And other implications of analyzing MV = PQ from first principles

About the writer

Madhumitha Harishankar is a PhD Candidate in Carnegie Mellon University studying the application of network economics in wireless networks for resource sharing. She is advised by Prof. Patrick Tague and works closely with Prof. Carlee Joe-Wong. She also advises blockchain-based IoT service provider Nodle on token design and modeling.

Special thanks to Prof. Carlee Joe-Wong and Sriram V Iyer for helping think through this analysis and proof-reading. Thanks to the team @ Nodle for their valuable feedback.

Introduction

The equation of exchange (EoE), derived by John Stuart Mill, has become a popular approach for valuing medium of exchange crypto tokens. With P as the average price level of final goods and services produced in the economy and Q as the quantity of these final goods and services (also called the real GDP), PQ represents the nominal GDP of the economy — i.e. the total economic value of goods and services produced. With M as the amount of money supply and V as the average amount of times unit money exchanges hands in the purchase of final goods and services, MV represents the total monetary spending in the economy on final products. The equation of exchange is then the identity: MV = PQ.

The EoE has been used to conclude that token velocity is bad for coin value. I will show otherwise in this analysis — indeed, 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. However, we also see that EoE is of limited use in modeling the complex dynamics of crypto token uses, and the conclusions it yields may not adequately characterize real-world phenomena. Ultimately, better models are required to understand the evolution of cryptocurrency valuations.

Interpreting EoE in its intended context

Firstly, MV is the effective buying power in the economy. Consider, for instance, an economy with $1. If this $1 is exchanged for an ice-cream, and the ice-cream seller now uses this $1 to buy $1 worth of chips, then this $1 resulted in a $1 * 2 => $2 worth of buying power (since it facilitated 2 transactions of value $1 each). In this extremely simplified instance, M = 1, V = 2.

The quantity theory of money (QTM), formulated by Irving Fisher, uses the equation of exchange to relate an economy’s GDP to the money supply. Fisher’s QTM says that changes in the money supply M yields proportional changes in the price level of goods and services P, keeping velocity V as constant and the production level Q as exogenous.

Now, Fisher’s QTM holds only under the following assumptions:

• The real GDP Q is exogenous (not dependent on the money supply) and does not change significantly. In other words, all resources in the economy are already being fully utilized and additional money supply will not increase this.
• It assumes that V is constant. V is a measured quantity of people’s spending behaviors and is believed by monetarists to be largely unchanging over time. The long-term effect of changes in M are hence claimed to be observed in P, holding V constant.
• This theory completely assumes that money is demanded for transactional purposes and not for its own sake; i.e. it has no speculative purposes (is not used for hoarding by people).

Lets take the example of the US economy. Though QTM makes questionable assumptions, it is nonetheless useful because it illustrates the relationship between money supply in the economy (as controlled by the Fed) and inflation. Rearranging EoE, we get M = PQ / V. An increase in M results in lower value of money (simply because there is more of it now than previously). Since money is less valuable now, the price of goods and services P increases to retain the same value as it did before. Hence we get a higher nominal GDP that reflects the inflation created by the newly increased money supply. Unlike Fisher’s assumption however, this may have an impact on V as well (as argued by the Keynesian Money Demand theory): V may decrease in since people have higher demand for money now (as more money is required to continue to buy the goods they desire at the increased price level). On the other hand, the Fed starts issuing bonds to decrease the money supply when inflation is deemed too high, thereby reducing M . Hence, money increases in value and the demand for money decreases, resulting in lower P for goods and services. However, V may increase as well: the Federal Reserve increases interest rates as well to combat inflation, causing banks to increase their interests, which increases the opportunity cost of holding money. Also per the EoE, if V is fixed but a country’s real GDP Q increases by x% every year, then an increase in M beyond x% per year will reflect in higher P (i.e. inflationary effects).

Hence, while EoE is an identity that holds true as long as money has no speculative purposes, its interpretations are widely debated by monetarists and Keynesians.

As an aside, note that V is not the cure-all for money problems; if (for a given money supply) higher and higher V is is sufficient to increase the nominal GDP PQ, then governments may well ban holding onto money and solve world poverty. The caveat is that V can only be as large as the value created by people in the economy that facilitates the use of money for transacting such value. V itself is a function of the GDP and hence must be interpreted with caution and not as an independent variable. For instance, consider that Alice creates a software program and exchanges it with Bob for $500, and then uses the $500 to buy a couch from Ikea, which may use the $500 to upgrade its showroom. Ultimately, Alice generated $500 worth of value by creating the program and hence was able to engage in these transactions using money as a proxy for that value, and Ikea created $500 of value by manufacturing the couch, which it exchanges for other services. PQ implicitly captures the value creation in the economy, hence V is not independent of that but rather a function of it (along with other variables).

Interpreting EoE in crypto context

Since cryptocurrencies (especially utility tokens) can be considered as a medium of exchange, we can try interpreting MV = PQ in this context and see what we get. Let P represent the average price level of goods and services produced by the crypto company (in $) and Q the total quantity of transactions the company has with customers. Hence, PQ represents the economic worth, in dollars, of the company’s services. Further, define α as the number of coins comprising the circulating supply in the market and let 1 coin equals β USD; the money supply in this economy is β*α USD.

In defining the velocity here, note that we must disambiguate between generic token transactions vs. token transactions that result in consumption of network services. With money as the medium of exchange, MV = PQ inherently assumes that any exchange of money that happens in this economy directly contributes to the GDP; i.e. a higher velocity indicates that the GDP is higher. In the case of tokens, however, a coin may be bought by a speculator, who may over time sell it to another, which still counts as two hand-exchanges for the coin but does nothing for the network GDP PQ as the coin did not participate in any of the company’s functions (i.e. services and goods produced by the company). Hence we define V’ as the average amount of times a token exchanges hands for speculative purposes and V as the average amount of time a token exchanges hands in a non-hodling transaction.

This means that an exchange of a token from the company to minter as reward/fees counts towards V. An exchange from minter to a speculator of the token does not count towards V. But if the speculator eventually sells it to a customer who uses it to exchange the company’s services, this flow counts towards V. Needless to say, the customer exchanging the token for the company’s services counts towards V as well. To summarize, V is the number of times a token exchanges hands towards GDP-contributing transactions, averaged across all tokens in the supply α. Strictly speaking, we may disambiguate V further to account for coin exchanges between customers and the company (i.e. transactions regd. the final goods and services of this economy) vs. coin exchanges between the company and the minters (as reward), and transactions on the exchange where customers acquire coins from speculators or minters. However, this disambiguation has no effect on the rest of this analysis since we assume that the transactions between customers and the company constitutes a fixed fraction of V; as such, we continue with this definition of V.

Then, we have our crypto-EoE: βαV = PQ.

Lets first retain Fisher’s assumptions as stated earlier and treat V and Q as constant. Note that in the case of crypto companies, both α and P are set and controlled by the company, unlike the analysis earlier where the Fed directly controls money supply but not the price level. We see that if we increase the supply of coins α, the price level P need not increase; in-fact, β decreases to β’ proportionally to reflect the higher available supply, and a larger amount of coins are required to transact at the price level P (i.e. we need P/β’ coins now instead of P/β). If the company increases P as well, then β either remains the same or increases (depending on the amount of increase in P). If, for instance, the underlying economic value of the goods/services produced by the company increases, then the company might charge a significantly higher P. Finally, if α is not changed but P is increased to reflect a higher quality of goods/services, then β increases.

Now lets move away from QTM and consider P and α as fixed while Q and V are subject to change. Rearranging for the exchange rate β, we get: β = PQ/αV. Consider what happens to β when V changes. Either users engage in more transactions with the company than they previously did (hence increasing V and Q) or users engage in less transactions with the company (hence decreasing V and Q). In either case, β remains unaffected by changes in V as long PQ changes by the same amount. Given a fixed α and P then, V must necessarily increase for Q to increase, which may increase β if the rate of increase in Q is greater than the rate of increase in V. Hence for a fixed α and P, velocity necessarily has to increase to facilitate higher Network GDP. Token velocity V is not our enemy. It, in-fact, increases the buying power of goods and services (which is what a medium of exchange is for) in the crypto-economy. It directly facilitates higher Network GDP (aka PQ).

The question is whether the rate of increase in Q (lets call it δ) is greater than rate of increase in V (lets call it η). The EoE is inherently a steady-state model which assumes these factors are in equilibrium and captures the resulting economic behavior. However, the exchange rate β is, in-fact, instantaneous and a function of real-time supply/demand dynamics at the exchange. We now step outside the scope of QTM in interpreting the equation of exchange here:

Firstly, as long as there are no exogenous factors affecting β, η cannot be greater than δ, since η refers not to an increase in supply or demand of coins in isolation but to a net increase in the rate of overall GDP-contributing coin transactions. Hence, if the overall rate of transactions in the system increases, the exchange rate β stays the same in the worst case, or increases, at best. If η increases beyond a certain limit, presumably, the demand is too high and exchange rate β starts to increase in response. This implies that a lesser number of coins are required per transaction for procuring services at the price level P, which increases the number of transactions Q facilitated by the coins α (higher V).

The above interpretation may be confusing because it would appear, as a result, that increasing exchange rates allow more network transactions, which may appear to contradict traditional supply/demand wisdom. Note, however, that the exchange rate and exchange transactions are in terms of the crypto-token, while the company’s price level for its services has been fixed to the market-competitive fiat rate. Therefore, as more people demand the service, the value of the token increases (i.e. equivalent to money becoming more valuable in the monetary context of QTM), which results in the price level in tokens droppings (i.e. less tokens required for the same fiat rate), and more transactions taking place.

But what causes V to change exogenously to begin with? Different market factors may cause changes in V and will have different effects. For instance, if the company’s underlying products and services offerings becomes more differentiated or increases in scope, customers may transact more frequently with it or new customers may flock to it. Another way V may change is via hodler behavior. A fundamental assumption in QTM is that money (or coins in this case) are not held for speculative purposes. In the Keynesian interpretation of EoE, money may be held for precautionary or convenience purposes, and the fraction of income that is not transacted (aka held) depends on the opportunity cost of holding money (i.e. the interest rates charged on alternative asset investments).

EoE quickly starts to break down for our purposes as tokens have a significant speculative aspect. As hodlers overall lose interest in holding the coin and become willing to part with it, or if customers believe the market for coins is relatively liquid and are willing to purchase them as needed on-demand instead of precautionarily holding up-ahead, they cause an increase in exchange supply. Note that the real market has participants with heterogeneous beliefs who may hodl/sell at different times; it may well be the case that some coins being made available just gets absorbed by other hodlers, yielding some temporarily higher V’ but not V. This is a good reason to consider alternate models to the EoE as it does not allows us to capture these dynamics and considers homogenous users, equilibrium market behavior, and non speculative money uses, whereas cryptocurrency markets are a multi-equilibrium game due to these complex factors (as noted by Vitalik Buterin as well).

That said, lets consider the scenario where most hodlers no longer see value in holding the coin (e.g. most of the market agrees that most of the coin value appreciation has already happened); in which case they create an additional supply of coins in the marketplace that is not absorbed by V’. This bulk of newly un-hodled coins can exogenously affect GDP-contributing velocity V.

However, speculators dumping their holdings on the exchange does not immediately result in an increase in GDP-contributing velocity V but rather increases the “GDP-contributing supply”, so to speak, that was hoarded earlier. When an argument considers V as having increased due to this, it implicitly assumes that the dump has been consumed by interested consumers and is being transacted for services, resulting in this higher V.

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.

Looking at the EoE then, it may be unclear what to conclude about V when hodled coins are released in this way without knowing if there is a market for these coins. However, it is impossible for V to decrease: these coins have now been made available for transactions with the company — hence, if they are purchased by customers at all, it only increases V since customers then exchange it for the company’s services and products. If there is no demand for the coins hodlers are making available, then coin velocity remains the same.

In this scenario then, what can we say about the relation between V, β and Q? Note that β = PQ/αV. If there is no earlier-untapped market for the bulk un-hodled coins, the exchange rate β of the coin decreases, which means more number of coins are required now for a transaction with the company since the company’s price floor is still $P. Hence existing customers will now consume more coins each to make the same level of transactions they did earlier. Hence velocity V increases, without any significant increase in Q since the market rate β changed externally (i.e. it changed exogenously due to speculator dynamics that are not captured by the EoE). On the other hand, if there is a market for the un-holded coins at the same rate β (the conditions under which this may hold true are interesting to think about), then Q and V increase together, and β either stays the same or increases, depending on η and δ as we saw earlier.

It is clear from what we’ve seen that we need better models of token dynamics to really understand what happens to their valuation over time under different kinds of market behavior. An interesting question to think about also is how the analysis above would change if the company had token-denominated prices for its services.

Contrasting with popular interpretation

There are a few popular ways that MV=PQ have been explained in the crypto context previously. I’ll review these below:

Chris Burniske and Vitalik Buterin’s versions:

Note that in Vitalik Buterin’s version, M and P are expressed in terms of tokens and are not fiat-dominated. Regardless, per their interpretation, V increases when long-term speculators start to sell their holdings back on the exchange, which appears inversely correlated with M and may hence result in a decrease in the market cap of the coin. This does not hold based on our analysis above. Speculators dumping their holdings on the exchange does not increase velocity but the “GDP-contributing supply”, so to speak, that was hoarded earlier. Whether this un-hodled supply meets with adequate demand or not is beyond the scope of the EoE. However, as we’ve shown, V increases regardless in this case. 1) If the exchange rate drops due to an excess supply and more coins are required per transaction now for the same fiat-based price level, thereby increasing V for same Q. Or 2) there exists previously unrealized demand for the company’s services which is realized now by consuming these coins, increasing the transactions Q, the network GDP, and either retaining the same exchange rate β or increasing it.

Kyle Samani’s version:

Kyle Samani defines velocity as the quantity of transactions/average network value; and concludes that the problem with utility tokens is that the quantity of transactions may increase multifold without necessarily the network value increasing since the velocity may be increasing proportionally. This is indeed a likely scenario; as we’ve seen above, rate of velocity growth η may well equal rate of transaction growth δ. However, this is not as much an issue as a feature of crypto-tokens, which I’ve discussed in detail in a separate post here. Coins buy privileges to access the services of a platform, not the platform itself. So it makes sense that their value will fluctuate with real-time dynamics of supply/ and demand for the platform’s services. Since tokens are not stocks and they do not pay dividends, it makes sense for their value to not grow directly with growth in demand. It grows with growth in instantaneous demand. As shown in my analysis in this post, if η grows beyond a certain market/application-driven threshold, then β starts to increase, and hence the market cap.

In Summary

• Token velocity is crucial for medium of exchange tokens. For a given supply of tokens and price level of services, higher token velocity means more transactions with the network, aka higher real GDP for the network and potentially higher exchange rates and market cap.
• However, the market rate and token velocity could change exogenously under different market conditions, and have different impacts on the Network GDP and token valuation that cannot be determined wholly by just the EoE.
• Speculators holding utility tokens and over-valuing them may cause the value of the token to increase beyond the transactional value of the token (i.e. its medium-of-exchange value). The dynamics of this is not characterized by the equation of exchange or the token velocity.

Token Velocity is Good! was originally published in Nodle on Medium, where people are continuing the conversation by highlighting and responding to this story.

Velocity-based valuation approaches may be fundamentally wrong

About the writer

Madhumitha Harishankar is a PhD Candidate in Carnegie Mellon University studying the application of network economics in wireless networks for resource sharing. She is advised by Prof. Patrick Tague and work closely with Prof. Carlee Joe-Wong. This work is in collaboration with blockchain-based IoT service provider Nodle.

Token staking and burn-and-mint mechanisms are used to incentivize users of a cryptocurrency to lock up their token holdings for periods of time, and have become popular ways of controlling token velocity. Lowering token velocity is considered necessary to ensure that the network’s valuation grows proportionally with transaction volume, especially for utility tokens. The argument typically goes something like this:

• Assume there are no speculators in the market; i.e., only the vendors providing services and customers interested in these services exist in the marketplace for a decentralized company called AppCoinCompany. AppCoinCompany may require vendors to charge a uniform price (e.g. if these are substitutable services) or allow different vendors to price differently. Assume the former for ease of argument.
• AppCoinCompany has x customers who purchase y AppCoins each to avail the vendors’ services and immediately exchange these coins for the services. Vendors then immediately go on the exchange to sell the AppCoins (x*y in total) and realize revenues in, for instance, USD.
• Assume a pure exchange economy; i.e., fixed total circulating supply. Let this total available supply of coins in the market be x*y.
• Now, say AppCoinCompany’s customers grow from x to 2x. Given that there are only x*y coins available, which are fully required by the x customers who were already present, one may expect the AppCoin to increase in (USD) value, simply because demand > supply now, assuming vendors’ prices (charged in AppCoins) remain fixed.
• However, this may well not be the case. Previously, when the system had x customers, suppose they availed the vendors’ services on a daily basis at some time t of the day, and the vendors put the coins back on the marketplace at time t+1, after their transactions with these x customers are done. Assume the newly added customers also have a daily demand for the vendors’ services. In that case, they could well come to the exchange everyday at t+2, and procure these coins at the same rate that the first x customers that came in at t procured them, since at t+2, the instantaneous demand is only x (instead of 2x) and the supply is still x*y.
• Hence, the problem seems to be: growth in the network use and cash flows do not translate directly into growth in coin value.

And hence the various mechanisms to force users to hold on to their AppCoins for a while (rather than dumping them straight on the exchange as soon as they are done using the company’s services), which lowers supply and presumably gives the marketplace a chance to reflect the increased demand.

But is this really a problem?

Decreasing token velocity in response to this “problem” appears inappropriate. For dividend-paying stocks, it indeed makes sense for the value of the stock to be proportional to the cash flows (and thereby reflect such growth in customers and service demand as in our previous example). However, tokens are not stocks, they do not pay dividends, and their value should not grow directly with growth in demand.

Coins buy privileges to access the services of a platform, not the platform itself. So it makes sense that their value will fluctuate with real-time dynamics of supply/ and demand for the platform’s services.

As we already see from the simple example discussed previously, AppCoin’s value should, in fact, grow with growth in instantaneous demand. As evident, this is harder to achieve than mere growth in overall product consumption and is one of key tradeoffs in considering a crypto-funding model vs. a stock model. Coin holders do not directly benefit monetarily from an increase in consumption (since they are not entitled to any dividends). Therefore, the consumption growth must be significant enough to raise the overall instantaneous coin demand for the token to have meaningful rise in its exchange rate. If it is likely that many consumers will ask for the platform’s services at the same time and continue to return with high frequency to procure services, then there is a good reason to believe instantaneous demand for AppCoins will grow and to therefore invest in it.

There is a second way that the token can organically increase in value as well. If the quality of the underlying services provided increases/the target consumer base is diversified (i.e. higher product differentiation), then the fiat value of the provided services increases. AppCoinCompany, in that case, should increase its price (in AppCoins) to reflect its evolved product quality under the competitive market. For instance, x customers may now require 2y coins instead of y to procure AppCoinCompany’s services, thereby increasing instantaneous demand for the coin (without necessarily increase in customers) and hence the coin value.

Is the mechanism design feasible?

The real market has several types of investors and speculators with largely unpredictable and possibly manipulative market behaviors. Further, most AppCoinCompanys have an increasing amount of total circulating supply (until all coins are minted) and hence they are not pure exchange economies that we saw in the simple example above. Given these, is it possible for an AppCoinCompany to influence its service pricing, rewards and coin policy such that AppCoin exhibits a meaningful growth in value against these two factors described above?

I will delve into this further in future work, but here is a (possible) broad-strokes answer:

• At any given time, the market should price AppCoins for the (discounted) final future customer-facing prices (in fiat) of AppCoinCompany’s final products and services and final future instantaneous demand. Consequently, AppCoinCompany should charge consumers the competitive fiat value of their services, dynamically priced in AppCoins since the market rate for AppCoin may be highly volatile.
• When all coins are mined, AppCoinCompany’s services are “final”, and customer base is more or less constant, speculators’ best strategy is likely to sell their coins at the current market rate since there is no reason to expect further coin appreciation (based on the two factors discussed above).
• At that time, AppCoinCompany may revert to a fixed-price model where its charged prices (in AppCoins) takes into account the total circulating supply (most of which should be highly liquid at this point).

In Summary

1. Tokens are not stocks. By putting in techniques to artificially decrease token velocity, AppCoinCompany forces AppCoins to appreciate directly with growth in cash flows which is inappropriate for a non-dividend model and lends AppCoins’ value uninterpretable. If AppCoins are used to access AppCoinCompany’s services but valued for a market cap that does not translate into dividends on the token, the resulting value of AppCoin is un-interpretable.
2. There is still reason good to invest in tokens if you believe customers will, at some point in future, be willing to pay more for underlying service than they do today. This could happen due to increase in long-term instantaneous demand or increase in product quality or both.
3. As an investor in AppCoins, what you are investing in/betting on is different compared to stock investments, though these are related. When you invest in stocks you are acquiring a stake in the company’s profits in the form of dividends. When you invest in tokens, you are acquiring a stake in the consumers’ willingness to pay function for the underlying product/services that can be redeemed with AppCoins.

Contesting Token Velocity as an Issue was originally published in Nodle on Medium, where people are continuing the conversation by highlighting and responding to this story.