Network Effects Models
Introduction
This paper provides a general introduction to network effects models along with a survey of the existing literature. The paper is organized as follows. In the first section a general overview of network effect models is presented. Stability and existence of equilibria are discussed.
In the next section we turn to welfare, and then follow that by dealing with the case of indirect network effects. Finally we close by providing some examples, which include several notable casees from the literature.
Network Effects Models
Introduction
At their most minimal network effects (NE) models contain the following:
A set of agents/consumers. This set is normally modelled by the interval $$[0,1]$$ and $$t \in [0,1]$$ is used to index consumers.
Platforms/Networks denoted by capitalized letters: $$A,B ...$$. Normally there are only two such networks $$A,B$$. Often use $$X$$ as a variable to refer to the network.
Agents are 'on' a particular network (or multiple networks). This is usually assumed to relate to purchasing decisions. The measure of agents on network $$X$$ is given by $$n_{X}$$.
The agent's utility function is of the following form: $$\textrm{utility of network X} = u(X) = u(X,t,n_{X},Z)$$ where $$Z$$ is a catch-all vector of other parameters^{1}. It has the following features:
- Network effects: $$u = u(n_{X})$$, i.e. agents utility depends upon the number of other agents on their network. This is the central feature of network effect models. If it does not exist and one can write $$u = u(t,X,Z)$$ then we no longer have a network effect model.
- Heterogeneous: $$u = u(t)$$ i.e. the function varies among agents. This feature is almost always found in a network effect model since otherwise the decisions of all agents will be the same (assuming those decisions are based on utility maximization).
- Network dependent $$u_{X}$$
Remarks
- Are introduction of network effects immediately introduces an 'externality' since agents' payoffs are now affected by the choices of other agents.
- Network effects may also usefully be divided into direct and indirect. Direct effects are those where the effect arises in a manner directly from the larger pool of other agents on your network, for example in telephony network from the number of other people that you can call. Indirect effects usually operate via the effect of other agents on the availability of goods to an individual agent. This distinction is explored further below
- In such models the central endogenous parameter of interest is the equilibrium size of networks given by $$n_{X}$$ the number of agents on that network. Related to this are questions of whether such an equilibrium will be optimal both in a pareto and welfare maximizing terms. The procedure for determining equilibrium is addressed in the next section
Equilibrium in Network Markets
To solve for equilibrium a sequence of activity must first be assumed.
- Agents become aware of the functional form of their utility function
- Exogenous parameters are made known
- Agents form expectations of network size, $$n_{X}^{e}$$.
- Agents then compute their expected utility on each network and choose the network that yields the greatest utility.
- Based on this actual network sizes and agents' payoffs are computed
Clearly, the method by which expectations of network size are formed at stage 4 is central to the model. Generally rational expectations are imposed which require $$n_{X}^{e} = n_{X}$$.
Footnotes
For future reference note that $$Z$$ will often be dropped and the utility function may also written $$u_{X}(t,n_{X},Z)$$↩