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→Part I
Absent new entry, if there was only one firm at White and one at Red we would expect them to price to compete against each other exactly in the Hotelling model (given these location choices). However, the Bertrand competition at the White end of the line reduces price to marginal cost, and only customer whose valuation of a White above marginal cost can buy from them. This gives the player at Red, potentially at least, the rest of the market without competiting with players at White - the customer whose valuation of White is marginal cost would have to have a valuation of Red greater than or equal to marginal cost for the entire market to be served. However, as the Red player is a monopolist over this area he would rather price above marginal cost and earn positive profits which he does.
As the cost of entry is extremely low, I will assume it to be effectively zero. Then firms will enter and while there are a finite number of firm they locate so that they have space around themselves on the line, charge prices about above marginal cost and reap profits above zero. Once an infinite number of firms have entered the line will be saturated all firms will charge marginal cost and earn zero profits.
Given that the battery innovation has perfect property right protection around it (for 20 years) by assumption, we expect the that firm will have a monopoly on it. Thus, if there is no effect between the markets, Bertrand competition will continue for the White phones which will be priced at marginal cost and earn their firms zero profits, and the battery life phones will be monopoly priced and earn monopoly profits. If there is an effect between the markets, that is because of a taste for variety a change in the Battery phone's price will have an effect on not only the demand for Battery phones but also for White phones, then we have a Dixit-Stiglitz (1977) type model, except that we do not have a free entry assumption. With free entry firms would earn zero profits, but with costly entry (i.e. the cost of coming up with a comparable innovation to the Battery innovation), the Battery firm will earn positive profits. However, because of the demand effect, the battery firm will not earn monopoly profits, but something less. With free entry the number of firms will increase but remain finite in this model - firms enter until the customer's taste for variety is satiated. Likewise with costly entry but excess taste for variety we could expect more entry and more positive profits (to cover the cost of entry) until satiation.
