Determine the payoffs (profits) for a firm

 

 

Question 2

The total industry output is represented by (Q = q_1 + q_2), and the industry demand curve is given by (P(Q) = 11 – 2Q). The cost functions for the two firms are (C(q_1) = 1q_1) and (C(q_2) = 2q_2). To determine the payoffs (profits) for each firm, we need to calculate the individual firm profits based on the given information.

Let’s calculate the profits for each firm:

– Firm 1 profit ((\pi_1)) = (P(Q)q_1 – C(q_1))
– Firm 2 profit ((\pi_2)) = (P(Q)q_2 – C(q_2))

Question 3

In a duopoly scenario, the Nash equilibrium occurs when both firms choose their optimal strategy, considering the other’s choice. The Nash equilibrium levels of output can be derived by analyzing the reaction of each firm to the other’s output choice. By finding the point where neither firm has an incentive to deviate from their chosen output level, we can determine the Nash equilibrium.

Question 4

The reaction functions for both firms indicate how each one adjusts its output level in response to the other firm’s output choice to maximize its own profit. By differentiating each firm’s profit function with respect to its own output, we can derive the reaction functions.

– Firm 1 reaction function: Find (\frac{d\pi_1}{dq_1} = 0)
– Firm 2 reaction function: Find (\frac{d\pi_2}{dq_2} = 0)

Question 5

If firms choose prices rather than output, the demand for good 1 would be a function of both prices ((P_1,P_2)). The demand for good 1 can be expressed as (Q = f(P_1,P_2)), where (Q) is the quantity demanded of good 1 depending on the prices set by both firms.

The relationship between the prices set by each firm and the resulting demand for good 1 will determine the equilibrium in a price-setting duopoly scenario. Analyzing how changes in prices affect demand will be crucial in understanding the implications of price competition between the two firms.