# Introduction

Currently, most of the blockchains adopt either a fixed-supply token model or inflation-supply token model. For example, Ethereum adopts inflation-supply token, although the Constantinople hard-fork will reduce block reward from 3ETH to 2ETH. Bitcoin employs fixed-supply token model, which halves the block reward in every 210,000 blocks and thus the limit of the supply is 21,000,000.

The next-generation blockchain will have multi-shard/multi-chain. By adding more shards/chains on-demand, the capacity of blockchain could be increased to meet the needs of more transactions per second (TPS) on blockchains. However, one tricky case is how to impose a proper token model on the shards/chains. In this article, we will discuss a fixed-supply token model with dynamically-added shards/chains.

# Fixed-Supply Model for Single Chain

Let us assume the block reward of the chain differs in different epoch (e.g., bitcoin’s epoch is about 4 years), the total block reward in the first epoch is r, and for every epoch, the block reward is decreased by a factor of q. Therefore, the limit of the supply is:

Equation of fixed-supply model

For example, bitcoin halves block reward for every 210,000 blocks with initial block reward 50 BTC. Thus, r = 10,500,000, q = 0.5, and S = 21,000,000.

# Fixed-Supply Model with Dynamically-Added Shards/Chains

Let us assume the sharded blockchain has n genesis shards. Block rewards of all shards are the same value r, and the block reward is decreased by a factor of q for each epoch. In addition, for every epoch, m number of shards are added to the blockchain to increase the capacity. Therefore, the equation of the limit of the supply becomes:

Equation of fixed-supply model with dynamically-added shards/chains

where the first term in Eq. (7) is the total block rewards collected by initial shards/chains, and the second term in Eq. (7) is the total block rewards collected by dynamically-added shards/chains.

For example, let us assume a sharded blockchain with n = 8 genesis shards, and for every epoch, only one shard/chain is added, and q = 0.9. Suppose the limit of the supply for mining is 4B, then the block reward of all shards in the first epoch becomes nr = 4B / (8 / (1–0.9) + 0.9 / (1–0.9) / (1–0.9)) * 8 = 188.2M. In addition, 80 / (80 + 90) block rewards are allocated to genesis shards, and 90 / (80 + 90) are allocated to new shards.

# Conclusion

In this short article, we briefly reviewed the fixed-supply token model of single blockchain (such as Bitcoin). Then, considering the number of shards/chains can be increased over time, we derived a fixed-supply token model and provided some examples based on the proposed token model.