Suppose that two identical firms produce widgets and that they are the only firms in the market. Their costs are given by \(C_{1}=60 Q_{1}\) and \(C_{2}=60 Q_{2},\) where \(Q_{1}\) is the output of Firm 1 and \(Q_{2}\) the output of Firm 2. Price is determined by the following demand curve: \\[ \begin{aligned} P &=300-Q \\ \text { where } Q=Q_{1}+Q_{2} \end{aligned} \\] 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?

Step by step solution

01

Find the Cournot-Nash Equilibrium

First, one assumes that each firm takes the output level of the other firm as given and chooses its output level in order to maximize its own profit. So, the profit function for each firm is: \[\begin{aligned} \pi_{i}=P \cdot Q_{i}-C_{i}=(300-Q) Q_{i}-60Q_{i} \end{aligned}\] Differentiating with respect to \(Q_{i}\), we can find the first order condition of each firm:\[\begin{aligned} \frac{d \pi_{i}}{d Q_{i}}=300-2Q_{i}-Q_{j}-60=0 \end{aligned}\] Since both firms are identical, at Cournot equilibrium, \(Q_{1}=Q_{2}=Q_{c}\). So, substituting \(Q_{1}\), \(Q_{2}\) with \(Q_{c}\) in the first order condition, we can find the output level \(Q_{c} = 60\). The price at equilibrium can be found from the demand curve as \(P_{c}=300-Q_{c}=240\). The profit for each firm can be calculated by substituting \(Q_{c}\), \(P_{c}\) into the profit function, which is \( \pi_{c}=P_{c} \cdot Q_{c}-C_{c}=240 \cdot 60 - 60 \cdot 60 = \$10800\)

02

Determine Output and Profit when Firms Form a Cartel

A cartel acts like a monopoly and produces the output level that maximizes joint profits. The sum of the cost functions gives us the total cost of the cartel, which is \(C_{C} = 60 (Q_{1}+Q_{2}) = 60 \cdot 2Q = 120Q\). Now, we substitute \(Q=Q_{1}+Q_{2}\) into the demand function and find the profit function of the cartel:\[\pi_{C}=(300-Q)Q-120Q\] Differentiating with respect to \(Q\) gives us \( \frac{d\pi_{C}}{dQ} = 0 \). Solving the equation would give us the cartel's output level and hence, the profit. Assume both firms split the output equally, so their profits would also be equal.

03

Analyze the Scenario where Firm 1 is the Only Firm in the Industry

Suppose Firm 1 is now the monopoly in the market. It will choose its output level to maximize its profit, which is \( \pi_{1}=(300-Q_{1})Q_{1}-60Q_{1} \). First order differentiation and setting it to zero allows one to solve for \(Q_{1}\), which is the monopolistic output level, and hence determine the monopolistic price and profit.

04

Analyze Scenario where Firm 1 Abides by Cartel Agreement but Firm 2 Cheats

Assuming Firm 1 sticks to the cartel output while Firm 2 deviates and maximizes its profit considering Firm 1's output as given, we have:\[\pi_{2}=(300-Q_{1}-Q_{2})Q_{2}-60Q_{2}\] Setting the first order condition equal to zero we can determine \(Q_{2}\). Plugging this value into demand equation, you can find the price which will help us find individual profits.

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