Question

Bright Future, Ltd (BF) is a nonprofit foundation providing medical treatment to emotionally distressed children. BF has hired you as a business consultant to design an employment policy that would be consistent with its goal of providing the maximum possible service given its limited financial resources. You have determined that the service (Z) provided by BF is a function of its medical staff input (M) and social worker staff input (S) which is given by: Z = M + 2S + MS - S2 BF’s staff budget for the coming year is $900,000. Annual employment costs are $60,000 for each social worker staff member (S) and $120,000 for each medical staff member (M).

(a) Using the Lagrangean multiplier approach calculate the optimal (i.e., service maximizing) combination of medical and social worker staff. Determine the optimal amount of service provided by BF.

(b)Calculate BF’s marginal cost. Explain your answer.

Answer 1

a. Output function: Z = M + 2S + MS - S^2 , Cost function: 60000S + 120000M = 900000

For langrangean function, we need to equate the derivative of output function wrt to M and S respectively with the MC wrt to M and S, we get

60000S = 60000n and 120000M = 120000n

S = n and M = 2n

60000n + 120000(2n) = 900000

60000n + 240000n = 900000

300000n = 900000

n = 1/ 3= 0.33

Thus S = 0.33 and M = 2(8) = 0.66

Put these values of S and M in Z, we get

Z = 0.33 + 2 (0.66) + 0.33(0.66) - (0.66)^2 = 0.33 + 1.32 +0.217 - 0.435 = 1.432

b. MCs = 60000 and MCm = 120000 . This shows the additional cost of S and M to produce the one additional amount of services