Network externalities & the steady state blog

Mr X is a self-important individual who seeks both a blog audience and the articulation of his idiosyncratic tastes. The utility Mr X gets from blogging depends on the degree to which he can be idiosyncratic and pursue his own preferences in his posting (measured by D, for distinctiveness of his average post) and the number of bloggers who visit X's site N. Suppose X gains utility U = U(N, D), an increasing function of both numbers and distinctiveness. Suppose X wants to maximise steady state U. Suppose, too, the evolution of N is logistic with:

N - N(-1) = I*N(-1)(K - N(-1)) - G(D)

where G(D), an increasing function of D (for simplicity take G= D^2), describes the disincentive effect of very distinctive blog postings on the resident population's demand for the blogsite. The potential population of blog posters is K and I is an index of how intrinsically interesting X is as a blogger (a parameter determined by Mr X's genes that X cannot control).

Of course there may not be an interior steady state here. People will only post to X's blog if others do (they don't just want to respond to X, they want others to read their responses) and initially there will be few who do post so he might have to bribe a few of his friends to get the ball rolling, advertise himself, or rely on prompts from other successful blog sites. A big problem is to generate enough initial positive externalities to make the blogsite viable. If there is an interior steady state equilibrium it will occur when D^2 = IN(K-N). Solving this for N we get two roots:

N= K/2-sqrt((K^2-4(D^2)/I)/2 and
N=K/2+sqrt((K^2-4(D^2)/I)/2.

The smaller root here is dynamically unstable so take N as the second, larger root. This is the equilibrium X can hope to get to provided he can generate enough network externalities initially. This equilibrium blog audience size depends positively on the size of the potential blogging population (K) and on how interesting X is (I). But it depends negatively on how idiosyncratic X's views are. Indeed an upper bound on the distinctiveness of his postings is provided by the fact that D must be less than or equal to K*sqrt(I)/2. For values of D greater than this X will never be able to secure a positive equilibrium blog audience. This upper bound is more likely to be satisfied the larger is the potential audience and the more interesting X is.

Substituting this root back into the expression for U and maximising U with respect to D gives the optimal average level of distinctiveness of a posting and substituting this back into the expression for the root gives the optimal steady state audience.

One could complicate this by considering non-steady state dynamics. Mr X might initially offer non-idiosyncratic posts with wide appeal to help generate the network externalities sought and then go for more idiosyncratic posts that suit his own tastes long-term. But the steady state blog-posting audience is independent of such shenanigans. These only determine whether or not the blog is viable or not.