Question

Corduroy Island is very rocky and very dangerous for passing ships. The island’s citizens have decided to build lighthouses to alert passing ships to the contours of the island and to avoid environmental disaster. Those on the west side of the island (“westerners”) have an inverse demand function for lighthouses given by P = 36 – 4Q and those on the east side of the island (“easterners”) have an inverse demand function for lighthouses given by P = 14 – 2Q. The total cost of building/maintaining lighthouses on the island is TC = 20Q.
a. Calculate is the marginal cost of a lighthouse.
b. If the westerners act on their own, calculate how many lighthouses they will build.
c. Calculate the westerners’ consumer surplus in part (b).
d. If the easterners act on their own, calculate how many lighthouses they will build.
e. Calculate the easterners’ consumer surplus in part (d).
f. Derive Corduroy Island’s total inverse demand function for lighthouses. (Hint: lighthouses are public goods, not private goods).
g. Based on your answer to part (f), calculate the socially-optimal number of lighthouses.
h. Calculate the net gain in consumer surplus by forcing the islanders’ to act together.

Answer 1

We have the following information

Westerners demand: P = 36 – 4Q where P is price and Q is quantity of lighthouses

Easterners demand: P = 14 – 2Q

Total Cost (TC) = 20Q

a) Marginal Cost (MC) = ΔTC/ΔQ = 20

b) If westerners act on their own then the equilibrium level of lighthouses will be determined by equating marginal revenue (MR) of Westerners with the MC

Total Revenue (TR) = Price × Quantity

TR = (36 – 4Q)Q

TR = 36Q – 4Q²

MR of Westerners = ΔTR/ΔQ = 36 – 8Q

Equating MR and MC

36 – 8Q = 20

8Q = 16

Q = 2, so Westerners will build 2 lighthouses.

c) The consumer surplus is given by the area of the triangle ABC

Area of triangle ABC = ½ × Base × Height

Consumer surplus = ½ × 2 × (36 – 20)

Consumer surplus = ½ × 2 × 16

Consumer surplus = 16

d) If Easterners act on their own then the equilibrium level of lighthouses will be determined by equating marginal revenue (MR) of Westerners with the MC

Total Revenue (TR) = Price × Quantity

TR = (14 – 2Q)Q

TR = 14Q – 2Q²

MR of Easterners = ΔTR/ΔQ = 14 – 4Q

Equating MR and MC

14 – 4Q = 20

4Q = – 6

Since, the value will be negative so Easterners will not build any lighthouse if they are to act on their own.