Profit Function where total revenue is re-spent on production?

Question

I'm trying to find a function that finds the net profit gain over multiple iterations. For example, if I produce a unit for 5 dollars , and can sell it for 8 dollars, I would have a net gain of 3 dollars per unit. However, let's say I have 50,000 dollars as a loan (for theory sake I don't have to repay it) - I can produce 10,000 units at a total profit of 30,000 dollars, which then means I have 80,000 dollars. I would then re-invest this ENTIRE quantity into producing far more units than originally (I would then be creating 16,000 units which would result in a re-investment of 128,000 dollars) and so on and so forth.

I'm assuming a perfect system where there's no price fluctuations on the sell price as well as no additional costs.

So far I can calculate the total profit of one cycle of the system, but other than that I have absolutely no idea where to start.

Answer 1

Let's derive an equation that gives you profits and total wealth after reinvesting for $T\ge 1$ iterations.

The per-unit profit $\pi$ depends on per-unit production cost $c$ and revenue $r$, $\pi=r-c$. Your initial endowment of wealth is $w_1$. In the first iteration, using all your wealth $w_1$ you can produce $w_1/c$ units (I assume for simplicity that this is an integer value), and the total profit from the first iteration is $$\Pi_1=\pi w_1/c=(r-c)w_1/c=rw_1/c-w_1.$$ In the second iteration, your wealth is $$w_2=w_1+\Pi_1=w_1+rw_1/c-w_1=rw_1/c.$$ The total profit in the second iteration is $$\Pi_2=\pi w_2/c=rw_2/c-w_2=r^2w_1/c^2-rw_1/c.$$ In the third iteration, your wealth is $$w_3=w_2+\Pi_2=r^2w_1/c^2,$$ and the total profit is $$\Pi_3=\pi w_3/c=rw_3/c-w_3=r^3w_1/c^3-r^2w_1/c^2.$$

You can go on like this, but using induction you will find that for iteration $T$, you get a profit of $$\Pi_T= r^Tw_1/c^T-r^{T-1}w_1/c^{T-1}.$$ The total money you have after iteration $T$ is $$w_{T+1}=\Pi_T+w_T=r^Tw_1/c^T.$$

In your case, plug in the values $c=5$, $r=8$, and $w_1=50,000$.

Just one non-mathematical remark: it is not plausible that conditions such as prices (revenue $r$) or costs $c$ do not change if you produce a lot as $T\to\infty$.