Question

suppose a perfectly competitive firm produces 2 outputs. The firm's price for the first output is Pi and the price for the second output is P2. The cost function is given by: C(Q1, Q2) = 20,2 + 2022 a) Give the profit function for the firm. b) Find the FOC's for profit maximization and interpret them economically. c) Find the SOC's.

Answer 1

D Here : P, a Price of first product & Quantity of first product P2 : Price of second products &z= guantity of second product 8. Total Revenue, TR, of the firm is given by? - TR = P, 8, + P2 G2 . a.) Profit function, I., for the firm will be : T = TR-C ») TI = Pig,+ 12 92 - (2012+282). >> ] = P, 8, + f2 82 - 2 (0,2 + ) = 0. The abone equation o gives us the profit fonction of the firm. 6). According to the first order conditions (f.0.c): . 1. 3T = Pi - 40, -0 - @ jgi og! >> 8,54

6 gives us the first F.O.C (ie, p =48) According to this equation, MR = MC e first product where MR, & Marginal Revenue for first product Twe have, [ TR = P, Q, + P2 Q2 . 1. Marginal Revenue of first product, MR, will be : MR = 1 = P1] . Jom MG = Marginal cost of first product [we have c = 28,2 + 2 Q2² imea = 480] Now, for the second product we have to DIT = P2 - 482 -0. . => P2= 482 -- li => 82 = e i ole life the Equation gives as the second roc(ie, P2 = 482). According to this equation, MR2 = MC, where, MR2 : Marginal Revenue of second product JQ2 =) P2 = 4 4 I . MR2 = JQ2 MC : Marginal cost of second product : I MC2 = 2 2 = 492

Thus, FO.C's establishes the fact that the profit is maximised for a quantity, where, Marginal Revenue (MR) = Marginal cost (MC). c). From the partial derivšatives we have T = P,-48. 327-430 — . w gives us the s.o. e for first first product 1. Again, we have: DTI = P2 -4 32 .. - D27 =-450 — ☺ gives as the S.O.C for second product. Thus, we can see that both the S.O.C's, for profit maximisation, are satisfied.