In this paper we study space-scarce economy in massively replicated open blockchain systems, such as Bitcoin. In these systems memory to hold a state needed to validate transactions becomes the most scarce resource eventually. The issue is more critical for blockchain systems used to store data~(votes, certificates, logs etc.). Uncontrolled state size growth could lead to security issues, such as denial-of-service attacks. Only technical solutions, not economic, have been proposed to tackle this problem to the moment. In contrast, we propose to add a new component to a transaction fee scheme, which is a fee with a mandatory minimum based on how much additional space will be needed for new objects created in result of transaction processing, and for how long they will live in the state. We provide three possible options towards implementing the new fee component, namely \textit{prepaid outputs}, \textit{postpaid outputs} and \textit{scheduled payments}. We provide an analysis of the model with respect to all the three options. We show that the state growth could be bounded by a fee factor, miners are getting additional rewards, and lost coins are being taken back into circulation eventually.