Exercise 5.1 [10 pts] The benefits of price discrimination. ABC is a small village that offers nothing to do except liedown and enjoy the sun. CBA in turn is a larger town with lots of business going on. There are bars and restaurantsin CBA, but also the city hall, a hospital, etc. Even though many people in ABC have a car, there is some demand forpublic transportation. This demand is served by a single bus company enjoying a monopolistic position in the market.The bus company faces a constant marginal cost of providing a bus ride for each passenger equal to $2. That means,their cost function can be written as c(y) = 2y, where y is the number of passenger-rides.There are two groups of consumers in ABC. First, youngsters have more time and are happy to hang out in BCA.However, they are dependent on their parents’ allowance, so their demand for rides to BCA is pretty elastic. A linearfunction describes their inverse demand function:p(y) = − 12 y + 5The second group of customers are adults. They have more money, and a trip to BCA is sometimes more of a necessitythan a whim for them. Their inverse demand is also linear, but less elastic:p(y) = −y + 20(i) [1 pt] Draw the inverse demand functions for both adults and youngsters in the same graph. Do they intersect?Is there a range of prices for which youngsters will not use bus services at all whereas adults will still demand somerides?(ii) [3 pts] Derive the market inverse demand function for bus rides and plot it along with individual market demands.(iii) [2 pts] Suppose the monopolist transit company has not heard about price discrimination and maximizes its profitsfacing the market demand. Write down the monopolist’s objective and solve for the optimal price and quantity. Isany of the youngsters served in this equilibrium?
Question
Exercise 5.1 [10 pts] The benefits of price discrimination. ABC is a small village that offers nothing to do except liedown and enjoy the sun. CBA in turn is a larger town with lots of business going on. There are bars and restaurantsin CBA, but also the city hall, a hospital, etc. Even though many people in ABC have a car, there is some demand forpublic transportation. This demand is served by a single bus company enjoying a monopolistic position in the market.The bus company faces a constant marginal cost of providing a bus ride for each passenger equal to $2. That means,their cost function can be written as c(y) = 2y, where y is the number of passenger-rides.There are two groups of consumers in ABC. First, youngsters have more time and are happy to hang out in BCA.However, they are dependent on their parents’ allowance, so their demand for rides to BCA is pretty elastic. A linearfunction describes their inverse demand function:p(y) = − 12 y + 5The second group of customers are adults. They have more money, and a trip to BCA is sometimes more of a necessitythan a whim for them. Their inverse demand is also linear, but less elastic:p(y) = −y + 20(i) [1 pt] Draw the inverse demand functions for both adults and youngsters in the same graph. Do they intersect?Is there a range of prices for which youngsters will not use bus services at all whereas adults will still demand somerides?(ii) [3 pts] Derive the market inverse demand function for bus rides and plot it along with individual market demands.(iii) [2 pts] Suppose the monopolist transit company has not heard about price discrimination and maximizes its profitsfacing the market demand. Write down the monopolist’s objective and solve for the optimal price and quantity. Isany of the youngsters served in this equilibrium?
Solution
(i) The inverse demand functions for both adults and youngsters can be plotted on a graph with price on the y-axis and quantity on the x-axis. The inverse demand function for youngsters is p(y) = -1/2y + 5, which is a downward sloping line starting at a price of 5 when quantity is zero. The inverse demand function for adults is p(y) = -y + 20, which is also a downward sloping line but starts at a price of 20 when quantity is zero. These two lines intersect at a price of 4 and quantity of 8. Therefore, there is a range of prices above 4 for which youngsters will not use bus services at all whereas adults will still demand some rides.
(ii) The market inverse demand function for bus rides is the horizontal sum of the individual inverse demand functions. This can be derived by adding the quantities demanded by adults and youngsters at each price level. The resulting market inverse demand function is p(y) = -3/2y + 25. This can be plotted on the same graph as the individual market demands.
(iii) The monopolist transit company maximizes its profits by setting marginal cost equal to marginal revenue. The marginal cost is constant at 13.5 and the optimal quantity is 7.67 rides. At this price, none of the youngsters are served because their maximum willingness to pay is $5, which is less than the price.
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