A firm has two plants that produce the same stuff

a)At what q does the average cost curve of each plant reach a minimum? Assume q can be produced in partial units, so decimal or fractional answers are ok. The average cost at plant 1 is minimized at 2 units and the average cost at plant 2 is minimized at 1 unit. To find the cost-minimization output level, first solve for the average cost function (total cost divided by ouput). By solving for this functions first derivative and setting equal to zero, you will then be able to find the point where average cost is minimized.

C1=10q−4q21+q31AC1=C1q1AC1=10q1−4q21+q31q1AC1=10−4q1+q21soAC′1=−4+2q1andAC′1=0−4+2q1=02q1=4q1=2andC2=10q−2q22+q32AC2=C2q2AC2=10q2−2q22+q32q2AC2=10−2q2+q22soAC′2=−2+2q2andAC′2=0−2+2q2=02q2=2q2=1C1=10q−4q12+q13AC1=C1q1AC1=10q1−4q12+q13q1AC1=10−4q1+q12soAC1′=−4+2q1andAC1′=0−4+2q1=02q1=4q1=2andC2=10q−2q22+q23AC2=C2q2AC2=10q2−2q22+q23q2AC2=10−2q2+q22soAC2′=−2+2q2andAC2′=0−2+2q2=02q2=2q2=1

b)To produce a total of 4 units of output q, how much should the firm produce at each plant, given that its goal is to minimize total costs? Assume that q cannot be produced in partial units, just whole number units. The firm should produce 3 units at plant 1 and 1 unit at plant 2. As shown in part (a), the firm will minimize their average costs at plants 1 and 2 at 2 and 1 units, respectively. Therefore, the first three units should be allocated accordingly. When output is 4 units, however, the cost-minimization allocation is unknown. To determine where the total cost is minimized, find the company's total costs (the total cost at plant 1 and the total cost at plant 2) when the fourth unit is produced at plant 1 and when it is produced at plant 2. As shown below, the average cost is lower then the fourth unit is produced at plant 1 ($30) than if it was produced at plant 2 ($32).

C=C1+C2C=(10q1−4q21+q31)+(10q2−2q22+q32)soC=(10(3)−4(3)2+(3)3)+(10(1)−2(1)2+(1)3)C=(30−36+27)+(10−2+1)C=21+9C=30andC=(10(2)−4(2)2+(2)3)+(10(2)−2(2)2+(2)3)C=(20−16+8)+(20−8+8)C=12+20C=32