Denote the following:
The average block fee F must be proportional to the block time T and money volume because on average the number of transactions of the block is proportional to block time and money volume is simply an artificial nominal setting. Therefore F = 21000000qT * I(N / nA) for some function I which describes the included N transactions among all the broadcast transactions nA because of the incentive to include higher fee transactions into the block.
Empirically the wealth distribution follows exponential distribution. Because the average BTC balance must be 21000000/A, f is of exponential distribution with average mean 21000000q/A/n.
While nAT transactions compete for the block space who has the capacity of NT transactions, F is contributed by the higher-fee NT transactions among the nAT transactions. In other words,
with a multiple factor of 1-Ln(N/n/A) to the average fee and F as a monotonically increasing function of N / nA:
whose graph is like this and maximized at 21000000qT. The function I(x) here in the exponential distribution is . Let V be money volume. The reason that the average block fee must be of the form for some function I is that
where g is the the re-scaled dimensionless random variable of f and and . Note that N is never greater than nA by definition and I(x) is increasing and I(0)=0 and I(1)=1.
When N is less than nA, n must become smaller to have the backlog cleared and results in a higher average fee 21000000q/A/n for transaction. Or, the number of base users, A, must be smaller to N/n. For example of a scenario, knowing A is 7 billion and N is 4 and q is 0.005 annually and n is 1.0 daily, q=0.005/365/86400=1.5855E-10 and n=1/86400=1.1574E-05 and F is initially 1.08E-03 with huge backlog with a multiple factor 10.9 to the average fee. Then competition will start and drive out some transaction and make n smaller to N/A which is 5.71E-10 and is equivalent to one transaction per 55.5 year for a person where F is finally 1.9977 and the fee per transaction for a person is 8.32E-04. Or the number of base users drops to only 345602 instead of 7 billion and 7 billion users may trust and use the base user as an exchange or other methods to conduct more transactions. Mathematically, no other solutions given N is constrained by security concern.
As to q, its value results from the stochastic context of money flow as mentioned in time preference. When Albert sells something for a customer's purchase next day, he faces the risk the committed money of production may cause his shortage if an irrelevant shit happens today, a surcharge of transaction fee to mitigate the risk of insolvency. Whatever the money is, the tendency for transaction and fee shall be the same; imagine an old fiat is replaced with another new fiat of different nominal by the government, all the tendency of transaction and fee shall be the same regardless the price of the fiat and its volume. Therefore q is supposed to be stable than other parameters and is treated as a constant; block fee shall be therefore stable unless the evolution of the money flow in the society becomes much certain, other than that, it is difficult to persuade people from changing its value because this might be Charles Darwin's call.
The average block fee is 1.9977 irrelevant with N and the price and the number of users, contrary to those who claim big blocks will have more block fee and those who claim the more users the higher the price; these intuitions are right when there is backlog but then it means the relevant parameters n and A are not yet settled down so the phenomenon is only temporary.