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# Optimal levels of p and q and calculating the Lagrangian multiplier

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Gabriella Inc. is a business that helps individuals on job with local employers. Gabriella does not hire individuals; rather it provides the service of calling individuals for employers who then hire them. A key feature of Gabriella's success is its ability to reach a large number of individuals in a short period of time by means of the telephone. In making these telephone calls, the company can use telephone operators; each using a single phone, and it can use computer operators, each using a computerized process with automatic dialing capability. A production function for the calling operation is as follows:
Y = 400p + 800q + 4pq
Where: Y = number of call made per day
p = number of telephone operators per day
q = number of computer operators using phone-computer combinations
Each phone operator using a single phone costs the firm an average of \$90 per day, and each computer operator matched with a phone computer combination costs \$180 per day. The total expense budget allocated to this project has been limited to \$3600 per day.

A. Use the Lagrangian multiplier technique to determine the optimal levels of p and q to maximize the total number of calls that can be made per day.
B. Calculate and interpret the value of the Lagrangian multiplier in this case.

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A. Use the Lagrangian multiplier technique to determine the optimal levels of p and q to maximize the total number of calls that can be made per day.

The problem is to max Y = 400p + 800q + 4pq
Subject to: 90p + 180q=3600
The Lagrangian equation is then:

L = 400p + 800q + 4pq - M (90p + 180q-3600) where M is the Lagrangian ...

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