Question

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. Explain, based on your understanding of Chapter 7 (The Cost of Production), 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 + a2Q1^2 + Q1^3and that of Plant 2 is C2 = b1Q2 + b2Q2^2 + Q2^3, where Q1 and Q2 denote, respectively, the outputs of Plant 1 and Plant 2.

If possible provide answer using calculus and the cost functions for each plant to arrive at the output each plant should produce.

Answer 1

Output of shoes needs to be allocated so as to reduce total cost of production. This is done by allocating a pair of shoe to be produced at a plant (factory) with marginal lower cost of of the two plants. Thus, firms will equate marginal costs of production across two firms.

Cost of Productin of Firm 1 C1 = a1Q1 + a2Q1^2+Q1^3.

To get marginal cost differentiate this with respect to Q1.

Marginal cost of production of firm 1 MC1 = a1 + 2a2Q1 + 3 Q1^2

Similarly, MC2 = b1 + 2b2Q2 + 3Q2^2

Firm equates this two costs, giving:

a1 + 2a2Q1 + 3Q1^2 = b1 + 2b2Q2 + 3Q2^2

We know that Q1 = 10,000 - Q2. Therefore we can substitute this in the equation above and solve for Q2 in terms of a1, a2,b1,b2 using the quadratic formula. If we knew values of a1,a2,b1,b2, we can solve for the numerical value of Q2 using quadratic formula and deduct this value from 10,000 to get value of Q1.