Question

Suppose that two identical firms produce laptops and they are the only firms in the market. Their cost functions are given by TCl = 60ql and TC2 = 60q2 where q1 and q2 are output of firm 1 and 2 respectively. Price is determined by the following demand curve:
P = 300 — Q where Q = q1 + q2.

i. Find the Cournot-Nash equilibrium. Calculate the profit of each firm at this equilibrium.

ii. What are the characteristics of a Cournot Model?

iii. Assuming that firm 1 moves first to choose the level of output to produce, how much output will each firm produce? What will be the equilibrium price? How much profit will each firm earn?

iv. Suppose the managers of the two firms collude, what will be the equilibrium quantities, price and profit for each firm?

Answer 1

Firm 1 $TC_l = 60Q_l$

Firm 2 $TC_2 = 60Q_2$

$P = 300—Q$

$Q = Q_1 + Q_2$

i. Cournot-Nash equilibrium

price=140 /quantity= $160$

firm 1 profit=$$6400$

firm 2 profit=$$6400$

explanation

Firm1 $TC_l = 60Q_l$

Firm2 $TC_2 = 60q_2$

$P = 300-Q$  

$Q = Q_1 + Q_2$

Firm 1

Find total revenue

$TR=P*Q_1$

$=300Q_1- (Q_1 + Q_2)Q_1$

$TR=300Q_1-Q^2-Q_1Q_2$

Marginal revenue

$MR= d TR/dQ_1$

$MR=300-2Q_1-Q_2$

Marginal cost

$MC=dTC/dQ$

$=d60Q_1/dQ_1$

$MC=60$

$MR_1=MC_1$

$60=300-2Q_1-Q_2$

$Q_1=120-0.5Q_2$

Firm 2

Find total revenue

$TR=P*Q_2$

$=300Q1- (Q_1 + Q_2)Q_1$

$TR=300Q_1-Q^2-Q_1Q_2$

Marginal revenue

$MR=dTR/dQ$

$MR=300-2Q_2-Q_1$

Marginal cost

$MC=dTC/dQ_2 =d60Q_2/dQ_2$

$MC=60$

$MR_2=MC_2$

$60=300-2Q_2-Q_1$

Q1=120-0.5Q2

Equilibrium calculated $Q_2$ putting $Q_1$

$Q1=120-0.5Q_1*(120-0.5Q_2)$

$3Q=240$

$Q1=80$

firm 1 quantity+firm 1 quantity

$=80+80$

 equilibrium quantity $=160$

Calculate price

$P=300-Q$

$=300-160$

$p=140$

equilibrium price$=140$

Firm 1 profit

$Profit=TR-TC$

$=PQ_1-(60Q_1)$

$= (140*80)-(60*80)$

$Profit=6400$    

     

Firm 2 profit

$Profit=TR-TC$

$=PQ_2-(60Q_2)$

$= (140*80)-(60/80)$

$Profit=6400$

ii) The Courton model of oligarchy assumes that rival companies produce a homogeneous product, and each chooses to maximize profit by selecting production. All firms simultaneously select output (quantity). The basic assumption is that each firm chooses its own volume given the volume of its competitors. The resulting equilibrium is a Nash equilibrium, called a court equilibrium

$iii) Firm 1$

$quantity= 120$

Firm 2 price=$$140$

Firm 2 profit=$$14400$

explanation

$P=300-Q_1$

$TC_1=60Q_1$

$Maximized/ output= MR=MC$

$TR=p*Q_1$

$= (300-Q_1) Q_1$

$TR=300Q-Q^2$

$MR=dTR/dQ_1 =d300Q-2Q_1/dQ_1$

$MR=300-2Q_1$

$MC=dTC/dQ=d60Q/dQ_1$

$MC=60$

$MR=MC$

$300-2Q_1=60$

$Q_1=120$

Price

$p=300-Q$

$p=300-120$

$p=180$

$Profit=TR-TC$

$=(120*180)-(120*60)$

$Profit=14400$

iv. $quntity=120$

price=$$180$

profit=$$7200$

explanation

The two firms act as a monopolist, where each firm produces an equal share of total output. Demand is given $by P = 300 – Q$  

$MR = 300 − 2Q,$

 $MC = 60.$

$M C = MR$  

$=300-2Q=60$

$Q = 120$

and $Q_1 = Q_2 = 60$ (each firm produce($120/2$ )

respectively. Therefore:

$p=180$

$π1 = π2 = 180 × 60 − 60 × 60$

=$ $7200.$