Search Frictions on Product and Labor markets: Money in the Matching Function

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1 Search Frictions on Product and Labor markets: Money in the Matching Function Etienne LEHMANN y CREST, Université Catholique de Louvain and IZA Bruno VAN DER LINDEN z Université Catholique de Louvain, FNRS, ERMES and IZA April 17, 2008 Abstract This paper builds a macroeconomic model of equilibrium unemployment in which rms persistently face di culties in selling their production and this a ects their decisions to create jobs. We show that the Competitive Search Equilibrium is not e cient and that in ation is detrimental to unemployment. Keywords: Equilibrium unemployment, Matching, In ation, Demand Constraints. JEL Code: E12, E24, E31, J63. We thank the editor William A. Barnett, the anonymous associated editor, the anonymous referee as well as participants at seminars in IIES-Stockholm University, ERMES-Paris 2, Università Cattolica del Sacro Cuore di Milano, Universitat Autònoma de Barcelona, CREST. We thank in particular Aleksander Berentsen, Pierre Cahuc, Lars Calmfors, Melvyn Coles, Jacques Drèze, Peter Fredriksson, John Hassler, Mathias Hungerbühler, Guy Laroque, Carsten Krabbe Nielsen, Torsten Persson, Guillaume Rocheteau, Henri Sneessens, Christopher Waller and Fabrizio Zilibotti for their valuable comments and suggestions on a rst draft of this paper (Lehmann and Van der Linden 2006). The authors acknowledge nancial support from the Belgian Federal Government (Grant PAI P6/07, Economic Policy and Finance in the Global Economy: Equilibrium Analysis and Social Evaluation ). The usual caveat applies. y CREST-INSEE, Laboratoire de Macroéconomie, Timbre J360, 15 Boulevard Gabriel Péri, Malako Cedex, France. etienne.lehmann@ensae.fr z Departement d Economie, Université Catholique de Louvain, 3 place Montesquieu, B1348, Louvain la Neuve, Belgium. bruno.vanderlinden@uclouvain.be 1

2 I Introduction In any market, search-frictions exist and make the matching between trading partners costly and time-consuming. This is particularly true for the analysis of labor markets where search frictions are at the root of the simultaneous presence of unemployed individuals looking for a job and of vacant jobs looking for workers (Mortensen and Pissarides 1999 and Pissarides 2000, henceforth MP). However, since the seminal paper of Diamond (1982), many economists believe that matching frictions matter on the product markets too, even if the problem might quantitatively be less severe than in the labor market. When a buyer and a seller start a transaction, one may be interested in the quantity of goods they exchange (the intensive margin). This is the question addressed by most of the literature (see e.g. Kiyotaki and Wright 1993, Shi 1997, Lagos and Wright 2005 or Berentsen Menzio and Wright 2008 (henceforth BMW)). The literature has however paid scant attention to the number of transactions (the extensive margin). This is surprising since matching frictions should rst of all a ect this number. In this paper, we enrich the MP model by studying the extensive margin due to matching frictions on the product markets. Frictions on the product market are captured by a matching function. This function relates the ow of goods purchased to two endogenous quantities: The stock of rms inventories (aggregate supply) and the consumers total resources available to buy goods (aggregate demand). This matching function is a convenient way of summarizing congestions on the product market. 1 It allows accounting for persistent di culties rms have to sell their production. These di culties are due to structural frictions in the trading technology, and not to price sluggishness that generates only transitory disequilibria, as in the New-Keynesian literature. An increase in aggregate demand reduces congestion externalities from the point of view of sellers and eventually increases the number of transactions. We assume that produced goods are exchanged against money because of a double coincidence problem of needs. 2 Therefore, consumers total resources available to buy goods equal the purchasing power of aggregate money holdings. We hence de ne tightness on the product market as the ratio of real money balances over rms inventories. As tightness increases, rms sell their production more rapidly while consumers need more time to nd goods. Aggregate demand, thereby tightness on the product market, are made endogenous through the determination of the price of goods. Various assumptions could be made about price formation on the product market. We are aware that the prices of some goods 1 Wasmer and Weil (2004) introduce a similar matching function on the credit market. 2 We also assume that transactions are anonymous to make money essential as a medium of exchange (Kocherlakota 1998). 2

3 are bargained over. However, as BMW, we think that price posting with directed search de ects some critiques to alternative mechanisms. A Competitive Search Equilibrium mechanism determines prices and thereby the endogenous level of tightness on the product market. 3 The equilibrium level of prices is the one that maximizes the private value of a good produced and hence the one that minimizes the level of equilibrium unemployment. We show that the competitive search equilibrium leads to an ine ciently high level of prices, thereby an ine ciently low level of aggregate demand. Because of the double coincidence problem, the private return of a good produced is obtained after two transactions on the product market: The entrepreneurs have rst to sell their production and second to buy goods with the money made by their sales. From a social viewpoint however, only a single transaction between a producer and a consumer generates utility. This discrepancy implies that prices at the competitive search equilibrium are too high. Intuitively, the posted prices maximize the private value of a produced good. The latter increases with the probability that money holdings (from the sales) nd an appropriate good to consume. Now, if prices are set below their equilibrium levels, aggregate demand increases, more transactions are conducted and hence social welfare increases. However, such a price deviation reduces the value of money holdings because it reinforces search-congestion from the viewpoint of consumers. This is the reason why a welfare-improving reduction in prices below their price-posting level decreases the private value of a good produced. We show that this ine ciency result remains in the case where money supply grows at a constant rate. Higher monetary growth implies higher in ation at the steady state. In the absence of a successful meeting, a given amount of current money holdings will be depreciated next period and so will enable to express a lower level of demand. Therefore, it is currently less valued by consumers. Higher in ation thus reduces rms incentive to create jobs and increases unemployment in the long run. In other words, our model exhibits an upward-sloping long-run Phillips curve, and money is not superneutral. Moreover, higher in ation increases the ine ciency of the competitive search equilibrium, since it increases the loss for the consumer in the absence of a successful meeting. We nally show that only the Friedman rule, according to which money supply growth rate is negative, can restore the social optimum. Other papers have already stressed that rms face di culties in selling their output because of search frictions on the product market. The seminal paper, Diamond (1982) considers a barter economy. Because the matching technology implicitly exhibits increasing returns to scale, there are multiple equilibria. Howitt (1985) replicates the argument 3 On the labor market, we stick on the usual assumption that wages are bargained over. 3

4 of Diamond in a model that distinguishes labor and product markets. Diamond (1984) extends Diamond (1982) to a monetary economy. Kiyotaki and Wright (1993) explain how the existence of frictions on the product market gives a convincing micro-foundation for the use of money as a medium of exchanges. Money has an essential role to play because of three things. 1) A double coincidence problem arises in an environment where trade is decentralized and time-consuming; 2) There is a lack of commitment and 3) there is private information concerning trading histories (Kocherlakota 1998). However, in these papers, money is indivisible and agents either own zero or one unit of money. Monetary policy issues require the introduction of divisible money, so the whole distribution of money holdings typically. These models can only be solved numerically (see Molico 2006) unless some simplifying assumptions are introduced. Shi (1997, 1998) considers a model where a large family is composed of di erent members and resources are pooled. Each of them produces, consumes and trades in decentralized frictional markets. But at the end of each period, money is redistributed equally across household members. Lagos and Wright (2005) propose an alternative. Agents act individually but in addition to frictional markets, there exists a frictionless market. Moreover, individuals have quasilinear preferences over the good exchanged in the latter market. Therefore, they leave this market with the same amount of money. These alternatives enable to reduce the distribution of money holding to a mass point. We propose another modeling strategy. In our model, any consumer can costlessly and simultaneously search for several goods but they cannot default on paying. So, her real money holdings imposes an upper limit on her level of demand. Then, a rise in individual money holdings eventually raises linearly the ow of purchased goods. This linearity makes the distribution of money holdings useless to de ne the equilibrium. In Kiyotaki and Wright (1993), Shi (1997), Lagos and Wright (2005), Molico (2006) and BMW among others, the probabilities of nding a trading partner on the decentralized product market are exogenous and only the intensive margin of exchange matters. Shi (1998) and Berentsen Rocheteau and Shi (2007) deal to some extent with the extensive margin by introducing costly search e ort. In Rocheteau and Wright (2005), the probabilities of nding a trading partner are functions of the relative numbers of buyers and sellers. An alternative is proposed by Diamond and Yellin (1990) where rms supply all their inventories. In this model, each worker can only search for a single unit of good at a time and only if she owns enough money. In our model, what matters is not only the number of agents but also their economic weight on the product market. The weight of each seller is her level of inventories, so only aggregate inventories matters as in Diamond and Yellin (1990). We treat consumers symmetrically. The weight of each consumer is her 4

5 purchasing power, so only total real money holdings matter. There have been di erent models where the growth rate of money supply has permanent e ect on unemployment. Friedman (1977) and Cooley and Hansen (1989) provide some empirical support to this conclusion. While non-employment is driven by labor supply decisions in this paper, Cooley and Quadrini (1999, 2004) introduce search-matching frictions on the labor market. However, in these three papers, money is introduced thanks to a Cash-In-Advance assumption. Shi (1998) introduces matching frictions on the labor market in his model with large families. BMW and Lehmann (2006) investigate the superneutrality of money on in ation in models with search unemployment and microfounded use of money along the monetary-search paradigm. They also nd that higher in ation increases unemployment. There are no search frictions on the product market in Lehmann (2006), unlike BMW. In their paper, the probability to nd a trading partner on a decentralized product market is exogenous while the amount of goods exchanged is endogenous. Conversely in our paper, the intensive margin of transaction in the product market is exogenous while the extensive margin is made endogenous through our matching function on the product market. The paper is organized as follows. The assumption of matching frictions on the product market is motivated in the next section. The environment is presented in Section III. The xed price equilibrium is described in Section IV. Assuming exogenous price is not a very relevant assumption for a steady-state equilibrium. We make this assumption essentially in order to build a general model step by step. Endogenous pricing is introduced in section V, through price posting. This model is further extended in Section VI, where money grows at a constant rate. There, we show that money is not superneutral. Section VII challenges the e ciency of the equilibrium, and the last section concludes. II Motivating matching frictions on the product market Do rms and consumers search respectively for clients and goods in an environment characterized by imperfect information? If they do, there is a similarity with what is going on in the labor market. There, many labor economists have argued in favor of a matching function on the basis of a similar environment for job vacancies and job-seekers. Empirical studies of search behavior and search costs on the product market can be divided in two groups according to whether they focus on online (i.e. internet) or o ine markets. On the latter, the number of stores visited by US customers for the purchase of a major appliance is reported to be about three (Beatty and Smith 1987). Sorensen (2001) uses detailed transactions data for 111 drug products sold by retail pharmacies in local markets in the US. He strongly rejects the hypothesis that the search probability can 5

6 simply be constrained to equal one. Moreover, consumers propensities to price-shop (i.e. phone to get price quotes) vary in a way consistent with search theory. On the internet, it is widely agreed that consumers search costs are reduced. Examining the online shopping behavior during the period from July 1997 through June 1998, Johnson et al (2004) look at the number of sites visited during a given session. On average 1.1 stores were searched for books, 1.3 for CDs and 1.8 for air travel tickets. The period covered by this analysis is however the early stage of e-commerce. Collecting data in the same way but looking at the period from July to December 2002, Zhang et al (2006) observe a sharp increase in the number of stores visited, ranging on average from 2.1 for a purchase of CDs to 3.3 for air tickets. The same number was found for computer hardware products. Hong and Shum (2006) consider homogeneous products (major books in economics and statistics). They collect their prices on shopbots (online price search agents or shopping robots ) in February Assuming that data arise as an equilibrium outcome of a non-sequential search model, they are able to estimate consumers search costs. For the Stokey-Lucas (1989) text, 52% of purchasers have search costs exceeding $2.32, while over 60% of the Lazear (1990) book purchasers have search costs exceeding $1.31. Given the prices of these books (respectively $66.6 and $35.7 on average), these costs appear to be nonnegligible. The rest of the sample however does not search in the sense that they purchase at the rst store they visit (probably, the one they are used to and/or which has the reputation of completing transactions without a hitch). Imperfect information a ects rms behavior, too. Diamond and Yellin (1990) consider capitalists who hold inventories. They assume that each worker and each unit bundle of consumer goods search at random for each other (p. 931). In addition, observed economic behavior provide clear evidence that rms spend considerable amounts of money on trying to attract new customers via advertisements for a wide audience. If rms had perfect information about whom is interested by their products, they could increase pro ts by replacing these advertisements by information targeted to the right consumers. 4 Even if price search in the internet has substantially reduced search frictions, Ellison and Ellison (2005) forcefully argue, both theoretically and empirically, that rms engage in practices that frustrate consumer search or make it less damaging to them. In sum, bilateral search on the product market is an empirically relevant market imperfection to consider. 4 On the interaction between advertising and search, see Robert and Stahl (1993) and Janssen and Non (2008). 6

7 III Environment Time is discrete. All agents are risk neutral, live forever and have a common discount rate r > 0. The analysis is conducted in steady state. Money is storable, divisible and is the unique medium of exchange. There is no credit market. 5 Trades in the labor and in the product markets are uncoordinated and time-consuming. We assume two types of agents: entrepreneurs and workers. There is a continuum of both types. Workers are either employed or unemployed and their total measure is normalized to 1. While employed, their earnings allow them to accumulate money holdings. Whether employed or unemployed, they use their money holdings to express a demand for produced goods. There is a large and exogenous number of entrepreneurs. They are simultaneously employers, sellers and buyers. Entrepreneurs post an endogenous number of vacancies. Posting a vacancy entails a ow disutility cost. An entrepreneur recruit only unemployed individuals. 6 A job is created when a vacancy matches an unemployed worker and the Nash bargain over the wage leads to an agreement. Produced goods are indivisible, storable and heterogeneous. Each good is unique and cannot be reproduced. Put di erently, a given type of good can only be produced once and in quantity one. During a period, a lled job produces a (large) number q of di erent goods. These produced goods are added to the rms inventories and supplied on the product market. Individuals, whether they are workers or entrepreneurs, derive utility whenever they nd a good they are looking for. Neither an entrepreneur nor an employed worker is interested in consuming the goods she has produced. Therefore, each entrepreneur has to supply her inventories on the product market, searching for a buyer that will be interested in one of the goods she is supplying. There is a double-coincidence problem: When an entrepreneur nds a good to consume, the producer of this good is not interested in consuming what the former entrepreneur produces. Trade is anonymous and there is no record keeping technology. Therefore, money is used as a medium of exchange (see Kocherlakota 1998). In case of a meeting, the money made by the sale is used for two purposes: To pay the worker and to express a demand on the product market. 5 The absence of credit in our model, which is coherent with the assumption that transactions are anonymous, is standard in the monetary search litterature (see e.g. Lagos and Wright 2005 or BMW). For an introduction of credit in the monetary search framework, see Berentsen Camera and Waller (2007). 6 We rule out the possibility that an entrepreneur becomes self-employed or recuit another entrepreneur. We also ignore on-the-job search. 7

8 IV The Fixed Price model In this section, the prices are xed on the product market while wages are bargained over. A given period is divided between a labor market (production) sub-period and a product market (consumption) sub-period. During the labor market sub-period, Firms decide whether or not they open vacancies. The aggregate number of vacancies at that moment is 7 1. Job destruction and Job creation take place simultaneously. The wage W is then bargained over. The wage is actually a kind of piece rate in the sense that it is proportional to the (random) quantity of goods sold in step 2c. Since workers and entrepreneurs are risk-neutral, they bargain over the expected (monetary) wage E (W ). In stage 1d each lled vacancy produces q units of goods. This ends the labor market subperiod. The product market sub-period starts with entrepreneurs supply decisions. Then, workers and entrepreneurs express their demand decisions. Trade takes place on the product market. Finally, entrepreneurs use part of the money made by the sales to pay their employees (see Figure 1). Labor market Product market 1a: Vacancy decisions 1c: Wage bargaining 2a: Supply decisions σ 2c: Trades T(Σ,D) 1b: Job Creation H(u,υ) and Job Destruction s 1d: Production q per job 2b: Demand decisions d 2d:Wage payments Figure 1: The timing of events in the Fixed Price Model This section rst describes the functioning of the labor (IV.1) and of the product (IV.2) markets. The emphasis is on the way we model frictions. Then, we turn to the behavior of workers (IV.3) and the one of entrepreneurs (IV.4). Next, we characterize the labor market equilibrium (IV.5) and nally the xed price equilibrium (IV.6). IV.1 Job Creation and Job destruction We start by describing the job creation and job destruction processes in the labor market. Firms open 1 vacancies during stage 1a. At the beginning of stage 1b, there are u 1 unemployed workers and 1 u 1 employed workers. We de ne tightness on the labor 7 In all the paper, values at the past (next) period are indexed 1 (+1): 8

9 market as the ratio 1 = 1 =u 1. To form a job, an unemployed worker and a vacancy have to meet. Let H (u; ) be the number of meetings between u unemployed workers and vacancies. The following assumption is standard in the labor-matching literature. Assumption AS 1 The (labor market) matching function is assumed to be continuously di erentiable, increasing and concave in both arguments, to yield constant returns to scale and to verify the following boundary properties for any (u; ): H (u; 0) = H (0; ) = 0 lim H (u; ) = u and lim H (u; ) = 7!+1 u7!+1 The monotonicity and boundary conditions imply that for any (u; ), H (u; ) min (u; ). Each vacant job meets an unemployed worker with probability 1 h () H ; 1 H (u; ) = This job lling probability h (:) is a decreasing function of. It varies from 1 to 0 when increases from 0 to +1. probability Symmetrically, each unemployed worker nds a job with h () H (1; ) = H (u; ) u = u H (u; ) This job nding probability is an increasing function of and varies from 0 to 1 when increases from 0 to +1. We assume that each job ends with an exogenous probability s 2 (0; 1). Hence, in each period, s (1 are created. Unemployment therefore evolves according to: u) jobs are dissolved and H (u; v) = u h () jobs u = u 1 + s (1 u 1 ) H (u 1 ; 1 ) = s (1 u 1 ) + (1 1 h ( 1 )) u 1 In steady state, the labor market ow equilibrium implies: u = s s + h () So, characterizing is su cient to obtain the steady-state value of unemployment. The tighter the labor market, the lower the unemployment rate. (1) IV.2 The working of the product market Trade in the product market requires a meeting between a consumer and an entrepreneur (a seller). With a certain probability, this meeting leads to trade, that is to the exchange of p units of money against a single unit of good. For the consumer, this exchange generates a utility level normalized to unity and measured at the end of the period. Consider an entrepreneur who owns units of inventories at Stage 2a of a given period. We assume that an entrepreneur can simultaneously supply many di erent goods. Let 9

10 be the number of goods she supplies. An entrepreneur is not allowed to supply goods that she has not yet produced. We therefore impose: 0 (2) We treat consumers in a symmetric way. Consider a consumer with money holdings m 0 at Stage 2b of a given period. This consumer can be an entrepreneur, an employed worker or an unemployed one. We assume that any consumer can simultaneously search for di erent types of goods. Let d 0 be the number of goods she is currently searching for or equivalently the number of units of demand she expresses. Expressing a unit of demand means searching for a certain type of good. Under our matching technology on the product market (to be described), there is always a positive, if small, probability that all units of demand will be satis ed. Furthermore, we assume that when a consumer nds the good she is looking for, she cannot default on paying. Hence, by demanding d units of goods, a consumer must be able to spend p times d units of money. In the absence of a credit technology, the demand expressed by a consumer has to satisfy: 0 p d m (3) Modelling the meeting process between demand for and supply of heterogeneous goods is beyond the scope of this paper. Our objective is instead to build an analytically tractable model that incorporates trading frictions in both markets. As each good is assumed to be unique, product market frictions do not qualitatively di er from those observed on the labor market. We therefore model trades in the product market and in the labor market in a similar way. Let and D be respectively the total amounts of supply and demand expressed at the beginning of stage 2c. Constraints (2) and (3) imply at the aggregate level that S and D M=p, where S and M denote respectively aggregate inventories and total money supply. We assume that the number of trades is given by a product market matching function T (; D). We introduce a matching e ectiveness parameter denoted 0, with 0 2 [0; 1]. This parameter is an implicit argument in function T (; D). Basically, one has T (; D) = 0 (; D), where function (:; :) is independent to 0. Assumption AS 2 The (product market) matching function T (:; :) is assumed to be continuously di erentiable, increasing and concave in both arguments, to exhibit constant returns to scales, and to verify the following boundary properties for any (; D) : T (; 0) = T (0; D) = 0 lim T (; D) = 0 D and lim T (; D) = 0!1 D!1 10

11 Since 0 1, for any (; D), we have T (; D) min (; D). Let = D= denote tightness on the product market. The demand (satisfying) probability 8 is de ned as the probability according to which a unit of demand is satis ed: 1 () T ; 1 T (; D) = D As the product market becomes tighter, the congestion-search externality makes the demand probability decreasing from 0 to 0. Symmetrically, the supply (satisfying) probability is de ned as the probability at which a unit of inventory is sold: () T (1; ) = T (; D) = D T (; D) D As the product market becomes tighter, supplied goods are sold more quickly. So, the supply probability increases from 0 to 0. Hence, for a given tightness, () and () are linear in 0. As 0 increases, matching in the product market become easier. We de ne the elasticity of the product market matching function as " () 0 () = () " () 0 () = () : (4) From above, we get that " () 2 (0; 1). The following technical assumption will appear very useful: Assumption AS 3 " () is nondecreasing. An example of matching function that satis es AS2 is the CES speci cation: with a low elasticity of substitution, i.e: h i T (; D) = D This speci cation further veri es Assumption AS3 when 2 (0; 1). Each of the 1 u jobs produces q units of goods per period. The ow of produced goods is therefore (1 u) q. A ow of T (; D) units of supply is purchased. Finally, through a depreciation process, we assume that each unit of inventory is dissolved with probability 0. This depreciation probability is exogenous. Aggregate inventories S therefore evolve according to: S +1 = (1 ) S + (1 u) q T (; D) = (1 ) S + (1 u) q () (5) 8 Our demand satisfying rate corresponds to the money velocity rate in monetary economics. 11

12 IV.3 The workers program The matching process on the product market being stochastic, there is a (small but) positive probability that an entrepreneur sells no good during stage 2c. If furthermore her money holdings are very small, she can be liquidity constrained and therefore unable to pay a xed wage at stage 2d. To avoid the complexities induced by a xed wage, we assume that the current total wage paid to an employee, W, is proportional to the quantity of inventories sold at stage 2c. We further assume that the bargaining is over the expected wage E (W ), so the rm commits to adjust the piece rate so that the expected wage of a worker is independent of her employer s demand and supply behaviors. At the beginning of stage 2b, a worker can be either employed or unemployed. Her maximized lifetime expected utility depends on her money holdings m at that moment. Let respectively V e (m) and V u (m) be the value functions of an employed (unemployed) worker at stage 1c. Consider rst the program of an unemployed worker with money holdings m. ignore unemployment bene ts. Because of frictions on the product market, the number of goods bought during the period is a random variable, denoted X. To characterize the distribution of X, we notice that each of the d units of demand expressed can be seen as a random trial with a probability of success of (). At an individual level, these trials are independent. Hence, X follows a binomial distribution with d trials and success probability (). Therefore, the discrete density function of X is given by Pr (X = x) = B (x; d; ()), where: B (x; d; ()) = d x ( ()) x (1 ()) d x In particular, we notice that Pr (X = d) > 0. So, there is always a positive probability that all the d units of demand expressed are satis ed. The expected purchase, conditional on expressing d units of demand, 9 is: We E (X) = () d (6) For any realization x of X, the unemployed worker enjoys x units of utility and will start the next period with m px units of money. Furthermore, at stage 1a of the next period, she will nd a job with probability h () and therefore gets an expected lifetime utility 10 of V e (m p x). Finally, with probability 1 h (), she stays unemployed. Her expected lifetime utility is then V u (m p x). Therefore, the value function of an unemployed 9 To ease notations, this conditionality does not appear in the notations. 10 Recall that we consider only steady states. Hence, value functions are time-invariant. 12

13 worker solves the following Bellman equation: V u (m) = X + h () Ve (m p X) + (1 h ()) V u (m p X) max E 0d s:t : p d m (7) Consider now an employed worker. She will receive a random wage W. For any realization x of X and n of W, the employed worker enjoys x units of utility. At the beginning of the next period, she will own m+n px units of money. With probability 1 s, she will be employed and will therefore get an expected lifetime utility of V e (m + n p x). With probability s, she will loose her job and will therefore bene t from an expected lifetime utility of V u (m + n p x). Hence, the value function of an employed worker solves the Bellman equation: V e (m) = X + (1 s) Ve (m + W p X) + s V u (m + W p X) max E (8) 0d s:t : p d m Since individuals preferences are linear with respect to the number of goods consumed, we guess that value functions are linear in m and of the form: V e (m) = V e + m A and V u (m) = V u + m A (9) where V e, V u and A are constant values. A stands for the marginal value of money. V e (V u ) is the value of being employed (unemployed). Under the speci cation (9), and given (6), the rst-order 11 and envelope conditions to programs (7) and (8) are identical for the unemployed and the employed. They write respectively: 0 = () 1 p A p A = A + where 0 is the (non-negative) Kuhn-and-Tucker multiplier associated to the constraint 12 m p d. Substituting from the rst-order conditions and multiplying the envelope conditions by p gives: p A = () + (1 ()) p A To interpret this asset equation, we notice that with p additional units of money at the start of stage 2b, the consumer can express one additional unit of demand. With probability (), this additional unit of demand will be satis ed, that is, the consumer will nd the good she is searching for, will buy and consume it. This will generate a utility level 11 We henceforth consider that d is a real number and not an integer, which simpli es the problem. 12 Since r > 0, the envelope condition implies > 0. Hence, the exclusion condition (m p d) = 0 implies d = m=p. Therefore, the constraint d 0 does not bind. 13

14 normalized to 1= (). With the remaining probability, the consumer ends stage 2c of the current period with p units of money, that are valued p A. The last equation can be rewritten to express p A as a function of the demand satisfying probability () and the discount rate r: p A = () 2 (0; 1) (10) r + () We call the right-hand side of (10) the rationing of demand term. This term accounts for the fact that it is better to consume a unit of good than to have p units of money. In the absence of discounting, the rationing of demand term is equal to 1. When conversely consumers discount the future, they have to wait at least one period before spending their money and the rationing of demand term is lower than 1. As tightness on the product market increases, the demand satisfying probability () declines and the rationing of demand term decreases. The decreasing curve in Figure 2 represents this phenomenon. G, θ ( ) φµ φ Rationing of supply r + δ + φµ ( φ) LL Rationing of demand µ ( φ) r + µ φ ( ) φ ~ φ Figure 2: Rationings on the Product market Furthermore, since pa < 1, a consumer always expects a positive net gain () (1 p A) when she expresses an additional unit of demand. Therefore, she expresses the maximum amount of demand, given her money holdings. So, d = m=p. Finally, using (9), Equations (7) and (8) imply: r V u = h () (V e V u ) and r V e = E (W ) A + s (V u V e ) (11) which are the usual asset equations for, respectively, the value of being unemployed and of being employed (see e.g. Pissarides 2000 or Cahuc and Zylberberg 2004). The only di erence is that wage W being random, only its expectation enters (11). In Appendix A.1, we verify that V e (m) = V e + A m and V u (m) = V u + A m solve the system of Bellman equations (7) and (8). 14

15 IV.4 The entrepreneurs program At the beginning of stage 1b, an entrepreneur owns m units of money, a level of inventory and employs l workers. She must choose how many units of demand d to express, how many units of supply to express and how many vacancies! to open, subject to constraints (2) and (3). Let V f (m; ; l) be her value function at that moment. As in the case of a worker, the number X of goods consumed, conditional on the level of demand d is a random variable that follows a binomial distribution. Symmetrically, let the random number of goods sold be denoted by Y. Each unit of good supplied will be sold according to the supply satisfying probability (). At the individual level, these trials are independent of X. Therefore, Y follows a binomial distribution with trials and a probability of success (). Hence, the density function is B (y; ; ()), with: E (Y ) = () (12) The worker and the rm bargain over the expected wage E (W ). For each unit of good sold, the rm pays w units of money per employee, with W = w Y. So: w = E (W ) E (Y ) = E (W ) () (13) The number of workers that will be hired during stage 1a of the next period is also a random variable denoted Z. Since the job lling probability is h (), Z follows a binomial distribution of! independent trials with success probability h (). The number of job dissolved is also a random variable R that follows a binomial distribution with l employment and a success probability s. Finally, the number of inventories depreciated is a random variable that follows a binomial distribution with trials and a success probability. We hence get: E (Z) = h ()! E (R) = s l E () = (14) For any realization x of X, y of Y, z of Z, r of R and of, the entrepreneur consumes x units of goods, which generates x units of utility. Opening! vacancies induces a total disutility of k!, where k > 0 is a parameter. She sells y of units of goods, thereby getting p y units of money. For each unit of goods sold, she pays w units of money to each of her l employees. Hence, her money holdings at the beginning of stage 1a of the next period will be m +1 = m + p (y x) W l, where W = w y: The output of l workers increases inventories by an amount of q l, of which y units are sold and " units are depreciated. Therefore her inventories at the beginning of the next period will be +1 = +ql y ". r jobs are dissolved, but z new workers are recruited. So, her future employment level after stage 1a of the next period will be l +1 = l r + z. Finally, her expected lifetime utility 15

16 at the start of stage 2a of the next period will be given by V f m +1 ; +1 ; l +1. Therefore, the entrepreneur s value function solves the following Bellman equation: ( ) X k! + Vf m +1 ; +1 ; l +1 V f (m; ; l) = max w;0pdm;0;0! E s:t : m +1 = m + p (Y X) W l +1 = Y + q l l +1 = l R + Z w = E (W ) E (Y ) Since entrepreneurs preferences are linear with respect to the number of goods consumed and the number of jobs opened, we guess that the value function is linear in m, and l and takes the form: (15) V f (m; ; l) = m A + G + l J (16) where A, G and J are constants that stand for the marginal values of respectively money, inventory and a lled job. Let and be the (non-negative) Kuhn-and-Tucker multipliers associated with constraints m p d and. Given (6), (12), (13), (14), and the speci cation (16), the rst-order conditions of program (15) with respect to individual demand d, supply and the number of vacancies! are respectively: 0 = () 1 p A p 0 = () p A G 0 k + h () J The envelope conditions over respectively money holdings m, inventories and employment l are 13 : A = A + G = 1 (1 s) J + q G E (W ) A G + J = Substituting from the rst-order condition with respect to d into the envelope condition with respect to money holdings m gives (10). Therefore, one has A = A which means that entrepreneurs and workers value money identically. Furthermore, as for workers, entrepreneurs express the maximum amount of demand permitted by their money holdings and d = m=p. Turning now to supply decisions, substituting from the rst-order condition with respect to supply into the envelope condition with respect to inventories gives: G = () p A + (1 ()) G The interpretation for this equation parallels the one of Equation (10). An additional unit of inventory will be sold during stage 2c with probability (). The entrepreneur 13 From the enveloppe conditions over respectively m and, we see that > 0 and > 0 so constraints (2) and (3) are binding and therefore, constraints 0 d and 0 are not bidding. 16

17 then gets p units of money, which are valued p A. With probability 1 () the good is neither sold nor depreciated. This asset equation can be rewritten as: G = () r + + () p A (17) We call the right-hand side of (17) the rationing of supply term. This term accounts for the fact that it is better to have p units of money than a unit of inventory. In the absence of discounting and depreciation, the rationing of supply term is equal to 1. Otherwise, this term is lower than 1. As tightness on the product market increases, the supply satisfying probability () rises, and the rationing of supply term increases. The increasing curve in Figure 2 represents this phenomenon. Furthermore, since G < p A, an entrepreneur always expects a positive net gain () (p A unit of good. Therefore, she supplies all her inventory: =. The envelope condition over employment l implies: G) from the supply of one additional (r + s) J = q G E (W ) A (18) During each period, a job produces q additional units of inventories, each of them being valued G. Furthermore, the rm has to pay an expected wage E (W ). Hence, the right hand side of (18) represents the expected current pro t on a lled job. Given the probability s of destruction, the lifetime value of a job discounts this current pro t at rate r + s. The rst-order condition with respect to vacancies! relates the cost of posting a vacancy k to the lifetime expected pro t from searching for a worker. The latter equals the job lling probability h () times the value of a lled job J. If the cost is larger than the expected pro ts, rms prefer not to search for workers. Then, the aggregate number of vacancies decreases. Tightness on the labor market therefore decreases (since the number of unemployed workers is predetermined) and so does the job lling probability h (). If h (0) J k, rms post no vacancies. Otherwise, the preceeding phenomenon continues until k = h ()J. This equality corresponds to the so-called free-entry condition in the standard labor matching model. Given (18) we get the usual free-entry condition (see e.g. Pissarides 2000): k h () = J (19) Appendix A.2 veri es that V f (m; ; l) = m A + G + l J solves the Bellman Equation (15). IV.5 The labor market equilibrium At stage 1c, each employed worker negotiate its expected wage E (W ) with her employer. We assume generalized Nash bargaining. Let 2 (0; 1) denote workers bargaining power. 17

18 From (9), the surplus extracted by the worker from employment V e (m) V u (m) is independent of her money holdings m. Symmetrically, from (16), the marginal value of a job J for an entrepreneur is independent of her money holdings m, inventories, or employment l. The expected wage E (W ) maximizes the generalized Nash Product: max E(W ) (V e V u ) J 1 taking workers outside option V u as given. Given (11) and (18), the rst-order condition leads to the following sharing rule: (1 ) (V e V u ) = J (20) Hence, the total surplus generated by a job V e V u + J is split in shares and 1 between the worker and her employer. From (11), we then have: Together with (18), we get: (r + s) (V e V u ) = E (W ) A h () (V e V u + J) (r + s + h ()) (V e V u + J) = q G Together with the free-entry condition (19) and the sharing rule (20), the last equation leads to the labor market equilibrium condition: r + s k h () + 1 = q G (21) Following Assumption AS1, the left-hand side of (21) is an increasing function of. When increases from 0 to +1, the left hand side increases from (r + s) k= (1 ) to +1. Therefore, if q G > (r + s) k= (1 ), there exists a unique equilibrium value of the labor market tightness that satis es (21). Otherwise, a lled job generates too few utility, so entrepreneurs prefer not to open vacancies and = 0. This is summarized in the following lemma Lemma 1 There exists a single equilibrium value for the labor market tightness. If G > (r + s) k= (q (1 )), this values is the unique solution of (21). Otherwise, = 0. The functioning of the product market a ects the labor market only through the value of a unit of inventory G. This is because the wage payment scheme is adjusted so that the total surplus generated by a job is split between the worker and her employer. Hence, the vacancy supply behavior only depends on the total surplus from a job. This surplus is proportional to the total value of the q units of goods produced by a job within a period. 18

19 Equilibrium labor market tightness is therefore an increasing function of the value G of a unit of inventory and of the productivity q of a job. We also obtain the usual comparative static properties with respect to the labor market parameters. A rise in the workers bargaining power, in the job separation probability s, or in the vacancy cost k decreases equilibrium labor market tightness. Finally, using (11) and (18), the sharing rule (20) implies E (W ) A = q G + (1 ) h () (V e V u ) Using again (20) with the free-entry condition (19) gives nally the value of the expected wage: E (W ) A = fq G + k g (22) IV.6 The xed price Equilibrium We have seen that workers and entrepreneurs choose to express as many units of demand as they are allowed to, given the constraint (3). At the macroeconomic level, aggregate demand D equals the real aggregate money stock M=p. Symmetrically, entrepreneurs choose to supply all their inventories. Hence, at the macroeconomic level, aggregate supply of product equals the total level of stock S. Hence, we have: = M p S (23) From (5) rewritten in steady state, we get the ow equilibrium condition on the product market: S = (1 u) q (24) + () The aggregate level of inventories S is an increasing function of the productivity level q and of the steady-state level of employment 1 u. The steady-state level of inventories decreases with the depreciation rate and the probability of selling supplied goods (). From D = M=p, we notice that our model amounts to putting Money in the Matching Function. It is worth stressing that the distribution of the money holdings does not a ect the level of demand for produced goods. This is because every agent values money identically. This property comes from the fact that the returns on money holdings are linear at the individual level: holding a twice bigger amount of money permits to simultaneously search for twice more goods and thereby to double the ow of consumed goods. We are now ready to de ne the xed price equilibrium. A xed price equilibrium is a list fa; G; ; ; S; u; E (W )g that veri es i) The asset equations for the marginal values of money (10) and inventories (17); ii) the labor market equilibrium condition (21); iii) the demand and supply behaviors, as summarized by (23); iv) The ow equilibrium condition 19

20 on the labor market, which implies (1); v) The ow equilibrium condition on the product market (24); vi) The wage equation (22). We now explain how to reduce this equilibrium to the analysis of two curves in the (; ) space. Combining (10) and (17), we get: G = () where () () r + + () () r + () (25) A unit of good produced is transformed into utility in two steps. not sold instantaneously. term de ning First, production is The imperfection of this process is represented by the rst (:), i.e., the rationing of supply term. Second, p units of money cannot instantaneously be used to buy a good. The imperfection of this process is represented by the second term, i.e. the rationing of demand term. Each of these two steps are uncertain and take time due to the presence of frictions on the product market. Function (:) summarizes the consequence of these two imperfections. following lemma: Lemma 2 Under assumptions AS2 and 3, function We prove in Appendix B the () is hump shaped, with zero limit for! 0 and! 1, It admits a unique maximum value for = ~ de ned by: r r + + ~ ~ 1 " ~ r + = (26) r + ~ " ~ where " (:) is the elasticity de ned in (4). When tightness on the product market is su ciently low ( < ~ ), a rise in relaxes more the rationing of supply than it reinforces the rationing of demand. That is G= (p A) increases more than p A decreases with. Then, increasing raises the pace at which a unit of production is transformed into utility Figure 2). (). The opposite holds when > ~. (See For a given value of, when the discount rate r increases, agents are more impatient and both values of inventories G and of money holdings A decrease. As a consequence, the three curves in Figure 2 shift downwards, and () decreases. The opposite hold when the product market matching function parameter 0 increases. Finally, a rise in the depreciation rate increases the depreciation ow of inventories. The rationing of supply term in (17) and therefore shift downwards in Figure 2. Combining (21) and (25), we can express the equilibrium labor market tightness as a function of the product market tightness through: r + s k h () + = q () (27) 1 20

21 The e ects of, 0 and on tightness on the labor market come through changes in the value of. Since the left-hand side of (27) is an increasing function of, these e ects can immediately be illustrated by rescaling the y-axis of Figure 2 in terms of. Equation (27) is represented in Figures 2 and 3 by the curve labelled LL. Following lemma 2, a rise in increases (decreases) if < e (resp. > e ). For any value of, a rise in or a decline in the scale parameter of the matching function 0 lower (:) and hence. Finally, an increase in the discount rate r decreases through two channels. In addition to the negative e ect on Pissarides 2000). (), there is the standard negative e ect on job creation (see The LL curve relates two endogenous variables, namely and. One therefore needs and additional relation, which in the xed price model is given by the ow equilibrium conditions on the labor (1) and product (24) markets. Combining these two conditions with (23) leads to: h () s + h () q = M p + () A rise in tightness on the labor market implies a higher steady-state level of employment 1 u. Therefore, more output is produced during each period, which raises inventories S. Since the aggregate Money supply M is xed and the price p is exogenous, the steady-state level of tightness on the product market decreases. This can be seen from the right-hand side of (28), which decreases with. Therefore, the two ow equilibrium conditions de ne as a decreasing function of at the steady-state. This second relation is denoted DD in Figure 3. (28) θ DD DD p > 0 Μ < 0 LL φ ~ φ Figure 3: The Fixed Price Equilibrium Assuming exogenous price is not a very relevant assumption for a steady-state equilibrium. We made it essentially for pedagogical reasons in order to build a more general 21

22 model step by step. We therefore do not devote too much attention to the study of this equilibrium. Neither existence nor uniqueness are guaranteed. However, a noticeable property is the ambiguous e ect of a decline in Aggregate Demand D = M=p on equilibrium, coming from either a rise in price p or a decrease in money supply M. A decrease in D has no e ect on the LL curve (see Equation 27). Considering the other curve, such a decline reduces the amount of goods exchanged on the product market T (S; D). For a given level of tightness on the labor market, thereby a given ow of produced goods q (1 u), the level of inventories S increases, which o set the reduction in T (S; D). Hence, from (23), the decrease in D and the increase in S lead to a decrease in (see 28). This is illustrated in Figure 3 by the shift of the DD curve to the left. Now, such a decline of has an ambiguous e ect on the value of inventories G, thereby on tightness on the labor market and on unemployment u. If <, e the decline in decreases () and we get the standard e ect: a rise in price p, or a decrease in money supply M decreases labor market tightness and increases unemployment u. However, if >, e we get the reverse e ects: the reduction in improves more the value of money holdings (by increasing the probability of nding a seller) than it deteriorates the value of inventories relative to the value of money holdings (by decreasing the probability of nding a customer). Therefore, the value of inventories increases, thereby increasing tightness on the labor market and decreasing unemployment. V Endogenous price Various assumptions could be made about price determination in the product market. We assume that sellers post prices, that is they annonce take-it-or-leave-it prices and commit to sell production at this price. To us, this seems to be the most natural assumption for price setting in the product market. However, we are aware that the price of certain goods are also bargained over. We build upon the price posting equilibrium of Moen (1997), also known as the Competitive Search Equilibrium (henceforth CSE). Prices are observable at no cost, search is directed and mobility of agents is perfect. The product market is now made of a continuum of identical submarkets indexed by i 2 I. Each submarket is characterized by a price p i. At the beginning of stage 2c of a given period, there are i units of supply and D i units of demand in market i. The matching technology is identical across submarkets. Then, a unit of demand (supply) in submarket i is satis ed (is sold) with probability ( i ) ( i ( i )) in submarket i, where i = D i = i is the tightness in the i th submarket. In a CSE, submarkets with a higher price attract relatively more supply and less demand. Tightness is therefore lower in submarkets with a higher price, which implies 22