Search Theory

Neoclassical economics considers a centralized market for exchange in which all kinds of information are perfectly available to all individuals. Walras (1874)

in-9

troduces a centralized auction to find a market clearing price that matches supply and demand in a perfect market.¹⁷ But usually, as Stigler (1961) notes in the early literature about search theory, there is no centralized market, no facility for a per-fect and costless allocation of resources, and no benevolent Walrasian auctioneer matching supply and demand:

“Prices change with varying frequency in all markets, and, unless a market is completely centralized, no one will know all the prices which various sellers (or buyers) quote at any given time. A buyer (or seller) who wishes to ascertain the most favorable price must canvass various sellers (or buyers)—a phenomenon I shall term ‘search’.”¹⁸

One implicit assumption derived from this specification is that trade should not be modeled between buyer / seller and ‘the market’ but directly between buyer and seller, in order to account for the time-consuming search for a trading part-ner. Of course, search is not restricted to just markets for goods or to financial markets, though these are the most obvious. Other examples are the labor mar-ket, housing marmar-ket, or even the marriage market.¹⁹ The central idea of search theory is summarized by Pissarides (2001, p. 13760) as follows:

“The economics of search study the implications of market frictions for economic behavior and market performance. ‘Frictions’ in this context include anything that interferes with the smooth and instan-taneous exchange of goods and services.”

Search frictions result in the expenditure of time, money, and other resources in order to learn about opportunities. For example, if individuals have incomplete information about the location of an item or a trading partner, potential buyers must search for the needed item and potential sellers cannot easily locate a poten-tial buyer for the item on sale. As a result, prices are influenced by such frictions, which cannot be eliminated with price adjustments. Trade is delayed and mar-kets do not clear at all times.²⁰

The characterization of search in a financial market can be taken further by re-garding it as a substitute for modeling liquidity. Harris (2003, p. 395) defines liquidity as follows:

17 See Neus (2013), pp. 85.

18 Stigler (1961), p. 213.

19 See Diamond (1984), pp. 1.

20 See Pissarides (2001), pp. 13760 and Diamond (2011), pp. 1045.

“Liquidity is the object of bilateral search. In abilateral search, buyers search for sellers, and sellers search for buyers. When a buyer finds a seller who will trade at mutually acceptable terms, the buyer has found liquidity. Likewise, when a seller finds a buyer who will trade at mutually acceptable terms, the seller has found liquidity.”

Market liquidity has many different dimensions. The three dimensions estab-lished by Kyle (1985, p. 1316) are commonly stated: “[...] ‘tightness’ (the cost of turning around a position over a short period of time), ‘depth’ (the size of an or-der flow innovation required to change prices a given amount), and ‘resiliency’

(the speed with which prices recover from a random, uninformative shock).”²¹ Kyle (1985, pp. 1316) himself refers to Black (1971, pp. 29), who describes the bid-ask spread as a measure for market tightness. Kyle (1985, p. 1317) summarizes the definition of “a liquid market as one which is almost infinitely tight, which is not infinitely deep, and which is resilient enough so that prices eventually tend to their underlying value.”

Sometimes a fourth dimension, ‘immediacy’ (referring to the time a trade takes), is added.²² Harris (2003, p. 399) defines “liquidity [as] the ability to quickly trade large size at low cost.” The key issue of liquidity is the ability to trade, which is the core element of search and matching theory. Harris (2003, p. 399) continues by characterizing liquidity as a function, stating “the probability of trading a given size at a given price, given the time we are willing to put into [.] search.” Search models imply a natural—though slightly different—liquidity measure. The cost of a time lag due to delayed trade can be measured by the expected time it takes to find a trading partner. Liquidity in this setting is characterized by the speed of finding a trading partner.²³

The following passage provides a short overview of important contributions to the research on search theory:

Stigler (1961) is one of the first to formally model the behavior of buyers in a com-modity market—from which the ‘fixed sample rule’ became known: First, a buyer chooses the optimal number of sellers to search for and then decides in favor of the seller quoting the lowest price. This model paved the way for the ‘optimal stopping rule’, and can be traced back to a model of labor markets by McCall (1970): First, a reservation price is specified, and the buyer then buys from the

21 Market ‘width’ is another name for ‘tightness’. See Harris (2003), pp. 398.

22 See Harris (2003), pp. 398 and Black (1971), p. 30.

23 See Vayanos and Wang (2007), p. 75.

first seller quoting a price equal to or less than the reservation price. Therefore, search intensity and time depend on the reservation price. An exogenous price distribution is assumed within this model and a partial equilibrium problem, i.e.

one-sided search with a take-it-or-leave-it price, is solved.²⁴

The next stage of development in search theory was reached by Diamond (1982a,b), Mortensen (1982a,b), and Pissarides (1984a,b, 1985). Their contri-butions are threefold: (1) They replaced the assumption about an exogenous price distribution with Nash bargaining, (2) they introduced two-sided search by means of an exogenous matching function, and (3) they considered the flow of agents to model a general equilibrium problem.²⁵

Search theory extends the classical, i.e. deterministic, theory of exchange by con-sidering uncertainty.²⁶ The matching function accounts for this uncertainty; it describes how individuals come into contact through search. One input parame-ter is the arrival rate of trading partners within a short time inparame-terval, modeled as a stochastic process. The simplest form describing these contacts is the first arrival timeτ of a Poisson process with a constant mean arrival rateλ. This arrival rate λis denoted as ‘search intensity’, which is usually costly to increase. In general, it has the following properties: λ → ^∞ in a world with no frictions,λ > _{0 in a} market with search frictions, andλ =0 with no search at all.²⁷

An equilibrium model is characterized by a search process that persists over time.

This flow of ‘new’ agents searching for trading partners can be modeled by either exogenous inflows, wherein matched agents leave the market, or, as in the model considered in chapter 3–7, by exogenous and independent idiosyncratic shocks to a fraction of the population. These shocks induce search and trading impulses and are commonly driven by a Poisson process.²⁸

The following section gives a short introduction to probability theory, with the aim of modeling uncertainty in search theory. Thereafter, matching functions, particularly with regard to the model discussed subsequently, are presented. Bar-gaining is introduced in chapter 2.2.

24 See Pissarides (2001), pp. 13761, McCall (1970), pp. 114, and McCall and McCall (2008), pp.

25 11.See Mortensen and Pissarides (1999a), p. 1173, The Royal Swedish Academy of Sciences (2010), p. 3, and McCall and McCall (2008), pp. 9 and 12.

26 See McCall and McCall (2008), p. 10.

27 See Diamond (1984), p. 9, Pissarides (2001), p. 13761, and Pissarides (2000), p. 127.

28 See Pissarides (2001), p. 13762.