Question

Work & Rest is an all inclusive hotel operating as a monopoly in a mid-sized community. The manager estimates that the hotel's total costs are described by the equation C = 0.05Q2 + 10Q + 15000, where Q is the number or persons serviced and P is the price paid per person. Although it is located in a mid-sized community, it is close to a major highway that runs along the east coast. Often times, travelers would use the services of Work & Rest as a stop in their commute between two major citys. As such, the market demand for its service is described by the equation P = 160 - 0.2Q. 1. Given the current market conditions, Work & Rest's profit-maximizing quantity is ___________ and the price charged per person is__________  ( 4 points) Consider that after a series of reports from special interest groups, the local government decides to regulate Work & Rest by using the optimal pricing policy. 2. Given the successful implementation of the government's policy, Work & Rest's profit-maximizing quantity is__________  (2 points) 3. Given the successful implementation of the government's policy, what is Work & Rest's profit at its profit maximizing quantity_____________?  (4 points)I 4. If the policy remains in place, and there are no further external shocks to the market, is it likely that West & Rest will continue to operate and provide service? (In the space provided enter will or will not) In the long run, West & Rest _______________ continue to provide service in this mid-sized community. (2 points)

Answer 1

Note that for a monopolist, profits are maximized where marginal revenue equals the marginal revenue. Marginal cost is the derivative of the total cost with respect to Q. Marginal revenue is the derivative of the total revenue with respect to Q, where the total revenue is the product of price and quantity.

Thus, TR= PQ= (160-0.2Q)Q= 160Q-0.2Q^2

MR= 160-0.4Q

TC= 0.05Q^2+10Q+15000

MC= 0.1Q+10

Setting MR=MC will give us: 160-0.4Q=0.1Q+10

0.5Q=150

Q= 150/0.5= 300

P= 160-0.2(300)=160-60=$100

1. Thus the profit-maximizing output is 300 and the price charged per person is $100.

2. When the optimal price policy is implemented, then the price would be set equal to the marginal cost. Thus, setting P equal to MC will give us

160-0.2Q=0.1Q+10

0.3Q=150

Q=500

3. Profit= Total revenue-Total cost

Thus at Q=500, profit= (PQ)-(TC)=(160*500-0.2*500*500)-(0.05*500*500+10*500+15000)=(80000-50000)-(12500+5000+15000)=-2500 which is a loss of $2500.

4. If the firm continues to make losses in the long run as we saw above, then it will not operate to provide services.