The mathematics of a contraction

The economy operates with minimal redundancy, resulting in rank reduction, as I point out so many times. The result is that the yield curve takes discrete jumps from position to position. So, consider an economy that attempts to maintain goods flow under two different rank settings.  I used a 7 and 6 length Fibonacci series.  But in the N=6 case, I have to equalize goods flow goods flow. Let jump below the graph and look at the yield curve equation

Tis si the standard aggregate generalized yield curve, I have added vi in the numerator to sum over all ecomponents operating at rates v.  Remember that the cargo size, -log(vi) is always uniformly dirstibuted modulo one, and the economy always works integers.  The value -vi*log(vi) is always one to the nearest integer, as in R:

Here is the scaled, rank seven F series:  0.038 0.038 0.077 0.115 0.192 0.308 0.500
My fastest transationrate is .5, the spectra is normalized to 1.0, meaning the Beta in the eqution is 1.0.

What is my equivalent to the economic quantity eqution? It is -log(g7) cargos, transacted g7 times over the longest inventory cycle, which becomes:

> round(-g7*log(g7)+.5)
[1] 1 1 1 1 1 1 1

All ones. In the completely normalized case, this is the quantity of total goods at each transaction rate, and the total good transacted is 7, over the longest inventory cycle in the distribution chain.

What happens if I drop rank?  The good transacted has to increase, I have fewer transactions over all. So the gamma in the exponent of the yield curve must go to 7/6, to preserve quantity, as in the economic standard:

From Wiki. Except of course, in channel theroy we are smart enough to eliminate money.  We would replace the equation above with the mutual entropy operator between two yield curves.

But, the point is, when we drop rank, we ship larger cargos and we have a total (sum but not mean) reduced transaction rate.  That is we have to adopt economies of scale.  Inside the firm or household we will see larger inventory levels.  The curve is not as steep, so gains from specialization are lower. Variance in inventory levels reduce, we are safer. But less focused on the long term.

But, bottom line, there are only so many discrete solutions. When oil gets too expensive, we will suddenly contract farther than any one really expects, unless they understand channel theory.

Why did I increase gamma in the yield curve eqution?  That goes back to the shannon theory:

 C/B = log(1 + SNR)  The C is the number of cargos I can carry in B trucks. -log(g6) is the information carraige, I have essentially multiplied all my cargo sizes by 7/6, and it is factored out of the maximum entrpy formu;a -g6*log(g6) which you ill see in the Huffman encoder. I grew my bit size over all symbols.