Promoters of a major college basketball tournament estimate that the demand for tickets for adults and by students are given by:

\begin{aligned} q_a&=5,000-10p_a\\ q_s&=10,000-100p_s\\ \end{aligned}

where $$a$$ represents adults and $$s$$ represents students. They estimate that the marginal and average total cost of seating an additional spectator is constant at 10. ## 1. The promoters wish to segment the market and charge adults and students different prices. ### a. For each segment of the market, find the inverse demand function and marginal revenue function. Take each market segment’s demand function that we are given, and solve for the respective price $$p$$ to get the two segments’ inverse demand functions. Once we have the inverse demand function, we simply double the slope to find the marginal revenue functions: #### For adults: \begin{align*} q_a&=5,000-10p_a\\ q_a+10p_a&=5,000\\ 10p_a&=5,000-q_a\\ p_a&=500-0.1q_a\\ \end{align*} With this inverse demand function, we double the slope to obtain marginal revenue: $MR(q_a)=500-0.2q_a$ #### For students: \begin{align*} q_s&=10,000-100p_s\\ q_s+100p_s&=10,000\\ 100p_s&=10,000-q_s\\ p_s&=100-0.01q_s\\ \end{align*} With this inverse demand function, we double the slope to obtain marginal revenue: $MR(q_s)=500-0.02q_s$ ### b. Find the profit-maximizing quantity and price for each segment. For each segment, set $$MR=MC$$ to obtain the profit-maximizing quantity of tickets to sell. We know the marginal cost for each segment is simply10. Then, since the firm has market power, raise the price to the maximum each segment is willing to pay (it’s demand at that quantity).

\begin{align*} MR(q_a)=MC(q_a)\\ 500-0.2q_a&=10\\ 500&=10+0.2q_a\\ 490&=0.2q_a\\ 2,450&=q_a^{\star}\\ \end{align*}

Plug this into the adults’ inverse demand curve to obtain the price:

\begin{align*} p_a&=500-0.1q_a\\ p_a&=500-0.1(2,450)\\ p_a&=500-245\\ p_a^{\star}&=\255\\ \end{align*}

#### For students:

\begin{align*} MR(q_s)=MC(q_s)\\ 100-0.02q_s&=10\\ 100&=10+0.02q_s\\ 90&=0.02q_s\\ 4,500&=q_s^{\star}\\ \end{align*}

Plug this into the students’ inverse demand curve to obtain the price:

\begin{align*} p_s&=100-0.01q_s\\ p_s&=100-0.01(4,500)\\ p_a&=100-45\\ p_a^{\star}&=\55\\ \end{align*}