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Token Economics Considering “Token Velocity”

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Token Economics Considering “Token Velocity” 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 be throttled 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. 1 (c) BSI SEZC Topic: animosity to velocity Contents 1 Economics in general 3 2 Addressing \On Medium of Exchange Tokens" 3 3 Addressing assertions in \New Models for Utility Tokens" 8 4 Sanity checks 9 5 A review of our statement on BAT price stability 10 Report: BSI-02 2 (c) BSI SEZC Topic: animosity to velocity 1 Economics in general \Beware of economists bearing Greek symbols" is generally good life advice[1]. Non- economists 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 to look like a random walk over most timescales of interest to human beings. For a real world example, consider the dollar exchange rate of Iceland's krona, a \token" for a small economy of about 340,000 people. While there are long term trends, they are not especially 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 for human decision makers. Invalid economic ideas can be safely ignored. By invalid eco- nomic ideas, I mean ideas that are not internally consistent, use inconsistent definitions, wrong equations, or other such faulty reasoning. The confusion over token velocity and its “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 = PT Here: • 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 in terms of the token; so it is actually the inverse of the currencys price Report: BSI-02 3 (c) BSI SEZC 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 each changes 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 price level 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 is that the total number of units of a token times the velocity of a token (token units per time) is equal to a price in dollars times a price per time (or a price per time). This is false. The definitions are also incorrect. I will correct Buterin's definitions below, starting with Fisher's equation of exchange[5]: b b b b Pt Tt = M Vt • M b is the total coin supply of token b b • Vt is the velocity of money: i.e., the average number of times that a coin changes hands per unit time, at time t b • Pt is the price level. This is the weighted average price of goods and services in terms of the token b at time t b • Tt 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 make sense 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 = TH The left term is quite simply the market cap. The right term is the economic value transacted per day, multiplied by the amount of time that a user holds a coin before using it to transact. Report: BSI-02 4 (c) BSI SEZC Topic: animosity to velocity 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 per day. 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, raising number 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 ex- change 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 eco- nomic output is also not the market cap in dollars. What is apparently desired here is to come up with something like a conversion rate in dollars per token. It's possible to do 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: b b b b Pt Tt = M Vt $ Pt If we multiply the left hand side by 1, aka $ , we get the following relation. Pt P b t (P $T b) = M bV b $ t t t Pt b Pt The term $ is the number of tokens per dollar (the inverse of the dollar/token Pt $ exchange rate) Sb , therefore: Report: BSI-02 5 (c) BSI SEZC Topic: animosity to velocity 1 (P $T b) = M bV b $ t t t 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: $ P $T b S b = t t t b b M Vt 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 the equation where it belongs so we can see what it means: $ b b b $ b M Vt St = Pt Tt The left hand side of the equation is the number of tokens transacted times the conversion rate of tokens to dollars.
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