Quantity of Sports App per Week: 1, 2, 3, 4, 5, 6
Marginal Utility of Sports App: 1200, 1000, 800, 600, 400, 200
Quantity of Game App per Week: 1, 2, 3, 4, 5, 6
Marginal Utility of Game App: 1700, 1400, 1100, 800, 500, 200
Refer to the table above. If the subscription price for a sports app is $1 per week, the subscription price of a game app is $2 per week, and a student has $13 per week to spend on these two items, at a consumer optimum she will purchase---sports apps and--- game apps.
To determine the consumer's optimum, we need to compare the marginal utility per dollar of each app. This can be found by dividing the marginal utility of each app by its respective subscription price.
For the sports app:
- Marginal utility per dollar for 1 sports app: 1200/1 = 1200
- Marginal utility per dollar for 2 sports apps: 1000/2 = 500
- Marginal utility per dollar for 3 sports apps: 800/3 ≈ 267
- Marginal utility per dollar for 4 sports apps: 600/4 = 150
- Marginal utility per dollar for 5 sports apps: 400/5 = 80
- Marginal utility per dollar for 6 sports apps: 200/6 ≈ 33.33
For the game app:
- Marginal utility per dollar for 1 game app: 1700/2 = 850
- Marginal utility per dollar for 2 game apps: 1400/4 = 350
- Marginal utility per dollar for 3 game apps: 1100/6 ≈ 183.33
- Marginal utility per dollar for 4 game apps: 800/8 = 100
- Marginal utility per dollar for 5 game apps: 500/10 = 50
- Marginal utility per dollar for 6 game apps: 200/12 ≈ 16.67
Considering the student has $13 per week to spend, let's calculate the maximum number of apps the student can purchase for each:
For sports apps:
- $1 per sports app
- $13 budget
- Maximum sports apps = $13 / $1 = 13
For game apps:
- $2 per game app
- $13 budget
- Maximum game apps = $13 / $2 = 6.5 (round down to 6)
Comparing the marginal utility per dollar, the consumer optimum is to purchase:
- Sports apps: 6 apps (maximum affordability)
- Game apps: 6 apps (maximum affordability)
Therefore, at the consumer optimum, the student will purchase 6 sports apps and 6 game apps.
To find the consumer optimum, we need to calculate the total utility received from each combination of sports apps and game apps, considering the budget constraint.
Let's start by calculating the total utility for each combination of apps:
For 1 sports app and 1 game app:
Total utility = Marginal utility of sports app + Marginal utility of game app
Total utility = 1200 + 1700 = 2900
For 1 sports app and 2 game apps:
Total utility = Marginal utility of sports app + Marginal utility of game app + Marginal utility of game app
Total utility = 1200 + 1400 + 1400 = 4000
For 1 sports app and 3 game apps:
Total utility = Marginal utility of sports app + Marginal utility of game app + Marginal utility of game app + Marginal utility of game app
Total utility = 1200 + 1400 + 1100 + 1100 = 3900
For 1 sports app and 4 game apps:
Total utility = Marginal utility of sports app + Marginal utility of game app + Marginal utility of game app + Marginal utility of game app + Marginal utility of game app
Total utility = 1200 + 1400 + 1100 + 800 + 800 = 5300
For 1 sports app and 5 game apps:
Total utility = Marginal utility of sports app + Marginal utility of game app + Marginal utility of game app + Marginal utility of game app + Marginal utility of game app + Marginal utility of game app
Total utility = 1200 + 1400 + 1100 + 800 + 500 + 500 = 4600
For 1 sports app and 6 game apps:
Total utility = Marginal utility of sports app + Marginal utility of game app + Marginal utility of game app + Marginal utility of game app + Marginal utility of game app + Marginal utility of game app + Marginal utility of game app
Total utility = 1200 + 1400 + 1100 + 800 + 500 + 200 + 200 = 4400
Now, let's calculate the maximum number of apps the student can purchase within the budget constraint.
With $13 per week, the student can buy at most 6 sports apps or 6 game apps, as the prices are $1 and $2 per week respectively.
Since the student can only spend $13 per week, her maximum expenditure on sports apps would be $6 and on game apps would be $12.
Comparing the total utilities, we find that the highest total utility is obtained when purchasing 1 sports app and 4 game apps, which yields a total utility of 5300.
Therefore, at the consumer optimum, the student will purchase 1 sports app and 4 game apps.
To find the consumer optimum, we need to compare the marginal utility per dollar spent for both the sports app and the game app.
First, let's calculate the marginal utility per dollar spent for each app. This can be done by dividing the marginal utility by the price of the app.
For the sports app:
Marginal utility per dollar spent = Marginal utility of sports app / Price of sports app = 1200 / 1 = 1200
For the game app:
Marginal utility per dollar spent = Marginal utility of game app / Price of game app = 1700 / 2 = 850
Now, we can determine the consumer optimum. Since the student has $13 per week to spend on the apps, we need to find the combination of sports apps and game apps that will maximize the total marginal utility within this budget.
Let's start with purchasing one sports app. The total cost would be $1 and the total marginal utility would be 1200.
With $12 left to spend, we can then purchase 6 game apps (since $2 per app) with a total cost of $12. The total marginal utility would be 200 * 6 = 1200.
Therefore, at the consumer optimum, the student will purchase 1 sports app and 6 game apps.