Download Money and Costly Matchmaking: A Study on Monetary Equilibria in a Random Matching Model and more Papers Physiotherapy in PDF only on Docsity!
Money, Search and Costly Matchmaking*
Gabriele Camera Dept. of Economics, Purdue University West Lafayette, IN 47907- E-mail: Gcamera@mgmt.purdue.edu
ABSTRACT
I examine the robustness of monetary equilibria in a random matching model where a more efficient mechanism for trade is available. Agents choose between two trading sectors: the search and the intermediated sector. In the former, trade partners arrive randomly and there is a trading externality. In the latter a costly matching technology provides deterministic double-coincidence matches. Multiple equilibria exist with the extent of costly matching endogenously determined. Money and “mediated” trade may coexist. This depends on the size of the probability of a trade, relative to the cost of deterministic matching. This outcome is inferior for an increasing returns externality. Under certain conditions regimes with only costly matching are welfare superior to monetary regimes with random matching.
Keywords: monetary economics, search, multiple equilibria, coordination failures. (JEL C62, D83, E40).
- This paper is based on material from my Ph.D. dissertation at the University of Iowa. I am indebted to Steve Williamson for valuable comments and support, and to Dean Corbae who suggested the paper's basic idea, and provided thoughtful remarks. The paper has greatly benefited from the comments of three anonymous referees, Andreas Blume, Narayana Kocherlakota, Yiting Li, Hanno Ritter, Alberto Trejos, Randy Wright, seminar participants in the 1996 SEDC and the Midwest Macroeconomics meetings, and at the University of Iowa. Marikah Mancini has provided useful research assistance. All errors are my responsibility.
1. Introduction This paper studies the role of alternative transaction mechanisms on equilibrium patterns of exchange. In particular, it explores the robustness of monetary equilibria to introduction of an improved trade mechanism in a prototypical absence-of-double-coincidence model. This innovation is modeled as a costly matching technology capable of ameliorating the trade frictions by providing deterministic double- coincidence matches. The welfare implications this new trade arrangement has on diverse trading regimes—both monetary and non-monetary—are also examined.
Kiyotaki and Wright (1993) formalize fiat money's medium of exchange function by adopting a search theoretic approach. Money can be endogenously valued if it ameliorates the search frictions stemming from existence of an “imperfect” trading technology, imperfect, that is, when compared to a standard Arrow- Debreu setting. Differentiated goods and pairwise random matching, impair an exchange process which may be thought of as suffering from an extreme degree of spatial separation of spot markets. Since usage of an intrinsically worthless medium of exchange may increase the likelihood of successful exchanges— essentially performing the role of a matchmaker—the model delivers fiat money’s valuation as an equilibrium phenomenon. A similar result characterizes the spatial model of Townsend (1980).
One natural question is whether money's value is susceptible to the availability of a costly trade innovation improving the degree of market integration. One suspects the existence of correlation between the degree of interconnectedness of traders in an economy and the types of assets which are used to facilitate exchange (see for example Townsend, 1983). Resorting to fiat money—thus improving on the existing trade technology—naturally has a beneficial welfare effect. May an improved costly trading technology prove superior to strict reliance on a monetary payment system? Townsend (1983) for example points out the existence of a welfare gain when autarky is replaced by a monetary trade regime. However he also notes that taking steps towards a more integrated financial regime—via a centralized credit-debit system—is welfare improving.
In the present model individuals with diverse tastes try to acquire and sell commodities or money by pairwise exchange. They may choose between two trading sectors characterized by different matching technologies. The search sector has a standard (and costless) bilateral matching technology providing random pairwise matches. Participation in the search process has positive external effects. A costly multilateral matching technology , provides deterministic double-coincidence matches in the intermediated
purely monetary regime proves to be superior to a mixed regime with both monetary exchange and costly trade arrangements.
The paper proceeds as follows. Section 2 presents the model, and section 3 the steady state equilibrium analysis. Welfare considerations are contained in section 4, section 5 extends the model with a more explicit matching technology, and section 6 concludes.
2. Environment The population is constant and there is a unit measure of identical infinitely-lived agents indexed by
j∈[0,1]. Time is discrete and continues forever.^3 Agents have heterogeneous preferences, constant across time, defined over a proper subset x ∈(0,1) of the [0,1] set of differentiated goods that can be produced in the economy. The subset x is agent-specific and it does not include the agent's own production good. That is, x denotes the probability that any trader j consumes the production of a randomly encounter party j'. Contingent on this event, I let y ∈[0,1] denote the probability that the randomly encountered party consumes the production of j. Thus x (1- y ) is the probability of single coincidence and xy is the probability of double coincidence in a random match where someone has a good to offer. By letting y = x , I thus can define the measure x as the proportion of people willing to consume any given good j, independent across time and matches (the special case y =0 is considered in section 5). Consumption of one unit of commodity j generates temporary utility payoff u > 0, if the commodity is in x , and zero otherwise, that is u(qj) = u qI{j∈ x }where, qj^ is the quantity of commodity j consumed. Each agent discounts the future at rate r>0.
Since goods are differentiated in terms of the utility they provide and preferences are not defined over individual output, agents cannot consume in autarky and trade is necessary for consumption to take place.
All individuals are initially located in the search sector of the economy. An exogenously determined fraction m ∈[0,1) is randomly endowed with one indivisible unit of fiat money, the remaining 1- m are endowed with a production opportunity. Agents with money may dispose of it, and costlessly obtain a production opportunity. Since their choice depends on the strategies adopted by the others, I let μ t ∈[0, m ] be
(^3) This is standard (see Kiyotaki and Wright 1989, for instance) and equivalent to a model with continuous time and transactions occurring at discrete points in time.
the endogenous proportion of individuals holding money in the economy at time t. Individual j with a production opportunity can use it once to costlessly produce one unit of indivisible output j. Besides the initial distribution, a production opportunity is obtained immediately after consumption has taken place. Given u >0, time discounting and costless production, production opportunities are used up as soon as they arise.
Commodities and money may be costlessly stored. The inventory technology has an upper bound set (conveniently) to one item. Since holding any inventory excludes obtaining a production opportunity, no one engages in production if carrying a good or money, and no one carries money and a good at the same time. This allows identification of traders according to their inventory. A money trader carries money and a commodity trader carries a commodity. The model is one of complete information and only individuals' trading histories are private information. This and the population assumption rule out the possibility of credit arrangements, the inventory and indivisibility restrictions are for the sake of tractability (they limit the dimensionality of the state space), while the assumptions on preferences and production technology motivate the need for trade.
There are two different matching technologies. Both match agents pairwise, one match per period. The bilateral matching technology matches traders according to a known random process as in Kiyotaki and Wright. By using the multilateral matching technology an agent incurs a cost, in utility terms, at the beginning of t in order to be paired with an appropriate partner during t. The technologies are operated in two different sectors of the economy, the search and the intermediated sector, respectively. In what follows I first introduce some notation and then describe these two technologies in more detail. At the end of each period t , individuals choose in which sector to trade the following period (see figure 1, more in Section 3).
[Figure 1 about here] Let et ∈[0,1] define the probability that, in a symmetric equilibrium, an average commodity trader
participates in the search sector in t +1. This is an endogenous variable, chosen at the end of t and corresponding to what I will later call the commodity trader's equilibrium market strategy. The equilibrium
b( et -1)), or to be unable to trade with his party (absence of mutual agreement) in which case he must wait
until t +1 to attempt a new trade.
Traders can overcome the random matching problem by choosing to use the improved multilateral matching technology, discussed next. At the end of each period t commodity traders decide whether to participate in the intermediated sector in the following period, t +1. In a symmetric equilibrium, the average commodity trader does so with probability 1- et (more in Section 3). Participating in the intermediated
sector generates a per capita cost τ( et ) = τ + C c ( et ) in utility terms, upon entrance in the sector. The
function is continuous with 0≤τ,C<∞, c ( et ):[0,1]→R+, c (0)≤ c (1), c '( et )≤0 (>0) for et ≤ et " ( et > et "), and
c ''( et )>0. Once again, notice that the functional form of the cost is exogenous, but its equilibrium size is
endogenous when C≠0.^6 These costs may be loosely interpreted as the expenses borne when resorting to the services of a trading intermediary. Once an individual has entered the intermediated sector she is matched with a partner who is holding the good she likes, provided she is not the only one to have chosen participation in that sector ( et ≠1). Costless exchange and consumption (followed by production) occur with
certainty at the beginning of the following period, then the agents separate. This specification is adopted to focus on the advantages generated by a technology which, much like money, is capable of lessening the trade frictions stemming from randomness in trade. That is why all other similarities between the two competing matching technologies are preserved (one full period for trade to take place, one match per period, etc.).
3. Symmetric Stationary Equilibria Consider active equilibria in which agents adopt time invariant strategies, the proportion of money traders is stationary and identical types act alike. The choice of trading sector and acceptance of money is
Inclusion of μ would not change the quality of the results, at the expense of clarity of discussion (see Section 3). (^6) While in a symmetric equilibrium the total cost is endogenous, τ( et )(1- et )(1-μ t ), the per capita cost is only a function of the strategy et (if C≠0). This rules out direct effects (on the multilateral matching technology) due to changes in liquidity. The convexity assumption seems to be reasonable for it has increasing returns (from participation in intermediated trade) only initially. One may interpret the subsequent decreasing returns as the consequences of an overcrowded intermediated sector.
made with the objective of maximizing the discounted expected utility from consumption.
Traders have a set of three possible choices in each period: which trading sector to choose (if the agent is holding a good), whether to accept money in exchange for her own inventory (if offered any), and whether to accept a good in exchange for her own inventory (if offered any). Consider the simplifying
assumption that a commodity is not accepted if it cannot be consumed.^7 Define the strategy profile of commodity trader j as σj, a set of state-contingent rules specifying which trading sector will be chosen and the probability of acceptance of money. Let these two components be denoted respectively as the market and the trading strategy. Taking as given everybody else's actions, each period the average commodity trader chooses her market strategy e ∈[0,1], that is the probability—on which individuals have symmetric beliefs—that she trades in the search sector in the following period. Let e j∈[0,1] denote the best response of individual j, so that if all other commodity traders are choosing e then commodity trader j chooses e j. Let π∈[0,1] define the trading strategy, the probability—on which individuals have symmetric beliefs—that an average commodity trader who is in the search sector accepts money. Let πj∈[0,1] denote the best response of individual j. Taking as given everybody else's actions, individual j chooses πj. Finally, let σ≡{ e ,π} and σj≡{ e j, πj}.
Next, consider an agent holding currency. In principle he can choose the trading sector and whether to dispose of money in any period. In a stationary equilibrium, however, money traders trade only in the search sector since monetary transactions do not occur anywhere else. A commodity trader who has been matched by means of the costly technology is assured consumption. By agreeing to a money-for-goods exchange he would have to sustain at least one additional trading round before consumption can take place. Since time is discounted, accepting currency is then a dominated action, thus only commodity traders enter the intermediated sector in equilibrium.
Because of stationarity the individual’s choice of disposing of money is considered only once, at the beginning of time. It determines the stationary fraction of money traders and depends on the trading and
(^7) By incorporating arbitrarily small transaction costs—as Kiyotaki and Wright do—one can rule out commodities being used as money: since goods have identical acceptability ( x ) no good provides exchange advantages over another. Jones (1977), Kiyotaki and Wright (1989) and Oh (1989) discuss the use of commodities as generally accepted media of exchange. Also, two money traders could swap inventory, an inconsequential action for the equilibrium analysis, and so is not considered.
At the beginning of period t , agent j may be in three different “states” depending on her trading history. She may be holding a commodity while being located either in the search or in the intermediated sector, with expected discounted lifetime utilities respectively given by Vs, t and Vi, t. Alternatively, she may be
holding money and be located in the search sector, with expected discounted lifetime utility Vm, t. The
sequence of actions during a trade round in period t is as follows (see figure 1). Immediately after the beginning of t a match is realized: with certainty if the individual is in the intermediated sector, and with probability b( e ) if she is in search. During t matched traders must choose whether to exchange their inventory for their partner's, while unmatched individuals do nothing. Before the end of t , all individuals must also choose where to trade in the following period. Finally, at the beginning of t +1 all individuals who mutually agreed to exchange (during t ) swap inventories, separate, and the recipients of commodities consume and produce. The ones who decided to participate in the intermediated sector enter it and suffer a
utility loss τ( e ). A new trade round then begins.^9
Agents choose the strategy profile which maximizes their expected discounted lifetime utility. In a steady state where individuals act symmetrically and take everybody else's strategies as given, the value functions are given by (derivation in Appendix)
rVs=b( e )pc(σ,μ) x^2 u +b( e )pm(σ,μ) x maxj π πj{Vm- max e j [ e jVs+(1- e j)(Vi-τ( e ))]}
- max e j (1- e j)[Vi-τ( e )-Vs] (4) rVm=b( e )pc(σ,μ) x π{ u + max e j [ e jVs+(1- e j)(Vi-τ( e ))]-Vm} (5) rVi= u I{ e ≠1}+ max e j [ e j(Vs-Vi)-τ( e )(1- e j)] (6)
Equation (4) shows that the expected flow return to a commodity trader in the search sector, has three distinct components. With probability b( e )pc(σ,μ) x^2 she is in a double coincidence match with another
commodity trader (both net u ). With probability b( e )pm(σ,μ) x , she is in a single coincidence match with a
(^9) This timing convention (attempt to get a match during the period, and exchange at the beginning of the following) is standard in the search literature, hence adopted to facilitate the comparison with similar models. It is also qualitatively inconsequential for the results (both matching and exchange could occur in the same period). The choice of sector could also be moved at the beginning of each period. The critical feature here is the presence of
money trader and must choose the probability of accepting the currency offered, πj. By agreeing to the transaction she ends up holding currency and becomes a money trader. Since her best alternative is keeping the commodity, her flow payoff is {Vm- max e j [ e jVs + (1- e j)(Vi-τ( e )) ]}. The third component vanishes for
all market strategies except when the agent strictly prefers trading in the intermediated sector ( e j^ = 0) in which case it represents the flow payoff she derives (Vi-τ( e ) -Vs). Equation (5) has a similar interpretation,
whereas equation (6) shows that a commodity trader who is in the intermediated sector (the fee has been assessed upon entrance, at the beginning of the period), receives utility u with certainty at the end of the trading round, if e <1 (0 otherwise). She can keep using the multilateral matching technology, paying τ( e ). Going back to the search sector is accounted for by the change in her value function, Vs - Vi.
Consider the optimal market strategy when μ and σ are taken as given. The choice of trading sector depends on whether switching sectors provides the commodity trader with a non-negative net payoff. Since moving to the intermediated sector is costly, her optimal market strategy is a decision rule e j^ : τ( e ) → [0,1], mapping the entrance cost into a probability
= τ <
∈ τ =
= τ >
0 () V V s
[ 0 , 1 ] () V V
1 () V V
i
i s
i s j
if e
if e
if e e (7)
In a similar manner, taking π, μ and e as given, a commodity trader accepts money depending on whether becoming a money trader gives her an advantage, when compared to the best alternative offered by her current inventory position. Her optimal market strategy is a decision rule πj: Vm → [0,1], mapping the
expected value from holding money, relative to holding a commodity, into a probability
= < −τ
∈ = −τ
= > −τ π 0 V max{V,V ( )}
[ 0 , 1 ] V max{V,V ()}
1 V max{V,V ()}
m s i
m s i
m s i j
if e
if e
if e (8)
time discounting together with trade frictions stemming from the restriction on coalition formation.
φ<0 and e j=0. Consequently the symmetric equilibrium mixed market strategy is defined on the open set (0,1). Similar considerations can be made for the symmetric equilibrium mixed trading strategy π which, as in Kiyotaki and Wright (1993), is defined on the open set (0,1).
To summarize, in a stationary symmetric equilibrium, given the others' strategies and μ, both the extent of trade “intermediation” and valuation of money are endogenously determined. Outcomes are fully described by the combination of market and trading strategies. They may be pure intermediated ( e =0), pure search ( e =1), or mixed search ( e ∈(0,1)), and—depending on the trading strategy adopted— non monetary (π=0), pure monetary (π=1), or mixed-monetary (π∈(0,1)).
3.2. Benchmark Equilibria Consider a benchmark formulation with fixed entrance cost and positive trading externality by letting C=0, so that τ( e )=τ>0, and b( e )=b e. The symmetric stationary equilibrium value functions, strategies and the proportion of money traders are obtained as follows. A market strategy e is conjectured. Taking it as given, I examine the possible optimal trading strategies π (and the associated μ). To confirm the existence of an equilibrium (i.e. a fixed point), I verify that the proposed e is optimal (i.e., that it satisfies (7)) and derive conditions supporting its existence, for each optimal π. This includes verifying Vs,Vm,Vi≥0 and
checking the sign of φ(σ, μ). This process is repeated for all types of market strategies. The market and trading strategy combinations, the set of equilibria, their nomenclature, and their existence are summarized
in table 1 where I retain the more general notation τ( e ) and b( e ).^11
[Table 1 about here] Case I. Pure Search Equilibria ( e =1) When e =1 all commodity traders stay in the search sector, and the economy resembles the one in Kiyotaki and Wright (1993). From (10) it is immediate that this outcome is always viable since φ(1,π,μ)> ∀π.
Proposition 1. There always exist equilibria with trade taking place in the search sector only. They
can be either monetary or non-monetary.
The three possible outcomes, labeled Sπ, π∈{1, x , 0}, are fundamentally equivalent to the three
equilibria in Kiyotaki and Wright (1993). Pure search equilibria are always possible due to the self- fulfilling nature of beliefs. If individuals believe no one trades in the intermediated sector, expected consumption from trade in that sector is zero. This makes entrance a dominated strategy even when access to an improved matching technology is totally free (τ=0), a coordination failure. As already spelled out in Kiyotaki and Wright money is not valued if agents hold the common belief that no one exchanges a commodity for money (π=0).
Case II. Pure Intermediated Equilibria ( e =0) When e =0 then b(0)=0. If the cost of accessing the multilateral matching technology is sufficiently low then e j=0, so that all individuals holding commodities avoid the search sector. Money traders are then unable to acquire commodities, thus freely dispose of money and produce. This is summarized in the following
Proposition 2. Money is not valued when trade is carried out only in the intermediated sector. The condition τ < u (1+r)-1^ is necessary for existence of this equilibrium, but not sufficient to guarantee money to be valueless in the economy.
Denote the outcome as I (for intermediated) and notice that the belief e =0 is sufficient to support this non-monetary outcome as long as it is consistent, that is if the entrance cost is smaller than the discounted temporary utility from consumption, u (1+r)-1. In what follows I will refer to this necessary requirement as a feasibility condition for intermediation to arise. The advantage to the society from using the multilateral matching technology is twofold. It eliminates the output loss due to the use of money (money drives out production opportunities) and it reduces the search costs, by speeding up the transaction process. While feasibility of τ is necessary for the existence of the pure intermediated equilibrium, it is not sufficient to guarantee money to be valueless in the economy: by Proposition 1 random search monetary equilibria are always possible. The coexistence of these two corner equilibria ( e =0,1) explains why the possibility of resorting to a deterministic and inexpensive matching technology may deprive money of value, although not
(^11) Proofs and algebra in Appendix. The examples' baseline is b( e )=b e , τ=0.89, x =0.3, m =0.35, r=0.05, u =1, b=0.9.
value of the difference between the (certain) payoff derived from intermediated trade ( u ) and the (expected) maximum payoff attainable from search trade ( u b x^2 ), to be smaller than the disutility suffered by resorting to costly matching (τ).
When commodity traders accept money with probability x , a unique e generates the mixed monetary equilibrium M x where search and intermediated trade coexist. The proportion of money traders, μ, must
now be taken into account in determining the conditions for the existence of equilibria. The cost of obtaining a match must exceed the present value of the (maximum) difference between the utility payoffs for the two sectors: u (intermediated sector) minus u b x^2 (1-μ) (search sector with full participation). Similar considerations can be made for pure monetary equilibria with intermediation, M 1 , which are also uniquely
determined.
The existence of each equilibrium is affected by trading externality, search frictions, and liquidity level.
The set defined by (^)
τπ , (^1) +r u (^) shrinks with increasing difficulty of search. As b or x fall, the set of τ's
supporting a mixed market strategy equilibrium shrinks to a singleton, u (1+r)-1. Individuals face a tradeoff between τ and the degree of differentiation of goods, x. If cheap intermediary services are available, some traders may search only if the bilateral matching technology promises a high likelihood of a match ( x large). Conversely, even a rather expensive deterministic matching technology may be attractive when tastes are
highly diverse ( x small).^12
This is illustrated in figure 2 which, for the baseline case, depicts the set of feasible {τ, x } pairs supporting the different types of interior outcomes.
[Figure 2 about here] The horizontal line originating at point A delimits the set of feasible τ, by marking the upper bound u (1+r)-1. All parameterizations lying in the space above that line support only the corner outcome with random search, e =1 (monetary or not). Below the horizontal line there exists a multiplicity of equilibria,
(^12) One can verify τ 1 ≤ τ x , τ 0 ≤ τ x , and τ 0 ≤ τ 1 if x ≥(1-μ)/(2-μ). If π= x money is least effective and τπ must be larger (vs. π=1) to "penalize" more intermediated trade. Similarly, τ 0 ≤ τ x since meeting commodity traders is less likely with money circulating. Last, τ 0 ≤ τ 1 for large x (money is least beneficial). In figures 2-3, τ x always binds first.
monetary and non monetary: the two corner outcomes e =0,1 always exist, and interior outcomes e ∈(0,1) may exist. The regions (1) through (5) indicate where interior outcomes exist and, contingent on that, whether they can be monetary or not. The {τ, x } pairs lying in area (1) (above the curve τ x ) support all
interior equilibria, monetary or not, since τ satisfies (11) for all π. The opposite is true in area (4), where no equilibrium e ∈(0,1) exists. Partial participation in intermediated exchange (sometimes monetary, sometimes not) is an equilibrium in the remaining areas. In region (2), between the τ 1 and τ 0 curves, money
and intermediated trade cannot coexist, and only M 0 exists. The opposite occurs in the area between the
curves going through points A and B (M 0 does not exist, but Mπ does for π= x ,1). Pure monetary and non-
monetary interior equilibria exist in area (5), enclosed by the three curves τ 0 , τ 1 , and τ x. The role of double
coincidence and costs for the existence of the different types of interior equilibria can be best understood by moving, alternatively, vertically and horizontally in the picture. Fix τ. As x increases the set of equilibria with some participation in intermediated exchange grows. While no such equilibria exist when the double coincidence problem is severe, the M 0 equilibrium first comes to exist as x grows, while coexistence of
money and intermediated trade (M 1 , and M x subsequently) is also possible as x heads further towards 1.
Now fix x and move south starting from the horizontal line originating at A. Coexistence of money and intermediated trade is possible for τ large. As τ becomes smaller, first the mixed monetary, and then the pure monetary outcomes, M x and M 1 , cease to exist. As τ shrinks further the only interior equilibrium
which still exists is non-monetary, but it disappears as τ becomes sufficiently low. That is, while money coexists with mediated trade when the latter is feasible but expensive, costly deterministic matching is strictly preferred to monetary exchange when the former is sufficiently cheap.
Figure 3 illustrates coexistence of money and intermediated trade for different money supplies. [Figure 3 about here]
As in figure 2, the areas of existence of different equilibria are marked and enclosed between the three curves τπ and the horizontal curve defining feasibility of τ. High liquidity levels reduce the set of τ's
supporting coexistence of intermediation: τπ→ u (1+r)-1^ as μ converges to one. An increase in money
negatively affects the probability of meeting a commodity trader. At high liquidity levels this also offsets the beneficial effects of monetary trade, hence only a larger τ keeps individuals indifferent between the two
depend on the size of e (whether it is located in the decreasing or increasing returns region of the cost
function).^14 Similar considerations apply to the case of a trading externality with decreasing returns on part of its domain. Condition (11) is sufficient (but not necessary), potentially multiple mixed market strategy equilibria exist, and the set of costs supporting them is smaller (b(1)<b implies τπ is larger than the
benchmark case). The comparative static results depend on the size of e but for a different reason: high levels of participation in search trade may now be the source of negative feedback for commodity traders in search.
Furthermore, one may consider a trading externality that is not only a function of the strategy profile (hence of the mass of commodity traders in search), but of the measure of all traders searching, including the ones holding money. While changes in the amounts of liquidity—under this different modeling choice— would affect both the liquidity level and trade externality, the conditions for existence of equilibria and the
comparative statics results (concerning changes in m ) would not.^15 Finally if the utility u were derived at the same time τ( e ) is incurred, the discounting factor r would not affect the upper bound of τ in Mπ
equilibria and changes in r would not influence e for π={0, x }. In such a case only the probability of a match vs. the cost of sure trade matters (either money is not used or traders are indifferent towards it). The equilibrium e would fall with higher r for π=1 and a positive trading externality. Obtaining money has a net positive payoff but the money cannot be used before one period has elapsed, so increased impatience lowers the lifetime utility of commodity traders in search; e must fall for the externality's feedback help maintaining indifference between the trade sectors.
4. Welfare Considerations To rank the outcomes, consider the ex-ante lifetime utility of an individual defining the welfare measure
( u - τ) < -b ex^2 u implies τ<τ(1+r), at the equilibrium e. (^14) Consider π=0 and a larger x benefiting search: e must drop when the trading externality is positive. However if τ( e ) is convex, the cost may increase for lower e. These diverse effects on matching costs—due to changes in e — will reflect differently on the relative value of lifetime utilities, and hence will differently affect the sign of ∂ e /∂ x. (^15) Substitute Q( e ,μ)=μ+(1-μ) e (the mass of search traders) instead of e in the trading externality. An increasing and low μ would improve even more the random matching process (vs. the original modeling choice). Hence e or τ would have to fall even more to restore the indifference balance. The opposite occurs for high μ.
W(k)=μVm+(1-μ)Vs, for e ∈(0,1] and k∈{Sπ,Mπ}, and W(I)≡(1+r)Vs= u^ −τr(^1 +r) for e =0.^16 I first
examine monetary and non-monetary outcomes separately, in order to compare welfare across different market strategies, and then compare outcomes across trading strategies (see table 2).
[Table 2 about here] A. Non-Monetary Regimes. Consider the benchmark model and notice that welfare is independent of the money supply when equilibria are non-monetary. For example in a pure search equilibrium W(S 0 )≡Vs=b x^2 ( u /r), i.e. the infinite
sum of temporary utilities from consumption from the second period of life on. When a positive trading externality exists, the following is shown
Proposition 4. Consider non-monetary equilibria. If the costly matching technology is sufficiently inexpensive, welfare is highest when all individuals participate in intermediated exchange. Otherwise welfare is highest when there is full participation in random search.
Only if the alternative trading mechanism is sufficiently expensive random matching is superior to deterministic matching. In that case W(S 0 )>W(M 0 )=W(I), and notice that welfare in the outcomes with
active intermediation (M 0 and I) is similar due to both the indifference across trading sectors and the
absence of money. The positive participation externality is instrumental since W(S 0 )<W(M 0 ) could result if
large participation in search activities were to produce a negative feedback (b( e ) strictly concave, and b( e )>b(1) on part of the domain). To illustrate the proposition consider a social planner maximizing welfare by directing traders in the sector providing the largest net payoff from trade. By construction all traders are in search before the initial period. They would consume for sure in period two if they opted for the intermediated sector in period one, while they would consume with probability b x^2 if they searched in period one. The planner recommends entrance in the intermediated sector (in t =1) if the expected discounted utility
(^16) This amounts to computing the expected lifetime utilities of a trader before the distribution of money takes place, at the beginning of t =1. Observe that μVm + (1-μ)[ e Vs+ (1- e )(Vi-τ)] is the ex-ante lifetime utility, equivalent to W(k) for both e =1 and e ∈(0,1) since Vi-τ=Vs. For e =0 the ex-ante lifetime utility is W(I)=−τ+Vi≡Vs(1+r) since all traders enter the intermediated sector in period one (suffering -τ), and start consuming only in the second period of life. Their lifetime utility is -τ plus the infinite discounted sum of net utility u - τ from period 2 on.
