Peter_R:
Great paper! The last paragraph got me thinking though, and I think I have come up with another way of visualizing the system that could generalize the result to the case where block reward is zero.
Mempool Demand Curve:
I think it simplifies things if block reward and transaction fees are treated as the same thing, where the block reward is simply treated as a transaction with a large fee (ie, coinbase transaction would be a very tall skinny triangle). The block reward can then be included in the mempool demand curve, causing it to pretty much start at R instead of 0.
Block Space Supply Curve:
Again, when considering the revenue per block, I will combine the reward and fees (R and M) and call it M
rev. The profit equation then becomes:
Profit = M
rev (h/H) e^(-τ/T) - ηhT
(Sorry for the rudimentary looking equation, I'm not sure how to enter it properly here)
In your paper, you base the supply curve on the "neutral profit". The problem with this is that the analysis breaks down when the block reward is zero. Instead of assuming a profitable empty block, I will simply solve for the total revenue (reward plus fees) needed to yield a profit. So, to get the block space supply curve, simply set the profit to 0 and solve for M_rev, yielding:
M
rev = ηHT e^(τ/T)
Similar to your formulations, this curve will curve upward as block size increases, but will intersect the y axis at ηHT. So for a miner to profit by mining an empty block, the block reward must be greater than ηHT.
Plotting the Supply and demand curve together looks like this:
(I have normalized both curves to a single miner’s point of view by multiplying by h/H)
https://i.imgur.com/mxgW7UE.jpgAnd the miner’s profit is the distance of the revenue curve above the cost curve.
The nice thing about plotting it this way is that we can also consider what would happen if transaction fees become a significant source of revenue and the block reward is not sufficient to profitably mine empty blocks:
https://i.imgur.com/Cd1Anik.jpgWe can see that in this case, it would only make sense for the miners to include enough transactions to be in the region of the graph where revenue exceeds cost. This would also work in the extreme case where R=0.