Strength of the Network Externality

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**Strong Network Externalities Facilitate Monopolies**

A result by Katz and Shapiro captures the effect of network externalities on an oligopoly:
"[In the case of complete incompatibility] a symmetric equilibrium with k active firms exists if and only
if `v(A/k) ≥ A/k`

."

Recall that *v(y)* is a function that reflects the strength of the network externality effect.
The greater *v(y)* for each *y*, the more utility each user derives from a particular
network, and the more each firm can capitalize on the size of its network to increase its output
and profit. Recall also that consumer willingness to pay is distributed uniformly up to the constant *A*,
which we fix at 15 throughout this tutorial.

We fix *v(x)* to be of the following form in all our experiments: `v(x) = β x`

. This means that by increasing ^{1/2}*β* we can increase *v(x)*, and thus strengthen the network externality effect.

In the light of our constraint on the network externality function, we can rewrite the proposition as follows:

```
When none of the three firms are compatible with each other, a monopoly can exist if and only
if β 15
```^{1/2} ≥ 15, and a duopoly can exist if and only if
β (15/2)^{1/2} ≥ 15/2.

In other words,

```
When β ≥ 3.87, a monopoly, a symmetric duopoly, and a symmetric triopoly are all possible. When
3.87 > β ≥ 2.74, a monopoly is not possible, but a duopoly and a triopoly still are.
When β < 2.74,
the only possible symmetric equilibrium is a triopoly.
```

This confirms our intuition from the previous slide that a strong network externality function facilitates the existence of a monopoly. In the next slide, we demonstrate this result visually using our Equilibrium Calculator.