Antecedentes a Nivel Internacional

Similarly, the expected benefit of searching off the platform iszD. Hence a buyer is willing to pay the membership fee and search on the platform if,

zL+Vb−Pb ≥zD.

2.4. Equilibrium Targeting Strategy

The equilibrium of the search model is defined as follows.

Definition 6 (Search Equilibrium). For a buyer-seller search model with platform facili-

tating search, given (Ps, Pb) the membership fee announced by the platform, a search equi-

librium is(pL, pD, zL, zD), wherepL: [m, m]→R∪ {∞}is a mapping from firms’ marginal

costs to its posted prices on the platform, and pD _{: [m, m] → R∪ {∞} a mapping from}

firm’s marginal costs to its posted prices off the platform,5 such that

(i) Each firm with marginal production cost m who is subscribed to the platform, posts

the optimal prices to maximize its aggregate profit from transaction on and off the platform,

(pL(m), pD(m)) = arg max p1,p2

πL(p1, p2;mj). 5_pL

For those not subscribed to the platform, the optimal prices are selected to maximize profit from direct search, i.e.,

pD(mj) = arg max

p πD(p;mj),

(ii) Firmj joins the platform if

max p1,p2

πL(p1, p2;mj) +Vs−Ps ≥max

p πD(p;mj).

(iii) Buyers join the platform if

zL+Vb−Pb ≥zD.

and adopt the optimal stopping rule stated in Lemma 2.3.1

One incentive for buyers to search online is lower prices. The following result confirms that on the buyer-seller search platform, prices charged are lower than prices through direct search, thus eliminating the showroom issues in Wang and Wright (2016) where buyers search for sellers on the platform but complete their transaction off the platform.

Lemma 2.4.1. In any search equilibrium when the platform operates,pL(m)≤pD(m),∀m.

Based on the proof of Lemma 2.4.1, in any equilibrium when the platform operates, the equilibrium pricespD_j , pL_j posted by firm j satisfy:

pD_j =mj+ 1−G(pD j +zD) g(pD_j +zD) , pL_j =mj + 1−G(pL_j +zL) g(pL j +zL) .

with demand from on the off the platform being:

XD(pDj ) =

BD

SD(1−ρD)

XD(pLj, pDj ) =

BL

SL(1−ρL)

(1−G(pL_j +zL)).

Therefore, the profits for firm j from transaction on or off the platform are respectively,

πL(pLj, pDj ;mj) = (1−G(pL_j +zL))2 g(p+zL) BL SL(1−ρL) +(1−G(p D j +zD))2 g(p+zD) BD S(1−ρD) πD(pDj ;mj) = (1−G(pD_j +zD))2 g(p+zD) BD S(1−ρD)

The incentive condition for firms’ entry thus depends on:

Bl SL(1−ρL) (1−G(pL_j +zL))2 g(pL j +zL) +Vs≥Ps (2.5)

Since (1−_g(p+z)G(p+z))2 is non-increasing inp, the lemma above yields an upper bound for posted prices on the platform for givenVs, Ps. The equilibrium strategy profile exhibits the follow- ing threshold structures in any search equilibrium.

Theorem 2.4.2 (Threshold Strategy). In any buyer-seller search equilibrium,

(i) the equilibrium prices for firms with marginal costsm ispL(m) =p(m, zL), pD(m) =

p(m, zD), where p(m, z) satisfies,

p(m, z) =m+ 1−G(p(m, z) +z)

g(p(m, z) +z)

(ii) there existsmL, s.t., any firms with production cost m < mL join the platform while

those with higher production cost m > mL choose to stay off the platform. Moreover,

BL SL(1−ρL) (1−G(p(mL, zL) +zL))2 g(p(mL, zL) +zL) +Vs=Ps. (iii) SL=S·F(mL)

(iv) zL satisfies, cL= 1 F(mL) Z mL m Z v≥p(m,zL)+zL (v−p−zL)dG(v)dF(m).

The success of the platform is determined by the amount of membership feesPs, Pb charged on both side, which in term are determined by the optimal thresholdmL. By including more sellers with high threshold mL, buyers enjoy positive externalities of the variety of choices and higher values of tradesvij in equilibrium. On the other hand, the price dispersion on the platform can be detrimental for the success of the platform, as buyers have to search extensively to find sellers with low pricespL

j. Therefore, it is not obvious that the middleman should attract all sellers to the platform in equilibrium.

The following theorem characterizes the platform’s optimal choice of thresholds mL by limiting the set of sellers available on the platform.

Theorem 2.4.3. Suppose

(i) f(m) is non-increasing inm, (ii) ∂C_∂s(BL, SL)≥Vs,

then the profit of the platform is a non-decreasing function of the thresholdmL, i.e., _∂m∂π0_L ≤ 0.

Condition (i) states that the density of sellers is a monotone function of its production costs. In particular, more sellers are equipped with production technologies with lower marginal costs. Condition (ii) implies that the cost of serving one more seller is at least that of the subscription benefit the seller received after joining the platform. For example, it can be satisfied if ∂C_∂s(b, s) =Vs. What is non-standard is that under the two conditions, the middleman can profit more by targeting fewer sellers. In other words, the middleman profits by acting as a gatekeeper that only attract the set of sellers with low marginal costs.