Woozles can be manufactured in either of two types of factory with the following

Question

Woozles can be manufactured in either of two types of factory with the following cost functions. A perfectly competitive manufacturer of woozles currently has two factories, one of each type:
    C1 = 0.2q12 + 5.2q1 + 25
       C2 = q22 + q2 + 9.8
   where Ci   = total cost function for type i factory, i = 1, 2.
        qi    = production of type i factory, i = 1, 2.
(a) Market price for woozles is currently 4. What is the profit-maximizing (loss-minimizing) production from each factory?
(b) The manufacturer wants to produce 16.5 woozles. What is the cost-minimizing allocation of this production between the two factories?

Explanation / Answer

(a) Profit1 = 4q1 - 0.2q1^2 - 5.2q1 - 25

dProfit1/dq1 = 4 - 0.4q1 - 5.2 = 0

or, q1 = 3

Profit2 = 4q2 - q2^2 - q2 - 9.8

dProfit2/dq2 = 4 - 2q2 - 1 = 0

or, q2 = 3/2

(b) MC1 = MC2

or, 0.4q1 + 5.2 = 2q2 +1

or, 2q2 - 0.4q1 = 4.2

or, 5q2 - q1 = 10.5

q2 + q1 = 16.5

Therefore, q2 = 4.5

q1 = 12

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