Question

2. Suppose a monopoly firm is allowed to price discriminate in 3 markets where the prices for the good in each market are given by: P1 = 63 - 401 P2 = 105 - 502 P3 = 75 - 603 where: Q = Q1 + Q2 + Q3 The cost of the output is (Q) = 20 + 15Q+Q2 a) Give the profit function for the firm. b) Find the FOC's and find the p*'s and Q*'s that maximize profit. c) Find the SOC's.

Answer 1

2) P = 63-48.. Pa = 105-592 :: - P₂ = 75-683. in Total Revenues, TR's will be given as . - TRI = P, Q, = (63-48.) 9,= 638, -48,2 TR2' = P292 - (105-582) 92 = 10582-5922. - TRB = P3 Q3 = (75-683) 93 = 7593 - 693

The cost function is given as: e = 20+ 158 +82, where, g = 8 + Q2t83 : Ç=20+1591 +1582 +1503+(91+22+83). 1. The profit function. It will be given by: 1 - TR, 1 TR + TR) - C.. -) = 638, -48,+10582-502 +7583-64a -20 + 158,+1582 +1583+(81+82 +83)2). | A T = 488,-49,²+9082-58 +6083-6Q3² -20 - (81+32+83) — ☺ I ☺ gives us the profit function & II, for the firm. D. Now, wording to the First Order conditions: 48 - 88,-2 (81 +82783) =0. => 48 -109,-2182783) = 0 => 24-581-82+83) = 0. >> 581 +82 +83 = 24 — . -590-1082-2(81+82+ 8 3) = 0. >> 90-1202-2181+83) = 0 . >> 45 – 682 -(81+93) = 0. =) 81 +682 +33 = 45 — dQ2

әл strict =60-1983–2(81+82+83) -0. 5) 60-1403-218,+82) = 0. >> 30-783-(81+82) - 0. >) 9,+82 +783 = 30 - ☺ and W. gives us the th Now, from these three equations, we can the following matrix and some for 8, 92 and 83, by Cramer's Rule 58, +42+ 83 = 24 : – 0 Bi +682 +22=45 - 8, + 32 + 783 = 30— . ... In matrix form is, r87 - li | 7 | 83 30 Now, ic 15 11 DE 116 11 = 5(42-1)-1(7-1)+1(1-6). Ti 17 - 205 -6 -5 -194 >> D=194

Again, 45 30 6 = 24(42-1) -4517-1) +30(1-6). ( 7 ) = 984 – 270-150 = 564. 2)D1=5643 Again, - 15 24 D2 i 45 11 30 "I 7 F clic 30)-1. (168-30) . -1425-138 -21 +1/24-45). ::. = 1266 . : >> D2=1266 ::: Again, on=151 241=51180-45)-1 (30-26) Ti 6 451...+1645-144). 11.301=-675-6-99.. ... = 570. =>D3:=570: :10, 2 beses = 2.907 units .= $. * 822 Dec 2016 = 6: 526 units .= 82* 1.892 Spor Toto = 2.938 uit = 935 or Pi = 63-64Q) = 63-(4x2-907) = 51.372 units =P,ms P2 = 105-582 = 105-15x6-526) = 72-37 units =P* px - 75-683-75-(6x 2.938) = 57372 units per 194

20,2 - () To find the S.O.Cs: T = 48 - 89, -2(0, +82 +83). 32 --8-2 = -10 <0 - i, 3.0.C for good I is satisfied Againy dm -90-1092 -2(6+82 +43) :: 323 -10-2-1230 80210 ...3.0. e for good 2 is satified Again, 27-60-1203-2(, +82 +83). és 727 so o date 3-12-2--14 <0 - Thus, s.o. c for good 3 is satisfied. Equation ④, ④ and gives us the S.O.C's. 893