A Search Theory of Money and Commerce with Neoclassical Production

Transcription

1 September 10, 2002 A Search Theory of Money and Commerce with Neoclassical Production Miquel Faig Department of Economics University of Toronto 150 St.George Street Toronto, Canada M5S 3G7 mfaig@chass.utoronto.ca Abstract This paper advances a tractable model with search theoretic foundations for money and Neoclassical production. The distinctive feature of the model relative to earlier work is that manufacturing and commerce are two separate activities. In manufacturing, goods are efficiently produced combining capital and labor. In commerce, goods are exchanged in bilateral meetings. The model is applied to study the effects of inßation on capital accumulation and welfare. Inßation has the following effects: (1) it reduces the congestion of buyers in the market, (2) it increases steady state output and capital, (3) it reduces the diversity of purchased goods, and (4) for realistic parameters it has a negative effect on welfare. Keywords: search, money, commerce, inßation, Neoclassical production, capital accumulation, optimum quantity of money. JEL ClassiÞcation codes: E40, E52, E13. This paper was elaborated while visiting the Universitat Pompeu Fabra in Barcelona during the academic year I beneþted greatly from reports of two referee and the editor Valerie Ramey. Also, I am grateful to to Randy Wright, Neil Wallace, Richard Rogerson, Shouyong Shi, Victor Rios-Rull, Xavier Cuadras, Jordi Gali, Belén Jerez, Mara Berman, and the attendants to the various seminars where this paper was presented for their comments. All errors and omissions are my own. 1

2 1 Introduction Most exchanges of Þnal products take place in a retail trade sector where buyers and sellers interact in bilateral meetings. This important feature of reality is abstracted from in models with Walrasian markets. In contrast, monetary search models build precisely on the assumption that trade takes place in bilateral meetings in order to study the properties of the media of exchange and to provide a motivation for the existence of money. 1 For tractability, initial research in the monetary search paradigm assumed indivisible money and abstracted from capital accumulation. In recent contributions, Shi (1997, 1999, and 2001), Lagos and Wright (2001), Aruoba and Wright (2002), and Molico and Zhang (2002) have provided tractable models that can incorporate divisible money, capital accumulation, or both. This paper builds on the earlier work of Shi, but separates the manufacturing sector where goods are produced from the commerce sector where goods are exchanged. Hence, the model advanced in this paper has an explicit commercial sector whose only activity is the exchange of goods in bilateral meetings in imitation of what we observe in retail trade. In a series of papers, Shi (1999 and 2001) integrates capital accumulation into a monetary search model. These papers adopt the construct of large households introduced in Shi (1997) in order to circumvent the difficulties caused by non-degenerate distributions of money and capital across individuals. 2 In these models, households allocate some members to buying goods (buyers), some to selling goods (sellers), and some to seeking leisure (leisure seekers). Each seller is given a portion of the capital stock which is an essential input in the production of goods. Similarly, each buyer is given a portion of the money available to the household. Buyers and sellers are matched bilaterally in a market place where everybody is anonymous. To establish the terms of trade, when a buyer meets a seller with a desirable product, the buyer makes a take-it-or-leaveit-offer, Shi (1999), or the buyer and the seller enter in a sequential bargaining game, Shi (2001). When an agreement that is mutually advantageous is reached, the seller produces a good which is immediately exchanged for the money of the buyer. The goods purchased by buyers can either be consumed or invested. The model of this paper follows Shi in assuming the existence of large households, the use of 1 An additional key assumption is the anonymity (i.e. private histories) of buyers. See Kocherlakota (1998). 2 See Molico (1998) for a search model that incorporates divisible money and goods without the representative household device. Also, see Taber and Wallace (1999) and Zhou (1999) for search models where individuals hold non-degenerate inventories of indivisible money. 2

3 capital and labor in the production of goods, the random matching of anonymous buyers and sellers, and the use of money as the medium of exchange. However, the model here differs fundamentally from Shi in that it assumes that production and selling activities are performed by distinct individuals, to be respectively denoted as producers and sellers. Producers combine their labor with the capital of the household to produce goods. This activity takes one period to be completed. At the beginning of each period, the stock of goods inherited from past production and investment is divided into two parts. One part is used as capital by the producers. The other part is the commercial inventories given to the sellers to exchange for money. In the basic version of the model, all sellers in a household share their commercial inventories, so that they can avoid ending a period with unsold commercial inventories. In the more sophisticated version of the model, sketched at the end of the paper, sellers can only sell the inventories they individually carry thus allowing for the possibility of unsold commercial inventories. The present model also differs from Shi in other, more subtle, characteristics. Firstly, sellers set prices instead of prices being set by buyers or through a bargaining process. Secondly, buyers have private information about their preferences. Thirdly, money is used only for the purchase of consumption goods excluding capital. One major advantage of the separation of the production and the sale of goods is that the properties of the search technology and the mechanisms for determining the terms of trade in bilateral meetings can be related to data from retail trade. For example, this paper uses the average commercial margin in retail trade to calibrate the model. An additional advantage of the separation of production and exchange activities is that it facilitates the tractability of several issues. For example, this separation allows us to combine into one model interesting elements, such as capital accumulation, price-setting by sellers, a preference for variety of goods, and commercial inventories, which until now had only appeared separately in the monetary search literature. 3 In the present model, price setting by sellers is compatible with a monetary equilibrium because sellers face buyers with unknown preferences. Consequently, buyers, in general, retain a fraction of the trading surplus even if sellers have the power to set prices in a take-it-or-leave-it offer. The preference for variety allows for an interesting endogenous 3 See Soler-Curtis and Wright (2000) and Camera and Delacroix (2001) for models where sellers set prices. Soller- Curtis and Wright assume that sellers set prices and focus on the coexitence of at most two prices for the same good in equilibrium. Camera and Delacroix focus on the endogenous determination of the terms of trade mechanism, that is price setting versus bargaining. Laing, Li, and Wang (2000) present a search monetary model with a preference for the variety of goods. Finally, Shi (1998) and Wang, Liu, and Shi (2000) study search monetary models with inventories of Þnal products. 3

4 determination of the size of transactions. In equilibrium, households balance the desire for diversity against the cost of performing extra transactions as the number of goods consumed increases. Finally, it is straightforward to introduce in the present model the possibility of unsold commercial inventories by assuming that each seller can only sell the commercial inventories that he carries along. 4 In its simplest form, examined in section 4, the seller either sells all or nothing of this inventory. As shown there, the opportunity cost of unsold inventories is a determinant of both the fraction of goods purchased and the congestion of buyers in the market (the fraction of buyers among traders). Furthermore, in future research the model could be enriched to allow for multiple matches in one period in which case sellers would have an interesting stock-out avoidance motive for holding commercial inventories. The efficiency properties of the model s equilibrium can be summarized as follows: (1) Buyers accept purchasing too few goods because they do not take into account the sellers surplus in making their decision. (2) The congestion of buyers is too low because of the market power that the sellers exercise through their ability to set prices. (3) Output is too low because these two inefficiencies discourage market production. (4) As in the standard Neoclassical model, the steady state capital-labor ratio employed in the production of goods is efficient and is uniquely determined by the condition that the net marginal product of capital is equal to the subjective discount rate. This last property holds due to the separation of production and sales into two sectors, and it is in sharp contrast to earlier contributions. In Shi (1999 and 2001), the utilization of capital and labor employed in the production of goods depends on how successful seller producers are in Þnding desirable matches with buyers. Hence, the marginal product of capital depends on market conditions, that is the composition of market participants and their willingness to trade. This is a crucial difference between those models and the model here, and it is responsible for many of the contrasting results. SpeciÞcally, this property is essential for Shi s extensive effect of inßation on total factor productivity and hence the marginal product of capital. In contrast, in the present model, even if market conditions have important effects on the pricing of goods and the amount of output produced, neither inßation nor market conditions have any effect on total factor productivity in the goods producing sector. In the present model, inßation increases the opportunity cost of the money carried by buyers while they search for a desirable match with a seller. This extra cost has two direct effects on incentives: First, buyers become more willing to accept a particular trading offer to speed up the 4 This possibility was suggested by one of the referrees. 4

5 circulation of money. 5 In the model, this is the counterpart of searching more intensely, which is found in some other search models such as Li (1995), Shi (1997), and Berentsen, Rocheteau, and Shi (2001). Second, the return from being a buyer falls with the extra cost of carrying money, so households send fewer buyers to the market and increase the number of individuals performing all other tasks. Because of the general equilibrium nature of the model, the Þnal outcome of inßation depends on the reactions of other households to the two direct effects mentioned above. In particular, sellers increase prices when they realize that buyers are more willing to accept offers. In the absence of unsold commercial inventories, this pricing response is such that buyers end up purchasing the same fraction of goods at all inßation levels. With unsold commercial inventories, the pricing response is not as strong, so the fraction of goods purchased ends up increasing with inßation. After all general equilibrium adjustments, inßation has the following effects: Inßation reduces the measure of buyers and increases the measures of home workers, sellers, and producers. Consequently, the economy moves to a steady state with more output and more capital. Despite this abundance of output, the effect of inßation on steady state welfare is theoretically ambiguous. Inßation has two welfare beneþts. First, it increases output and capital. Second, in the version of the model with unsold inventories, it increases the fraction of purchased goods. However, inßation also has two welfare costs. First, it reduces the diversity of the goods consumed as a result of the households decision to send fewer buyers to the market. Second, it reduces the congestion of buyers, which reduces the efficiency with which matching takes place in the commercial sector. In a numerical calibration of the model, inßation is found to be detrimental to welfare. The paper is organized as follows. Section 2 analyzes a simple version of the model in which the sellers of a household share their commercial inventories. Section 3 analyzes the effects of inßation on capital accumulation and welfare and describes the welfare properties of an equilibrium. Section 4 examines a more complicated version of the model in which the sellers of a household do not share their commercial inventories, so they face the risk of carrying unsold commercial inventories when they fail to Þnd a desirable match during a period. Finally, section 5 concludes with a summary of the main results and their robustness to several extensions of the model. 5 In more detail, inßation reduces the demand for money, which depresses the value of the current stoock of money, which in turn makes buyers more willing to accept a particular trading offer. 5

6 2 The Basic Model The economy is populated by a [0, 1] continuum of inþnitely lived households who produce and consume differentiated goods. Each household is composed of a large (inþnite) countable number of individuals. 6 Subsets of individuals in a household are measured by the fraction of their size over the size of the household, so the measure of individuals in a household is normalized to be one. Time is discrete. Households do not consume the goods they produce so they need to trade. In addition to consuming goods, households enjoy home services, or equivalently leisure activities, which can neither be traded nor stored. In each period, the individuals of a household are assigned to one of four different tasks: production of market goods (producers), production of home services (home workers), purchasing of commodities (buyers), and sale of market goods (sellers). No individual can perform more than one of these tasks in the same period. Assume that the production of market goods, the production of home services, and the exchange of goods are performed in different locations. Furthermore, the goods consumed and the goods produced by a particular household are traded in different markets. However, consumption of both goods and home services is shared equally by all members of the household so there is no contradiction between the objectives of a household and any of its members. The exchange of goods is performed in decentralized markets where buyers and sellers interact. All trades in the market are bilateral and must be mutually beneþcial to both parties. Market participants are anonymous so exchanges must be quid pro quo. The goods produced by a household either have to be traded for others to consume in the same period or have to be invested as productive capital by the same household who has produced them. This assumption is convenient to study equilibria with Þat money because it precludes the use of goods as media of exchange. 7 There are H typesofhouseholdsandh typesofgoods(indexedbothbyh H), where H 3. Each good h comes in a continuum of varieties (indexed by i [0, 1)) distributed around a unit circle. Household hi produces good h variety i and can consume all varieties of good h +1 modulus 6 The reason for assuming that the number of individuals in a household is countable is to ensure that all buyers in a household purchases different varieties of the good consumed almost surely. This assumption plays a role in equations (4) to (6). 7 Goods that are not invested as capital last for a single period. Goods that are invested as capital cannot simultaneously by carried by buyers as a means of payment. When goods are assumed to be durable in all conditions, the fact that production if done prior to exchange allows for an endogenous determination of the media of exchange as in Kiyotaki and Wright (1989). 6

7 H. However, it does not like all these varieties the same. For concreteness, let us say that household hi likes variety i of good h +1 the best. The utility of the other varieties declines the further apart they are from i. Households are evenly spread over the set of goods and varieties they produce. Each one of the H goods has a different market place where all varieties of that good are traded. Since H 3, two households are unable to mutually satisfy their consumption needs in a barter exchange. The absence of barter combined with the impossibility of goods being used as media of exchange implies that all exchange in this economy must use money. 8 Money consists of storable objects, referred to as dollars, that are useless for consumption or production. These objects are perfectly divisible and can be created without cost by the government, who has the monopoly to do so. 2.1 Households Decisions Without loss of generality, I describe the actions of household hi. In this description, I adopt the following notation. Lower-case letters denote household s hi decision variables. Upper-case variables denote the decisions of other households, which are taken as given by household hi. In a symmetric equilibrium, lower-case letters are equal to the corresponding upper-case letters. A typical day in the life of household hi proceedsasfollows. Inthemorningofdayt, the household starts with a given stock of good hi to be denoted a t. This stock is divided in two piles. One pile is allocated to be the capital stock k t, used for the production of market goods. The other pile is the commercial inventories v t which are transferred to a warehouse to which all the household s sellers have access. The individuals of the household are then assigned to be producers, n t, home workers, l t, buyers, b t, or sellers, s t. The producers of market goods use the capital stock, k t, to generate the output that is going to be available the next day. The production technology forces the packaging of output in units of identical size q t+1, although households can optimally choose this size. 9 (For example, consider a newspaper where the number of pages is endogenous but must be the same for all units). Sellers go to the market for good h and, if they meet a buyer, they announce a take-it-or-leave-it offer. These offers specify the quantity of good to be supplied, q t (predetermined by the production of the day before) and the monetary payment demanded, z t (hence, the output price is p t = z t /q t ). Offers are made without prior knowledge of the most 8 Buyers cannot issue a debt promise or pledge future returns on the capital of the household they belong to because they are anonymous. 9 This assumption greatly simpliþes the analysis. See Faig and Jerez (2002) for a version of the model where this constraint is eliminated and sellers can precommit to a price schedule. 7

8 preferred variety of the buyer faced in a meeting. 10 Each buyer gets an equal share of the money of the household and travel to the market for good h +1. Upon meeting a seller, they observe the variety for sale, receive an offer (Q t,z t ), and decide whether to accept the offer or not. Home workers perform the home services without any need of capital. In the evening, all the individuals of the household get together, and equally share in the consumption of the market goods purchased and home services produced during the day. Buyers and sellers are matched according to a matching technology speciþed by the two functions B and S. Letφ t be the congestion of buyers in the market: φ t = B t B t + S t. (1) The function B : < 3 + [0, 1] maps b t and φ t onto the measure of buyers in the household paired with a seller of good h +1. This seller carries with equal probability any one of the varieties of good h +1. Analogously, the function S : < 3 + [0, 1] maps s t and φ t onto the measure of sellers in the household paired with a buyer of good h. This buyer prefers with equal probability any one of the varieties of good h. For concreteness, I assume that all individuals in a market are randomly paired once and only once each day, so applying the Law of Large Numbers we have that B (b t, φ t )=b t (1 φ t ), and S (s t, φ t )=s t φ t. (2) The model could be easily generalized to allow for alternative functional forms for the matching technology. The objective of the household is to maximize the utility from the consumption of goods and home services. This utility is additively separable with discount factor β: X β t [U(c t )+V(l t )], 0 < β < 1. (3) t=0 The function U : < + < is assumed to be logarithmic to greatly simplify the algebra. The function V : < + < mapsthefractionofhomeworkersinthehouseholdintotheutilityfrom the services they produce (or equivalently the leisure they enjoy). This function is increasing, concave, differentiable, and V 0 (0) =. The variable c t is a hedonic measure of consumption that 10 The trade meeting is assumed to proceed in such a way that the preferred variety of the buyer is not revealed. For example, this is the outcome if the buyer moves Þrst with an action that consists in either showing an unequivocal mark that reveal his type or declining to do so. Next, the seller makes an offer. Finally, the buyer decides accepting the offer or not. Given this sequence of events, the seller captures the whole trade surplus if he infers the type of the buyer. Consequently, the buyer has no incentive to reveal his type. 8

9 combines the Kiyotaki and Wright (1991) preferences with the Dixit-Stiglitz aggregator. measure depends on the quantities acquired q jt by each buyer j in the household and the variety acquired in each purchase: 11 JX c t = j=1 (1 θ jt ) q 1 σ jt J σ This, σ (0, 1). (4) Here, J is the number of buyers of the household matched with a seller, and J 0 is the number of individuals in the household. The parameter σ measures the preference for diversity. When σ 0, the preference for diversity vanishes. The variable θ jt is the distance (measured clockwise) between the variety acquired by j at t and the most preferred variety. The household gets maximum utility when consuming the most preferred variety, in which case θ jt =0. This utility declines linearly to 0 as the variety consumed is further away from their most preferred variety. Since we assume that J 0, the deþnition of c t in (4) can be conveniently simpliþed. The Law of Large numbers implies lim J 0 J/J 0 = B (b t, φ t ). Therefore, c t = B (b t, φ t )lim JX J j=1 (1 θ jt ) q 1 σ jt J 1 1 σ. (5) Moreover, in a symmetric equilibrium where all sellers from the other households offer the same pair (Q t,z t ), the variable q jt takes only two values: Q t if j is a buyer accepting an offer at time t and 0 otherwise. In this environment, the optimal strategy for a buyer is to accept offers for which θ jt is not greater than a reservation distance x t and reject all other offers. Because buyers and sellers are randomly matched, θ jt is the realization of a random variable uniformly distributed from 0 to 1. Consequently, applying the Law of Large Numbers to (5), we obtain c t = Q t B (b t, φ t ) Z xt = Q t B(b t, φ t )x t µ σ (1 θ)dθ 1 x t σ. (6) Hence, the hedonic measure of consumption c t is proportional to the quantity purchased in each transaction Q t.also,c t is a strict convex function of the measure of purchases B(b t, φ t )x t made by the household. This convexity is the result of the preference for variety implicit in (4). Finally, c t declines with the average distance of the goods purchased, x t /2, to the most desirable variety. 11 These preferences combine those in Kiyotaki and Wright (1991) with the Dixit-Stiglitz aggregator. 9

10 The stock of goods available at the beginning of period t is equal to the capital surviving from the previous period plus the newly obtained production: a t = k t 1 (1 δ)+y t, δ (0, 1), (7) where δ is the depreciation rate. Production depends on the capital and labor employed in the previous period: y t = F(k t 1,n t 1 ). (8) The function F : < 2 + < + maps capital and labor used in period t 1 onto the output obtained in period t. This function is assumed to be continuously differentiable, increasing in both arguments, concave, and homogeneous of degree one. Also, the Inada conditions for an interior solution are assumed to apply. The stock of goods available at the beginning of the period is divided into capital k t and commercial inventories for sale v t : a t = k t + v t. (9) Because there is no aggregate uncertainty in a household and all sellers share their commercial inventories, the whole stock v t is sold during period t, sov t must be equal to sales (or commercial expenditures) in period t. (See section 4 for an extension of the model with unsold commercial inventories.) Sales in a period are equal to the buyers contacted by the sellers of a household times the fraction of buyers accepting the offer (q t,z t ) and times the quantity of goods sold in a transaction. Therefore, v t = S(s t, φ t ) X t (q t,z t )q t, (10) where the function X t : < 3 + [0, 1] maps the vector (q t,z t ) onto the fraction of buyers in the market of good h that are expected to accept an offer (q t,z t ) in period t. The function X t is assumed to be increasing in q t,decreasinginz t, and continuously differentiable. As we will see, these assumptions about X t are consistent with the buyers behavior in a symmetric equilibrium. The household must satisfy the following monetary budget constraint: m t+1 = m t + τ t + z t X t (q t,z t )S(s t, φ t ) Z t x t B(b t, φ t ) (11) The money holdings at the beginning of t +1, m t+1, are equal to the money holdings at the beginning of t, m t, plus the monetary lump-sum transfer received at the beginning of t, τ t, plus the revenue from sales minus the money spend in purchases. Money balances can only be positive: m t

11 The measures of individuals allocated to the four different tasks in the household must add up to one: b t + s t + n t + l t =1. (12) In addition, these measures must be non-negative, and the measure of buyers is limited by the money available at the beginning of the period: b t Z t m t. (13) That is, each buyer must be given at least the quantity of money Z t for his/her activity to be of any use. The household hi maximizes (3) subject to constraints (7) to (13). Constraint (12) can be eliminated by substitution of 1 b t s t n t for l t into the other constraints. Likewise, expressions (7), (8), and (10) can be substituted into (9) to form a single goods resource constraint. Therefore, we are left with the following set of choice variables {x t,z t,q t,n t,s t,b t,k t,a t+1,m t+1 } t=0 that must satisfy the constraints (9), (11), and (13) and the non-negativity of all variables. Using Lagrange multipliers λ t β t,µ t β t, and ν t β t for these three constraints, the Þrst order conditions for an optimum when the non-negativity constraints are not binding are: " U 0 (c t )c σ t U 0 (c t )c σ t Q 1 σ t 1 σ (1 x t)=λ t Z t, (14) µ t q t X tz (q t,z t )=λ t [X t (q t,z t )+z t X tz (q t,z t )], (15) µ t [X t (q t,z t )+q t X tq (q t,z t )] = λ t z t X tq (q t,z t ), (16) Q 1 σ t 1 σ µ t+1 βf n (k t,n t )=V 0 (l t ), (17) (λ t z t µ t q t ) X t (q t,z t )φ t = V 0 (l t ), (18) Ã! # λ t Z t x t x t x2 t 2 (1 φ t ) ν t Z t = V 0 (l t ), (19) µ t+1 β [1 δ + F k (k t,n t )] = µ t, (20) λ t 1 =(ν t + λ t ) β and ν t (m t b t Z t )=0, and (21) lim t β t µ t k t =0. (22) Condition (14) equates the utility from and the cost of accepting an offer when the variety of the good for sale is at the reservation distance x t from the variety the household likes most. Conditions (15) and (16) equate the marginal costs of increasing q t and z t to the marginal revenues 11

12 these increases generate. Condition (17) equates the value of the marginal productivity of labor in the production of goods and the production of home services. Condition (18) equates the surplus generated by a seller to the cost of the seller s labor, which is equal to the marginal utility of home services. Condition (19) equates the utility of the purchases made by the marginal buyer net of the value of the money spent and the opportunity cost of holding this money to the opportunity cost of the buyer s time. Condition (20) equates the discounted value of the marginal productivity of capital to the value of a good today. Condition (21) equates the cost of acquiring one dollar yesterday with the discounted beneþts this dollar brings today not only for its purchasing power but also for allowing extra buyers into the market. Finally (22) is a standard transversality condition. 2.2 Equilibrium DeÞnition: The set {x t,q t,a t,k t,y t,v t,b t,s t,n t,l t,m t,z t } t=0 equilibrium (equilibrium for short) if is a diversiþed symmetric monetary 1. These paths solve the household optimization problem. That is, they maximize (3) subject to all choice variables being non-negative and the constraints (7) to (13) being satisþed for a given a 0 and given paths for the set of variables {Q t, Z t, B t, S t, and τ t } t=0. 2. Sellers have rational expectations about the acceptability of their offers. SpeciÞcally, the function X t (q t,z t ) is consistent with the choice of x t by households. 3. Aggregate variables are consistent with individual optimization: Q t = q t, Z t = z t, B t = b t, and S t = s t. 4. The fractions n t,l t,s t, and b t are all positive (diversiþcation inside a household applies). 5. Money has value: q t > 0. As is common in monetary search models, there is a trivial non-monetary equilibrium where 1 to 3 in the previous deþnition hold but 4 and 5 do not. That is, if individuals believe that money will have no value next period, the solution to the household s optimization problem is to revert to autarchy with x t = q t = v t = b t = s t = n t = k t = y t =0, and l t =1. For a combination of interest and tractability, this section focuses on equilibria where money has value and risk is diversiþed at the household level, so 4 and 5 hold. In such equilibria, the Þrst order interior conditions (14) to (22) apply. These conditions together with the constraints (7) to (13) and symmetry deþne a system of difference equations that 12

13 determines the equilibrium path. The complexity of this system depends on monetary policy. When the government adjusts the transfer τ t to achieve a constant and positive opportunity cost of holding money, this system simpliþes to two difference equations with similar dynamics to those of the Neoclassical model. To solve for an equilibrium, the Þrst step is to Þnd the function X t. 12 Without loss of generality, consider a seller from household h0 meeting a buyer from a household of type (h 1) whose most desirable variety is at a distance X t +,where is sufficiently close to zero. If the buyer s household accepts the varieties in the interval [0, ] with the terms of trade (q t,z t ), then the marginal increase of utility net of the marginal cost of money spend on these purchases is the following expression: " Z # 1 Xt U 0 (C t ) C 1 σ t + B t φ t qt 1 σ + 1 σ (1 i)di) C t B tφ t Λ t z t (23) X t Dividing this expression by and taking the limit as 0, we obtain that (23) is non-negative ifandonlyif X t 1 (1 σ) Λ t Ct 1 σ qt σ 1 z t. (24) Since the buyer s household accepts all trades which satisfy the above inequality, we have X t (q t,z t )=1 (1 σ) Λ t Ct 1 σ qt σ 1 z t. 13 (25) Equation (25) implies that the derivatives of X t obey: q t X tq (q t,z t )=(1 σ)[1 X t (q t,z t )] and z t X tz (q t,z t )= [1 X t (q t,z t )]. Substituting these expressions in (16) and (15), and using the fact that in an equilibrium X t (q t,z t )=x t, we obtain: λ t z t µq t λ t z t = σ, and (26) x t = σ 1+σ. (27) As expected, given the isoelastic aggregator (4), the commercial margin, λ t z t µq t,isaconstant fraction, σ, ofthevalueofsales,λ t z t. Moreover, the endogenous fraction of purchased goods, x t, is constant that depends only on σ in equilibrium. DeÞne the opportunity cost of holding money as R t =[1 δ + F k (k t 1,n t 1 )] (1 + π t ) 1, where π t = p t /p t 1 1. Using conditions (20), (21), and (26), this opportunity cost obeys: R t = λ t 1 λ t β 1=ν t. (28) λ t 12 I am thankful to an anonymous referee that suggested the formal derivation of X t that follows. 13 With the functional form of X t in (25), it is easy to check that the second order conditions for the households optimization problem are satisþed. 13

14 Therefore, as long as R t is positive, condition (13) holds with equality. Using (2), (14), (18), (26) and (28), the Þrst order condition (19) in a symmetric equilibrium simpliþes to: x t (1 φ t ) 2(1 x t ) R x t t = φ t. (29) 1 x t Hence, 1 x 1 2R t t x φ t = 2 t. (30) 1+2σ 1 xt x t Since, in equilibrium, x t is constant that belongs to the interval (0, 1), the congestion of buyers, φ t, is a decreasing function of R t. This function will be denoted φ, thatisφ t = φ(r t ), φ 0 < 0. Intuitively, an increase in R t increases the opportunity cost of the money buyers must carry to purchase goods, so an increase in R t reduces the return of being a buyer (LHS in [19]). As a result, households decide to send fewer buyers to the market relative to the number of sellers they are sending. A negative relation between φ t and R t is also found in Shi (2001). However, in Shi (1999) inßation increases the congestion of buyers in the market in a comparison across steady states. This result is due to the extensive effect that inßation has on the return on capital in Shi s model. Due to this effect, higher inßation gives an incentive for households to increase their consumption, which can only be accomplished by increasing the number of buyers. These increases are sustainable in a steady state because the increase in the number of buyers raises the fraction of sellers that employ the capital and labor at their disposal. These effects are precluded here because of the separation between the production and the exchange of goods into two different sectors. The fact that in equilibrium the fraction of purchased goods x t is constant, and so independent from the opportunity cost of holding money R t, is counterintuitive. When R t increases, the value of money, λ t and Λ t, falls. Consequently, households have an incentive to accelerate the velocity of their money holdings by accepting more offers (see 25). 14 However, the acceptance of trade offers depends also on the terms of trade (q t,z t ). In equilibrium, sellers are aware that with a higher R t buyers are more willing to accept an offer. As a result, they offer less desirable trading conditions to the buyers they encounter, to the point that x t remains invariant from R t. The constancy of x t is very convenient to simplify the analysis of the model. However, it is not a robust feature. The fraction of purchased goods x t depends positively on R t in the variation of the model studied in section,4 where commercial inventories cannot be shared among all sellers of a household. 14 Increasing the measure of buyers does not help to accelarate the velocity in which money is spent because buyers do not share their money holdings. Consequently, each buyer must be provided with the money necessary to accept an offer. 14

15 Using (27) to (28), we can combine and simplify the Þrst order conditions (14), and (18) to (20), to obtain the following system of equations: 15 V 0 (l t )s t = 2 σ 2+σ 1 σ, (31) V 0 F n (k t,n t ) (l t )= 1 δ + F k (k t,n t ) µ t, (32) v t = 2 2+σ µ 1 t, and (33) q t = v t s t φ t x t (34) Equations (31) and (32) together with (30) and the resource constraints (9) and (12) determine the equilibrium values for b t, s t, l t, n t, v t and k t as a function of µ t, R t, and a t. Using these values and (30), equilibrium q t is determined by (34). To complete the dynamic system that determines the equilibrium path, we need the laws of motion for R t, µ t, and a t.iassumer t to be the target of monetary policy, which manipulates τ t to achieve a predetermined path for R t.thelawsofmotionforµ t and a t are determined by (20) and a combination of (7) and (8): µ t+1 = µ t [1 δ + F k (k t,n t )] 1 β 1 and (35) a t+1 = F(k t,n t )+k t (1 δ). (36) Finally, the two side conditions to determine the equilibrium path are the transversality condition (22) and the initial stock of goods a Existence of an Equilibrium The existence of an equilibrium is easily proved using the method that Stokey, Lucas, and Prescott (1989) denotes as the indirect approach. The strategy of this indirect approach is to prove the existence of the optimal path of a pseudo-economy that matches the equilibrium path we are interested in. The following proposition formalizes this idea. A sketch of the proof is in the Appendix. 15 Equation (31) is obtained combining (6), (14), (18), and (26), and using symmetry, (2), and (27) to simplify the resulting expression. Equation (32) is obtained combining (20) and (17). Equation (34) is obtained combining (6), (10), (14), and (26), and using symmetry, (1), (2), and (27) to simplify the resulting expression. 15

16 Proposition 1 Anequilibriumwheremoneyisonlyheld as a medium of exchange exists if 0 < R t < σ 2 / [2(1 + σ)] for all t 0 and a 0 (0, ba), where ba is implicitly deþned by δba = F(ba, 1). Moreover, the equilibrium path for {v t,b t,n t,l t,s t,k t,a t+1 } t=0 in our model is identical to the optimal path of a pseudo-economy with the following characteristics: There is a representative consumer whose preferences are: where X β t [ω 1 ln(v t )+ω 2t ln(b t )+V(l t )], (37) t=0 ω 1 = 2 2+σ ω 2t = 2 2+σ σ 1 σ and (38) 1 1 φ t. (39) The feasible paths are constrained by (7) to (9), (12), φ t satisþes (30), and s t = b t φ t /(1 φ t ), and the given initial stock of goods a 0. The condition 0 <R t is necessary to ensure that constraint (13) is binding and so money is only held for the purpose of giving it to the buyers to purchase goods. The condition R t < σ 2 / [2(1 + σ)] is necessary to ensure that b t and s t are positive. Proposition 1 not only proves the existence of an equilibrium but also provides a simple method to calculate it. To Þnd an equilibrium path, we simply need to solve a simple optimization program. 16 This procedure should be especially useful for stochastic extensions of the model. 2.4 Dynamics For tractability, the analysis of the equilibrium dynamics are limited to the case in which R t is constant. In each period, the system of equations (9), (12), (27), (29), (31), (32), (34), and (33) determines the endogenous variables x t,q t,b t,s t,l t,n t,k t, and v t as a function of µ t and a t. Using the implicit function theorem, the following proposition follows. (See the analysis of this dynamic system in the Appendix). Proposition 2 The signs of the derivatives of the most important endogenous variables of the 16 To Þnd the utility of the representative household, we must use (3) and (6) once the equilibrium path is found because utility levels differ between the two economies. 16

17 model with respect to a t and µ t are summarized in the following panel: Endogenous Variables x t q t b t s t l t n t k t v t y t+1 b t /s t k t /n t Derivative with respect to: µ t 0? ? a t For a given initial stock of goods a t, an increase in the value of goods produced µ t induces a shift of labor away from home services and into both market production and trading. Also, an increase in µ t leads to a decline of goods for sale in favor of capital. For a given value µ t,anincreaseina t is fully employed as capital. This attracts labor into the production of market goods from all other activities because of the complementarity between k t and n t. Moreover, to continue exchanging the same amount of goods with fewer traders, the size of each transaction must increase. The dynamic system has two stationary equations, one for µ t+1 = µ t and the other for a t+1 = a t : F k (k t,n t ) δ = β 1 1 (40) F(k t,n t )=v t + δk t (41) Using the implicit function theorem, we can characterize the slope and the properties of the two lines described by these two equations (see the Appendix). In the plane (µ t,a t ), the line (a t+1 = a t ) is downward sloping. The line µ t+1 = µ t may be upward or downward sloping, but in any case its slope surpasses that of the line (a t+1 = a t ). It can be shown using standard arguments that the two lines cross once and only once. The phase diagram of the system is represented in Figure 1. As the Þgure shows, the system is saddle path stable, and the stable arm is downward sloping. 17 In the steady state, where the two stationary lines cross, the marginal product of capital is equal to the subjective discount rate. During the process of accumulation of the stock of goods a t, capital increases, and the value of an extra good produced µ t falls. During this adjustment, both the portion of goods destined for sale and consumption increase, the measure of individuals engaged in trade, b t and s t, fall, and the size of each transaction, q t, increases. The effect of capital accumulation on the measure of producers and home workers is ambiguous because the changes in a t and µ t tend to move these variables in opposite directions. 17 Because the present model is in discrete time, convergence may not be monotonic. For monotonic convergence the smallest eigenvalue of the dynamic system must be between 0 and 1. In general, checking this condition is analytically intractable. In the special case that V is linear, this condition can be easily checked around a steady state, and it is satisþed. In all numerical examples computed by the author with a logarithmic V, convergence is also monotonic. 17

18 2.5 The Demand for Money In an equilibrium where money is only wanted as a medium of exchange, (13) holds with equality. As in a cash-in-advance economy, the maximum quantity buyers can purchase in period t is limited by the money held from period t 1 to period t. However, in the present model only a fraction of this money is spent in equilibrium. To spend their money, buyers must Þnd a suitable seller that makes an acceptable offer. For this reason, the velocity of money with respect to the quantity of goods actually purchased by households is not constant. Using (2), (10), (13), and the deþnition of p t, we obtain: Velocity at t p tv t = x t (1 φ m t ). (42) t The velocity of money with respect to v t (sales or consumption expenditures) depends on how willing buyers are on departing from their most preferred variety and how easy it is for them to Þnd a seller. Using (27) and (30), the following proposition follows. Proposition 3 The velocity of money with respect to consumption expenditures in period t is an increasing function of the opportunity cost of holding money from period t 1 to period t: Velocity at t = σ 2 1+σ + R t. (43) 1.5σ Since x t is constant, R t increases the velocity of money because it reduces φ t, so buyers Þnd sellers more easily. In section 4, R t also affects the velocity through an increase in x t. 3 The Optimum Quantity of Money and Welfare This section analyzes substantive issues of monetary theory in an economy with Þat money. Specifically, it studies the interaction between the rate of growth of money and capital accumulation, the optimum quantity of money, the welfare properties of a monetary equilibrium, and the welfare costs of inßation. 3.1 Money and Capital Accumulation Money in the model is neutral. A once-and-for-all increase in the quantity of money distributed equally to all households increases monetary payments made in transactions proportionately without any real effect. Money, however, is not superneutral. Changes in the rate of growth of the money supply induce changes in the rate of inßation and thereby in the opportunity cost of holding 18

19 money, R. This subsection investigates how changes in R affect capital accumulation and the other real variables in the economy. For tractability, this analysis is centered in a comparison across steady states in which case the changes in R are equal to the changes in both the rate of growth of the money supply and the rate of inßation. For brevity in notation, time subscripts are dropped. In the system of equations that describes an equilibrium, the opportunity cost of holding money only enters equation (29). As implied by this equation, the direct effectofanincreaseinr is a drop in the congestion of buyers φ. This drop has an indirect effect on the other variables of the model as summarized in the following proposition. (For the proof see the comparative statics across steady states in the Appendix): Proposition 4 When V is strictly concave, the signs of the derivatives with respect to R in comparisons across steady states are: Endogenous Variables x q b s l n k y v a µ k/n s/n Derivative with respect to R An increase in R induces households to reduce the fraction of buyers they send to the market as explained in 2.2. The labor that is freed as a result of the drop in the number of buyers is spread over the three alternative activities in the model: s, l, andn. Since the ratio k/n remains equal to the subjective discount rate, capital, output, and sales increase in the same proportion as n. Consequently, an increase in R due to an increase in the rate of growth of the money supply leads to an increase in the capital stock. The quantity exchanged at each trade, q, isincreasingwithr. In the present model, when R increases buyers are more willing to accept trading offers and sacriþce variety of the goods purchased. Sellers exploit this by forcing them to purchase larger quantities at higher prices. For this result, it is crucial that sellers must sell the same quantity at the same price to a set of heterogeneous buyers. 18 Curiously, when V is linear, the model presents the Neoclassical Dichotomy property emphasized in Aruoba and Wright (2002). That is, inßation has no effect on capital, output, consumption expenditures, and employment in the production and the sale of goods. The only effect of inßation 18 Using the arguments in Berentsen, Rochetau, and Shi (2001), one can show that the quantity exchanged at each trade would be decreasing with R, as is common is earlier monetary search models, if the terms of trade were determined by a generalized Nash bargaining game with complete information. 19

20 is a shift of labor from purchasing goods into home services, which induces a sacriþce in the variety of goods consumed (b falls and l and q increase in such a way that qb and b + l are unaffected). This property applies not only to the steady state but also to the transitional paths The Optimum Quantity of Money This subsection investigates the conditions for the optimality of Friedman s (1969) prescription to reduce the opportunity cost of money to zero. Using (1), (2), (3), and (6), together with symmetry, the effect of a small increase in R on the one period utility of a representative household, to be denoted W,is: dw = 1 q dq σ + V0 dl. (44) Using the implicit function theorem results used to elaborate Proposition 4, we obtain the following derivative as long as R is positive (see the comparative analysis across steady states in the Appendix): dw dr = [(σs b)(b + s)+σsn)] + V 0 [V 0 b (b + s)(1 σ) σs] V00 b (1 φ)(1 σ)[v 0 V 00 φ 0. (45) (1 l)] The denominator in (45) is always positive and φ 0 < 0. Hence, the condition for R 0 20 to maximize steady state utility is a negative numerator. The second summand of the numerator can be shown to be always negative(see the Appendix). Hence,a sufficient condition for the numerator to be negative is: V 00 [(σs b)(b + s)+σsn] 0. (46) Condition (46) is satisþed in the case that σ 0.5 because s 2b (see equation [30]) and V Consequently, dw/dr is negative if the desire for diversity is strong (σ 0.5), the supply of labor is perfectly elastic (V 00 =0), or the total number of traders (b + s) isnottoolargerelativeto producers (n). Conversely, for the derivative dw/dr to be positive, we require a low commercial margin σ. Moreover, we require a large fraction of traders relative to producers. When σ is low, the fraction of traders relative to producers can only be large if traders take a long time to meet one another, that 19 For steady state comparisons, this property follows from (95). For transitional paths, note that the line µ t = µ t+1 infigure1ishorizontalwhenv 00 =0, so µ t is always at the steady state value. Therefore, µ t is independent from R t, and so the Neoclassical Dichotomy follows from the system (27) to (34). 20 When R =0, the constraint (13) is not binding which opens the possibility to multiple equilibria. With R 0, I indicate that of these equilibria I use the one where money is not held as store of value for the comparisons across steady states in this section. 20

21 is if the length of the period is long. For example, with the baseline parameters of the following subsection (except for σ and T ), the length of the period must be over 78 years (T < 1/78) for dw/dr > 0 if the commercial margin is a minute The length of the period must be over 544 years for the same purpose if the commercial margin is 0.1. For realistic commercial margins 0.2 and over, there is no period shorter than one million years that yields dw/dr > 0. As long as R remains positive and dw/dr is negative, reducing R improves steady state utility. Transitional dynamics further reinforce the beneþts of reducing R because this reduction leads to an economy with a lower capital stock. Therefore, with a lower R households enjoy not only a higher steady state utility, but also the beneþt ofhavingtoinvestlittleduringthetransitiontothe new steady state. 3.3 Optimal Allocations This subsection studies the welfare properties of the equilibrium paths described in section 2. To this end, it characterizes the symmetric optimal path that maximizes the utility of a representative household when the decisions about the extent of search x t, production q t, investment (k t k t 1 ), and labor allocation (b t,s t,n t,l t ) are made by a benevolent central authority. Following standard practice in this literature, this central authority is not bound to using money in exchange. However, it is restricted by the resources available in the economy, the technological constraints in production, and the bilateral matching among traders. The symmetric optimal path maximizes the following objective: " Ã!# 1 X β t q t M(b t,s t ) x t x2 1 σ t + V(l t ) 2, (47) ³ where M(b t,s t )=B b t, t=0 ³ b t b t +s t = S s t, b t b t +s t is the probability that a buyer meets a seller in the appropriate market. The following two constraints bind the optimal path: F(k t 1,n t 1 )+(1 δ) k t 1 q t x t M(b t,s t ) k t =0and (48) 1 l t n t b t s t =0, (49) together with the non-negativity constraints on the choice variables: x t,q t, b t,s t,n t,l t, and k t. Denoting µ t β t the Lagrange multiplier for constraint (48), the Þrst order conditions for an interior solution of the optimal paths are: U 0 (c t )c σ t q 1 σ t 1 σ (1 x t)=µ t q t, (50) 21