Moduleco Annex 1 - Two part Tarif Competition with consumption externality : (b) the user interface

The customers are "blue" and "red" people. white people doesn't adopt. The two networks are not interconnected (standardized)

A is the fixed part (Subscription), p the price of one unit of the usage ;epsilon is the price elasticity for the usage

• Set N the percentage of the adopters in the neighbourhoud (of size V) : N € {0, 1/V,2/V...(V-1)/V, 1}

• Set W the customer willingness to pay given the percentage of the adopters in the neighbourhoud  and a given level q of usage for the services (N,q) :
  (1)    W(N,q) = e/(e-1).q^a.N.q^(e-1)/e

• Set S the consumer quasi linear Surplus : (2) S = W(N,q) -(P.q + A) ; where P is the usage price and A the subscription price per period.

• For a given {N,P,A}, the optimal consumer usage q, maximize this Surplus :
  (3)        q^d(N,P) € argMax{W(N,q) -(P.q + A)}
  (3 bis)  q^d(N,P) = P^-e.(q^a.N)^e
  remark 1 : optimal q^d(N,P) is directly independent of A (but it would be indirectly through N)
  remark 2 : e is the elasticity of the individual demand for usage.

• We can rewrite the indirect willingness to pay as a function of (N,P)
  (1 bis)    W(N,q) = e/(e-1).[P^1-e.(q^a.N)^e]

• Set q_min(N) the idiosyncratic parameter of the marginal consumer just indifferent between to be a subscriber or not to be a subscriber ie :
  (4)    q_min(N) = [((e-1).A)/(P^1-e.N^e)]^1/(a.e)

• for a given A, under the assumptions of this model, equation (4) exhibit an implicit relationship between A and N.

Denis.Phan@enst-bretagne.fr ;   Antoine Beugnard@enst-bretagne.fr