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I'm trying to put the following situation into one simple formula:

Imagine a store that sells a product for X dollars. They pay Y for it and profit Z (which may or may not be greater than Y). The store receives the payment A days after the sale.

What is the formula that will calculate how much money has been generated if every single penny is reinvested into more items to sell as soon as the payment is received and if the store invests in N items per day until its funds F run out (which always happens before M) and the profits are reinvested for M days, which is always greater than or equal to A? Any money still tied to products that have not yet been paid for after M days can be included as well.

So, let's say a store sells a product for 120 dollars, for which they pay 100 and from which they profit 20 dollars. The store only receives payment after 30 days so when they deliver a product they are out 100 dollars. If they sell 5 products a day, they will be out 15000 dollars when they start receiving back their money along with their profits 30 days later. They will continue investing 500 USD a day on new products until their initial funds of 10,000 USD run out.

So, to clarify:

Day 1 - 5 items invested in and sold

Day 2 - 5 items invested in sold

Day 3 - 5 items invested in sold

...

Day 20 - Funds run out as last 5 items are invested in and sold.

...

Day 31 - the 5 first items are paid for by the client, everything is reinvested and 6 items sold

Day 32 - the second 5 items are paid for by the client, everything is reinvested and 6 items sold

...

The things is, as soon as they receive their first 120 USD * 5 products = 600 USD, they will reinvest it into 6 more products and sell them immediately (with 100% of certainty) and so on so the store is always growing. Any money that is not enough to buy a product to sell is saved until there is sufficient money. How much would they make in a year?

I tried all sort of approaches for like 5 hours but putting this into one single formula is, unfortunately, beyond me at the moment.

Can it be done (I imagine so)?

amWhy
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  • Not sure this is clear. How does $A$ differ from $X$? Are you assured of selling $100%$ of available inventory each day? – lulu Aug 19 '21 at 17:53
  • X dollars I meant, please excuse me. Yes, 100% sure of selling it 100% immediately – JoeGeddsMe Aug 19 '21 at 18:16
  • I would not expect a sensible closed formula. Instead, I'd write a program. Just keep track of the available funds per day. Given that everything is deterministic (no probabilities involved) the program should not be too unpleasant. – lulu Aug 19 '21 at 18:27
  • Wow, really? This REALLY feels like there should be a closed formula. I thought about writing a program, which I could do very easily, but that's not anywhere near as elegant or fun.

    Would this situation not be kind of like compounded interest except that the interest amount is not usually equal to the price of a new product so it needs to add up until it too can contribute towards generating more interest? So, for example. If they invest 100 and get 20% back as profit, it would take one product 5 sales cycles to buy a new item with the profits but it would only take 5 products one cycle.

    – JoeGeddsMe Aug 19 '21 at 18:37
  • Integrality poses a problem. I presume that if you have $$10$ to spend and each unit costs $$3$, you can only make $3$ units, leaving a dollar available for the next day. That kind of phenomenon is very hard to capture with a simple, universal formula. Effortless with a program, of course. – lulu Aug 19 '21 at 18:40
  • Well then I am at least glad I am not as incompetent as I thought because it seemed like it should be easy while I tried but I never managed to get it. It still does feel like there should be a solution, though. I am lousy at math so I may be talking nonsense but wouldn't modular arithmetic have something to do with it since there will always be an amount between 0 and the product's cost remaining? Also, wouldn't this extra money tend to become irrelevant as the number of products increases? – JoeGeddsMe Aug 19 '21 at 18:45
  • Even for compound interest problems...if you imagine a cash flow that has withdrawals according to some formula and staggered interest payments, you will need a program. Of course simple cases do admit simple formulas. That's true in the present problem as well. – lulu Aug 19 '21 at 18:47

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