Intermediate Microeconomics and Its Application 12th Edition by Walter Nicholson,Christopher Snyder

Exercise 14

Consider a two-period model with two firms, A and B. In the first period, they simultaneously choose one of two actions, Enter or Don't enter. Entry requires the expenditure of a fixed entry cost of 10. In the second period, whichever firms enter play a pricing game as follows. If no firm enters, the pricing game is trivial and profits are zero. If only one firm enters, it earns the monopoly profit of 30. If both firms enter, they engage in competition as in the Bertrand model with homogeneous products.
a. Using backward induction, fold the game back to the first period in which firms make their choice of Enter or Don't enter. Write down the normal form (a 2 by 2 matrix) for this game.
b. Solve for the mixed-strategy Nash equilibrium of this game (see Chapter 6 for a discussion of mixed strategies).
c. Compare the results from the mixed-strategy Nash equilibrium to the Bertrand Paradox.