A Pretty Curve Is Not an Economic Theory
Why BTC “power law” charts prove much less than their believers think
Keywords: BTC, Bitcoin power law, log regression, R-squared, econometrics, speculative bubbles, Ponzi schemes, tulip mania, housing bubble, spurious regression, power-law distributions, financial markets
The BTC power-law argument usually arrives dressed as mathematics and behaves like theology.
The claim is familiar: BTC price, when plotted on a log-log chart against time, appears to follow a clean line. The faithful then announce that this is not merely a historical pattern but a discovery of destiny. Price has followed the curve; therefore price will continue to follow the curve. The chart becomes prophecy. The regression becomes scripture. The R² becomes a rosary bead.
This is bad economics.
It is also bad econometrics.
A curve fitted to historical prices is not a theory of value. A high R² is not evidence of causality. A log regression against time is not a monetary model. And a power-law-looking chart is not remotely unique to BTC. Product adoption curves, infrastructure buildouts, speculative bubbles, and even Ponzi account statements can all produce extremely smooth logged relationships over finite intervals.
The error is not subtle. It is elementary.
1. What the BTC power-law model actually is
The basic BTC power-law model is usually expressed as a regression of logged price on logged time.
In ordinary web-readable notation:
ln(Pt) = α + β ln(t) + εt
where:
Pt = BTC price at time t
t = elapsed time since some chosen starting point
α = intercept
β = slope or power-law exponent
εt = residual error term
Back-transforming gives:
Pt = exp(α) × t^β × exp(εt)
If residuals are ignored, the fitted curve becomes:
Pt = A × t^β
where:
A = exp(α)
That is the entire trick.
It is not magic. It is a linear regression after taking logarithms.
The model says that price is a function of time. Not transaction demand. Not settlement volume. Not merchant use. Not velocity. Not real output. Not discounted cash flow. Not productivity. Not legal integration. Not liquidity depth. Not marginal utility.
Time.
That is why the model is economically barren. It does not explain why value arises. It merely says that, historically, price moved in a way that can be approximated by a curve.
That is chronology pretending to be economics.
2. What R² actually measures
R², or the coefficient of determination, is an in-sample goodness-of-fit statistic. It measures the proportion of variance in the dependent variable explained by the fitted model.
For an ordinary least squares regression:
R² = 1 − [Σ(yi − ŷi)² / Σ(yi − ȳ)²]
where:
yi = observed value
ŷi = fitted value
ȳ = mean observed value
Σ(yi − ŷi)² = residual sum of squares
Σ(yi − ȳ)² = total sum of squares
If the regression is run on logged price, then R² measures the proportion of variation in ln(price) explained by the model. It does not measure the proportion of actual price explained in ordinary dollar terms.
This matters.
A high R² means that the fitted line tracks the historical transformed data well. It does not mean that the model is causal. It does not mean the model is stable. It does not mean the model predicts out of sample. It does not mean the model has economic content.
A regression of one trending variable against another trending variable can produce a high R² even when the relationship is economically meaningless. Granger and Newbold (1974) made this point half a century ago in their work on spurious regressions. Phillips (1986) later formalised why non-stationary time series can generate impressive-looking nonsense.
So when someone points at a BTC power-law chart and says, “Look at the R²,” the correct response is not awe.
The correct response is: “What is the mechanism, and has it survived out-of-sample testing without parameter fiddling?”
3. Log regression is not one thing
The phrase “log regression” is often used sloppily. There are several related but distinct models.
Log-linear model
ln(y) = α + βt + ε
Back-transforming:
y = exp(α) × exp(βt) × exp(ε)
This is an exponential growth model. It is common in finance, demography, diffusion, inflation, and bubble dynamics.
Log-log model, or power-law regression
ln(y) = α + β ln(t) + ε
Back-transforming:
y = exp(α) × t^β × exp(ε)
This is the form usually invoked by BTC power-law advocates.
Log-periodic power-law model
Bubble researchers often use more complicated models. A simplified log-periodic power-law form is:
ln(p(t)) = A + B(tc − t)^m + C(tc − t)^m cos[ω ln(tc − t) − φ] + εt
where:
tc = critical time
m = power-law exponent
ω = angular log frequency
φ = phase parameter
This type of model is associated with Johansen, Ledoit, and Sornette (2000), who studied crashes as critical points. Whether one accepts that literature in full or not, the point is simple: serious bubble modelling is not the same as drawing a line through a log-log price chart and declaring victory.
4. Power-law distributions are not the same thing as BTC price over time
There is another confusion.
A power-law distribution concerns the probability distribution of a variable, often written:
Pr(X ≥ x) ∝ x^−α
or, in density form:
p(x) = Cx^−α
That is not the same as saying “BTC price follows a power law in time.”
Clauset, Shalizi, and Newman (2009) showed that power-law claims in empirical data require careful testing. Visual straight lines on log-log plots are not sufficient. Least-squares fits on log-log plots can produce misleading parameter estimates. Proper testing requires maximum likelihood estimation, goodness-of-fit tests, and comparison against alternatives such as lognormal, exponential, stretched exponential, and truncated power-law distributions.
BTC price-time charts usually do not meet that standard. They are not testing whether a distribution follows a power law. They are fitting price against elapsed time.
That is a much weaker claim.
It is also much easier to fake with any sufficiently strong historical trend.
5. The calculation method
For the simple power-law examples below, the model used is:
ln(y) = α + β ln(t) + ε
The calculation proceeds as follows:-
Select a positive-valued time series. Logs cannot be taken of zero or negative values.
-
Define t as 1, 2, 3, …, n from the first observation in the sample.
-
Compute ln(y).
-
Compute ln(t).
-
Estimate α and β by ordinary least squares.
-
Compute fitted values:
ln(ŷ) = α + β ln(t)
-
Compute R²:
R² = 1 − [Σ(ln(y) − ln(ŷ))² / Σ(ln(y) − mean(ln(y)))²]
This R² is calculated in logged space. It tells us how much of the variance in ln(y) is explained by ln(t). It does not tell us whether the relationship is causal.
For the housing-bubble example, I also use the log-linear form:
ln(y) = α + βt + ε
That is appropriate because asset bubbles often resemble exponential or super-exponential growth over the inflation phase. Forcing every bubble into the BTC log-log format would be bad modelling. The point is not that every bubble must use the exact BTC formula. The point is that high in-sample logged curve fits are common and therefore do not prove much.
6. Data sources used
The data examples use public economic series.
Internet adoption
Series: Internet users for the World
Source: World Bank, World Development Indicators, accessed through FRED
Series ID: ITNETUSERP2WLD
Unit: internet users per 100 people
Frequency: annual
Sample used here: 1995–2023
The sample begins in 1995 because global internet usage before that is extremely close to zero. Positive values are required for logarithms.
Mobile-phone adoption
Series: Mobile Cellular Subscriptions in World
Source: World Bank, World Development Indicators, accessed through FRED
Series ID: ITCELSETSP2WLD
Unit: subscriptions per 100 people
Frequency: annual
Sample used here: 2005–2025
The early mobile series contains zeros and missing values. The sample used here begins after the series becomes positive and consistently reported.
U.S. railroad expansion
Series: Miles of Railroad Built for United States
Source: National Bureau of Economic Research Macrohistory Database, accessed through FRED
Series ID: A02F2AUSA374NNBR
Unit: miles built per year
Frequency: annual
Sample used here: 1830–1890
For the regression, I convert annual miles built into cumulative miles built from 1830 onward:
Cumulative miles in year T = Σ annual miles built from 1830 through T
That cumulative stock is what corresponds to network buildout.
U.S. housing bubble
Series: S&P Cotality Case-Shiller U.S. National Home Price Index
Source: S&P Dow Jones Indices, accessed through FRED
Series ID: CSUSHPINSA
Unit: index, January 2000 = 100
Frequency: monthly
Sample used here: January 1997–December 2006
This captures the inflation phase and peak of the U.S. housing bubble.
BTC comparison
BTC power-law advocates commonly report R² values around 0.95 to 0.96 for price regressed against time on log-log axes. Those are their claims, not mine. The point here is that such R² values are not unique, not economically decisive, and not sufficient to prove causality.
7. Fitted examples
Using the model:
ln(y) = α + β ln(t)
the following fits result.
World internet adoption, 1995–2023
ln(y) = −0.6857 + 1.4429 ln(t)
R² = 0.9955
That is higher than the usual BTC power-law claim.
It does not mean the internet is metaphysically destined to expand forever. It means that, over this finite interval, internet adoption followed a very smooth diffusion path.
The economic explanation is straightforward: falling access costs, infrastructure rollout, network effects, rising household and business demand, and increasing digital utility.
The regression describes the pattern. It does not create the pattern.
Cumulative U.S. railroad miles built, 1830–1890
ln(y) = 3.0039 + 2.1097 ln(t)
R² = 0.9827
Again, this is an extremely strong log-log fit.
The economic mechanism is also obvious: railway expansion created transport capacity, lowered freight costs, connected markets, supported westward expansion, attracted capital, and produced measurable demand for passenger and freight services.
The curve reflects an infrastructure network being built.
It is not a law of nature. It is capital formation, geography, industrialisation, and network economics.
Global mobile-phone adoption, 2005–2025
ln(y) = 3.5393 + 0.4126 ln(t)
R² = 0.9697
This also beats or matches many BTC power-law claims.
Again, nobody sensible says mobile subscriptions must rise forever because a curve fit says so. Mobile adoption eventually saturates. Subscriptions per 100 people can exceed 100 because individuals may hold multiple subscriptions, but even that process has limits.
The curve is a finite-period adoption pattern.
U.S. housing bubble, 1997–2006
For the housing bubble, the log-linear model is more appropriate:
ln(y) = α + βt + ε
Fitted result:
ln(y) = 4.3555 + 0.007293t
R² = 0.9866
That is the point.
A bubble can produce a beautiful logged fit.
U.S. housing prices in the bubble years rose in a way that looks clean in logged form. That did not make the housing bubble permanent. It did not make the fitted trend a law. It did not prevent collapse. It did not mean the regression had discovered the future.
It meant that credit expansion, loose lending, securitisation, leverage, refinancing incentives, speculative demand, and narrative economics produced a smooth historical rise until the mechanism failed.
Beautiful fit. Terrible inference.
8. Ponzi schemes can produce perfect log fits
A Ponzi scheme can produce an even better curve than BTC.
Suppose a fraudulent operator reports a constant return r each period. The reported account balance evolves as:
A(t) = A0(1 + r)^t
Taking logs:
ln(A(t)) = ln(A0) + t ln(1 + r)
That is a perfect straight line in logged account value against time.
R² = 1.0000
By construction.
That does not mean the scheme is economically productive. It means the reported balances were manufactured according to a compounding formula.
Madoff is the obvious modern example. The SEC’s own post-mortem described years of warnings about unusually consistent returns and secretive operations. The very smoothness of the reported returns was not evidence of genius. It was a red flag.
This is the simplest way to expose the childishness of chart-worship.
A fraudulent return stream can produce a better fit than BTC.
Therefore fit is not proof of legitimacy.
9. Tulip mania and the problem of sparse data
Tulip mania is harder to treat as a clean numerical series because the surviving seventeenth-century price data are sparse, contract-specific, and institutionally unusual. Garber (1989) compiled individual bulb-price series and argued that much of the rare-bulb price behaviour was explicable by fundamentals of bulb propagation and rarity. Thompson (2007) later argued that the legal conversion of forward contracts into option-like claims helps explain the extreme contract pricing.
That dispute is important.
It means one should not lazily wave at “tulips” without understanding the contract structure.
But it also strengthens the argument here. The tulip episode shows that historical price curves can look spectacular while the underlying institutional explanation is disputed, conditional, and dependent on market microstructure.
A curve alone is not enough.
You need the legal form of the contract. You need the delivery mechanism. You need the enforcement rule. You need the market participants. You need the cash-settlement logic. You need the option-like payoff. You need the asset’s production constraints.
In other words, you need economics.
Not a pretty chart.
10. Why all these curves can look good
Many economic and financial processes generate smooth logged relationships over limited windows.
Diffusion
Adoption often starts slowly, accelerates, and then saturates. Early and middle phases can appear exponential or power-law-like.
Examples:
Internet users
Mobile phones
Electricity adoption
Railway networks
Software platforms
Diffusion models such as the Bass model explain this through innovators, imitators, network effects, and changing adoption probabilities.
Infrastructure buildout
Network infrastructure can produce compounding gains for a period. Each new connection increases the value of adjacent connections. But expansion eventually meets cost, geography, regulation, saturation, and diminishing marginal returns.
Railways are the clean historical example.
Speculative bubbles
Bubbles can produce smooth logged price acceleration because price appreciation itself becomes the sales pitch. Rising prices attract new buyers. New buyers push prices higher. Higher prices validate the narrative. The process continues until liquidity, leverage, belief, or credit fails.
That is not adoption. It is reflexivity.
Ponzi finance
Ponzi schemes can report stable compounding returns because the returns are administratively generated, not economically earned. The accounting path can be made smoother than any real market process.
A perfect curve can be fraud.
That point should not be difficult.
11. Why BTC price is not adoption
BTC proponents often slide between different variables.
When price rises, they call it adoption.
When speculators buy, they call it use.
When custodians hold, they call it monetary success.
When ETFs gather assets, they call it network effect.
Those are not the same thing.
Price is not transaction demand.
Custody is not monetary circulation.
Speculation is not payment utility.
ETF exposure is not peer-to-peer electronic cash.
A monetary system would be evaluated by variables such as:
transaction volume for goods and services
merchant acceptance
unit-of-account usage
contract denomination
wage denomination
settlement use
fee stability
velocity
liquidity for actual commerce
legal and accounting integration
payment finality
cost per transaction
scalability under real demand
A BTC power-law chart uses none of these.
It models price as a function of time.
That is not a monetary theory.
It is a trend line.
12. The fatal problem: no independent economic variable
A serious economic model has explanatory content.
For example:
Demand for money depends on income, prices, interest rates, transaction needs, uncertainty, and institutional use.
A productive asset can be valued through expected cash flows, discount rates, risk premia, and growth assumptions.
A network service can be analysed through user adoption, marginal utility, congestion, switching costs, and complementary goods.
The BTC power-law chart contains none of this.
It says:
Price = f(time)
That is not enough.
Time does not cause value.
If a model contains no independent economic variable, it cannot distinguish between:
productive adoption
speculative mania
liquidity bubble
Ponzi accounting
market manipulation
monetary use
pure narrative momentum
All of them can produce attractive historical curves.
That is why the BTC curve proves almost nothing by itself.
13. The parameter problem
Power-law fits are sensitive to modelling choices.
Change the start date and t changes.
Change the price source and Pt changes.
Use daily closes, weekly averages, monthly averages, market cap, realised cap, or all-time lows, and the fitted line changes.
Include or exclude early illiquid data and the slope changes.
Use different exchange data and the residuals change.
Use price rather than transaction demand and the interpretation changes.
A robust structural law should not depend so heavily on arbitrary choices.
If the exponent β changes materially when the time origin is shifted, then the model is not a deep law. It is a fitted convention.
That is one reason Clauset, Shalizi, and Newman warned against casual power-law claims. Log-log straightness is easy to see and easy to overstate.
14. The out-of-sample test
The only serious question is not:
“Did the historical data fit?”
The serious question is:
“Did the model predict new data without being recalibrated?”
In-sample fit is cheap.
Out-of-sample performance is expensive.
If a BTC power-law model is estimated using data only up to 2014, it should forecast 2015–2020 without moving the goalposts.
If estimated up to 2018, it should forecast 2019–2024 without new bands, new exclusions, new caveats, or new cycle adjustments.
If every failure leads to redrawing the curve, widening the bands, shifting the origin, excluding outliers, adding halving corrections, or inventing a new auxiliary model, then the model is no longer a simple discovered law.
It is a story being protected from reality.
15. What the examples prove
The examples above prove a narrow but decisive point.
High R² log regressions are common.
Internet adoption can produce R² = 0.9955.
Cumulative railway buildout can produce R² = 0.9827.
Mobile-phone adoption can produce R² = 0.9697.
The U.S. housing bubble can produce R² = 0.9866 under a log-linear bubble-phase model.
A Ponzi account balance can produce R² = 1.0000 by construction.
Therefore BTC having an R² around 0.95–0.96 in a log-log price-time model is not extraordinary evidence of anything.
It is a fitted historical relationship.
That is all.
16. The economist’s conclusion
Economics is not curve admiration.
Economics asks what mechanism generates the observed data.
For railways, the mechanism was transport demand, freight revenue, capital formation, industrial expansion, and network infrastructure.
For internet adoption, the mechanism was communication utility, falling access costs, digitisation, and network effects.
For mobile phones, the mechanism was direct user utility, coverage expansion, cheaper devices, and network demand.
For housing bubbles, the mechanism was credit, leverage, collateral valuation, securitisation, and speculative belief.
For Ponzi schemes, the mechanism is fraudulent reporting and new money used to satisfy old claims.
For BTC power-law charts, the mechanism is usually not specified. The curve is presented as its own explanation.
That is the fraud of the argument.
Not criminal fraud in the legal sense. Intellectual fraud in the analytical sense.
The BTC power-law argument points at a historical price chart and pretends that a fitted curve is a theory of money. It is not. It is numerology with regression output.
A high R² can describe adoption.
A high R² can describe a bubble.
A high R² can describe a Ponzi.
A high R² can describe almost any smooth historical trend over a carefully chosen finite window.
The curve does not tell you which one you are looking at.
For that, you need economics.
And that is precisely what the BTC chart priests do not have.
References
Bass, F. M. (1969). A new product growth model for consumer durables. Management Science, 15(5), 215–227.
Clauset, A., Shalizi, C. R., & Newman, M. E. J. (2009). Power-law distributions in empirical data. SIAM Review, 51(4), 661–703.
Federal Reserve Bank of St. Louis. (2026). FRED economic data: Internet users for the World, ITNETUSERP2WLD. Source data from the World Bank, World Development Indicators.
Federal Reserve Bank of St. Louis. (2026). FRED economic data: Mobile Cellular Subscriptions in World, ITCELSETSP2WLD. Source data from the World Bank, World Development Indicators.
Federal Reserve Bank of St. Louis. (2026). FRED economic data: Miles of Railroad Built for United States, A02F2AUSA374NNBR. Source data from the National Bureau of Economic Research Macrohistory Database.
Federal Reserve Bank of St. Louis. (2026). FRED economic data: S&P Cotality Case-Shiller U.S. National Home Price Index, CSUSHPINSA. Source data from S&P Dow Jones Indices LLC.
Garber, P. M. (1989). Tulipmania. Journal of Political Economy, 97(3), 535–560.
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Kindleberger, C. P., & Aliber, R. Z. (2011). Manias, panics, and crashes: A history of financial crises (6th ed.). Palgrave Macmillan.
Minsky, H. P. (1986). Stabilizing an unstable economy. Yale University Press.
Phillips, P. C. B. (1986). Understanding spurious regressions in econometrics. Journal of Econometrics, 33(3), 311–340.
Securities and Exchange Commission, Office of Inspector General. (2009). Investigation of failure of the SEC to uncover Bernard Madoff’s Ponzi scheme: Executive summary.
Shiller, R. J. (2015). Irrational exuberance (3rd ed.). Princeton University Press.
Shiller, R. J. (2019). Narrative economics: How stories go viral and drive major economic events. Princeton University Press.
Thompson, E. A. (2007). The tulipmania: Fact or artifact? Public Choice, 130, 99–114.