Appendix 2.A Proofs
4.5 Convergence of F n
5.2.2 Market structure (B); Large firm and small firms in two
I next consider price and expected utilities when the small firms are located to-gether in one location but separately from the large firm. I proceed to find equi-librium prices and expected utilities in this case.
Assume that fraction z of buyers go to small firms and fraction 1−z go to the large firm. Then the small firms set their price using a mixed strategy with support9 [a, A], and the large firm quotes price PB. As before, it is clear that F(A) =F(1) = 1.
A small firm quoting price1can then expect to get(1−H(1
z)). As the expected profit must be the same over the support one easily sees that the lower bound of
8I derive the expression for the expected price paid under market structure(A)or (EPPA) in the appandix.
9It is clear that A=1 as a small firm pricing A would otherwise have a deviation to 1.
the support isa=1−H(1z),as a seller quoting this price trades with certainty.
Any price ρ ∈ [a,1] in the support of the small firms yields the same profit 1−H1
z
or more formally
ρ
1−H F(ρ)
z
= 1−H 1
z
. (5.19)
From Eq. (5.19) I solve the mixed strategy of the small firms.
F(ρ) =zH−1
ρ−1−H1
z
ρ
. (5.20)
The expected profit of the large firm is simply its price multiplied by the expected number of trades it makes. This can be expressed as
PB
m 0
(1−z)θh(θ)dθ= Π(PB). (5.21) To continue I first look at the expected utilities of buyers that go to the small firms. The small firms set their prices using a mixed strategy. To make calculations easier I follow Kultti (2008) and assume that all sellers charge the virtual pricer described below. I denote the expected utility of buyers visiting the small firms byu(z, F).A buyer going to a large seller knows that he can expect to get1−PB.
In equilibrium the following must hold:
u(z, F) = 1−PB (5.22)
and
Π(PB) = 0. (5.23)
To solve this set of equations I begin by looking at the small firms’ pricing decision. To simplify I let every small firm asks the same pricer, wherercan be thought of as the virtual price that gives the sellers the same expected profit as
the sellers would get using the mixed strategyF derived above. I get
In the first term on the LHS I integrate over levels of demand when there are fewer buyers than small sellers. The term is the expected profit of the small firm when there are less buyers than small firms. The second term is the expected profit of a small firm when demand is higher than the number of small firms, in which case all sellers trade. Forcing the LHS to equal the expected profit from the mixed strategy, i.e., the RHS of Eq. (5.24) I solve for the small seller’s virtual price
r=
Thus the buyers’ expected utility from visiting the small sellers is given by u(z, F) = (1−r)
where g(θ) = θh(θ)E(Θ) is a buyer’s expectation of the overall demand given that he exists and E(Θ) = 0m
0 θh(θ)dθ. If the buyer acquires the good he expects to get 1−r. Whenever less than 1z other buyers visit the small firms the buyer is certain to acquire the good. The probability that demand is small in this sense is 01
0zg(θ)dθ. When demand is high and more than 1z buyers visit the small sellers then only some of the buyers get served, this probability is simply the measure of goods divided by the measure of buyers visiting the small sellers, i.e., zθ1. The probability of acquiring a good is thus0m
1 z
1
zθg(θ)dθ.
The buyers’ indifference condition can be stated as (1−r) The LHS is the buyers’ expected utility from visiting the small firms’ location, the RHS is the buyers expected utility from visiting the large firm. It is clear that PB≤1 as no buyer would otherwise visit the large firm. Before solving the large
firm’s optimization problem I rewrite the buyers indifference condition. Using Eq.
Partially integrating the first term in brackets, the last equation the expression becomes
I can now rewrite the buyers indifference condition Eq. (5.26) as 1−PB=
0zH(θ)dθand simplifying the buyers indifference condition
becomes
The large firm’s objective is to maximize its revenue, which is just its price multi-plied with its share of the expected demand or more formally
maxPB
B by totally differentiating the buyers indifference condition Eq. (5.32) or zE(Θ)(1−PB)−
Substituting Eq. (5.37) into the Eq. (5.34) the FOC can be expressed as
m
and simplified to
I use the simplified FOC to solve for the large firm’s price PB=E(Θ)0m
Determining the sign of the second order condition is cumbersome for a general distribution H. Therefore I again assume that H is uniformly distributed. The key equations are then as follows.
The virtual price of the small firms is r=
indifference condition under uniformH is 1−PB− 1
m2z2 = 0. (5.42)
The objective function of the large firm is
max
PB
m 0
(1−z)θh(θ)dθ·PB= max
PB
(1−z)m 2 ·PB. The first order condition is
(1−z)m 2 − dz
dPB m
2PB= 0, where I get dPdz
B = m2z2−P mm2z32z2+1 by totally differentiating the buyers’ indifference condition, which is given by Eq. (5.42) asH is uniform. The FOC thus simplifies to
(1−z)m
2 − m2z3
m2z2−PB·m2z2+ 1 m
2PB= 0, which gives me
PB= (1−z+m2z2−m2z3)
m2z2 . (5.43)
Asm >1this is defined for allz∈(0,1).
I next show uniqueness when H is uniform. To solve forz I substitutePB in the buyers’ indifference condition and solve forz. The only real root is then given by
z∗=− 1
3m2Υ + Υ, (5.44)
where Υ = 233
1
m4+27m1 6+m12. There exists a uniquez∗(m) that is convex and decreasing in the interval[0,1]whenm >1.
It is clear from the expected profit of the large firmE(πL) = (1−z)E(Θ)·PB, coupled with the buyers’ indifference condition (5.42) thatPB= 0andPB= 0lead to profits of zero. Then asE(πL)is continuous and differentiable and the bordered Hessian of the second order conditions is positive, as shown in the appendix, Eq.
(5.43) describes the maximizing price of the large firm.
Claim 5.4 When H is uniform the large firm has a unique price PB∗ which is a function of z andm and is defined in Eq. (5.43). The small firms have a unique mixed strategyF(ρ)which is a function ofz∗ andmand is defined in Eq. (5.20).
Proof. The proof is by construction and can be found above. The second order
condition is in the appendix.
The expected profitE(πL)of the large firm is E∗(πL) = (1−z)E(Θ)·PB∗
= (1−z)2 m
2 + 1 2
1 z2
1 m
, (5.45)
wherezis defined by Eq. (5.44).
Expected utility under market structure (B)
A buyer visiting the large firm trades with certainty and by the buyers’ indifference condition a buyer is, in equilibrium, indifferent between the small sellers and the large seller. Thus the expression for the buyers’ expected utility is very simple, it is just1minus the large firm’s prize or,
EUB = 1−PB∗, wherezis given by Eq. (5.44).