Diagramas de secuencia 50

The set of players is the set of standardised firms Ns and customised firms Nc = M. Each firm i ∈ Nc maximises its profit. Firm types (s,c) are common knowledge at the beginning of the pricing stage.

At this stage all players have complete information. As standard- ised firms can choose a different price for each local market, and cus- tomised firms only operate in one market, I can analyse the problem for each market separately. I focus on a given market in the post-entry sub- games. However, a similar game is played in each market. Thus, I drop the subscript indicating the local market in the following two sections unless required for clarity.

In this stage, the customised firm chooses the set of profit max- imising prices as a function of the number of standardised firms ship- ping to the local market ,Ns, their location in the product spacea(a1, ...aNs),

their cost parameterk(k1, ...kNs)and their pricesp(p1, ...p_Ns). Given this

information, customised firms choose a continuum of prices, as they are able to choose a different price for each variety.

Let the set of consumers that purchase from the customised firm jin a given market be denoted by ϕc_m,j(p). The profit of customised firm jin market mis thus given by:

Πj = Z 1 0 (pj,l−κ)Fj,l(p) dl where Fj,l(p) =    1 ifl ∈ ϕc_m,j(p) 0 otherwise

Market Shares and Profits

Suppose that at least one standardised firm ships to market m. Label the customised firm as c. Starting at a random point on the circle, la- bel each standardised firm i = {1, ...,N} such that firm i’s neighbour in the clockwise direction is labeled as i+1 and in the anti-clockwise direction as i−1. As explained in the previous section, to make the analysis tractable, I consider only the case where no standardised firm is undercut by another standardised firm in each market it operates in, i.e. all standardised firms sell to a positive mass of consumersconditional

on shipping to a given market unless they are undercut by the customised

firm. I start by analysing which consumers located between firm i and i+1 choose to purchases from the customised firm and then aggregate this over all pairs of standardised neighbours in the market to obtain the total market share of the customised firm c.

Denote the location of firm i andi+1 byai and ai+1 respectively. First suppose that in the absence of a customised firm operating in the market, there existed a consumer xi,i+1located between ai andai+1such that any consumer with a larger distance from firm i purchases from firm i+1, while any consumer with a distance smaller purchases from firm i. Call this consumer the marginal consumer. This marginal consumer is relevant for the analysis in this section to understand which firm the customised firm is competing with.

In this case, each consumer located betweenaiand xi,i+1will pur- chase from firm c if:

Similarly, suppose that there is also a marginal consumer xi−1,i

located between firm i and firm i-1. In this case, any consumer located between xi−1,i and ai will purchase from firm c if:

pc(l)l∈[xi−1,i,ai] ≤ pi+θ|l−ai|

Note that as the customised firm can choose a different price for each variety, we cannot say more at this point about the set of consumers who purchase from the customised firm without restricting the set of prices. Note for example, that the customised firm does not necessarily sell to a compact set of consumers between firm i and i+1.

Profits of the customised firm in this case become:

Πc=∑N s i=1 (_x i,i+1 R xi−1,i pc(l)−κ ·Fc,i(l)dl ) whereFc,i(l) =    1 ifpc(l)≤ pi+θ|l−ai| 0 otherwise

Note that if no standardised firm ships to market, any consumer will purchase from firm c if pc(l)≤vm

Pricing

Note that the customised firm can choose a different price for each vari- ety and thus chooses an infinite number of prices. It can be shown that if at least one standardised firm is active in market m, the optimal pricing schedule of the customised firm as a function of the preferred variety of consumer l follows the following proposition.

Proposition 3 The optimal pricing schedule of the customised firm for each

variety l ∈ [ai,xi,i+1] for any firm i and i+1 is given by:

pc(l) =max{pi+θ|l−ai|,κ} (C.1) Proof: To show that this pricing schedule is optimal, suppose that l ∈

[xi−1,i,xi,i+1] is such that piθ|l−ai| > κ and therefore, according to the

the tie-breaking rule that has been assumed, in this case, consumer l purchases the good from firm c. Profits from this particular consumer l are then given by πc(l) = piθ|l−ai| −κ, which are by assumption

greater or equal to zero. I show that there is no profitable deviation: If firm c raises the price for consumer l, the consumer would then switch to purchase from firm i, according to the preferences outlined. In this case, profits would be equal to zero. If on the hand, firm c lowers it’s price, consumer l continues to purchase from firm c, but profits are strictly decreased. Thus there is no profitable deviation for l ∈ [xi−1,i,xi,i+1] if

piθ|l−ai| ≥κ.

Now suppose that piθ|l−ai| < κ and thus the customised firm c sets pc(l) = κ accordingly. In this case, the consumer will purchase from firm i, as piθ|l−ai| < pc(l). Thus Fc,i(l) = 0 and as a result

profits of firm c are equal to zero. If firm c now raises its prices, the consumer will continue to purchase from firm i and profits remain at zero. If firm c, however, lowers pc(l), either profits remain equal to zero

if pc(l) > piθ|l−ai|,as the consumer again continues to purchase from

firm i or if firm c lowers its prices such that pc(l) > piθ|l −ai| , the

consumer switches to purchase from firm c. In this case, though, profits become negative as now prices are below marginal costs i.e. pc(l) ≤ κ.

The same argument follows for consumers located between the marginal consumers of any firm i ∈ Ns. Thus, there exist no profitable deviation from the pricing schedule as outlined by Proposition 6.

If no standardised firm is active in market m, firm c sets its price equal to the reservation price of the consumer pc(l) = vm. Thus as

a result, the market share of the customised firm is equal to one and profits are given by Pic,m =vm−κ.