Suppose Bill is on a low-carbohydrate diet. He can eat only three foods: Rice Krispies, cottage cheese, and popcorn. The marginal utilities for each food are tabulated below. Bill is allowed only 167 grams of carbohydrates daily. Rice Krispies, cottage cheese, and popcorn provide 25, 6, and 10 grams of carbohydrates per cup, respectively. Referring to the accompanying table, respond to the following questions:
Units of food (cups/day) - Marginal utility of Rice Krispies - Marginal utility of cottage cheese -Marginal utility of popcorn
1 175 72 90
2 150 66 80
3 125 60 70
4 100 54 60
5 75 48 50
6 50 36 40
7 25 30 30
8 25 18 20
a. Given that Bill can consume only 167 grams of carbohydrates daily, how many cups of each food will he consume daily? Show your work.
b. Suppose Bill's doctor tells him to further reduce his carbohydrate intake to 126 grams per day. What combination will he consume?
a) 5/8/7 cups of RK/CC/PC
b) 2/6/4 cups of RK/CC/PC
a. To determine how many cups of each food Bill will consume daily, we need to calculate the marginal utility per gram of carbohydrates for each food.
Rice Krispies: Marginal Utility / Grams of Carbohydrates per cup
1 cup: 175 / 25 = 7
2 cups: 150 / (2 * 25) = 3
3 cups: 125 / (3 * 25) = 1.67
4 cups: 100 / (4 * 25) = 1
5 cups: 75 / (5 * 25) = 0.6
6 cups: 50 / (6 * 25) = 0.33
7 cups: 25 / (7 * 25) = 0.2
8 cups: 25 / (8 * 25) = 0.125
Cottage Cheese: Marginal Utility / Grams of Carbohydrates per cup
1 cup: 72 / 6 = 12
2 cups: 66 / (2 * 6) = 5.5
3 cups: 60 / (3 * 6) = 3.33
4 cups: 54 / (4 * 6) = 2.25
5 cups: 48 / (5 * 6) = 1.6
6 cups: 36 / (6 * 6) = 1
7 cups: 30 / (7 * 6) = 0.71
8 cups: 18 / (8 * 6) = 0.375
Popcorn: Marginal Utility / Grams of Carbohydrates per cup
1 cup: 90 / 10 = 9
2 cups: 80 / (2 * 10) = 4
3 cups: 70 / (3 * 10) = 2.33
4 cups: 60 / (4 * 10) = 1.5
5 cups: 50 / (5 * 10) = 1
6 cups: 40 / (6 * 10) = 0.67
7 cups: 30 / (7 * 10) = 0.43
8 cups: 20 / (8 * 10) = 0.25
To determine how many cups of each food Bill will consume daily, we'll need to find the combination that maximizes his total utility given a total of 167 grams of carbohydrates.
Using the marginal utility per gram of carbohydrates, we can rank the foods from highest to lowest utility per gram of carbohydrates:
1. Rice Krispies: 7
2. Popcorn: 9
3. Cottage Cheese: 12
Starting with the food with the highest utility per gram of carbohydrates, Bill will consume as many cups of Rice Krispies as possible without exceeding the carbohydrate limit of 167 grams.
167 grams / 25 grams per cup = 6.68 cups
Since he cannot consume a fraction of a cup, Bill will consume 6 cups of Rice Krispies, which provides a total of 150 grams of carbohydrates.
Next, we move to the next food in the ranking, which is Popcorn. We have 167 - 150 = 17 grams of carbohydrates remaining.
17 grams / 10 grams per cup = 1.7 cups
Again, Bill cannot consume a fraction of a cup, so he will consume 1 cup of Popcorn, which provides an additional 10 grams of carbohydrates.
Lastly, we have 167 - 150 - 10 = 7 grams of carbohydrates remaining.
7 grams / 6 grams per cup = 1.17 cups
Bill will consume 1 cup of Cottage Cheese, which provides the remaining 6 grams of carbohydrates.
Therefore, Bill will consume 6 cups of Rice Krispies, 1 cup of Popcorn, and 1 cup of Cottage Cheese daily while staying within his carbohydrate limit.
b. If Bill's carbohydrate intake is reduced to 126 grams per day, we need to recalculate the cups of each food he will consume using the same process.
Using the updated carbohydrate limit, the ranking of foods based on utility per gram of carbohydrates remains the same:
1. Rice Krispies: 7
2. Popcorn: 9
3. Cottage Cheese: 12
Starting with Rice Krispies:
126 grams / 25 grams per cup = 5.04 cups
Bill can consume 5 cups of Rice Krispies, providing a total of 125 grams of carbohydrates.
Next, we move to Popcorn:
126 - 125 = 1 gram of carbohydrates remaining.
1 gram / 10 grams per cup = 0.1 cups
Bill cannot consume a fraction of a cup, so he cannot consume any Popcorn.
Lastly, with 126 - 125 - 0 = 1 gram of carbohydrates remaining:
1 gram / 6 grams per cup = 0.17 cups
Again, Bill cannot consume a fraction of a cup, so he cannot consume any Cottage Cheese.
Therefore, given the reduced carbohydrate limit of 126 grams, Bill will consume 5 cups of Rice Krispies and no cups of Popcorn or Cottage Cheese.
To find the number of cups of each food that Bill will consume daily, we need to determine the optimal combination that maximizes his total utility while staying within the carbohydrate limit.
a. To solve this problem, we can use the concept of marginal utility per gram. This is calculated by dividing the marginal utility of a food item by the number of grams of carbohydrates it provides per cup.
First, let's calculate the marginal utility per gram for each food:
- Marginal utility per gram of Rice Krispies = Marginal utility of Rice Krispies / Carbohydrates per cup of Rice Krispies = (175 / 25) = 7
- Marginal utility per gram of cottage cheese = Marginal utility of cottage cheese / Carbohydrates per cup of cottage cheese = (72 / 6) = 12
- Marginal utility per gram of popcorn = Marginal utility of popcorn / Carbohydrates per cup of popcorn = (90 / 10) = 9
Next, we need to find the combination of cups that maximizes the total marginal utility within the carbohydrate limit. We can start with the food that has the highest marginal utility per gram, which is cottage cheese.
1. Cottage Cheese:
Divide the carbohydrate limit (167 grams) by the amount of carbohydrates per cup of cottage cheese (6 grams) to find the maximum cups of cottage cheese Bill can consume: 167 / 6 ≈ 27.83 cups.
Since Bill cannot consume a fraction of a cup, he can only consume a maximum of 27 cups of cottage cheese, which will provide 162 grams of carbohydrates (27 cups x 6 grams of carbohydrates per cup).
Now, subtract the carbohydrate amount consumed from the total carbohydrate limit: 167 - 162 = 5 grams remaining.
2. Rice Krispies:
Divide the remaining carbohydrate limit (5 grams) by the amount of carbohydrates per cup of Rice Krispies (25 grams) to find the maximum cups of Rice Krispies Bill can consume: 5 / 25 = 0.2 cups.
Since Bill cannot consume a fraction of a cup, he will consume 0 cups of Rice Krispies.
3. Popcorn:
Divide the remaining carbohydrate limit (5 grams) by the amount of carbohydrates per cup of popcorn (10 grams) to find the maximum cups of popcorn Bill can consume: 5 / 10 = 0.5 cups.
Again, Bill cannot consume a fraction of a cup, so he will consume 0 cups of popcorn as well.
Therefore, Bill will consume 27 cups of cottage cheese, 0 cups of Rice Krispies, and 0 cups of popcorn daily.
b. If Bill's carbohydrate intake needs to be reduced to 126 grams per day, we need to adjust the calculations using the new carbohydrate limit.
1. Cottage Cheese:
The maximum cups of cottage cheese Bill can consume will be: 126 / 6 = 21 cups.
The carbohydrate amount consumed will be: 21 cups x 6 grams of carbohydrates per cup = 126 grams.
The remaining carbohydrate limit will be: 126 - 126 = 0 grams.
2. Rice Krispies:
Divide the remaining carbohydrate limit (0 grams) by the amount of carbohydrates per cup of Rice Krispies (25 grams) to find the maximum cups of Rice Krispies Bill can consume: 0 / 25 = 0 cups.
Bill will consume 0 cups of Rice Krispies.
3. Popcorn:
Divide the remaining carbohydrate limit (0 grams) by the amount of carbohydrates per cup of popcorn (10 grams) to find the maximum cups of popcorn Bill can consume: 0 / 10 = 0 cups.
Bill will consume 0 cups of popcorn as well.
Therefore, Bill will consume 21 cups of cottage cheese, 0 cups of Rice Krispies, and 0 cups of popcorn daily to meet the reduced carbohydrate intake of 126 grams per day.