The demand function for football tickets for a typical game at a large mid western university is D (p) = 200, 000−10, 000p. The university has a clever and avaricious athletic director who sets his ticket prices so as to maximize revenue. The university’s football stadium holds 100,000 spectators.
(a) Write down the inverse demand function. P (q) = 20 – q /10, 000.
(b) Write expressions for total revenue _______and marginal revenue _______ as a function of the number of tickets sold.
(c) On the graph below, use blue ink to draw the inverse demand function and use red ink to draw the marginal revenue function. On your graph, also draw a vertical blue line representing the capacity of the stadium.
(d) What price will generate the maximum revenue? ______. What quantity will be sold at this price? ________.
(e) At this quantity, what is marginal revenue? ________. At this quantity, what is the price elasticity of demand? _____. Will the stadium be full? _______.
(f) A series of winning seasons caused the demand curve for football tickets to shift upward. The new demand function is q (p) = 300, 000 − 10, 000p. What is the new inverse demand function?
(g) Write an expression for marginal revenue as a function of output. M R (q) = ________. Use red ink to draw the new demand function and use black ink to draw the new marginal revenue function.
(h) Ignoring stadium capacity, what price would generate maximum revenue? _______. What quantity would be sold at this price?______.
(i) As you noticed above, the quantity that would maximize total revenue given the new higher demand curve is greater than the capacity of the stadium. Clever though the athletic director is, he cannot sell seats he hasn’t got. He notices that his marginal revenue is positive for any number of seats that he sells up to the capacity of the stadium. Therefore, in order to maximize his revenue, he should sell ________ tickets at a price of _________.
(j) When he does this, his marginal revenue from selling an extra seat is _______. The elasticity of demand for tickets at this price quantity combination is _______.