Introduction
An economic problem in recent times has been the development of an electronic version of cash, which would achieve the convenience of electronic transactions without the need for costly trusted third parties. A solution to this problem emerged with the development of Bitcoin \citep{nakamoto08}. Bitcoin is a decentralized digital cryptocurrency that operates a electronic peer-to-peer payments system in January 2009. The key innovation of Bitcoin is the “block chain”, a public record of all past transactions on the network, eliminating the need for the facilitation of payments by trusted third parties. \cite{kocherlakota98} showed that money is an imperfect form of memory, where money is taken to be physical cash, and memory is defined as a perfect public record of all past transactions. Bitcoin’s block chain is seemingly an implementation of Kocherlakota’s notion of memory. However, it shares the properties of privacy and anonymity of cash that make it imperfect memory. In order words, Bitcoin is electronic cash.
Bitcoin mining is the process used to enforce and secure the block chain. It is a competition between users on the network known as “miners” to solve a cryptographic puzzle. Transactions are recorded on the block chain each time a puzzle is solved, with the solver being rewarded with newly issued bitcoins. The puzzle is a proof-of-work function which requires a set number of computations to be completed before a solution is reached. The difficulty of the puzzle is set dynamically according to Bitcoin protocol, which determines the number of computations required to ensure that transactions are confirmed every 10 minutes on average. The number of calculations a miner can compute within a given time interval increases with their technology. However, the proof-of-work function is a random process, so there is no guarantee that the miner in the network with the most computational power will be the first to solve the puzzle.
The model builds on similarities in the process of Bitcoin mining and R&D models. It extends the R&D model described in \cite{loury79} to reflect the occurrence of multiple events before a reward is attained in Bitcoin mining.