curve is in blue and the LM curve is gray.
The three points on the $\Delta M=0$ curve represent the same three points in
Figure 6. An LM curve after a shift by $+\Delta M$.

####  3.3.1 Long and short term interest rates

The short term interest rate is empirically given by the same model with the
same parameters (see Fig. 8); the difference is that the full monetary base
including central bank reserves is used instead of just the currency
component. These are FRED, (2015) series AMBSL (call this variable $MB$) and
MBCURRCIR, respectively. The full market for the long $i_{l}$ and short
$i_{s}$ term interest rates would be:

$\displaystyle i_{l}^{k_{i}}$ $\displaystyle:$ $\displaystyle
N\rightleftarrows M$ (3.11) $\displaystyle i_{s}^{k_{i}}$ $\displaystyle:$
$\displaystyle N\rightleftarrows MB$ (3.12)

where $k_{i_{l}}=k_{i_{s}}=k_{i}$ and $k_{p_{l}}=k_{p_{s}}=k_{p}$ (i.e. the
parameters for both models are the same).

Figure 8: The model of US long and short interest rates. The short interest
rate model using N = NGDP and M being the full monetary base(including
reserves) is shown in purple. The long interest rate is as in Fig. 5. The gray
dotted line is the short term interest rate data from FRED, (2015); the short
interest rate data is taken to be the 3-month secondary market rate, series
TB3MS.

The theoretical reason both the long and short term interest rate are given by
the same model simply by exchanging currency (monetary base minus reserves)
for the full monetary base (including reserves) is not immediately obvious. As
the relationship was observed in empirical data, we can only provide a hand-
waving argument based on the properties of central bank reserves (which are
purely electronic) as opposed to currency which manifests as physical pieces
of paper. Reserves may be seen as temporary by the market - they only exist in
the short run. Therefore they need to be included as part of the supply of so-
called high powered money for short term interest rates. Physical currency in
circulation may be seen as more permanent by the market, and therefore
represent the proper supply of high powered money for long term interest
rates. This argument is speculative and involves the expected path of the
monetary base, something not empirically measurable.

####  3.3.2 Assumptions in the IS-LM model

One useful property of the information equilibrium approach is that is makes
explicit several assumptions in the IS-LM model.

  * $\bullet$

It is a partial equilibrium model and we use the partial equilibrium solutions
to the information equilibrium equation Eq. (2.3).

  * $\bullet$

No distinction is made between real and nominal quantities (all quantities are
treated as nominal). Since we have partial equilibrium, $N$ is assumed to be
slowly varying which implies that if $N=PY$, $P$ must be slowly varying unless
$P$ and $Y$ conspire to make $N$ slowly varying.

  * $\bullet$

If the price of money is scaled by a constant factor $p\rightarrow\alpha p$,
the only change to the model is a change in the value of the constant
$k_{p}\rightarrow k_{p}/\alpha$.

###  3.4 Price level and inflation

Let us begin our discussion of the price level with the market
$P:N\rightleftarrows M$ described in the AD-AS model in section 3.1 with $N$
being NGDP, the information source, and $M$ being the monetary base minus
reserves and return to the differential equation (2.3). Assuming ideal
information transfer we have

$P=\frac{dN}{dM}=k\;\frac{N}{M}$ (3.13)

Let us allow $k=k(N,M)$ to be a slowly varying function of $N$ and $M$, i.e.

$\frac{\partial k}{\partial N}\mbox{ , }\frac{\partial k}{\partial M}\approx
0$ (3.14)

We can approximately solve the differential equation (3.13) by integration
such that

$\displaystyle\int_{N_{0}}^{N}\frac{dN^{\prime}}{N^{\prime}}$
$\displaystyle\approx$ $\displaystyle
k(N,M)\int_{M_{0}}^{M}\frac{dM^{\prime}}{M^{\prime}}$ (3.15)
$\displaystyle\rightarrow\;\frac{N}{N_{0}}$ $\displaystyle=$
$\displaystyle\left(\frac{M}{M_{0}}\right)^{k(N,M)}$ (3.16)

so that, using Eq. (3.13) again, we obtain the price level as a function of
$N$ and $M$

$P(N,M)\approx\alpha\;k(N,M)\left(\frac{M}{M_{0}}\right)^{k(N,M)-1}$ (3.17)

where $\alpha$ is an arbitrary constant (because the normalization of the
price level is arbitrary).

Now the information transfer index $k$ is related to the number of symbols
$s_{N}$, $s_{M}$ used by the information source and information destination,
specifically:

$k=\frac{K_{0}\log s_{N}}{K_{0}\log s_{M}}=\frac{\log s_{N}}{\log s_{M}}$
(3.18)

Let us posit a simple model where $s_{N}$ and $s_{M}$ are proportional to $N$
and $M$

$\displaystyle s_{N}$ $\displaystyle=$ $\displaystyle N/(\gamma M_{0})$ (3.19)
$\displaystyle s_{M}$ $\displaystyle=$ $\displaystyle M/(\gamma M_{0})$ (3.20)
$\displaystyle k(N,M)$ $\displaystyle=$ $\displaystyle\frac{\log N/(\gamma
M_{0})}{\log M/(\gamma M_{0})}$ (3.21)

where we have introduced the new777We have simply traded the parameter degree
of freedom $k$ in the constant information transfer index version of the model
for $\gamma$; we have not increased the number of parameters in the model.
dimensionless parameter $\gamma$. This functional form meets the requirement
that $k(N,M)$ is slowly varying with $N$ and $M$:

$\displaystyle\frac{\partial k}{\partial N}$ $\displaystyle=$
$\displaystyle\frac{\gamma M_{0}}{N\log M/(\gamma M_{0})}\approx 0$ (3.22)
$\displaystyle\frac{\partial k}{\partial M}$ $\displaystyle=$
$\displaystyle-\frac{\log N/(\gamma M_{0})}{M(\log M/(\gamma
M_{0}))^{2}}\approx 0$ (3.23)

for $N,M\gg 1$. The rationale for introducing such a model for a changing
information transfer index $k$ is that the units of $N$ and $M$ are the same:
the national unit of account. Therefore the information content of e.g. $ 1
billion of nominal output depends on the size of the monetary base – and vice
versa, and so we should expect $k=k(N,M)$. However, we will see in Section 4
that this functional form is a good approximation to the case where we
consider $n\gg 1$ markets with a distribution of constant values of $k$,
meaning $k=k(N,M)$ effectively describes _emergent_ properties of the
macroeconomy. There is an additional benefit of introducing this functional
form and constant $\gamma$ that may assist in cross-national comparisons that
we discuss in Appendix C.

The full price level model is

$P(N,M)\approx\alpha\;\frac{\log N/(\gamma M_{0})}{\log M/(\gamma
M_{0})}\left(\frac{M}{M_{0}}\right)^{\frac{\log N/(\gamma M_{0})}{\log
M/(\gamma M_{0})}-1}$ (3.24)

with free dimensionless parameters $\alpha$ and $\gamma$ along with $M_{0}$,
which has dimensions of currency. If we fit these parameters using data from
FRED, (2015) for $P$ being so-called core price level of Personal Consumption
Expenditures (PCE price level, less food and energy, series PCEPILFE), $N$
being nominal gross domestic product (series GDP), and $M$ being the currency
component of the monetary base (series MBCURRCIR), performing a LOESS
smoothing (of order 2, with smoothing parameter 1.0, see Appendix B) on the
inputs $N$ and $M$ we arrive at Figures 9 and 10. The empirical accuracy of
the model is on the order of the $P^{*}$ model of Hallman, (1989) (see
Appendix A for fit parameters).

Figure 9: The model of US price level with N being NGDP and the M being
monetary base minus reserves (MBCURRCIR) is shown in blue. Price level data
(core PCE, with 2011 = 100) is in green. Figure 10: The model of US inflation
using N = NGDP and the monetary base minus reserves is shown in blue.
Inflation data (core PCE) is in green. The blue bands represent 1-$\sigma$
error bands for the residuals.

If we look at Eq. (3.24) we can see that when $k=2$, we have

$\displaystyle P(N,M)$ $\displaystyle=$ $\displaystyle
2\alpha\;\left(\frac{M}{M_{0}}\right)^{2-1}$ (3.25) $\displaystyle P$
$\displaystyle\sim$ $\displaystyle M$ (3.26)

so that price level grows proportionally with the monetary base, the essence
of the quantity theory of money. Additionally, when $k=2$ we have, using Eq.
(3.19),

$\displaystyle 2(\log M-\log\gamma M_{0})$ $\displaystyle=$ $\displaystyle\log
N-\log\gamma M_{0}$ (3.27) $\displaystyle M^{2}$ $\displaystyle\sim$
$\displaystyle N$ (3.28)

If we use the fact that $M,N\gg 1$. If we take $M$ and $N$ to be exponentially
growing with growth rates $m$ and $n$ (i.e. $M\sim\exp mt$), respectively,
$2m=n$. In general, we have (introducing the inflation rate $\pi$)

$\displaystyle\pi$ $\displaystyle\simeq$ $\displaystyle(k-1)m$ (3.29)
$\displaystyle n$ $\displaystyle\simeq$ $\displaystyle km$ (3.30)

Defining a real growth rate $r\equiv n-\pi$, then for large $k\gg 1$ we have

$\frac{n}{\pi}=\frac{km}{(k-1)m}\simeq
1\simeq\frac{\pi+r}{\pi}\sim\frac{\pi}{\pi}$ (3.31)

which implies large $k$ means high inflation. In contrast, $k\approx 1$ means
that $\pi\approx 0$. When $k\approx 1$, the IS-LM model becomes a better
approximation since changes in $M$ do not result in strong changes in the
price level $P$ since $P\sim M^{k-1}\sim M^{0}=1$. We will discuss this more
in Section 4.1.

###  3.5 Solow-Swan growth model

Let us assume two markets $p_{1}:N\rightleftarrows K$ and
$p_{2}:N\rightleftarrows L$:

$\displaystyle\frac{\partial N}{\partial K}$ $\displaystyle=$ $\displaystyle
k_{1}\;\frac{N}{K}$ (3.32) $\displaystyle\frac{\partial N}{\partial L}$
$\displaystyle=$ $\displaystyle k_{2}\frac{N}{L}$ (3.33)

The economics rationale for equations (3.32) are that the left hand sides are
the marginal productivity of capital/labor which are assumed to be
proportional to the right hand sides – the productivity per unit
capital/labor. In the information transfer model, the relationship follows
from a model of aggregate demand sending information to aggregate supply
(capital and labor) where the information transfer is “ideal” i.e. no
information loss. The solutions are:

$N(K,L)\sim f(L)K^{k_{1}}$ $N(K,L)\sim g(K)L^{k_{2}}$

and therefore we have

$N(K,L)=AK^{k_{1}}L^{k_{2}}$ (3.34)

Equation (3.34) is the generic Cobb-Douglas form. In the information
equilibrium model, the exponents are free to take on any value (not restricted
to constant returns to scale, i.e. $k_{1}+k_{2}=1$). The resulting model is
remarkably accurate as seen in Figure 11.

(a) Output level (b) Growth rate

Figure 11: Left: Nominal output using the Cobb-Douglas production function.
Right: Growth rate of nominal output using the Cobb-Douglas production
function.

It also has no changes in so-called total factor productivity ($A$ is
constant). The results above use nominal capital and nominal GDP $N$ rather
than the usual real capital and real output (RGDP, $R$). We use the FRED,
(2015) data series RKNANPUSA666NRUG for the real capital stock (capital stock
at constant prices) and inflate to nominal capital stock via CPI less food and
energy (CPILFESL).

Let us assume two additional information equilibrium relationships with
capital $K$ being the information source and investment $I$ and depreciation
$D$ (include population growth in here if desired) being information
destinations. In the notation we have been using: $K\rightleftarrows I$ and
$K\rightleftarrows D$. This immediately leads to the solutions of the
differential equation Eq. (2.5):

$\frac{K}{K_{0}}=\left(\frac{D}{D_{0}}\right)^{\delta}$
$\frac{K}{K_{0}}=\left(\frac{I}{I_{0}}\right)^{\sigma}$

Therefore we have (the first relationship coming from the Cobb-Douglas
production function)

$N\sim K^{\alpha}$ $I\sim K^{1/\sigma}$ $D\sim K^{1/\delta}$

If $\sigma=1/\alpha$ and $\delta=1$ we recover the original Solow model, but
in general any $\sigma>\delta$ allows there to be an equilibrium. Figure 12
represents a generic plot of the relationships above.

Figure 12: The Solow growth model as an information equilibrium model. Output
is in blue, while investment is in red and depreciation is dashed red.

Assuming the relationships $K\rightleftarrows I$ and $K\rightleftarrows D$
hold simultaneously gives us the equilibrium value of $K=K^{*}$:

$K^{*}=K_{0}\exp\left(\frac{\sigma\delta\log
I_{0}/D_{0}}{\sigma-\delta}\right)$

This equilibrium value represents simultaneous information equilibrium in the
two markets $K\rightleftarrows I$ and $K\rightleftarrows D$. Fluctuations in
the value of capital $K$ away from $K=K^{*}$ will experience an entropic force
to return $K^{*}$, so the equilibrium $K^{*}$ would be stable. Entropic forces
will be discussed in more detail in Section 4.1.

As a side note, the small $K$ region in Figure 12 does not appear because it
is not a valid region of the model. The information equilibrium model is not
valid for small values of $K$ (or any process variable). That allows one to
choose parameters for investment and depreciation that could be e.g. greater
than output for small $K$ – a nonsense result in the traditional Solow model,
but just an invalid region of the model in the information equilibrium
framework. Another useful observation is that $N$ and $I$ have a supply and
demand relationship in partial equilibrium with capital being demand and
investment being supply since by transitivity (see Appendix D) they are in
information equilibrium (i.e. $N\rightleftarrows K$).

There might be more to the information equilibrium picture of the Solow model
than just the basic mechanics – in particular we might be able to analyze
dynamics of the savings rate relative to demand shocks. We have built the
model:

$N\rightleftarrows K\rightleftarrows I$

Where $N$ is output, $K$ is capital and $I$ is investment. Since information
equilibrium is an equivalence relation (see Appendix D), we have the model:

$p_{I}:N\rightleftarrows I$

with abstract price $p_{I}$. If we write down the differential equation
resulting from that model

$p_{I}=\frac{dN}{dI}=\frac{1}{\eta}\;\frac{N}{I}$ (3.35)

There are a few things we can glean from this that are described below using
general equilibrium, partial equilibrium, and making a connection to interest
rates.

####  3.5.1 General equilibrium in the Solow model

We can solve equation (3.35) under general equilibrium giving us $N\sim
I^{1/\eta}$. Empirically, we have $\eta\simeq 1$. Combining that with the
results from the Solow model, we have

$N\sim K^{\alpha}$ $K\sim I^{\sigma}$ $N\sim I$

which tells us that $\alpha\simeq 1/\sigma$ – one of the conditions that gave
us the original Solow model result.

####  3.5.2 Partial equilibrium in the Solow model

Since $N\rightleftarrows I$ we have a supply and demand relationship between
output and investment in partial equilibrium. We can use equation (3.35) and
$\eta=1$ to write

$I=(1/p_{I})N\equiv sN$

where we have defined the saving rate as $s\equiv 1/(p_{I}\eta)$ to be (the
inverse of) the abstract price $p_{I}$ in the investment market. A shock to
aggregate demand would be associated in a fall in the abstract price and thus
a rise in the savings rate. Overall, an economy does not always have pure
supply or demand shocks, so there might be some deviations from a pure demand
shock view. In particular, a "supply shock" (investment shock) should lead to
a fall in the savings rate.

####  3.5.3 Interest rates in the Solow model

If we add the IS-LM model from section 3.3 to include the interest rate ($i$)
model using $N\rightleftarrows I\rightleftarrows M$ written in terms of
investment and the money supply/base money:

$(i\rightleftarrows p):I\rightleftarrows M$

where $p$ is the abstract price of money (which is in IE with the interest
rate), we have a pretty complete model of economic growth that combines the
Solow model with the IS-LM model. The interest rate model in Figure 5 joins
the empirically accurate Cobb-Douglas production function in this section
Figure 11.

###  3.6 A note on constructing models

In the previous sections we have used simultaneous markets that look formally
the same. However, they are interpreted differently:

  * $\bullet$

In the Solow-Swan model, we used $N\rightleftarrows K$ and $N\rightleftarrows
K$ to define the production function. These are taken to be _independent
equations_ in general equilibrium. The represent two channels of information
flow to two destinations as shown in Figure 14.

  * $\bullet$

In the Solow-Swan model, we also used $K\rightleftarrows I$ and
$K\rightleftarrows D$. These were taken to be _simultaneous equations_ in
general equilibrium. The information transfer figure would look like a single
channel with two destinations as shown in Figure 13.

  * $\bullet$

In the IS-LM model, we used $N\rightleftarrows M$ and $N\rightleftarrows S$.
These are taken to be simultaneous equations in _partial equilibrium_ (i.e.
$N$ moves slowly). The information transfer figure would look like the single
channel Solow-Swan model diagram in Figure 13.

Figure 13: The Solow-Swan production functio as an information equilibrium
model. Figure 14: The Solow growth model as an information equilibrium model.

##  4 Statistical economics

Analogies between physics and economics only have merit inasmuch as they are
useful. In this section we will take some initial steps toward defining the
“statistical economics” of Smolin, (2009) analogous to statistical mechanics.
Consider a collection of individual market information sources $\\{N_{i}\\}$.
In the following we will work in “natural units” and take
$n_{i}=N_{i}/N_{i,ref}$ and $m=M/M_{ref}$. The $n_{i}$ are the demands in the
individual markets and $m$ is the money supply (it does not matter which
aggregate at this point). The individual markets are the solutions to the
equations:

$\frac{dn_{i}}{dm}=a_{i}\frac{n_{i}}{m}$ (4.1)

following from the introduction of the money-mediated information transfer
model $n_{i}\rightleftarrows m\rightleftarrows s_{i}$ as was shown in Section
3.1. One interesting thing is that the defining quality of these individual
markets – equation (4.1) leads to supply and demand diagrams – is homogeneity
of degree zero in the supply and demand functions (as noted in Section 2.3),
which is one of the few properties that survive aggregation in the
Sonnenschein-Mantel-Debreu theorem.

Now consider the sum (defining aggregate nominal output or NGDP across all the
markets)

$N(m)=\sum_{i}n_{i}=\sum_{i}m^{a_{i}}=\sum_{i}e^{a_{i}\log m}$ (4.2)

This has a form similar to a partition function888Partition functions
represent maximum entropy probability distributions; the mathematical
formalism is similar to random utility discrete choice models.

$Z(\beta)=\sum_{i}e^{-\beta E_{i}}$ (4.3)

Proceeding by analogy, we will define the macroeconomic partition function to
be:

$Z(m)\equiv\sum_{i}\frac{1}{n_{i}}=\sum_{i}m^{-a_{i}}=\sum_{i}e^{-a_{i}\log
m}$ (4.4)

With this partition function, the ensemble average (or expectation value,
denoted with angle brackets) of the exponent $a_{i}$ is:

$\langle a\rangle=-\frac{\partial\log Z(m)}{\partial\log
m}=\frac{\sum_{i}a_{i}e^{-a_{i}\log m}}{\sum_{i}e^{-a_{i}\log m}}$ (4.5)

which corresponds to the aggregate information transfer index $k=\langle
a\rangle$. Additionally, the nominal economy will be the number of markets
$N_{0}$ times the ensemble average of an individual market $m^{a_{i}}$, i.e.

$\displaystyle\langle N(m)\rangle$ $\displaystyle=$ $\displaystyle
N_{0}\frac{\sum_{i}m^{a_{i}}e^{-a_{i}\log m}}{\sum_{i}e^{-a_{i}\log m}}$ (4.6)
$\displaystyle=$ $\displaystyle N_{0}\frac{\sum_{i}1}{\sum_{i}e^{-a_{i}\log
m}}=\frac{N_{0}^{2}}{Z(m)}$ (4.7)

Equation (4.7) simplifies when $M=M_{ref}$ ($m=1$) to

$\langle N(1)\rangle=N_{0}^{2}/N_{0}=N_{0}$

First there is an interesting new analogy with thermodynamics: $\log m$ is
playing the role of $\beta=1/kT$, the Lagrange multiplier (thermodynamic
temperature). As $m$ gets larger the states with higher $a_{i}$ (high growth
markets) become less probable, meaning that a large economy (with a large
money supply) is more like a cold thermodynamic system. The meaning of large
here is measured by $M_{ref}$. As an economy grows, it cools, which leads to
slower growth – going by the terms "the great stagnation" in Cowen, (2011) or
"secular stagnation" in Summers, (2013) – and as we shall see a bending of the
price level vs money curve (low inflation in economies with large money
supplies).

Let us take $N_{0}=$ 100 random markets with normally distributed $a_{i}$ with
average $\bar{a}=1.5$ and standard deviation $\sigma_{a}=0.5$ and plot 500
Monte Carlo runs of the information transfer index $\langle a\rangle$, the
price level $\langle am^{a-1}\rangle$ and the nominal output $\langle
N(m)\rangle$.

Figure 15: Partition function calculation of the ensemble average of the
information transfer index from Eq. (4.5). Figure 16: Partition function
calculation of the ensemble average of the price level $\langle
am^{a-1}\rangle$. Figure 17: Partition function calculation of the ensemble
average of nominal output from Eq. (4.7).

In Figure 15 we can see the economies start out well described by the quantity
theory ($k\approx 2$) and move towards lower $\langle a\rangle$ as the money
supply increases. We can see the bending of the price level versus money
supply in Figure 16. In Figure 17, we can see the trend towards lower growth
relative to the growth in the money supply.

The question now is: how well does this oversimplified picture work with real
data? After normalizing the price level and scaling the money supply, the
function $P=\langle am^{a-1}\rangle$ almost exactly matches the information
transfer model for the price level in Section 3.4. The information transfer
model of Section 3.4 and the partition function version above are graphed in
Figure 18. There are only small deviations.

Figure 18: Model of price level from the partition function approach and the
approach of Section 3.4 using core CPI data from FRED, (2015). The deviations
are small between the models.

We apply the ensemble average result calculated using Eq. (4.7) and presented
in Figure 17 to empirical data for the US and show it in Figure 19. This
general trend is frequently encountered in the data for several countries as
part of a growing survey, see Smith, 2015a , and will be explored in future
work.

Figure 19: Model of nominal output.

With the partition function approach, we can see that reduced inflation with a
large money supply (a thermodynamically colder system) as well as reduced
growth in Figure 19 are _emergent_ properties. They do not exist for the
individual markets; it is important to emphasize this aspect of the model. An
economy with a larger money supply is more likely to be realized as a large
number of lower growth states (higher entropy) than a smaller number of high
growth states999A physical example: a state consisting of many low energy
photons has higher entropy than a state of equal energy consisting of a few
high energy photons. Therefore the blackbody radiation spectrum tends to be
produced by a distribution consisting of more photons with lower energy than
one with a few photons of higher energy.. One can think of the exponents
$a_{i}$ as growth states where the available $a_{i}$ depend macroeconomic
conditions.

###  4.1 Entropic forces and emergent properties

There are several novel interpretations of observed or theorized macroeconomic
effects that come from this partition function treatment. First, partition
functions are maximum entropy distributions so macroeconomic equilibrium may
be thought of as a maximum entropy state. Second, while the trend towards
lower $\langle a\rangle$ is apparent in the ensemble average, there is no
microeconomic rationale. Lower growth as an economy increases in size is an
“emergent” property of economies. Larger economies are “cold”, but small
economies without asymptotically large $N_{0}$ do not have a well-defined
“temperature” and individual markets may dominate output.

An additional emergent property is the slow decline in the response of the
price level to changes in the monetary base since

$\log P\sim(\langle a\rangle-1)\log M$ (4.8)

Using the interest rate model of section 3.3 to connect $M$ to $i$ means that
the impact of monetary policy is reduced for larger, “colder” economies.
Lowering interest rates (expanding the base) has a smaller and smaller effect
as $\langle a\rangle$ falls. This idea of ineffective monetary policy is
similar to the concept of the liquidity trap, see e.g. Krugman, (1998),
however there are some key differences:

  * $\bullet$

This _information trap_ does not depend on the zero lower bound for interest
rates. However, lower interest rates are related. As $\langle
a\rangle\rightarrow 1$, $\partial P/\partial M\rightarrow 0$ so that if
$i^{k_{i}}\sim N/M=PY/M$, increasing $M$ will lower interest rates (this
describes the liquidity effect). Therefore $\langle a\rangle\approx 1$ will
tend to be associated with lower interest rates.

  * $\bullet$

The information trap does not have a sudden onset, but is part of a gradual
trend towards lower interest rates. The onset may appear sudden in economies
that use monetary policy for macroeconomic stabilization when a large shock
hits (for example, the global financial crisis of 2008) and monetary policy
appears more ineffective than during previous shocks.

  * $\bullet$

There is no microeconomic rationale for the information trap. One mechanism
for the liquidity trap is based on the idea that at the zero lower bound for
interest rates, there is no difference between short term treasury securities
and zero-interest money. In the information equilibrium approach, the
information trap is an emergent property dependent on an ensemble of markets.

Finally, interpreting the $a_{i}$ realized in an ensemble of markets as the
occupied growth states in an economy lets us form a novel hypothesis: nominal
rigidity is an entropic force. Entropic forces in thermodynamics are forces
that have no microscopic analogy, yet have observable macroscopic effects. One
of the most commonly encountered physical entropic forces is
diffusion101010Gravity may be an entropic force as well, and if it is true
would actually be the best example. See Verlinde, (2011).. If molecules are
initially distributed on one side of a container, in short order they become
uniformly distributed throughout the container. There is no microscopic force
on a molecule proportional to local deviations in density $\rho(r)$ from
average density $\rho_{0}$ (the volume is $V$)

$F(r)\sim\rho(r)-\rho_{0}$ $\rho_{0}\equiv\frac{1}{V}\int d^{3}r\rho(r)$

However molecules behave _in aggregate_ as if such a microscopic force
existed, evening the distribution of molecules and producing a uniform
distribution. Individual molecules feel no such force. If the distribution is
perturbed away from a uniform distribution, it will feel an entropic force to
return to the original uniform distribution. Thus in thermodynamics we say
there exists an entropic force (diffusion) maintaining a uniform distribution.

Returning to our ensemble of markets we can imagine an equilibrium
distribution of growth states $a_{i}$. Analogous to the physical system,
markets will feel an entropic force to maintain the distribution of growth
states set by macroeconomic observables. Growth states will not spontaneously
over-represent the negative (or simply sub-inflation) growth states and will
behave as if there was a force keeping them in the distribution. Specifically,
while the distribution of prices (or wages) in the economy may not adjust to
adverse shocks as an aggregate (e.g. the price level will not fall),
individual prices may fluctuate by a large amount, for example see Eichenbaum,
(2008). Microfounded mechanisms like Calvo pricing (e.g. menu costs) enforcing
nominal price or wage rigidity would be analogous to the fictitious density
dependent force mentioned above for diffusion.

The physical concept of entropic forces is similar to the economic concept of
_tâtonnement_. In the case of entropic forces, individual molecules restore
equilibrium by random chance because equilibrium is the most likely state. The
molecules are “coordinated” by entropy. In the case of tâtonnement, individual
agents restore equilibrium by announcing their guesses at equilibrium prices
to the Walrasian auctioneer, who coordinates agent prices until equilibrium
(zero excess demand/supply) is achieved. Jaynes, (1991) referred to this
process as “dither” and noted its relevance for economics.

We can take this entropic description further by analogy with physical
systems. In the beginning of Section 4 we connected $\log M$ (where M is the
currency supply) with $\beta=1/k_{B}T$ in the partition function. If we take
the thermodynamic definition of temperature:

$\frac{1}{k_{B}T}=\frac{dS}{dE}$

as an analogy (where $S$ is entropy and $E$ is energy), we can write

$\log\frac{M}{\gamma M_{0}}=\frac{dS_{e}}{dN}$ (4.9)

where we have used the correspondence111111In the model of the ideal gas in
Fielitz and Borchardt, (2014), the information source is the energy of the
system. In our case that is aggregate demand $N$. of the demand (NGDP, or $N$)
with the energy of the system. We do not assume what the economic entropy
$S_{e}$ is at this point. However, if we take

$N\sim M^{k}$

then we can write down

$\frac{1}{k}\log\frac{N}{\gamma M_{0}}=\frac{dS_{e}}{dN}$

So that, integrating both sides (with $k$ being a slowly varying function of
$N$), we obtain

$S_{e}=\frac{1}{k}\;\frac{N}{\gamma M_{0}}(\log\frac{N}{\gamma M_{0}}-1)+C$

Using Stirling’s approximation for large $N/\gamma M_{0}\gg 1$ allows us to
write (dropping the integration constant $C$)

$S_{e}\simeq\frac{1}{k}\log\;\left(\frac{N}{\gamma M_{0}}\right)!$ (4.10)

If we compare this equation with the Boltzmann definition of entropy

$S=k_{B}\log W$

We can identify $(N/(\gamma M_{0}))!$ with the number of microstates in the
economy and $1/k$ being the ‘economic Boltzmann constant’. The factorial $N!$
counts the number of permutations of $N$ objects and one possible
interpretation is that $\gamma M_{0}$ adjusts for the distinguishability of
given permutations – all the permutations where dollars are moved around in
the same firm or industry are likely indistinguishable or approximately so.
This could lend itself to an interpretation of the constant $\gamma$ across
countries discussed in Appendix C: large economies are diverse and likely have
similar relative sizes of their manufacturing sectors and service sectors, for
example. Once you set the scale of the money supply $M_{0}$, the relative
industry sizes (approximately the same in advanced economies) are set by
$\gamma$. This picture provides the analogy that a larger economy ($N$) has
larger entropy (economic growth produces entropy) and lower temperature
($1/\log M$).

For small changes in $N\rightarrow N+\Delta N$, we can show

$\Delta S_{e}\simeq\frac{\Delta N}{k\gamma M_{0}}\log\frac{N}{\gamma M_{0}}$
(4.11)

Economic growth represents a rise in economic entropy. If the second law of
thermodynamics applied to economic systems, then one would expect that $\Delta
N>0$. However this is not true in real macroeconomic systems. In particular,
one heuristic indicator for a recession is two consecutive quarters of falling
NGDP. The second law of thermodynamics is statistically violated on small
scales per the so-called fluctuation theorem, see e.g. Evans, (2002), however
this would imply a specific form of the violation in terms of the
probabilities $P$

$\frac{P(+\Delta N)}{P(-\Delta N)}=e^{\Delta N}$

The tail of the actual distribution of changes in NGDP is over-represented
relative to a naive application121212This is not intended as a rigorous
argument, but rather simply to motivate the idea that falling NGDP in a
recession is not a random event. of this theoretical distribution as can be
seen in Figure 20. This is not a new observation; the fact that the
distribution of changes in NGDP (and other markets) does not have exponential
tails is a stylized fact of macroeconomics.

Figure 20: The distribution of quarterly changes in NGDP 1947-2015 (gray
bars). Data from FRED, (2015) series GDP. Heuristic estimate of the
probability tail from an application of the fluctuation theorem is shown as a
blue line.

However there is another way an economic system could violate the second law
of thermodynamics that is not available to a physical system composed of
molecules: coordination among the constituents. An ideal gas that changes from
a state where molecules have randomly oriented velocities to a state where
velocities are aligned represents a large fall in the entropy of that ideal
gas. This will not spontaneously happen with meaningful probability in large
physical systems. In economic systems, agents will occasionally coordinate
(for example, so-called “herd behavior”), and this may be the source of the
fall in economic entropy – and hence output – associated with recessions. It
is also extremely unlikely that economic agents will re-coordinate themselves
in order to undo the fall in NGDP. Absent reactions from the central bank or
central government (monetary or fiscal stimulus), the return to NGDP growth
will continue at the previous growth rate.

##  5 Summary and conclusion

We have constructed a framework for economic theory based on the concept of
generalized information equilibrium of Fielitz and Borchardt, (2014) and used
it to recover several macroeconomic toy models and show they are empirically
accurate over post-war US economic data. A question that comes to the
forefront: does the model work for other countries? The answer is generally
yes131313Some care is needed when looking at interest rates for e.g. the UK
and Australia where foreign-currency denominated debt (in this case, US
dollar) appears to cause countries to “import” the foreign interest rate.
(albeit with different model parameters), although a complete survey is
ongoing (Smith, 2015a ). Several examples appear in Figure 21.

(a) Cobb-Douglas function for Mexico (b) Price level for Japan (c) Inflation
rate for the EU (d) Interest rates for the UK

Figure 21: Application of information equilibrium to other countries. Nominal
growth from the Cobb-Douglas production function (Mexico) in the Solow model,
price level (Japan), inflation rate (EU) and long- and short-term interest
rates (UK).

This framework gives us a new perspective from which to interpret
macroeconomic observations and tells us that sometimes macroeconomic effects
are emergent and may not have microeconomic rationales141414This does not mean
they cannot be constructed as microeconomic interactions; they just do not
need to be.. Microfoundations, like Calvo pricing, may be an unnecessary
theoretical requirement. However the information equilibrium may also be seen
as satisfying the famous Lucas critique by utilizing information theoretic
constraints to analyze empirical regularities in macroeconomic systems.

In general, the information equilibrium approach is agnostic about what
mediates macroeconomic activity at the agent level or precisely how it
operates. This may be unsatisfying for much of the field. However a useful
analogy may be seen in physics. When Boltzmann developed statistical
mechanics, the atoms he was describing – although he believed they existed –
had not been established scientifically. The present approach can be thought
of as looking at the economy from a telescope on a distant planet and treating
economic agents as invisible atoms.

Even if it does not lead any further than the models presented here, the
information equilibrium framework may still have a pedagogical use in
standardizing and simplifying the approach to Marshallian crossing diagrams,
partial equilibrium models and common classroom examples. A future paper
Smith, 2015b  will look into the connection between the utility maximization
approach and an entropy maximization approach including: re-framing utility
maximization as entropy maximization and interpreting the Euler equation and
the asset pricing equation as maximum entropy conditions.

## Acknowledgment

We would like to thank Peter Fielitz, Guenter Borchardt and Tom Brown for
helpful discussions and review of this manuscript.

##  Appendix A Appendix

We have shown that several macroeconomic relationships and toy models can be
easily represented using the information equilibrium framework, and in fact
are remarkably accurate empirically. Below we list a summary of the
information equilibrium models in the notation

$detector:source\rightleftarrows destination,$

i.e. $price:demand\rightleftarrows supply$. Also the information equilibrium
models that do not require detectors are shown as

$source\rightleftarrows destination.$

All data for the US is available at FRED, (2015), including the Solow model
data for Mexico (real capital is inflated using the CPI less food and energy).
The UK data is from the Bank of England website and FRED. The Japan data is
from the Bank of Japan website and FRED. The Eurozone data is from the
European Central Bank website and FRED. The models shown in Section 3 are:

AD-AS model

$P:N\rightleftarrows S$

Labor market (Okun’s law)

$P:N\rightleftarrows H\;\;\;\;\mbox{or}$ $P:N\rightleftarrows L$

Model parameters for the US

$\displaystyle k_{H}$ $\displaystyle=$ $\displaystyle 0.43\mbox{ h/G\$}$

IS-LM model

$(i\rightleftarrows p):N\rightleftarrows M$ $i:N\rightleftarrows S$

Model parameters for the US interest rates (simultaneous fit)

$\displaystyle k_{i}$ $\displaystyle=$ $\displaystyle 3.49$ $\displaystyle
k_{p}$ $\displaystyle=$ $\displaystyle 0.124$

Model parameters for the UK interest rates (separate long, short fit)

$\displaystyle k_{i}$ $\displaystyle=$ $\displaystyle 2.71$ $\displaystyle
k_{p}$ $\displaystyle=$ $\displaystyle 0.0344\mbox{ (long)}$ $\displaystyle
k_{i}$ $\displaystyle=$ $\displaystyle 1.93$ $\displaystyle k_{p}$
$\displaystyle=$ $\displaystyle 0.0357\mbox{ (short)}$

Solow growth model

$N\rightleftarrows K\rightleftarrows I$ $K\rightleftarrows D$
$N\rightleftarrows L$ $1/s:N\rightleftarrows I$ $(i\rightleftarrows
p):I\rightleftarrows M$

Model parameters for Mexico

$\displaystyle k_{1}$ $\displaystyle=$ $\displaystyle 0.51$ $\displaystyle
k_{2}$ $\displaystyle=$ $\displaystyle 0.90$ $\displaystyle A$
$\displaystyle=$ $\displaystyle 0.0045$

Model parameters for the US

$\displaystyle k_{1}$ $\displaystyle=$ $\displaystyle 0.44$ $\displaystyle
k_{2}$ $\displaystyle=$ $\displaystyle 0.84$ $\displaystyle A$
$\displaystyle=$ $\displaystyle 0.0024$

Price level and inflation/quantity theory of money

$P:N\rightleftarrows M$

Model parameters for the US, using the PCE price level $PCE(2009)=1$

$\displaystyle M_{0}$ $\displaystyle=$ $\displaystyle 603.8\mbox{ G\$}$
$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle 0.641$
$\displaystyle\gamma$ $\displaystyle=$ $\displaystyle 5.93\times 10^{-4}$

Model parameters for Japan, using the core CPI price level 2010 index

$\displaystyle M_{0}$ $\displaystyle=$ $\displaystyle 12117.2\mbox{ G\yen}$
$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle 0.673$
$\displaystyle\gamma$ $\displaystyle=$ $\displaystyle 1.17\times 10^{-5}$

The definitions for the variables for all of these models are:

$\displaystyle N$ nominal aggregate demand/output (NGDP) $\displaystyle M$
monetary base minus reserves $\displaystyle H$ total hours worked
$\displaystyle L$ total employed persons $\displaystyle S$ aggregate supply
$\displaystyle P$ price level (core CPI or core PCE) $\displaystyle i$ nominal
long term interest rate (10-year rate) $\displaystyle p$ price of money
$\displaystyle K$ nominal capital stock $\displaystyle D$ nominal depreciation
$\displaystyle I$ nominal investment $\displaystyle s$ savings rate

##  Appendix B Appendix

In this appendix we show the numerical codes for the optimizations in Sections
3 and 4. They are written in _Mathematica_ using versions 8, 9 and 10.
_Mathematica_ does not have its own local weighted regression (LOESS or
LOWESS) smoothing function so we wrote one; the code is shown in Figure 22.

Figure 22: _Mathematica_ code for performing LOESS smoothing.

The parameter fits were accomplished by minimizing the residuals using the
_Mathematica_ function $FindMinimum$ using the method $PrincipalAxis$, a
derivative-free minimization method. The functions of the form $M0[yy]$ are a
_Mathematica_ interpolating function with interpolation order set to linear
using FRED, (2015) data as input. $M0[yy]$ is the monetary base minus reserves
(currency component), FRED series MBCURRCIR. $GDP[yy]$ is nominal gross
domestic product FRED series GDP. $PCE[yy]$ is the personal consumption
expenditures price level, excluding food and energy. $MB[yy]$ is the monetary
base, FRED series AMBSL. $R03[yy]$ is the three month treasury bill secondary
market interest rate, FRED series TB3MS.

Figure 23: _Mathematica_ code for fitting the price level. Figure 24:
_Mathematica_ code for fitting the interest rate. The labor market model was
fit using the similar code leaving out the parameter variable $aa$.

Figures 15, 16 and 17 in Section 4 were generated with the code in Figure 25.
The fits to the price level and nominal output used the code in Figure 26.

Figure 25: Partition function calculation of the ensemble average of the price
level. The 500 curves were generated by replacing $\\{1\\}$ with
$\\{500\\}$.The different ensemble averages plotted the variables $aavg$,
$navg$ and $pavg$. Figure 26: Fitting the ensemble average to the price level
and nominal output data.

##  Appendix C Appendix

If we keep the parameter $\gamma$ constant across countries, it can aid cross-
national comparisons as we show in this appendix. First, set up the variables

$\displaystyle\kappa$ $\displaystyle=$ $\displaystyle 1/k(N,M)=\frac{\log
M/C_{0}}{\log N/C_{0}}$ (C.1) $\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\frac{M}{M_{0}}$ (C.2)

setting up the constant $C_{0}$. I call these the information transfer index
(from the original theory) and the normalized monetary base, respectively.
Defining the constant

$\alpha=\frac{N_{0}}{M_{0}}$

we can write

$P=\alpha\frac{1}{\kappa}\sigma^{1/\kappa-1}$

Calculating the derivative above (after dividing by $\alpha$), one obtains

$\frac{\partial
P(\kappa,\sigma)}{\partial\sigma}=\frac{\partial}{\partial\sigma}\frac{\log
N/C_{0}}{\log\sigma M_{0}/C_{0}}\sigma^{\frac{\log N/C_{0}}{\log\sigma
M_{0}/C_{0}}-1}=0$ $\frac{P(\kappa,\sigma)}{\sigma}\left[\frac{\log
N/C_{0}}{\log\sigma M_{0}/C_{0}}\left(\frac{\log\sigma}{\log\sigma
M_{0}/C_{0}}-1\right)+\log\sigma M_{0}/C_{0}+1\right]=0$

The bracketed term must be zero since the piece outside the bracket is
positive, so therefore, after some substitutions

$-\frac{1}{\kappa}\log\left(\frac{M_{0}}{C_{0}}\right)+\log\left(\frac{\sigma
M_{0}}{C_{0}}\right)+1=0$

And we arrive at

$\sigma=\frac{C_{0}}{M_{0}}\exp\left(-\frac{\kappa+\log\frac{C_{0}}{M_{0}}}{\kappa}\right)$

Note that this function is only of $\sigma$, $\kappa$ and
$C_{0}/M_{0}\equiv\gamma$. This means if we use the parameters for one country
to find $\gamma$, we can then constrain the subsequent fits for Japan (and
other countries) to maintain $C_{0}=\gamma M_{0}$ (reducing one degree of
freedom). This constrains the fits so that the ridge lines where $\partial
P/\partial M=0$ coincide.

##  Appendix D Appendix

Information equilibrium is an equivalence relation. If we define the statement
$A$ to be in information equilibrium with $B$ (which we’ll denote
$A\rightleftarrows B$) by the relationship (i.e. ideal information transfer
between $A$ and $B$):

$\frac{dA}{dB}=k\frac{A}{B}$ (D.1)

for some value of $k$, then, first we can show that $A\rightleftarrows A$
because

$\displaystyle\frac{dA}{dA}$ $\displaystyle=$ $\displaystyle k\frac{A}{A}$
(D.2) $\displaystyle 1$ $\displaystyle=$ $\displaystyle k\cdot 1$ (D.3)

and we can take $k=1$. Second we can show that $A\rightleftarrows B$ implies
$B\rightleftarrows A$ by re-deriving the relationship 2.5, except moving the
variables to the opposite side:

$\frac{dB}{dA}=\frac{1}{k}\;\;\frac{B}{A}=k^{\prime}\;\frac{B}{A}$ (D.4)

for some $k^{\prime}$ (i.e. $k^{\prime}=1/k$). Lastly we can show that
$A\rightleftarrows B$ and $B\rightleftarrows C$ implies $A\rightleftarrows C$
via the chain rule:

$\displaystyle\frac{dA}{dB}$ $\displaystyle=$ $\displaystyle a\frac{A}{B}$
(D.5) $\displaystyle\frac{dB}{dC}$ $\displaystyle=$ $\displaystyle
b\frac{B}{C}$ (D.6)

such that

$\frac{dA}{dC}=\frac{dA}{dB}\;\frac{dB}{dC}=ab\;\frac{A}{B}\frac{B}{C}$ (D.7)
$\frac{dA}{dC}=k\;\frac{A}{C}$ (D.8)

with information transfer index $k=ab$. That gives us the three properties of
an equivalence relation: reflexivity, symmetry and transitivity.

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