Question

15. Suppose you own a firm in the transportation business that is looking to expand its services to three cities. You hire a research firm to conduct some market research and they report the following demand for bus tickets in the past to you. Halle 2011 Leipzig 2011 Dresden 2011 2010 2009 2010 2009 2010 2009 City Year Bus tickets sold Price per ticket 80 6040 80 60 90 80 60 90 20€ 40€ 60€ 40€ 80€ 20€ 10€ 20€ 5€ 1. Find the demand functions for each city. ii. Calculate the total maximal revenue if you can set different optimal prices in each city. iii. Find the total demand function for all cities combined and calculate total maximal revenue if you can only charge one price in all 3 cities. Following your company's current cost structure, you can sell bus tickets according to a supply function of Q.= 2P and each city represents an independent market again. iv. To which city would you expand first, second and third if you wanted to maximize producer surplus at each step? A few years after expanding to all three cities, you read in the newspaper that the municipalities Halle, Leipzig and Dresden intend to increase their revenues in order to build more roads and to improve road quality. To do so, they plan to levy a 5€ fee on every bus ticket sold that has to be paid for by the buyer. v. In which city will you expect the fee to have the largest relative impact on your producer surplus? Calculate the relative post-tax levels of producer surplus for each city. A few more years down the road, the tax is removed and the 3 cities become a unified market with one aggregate demand function consisting of the 3 previous demand functions. vi. Calculate the equilibrium price and quantity in the unified market. vii. If the government introduces a price ceiling of 50€ per ticket, what will be the precise effect on consumer surplus? Do consumers gain or lose welfare?

Answer 1

(i) For City Halle, For City Leipzig, For City Dresden,

In 2011, Q_d11 = 80 - 20$\euro$ In 2011, Q_d21 = 80 - 40$\euro$    In 2011, Q_d31 = 80 - 10$\euro$

In 2010, Q_d12 = 60 - 40$\euro$ In 2010, Q_d22 = 60 - 80$\euro$ In 2010, Q_d32 = 60 - 20$\euro$

In 2009, Q_d13 = 40 - 60$\euro$   In 2009, Q_d23 = 90 - 20$\euro$ In 2009, Q_d33 = 90 - 5$\euro$

(ii) Total Maximal revenue of city Halle = (80 - 20$\euro$) + (60 - 40$\euro$) + (40 - 60$\euro$) = 180 - 120$\euro$

Total Maximal revenue of city Leipzig = (80 - 40$\euro$) + (60 - 80$\euro$) + (90 - 20$\euro$) = 230 - 140$\euro$

Total Maximal revenue of city Dresden = (80 - 10$\euro$) + (60 - 20$\euro$) + (90 - 5$\euro$) = 230 - 35$\euro$

(iii) Total demand function for all cities combined = 640 - 295$\euro$

# We can assume that the only one price that can charge in all three cities is 40$\euro$​​​​​​​.

Now, the total maximal revenue of city Halle =  (80 - 40$\euro$) + (60 - 40$\euro$) + (40 - 40$\euro$) = 180 - 120$\euro$

the total maximal revenue of city Leipzig = (80 - 40$\euro$) + (60 - 40$\euro$) + (90 - 40$\euro$) = 230 - 120$\euro$

the total maximal revenue of city Dresden = (80 - 40$\euro$) + (60 - 40$\euro$) + (90 - 40$\euro$) = 230 - 120$\euro$

(iv) Now the Supply function is Qs = 2P

If I wanted to maximize producer surplus at each step then I should expand the City Dresden firstly, secondly I should expand City Halle and thirdly the City Leipzig.

(v) To build more roads and to improve road qualities, the municipalities of all three cities will plan to levy a 5$\euro$ on every bus ticket sold. So now the total maximal revenue of each cities can be written as follows,

For City Halle, it will be (180 - 120$\euro$) + 5$\euro$ = 180 - 115$\euro$

For City Leipzig, it will be (230 - 140$\euro$) + 5$\euro$ = 230 - 135$\euro$

For City Dresden, it will be (230 - 35$\euro$) + 5$\euro$ = 230 - 30$\euro$

* In City Leipzig, I will expect that the fee to have the largest relative impact on my producer surplus.

* In each city the relative post tax levels of producer surplus are as follows,

For City Halle, it will be (180 - 115$\euro$), For City Leipzig, it will be (230 - 135$\euro$) and For City Dresden, it will be (230 - 30$\euro$)

(vi) After few more years, the tax is removed and three cities become a unified market with one aggregate demand function consisting of the 3 previous demand functions. So the Aggregate Demand function will be = Q_d = 640 - 295$\euro$ (after removing the tax from all three cities)

Now we can determine the equilibrium price and quantity with the help of equilibrium condition.

Therefore, Q_d = Q_s

or, 640 - 295$\euro$ = 2P [ Here the P = 40$\euro$ ]

or, 640 - 295$\euro$ = 2 x 40$\euro$ or, 640 = 295$\euro$ + 80$\euro$ or, 640 = 375$\euro$

Hence, the equilibrium price is 375$\euro$ and equilibrium quantity is 640