Token Economics
Considering “Token Velocity”
Scott Locklin (for Brave Software)
February 10, 2021

Abstract
Various assertions about the “effect” of token velocity on token exchange rate
are considered. The claim that token velocity is an extrinsic quantity that can bethrottled to raise token exchange rate, or will naturally increase and lower token
exchange rate is examined and rejected. Heuristics are given for future rejection of
obviously ridiculous ideas.

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Contents
1 Economics in general3

2 Addressing “On Medium of Exchange Tokens”

3

3 Addressing assertions in “New Models for Utility Tokens”

8

4 Sanity checks

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5 A review of our statement on BAT price stability

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Economics in general“Beware of economists bearing Greek symbols” is generally good life advice[1]. Noneconomists attempting to do economics are even more fraught. Anyone attempting to
do economics with unclear or incorrect definitions, or with equations that fail basic
dimensional analysis or defy laws of addition and division, can be summarily rejected,
regardless of who makes the argument.
Economics as a field of thought is occasionally useful, but it is only useful as a model :a toy or tool for reasoning about what may or may not happen in an economic system.
The power of economic models as tools for prediction is fairly low. Even for large, stable
and simple economic systems, something like an exchange rate is necessarily going tolook like a random walk over most timescales of interest to human beings. For a realworld example, consider the dollar exchange rate of Iceland’s krona, a “token” for asmall economy of about 340,000 people. While there are long term trends, they are notespecially predictable, and most of the time the exchange rates look like a random walk.
If economic theories were any good at point prediction, there would be no reason for
financial services. We could fulfill the dreams of mid-20th century economists such as
Bill Phillips[2] and the Soviets[3] to optimally allocate resources without using markets.
Whether to believe the result of a valid economic speculation at all is a matter forhuman decision makers. Invalid economic ideas can be safely ignored. By invalid economic ideas, I mean ideas that are not internally consistent, use inconsistent definitions,wrong equations, or other such faulty reasoning. The confusion over token velocity andits “influence” on token exchange rate is the result of numerous incorrect economic
ideas.

2

Addressing “On Medium of Exchange Tokens”

This “velocipedian” idea seems to originate with a Vitalik Buterin blog post[4] called“On Medium of Exchange Tokens” in October 2017. This blog post contains various
overt errors and misunderstandings, which I document below. Quoting relevant sections:
Traditional macroeconomics has a simple equation to try to value a
medium of exchange:
MV = PTHere:• M is the total money supply; that is, the total number of coins• V is the velocity of money; that is, the number of times that an
average coin changes hands every day• P is the price level. This is the price of goods and services interms of the token; so it is actually the inverse of the currencys
price

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Topic: animosity to velocity• T is the transaction volume: the economic value of transactions
per day
The proof for this is a trivial equality: if there are N coins, and eachchanges hands M times per day, then this is M * N coins worth of
economic value transacted per day. If this represents $ T worth of
economic value, then the price of each coin is T / (M * N), so the pricelevel is the inverse of this, M * N / T.
There are several errors here already. Equalities in the world of matter, including
the world of economic reality, must have commensurate units. Buterin’s assertion isthat the total number of units of a token times the velocity of a token (token units pertime) is equal to a price in dollars times a price per time (or a price per time). Thisis false. The definitions are also incorrect. I will correct Buterin’s definitions below,
starting with Fisher’s equation of exchange[5]:Ptb Ttb = M b Vtb• M b is the total coin supply of token b• Vtb is the velocity of money: i.e., the average number of times that a coin changes
hands per unit time, at time t• Ptb is the price level. This is the weighted average price of goods and services interms of the token b at time t• Ttb is the quantity of goods/services purchased in terms of tokens b at time t
These definitions are important. The Fisher equation of exchange doesn’t makesense without these distinctions. The misunderstanding of the definitions inherent in
the equation of exchange causes worse confusion further down:
For easier analysis, we can recast two variables:• We refer to 1/V with H, the time that a user holds a coin before
using it to make a transaction• We refer to 1/P with C, the price of the currency (think C =cost)
Now, we have:
M/H = T/C
MC = THThe left term is quite simply the market cap. The right term is the
economic value transacted per day, multiplied by the amount of timethat a user holds a coin before using it to transact.

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This is a steady-state model, assuming that the same quantity of users
will also be there. In reality, however, the quantity of users may change,and so the price may change. The time that users hold a coin may
change, and this may cause the price to change as well.
The inverse of (average) token velocity is not average holding time. For example, let
us postulate a money supply of 10 tokens in an economy with a velocity of 10 times perday. If 9 of the tokens are traded once every 1000 days, and one of the tokens 99.991
times a day, this gives mean token velocity 10 times per day. However, the average
holding time for a coin in this ecosystem is 900.001 days, not 1/10 day per transaction.
Getting things like this right is key to understanding the use of the equation of exchange.
The skewed holding time described above is also closer to the present reality for virtually
all cryptocurrencies. The distribution of token holding times is skewed, and most tokens
are held in expectation of future returns.
Similarly, while Fisher’s equation of exchange is an equilibrium model (which I
suppose could be called “steady state”), it does not depend on the number of users.
There is no number of users in the equation. Where the “quantity of users” comes
into play is never revealed; in fact the number of users is irrelevant to the equation of
exchange. Number of users can have influence on the variables in the equation, but it
doesn’t change the result or hinder its applicability. If we assume that average users
contribute an approximately fixed amount of economic output to the system, raisingnumber of users will generally grow the economy, but the equation of exchange is valid
for an economy of any size.
Finally, the inverse of the weighted average of a token-denoted price is not the exchange rate of a token in terms of dollars. The holding time (as stated above, inverse
velocity is not holding time) or the inverse velocity times the token-denominated economic output is also not the market cap in dollars. What is apparently desired here isto come up with something like a conversion rate in dollars per token. It’s possible todo this, but the above statements do not do this, and are a mass of confusion and error.
In appendix-3 of the original BAT whitepaper, this operation was done in one line,
as I thought writing the actual conversion details were obvious and uninteresting. In
hindsight, I realize this was an error in didactics, which may be the origin of the problems
above. Starting again from the above Fisher equation of exchange:
Ptb Ttb = M b Vtb
If we multiply the left hand side by 1, aka

Pt$, we get the following relation.
Pt$

Ptb $ b
(Pt Tt ) = M b Vtb
Pt$
The term

Ptb
Pt$

is the number of tokens per dollar (the inverse of the dollar/token

exchange rate) Sb$ , therefore:
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1(Pt$ Ttb ) = M b Vtb$
Sb
We can rearrange the values to look like the following, as was done in the whitepaper,
though the transformation of the economic output to dollars was swept under the rug:
Pt$ Ttb
M b Vtb$

Stb =The important thing to note here is that if the token is used to exchange more goods
and services valued in dollars, the token is also more valuable in terms of dollars.
The denominator is apparently scary to some people, but it can only frighten if you
forget what the denominator means. The denominator is just the number of tokens
used in exchanges for a given economic system. Let’s put it on the other side of theequation where it belongs so we can see what it means:$

M b Vtb Stb = Pt$ Ttb
The left hand side of the equation is the number of tokens transacted times theconversion rate of tokens to dollars. The right hand side of the equation is the economic
output of the token economy converted to dollars. It is a trivial relation. Apparently
understanding of this is non-trivial. There is exactly one thing central planners cancontrol here: the number of tokens. The right hand side; economic output, is somethingthat economic participants do. If nobody chooses to use the platform, or the software
hasn’t been written yet, this value is zero and, as shown below, the exchange rate for atoken is an entirely speculative position on future network activity. In a token market
which is freely traded, the exchange rate is not set by central planners; it is set bymarket participants, participants in the token economy, hedgers and speculators.
Belaboring this point further, it helps to go back to Fisher’s original statement on
the equation of exchange to make it completely concrete and obvious and measured in
actual units. We can make a toy example such as in chapter 6 of his book[5], which canbe found online. It is an excellent book filled with clear thinking, and should be thefirst thing read by any would-be economist working with the equation of exchange.
Imagine you are on a desert island with seashells used as money. The island has
turtles (cost: four seashells each), coconuts (cost: half a seashell) and seagull eggs (cost:
one seashell). In a given month, 200 seagull eggs, 200 turtles and 1000 coconuts arebought and sold. So the total monetary value in seashells for this month is

Ptb Ttb = 200 eggs

1 seashell
4 seashell
0.5 seashell+200 turtles+1000 coconut= 1500 seashellegg
turtlecoconut

If our island has a total money supply of 10 seashells, the seashell velocity is 150 ina monthly period. Now, if someone on a motorboat shows up with some large amountof dollars and wants to buy some coconuts (assuming no arbitrage and a $2 per seashellconversion rate), thenReport: BSI-02

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Pt$ Ttb = 1500 seashell ∗Topic: animosity to velocity$$2$2= M b Vtb Stb = 10 seashell ∗ 150 ∗seashellseashellIt’s obvious from looking at the above that increasing the economic output (i.e.,
more things bought and sold aka Pt$ Ttb ) will make seashells worth more, both in termsof dollars and in local purchasing power. That’s the fundamental insight the pricing
equation gets us.
Skipping over more false irrelevancies, we come to the source of the velocipedianidea: the 1854 Broad Street cholera outbreak itself.
Another, and perhaps even more important, conclusion is that the market cap of an appcoin depends crucially on the holding time H. If someone creates a very efficient exchange, which allows users to purchasean appcoin in real time and then immediately use it in the application, then allowing sellers to immediately cash out, then the market
cap would drop precipitously. If a currency is stable or prospects arelooking optimistic, then this may not matter because users actually see
no disadvantage from holding the token instead of holding somethingelse (ie. zero de-facto fee), but if prospects start to turn sour then sucha well-functioning exchange can acelerate its demise.
The statement above is false. Again, the inverse of velocity is not average holding
time. Worse, the following statement subordinated to his holding time statement is a
giant non sequitur. Imagine we go back to my desert island with an economy consistingof seashells, eggs, turtles and coconuts. Now, we’re going to crush the seashells into
powder and issue 10 SEASHYLL tokens. Because it is a small island, they’ll be issued
from a central authority. I know the guy who wrote NASDAQ’s Xstream crossing
engine, so let’s say we have a crossing engine as fast as that and it’s solar powered,donated by the UN and run feeless by an NGO so it doesn’t cost anything to use. Will
this increase the velocity of money? Of course it won’t. It won’t increase if you put
it on a perfect and frictionless blockchain either. The only thing that will increase thevelocity of a fixed supply of money is if you increase the economic output of theisland. Maybe you can lure more turtles with the monitor glow from your NASDAQ
crossing engine.
It is possible that velocity can be limited technologically, however. If we use, insteadof NASDAQ Xstream, a blockchain that crosses one transaction, once a month, everyone
on the island will starve to death. This doesn’t increase the value of the SEASHYLL
token, despite the Fisher equation of exchange; it simply creates an artificial lack ofuseful money, equilibrium not achieved, and an equation no longer valid. The island
dwellers would be better off finding some kind of rock or wampum to use instead of
SEASHYLL, as they probably will. Such things have historically happened; the Long
Depression of 1873 was caused in part by a lack of useful money, and was in part
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ameliorated by the issuance of private scrip currencies by banks, the distant ancestorsof blockchain money.
There is a price to pay in holding tokens for use in the payment network. Our
tokens are potentially long-term deflationary, particularly as stated, if the economy
grows. If you use them to buy something, you might be able to buy more things at
a later date if you hold the token. As Buterin notices in his essay, this acts like anextra fee on networks. Similarly, short term price volatility might also be considered an
extra fee on utility token intermediated networks. Neither is relevant to our use of theequation of exchange. The equation of exchange is not a statement about convertibilityof currencies; it is a statement about using a token to mediate economic activity. If wego look at the example of, say, bitcoin, we can immediately see the problem. People
are using it as a store of value because it often goes up in value versus the dollar.
People also hold volatile tokens in hopes of future return. People also use such tokens
to intermediate economic activity where it is useful to do so. Money is useful for buying
things, after all. The idea that people will want to cash out immediately in a deflationaryor volatile environment just because we have a frictionless exchange somewhere is silly
and self-contradictory. People are more inclined to hold money when it goes up in value.

3

Addressing assertions in “New Models for Utility Tokens”

Since the above assertions on the subject of velocity are incorrect, there seems little
reason to examine subsequent claims by Kyle Samani in his blog post[6] “New Models for
Utility Tokens”. While the above blog post is a tragedy of misunderstanding, Samani’s
assertions are a disaster of confusion.
As I noted in Understanding Token Velocity, the V in the equationof exchange is a huge problem for basically all proprietary payment
currencies. Proprietary payment currencies are, generally speaking,susceptible to the velocity problem, which will exert perpetual downwards price pressure. Due to this effect, I expect to see utility tokens
that are just proprietary payment currencies exceed a velocity of 100.
Velocities of 1,000 are even possible.
There is, as demonstrated above, no “velocity problem.” There is also no reason to
expect a numeric velocity for an arbitrary utility token made for exchanging goods andservices to be 100, 1000 or the square root of the kitchen sink. The two thoughts are
not connected. It makes no sense putting the sentences next to each other as if there issome relationship between them.
Rolling back to the cited blog post[7] he mentions, “Understanding Token Velocity”,
Samani fails to define token velocity correctly in his first mathematical statement:

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Heres where velocity comes in. Its defined as:
Velocity = Total Transaction Volume / Average Network Value
Therefore:
Average Network Value = Total Transaction Volume / Velocity
None of these statements are true. Samani then goes on to make the statementthat burning tokens will reduce velocity. This statement is also false. Samani thengoes on to make other ill considered assertions about staking, lock up, mint and burn,gamification, etc. Samani also thinks velocity will shrink if you continually shrink themoney supply. This is, of course, obviously incorrect.
Going back to the desert island I’d like to leave Samani on, the attentive reader willremember we had a money supply of 10 seashells and a monthly economic output of
1500 seashells, giving a velocity of money of 150 seashells per month. If we were toburn, bury or otherwise somehow lock up 9 of the 10 seashells, the economy is the same
size; people still want to trade the same number of coconuts, eggs and turtles. Now, thevelocity of money is 1500 seashells per month. When my boat full of dollars shows up,
the seashells are of course worth more, which I suppose is what is desired here. But allalong for some reason, Samani keeps talking about reducing velocity.
As for the rest of “New Models for Utility Tokens”, these are not models. They are
varying assertions based on faulty understanding of how economics, ratios or much ofanything works. There is no reason to address any of them.

4

Sanity checks

The above exercise is painful and unnecessary.
The lack of self consistent units or definitions in these blog posts means, for any
sane person who cares about numbers in the corporeal world, that they may be safely
ignored. If your citation is an impenetrable word salad of falsehoods and clodplated
babbling, there is no reason to pay attention to the article. It doesn’t matter who
the author is, as “many people have webpages.” As far as can be told, the assertionsin these articles were widely, credulously accepted, and nobody, including the authors,
thought about these ideas in any reasonable or careful way. Performing the seashell
exercise would have helped.
The idea that one can raise exchange rates by lowering velocity of money is immediately obviously false. Limiting cases are a useful general technique. Imagine you ran
a general payment company that used one token. There is no reason to have more than
one token; token size is arbitrary. The company transacts a million dollars in businessper day by having a very large token velocity. What happens if you reduce the token
velocity to one transaction a year? Is there any way for it to do two million dollars a
day in business without increasing token velocity? Will the price of the token go up or
down if the payment company does twice the business?
To make the example more obvious: at present, blockchains are constrained to alimited number of transactions per block and a limited number of blocks per year. If
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we increase the potential number of transactions per block, or provide a micropayment
solution (meaning the average token velocity could go up enormously), would we expect
the token for the blockchain to go up or down in price? For those who assert that thetoken for the blockchain would become less valuable in the presence of potential forhigher velocity: if this were true, everyone should use blockchains that only cross 10transactions once per year. Volume limits achieved; we’ll all be rich. A blockchain that
doesn’t function at all, or allow any transactions (aka the velocity will be zero) will be
infinitely valuable.
Finally, it is also empirically obvious none of these things apply to utility tokens
that exist in the actual world, such as BAT, BTC, 0x, ETH or any other value carrying
token. While speculation and correlation with the rest of the cryptocurrency sector hascaused a fair amount of volatility to a given token’s dollar exchange value, the increased
velocity, which has been increasing for the BAT donation system, hasn’t caused theprice to fall. The same is true for 0x, which has, roughly speaking, appreciated alongwith the economic transactions in its ecosystem, just as the correct model, as opposedto the ridiculous “inverse velocity model”, predicts. The reality of what drives token
price is explicated (again) below.

5

A review of our statement on BAT price stability

Speaking of correct economic models, let us review again Appendix-3 of the BAT
whitepaper[8].
Our model for virtual currency exchange rates was postulated by Dutch economists
von Oordt and Bolt in 2016[9]. The model postulates that the value of virtual currencies
consists of three major factors: the utility of the virtual currency to make payments, thedecision of forward-looking speculators to regulate the supply of virtual currency, andthe elements that drive user adoption and merchant acceptance of a virtual currency.
The argument originates with Fisher’s 1911 observation that speculators may effectively limit the money supply by withdrawing money from circulation in anticipationof higher future utility. Since this dynamic particularly applies to limited issuance currencies such as bitcoin or BAT, it can be an important factor in the pricing for token
sales and stability analysis of virtual currencies.
For a simple economic system with fixed quantity of currency tokens M BAT , we canwrite down a transaction quantity relationship:
PtBAT TtBAT = M BAT VtBAT
Where VtBAT is velocity of BAT, the average number of times each unit of BAT isused to purchase services within the defined period of time t. TtBAT is the quantity ofservices purchased with BAT over the period of time t and PtBAT is the weighted priceof the services.
Inserting the exchange rate in terms of $PtBAT $ BAT= M BAT VtBAT
Pt Tt
$
Pt
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Since we can assume the legacy fiat currency is the accounting unit for all parties$

involved, we define the exchange rate StBAT , and substitute in the above equation to
give$

StBAT =

TtBAT*M BAT VtBAT

Where TtBAT* is now considered to be the economic transactions which are done interms of BAT valued in dollars (aka Pt$ TtBAT ). If we consider the fraction of currency
which is not used in transfer of services, we can postulate a velocity of the fraction of
BAT . Defining Z BAT to be the number
currency which is actually used for settlement Vdttof BAT units not used in transactions.
Since the entire velocity of money in our economy VtBAT is an average between thecurrency units used and the units unused for transfer of services,
VtBAT =M BAT − ZtBAT d
VtBAT
M BAT

Combining these into the exchange rate
BAT*
Ttd

$

StBAT =

(1)BAT
(M BAT − ZtBAT )Vdt

The exchange rate for BAT tokens is therefore proportional to the volume of services
purchased and inversely proportional to the currency not used in transactions for thetime period t. This equation encapsulates the insight that a lack of money in circulation
will raise the exchange rate.
We now turn our attention to the fraction of BAT which is not used for exchange.
Some of the ZtBAT tokens may be the result of users forgetting about the small numberof tokens they hold. Some may be due to exchange delays in settlement for legacy currencies. Overall though, the holders of inactive tokens have standard ways of evaluating
future utility of the tokens in terms of modern risk management theory.
Rational token holders expect future returns from a position in BAT to be proportional to the volatility of the position over the time period in question, scaled by a risk
aversion term γ,$

BATγσ 2 (St+1)ztBATIf we consider the future expected exchange rate:$
BAT||St+1||The period t exchange rate is discounted by the risk free exchange rate operator (R)to make it comparable to the future expected exchange rate, we get the time discounted
exchange rate:

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Topic: animosity to velocity$−RStBATThe difference in these two values are equivaluent to the expected returns. We reachthe efficient frontier from modern portfolio theory thusly:
$

$

$

BAT
BAT
||St+1|| − R(StBAT ) = γσ 2 (St+1)ztBAT

Using this standard result, we can solve for the optimal number of tokens held byan individual during a given time period.
$

ztBAT =$

BAT||St+1|| − R(StBAT )$BAT)γσ 2 (St+1

If we consider all of the people holding BAT at a given time interval t to have thesame risk preferences, we get the economically efficient number of BAT held for later
use.
$

ZtBAT = Nt ztBAT =$BAT||St+1||ztBAT − R(StBAT )
$

γ 2
BAT
Nt σ (St+1 )Since this value can’t be negative, we assume that people who hold BAT have theposition that expected future exchange rate is
$

$

BAT
||St+1|| ≥ R(StBAT )hence, using our above relationship, we get the relationship between the expected
future value of the BAT, the interest rate and the velocity of transfers in the BAT
economy:$BAT
R−1 (||St+1||) ≥TtBAT*M BAT VtBAT

So, people hold BAT if the discounted expected value exceeds the hypothetical valueof the current exchange rate. So, the exchange rate as a function of future expectedvalue of BAT is
$

$

BAT
StBAT = R−1 (||St+1|| −$γ BAT 2 BAT
Zt σ (St+1 ))Nt(2)Thus, the BAT holdings are the discounted expected future exchange rate minus
the risk premium for the uncertainty in future value of the BAT, as expected.
If the model holds, 1 and 2 can be used to define supply and demand for BAT. Since
M BAT is not time dependent in the case of BAT, the time varying exchange rate canbe readily understood in terms of BAT transactions and opinions on future utility of
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by the transactions rather than future expectations of utility. This dynamic has been
observed in maturing virtual currencies as well as various other in-house token systems.
While models are imprecise, this model argues for long term price stability in a
token mediated economy.

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References[1]

Emmanuel Derman. “Beware of Economists Bearing Greek Symbols”. In: Harvard
Business Review (Oct. 2005). url: http : / / www . emanuelderman . com / media /beware.hbr.pdf.

[2]

Reserve Bank of New Zealand Museum. Bill Phillips and the MONIAC. 2007. url:
https://www.rbnz.govt.nz/research-and-publications/fact-sheets-andguides/factsheet-bill-phillips-and-the-moniac.

[3]

Leonid V. Kantorovich. The best use of economic resources. Trans. by P.F. Knightsfield. Pergamon Press, 1965. url: https://drive.google.com/file/d/1h8VXbMkPRWehB7khZr2lnATdOC
view.

[4]

Vitalik Buterin. on medium of exchange tokens. 2017. url: https://vitalik.ca/
general/2017/10/17/moe.html.

[5]

Irving Fisher. The purchasing power of money: Its determination and relation tocredit, interest and crises. Macmillain, 1911. url: https://www.econlib.org/
library/YPDBooks/Fisher/fshPPM.html?chapter_num=6#book-reader.

[6]

Kyle Samani. new models for utility tokens. 2018. url: https : / / multicoin .
capital/2018/02/13/new-models-utility-tokens/.

[7]

Kyle Samani. understanding token velocity. 2017. url: https : / / multicoin .
capital/2017/12/08/understanding-token-velocity/.

[8]

Brave Software International. Basic Attention Token Whitepaper. Tech. rep. Mar.
2018. url: https://basicattentiontoken.org/index/BasicAttentionTokenWhitePaper4.pdf.

[9]

Wilko Bolt and Maarten van Oordt. On the Value of Virtual Currencies. Tech. rep.
Working Paper No. 2016-42. Bank of Canada, Apr. 2016.

The author wishes to thank Pushkar Kopparla for a valuable correction to the efficient frontier section.
D:20210210185841Z

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