Moving toward a proof of the cash-inventory equivalency, another factor that has to be considered—in addition to money velocity, production velocity, money supply, and inventory—is the maximum rate of spending.
Does money supply limit the velocity of money? Cannot one dollar spent a million times over clear the same amount of inventory as a million dollars spent once? Indeed it can.
If we are at a dollar bazaar with a number of venders all with goods at a dollar a piece, a money supply of one dollar can see all of those goods sold, eventually. If everyone there had a million dollars divided among themselves, then a million dollars worth of inventory could be cleared instantly. But with just one dollar, the rate inventory clears is dependent on the rate of spending.
Now, factor in production velocity. Say a new good enters the bazaar every five minutes during the hours it is open. If that one dollar can be spent once a minute, it can clear this new addition of inventory as well as the old inventory. If new inventory is added every minute, then that dollar can keep up, though the inventory will not change in size—it won't shrink, but at least it will not grow. If new inventory is added every 30 seconds, and if on average that dollar can be spent no faster than once a minute, then inventory will accumulate.
If a second dollar is added to the money supply at the bazaar, then the inventory can still remain in balance with new goods added every 30 seconds. A third dollar added to the money supply puts it in good shape: not only can the new inventory be accommodated, the old inventory can also clear. Once old inventory clears, production can step up to a new item every 20 seconds. If this is the maximum rate of spending, if buyers can move dollars around no faster than once per minute, then new production is limited by the $3 money supply, in a balanced economy.
In each "spend cycle" of one minute—the duration it takes for the money supply to be spent—no more than $3 of inventory can be added, or it will not clear. If $4 of inventory is added one cycle, then $1 of it would be left over, so only $2 could be added in a later cycle for things to balance out.
In the $3 economy, if two vendors are very close to one another, then the same dollar might be spent twice in a minute, whereas if sometime needs an item from a vendor far away, then a minute might go by and the dollar in transit is not spent at all. The maximum rate of money turnover should be regarded as an average.
Once money velocity hits its maximum stride, then and only then does money supply limit the amount that can be spent. If money is being spent as fast as it can, the cash-inventory equivalency can be mathematically demonstrated.