The Denver Athlete’s Club (DAC) is a private, not-for-profit athletic club located in Denver, Colorado. DAC currently has 3,500 members but is planning on a membership drive to increase this number significantly. An important issue facing Jessica Nicholson, DAC’s administrative director, is the determination of an appropriate membership level. In order to efficiently employ scarce DAC resources, the board of directors has instructed Nicholson to maximize DAC’s operating surplus, defined as revenues minus operating costs. They have also asked Nicholson to determine the effects of a proposed agreement between DAC and a neighboring club with outdoor recreation and swimming pool facilities. Plan A involves paying the neighboring club $100 per DAC member. Plan B involves payment of a fixed fee of $400,000 per year. Finally, the board has determined that the membership fee for the coming year will remain constant at $2,500 per member irrespective of the number of new members added and whether Plan A or plan B is adopted. In the calculations for determining an optimal membership level, Nicholson regards price as fixed; therefore, P=MR=$2,500. Before considering the effects of any agreement with the neighboring club, Nicholson projects total and marginal cost relations during the coming year to be as follows:
TC= $3,500,000 + $500Q + $0.25Q^2 (Q^2 means squared)
Where Q is the number of DAC members.
(A) Before considering the effects of the proposed agreement with the neighboring club, calculate DAC’s optimal membership and operating surplus levels.
(B) Calculate these levels under plan A.
(C) Calculate these levels under plan B.