Question

On Sundays people in Los Angeles consider taking a boat to Catalina Island to spend the day on the beach there. The utility that a person gets from visiting Catalina is 1 − [n/80] − p, where n is the number of visitors on the island and p is the price of round-trip transportation (by boat). (Note that a visitor obtains more satisfaction if there are fewer other visitors on the island.) The utility of staying home is zero.

Part A: In equilibrium, how many people visit the island on a given Sunday? (Your answer should depend on p.)

Part B: Suppose that the boat companies can transport people to and from the island at zero cost. If the boat transportation market is perfectly competitive, what is the equilibrium price p∗ and what is the number of visitors n∗?

Part C: Compute the total surplus in this market in the equilibrium of part (b).

Part D: How does an externality arise in this problem and is it positive or negative?

Part E: Now suppose that the demand for boat transportation is served by a monopoly firm. That is, a single firm sets the price p. What price will the monopolist set and how many people will visit the island?

Part F: Compute the total surplus for the outcome of part (e). Is the monopoly good or bad for the economy? Explain.

Answer 1

• We have U = 1 – (n/80) – p

  A person will visit the island as long has U >0, where 0 is his utility from staying home.

  We have, 1 – (n/80) – p =0,

  i.e. n* = 80(1-p)

• We are given that the cost of transportation is zero. Hence in a perfectly competitive market, the price p* = mc* = 0, i.e. the boat companies will compete against each other by cutting prices till price equals the marginal cost which in this case is 0.

  We already have, n* = 80(1-p) = 80(1-0) =80.

  Hence there will be 80 visitors.

• Consumer surplus = Utility to each person net of price paid times the number of visitors = 80*(1-(80/80)- 0) = 80*(1-1-0) = 0

  Profit for boat companies = 0 because they are perfectly competitive

  Hence Total surplus = CS + profits = 0

• In this case there is an externality in terms of the negative utility an individual has if an additional person visits the island. Hence each individual who visits the island is exerting a negative externality on other visitors. Here each person decides to visit as long as they have positive utility, i.e. utility greater than 0. Hence all 80 will visit as the price p=0.

• We need to first identify the demand curve for the monopolist.

  We have, n* = 80(1-p),

  i.e. p = 1- (n/80)

  Hence the profit function for monopolist is: π = p*n – C = p*n = n – n²/80

  (since costs are zero)

  The monopolist maximizes profits and the first order condition for doing so is:

  dπ/dn =0,

  i.e. 1 – n/40 =0,

  i.e. n^M = 40.

  Hence p = 1 – (40/80) = ½

  Hence the monopolist will set p= ½ and there will be 40 visitors.

• Consumer surplus = Utility to each person net of price paid times the number of visitors = 40*(1-(40/80)- 1/2) = 40*(1-1) = 0

  Profit for monopoly = p*n = ½*40 = 20

  Hence total surplus = 0+20 =20

  Here the monopoly is better in terms of social welfare. The monopoly leads to less number of visitors and hence reduces the extent of negative externality. In this scenario, having a monopolist is better.

  
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