Reworking Paul Samuelson's Consumption/Loan model with hyperbolics and Lagrange

 This one on the Exact Solution to overlapping generations.

We can greatly simplify his whole effort using hyperbolic discounting. The idea is to replace savings and consumption with loans to deposit ratios, then find out where the endowment comes from and split that among the generations under no arbitrage conditions.  We let the endowment flow  by slight changes in the term lengths.

Samuelson assumes that the retired person has only savings and no loans, that is hyperbolic angle zero.  Under these conditions he finds that the periods go on forever because there is no uncertainty and Tanh -> 1.0, but never actually arrives.  Thus infinit generations and no solution.

What is uncertainty?

Uncertainty is the probable arrival rate of the endowment.  For example, the youngest workers are digging for energy in the ground, but the certainty of finding energy is finite, but bound.  Under these conditions, then, the retired miners never really set their loan balances to zero, but operate at some slight positive hyperbolic angle in which he keeps a small loan balance thus maintaining his constant consumption by the occasional debt.  So the retired person and all people between him and the young miner share both the uncertainty of energy flow evenly, and they all have constant consumption.

Assume population growth is steady, and generational population are equal, and get the first Lagrange.  Assume some death ratio, and more children born then needed for constant population, and get the second Lagrange. Define the probability of finding energy in any given dig, and you define the number of generations supported at any one time.

Formula:

The population pyramid determines your Lagrange number. It appears in your Tanh'' in the hyperbolic flow constraint as,
(1/L)  * (1/2) * sinh(n*ln(r))/cosh(n*ln(r))^3. I think I have that right, economics profs need to check me on this.  The n is the nth generation, and the r is the additive ratio associated with the Lagrange.  The borrowing and lending between generations is a sequence:  Kn + b* Kn-1 = Kn+1.  B is the fan out so that r-1/t = b, and integer.  The total endowment is the sum of the n tanh'' forms above.  Anyway, I did this from scratch and need to be math checked.

The periods are the number o generations, Paul misnamed this. The projected lookahead, cosh^2 - sinh^2 = L,  is the adaptive process constraint.  All the periods are overlapping, so the model is more like each generation borrowing from a common pool, the currency banker. Terms are not fixed, there is no time. And the endowment gets pumped down the chain by adjustments in the loan deposit ratio that are within the uncertainty bounds. This is adiabatic, a Weiner process and no arbitrage. And it is price neutral. The slight variability in flow results because, in reality, r, the additive ratio, is a finite approximation to the Lagrange number. Its the theory of everything!

This all works, I am sure, use the flow constraints.  Really, the hyperbolics is real, and simplifies a whole bunch of economics.