show detailed solutions for the lagrange multiplier answer and give its economic interpretation

Example 3: Cost minimization A rm produces two goods, x and y. Due to a government quota, the rm must produce subject to the constraint 3: + y = 42. The rm's cost functions is C(m, y) = 8:1:2 my + 1292 The Lagrangian is L=Sm2a:y+12y2+A(42my) The rst order conditions are L3 16xyA=0 Ly = a:+24yA= L; = 42my=0 (1) Solving these three equations simultaneoust yields 33:25 y=17 A=383