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The authors have declared that no competing interests exist.

Contributed reagents/materials/analysis tools: MG KK. Wrote the paper: MG KK.

We analyze endogenous capacity formation in a large frictional market with perfectly divisible goods. Each seller posts a price and decides on a capacity. The buyers base their decision on which seller to visit on both characteristics. In this setting we determine the conditions for the existence and uniqueness of a symmetric equilibrium. When capacity is unobservable there exists a continuum of equilibria. We show that the “best” of these equilibria leads to the same seller capacities and the same number of trades as the symmetric equilibrium under observable capacity.

We analyze endogenous capacity formation in a large market with frictions when the good for sale is perfectly divisible. The sellers post prices and decide on costly capacities. Buyers individually decide which seller to visit based on what is posted. This leads to the usual coordination frictions as the buyers don’t know which sellers the other buyers visit. This approach is called directed search. Standard references include

We determine the conditions that guarantee the existence and uniqueness of a symmetric equilibrium both under free entry and when the measure of sellers is fixed. When capacities are observable, both price posting and auctions give rise to the same equilibrium quantities. When only prices are observable before the matching takes place, there is a continuum of equilibria. We show that the “best” of these equilibria yields the same seller capacity as the case of observable capacities and leads to the same number of traded goods. All equilibria under unobservable capacities give the sellers positive expected profits. Free entry of sellers therefore leads to a very large number of sellers each offering very small quantities. This is clearly inefficient and different than under observable capacity.

Two assumptions let us simplify the analysis compared to earlier papers. First, we focus on perfectly divisible goods. This is in contrast to several recent papers analyzing frictional markets and seller capacity for goods that are sold in units. (For example

The environment consists of a unit interval of buyers and a large continuum of potential sellers of which

The utility function of the buyers is linear,

The order of events is as follows: At stage 1, each active seller chooses a capacity

We capture the frictions by focusing on symmetric equilibrium strategies for the buyers. We further assume that the strategies of both the sellers and the buyers are anonymous so that sellers with the same capacity and the same price are treated identically by the buyers and all the buyers are treated identically by the sellers.

When there are different capacity-price pairs the buyers adjust their behavior so that they are indifferent between visiting the different types of sellers and expect the market utility

At stage 1, all sellers choose their capacity

At stage 2, all sellers choose their price

At stage 3, buyers maximize their expected utility. Given a distribution of different capacity-price pairs

We begin by analyzing the second stage of the game where sellers have chosen their capacity. To find a symmetric equilibrium price, we first make the assumption that all sellers have the same capacity

The LHS is the expected utility of a buyer visiting a deviating seller. As described in the set up, the linearity of the utility function allows us to write the expression for the expected utility as

The first order condition of the deviating seller's problem is

To find out how the queue length is affected by the price we totally differentiate the indifference condition of the buyers with respect to

In equilibrium

We show in the appendix that there are no profitable deviations from (2). Thus it is the equilibrium price.

The equilibrium price above in (2) depends only on the overall market tightness

When all the sellers have capacity

The buyers' expected utility is then

When the terms of trade are decided by auction at the sellers' locations a single buyer would bid zero and still acquire quantity

It turns out that the equivalence result also holds when there are sellers with differing capacities, just as in

With perfectly divisible goods it is straightforward to determine the conditions for existence and uniqueness of the symmetric equilibrium and to find the equilibrium. This is the main advantage compared to the approach with integer capacities. We proceed as in the last subsection. Namely, we assume that all sellers have capacity

keeping in mind that the queue length cannot be negative. Simple algebra allows us to write the queue length

A seller deviating to capacity

In order to solve the FOC we first derive

The necessary condition for equilibrium thus gives us

Unfortunately, it turns out that we need to make an extra assumption on the cost function in order to guarantee that the sufficient conditions hold, i.e., to show existence and uniqueness of the symmetric equilibrium. The reason is that the MR function of a potential deviator is increasing and concave in capacity. Therefore convexity of the cost function is not enough to guarantee that there are no profitable deviations. (The MC curve might be increasing and convex as well and might therefore cross the MR curve any number of times.) One way to guarantee existence and uniqueness is to assume that the MC increases more steeply than the MR curve after the potential equilibrium, but this is rather ad hoc as the condition then depends on the parameter values. A better way is to assume that

It is somewhat surprising that Assumption A is needed for existence and uniqueness. The reason is that a seller can increase his queue length, and therefore the probability of trading, by deviating to a higher capacity. This in turn implies that the deviator has an increasing and convex revenue function. If the cost function is not convex enough, there might exist a profitable deviation to a high enough capacity. With a linear cost function no equilibrium exists as there is always a profitable deviation to a higher capacity just as in

With Assumption A the equilibrium is straightforward to derive and easy to analyze. We elaborate with the following example.

The sellers' profit from capacity

One notices that

Even though no equilibrium with symmetric quantities for the sellers exist when

We analyze the efficiency of the decentralized equilibrium in a standard way by comparing it to the choice of a benevolent planner that maximizes overall utility. The planner chooses the capacities and the proportion of sellers offering each capacity and allocates the buyers over the sellers (see e.g.

We define social welfare directly as the expected value of the trades minus the sellers' capacity cost. We show in the appendix that the planner has no solution in which there are sellers with different (positive) capacities. When all sellers have the same capacity the social welfare is

The planner maximizes welfare by choosing the capacity of the sellers. As noted above, she can’t base her decisions on the identities of the agents. Assuming that she chooses the same capacity for all sellers, the planner's problem is

The FOC is

Solving for

To see that

Equilibrium is constrained efficient when the number of sellers is determined by free entry. In this case the measure of sellers,

From Eq. (8) we know that in any equilibrium

With the two equations we can solve for the free entry equilibrium. Returning to our example with cost function

We find that

When the social planner is free to chose both the overall market tightness and the capacity of the sellers the social optimum is given by

The FOC's are

Rearranging we get

As (17) and (18) are identical to (8) and (15) the free entry equilibrium is constrained efficient.

Constrained efficiency is not a surprising result. It is almost a defining property of directed search and has been demonstrated several times with fixed capacities and free entry. Here capacity is not fixed but it is observable and there is an optimal price for each capacity. Thus capacity choices are reflected in the queue lengths and hence in the trading probabilities of the sellers. With free entry the sellers fully internalize the effect their decisions have on welfare. Analogous results can be found for example in

Our results on the constrained efficiency of equilibrium are a useful benchmark in the following section where capacities are unobservable.

In this section we let the sellers' capacities be unobservable before matching takes place. The definition of equilibrium from section 2.1 needs to be changed accordingly. To derive the queue lengths we need to describe the beliefs. The standard way is to impose strict beliefs of the type that all sellers that post price

The equilibrium capacity

Unfortunately there is a continuum of equilibria satisfying these beliefs as any

Let us first focus on the equilibrium capacity that maximizes social welfare. We define social welfare as the overall utility from trade minus the costs of production or

The FOC is

The equilibrium capacity is then given by

When trade is determined by auction (without reserve price) the equilibrium capacity is simply

The unique symmetric equilibrium capacity under auctions is given by (27). It is lower than (25), the capacity in the “best” equilibrium. To further analyze the differences between the different cases we again let the cost function be given by

This is not surprising as the gains from trade are divided more equally between the market participants under auctions but the capacity costs are still borne by the sellers. It is likewise clear that the expected profits of the sellers are higher in the “best” price posting equilibrium

The buyers' expected utility is, on the other hand, zero in the “best” price posting equilibrium whereas it is positive under auctions.

The “best” equilibrium under price posting and unobservable capacity achieves the same welfare as the symmetric equilibrium under observable capacities. In doing so it allocates all the gains from trade to the sellers. Auctions result in lower equilibrium capacities than the “best” price posting equilibrium, but the benefits of trade are more evenly distributed by the market participants.

By substituting the equilibrium capacity under auctions

Interestingly

Above we analyze capacity choice when only prices are observable. To do so we impose strict beliefs. The downside of this assumption is that it kills any interesting link between queue length and capacity. In addition, it gives rise to a continuum of equilibria of which we focus on two. The “best” equilibrium maximizes social welfare and leads to the same capacities as under observable capacity. The reason is that now the whole surplus of trade befalls the sellers and hence they fully internalize the effect of their capacity decisions. The equilibrium is therefore constrained efficient with a fixed number of sellers.

In the second equilibrium the trades are determined by auction. The equilibrium leads to too small capacities compared to observable capacity. Somewhat interestingly it is outcome-wise equivalent to one where sellers post the same price as under observable capacity.

Directed search is a standard method to analyze frictional markets. At its core is the trade-off that sellers face between asking a higher price and attracting fewer buyers; hence trading more slowly. Typically, all sellers are assumed to have a fixed capacity, often one unit. Several recent papers relax this assumption by allowing the sellers to choose their capacity. This makes it possible to compare markets with a few large sellers to markets with many small sellers in terms of welfare and to find the equilibrium size and number of sellers given the cost function. In the realistic setting where production takes place before trading these models usually yield equilibria that can be analyzed only numerically (see

Were we to relax the assumption of linear demands, for example, by assuming that buyers have diminishing marginal utilities even the observable capacities case would be quite cumbersome to analyze as can be seen e.g. in

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