Question

suppose that the hotel acts as a monopolist whose manager chooses what quantity q of rooms to offer for rent. We want to determine the hotel’s optimal choice of quantity on game days.

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.1 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.

Find the hotel’s total revenue on game days as a function of its quantity choice q. (Recall that total revenue equals price times quantity, where in this case price is described by the inverse demand curve.)

(e) Assuming the hotel maximizes profit, show that it will supply quantity q = 40 on game days.

(f) What will be the hotel price on game days? And what will be the hotel’s game-day profits?

(g) Still focusing on game days, graphically illustrate the demand curve, the hotel’s marginal revenue curve, and the hotel’s marginal cost curve. Indicate the hotel’s optimal quantity and price choices on the graph.

Answer 1

Total Cost = 20q

​​​​​​Demand Equation in game days : q = 100 - p

So inverse demand function : p = 100 - q

Total Revenue = P*Q

= (100 - q) * q

= 100q - q^{​​​​​​2}

So, Hotels total revenue on game days is 100q - q_​​^{​​​​​​2} ​​​​​​

Now,

Marginal cost ( MC) = d (Total cost) /dq

=20

Marginal revenue (MR) = d ( Total revenue)/dq

= 100 - 2q

e) Profit maximizing equilibrium is at :-

MC = MR

20 = 100 - 2q

2q = 100 - 20

2q = 80

q = 40

So q = 40 at profit maximizing output on game days.

f) From inverse demand function p = 100 - q on game days

So p = 100 - 40

= 60

So p = 60 on game days.

Profit = Total Revenue - Total Cost

= (100q - q^{​​​​​​2} ) - 20q

= 80q - q^{​​​​​​2}

At q = 40 i.e. game days profit-maximizing output.

Profit = (80*40) - 40²

= 3200 - 1600

= 1600

Hotels game day profit is 1600

g) Following graph represents the Demand curve (D), MR curve, MC curve and the Equilibrium(E) :-

Point E represents MC = MR.

Equilibrium Price is determined by the demand curve.