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In this paper we provide a mathematical derivation that links traditional time-value-of-money concepts to Metcalfe value, and use Bitcoin, Facebook as numerical examples of the proof. There is compelling evidence that suggests that the growth and price of bitcoin and other cryptocurrencies are likely to proceed according to a relatively straightforward mathematical model similar to the growth curves of Facebook and other networks. Using observed data for Bitcoin, we derive the relationships between price, number of users, and time, and show that the resulting market prices likely follow a Gompertz sigmoid growth function. This function, historically used to describe the growth of biological organisms like bacteria, tumors, and viruses, likely has some application to network economics. We conclude that the long-term growth rate in users has considerable effect on the long-term price of bitcoin.