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The Katz and Shapiro Model of Network Externalities

"Network Externalities, Competition and Compatibility" by Michael L. Katz and Carl Shapiro (The American Economic Review, Volume 75, Issue 3, June 1985, 424-440) offers a mathematical model of a market in which network externalities play an important role. It captures some of the basic complexity that arises when consumers value a firm's product more if the firm belongs to a larger compatibility network.

Katz and Shapiro start with a basic economic model of supply and demand. Several firms try to sell versions of the same type of product to consumers (e.g. versions of a personal computer). Each consumer has a certain basic willingness to pay for the product, even in absence of network effects. This is the value of the product to the consumer if nobody else uses it. We assume that the basic willingness to pay is uniformly distributed among consumers between negative infinity and some constant, A. Throughout this tutorial, we fix A at 15; perhaps a certain word processor is valued at up to 15 dollars in absence of network effects (The uniform distribution results in a linear demand curve, simplifying calculations). Furthermore, it is assumed that two firms have non-zero outputs only if they offer the same consumer surplus per unit of product (otherwise every consumer will select the product of the firm that offers higher surplus).

On top of these basic assumptions, the authors embed the notion of network externalities. Define the network externality function v(y), as the additional value a consumer derives from a product when she buys it from a network of size y. In other words, the surplus to each consumer who buys this product is

r + v(y) - p,

where r is the basic willingness of the consumer to pay for the product, y is the total sales volume of the firm's network, and p is the price the firm charges. In this model, firms can join to form networks, increasing the value of their products to the consumers.

Throughout this tutorial, Complete Incompatibility means that all the firms belong to their own separate networks, Full Compatibility designates a situation in which there is a single network that contains all firms, and Partial Compatibility corresponds to an intermediate scenario.

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