In chapter 15 of the book Wireless Communications by Andrea Goldsmith, Goldsmith gives a formula for an estimate of the signal-to-interference-ratio for the node on the edge of a cell in a cellular system
SIR = 0.167 (3N)^(ɣ/2)
where N is the number of channels and the path loss is modeled as P = P_t d^-ɣ. There are some simplifications in this derivation but it roughly makes sense; the SIR should improve according to the path loss model, and the distance to co-channel cells should increase as the square root of the number of channels.
If you then try to calculate the capacity at this node, you get
C = (1/N) log2(1 + SIR)
For reasonable values of gamma (from table 2.2 in the same book, "urban macrocells" should have gamma between 3.7 and 6.5), the maximum capacity is found when N = 1.5, and C(N=1) is roughly equal to C(N=2) and is greater than C(N > 2). And if you consider beamforming (multiplying the SIR by the number of antennas), then the maximum capacity is found closer and closer to N = 1.
N = 1 is of course the scenario where each cell uses the entire band and there's no need to assign different channels to different cells. If this either maximizes area capacity (or is close enough to the maximum), then why isn't this done in practice? My understanding is that cellular systems do assign different channels to adjacent cells. So what am I missing in this analysis?
