## Markets with Network Externalities

Let us try to model network externalities using a simple demand and supply model. Suppose that there are 1000 people in a market for some good and we index the people by v — 1,..., 1000. Think of v as measuring the

5 More generally, a person's utility could depend on the identity of other users; it is easy to add this to the analysis.

### MARKETS WITH NETWORK EXTERNALITIES 659

reservation price for the good by person v. Then if the price of the good is p, the number of people who think that the good is worth at least p is 1000 — p. For example, if the price of the good is \$200, then there are 800 people who are willing to pay at least \$200 for the good, so the total number of units sold would be 800. This structure generates a standard, downward-sloping demand curve.

But now let's add a twist to the model. Suppose that the good we are examining exhibits network externalities, like a fax machine or a telephone. For simplicity, let us suppose that the value of the good to person v is vn, where n is the number of people who consume the good—the number of people who are connected to the network. The more people there are who consume the good, the more each person is willing to pay to acquire it.6 What does the demand function look like for this model?

If the price is p, there is someone who is just indifferent between buying the good and not buying it. Let v denote the index of this marginal individual. By definition, he is just indifferent to purchasing the good, so his willingness to pay for the good equals its price:

Since this "marginal person" is indifferent, everyone with a higher value of v than v must definitely want to buy. This means that the number of people who want to buy the good is n = 1000 - t). (35.6)

Putting equations (35.5) and (35.6) together, we have a condition that characterizes equilibrium in this market:

This equation gives us a relationship between the price of the good and the number of users. In this sense, it is a kind of demand curve; if there are n people who purchase the good, then the willingness to pay of the marginal individual is given by the height of the curve.

However, if we look at the plot of this curve in Figure 35.1, we see that it has quite a different shape than a standard demand curve! If the number of people who connect is low, then the willingness to pay of the marginal individual is low, because there aren't many other people out there that he can communicate with. If there are a large number of people connected, then the willingness to pay of the marginal individual is low, because everyone else who valued it more highly has already connected. These two forces lead to the humped shape depicted in Figure 35.1.

6 We should really interpret n as the number of people who are expected to consume the good, but this distinction won't be very important for what follows.

Network externalities. The demand is given by the curved hump, the supply by the horizontal Hue. Note that there are three intersections where demand equals supply.

Now that we understand the demand side of the market, let's look at the supply side. To keep things simple, let us suppose that the good can be provided by a constant returns to scale technology. As we've seen, this means that the supply curve is a flat line at price equals average cost.

Note that there are three possible intersections of the demand and supply curve. There is a low-level equilibrium where n* = 0. This is where no one consumes the good (connects to the network), so no one is willing to pay anything to consume the good. This might be referred to as a "pessimistic expectations" equilibrium.

The middle equilibrium with a positive but small number of consumers is one where people don't think the network will be very big, so they aren't willing to pay that much to connect to it—and therefore the network isn't very big.

Finally the last equilibrium has a large number of people, Here the price is small because the marginal person who purchases the good doesn't value it very highly, even though the market is very large.