Defining the hyperbolic banker in terms of simple interest

I want to use simple interest in defining the currency banker, then show that the equilibrium condition  meets the hyperbolic condition. Then we can use simple interest on loans and deposits, but have the option of converting to the hyperbolic 'accumulant' and use all the rules of hyperbolic calculus.

So lets call deposit balances D and the deposit rate d, and for loans we have L and l. Then over a two period projection we have deposit liabilities is:
D*(1+d)^2, and loan assets become L*(1+l)^2. Much simpler.

Now are target says that D^(1+d)^2 - L*(1+l)^2 = cash outstanding. Now everything is in simple interest, and the period is definable by the banker.  But equilibrium is defined as D = L = cash outstanding. And when that is met we have:

(1+d)^2 - (1+l)^2 = 1, and that meets the hyperbolic condition. So at equilibrium we can define (1+l)/(1+d) = tanh(a). Compute a, and use all the integration and differentiation rules of hyperbolics. Much simpler.

Call (1+l)/(1+d) the yield ratio, Y.   And, without proof, say that Loans = Deposits = Cash.  Then we want Y less than 1. Why? Because that is slightly inflationary, the currency banker has less flow coming in and more flow going out.
D and L are the balances of the member banks. What is cash outstanding? I guess the amount of net cash member banks have, to be defined later.  But when D/Cash < 1 the banker bot raises the deposit rate. When L/Cash < 1, the banker bot lowers loan rates.  Now, I suppose one year out might be two periods, so the projection is over the full seasonal year, and the base period is two quarters.   The rates can be converted to daily rates and posted and computed daily.  How and when the banker might change rates depends on the approach to equilibrium, to be defined later.

Slightly inflationary?

Well, member banks have to exist, they should earn the ATM fee with member loan fees slightly less than deposit rates, maybe a quarter point difference. Member banks are contracting agents of banker bot, they are a cost of using money.  Beyond their mere existence, member banks are free to find investment opportunities and earn more money as real productivity increases. Banker bot will always adjusts back to zero inflation.