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.

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 [tex]c_1 = a_1Q_1 ^2 +a_2Q_1^2+Q_1^3[/tex]

Differentiating with respect to [tex]Q_1[/tex] to determine the marginal cost;

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

For firm 2; the marginal cost [tex]MC_2 = b_1 +2b_2Q_2+3Q_2^2[/tex]

Equating both from above:

[tex]a_1 +2a_2Q_1+3Q_1^2 = b_1 +2b_2Q_2+3Q_2^2[/tex]

Recall that:

[tex]Q_1 = 10000 - Q_2[/tex]

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

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