Question

X Text Question 4.3 Question Help Suppose a nonlinear price discriminating monopoly, can set three prices, depending on the quantity a consumer purchases. The firm's profit is T=P1 (Q1) +P2 (Q2-Q1) +P3 (Q3 - Q2) – mQ3, where p, is the high price charged on the first Q, units (first block), P, is a lower price charged on the next Q, -Q, units, P3 is the lowest price charged on the Q3-Q, remaining units, Q, is the total number of units actually purchased, and m= $30 is the firm's constant marginal and average cost. Use calculus to determine the profit-maximizing P1, P2, and p3- Let demand be p = 210 - Q. The profit-maximizing prices for the nonlinear price discriminating monopoly are Pa =$ P2 = $_], and P3 = $ (Enter numeric responses using real numbers rounded to two decimal places.)

Answer 1

Profit is given by n=plql + p2(q2-21) + p3(q3 - 42) – mq3 We know that P= 210 - Q. This implies q1 = 210 - p1, 42 = 210 - p2 and q3 = 210 - p3. Hence 22-q1 = 210 - p2 – 210 + pl = pl - p2. Also, q3 – 42 = 210 - p3 – 210 + p2 = p2 - p3. This suggests that the profit function will now become t=p1(210-pl) +p2(pl - p2) + p3(p2 - p3) - 30(210- p3) = 210p1 - p1^2 + p1p2 – p2^2 + p2p3 - p3^2 - 6300 + 30p3 We now find t'(pl)= 0, 1'(p2) = 0 and it'(p3) = 0 210 – 2p1 + p2 = 0... (which gives p1 = 105 +0.5p2) pl - 2p2 + p3 = 0 p2 – 2p3 + 30 = 0... (which gives p3 = 15 +0.5p2) We have 105 + 0.5p2-2p2 + 15 + 0.5p2 = 0 or p2 = 120. Now we have pl = 165 and p3 = 75 Hence, p1 = 165, p2 = 120 and p3 = 75