1 Introduction

This Element surveys the theoretical economic literature on middlemen (i.e., intermediaries in the exchange process). This is relevant because, as is well recognized, intermediation plays a big role in many (if not most) markets for consumption goods, productive inputs, and assets, yet the activity is absent in standard general equilibrium theory. We present modern developments using search theory to study middlemen in markets with explicit frictions. The goal is not to discuss every paper in detail, but to develop a consistent framework that can be used to illustrate various models and ideas.Footnote ¹

Starting with Rubinstein and Wolinsky (Reference Rubinstein and Wolinsky1987), this research builds models where intermediation emerges because certain agents have comparative advantages along some dimension. A general version of the Rubinstein–Wolinsky model is presented in Section 3, but here is an outline. There are consumers and producers of something called $x$ , plus other agents that neither consume nor produce $x$ , but potentially could act as middlemen by buying it from producers and selling it to consumers. In many (but not all) models agents meet bilaterally at random. In the original formulation, these other agents perform a middleman function when they have an advantage in search (i.e., they meet consumers faster than producers meet consumers).

Further studies generalize the framework’s technical specification by allowing different bargaining powers (in the original specification all agents have the same bargaining power); general populations (in the original the measures of buyers and sellers are the same); endogenous entry (participation is fixed in the original); production, search, and storage costs (these are absent from the original model); goods that are divisible (goods are indivisible in the original); and payment frictions (the original has transferable utility).

Further studies also explore different assumptions about the kinds of advantage middlemen might have, including superior information that lets them better recognize quality, a technology that lets them hold larger or more diverse inventories, and a superior ability to better enforce debt repayment. A main goal is to characterize the set of parameters consistent with the existence of equilibria with active intermediaries. Papers also investigate whether there is uniqueness or multiplicity of equilibria, as well as how intermediation affects efficiency. They also analyze if it attenuates or accentuates volatility. Of particular interest is finding conditions under which there emerge different patterns of exchange – only direct trade, only indirect trade, or both. Some papers also consider how multiple middlemen enter into intermediation chains. We review all of these in what follows.

While this Element mainly concerns theory, some facts help the motivation. In an early contribution, Spulber (Reference Spulber1996a) documented that intermediated exchange accounted for over 25% of GDP in the US in 1993, including retail trade (9.33%), wholesale trade (6.51%), finance/insurance (7.28%), and selected services (1.89%). Updating this from 1993 to 2025, 25% increased to 35.1% (BEA, US Department of Commerce, Federal Reserve Bank of St. Louis). For food, from 1993 to 2023, the share of dollars going to farmers and manufacturers declined from about 34% to 29%, while the share going to wholesale (distribution), retailing (supermarkets and grocery stores), and food services (restaurants, cafeterias, etc.) rose from 48% to 58% (USDA ERS Food Dollar Series). Switching from food to drink, in 2017 direct-to-consumer wine sales in the US were around 10% of the market, with the rest accounted for by retailers (Rhodes et al. Reference Rhodes, Watanabe and Zhou2021). In real estate, intermediated trade accounts for 91% of sales. In the labor market, recruitment and staffing firms account for two-thirds of all postings and attract most of the applications (Davis and de la Parra Reference Davis and de la Parra2025).

Philippon (Reference Philippon2015) discusses the share of financial intermediation in GDP and how it changes over time. As Lagos and Rocheteau (Reference Lagos and Rocheteau2006) report, different asset markets feature different microstructures – that is, while intermediated trade in the fed funds market is about 40%, NASDAQ is closer to 100%, and many OTC (over-the-counter) markets are in between, including markets for corporate, municipal, and emerging-market debt. In international trade, the impact of intermediation shows up in various ways, such as the fact that in the US wholesale and retail firms account for approximately 11% and 24% of exports and imports (Bernard et al. Reference Bernard, Jensen, Redding and Schott2007). The use of intermediary firms is especially important in developing economies, especially in Asia – for example, in the 1980s 300 trading (nonmanufacturing) Japanese firms accounted for 80% of trade (Rossman Reference Rossman1984).

In China today, Ahn et al. (Reference Ahn, Khandelwal and Wei2011) show 22% of exports are handled by intermediaries. In firm-level data, they also show that small or less productive firms rely disproportionately on intermediaries to access foreign markets. Those firms using intermediaries are significantly more likely to eventually become direct exporters, implying intermediaries lower entry barriers and boost overall trade. In the used-car market, Biglaiser et al. (Reference Biglaiser, Li, Murry and Zhao2020) document that dealer-mediated transactions command a consistent price premium over private sales, especially for older vehicles, and that cars sold through dealers exhibit higher quality. Together, all these findings demonstrate that middlemen can affect both the quantity and the quality of trade, and that suggests they are worth studying seriously.

Monieson (Reference Monieson2010) provides historical perspective on middlemen. While there are many interesting facets of this history that could be mentioned, one issue going back a long way is this: Do they provide a useful service, or simply profit from buying low and selling high? An extreme view is epitomized by Benjamin Disraeli, who said “It is well-known what a middleman is: he is a man who bamboozles one party and plunders the other.”Footnote ² An alternative possibility is that they facilitate the process of exchange, which can enhance efficiency and welfare. As Ayn Rand (1971, p. 313) put it, “The shortest distance between two points is not a straight line – it’s a middleman.”Footnote ³

Reality may be somewhere between these extremes. While theory alone does not definitively resolve this debate, modeling intermediated trade formally in economic theory helps us understand the relevant factors. Before getting into formal models, we offer this story from Turner (Reference Turner1836, pp. 115–16), quoting a source from the eleventh century on the practice of buying low and selling high:

The rest of the Element is organized as follows. Section 2 presents a rudimentary environment by way of introducing some basic terminology, notation, and ideas. Section 3 extends this to a generalization of the environment in Rubinstein–Wolinsky. Section 4 considers alternative assumptions and analyzes the possibility of multiple equilibria and endogenous dynamics. Sections 5 and 6 discuss information frictions and intermediation chains. Section 7 considers papers on inventories and directed rather than random search, while Section 8 summarizes some work focusing on OTC asset markets. Section 9 focuses on the relationship between middlemen, money, and credit. Section 10 mentions papers that do not fit elsewhere in the Element but are still relevant. Section 11 concludes.

2 A Simple Model

Before discussing the literature, it is useful to construct a simple two-period model that captures some of the main ideas, and, in particular, provides a clean version of the classic result in Rubinstein and Wolinsky (Reference Rubinstein and Wolinsky1987). It also lets us introduce notation, labeling, etcetera.

There are three agents – a producer $P$ , a middleman $M$ , and a consumer $C$ – that are interested in trading an indivisible good $x$ . The producer $P$ produces one unit of $x$ at cost $c$ , which we set to zero for now (for some applications it is better to think of $P$ as being endowed with $x$ ). The consumer $C$ derives utility $u>c$ from consuming one unit (for some applications it is better to think of $x$ as an input or asset and $u$ as $C$ ’s profit or return to acquiring it). The middleman $M$ neither produces nor consumes $x$ , but could acquire it from $P$ and resell it to $C$ ; that is intermediated trade.

There are two periods, or stages, to the trading process, with $C$ indifferent between getting $x$ in the first or second stage. Trade is bilateral, with the terms of trade determined here by generalized Nash bargaining, where the share of agent $i$ when bargaining with $j$ is denoted ${\theta }_{ij}$ . There is at most one meeting per period, and the probability agent $i$ meets agent $j$ in each period is ${\alpha }_{ij}$ , or equivalently ${\alpha }_{ji}$ , by the bilateral nature of meetings. For now the $\alpha$ ’s are parameters; later, they come from an underlying meeting technology. If $P$ meets $C$ in the first stage, they trade, and the game is over. What is to be determined: Do $P$ and $M$ trade if they meet in the first stage?

Let there be returns to holding $x$ for $P$ and $M$ in the second stage, denoted ${\rho }_p$ and ${\rho }_m$ , where ${\rho }_j<0$ stands for storage costs in goods markets, while ${\rho }_j>0$ stands for dividends in asset markets. For now we assume ${\rho }_j<0$ (but see Section 4), with the cost incurred between the first and second stage, so it is sunk when $P$ or $M$ with $x$ meets $C$ in stage 2. In addition to $x$ , there is a second tradable object, $y$ , that is divisible and serves as a payment instrument. The most generous way to think about this is the following: Any agent can produce or consume $y$ at constant marginal cost, or constant marginal utility, both normalized to $1$ , and when $i$ acquires $x$ from $j$ , the former makes a payment $y_{ji}$ that increases $j$ ’s payoff and decreases $i$ ’s payoff by the same amount.

This way of sharing the trade surplus, using $y$ as a means of payment, is essentially what is usually called transferable utility. Our interpretation is generous in the sense that the direct transfer of utility is problematic. Consider Binmore (Reference Binmore and Health1992):

We agree that “Utils are not real objects and so cannot really be transferred,” although we are less sure that “only physical commodities can actually be exchanged” since it seems clear that, for example, information or ideas can also be exchanged. A bigger issue concerns the claim that transferable utility is the same as paying with money. In serious monetary models, agents have a value function, or indirect utility function, over money holdings, but it is not generally linear (see surveys by Lagos et al. Reference Lagos, Rocheteau and Wright2017 and Rocheteau and Nosal Reference Rocheteau and Nosal2017). Well, in some models, sometimes money actually does enter linearly but up to a point, which does not correspond to transferable utility for the simple reason that buyers tend to run out of money, while in transferable utility models they never run out of utils.

Therefore, instead of pretending that $y$ is money, it is better to say there is a good $y$ that anyone can consume and produce. This avoids Binmore’s justified complaint about utils being transferred without doing a disservice to monetary/payment economics. At the risk of sounding pedantic, we emphasize this because work in the area is concerned with microfoundations, so it is important to think through such details, although we try not to dwell on it too much in what follows.

So, when $x$ changes hands, $C$ pays $y_{pc}$ to $P$ , $C$ pays $y_{mc}$ to $M$ , and $M$ pays $y_{pm}$ to $P$ . Interpreting these as prices, we call $y_{pc}$ the direct price, $y_{pm}$ the wholesale price, and $y_{mc}$ the retail price. However, this is not as obvious as it may appear. First, if $x$ is divisible, and endogenous, as it is in some of the models presented in Sections 5, 7, and 9, in principle it might be better to call $y_{ij}/x_{ij}$ the price. Having said that, in practice it could be that we see (in the data) $y_{ij}$ but not $x_{ij}$ , which is especially relevant when $x$ corresponds to quality rather than quantity. Second, while $y_{ij}/x_{ij}$ is the price of $x$ in terms of $y$ , it makes as much sense to say $x_{ij}/y_{ij}$ is the price of $y$ in terms of $x$ .

In standard usage the price refers to the amount of money a buyer pays to get $x$ , which is hard to understand in models without money. Indeed, without money changing hands, it is not at all clear who is the buyer and who is the seller, any more than those labels make sense when $i$ gives $j$ apples in trade for bananas. This may or may not matter, depending on context (see Wright and Wong Reference Wright and Wong2014 for an extended discussion). In any case, if we call the $y$ s prices, there arise several other interesting variables, like the spread $y_{mc}-y_{pm}$ and markup $y_{mc}/y_{pm}$ .

In the first period, $P$ is assumed to produce $x$ before meetings occur. If $P$ meets $C$ they always trade, as mentioned. If $P$ meets no one, or $P$ meets $M$ and they decide not to trade, $P$ and $C$ proceed to stage 2, while $M$ is assumed to drop out, which, by assumption, does not affect the probability $P$ and $C$ meet. If $P$ meets $M$ and they trade, $M$ and $C$ proceed to stage 2 while $P$ drops out, since, by assumption, $P$ can only produce once here. Note that in a stage 1 meeting, $P$ and $M$ can differ along three dimensions: ${\alpha }_{ic}$ , their probability of meeting $C$ in stage 2; ${\theta }_{ic}$ , their bargaining power; and ${\rho }_i$ , their return to holding $x$ , which for the purposes of this discussion satisfies ${\rho }_i\le 0$ (but see Section 4).

Let $V_p$ , $V_m$ , and $V_c$ be the value functions of $P$ , $M$ , and $C$ in the second stage. Then, the surpluses $S_{ij}$ when $i$ trades with $j$ , using the labels direct, retail, and wholesale suggested earlier, are:

$\begin{array}{rl}{S}_{cp}& =u-y_{pc}\text{and}{S}_{pc}=y_{pc},\end{array}$ (1)

$\begin{array}{rl}{S}_{cm}& =u-y_{mc}\text{and}{S}_{mc}=y_{mc},\end{array}$ (2)

$\begin{array}{rl}{S}_{mp}& =V_m-y_{pm}\text{and}{S}_{pm}=y_{pm}-V_p.\end{array}$ (3)

In each trade $y$ solves the bargaining problem

$y_{ij}=\arg \underset{y}{\max }{S}_{ij}{(y)}^{{\theta }_{ij}}{S}_{ji}{(y)}^{{\theta }_{ji}}.$ (4)

However, with transferable utility (in the sense discussed earlier, where $i$ produces and $j$ consumes it with equal marginal cost and utility), the bargaining protocol, as long as it is reasonable, does not matter. So, we simply say $i$ gets a share ${\theta }_{ij}$ of total surplus $S_{ij}+S_{ji}$ , which leads to

$y_{pc}={\theta }_{pc}u,y_{mc}={\theta }_{mc}u\text{and}{y}_{pm}={\theta }_{pm}{V}_m+{\theta }_{mp}{V}_{p\text{.}}$ (5)

Let the probability $P$ and $M$ trade at stage 1 be $\tau$ . Then the second-stage value functions are

$\begin{array}{rl}{V}_c& =\left(1-\tau \right){\alpha }_{cp}{S}_{cp}+\tau {\alpha }_{cm}{S}_{cm},\end{array}$(6)

$\begin{array}{rl}{V}_p& ={\alpha }_{pc}{S}_{pc}+{\rho }_p,\end{array}$(7)

$\begin{array}{rl}{V}_m& ={\alpha }_{mc}{S}_{mc}+{\rho }_m.\end{array}$(8)

In other words, these values are the product of meeting probabilities and surpluses, minus storage cost. Note that for now it is taken for granted that all agents participate in the market, but that will be checked in Section 3.

Then, after inserting the $S$ s and $y$ s, we have

$\begin{array}{rl}{V}_c& =(1-\tau ){\alpha }_{cp}{\theta }_{cp}u+\tau {\alpha }_{cm}{\theta }_{cm}u,\end{array}$(9)

$\begin{array}{rl}{V}_p& ={\alpha }_{pc}{\theta }_{pc}u+{\rho }_p,\end{array}$(10)

$\begin{array}{rl}{V}_m& ={\alpha }_{mc}{\theta }_{mc}u+{\rho }_m.\end{array}$(11)

Note that $P$ and $M$ do not enter the second stage simultaneously. At least one of $V_p$ or $V_m$ is an off-equilibrium value. Nonetheless, we need to track both in order to determine $\tau$ , the probability of wholesale trade. Here $\tau$ only depends on the sign of the total surplus $S_{mp}+S_{pm}$ ; that is, whether there are gains from trade between $P$ and $M$ . What we call the best response conditions are then:

$\tau =\left\{\begin{array}{ll}1& \text{if}{V}_m>V_p\\ [0,1]& \text{if}{V}_m=V_p\\ 0& \text{if}{V}_m<V_p\end{array}\right..$(12)

When all agents participate, $\tau$ is the only decision. Thus, an equilibrium is given by $V$ ’s and $\tau$ satisfying (9)–(12), from which other variables, like the $y$ s, are easily determined.

Figure 1 shows the structure of the model. It is easy to see that equilibrium exists and is unique. Letting $\mathrm{\Gamma }={\alpha }_{mc}{\theta }_{mc}+{\rho }_m/u-{\alpha }_{pc}{\theta }_{pc}-{\rho }_p/u$ , in equilibrium $\tau =1$ if $\mathrm{\Gamma }>0$ and $\tau =0$ if $\mathrm{\Gamma }<0$ .Footnote ⁴ Intuitively, $\tau =1$ means $M$ is active, intermediating by buying $x$ from $P$ and selling it to $C$ , if and only if $M$ has some advantage over $P$ . This in general means $M$ has some combination of being better at search, ${\alpha }_{mc}>{\alpha }_{pc}$ , or bargaining, ${\theta }_{mc}>{\theta }_{pc}$ , or storage, ${\rho }_m>{\rho }_p$ . Although their environment is more complicated in some ways, Rubinstein–Wolinsky (Reference Rubinstein and Wolinsky1987) is, in our notation, the special case where ${\rho }_p={\rho }_m$ and ${\theta }_{pc}={\theta }_{mc}$ , which gives their classic result: $M$ is active if and only if ${\alpha }_{mc}>{\alpha }_{pc}$ . More generally, $M$ can be fundamentally inferior to $P$ in terms of search or storage, ${\alpha }_{mc}<{\alpha }_{pc}$ or ${\rho }_m<{\rho }_p$ , and equilibrium can still have $\tau =1$ , if $M$ has an advantage over $P$ squeezing surplus out of $C$ , which looks like inefficient rent-seeking activity.

Figure 1

Structure of the simple model.

To be precise, we can measure welfare by the sum of the $V$ s, which equals the expected utility of $C$ minus the cost of $P$ or $M$ delivering the goods. The optimal wholesale trade maximizes welfare:

${\tau }^{*}=arg\underset{\tau }{max}(1-\tau )\left({\alpha }_{pc}u+{\rho }_p\right)+\tau \left({\alpha }_{mc}u+{\rho }_m\right).$

Letting ${\mathrm{\Gamma }}^{*}={\alpha }_{mc}+{\rho }_m/u-{\alpha }_{pc}-{\rho }_p/u$ , it follows that ${\tau }^{*}=1$ if ${\mathrm{\Gamma }}^{*}>0$ and ${\tau }^{*}=0$ otherwise. Note that $\mathrm{\Gamma }$ depends on the ${\theta }_{ic}$ s while ${\mathrm{\Gamma }}^{*}$ does not. Given $\mathrm{\Gamma }$ and ${\mathrm{\Gamma }}^{*}$ , we can evaluate when equilibrium is efficient. In the special Rubinstein–Wolinsky (Reference Rubinstein and Wolinsky1987) case, ${\rho }_p={\rho }_m$ and ${\theta }_{pc}={\theta }_{mc}$ , equilibrium and efficiency coincide: Both imply middlemen should be active if and only if ${\alpha }_{mc}>{\alpha }_{pc}$ . However, going beyond their special case, we can have $\tau =1$ when ${\tau }^{*}=0$ or vice versa, depending on parameters, and in particular depending on the $\theta$ s.Footnote ⁵

The aforementioned analysis is predicated on $P$ and $M$ participating in the market, which must be checked. For stage 1, everyone participates, since it is costless. For stage 2, while $C$ still participates for free, for $j=P,M$ we need $V_j\ge 0$ , or, in terms of primitives, ${\alpha }_{jc}{\theta }_{jc}u\ge -{\rho }_j$ . In general, there are four possibilities for the equilibrium pattern of exchange, or what we call the equilibrium regime, in stage 2: Regime N, for no trade, occurs if ${\alpha }_{jc}{\theta }_{jc}u<-{\rho }_j$ for both $j=P,M$ , so expected profit does not justify the cost. Regime D, for direct trade, occurs if ${\alpha }_{jc}{\theta }_{jc}u\ge -{\rho }_j$ for $P$ but not $M$ . Regime I, for indirect trade only, occurs if ${\alpha }_{jc}{\theta }_{jc}u\ge -{\rho }_j$ for $M$ but not $P$ . Regime B, for both direct and indirect trade, occurs if ${\alpha }_{jc}{\theta }_{jc}u\ge -{\rho }_j$ for $P$ and $M$ . These four regimes appear in several other models discussed in Sections 4, 5, 6, and 9.

Equilibrium features a classic holdup problem since the search/storage cost is sunk when $P$ or $M$ meets $C$ at stage 2. Likewise, the wholesale price $y_{pm}$ at which $M$ gets $x$ from $P$ in stage 1 is sunk when $M$ meets $C$ . Agents cannot recoup sunk costs in bargaining for the usual reason: These costs are paid whether or not they trade, so they cancel out of the surplus (the payoff to trade minus the payoff to no trade). This distorts equilibrium as follows: In equilibrium $M$ and $P$ participate as stage 2 sellers when ${\alpha }_{jc}{\theta }_{jc}u\ge -{\rho }_j$ , while efficiency suggests they should participate when ${\alpha }_{jc}u\ge -{\rho }_j$ . For the equilibrium and efficiency conditions to coincide for all values of the parameters we need ${\theta }_{ij}=1$ .Footnote ⁶

Holdup problems exist in many economic models with bargaining. One take on this is that they can be avoided by having agents contract the terms of trade before incurring sunk costs. That is sometimes ruled out exogenously. In search theory, ruling it out seems less ad hoc once one recognizes that you cannot contract with someone before you contact them. We can eliminate part of the problem by an assumption on exogenous parameters, ${\rho }_j=0$ , but the endogenous wholesale price $y_{pm}$ is still sunk when $M$ and $C$ meet. Rubinstein and Wolinsky (Reference Rubinstein and Wolinsky1987) propose a consignment procedure to deal with that: $M$ only pays $P$ after $C$ pays $M$ . Whether this is feasible depends on assumptions – can $P$ and $M$ stay in contact while waiting for $C$ ? In any case, it does not affect their main result: Given ${\theta }_{mc}={\theta }_{pc}$ and ${\rho }_{mc}={\rho }_{pc}$ , activity by $M$ depends solely on the sign of ${\alpha }_{mc}-{\alpha }_{pc}$ .

In what follows we consider other advantages middlemen may have. We also consider going beyond two periods, sometimes to an infinite horizon, which is interesting and even crucial in some applications (e.g., introducing money or endogenous debt limits). A goal is to characterize the equilibrium set to see when we can support different regimes. Another is to discuss existence, uniqueness or multiplicity, and dynamics in various extensions of the basic framework.

3 Extending the Simple Model

We now present (a generalized version of) the framework used in Rubinstein and Wolinsky (Reference Rubinstein and Wolinsky1987). Time $t$ is continuous and the horizon is infinite. A continuum of agents come in three types, $P$ , $C$ , and $M$ , acting in the roles played in Section 2. They all discount future payoffs at rate $r$ . While different versions presented in this section make different assumptions about this, in the original specification type $M$ stay in the market forever while $P$ and $C$ exit after one trade, with an exogenous inflow of new $P$ and $C$ agents so the market can be open in the long run.Footnote ⁷

At any $t$ the state of the system is given by the distribution of inventories across type $M$ . With $x$ indivisible, an individual $M$ has inventory $I\in \{0,1,\dots ,\hat{I}\}$ , where $\hat{I}$ may be finite or infinite. Many papers (though not all) set $\hat{I}=1$ , so there are just two kinds of middlemen; let us call them $M_1$ and $M_0$ , with the subscript indicating inventory. Hence, at any $t$ a measure $N_1$ are type $M_1$ with $1$ unit of $x$ , and a measure $N_0$ are type $M_0$ with $0$ inventory, where $N_0+N_1=N_m$ . Restricting $I\in \{0,1\}$ is a technological assumption that $M$ can store at most one unit of $x$ , and $P$ can produce and $C$ can consume at most one unit at a time. This assumption has precedent in search theory,Footnote ⁸ and Section 7 discusses papers that relax it.

Given $I\in \{0,1\}$ , the return to $i=P,M$ from holding a unit of $x$ is again ${\rho }_i$ . We can also add a fixed entry cost ${\kappa }_i$ for type $i$ , different from ${\rho }_i$ in that it is paid once and not each period, but ${\kappa }_i=0$ for now. Meetings are characterized by Poisson processes with arrival rates ${\alpha }_{ij}$ denoting the probability per unit time that type $i$ meets $j$ , and bilateral meetings imply the identities $N_i{\alpha }_{ij}=N_j{\alpha }_{ji}$ .

The meeting process in Rubinstein–Wolinsky can be interpreted in terms of spatial separation, even if they were not specific about it. A significant feature of their formulation is that the measure of type $i$ does not affect the probability $j$ meets $k$ when $j\ne i$ and $k\ne i$ . Other papers proceed differently, and a common specification has the probability that $i$ meets $j$ proportional to the fraction of type $j$ among the total population of participating agents in the market. This is sometimes called uniform random meetings, and implies the measure of $M$ in the market, for example, affects the probability $P$ meets $C$ , which can be understood as a congestion effect that is assumed away in the original model.

Let ${\tau }_{ij}$ be the probability of trade when $i$ meets $j$ . With transferable utility, as described earlier, $i$ wants to trade with $j$ if and only if $j$ wants to trade with $i$ if and only if the joint surplus is positive, so we can use either ${\tau }_{ij}$ or ${\tau }_{ji}$ to indicate the probability they trade. Some of the $\tau$ s are trivial; for example, ${\tau }_{c0}={\tau }_{p1}=0$ is automatic since when $C$ meets $M$ with inventory $0$ , or $P$ meets $M$ with inventory $1$ , trade is impossible. The rate at which $M$ switches from $M_1$ to $M_0$ is ${\alpha }_{1c}{\tau }_{1c}$ , and the rate at which $M$ switches back is ${\alpha }_{0p}{\tau }_{0p}$ . In general, both are the product of chance (a meeting) and choice (a trade).

Let $V_p$ and $V_c$ be the value functions for $P$ and $C$ , and let $V_0$ or $V_1$ be the value functions for $M$ holding $0$ or $1$ unit of $x$ . As mentioned, here $C$ and $P$ leave after trading while $M$ stays. The surplus $S_{ij}$ when $i$ trades with $j$ is:

$\begin{array}{rl}{S}_{cp}& =u-y_{pc}-V_c\text{and}{S}_{pc}=y_{pc}-c-V_p,\end{array}$ (13)

$\begin{array}{rl}{S}_{cm}& =u-y_{mc}-V_c\text{and}{S}_{mc}=y_{mc}-V_1+V_0,\end{array}$ (14)

$\begin{array}{rl}{S}_{mp}& =V_1-V_0-y_{pm}\text{and}{S}_{pm}=y_{pm}-c-V_p,\end{array}$ (15)

where $P$ is assumed to produce upon meeting as in Rubinstein–Wolinsky.

For the terms of trade, with transferable utility, reasonable bargaining solutions all give similar results. Here we use generalized Nash bargaining: In each trade, the payment $y$ solves

$y_{ij}=\arg \underset{y}{\max }{S}_{ij}{(y)}^{{\theta }_{ij}}{S}_{ji}{(y)}^{{\theta }_{ji}}.$ (16)

This yields:

$\begin{array}{rl}{y}_{pc}& ={\theta }_{pc}\left(u-V_c\right)+{\theta }_{cp}\left(c+V_p\right),\end{array}$ (17)

$\begin{array}{rl}{y}_{mc}& ={\theta }_{mc}\left(u-V_c\right)+{\theta }_{cm}(V_1-V_0),\end{array}$ (18)

$\begin{array}{rl}{y}_{pm}& ={\theta }_{pm}\left(V_1-V_0\right)+{\theta }_{mp}\left(c+V_p\right).\end{array}$ (19)

Thus, payment $y$ is a weighted average of the benefit to the agent receiving $x$ and the cost to the agent giving it up; for example, when $C$ gets $x$ from $M$ the benefit to the former is $u-V_c$ while the cost for the latter is $V_1-V_0$ .

Also, with transferable utility, the best response conditions for whether $i$ and $j$ trade are given by:

${\tau }_{ij}=\left\{\begin{array}{ll}1& \text{if}{S}_{ij}+S_{ji}>0\\ [0,1]& \text{if}{S}_{ij}+S_{ji}=0\\ 0& \text{if}{S}_{ij}+S_{ji}<0\end{array}\right..$ (20)

The original Rubinstein–Wolinsky setup has new $C$ and $P$ agents flowing into the market at an exogenous rate $E$ , making the stocks $N_c$ and $N_p$ endogenous. The stocks $N_1$ and $N_0$ are also endogenous, but of course we only need to keep track of $N_1$ since $N_0=N_m-N_1$ with $N_m$ fixed. Hence, the relevant laws of motion or the state of the system are

$\begin{array}{rl}{\dot{N}}_c& =E-\left({\alpha }_{cp}+{\alpha }_{c1}\right)N_c,\end{array}$(21)

$\begin{array}{rl}{\dot{N}}_p& =E-\left({\alpha }_{pc}+{\alpha }_{p0}\right)N_p,\end{array}$(22)

$\begin{array}{rl}{\dot{N}}_1& =\left(N_m-N_1\right){\alpha }_{0p}{\tau }_{mp}-N_1{\alpha }_{1c}{\tau }_{mc},\end{array}$(23)

where the $\dot{N}$ s are derivatives with respect to time.

The value functions expressed in terms of $S_{ij}$ satisfy

$\begin{array}{rl}r{V}_c& ={\alpha }_{cp}{\tau }_{cp}{S}_{cp}+{\alpha }_{c1}{\tau }_{c1}{S}_{cm}+{\dot{V}}_c,\end{array}$(24)

$\begin{array}{rl}r{V}_p& ={\alpha }_{pc}{\tau }_{pc}{S}_{pc}+{\alpha }_{p0}{\tau }_{p0}{S}_{pm}+{\rho }_p+{\dot{V}}_p,\end{array}$(25)

$\begin{array}{rl}r{V}_1& ={\alpha }_{1c}{\tau }_{1c}{S}_{mc}+{\rho }_m+{\dot{V}}_1,\end{array}$(26)

$\begin{array}{rl}r{V}_0& ={\alpha }_{0p}{\tau }_{0p}{S}_{mp}+{\dot{V}}_0.\end{array}$(27)

Consider (24). In words, it says the flow value $r{V}_c$ is the rate ${\alpha }_{cp}$ at which $C$ meets $P$ , times the probability ${\tau }_{cp}$ they trade, times $C$ ’s surplus from that trade; plus the rate ${\alpha }_{c1}$ at which $C$ meets $M$ with inventory, times the probability ${\tau }_{c1}$ they trade, times $C$ ’s surplus from that trade; plus the pure rate of time change ${\dot{V}}_c$ , which is $0$ in steady state but not in general. The other equations have similar interpretations.

With dynamic models we have to be a little more careful in defining things. An equilibrium here consists of paths for: the value functions $\mathbf{V}=\left(V_p,V_c,V_0,V_1\right)$ , the terms of trade, which with $x$ indivisible are given by $\mathbf{y}=\left(y_{pc},y_{pm},y_{mc}\right)$ , the trading strategies $\mathit{\tau }=\left({\tau }_{pc},{\tau }_{p0},{\tau }_{1c}\right)$ , and the state variables $\mathbf{N}=\left(N_c,N_p,N_1\right)$ , satisfying the dynamic programming equations, bargaining solutions, best response conditions, and laws of motion. Equilibrium must also satisfy an initial condition saying where the system starts in terms of N, plus the usual nonnegativity and boundedness conditions.Footnote ⁹ A stationary equilibrium is a special case where $\mathbf{V}$ , $\mathbf{y}$ , and $\mathit{\tau }$ are time-invariant functions of $\mathbf{N}$ (they can depend on the state but not the date). A steady state is a solution to the equilibrium conditions other than the initial condition where endogenous variables are constant.

In Rubinstein–Wolinsky, as mentioned earlier, the arrival rate ${\alpha }_{ij}$ depends on $N_i$ and $N_j$ , the measures of $i$ and $j$ , but not on other $N$ s. This rules out third-party congestion in the meeting process, which is convenient, but the microfoundations may not be obvious. Figure 2 depicts a market structure consistent with their assumptions, featuring spatial separation, as in Gong et al. (Reference Gong, Qiao and Wright2024).

Figure 2

Market structure consistent with Rubinstein–Wolinsky (Reference Rubinstein and Wolinsky1987).

In Figure 2 different types are located at distinct locations represented by nodes on a triangle. Agents visit both of their nearby markets, represented by the edges, but not the third – it’s just too far. One can interpret this as saying there are three submarkets labeled as follows: a direct market (DM) with $C$ and $P$ ; a wholesale market (WM) with $P$ and $M_0$ ; and a retail market (RM) with $M_1$ and $C$ . This spatial separation is consistent with the Rubinstein–Wolinsky paper.

Now is a good time to discuss meeting technologies. Following textbook methods (e.g., Pissarides Reference Pissarides2000), consider a two-sided market with measures $N_b$ of buyers and $N_s$ of sellers. The number of meetings is assumed to be an increasing, concave function $\mu \left(N_b,Ns\right)$ , implying arrival rates ${\alpha }_i=\mu \left(N_b,Ns\right)/N_i$ . A common assumption that we adopt is that $\mu$ displays constant returns to scale, which implies the arrival rates depend only on market tightness, $N_b/N_s$ . As mentioned earlier, and as Figure 2 shows, agents participate in both of their nearby markets but not the third.Footnote ¹⁰

This allows us to use a general two-sided meeting technology in each of the submarkets, which is convenient because it is not clear how to use a general meeting technology in a three-sided market (more on this in Sections 4, 6, 8, and 9). To proceed, let the DM, WM, and RM meetings technologies be ${\varphi }_j\mu \left(\cdot \right)$ , for $j=D,W,R$ , where the $\varphi$ s are constants describing search efficiency. Then, as in Rubinstein–Wolinsky, assume ${\varphi }_W={\varphi }_R$ , but ${\varphi }_D$ can be different, and in particular $M$ is better than $P$ at meeting $C$ based on the fundamental technology when ${\varphi }_R>{\varphi }_D$ , although the equilibrium arrival rates depend on tightness.

Now, as in Rubinstein–Wolinsky, we focus on steady states that are symmetric in the sense that $N_p=N_c$ , which implies $N_1=N_0$ , ${\alpha }_{cp}={\alpha }_{pc}$ , ${\alpha }_{c1}={\alpha }_{p0}$ , and ${\alpha }_{1c}={\alpha }_{0p}$ . Also, as in their original model, we set $c={\rho }_p={\rho }_m=0$ . The goal is to determine the trading pattern given by the $\tau$ s. It is clear that $C$ and $P$ always trade ( ${\tau }_{cp}={\tau }_{pc}=1$ ). The more interesting questions are whether $M$ without inventory trade with $P$ (determined by ${\tau }_{p0}={\tau }_{0p}$ ), and whether $M$ with inventory trade with $C$ (determined by ${\tau }_{1c}={\tau }_{c1}$ ). Clearly, if ${\tau }_{p0}=1$ then ${\tau }_{1c}=1$ , since a buy-and-hold strategy for $M$ is not a good idea when ${\rho }_m=0$ . What remains to determine is whether $M$ and $P$ trade, as determined by ${\tau }_{mc}$ .

Although the Rubinstein–Wolinsky paper does not mention existence or uniqueness, from Gong et al. (Reference Gong, Qiao and Wright2024) we know that a symmetric steady state exists and is unique. This is illustrated in Figure 3. In the panel (a), on the vertical axis is $M$ ’s bargaining power, with the horizontal dashed line giving $P$ ’s bargaining power, so that above the line $M$ is better than $P$ at extracting surplus from $C$ . On the horizontal axis is $M$ ’s search efficiency, defined by ${\varphi }_R$ in meeting technology, ${\varphi }_R\mu \left(\cdot \right)$ , while the vertical dashed line represents ${\varphi }_R={\varphi }_D$ –so to the right of this line $M$ has a fundamental advantage in search. Panel (b) is similar but drawn in $\varphi$ space.

Figure 3

Equilibrium set as a function of parameters.

Figure 3 shows there are regions of parameter space where $M$ and $P$ trade with probability ${\tau }_{p0}=1$ , where they trade with probability ${\tau }_{p0}=0$ , and where they trade with probability ${\tau }_{p0}\in \left(0,1\right)$ . In terms of the language introduced earlier, there are two possible outcomes: Regime D or B emerges when ${\tau }_{p0}=0$ or ${\tau }_{p0}>0$ . (Regimes $N$ and $I$ for now cannot emerge because $P$ is always in the market and happy to trade with $C$ for now – but see Sections 4 and 9) Related to Section 2, the result can be stated as follows: ${\alpha }_{1c}{\theta }_{mc}>{\alpha }_{pc}{\theta }_{pc}$ implies ${\tau }_{p0}=1$ , so $P$ and $M$ trade whenever they meet; ${\alpha }_{1c}{\theta }_{mc}<{\alpha }_{pc}{\theta }_{pc}$ implies $P$ and $M$ never trade when they meet; and ${\alpha }_{1c}{\theta }_{mc}={\alpha }_{pc}{\theta }_{pc}$ implies they sometimes trade when they meet.Footnote ¹¹

This is a generalization of the Rubinstein–Wolinsky result, but as intuitive as it may be, the result is not totally satisfactory because it gives a relationship between the $\alpha$ s and $\tau$ s, both of which are equilibrium outcomes. Hence, it must be interpreted carefully. In particular, it is possible for $M$ to face a less efficient meeting technology, in the sense that ${\varphi }_m$ is low, but due to endogenous tightness ${\alpha }_{mc}$ can still be high and hence $M$ still active. Indeed, as panel (a) of Figure 3 shows, $M$ is active when $M$ and $P$ have the same meeting technology and bargaining power, at the intersection of the two dashed lines, so by continuity $M$ is active with a moderate disadvantage in their meetings and bargaining. Similarly, $M$ might have a better $\varphi$ and better $\theta$ than $P$ , yet ${\tau }_{p0}<1$ .

So while the result is correct, it could easily be misinterpreted. However, the bigger point is that we now have a characterization of the $\alpha$ s and $\tau$ s in terms of parameters, as shown in Figure 3. In other words, there is a relationship between the $\alpha$ s and $\tau$ s, but it is not causal, since both are functions of other, more fundamental, factors.

Having made that point, beyond characterizing $M$ ’s activity, the theory has implications for how the terms of trade depend on fundamentals. One can check that both $y_{pc}$ and $y_{mc}$ decrease with an improvement in the meeting technology indexed by the $\varphi$ s, which seems natural. However, the average $y$ paid by $C$ is nonmonotone because of composition effects – namely, faster search encourages participation by $M$ , and since $M$ charges more than $P$ , the average price paid by $C$ might increase. One can also check that price dispersion (as measured by, e.g., the coefficient of variation) can be nonmonotone.Footnote ¹²

We close this part of the discussion with an important modeling detail. The original Rubinstein–Wolinsky formulation has long-lived $M$ (they stay in the market forever) but short-lived $P$ and $C$ (they leave after one trade), with an exogenous inflow $E$ of the short-lived types. Consider an environment that is similar except $C$ and $P$ , as well as $M$ , stay forever, and in the interest of stationarity set $E=0$ . This makes the surpluses simpler because continuation values cancel with threat points,

$S_{cp}=u-y_{pc},S_{pc}=y_{pc}-c,S_{cm}=u-y_{mc},\text{and}{S}_{pm}=y_{pm}-c,$ (28)

and the payments become

$\begin{array}{rl}{y}_{pc}& ={\theta }_{pc}u+{\theta }_{cp}c,\end{array}$ (29)

$\begin{array}{rl}{y}_{mc}& ={\theta }_{mc}u+{\theta }_{cm}(V_1-V_0),\end{array}$ (30)

$\begin{array}{rl}{y}_{pm}& ={\theta }_{pm}\left(V_1-V_0\right)+{\theta }_{mp}c.\end{array}$ (31)

Having all agents in the market forever has consequences. If, for example, type $P$ agents leave after trading, when they meet $M$ they may pass on trade to wait for a meeting with $C$ ; but if type $P$ stays after trading, at least with $c={\rho }_p={\rho }_m=0$ , we must have ${\tau }_{pm}=1$ , allowing us to shift the focus elsewhere, like endogenous entry. Due to its relative tractability, several models presented herein have all agents in the market forever. A general point that one should always keep in mind is that modeling details like this can matter when exploring microfoundations.Footnote ¹³

4 Multiplicity and Dynamics

Some papers are concerned with whether we get uniqueness or multiplicity of steady state, and whether there are dynamic equilibria with fluctuations based solely on beliefs. This issue is obviously related to the venerable notion that intermediation, perhaps especially financial intermediation, might engender instability or volatility.Footnote ¹⁴ Now models based on Rubinstein–Wolinsky have dynamics: If the initial condition for $\mathbf{N}$ is not at its steady state value, then the unique equilibrium is stationary and converges to steady state, so there is no instability or volatility. What happens if we deviate from the original formulation?

Nosal et al. (Reference Nosal, Wong and Wright2019) study a three-sided market where $P$ , $M$ , and $C$ interact via a uniform random meeting specification: conditional on $i$ meeting someone, the probability it is type $j$ is proportional to the fraction of type $j$ in the market. Normalizing $N_p+N_c+N_m=1$ , the rate at which type $i$ meets $j$ is ${\alpha }_{ij}=\alpha N_j$ , where $\alpha$ is a baseline arrival rate that is the same for all agents. Notice that rules out something that was the focus of the above analysis: It means ${\alpha }_{mc}={\alpha }_{pc}$ and therefore $M$ cannot have an advantage over $P$ in finding type $C$ .Footnote ¹⁵

Even without an advantage in finding type $C$ , $M$ can still be active for other reasons in Nosal et al. (Reference Nosal, Wong and Wright2019). In that paper, all agents stay in the market forever, which, as mentioned earlier, means that $P$ always trades with $M$ since there is no opportunity cost. This might suggest that $M$ agents are doing something socially valuable: When $P$ trades $x$ to $M$ they both become sellers, increasing the probability $C$ gets $x$ . However, rather than having agents acting as $M$ to exploit this idea, it might be better to have them act as $P$ , because the best $M$ can do is to give $x$ to $C$ if $M$ has it in its inventory, while $P$ can give $x$ to $C$ whenever they meet. To pursue this, Nosal et al. (Reference Nosal, Wong and Wright2019) incorporate occupational choice: Anyone who is not type $C$ can choose to act as $P$ or $M$ , or to opt out of the market entirely.

That paper also plays up the distinction between markets for goods and markets for assets by identifying the former with ${\rho }_j<0$ and the latter with ${\rho }_j>0$ (although that is less relevant given an extension discussed later in this section). It is further assumed that $P$ produces a new unit of $x$ immediately after trading, before the next meeting, as opposed to producing in a meeting. Hence, $P$ as well as $M$ agents carry inventory. It is useful to have inventories depreciate – that is, vanish – at rate ${\delta }_j$ . Thus, a fundamental advantage for $M$ could be captured by ${\rho }_m>{\rho }_p$ or ${\delta }_m<{\delta }_p$ , although for simplicity we set ${\delta }_j=0$ .

Let us start with ${\rho }_j<0$ . Focusing for now on steady state, there are three possible outcomes: Regime $N$ , where the market shuts down; Regime D, where the market is open but no one chooses to act as $M$ , so there is only direct trade; and Regime B, where some agents choose to act as $M$ , so there is both direct and indirect trade. (Regime I is ruled out here because $M$ can only be active if some agents act as $P$ , since $M$ gets inventory from $P$ .) It can be shown that equilibrium exists uniquely: There is a unique steady state, and if we start away from steady state there is a unique transition path converging to it.

To give more detail, consider parameter space summarized by storage costs $\left(-{\rho }_p,-{\rho }_m\right)$ of $P$ and $M$ . It partitions into three regions: When $\left|{\rho }_j\right|$ is big for both $j=P,M$ , Regime N emerges, since storage costs are too high to make participation in the market worthwhile; when $\left|{\rho }_p\right|$ is below a threshold and $\left|{\rho }_m\right|$ above a (generally different) threshold, Regime D emerges; and when $\left|{\rho }_p\right|$ is somewhat higher and $\left|{\rho }_m\right|$ lower, Regime B emerges with both direct and indirect trade.

Now consider ${\rho }_j>0$ . This turns out to be rather different. First, Regime N cannot emerge for a production cost $c=0$ , or, more generally, for any $c$ that is not too big, since $P$ can always produce and hoard $x$ for its return ${\rho }_p$ . Therefore $P$ must produce, and when $P$ meets $C$ they must trade, since $P$ can produce again. The possible outcomes are shown in panel (a) of Figure 4.Footnote ¹⁶

Figure 4

Multiplicity and dynamics.

In the graph, there are steady states with $N_m=0$ , which is labeled as region $D^K$ or $D^T$ , where $D$ indicates direct trade and the superscript indicates what happens off the equilibrium path: If there were a type $M$ with $x$ , superscript $T$ says $M$ would trade it to $C$ , while superscript $K$ says $M$ would keep it (while these are the same on the equilibrium path, to show it is an equilibrium, as usual one has to know what happens off the equilibrium path).

Now consider steady states with $N_m>0$ . These are labeled $B^T$ , $B^K$ , and $B^R$ , where the $B$ indicates there is both direct and indirect trade, and the superscripts mean this: $T$ says $M$ trades $x$ to $C$ ; $K$ says $M$ keeps $x$ ; and $R$ says $M$ randomizes. Naturally, $B^K$ obtains when ${\rho }_m$ is big and $B^T$ obtains when ${\rho }_m$ is small. What is interesting is that over some range, where ${\rho }_m$ is neither too big nor too small, there is multiplicity: two steady states coexist in pure strategies, one with ${\tau }_{mc}=0$ and one with ${\tau }_{mc}=1$ , and, as usual, when there are two pure strategy equilibria there is also a mixed strategy equilibrium with ${\tau }_{mc}\in \left(0,1\right)$ .

This multiplicity arises only for some parameters; for others there is a unique steady state with $N_m>0$ , which is $B^T$ if ${\rho }_m$ is small, $B^K$ if ${\rho }_m$ is big, and $B^R$ if ${\rho }_m$ is in between. But at least for some nondegenerate set of parameters there is multiplicity. Why? There are two necessary ingredients: ${\rho }_m>0$ , and $N_m$ endogenous.