Suppose there are two firms having equal market share in an oligopoly market. The market demand function, [tex]P=420-10\left(Q_1+Q_2\right)[/tex] and cost functions, [tex]TC _1=20 Q _1[/tex] and [tex]TC _2=60 Q _2[/tex]. Determine price, output and profit of low cost firm. Is this price leadership acceptable to the high cost firm?
Answer (2)
Firm 2's reaction function is derived by maximizing its profit, resulting in Q 2 = 18 − 2 1 Q 1 .
Firm 1 maximizes its profit by substituting Firm 2's reaction function into its profit function and solving for Q 1 , yielding Q 1 = 22 .
Substituting Q 1 into Firm 2's reaction function gives Q 2 = 7 , and the market price is calculated as P = 130 .
Firm 1's profit is π 1 = 2420 , and Firm 2's profit is π 2 = 490 . Price leadership is not acceptable to Firm 2, as its profit would be higher under Cournot competition.
P = 130 , Q 1 = 22 , π 1 = 2420
Explanation
Problem Setup We are given an oligopoly market with two firms. Firm 1 has a cost function T C 1 = 20 Q 1 and firm 2 has a cost function T C 2 = 60 Q 2 . The market demand function is P = 420 − 10 ( Q 1 + Q 2 ) . We need to determine the price, output, and profit of the low-cost firm (Firm 1), assuming it acts as a price leader. We also need to assess whether this price leadership is acceptable to the high-cost firm (Firm 2).
Firm 2's Reaction Function First, we need to find Firm 2's reaction function. Firm 2's profit is given by:
π 2 = T R 2 − T C 2 = P ⋅ Q 2 − 60 Q 2 = [ 420 − 10 ( Q 1 + Q 2 )] Q 2 − 60 Q 2
To maximize Firm 2's profit, we take the derivative with respect to Q 2 and set it equal to zero:
d Q 2 d π 2 = 420 − 10 Q 1 − 20 Q 2 − 60 = 0
Solving for Q 2 , we get Firm 2's reaction function:
20 Q 2 = 360 − 10 Q 1
Q 2 = 18 − 2 1 Q 1
Firm 1's Profit Function Now, we can determine Firm 1's profit-maximizing output. Firm 1's profit is given by:
π 1 = T R 1 − T C 1 = P ⋅ Q 1 − 20 Q 1 = [ 420 − 10 ( Q 1 + Q 2 )] Q 1 − 20 Q 1
Substitute Firm 2's reaction function into Firm 1's profit function:
π 1 = [ 420 − 10 ( Q 1 + 18 − 2 1 Q 1 )] Q 1 − 20 Q 1
π 1 = [ 420 − 10 Q 1 − 180 + 5 Q 1 ] Q 1 − 20 Q 1
π 1 = [ 240 − 5 Q 1 ] Q 1 − 20 Q 1
π 1 = 240 Q 1 − 5 Q 1 2 − 20 Q 1
π 1 = 220 Q 1 − 5 Q 1 2
Firm 1's Output To maximize Firm 1's profit, we take the derivative with respect to Q 1 and set it equal to zero:
d Q 1 d π 1 = 220 − 10 Q 1 = 0
Solving for Q 1 , we get:
10 Q 1 = 220
Q 1 = 22
Firm 2's Output Now, we can find Firm 2's output by substituting Q 1 = 22 into Firm 2's reaction function:
Q 2 = 18 − 2 1 ( 22 ) = 18 − 11 = 7
Market Price Next, we calculate the market price using the market demand function:
P = 420 − 10 ( Q 1 + Q 2 ) = 420 − 10 ( 22 + 7 ) = 420 − 10 ( 29 ) = 420 − 290 = 130
Firm 1's Profit Now, we can calculate Firm 1's profit:
π 1 = ( P ⋅ Q 1 ) − ( 20 ⋅ Q 1 ) = ( 130 ⋅ 22 ) − ( 20 ⋅ 22 ) = 2860 − 440 = 2420
Firm 2's Profit And Firm 2's profit:
π 2 = ( P ⋅ Q 2 ) − ( 60 ⋅ Q 2 ) = ( 130 ⋅ 7 ) − ( 60 ⋅ 7 ) = 910 − 420 = 490
Acceptability of Price Leadership Finally, we need to determine if price leadership is acceptable to Firm 2. To do this, we can compare Firm 2's profit under price leadership to its profit in a Cournot competition scenario. In a Cournot scenario, both firms choose their output simultaneously. The profit functions are:
π 1 = [ 420 − 10 ( Q 1 + Q 2 )] Q 1 − 20 Q 1 π 2 = [ 420 − 10 ( Q 1 + Q 2 )] Q 2 − 60 Q 2
The reaction functions are:
Q 1 = 20 − 2 1 Q 2 Q 2 = 18 − 2 1 Q 1
Solving these simultaneously:
Q 1 = 20 − 2 1 ( 18 − 2 1 Q 1 ) = 20 − 9 + 4 1 Q 1 4 3 Q 1 = 11 Q 1 = 3 44
Q 2 = 18 − 2 1 ( 3 44 ) = 18 − 3 22 = 3 54 − 22 = 3 32
P = 420 − 10 ( 3 44 + 3 32 ) = 420 − 10 ( 3 76 ) = 420 − 3 760 = 3 1260 − 760 = 3 500
π 2 = ( 3 500 ⋅ 3 32 ) − ( 60 ⋅ 3 32 ) = 9 16000 − 3 1920 = 9 16000 − 5760 = 9 10240 ≈ 1137.78
Since Firm 2's profit under Cournot competition (approximately 1137.78) is greater than its profit under price leadership (490), price leadership is NOT acceptable to the high-cost firm.
Examples
Understanding oligopoly models and price leadership is crucial in various real-world scenarios. For instance, consider the airline industry, where a dominant airline with lower operating costs might set prices, influencing smaller airlines to adjust their fares accordingly. Analyzing these strategic interactions helps businesses make informed decisions about pricing, output, and market entry, ultimately impacting profitability and market share. This type of analysis is also relevant in industries like telecommunications and automotive manufacturing, where a few large players dominate the market.
Firm 1 (low-cost firm) produces 22 units at a price of 130, earning a profit of 2420, while Firm 2 (high-cost firm) produces 7 units with a profit of 490. Price leadership is deemed unacceptable for Firm 2 as it could earn more under Cournot competition. Therefore, Firm 2's potential profit in a competitive environment outweighs the gains from accepting Firm 1's price leadership.
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