تعادل رقابتی با اصطکاک های جستجو: رویکرد تعادل عمومی

When the trading process is characterized by search frictions, traders may be rationed so markets need not clear. We build a general equilibrium model with transferable utility where the uncertainty arising from rationing is incorporated in the definition of a commodity, in the spirit of the Arrow–Debreu theory. Prices of commodities then depend not only on their physical characteristics, but also on the probability that their trade is rationed. The standard definition of competitive equilibrium is extended by replacing the market clearing condition with an exogenous matching function which describes a trading technology that is not frictionless. As is typical in search models, the matching function relates the rationing probabilities of buyers and sellers to the ratio of buyers to sellers in the market. When search frictions vanish, our model is equivalent to the continuous assignment model of Gretsky et al. [20]. We adapt their approach, which uses linear programming techniques and duality theory, to show that a competitive equilibrium exists and is constrained efficient in our environment. Our competitive equilibrium notion is equivalent to that of directed (or competitive) search. The strength of our formulation and the linear programming approach is that they allow us to generalize the constrained efficiency and existence results in the directed search literature to a much broader class of economies. Our framework also opens the door to the use of existing algorithms for computing equilibria and taking these models to the data.

The Arrow–Debreu model is the cornerstone for the analysis of competitive markets. In the classical theory of Arrow [4] and Debreu [7], trade is represented as a costless process. Any agent seeking to buy or sell a good at a given point in time can do so at the equilibrium market price. Trade involves no further costs in terms of time and resources. Search theory, on the other hand, highlights the costly nature of the trading process. In particular, this theory has become the dominant paradigm to study labor markets since the seminal work of Diamond [9] and [10], Mortensen [38] and [39] and Pissarides [44] and [45]. In these markets workers usually take time and spend resources in order to find a suitable employer, and vice versa. Also, rationing arises in the form of unemployment, which tends to coexist with unfilled job vacancies. The key assumption of the Diamond–Mortensen–Pissarides model is that workers and firms must search for trading opportunities, and the outcome of their search is uncertain. Search frictions are typically modeled via an exogenous matching function which describes a random meeting process between workers and firms.2 This random process implies that at any point in time some agents will manage to trade and others will not. Hence, unlike in the classical Arrow–Debreu model, agents may be rationed in equilibrium (so markets need not clear), and in general it will take time to trade. In this paper we use a notion of competitive equilibrium in the spirit of Arrow and Debreu to study a general class of exchange economies with search frictions and transferable utility3 (e.g. with a large variety of different goods, general distributions of buyer and seller types, with or without complementarities, general matching technologies which may differ across markets,...). We show that a competitive equilibrium exists and is constrained efficient. The definition of constrained efficiency takes into account the fact that the social planner, just like the market, is restricted by the matching technology. Our equilibrium notion is equivalent to that of directed (or competitive) search.4 We elaborate on this below, but let us already mention at this point that our results generalize earlier results in the directed search literature in a significant way. It is well-known that directed search equilibria are constrained efficient both in simple environments with homogeneous buyers and sellers (see Moen [35] and Shimer [53]) and in several environments with heterogeneity (e.g. Shi [50], Shimer [52] and Eeckhout and Kircher [11]). Yet only Eeckhout and Kircher [11] study a comparable environment with rich two-sided heterogeneity. These authors derive necessary and sufficient conditions for assortative matching. They then show that under these conditions—which regard the degree of supermodularity of the match value function and imply that there are sufficiently strong complementarities5—an equilibrium exists and is constrained efficient. Our existence and efficiency results are more general as they hold regardless of the matching pattern displayed by the equilibrium allocation. Neither do they require the differentiability and concavity assumptions typically imposed in the literature. All we assume is continuous utility and matching functions, and compact sets of physical goods and agents types. Moreover, an important novelty of our model with respect to the directed search literature is that it allows for multidimensional types and multidimensional (hedonic) goods. This is important since workers, for instance, differ in several aspects (e.g. education, experience, ability...) which affect the value in a match with a firm, and vice versa. Our key modeling choice is to incorporate the uncertainty arising from rationing in the definition of a commodity, in the spirit of the Arrow–Debreu theory. Prices of commodities then depend not only on their physical characteristics, but also on the probability that their trade is rationed (i.e., their “search characteristic”). In a competitive equilibrium agents—who are infinitesimal relative to the size of the economy—take prices as given. They also take as given rationing probabilities, which are part of the description of a commodity. Markets are anonymous, so prices and rationing probabilities do not depend on the identities of the traders. The departure from the standard definition of competitive equilibrium is that market clearing is replaced with a matching condition which describes a trading technology that is not frictionless. As is typical in search models, this condition relates the trading probabilities of buyers and sellers to the ratio of buyers to sellers in each market via an exogenous matching function with constant returns to scale. In doing so, it ensures that the measures of buyers and sellers who trade a given commodity are equal (even though the measures of buyers and sellers in the market for the commodity may differ). In equilibrium, the price system adjusts so that the optimal decisions of the agents are consistent with the matching condition. The matching function captures the presence of external congestion effects in the trading process. Intuitively, as the ratio of buyers to sellers (or market tightness) increases, the probability that each buyer trades falls and the probability that a seller trades increases. In other words, agents seeking to trade impose a negative congestion externality on traders on the same side of the market and a positive externality on traders on the other side of the market. In a competitive equilibrium these external effects are internalized, so the equilibrium allocation is constrained efficient. Because rationing probabilities are specified in the definition of a commodity, they are explicitly priced. Suppose, for instance, that the same physical good trades in two locations. Suppose also that there are fewer buyers per seller in the first location in equilibrium. If buyers and sellers are expected utility maximizers who are free to choose the location where they trade, the good should trade at a higher price in the first location (because there the probability of trading is higher for buyers and lower for sellers). The fact that the price of the good differs across locations is not surprising from the general equilibrium perspective, since the objects traded are formally two different commodities (described by their physical characteristics and the market tightness at the trading location). The standard definition of directed search equilibrium in this kind of environment assumes that sellers choose prices. Buyers in turn choose the good they want to buy and a price offer. Put differently, agents choose to trade in a particular “submarket,” described by the good's physical characteristics and a price. All traders form beliefs about the tightness (and thus the probability of trading) in each submarket. The intuition is that, for a given good, lower prices attract relatively more buyers, increasing the sellers' trading probability and decreasing the buyers' trading probability.6 In equilibrium beliefs are rational in all active submarkets, meaning that the measures of buyers and sellers who trade in these submarkets generate the tightness levels that the traders take as given. On the other hand, beliefs in inactive submarkets are assumed to be always well-defined and common for all traders.7 Specifically, traders believe that there are many buyers and sellers in all submarkets, even if they are inactive in equilibrium. Also, a particular restriction, known as the “market utility property”, is imposed on beliefs. For environments where traders on one side of the market are homogeneous, the restriction implies that these traders are indifferent between active and inactive submarkets; i.e., they get their “market utility level” in all submarkets. Say all buyers are homogeneous. One can then characterize the equilibrium by assuming that sellers choose a price and a market tightness level so as to maximize their expected utility subject to the constraint that buyers get their market utility level. Hence, the equilibrium is constrained efficient.8 Eeckhout and Kircher [11] show that models with two-sided heterogeneity are more involved. In particular, the market utility assumption in this case says that the beliefs about the tightness in each inactive submarket are determined by the indifference condition of the buyer type who is most eager to trade there.9 Our formulation and that of directed search are essentially two sides of the same coin. In directed search models agents choose a submarket (i.e., a good and a price) given their beliefs about the tightness in all submarkets. In our model they choose a commodity (i.e., a good and a tightness level) taking as given the prices of all commodities. As we shall see, our matching condition is the equivalent of the condition that beliefs are rational in all active markets. The equivalent of the assumption that all traders have the same beliefs in our model is that all traders face the same prices. And the equivalent of the assumption that beliefs in all markets are well-defined is that all commodities are priced, even if they are not traded in equilibrium. The latter feature is standard in general equilibrium models with a continuum of commodities, like ours, where the prices of commodities which are not traded in equilibrium are indeterminate (e.g. see Mas-Colell and Zame [32]). The difference between the two formulations has to do with the market utility property. In our model prices in inactive markets keep buyers and sellers out of these markets in equilibrium, but need not provide any agent with his/her equilibrium utility level. So our definition is in principle less restrictive. Yet it is common to get rid of the price indeterminacy by selecting the infimum of the set of supporting price systems, that is, the “cheapest prices” supporting the equilibrium allocation (e.g. see Gretsky et al. [21]). We shall show that this is equivalent to the market utility assumption in Eeckhout and Kircher [11], since at these prices the buyer type who has the highest willingness to pay for a commodity that is not traded in equilibrium is indifferent between trading or not in the market for that commodity. Our model is closer in spirit to classical general equilibrium theory in that it fleshes out the allocating role of prices.10 “Flipping things over” and laying out the search model in this more standard form is very useful for our purposes, for the following reason. In the absence of search frictions, our model is isomorphic to that of Gretsky et al. [20]. These authors show that the results of Shapley [48] and Shapley and Shubik [49] on the classical assignment model extend to economies with a continuum of agents and commodities. Namely, their first result is that efficient allocations solve a linear programming (LP) problem, known in mathematics as the optimal transportation problem. Like any LP problem, this problem has a “dual”, which is also linear. Second, the set of optimal solutions to the LP problem and its dual is equivalent to the set of competitive equilibria of the large economy. In addition, the set of optimal dual solutions is equivalent to the core (and to the set of stable matches). Gretsky et al. [20] also show that the two LP problems have optimal solutions under very general conditions, thus establishing the existence of the equivalent solution concepts (i.e., competitive equilibria, the core, and a stable match).11 Their analysis is rather involved because, unlike in the Shapley–Shubik finite assignment game, the LP problems are infinite dimensional. As we shall show, constrained efficient allocations in our search environment can also be determined as optimal solutions to an infinite dimensional LP problem. This allows us to adapt the methodology in Gretsky et al. [20] and use essentially the same argument to derive our efficiency and existence results in a very general search environment.12 Our proof of existence is slightly different though (mainly because the LP problem in this paper, while similar, is not an optimal transportation problem).13 Interestingly, the linear programming framework opens the door to the use of existing algorithms for computing equilibria. So our model should be particularly useful for quantitative studies of search markets (where there is typically a lot of heterogeneity in prices and outcomes). In recent work, Galichon and Salanié [18] and [17] use the optimal transportation model to derive a general method of structural estimation of frictionless matching (and hedonic) models in the presence of unobservable heterogeneity.14 Given that the LP problem in this paper is a close relative of the optimal transportation problem, it seems likely that their methodology can be adapted to take directed search models to the data. To the best of our knowledge, the linear programming approach has not been used before to study economies with search frictions.15 The paper is organized as follows. Sections 2 and 3 describe the environment and the general equilibrium model, respectively. A competitive equilibrium is defined in Section 4. In Section 5 we describe the LP problem and its dual, show that both problems have optimal solutions, and characterize constrained efficient allocations via the complementary slackness theory of linear programming. In Section 6 we show that the welfare and existence theorems follow directly from the LP formulation in Section 5. In Section 7 we first illustrate the equivalence between a competitive equilibrium and a directed search equilibrium using examples, and then present the equivalence result formally. The technical details of the LP formulation and the main proofs are presented in Appendix A and Appendix B.

This paper shows that the tools and intuitions of general equilibrium theory can be applied to study competitive economies with search frictions. Specifically, there is a well-established mathematical literature on infinite dimensional linear programming which has been used by Gretsky et al. [22] to study frictionless matching markets, and which can be applied to these search environments. The strength of our general equilibrium formulation and the linear programming approach is that they allow us to generalize the constrained efficiency and existence results in the directed search literature to a much broader class of economies. This is accomplished by showing that constrained efficient allocations can be determined as solutions to an LP problem, and by bringing to light a direct mapping between a competitive equilibrium and the optimal solution to the LP problem. Interestingly, our framework opens the door to the use of existing algorithms for computing equilibria. An important novelty with respect to the directed search literature is that it allows for multidimensional types. As noted in the empirical literature, there is evidence that matching is indeed multidimensional (e.g. see Galichon and Salanié [18]). Workers, for instance, differ in several characteristics which employers typically take into account in the hiring process (e.g. education, experience, ability,...), and vice versa. Galichon and Salanié [18] and [17] use the optimal transportation model to estimate frictionless matching models. To this aim, they allow for some characteristics of the types to be unobservable to the econometrician. The introduction of unobservable heterogeneity implies that, while the social surplus (or total welfare) is linear in the assignment of different agent types, it is non-linear in the assignment of observable types (which is what is available in the data). The authors impose identifying assumptions that render this non-linear problem convex. They then use the tools of convex analysis and duality theory to identify and estimate the social surplus of matching different observable types, and the agents' equilibrium utilities. Decker et al. [8] present a related independent analysis also based on convex analysis and duality. Galichon and Salanié [17] also present a powerful algorithm for computing the optimal observable assignment. Whether their methodology can be adapted to estimate directed search models is an open question, but the fact that the starting point of their analysis is a close relative of the LP problem in this paper is definitely promising. Chiappori et al. [6] extend the analysis of Gretsky et al. [22] to a production economy building on results from the optimal transportation literature, and derive conditions for uniqueness of equilibrium. They also provide conditions for the equilibrium to be “pure”, meaning that buyers of the same type are not matched with different seller types, and vice versa. While we do not investigate these issues here, it is again likely that their methodology can be adapted to our framework. Our framework embeds most static models in the directed search literature as particular cases,41 and extends easily to dynamic models which are stationary or have a finite horizon. Yet it does not apply to nonstationary infinite horizon dynamic models (see, however, Shi [51] and Menzio and Shi [34]). This is because our proofs rely on the fact that one can restrict to a compact subset of the set of commodities without loss of generality (which is not true with an infinite horizon). We leave the analysis of this extension for future research.