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Suppose that two identical firms produce widgets and that they are the only firms in the market. Their costs are given by C1 = 60Q1 and C2 = 60Q2, where Q1 is the output of Firm 1 and Q2 the output of Firm 2. Price is determined by the following demand curve:
P = 300 – Q
where Q = Q1 + Q2.

a. Find the Cournot-Nash equilibrium. Calculate the profit of each firm at this equilibrium.
b. Suppose the two firms form a cartel to maximize joint profits. How many widgets will be produced? Calculate each firm’s profit.
c. Suppose Firm 1 were the only firm in the industry. How would market output and Firm 1’s profit differ from that found in part (b) above?
d. Returning to the duopoly of part (b), suppose Firm 1 abides by the agreement, but Firm 2 cheats by increasing production. How many widgets will Firm 2 produce? What will be each firm’s profits?

Respuesta :

cancinodavidq

Answer:

Consider the following calculations

Explanation:

a. Ο€1 = P Q1 βˆ’ C1 = (300 βˆ’ Q1 βˆ’ Q2 )Q1 βˆ’ 60Q1 = 300Q1 βˆ’ Q1^2 βˆ’ Q1 Q2 βˆ’ 60Q1

Ο€2 = P Q2 βˆ’ C2 = (300 βˆ’ Q1 βˆ’ Q2 )Q2 βˆ’ 60Q2 = 300Q2 βˆ’ Q1 Q2 βˆ’ Q2^2-60Q2

Take the FOCs:

βˆ‚Ο€/(βˆ‚Q1)= 300 βˆ’ 2Q1 βˆ’ Q2 = 0 β‡’ Q1 = 120 βˆ’ 0.5Q2

βˆ‚Ο€/(βˆ‚Q2)= 300 βˆ’ Q1 βˆ’ 2Q2 = 0 β‡’ Q2 = 120 βˆ’ 0.5Q1

Q1 = 120 βˆ’ 0.5[120 βˆ’ 0.5Q1 ] = 60 βˆ’ 0.25Q1 β‡’ Q1 = 80

Similarly find Q2 = 80 such that Ο€1 = Ο€2 = 6, 400.

b. The two firms act as a monopolist, where each firm produces an equal share of total output. Demand is given by P = 300 βˆ’ Q, M R = 300 βˆ’ 2Q, and M C = 60. Set M C = M R tofind that Q = 120 and Q1 = Q2 = 60, respectively. Therefore:

Ο€1 = Ο€2 = 180 Γ— 60 βˆ’ 60 Γ— 60 = 7, 200.

c. It would be higher because they could make more money.

d. Firm 2 knows that Q1 = 60 and given the reaction function derived in part (a) firm 2 sets Q2 = 120 βˆ’ 0.5 Γ— 60 = 90. Overall, QT = 150 and P = 300 βˆ’ 150 = 150. Hence:

Ο€1 = 150 Γ— 60 βˆ’ 60 Γ— 60 = 5, 400

Ο€2 = 150 Γ— 90 βˆ’ 60 Γ— 90 = 8, 100.

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