Mathematical question on Structures of production in ABCT

Economics Economics Questions
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1

I understand ABCT but I can’t understand some aspects of Structure of Production diagramm (used in Hayek’s “Prices and Production” and other books on ABCT(de Soto for example)): how to get from fig.2 to fig.3 (see image below)?

Fig.3 shows how the structure of production changes when people decide to save 25% of their income.

I see that all numbers on second diagramm match each other, but how do we get these numbers from first diagramm knowing only about rise in savings and fall in consumption? How do we know, that there will be 7 stages?

I assume that there can be a lot of mathematical solutions of that task even with same 5 stages. In this case, how that example can be used as a proof? Because, as I see from the context both of “Price and Production” and “Money, Bank Credit, and Economic Cycles” (J.H.de Soto), these diagramms are used not as illustration but as a logical proof.

What do I miss?

2

I suspect its just an example. The only thing that changes form fig.2 to 3 is that the ‘step’ between stages falls from eight to 4.3, meaning that the same amount of original factors are spread wider. Perhaps, in a sounder mathematical example, stages would be factors of each other (stage0*f=stage1, and so on), not additions.

3

It’s important not to get hung up on the actual number of stages. The so-called “stages of production” serve only to elucidate the phenomena associated with a reduction in the aggregate demand for final goods and services (i.e., a higher savings rate); it is a theoretical construct. The elongation of the structure of production does occur in the real world, but actual firms don’t fit neatly into “stages” that can be accurately depicted by various rectangles.