Problem $3(1\mathrm{.}1$ pts$)$

Consider selling tickets for Berkeley and Stanford football game to be held at a stadium with $60,000$ seating capacity. Demand curves for general public and students are given by

$d_g(p_g)=(90,000-3,000p_g{)}^{+},d_s(p_s)=(20,000-1,250p_s{)}^{+}.$

a$)(0\mathrm{.}3$ pts$)$ Determine the maximum willingness to pay for each segment?

b$)(0\mathrm{.}4pts)$ Formulate the constrained optimization problem if the manager wants to charge separate prices for students and the general public $($see slide #$21$ in Ch$\mathrm{.}5$ slides$)?$

c$)(0\mathrm{.}4$ pts$)$ Consider the optimization problem you formulated in part b$)$ Determine $p_g^{**}$ and $p_s^{**},$ and also the resulting total revenue when these two prices are charged $($see slides $22$ and $23$ in Ch$.5$ slides$)?$

Problem $4(1\mathrm{.}3$ pts$)$

We return to the Stanford Stadium pricing problem in Ch$\mathrm{.}5($slide #$19),$ with $60,000$ seating capacity and demand curves for general public and students are given by $d_g(p_g)=(120,000-3,000p_g{)}^{+},d_s(p_s)=(20,000-1,250p_s{)}^{+}.$

Suppose that an earthquake damages Stanford Stadium so that only $53,000$ seats are available for the Big Game.

a$)(0\mathrm{.}4$ pts$)$ What is the optimal single price to charge and the corresponding total revenue?

b$)(0\mathrm{.}6$ pts$)$ What are the optimal separate prices to charge for students and the general public and the corresponding total revenue?

c$)(0\mathrm{.}3$ pts$)$ What is the "opportunity cost" of having the capacity of $53,000$ as opposed to $60,000$ in both cases $($charging a single price and separate prices$)?$