Technical Appendix

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Complete Incompatibility

In the case of complete incompatibility, m=1 for every firm. Given equations [1], [2], and [3], this means that every firm has the same equilibrium reaction correspondence:


Which is plotted in the graph below:

Figure 1: The Equilibrium Reaction Correspondence Curve

Note that the curve includes the f(x) axis for f(x) > A = 15. This corresponds to equilibria in which other networks' combined output is so high that firm i doesn't find it profitable to produce any products at all.

We employ a variation of Newton's Method to gradually shift the outputs of every firm in the system to points on their respective equilibrium reaction correspondence curves. Details of what happens from the point of view of a single firm follow.

Consider a particular firm i, whose initial output is x0. Furthermore, let the initial combined output of all the firms outside of i's network be y0:

Figure 2: A Firm's Position with respect to the Curve

The goal is for the firm to change its output x so that it's on the above equilibrium curve, as f(x) remains constant at y0. One way to do that would be to simply find the inverse of f(x) - f-inverse(y), and set x to f-inverse(y0). However, in the case of even the simplest externality function v(y) it's very difficult to find f-inverse. We offer an alternative method. We linearize f(x) at x0, then see where the resulting tangent line crosses f(x) = y0, and set x to x1, the value at the point of intersection:

Figure 3: An Iteration of Newton's Method

With every such iteration of the Newton's Method, the firm's output gets closer to the one corresponding to its equilibrium reaction correspondence curve, given that the other firm's outputs stay constant. We go through several of these iterations at a time, to bring the firm's output even closer to the curve. However, when we perform the same procedure on the other firms, their output will change correspondingly, and firm i will be further away from the curve once again:

Figure 4: Change in Combined Output of Other Firms

Fortunately, as we allow this process to iterate over all the firms many times, each one gets closer and closer to its equilibrium reaction correspondence curve, and the steps become smaller and smaller. Eventually, the outputs of all firms converge to some configuration in which each firm locates itself right on its curve, and further iterations of Newton's Method don't lead to significant changes.

Partial and Complete Compatibility

For any FECE with partial or complete compatibility, "Network Externalities" shows that all firms within a network produce the same output. On the other hand, equation [1] from the previous slide still holds. Thus, we append our Newton's Method procedure in a following way: if in a particular network the firms' outputs aren't equal, we first set each output to the network average. This way of equalizing the outputs within a network conserves the total output of the network. Consequently, for all the firms outside the network, the y-coordinate on the Equilibrium Reaction Correspondence graph doesn't shift as a result of the equalization procedure. Then, when all the firms' outputs inside a network are equal, we enact the Newton's Method procedure as described in the previous slides.

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