Velocity and Demand for Money Models
Written by Michael Zochowski and Joe Alie
We’ve talked in the past about using the quantity theory of money or (a.k.a. Equation of Exchange) model to value cryptoassets. We labeled this the “base” model, since it is broadly applicable to all networks that have transactions denominated in a native token, which covers almost all networks.
One issue with the Equation of Exchange model is that it depends on assumptions around velocity and its relationship with the other variables in the formula M*V = P*Q. Typically, some constant velocity or functional form that is independent of the other variables is assumed. Note that this is not an unusual assumption for economic analysis, but it does impose some limitations on the model. Here’s what noted economist Greg Mankiw says about constant velocity:
“As with many of the assumptions in economics, the assumption of constant velocity is only a simplification of reality. Velocity does change if the money demand function changes. For example, when automatic teller machines were introduced, people could reduce their average money holdings, which meant a fall in the money demand parameter k and an increase in velocity V. Nonetheless, experience shows that the assumption of constant velocity is a useful one in many situations.”¹
The admitted difference between this standard economic analysis and forecasting done in the EoE cryptoasset valuation model is that the former has pretty high confidence around predominant levels of all the variables, while the latter involves substantially more uncertainty.
It’s been noted in several places, including one of our recent articles, that empirically observed velocities of cryptoassets are quite low, particularly when accounting for “self churn” transactions — often lower than fiat currency velocity, despite arguments to the contrary. Nevertheless, we are still in the extreme early stages of crypto adoption and infrastructure, so current empirical estimates have limited utility in informing our posterior expectations.
We’ve also argued before from first principles that there are many reasons to believe that velocity will not become arbitrarily high but instead will stick in the 1 to 100 range. While I believe there is little empirical or logical evidence for another result, such arguments are inherently informal.
The natural question that arises is: Are there any more rigorous models that can give us more insight into realistic values of velocity going forward?
In this article, we’ll explore some common economic models for demand for money (which directly determines velocity), empirical evidence for these models in real economies, and implications for crytpoasset valuations.
Transaction Models of Monetary Demand
Velocity ultimately is determined by the demand for money. Demand for money is a well-studied topic in economics, with a variety of proposed approaches. The most directly relevant class of models for the purposes of modeling cryptoasset velocities are inventory models for transactions demand.
The Baumol-Tobin Model
The most commonly used inventory model is the Baumol-Tobin model. Alex Evans, among others, has previously proposed the Baumol-Tobin model as a means of estimating velocity.
The idea is that individuals who know they are going to spend Y dollars over a period will optimize the amounts held in their interest-bearing savings account and in cash.² The money in the savings account earns interest i, while each withdrawal costs C (with the biggest component of cost typically being the value of someone’s time). Given this parameterization, we can use calculus to solve for the optimal number of withdrawals, N.
The end result of the Baumol-Tobin model is the average money holdings (adjusted for price level):
H = M/P = (C* Y / (2 * i))^(½)
The logic around an individual decision maker can be generalized to the entire economy, assuming uniform behavior (more on this in a bit), so the above equation also applies generally to a token GDP of Y = P*Q.
Given the average money holdings, we can easily solve for velocity by dividing Y by H.
There are a couple of issues with using Baumol-Tobin and other models. First, it recasts a single variable (V) in the M*V = P*Q model into multiple variables (C, i) for which we don’t necessarily have higher estimate precision. Second, the meta-assumptions of the Baumol-Tobin model — and any other monetary demand model, for that matter — end up being quite large and impactful in any conclusions. Of particular consequence is the income elasticity of demand, as noted by Greg Mankiw:
“The Baumol–Tobin model, for example, makes precise predictions for how income and interest rates influence money demand. The model’s square-root formula implies that the income elasticity of money demand is 1/2: a 10-percent increase in income should lead to a 5-percent increase in the demand for real balances. It also says that the interest elasticity of money demand is 1/2: a 10-percent increase in the interest rate (say, from 10 percent to 11 percent) should lead to a 5-percent decrease in the demand for real balances. Most empirical studies of money demand do not confirm these predictions. They find that the income elasticity of money demand is larger than 1/2 and that the interest elasticity is smaller than 1/2. Thus, although the Baumol–Tobin model may capture part of the story behind the money demand function, it is not completely correct.”³
The effect of these meta-assumptions is that Baumol-Tobin can give some crazy velocity numbers in a cryptoasset valuation model. In Alex Evans’s model, for example, velocity ranges from 105 to 37,962!
Income Elasticity of Money Demand
What exactly is going on here? It turns out that income elasticity of money demand is a relatively poorly understood economic variable, and model-indicated values often differ substantially from empirical evidence in mature economies. Ultimately, it is this discrepancy that limits the value of these models — we are trying to develop better precision around velocity, but we’re no better off if the model we introduce has even less certainty!
Understanding the implications of income elasticity of money demand requires a bit of math. Elasticities measure the percentage change in a dependent variable Y for a unit percentage change in another variable X. This allows us to measure the unitless sensitivity of an output to its inputs.
Mathematically, elasticity is the partial derivative of the logarithmic variables:
With any exponential function Y = X^b, similar to the Baumol-Tobin money demand equation referenced earlier, elasticity will be equal to the exponent b. Solving the above equation:
Transforming the Baumol-Tobin equation H = (C*Y / 2*i)^(½) into logarithmic terms gives
log(H) = (½)log(Y) + (½)log(C) — (½)log(2i)
The income elasticity of money demand in the Baumol-Tobin model is given by the coefficient of log(Y), and therefore the model implies an income elasticity of ½. Other variants of the inventory model (e.g. Miller & Orr 1966) suggest an income elasticity range between ⅓ and ⅔, yet empirical observations of income elasticity of money demand indicate much higher levels (see chart below, A Meta-Analysis of Money Demand).
We can therefore conclude a reasonable range for income elasticity is ⅓ on the low end (given by analytical models) and 1.34 on the high end (given by empirical observations for the EU). This seems to be a relatively small range, but, as we will see in the next section, even small differences in assumed income elasticity lead to massively different estimates of velocity.
We can adjust the Baumol-Tobin equation to reflect this range in income elasticity by exponentiating Y by the desired elasticity ε rather than the constant ½.
Velocity under different inventory model assumptions
The following table shows velocities implied by the range of model-derived and empirically observed income elasticities of money demand.
As we can see, velocity is extremely sensitive to the income elasticity of demand. While the model-derived elasticities suggest velocities that are very high, empirically observed elasticities suggest velocities that are extremely low. This variance in velocities directly corresponds to high variance in modelled cryptoasset value since value is inversely proportional to velocity.
We can replicate this analysis for other input variables in the Baumol-Tobin model, such as the cost of conversion from reference asset to cryptoasset, C.
What causes these outsized changes in velocity for small changes in income elasticity? Elasticity manifests itself as an exponent in money demand models, which means it is extremely sensitive to small perturbations in elasticity.
Incidentally, the velocity of the U.S. M1 Money Supply has hovered between 5 and 10, which is roughly in line with the implied velocities corresponding to the USA Empirical income elasticity of 0.84 in the sensitivity tables above.
Conclusions
Due to the large uncertainty around what level of income elasticity is appropriate to use (let alone other assumptions like conversion costs and interest elasticity) and the extreme sensitivity of the model to this assumption, it is quickly apparent that money demand models are very limited in their ability to provide additional insights into velocity. In addition to the sensitivity, these models are further limited due to the fact they replace one uncertain variable (velocity) with multiple, equally or more uncertain variables.
At best, they can providing cross-validation and a sanity check of prior assumptions around velocity.
[1] N. Gregory Mankiw, Macroeconomics, 2010, Section 4–2.
[2] The model logic can easily be extended to two return-generating alternative assets.
[3] Mankiw, Section 19–2.
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