Increasing Compatibility Among Three Firms

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Three Firms with Incompatible Products

Note: in order to run the simulation referred to in this slide, go here, to the Java Applet version. You will be directed to download the latest version of the Java plug-in.

In the graphic to the left, we demonstrate four different equilibria in a scenario where three firms are fully incompatible with each other. Each of the four buttons "A", "B", "C", and "D" sets up the society to find one of these. For instance, press "A" and then "Go" to achieve the monopoly equilibrium, in which only one firm has positive output. Initializing with "B" or "C", on the other hand, yields the symmetric two-firm and three-firm equilibria, respectively. Finally, "D" sets up the initial conditions for an asymmetric equilibrium in which one firm ends up with zero output, and the other two firms have different positive outputs.

Unfortunately, our procedure doesn't allow us to find the full set of possible equilibria under these conditions - two more equilibria actually exist. Both have all three firms at non-zero, asymmetric outputs. Both are unstable equilibria under the variant of Newton's Method we use.

For reference in future slides, here are the equilibrium output values for the four initial states above:

  A: 0, 0, and 18.15
  B: 0, 10.36, and 10.36
  C: 7.07, 7.07, and 7.07
  D: 0, 0.63, and 17.71

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