If I have no spare money, then my slack is… undefined? The formula would seem to require division by zero in that case; that seems strange. (I assure you, it is possible to have no spare money at all, so this is not merely an academic edge case.)
Units?
The question of units presents several problems:
What are the units of “spare time”, “spare money”, and “spare energy”? This is important, because the choice of units determines where the inflection points of the parenthesized components are—which, in turn, affects how the components trade off against one another.
The inverse of “spare time”, presumably, is “spare speed”. The square of “spare speed” must then be “spare acceleration”. Fair enough. However, it is less clear what the inverse of “spare money” is (or its square), or what the inverse of “spare energy” is (or its square).
Is “spare energy” being measured in joules, or (as seems more likely) are we speaking of metaphorical “spare energy”—willpower, perhaps, or something like it? I am not aware of an accepted way of measuring such a thing; is there such?
How does one add quantities of these disparate units? Is there a conversion rate? (The “exchange rates” described in the post do not seem to suffice to add squared inverse units. Is the idea that we first convert everything into one units, and then apply the formula? But that makes the formula largely redundant, does it not?) What do the units of “slack” (and of “scarcity”) end up being? (You mention “distance from binding constraints”, but that doesn’t quite clarify; is there a defined metric space, here?)
Terminology
It would seem that the term “scarcity” is being used here in something other than its usual meaning (or, I should say, any of its usual meanings—whether colloquial or well-known technical). However, unless I missed it, the term is not defined (except in the formula, which merely says that it’s equal to 1 divided by “slack”… but that doesn’t make the matter any less murky). What is the meaning of “scarcity” in this context?
In response to the Division by Zero question, I think the right answer is that slack is zero in that case.
Here’s why. Take one of the components, say spare money. As spare money tends to zero, scarcity diverges off to infinity. Therefore slack, which is 1/scarcity, tends to zero as well. So this formula is defined for all values of spare money except zero, and at that point the limit is defined. I would recommend simply filling in the gap.
I have questions about your formula:
Division by zero?
If I have no spare money, then my slack is… undefined? The formula would seem to require division by zero in that case; that seems strange. (I assure you, it is possible to have no spare money at all, so this is not merely an academic edge case.)
Units?
The question of units presents several problems:
What are the units of “spare time”, “spare money”, and “spare energy”? This is important, because the choice of units determines where the inflection points of the parenthesized components are—which, in turn, affects how the components trade off against one another.
The inverse of “spare time”, presumably, is “spare speed”. The square of “spare speed” must then be “spare acceleration”. Fair enough. However, it is less clear what the inverse of “spare money” is (or its square), or what the inverse of “spare energy” is (or its square).
Is “spare energy” being measured in joules, or (as seems more likely) are we speaking of metaphorical “spare energy”—willpower, perhaps, or something like it? I am not aware of an accepted way of measuring such a thing; is there such?
How does one add quantities of these disparate units? Is there a conversion rate? (The “exchange rates” described in the post do not seem to suffice to add squared inverse units. Is the idea that we first convert everything into one units, and then apply the formula? But that makes the formula largely redundant, does it not?) What do the units of “slack” (and of “scarcity”) end up being? (You mention “distance from binding constraints”, but that doesn’t quite clarify; is there a defined metric space, here?)
Terminology
It would seem that the term “scarcity” is being used here in something other than its usual meaning (or, I should say, any of its usual meanings—whether colloquial or well-known technical). However, unless I missed it, the term is not defined (except in the formula, which merely says that it’s equal to 1 divided by “slack”… but that doesn’t make the matter any less murky). What is the meaning of “scarcity” in this context?
In response to the Division by Zero question, I think the right answer is that slack is zero in that case.
Here’s why. Take one of the components, say spare money. As spare money tends to zero, scarcity diverges off to infinity. Therefore slack, which is 1/scarcity, tends to zero as well. So this formula is defined for all values of spare money except zero, and at that point the limit is defined. I would recommend simply filling in the gap.