Quantity of Sports Appper 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 $2 per week, the subscription price of a game app is $1 per week, and a student has $9 per week to spend on these two items, at a consumer optimum she will purchase ---sports apps and---game apps. Consumer utility will be ---utils.
To find the consumer optimum, we need to compare the marginal utilities per dollar spent on each app.
First, let's calculate the marginal utilities per dollar spent on sports app:
Marginal Utility per Dollar of Sports App:
= Marginal Utility of Sports App / Price of Sports App
= 1200 / 2 = 600 utils/dollar
Similarly, let's calculate the marginal utilities per dollar spent on the game app:
Marginal Utility per Dollar of Game App:
= Marginal Utility of Game App / Price of Game App
= 1700 / 1 = 1700 utils/dollar
Now let's compare the marginal utilities per dollar spent on each app.
For the sports app:
600 utils/dollar
For the game app:
1700 utils/dollar
The game app provides higher utility per dollar spent compared to the sports app. Therefore, the consumer should allocate more of their budget towards the game app.
To determine how many apps of each type should be purchased, we will compare the total utility gained from purchasing different quantities of each app:
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
By comparing the marginal utility of each app for different quantities, we can determine the optimal quantities that maximize utility.
Let's calculate the marginal utility per dollar spent for each quantity of sports app:
Quantity 1:
Marginal Utility per Dollar of Sports App = 1200 / 2 = 600 utils/dollar
Quantity 2:
Marginal Utility per Dollar of Sports App = 1000 / 4 = 250 utils/dollar
Quantity 3:
Marginal Utility per Dollar of Sports App = 800 / 6 = 133.33 utils/dollar
Quantity 4:
Marginal Utility per Dollar of Sports App = 600 / 8 = 75 utils/dollar
Quantity 5:
Marginal Utility per Dollar of Sports App = 400 / 10 = 40 utils/dollar
Quantity 6:
Marginal Utility per Dollar of Sports App = 200 / 12 = 16.67 utils/dollar
Now let's calculate the marginal utility per dollar spent for each quantity of the game app:
Quantity 1:
Marginal Utility per Dollar of Game App = 1700 / 1 = 1700 utils/dollar
Quantity 2:
Marginal Utility per Dollar of Game App = 1400 / 2 = 700 utils/dollar
Quantity 3:
Marginal Utility per Dollar of Game App = 1100 / 3 = 366.67 utils/dollar
Quantity 4:
Marginal Utility per Dollar of Game App = 800 / 4 = 200 utils/dollar
Quantity 5:
Marginal Utility per Dollar of Game App = 500 / 5 = 100 utils/dollar
Quantity 6:
Marginal Utility per Dollar of Game App = 200 / 6 = 33.33 utils/dollar
Now, we can determine the optimal quantities of each app.
Given that the student has $9 per week to spend on the apps, let's find the combination that maximizes utility.
Starting with 1 sport app and 1 game app:
Total spent = 2 + 1 = $3
Total utility gain = 1200 + 1700 = 2900 utils
1 sport app and 2 game apps:
Total spent = 2 + 2 = $4
Total utility gain = 1200 + 1400 + 1100 = 3700 utils
1 sport app and 3 game apps:
Total spent = 2 + 3 = $5
Total utility gain = 1200 + 1400 + 1100 + 800 = 4500 utils
1 sport app and 4 game apps:
Total spent = 2 + 4 = $6
Total utility gain = 1200 + 1400 + 1100 + 800 + 500 = 5000 utils
1 sport app and 5 game apps:
Total spent = 2 + 5 = $7
Total utility gain = 1200 + 1400 + 1100 + 800 + 500 + 200 = 5200 utils
1 sport app and 6 game apps:
Total spent = 2 + 6 = $8
Total utility gain = 1200 + 1400 + 1100 + 800 + 500 + 200 + 200 = 5400 utils
Therefore, the optimal combination is 1 sport app and 6 game apps. The student will purchase 1 sports app and 6 game apps. The consumer utility will be 5400 utils.
To determine the optimal quantity of sports and game apps that a student should purchase, we need to consider the marginal utility per dollar obtained from each app.
1. First, let's calculate the marginal utility per dollar for sports app:
- Marginal Utility per Dollar of Sports App = Marginal Utility of Sports App / Subscription Price of Sports App
- Marginal Utility per Dollar of Sports App = 1200 / 2 = 600 utils per dollar.
2. Next, let's calculate the marginal utility per dollar for game app:
- Marginal Utility per Dollar of Game App = Marginal Utility of Game App / Subscription Price of Game App
- Marginal Utility per Dollar of Game App = 1700 / 1 = 1700 utils per dollar.
3. Now, let's determine how many apps should be purchased to maximize utility within the budget of $9 per week. We need to compare the marginal utility per dollar for both apps.
- Start by comparing the marginal utility per dollar for the first app purchased. Since the sports app has a higher marginal utility per dollar (600 utils per dollar) compared to the game app (1700 utils per dollar), it is more beneficial to purchase a sports app first.
- After purchasing the first sports app, the total spending will be $2, leaving a remaining budget of $7 ($9 - $2).
- Now, compare the marginal utility per dollar for the second app purchased. The remaining budget allows for purchasing three more sports apps or seven game apps.
- For the sports app, the marginal utility per dollar is 600 utils per dollar.
- For the game app, the marginal utility per dollar is 1400 utils per dollar.
- Since the game app has a higher marginal utility per dollar, it is more beneficial to purchase a game app as the second app.
- After purchasing the second app (game app), the total spending will be $3, leaving a remaining budget of $6 ($7 - $1).
- Continue comparing the remaining budget to the marginal utility per dollar for each app until the budget is exhausted.
4. Now let's determine the quantities of sports and game apps that should be purchased:
- With a remaining budget of $6, we compare the marginal utility per dollar for the sports and game app:
- For the sports app, the marginal utility per dollar is 800 utils per dollar.
- For the game app, the marginal utility per dollar is 1100 utils per dollar.
- Since the game app has a higher marginal utility per dollar, it is more beneficial to purchase a game app as the third app.
- After purchasing the third app (game app), the total spending will be $4, leaving a remaining budget of $5 ($6 - $1).
- With a remaining budget of $5, we compare the marginal utility per dollar for the sports and game app:
- For the sports app, the marginal utility per dollar is 600 utils per dollar.
- For the game app, the marginal utility per dollar is 800 utils per dollar.
- Again, it is more beneficial to purchase a game app.
- After purchasing the fourth app (game app), the total spending will be $5, leaving a remaining budget of $4 ($5 - $1).
- With a remaining budget of $4, we compare the marginal utility per dollar for the sports and game app:
- For the sports app, the marginal utility per dollar is 400 utils per dollar.
- For the game app, the marginal utility per dollar is 500 utils per dollar.
- Once again, it is more beneficial to purchase a game app.
- After purchasing the fifth app (game app), the total spending will be $6, leaving a remaining budget of $3 ($4 - $1).
- With a remaining budget of $3, we compare the marginal utility per dollar for the sports and game app:
- For the sports app, the marginal utility per dollar is 200 utils per dollar.
- For the game app, the marginal utility per dollar is 200 utils per dollar.
- In this case, the marginal utility per dollar is the same for both apps.
5. As we can see, all remaining budget of $3 will be spent on either sports or game app. Since their marginal utility per dollar is equal, we can choose either the sports app or the game app as the final purchase.
In conclusion, at a consumer optimum with a budget of $9 per week, the student will purchase 1 sports app and 5 game apps. The consumer utility will be the total marginal utility obtained from these apps, which is equal to 1200 (marginal utility of sports app) + 1700 (marginal utility of the first game app) + 1400 (marginal utility of the second game app) + 1100 (marginal utility of the third game app) + 800 (marginal utility of the fourth game app) + 500 (marginal utility of the fifth game app) = 6600 utils.
To find the consumer optimum, we need to compare the marginal utility per dollar spent on each app. We calculate this by dividing the marginal utility of each app by its price.
For the sports app:
Marginal utility per dollar = Marginal utility / Price = 1200 / $2 = 600 utils per dollar
For the game app:
Marginal utility per dollar = Marginal utility / Price = 1700 / $1 = 1700 utils per dollar
Now, we need to compare the marginal utility per dollar for each app and allocate the available budget accordingly. Since the game app offers a higher marginal utility per dollar, the student should allocate more of her budget towards the game app.
Let's calculate how many game apps she can afford with her budget of $9 per week:
Game apps affordability = Budget / Price = $9 / $1 = 9 game apps
Now, let's compare the marginal utility for the 9th game app and the marginal utility for the 6th sports app:
Marginal utility of 9th game app = 200
Marginal utility of 6th sports app = 200
Since both have the same marginal utility, our consumer optimum will be to purchase 6 sports apps and 9 game apps.
Therefore, the student will purchase 6 sports apps and 9 game apps. Her consumer utility will be the sum of the marginal utilities of those apps:
Consumer utility = (1200 + 1000 + 800 + 600 + 400 + 200) + (1700 + 1400 + 1100 + 800 + 500 + 200)
Consumer utility = 5400 + 4600
Consumer utility = 10000 utils