Decision Making Assignment: Solutions to the exercises in chapter "Monopoly and Cournot Markets" of the book of Charles Holt "Markets, Games and Strategic Behavior"

Solved by Desirée Suárez

Exercise 1: Use table 6.2 to show that outputs of 6 and 3 do not constitute a Nash equilirbium for the duopoly model that is the basis for that table.

In the next table we can observe the profits of each firm depending on their respective production quantities:

If the entrant produces an output of 3, the incumbent’s best response will be to produce an output of either 4 or 5.

If the entrant produces an output of 6, the incumbent’s best response will be to produce an output of 3.

If the incumbent produces an output of 3, the entrant’s best response will be to produce an output of either 4 or 5.

If the incumbent produces an output of 6, the entrant’s best response will be to produce an output of 3.

There is no Nash Equilibrium for outputs of 6 and 3.

Exercise 2: Use table 6.2 to show that outputs of 3 for each firm do not constitute a Cournot/Nash equilibrium.

If the entrant produces an output of 3, the incumbent’s best response will be to produce an output of either 4 or 5.

If the incumbent produces an output of 3, the entrant’s best response will be to produce an output of either 4 or 5.

There is no Nash/Cournot Equilibrium for outputs of 3 for each firm.

Exercise 3: Find an asymmetric Cournot/Nash equilibrium in Table 6.2 with the property that the total quantity is 8 but one seller produces more than the other. Therefore, you must specify what the two outputs are, and you must show that neither seller has a unilateral incentive to deviate. (This asymmetric equilibrium is an artifact of the discrete nature of the quantity choices, which are constrained to be integers.)

1) If the incumbent produces an output of 6, the entrant’s best response will be to produce an output of 3.

   If the entrant produces an output of 2, the incumbent’s best response will be to produce an output of 5.

2) If the incumbent produces an output of 2, the entrant’s best response will be to produce an output of 5.

   If the entrant produces an output of 6, the incumbent’s best response will be to produce an output of 3.

3) If the incumbent produces an output of 5, the entrant’s best response will be to produce an output of either 3 or 4.

   If the entrant produces an output of 3, the incumbent’s best response will be to produce an output of either 4 or 5.

4) If the incumbent produces an output of 3, the entrant’s best response will be to produce an output of either 4 or 5.

   If the entrant produces an output of 5, the incumbent’s best response will be to produce an output of either 3 or 4.

There are 2 asymmetric equilibria: one when the incumbent produces an output of 5 and the entrant 3, and another one when the incumbent produces 3 and the entrant 5.