Question

2 Variable Game Pricing Consider that the Minnesota Vikings have two games left in their season. The first game is against the Chicago Bears and the game determines who wins the NFC North Division. The second game is against the lowly Green Bay Packers and their washed up quarterback. The game has no playoff or division implications The aggregate inverse demand curve for the game against the Bears is P- 300- 2Q and the inverse demand for the game against the Packers is P = 240-2Q, where Q is in thousands. The marginal cost of supplying a seat is zero 1. First, consider that the Vikings' front office is required to charge the same price for all tickets to both games (a) Write down the Vikings' profit maximization problem (b) What price maximizes profit? (c) What is total attendance for the two games? (d) What is total profit at the optimal price? (e) What is total Consumer surplus at the optimal price? 2. Now consider that the Vikings can choose a different price for each game. (a) Write down the Vikings' profit maximization problem (b) What prices maximizes profit? (c) What is total attendance for the two games? (d) What is total Profit at the optimal prices? (e) What is total Consumer Surplus at the optimal price? 3. Compare consumer surplus between single price and variable pricing. Explain the intuition behind the result.

Answer 1

(1)

For Bears,

P = 300 - 2QB (Where QB: Number of seats for Bears game)

2QB = 300 - P

QB = 150 - 0.5P

For Viking,

P = 240 - 2QV (Where QV: Number of seats for Viking game)

2QV = 240 - P

QV = 120 - 0.5P

(a) With single pricing model,

Total demand (Q) = QB + QV = 150 - 0.5P + 120 - 0.5P

Q = 270 - P

P = 270 - Q

Profit (Z) = Revenue - Cost = (P x Q) - (MC x Q) = 270Q - Q² - 0 [Since MC (Marginal cost) is zero] = 270Q - Q²

The profit-maximization problem is:

Maximize Z = 270Q - Q²

Subject to Q >= 0

(b) Profit is maximized when dZ/dQ = 0 and d²Z/dQ² < 0.

dZ/dQ = 270 - 2Q = 0

270 = 2Q

Q = 135

P = 270 - 135 = 135

d²Z/dQ² = d/dQ(270 - 2Q) = -2 < 0, so second order condition is satisfied.

(c) Profit maximizing attendance (Q) is 135 seats.

(d) Profit = (270 x 135) - (135 x 135) = 36,450 - 18,225 = 18,225

(e) From market demand function, when Q = 0, P = 270 (Reservation price and vertical intercept of demand curve).

Consumer surplus = Area between demand curve and market price = (1/2) x (270 - 135) x 135 = 67.5 x 135 = 9,112.5

NOTE: As per Answering Policy, 1st part with multiple sub-parts is answered.