Optimum savings ratios

 They exist, but only in fair traded pits, under no arbitrage conditions.

We are not a fair trade currency system, as it obvious from the Fed balance sheet and cash flow. But private goods tend to be more fair traded.   The optimum savings ratio still exits, in units of goods, but prices are erratic when the currency banker is not fair traded.

Finite systems, bounded error, self sampled, there will be one fractional approximation that minimizes transactions.  In essence it is the optimum use of savings and credit in that you will almost never be third in line.

The standard Euler assumption generates optimum ratios. They assume error zero and everyone sufficiently sample. The two should follow each other, the Eurler condition just gives you the irrational ratio. For large N it is not a problem.

The uncertain assumption is risk equalized traders.  A very similar assumption as ergodicity. In sandbox, sufficient mixing is when there are pits for every level of risk. The Euler assumption is that each sector has sufficiently liquidity to keep up.

I have nothing against Euler, and Menzie uses it properly because log of growth shows the break points more clearly on the graph. They get amplified when one assumes sufficient liquidity always, but it is a good idea.. The trick is to get enough before and after around the break point you want to emphasize.

Being more ergodic means getting the Fed a long term contract and having Treasury manage inflation.  The Fed then obtains a fair sample of economic risk and the pits work toward the optimum ratio.