The Principle of Economic Value Conservation: A Quantitative Analysis of Fiat-to-Bitcoin Dynamics

Abstract

This paper presents a theoretical framework for understanding the conservation of economic value in systems transitioning between fiat currency and Bitcoin. We propose that under idealized conditions, the total economic value of a closed system remains conserved, though its distribution across different forms of money evolves according to specific economic parameters.

Mathematical Formulation

In an isolated economic system where external interference is minimal, we observe a conservation principle that can be expressed as:

Where:

• E_f represents the effective economic energy stored in fiat currency
• E_B represents the effective economic energy stored in Bitcoin

These energies can be further decomposed into:

Where:

• m_f = Monetary mass of fiat (total units in circulation)
• v_f = Velocity of fiat (transaction frequency)
• α_f = Purchasing power coefficient of fiat (declining with inflation)
• m_B = Monetary mass of Bitcoin (capped at 21 million)
• v_B = Velocity of Bitcoin (adoption rate and transaction frequency)
• α_B = Purchasing power coefficient of Bitcoin (subject to market dynamics)

Theoretical Implications

1. Value Migration Under Monetary Debasement

When central banks engage in quantitative easing, increasing m_f without corresponding economic growth, α_f decreases proportionally. This creates a potential energy gradient that drives capital toward Bitcoin, increasing α_B and v_B.

2. Inelastic Collisions in Monetary Systems

Economic shocks (e.g., banking crises, hyperinflation events) can be modeled as inelastic collisions where purchasing power is transferred from fiat to Bitcoin with minimal overall system energy loss. This explains the observed phenomenon of Bitcoin price appreciation during periods of significant fiat instability.

3. Time-Preference Arbitrage

The conversion between fiat and Bitcoin creates a time-preference arbitrage opportunity. Agents with lower time preference accumulate Bitcoin, effectively transferring future purchasing power from high-time-preference fiat holders to themselves.

4. Entropy and Monetary Systems

The Second Law of Thermodynamics has an economic analog: closed monetary systems tend toward increasing entropy. Fiat systems, with their inherent expansion mechanisms, exhibit higher entropy generation rates than Bitcoin's fixed-supply system.

Empirical Observations

The theory predicts several observable phenomena:

1. Fiat currencies with higher inflation rates (α_f decline) should correlate with higher Bitcoin adoption rates in those regions.
2. The ratio of Bitcoin market capitalization to fiat monetary base should increase during periods of significant monetary expansion.
3. The volatility of Bitcoin should decrease as its market capitalization grows, consistent with the conservation principle stabilizing as the system reaches equilibrium.

Conclusion

The conservation principle described herein provides a quantitative framework for understanding the relationship between fiat currencies and Bitcoin. Though simplified, this model offers testable predictions about capital flows and value preservation in the evolving monetary landscape.