Middlemen: A Directed Search Equilibrium Approach

1 Introduction

This paper presents a simple search model which allows us to study explicitly the dependence of middlemen’s premium on their immediacy service backed by inventory capacity. It is not only that middlemen or dealers are viewed as the suppliers of immediacy to the market as in Stigler (1964), but also that the compensation to middlemen is captured by “the markup that is paid for predictable immediacy of exchange in organized markets; in other markets, it is the inventory markup of retailer or wholesaler.”^[1]

Using a directed search approach, Watanabe (2010) considers retail markets that are operated by middlemen and sellers who differ in the selling capacity. However, Watanabe (2010) focuses attention on turnover behavior of sellers to become middlemen, and so the behavior of retail premium has never been analyzed explicitly.^[2] In this economy, buyers have limited search time and their search activities are on an uncoordinated basis. Under these frictions, the middlemen’s advantage in the selling capacity provides buyers with a lower likelihood of experiencing a stockout. Thus, the price difference between sellers and middlemen determines the retail premium charged for the more sure service rate, i. e., the immediacy service.

My approach offers a natural link between inventory capacity and immediacy. I show that the middlemen’s inventory capacity has non-trivial effects on prices. On one hand, a larger inventory makes it less likely that excess demand occurs at individual middlemen, generating a downward pressure on the middlemen’s price. On the other hand, it influences the search decision of buyers: a larger capacity of a middleman can attract more buyers. This effect creates a tighter market that allows to charge a higher price, since the middleman knows that buyers receive zero payoff in the event of a stockout. These conflicting effects cause a non-monotonic response of the price to capacities: it goes up (down) when the initial total supply is scarce (abundant). This is because the stockout probability is initially high (low) in the former (latter) situation, thereby buyers are prepared to pay a higher premium for a larger inventory when the the initial scarcity of resources is higher.

The empirical relationship between middlemen’s premium and their inventory capacity can be positive in some markets (e. g., rental video shops, used-car dealers)^[3] and negative in other markets (e. g., supermarkets, theater tickets offices).^[4] Our result suggests that the former (latter) is likely to occur when the initial capacity of individual middlemen and total supply are relatively low (high).

The main insight can be generalized by considering the extensive margin of middlemen. With fixed supply in the middlemen’s market, a larger individual capacity should accompany fewer middlemen. This setup allows me to show that the concentration of middlemen’s market, i. e., having fewer middlemen, each with larger capacity, leads to a higher retail premium when the total supply is small, but can lead to a lower retail premium when the total supply is relatively large. With free entry of middlemen, when the inventory cost is low, there are many operating middlemen. This creates a situation of abundant total supply when the individual capacity level is high enough. In this situation, buyers would not appreciate much higher capacities and so the premium they are willing to pay for middlemen can decrease as the middlemen’s market gets more concentrated. Conversely, when the capacity level is low, the total supply can be scarce enough for the premium to be increasing. Hence, with free entry, the retail premium can be non monotone in the capacity when the inventory cost is low.

The rest of the paper is organized as follows. This introductory section closes with more detailed discussions on the related literature. Section 2 presents the basic setup and studies the steady state equilibrium allocations. Section 3 provides the characterization of the retail price differentials. Section 4 extends the analysis to allow for the free entry of middlemen. Section 5 concludes. All the proofs are in the Appendix.

2 Model

Consider an economy inhabited by a continuum of homogeneous buyers, sellers and middlemen, indexed b, s and m, respectively. The population of buyers is normalized to one, and the population of sellers and middlemen are denoted by S and M, respectively. These population parameters are constant over time. All agents are risk-neutral and infinitely lived. Time is discrete and each period is divided into two subperiods. During the first subperiod, a retail market is open for a homogeneous, indivisible good to buyers. The good is storable. This retail market is operated by sellers and middlemen, and is subject to search frictions as described in detail below. Each period, each buyer has unit demand while each seller has a capacity k_s = 1, and each middleman has a capacity k_m ≥ 1 units of the good. The selling capacity of suppliers k_i is exogenously given, for both i = s, m. The consumption value of the good is normalized to unity. If a buyer successfully purchases in the retail market at a price p, then he obtains the per-period utility of one. Otherwise, he receives zero utility that period. A seller or a middleman who sells z units at a price p obtains the revenue zp during the first subperiod.

During the second subperiod, a wholesale market opens. This market is operated by sellers, and middlemen can restock their units for the future retail markets.^[9] There is no search friction in the wholesale market and the price is determined competitively. Sellers decide whether to produce for the current wholesale market and for the future retail market. Sellers can produce any units they wish for any purposes. The production cost is measured in terms of utility and is given by c < 1 per unit. Agents discount future payoffs at a rate β∈[0, 1) across periods, but there is no discounting between the two sub-periods.

In each retail market, sellers and middlemen simultaneously post a price which they are willing to sell at. Observing the prices, all buyers simultaneously decide which seller or middleman to visit. Each buyer can visit one seller or one middleman. If more buyers visit a seller or middleman than its selling capacity, then the unit or units are allocated randomly. Assuming buyers cannot coordinate their actions over which seller or middleman to visit, a directed-search equilibrium is investigated where all buyers use an identical strategy for any configuration of the announced prices. I focus my attention on a stationary directed-search equilibrium where all sellers post the identical price p_s and all middlemen post the identical price p_m every period.

In any given period, each individual seller or middleman is characterized by an expected queue of buyers, denoted by x. The number of buyers visiting a given seller or middleman who has expected queue x is a random variable, denoted by n, which has the Poisson distribution, Prob·(n=k)=e−xxkk!. In a directed-search equilibrium where x_i is the expected queue of buyers at i, each buyer visits some seller (and some middleman) with probability Sx_s (and Mx_m). They should satisfy the adding-up restriction,

requiring that the number of buyers visiting individual sellers and middlemen be summed up to the total population of buyers. The buyer’s probability of being served by a supplier i, conditional on visiting it, depends on the queue x_i and the selling capacity k_i. Denote this probability by η(xi, ki). It is computed as follows:^[10]

where Γ(k, x)=∫x∞tk−1e−tdt. η(⋅) is strictly decreasing (increasing) in x_i (in k_i). For i = s, η(xs, 1)=1−e−xs/xs is a standard urn-ball matching function.

In steady state, each seller holds k_s = 1 unit and each middleman holds k_m ≥ 1 units at the start of every period. As the wholesale market is competitive, middlemen can restock at the sellers’ reservation price c.^[11] Simultaneously, sellers produce another unit for the next retail market, if the production cost is not too high, and if they have successfully sold in the retail market – holding more than one unit is never optimal since sellers only have the selling capacity of k_s = 1 in the retail market.

I now characterize the equilibrium retail prices. In any equilibrium where V^b is the value of a buyer and where middlemen restock at the sellers’ reservation price c, the optimal retail price of a supplier i who has a capacity k_i and an expected queue of buyers x, denoted by p_i(V^b), satisfies

subject to

For i = s, a seller sells its good at price p with probability xη(x, 1), and produces a new unit with cost c for the next period. If unsuccessful in the retail market, then the seller’s payoff is zero: if it sells the current unit to a middleman in the second subperiod, then it receives c from the middleman and produces a new unit for the next period with cost c; otherwise, the seller receives nothing and carries its unit into the next period. For i = m, a middleman’s expected number of sales is xη(x, km), and it restocks at the competitive price c. (2) says that the supplier i with a price p and an expected queue x must deliver to buyers at least their market payoff V^b, and clearly it does not deliver more. While V^b is an equilibrium object (i. e., determined below by equilibrium), it is taken as given by individuals. Notice that (2) generates tradeoffs between p and x: buyers can get the same V^b from higher p if x is lower. A buyer choosing p is served with probability η(x, ki) in which case he obtains per-period utility 1−p. If not served by the seller or middleman, the buyer’s payoff is zero that period. Irrespective of whether or not to be served, he enters the next period and his continuation value is βVb. The situation is the same for all the other buyers.

Substituting out p using (2), p=1−1−βη(⋅) Vb, the objective function of a supplier i, denoted by Π_i(x), can be written as

Note that pricing and restocking are separate decisions, and one can find an optimal price by maximizing the objective function Π_i(x) with respect to x. The first-order condition is

for i = s, m.^[12] Rearranging the first order condition above using (2) and

one can obtain the optimal price of the seller (if i = s) or the middleman (if i = m),

where

is the elasticity of the matching rate of buyers, which represents the supplier i’s share of the net trading surplus 1−c.

The analysis above has established the equilibrium prices p_i(V^b) given V^b. Equilibrium implies buyers are indifferent between any of the individual suppliers i = s, m, leading to

where x_i is the equilibrium queue of buyers at p_i = p_i(V^b), i = s, m. Buyers successfully purchase the good from the seller or middleman with probability η(xi, ki) each period. The value of sellers and middlemen are given by

respectively. Sellers produce with cost c whenever needed, and middlemen restock at the competitive price c each period. To solve for the equilibrium, it is important to observe that the indifference conditions (4) and (5) can be reduced to the following simple form: applying (3) for i = s to (4) with a rearrangement,

similarly, applying (3) for i = m to (5) with a rearrangement,

these two equations together imply

The adding-up restriction (1) and the indifference condition (8) identify an equilibrium allocation x_s, x_m > 0.

3 Retail Market Prices

In this section, I study the behaviors of the retail prices. I begin by showing that the usual market-tightness effect leads to a lower price.

Figure 1:

Retail price of middlemen.

Figure 2:

Retail price differentials.

Figure 3:

Concentration of middlemen’s market.

4 Free Entry Equilibrium

In this section, I allow for the number of middlemen to be determined endogenously by free entry. So far, I have implicity assumed that middlemen hold the initial endowment k_m at the start of their lifetime period. Suppose now that the initial endowment can be obtained by paying ck_m > 0 from the sellers’ market, whereas before c > 0 represents the wholesale price. Middlemen possess the inventory management technologies that enable them to operate with a relatively large selling capacity in the retail market. Denote by c_k the latter cost of inventory management per unit. This is a flow cost payed period by period. An agent chooses to be a middlemen if the value of being a middlemen is non-negative, −(c+ck/1−β)km+Vm≥0, given values of k_m ≥ 1 and M > 0. In a free entry equilibrium, entry and exit occur until the middlemen operating in the markets earn zero expected net profits, just to cover the cost. The equilibrium number of middlemen M > 0 is determined by the free entry condition, Vm=(c+ck/1−β)km, or

where the L.H.S. represents the per-unit profit of middlemen (apply the price in (3), i = m, to the value V^m in (7)) and the R.H.S. the lifetime per-unit cost. As before, we consider low production costs, c∈(0, c¯], where c¯=c¯(M¯)∈(0, β) at some M¯<∞ (see the proof of Theorem 1), guaranteeing that seller produce for future sale under free entry of middlemen. Define the total per-unit cost per period as C≡c(1−β)+ck/1−c and its upper bound as C¯≡limM→01−Γ(km+1,xm)/Γ(km+1)∈(0,1).

Figure 4:

Free entry equilibrium.

Figure 5:

Retail price/premium with free entry.

5 Conclusion

This paper proposed a simple theory of middlemen using a standard directed search approach. It offers wide applicability and economic insights into many empirically relevant forms of middlemen. Middlemen’s inventories can provide buyers with immediacy service under market frictions, thereby the retail price of middlemen includes a premium to buyers. The model generates two important effects of middlemen’s inventories that serve as the critical determinant of the retail premium. On the one hand, middlemen can attract more buyers with a larger selling capacity, which allows them to charge a higher price. On the other hand, it puts downward pressure on their retail price. These conflicting effects cause non-monotonic responses of the retail price/premium to changes in their inventories. In particular, the premium goes up when the initial total supply is scarce, because buyers appreciate much the capacity increase and are prepared to pay a higher premium for a larger inventory. This result may help us understand why popular items are often sold at a larger premium by larger scaled intermediaries. The main insight survives with free entry of middlemen. With fixed supply in the middlemen’s market, the concentration of middlemen’s market, i. e., having fewer middlemen, each with larger capacity, leads to a higher retail premium when the total supply is small, but can lead to a lower retail premium when the total supply is relatively large.

For future research, it would be interesting to extend the model to allow for the middlemen’s choice of payment method. In history, merchants and bankers, or enforcers, often share the same root. One implication of the retail premium analyzed here is that if buyers have to use cash for their retail transactions, they have to hold more cash to visit the middlemen’s market than to visit the sellers’ market. Hence, to the extent that cash holdings are costly, due to inflation or the cost of fraud, or because cash is susceptible to theft, middlemen have an incentive to introduce credit as a means of payment. In such an extended model, it would also be interesting to study the benefit and cost of making retail prices publicly observable, because it would determine the cash holdings of buyers in each market.