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You are the manager of two plants (factories) in Mexico that manufacture shoes. The combined monthly output of both plants is to be 10,000 pairs of shoes. How you would best divide this output of 10,000 pairs of shoes between the two plants. You may make your arguments in words with the aid of diagrams, i.e., without the use of math. On the other hand, if you are comfortable with math (calculus), the following additional information may be made use of by you. The cost function of Plant 1 is C1 = a1Q1 + a2Q12 + Q13and that of Plant 2 is C2 = b1Q2 + b2Q22 + Q23, where Q1 and Q2 denote, respectively, the outputs of Plant 1 and Plant 2.

User Philip Atz
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Answer:

Step-by-step explanation:

The purpose of allocating the output of the shoes is to diminish the total cost of production. The process is achieved by assigning a pair of shoes that requires production at the factory with a marginal lower cost of the two plants. Afterward, the firms will have to equate the marginal cost of production across the two firms.

For firm 1:

The cost of production
c_1 = a_1Q_1 ^2 +a_2Q_1^2+Q_1^3

Differentiating with respect to
Q_1 to determine the marginal cost;

For firm 1, the Marginal cost
MC_1 = a_1 +2a_2Q_1+3Q_1^2

For firm 2; the marginal cost
MC_2 = b_1 +2b_2Q_2+3Q_2^2

Equating both from above:


a_1 +2a_2Q_1+3Q_1^2 = b_1 +2b_2Q_2+3Q_2^2

Recall that:


Q_1 = 10000 - Q_2

Thus, we can replace the value of
Q_1 into the above equation to determine the value of
Q_2 in terms of
a_1, a_2, b_1, b_2 by applying a quadratic formula.

Assuming we knew the values of
a_1, a_2, b_1, b_2, we can estimate the numerical value of
Q_2, then replace it into the equation
Q_1 = 10000 – Q_2 to find the numerical value for
Q_1.

User Ankit Bhatia
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