New bitcoins are minted with every new block in the Bitcoin blockchain, called “block rewards”, in order to incentivize people to mine and increase the security of the network. This inflates Bitcoin’s supply in a predictable manner. The inflation rate halves every 4 years, decreasing geometrically.

There have been some confusion of the terminology, like people calling Bitcoin deflationary. Bitcoin is in fact not deflationary—that implies a negative inflation rate. Bitcoin rather has negative inflation curvature: Bitcoin’s inflation rate decreases monotonically.

An analogy from elementary physics should clear things up: Speaking strictly in terms of monetary inflation,

• displacement is analogous to inflation/deflation, as in total money minted/burned, without considering a time period. Dimensions: $[M]$.
• Velocity is analogous to inflation rate, which defines total money minted/burned in a given period. Dimensions: $[M/T]$.
• Acceleration is analogous to inflation curvature, which defines the total change in inflation rate in a given period. Dimensions: $[M/T^2]$.

Given a supply function $S$ as a function of time, block height, or any variable signifying progress,

• inflation is a positive change in supply, $\Delta S > 0$, and deflation, $\Delta S < 0$.
• Inflation rate is the first derivative of supply, $S’$.
• Inflation curvature is the second derivative of supply, $S’’$.

In Bitcoin, we have the supply as a function of block height: $S:\mathbb{Z}_{\geq 0} \to \mathbb{R}_+$. But the function itself is defined by the arithmetic¹ initial value problem

\[S'(h) = \alpha^{\lfloor h/\beta\rfloor} R_0 ,\quad S(0) = 0 \tag{1}\]

where $R_0$ is the initial inflation rate, $\alpha$ is the rate by which the inflation rate will decrease, $\beta$ is the milestone number of blocks at which the decrease will take place, and $\lfloor \cdot \rfloor$ is the floor function. In Bitcoin, we have $R_0 = 50\text{ BTC}$, $\alpha=1/2$ and $\beta=210,000\text{ blocks}$. Here is what it looks like:

Bitcoin inflation rate versus block height.

We can directly compute inflation curvature:

\[S''(h) = \begin{cases} \frac{\ln(\alpha)}{\beta} \alpha^{h/\beta} & \text{if}\quad h\ \mathrm{mod}\ \beta = 0 \quad\text{and}\quad h > 0\\ 0 & \text{otherwise}. \end{cases}\]

$S’’$ is nonzero only when $h$ is a multiple of $\beta$. For $0 < \alpha < 1$, $S’’$ is either zero or negative, which is the case for Bitcoin.

Finally, we can come up with a closed-form $S$ by solving the initial value problem (1):

\[\begin{aligned} S(h) &= \sum_{i=0}^{\lfloor h/\beta\rfloor -1} \alpha^{i} \beta R_0 + \alpha^{\lfloor h/\beta\rfloor} (h\ \mathrm{mod}\ \beta) R_0 \\ &= R_0 \left(\beta\frac{1-\alpha^{\lfloor h/\beta\rfloor}}{1-\alpha} +\alpha^{\lfloor h/\beta\rfloor} (h\ \mathrm{mod}\ \beta) \right) \end{aligned}\]

Here is what the supply function looks like for Bitcoin:

Bitcoin supply versus block height.

And the maximum number of Bitcoins to ever exist are calculated by taking the limit

\[\lim_{h\to\infty} S(h) = \sum_{i=0}^{\infty} \alpha^{i} \beta R_0 = \frac{\beta R_0}{1-\alpha} = 21,000,000\text{ BTC}.\]

Summary

The concept of inflation curvature was introduced. The confusion regarding Bitcoin’s inflation mechanism was cleared with an analogy. The IVP defining Bitcoin’s supply was introduced and solved to get a closed-form expression. Inflation curvature for Bitcoin was derived. The maximum number of Bitcoins to ever exist was derived and computed.

1. Because $S$ is defined over positive integers. ↩