Technical Appendix

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Equilibrium Modes

Most of the time, each firm can only tend towards one possible equilibrium point. However, in the gray area of the Cartesian plane below, there are three equilibrium outputs that a firm can approach:

Figure 5: The 3-Equilibrium Area

Because of this, we define three Equilibrium Modes: 0, 1, and 2. A firm in mode 0 will tend to the 0-output equilibrium inside the gray area. One in mode 1 will tend to the low-output equilibrium. Finally, a firm in mode 2 will tend to the high-output equilibrium. Outside the gray area, firms in all three modes must tend to the same equilibria: either the 0-output one, or to the high-output one.

Equilibrium Ranges

To make these equilibrium modes function, we sometimes need to shift a firm's output “manually” into an appropriate range in order to make sure the Newton's Method procedure works correctly. Consider, for example, the figure below. Let us say the output of the firm under consideration is at x0. If we apply Newton's Method at this point, the firm will tend to the low-output equilibrium value, which is great if it is in Equilibrium mode 1. However, if the firm is in mode 2, we first shift its output slightly beyond x-max, to some value x1. Then, applying Newton's Method will bring it closer to the high-output equilibrium.

Figure 6: Shifting into the Appropriate Equilibrium Range