A Monopoly Platform
This framework doesn’t apply to most of the examples of two-side markets but there are a few application. For example, yellow pages directories are often a monopoly of the incumbent telephone company, nightclubs are sometimes far enough away from others that the monopoly paradigm might be appropriate, and sometimes there is only one newspaper or magazine in the relevant market.
Suppose there are two groups of agents, denoted and . A member of one group cares about the number of the other group who join the platform. For simplicity, I ignore the possibility that agents care also about the number of the same group who use the platform. Suppose that the utility of an agent can be defined as follows: if and members of the each group join the platform, the utility of a group is,
is the platform’s price to the each group. The parameter measures the benefit that a group agent derived from interacting with the other group. Therefore the utility is a function of the numbers of agents who participate and price. Suppose the numbers who join the platform are a function of utility,
where is increasing function of utility.
For the platform side, suppose that the platform has a per-agent cost for serving group . Then, the platform’s profit function is,
If we consider the platform to be offering utilities instead of with the implicit price for each group, then, the platform’s profit is,
The profit maximizing prices can be derived from the first order conditions,
Thus, the profit-maximizing price for each group is equal to the cost of providing service adjusted downwards by the external benefit and adjusted upwards by a factor related to the elasticity of the group’s participation. In addition, we can derive the Lerner conditions, thus, the profit-maximizing pair of prices satisfies
for a group’s price elasticity of demand for a given level of participation by the other group. It is possible that the profit-maximizing outcome might offer a subsidized service, for example,
suggests that this can happen if the group’s elasiticity of demand is high and if the external benefit enjoyed by the other group is large.
Now, comparing the profit maximizing solutions to the social optimum, let the aggregate consumer surplus be where the envelope condition, holds. Then total welfare, the sum of profit and consumer surplus, is
Thus, the welfare maximizing outcome is,