Four roommates are planning to spend the weekend in their dorm room watching old movies, and they are debating how many to watch. Here is their willingness to pay for each film:
Film Orson Alfred Woody Ingmar
1st. $7 $5 $3 $2
2nd. 6 4 2 1
3rd. 5 3 1 0
4th. 4 2 0 0
5th. 3 1 0 0
a. Within the dorm room, is the showing of a movie a public good? Why or why not?
b. If it cost $8 to rent a movie, how many movies should the roommates rent to maximize total surplus?
c. If they choose the optimal number from part(b)and then split the cost of renting the movies equally, how much surplus does each person obtain from watching the movies?
d. Is there any way to to split the cost to ensure that everyone benefits? What practical problems does this solution raise?
e. Suppose they agree in advance to choose the efficient number and to split the cost of the movies equally. When Orson is asked his willingness to pay, will he have an incentive to tell the truth? If so, why? If not, what will he be tempted to say?
f. What does this example teach you about the optimal provision of public goods?
a. Within the dorm room, the showing of a movie is not a public good. A public good is a good that is non-excludable, meaning it cannot be denied to anyone, and non-rivalrous, meaning its consumption by one person does not diminish its availability to others. In this case, the movie showing is a private good because only the roommates can enjoy it, and they can exclude others from entering their dorm room.
b. To determine how many movies should be rented to maximize total surplus, we need to find the point where the marginal benefit equals the marginal cost. Let's calculate the total surplus for each possible number of movies rented:
For 1 movie:
- Total willingness to pay = $7 + $5 + $3 + $2 = $17
- Total cost = $8
- Total surplus = $17 - $8 = $9
For 2 movies:
- Total willingness to pay = $7 + $5 + $3 + $2 + $6 + $4 + $2 + $1 = $30
- Total cost = $16
- Total surplus = $30 - $16 = $14
For 3 movies:
- Total willingness to pay = $7 + $5 + $3 + $2 + $6 + $4 + $2 + $1 + $5 + $3 + $1 + $0 = $39
- Total cost = $24
- Total surplus = $39 - $24 = $15
For 4 movies:
- Total willingness to pay = $7 + $5 + $3 + $2 + $6 + $4 + $2 + $1 + $5 + $3 + $1 + $0 + $4 + $2 + $0 + $0 = $43
- Total cost = $32
- Total surplus = $43 - $32 = $11
For 5 movies:
- Total willingness to pay = $7 + $5 + $3 + $2 + $6 + $4 + $2 + $1 + $5 + $3 + $1 + $0 + $4 + $2 + $0 + $0 + $3 + $1 + $0 + $0 = $46
- Total cost = $40
- Total surplus = $46 - $40 = $6
The number of movies that maximizes total surplus is 3, with a surplus of $15.
c. If they choose to rent 3 movies and split the cost equally, each person would pay $8/4 = $2. The surplus obtained by each person can be calculated by subtracting their equal share of the cost from their willingness to pay:
- Orson's surplus = $7 + $5 + $3 - $2 = $13
- Alfred's surplus = $5 + $3 + $1 - $2 = $7
- Woody's surplus = $3 + $1 + $0 - $2 = $2
- Ingmar's surplus = $2 + $0 + $0 - $2 = $0
d. To split the cost in a way that ensures everyone benefits, the cost can be allocated proportionally to each person's willingness to pay. However, this solution raises practical problems as it requires accurate information about each person's willingness to pay, and it may be challenging to determine precise values.
e. If they agree to choose the efficient number and split the cost equally, Orson would have an incentive to tell the truth about his willingness to pay. Since the cost is divided equally, it wouldn't directly affect his payment regardless of his stated willingness to pay. Therefore, he has no temptation to be dishonest.
f. This example illustrates the importance of considering the marginal benefit and marginal cost in determining the optimal provision of public goods. By aligning the quantity of movies rented with the point where marginal benefit equals marginal cost, total surplus can be maximized. Additionally, it highlights the challenges in cost allocation and the potential for free-riding in public goods provision.