Bright Future, Ltd (BF) is a nonprofit foundation providing medical treatment to

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 sound service input (S) which is given by:
Z = M + .5S + .5 MS - S2
BF%u2019s 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.
(32 points)
(b) Calculate BF%u2019s marginal cost. Explain your answer.
(c) Using Excel-Solver verify your answer to (a).
(8 points)
(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.)

Explanation / Answer

Let Q=output, L=labor input and K=capital input where Q = L2/3K1/3. The cost of resources used is C=wL+rK, where w is the wage rate and r is the rental rate for capital.

Problem: Find the combination of L and K that maximizes output subject to the constraint that the cost of resources used is C; i.e., maximize Q with respect to L and K subject to the constraint that vL+rK=C.

Note that maximizing a monotonically increasing function of a variable is equivalent to maximizing the variable itself. Therefore ln(Q)=(2/3)ln(L)+(1/3)ln(K), a more convenient expression, is the same as maximizing Q. Therefore the objective function for the optimization problem is ln(Q)=(2/3)ln(L)+(1/3)ln(K).

Step 1: Form the Langrangian function by subtracting from the objective function a multiple of the difference between the cost of the resources and the budget allowed for resources; i.e.,

G= ln(Q) - %u03BB(wL+rK-C)

G= (2/3)ln(L) + (1/3)ln(K) - %u03BB(wL+rK-C)

where %u03BB is called the Lagrangian multiplier. In effect, this method imposes a penalty upon any proposed solution that is proportional to the extent to which the constraint is violated. By choosing the constant of proportionality large enough the solution can be forced into compliance with the constraint.

Step 2: Find the unconstrained maximum of G with respect to L and K for a fixed value of %u03BB by finding the values of L and K such that the partial derivatives of G are equal to zero.

%u2202G/%u2202L = (2/3)(1/L) - %u03BBw = 0

%u2202G/%u2202K = (1/3)(1/K) - %u03BBr = 0

Step 3: Solve for the optimal L and K as function of %u03BB; i.e.,

(2/3)(1/L)= %u03BBw so L = (2/3)/(%u03BBw)

(1/3)(1/K)= %u03BBr so K = (1/3)/(%u03BBr)

Step 4: Find a value of %u03BB such that the constraint is satisfied. This is accomplished by substituting the expressions for L and K in terms of %u03BB into the constraint and solving for %u03BB.

wL + rK = (2/3)(1/%u03BB) + (1/3)(1/%u03BB) =

1/%u03BB = C so %u03BB = 1/C.

Step 5: Use the value of %u03BB found in Step 4 in the expressions for L end K found in Step 3 to determine the optimal value of L and K.

L* = (2/3)(C/w)

and

K* = (1/3)C/r).

Step 6: Use the optimal values of L and K found in Step 5 to compute the optimum level of the objective function.

Step 7: Note that the value of %u03BB is equal to the partial derivative of the objective function with respect to the size of the constraint. In this case

%u2202G/%u2202C = %u2202(ln(Q))/%u2202C = %u03BB

Related Questions

Navigate

• Browse (All)
• Browse B
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.