Seller’s problem: Prices

Now, for any partition{B_∅, B₁, B₂, B_1,2}, the profit Πⁱto the selleriis decomposed as Πⁱ(pⁱ) =

B_i+B_1,2(1−F⁻ⁱ(pⁱ)) pⁱ.

Given a way the buyers search (as captured by the partition intoB_∅, B₁, B₂andB_1,2), the demand has thus a price-insensitive part (the 1st term in the brackets on the rhs) and a price-sensitive part (the 2nd term in the brackets on the rhs). The captive buyers B_i, who have only the price quotepⁱ, buy from selleri= 1,2 whatever the pricepⁱ. The shoppers B_1,2, who have discovered both pricep¹ and pricep², buy from seller i= 1 iff p²is abovep¹; it takes place for probability (1−F²(p¹)).

Note particularly that, as in Varian (1980); Stahl (1989) and, say, as in Wilson (2010), search affects profit only through the number of captive buyersB_i, wherei= 1,2, and the number of shoppersB_1,2. Moreover, the numbers of captive buyersB_iand shoppers B_1,2are affected only by theexpected pricesF= (Fⁱ, F⁻ⁱ) but not by therealized prices p= (pⁱ, p⁻ⁱ); the buyer’s search problem is non-trivial only when no price is found yet.

Once that happens, they of course to switch the seller.

Hence, after the ratesθhave been set, the equilibrium of the subgame that follows is a fixed point between the maximal sequential search strategiest∈BR(F) for the buyers and the maximal randomized pricing strategies F ∈ BR(t) for the stores. This entails that the analysis of equilibrium pricing strategies goes along the same lines as in Varian (1980) and Stahl (1989) and, basically, as in Wilson (2010). The difference is only that now the numbers of captive buyers and shoppers are determined in equilibrium: In the preceding literature, they were given by a parameter.²³

Lemma 13 AssumeB_i>0, either for selleri= 1or selleri= 2, andB_1,2>0. Then, the following hold true in any equilibrium:

1. The sellers use randomized pricing strategies: F¹andF². 2. Both F¹ and F² have the same interval support supp(F) =

p,p¯

, where 0< p <

¯ p= 1.

3. Neither has an atom atp∈ p,1

: lim_x→p−Fⁱ(x) =Fⁱ(p)for allp <1andi= 1,2.

4. IfF¹ has an atom at p= 1,F² has not and, if F² has an atom atp= 1, F¹ has not.

Proof. See Appendix.

23The support is defined assupp(F) =cl{x|f(x)>0}, wherecldenotes a closure of a set andfis a point probability or a density function.

Observe also that, ifB_1,2 = 0 (no shoppers; would arise underθ= (0,0),θ= (a,0) andθ= (0, a) fora≥0), the sellers use a pure strategypⁱ= 1 (p:M R(p) =M C(p), this is basically the Diamond outcome) or, if B_1,2 >0 butB₁ =B₂= 0 (no captive buyers;

would arise underθ= (∞,∞)), the sellers use a pure strategypⁱ= 0 (p=M C, this is basically the Bertrand outcome). That is, our model nests the Diamond outcome and the Bertrand outcome as special cases for appropriateθ.

Lemma 14 Considerθ= θⁱ, θ⁻ⁱ

andt= t¹, t²

such thatB¹≥B² andB_1,2>0.

Then, there exists a unique equilibrium price distributionF= F¹, F²

The store with more captive buyers has higher profit and prices. It mixes between using random discount pricesp¹ <1, to compete with the other store over the shoppers, and the monopoly pricep¹= 1, to tax its numerous captive buyers. The other store, who has fewer captive buyers, is randomizing only the size of the discount,p²<1.

In other words, the stores’ equilibrium pricing strategies are wired so as to let them specialize in different groups of buyers. This aligns the sellers’ payoffs and helps to relax the price competition. The profit to the high-profit seller, Π¹, equals the number of captive buyers it attracts,B₁, whereas the profit to the low-profit seller, Π², is a weighted average of its own captive buyers,B₂, and the other store’s captive buyers,B₁.

The weights, p= _B ^B1

1+B1,2 and 1−p = _B^B1,2

1+B1,2 could be taken as a measure of how close the market is to the Bertrand outcome (arising forB_1,2 >0,B₁ =B₂ = 0) and to the Diamond outcome (arising forB_1,2= 0,B₁>0, B₂≥0) – or the competitiveness and the relative standing of the sellers and the buyers in the market.

This entails that, if the sellers’ have high ”bargaining power”, as captured by a high p, the sellers’ have less aligned preferences (they compete more fiercely) but, if the sellers’

have low ”bargaining power”, a lowp, they have more aligned preferences (they compete less fiercely). As it later turns out, the outcome that obtains can therefore be regarded as a compromise of some sort between the two stores and the buyers.²⁴

24In particular, we show that in equilibriump= 1/2.

It is now straightforward to calculate the expected prices that we need:

The firm who is capable of attracting more captive buyers extracts a higher profit and has an incentive to set higher prices in expectation. While it tends to offer a lower discount price, when it does so, ^Π2

B1,2ln ¹

p

≤ _B^Π_1,2¹ ln ¹

p

, it also uses the monopoly price more often,α≥0. It is next an easy three line homework to show that the latter effect offsets the former. To calculate the expected minimum of the two pricesE(p|F_min), note that the distribution functionF_minis given by

1−F_min= A direct calculation results in²⁵

E(p|F_min) = determined byθandtuniquely whereas, by Lemma 14,Fis dependent onθandtonly through B₁, B₂ and B_1,2. This allows us to construct a hypothetical price distribution F(θ,t) for any (θ,t) by first calculating the associatedB₁(θ,t), B₂(θ,t) and B_1,2(θ,t) and then the inducedF(B₁, B₂, B_1,2). We next use this property to characterize the fixed point between optimal search and optimal prices.

Proposition 17 For anyθ, there exists a unique fixed point in search and prices(t,F) whereF=F(θ,t)andt=t(θ,F).

25Note that a linear approximation of ln(p) around p = 1 yields ^1−p_p ≥ ln

1

B₁> B₂= 0andE(p|F¹)> E(p|F²), and 2. if θ¹

1−E(p|F¹(θ,(1,0)))

< θ²

1−E(p|F²(θ,(1,0)))

, then t¹ = 1−t² < 1, B₁≥B₂>0.

In this latter case,t=t(θ,F)is the unique solution to θ²

θ¹ =1−E(p|F¹(θ,t))

1−E(p|F²(θ,t))= 1−α(θ,t).

Proof. See Appendix.

Corollary 5(Effects of frictions on search efficiency)The buyers search efficiently if the sellers are either distinctly different in terms of their frictions,θ¹

1−E(p|F¹(θ,(1,0)))

≥ θ²

1−E(p|F²(θ,(1,0)))

, or exactly similar,θ¹=θ².

Corollary 6 (Effects of frictions on seller prominence) Lower frictions grant a seller more prominent position in search and thus higher prices and profit: if θⁱ ≥ θ⁻ⁱ, then B_i≥B_−iimplyingΠⁱ≥Π⁻ⁱandE(p|Fⁱ)≥E(p|F⁻ⁱ).

In other words, candidate equilibria are of two kinds: If the sellers are equally fast, the equilibrium is symmetric but, if seller one is faster than seller two, the equilibrium is asymmetric.