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Mathematics & Economics
English (U.S.)
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Case Problem: The Warren and Smith Company (Coursework Sample)


The Warren & Smith Company manufactures commercial zippers of two kinds, kind X and kind Y. Its production department estimates that the average cost function of the firm is

(a) The manager of the firm would like to know the level of the output of zipper X and zipper Y at which the average cost of the firm is minimized, and the level of this minimum average cost.

(b) The firm expects an order that will require it to produce a total output of 6 units of both kinds of zippers (each unit may be a large number of zippers), and so the manager would also like to know how many of each type of zipper the firm must produce to minimize its average cost, and what its minimum average cost would be if it receives the order. The manager gives this assignment to two researchers who use different methods to obtain their answers.

While the firm expects the order to be of 6 units, it may be as large as 7 units or as small as 5 units. Determine the minimum average cost the firm with these different order sizes.


Case Problem
The average cost function of the firm, Warren & Smith Company is given by
AC=X2+2Y2-2XY-2X-6Y+20…… (1)
X and Y represents the level of output for zippers X and Y respectively
* Finding the minimum average cost
dACdX=2X-2Y-2=0…… (2)
dACdY=- 2X+ 4Y-6=0……(3)
Solving the above equations (2) & (3), we get
X= 5 and Y=4
Minimum average cost is computed by using the above level of output in (1), we get
Minimum average cost= 3
* Finding the minimum average cost to complete the order
Given: X + Y = 6…… (4)
As stated, the manager assigns the task to two different researchers, who used different methods to find the solution.
First method
The first method which can be used is the Substitution method
Equation (4) can be rearranged as
X= 6 –Y …… (5)

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