There are two firms in an oligopolistic industry competing in prices and selling a homogeneous product. Total cost of production for firm i is C_i(q_i) = 10q_i, i = 1,2; where q_i... There are two firms in an oligopolistic industry competing in prices and selling a homogeneous product. Total cost of production for firm i is C_i(q_i) = 10q_i, i = 1,2; where q_i is the quantity produced by firm i. Suppose firm i sets price p_i and firm j sets price p_j. The market demand faced by firm i is given by q_i(p_i,p_j) = 100 - p_i if p_i < p_j; 0 if p_i = p_j; (100 - p_i) / 2 if p_i > p_j. Price can only take integer values in this market. Nash equilibrium/equilibria is/are given by.
Understand the Problem
The question is asking to analyze a scenario in an oligopolistic market involving two firms. It involves understanding their cost functions, market demand, and Nash equilibrium related to price settings by the firms.
Answer
Nash equilibria occur at $p_1 = p_2 = 55$ with quantities approximately $(22, 23)$.
Answer for screen readers
The Nash equilibria occur when both firms set prices at $p_1 = p_2 = 55$ yielding quantities of $q_1 = q_2 = 22.5$. Rounding results in possible quantities of either $(22, 23)$ or $(23, 22)$.
Steps to Solve
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Understand the cost function The total cost for each firm is given by $C_i(q_i) = 10q_i$. This indicates that the marginal cost (MC) for each firm is constant at $10$.
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Set up the profit function The profit for each firm can be expressed as: $$ \pi_i = p_i q_i(p_i, p_j) - C_i(q_i) $$ Substituting the cost function: $$ \pi_i = p_i q_i(p_i, p_j) - 10q_i $$
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Determine demand for each pricing case Consider the three scenarios for the demand:
- If $p_i < p_j$, then $q_i = 100 - p_i$.
- If $p_i = p_j$, then $q_i = 0$.
- If $p_i > p_j$, then $q_i = \frac{100 - p_i}{2}$.
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Analyze case when $p_i < p_j$ Substituting $q_i = 100 - p_i$ into the profit function: $$ \pi_i = p_i (100 - p_i) - 10 (100 - p_i) $$ $$ = p_i (100 - p_i) - 1000 + 10p_i = -p_i^2 + 110p_i - 1000 $$
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Find the best response functions To find the equilibrium prices, set the derivative of the profit function with respect to $p_i$ to zero: $$ \frac{d\pi_i}{dp_i} = -2p_i + 110 = 0 $$ Solving gives $p_i = 55$. Use this to find $p_j$ using symmetry.
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Check scenarios for Nash equilibrium If both firms set $p_i = p_j = 55$, then $q_i = \frac{100 - 55}{2} = 22.5$. Round to the nearest integers, which would lead to $(22, 23)$ or vice versa. Verify best responses and check for other integer combinations.
The Nash equilibria occur when both firms set prices at $p_1 = p_2 = 55$ yielding quantities of $q_1 = q_2 = 22.5$. Rounding results in possible quantities of either $(22, 23)$ or $(23, 22)$.
More Information
This scenario reflects a well-known principle in game theory where firms in an oligopoly need to consider the pricing strategy of their competitor while deciding their prices. The equilibrium price of $55$ minimizes the likelihood of price wars while allowing both firms to maintain positive sales.
Tips
- Ignoring marginal cost: Ensure that the marginal cost is factored into profit calculations.
- Not considering integer constraints: Remember that prices can only take integer values; always round after calculating prices.
- Misapplying demand function: Be careful with the conditions for demand; ensure that the conditions $p_i < p_j$, $p_i = p_j$, and $p_i > p_j$ are correctly used.
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