The functional form of network effects

Abstract Metcalfe’s Law states that aggregate network value is proportional to the square of network size, and that implies the individual user’s utility is a linear function of network size. This paper explores the functional form of network effects in a simple model of a telephone network, and finds that it can be linear, but only under strong conditions. The paper establishes weaker conditions under which a representative subscriber’s utility is a linear function of network size. When these linearity conditions are not satisfied, we show that under some assumptions, the value to a typical subscriber will be an s-shaped function of network size.