text:
\section*{Introduction}

Understanding the fundamental nature of money is paramount for macroeconomic models. Indeed, over the past
century the description of money was central to the building of
simplified macroeconomic models, and monetary economics became a
field of macroeconomics in itself. The {\it classicals} first
postulated that money was neutral and avoided the need for any
suitable description. The {\it new classicals}, and most prominently
the monetarists reached a milder version with the quantity theory of money,
stating that money was neutral in the long run. Again, the debate was
not on the structure of the monetary system but rather on its
implications for macroeconomic theories. The debate between
neoclassicals and Keynesians led instead to theories of money being
rather theories of interest rates.

There is certainly no unique theory of money, since money is not a universal concept. Indeed, any monetary arrangement in a society is
different and leads to a specific description. We should describe the various possibilities of {\it  monetary systems} rather
than talk about a unique {\it theory of money}, including those which have
been or are realized, but also allow for the description of possible
systems not yet realized in practice, as required by the general
democratic debate. This is nicely summarized by~\citet[p1]{Minsky1996a}: {\it ``[\dots] relevant theory is not a
  compendium of propositions derived from axioms assumed to be
  universally true: economic theory is not a subdivision of mathematics. Relevant theory is the result of the exercise of imagination and logical powers on observations that are due to
experience: it yields propositions about the operation of an actual economy"} .

The monetary systems can be gathered in two major categories: metallism and
chartalism. Oversimplifying, one can say that metallism is the description of monetary systems in which money
is backed by a real asset, often gold, whereas chartalism is the
description of monetary systems based on pure fiat money that
originates from the state's ability to raise taxes (see \citet{StephanieBell2001} for a historical perspective and~\citet{Innes1913,Lerner,Knapp} for seminal papers). Since the
end of the Bretton-Woods system in 1971, which was already a remote form
of convertibility, this latter descriptive framework has received renewed
attention and is often named neo-chartalism in that respect~\citep{Wray1990}.

In this article, we develop a graph representation of monetary arrangements, based on the fact that accounting is performed
with double entry balance sheets. As it was formulated by~\citet[p12]{Minsky1992a}: {\it ``A capitalist economy
  can be described by a set of interrelated balance sheets and income
  statements''}, implying that graphs are the natural theoretical tool to
describe the monetary systems. Such graphs are rather common in
representations of flows in macroeconomics, but are generally absent
for the description of stocks, although they have already been extensively
used to study the systemic risks in networks of commercial banks~[see
e.g.~\citet{SheldonMaurer,UpperWorms,Upper2011,Gai,Wells2004}].
In this article, we argue that this is the natural language in which the discussions
about the nature of monetary systems should be carried out. By
representing a complex structure in a very compact form, it thus
carries a much clearer information, just like a very deep equation
carries a powerful meaning in just one line. It opens the possibility to discuss on a clear basis the
major topics associated with monetary systems.

\section{General formalism for graphs of monetary arrangements}\label{SecGenForm}

\subsection{Commodities and units of account}

\revision{A general monetary arrangement is a collection of debts
  between the various members of the system. However, before describing how this
  system of debts is organized,  we must first specify what is owed in
  these debts, and we thus need an appropriate description for ownership. The simplest
  thing which can be possessed and then owed is a commodity} and in that case, the unit of account (UoA) is any natural unit for that specific commodity, e.g. a unit of mass for gold.

\revision{The simplest description consists in representing the owners
  together with the amounts they own. However, in order to unify the
representation of ownership with the representation of debts that we will include later, it proves useful to base our representation of
ownership on a graph by representing} the totality of a given existing commodity on a single
vertex, and adding edges departing from it toward the various owners. \revision{Both
representations are depicted in the left plot of Fig.~\ref{fig4}. In
the graph representation, the edges oriented from the commodity vertex
toward the individual owners are weighted by the
corresponding amounts owned.} The resulting
graph structure is thus a weighted and oriented graph with in fact a
simple tree structure. The oriented edges can be viewed
as arrows, and when departing from a vertex they represent a liability, while when incoming on a vertex they represent an asset.

A commodity can \revision{even} be considered to be completely stored in a vault, and the liability of
the commodity can be viewed as the liability of the vault itself. Since the vault is nobody's, the corresponding assets of members
are net assets as they are not the liability of somebody else. If the concept of a vault is not a purely theoretical simplification,
and if the commodity is really stored somewhere with a guard, then we
can reach a more complicated structure of ownership, where it is the
vault keeper which acts as a bank safe and possesses all the
commodity, but in turn issues the corresponding liabilities to the
customers.
The members of the system can thus in general
either possess directly the UoA, or own it indirectly
through a bank, having a (positively credited) account in that bank. In that case, we say that the bank issues an {\it I Owe
  You} (IOU), recording the gold placed into the vault. In the first case the customers need to physically
exchange the commodity, whereas in the second case they only exchange the
ownership certificates, that is IOUs. A possible
extension is then to reach a system in which this indirect ownership
is made through several banks, as in free banking systems. We
illustrate this in Fig.~\ref{fig4}.

\begin{figure}[!htb]
 \includegraphics[height=0.32\textwidth]{Plots/Fig40.pdf}
 \includegraphics[height=0.32\textwidth]{Plots/Fig41.pdf}
\includegraphics[height=0.32\textwidth]{Plots/Fig42.pdf}
    \caption{Left top: \revision{direct commodity ownership with
      a graph representation. Left bottom: standard representation of
      direct commodity ownerships without a graph.} \revisionbis{From
      that representation, we infer e.g. that the third customer owns
      $g_3$ UoA of the commodity}. Middle: indirect
      ownership of a commodity through a single bank. \revisionbis{The bank {\it owns}
      $g_B$ UoA, and it {\it owes} e.g. $g_{B3}$ UoA to the third customer.} Right: indirect
      ownership through several banks, with each bank owning directly the commodity.}
      \label{fig4}
\end{figure}

\subsection{Emergence of central banks}

The system of keeping track of ownership with the actual commodity
stored in a vault is a primitive system of bank, and it is likely that
several banks will appear. Depending on the degree of cooperation several configurations
are bound to appear once customers of different banks start to
interact in the economic world and need to exchange to settle
payments.
\begin{itemize}
\item {\it Independent banking.} The banks can ignore each other. This would force every customer
  to have a bank account in each bank. Indeed, once a person accepts a payment from somebody having its assets in a given bank,
  this person would be forced to open a bank account in the same bank to be able to transfer the liability. Eventually we just have
  several copies of the same system, which could be merged (all banks
  gathered to one, and the bank account of a given person in the
  different banks can then be summed).
\item {\it Decentralized banking}. In an intermediate system of cooperation, each customer would
  still have only one bank account in a given bank, but the banks
  themselves would possess a bank account in other banks, that is
  there would be oriented edges between the banks.
In this system, a bank $A$ owes gold to its customer, either because it has gold
  in its vault, but also possibly because another bank $B$ owes gold
  to bank $A$.
All banks have to keep tracks of  the claims they have on other banks, and will for sure be reminded
  what they owe.
\item {\it Centralized banking}.
  The previous situation can become rather complex, with $N(N-1)/2$
  claim-debt relations between $N$ banks. \revisionbis{Indeed any weighted
    oriented graph with $N$ vertices can be represented by a $N\times N$
    antisymmetric matrix which has $N(N-1)/2$ degrees of
    freedom. This point is further developed in \S~\ref{SecMatrix}.} It is actually a general
  theorem that whenever there are  strictly  more that $N-1$ edges between $N$ connected actors, there must be loops. It can thus become optimal to simplify the networks of
  debts and claims between banks, by removing all unnecessary loops. The simplest solution is to use a tree graph of unit depth, with one bank at
  the root acting as a central bank. It is obvious that in that
  case there would be only $N-1$ edges in this network of $N$ banks.
 As explained in the next section, any new debt added between two
 actors in a tree creates a new loop which can be removed to preserve \revision{an} optimal tree structure.
Instead of all banks keeping positions with all banks, the
  banks keep only a net position with a central bank.
In a last round of cooperation, the banks can decide to send their gold in the vault of the central bank, and keep a claim
  on it, so as to optimize the cost of gold keeping.
\end{itemize}

Note that in the case of independent banking and decentralized
banking, there is actually a central structure hidden through the
possession of gold. In decentralized banking, if all banks
settle their debts, they are back to independent banking, and they are
united through the claims they have on the commodity. So even if
there is no central bank, or no central system, the fact that only one
commodity is used is functionally close to a system with a central
bank. Gold acts as the root of a tree graph. The centralized
banking system is just there to avoid displacing physically heaps of
gold thanks to a common storage at the central bank. The use of gold, with its physical exchange, is rooted in the
lack of a central organization, as it effectively replaces it.

\subsection{Payments in a tree graph}\label{SecPay}

\revision{If} the tree-structure has been clearly identified
\revision{with a central bank and all other banks directly related to
  it}, then payments \revision{between customers of these banks} are
made extremely easy. For each pending payment between customers there
is a unique loop created, because the payment at this stage is the
addition of a new debt. We remind that a tree with $N$ vertices, has
$N-1$ edges, so when one edge is added, there is necessarily one loop.
In order to maintain the tree structure and to avoid direct claims between customers or between banks, this loop needs to be
removed.  We simply need to subtract the amount of the
payment in all lines of the loop as illustrated in
Fig.~\ref{fig5}. \revision{Note also that commercial banks can decide
  to keep direct claims as in decentralized
  banking. In that case, one would have a loop displacement rather
  than a loop removal, as the direct claim between customers would be
  transformed into a direct claim between commercial banks. It amounts
  to a partial settlement since it maintains the tree structure below each bank but
not among banks. In the remainder of this article we do not
investigate in more details the interbank structure and for simplicity
we assume that payment are fully settled.}
At any time step, there will be payments, and by successively
incorporating them, we effectively build the graph of stocks as an
addition of payments from an initial condition. Rephrased
differently, the graph of payments at a given time is the time
derivative of the graph of stocks at that given time. Graphs of stock
are the position of the monetary system, and graphs of flows are their variation.

In any case, a payment between customers requires to identify a loop that needs
to be eliminated. For that purpose, when a customer $a$ pays a customer $b$,
it needs to provide its bank $A$ with the name of the bank $B$ of
customer $b$. \revision{If the monetary system was previously in a tree
form as would be the case if commercial banks do not keep claims
between each other, this loop is unique. However if banks perform only
partial settlements of payments and thus always keep claims among each
other, there can be several possibilities to remove the extra loop introduced by the
payment of one customer to another one.}

\begin{figure}[!htb]
\includegraphics[height=0.38\textwidth]{Plots/Fig50JPKE.pdf}
\includegraphics[height=0.38\textwidth]{Plots/Fig51JPKE.pdf}
\includegraphics[height=0.38\textwidth]{Plots/Fig52JPKE.pdf}
    \caption{\revisionbis{Notation: $C$ stands for the central bank, $B_1$ and $B_2$ stand for the commercial banks and
      $1,2,3,4$ are the various customers of the system.  All
      debts/claim relations are reported as weights on the edges of
      the graphs. They are expressed in the UoA of the underlying
      commodity fully stored in the central bank. The orphan edge
      emerging from the total gold commodity stands for the gold
      possibly lost, not yet found or held in foreign countries.} Left: a pending payment results in a new short-term
      debt which creates a new loop. Middle: this loop is emphasized. Right: after the loop is
      removed the tree structure is recovered, and the new balances of
      debts-claims have been adjusted. \revisionbis{Note that we are actually
      stacking two types of graphs. The red graph represents commodity
      ownership relations, and the black  graph is made of IOUs for that
      commodity.} }
      \label{fig5}
\end{figure}

\subsection{Representing debts}\label{SecDebts}

After a payment, nothing prevents a customer from having a negative asset, that is from having a liability toward its bank.
It would be tempting to make more payments than the assets owned, and this
infinitely. There are several possibilities to prevent this from happening. The simplest one is to simply forbid
spending if it leads to negative assets.  This is clearly too simple,
and there are several reasons for allowing people to go into debt, as
investment is the key to economic growth. The usual solution to allow for customers to go into debt, but still keeping
control of it, is to charge interests and to set a time limit (the maturity of the debt). There are several ways one can
charge interests. But they are all based on the same principle which
is to lend an amount and to ask for a higher repayment at maturity. The difference between the initial amount lent $S$, and the repaid
amount can be converted into an interest rate $i$ for a given unit of time, using usual composition
of interests for the duration of the maturity $D$.
Any complicated loan structure can always be considered as a set of
 independent loans which have this structure.
In the graph representation, we indicate a debt by $(S,i,D)$
where $S$ is the current amount, $i$ the interest rate and $D$ its
maturity.
After every unit of time, $(S,i,D)$ is replaced by $(S\times(1+i),i,D-1)$, and when the maturity becomes null, it is
transformed into an immediate payment of amount $S\times(1+i)^D$ whose
loop needs to be removed as described in
~\S~\ref{SecPay}. \revisionbis{Because the nature of each debt is
  different due to variations in interest rates and maturities, this
  means that we are now considering several types of assets and
  liabilities. By representing several types of debts on
  the same financial graph we are actually stacking several weighted
  graphs on the same set of vertices.} The representation of debts is used in \S~\ref{SecLoanDeposit} where we
detail credit creation.

\subsection{Conservation law and analogy with physics}\label{SecConsLaw}

The concept of oriented graphs, with local conservation of charge has
long been known in physics~\citep{Kirchhof1847}. It is used in
electric circuits, with conservation of current at internal vertices, or in
Feynman graphs, with conservation of momentum at internal interaction vertices.
The same concept applies here to graphs of monetary arrangements.
At every vertex, the sum of all lines should vanish. Or if we separate
into assets (claims) and liabilities (debts), the sum of assets at one
vertex should be equal to the liabilities of that vertex. The assets
are similar to positive charges whereas liabilities are similar to
negative charges, and vertices have to be financially neutral. The double entry book-keeping of any company or bank is thus designed
to preserve this neutrality. Following this analogy, any oriented edge
joining two vertices is similar to a charge displacement or a charge polarization, as it is a liability on
one side (a negative charge) and an asset on the other side (a
positive charge). Any immediate payment or any new debt-claim relation
conserves the financial charge, and is simply a displacement. \revision{However,
the accumulation of interests described in the previous section would break the neutrality of vertices, and we shall
need to introduce the net worth of institutions in \S~\ref{SecNW} to
properly include interests in our graph description.}

\subsection{Money and net money}\label{SecNetMoney}

This conservation law calls for a clear difference between net money and
money, which are respectively the analogues of total charge and charge
displacement. The IOUs of the bank are just the promise to convert them into IOUs of the
central bank (and if there is convertibility, these can in turn be
converted into gold). These financial assets are equivalent to interest-less debts. By introducing the
possibility to have debts, we see that we will have a mixed system in which we have two categories
of IOUs. In the first category, we have the IOUs of the
central bank, and the IOUs of commercial banks which are promises to
deliver IOUs of the central banks, and in the second category we have the IOUs of customers who have
borrowed and which promise to realize a payment in the future using IOUs of the first category. Throughout this paper we will make a
distinction between {\it money} and {\it net money} when referring to
these IOUs:
\begin{itemize}
\item money \revisionbis{held by a customer}  of the system consists in the financial assets, and it
  is the reflection of IOUs of commercial banks and of the central bank
  toward the customer;
\item net money \revisionbis{held by a customer} consists in money from which we subtract the
  liabilities, hence it is made of its financial assets minus its financial
  liabilities, or net assets.
\end{itemize}
\revisionbis{These definitions are straightforwardly extended to a group of members
of the system by direct summation.} All amounts, be it for money or for net money, are expressed in the same UoA, e.g. the
national UoA set by the central bank. As we shall detail in \S~\ref{SecLoanDeposit}, any new loan modifies
the total money, but conserves the total net money and one should
always be very careful about the words used.
\ifmoneyagg
This distinction between
money and net money is also related to the various definitions of monetary
aggregates that we discuss in \S~\ref{SecMi}.
\else
\fi

\subsection{Consolidations and sub-graphs}\label{SecConsolidations}

Let us draw a closed line in the monetary graph to define a sub-region. We assume at first that there are no customers sitting
inside this sub-region. Since all vertices inside it are by construction
neutral, and since we can ignore the internal financial relations as they net out, we can replace the subregion by a single
vertex whose assets and liabilities are given by the incoming and outgoing lines of the
region. In practice, it means that if we do not need to consider the
internal financial structure inside the region, we can ignore it and simplify
the graph. In accounting, this is called a {\it consolidation} and helps in reading the structure of graphs. One can for instance decide to consolidate the
financial sector, and thus to reduce all banks to just one bank like
in Fig.~\ref{fig7}.

\revision{Alternatively, instead of erasing all the details of the
  sub-region, we can focus on it by eliminating the
  details of the outside region. This amounts to considering a
  sub-graph, with externals sources and sinks. For instance, in the
  left plot of Fig.~\ref{fig7}, we could focus on the banking sector
  which sits inside the closed curve. This is exactly what is done by \citet{SheldonMaurer,UpperWorms,Upper2011,Gai,Wells2004} where the emphasis is on the stability of the banking
  sector itself.}

\begin{figure}[!htb]
\includegraphics[width=0.42\textwidth]{Plots/Fig70.pdf}
\includegraphics[width=0.42\textwidth]{Plots/Fig71.pdf}
\caption{Left: we identify a closed curve for a consolidation of banks
  located inside \revisionbis{($B_1$, $B_2$ and $B_3$)}. Right: the result of consolidating the
  commercial banking sector. \revisionbis{For simplicity, the commodity held by the
central bank $C$ is not represented.}}
   \label{fig7}
\end{figure}

\subsection{\revisionbis{Relating balance sheets and financial graphs}}\label{SecMatrix}

\revisionbis{We have exposed basic tools to represent a financial system with
graphs. Since this can also be represented using balance sheets, it
appears natural to wonder how these two approaches are related.
The starting point is to note that a weighted graph is equivalent to
an antisymmetric matrix $W$, if the entries of the matrix correspond
to the weights of the graph. For instance, if we have
$W_{ij}=-W_{ji}>0$, then this value corresponds to the weight of the directed edge from the vertex $i$
toward the vertex $j$. Balance sheet representations are then just a
special way to visualize such antisymmetric matrices. It consists in
detailing the matrix column by column, separating the positive and
negative entries. For a chosen vertex $j$, the entries of the column
are the $W_{ij}$ where $i$ runs over all vertices. If $W_{ij}>0$, then
the financial relation is an asset as it corresponds to an IOU from
vertex $i$ to vertex $j$, and if $W_{ij}<0$ it is a liability since it corresponds to an IOU from vertex
$j$ to vertex $i$. Said differently, for a given vertex the outgoing
edges (resp. the ingoing edges) are the liabilities of that vertex
balance sheet (resp. the assets of that vertex balance sheet). For each graph representation, the balance sheet picture can
be inferred trivially by the examination of the edges attached to each vertex. A
graph can be considered as the simplest tool to visualize a set of balance sheets.

We must note that if we have several different types of financial relations, then we have several types of
antisymmetric matrices and thus several types of weighted graphs and
several types of entries in the balance sheets. For
instance in Fig.~\ref{fig5}, we have one graph for the ownership of
gold which is partially represented, and one graph for IOUs on gold.
Just like matrices can be added, graphs can be stacked, and balance sheets
can be combined so as to lead to a global representation of the
financial relations.
}

\section{Private banking and endogenous money creation}\label{SecEndo}

Having all the tools to represent the graphs of financial stocks, we
are now ready to investigate in details the general organisation of monetary
systems. In this part, we focus on the specificities of the banking sector and
postpone the role of the Treasury to \S~\ref{SecState}.

\subsection{Net worth of banks}\label{SecNW}

In this section, we define the net worth of banks so as to be able to investigate in details the mechanism of
money creation through credit creation in the next section.
An immediate issue when allowing for debts, is the possibility that
the borrower might not be able to repay when it falls due. Indeed, the borrower thinks that he will be able to repay in the future
because he is going to exchange his work against money, that in turn
he will use to reimburse. Once the debt is turned into
an immediate payment at maturity as explained in \S~\ref{SecDebts},
the borrower must have claims of at least the same amount on the
banks. There are then obvious reasons why the borrower might not be able to work to find the money needed. This
ranges from losing his job, having an accident, or simply being dead,
among many other reasons.

From a graphical point of view this would be equivalent to a
unilateral removal of the debt from a customer to its bank. However,
our formalism starts to be inconsistent if we allow for such
possibility, because the bank vertex will not remain neutral once this happens. In order to guarantee the neutrality of all bank
vertices, even in the case of a defaulting borrower, one thus needs to consider the {\it net worth} of the banker.

So far the bank was just a node or a vertex in the graph structure,
but behind a bank there is a banker\footnote{More
  realistically the physical group of persons made of
  stockholders. But throughout this paper we symbolically describe
  these as {\it the banker}.}. If a borrower defaults, the banker
should pay the corresponding debt for him. The banker is thus
considered as one special customer of the bank. In order to differentiate him from other customers, it
is useful to draw him on a different level. We will refer to it as the bank {\it net
  worth}. If a customer defaults, then instead of just removing the
corresponding line of the debt, we consider that the banker
(unvoluntarily) pays the borrower by reducing the bank net worth, so that he can
remove his debt. Conversely, when the borrower reimburses but also
pays interests, it does so to the banker and this means that the bank
net worth is continuously increased by the interests, \revision{and
  this is also another reason why the introduction of a net worth is
  necessary in the formalism}. With this notation, the bank vertex is by construction always neutral,
and the risk of defaulting together with the benefit of charging
interests, are reported on the bank net worth.

The notation of consolidation needs to be extended to take into account
 the net worth of banks or companies. The net worth in a sense is
 nothing but the total charge of a bank or of a company, that we have
 put outside of the central vertex, to comply with our habits of
 neutrality at vertices. If we draw a closed region that we want to
 consolidate, we just need to add all the values of the net worths
 which are inside, and they will add up to the equivalent net worth,
 as when we integrate the electric charge density in a given region in order to compute
 the total electric charge in it.

\subsection{Loans, deposits and banking}\label{SecLoanDeposit}

In this section, we detail the process of credit creation, and
illustrate the well known fact that it is at the origin of money, as
bank loans create deposits~\citep{Tobin}. When a banker grants a loan, it is equivalent to the addition of two arrows going in
opposite directions (see Fig.~\ref{fig8}, left plot). One is going from the commercial bank to the
borrower and corresponds to an increase of its assets (of its
money). But in exchange of this, the
other arrow is the debt going from the borrower to the bank. It has
the same amount, but it bears interest up to a maturity.
We see that the total of the assets possessed by a customer in his bank has increased, but of course the total
of its liabilities has increased as well at the same time by the same
amount. The borrower then spends what he has borrowed by paying
another customer (say of the same bank for simplicity). After this
payment is settled, the sum of all customers' assets and liabilities toward the
bank is unchanged. Indeed, if we were to consolidate the customers into only one customer, then the
total net assets, that is the total debt of the banking sector toward the
consolidated customer is unchanged. But since we have decided to call
money the financial assets of customers, and not the
net assets, we can say that there has been money creation as a result
of the act of borrowing.  This is called the {\it endogenous } nature of money

\begin{figure}[!htb]
\includegraphics[height=0.315\textwidth]{Plots/Fig80JPKE.pdf}
\includegraphics[height=0.315\textwidth]{Plots/Fig81JPKE.pdf}
\caption{\revisionbis{For simplicity we do not represent the commodity held by the
  central bank.} {\it Left}: we illustrate the basic steps of credit creation. {\it First graph}: the customer contracts first a loan for an amount $S$,
 with interest rate $i$, and maturity $D$ \revisionbis{which is represented on the
 edge going from customer $2$ to bank $B$. As a result the liabilities
 from bank $B$ to customer $2$ have been increased from $M_2$ to
 $M_2+S$. The bank $B$ has also liabilities toward customer $1$ for an
 amount $M_1$. This is reflected into the liabilities from the central
 bank $C$ to the bank $B$ which sum up to $M_1+M_2$.} We show in dashed
  arrow the payment that customer $2$ intends to make to customer
  $1$.  {\it
  Second graph}: \revisionbis{After payment from customer $2$ to
  customer $1$ for a service or an investment, customer $2$ is left
  with an extra liability, and customer $1$ has an
increased asset.} The loan has preceded the creation of a deposit.
{\it Right}: we illustrate loan repayment after $D$ units of time have
elapsed. \revisionbis{For simplicity we assume $M_2=0$.} {\it First graph}: the
borrower needs to work so as to receive a sufficient payment. Since
the borrower owes the principal plus interests, the net worth of the
banker has increased by an amount $I$ equal to the planned interests. {\it
  Second graph}: the borrower receives effectively the payment $P$ as a
claim on the bank. {\it Third graph}: it repays the loan and credit
money is destroyed. \revisionbis{The net worth of the bank has been increased by
the interests $I$ which has been subtracted from the total of the
customers' assets. The liability from the central bank to the
commercial bank is unchanged.
}
\label{fig8}
\end{figure}

\subsection{Interbank payments and financial institutions}\label{SecIntBank}

In the previous section, we have considered customers who would pay each other
easily as they have accounts in the same bank. If the borrower spends
by paying someone in another bank, this creates a loop in the
banking tree structure, which is removed when the payment
is actually settled. The bank of the borrower might then be indebted
toward the central bank if it does not have enough central bank money, leading
to two possibilities which are illustrated in Fig.~\ref{fig11}. {\it i)} Either the central bank allows for the
first bank to be in debt, and it will charge interests for
this. Then the central banks acts with banks as a bank acts with its
customers. This is not a surprise since this is the purpose of the
underlying tree structure. {\it ii)} Or the central bank does not allow
for such possibility and the second bank has to
accept a direct debt from the first one. We recover a decentralized
banking system in which they all have mutual debt \revision{as there
is no tree structure in the commercial banking sector.}

The way the monetary system works is actually a mix
of the two. First, the central bank guarantees the payment of an
interest on the deposits (possibly zero) of commercial banks at the
central bank. But it also guarantees that the commercial banks can borrow from the
central bank at the discount rate which is of course higher than the
rate paid on deposits. This means that no commercial bank would loan
its excess of central bank money at a rate lower than the deposit
rate, and would borrow at a rate larger than the discount rate. The central bank thus
defines a corridor of rates in which the commercial banks negotiate
interbank loans. The interbank rate evolves on a daily basis and the
central bank tries to influence it with its monetary policy (see ~\S~\ref{SecMonetary}).

If the bank of the borrower had enough central bank money (due to
previously existing depositors) to settle
the interbank payment, we say that it acts like a {\it financial
  institution}, if we say that it lends the previous deposits of customers. But if the
depositing customers decide to withdraw their assets before the loans
made are repaid, then the balance sheet of this institution looks
again like a usual bank.

Finally, we note that the power lies in the institution which
allows to be indebted. When a bank goes into debt, the central bank
sets the conditions of the loan. In a tree structure, we have a hierarchy with the upper vertices holding a form of power on those situated
below.

\begin{figure}[!htb]
\includegraphics[height=0.335\textwidth]{Plots/Fig110.pdf}
\includegraphics[height=0.335\textwidth]{Plots/Fig111.pdf}
\caption{\revisionbis{Notation: $C$ is the central bank, $B_1$ and $B_2$ are commercial
  banks, and $1,2,3$ are various customers. On all edges are written
  the amounts of the debt/claim relations between the various actors. They
  are expressed in the UoA of the commodity stored in the central bank
  which is not represented there for simplicity.} {\it Left}: First, the banker who has already received the deposits
  of some customers, grants a loan to a given customer. Then the customer spends in another bank, and if the
  amount lent is less than the deposits received earlier by the institution, it still holds a claim on  the central bank. This way of lending is
  typical of financial institutions.
{\it Right}: First the banker grants a loan, but has never received any deposit
from any saver. It makes sure to ask for an interest rate higher than
the discount rate. Then the borrower spends in another bank, forcing his bank to borrow at the central
bank. Finally, since the second bank now has an excess of central bank
money bearing only the deposit rate, it might prefer to lend
it directly to the first bank at a larger rate which is lower than the
discount rate. We have not shown the net worth of bankers for
simplicity, assuming they vanish, but they will evolve because of
interests payments.}
\label{fig11}
\end{figure}

\subsection{\revisionbis{Bank runs and the nature of money}}\label{SecBankRun}

Let us show why once credit has been included in a
monetary system, it becomes incompatible with gold convertibility.
We remind that in that case, the central bank IOUs are the promise to
deliver gold. In a monetary arrangement, then thanks to conservation of charge at each
vertex, the sum of the leaves of the tree, that is the sum of IOUs
possessed by all customers, reflects all the IOUs issued by the
central bank. But it is the total sum of leaves, including assets
(claims on the banks) and liabilities (debt toward the banks) which is
the net money and reflects the total commodity in the central bank's
vault. If there is no debt at all, and customers have only assets,
then everybody can ask at the same time to convert their wealth into
gold. The entire tree would vanish, and we would be back to a
situation where everybody has physically a share of the commodity.
But if we allow for debt among the customers then we must be careful with
the fact that money (positive assets) is not net money. If we allow all
positive assets to be converted to gold, then we have more claims on
gold than gold itself, and the system fails. How is it possible if we
have been so careful to ensure charge conservation at every stage of
our monetary system? It is extremely simple and reflects the true
nature of money with respect to net money. Money is now partially
claims on the gold of the central bank, and claims on the debts of
the borrower. It has shifted from its {\it commodity nature} to its
{\it credit nature}. In order to see this we need to look at the outside
vertices of the monetary structure. This includes the assets of the
customers but also their liabilities to the system. And these debts are not available immediately since the
borrowers have to work to be able to repay them. This is illustrated
in Fig.~\ref{fig10bis}. Money is partially a right to get gold, and partially a right to ask for the
borrowers to work by purchasing their workforce. Indeed when a
customer with positive assets pays a borrower in exchange of work, the reduction of the loop reduces the assets of the payer,
and reduces the liabilities of the borrower. There is net
money conservation, but there is money destruction.

\begin{figure}[!htb]
\includegraphics[height=0.40\textwidth]{Plots/Fig102.pdf}
\includegraphics[height=0.40\textwidth]{Plots/Fig103JPKE.pdf}
\caption{{\it Left}: \revisionbis{The central bank holds $M$ units of gold which
  are reflected  into the central bank IOUs toward the commercial banks.} The total assets of customers are reflected partially by
the gold held at the central bank, and partially by the debts of the
borrowers. \revisionbis{The orphan edge emerging from the total gold commodity
stands for all the gold held outside, e.g. in foreign countries, or
held privately, and even the gold which is lost or not yet discovered.} If customers ask to convert their assets into gold,
they cannot convert all of it. At some point, the central bank needs to stop the
convertibility. {\it Right}: if the central bank stops gold
convertibility, then customers can only convert their assets into
central bank IOUs, usually in the form of anonymous paper money.
}
 \label{fig10bis}
\end{figure}

It is because of this incompatibility between money and net money, the fact that
the first one is only partially a claim on the assets of the central
bank, but also partially a claim on the future workforce of borrowers,
that convertibility must be abandoned. By allowing customers
to go into debt, but by incorporating these debts in the monetary
structure, we have shifted the nature of the IOUs, in an irreversible
manner which calls for the end of convertibility. Conversely, if no
bank had accepted to polarize money, that is to generate money endogenously, and if loans where made by
customers to customers in the form of bonds that would not be
encapsulated into the tree structure, then the IOUs of the banks would
still be claims on the central bank gold reserves. One would have two types of
currencies. One would be the IOUs of banks which are eventually
redeemable in central bank assets (gold), and then the IOUs of
individuals that we could also exchange to settle payments. In some cases we would exchange claims on gold, and in other cases claims
on future workforce, but nothing guarantees that they would be traded at par. By incorporating the loans, or at least a major
part of the loans in the tree structure, we have obtained
a much more liquid system, at the price of abandoning
convertibility. As we explain in \S~\ref{SecPeg}, the same process
happens with a currency pegged to an external one, since the external currency acts like gold. Allowing to convert money instead of just net
money in the external currency, contains the same internal contradiction leading to an
unavoidable breakdown of the currency peg. In the remainder of this
article, we assume that gold convertibility by the central bank is no
more possible, as is the case for the current monetary systems.

\subsection{From commodity to fiat money}

Before we go any further, a comment is in order about the history of
monetary systems. The story we presented for the integration of banks
in a network is only a pedagogical story, and is by no means {\it
  history}.
The presentation adopted here would imply, if taken at face value, that there was first barter,
then in a sophistication, there was exchange of commodities, and only
after came a mild form of fiat money as a claim on commodities (gold)
toward banks. This story implies that there is a positive historic movement toward liquidity,
but where the value of claims is backed by gold. This point of view is
essentially the {\it metallist}  point of view or the {\it commodity
  theory of money}. If this might have been true for some societies at some times, there is no proof that
this has been a general feature. And as we have shown in
\S~\ref{SecBankRun}, it is inconsistent when the banking system allows
for the creation of credits.

It is argued by~\citet{Graeber2011} that this is rather the exception in history, and that the most
generic type of money is credit money which finds its origins in the quantification of moral debts in units of
accounts. The gold standard money appears in that perspective as an
exception driven by the industrial revolution, and ended in 1971.
These general ideas are rooted in the {\it chartalist} description of the origin of money, where the UoA for the
debts is actually set by the state. It dates back from Mitchell-Innes~\citep{Innes1913} who emphasized
that money is a standard of deferred payment. As the state spends in a
chosen and arbitrary UoA, it also sets that this unit
should be used for debt repayment, as all individual debts toward society which take the form of taxes have to be redeemed in
that state UoA. In that case, this pure fiat money does
not derive its value from the market as a commodity, but is rather credit money whose value is
initiated by the sovereign states. These ideas have been further developed by~\citet{Lerner} and \citet{Knapp} and are currently
revived under the name {\it neo-chartalism}~\citep{Wray2004}.

In order to shift from a commodity money to a pure fiat money we just
need the central bank to give up on its commitment to convert its
IOUs into gold.
In that case, any customer can ask for its assets to be converted into
central bank's IOUs, usually paper money, but not into gold. The
{\it chartalist} theory of money argues that everything would behave
as usual as long as the state continues to tax in that
UoA. It thus argues that taxation is a {\it sufficient condition} to
impose that the state IOUs are used for debt repayments. The true amount of commodity inside the state issued coins does not matter, as long as the state has the monopoly on
coinage and the ability to tax in this unit.
The power of states lies in their ability to tax and neo-chartalists
thus refer to such a fiat money system as a {\it sovereign currency}.

\citet{Forstater2003} [see also~\citet{ClassicalEconophysics}] has recently explored the monetization in colonies by colonizing countries, and
showed that it was performed exactly following that logic. In many
cases, the colonizing countries have spent their currency in the colonies, and enforced its use by
promising to tax in that same currency. When the US landed in Europe
on 6 June 1944, they wanted to impose such a chartalist monetization, and it was strongly opposed by the French
government in exile,  precisely on the ground that this would amount to colonization~\citep{deGaulle} and not liberation.

This controversy about the nature of money is extremely pregnant in
the macroeconomic debate. But as we have argued already, there is no
unique theory of money.
Given that the gold convertibility of the Bretton-Woods agreement has been abandoned in 1971, there is no
doubt that the current international system is a chartalist system.
In the next part, we  detail the role of the state in the
monetary arrangements based on such pure fiat money, and we discuss the budgetary tools.

\section{State representation}\label{SecState}

\subsection{Borrowing and spending: Budgetary policy}

\subsubsection{Borrowing methods}\label{SecBoCen}\label{SecBorrowBanks}

The state has a bank account at the central bank, called the {\it
  Treasury}, and it feeds it by forcing the payment of taxes. Any taxpayer, creates a new loop when he is asked to pay the Treasury, and
by the usual removal of the loop described in~\S~\ref{SecPay}, the Treasury account at the
central bank increases, whereas the customer's account and the
account of its bank at the central bank decreases.
It is formally equivalent to taxpayers converting their assets into
central bank paper money and handing them over to the state. From a
chartalist perspective, this taxation drains central bank IOUs out of
the private sector, enforcing the need to use and hold them.

By contrast, public spending goes from the Treasury account to
individuals. For sure, the state can always tax more than it
spends. However, if the state wants to spend more than what it taxes,
it needs to create a public debt which can take various forms.
\begin{itemize}
\item {\it Money printing}: This is the easiest possibility.
It consists in increasing unilaterally the amount on the Treasury
account. Since there is no more convertibility this is technically possible, because the IOUs of the central bank are not exchangeable
with anything.

\item {\it Directly borrowing from the central bank}: Another possibility would be not to print money, that is to put it on
the Treasury account out of thin air, but to lend it to the Treasury. This is
exactly as when money is printed, except that now the state also
issues a Treasury bond (T-bond) to the central bank. Such a situation can only really work
if both the Treasury and the central bank obey the same sovereign
power, so that the state actually controls both sides of the deal, and
it is only a formal arrangement. If this is the case, then it can
borrow as much as it wants. This is fundamentally different from a country
borrowing in a foreign currency, and it is the essence of a sovereign
currency. And when it needs to reimburse, it can just borrow again
what is needed.
Eventually, as the bonds remain and are extended or replaced by similar bonds, this is equivalent to money printing.
The process of directly borrowing from the central bank is depicted in
Fig.~\ref{fig12A} (left plot).  What we realize is that initially the net money of
customers and bankers is $M$ and after the Treasury has borrowed and
spent $S$, the total net money of customers and bankers is $M+S$. This
increase can be traced to the central bank as we see that the
liabilities of the central bank toward the commercial banks now include the spending of the Treasury. Everything happens as if the T-bonds were the new gold
from which the central bank originates its liabilities. If we consider
the consolidated state, which would be the Treasury and the central
bank merged, there is creation of total net money.

\begin{figure}[!htb]
\includegraphics[height=0.40\textwidth]{Plots/Fig12A.pdf}
\includegraphics[height=0.40\textwidth]{Plots/Fig12B.pdf}
\caption{{\it Left}: directly borrowing from the central bank leads to an increase of
  the total net money of customers and bankers. {\it Right}: Initially the bank already has a T-bond in its
  assets. Then, the bank uses it as a collateral to borrow from the
  central bank. The maturity of the loan is smaller and for this to be
interesting, the interest rate needs also to be smaller than the
interest served on T-bonds. The repurchase agreement is
enclosed in orange dashed lines. Instead of holding directly the
collateral, the commercial bank holds a claim on it. Finally, after
lending to the Treasury, the commercial bank receives a new Treasury
bond. The subsequent spending of the Treasury is depicted below in Fig.~\ref{fig12}.
}
\label{fig12A}
\end{figure}

\item {\it Borrowing from banks}: In this last case, the state is
  forbidden to borrow directly from the central bank and needs to
  borrow from commercial banks.
The result is that in exchange of a payment to the Treasury,
the state will issue a bond, that is an IOU toward those who have paid the Treasury. It can be individuals, but it is not
very common. Commercial bankers either use their net worth if it is
positive, or they simply borrow from the central bank what they want to lend to the
Treasury. This latter method is slightly more complicated as banks need to
provide a collateral, for instance a previously owned T-bond,
in a repurchase agreement illustrated in Fig.~\ref{fig12A} (right plot). At first the commercial bank sells a bond to the
central bank against a smaller amount of central bank money, and then it buys it
back at its value, the difference between the two amounts being
effectively an interest paid. The collateral is there to make sure the
bank will repay the loan in one form or another. Either it repays by
buying back the collateral, or it defaults and the central bank keeps it.
\end{itemize}

\subsubsection{Net money and financial wealth}

When the commercial banks pay the Treasury, the amount of IOUs possessed by the
bankers remains constant, but the nature has shifted from central bank
IOUs which promise nothing, to Treasury IOUs which promise central
bank IOUs in the future with interests. If the bank had to borrow the
amount lent to the Treasury, it must only make sure that the interest
received is larger than the interest paid. Nearly immediately after this, the
Treasury spends in the monetary system the amount of central bank
money it has borrowed. This situation is depicted in Figs.~\ref{fig12}
(left plot). This new T-bond held in the banks net worth
can then in turn be used as a collateral for further borrowing at the
central bank, and further lending to the Treasury, without any
theoretical limit.

\begin{figure}[!htb]
\includegraphics[height=0.42\textwidth]{Plots/Fig120JPKE.pdf}
\includegraphics[height=0.42\textwidth]{Plots/Fig130.pdf}
\caption{{\it Left}: First we show the simplified monetary system with a consolidated bank
  sector and a consolidated customers sector, where the net money of
  the customers and the bankers is $M$. Then we illustrate when the Treasury
  borrows an amount $S$ to the banks. Finally the state spends what it has borrowed. If the initial net
worth of the bankers is initially $m=0$, we realize that now the net
money of the customers is $M+S$ instead of $M$ initially. If we
consider both the customers and the net worth of the banking sector, the
net money made of IOUs which are convertible in central bank money has
not increased, but the financial wealth has increased as the net worth
of banks is also made of T-bonds. {\it Right}:
Several possible consolidations schemes lead to different
  interpretations. First, if we look at customers but also the net
  worth of bankers, there is no net creation of money induced by the
  public debt, but the financial wealth has increased as the net worth of bankers is made
  of T-bonds. However, if we look only at the customers, there is net money creation induced by the new public debt.
}
\label{fig12}
\end{figure}

The total net money of customers is increased by the amount
spent by the Treasury and it has a double origin. Some comes from the liabilities of the central bank, and some
comes from the liabilities of the bank sector net worth. But the net
worth of the bank sector remains unchanged since it also receives the
IOUs of the Treasury. Effectively, the net worth of bankers converts the IOUs of the Treasury to IOUs which are
redeemable in central bank money. Finally, we conclude that when the
Treasury borrows directly from the central bank, the T-bonds are
converted into net money by the central bank, whereas when the
Treasury borrows from a commercial bank, this conversion is made by the
commercial banks.
Consolidating the Treasury and the central bank helps to visualise it
more thoroughly.
The total net money of customers has increased by the amount spent by the state. In the first
case the Treasury IOUs are internal to the consolidated state, whereas by
borrowing to banks, the T-bonds now leak out of the
consolidated state sector as they enter the bankers' net worth.

Now the discussion about the creation of money is a discussion about the net worth of bankers. Should we look
only at the total net money of customers, ignoring the structure of
the bankers' net worth? If yes, then we conclude that there has been
net money creation, which finds its origin in the bankers' net worth. Both
the central bank and the commercial banks are responsible for the
total net money of customers.  But if we consider that there is no
reason to separate the net worth of the bankers, which are just a
special type of customers, then we need to subtract the amount owed by
the net worth to the bank vertex, and this amount exactly compensates
for the increase of assets in the customers sector. However, we should not forget that there
is another part in the net worth, which is an asset for the bankers, and which consists in the
T-bonds. So in this second interpretation, we would conclude
that the total net money originated by the central bank is the same as
initially - it is a reflection of the central bank liabilities which
are unchanged - but the system has been supplemented by T-bonds
which have a financial value as well. With this interpretation, one would say that no net money has been created, but the financial wealth has
increased. In Fig.~\ref{fig12} (right plot) we illustrate with graphs the process
of Treasury borrowing from banks and spending in the private sector, but also the possible
interpretations which can be made, depending on the consolidations
chosen.

\subsection{Open market operations} \label{SecMonetary}

It can be argued that it would be much more democratic if the
Treasuries were allowed to borrow directly from their central bank.
By electing a government on a program, we would know what deficit it
intends to run and thus how much it will be willing to
print, which in the long run is a debate about the possible level of
inflation. Instead, it has been argued that decisions made on democratic grounds
might be unstable as they are affected by elections.
However, the independence of central banks is also serving the
interest of commercial bankers as we argue now.

In practice, the central bank buys and sells bonds in open market operations. At least it is always doing
so with short term T-bonds as part of the conventional monetary
policy, and it might decide sometimes to do it as well with longer maturity T-bonds as part of the unconventional
monetary policy. This blurs the lines between a model where the
central bank directly finances the Treasury, and a model where this is
done by commercial banks since they result in the same final
situation. Indeed, before an open market operation the Treasury owes central bank
money to a commercial bank, and in the final situation it owes it to
the central bank itself, and the central bank money held by the
commercial bank has been increased accordingly.

The commercial bank has accepted to get rid of an IOU which bears
interest, in exchange of a central bank IOU which bears no interest.
However the Treasury will never default on its debt, because the state also runs the central bank which can buy an
infinite amount of T-bonds. Said differently, if the interest rates for short term
T-bonds start to increase as the commercial banks become more
and more reluctant to buy these, the central bank needs to buy as many
short term bonds as necessary to ensure the short term interest rates
on T-bonds remain at the targeted level.
By using these open market operations a sovereign state running a
sovereign currency has the means to ensure that the banks are always
willing to buy T-bonds, whatever the deficit is.

However, this system has a drawback. First when the
commercial bank bought the T-bond, it had to pretend that it was worried
the state might never reimburse, so as to ask for interests rates
which are at least slightly higher than the interest rate at which
they can borrow from the central bank, and make a profit on  the difference. Of course the banks knew they would
always be reimbursed, because the central bank always stands ready to
buy bonds. As the interest rates departed from the target chosen by the
central bank, the latter bought short term bonds to prevent the short
term rate from increasing. In order to convince a
commercial bank to get rid of a financial instrument which is not
risky and which bears interest, the only solution is to pay more than
the current value of the bond, which amounts to a decrease of the
interest rate on those bonds. The bank thus makes an immediate profit
instead of a larger profit later.
This difference goes directly into the net worth of the banker and amounts to money creation.

To conclude, we reach the same stage as if the Treasury had
sold directly its bond to the central bank, except that now we have increased
by a small amount the net worth of the bankers. By first selling the
bonds to the commercial banks, instead of selling directly to the
central bank, the bankers were able to realize a small profit. But this
profit is an immediate and easy one. So they have on one side to
pretend they do not like when the Treasury goes into debt, so as to be
able to ask for the highest possible interest rate, and secretly enjoy
it since either they make a profit when it falls due, or even
better immediately if the central bank buys the bonds to control
the interest rates.

The commercial banks will always end up with a part of their assets
denominated directly in central bank money, which bears no interest,
and T-bonds, which bear interest. If we adopt a consolidated
state point of view, where we merge the Treasury and the central bank,
then the commercial banks have two types of accounts. Deposits which
bear no interests, and saving accounts which generate interests, just
like everybody. In order to control the interest rate, the
consolidated state shifts the amounts from the interest-less to the interest-bearing account and
vice-versa.

\subsection{State consolidation}

This point of view of consolidating the Treasury and the central bank
is usually criticized because the central bank is supposedly fully independent.
On the contrary, we can argue that it is also a creature of the state and independence is only an
apparent one since it is decided by the state itself through legal
dispositions. Furthermore, it must react automatically with open
market operation whenever a new public debt is issued to control
interest rates, so that it serves the interest of the Treasury
and thus the state.

We must recall that there are two types of short
term interest rates that the central bank can control:
\begin{enumerate}
\item the interbank rate (called the Fed funds rate in the US), which is the
  rate at which the banks can borrow with collaterals at the central
  bank. This is depicted in the middle of Fig.~\ref{fig12A} (right plot). It is
  necessarily capped by the discount rate which is the maximum rate at
  which they can borrow, and above the rate paid on
  commercial bank deposits, if it exists;
\item the short term bonds rate which is affected by open market operations.
\end{enumerate}
If a bank decides to control the short term bonds rate, then when
conducting outright purchases of short term T-bonds, it increases the central
bank liabilities (central bank money), which decreases the
interbank rate. So controlling the short term bonds rate implies to control also
the interbank rate.

However, a central bank can decide to control only the interbank
rate and not the rate on short term bonds. For this, it will only conduct repurchase operations, or
collateralized loans to commercial banks. This would affect the rate
at which the commercial banks can borrow, but in general it will not affect directly the
rate of the short term T-bonds. Indeed, the Treasury would then
borrow like any customer, and banks are free to set the conditions. Nearly all developed countries, except the Eurozone, control both
rates and run fully {\it sovereign currencies}. However, the
European Central Bank (ECB) is in general unable to control interest rates on T-bonds. This is why the ECB performs only repurchase
agreements to control the interbank rate, but does not control the
short term rate on bonds via outright purchases of T-bonds,
contrary to the Fed monetary policy. As there are several
types of bonds, each one of them issued by a different government, the
ECB cannot decide which one to buy as it would amount to a form of
financial solidarity between the various European states, and this is
intentionally avoided in the European Union construction. As the various Treasuries are not helped by the ECB to issue low
interest bonds, everything happens as if they were borrowing in a foreign
currency, where the interest rates are set by the bankers, just like
for any standard customer. The bankers lending to the Eurozone
Treasuries decide what should be the markup rate, that is the difference between the rate at which they
borrow, which is effectively the interbank rate set by the central bank, and the rate at
which the lend to the various Treasuries. On the contrary, the United Kingdom (UK) or the USA are in the
configuration where the central bank buys whatever is necessary to
control the interest rate on bonds, and things are as if the Treasury was
borrowing from the central bank, except for the small profit made by
bankers due to their intermediation.

To conclude, if the Treasury borrows directly from the central bank, it
makes sense to consolidate the Treasury and the central bank. If it
borrows from commercial banks, but the central bank controls the short
term interest rates on the bonds, the effective theory is nearly the
same, and it also makes sense to consolidate the Treasury and the
central bank.
However, if like in the Eurozone the central bank stops
controlling the interest rate on bonds but controls only the interbank
rate, the Treasury is treated like a standard customer, as it is
treated by the central bank as if it was a foreign Treasury.
The consolidation might still be possible formally, since any
consolidation can be made as it is just an accounting simplification, but it hides some
salient features. Sometimes the commercial banks would add only a
small markup rate to the T-bonds and one would not see the
difference, but in case of crisis the markup rates can start to be
huge, on all maturities, even reaching the point where the commercial
banks stop buying T-bonds like in the recent Eurozone debt
crisis.

\subsection{Monetary policy}

The instruments of the monetary policy are very complex and can be
very different from one country to another. It is by no means the goal
of this paper to present them all. The main conventional tools are the control of
short term interest rates, and the reserves requirements.

\begin{itemize}
\item Conventional monetary policy is about controlling the short term
  interest rates. However, controlling the interbank rate is not sufficient to control the rate at which customers borrow. Indeed, when commercial
banks lend, they need to apply at least the interbank rate as it is the rate at which they need to borrow when the loan is used to
pay outside of the commercial bank, or when a part of the amount
loaned is transformed into cash, that is into central bank money.
But they also need to apply a markup rate to this basic interbank rate at
which they borrow for several reasons. First, the borrowers might default and
the bank needs to make sure that it generates enough profits from non
defaulting loans to compensate for the defaulting ones\footnote{When
  the loan is an investment loan, such as when customers buy houses,
  the investment in real assets is also used as a guarantee. If the borrower fails
  to repay the loan, the property of the real asset, e.g. the house, is transfered to the bank.}. Second the
commercial banks need to make a profit to cover their running costs as they need
to pay at least the salaries of their employees. Finally, since they
have borrowed on a short term basis, but they lend on a longer term
basis, they need to have a security margin in case the central bank
increases the interbank rate. For all these reasons, the effective
rate at which the economy is functioning, is different from the basic
interbank rate chosen by the central bank. If the central bank wants to foster
credit with low interest rates, it is as important to set a low interbank rate as
to communicate about the fact that this interest rate shall remain low, so
as to decrease as much as possible the markup rate. Finally, we must
stress that the interest rate is not the only criteria to ask for a credit,
as decisions are made on much more fundamental economic grounds. Even if the markup rate remains constant, the interest rate set
by the central bank is only an indirect tool to control credit and thus the total money.

\item The second most common monetary tool is made of the reserve
  requirements for commercial banks. In the theory of the money multiplier, the
  central money held by banks should be a fraction of its liabilities
  toward its customers, and this fraction should be set by the
  monetary authorities. It is then assumed that by controlling the
  amount of central bank money held by the banks, and by fixing the
  ratio, the central bank could control the total credit, and thus
  the total money. But this tool cannot be efficient, because the
  amount of central bank money held by banks is not exogenously set.
Indeed what is counted as a reserve for a commercial  bank is not its net central money, but its central money. So if the
  bank does not have enough central money reserves, for instance
  because it has granted too many loans, it can borrow the reserves needed at most at
  the discount rate, and more probably at the interbank rate.
When the bank does so, at the same time it receives central bank money
and increases its reserves to comply with its legal obligation, but it
owes it as well at a later date, and this does not count negatively in
the reserves. We see that the difference between net central bank money and central bank money is very important. The
central bank can control its net liabilities toward the banks, e.g. by
performing outright purchases of T-bonds, but it cannot control
its liabilities, as these are endogenously determined by the needs of
the commercial banks. In practice, banks lend whenever they think it
is profitable for them, and if they fail to meet their reserve
requirements at the end of the day, they just borrow (directly to the
central bank at most at the discount rate or to other banks) what is
needed.  In a few developed countries (Canada, Australia, UK, Sweden, Norway) there are no more fractional
reserves and nothing special happens. As long as the required
reserves are not net reserves, they are entirely useless. Furthermore,
even if the fractional reserve system was applied to net central
money held by banks, this would only set an upper limit to credit, but
it could not increase automatically the amount of credit if it is below the cap.

In fact the net assets of the banks, that is their net worth which includes the capital which has been given in by the stockholder,
is instead extremely important. These are true reserves which are
going to be used whenever a bank suffers losses. It is thus no wonder that after the 2008 financial crisis, the rules for the fractional reserves
have not been modified, whereas the capital requirements have been radically increased in the third Basel Accord.
\end{itemize}

\subsection{Should it stay or Schuld it go: the clash}

There is a huge debate on the nature of money and the nature of
public debts. Apart from the fact that T-bonds bear an
interest, they have an intrinsic difference with central bank
IOUs, which is that they have a maturity, and are thus bound to
disappear through the reimbursement of the Treasury's debt.
But we have seen that for the customers, the monetary system has
resulted in an apparent conversion of the Treasury's IOUs into normal
bank deposits for customers. Let us take a simple case where $20\%$ of
the net deposits is a reflection of the central bank IOUs, and the rest comes from government debt. The good news is that since this mixing between the different origins has been
made possible thanks to the banking system, it means that by
construction the customers have always more assets than the debt of
the state. So the state could always, at least in theory, run a huge
temporary tax on capitals which would reduce the assets of customers
and increase the Treasury account so as to allow debt repayment. For an order of magnitude this
would be one year of GDP since nowadays public debts are of that
order. See Fig.~\ref{fig15} (left plot) for an
illustration. As a result, the net money would be reduced by that
amount. But if initially around $80\%$ of the customers net financial wealth was the
reflection of public debt, then it means that the financial wealth on
deposit and saving accounts has been reduced by the same
amount. If we imagine that people will feel poorer they will stop
consuming and this will cast the economy into an extreme recession
initiated by a deflation. But the Treasury could be proud of having eliminated all of its debt. Since 2011, this is
what has been done in European countries where it has been decided to
reduce the public debt at any cost.
People could be fine with the reduction of their financial
wealth, being prepared that in the long run prices should
decrease, or even changing their saving habits so that it does not happen. There is in principle nothing wrong and one could imagine
that some countries might cope with it in happiness. But we think that
it is more likely that the private sector sticks at least partially to
its previous saving habits, meaning that everybody tries to spend less, causing
deflation and possibly after recession.
So by ignoring this, the goal of debt reduction can be extremely
harmful to the economy. Actually whenever a state runs a surplus, this never
lasts more than a few years~\citep{Wray1990,Wray2004}, and then recession enters the game to
generate new deficits. If public debts have been reduced when
compared to the GDP during some period of history, they are nearly always constantly
increasing in nominal value, because the total net money of customers
needs to go more or less at the same pace as GDP growth plus inflation
to satisfy the habits.

However, all our preconceived ideas run counter proper thinking. For
instance, in German
money is named {\it Geld}, and it is a derivative of gold. It
thus carries a meaning which goes beyond the real nature of money where
convertibility has been abandoned. On the contrary, debt is called {\it Schuld}, but this is the same word which
is used for guilt or fault\footnote{As noticed and analyzed in~\citet{Graeber2011}, this is also the case in many ancient
  languages (Sanskrit, Hebrew, Aramaic).}, depending on how we translate. So the
denomination hampers a proper thinking, since as the Germans pronounce the
word {\it debt}, they immediately mean what they should do about it, that is
getting rid of it, as it is morally bad.
By drawing the consolidated graph structure, we are able to remove the words which convey too much
ideology, as only asset/liabilities relations of different types appear.

\begin{figure}[!htb]
\includegraphics[height=0.39\textwidth]{Plots/Fig150.pdf}
\includegraphics[height=0.39\textwidth]{Plots/Fig160.pdf}
\caption{{\it Left}: First the state levies a huge tax. Then the tax is effectively paid to the
  Treasury. Finally, the state settles its debt. {\it Right}:
The left graph illustrates the point of view that when the state issues a debt the
customer do not feel the need to reimburse it. Conversely, the right
graph illustrates the belief that for any new debt, the customers anticipate that they have a debt toward the state,
with the same value, interest and maturity. This is the Ricardo-Barro
equivalence. On both graphs we circle the consolidated sector of
customers and bankers. In the right graph, the Ricardo-Barro
equivalence reduces the total financial wealth. Actually, in this
point of view, no matter the debt of the state, the financial wealth
can only remain constant.}
  \label{fig15}
\end{figure}

\subsection{Ricardo equivalence against an asymptotic definition of money}

In the end, the debate is about what we think the debt will become in
the very long run. Indeed, in a system which has abandoned
convertibility, central bank money is an eternal debt. The central
bank owes gold, but it owes it in an infinite time because it has
abandoned convertibility. It would thus be tempting to define money as a debt which is never
reimbursed. Interestless money in that definition would thus be just
one type of money. A concrete example to apply such definition is to
consider the paper debts (Demand Notes) which have been issued by the US federal
government during the Civil War. At that time, they were issued having
the status of a debt, in order to finance the war, but they kept being exchanged and were
subsequently considered as currency, without being redeemed into gold, but being used to pay taxes.

The problem with such definition of money is that it is not local in
time. Indeed, we need to know the full future of the debt to deduce if
it was actually money or if it really was a debt which has been
reimbursed.
So we have to rely on a lighter and imperfect definition,
where we would consider that a debt is money if there is a consensus
among customers that it will never be reimbursed (or reimbursed by
issuing an equivalent debt). This deeply depends on the point of view. For any debt issued by the Treasury, there are thus always
two extreme point of views. For the ones who believe that since states
always run deficits, it is for sure a type of money. We call this
point of view the {\it should} point of view since in this point of
view we should run deficits, at least in pace with growth and expected
inflation. Conversely, for those who think that debts must be reimbursed, no matter the state
of the future economy, the public debt cannot be considered as money. We can call this point of view the {\it Schuld}
point of view. The debate between {\it should} and {\it Schuld} is
reminiscent of the debate about rational expectations. If the state
issues a new debt, does it mean that there is an actual debt of the
customers toward the state? Those who think that this debt exists no
matter what are in the {\it Schuld} sector, whereas those who ignore such possibility, by arguing that people do not look at
national accounting for the personal wealth, are in the {\it
  should}. For the {\it should} sector, since the state is everybody
it is nobody and lies outside. For the {\it Schuld} sector, since the state is nobody,
then it means that it is everybody and it lies inside. The {\it Schuld} point of view was first
formulated formally as the  Ricardo-Barro equivalence, according to which
taxpayers exactly anticipate future taxes from current deficits. The
two point of views are summarized in Fig.~\ref{fig15} (right plot). If the
Ricardo-Barro equivalence is invoked, the financial wealth of
customers and bankers always remains unchanged, even when the Treasury
borrows, and remains $M$, that is the total liabilities of the central
bank, whereas if the equivalence is ignored (as we think it should
be), the financial wealth is increased by public deficits $S$.

It is hard to be convinced that people will take decisions thinking about a possible debt they owe to the
Treasury. We think that in order to find a convincing answer in this
debate we must look at the past behaviour of major western states. And
what we find is that, apart for a few years, they run constant deficits and the
sovereign debts keep growing in nominal values. Only on some
occasions, some governments manage to run a surplus but this never
lasts very long. For the US history, this is certainly the case, where
there was just occasional years of surplus in an ocean of
deficits~\citep{Wray2004}.

\section{Foreign currencies and exchange rates}\label{SecFor}

There are inevitably several sovereign states, each with its own currency, that is its own central bank and its
own Treasury, using its own UoA. A complete description of
a general monetary system must thus allow for several currencies and their interactions.

\subsection{Fixed exchange rate system}\label{SecExRate}

We first focus on fixed exchange rates systems which
were more common in the past. In this section we describe the
interactions of various monetary systems, and in the next section we
argue clearly why this can never be maintained.

Whenever a national citizen, is paid by a foreign citizen in a
foreign currency, he might prefer an asset denominated in his national
currency rather than an asset in a foreign denominated unit of
account. There is no universal law behind this, and all possible
choices are in principle possible. He might find it perfectly fine and
preferable to hold foreign currencies.

In the case where the citizen indeed wants to get rid of a foreign
asset, he will ask his bank to exchange it for a national asset.
The commercial bank will take the foreign currency for itself, and will create a deposit denominated in the national
UoA according to the fixed exchange rate. The process might then be repeated
between the commercial bank and the central bank.
In that case, the central bank will take the foreign central bank money as an asset and will increase
the  national central bank money deposit of the commercial
bank. As a final result, it is the national central bank which directly possesses the liability of the foreign central bank, that is which has
a deposit at the foreign central bank. All these steps are illustrated
in Fig.~\ref{fig18}. If the whole procedure is reversed and the customers
prefer to hold assets denominated in a foreign currency, this is
usually called {\it capital flight}.

\begin{figure}[!htb]
\includegraphics[height=0.30\textwidth]{Plots/Fig180.pdf}
\includegraphics[height=0.30\textwidth]{Plots/Fig181.pdf}
\caption{{\it Left}: first a national citizen is paid with foreign
  currency. Then he asks to its commercial bank to transform
  this foreign asset into a national currency asset. {\it Right}:
  the national central bank prefers to hold the foreign central bank IOUs
than foreign commercial bank IOUs. Then the national bank asks its central bank to convert its asset denominated in a foreign
currency to an asset in the national currency.}
  \label{fig18}
\end{figure}

\subsection{Fixed exchange rate limitations}\label{SecPeg}

The resulting situation is extremely similar to what would have happened if the
national citizen had been paid in gold in a regime of convertibility,
and if at every stage the gold had been passed to the higher
level (from the citizen to its commercial bank and then to the central
bank).
In a regime of fixed exchange rate,
foreign currencies are treated like gold in a regime of
convertibility. If payments are made in gold or in  foreign
currencies, they can be passed over to the central bank which creates
the corresponding net money. Conversely when gold or foreign
currencies are asked, the central banks reverts the situation and
detaches the asset from its balance sheet, but in the process it also
reduces its liabilities in central bank money, effectively reducing
the net national money.

In a regime of convertibility between money and gold, we have already
argued in \S~\ref{SecBankRun} that the system cannot hold because of
credit creation. The problem is exactly the same in a regime of fixed
exchange rate. It is always possible to break down the fixed
exchange rate by asking the conversion of national currency into
foreign currency, and given that there is more money than net money,
there is always a way to reach the point at which the central bank
reserves are depleted.

The only possibility for the system to hold would be if the national
central bank had the right to borrow foreign central money to the
foreign central bank. But if this right was granted, then the situation
would be rather similar to the case of commercial banks having the
right to borrow from a central bank, and this would ensure that all
liabilities can be cleared at par, that is with a fixed exchange
rate. This was exactly the European situation between 1999 and 2002,
when all eurozone currencies had a fixed exchanged rate with the euro
foreign currency, and all national central banks were granted the right to borrow from the ECB as much as they
needed. Eventually, as planned in the 1992 Maastricht treaty, all national currencies were abandoned and the
system was further integrated into a single currency, as this was just
a technical intermediary situation. But in general a foreign central
bank has no particular interest in granting the right to borrow to the
national central bank.

We realize again that granting the right to go in debt is the key of
financial power. At every vertex of the tree structure, granting the
right to go in debt to vertices which are lower is a form of power on
them. The household is afraid that its commercial banker might not grant a
loan, and needs to accept the conditions set by the banker, the
commercial bank is told at the higher level at which rate, under which
conditions, for which maturity, it can borrow central bank money. And
if the national central bank now wants to borrow a foreign currency, it
will have to accept the conditions of the foreign central bank.

If an economy starts to fix its currency at parity with a foreign
currency, its central bank somehow inserts itself in the tree
structure of the foreign country, and instead of being at the apex of
the system, it is now situated below the foreign central bank and
needs to ask for the permission to go into debt. Even though this type of
power of the foreign central bank onto the national central bank is
formally similar to the power a commercial bank has on its customers, it is
also extremely different.
A central bank possesses a form of power on the national financial
system as it sets the condition of debts in commercial banks and thus
below for national customers/citizens. At least in principle, it is acting for the interest
of the national economy as it is a creature of the state. Instead,
when a foreign central bank grants the right to a national bank to
borrow foreign central bank money, it has absolutely no reason to act
in the interest of the national citizens.

If a weak economy wants to fix its currency to a strong economy, and does not have a full constitutional right to borrow in this
external currency, as is always the case, there are essentially two possible situations. Either the country ensures that it will never need foreign
currencies to defend the exchange rate. In order to achieve this it
would need  a strong commercial surplus. Or it needs to borrow the
foreign currency, e.g. because of a current account deficit, and needs to
comply with the politics imposed by the foreign country. The alternative
is thus either to work for free (this is the essence of a permanent
commercial surplus for which the dominant foreign countries pays simply by
public deficits) or to be told how to work and what to sell (structural reforms
imposed by creditors).

\ifmoneyagg
\section{Analyzing stocks and their associated flows}

\subsection{Monetary aggregates}\label{SecMi}

Usually the amount of money in a given monetary system is estimated
through monetary aggregates. They all correspond to using a subregion of the monetary system for consolidation purposes. The total
net asset consolidated for that particular region is a particular
aggregate. In this section we present the various aggregates and
illustrate them in Fig.~\ref{fig22}.

\subsubsection{$M_0$ and $M_B$}

The simplest monetary aggregate is $M_0$ and is composed of the assets
directly held by the customers at the central bank, in the form of
coins and paper money. Whenever a customer deposits money at its
commercial bank, this amount is reduced, and conversely it is
increased when deposits at commercial banks are withdrawn in cash.

If we then include all net deposits held by commercial banks at the
central banks (including the reserves they have at the central bank
but subtracting their debt toward the central bank), we just need to extend the closed region to include the
commercial banks with their underlying tree-structure of assets and
liabilities made of deposits and loans. This defines the monetary
basis $M_B$. We have made sure to exclude everything  which
is related to the public debt issuing.
This ensures that the entering lines of the contour are only made of bank deposit at the central bank plus
customer account possessed in the form of coin and paper money. We
already see that since we have split the net worth of bankers in two
parts, excluding only T-bonds, but including other assets, this
definition of the monetary basis will always vary whenever the central
bank performs open market operations. Indeed, when the central bank buys bonds
held by the bankers, it does not modify the net worth of bankers, but the
part of the net worth which has to be included in the
contour has to be modified. Put in other words, the aggregate $M_B$ counts only assets
which enter through central bank IOUs, but not the asset which have
entered from Treasury IOUs. Finally note that our definition for $M_B$ takes
into account only the net deposits of the commercial banks at the
central bank, and not just the deposits. This is different from the
usual definitions of the monetary basis. With our definition, any central money borrowed at the
discount window does not affect the monetary basis $M_B$, whereas it would
affect the usual definitions of the monetary basis.

\subsubsection{$M_{\rm net}$ and $M_{\rm CB}$}

This arbitrariness leads to consolidate the full aggregate of
customers and bankers $M_{\rm net}$. This aggregate now counts the
Treasury debts as well, as it includes all types of assets entering
the bankers net worth. Whenever the central bank engages in open
market operations, buying or selling bonds, this monetary aggregate
does not change. It corresponds to the net financial assets held by the
whole economic system and this is what we have called the {\it net money}.
An operation of quantitative easing, which is just a massive outright
purchase, changes $M_B$ but not $M_{\rm net}$.
If we use the ambiguous word {\it money} for different aggregates, we would unavoidably disagree
on the effect of open market operations. Those looking at $M_B$ would
certainly agree that it increases the money  supply, whereas those
looking at $M_{\rm net}$ would find no variation in their definition
of the money supply. The use of graphs in representing what is
considered considerably clarifies the debate.

However, it should be noted that $M_{\rm net}$ is not conserved
whenever the Treasury spends by increasing its debt. We can thus
define a central bank aggregate $M_{\rm CB}$ which encompasses
everything but the central bank, and this one will remain constant,
whatever the public debt. This aggregate is the reflection of all gold
and foreign reserves of the central bank (plus the foreign money
possessed directly by national citizens).

\subsubsection{$M_i$}

Further complication can also be introduced if we now decide to exclude
the liabilities of customers, but to include only some assets, in an
attempt to extend $M_B$. This leads to the various definitions
$M_1, M_2,\dots$, that we gather collectively as $M_i$. The more
assets of customers are included (with all liabilities excluded in all
cases), the larger the aggregate $M_i$ is. We thus have a hierarchy $M_1
< M_2<\dots$. Any loan issued by a bank would surely affect some of the $M_i$
as it would increase the assets held by customers. Depending on the
nature of the asset, it would count in some $M_i$ and not in
others. These definitions of the aggregates are a way to estimate the amount of lending made by
commercial banks. More precisely the ratio between $M_i$ and $M_{\rm
  net}$ can be used the estimate the amount of credit in the origin of money.

\begin{figure}[!htb]
\includegraphics[height=0.42\textwidth]{Plots/Fig220.pdf}
\includegraphics[height=0.42\textwidth]{Plots/Fig221.pdf}
\caption{Schematic representation of $M_0$, $M_B$, $M_{\rm net}$, $M_i$ and $M_{\rm CB}$.}
  \label{fig22}
\end{figure}

\subsection{Sectoral financial balances}

\subsubsection{General construction}

As we have just emphasized, defining aggregates on some closed regions
of a monetary system allows to discuss the situation of stocks at a
given time. It also allows, by looking at the variation of the stock,
to examine the flows for these regions. One would for instance rather
look at the variations of $M_{\rm net}$, than $M_{\rm net}$ itself.

This idea of dividing the monetary system into major consolidated
areas so as to examine their flow relations led to the so-called
sectoral financial balances (SFB), which were popularized as a
tool of macroeconomic analysis by Wynne Godley~\citep{Godley1999,Godley2000,Godley2007}. The usual SFB analysis
consists in dividing the monetary system into three global regions,
which form a partition of the total system (that is such that their
union covers the whole system). The first sector is the Treasury
consolidated with its central bank, the second sector is the total
foreign sector, and the third sector is the domestic sector. See
Fig.~\ref{fig23} for an illustration.

\begin{figure}[!htb]
\includegraphics[height=0.42\textwidth]{Plots/Fig230.pdf}
\includegraphics[height=0.42\textwidth]{Plots/Fig231.pdf}
\caption{Top: typical representation of the three main
  sectors. Consolidated government, Foreign sector and domestic
  sector. Bottom: we have added the real assets which do not sum up to zero
  and are not conserved, since they are nobody's liability. Foreign assets cam be possessed by the domestic or the foreign sector, just
  like domestic real assets.}
  \label{fig23}
\end{figure}

The main interest in performing a partition is that it enforces the
global conservation of flows. In details, these aggregates evolve as follows.
\begin{itemize}
\item The government aggregate variation is noted $T-G$, and its stands
  for the difference between what it has taxed and what it has
  spent. The difference being necessarily in form of increased public
  debt if it is negative. From this aggregate perspective, any tax $T$ has
  reduced the liabilities of the central bank toward the commercial
  banks, as it led to credit the Treasury account at the central
  bank. Any spending $G$ reverts this process. Spending by public
  deficit affects the aggregate, in two possible ways. Either it increases the liabilities of the central bank
  toward the commercial banks if the central bank holds the new
  Treasury bonds, or it affects the bank net worth if the Treasury
  bonds are bought by the commercial banks.

\item The total foreign sector variation comes from the current
  account position of the domestic sector. If the national system runs
  a current account surplus, it drains IOUs from outside. From a
  graph perspective, this means that the number of IOUs from outside toward
  either the national central bank, the commercial bank, or simply
  directly the national citizens if they hold foreign paper money,
  increases. So the evolution of the foreign aggregate is $-NX$, where $NX$ stands for the current account balance of the national
  monetary system (net exports).
\item Finally the domestic sector aggregate is noted $S-I$, where $S$
  stands for the variation of the total of assets and $I$ is the
  variation of the total of its liabilities. $S-I$ is simply the net
  variation of the domestic aggregate in terms of financial assets.
\end{itemize}
By construction the sum of the aggregates variations should vanish and
we thus find the usual accounting identity
\begin{equation}
(T-G)+(S-I)-NX=0 \quad \Rightarrow \quad S-I = NX+(G-T)\,.
\end{equation}

\subsubsection{SFB analysis and discussion}

The main conclusion from this analysis is that the domestic sector can
only save {\it financial assets} if the government runs a deficit or
if the current account is in surplus.

The main shortfall of such analysis is that it ignores the variation
of the real assets, as it focuses only on financial assets. That is it
considers only the variation of assets which are claims on a
debt. It is true that all financial liability/asset relations
should balance, as any outgoing relation (a liability) has its ingoing
counterpart somewhere (the asset). But the picture would be more
complete if the evolution of real assets was also examined at the same
time.
The amount of real assets can evolve as companies are created or
invest in production goods, new houses are built, new discoveries are
valued as assets through patents etc... And the owners of the real
assets might evolve as the exchange of real assets can be the
counterpart of the exchange of financial assets, when these real assets are purchased.
If we model simply the assets by two categories, one being the
national real assets, and the other one the foreign real assets, the
SFB analysis can be extended and this is depicted in Fig.~\ref{fig23}.

The SFB analysis does not lead to too much controversy per se, as it is
an accounting identity. It is rather how it is then used to predict
the behavior of economic agents and thus to predict the evolutions of
an economy that the SFB analysis leads to tough debates between the
various theories of economy (see for instance~\citet{Fiebiger2013}
for a critic of this type). Can we guess the decisions on spending,
that is roughly speaking the aggregate demand, just by examining
$S-I$?
One should not however be too pessimistic about economics and throw
away all econometric analysis based on aggregated indicators. We
should always remember that they are here to help describing a given
situation, and not to forecast the future.

For instance, one cannot say that if $S-I$ decreases then the private sector will
underspend. One can only say that if $S-I$ decreases, and if the
private sector {\it has preferences, and maintains them}, for a given amount of net
financial saving, then it will underspend since everybody will try, and
necessarily fail, to maintain its net financial saving. The difference
between the two propositions is that we assumed that some behavior of
agents (the amount of net financial saving desired) is
conserved. Taking decisions on aggregated indicators is thus a bet on
the evolution or the constancy of behaviors with respect to these
arbitrary indicators. Those who wanted to reduce the public deficits
while still in the midst of the global financial crisis, have bet that
the private sector is happy to reduce its net financial wealth (for
instance assuming they would prefer to increase the wealth located in
real assets). It is for instance the bet which has been done in
Europe. After years of stagnation and at the brink of deflation, it is
now obvious that it was as good as betting
on a lame horse.
\else
\fi

\section*{Conclusion}

We have argued that graphs for financial stocks are natural tools to describe and discuss the
theories of money. By drawing together the balance sheets in a graph rather than explicitly writing one after
another the balance sheets of each institution [e.g. in the figures
of~\citet{BoEMoney} or in the balance sheets explanations of~\citet{Keen2014}], we found that a given financial arrangement can be described more easily, and this
simplifies the argumentation about the monetary systems.

For pedagogical reasons and concision, we have overlooked some building blocks which
were not essential for understanding the general method.
We have not mentioned the description of real assets as we intentionally restricted our description to the financial part made of
asset/liability relations. We also ignored the representation of private companies, but they can
easily be incorporated as a set of financial assets, real assets and
liabilities, which induce a net worth possessed by the owners.
Instead we developed in details how these tools can be used to
analyze the structure of the state, with its central bank and its
Treasury, and understand the nature of money. We argued that in
sovereign states running sovereign currencies, the Treasury and the
central bank can be meaningfully consolidated, as we showed how their actions are coordinated thanks to the monetary
policy. We explained that the state only controls exogenously the net money
through public deficits, that is it sets the boundary conditions for
the financial structure of the private sector, but it does not control the amount of credit
which is endogenously determined.

The financial relations between all actors of a monetary system are complex and simple at the same time. They are simple
because double-entry bookkeeping is understood at all levels of the
system, given that it is extremely simple, and it is responsible for the conservation laws in the
financial system. At every vertex of the graph, that is for every
institution, the local laws of financial accounting necessarily hold,
and they enforce the neutrality of the vertices \revision{and thus determine the
values of the net worths}. In a physical system with a high number of particles, there are complex structures which
emerge from the simple local laws, and they are quite often
very difficult to understand. Quite similarly, complex
financial structures emerge out of the simple local laws of
accounting, due to a high number of financial interactions.
As a result, if the graph description is well understood, the different schools of economic
thought should agree on the graph representation of any monetary
structure, as they should agree on the local accounting at each
vertex. Indeed, each given financial situation can be
summarized by drawing the corresponding graph. But it will then appear that the disagreements always lie in the interpretation, that is
on the words they want to drape around a given graph. This would be related to the various possible consolidations which can be made in
a monetary structure, when trying to grasp the complex emerging structures.\\

\ifrevtex

\ifanonymous
\else
  \begin{acknowledgments}
  The author thanks D. D\'efossez-Carme, L. Gastard, M. Lavoie, E. Le H\'eron, S. Renaux-Petel, E. Rolet and  J.-P. Uzan for
  discussions and comments.
  \end{acknowledgments}
\fi

\else

\ifanonymous
\else
  {\it Acknowledgments}: The author thanks D. D\'efossez-Carme,
  L. Gastard, M. Lavoie, E. Le H\'eron, S. Renaux-Petel, E. Rolet and  J.-P. Uzan for discussions and comments.
\fi
\fi

\ifrevtex
\else
\section*{References}
\fi

redpajama_set_name: RedPajamaArXiv