Consider a duopoly market where Firm 1 and Firm 2 produce differentiated products such that the demand function of each firm is given by :
q1(p1, p2) = 18 - p1 + p2
q2(p1, p2) = 18 + p1 - p2
Here, q1 and q2 are the outputs produced by Firm 1 and Firm 2, respectively, and p1 and p2 are the corresponding per unit prices.
Cost of production for the ith firm is given by Ci(qi) = 2qi ∀ i = 1,2
The firms compete in prices. The price set by Firm 2 such that the market is in Nash equilibrium will be ________ (in integer).
q1(p1, p2) = 18 - p1 + p2
q2(p1, p2) = 18 + p1 - p2
Here, q1 and q2 are the outputs produced by Firm 1 and Firm 2, respectively, and p1 and p2 are the corresponding per unit prices.
Cost of production for the ith firm is given by Ci(qi) = 2qi ∀ i = 1,2
The firms compete in prices. The price set by Firm 2 such that the market is in Nash equilibrium will be ________ (in integer).
Correct Answer: 20
Solution and Explanation
To find the Nash equilibrium in this differentiated products duopoly setting, we use the concept of Bertrand competition, where each firm chooses its price to maximize profit given the price of the other firm.
Step-by-step Solution:
1. **Demand Functions:** Given demand functions, for Firm 1: q1 = 18 - p1 + p2 and for Firm 2: q2 = 18 + p1 - p2.
2. **Profit Functions:** The profit for each firm is revenue minus cost. Thus for Firm 1: π1 = p1q1 - 2q1 and for Firm 2: π2 = p2q2 - 2q2.
3. **Substituting Demand into Profit:**
For Firm 1: π1 = p1(18 - p1 + p2) - 2(18 - p1 + p2)
For Firm 2: π2 = p2(18 + p1 - p2) - 2(18 + p1 - p2).
4. **Maximizing Profit and Finding Best Response Functions:**
- Differentiate π1 with respect to p1 and set to zero to find Firm 1's best response: 18 - 2p1 + p2 = 0. Solving, p1 = 9 + 0.5p2.
- Differentiate π2 with respect to p2 and set to zero: 18 + p1 - 2p2 = 0. Solving, p2 = 9 + 0.5p1.
5. **Finding Nash Equilibrium:**
Substitute the expression for p1 into the expression for p2:
p2 = 9 + 0.5(9 + 0.5p2).
Solving, p2 = 12.
Substitute p2 back to find p1:
p1 = 9 + 0.5(12) = 15.
Final Result: The price set by Firm 2 in Nash equilibrium is 12, which falls within the expected range of 20,20.
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