Sam Hitchem owns the dance hall in Lonely Pine, Wisconsin. He holds a dance every week. This dance hall is one of the few suitable locations in the county for unattached men and women to meet each other. Sam charges admission fee Pm for men and admission fee Pw for women. At any price, individuals are more likely to come if they expect to see more persons of the opposite sex. Sam has discovered that if he charges price Pm to men and if they expect Qw women to be in attendance, then the number of men who will attend is
Qm = 40 − Pm + 3/4 Qw.
If he charges Pw to women and if they expect Qm men to be present, the number of women who will attend is
Qw = 40 − Pw + 1/4 Qm.
(a) Rearrange the two demand equations above to produce “inverse demand functions” in which prices are expressed in terms of quantities. We have
(b) Sam chooses prices so as to maximize his total revenue PmQm + PwQw. Using your answer to the previous question, you can write Sam’s revenue simply as a function of the two variables Qm and Qw. Write this expression here.
(c) Sam chooses admission prices so as to maximize his revenue. We can solve for these admission prices by first finding the values of Qm and Qw that maximize revenue and then substituting these quantities into the demand functions to find the prices at which these quantities will be demanded. You have found an expression for revenue as a function of Qm and Qw. Set the partial derivatives of this expression with respect to Qm and Qw equal to zero and solve these two equations for the two variables Qm and Qw. Sam will maximize his revenue if the number of men who attend is __________ and the number of women who attend is ________ Therefore he will maximize his profits if he sets the admission price for men equal to ____________ and the admission price for women equal to _________