**
3.2.1 Perfect Competition**

As we have noted earlier, network externalities arise out of the complementarity of different
network pieces. Thus, they arise naturally in both one- and two-way networks, as well as in
vertically-related markets. The value of good *X* increases as more of the complementary good
*Y* is sold, and vice versa. Thus, more of *Y* is sold as more *X* is sold. It follows that the
value of *X* increases as more of it is sold. This positive feedback loop seems explosive, and
indeed it would be, except for the inherent downward slope of the demand curve. To understand
this better, consider a fulfilled expectations formulation of network externalities as in Katz and
Shapiro (1985), Economides (1993b, 1996a),
and Economides and Himmelberg (1995). Let the
willingness to pay for the *n*th unit of the good when *n ^{e}* units are expected to be
sold be

** Figure 5: Construction of the fulfilled expectations
demand**.

To avoid explosions and infinite sales, it is reasonable to impose lim_{(n-> infinity)} *p*(*n, n*) = 0; it
then follows that *p*(*n, n*) is decreasing for large *n*. Economides and Himmelberg (1995) show
that the fulfilled expectations demand is increasing for small n if either one of three conditions
hold: (i)* the utility of every consumer in a network of zero size is zero*, or (ii)* there are
immediate and large external benefits to network expansion for very small networks*, or (iii)
*there is a significant density of high-willingness-to-pay consumers who are just indifferent on
joining a network of approximately zero size*. The first condition is straightforward and
applies directly to all two-way networks. The other two conditions are a bit more subtle, but
commonly observed in networks and vertically-related industries.

When the fulfilled expectations demand increases for small n, we say that *the
network exhibits a positive critical mass under perfect competition*. This means
that, if we imagine a constant marginal cost *c* decreasing parametrically, the network will start
at a positive and significant size *n*^{o} (corresponding to marginal cost *c*^{o}).
For each smaller
marginal cost, *c* < *c*^{o}, there are three network sizes consistent with marginal cost pricing: a
zero size network; an unstable network size at the first intersection of the horizontal through
*c* with *p*(*n, n*); and the Pareto optimal stable network size at the largest intersection of the
horizontal with *p*(*n, n*). The multiplicity of equilibria is a direct result of the coordination
problem that arises naturally in the typical network externalities model. In such a setting, it is
natural to assume that the Pareto optimal network size will result.^{15}

In the presence of network externalities, it is evident that perfect competition is inefficient: The marginal social benefit of network expansion is larger than the benefit that accrues to a particular firm under perfect competition. Thus, perfect competition will provide a smaller network than is socially optimal, and for some relatively high marginal costs perfect competition will not provide the good while it is socially optimal to provide it.

One interesting question that remains virtually unanswered is how to decentralize the welfare maximizing solution in the presence of network externalities. Clearly, the welfare maximizing solution can be implemented through perfect price discrimination, but typically such discrimination is unfeasible. It remains to be seen to what extent mechanisms that allow for non-linear pricing and self-selection by consumers will come close to the first best.