Bertrand Competition.In the market for Christmas trees, demand is given by P=100-Q. There are two firms operating in this market and they both have average variable costs equal to $50 and zero fixed costs.

1. Characterize the best responses to the competitor's price in this case (hint: start assuming one firm sets a given price P>MC and discuss how its competitor will best respond to maximize profits analyzing also alternative strategies and why they are not optimal).
2. Find the Bertrand equilibrium. How much does each firm produce? What are the profits?
3. Assume now firm 1 innovates and lowers its average variable cost to 45. Does the Bertrand equilibrium change? If your answer is affirmative, find the new equilibrium.