# On unique tensor rank decomposition of 3-tensors

Pavel Gubkin  Pavel Gubkin<EMAIL_ADDRESS>St. Petersburg State
University  Universitetskaya nab. 7-9, St. Petersburg, 199034, Russia

###### Abstract.

We answer to a question posed recently in [7], proving the conjectured
sufficient minimality and uniqueness condition of the 3-tensor decomposition.

The work is supported by Ministry of Science and Higher Education of the
Russian Federation, agreement № 075–15–2019–1619.

##  1\. Introduction

Let $\mathbb{F}$ be a field, $m$ be a positive integer and $V_{1}$,…, $V_{m}$
be vector spaces over $\mathbb{F}$. We say that tensor rank
$\operatorname*{rank}(v)$ of $v\in V_{1}\otimes V_{2}\otimes\ldots\otimes
V_{m}$ is the minimal number such that $v$ can be expressed as a sum of $n$
product tensors, i.e., tensors of the form $v_{1}\otimes v_{2},\ldots\otimes
v_{m}$. Also for a collection of vectors $u_{1},\ldots,u_{r}$ $k$-rank is
defined as the largest number $k$ such that for any $k$ vectors
$u_{i_{1}},\ldots,u_{i_{k}}$ we have
$\dim\operatorname*{span}(u_{i_{1}},\ldots,u_{i_{k}})=k$.

In 1977 J. Kruskal proved [5] that a tensor of a form

$\displaystyle\sum_{i=1}^{n}x_{i}\otimes y_{i}\otimes z_{i}$

constitutes a unique representation as a sum of $n$ product tensors if
$k$-ranks of the sets $\\{x_{1},\ldots,x_{n}\\}$, $\\{y_{1},\ldots,y_{n}\\}$
and $\\{z_{1},\ldots,z_{n}\\}$ are large enough. Proof of this theorem in more
general form can be found in [8]. There are some other conditions that imply
uniqueness of the tensor rank decompositions in the literature. In this paper
we focus on so-called U-condition which was originally introduces in [2], [3];
we will use the definition from [7].

Denote $[n]=\\{1,\ldots,n\\}$; $[n..m]=\\{n,\ldots,m\\}$. By $\langle
A\rangle$ denote the linear span of a set $A$ in a linear space. We also write
$\langle x_{1},\ldots,x_{k}\rangle$ meaning
$\langle\\{x_{1},\ldots,x_{k}\\}\rangle$. For a sequence
$\alpha=(\alpha_{1},\ldots,\alpha_{n})\in\mathbb{F}^{n}$ let $\omega(\alpha)$
be the number of indices $i$ for which $\alpha_{i}\neq 0$. We say that a set
of tensors {$x_{i}\otimes y_{i}\otimes z_{i}$}, $i\in[n]$ satisfies the
Condition U if for any $\alpha\in\mathbb{F}^{n}$

(1)
$\displaystyle\operatorname*{rank}\left(\sum_{i=1}^{n}\alpha_{i}y_{i}\otimes
z_{i}\right)\geqslant\min(\omega(\alpha),n-d+2),$

where $d$ is the dimension of $\langle x_{i}|i\in[n]\rangle$. The aim of this
paper is Question 38 from [7], we prove the following

######  Theorem 1.

Let $\mathbb{F}$ be a field; $\mathcal{X},\mathcal{Y},\mathcal{Z}$ be linear
spaces over $\mathbb{F}$; $x_{1},\ldots,x_{n}\in\mathcal{X}$;
$y_{1},\ldots,y_{n}\in\mathcal{Y}$; $z_{1},\ldots,z_{n}\in\mathcal{Z}$. Assume
that

  * •

$x_{i}\neq 0$ for all $i$ and also $x_{i}$ and $x_{j}$ are not parallel for
all $i\neq j$;

  * •

the set of tensors {$x_{i}\otimes y_{i}\otimes z_{i}$}, $i\in[n]$ satisfies
the Condition U.

Then for $\theta:=\sum_{i=1}^{n}x_{i}\otimes y_{i}\otimes z_{i}$ we have
$\operatorname*{rank}\theta=n$ and the representation of $\theta$ as a sum of
$n$ product tensors is unique.

######  Remark 2.

The theorem holds true if we demand (1) not for all $\alpha\in F^{n}$, but
only for $\alpha$ such that $\alpha=(\eta(x_{1}),\ldots,\eta(x_{n})$ for some
linear functional $\eta\in\mathcal{X}^{*}$. In other words, following the
notations from [7], the theorem holds with Condition U replaced by condition
W.

##  2\. Proof of the Theorem 1

We will use the following

######  Lemma 3 (Lemma 12.5.3.5, [6]).

Let $W$ be a $d$-dimensional linear space over a field $\mathbb{F}$, and
$\\{x_{i}\\}$, $\\{v_{i}\\}$, $i\in[n]$ be two sets of vectors such that

  * •

$\langle v_{i}|i\in[n]\rangle=W$;

  * •

$x_{i}$ and $x_{j}$ are not parallel for any two distinct indices $i,j$;

  * •

for any hyperplane $H$

$\displaystyle\\#(H\cap\\{v_{i}\\}_{i\in[n]})\geqslant
d-1\Longrightarrow\\#(H\cap\\{x_{i}\\}_{i\in[n]})\geqslant\\#(H\cap\\{v_{i}\\}_{i\in[n]}).$

Then there exists a permutation $\pi$ of indices $[n]$ and the non-zero
scalars $\lambda_{i}\in\mathbb{F}$ such that $v_{i}=\lambda_{i}x_{\pi_{i}}$
for all $i\in[n]$.

###### Proof.

Let us outline the idea of the proof. Proceeding by induction one has to prove

  * •

for any $m$ \- dimensional subspace $H_{m}$

$\displaystyle\\#(H\cap\\{v_{i}\\}_{i\in[n]})\geqslant
m\Longrightarrow\\#(H\cap\\{x_{i}\\}_{i\in[n]})\geqslant\\#(H\cap\\{v_{i}\\}_{i\in[n]}).$

Notice that this assertion for $m=d-1$ is given in the lemma and for $m=1$
this immediately implies the lemma. For details see Section 12.5.4, [6]. ∎

###### Proof of the Theorem 1.

Assume that $\theta=\sum_{i=1}^{n}v_{i}\otimes u_{i}\otimes w_{i}$ for certain
$v_{i}\in\mathcal{X}$, $u_{i}\in\mathcal{Y}$, $w_{i}\in\mathcal{Z}$. We need
to show that the multisets $\\{x_{i}\otimes y_{i}\otimes z_{i}|i\in[n]\\}$ and
$\\{v_{i}\otimes u_{i}\otimes w_{i}|i\in[n]\\}$ coincide. We do it in two
steps. At the first step, we apply Lemma 3 to prove that

$(\heartsuit)$ there exists a permutation $\pi$ of indices $[n]$ and the non-
zero scalars $\lambda_{i}\in\mathbb{F}$ such that
$v_{i}=\lambda_{i}x_{\pi_{i}}$ for all $i\in[n]$.

In order to prove $(\heartsuit)$, take an arbitrary linear functional
$\eta\in\mathcal{X}^{*}$. We get

(2) $\sum_{i=1}^{n}\eta(x_{i})y_{i}\otimes
z_{i}=\sum_{i=1}^{n}\eta(v_{i})u_{i}\otimes w_{i}=:\Theta(\eta).$

Notice that for $\eta\in\langle v_{i}|i\in[n]\rangle^{\perp}$ one has
$\Theta(\eta)=0$ and the Condition U implies that $\eta(x_{i})=0$ for every
$i\in[n]$. Therefore,

$\langle x_{i}|i\in[n]\rangle\subset\langle v_{i}|i\in[n]\rangle=:W$

and without loss of generality we can now assume $\mathcal{X}=W$.

For a hyperplane $H$ denote the orthogonal projection onto $H^{\perp}$ by
$\eta_{H}$. Consider hyperplane $H$ such that

(3) $\displaystyle\\#(H\cap\\{v_{i}\\}_{i\in[n]})\geqslant\dim W-1.$

To apply Lemma 3 we need to show that

$\displaystyle\\#(H\cap\\{x_{i}\\}_{i\in[n]})\geqslant\\#(H\cap\\{v_{i}\\}_{i\in[n]}),$

or equivalently

$\displaystyle\\#\\{i|\eta_{H}(x_{i})=0\\}\geqslant\\#\\{i|\eta_{H}(v_{i})=0\\}$

Denote the two latter sets by $I$ and $J$ respectively then

$\operatorname*{rank}\Theta(\eta_{H})\leqslant n-|J|\leqslant n-\dim
W+1\leqslant n-d+1,$

where $d$ is the dimension of $\langle x_{i}|i\in[n]\rangle$. On the other
hand, from the condition U we kwow

$\displaystyle\operatorname*{rank}\Theta(\eta_{H})\geqslant\min(|\\{i:\eta_{H}(x_{i})\neq
0\\}|,n-d+2).$

If the latter minimum equals $n-d+2$, this is a contradiction. Thus

$n-|J|\geqslant\operatorname*{rank}\Theta(\eta_{H})\geqslant|\\{i:\eta_{H}(x_{i})\neq
0\\}|=n-|I|.$

Therefore $|I|\geqslant|J|$ for any hyperplane $H$ satisfying (3) and we get
$(\heartsuit)$ by Lemma 3.

Now, permuting indices and using $(\lambda x)\otimes u\otimes
v=x\otimes(\lambda u)\otimes v$ we may suppose that $v_{i}=x_{i}$ for all $i$.
Without loss of generality we may suppose that $x_{1},\ldots,x_{d}$ is a basis
of $\langle x_{i}|i\in[n]\rangle$. Denote
$x_{i}=\sum_{j=1}^{d}\alpha_{ij}x_{j}$ for $i\in[d+1..n]$. We get

$T_{j}:=y_{j}\otimes z_{j}+\sum_{i=d+1}^{n}\alpha_{ij}y_{i}\otimes
z_{i}=u_{j}\otimes w_{j}+\sum_{i=d+1}^{n}\alpha_{ij}u_{i}\otimes w_{i},\quad
j=1,\ldots,d.$

By Condition U, we get

$\displaystyle\operatorname*{rank}T_{j}=1+|\\{i:\alpha_{ij}\neq 0\\}|.$

Denote

$\displaystyle J_{j}=\\{j\\}\cup\\{i:\alpha_{ij}\neq 0\\}.$

By the above maximal rank condition we get that the vectors $\\{y_{i}:i\in
J_{j}\\}$ are linearly independent. Analogously, so are the vectors
$\\{z_{i}:i\in J_{j}\\}$, $\\{u_{i}:i\in J_{j}\\}$, $\\{v_{i}:i\in J_{j}\\}$.
Next, assume that $k\in J_{j}$ but $y_{k}\notin\langle u_{i}:i\in
J_{j}\rangle$. Then there exists a projector $P\colon\mathcal{Y}\to\langle
u_{i}:i\in J_{j}\rangle$ for which $Py_{k}=0$. We get $(P\otimes
Id)T_{j}=T_{j}$ since $Pu_{i}=u_{i}$ for all $i\in J_{j}$, but

$\displaystyle(P\otimes Id)T_{j}=(Py_{j})\otimes
z_{j}+\sum_{i>d}\alpha_{ij}(Py_{i})\otimes z_{i}$

is a representation of $T_{j}$ as a sum of less than $|J_{j}|$ product
tensors. Thus all $y_{k}$ with $k\in J_{j}$ belong to the subspace $\langle
u_{i}:i\in J_{j}\rangle=:\mathcal{Y}_{j}$. Since $|J_{j}|\leqslant n-d+1$, and
any $n-d+1$ (and even any $n-d+2$) $y_{j}$’s are linearly independent by
Condition U, we get $\mathcal{Y}_{j}=\langle y_{i}:i\in J_{j}\rangle$. Next,
$\dim(\mathcal{Y}_{j}+\mathcal{Y}_{k})=|J_{j}\cup J_{k}|$ whenever $j,k\in[d]$
and $j\neq k$, that’s because $|J_{j}\cup J_{k}|\leqslant n-d+2$ and
corresponding $y_{i}$’s with $i\in J_{j}\cup J_{k}$ are linearly independent.
But $\mathcal{Y}_{j}+\mathcal{Y}_{k}$ is also generated by $u_{i}$’s with
$i\in J_{j}\cup J_{k}$. Thus, these $u_{i}$’s are also linearly independent.

Let $s\in[d+1..n]$. Since no two $x_{i}$’s are parallel, there exist at least
two indices $j,k\in[d]$ for which $\alpha_{sj}\neq 0$ and $\alpha_{sk}\neq 0$.
In other words, $s\in J_{j}\cap J_{k}$. Then, $y_{s}$ belongs to
$\mathcal{Y}_{j}\cap\mathcal{Y}_{k}$. This space is generated by $u_{i}$ with
$i\in J_{j}\cap J_{k}\subset[d+1..n]$. So, we have just proved that

$\displaystyle y_{s}=\sum_{i=d+1}^{n}f_{si}u_{i}$

for certain scalars $f_{si}$. Now fix $s,k\in[d+1..n]$ such that $s\neq k$.
Since $x_{s}$ and $x_{k}$ are not parallel, we may without loss of generality
assume that $\alpha_{s1}\alpha_{k2}\neq\alpha_{k1}\alpha_{s2}$. Consider the
tensor $\alpha_{k2}T_{1}-\alpha_{k1}T_{2}$. It equals

$py_{1}\otimes z_{1}+qy_{2}\otimes z_{2}+\sum_{i=d+1}^{n}r_{i}y_{i}\otimes
z_{i}=pu_{1}\otimes w_{1}+qu_{2}\otimes
w_{2}+\sum_{i=d+1}^{n}r_{i}u_{i}\otimes w_{i}$

for certain coefficients $p,q,r_{d+1},\ldots,r_{n}$, where
$r_{i}=\alpha_{k2}\alpha_{i1}-\alpha_{k1}\alpha_{i2}$. In particular,
$r_{k}=0$ but $r_{s}\neq 0$. As above, we conclude that $y_{s}$ belongs to a
span of $u_{1},u_{2}$ and $\\{u_{i}:r_{i}\neq 0\\}$. Note that
$U:=\\{u_{1},u_{2}$, $u_{d+1},\ldots,u_{n}\\}$ are linearly independent, since
$y_{1},y_{2},y_{d+1},\ldots,y_{n}$ lie in their span (and any $n-d+2$ vectors
$y_{i}$’s are linearly independent). Therefore, the expansion of $y_{s}$ in
the system $U$ does not contain $u_{k}$. This yields $f_{sk}=0$.

So, we proved that $u_{i}$’s are parallel to $y_{i}$’s for $i>d$. Analogously,
$w_{i}$ are parallel to $z_{i}$ for $i>d$, and we get $u_{i}\otimes
w_{i}=\lambda_{i}y_{i}\otimes z_{i}$ for $i>d$. Fix $i>d$ and find $j\in[d]$
for which $\alpha_{ij}\neq 0$. Looking at $T_{j}$ we get

$y_{j}\otimes z_{j}+\sum_{i=d+1}^{n}(1-\lambda_{i})\alpha_{ij}y_{i}\otimes
z_{i}=u_{j}\otimes w_{j}.$

If there are at least two non-zero coefficients in the left hand side, this
contradicts to condition (ii). Therefore $\lambda_{i}=1$. This holds for all
$i=d+1,\ldots,n$. So, $u_{i}\otimes w_{i}=y_{i}\otimes z_{i}$ for all $i>d$.
Looking at two expressions for $T_{j}$ ($j=1,\ldots,d$) we get the same for
$j\leqslant d$. ∎

##  3\. Field dependence of the Condition U

It is well known that tensor rank is field dependent (see section 3.1 in [4],
or section 2 in [1]). Here we show that U-condition is also field dependent.

Example. Consider three tensors over $\mathbb{F}_{2}$:

$\displaystyle a\otimes a,\quad b\otimes b,\quad(a+b)\otimes(a+b).$

Linear combinations over the field $\mathbb{F}_{2}$ are just sums of subsets,
and one can check that

(4) $\displaystyle\operatorname*{rank}(\alpha_{1}a\otimes a+\alpha_{2}b\otimes
b+\alpha_{3}(a+b)\otimes(a+b))=\min(\omega(\alpha),2).$

On the other hand, we can consider the same set of tensors over
$\mathbb{F}_{4}=\mathbb{F}_{2}[\xi]$, where $\xi^{2}=\xi+1$. We have

$\displaystyle(1+\xi)a\otimes a+b\otimes b+\xi(a+b)\otimes(a+b)=$
$\displaystyle=a\otimes a+(1+\xi)b\otimes b+\xi a\otimes b+\xi b\otimes a=$
(5) $\displaystyle=(a+\xi b)\otimes(a+\xi b).$

Now consider 3 following 3-dimensional tensors:

(6) $\displaystyle a\otimes a\otimes a,\quad b\otimes b\otimes b,\quad
c\otimes(a+b)\otimes(a+b).$

We have $n=d=3$. Hence (4) and (5) imply that condition U holds over
$\mathbb{F}_{2}$ and does not hold over $\mathbb{F}_{4}$. Also from Theorem 1
and the statement below it follows that the uniqueness of tensor decomposition
of sum of tensors in (6) holds over $\mathbb{F}_{2}$ and does not hold over
$\mathbb{F}_{4}$.

######  Statement 4.

If $n=d$, then the Condition U is not only sufficient for uniqueness of the
tensor rank decomposition, but also a necessary one.

###### Proof.

Indeed, for $n=d$ Condition U takes the form

$\displaystyle\operatorname*{rank}\left(\sum_{i=1}^{n}\alpha_{i}y_{i}\otimes
z_{i}\right)\geqslant\min(\omega(\alpha),2).$

Assume that Condition U does not hold. Then there exists $\alpha$ such that
$\omega(\alpha)\geqslant 2$ and

$\displaystyle\operatorname*{rank}\left(\sum_{i=1}^{n}\alpha_{i}y_{i}\otimes
z_{i}\right)\leqslant 1,$

equivalently

$\displaystyle\sum_{i=1}^{n}\alpha_{i}y_{i}\otimes z_{i}=y\otimes z$

for some vectors $y\in\mathcal{Y}$ and $z\in Z$. Without loss of generality,
assume that $\alpha_{1}\neq 0$, then

$\displaystyle y_{1}\otimes z_{1}=\alpha_{1}^{-1}\left(y\otimes
z-\sum_{i=1}^{n}\alpha_{i}y_{i}\otimes z_{i}\right)$

and

$\displaystyle\sum_{i=1}^{n}x_{i}\otimes y_{i}\otimes
z_{i}=\alpha_{1}^{-1}x_{1}\otimes\left(y\otimes
z-\sum_{i=1}^{n}\alpha_{i}y_{i}\otimes z_{i}\right)+\sum_{i=2}^{n}x_{i}\otimes
y_{i}\otimes z_{i}=$ $\displaystyle=\alpha_{1}^{-1}x_{1}\otimes y\otimes
z+\sum_{i=2}^{n}(x_{i}-\alpha_{1}^{-1}\alpha_{i}x_{1})\otimes y_{i}\otimes
z_{i},$

therefore the uniqueness does not hold. ∎

##  4\. Acknowledgements

I am grateful to B. Lovitz for pointing my attention to Lemma 3, which allowed
to simplify the argument in the case of finite field dramatically.

## References

  * [1] George M. Bergman.  Ranks of tensors and change of base field.  J. Algebra, 11:613–621, 1969.
  * [2] Ignat Domanov and Lieven De Lathauwer.  On the uniqueness of the canonical polyadic decomposition of third-order tensors—Part I: Basic results and uniqueness of one factor matrix.  SIAM J. Matrix Anal. Appl., 34(3):855–875, 2013.
  * [3] Ignat Domanov and Lieven De Lathauwer.  On the uniqueness of the canonical polyadic decomposition of third-order tensors—Part II: Uniqueness of the overall decomposition.  SIAM J. Matrix Anal. Appl., 34(3):876–903, 2013.
  * [4] Tamara G. Kolda and Brett W. Bader.  Tensor decompositions and applications.  SIAM Rev., 51(3):455–500, 2009.
  * [5] Joseph B. Kruskal.  Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics.  Linear Algebra Appl., 18(2):95–138, 1977.
  * [6] Joseph M. Landsberg.  Tensors: geometry and applications, volume 128 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, 2012.
  * [7] Benjamin Lovitz and Fedor Petrov.  A generalization of Kruskal’s theorem on tensor decomposition.  arXiv 2103.15633, 2021.
  * [8] John A. Rhodes.  A concise proof of Kruskal’s theorem on tensor decomposition.  Linear Algebra Appl., 432(7):1818–1824, 2010.


# Elastic Cash

Anup Rao
University of Washington
<EMAIL_ADDRESS>
###### Abstract

Elastic Cash is a new decentralized mechanism for regulating the money supply.
The mechanism operates by modifying the supply so that an interest rate
determined by a public market is kept approximately fixed. It can be
incorporated into the conventional monetary system to improve the elasticity
of the US Dollar, and it can be used to design new elastic cryptocurrencies
that remain decentralized.

##  1 Introduction

Money is as old as recorded history, and yet it continues to evolve. Even the
mighty US Dollar has been repeatedly updated over the last 200 years. The
recent emergence of Bitcoin and other cryptocurrencies is another step in that
evolution, and it prompts us to revisit the mechanisms that ensure the
desirable properties of money. An important property of money is its
_elasticity_ : money is elastic if the money supply increases in response to
demand. In this work, I present a new decentralized mechanism to ensure the
elasticity of money.

The US Dollar is elastic, but it uses a convoluted system to achieve its
elasticity. The Dollar system was not designed from first principles; it was
iteratively amended in response to financial crises. Elasticity is currently
achieved by the combined actions of many institutions: banks and non-banks,
public and private, domestic and international. No single entity is entirely
in charge of the money supply, and a relatively small number of investor-owned
institutions have undue influence. This system computes the demand for money
without much transparency, and the money that is created is not distributed
fairly. I discuss the mechanics of the US Dollar system and its limitations in
Section 2.

In contrast, Bitcoin was designed to be inelastic. Bitcoin caps the total
possible supply at $21$M, and the available supply, currently about $19.2$M,
will slowly increase until it hits this limit. Satoshi Nakamoto, the creator
of Bitcoin, criticized conventional mechanisms for achieving elasticity in an
early forum post:

> The root problem with conventional currency is all the trust that’s required
> to make it work. The central bank must be trusted not to debase the
> currency, but the history of fiat currencies is full of breaches of that
> trust. Banks must be trusted to hold our money and transfer it
> electronically, but they lend it out in waves of credit bubbles with barely
> a fraction in reserve.

Nakamoto eliminated the need for the kind of infrastructure used to ensure
Dollar elasticity by simply choosing to make Bitcoin inelastic. Additional
technological innovations in Bitcoin further eliminated the need for any
trusted central authority to carry out transactions. Bitcoin inspired many
other cryptocurrencies, incuding stablecoins that are pegged to the US Dollar.
Some of these currencies are elastic, but their elasticity is ultimately based
on the elasticity of the US Dollar, so they inherit the limitations of US
Dollar elasticity. I discuss elasticity in the context of cryptocurrencies in
greater detail in Section 3.

To illustrate the negative consequences of inelasticity, consider the
trajectory of home prices denominated in Bitcoin over a long period. As the
population grows, the demand for houses is sure to grow. Even if the supply of
houses keeps pace, home prices must fall, because the supply of Bitcoin cannot
keep pace. Certainly, if the number of concurrent transactions involving home
sales grows and the prices of houses remain stable, the amount of Bitcoin
involved in these transactions would have to grow, yet it cannot grow beyond
$21$M. So, a fixed money supply leads to falling prices in a growing economy,
even if the underlying supply and demand of items keep pace with each other.

At the other extreme, if the money supply is increased excessively, the
currency is debased; the excess supply leads to rising prices and inflation.
So, a mechanism for correctly setting the money supply is essential, because
this is the foundation upon which stable prices are built. As much as
possible, prices should reflect the tension between the supply and demand for
items, and nothing else. But how much money is too much? It is not just a
matter of trust, the problem is that the appropriate supply is difficult for
anyone to calculate! Is there a principled method to compute the supply, and a
fair way to create new money? This work gives my answers to these important
questions.

Ideally, a mechanism for achieving elasticity should be transparent. It should
not rely on the judgment or integrity of small groups of people, or a few
institutions. I will describe a new decentralized mechanism called _Elastic
Cash_ that enjoys these features. Elastic Cash solves two problems at the same
time:

  * •

It gives a fully decentralized mechanism to compute whether the demand for
money has increased or decreased.

  * •

It gives a fair mechanism to increase and decrease the supply of money.

Here I will give a high-level description of Elastic Cash; a full description
is in Section 4. I will refer to the central bank and the algorithm as _cash
authorities_. At the heart of the mechanism is a new financial contract issued
by the cash authority called a _cashbond_ , and a public market that allows
anyone to buy and sell cashbonds using public auctions in the market. Each
cashbond can be redeemed for $1 on a specific date, so it executes a risk-free
loan from the holder of the cashbond to the cash authority. The loans are
risk-free because the cash authority can always create new money to repay the
loans. Perhaps it is counterintuitive, but the purpose of the market in
cashbonds is _not_ to give the cash authority a way to borrow money; instead,
the function of the market is to use the trading activity of participants to
compute the _risk-free rate of interest_ , which can be computed from the
prices of cashbonds in the market. The mechanism requires that cashbonds can
only be transfered by selling and buying them in the market. For example, no
entity should be permitted to use cashbonds as collateral to borrow money.
This forces participants to liquidate cashbonds when they need money, and
keeps the mechanism informed about the demands for money.

It is market participants who determine the money supply in Elastic Cash. The
supply is increased or decreased according to transparent rules ensuring that
the risk-free rate of interest remains approximately fixed. Trade in cashbonds
leads to fluctuations in the rate of interest, and the mechanism responds by
creating corresponding fluctuations in the money supply. Intuitively, the
risk-free rate of interest encodes the cost of renting money. By regulating
the supply of money to keep the cost fixed, the mechanism ensures that the
supply stays in equilibrium with the demand for liquidity. When the supply is
increased, market participants acquire any newly-generated money. The supply
is decreased by incentivizing market participants to exchange their money for
cashbonds. Anyone can participate in this market, and money is distributed to
or taken from participants according to transparent rules, so no single entity
can control the flow of money.

Elastic Cash can be implemented within the framework of conventional
currencies like the US Dollar by creating regulations that require the central
bank to implement the market for cashbonds and generate money according to the
rules of the mechanism. It can be implemented in the framework of
cryptocurrencies by setting up a distributed algorithm to implement the market
for cashbonds using the blockchain. Money is generated by the algorithm
according to the rules of the mechanism. So, one can obtain the positive
features of traditional currencies and cryptocurrencies in addition to the
elasticity of Elastic Cash.

#### Outline of this paper

In Section 2, I give more details about how the US Dollar achieves its current
elasticity, before turning to explain how known elastic cryptocurrencies
inherit the limitations of the Dollar in Section 3. I describe the new
mechanism in Section 4. I discuss how the new mechanism can be incorporated
into the Dollar system in Section 5 and how it can be implemented on the
blockchain to give new cryptocurrencies in Section 8.

##  2 US Dollar elasticity: the Fed, banks, and shadow banks

Figure 1: M2, a measure of the supply of US Dollars

Figure 1 shows how the supply of US Dollars held as deposits has changed over
time. In this section I briefly review the history and mechanics of the US
Dollar system. I recommend the following excellent sources for additional
background on the history of the Dollar system [1, 7], and the book [5] for a
longer history of the technology of money.

The US Dollar is the fiat currency with the most advanced infrastructure for
maintaining elasticity. The supply of US Dollars has consistently been deemed
too important to be left entirely in the control of a government agency.
Instead, we have tried to decentralize the creation of money by developing a
system where money is created by investor-owned private entities. These
include banks and so-called shadow banks — non-bank financial institutions
that are able to create Dollars. This privately created money is then
backstopped by the central bank, which is the Federal Reserve, or Fed in the
US. The Fed is governed by laws written by Congress, but these laws are
somewhat ambiguous about the Fed’s powers, and Congress has repeatedly made
adjustments.

There are three major ways in which the Fed can influence the money supply:

  1. 1.

It can change the Fed funds rate, which influences all borrowing in the
financial system, and changes the incentives for private entities to create
money.

  2. 2.

It can elevate a new kind of asset to the stature of the US Dollar, which
gives a new class of institution the power to create Dollars.

  3. 3.

It can directly buy or sell assets under a Quantitative Easing/Tightening
program.

Today, there are at least four kinds of private institutions that are able to
generate US Dollars. It is helpful to view the evolution of the US Dollar
system according to the events that elevated the instruments created by these
institutions to the stature of US Dollars:

Commercial bank deposits

By the early 1900s, deposits of US Dollars were being issued by a number of
investor-owned private banks. These deposits were treated by depositors as
equivalent to US Dollars, even though the banks were issuing loans by creating
deposits that did not correspond to cash reserves. This led to a run on the
banks in 1907, and the Fed was created in 1913 to solve the problem. The Fed
was given the power to backstop the money created by commercial banks by
lending to the banks. This meant that deposits could always be exchanged for
money created by the Fed. This solved the problem of bank runs, but also
elevated bank deposits to the stature of cash issued by the Fed, and
effectively meant that commercial banks were given the authority to create
legitimate US Dollars in the process of issuing loans. Currently, about
$\$17$T is held in commercial bank deposits.

Reverse purchase agreements (repos)

In the 1950s, broker dealers began to enter the banking industry using a
financial instrument called reverse purchase agreements, or repos. Repos allow
dealers to borrow money from cash providers using government securities as
collateral. Cash providers began to treat the repos they had purchased from
dealers as equivalent to cash. By itself, such an arrangement does not create
any new money, because if repos crash in value, then they cannot actually be
exchanged for cash. However, the properties of repos were substantially
altered when the Fed decided that the repo market was important enough to
backstop dealers by giving them access to overnight loans. This meant that the
private holders of repos were guaranteed that repos could always be exchanged
for US Dollars via the Fed’s repo facility, and so repos were elevated to the
same stature as cash and bank deposits. In 1991, Congress reduced restrictions
on the Fed to make it even easier to backstop the repo market. Effectively,
broker dealers were given the power to create new US Dollars. Currently, the
size of the repo market is about $\$4$T.

Eurodollars

Foreign companies including both banks and non-banks (e.g. insurance
companies) have been issuing financial instruments called eurodollars that can
be redeemed for US Dollars. These eurodollars were not initially backed by
actual Dollars. The oil shock of 1973-1974 led to a run on eurodollars that
eventually brought down Franklin National Bank, a large domestic US bank that
had access to the Fed’s lending facilities. The Fed responded by promising to
backstop eurodollars by providing actual US Dollars in the form of loans to
foreign central banks. This announcement ended the run on eurodollars, but
effectively permitted the banking systems of other countries to create
deposits that could be exchanged for US Dollars. The eurodollar market has
grown substantially since then. The size of the eurodollar market was
estimated at about $\$13$T in 2016 [6].

Money market mutual funds

These funds emerged in the 1970s as investments whose share price was pegged
at $1. In reality, these funds held assets whose value could drop below the
peg, and so it was not possible to guarantee the peg during a financial
crisis. In response to the great financial crisis of 2008, the Fed began to
backstop these funds using its Money Market Mutual Fund Liquidity Facility,
and so elevated deposits in these funds to the stature of US Dollars. Total
assets in these funds is about $\$4.8$T.

In addition to recognizing new forms of the US Dollar, the Fed has resorted to
buying assets in order to inject liquidity under the Quantitative Easing
program. During the Great Financial Crisis of 2008, the Fed kicked off this
program by buying mortgage-backed securities and treasuries, which are loans
to the US Treasury. Throughout the last decade, the Fed has continued to
expand its balance sheet, mostly with treasuries. In 2020 the Fed once again
bought significant quantities of these assets. Currently the Fed holds about
$\$8.5$T on its balance sheet.

###  2.1 Limitations

The history of the US Dollar is full of ad hoc amendments to maintain
stability in the face of financial crises. Here are some limitations of the
current system:

  1. 1.

At its heart, the problem is that there is currently no principled way to
regulate the supply of money. This becomes apparent during times of financial
crises, but the imbalance in supply is probably always brewing, even in normal
times.

  2. 2.

Although the Fed is nominally not in charge of the money supply, if the Fed
funds rate has been lowered to $0\%$ and the Fed decides that more money is
needed, the Fed has repeatedly chosen to inject money by buying assets and
sometimes propping up new kinds of financial assets. This goes against the
principle of having a distributed system of money creation, and interferes in
the functioning of financial markets.

  3. 3.

The Fed does not have the tools to distribute the money that it creates
fairly. For example, money that is created via the Quantitative Easing program
largely enters the system in the form of government spending and subsidized
mortgages, and the decision to backstop repo protected broker dealers. There
is no reason why newly created money should be tied directly to government
spending, the mortgage industry or broker-dealers; this is an artifact of bad
design.

  4. 4.

It makes sense to decentralize the power to create money, but the current
system does this in a half-hearted way: the power remains concentrated in the
hands of big players in the financial industry, who often get first access to
the newly-created money.

  5. 5.

Because private financial institutions with the power to create Dollars know
that the Fed will protect their financial instruments from the most negative
consequences of their choices, their incentives are not aligned with creating
the appropriate amount of money.

Elastic Cash is meant to provide a clean, transparent, and principled
mechanism to achieve robust elasticity by giving a truly decentralized way to
compute the appropriate supply as well as a fair way to distribute new money.
We do not need to cede control of the money supply to private institutions or
foreign banks. The mechanism will generate new US Dollars when required, and
financial institutions can obtain liquidity by participating in the mechanism,
just like everyone else. Even if we want to leave the powers of the Fed to
create new money undiminished, the scheme of Elastic Cash, at the very least,
gives a fair way to distribute new money. It removes the ability of the
finance sector to control the flow of the new money, and puts everyone on the
same playing field.

Extricating ourselves from the current system and its vested interests is
likely to be challenging, to say the least. Nevertheless, I describe a path to
incorporating the new mechanism in Section 5.

##  3 Elasticity in Cryptocurrencies

Stablecoins are cryptocurrencies that are pegged to the Dollar by holding
actual Dollars. These include Tether, USD Coin, and True USD. Every coin is
backed by a US Dollar and new coins can be created backed by new US Dollars.
So, the elasticity of these coins is ultimately tied to the elasticity of the
US Dollar, and an excessive increase in the supply of Dollars can lead to
corresponding increases in the supply of these coins. These coins inherit the
drawbacks of the US Dollar discussed in Section 2.

The Maker protocol [4] is a slightly different approach, where the currency is
_soft-pegged_ to the US Dollar. Users are able to vote on what constitutes
valid collateral, and currency is created against this collateral. Of course,
the US Dollar or other currencies are involved in the peg, and so once again
the supply of this money is tied to the supply of Dollars and other fiat
currencies.

Similarly, the Frax system [2] provides cryptocurrencies with _fractional_
collateral, where the currency is not backed Dollars $1:1$, but still attempts
to maintain a peg to the US Dollar. Once again, the supply of FRAX is
ultimately tied to the supply of US Dollars. Another approach in the Frax
ecosystem is the Frax Price Index (FPI) stablecoin [3]. FPI attempts to
maintain a peg to the consumer price index, which tracks the prices of a
basket of goods denominated in US Dollars. Short of actually holding the
constituent goods, there is no principled way to maintain this peg, and so it
is possible that the coin will deviate from the peg during times of price
instability.

As far as I am aware, there is no cryptocurrency that is currently equipped
with a native mechanism like Elastic Cash to maintain elasticity. I believe
one can implement Elastic Cash in a cryptocurrency and create the first
cryptocurrency with a self-contained decentralized mechanism for elasticity!

##  4 Elastic Cash: the details

Elastic Cash uses trade in cashbonds to determine a risk-free rate of
interest. The money supply is regulated to ensure that this interest rate
remains approximately fixed. Cashbonds are issued by the central bank (in the
case of conventional currencies) or by the distributed algorithm (in the case
of cryptocurrencies). I refer to these entities as _cash authorities_.

The contract $\mathsf{cashbond}(d)$ promises that the cash authority will pay
the holder of the contract $1 on the date $d$. Let us reserve $d_{0}$ to
denote the current date. On date $d_{0}$, the cash authority pays each holder
of $\mathsf{cashbond}(d_{0})$ $1, and these contracts expire. Elastic Cash
requires that the cash authority implement a public market in cashbonds. On
the date $d_{0}$, contracts of the type $\mathsf{cashbond}(d)$ for $d>d_{0}$
will be available in the market for cashbonds maintained by the cash
authority.

Cashbonds are a special class of asset, and they should not be treated like
other securities. Elastic Cash requires that cashbonds can be generated and
traded _only_ in the public market that is administered by the cash authority.
Cashbonds are not transferable, meaning they cannot be exchanged outside of
the public market, and they cannot be used as collateral for loans. Because of
these restrictions, cashbonds cannot themselves play the same role as money.
The purpose of these rules is to ensure that holders of cashbonds who desire
liquidity will sell their cashbonds in the market and so keep the mechanism
informed about the demand for liquidity. For the same reasons, trade in
cashbonds should not be taxed. The transactions of buying and selling
cashbonds should be viewed as similar to transactions that move money between
savings accounts paying varying rates of interest, and treated similarly under
the law.

###  4.1 Risk-free rate of interest

The price at which cashbonds trade implies interest rates for risk-free loans
of varying durations. Let $\mathsf{rate}(t)$ denote the interest rate for
duration $t$, where $t$ is measured in units of years. Note that $t$ can take
fractional values. Let us write $\mathsf{price}(d)$ to denote the price at
which $\mathsf{cashbond}(d)$ last traded in the market. Then, if the current
date is $d_{0}$ (again measured in years), the prices of cashbonds can be used
to compute implied interest rates according to the formula:

$\mathsf{price}(t+d_{0})\cdot(1+\mathsf{rate}(t))^{t}=1,$

which implies that the interest rate can be expressed as

$\mathsf{rate}(t)=\mathsf{price}(t+d_{0})^{-\frac{1}{t}}-1.$

Because the loans executed by cashbonds are risk-free, the values
$\mathsf{rate}(t)$ capture something about the market’s belief about the
opportunity cost of making risk-free loans for duration $t$. Generally, one
would expect $\mathsf{rate}(t)$ to be a monotone function of $t$, meaning that
$\mathsf{rate}(t)>\mathsf{rate}(t^{\prime})$ if $t>t^{\prime}$, because loans
of longer duration usually command higher interest rates. Moreover, if $t$ is
much larger than $t^{\prime}$, then we might expect $\mathsf{rate}(t)$ to have
higher variance than $\mathsf{rate}(t^{\prime})$, because predictions about
the distant future can diverge much more than predictions about the immediate
future. It is tempting to make inferences about the values of
$\mathsf{rate}(t)$ based on the current treasury market, but I believe these
markets are significantly different from each other, and so historic data
about treasury yields is unlikely to capture how the risk-free rate of
interest would behave in the cashbond market. I discuss the differences in
Section 4.3.

These rates encode important information about the demand for liquidity. The
goal of the mechanism is to regulate the money supply so that one of these
rates is held approximately fixed. It makes the most sense to pick a rate for
a relatively short duration, because these rates are likely to have the least
variance. With that in mind, let $\tau$ denote a short time period, say $1$
week, so in years we have $\tau=7/365$. The goal of the mechanism will be to
keep

$\mathsf{rate}(\tau)\approx 0.02.$

There is nothing special about $0.02$, except that it is convention for
central banks around the world to use $2\%$ as the target rate of long term
inflation.

Let us set

$p_{-}=(1+0.021)^{-\tau},$

and

$p_{+}=(1+0.019)^{-\tau}.$

The goal of the mechanism will be to regulate the money supply so that

$p_{-}\leq\mathsf{price}(\tau+d_{0})\leq p_{+},$

where again $d_{0}$ is the current date. This corresponds to keeping

$0.019\leq\mathsf{rate}(\tau)\leq 0.021.$

###  4.2 Using the market to regulate the money supply

Participants in the cashbond market can put in orders to sell a specific
number of cashbonds that they hold at a specific price, and can also put in
orders to buy a specific number of $\mathsf{cashbond}(d)$ at a specific price.
The cash authority acts as a _market maker_ to match buy orders to sell orders
and so conduct transactions at a specific price between market participants.
Ideally, the market for cashbonds will support auctions111I will not commit to
a specific style of auction here, though any implementation must carefully
specify how the cash authority behaves as a market maker and what the rules of
the auctions are. for sellers to sell their cashbonds when needed.

The cash authority will itself participate in this public market by buying and
selling cash bonds in prescribed ways. The goal of the mechanism is to
maintain $\mathsf{rate}(\tau)$ approximately fixed, and to keep the market in
cashbonds sufficiently liquid, so that the money supply can be quickly
adjusted based on changes to $\mathsf{rate}(\tau)$.

It makes sense to pick a particular target distribution on outstanding
cashbonds that is maintained during normal times. If the current date is
$d_{0}$, we say that $\mathsf{cashbond}(\delta+d_{0})$ has duration $\delta$.
For example, the cash authority might aim to maintain the invariant that at
any point in time, $1/4$ of the outstanding cashbonds have duration between
$0$ and $1$ month, $1/4$ have duration between $1$ month and $1$ year, $1/4$
have duration between $1$ year and $4$ years, and $1/4$ have duration between
$4$ years and $10$ years. I discuss some considerations for how this
distribution should be picked in Section 4.3.

Here is the proposed scheme for buying and selling cashbonds:

  1. 1.

The cash authority will buy and sell cashbonds to keep
$\mathsf{rate}(\tau)\approx 0.02$. The cash authority will place a standing
order to buy an infinite number of contracts $\mathsf{cashbond}(\tau+d_{0})$
at price $p_{-}$, and a separate standing order to sell an infinite number of
$\mathsf{cashbond}(\tau+d_{0})$ contracts at price $p_{+}$.

Because the cash authority is able to generate arbitrary amounts of both money
and cashbonds, it will always be able to satisfy any of the resulting
transactions. This ensures that

$p_{-}\leq\mathsf{price}(\tau+d_{0})\leq p_{+},$

as discussed above.

  2. 2.

When cashbonds are redeemed for money, the cash authority will need to sell
new cashbonds to restore the balance between money and cashbonds. The redeemed
cashbonds should be replaced by selling new cashbonds at auction, picking the
dates of the new cashbonds so that the overall distribution on duration is
restored to the target distribution on durations, as much as possible.

  3. 3.

When the demand for money is high, we are likely to reach the point where all
of the available bonds $\mathsf{cashbond}(\tau+d_{0})$ have been purchased by
the cash authority. In such times, the mechanism has run out of the means to
inject money into the financial system at a fast enough pace according to rule
1. This can be resolved by selling large quantities of
$\mathsf{cashbond}(2\tau+d_{0})$ contracts at auction in the market. Market
participants will be incentivized to buy these cashbonds and then sell them
back after time $\tau$; at that time the cash authority itself will be willing
to buy the cashbonds at price $p_{-}$. The net effect will be to inject money
into the system, while preserving the number of outstanding cashbonds.

The number of cashbonds sold in this process is a design choice. The goal is
to inject significant liquidity, so I would favor an exponentially escalating
volume of sales. For example, the cash authority might first sell a quantity
that corresponds to $1\%$ of all outstanding cashbonds, and a week later
escalate it to $2\%$, then $4\%$, and so on until the cashbond market returns
to the state where market participants are no longer willing to sell back the
cashbonds of duration $\tau$ to the cash authority at $p_{-}$. These actions
may temporarily distort the distribution on the durations of outstanding
cashbonds, but the distribution will be quickly restored when the new
cashbonds are redeemed and rule 2 is applied.

These rules allow the money supply to rapidly adjust to the demand for
liquidity. An actual implementation of Elastic Cash would need to resolve many
smaller technical details. Let me now make a few comments and observations
about the Elastic Cash mechanism as I have defined it.

###  4.3 Discussion

Here I want to make a few observations about Elastic Cash as I have defined
it:

Elastic Cash provides is a truly new mechanism

The Fed funds rate is a rate set by the Fed, while Elastic cash allows the
risk-free rate of interest to float freely in a narrow range, and injects
money only when needed to change the rate. This is a significantly different
process. Elastic Cash is quite different from a system where the central bank
simply allows deposits for all with interest rate $2\%$—such a scheme does not
give the central bank a method to inject large amounts of money when the
liquidity is needed. History has shown that the Fed needs a tool like Elastic
Cash to inject liquidity into the financial system, since interest rates have
proven too weak as a tool to inject large quantities of liquidity. As we
discussed in Section 2, this has led to the Fed buying assets or propping up
assets that were liable to crash in value. In doing so, the Fed is forced to
pick and choose between market participants who get first access to the new
liquidity that it provides.

Even partial implementations make sense

Central bankers should not be attempting to directly reason about the demands
for liquidity; they do not have enough data to make those decisions. But if
they must make dramatic unilateral changes to the money supply, the scheme of
Elastic Cash at the very least gives a fair way to do it by trading cashbonds
along the lines I have suggested above. This can be done even if rule 1 is not
enforced. The cashbond market would allow the Fed to inject money by buying
cashbonds at auction. This removes the ability of the finance sector to
control the flow of the new money. It is also preferable to having the Fed buy
treasuries or mortgage backed securities, because it disentangles the actions
of the Fed from the needs of the Treasury and the mortgage industry. There is
no need to tie increases in the money supply to increases in government
spending or mortgage rates.

Cashbond market $\neq$ US treasury market

Cashbonds should not be confused with conventional government securities like
US treasuries. These instruments are significantly different from each other,
and one cannot make inferences about the cashbond market, which does not yet
exist, based on the behavior of the US treasury market. Let me highlight some
key differences:

  1. 1.

The issuance of cashbonds is controlled by strict and transparent rules, and
is not tied to the spending of the US government.

  2. 2.

There is no analogue of debt ceilings, or any chance that the central bank
will default.

  3. 3.

Cashbonds cannot be used as collateral for loans and cannot be transferred
outside of the Elastic Cash market.

  4. 4.

Trade in cashbonds is not taxed.

For these reasons, I expect the cashbond market to be much more liquid than
the current US treasury market, and I expect the risk-free rates of interest
to be better behaved.

Distribution on cashbonds

It is important for the functioning of Elastic Cash to maintain a large volume
of outstanding cashbonds of varying durations. Ideally, we would like there to
be broad participation in the cashbond market from all kinds of financial
entities: banks, companies, pension funds, and individuals. Because these
participants will be willing to trade at different durations, participation
will be increased if a wide range of durations are available, and the market
is liquid at all durations. Even though the cash authority only regulates the
interest rate for duration $\tau$, this action will affect the rates for all
durations. One would expect that banks and other sophisticated players will
trade cashbonds of shorter duration, and perform the arbitrage necessary for
information about demands for liquidity of all durations to propagate to the
shorter durations. I suspect that there is a principled way to choose the
ideal distribution on durations of outstanding cashbonds, but I have not yet
been able to convince myself about what it ought to be.

##  5 Adopting Elastic Cash in the US Dollar system

As discussed in Section 2, the Dollar system involves many different kinds of
institutions that are currently creating instruments that can be exchanged for
US Dollars. Changing the system is not going to be straightforward. However, I
do believe that there is a path to making the change somewhat gradually, so
that all the parties involved have time to adapt to the new system. Here is a
proposed sequence of steps for adopting Elastic Cash into the Dollar system:

  1. 1.

The Fed begins to populate the cashbond market by gradually selling cashbonds
of varying duration. Cashbonds are held at accounts maintained by the Fed,
which allows the Fed to enforce that cashbonds cannot be transferred outside
the cashbond market. At this point, cashbonds that expire are replaced
according to the rules of Elastic Cash, but the risk-free rate of interest is
allowed to float freely. I would expect this floating rate to converge close
to the current Fed funds rate.

  2. 2.

Once the market for cashbonds is running at significant scale, regulations
should be enacted to curtail the private creation of US Dollars. This can be
done gradually by raising the interest rate at which the Fed lends to private
entities. At the same time, the Fed should begin to put upper and lower bounds
on the risk-free rate by trading in the cashbond market. Once the cost of
borrowing from the Fed far exceeds the risk-free rate in the cashbond market,
private entities like banks will be incentivized to participate in the
cashbond market and raise money there. The current creators of US Dollars can
be handled as follows:

    1. (a)

Commercial banks should be barred from creating new deposits that are not
backed by cash reserves. Banks should fund new lending activity by selling
corporate bonds instead. As the interest rate for these bonds goes up, the
risk-free rate of interest in the cashbond market will also go up via the
trading in the cashbond market, and lead to the creation of new money that is
funneled to loans. So, the money required to support loans will still be
created via the new system.

    2. (b)

The Fed’s repo facility and money market fund facility should be closed.

    3. (c)

The eurodollar market is, perhaps, a bigger problem, both because of its size
and the fact that the institutions cannot be regulated by US law. Still, the
Fed can wind down its swap lines with foreign central banks gradually, until
eurodollars lose their Fed backing. Foreign central banks and governments
should be allowed and encouraged to participate in the cashbond market to
obtain liquidity.

  3. 3.

The inevitable tantrums in the financial sector should be treated with
stoicism.

It is an understatement that moving from our current system of private money
creation to Elastic Cash would be a dramatic change. There are likely to be
many challenges that need to be overcome to implement it, not least the
resistance of the finance industry, whose raison d’être is to control the flow
of money. Elastic Cash represents a significant loss of control for financial
firms, and a democratization of the flow of money. For these reasons, it is
perhaps more easily implemented in a cryptocurrency, as I discuss next.

##  6 Elastic Cash in cryptocurrencies

A major advantage of Elastic Cash over conventional mechanisms for elasticity
is that it can be implemented in a truly decentralized way, without _any_
trusted central authority. Bitcoin made a technological leap when it
introduced the concept of a blockchain. Since then, a number of
cryptocurrencies have emerged, with different ways to implement the
blockchain. Any of these systems can be used to implement Elastic Cash, so
here I will keep the discussion at a high level, only talking about how the
blockchain can be utilized. Because Elastic Cash involves making significant
changes to the money supply, I do believe that implementing it requires _new_
cryptocurrencies. I do not think it can be implemented using a layer built on
top of Bitcoin, for example.

Here is how one can implement Elastic Cash on a blockchain at a high level:

  1. 1.

At any point in time, each user of the cryptocurrency is known to hold some
amount of money, as well as various cashbonds.

  2. 2.

Users of the currency can announce transactions of money, as well as orders
placed in the cashbond market. The orders can be placed with a specific expiry
date.

  3. 3.

Miners will add both money transactions and orders in the cashbond market to
the next block of the blockchain. To implement the market in cashbonds:

    1. (a)

Miners will act as market makers to map buy orders to sell orders and so
execute the trade in cashbonds. There are some subtle issues that need to be
addressed here. For example, a miner may be incentivized to choose some orders
over others to include on the blockchain, insert their own orders in the final
block, or choose to ignore some orders when acting as a market maker. In
particular, miners can always insert orders to themselves obtain the spread
between two crossing buy and sell orders, so it makes sense for miners to be
paid this spread as a transaction fee to carry out their market making
function. This removes the incentives to manipulate the orders that are added
to the most recent block.

    2. (b)

Miners will also execute the algorithm to simulate the activities of the cash
authority in the cashbond market. New money and cashbonds will be created
according to the rules of the mechanism, and these will be traded with users
based on the orders that have been added to the blockchain.

##  7 Conclusions and Questions

It is an exciting time to think about the technology of money. The US Dollar
is experiencing a once-in-a-lifetime contraction (see Figure 1), and the
demands for a stable global currency have never been larger. Elastic Cash is a
broad scheme to enable elastic money. I have purposefully left the mechanism
underspecified, because I believe that more work is required to understand the
details and trade-offs involved in the particulars of the mechanism.

Here are some important questions that I feel remain unanswered:

  1. 1.

How should the market maker behave in the cashbond market? In the context of
conventional currencies, can private entities function as market makers? In
the context of cryptocurrencies, how should the algorithm be set up so that
miners do not have an incentive to behave dishonestly when they are carrying
out the role of market maker?

  2. 2.

What style of auction would give the best results for the cashbond market?

  3. 3.

What is the ideal target distribution on cashbonds? If the cashbonds are
concentrated on very short durations, this gives the most power for the
mechanism to inject large quantities of money, but it also means that the
market loses information about the demand for liquidity over long durations.
So, there is a trade-off between various choices for distributions on
durations.

  4. 4.

How can we gradually transition the current US Dollar system to such a
mechanism? The steps I discussed in Section 5 are likely to be difficult to
execute. Perhaps there is a way to use cashbonds and incentivize the large
players in the financial system to adopt Elastic Cash without being forced to
do it. What is needed is a mechanism to transition to the new mechanism!

  5. 5.

How should we expect the free floating rate curve $\mathsf{rate}(t)$ to behave
as a function of $t$ during normal times? I would expect this function to be
monotone, but I am not sure how to reason about it beyond that.

##  8 Acknowledgements

Thanks to Paul Beame, Siddharth Iyer, Travis Kriplean, James Lee, Noam Nisan,
Darcy Rao, Tim Roughgarden, Eli Ben-Sasson, Oscar Sprumont, Michael Whitmeyer
and Amir Yehudayoff for many helpful and entertaining conversations about
money.

## References

  * [1] Fed history overview.  https://www.federalreservehistory.org/time-period.
  * [2] Frax.  https://frax.finance/.
  * [3] Frax price index.  https://docs.frax.finance/frax-price-index/overview-cpi-peg-and-mechanics.
  * [4] Makerdao.  https://makerdao.com/en/.
  * [5] Christine Desan.  Making Money: Coin, Currency, and the Coming of Capitalism.  Oxford University Press, 2014.
  * [6] Neels Heyneke and Mehul Daya.  The rise and fall of the eurodollar system.  https://www.nedbank.co.za/content/dam/nedbank-crp/reports/Strategy/NeelsAndMehul/2016/September/TheRiseAndFallOfTheEurodollarSystem_160907.pdf, 2016\.
  * [7] Lev Menand.  The Fed-Unbound: Central Banking in a Time of Crisis.  2022\.