Beyond smart beta : a dynamic statistical risk budgeting approach in portfolio construction

Bitcoin miners compete to validate users’ transactions via a proof-of-work mechanism using computing power. Thanks to their efforts, miners are compensated with a predetermined number of newly minted Bitcoins plus transaction fees whenever the verified users’ transactions are successfully added to the blockchain. One of the main concerns about Bitcoin mining is its electricity consumption. Many researchers have reported that nowadays, the electricity consumed annually by Bitcoin mining is even higher than that of medium-sized countries. The trend of electricity consumption is even more worrisome. As we have witnessed a tremendous increase in Bitcoin price and mining revenue in terms of the US dollar in the past decade, substantial new miners are lured into joining the mining business. Consequently, both the electricity consumption and the Bitcoin computing power measured by hashrate have grown up rapidly. There is relatively little literature on the economic modeling of computing power in the Bitcoin network. An exception is Prat and Walter (2021), in which a novel industry equilibrium model is developed by first introducing technology innovation to the model in Caballero and Pindyck (1996) and then calibrating the model to capture the evolution of miners’ computing power. However, they mainly focus on examining the entry rule of the miners, and the hashrate in their primary model cannot decrease over time. In addition, for an entry-only model, the resulting electricity consumption must be nondecreasing, and the ratio of electricity consumption to revenue may grow to infinity. We propose a dynamic industry equilibrium model for Bitcoin electricity consumption in a general framework, including Bitcoin miners’ optimal entry and exit with technology innovation. By adopting average operating costs as an approximation to the actual operating costs, we overcome the difficulty of strong path-dependency due to the interaction among entry, exit, and technology innovation. We also formulate the problem from a social planner’s perspective and show that his optimal strategy is an industry equilibrium under a competitive market. The penalty method is then applied to find the optimal strategy numerically. By calibrating the model, we can capture both the upside and downside co-movements of miners’ computing power, electricity consumption, and mining revenue. Our model shows that the Bitcoin electricity consumption will not grow indefinitely, with the ratio of Bitcoin electricity consumption to the miners’ revenue fluctuating within a rang