In the quantity-price contract example in Figure, we noted that the order quantity, Q = 20, is efficient. We can demonstrate that seemingly reasonable contracting methods can lead to inefficient results: too little output being produced and sold. As before, let benefits and costs be: B = 3Q - Q2/20 and C = Q2/40, respectively. Each side knows the other€™s benefit or cost function. The contracting method is as follows:
The seller names a price for its output, and the buyer chooses the quantity to purchase at this price.
a. Find the buyer€™s profit-maximizing purchase quantity as it depends on the seller€™s quoted price P. (The buyer sets Q to maximize Ï€B = B - PQ = 3Q - Q2/20 - PQ. Treating P as a parameter, set dÏ€B/dQ equal to zero. You should find that P = 3 - Q/10 or, equivalently, Q = 30 - 10P.)
b. Find the seller€™s optimal price. (The easiest approach is to use the price equation, P = 3 - Q/10, treat Q as the decision variable, and set MR = MC to find optimal quantity and price.)
c. Explain why an inefficient outcome results when the seller quotes a take-it-or-leave-it price perunit.

Payment (R) Optimal Quantity Q 12 quantity Q 20 Tg=30 40 B7.2 s 18 s 0 12 20 Ts30 10 10 20 25 30 Quantity (Q) 15