The recent development of private cryptocurrencies has created a need to extend existing models of private currency provision and currency competition. The outcome of cryptocurrency competition should be analyzed in a model which incorporates important features of the modern cryptocurrencies. In this paper I focus on two such features. First, cryptocurrencies operate according to a protocol - a blockchain - and are, therefore, free from the time-inconsistency problem. Second, the operation of the blockchain costs real resources. I use the Lagos-Wright search theoretic monetary model augmented with privately issued currencies as in Fernandez-Villaverde and Sanches (2016) and extend it by linear costs of private currency circulation. I show that in contrast to Fernandez-Villaverde and Sanches (2016) cryptocurrency competition 1) does not deliver price stability and 2) puts downward pressure on the ination in the public currency only when the costs private currency circulation (mining costs) are suciently low.