A power law is not a theory of Bitcoin price

Price scaling or distribution tail impacts?

Bitcoin's historical price scaling is difficult to miss, or dismiss. That is a useful place to begin because I was skeptical when I wrote about the Bitcoin power law in early 2025.

I still think the strong Bitcoin power law claims go well beyond the evidence but I would now state part of my earlier statistical criticism differently. The historical price relationship itself is real, persistent, and interesting. The harder question is what, economically, we are entitled to infer from it.

My earlier discussion leaned heavily on the Clauset-Shalizi-Newman methodology for identifying power law distributions. That methodology asks whether the tail of a probability distribution follows a power law. In their recent Nonlinear Science article, Giovanni Santostasi and Stephen Perrenod are asking a different question: whether Bitcoin's price scales with elapsed time according to a relationship of the form

P(t) = Ct^β,

where P is price at time t, C is a scale coefficient, and β is a price coefficient.

A relationship of this form can legitimately be described as power law scaling. Economists would recognize it as a power function or as a constant-elasticity relationship once both variables are expressed in logarithms.

Bitcoin's historical price can clearly be described by a power function; the question is whether that empirical relationship has been elevated into something it does not yet support – a law of Bitcoin price formation. Santostasi and Perrenod's article makes the strongest version of that case I have seen and it deserves to be taken seriously because it moves the debate beyond drawing a line through a log-log chart, toward a proposed economic mechanism.

What Santostasi and Perrenod actually show

Santostasi and Perrenod analyze 5,696 daily observations from July 2010 through February 2026. They estimate that Bitcoin's USD price has followed approximately

P(t) ∝ t^5.69,

with an R-squared of 0.961. Across roughly six orders of magnitude in price, a simple regression of log price on log time therefore captures most of the historical variation. That is a striking empirical result.

They then ask where the estimated 5.69 price-time exponent comes from. They first estimate that the number of Bitcoin addresses holding a non-zero balance grows approximately as

N(t) ∝ t^3.046.

They next estimate that price scales with the non-zero-balance address count as

P(N) ∝ N^1.838.

Combining those two relationships implies that price should grow approximately as time raised to the product of the two exponents:

3.046×1.838=5.60.

The implied exponent is close to the directly estimated price-time exponent of 5.69, an interesting correspondence.

Their empirical case is broader than this single calculation. The paper includes rolling-window tests, residual analysis, direct scale-invariance tests, cross-asset comparisons, and sequential Bayesian analysis.

Note, however, that their Bayesian argument against structural breaks is particularly weak because posterior variance shrinks approximately n^(-0.5) by construction under their fixed-variance conjugate model, not because the data independently demonstrate an absence of structural change.

So, it may not be fair to dismiss the Bitcoin power law as little more than log-log curve fitting. There is a persistent regularity here that needs explaining.

My disagreement begins with the interpretation. Santostasi and Perrenod argue that the address growth exponent reflects a process analogous to epidemic spreading through a heterogeneous network, while the price-address exponent reflects generalized Metcalfe scaling. The two relationships are then combined as a mechanistic derivation of Bitcoin's long-run price power law. I think that conclusion asks the empirical results to carry far more weight than they can currently bear.

Price formation still needs a supply side

There is nothing wrong with estimating a historical relationship between Bitcoin price and time without putting supply into the regression. If the question is simply whether price has scaled with time, supply does not need to appear explicitly. The standard changes once the claim becomes mechanistic – an explanation of why price follows that path – because price is the variable that clears a market.

For a conventional commodity, rising demand can be absorbed partly through quantities. Higher prices bring marginal mines into production, induce more planting or construction, draw down inventories, and encourage substitution. Bitcoin does not have that adjustment margin in the same way. Its protocol controls issuance and miners cannot collectively respond to a higher price by creating Bitcoin faster than the protocol permits over the long run. Bitcoin's block reward schedule and difficulty adjustment prevents a conventional commodity supply response.

Bitcoin's highly inelastic protocol supply is thus economically important because it cannot respond much to price. When demand changes, a larger share of the adjustment has to occur through price fluctuation. Any claimed law of Bitcoin price formation that assigns the entire long-run mechanism to network growth is incomplete unless it explains how this supply constraint enters the market-clearing process.

Supply is harder than the 21-million cap

Protocol supply - 21 million Bitcoin - is only the starting point. Knowing how many Bitcoin exist now and how new issuance will evolve does not tell us how much additional demand the market can absorb at the current price before marginal sellers or counterparties require a higher one. The supply relevant to price formation depends on holder willingness to sell and on the institutions through which Bitcoin exposure is created, financed, hedged, and settled.

Derivatives and rehypothecation increase the challenge. Derivatives change the mapping between demand for Bitcoin price exposure and immediate demand for spot bitcoin; the resulting spot effect depends on hedging, collateral, settlement and arbitrage arrangements.

I now think that liquid supply is too simple a term if it is interpreted as a directly measurable stock of coins. A more useful concept is market-clearing availability: the amount of additional demand that the combined spot, derivatives, lending, custody, and holder system can absorb at the current price before the marginal price has to move materially. That state is behavioral and institutional as well as on-chain, and there is no comprehensive global ledger from which we can observe it directly.

The measurement problem argues for caution rather than omission. If the relevant supply state cannot be observed cleanly, we should be careful about how precisely we model it. We should be much more careful about claiming that a model which leaves supplyh out has discovered a law of price formation.

Addresses are not adoption

The paper's other major vulnerability is its adoption variable. Santostasi and Perrenod use the number of Bitcoin addresses with a non-zero balance and acknowledge the familiar limitations: one person can control many addresses; abandoned addresses can retain dust balances; and custodial concentration can cause address counts to diverge from the number of actual participants.

I think this problem is deeper than ordinary proxy error because Bitcoin's architecture is changing the relationship between base-layer addresses and economic participation over time. The traditional reasons carry weight: a self-custody user can control many addresses; a custodian can represent many users through relatively few addresses; ETF investors can gain Bitcoin exposure without creating a Bitcoin address; and Lightning or other Layer 2 systems can support repeated economic activity without placing each payment on the base chain.

Non-zero-balance addresses do not directly measure economic adoption, and the direction and size of the measurement error can change as Bitcoin matures. That is structural measurement drift.

The safest interpretation of the estimated 3.046 coefficient is that it is an address growth exponent. Calling it an adoption exponent requires another empirical step: evidence that the changing address series remains a stable proxy for the changing number, intensity, and economic significance of Bitcoin participants.

Santostasi and Perrenod effectively acknowledge this in their limitations section when they note that direct measurement of the Bitcoin adoption network's degree distribution would be needed to establish the epidemic-spreading interpretation rigorously. I read that as a substantive limitation because it concerns the first half of the proposed mechanistic derivation, and is not simply a measurement footnote.

When should the power law hold?

Rather than arguing over whether the power law model is simply right or wrong, I think the more productive question is:

Under what economic conditions would the Santostasi-Perrenod price relationship emerge from a supply-and-demand framework?

Dennis Porter and I published two articles in the Journal of Risk and Financial Management (https://doi.org/10.3390/jrfm18020066; https://doi.org/10.3390/jrfm18100570) on such a Bitcoin supply and demand framework. They both examine potential Bitcoin price trajectory formation from first-principles supply and demand. The framework (rather than a Bitcoin price model) begins with market clearing and allows flexible assumptions regarding demand growth, demand elasticity, available supply, withdrawals, adoption, time preference, and uncertainty. Different assumptions generate different price paths, which is intentional: the framework is meant to identify the economic relationships that matter and allow empirical assumptions and models to evolve and improve as better evidence becomes available.

A Santostasi-Perrenod-style power law thus appears as a special case under particular restrictions in that framework rather than as a primitive law imposed from outside the market-clearing model.

Deriving the power law as a special case

This section is equation-heavy: nontechnical readers can skip this section without losing the main argument. The mathematical point is that an exact power law price path can arise from conventional market clearing once demand, adoption, supply availability, and price responsiveness are given particular functional forms.

Apologies for the crude math script (there are formatting limits in the Ghost platform).

Start with a simple constant-elasticity demand relationship:

Q_D (t) = A(t)P(t)^(-η).

Here, Q_D is the quantity of Bitcoin demanded, P is price, A is a demand shifter that captures changing willingness to hold Bitcoin, and η tells us how responsive buyers are to price.

For the simplest of examples, represent market-clearing availability with quantity S_L (t). This should not be read as a claim that "true liquid supply" can actually be observed - it is a compact analytical stand-in for the supply side of the market.

At equilibrium,

Q_D (t) = S_L (t).

This simplified derivation treats market-clearing availability as given at each point in time. A richer model would allow supply availability itself to respond to price, without changing the central point that a fixed price-time exponent requires restrictions on the underlying market-clearing process.

Solving for price in the basic relationship gives

P(t) = [A(t) / (S_L (t) )]^(1/η).

Now suppose economic adoption itself follows a power function:

N(t) = N_0 t^a.

Suppose adoption shifts aggregate demand according to

A(t) = A_0 N(t)^δ.

and suppose the simplified supply term follows

S_L (t) = S_0 t^s.

Substitution gives

P(t)∝ t^((aδ-s)/η).

so the price exponent, β, can be calculated as

β = (aδ-s)/η.

This is the central analytical result: a Bitcoin price power law is completely compatible with ordinary supply-and-demand economics but the observed price exponent can contain contributions from adoption-driven demand, supply conditions, and price elasticity at the same time. The price-time relationship alone cannot tell us that the whole exponent comes from network effects.

The Metcalfe exponent is likely a composite

The same identification problem applies to the paper's estimate

P(N) ∝ N^1.838.

Santostasi and Perrenod interpret 1.838 as a generalized Metcalfe exponent – the increase in price associated with growth in the network. However, once price is treated as a market-clearing variable, the elasticity of price with respect to adoption can reflect more than the demand effect of adoption alone.

What's inside 1.838?

Under the simplified equilibrium model, suppose

A(N) ∝ N^δ.

Let s_N denote the elasticity of market-clearing availability with respect to adoption:

s_N = (dlnS_L)/dlnN.

Then

dlnP/dlnN = (δ-s_N)/η.

In plain English, the observed price response to adoption combines three possible effects: (1) adoption can shift demand; (2) market-clearing availability can itself change as adoption rises; and (3) marginal buyers can differ in their price sensitivity. If Bitcoin becomes progressively harder to acquire as adoption expands, part of the estimated 1.838 coefficient can reflect scarcity rather than network value alone.

The regression can still be statistically correct - the problem is interpretive. The value 1.838 may be a reduced-form composite parameter rather than a clean estimate of a single network mechanism.

This decomposition does not falsify the empirical Metcalfe-type relationship. It shows that the observed coefficient does not uniquely identify Metcalfe network value as its economic source.

Stable scaling can hide changing economics

An apparently stable price exponent does not require the underlying economics to remain stable. Adoption-driven demand growth could slow as Bitcoin matures while market-clearing availability tightens as more Bitcoin moves into long-term holdings, institutional vehicles, corporate treasuries, strategic reserves, or other hands with a low willingness to sell. The first force pushes the effective growth rate down; the second pushes it up.

If they offset, the fitted exponent can remain close to 5.7 even while the economic mechanism underneath it changes materially.

Under that interpretation, the power law is an emergent reduced-form relationship rather than a structural constant - a "law." Temporal stability in the fitted coefficient is evidence about the net outcome of several forces, not necessarily evidence that each force has remained unchanged.

The effective exponent

Define the effective price exponent at any point in time as

β_eff (t) = dlnP/dlnt.

Under the simplified model and constant demand elasticity,

β_eff (t) = (g_A (t) - g_S (t))/η,

where gA(t) is the local elasticity of the demand shifter with respect to elapsed time, and g_S(t) is the corresponding elasticity of market-clearing availability.
A constant observed β_eff does not require either component to be constant.

Demand growth can weaken while scarcity intensifies, leaving their net contribution to the observed price exponent almost unchanged. A stable fitted exponent therefore does not, by itself, establish a fixed underlying law.

By now, you can probably see that the apparent stability of the power law coefficients requires several moving parts to keep balancing in a fairly particular way.

Why I suspect an eventual break could be upward

This section is my speculative hypothesis - to be developed further in the future - rather than a result.

If the historical power law relationship eventually fails because of Bitcoin's own market dynamics rather than a destructive external shock, my current belief is that an upside break will more likely than a permanent downside break. The reason is supply.

Suppose adoption and capital inflows continue while market-clearing availability becomes progressively tighter. If more Bitcoin is held by entities with little willingness to sell, new demand has to move price further to locate marginal sellers. In that environment a fixed historical exponent can become too shallow because the market does not know or care where a fitted power law corridor says Bitcoin 'should' trade. It clears where willing buyers meet willing sellers and counterparties.

My supply-demand work has already shown that highly nonlinear and, under restrictive assumptions, hyperbolic price paths can emerge when modeled available supply becomes small (< in the 1-2 million Bitcoin float range) and behavioral mechanisms fail to release enough supply back to the market. Those outcomes are relatively uncommon in our later work and depend on demanding assumptions. The mechanism is nevertheless economically coherent: if substantially more price movement is required to release supply, the effective price exponent rises.

A break to the upside - and more volatility, not less as the power law implies - comprise a family of outcomes; power law appreciation and diminishing returns comprise another.

A downside break has equally coherent mechanisms. Adoption can saturate, marginal buyers can become more price sensitive, or long-term holders can return substantial supply to the market. Regulation, technological substitution, or a severe security failure could also weaken demand.

Quantum computing is an extreme case because a genuine cryptographic break could reduce confidence sharply while also making some previously dormant Bitcoin vulnerable to movement, affecting supply and demand at the same time (see Section 4.4 of our working paper for more on that - our quantum hack example was removed in the final article due to space constraints). The movement of old coins would then be secondary to the reason they became movable: a one-time supply shock can be absorbed more readily than a collapse in confidence in Bitcoin's security.

The historical power law does not have to break upward. Once price sits inside a market-clearing framework, however, either direction of break has an economic explanation, and those explanations give us something concrete to monitor.

Weak structure can still produce strong forecasts

A May 2026 preprint by Carlos Baquero and Raquel Menezes adds an important piece to the discussion. Their working paper, Bitcoin's Power Law: Weak Structure, Strong Forecasts, is skeptical of a strong structural interpretation. They report that the fitted time exponent changes materially when the assumed time origin shifts, that flexible multi-component sigmoid models can fit Bitcoin's history better, and that some proposed scale-invariance diagnostics do not uniquely distinguish a power law from alternative historical processes.

Their forecasting results cut in a different direction. In walk-forward comparisons, the simple power law performs poorly against naive models at short horizons but does remarkably well at longer horizons; at 12- to 24-month horizons it beats the standard forecasting baselines they test. That is useful evidence in favor of the model as a forecasting benchmark even if we remain skeptical of its structural interpretation.

I think the debate becomes much clearer if we keep three questions separate:

1. Does a model describe the historical data well;
2. Does it forecast future prices well; and
3. Does it correctly identify the mechanism producing those prices?

A model can succeed at the second without succeeding at the third. Economists encounter this problem constantly. A time series model, for instance, can explain every historical price cycle and then forecast badly because the detailed historical pattern does not repeat. A simpler model can be biased as a structural description but robust as a long-horizon forecasting rule. Baquero and Menezes give us good reason to take that possibility seriously for Bitcoin.

What investors should – and should not – do with the power law

I do not think investors should throw away the Bitcoin power law (unlike my opinion about the stock-to-flow model). A simple model that has tracked Bitcoin's long-run trajectory for more than fifteen years and appears to perform well in long-horizon pseudo-out-of-sample forecasts deserves to be monitored. It can be a useful benchmark.

A benchmark should not be confused with destiny. The historical relationship does not give us a guaranteed long-run price floor, an invariant compound annual growth rate, proof that large deviations must mean-revert, or a sound basis for leverage on the assumption that the future path is constrained. Those conclusions require confidence in the underlying mechanism, not simply confidence in historical fit.

The more useful investor question is:

Are the economic conditions that produced the historical scaling relationship still present?

That shifts attention away from the price corridor alone.

I would watch whether economic adoption is still growing at roughly the rate implied by the model, whether base-layer addresses are drifting further away from actual participation, whether institutional custody and financial intermediation are changing market-clearing availability, and whether leverage or derivatives are absorbing demand that would otherwise reach the spot market. I would also watch holder willingness-to-sell and the effective price exponent itself for persistent change.

A fixed price-time equation cannot answer those questions. A first-principles supply and demand framework does not answer them automatically either, but it does tell us why they matter and what kinds of evidence would change the price dynamics.

Where I now stand

My position on the Bitcoin power law debate has changed somewhat but not in the direction of accepting a law of price formation. I would now separate three claims.

1. Bitcoin's historical price exhibits an exceptionally persistent scaling relationship with time and the empirical evidence for power law scaling is strong;
2. The relationship may also be useful for long-horizon forecasting - Baquero and Menezes (2026) provide evidence that deserves attention; and
3. Neither result establishes that Bitcoin's price trajectory is generated by a fixed network law with an invariant exponent.

Santostasi and Perrenod have provided a more serious and testable explanation than earlier versions of Bitcoin power law advocacy but I still think the mechanism is incomplete.

1. The proposed adoption variable measures non-zero-balance addresses rather than economic adoption;
2. The generalized Metcalfe interpretation does not separately identify the contribution of supply to price; and
3. A stable fitted exponent can emerge from economic mechanisms that change over time but offset one another.

I thus find it more useful to treat the Bitcoin power law as an empirical property of a market regime. A conventional supply-and-demand framework can generate exactly such a relationship when demand growth, market-clearing availability, and price sensitivity combine in sufficiently stable proportions. On that reading, the power law is not outside economics: it is nested inside it.

The research question I would now put at the center of the debate is:

What combination of adoption, demand, scarcity, financial intermediation, and holder behavior has produced the observed scaling relationship – and what evidence would tell us that those conditions are changing?

A power law can be part of a theory of Bitcoin price. It is not the theory itself.

References

Baquero, C., and R. Menezes. 2026. Bitcoin's Power Law: weak structure, strong forecasts. arXiv:2605.21316v1. https://arxiv.org/abs/2605.21316

Clauset, A., C. R. Shalizi, and M. E. J. Newman. 2009. Power-Law distributions in empirical data. SIAM Review 51(4): 661-703. https://doi.org/10.48550/arXiv.0706.1062

Rudd, M. A., and D. Porter. 2025. A supply and demand framework for Bitcoin price forecasting. Journal of Risk and Financial Management 18(2): 66. https://doi.org/10.3390/jrfm18020066

Rudd, M. A., and D. Porter. 2025. Bitcoin supply, demand, and price dynamics. Journal of Risk and Financial Management 18(10): 570. https://doi.org/10.3390/jrfm18100570

Santostasi, G., and S. Perrenod. 2026. A mechanistic derivation of the Bitcoin price Power Law: network adoption dynamics and generalised Metcalfe scaling. Nonlinear Science 8: 100172. https://doi.org/10.1016/j.nls.2026.100172

I run detailed AI-assisted summaries for most research papers I may use. You can view my research review notes here: Baquero, C., and R. Menezes. 2026; and Santostasi, G., and S. Perrenod. 2026. Access requires a subscription (free).