a) Ben will sell 750 smoothies and Will will sell 250 smoothies.
b) The demand function that Will faces is Q = 250(p2 - p1).
c) The equilibrium prices, quantities sold, and profits net of the $250 license fee are:
p = 1.25
Q = 500
π = 375
d) The customer who is indifferent between going to Ben's or to Will's store is located at 250 miles from the West end of Wall Street.
e) Ben's price is $1.25 and his profit is $375. Will's price is $1.40 and his profit is $250.
(a) If both Ben and Will charge $1 per smoothie, then customers will be indifferent between the two stores, regardless of their location on Wall Street. This is because the price of the smoothie is the same at both stores, and the travel expenses are the same for all customers. Therefore, each store will sell 500 smoothies, for a total of 1,000 smoothies sold per day.
If Ben charges $1 per smoothie and Will charges $1.40, then customers will prefer to buy their smoothies from Ben. This is because the price of the smoothie at Ben's store is lower than the price at Will's store, plus the travel expenses. The customer who is indifferent between the two stores is located at 5 miles from the West end of Wall Street. This is because the customer's travel expenses to Ben's store will be $2.50, and the customer's travel expenses to Will's store will be $3.50. Therefore, Ben will sell 750 smoothies and Will will sell 250 smoothies.
(b) If Ben charges $p1 and Will charges $p2, then the customer who is indifferent between going to Ben's or to Will's store is located at:
x = 500 - 250(p2 - p1)
This is because the customer's travel expenses to Ben's store will be $2.50 - (5 - x)p1, and the customer's travel expenses to Will's store will be $2.50 + (5 - x)p2. Therefore, the number of customers who go to Will's store is:
250(p2 - p1)
And the number of customers who go to Ben's store is:
750 - 250(p2 - p1)
The demand function that Ben faces is:
Q = 750 - 250(p2 - p1)
And the demand function that Will faces is:
Q = 250(p2 - p1)
(c)The marginal cost of a smoothie is constant and equal to $1 for both Ben and Will. In addition, each of them pays Tuftsville $250 per day for the right to sell smoothies. Therefore, the profit net of the $250 license fee for each store is:
π = (p - 1)Q - 250
The equilibrium prices, quantities sold, and profits net of the $250 license fee are:
p = 1.25
Q = 500
π = 375
(d) The customer who is indifferent between going to Ben's or to Will's store is located at 250 miles from the West end of Wall Street.
(e) The number of customers who go to Will's store is 250 and the number of customers who go to Ben's store is 750.
Ben's price is $1.25 and his profit is $375. Will's price is $1.40 and his profit is $250.
Question
In New York City, everyone lives along Wall Street, which is 10 miles long. There are 1,000 people uniformly spread up and down Wall Street, and every day they each buy a fruit smoothie from one of the two stores located at either end of Wall Street. Customers ride their motor scooters to and from the store using $0.50 worth of gas per mile. Customers buy their smoothies from the store offering the lowest price, which is the store’s price plus the customer’s travel expenses getting to and from the store. Ben owns the store at the West end of Wall Street and Will owns the store at the East end of Wall Street.
(a) If both Ben and Will charge $1 per smoothie, how many will each of them sell in a day? If Ben charges $1 per smoothie and Will charges $1.40, how many smoothies will each sell in a day?
(b) If Ben charges p1 and Will charges p2 , what is the location of the customer who is indifferent between going to Ben’s or to Will’s store? How many customers go to Will’s store and how many go to Ben’s store? What are the demand functions that Ben and Will face?
(c) Assume that the marginal cost of a smoothie is constant and equal to $1 for both Ben and Will. In addition, each of them pays Tuftsville $250 per day for the right to sell smoothies. Find the equilibrium prices, quantities sold, and profits net of the $250 license fee. From now, assume that Ben sets his price first and then Will sets his price p2. After the prices are posted consumers get on their scooters and buy from the store with the lowest price including travel expenses.
(d) What is the location of the customer who is indifferent between going to Ben’s or to Will’s store?
(e) How many customers go to Will’s store and how many go to Ben’s store? What are Ben and Will’s prices and profits? Compare with your answer in part (c).