A different approach to the equation of exchange for medium of exchange token dynamics modelling
I became interested in token valuation working on a project for OpenZeppelin. In this exploration I found that the equation of exchange (EoE) is widely used to model the dynamics of these tokens and help build a model for valuation.
So I decided to dive deep into the origins and implications of the EoE and tried to rethink this formula adapted to the crypto environment. Doing so, I found it interesting that the equation is subject to interpretation at many levels, and that there have been many critiques to it. Nevertheless I knew that the most intelligent minds behind cryptoasset valuation were using it so I felt the need to break it up. I began with a review of recurrently mentioned articles on the subject.
Vitalik had written some interesting concepts regarding a particular kind of medium of exchange token — those which accrue value by enforcing its use on a particular network. He provides an adapted form for the EoE which makes It easier for us to understand the effect of holding and minting tokens in its price. The equation used by Vitalik is the following:
MC = TH
- M = amount of coins in circulation
- C = price of each coin
- T = total transacted value per period (say, one day)
- H = average time (in days) a coin is held before it is used for a transaction
To understand this equality we can observe that it states the trivial fact that the value of the transactions made in an economy is equal to the value of the coins spent to make those transactions. Say for example that H = 3 days, and T = 5.000 usd/day, you will need a total market capitalization of 15.000 to satisfy the needs and behaviour of this economy. The simplicity of this equivalence hides the many layers of analysis that we can develop around it.
Vitalik remarks the importance of the holding period H and mentions how a frictionless exchange could accelerate a token’s demise in case users don’t have incentives to hold it. I realized that understanding the concept of velocity clearly was necessary to build models on top of the EoE, and felt that I was missing something.
The application of EoE for valuation
The first widely referenced article which describes a model for token valuation which uses EoE (in its traditional form, MV = PQ) is Chris Burniske's Cryptoasset Valuations. Chris makes clear that this model is not intended to calculate the price of tokens since we need more understanding of cryptoeconomics and markets are not efficient enough to respond to the models.
He describes a fictional network where bandwidth is sold at a determined price. In his interpretation of the EoE, the term “PQ” represents the GDP of this economy and only transactions which correspond to the purchase of bandwidth account for “Q”. Transactions occurring between exchanges should not be taken into account (since these don’t increase GDP).
Then comes velocity. Chris recognises that this is the most important assumption of the whole model, and chooses a V of 20 for tokens on the float (after bonders and hodlers), taking bitcoin’s velocity as benchmark and adjusting it based on his research of bitcoin user profiles.
EoE for payments — multimarket economy
In the case of a medium of exchange token which drives an open p2p network where different kinds of goods and services are sold, it could be difficult to determine the difference between “transactions which contribute to GDP” and any other transaction. For instance, any purchase of tokens with fiat money can represent a transaction in the network which accounts for the purchase of fiat money. Considering fiat money “a good which is sold in the network” every transaction between two users can then be considered on the calculus of P.Q and thus, V.
Michael Zochowski extended the concepts of EoE for token valuation to payment tokens which could address multiple markets. In later articles he provides some insights on velocity, where I highlight the concepts of “minimization of working capital” and that the main driver of velocity is consumer habits. I believe the key to understand velocity is here: to identify user profiles in terms of their holding behaviour and quantitatively predict this behaviour for each profile. I refer to holding behaviour not only as keeping coins as a store of value, but also as the relationship between average value in coins possessed by a user and the amount of value the user transacts to purchase goods and services.
Velocity from a behavioral perspective
My interpretation is that velocity works as the binder of the other variables and consolidates all the intricate concepts of behavioral economics that affect the dynamics, so that the EoE works. Velocity represents the incentives and ultimately the value of the coin as a product, in contrast with P.Q which represents the value of the market which the coin addresses, which in the case of many tokens (specially payment tokens) will be there even if the implementation of a coin to have access to it is mediocre. Even though we can calculate velocity on an existing network and define it as V=PQ/M, doing so invokes the tautologic aspects of the formulation and hides from us some interesting implications.
Say a market exists and we want to build a cryptoasset as a medium of exchange to provide access to it, with the price of the coin subject to supply and demand. Say supply is fixed and PQ is fixed, looking at EoE we deduce that the price of the token depends solely on V. So V provides a representation of this “Demand”. A term which consolidates the desire to buy, and the desire to hodl. And what creates this desire to buy and hodl?
- Expected token price appreciation (investors/speculators).
- Expected stability + real utility (consumers).
- Friction of exchanging the asset (economic, operative and cognitive).
Velocity in centralized applications
One approach to understand velocity for payment tokens is to benchmark the behaviour of users in other networks where money must be “loaded” to a card / application / wallet to have access to features of that network.
As an example, I observed the public transport card in Argentina. A plastic which can be charged in any train or subway station and must be used to pay for transport. I use 4 transports a day on average, which cost 20 pesos and I usually charge my wallet with about 400 pesos which accounts for 20 trips, equivalent to 5 days on average. So I charge my card once a week. I don’t actually calculate this, it just feels like an appropriate amount based on the amount of time I need to charge the card, the probability of losing it and the effort that should be made to get the credit back if I do, the consequences of having no money charged when I need it, the opportunity cost of my money held in the card (interest rate), and probably some other unconscious factors.
When a service to charge the card with NFC from my phone was launched, eliminating the need to wait in line to reload, the average amount of trips charged in my account went down (Reduced friction → reduced average holding → increase velocity).
As another example, Argentina’s main Amazon-like marketplace started offering a wallet service where to hold pesos and spend them via QR code in many stores. They added a mechanism by which the money held in that wallet is invested in a fund at an interest rate similar to Argentina’s inflation, (which is already interesting over here). This provides some sense of “stability” to the coins in the wallet (Argentinian pesos) which increases the average balance and reduces velocity.
In these two examples, the currency used in the wallet is the Argentinian peso, which we are pretty certain it will not accrue value any time soon. But for cryptoeconomic environments the speculative component plays a crucial role in velocity.
Can velocity be good?
I guess this depends on the interpretation we give to the EoE. If we stick to my approach of considering V as the result of exogenous aspects which affect behavior in terms of demand of coins and desire to hold, higher velocity can only be detrimental to the token’s value, but probably beneficial for the economy as a whole, or for the enterprise which created the token.
Madhumitha Harishankar does a great job contextualizing the EoE in this article. She recognises the limitations of the formula and develops an adaptation for the crypto environment. As the title suggests, she questions the common belief that velocity is detrimental for token value.
“…the equation of exchange implies that for a given supply of coins and price level of services, higher token velocity yields higher Network GDP and possibly increasing coin value.”
When analyzing the economy of a country we could agree that the holding behaviours of its coin can be altered by printing money. Inflation incentivizes consumption, as people expect prices to rise in the future and don’t want to hold a devaluating asset. This is a known strategy used by governments to increase GDP and create growth which I believe that works because a great portion of the people don’t have the tools or knowledge to move to other financial assets to preserve their purchasing power. In the economy of cryptoassets we will have to ask ourselves if the network we are analyzing responds to this behaviour.
Ultimately, to understand the effects of each parameter and answer questions like this we should analyze the exogenous factors which cause perturbations to each variable independently. Then, when we ask “Is velocity good?” we must agree we are talking about the effect of those factors. The same applies to all the other parameters in the EoE.
If we wanted to make a deep analysis of expected velocity in any network we should define user profiles and study their expected behaviour separately. A non-exhausting list of exogenous factors that affect the behaviour of users in terms of velocity can start with:
- Change in transaction fees.
- Launch of better applications where to make transactions.
- Implementation of mechanisms that provide stability to the value.
- News about regulation.
- Publicity / marketing of the coin (expected price appreciation increased).
- Introduction of new products which alter the frequency of purchases.
- Macroeconomic context (acquisitive power of coin).
- Modification in conditions of alternative investments (interest rates, friction, etc).
- Launch of new utilities where staking coins is useful.
- Opening / closing of voting periods where staking is needed to vote.
User profiles will depend on each network and could be defined in terms of the goods they purchase, the technical skills of the users, their purchasing habits, their acquisitive power, etc. Among these we would have the “hodler” profile, with an expected velocity of 0.
I believe a model which integrates these variables could be developed to estimate the holding behaviour of users and thus, velocity.
Effect of exogenous perturbations in velocity
An interesting scenario is analyzed by Madhumita regarding a perturbation in holding behaviour: speculators dumping coins in an exchange because they no longer see value in the coin.
“If we are concerned about whether there will be a market of customers who would like to use these newly available coin supply on the exchange to avail the company’s services, the Equation of Exchange can do nothing to answer this. When instantaneous coin supply suddenly increases due to hodlers dumping, the exchange rate of the coin may well decrease if this yet-untapped market does not exist. This is not captured by velocity. Equation of Exchange does not characterize spontaneous supply/demand dynamics and is too static a model for this kind of analysis.”
Even thought it is true that EoE cannot tell us if there exists a market willing to purchase this dumped coins, I believe the EoE can provide useful information about this dynamics in a multimarket economy if we observe velocity in terms of the “holding profiles” of users. To explain my interpretation let’s study a simple model of an economy fueled by a token which is valued solely by supply and demand.
Suppose a stationary model for an economy which transitions between states of equilibrium. In our economy, 4 actors (A,B,C,D) exchange goods and services for tokens. Transaction fees are non-existent and this is not necessarily a cryptoeconomic network.
Using Madhumita’s formulation where α = token supply, and β = token price in usd, we set α=8, P=1, Q=4, V=0,5. Thus, β=1. Q and V are variables subject to a period of time t. We define “holding profile” as the amount of value transacted by a user over the average amount of value possessed by a user within the same period.
The following diagram shows transactions of coins as arrows between actors and the balances of coins of each actor at the beginning of the period. In this file you can see the calculations for each period and download it to play with the variables for different scenarios.
In period t1 each actor spends one coin, with a value of 1 usd each for a total of 4 products transacted. Each user is also getting back a coin to spend in the next period, so we have a stable equilibrium. In this example each actor possesses an average of 2 coins. We can observe that 4 of the total coins are transacted one time in each period, and the other 4 are not transacted, so the ‘average transactions per coin per period’ is 0,5. This stable equilibrium allows for the transaction of 4 usd worth of value.
Now, for period t2 consider the modification of the holding profile of A and B, who do not wish to hold their ‘static’ coin in their wallets anymore while the amount of goods demanded in the economy and its price in usd is unaffected (remains 4 units a week at 1 usd each). If C and D don’t wish to absorb these coins and hold them as ‘static’ coins (in other words, they do not wish to change their holding profile), the price β will start to drop until it matches a demand. Now, every user in this economy will need more coins to purchase the same amount of goods. So if C and D wish to maintain their holding profile, they will eventually need to purchase the coins from A and B. EoE helps to explain how the price is adjusted to support the economy with the new holding profiles.
The stable model for the ecosystem with the new holding profiles becomes:
With variables in equilibrium at:
α=8, P=1, Q=4, V=0,66…, β=0.75
This elemental example allows us to understand the effects on token price based on the holding profiles, which can be modelled in terms of real life facts. If we explore these dynamics further the EoE could provide some light in the modelling of Medium of Exchange tokens economics and possibly serve as a tool to predict the effects of the aforementioned exogenous factors in token price and GDP.
Effect of payment channels in velocity
Payment channels affect velocity in both directions: reducing friction and costs for frequent transactions creates incentives to spend more and hold less. Part of this incentives is compensated by an increase in P.Q. (which we do not consider an exogenous perturbation in V) and other part would also have an effect in holding behaviour since users might reduce their need for ’backup’ tokens. Interestingly, the requirement to bond tokens to open channels and the subsequent cost (economic, operative and cognitive) of closing them reduces incentive to get rid of tokens, reduces the amount of liquid coins available for trade and thus reduces velocity.
Considering my interpretation of velocity as the term that describes user’s behaviour regarding the demand to buy and hodl coins, it seems clear that payment channels as they work today contribute to a reduction of velocity.
- EoE is subject to interpretation and it could be useful for the industry to define and agree a standard for the calculus of each parameter in the equation.
- We can develop a model which allows for the correct classification and quantification of transactions and velocity.
- If we achieve this in a real-time automatic way, we could actually design mechanics that provide stability to the value of the native token of the cryptoeconomic environment.
- When designing token mechanics for medium of exchange tokens, velocity should be taken into account and considered dependant on the expected holding profiles of users and their behaviour. This behaviour could be expressed as the average amount of value possessed in coins over the average value of a product sold in the industry, for each holding profile.
- When trying to project velocity of a network in the design phase we could benchmark velocity in other coins that have similar utilities. We will have to be careful in using bitcoin as benchmark since its positioned as a uniquely good store of value coin, incentivizing specific holding behaviours which cannot be projected accurately to minor coins. Maybe in the absence of an existing coin for our target market, analyzing the behaviour of different holding profiles of bitcoin users could do the trick.
Understanding velocity and its effects on MoE token dynamics was originally published in Coinmonks on Medium, where people are continuing the conversation by highlighting and responding to this story.