Question

B. A firm produces and sells two commodities. By selling x tons of the first commodity the firm gets a price per ton given by p = 96 – 4x. By selling y tons of the other commodity the price per ton is given by q = 84 – 2y. The total cost of producing and selling x tons of the first commodity and y tons of the second is given by C(x, y) = 2x2 + 2xy + y2. (a) Show that the firm's profit function is P(x, y) = –6x2 – 3y2 – 2xy + 96x + 84y. (b) Compute the first-order partial derivatives of P, and find its only critical point. (c) Suppose that the firm's production activity causes so much pollution that the authorities limit its output to 11 tons in total. Solve the firm's maximization problem in this case. Verify that the production restrictions do reduce the maximum possible value of P(x, y).

Answer 1

Ansuer Arc Brolirg to colet rom the data > let us fonouzd mat the claf a aj assume ux have to seling x-tons can be firtt Comırodity jn the Price pe Hhen p=96- 4* Jäen Cy)= zx*27y+y 2-94-29 To show that he ffm Proft -fanchon PM=-6x-3y 2xyt9i Rhet Now p,)S- ca 6x-4t+84y -2y-242ny +9) PC)-6x3y-2zy +96uy To compule he first-8dy partial duivahivg P can be Pollution ic autheie To find out Thc Statonary Points a NoW Qnd P 63-22yt9(x +94) Py -3-2xy++4y) -6y -2xt84 Px-127-2y + 6/->@ Py wc krow Ttat the above Naw equahions aril -122-2y+96 -cy 2+84 > and 84 2x6 16-12 2 42 3 48 6 and e cihal Point The Polluhbn wil auhovity limit output lI-tes on total To vevity tke Produchion reshichon to e duce Ponible PC the Manimum

Now 1 TO Maximie P CH9)6-3y- 2y +9684y S.t the n we have Pd 122y+96 -Cxty-) -dCx+yD - 6y-27+8y= 12-2y, + 94 = dC+o+0) -6y -2x8y =d 127-2y+96 = A oxtuy- -12-2y + 96 =) -6y 2 4 5-2y 6 from a wegt 2z+222 7X0 28 value substute fom y--y dy- 7 4t y , y) = (y, Put and y' v^lue subihk 6-3C1)2(4)+ Cy) 675 Ply,) Now y6,12) we Jef subshtc PCG,12) GC)-31)26)C) +9684 C1)792 is Profit orthout prducon yeshcon is 2 greate ean pYofit is 673 THE END