# Marginal Utility Problem

The local mall has a make-your-own sundae shop. They charge customers 35 cents for each fresh fruit topping and 25 cents for each processed topping. Barbara is going to make herself a sundae. The total utility that she receives from each quantity of topping is given by the following table:

If money is no object, how many fresh fruit toppings and processed toppings will Barbara purchase to maximize utility?

I'm guessing "money is no object" means unlimited money. The answer is 5 fresh fruit toppings & 2 processed toppings. However, at the 6th topping it does not make sense to purchase any toppings because both give negative marginal utility. Isn't the answer 3 fresh fruit toppings and 2 processed toppings?

• Can you please explain why you think it's 3 fresh fruit toppings? Commented Sep 13, 2022 at 9:22
• First, you are indifferent between the first unit of each topping (i.e., the marginal utility of an extra unit of either topping when you have 0 is 10). Your understanding of marginal utility is a bit off. The marginal utility for the fresh fruit toppings is negative at the 5th unit, not the 6th unit. That is, if you go from 5 to 6 units you experience a marginal disutility of 2. I also don't understand why you think you have to consume the toppings in a particular order. Regardless,, you stated that you stop consuming when the marginal utility is negative, so again, why 3 toppings of fruit? Commented Sep 13, 2022 at 10:04
• @EliJ the OP seems to mean that you'd purchase number 1 and 2 processed toppings and then number 3, 4 and 5 fresh fruit toppings, because this sequence has maximal marginal utility at each step. However, this is of course a misunderstanding of the table presented. Commented Sep 13, 2022 at 12:10

If money is no object, Barbara can afford any number of toppings of either kind.

So all she needs to do is find the row in each table that gives her the maximum utility.

• For fresh fruit toppings, 30 is the maximum utility and that corresponds to 5 toppings.
• For processed toppings, 20 is the maximum utility value and that corresponds to 2 toppings.

If Barbara had a limited amount of money then she might not be able to afford all the toppings she wanted and then we'd need to try to find the maximum total utility she could obtain given the money that she had.

• Could I also compare marginal utilities & always choose the bigger marginal utility until both topping types have negative marginal utilities & then it wouldn't make sense to buy any more toppings? For example, for my first topping, since I don't care about money I can just look for which has the greater marginal utility. Both the 1st fresh fruit topping gives 10 as well as the 1st processed topping. So I could choose either. Commented Sep 13, 2022 at 21:24
• Let's say I choose fresh fruit. I can then choose to buy a 2nd fresh fruit topping (a marginal utility of 8) or buy my first processed topping (a marginal utility of 10). 10>8 so my 2nd topping should be processed. So there are many several ways to optimize utility right since I could have chosen to have my first topping be processed. Commented Sep 13, 2022 at 21:24
• If I do have a limited budget, I would have to compare the marginal utilities per dollar right? Commented Sep 13, 2022 at 21:25
• No, because there's no guarantee that you've found the global maximum, you may have only found a local one. We seem to be digressing into other questions that you should ask as new questions. Commented Sep 13, 2022 at 21:26

That's a strange question. The toppings have positive prices but "money is no object". Well, if you take these assumptions seriously, then the question how many toppings she would purchase doesn't have a unique answer. Since free disposal is usually (tacitly) assumed, she will purchase any number $$p\ge 2$$ of processed toppings and any number $$f\ge 5$$ of fresh fruit toppings, only to throw away all but $$2$$ processed and all but $$5$$ fresh fruit toppings for a maximized total utility of $$50$$ (assuming additive utility, which is not even mentioned).