Firms A and B choose how much of a a homogenous good to produce at a marginal cost of1. The inverse demand function is p = 25 − qA − qB where qA is the output choice of firm A and qBis the output choice of firm B. A firm’s action is its output choice and can be any number greaterthan or equal to zero.Eliminating all strongly dominated actions by the two firms in the first round. In the gamethat remains eliminate all strongly dominated actions in a second round. What is the set of actionsthat are still available to each of the two firms after the second elimination round?[Write your answer as an interval; e.g.: [0.2, 0.4] or (0.3, 0.5], etc.
Question
Firms A and B choose how much of a a homogenous good to produce at a marginal cost of1. The inverse demand function is p = 25 − qA − qB where qA is the output choice of firm A and qBis the output choice of firm B. A firm’s action is its output choice and can be any number greaterthan or equal to zero.Eliminating all strongly dominated actions by the two firms in the first round. In the gamethat remains eliminate all strongly dominated actions in a second round. What is the set of actionsthat are still available to each of the two firms after the second elimination round?[Write your answer as an interval; e.g.: [0.2, 0.4] or (0.3, 0.5], etc.
Solution
This problem involves game theory and the concept of dominant strategies.
Step 1: Identify the profit function for each firm.
The profit function for each firm is given by π = pq - c*q, where p is the price, q is the quantity, and c is the marginal cost. Substituting the given inverse demand function and marginal cost into the profit function, we get:
πA = (25 - qA - qB)qA - qA = 24qA - qA^2 - qAqB πB = (25 - qA - qB)qB - qB = 24qB - qB^2 - qAqB
Step 2: Identify the best response functions for each firm.
The best response function for each firm is the output level that maximizes its profit given the output level of the other firm. To find this, we take the derivative of the profit function with respect to the firm's output level and set it equal to zero.
For firm A, this gives us: 24 - 2qA - qB = 0 Solving for qA, we get: qA = (24 - qB)/2
Similarly, for firm B, we get: 24 - 2qB - qA = 0 Solving for qB, we get: qB = (24 - qA)/2
Step 3: Identify the strongly dominated strategies.
A strategy is strongly dominated if there is another strategy that always gives a higher payoff, regardless of what the other firm does.
From the best response functions, we can see that for any given output level of the other firm, a firm can always increase its profit by producing less. Therefore, any output level greater than (24 - q)/2 is strongly dominated for each firm.
Step 4: Eliminate the strongly dominated strategies.
After the first round of elimination, the set of actions available to each firm is [0, (24 - q)/2].
In the second round of elimination, we again eliminate any output levels that are strongly dominated. However, since the best response functions are the same in each round, the set of actions available to each firm remains [0, (24 - q)/2].
So, the set of actions that are still available to each of the two firms after the second elimination round is [0, (24 - q)/2].
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