A Proof of Lemma 1

Define q¯ such that u(q¯)=q¯c′(q¯), i.e., B(q¯)=0, and p¯=c′(q¯). We summarize the seller’s optimal ex-post choice as follows:

It is important to highlight that both of these solutions imply a monotone relationship between quantity and price. Notice the following

In a competitive search equilibrium with Θ∈(0,∞) , the positive measure of deviating sellers can choose a price that is either in (0,p¯] and the constraint is not binding, or in (p¯,∞) under a binding constraint. Define the first possible deviation as p1=c′(q~1) and the second possible deviation as p2=u(q~2)/q~2. It is easy to show that the sellers’ expected payoffs are

while for buyers we have

It is clear that B~(p2)=0 holds when the buyer’s surplus is fully extracted. Thus no buyers would participate in a deviating submarket that offers p2 and q~2. Buyers fully anticipate that the sellers’ best ex-post choice is to fully extract all of their surplus. As a result, it must be the case that the competitive search equilibrium price is p∈(0,p¯]. The optimal ex-post choice is then p1=c′(q~1)∈(0,p¯]. In other words, buyers anticipate that q~1(p1) is the equilibrium sellers’ response. ◼

B Proof of Proposition 1

To show existence, first, rewrite eq. (8) as

This condition characterizes q(Θ). According to Lemma 1, p=c′(q). Since B(q)=u(q)−c′(q)q , we find B(0)=0=S(0). Then, ∃!q¯>0 such that B(q¯)=0. In addition, ∃!q_∈(0,q¯) such that B′(q_)=0. For q∈(q_,q¯),

Thus, under these sufficient conditions, we have that B(q) is downward concave. Similarly, we find that for S(q)=c′(q)q−c(q) ,

Hence, under these sufficient conditions we have that S(q) is upward convex. We also have that χ(q_)=1 andχ(q¯)=0, with χ(q)>0, ∀q∈[q_,q¯). Since we focus on matching technologies with constant returns to scale, it follows that ε(Θ)∈(0,1), ∀Θ∈(0,∞). Therefore, given ε(Θ), ∃q(Θ)∈(q_,q¯) such that eq. (16) holds and a symmetric equilibrium exists.

To prove uniqueness, we show sufficient conditions for χ′(q)<0, ∀q∈(q_,q¯). Using eq. (16), we have the following (omitting q as an argument)

Thus we have that, ∀q∈(q_,q¯), B′(q)<0, S′(q)>0, B′′(q)<0, and S′′(q)>0, if c′′′(q)≥0 or c′′′(q)<0 not too negative by assumption. For χ(q) to be monotonically decreasing in q over (q_,q¯) , we need to satisfy S′2−SS′′>0. This condition holds if

where ηs(q)=S′(q)q/S(q)>1, due to the convexity of S(q). Hence, we also need c′′′(q)≥0, not too large. Therefore, as long as c′′′(q)≥0 not too positive and c′′′(q)<0 not too negative to preserve the properties of B(q) and S(q), we have χ′(q)<0, ∀q∈(q_,q¯). As a result, there exists a unique equilibrium q∈(q_,q¯), ∀Θ∈(0,∞).

Notice that we cannot have an equilibrium with q∈(0,q_). Since B′(q_)=u′(q_)−c′(q_)−c′′(q_)q_=0, we have u′(q_)>c′(q_), q_<qe, where u′(qe)=c′(qe), and B′(q)>0 over (0,q_). In any competitive search equilibrium, if q∈(0,q_), a positive measure of sellers can deviate and increase q, and increase buyers’ surplus B(q) while sellers’ surplus S(q) also increases.

On the other hand, for any competitive search equilibrium with q∈(q¯,q^], where u(q^)=c(q^), B(q)<0 over this interval and S(q)=0, since the implied queue length Θ=0. A positive measure of sellers can deviate and post q1=q¯−ϵ, attracting a positive measure of buyers by offering them B(q1)>0, and get positive surplus S(q1)>0. Hence, we cannot have q∈(q¯,q^] in any competitive search equilibrium.

We are then left to show that for any Θ∈(0,∞), neither q_ nor q¯ is an equilibrium. Consider the Lagrangian of the competitive search problem:

which yields the following necessary (and sufficient) conditions

These conditions lead to eq. (16) and are valid for any θ=Θ∈(0,∞). Now transform the second condition as

with ε(θ)=α′(θ)θα(θ). Substituting into the first condition, we have that

Since B′(q_)=0,

and B(q¯)=0 implies

Therefore, neither q_ nor q¯ is an equilibrium for θ=Θ∈(0,∞).

Finally, recall that ε′(Θ)<0 , which holds for a large set of matching technologies. From eq. (16), we have that

Therefore, any increase in Θ leads to higher q and p over (q_,q¯). ◼

C Proof of Proposition 2

Recall that for sellers, the free entry condition is given by

It is easy to show that the left-hand side of the previous equation is strictly increasing in Θ∈(0,∞) for q(Θ)∈(q_,q¯). Only for a very specific value of the entry cost, kc, we have that q(Θc)=qe. In this case, entry is efficient. However, entry is inefficiently low whenever k>kc⇒Θ>Θc and q(Θ)>qe, or excessively high when k<kc⇒Θ<Θc and q(Θ)<qe . When sellers face low entry costs, the equilibrium prescribes larger surpluses for buyers. In contrast, for very large entry costs, sellers are able to extract almost all of the surplus of the match. A low entry cost facilitates entry and tilts the bargaining power afforded by the market (via Θ) towards buyers. The reverse is observed when sellers face large entry costs. ◼

D Proof of Lemma 2

Define q¯ such that c(q¯)=q¯u′(q¯), i.e., S(q¯)=0, and p¯=u′(q¯). We summarize the buyer’s optimal ex-post choice as follows:

It is important to highlight that both of these solutions imply a monotone relationship between quantity and price. Notice the following

In a competitive search equilibrium with Θ∈(0,∞) , the positive measure of deviating sellers can choose a price that is either in [p¯,∞) and the constraint is not binding, or in (0,p¯) under binding constraint. Define the first possible deviation as p1=u′(q~1) and the second possible deviation as p2=c(q~2)/q~2. It is easy to show that the buyers’ expected payoffs are

while for sellers we have

It is clear that π2=0 holds when sellers receive no surplus. Thus no sellers would deviate and offer p2 and q~2 in a submarket. Sellers fully anticipate that the best ex-post choice of buyers is to fully extract all of their surplus. It must be then that the competitive search equilibrium price is p∈[p¯,∞). The optimal ex-post choice is then p1=u′(q~1)∈[p¯,∞). In other words, sellers anticipate that q~1(p1) is the equilibrium buyers’ best response. ◼

E Proof of Proposition 3

The proof is very similar to the one of Proposition 1. First, to show existence, we use again the fact that

where under p=u′(q), S(q)=u′(q)q−c(q) and B(q)=u(q)−u′(q)q. ∃!q_>0 such that S′(q_)=0, which implies u′(q_)−c′(q_)=−u′′(q_)q_>0⇒q_<qe (the efficient quantity). Also ∃!q¯>0 such that S(q¯)=0. For q∈(q_,q¯), we find (omitting q as an argument) the following

Thus, under these sufficient conditions we have that S(q) is downward concave. For buyers,

Hence, under these sufficient conditions we have that B(q) is upward convex. We also have that χ(q_)=0 and χ(q¯)=1, with χ(q)>0, ∀q∈(q_,q¯). Again, since ε(Θ)∈(0,1), ∀Θ∈(0,∞), there exists a q∈(q_,q¯) satisfying eq. (18).

To prove uniqueness, we need χ′(q)>0, ∀q∈(q_,q¯). Using eq. (18) and the properties of B(q) and S(q), we can derive the following sufficient condition

where ηb(q)=B′(q)q/B(q)>1, due to the convexity of B(q). Note that if u′′′ is negative while preserving the properties of B(q) and S(q) as above, or u′′′≥0 but not too positive, this condition holds and we have χ′(q)>0, ∀q∈(q_,q¯). Then, there exists a unique equilibrium q∈(q_,q¯), ∀Θ∈(0,∞).

Notice that we cannot have an equilibrium with q∈(0,q_). Since S′(q_)=u′(q_)−c′(q_)+u′′(q_)q_=0, we have u′(q_)>c′(q_) and q_<qe where u′(qe)=c′(qe), and S′(q)>0 over this interval. In any competitive search equilibrium, if q∈(0,q_), a positive measure of buyers can increase q, and increase sellers’ surplus S(q) while buyers’ surplus B(q) also increases.

On the other hand, for any competitive search equilibrium with q∈(q¯,q^], where u(q^)=c(q^), S(q)<0 over this interval and B(q)=0, since sellers will not produce for negative surplus. A positive measure of sellers can deviate and post q1=q¯−ϵ, attracting a positive measure of buyers by offering them B(q1)>0, and get positive surplus S(q1)>0. Hence, we cannot have q∈(q¯,q^] in any competitive search equilibrium.

We are left to show that for any Θ∈(0,∞), neither q_ nor q¯ is an equilibrium. Consider the Lagrangian of the competitive search problem:

which yields the following necessary (and sufficient) conditions

These conditions lead to eq. (18) and are valid for any θ=Θ∈(0,∞). As in the proof of Proposition 1, we can derive

Since S′(q_)=0,

and S(q¯)=0 implies

Therefore, neither q_ nor q¯ is an equilibrium when θ=Θ∈(0,∞).

Finally, by assumption we have ε′(Θ)<0. From eq. (18)

Therefore, any increase in Θ leads to lower q and higher p over(q_,q¯). ◼

F Proof of Proposition 4

The free entry condition faced by a seller is given by

It is easy to show that the left-hand side of the previous condition is strictly increasing in Θ∈(0,∞) for q(Θ)∈(q_,q¯). Note that entry is efficient only for a very particular value of kc, which implies a specific market tightness, Θc, such that q(Θc)=qe. Otherwise, entry is inefficient. For k>kc, we have Θ>Θc and q(Θ)<qe, while for k<kc we have that q(Θ)>qe. For low cost of entry, the equilibrium is such that sellers’ surplus is small (buyers large), while for very large entry cost, sellers obtain large surpluses (buyers small). These findings echo the entry results under marginal cost pricing of the previous section. However, there are some important differences. First, in equilibrium now we have q′(p)<0 with marginal utility pricing. Moreover, the direction of inefficiency is reversed when compared to marginal cost pricing. Therefore, we can conclude that for efficiency properties, it matters if quantity is chosen ex-post by sellers or by buyers. ◼

G Proof of Proposition 5

The proof follows from the text.