Comparative statics
\begin{equation}
\par
\pi(W_{i};K^{*},x^{*})=\frac{1}{N}\,.\par
\nonumber \\
\end{equation}
We allow free entry into the mining race and derive comparative statics to examine the nature of the long run equilibrium of the Bitcoin mining network. In equilibrium, the symmetry of the game means that all miners must choose the same technology \(x^{*}\). Therefore, the probability of a miner \(i\) being the first to solve the puzzle and win the prize is
\begin{equation}
\par
U_{i}(x^{*})=\frac{P}{N}-c(x^{*})=0\,,\par
\nonumber \\
\end{equation}
Allowing free entry into the game means that all miners earn zero profits on average in equilibrium since they must be indifferent between entering and exiting
where \(x^{*}\) satisfies (\ref{foc}) for all \(i\). There are zero aggregate profits in equilibrium.
Therefore, the equilibrium technology when there is free entry is characterized by
\begin{equation}
\label{x}x^{*}=c^{-1}\left(\frac{P}{N}\right)\,.\\
\end{equation}
.\label{dxdn}
The equilibrium technology decreases as miners enter the race.
Proof.
Recall that \(c^{\prime}(\cdot)>0\) by assumption, which implies \((c^{-1})^{\prime}(\cdot)>0\). Differentiating (\ref{x}) with respect to \(N\)
\begin{equation}
\frac{\partial x^{*}}{\partial N}=\frac{\partial c^{-1}\left(\frac{P}{N}\right)}{\partial\frac{P}{N}}\frac{\partial\frac{P}{N}}{\partial N}=\frac{\partial c^{-1}\left(\frac{P}{N}\right)}{\partial\frac{P}{N}}\left(\frac{-P}{N^{2}}\right)<0\,,\nonumber \\
\end{equation}
for \(N,\frac{P}{N}\neq 0\)
∎
The incentive to rent technology decreases with increased competition in the network. As more miners enter the race, the probability that a miner \(i\) will be the first to solve the puzzle falls, decreasing the expected value of the prize.
By Proposition (\ref{eqmdelta}), there exists a unique equilibrium \((K^{*},x^{*})\) for every \(N\). If at any stage of the game, there is entry or exit, the long run equilibrium of Bitcoin mining game changes. For instance, if the number of miners increases to \(N^{\prime}\), then the network moves towards a new equilibrium \(({K^{\prime}}^{*},{x^{\prime}}^{*})\). Whether the new equilibrium difficulty levels are higher or lower is ambiguous and depends on the relative sizes of the changes in \(N\) and \(x^{*}\).
.\label{dxdp}
The equilibrium technology increases as the value of the prize increases.
Proof.
Differentiating (\ref{x}) with respect to \(P\)
\begin{equation}
\frac{\partial x^{*}}{\partial P}=\frac{\partial c^{-1}\left(\frac{P}{N}\right)}{\partial\frac{P}{N}}\frac{\partial\frac{P}{N}}{\partial P}=\frac{\partial c^{-1}\left(\frac{P}{N}\right)}{\partial\frac{P}{N}}\frac{1}{N}>0\nonumber \\
\end{equation}
since \((c^{-1})^{\prime}(\cdot)>0\) for \(N,\frac{P}{N}\neq 0\).
∎
The incentive to rent technology increases as the expected value of the prize increases. In the Bitcoin network, the prize prescribed by the network of newly minted bitcoins halves for every 210,000 blocks that are verified until the supply of all 21 million bitcoins has been exhausted. When the prize next halves, short run profits will fall below zero and we would expect to see miners respond by decreasing their computing technology.
In addition to the prescribed path of the bitcoin reward, the dollar value of the prize is affected by the exchange rate. To include exchange rate fluctuations in our model, let the prize \(P(e)\) be a function of the exchange rate \(e\), with \(P^{\prime}(e)>0\). If the exchange rate appreciates, then the dollar value of the prize increases, and vice versa.
.\label{dxde}
The optimal equilibrium technology increases as the exchange rate appreciates.
Proof.
Using the chain rule and the result from Proposition \ref{dxdp}, we see that
\begin{equation}
\frac{\partial x^{*}}{\partial e}=\frac{\partial x^{*}}{\partial P(e)}P^{\prime}(e)>0\,.\nonumber \\
\end{equation}
∎
Once the network is at some long run equilibrium \((K^{*},x^{*})\), miners are indifferent between entering and exiting the mining race since there are zero profits. Therefore, any movements away from an equilibrium will be due to changes in the prize \(P\).
The value of the prize is deterministically prescribed by the protocol of the Bitcoin network, and changes relatively infrequently. This suggests that the exchange rate is the most important factor in determining which equilibrium is realized, since it is exogenously determined and has been historically volatile.
By Proposition \ref{eqmdelta}, there exists a unique equilibrium for every \(P\). As the exchange rate stabilizes, the network will stabilize around the equilibrium for the given \(P\).
.
The equilibrium difficulty level \(K^{*}\) is increasing in the prize \(P\).
Proof.
By Proposition \ref{dxdp}, the equilibrium technology \(x^{*}\) is increasing in \(P\), and by Lemma \ref{xk}, \(x^{*}\) is increasing in the difficulty level \(K\) given a fixed \(N\). Therefore, it must be the case that the unique equilibrium difficulty level \(K^{*}\) is higher for a larger prize \(P\), holding \(N\) constant.
∎
If the prize increases, technology cannot increase immediately in the short run and there are positive profits. In the long run, by Proposition \ref{dxdp}, we know that if the prize increases, then the equilibrium technology and cost increases. Subsequently, the rate at which puzzles are solved increases until the next period when the difficulty level increases to maintain the solution target \(\delta^{*}\).