Question

Let’s return to Tallahassee hotel market we considered in Problem Set 1, but now from the perspective of a hotel manager. Consider a hotel which can supply an unlimited number of hotel rooms at the constant marginal cost c = 20 per room per night, so that the hotel’s total cost function is given by C(q) = 20q. Assume that demand for hotel rooms in Tallahassee takes two possible values: on game days, demand is described by the demand curve q = 100−p, while on non-game-days demand is described by the demand curve q = 60 − 2p. Next suppose that the hotel acts as a monopolist whose manager chooses atwhat price to offer hotel rooms for rent. We want to determine the hotel’s optimal price on non-game-days.

(h) Find the hotel’s profit π(p) as a function of its price p. (Hint: recall that if a firm has constant marginal costc, we may write its profit as π(p) =p·q(p)−c·q(p). Just plug in for c and q(p) and simplify!)

(i) Find the derivative of π(p) with respect to p. (If you haven’t already, it may help to simplify your expression for π(p) as much as possible first.)

(j) Show that the hotel’s optimal price is p= 25. What quantity is demanded at this price, and how much profit does the hotel earn? Bonus 1:Show that the hotel’s optimal price and quantity choices satisfy the margin-elasticity rule on both game days and non-game-days.

Bonus 2:Suppose I tell you that another hotel (a profit-maximizing localmonopolist in Thomasville, Georgia) sets a game-day price of p= 40, at which the price elasticity of demand is c =−4/3. What is this hotel’s marginal cost?

Answer 1

h. The profit function for the hotel on non game days is:

i. The derivative of the profit function with respect to p is:

j. At the optimal price, the derivative of the profit function with respect to p will be zero:

Quantity demanded at this price can be computed by plugging this price into the demand function for non game days: