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. Calculate the profit for each firm

Answer 1

Solution

Demand function

$Q=120-P$

So,

$P=120-Q$ (1)

Since there are two firms in the market:

$Q=q_1+q_2$

Cost Function

$C=30Q$

Firm 1 Cost: $30q_1$ (2)

Firm 2 Cost: $30q_1$ (3)

Revenue: $Q*P$

Firm 1 Revenue:

$R= q_1 (120-(q_1+q_2))=120q_1-q_1^2-q_1q_2$ (4)

Firm 2 Revenue:

$R= q_2 (120-(q_1+q_2))=120q_2-q_2q_1-q_2^2$ (5)

Profit: $Revenue -Total Cost=R-C$

Firm 1 Profit: (4) minus (2)

$π(q_1,q_2)=(120q_1-q_1^2-q_1q_2)-30q_1$

$π(q_1,q_2)=90q_1-q_1^2-q_1q_2$ (6)

Firm 2 profit:

$π(q_1,q_2)=(120q_2-q_1q_2-q_1^2)-30q_2$

$π(q_1,q_2)=90q_2-q_2q_1-q_2^2$ (7)

Derivative of $π$ with respect to $q_1$ for (6)

$dπ/(dq_1 )=90-2q_1-q_2=0$

$90-q_2=2q_1$

$q_1=(90-q_2)/2$ (8)

Derivative of $π$ with respect to $q_2$ for (7)

$dπ/(dq_1 )=90-q_1-2q_2-=0$

$90-q_1=2q_2$

$q_2=(90-q_1)/2$ (9)

Substitute (9) into (8)

$q_1=(90-(90-q_1)/2)/2$

$q_1=(90+q_1)/4$

$90+q_1=4q_1$

$90=3q_1$

$q_1=30$ units

Substitute $q_1$ with 30 in (9)

$q_2=(90-30)/2=60/2$

$q_2=30$ units

Each Firm Profit

Substitute the value of $q_1$ and $q_1$ into (6) and (7)

Firm 1 profit:

$π(q_1,q_2)=90(30)-(30)^2-30(30)=$ $$900$

Firm 2 profit:

$π(q_1,q_2)=90(30)-30(30)-(30)^2=$ $$900$