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Lagrangian Multipliers

1.
Fixed capital and labor expenses = 1.2million/year
Variable expenses = 2,000/unit of output
Demand: Q = 1000 - 0.1P

A. Calculate profit-maximizing output, price, and profit levels
B. Using the Lagrangian multiplier method, calculate profit maximizing output, price, and profit levels in light of a parts shortage that limits output to 300 this year
C. Calculate and interpret Lagrangian multiplier
D. Calculate the value of having parts shortage eliminated

2.
Employment Budget = 350,000 on tax preparers and accountants
Salary = Accountant: 50,000 and Tax Preparer: 25,000
Q = function of the number of accountants (A) and trained tax preparers (T)
the function of persons assisting in office is given by:
Q = 250A + 100T + 25AT

A. Using the Lagrangian technique, what is the optimal combination of accountants (A) and tax preparers (T), per office, if the objective is to maximize the number of individuals assisted (q)
B. Under the annual budget of 350,000, what is the maximum number of individuals that can be assisted
C. At the optimal use of accountants and tax preparers, how many individuals can be assisted by a $1.00 increase in budget?

3.
Production Function: Q = M + 0.5S + 0.5MS - S^2
M = number of medical staff
S = number of social services
Annual Budget = 1,200,000
Annual Employment Costs = 30,000 for each social service staff and 60,000 for each medical staff

A. Using the Lagrangian technique to determine the optimal combination of social service staff and medical personnel to employ, if the objective is to maximize the number of patients served
B. Under the annual staff budget of 1,200,000, what is the maximum number of patients that can be serviced?
C. At the optimal use of medical and social service staff members, how many extra patiens could be assisted by a $1.00 increase in budget

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