Final answer:
a) The best-response functions for each firm are Qa = (200 − Qb − 80) / 2 and Qb = (200 − Qa − 80) / 2. b) The Nash-Cournot equilibrium level of quantity for each firm is Qa = 60 and Qb = 60. c) The market quantity is 120 units and the price is $80.
Step-by-step explanation:
a) To determine the best-response functions for each firm, we need to consider how each firm will respond to changes in their own quantity. In a Cournot duopoly, each firm believes its competitor will keep its quantity constant. So, we can solve for the best-response function by setting the derivative of each firm's profit with respect to its own quantity equal to zero.
For Firm A: πA = (200 − (Qa + Qb))Qa − 80Qa. Taking the derivative with respect to Qa and setting it equal to zero, we find that the best-response function for Firm A is: Qa = (200 − Qb − 80) / 2.
Similarly, for Firm B: πB = (200 − (Qa + Qb))Qb − 80Qb. Taking the derivative with respect to Qb and setting it equal to zero, we find that the best-response function for Firm B is: Qb = (200 − Qa − 80) / 2.
b) To find the Nash-Cournot equilibrium level of quantity, we can substitute the best-response functions into each other and solve for Qa and Qb simultaneously. Substituting Qb = (200 − Qa − 80) / 2 into the best-response function for Firm A, we get: Qa = (200 − ((200 − Qa − 80) / 2) − 80) / 2. Simplifying this equation gives us Qa = 60.
Using the same process, we can substitute Qa = (200 − Qb − 80) / 2 into the best-response function for Firm B, which results in Qb = 60.
c) The market quantity is the sum of the quantities produced by each firm: Qmarket = Qa + Qb = 60 + 60 = 120 units. To find the price, we can substitute Qmarket into the market demand function Q = 200 − P. So, 120 = 200 − P, which gives us P = 80.