Two duopoly firms face a market demand of Y 50 05P Assumin

Two duopoly firms face a market demand of Y = 50 - 0.5P. Assuming the firms compete in quantities and face a marginal cost of 10 per unit of output, calculate: i) the outputs of the two firms ii) the equilibrium price iii) the profits of the two firms

Solution

Y=50-0.5P

P=100-2Y=100-2y1-2y2

Profit for Firm 1=P*y1-10y1=(100-2y1-2y2-10)y1

=90y1-2y1^2-2y2*y1

Differentiating profit with respect to y1 then FOC would be equal to

=90-4y1-2y2=0

45-2y1=y2

Now Profit for Firm 2

=(90-2y1-2y2)*y2

SImilalry both the firms are symmetric we get

45-2y2=y1

45-2(45-2y1)=y1

3y1=45

y1=15=y2

Price=100-2(y1+y2)=100-60=40

Profit for firm 1=40*15-10*15=450

Profit for firm 2=450