(iii) 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 sound service input (S) which is given by:
Z = M + .5S + .5 MS - S2
BF’s staff budget for the coming year is $1,200,000. Annual employment costs
are $30,000 for each social service staff member (S) and $60,000 for each medical
staff member (M).2
(a) Using the Lagrangean multiplier approach calculate the optimal (i.e.,
service maximizing) combination of medical and social staff. Determine
the optimal amount of service provided by BF.

(b) Calculate BF’s marginal cost. Explain your answer.
(c) Using Excel-Solver verify your answer to (a).

(Show your work. Show the spreadsheets in detail. Provide print outs
with Solver window. To print the solver window, use print screen
command on your key board and then create a MS Word document
using paste.)

(iv) The own price elasticity of demand for a pack of cigarettes is estimated to be -.4.
Current price and consumption are $4.00 and 2 million units per year.
Assuming a linear demand relationship determine the demand equation for
cigarettes. Show all your calculations.

(iii) To find the optimal combination of medical and social staff, we need to maximize the service provided by BF while considering its financial constraints. We can use the Lagrangean multiplier approach for this problem.

First, let's set up the optimization problem:
Maximize Z = M + 0.5S + 0.5MS - S^2

Subject to the budget constraint:
$60,000M + $30,000S ≤ $1,200,000

Now, let's define the Lagrangian function:
L(M, S, λ) = Z - λ($60,000M + $30,000S - $1,200,000)

To find the optimal solution, we need to take partial derivatives of L with respect to M, S, and λ, and set them equal to zero:

∂L/∂M = 1 + 0.5S - 2Sλ - $60,000λ = 0
∂L/∂S = 0.5 + 0.5M - 2Sλ - $30,000λ = 0
∂L/∂λ = $60,000M + $30,000S - $1,200,000 = 0

Solving these equations simultaneously will give us the optimal combination of medical and social staff, as well as the maximum service provided by BF.

(b) Marginal cost is the additional cost incurred when producing one more unit of output. In this case, we want to calculate BF's marginal cost.

The marginal cost (MC) can be calculated using the total cost (TC) function:
MC = ∂TC/∂Q

To determine the demand equation for cigarettes, we need more information. Specifically, we need the price and quantity demanded at a different point to estimate the slope of the linear demand relationship. Could you provide additional data?