Answer to Question #254816 in Microeconomics for Tami

Question

A firm has two plants that produce identical output. The cost functions are C₁ =10q-4q² +q³. a.At what output level does the average cost curve of each plant reach its minimum? b. If the firm wants to produce four units of output,how much should it produce in each plant?

Answer 1

a.

$TC_1 = 10q - 4q^2 + q^3\\ AC_1 = \frac{TC_1}{q}\\ AC = 10 - 4q + q^2\\$

Average cost will be minimum when it's derivative is set = 0

$\frac{d}{dq} (AC) = -4 + 2q = 0\\ q = 2 \space units\\ TC_2 = 10q - 2q^2 + q^3\\ AC_2 = 10 - 2q + q^2\\ \frac{d}{dq }(AC) = -2 + 2q = 0\\ q = 1$

q = 2 for first power plant and q=1 for second power plant the average cost is minimum. 

 

$b) \\ MC_1 = 10 - 8q +3q^2 \\ MC_2 = 10 - 4q + 3q^2$

If the firm wants to produce four units of output, to find how much should it produce in each plant

Let the number of units in 1st plant be q1

And the number of units in the second plant be q2

Since, total number of units produce is 4

$q_1+q_2=4 ....... (i)$

In equilibrium

$MC_1=MC_2\\ 10 - 8q_1 +3q_1^2 = 10 - 4q_2 + 3q_2^2$

solving this equation will give 

$q_1 = 2q_2$

substituting in equation (i) 

$2q_2 + q_2 = 4\\ 3q_2 = 4 \\ q_2 = \frac{4}{3}\\ q_1 = 4 - \frac{4}{3}\\ q_1 = \frac{8}{3}\\$

Hence, $\frac{8}{3}$  units are produce in 1st plant 

And $\frac{4}{3}$  units are produce in 2nd plant