OK, looking forward to your next installment, as clearly this is a monopoly. Two things: you will need to introduce the concepts of marginal revenue and marginal cost if you want to describe the market effectively. Lu will not artificially limit supply if there is excess demand. If he can't produce more he will raise his price, which a monopolist can do, until the quantity demanded drops far enough that MR = MC. He'll raise his price just high enough to where there is tepid demand for that last cake.

Second, "King-Lu limits purchases to one cake per person to ensure that their customers do not experience diminishing marginal utility as quickly." I've seen cases where purchases were limited when supply exceeded demand, but that was not to maintain marginal utility, it was more of a social control, like when Costco limited purchases of toilet paper to one bag per visit at the beginning of the COVID crisis. And that is very rare with private companies, it more often happens via government controls, which puts it in the realm of socialism. I've yet to hear a businessman talk about marginal utility, that is rarely something discussed on the business side. Maybe it is mentioned in the movie, but I don't think that's right. Marginal utility is a difficult thing to measure. One person's MU at 4 cakes may exceed another's at 1, so the artificial limit reduces total utility.

But maybe this is addressed further in your next installment. Good job.