Previously, I said an actual proof of the "cash-inventory equivalency" was probably beyond my mathematical ability, but this post is the start of an attempt.
Money supply and inventory are hard to compare. Money supply is fixed—who is holding on to a given dollar will change, but the total supply of money, excepting hyperinflationary situations, remains generally stable from month-to-month. When it does change from printing, credit expansions, or credit contractions, the change occurs slowly.
By contrast, things for sale are in constant flux. They enter the market when produced or put up for resale, and then leave it once purchased, and so what is for sale is always shifting in a healthy economy, and ideally nothing remains on shelves for long. To remain in balance, that which is sold must equal that which is produced or put up for resale.
In a productive economy, inventory has to clear at the rate it is produced or added, otherwise it accumulates if produced faster, or disappears if sales are faster. If sales stop but production continues, then the producer is working for free, which is an unsustainable condition. But if production stops and inventory stagnates then prices could be anything. However inventory will clear faster and production can resume if prices are lower.
So: money velocity = production velocity, where money velocity is dollars spent on inventory over time, and production velocity is the price of goods produced over time.
They do not have to equate—producing a unit does not imply it will be bought, and excess inventory can be bought up faster than it is added—but over time sales has to balance with production, otherwise inventory accumulates if production velocity exceeds money velocity, and vanishes to nothing if the reverse.
Further posts will consider how money velocity and production velocity relate to money supply and total inventory.