Question

AC = 3S2 + 2U2 - 2SU - 2S - 6U + 20 Refer to the AC function. Suppose that union negotiations result in an agreement that requires the firm to employ skilled and unskilled labor in equal proportions; ie., S/U = 1. Now determine the quantities of the two inputs that will result in production at the lowest possible average cost and determine the associated minimum average cost.

Answer 1

Consider the given problem here the “AC” is given by.

=> AC = 3*S^2 + 2*U^2 – 2*S*U – 2*S – 6*U + 20, where S/U = 1, => S = U.

=> AC = 3*S^2 + 2*S^2 – 2*S^2 – 2*S – 6*S + 20 = 3*S^2 + 2*S^2 – 2*S^2 – 2*S – 6*S + 20.

=> AC = 3*S^2 – 8*S + 20, => FOC require d(AC)/dS = 0.

=> 6*S – 8 = 0, => S = 8/6 = 4/3, => S = 4/3.

Now, d²(AC)/dS² = 6 > 0, => at “S = U = 4/3” the “AC” is minimum.

=> AC = 3*S^2 – 8*S + 20 = 3*(4/3)^2 – 8*(4/3) + 20= 3*1.78 – 8*1.33 + 20 = 14.7.

=> AC = 14.7, the minimum “AC” is given by, “AC = 14.7”, where “S = U = 4/3”.