Question

A seller faces two buyers. The big buyer has inverse demand P = 15 - Q and the small buyer has inverse demand P = 10 - Q. The seller knows these inverse demands but cannot tell in advance which buyer is big and which is small. Assume resale is impossible. Marginal cost is constant at $1. Which pricing strategy maximizes producer's surplus?

Buyer's choice: Pay a membership fee of $8 followed by a constant price per unit of $6, or a membership fee of $78 followed by a constant price per unit of $1. Why is this answer?

Answer 1

1st plan. At a price of 6, Q1 is 15 - 6 = 9 units and Q2 is 10 - 6 = 4 units. Total sales = 13 units. Use consumer surplus to find if both buyers will agree to buy or not. CS = 0.5*(maximum price - price paid)*quantity bought.

CS for 1st buyer = 0.5*(15 - 6)*9 = 40.5. CS for 2nd buyer = 0.5*(10 - 6)*4 = $8. Since both have a higher consumer surplus, both will pay.

Producer surplus = profit from fees + (price - minimum acceptable price or marginal cost)*quantity. This makes a surplus to producer equivalent to 8*2 + (6 - 1)*13 = 81

2nd plan. At a price of $1, there is no profit on sales of unit. Q1 = 15 - 1 = 14 units and Q2 = 10 - 1 = 9 units. Entire profit is made with fee.Use consumer surplus to find if both buyers will agree to buy or not. CS = 0.5*(maximum price - price paid)*quantity bought.

Under this scheme, the consumer surplus for 1st buyer before fee payment is 0.5*(15 - 1)*14 = 98. For 2nd buyer CS = 0.5*(10 - 1)*9 = 40.5. Note that only 1st buyer can pay a fee of 78 since its consumer surplus is higher than 78. Hence producer surplus is 78.

1st plan has a higher producer surplus (81 vs 78).