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Token velocity is a difficult topic for most people, since we spend our time embedded into monetary systems which are kept opaque and are hard to reflect on — unless you studied economics. And, let’s face it, most economists set the jargon shields to the maximum, so that they can keep doing what they do.

I am not an economist. I got trained as one, but decided to become a mathematician instead, and have spent the best part of three years trying to build an abstract model of a ledger. In that journey, I learned a few things about money, coins and tokens and this article is a technical take on token velocity (based on my research insights) but written for a non-technical audience. It’s easy enough to follow if you can relate to the problem of getting home while drunk (also known as a random walk).


A useful way to think of the money in your wallet (or your bitcoins, or any of the money/coin/token systems in the world — I’ll just call them tokens from here on) is that they are in constant random motion hopping from economic agent to economic agent. At the micro-level we have a limited amount of agency over the part of the token system that we own: We can decide where it will go next in the walk. And, possibly, we have agency in deciding the last person from whom we receive our tokens . So, we can choose the people that we pay and the people that we work for, but those are the hard limits.

From a macro perspective, these random walks accumulate into the movement of tokens in time, and the distribution of money in “wallet” space. At any given point in time, we can, in principle, measure the distribution (it’s easier on a public ledger) and over a given period of time we can measure the “flow” of tokens or the number of token hops — This is what is meant (in a way) by token velocity.

Token velocity reflects, in a broad sense, the differential between tokens as a store of value and as a means of exchange. Naively, as tokens lose value relative to the resources they represent the velocity goes up because as word spreads we’re more likely to want to dump the token for the resource. On the flip side, if we expect that the value of a token will go up and that it will represent “more” resources in the future, then it’s likely that on aggregate more agents will hold the token instead of selling it.



In (simple) blockchain payment ledgers (with a maximum of one movement per token per block), we can think of each block as a verified set of token movements and, over a number of blocks, we can ask questions about token velocity. A useful expression is the ratio of tokens that get transferred to a new wallet per block to the total number of tokens in the system. Let’s call this x(r). For example, if x(r)=0.1, then that just means that 10 out of every 100 tokens that exist get moved, in block r, from their current wallet to a new one. At most, all the tokens are moved and x(r)=1 and, at least, none of the tokens are moved and x(r)=0. Now, we can think of x(r) as the velocity in a given block or the probability that a randomly picked token has moved in block r.

What happens over several blocks? If we sum the value of x(r) over several blocks and divide by the number of blocks: Say, (x(1)+x(2)+…+x(k))/k. This gives us an average token velocity. For example, if x(1)=0.15, x(2)=0.2, x(3)=0.4, then the average velocity is just 0,25 over those three blocks. This just says that a token picked at random has on average a 25% chance to have “jumped” to another wallet during these three rounds.

What does this have to do with monetary economics?

Economics usually deals with multiple ledgers at the same time, so for this explanation we’ll need to consider two sets of tokens: Let’s call them “Nicknacks” and “Doodads” for amusement.

Let’s suppose that people create new nicknacks and destroy old ones, but the total supply of doodads is static. The ledgers are synchronised and anti-correlated, so that nicknacks jumping wallets seems to cause doodads to jump wallets in the opposite direction and visa versa.

Also, suppose that n(r) is the nicknack rate of movement per block, and that d(r) is the doodad rate of movement per block. Let N(r) be the total nicknack supply for block r and let D(r)=D be the constant value for the doodad supply. Now let’s go three rounds.

Block 1. N(1)=100, D(1)=100 and n(1)=0.2, d(1)=0.2.

Block 2. N(2)=200, D(2)=100 and n(2)=0.2, d(2)=0.2.

Block 3. N(3)=50, D(3)=100 and n(3)=0.2, d(3)=0.2.

In the first block, nicknacks move at a rate of 0.2 and doodads at a rate of 0.2. So, 20 nicknacks have changed wallets, and 20 doodads have changed wallets. Now, this gives us a way of reverse engineering the price of nicknacks in terms of doodads — just multiply the velocity into the total coins to get the expected number of coins that have moved, and compare the ratio.

Block 1. 1N=1D.

Block 2. 2N=1D.

Block 3. 0.5N=1D.

What was the magic trick? We didn’t alter anything except the supply of nicknacks. The velocity of tokens (probability of wallet jumps) stayed the same, but altering the supply changes the ratio at which the one converts into the other — the “price”. The basic idea is that price can be interpreted using random walks (velocity/probability) and supply in as much as it can be interpreted using demand and supply curves.

Next, suppose that we leave the supply alone and just alter the velocity of nicknacks.

Block 1. N(1)=100, D(1)=100 and n(1)=0.2, d(1)=0.2.

Block 2. N(2)=100, D(2)=100 and n(2)=0.4, d(2)=0.2.

Block 3. N(3)=100, D(3)=100 and n(3)=0.1, d(3)=0.2.

The trade volume per block of doodads is fixed at 20 (0.2*100); but for nicknacks this varies as n(1)=20, n(2)=40 and n(3)=10.

Block 1. 1N=1D.

Block 2. 2N=1D.

Block 3. 0.5N=1D.

It’s exactly the same “price” structure as before, but now it’s happened because of changes in spending patterns or rather in the hold/sell decision making of individual agents.

In fiat economies, doodads can be thought of as real goods which tend to grow slowly and move at similar rates year on year. On the other hand, nicknacks correspond to fiat money which is subject to (1) central bank supply decisions (eg. quantitative easing) and (2) interest rates. And, it’s a combination of (1) and (2) that central banks rely on to regulate economic systems through “money”.

Why should you be afraid of quantitative easing?

In 2008 the banking system collapsed, the main culprit was the networked nature of debt that had been labelled as “good” because too few people understood the credit instruments they were buying and selling, and banks are exposed via their networked relationships to the risks of other banks. It was a systemic, unpredicted failure that happens sometimes to all complex systems. The solution to the crises or part of it is a policy called quantitative easing which basically means — the central bank is going to create more money to make the system liquid again. 

Here is why that’s dangerous…

We don’t often think of money/tokens/coins as information, but that’s all that they are. It’s not your bank balance that matters, but the proportional value of your balance relative to the total system. Token systems act as a ring-fence around resources, and the tokens you own represent your claim on the resources. Now, if the resources increase and the tokens increase at the same rate, then your claim as a proportion of the whole doesn’t change. If the tokens increase faster than the resources do, then your proportion of the whole is less. So, it’s never a good idea to increase the money supply in a world economy that’s slowing down. But that’s what the world’s two largest economies (The USA and the European Union) are currently planning on doing once again.

How many times do we need to learn the lessons of Weimar and Zimbabwe before we return to more stable economic policies?

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This article is an opinion piece by mathematician Viroshan Naicker, and it is to be viewed objectively, not taken as certitude.  Viroshan is a freelance “mathematician for hire” who develops algorithms, studies tokenomics, and write philosophically. Feel free to contact Viroshan via LinkedIn.

Image by Gerd Altmann from Pixabay

Viroshan's background is in pure and applied maths, economics and physics. He completed a doctoral degree in graph theory (the formal study of networks) in 2014. After working as an academic, Viroshan moved into business as a consultant, and has since disappeared down the crypto-economics rabbit hole: He wrote both the yellow papers for Project UBU - an advertising based UBI system, and has since developed the idea of a Token Exchange Game - a formal game theoretic model for a ledger. Reach out to Viroshan if you are interested in the crypto-economic space, industrial and applied mathematics, or unusual and creative business problem solving.