2. Consider a monopolist facing the demand curve p = 10 Q. The monopolist can ch

Question

2. Consider a monopolist facing the demand curve p = 10 Q.
The monopolist can
choose either of the following cost functions:
c1(q) = 3q
or
c2(q) = 10 + q.
a) Which cost function does the monopolist choose?
b) Now suppose that there is a second rm with cost function c1 (q) above. Whichever
cost function rm 1 chooses, the second rm will observe this choice and then have the
option of entering the market or not. If he does not enter, rm 1 remains a monopolist
with the chosen cost function. If rm 2 does enter, he pays an entry cost of $4 and the
two rms compete as Cournot duopolists. More precisely, whichever cost function rm 1
chose, we have a pure strategy Nash equilibrium in quantity choices. Which cost function
does rm 1 choose now? Put dierently, what is the subgame perfect equilibrium of this
game?

Explanation / Answer

If both firms enter the market, and they collude, they will face a marginal revenue curve with twice the slope of the demand curve: MR = 10 - 2Q. Setting marginal revenue equal to marginal cost (the marginal cost of Firm 1, since it is lower than that of Firm 2) to determine the profit-maximizing quantity, Q: 10 - 2Q = 2, or Q = 4. Substituting Q = 4 into the demand function to determine price: P = 10 - 4 = $6. The profit for Firm 1 will be: The profit for Firm 2 will be: Total industry profit will be: p1 = (6)(4) - (4 + (2)(4)) = $12. p2 = (6)(0) - (3 + (3)(0)) = -$3. pT = p1 + p2 = 12 - 3 = $9. If Firm 1 were the only entrant, its profits would be $12 and Firm 2’s would be 0. If Firm 2 were the only entrant, then it would equate marginal revenue with its marginal cost to determine its profit-maximizing quantity: 10 - 2Q2 = 3, or Q2 = 3.5. Substituting Q2 into the demand equation to determine price: P = 10 - 3.5 = $6.5. The profits for Firm 2 will be: p2 = (6.5)(3.5) - (3 + (3)(3.5)) = $9.25 In order to determine how much Firm 1 will be willing to pay to purchase Firm 2, we must compare Firm 1’s profits in the monopoly situation versus those in an oligopoly. The difference between the two will be what Firm 1 is willing to pay for Firm 2. Substitute the profit-maximizing quantity from part a to determine the price: The profits for the firm are determined by subtracting total costs from total revenue: We know from part b that the profits for Firm 1 in the oligopoly situation will be $5; therefore, Firm 1 should be willing to pay up to $7, which is the difference between its monopoly profits ($12) and its oligopoly profits ($5). (Note that any other firm would pay only the value of Firm 2’s profit, i.e., $1.) Note, Firm 1 might be able to accomplish its goal of maximizing profit by acting as a Stackelberg leader. If Firm 1 is aware of Firm 2’s reaction function, it can determine its profit-maximizing quantity by substituting for Q2 in its profit function and maximizing with respect to Q1: Therefore Chapter 12: Monopolistic Competition and Oligopoly Substituting Q1 and Q2 into the demand equation to determine the price: Profits for Firm 1 are: and profits for Firm 2 are: Although Firm 2 covers average variable costs in the short run, it will go out of business in the long run. Therefore, Firm 1 should drive Firm 2 out of business instead of buying it. If this is illegal, Firm 1 would have to resort to purchasing Firm 2, as discussed above. In the Cournot model, Firm 1 takes Firm 2’s output as given and maximizes profits. The profit function derived in 2.a becomes p1 = (10 - Q1 - Q2 )Q1 - (4 + 2Q1 ), or Chapter 12: Monopolistic Competition and Oligopoly Setting the derivative of the profit function with respect to Q1 to zero, we find Firm 1’s reaction function: Similarly, Firm 2’s reaction function is To find the Cournot equilibrium, we substitute Firm 2’s reaction function into Firm 1’s reaction function: Substituting this value for Q1 into the reaction function for Firm 2, we find Q2 = 2. Substituting the values for Q1 and Q2 into the demand function to determine the equilibrium price: P = 10 - 3 - 2 = $5. The profits for Firms 1 and 2 are equal to p1 = (5)(3) - (4 + (2)(3)) = 5 and p2 = (5)(2) - (3 + (3)(2)) = 1.

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