The idea of banker bot is to lose money during periods of growth. How does the currency banker do that? Well, for now, let us say we want liabilities to the economy to increase and assets to decrease. So consider the ratio assets/liabilities = R. We want that to become assets/[liabilities * (1+g)] or we want R * 1/(1+g). For a 2% growth we get .98 is the outcome for R'.
Lets use our hyperbolic differential, without understanding why!
R * R' + (1/2)R'' = 0.
R,R' and R'' change as member banks makes loans and deposits.
R' is .98, and we know that to be 1-R^2, which should be .98. So right away we get that R must be around .14, a very small ratio. Hence the third term (1/2)R'' will be about -.07. That last term is the variation in growth we will allow, it is pricing counter pressure we need to keep the banking network in one piece. We measure the first difference in R, (R') and the difference in R' (R'') on a daily basis. As the bankers get the growth funded we want R to decrease and R' to head higher toward 1, and the acceleration of funding, R'', should get smaller, as bankers reach their goal.
The currency banker will change rates daily to make that happen.
How do we know that growth is going to happen? We don't, we just change rates to make the equation true, allowing some small adjustment time for the differences to settle. But the currency banker operates at the smallest hyperbolic angle with a very low asset to liability ratio, and R' will always be near one while R'' near zero, and the term period is very short. Hence, balances equilibriate quickly and deposits deliver free and unencumbered money quickly.
These are the boundary conditions. f'(0) = 1, and the currency banker operates with f' around .95 to 1, the derivative is with respect to the accumulation angle, not the simple interest rates, the simple rates come out as:
D*(1+d)^2 - L*(1+l)^2 = 1 Where D is the deposit balance and d the deposit rate and so on. The ratio, R, includes principle and interest. I think this is mostly right, but always check me. The currency banker does not make a lot of loans, the member banks keep money on deposit for sudden growth.
The higher hyperbolic angles should be longer term loans and assets/liabilities get close to 1, like home loans. The currency banker is not involved, that is four of five links down the banking chain. The real danger is that the currency banker runs with the ratio a bit low and a sudden growth spurt drives the deposit rate to near zero, the member banks cannot get enough free money to fund the growth.
As long as banker down the chain obey the Lucas rules in my previous post then
flow is conserved and no sudden price crashes are cause by the bankers. Government is free to sign up as member banks, Social Security, Fanny and Freddy can have member bank accounts, even Treasury. But currency banker is a spread sheet, a web bot. It earns no pay and only cares about meeting the flow equation above.
Where is time?
Dunno, working the Brownian motion thing is still in progress. Likely Lucas polynomials have a mirror image based on a different alternating sinh and cosh and they meet the condition. Hence my Lucas angles in the previous post have a second additive stream that makes the missing half of tanh along the angles. These likely provide the extra condition for Brownian motion.
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