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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.
Output function: Z = M + 0.5S + 0.5MS - S^2 , Cost function: 30000S + 60000M = 1200000
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
30000S = 30000n and 30000M = 60000n
S = n and M = 2n
30000n + 60000(2n) = 1200000
30000n + 120000n = 1200000
150000n = 1200000
n = 120/15 = 8
Thus S = 8 and M = 2(8) = 16
Put these values of S and M in Z, we get
Z = 16 + 0.5 (8) + 0.5 (16*8) - 8^2 = 16 + 4 + 64 - 64 = 20
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