Question

Part II: Long problems 1. (20 points) A firm produces two commodities, A and B. The inve demand functions are: pa = 900 - 2x - 2y and pb = 1400 - 2x - 4y respectively, where the firm produces and sells x units of comm A and y units of commodity B. Its costs are given by: CA = 7000 + 100x + x2 and CB = 10000 + 6y2 (a) What is the firm's profit function 7(x,y)?! (b) Find the values of x and y that maximize it(x,y). (c) Use the second-derivative test to verify the critical found in (b) is a maximum.

Answer 1

(4) п = пE PA - x+ Реч - CA - Се (900- 2 -290 x + Січоо -21-99) 9 - (Зооо + loox + - todoo + (11) л: 96ox - 2 - 2x y + 1Чoo9 - 2x y - Чу2 - 2000 - 106 - x 2 - jooo6 - 22 п= 860 - 2x? - Чxy + 1400 - 101 - 1906 о Е) ОТ = 800 - 6ҡ - чЯ оп = – чи + 19 оо - 29 put DIT = o or to 860 - 6 - 99 =. gsъ = 6 + ч —D) - Чи + 1чоо - 269 2 о 19oo = (х + 2o1 — 0 x 5 6х + 49 = 800 Чй 1207 = 19 оо 2 Зох + Чи t = Чооо • = чоо 2cx = 26со

м 26 со 26 х = 10ъ a=100 in put ega-③ ( + 99 = 8 sъ 6ci sъ2 + 9 = 8 со 928 so - 6 со Ч = 26 (c) 20 - 20 г. Soc: 2*2 = - 4 х? 22те Ox 29 2? = 392 от = 392x - ч – у Now ( 22 ) ( 32 ) Съ92/ - [ът ) = Тъд 32 74 = (-621-26) - (-422 120 - 16 = 10ч 20 siмля ( 2 ) ( ) - ( 9 ) 7. and a² con x 2 Thus, x=100, y = 50 maximizing values of are profit profit function.