Question

In Nyeri town there are only two milk processors. The local inverse demand for milk is given by: Q = 120− P, where P denotes price, Q denotes the total quantity measured in cartons. Both milk processors have the same cost function given by C = 30Q, where C is total cost and Q is output measured in cartons. What is the industry output (Q)?

Answer 1

Given,

$Q=120 - P$

$P=120 - (Q_1+Q_2)$

$C1 = 30(Q_1) , C2 = 30(Q_2)$

for Firm 1,

$\pi_1=TR_1-TC_1=[{120-(Q_1+Q_2)}]Q_1-30Q_1$

$=(120)Q_1-(Q_1)^2-(Q_2)Q_1-30Q_1$

$=9Q_1-(Q_1)^2-(Q_2)Q_1$

$\frac{\delta\pi_1}{\delta Q_1}=90-2Q_1-Q_2=0$

$90=2Q_1+Q_2$ ................eq.1

For Firm 2,

$\pi_2=TR_2-TC_2=[{120-(Q_1+Q_2)}]Q_2-30Q_2$

$=(120)Q_2-(Q_2)^2-(Q_1)Q_2-30Q_2$

$=9Q_2-(Q_2)^2-(Q_1)Q_2$

$\frac{\delta\pi_2}{\delta Q_2}=90-2Q_2-Q_1=0$

$90=2Q_2+Q_1$ ................eq.2

On Solving eq.1 and eq.2,

we get ;-

$Q_2=30 , Q_1=30$

hence, each of them produce 30 units.