Question

2. Let F(n, m) denote the most amount of money you can spend on stuff from the store for a given n, m, and p. Give a recursive definition of F:

(a) Base case: (Hint: your base case will occur when m = 0 since you have no items left to consider buying). F(n, 0) = if n ≥ 0 F(n, m) = −[infinity] if n < 0 (helpful case for the part

(b)) (b) Recursive case: (Hint: compare the total you can spend if you buy the mth item to the total you can spend if you don’t buy the mth item and take the maximum of the two options). F(n, m) = max(

Answer 1

The function F(n, m) provides a recursive definition for determining the maximum amount of money that can be spent given a budget n and m items. The base case occurs when there are no items to consider, while the recursive case compares spending outcomes with or without the last item, leveraging the marginal utility per dollar to maximize total utility.

Step-by-step explanation:

The student's question concerns the recursive definition of a function F(n, m) used to determine the most amount of money that can be spent on items from a store given a budget of n and m items to consider. The base case of this recursive function is when m = 0, meaning there are no items left to consider for purchase. In this case:

• If n ≥ 0, then F(n, 0) = 0, since you cannot spend any money if there are no items to buy.
• If n < 0, then F(n, 0) = -∞, representing an invalid scenario as you cannot have a negative budget.

The recursive case involves comparing the total spent if you include the mth item versus if you don't, which can be expressed as:

F(n, m) = max(F(n - Pm, m - 1) + Qm, F(n, m - 1))

where Pm and Qm represent the price and quantity of the mth item, respectively. You are essentially deciding whether to spend money on the mth item (subtract its price from your budget and add its quantity) or skip it entirely. The recursive case capitalizes on the concept of maximizing marginal utility per dollar spent until the entire budget is exhausted or all items are considered.