Question

27 Optimization of functions PL5. FAVOURITE agricultural firm makes profit from selling two products: cotton (X) and corn (V). Both products face the following inverse linear demand (where X is in tons of cotton and Y is in dozen bushels of corn): Px = 56 - 4X Py = 48 - 2Y The cost of providing these products is: C(X,Y)= XP + YP + 6XY. Set the profit function, find the combination x* and y* that maximizes profit and check that the point is a relative maximum. Finally, calculate the maximum profit and the prices for X and Y.

Answer 1

demand P = 56-4x ; demand Py = 48-24 revenue R= No. of units * demand Rx = x(56-4*) = 56x-4*? Ry = Y(48-2y) = '487 - 2y2 Total Revenue R(x, y) = 568 + 487 - 4x²-zy? (1) Told cost (x,y) = +y + 6xy (2) Profil P(x,y) = R(x, y) - Cley) p(2.) = 56x + 487 - 2 -2y2-x-y-Cry Plx, y = 56x + 487 -582 - 3y2 -669 -(3) Px = = 56-108-67 Py = D1 = 48–67 - 6x To get critica point Px=of Pyzo -10X-6y +56=0 ? 62 -69+48=0) (X,Y)= (3,61 * Per = -10; Pyy = -6; Pxy = -6 Hesian H = Pexe Yyy - Prešo At (2,6) kex = -10; Pyy = -3; Pxy = -6

| H - (--)(- ) --s* - Go - 3 = 2x > 0 - Hoo; Pxxco; Pyrco (2,6) is maximum, p(2,61 = 200 maximum profit is 200 at x=2, Y=6