Third-degree price discrimination often comes up in the context of discountsfor certain groups to some form of entertainment (e.g., a play, movie or a sporting event).Consider an event for which there are two audiences (e.g., students and non-students)and assume that the seller’s additional expenses (i.e., marginal cost) associated withhaving an additional seat occupied are essentially zero but the capacity of the venue islimited to K.(a) To make everything a bit more concrete, let the students’ and non-students’ inversedemands beP1 (x1) = 40 − x1 and P2 (x2) = 100 − x2.Solve for the single monopoly outcome (that is, no discrimination) without a capacityconstraint (i.e., K is a very large number).(b) Suppose now that the monopolist engages in third-degree price discrimination. Findthe optimal quantity sold to both groups of consumers, the price and the monopolist’sprofit. How does this compare to the answer you found in part (a) ?In the next two questions assume K = 50.(c) Suppose the monopolist does not engage in price discrimination. Find the profitmaximizing quantities, the price and the monopolist’s profit.(d) Suppose now that the monopolist engages in third-degree price discrimination. Findthe optimal quantity sold to both groups of consumers, the price and the monopolist’sprofit. How does this compare to the answer you found in part (c) ?2

(a) In a single monopoly outcome without discrimination, the monopolist will maximize profit by equating marginal cost (MC) to marginal revenue (MR). Given that MC is essentially zero, the monopolist will set MR equal to zero. The total demand is the sum of the students' and non-students' demands, which is P(x) = 40 - x + 100 - x = 140 - 2x. The total revenue is then TR = P(x)*x = (140 - 2x)x. The marginal revenue is the derivative of the total revenue with respect to x, which is MR = 140 - 4x. Setting MR = MC = 0, we get 140 - 4x = 0, which gives x = 35. Substituting x = 35 into the demand function, we get P = 140 - 235 = 70. So, the single monopoly price is 70 and the quantity sold is 35.

(b) In third-degree price discrimination, the monopolist will maximize profit by equating the marginal revenue of each group to the marginal cost. The demand functions for the students and non-students are P1(x1) = 40 - x1 and P2(x2) = 100 - x2, respectively. The total revenue for each group is TR1 = P1(x1)*x1 = (40 - x1)x1 and TR2 = P2(x2)x2 = (100 - x2)x2. The marginal revenues are MR1 = 40 - 2x1 and MR2 = 100 - 2x2. Setting MR1 = MR2 = MC = 0, we get x1 = 20 and x2 = 50. Substituting these into the demand functions, we get P1 = 40 - 20 = 20 and P2 = 100 - 50 = 50. So, the optimal prices are 20 for students and 50 for non-students, and the quantities sold are 20 and 50, respectively. The monopolist's profit is TR1 + TR2 - MC = 2020 + 5050 - 0 = 400 + 2500 = 2900. This is higher than the profit in part (a), which is 7035 - 0 = 2450.

(c) If K = 50, the monopolist cannot sell more than 50 units. The profit-maximizing quantities and price are the same as in part (a), but the quantity cannot exceed 50. So, the quantity sold is 50 and the price is 70. The monopolist's profit is 70*50 - 0 = 3500.

(d) If the monopolist engages in third-degree price discrimination with K = 50, the quantities sold to the students and non-students cannot exceed 50 in total. The optimal quantities and prices are the same as in part (b), but the total quantity cannot exceed 50. So, the quantities sold are 20 to students and 30 to non-students, and the prices are 20 and 50, respectively. The monopolist's profit is 2020 + 5030 - 0 = 400 + 1500 = 1900. This is lower than the profit in part (c), which is 3500.

(a) In a single monopoly without discrimination, the monopolist will maximize profit by equating marginal cost (MC) to marginal revenue (MR). Given that MC is essentially zero, the monopolist will set MR equal to zero. The total demand is the sum of the students' and non-students' demands, which is P(x) = 40 - x + 100 - x = 140 - 2x. The total revenue is then TR = P(x)*x = (140 - 2x)x. The marginal revenue is the derivative of the total revenue with respect to x, which is MR = 140 - 4x. Setting MR = MC = 0, we get 140 - 4x = 0, which gives x = 35. Substituting x = 35 into the demand function, we get P = 140 - 235 = 70. So, the single monopoly price is 70 and the quantity sold is 35.

(b) In third-degree price discrimination, the monopolist will maximize profit by equating the marginal revenue of each group to the marginal cost. The demand functions for the students and non-students are P1(x1) = 40 - x1 and P2(x2) = 100 - x2, respectively. The marginal revenues are MR1 = 40 - 2x1 and MR2 = 100 - 2x2. Setting these equal to MC = 0, we get x1 = 20 and x2 = 50. Substituting these into the demand functions, we get P1 = 20 and P2 = 50. The total profit is then π = P1x1 + P2x2 = 2020 + 5050 = 400 + 2500 = 2900. This is higher than the profit in part (a), which is π = Px = 7035 = 2450.

(c) If the capacity is limited to K = 50, the monopolist will sell to the group with the higher willingness to pay until the capacity is reached. The non-students have a higher willingness to pay, so the monopolist will sell to them first. The demand function for the non-students is P2(x2) = 100 - x2. Setting MR2 = MC = 0, we get x2 = 50. Substituting this into the demand function, we get P2 = 50. The profit is then π = P2x2 = 5050 = 2500.

(d) If the monopolist engages in third-degree price discrimination with a capacity of K = 50, the monopolist will sell to each group until the marginal revenue of each group is equal to the marginal cost. The demand functions for the students and non-students are P1(x1) = 40 - x1 and P2(x2) = 100 - x2, respectively. The marginal revenues are MR1 = 40 - 2x1 and MR2 = 100 - 2x2. Setting these equal to MC = 0, we get x1 = 20 and x2 = 50. However, the total quantity cannot exceed the capacity of 50, so the monopolist will sell 20 units to the students and 30 units to the non-students. The prices are then P1 = 20 and P2 = 70. The profit is then π = P1x1 + P2x2 = 2020 + 7030 = 400 + 2100 = 2500. This is the same as the profit in part (c).