Game Theory and Profit

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Chapter 11
Consider two firms facing the demand curve P = 50 - 5Q, where Q = Q1 + Q2. The firms’ cost functions are C1(Q1) = 20 + 10Q1 and C2(Q2) = 10 + 12Q2.
a.Suppose both firms have entered the industry. What is the joint profit-maximizing level of output? How much will each firm produce? How would your answer change if the firms have not yet entered the industry?

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 = 50 - 10Q.
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:
50 - 10Q = 10, or Q = 4.
Substituting Q = 4 into the demand function to determine price:
P = 50 – 5*4 = $30.
The question now is how the firms will divide the total output of 4 among themselves. Since the two firms have different cost functions, it will not be optimal for them to split the output evenly between them. The profit maximizing solution is for firm 1 to produce all of the output so that the profit for Firm 1 will be:

1 = (30)(4) - (20 + (10)(4)) = $60.
The profit for Firm 2 will be:
2 = (30)(0) - (10 + (12)(0)) = -$10.
Total industry profit will be:
T = 1 + 2 = 60 - 10 = $50.
If they split the output evenly between them then total profit would be $46 ($20 for firm 1 and $26 for firm 2). If firm 2 preferred to earn a profit of $26 as opposed to $25 then firm 1 could give $1 to firm 2 and it would still have profit of $24, which is higher than the $20 it would earn if they split output. Note that if firm 2 supplied all the output then it would set marginal revenue equal to its marginal cost or 12 and earn a profit of 62.2. In this case, firm 1 would earn a profit of –20, so that total industry profit would be 42.2.

If Firm 1 were the only entrant, its profits would be $60 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:
50 - 10Q2 = 12, or Q2 = 3.8.
Substituting Q2 into the demand equation to determine price:
P = 50 – 5*3.8 = $31.
The profits for Firm 2 will be:
2 = (31)(3.8) - (10 + (12)(3.8)) = $62.20.
b.What is each firm’s equilibrium output and profit if they behave noncooperatively? Use the Cournot model. Draw the firms’ reaction curves and show the equilibrium.
In the Cournot model, Firm 1 takes Firm 2’s output as given and maximizes profits. The profit function derived in 2.a becomes
1 = (50 - 5Q1 - 5Q2 )Q1 - (20 + 10Q1 ), or

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 = 2.4.
Substituting the values for Q1 and Q2 into the demand function to determine the equilibrium price:
P = 50 – 5(2.8+2.4) = $24.
The profits for Firms 1 and 2 are equal to
1 = (24)(2.8) - (20 + (10)(2.8)) = 19.20 and
2 = (24)(2.4) - (10 + (12)(2.4)) = 18.80.
c.How much should Firm 1 be willing to pay to purchase Firm 2 if collusion is illegal but the takeover is not?
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. From part a, profit of firm 1 when it set marginal revenue equal to its marginal cost was $60. This is what the firm would earn if it was a monopolist. From part b, profit was $19.20 for firm 1. Firm 1 would therefore be willing to pay up to $40.80 for firm 2.

3. A monopolist can produce at a constant average (and...
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