Intensve vs Extensve Margn Tradeo s n a Smple Monetary Search Model Sébasten Lotz y Unversty of Pars 2 Andre Shevchenko z Mchgan State Unversty Aprl 2006 hrstopher Waller x Unversty of Notre Dame Abstract We ntroduce ex-post heterogenety nto monetary search models wth lotteres. Heterogenety allows lotteres over goods to exst n equlbrum. These lotteres over goods create an ntensve margn (expected producton n a match) that s non-exstent n all ndvsble goods monetary search models. We then show there can be a tradeo between the ntensve margn and extensve margn (number of matches) when choosng the optmal monetary stock. We would lke to thank the partcpants of the 1 st European Workshop on Monetary Theory (Unversty of Nanterre, 2004) for ther comments on ths paper. We also thank the Federal Reserve Bank of leveland and ERMES at Pars 2 for research support. y ERMES (NRS), 12 place du Panthéon, 75005 Pars, France. E-mal: lotz@u-pars2.fr z Department of Economcs, 101 Marshall Hall, East Lansng, MI 48824. E-mal: shevchen@msu.edu x Department of Economcs and Econometrcs, 441 Flanner Hall, Notre Dame, IN 46556. E-mal: cwaller@nd.edu 1
1 Introducton In the Kyotak and Wrght (1991, 1993) search model of money, both goods and money are ndvsble and trade at the prce of one-for-one. Subsequently, Berentsen, Molco and Wrght (2002), examned the use of lotteres (randomzed tradngs) over money and goods to allow the prce, p, n a match to be determned endogenously. Berentsen et al. (2002) showed that the only equlbrum n whch p d ers from one s when money s exchanged wth a probablty less than one, and goods exchange wth probablty equal to one. Ths means the quantty of goods traded wthn a match s constant and ndependent of the money stock. onsequently, changes n the money stock only a ect the extensve margn (the number of trades that occur). In ths note, we show that by ntroducng ex post heterogenety n producton costs, monetary equlbra exst n whch goods are exchanged wth probablty less than one. Ex-post heterogenety means sellers experence match spec c..d. cost shocks. 1 We show that n these equlbra there s a tradeo between the extensve and ntensve margns ncreasng the money stock can expand the extensve margn but reduces the probablty of goods trades wth hgh cost sellers, thereby lowerng the ntensve margn. 2 Envronment The envronment s smlar to Kyotak and Wrght (1993). There are a contnuum of n ntely lved agents on the unt nterval. Agents dscount at rate r. They are also specalzed n producton and consumpton such that there s the usual double concdence of wants problem makng barter mpossble. The probablty of meetng an agent who can produce one s desred consumpton good s : Goods are ndvsble. Agents get utlty U from consumng ther desred consumpton good and ncur costs from producng ther producton good. For trade to occur we need U >, whch we assume from here on. Due to the absence of a double concdence of wants, agents need money to trade. Let M be the stock of money n the economy, and agents are constraned n ther nventory of money holdngs such that they can hold one ndvsble unt of money. onsequently, M s the fracton of agents holdng money n the economy. We call these agents buyers whle sellers are those wthout money. We only consder statonary equlbra such that the value of holdng money and goods s constant over tme. Let V 1 denote the value functon for an agent holdng one 1 Thus, at the begnnng of every perod agents are dentcal ex ante, as n Shevchenko and Wrght (2004) or urts and Wrght (2004) 2
unt of money and V 0 denote the value functon for an agent wthout money. 3 Homogeneous Agents In ths secton, we bre y revew the Berentsen et al. (2002) model wth buyer-take-all barganng as a benchmark. Ths s wthout loss of generalty snce all of ther results hold, and are strengthened, wth ths form of barganng. We assume that when a buyer meets an approprate seller he makes a take-t-or-leave-t o er to the seller. The o er conssts of a par of probabltes (; ) where s the probablty goods are traded whle s the probablty money changes hands. 2 The buyer s problem s: max ; [U (V 1 V 0 )] s:t: + (V 1 V 0 ) 0; 1; 1: It means that the buyer wants to maxmse the d erence between the utlty he gets for consumng the good (whch happens wth probablty ) and hs change of state from buyer to seller (whch happens wth probablty ). The constrant means that the seller must be wllng to produce for an exchange to take place. Snce the buyer wll extract the entre surplus of the seller, the constrant holds wth equalty. Usng the constrant to substtute out for n the objectve yelds: max (U ) s:t: = 1 V 1 V 0 It s clear that = 1 s the soluton as long as money s hghly valued or V 1 V 0. Ths smply says that acqurng money has greater value than the cost of producng. However, for > V 1 V 0, money has a low value and the constrant s volated. Nevertheless, the buyer can stll o er an acceptable lottery: he o ers = 1 and = (V 1 value of snce hs surplus s stll postve and gven by (U and thus s assumed to accept the o er. V 0 ) = 1. The buyer wll choose to o er ths ) > 0, whle the seller s nd erent The ntuton for ths s qute clear; f money s valued su cently hgh, then the buyer demands the good wth probablty one and o ers the money wth probablty less than one. Ths allows buyers to gve up a fracton of a unt of money on average for the good. However, when the value of money s low, the buyer must gve up the money wth probablty one and ask for goods wth a probablty less than one. Expected monetary prces are gven by p = =. Thus, when money s hghly valued p < 1 and when money has low value p > 1: 2 Lotz, Shevchenko and Waller (2006) look at a more general barganng problem where dependence between the two lotteres s possble. We show that the optmal lottery structure s what we have here. 3
Now, we have to prove exstence of the two prevous possble equlbra. Snce the buyer makes an o er that extracts the entre expected surplus of the seller, V 0 = 0 and the buyer s value functon s gven by: V 1 = (1 M) [U (V 1 V 0 )] (1) where = =r. We rst consder equlbra where = 1 and 1. It follows that V 1 = (1 M) (U ) ; = = (1 M) (U ) where V 1 s monotoncally decreasng n M. We assume that at M = 0, jm=0 < 1. Snce s ncreasng n M and approaches n nty as M! 1, there s a unque value of M < 1; such that = 1 and V 1 =. For values of M 2 (0; M), V 1 > and < 1. From an economc pont of vew, ths seems ntutve; when there s a su cently low quantty of money n the economy, money wll be hghly valued. Thus, tradng one unt of money for one unt of goods - whle bene cal to the buyer - nvolves a prce that s too hgh,.e p = 1. So the buyer explots hs barganng power to o er the seller a lottery over the money n order to trade at a lower expected prce, p = < 1, whch the seller s wllng to accept. On average, buyers spend less than a unt of money and t s n ths sense that money s dvsble. Now consder the case where M 2 M; 1. From the barganng problem, we know that = 1 and = V 1 = 1. Substtutng ths nto (1) yelds V 1 = (1 M) (U ) V 1 mplyng V 1 = 0 unless = (1 M) (U ). The value of M that solves ths expresson s M. For M > M, the only soluton s V 1 = 0 and so a monetary equlbrum does not exst. Ths s the result n Berentsen et al. (2002) monetary equlbra do not have lotteres over goods. 4 ost heterogenety In ths secton, we ntroduce ex post heterogenety across agents by assumng there s a contnuum of cost types j. By ex post heterogenety, we mean that agents experence match spec c..d. cost shocks. Thus, at the begnnng of every perod agents are dentcal ex ante. 3 By assumpton, nformaton s symmetrc,.e. j s observable by the buyers. Let j F () wth densty df (), 3 The advantage of assumng ex post nstead of ex ante heterogenety s that t has no dstrbutonal consequences n terms of money balances across agent types, and so t smpl es the model wthout a ectng our basc conclusons. To understand how ex ante heterogenety would a ect the model, see Lotz, Shevchenko and Waller (2006). 4
where F s contnuous n, wth 2 L ; H and expectaton e = R H df (). It follows that the probablty a buyer meets an approprate seller wth producton costs less than some value ^ s (1 M) R ^ L df (). We assume that all agents get the same utlty, U, from consumng ther desred consumpton good where U > H. Ths ensures that there are gans from trade wth all sellers who produce the buyer s desred consumpton good. It also elmnates market partcpaton ssues present n Johr (1999), and amera and Vesely (2006). Snce agents are dentcal before tradng begns, let V 1 denote the value functon for an agent holdng one unt of money, and V 0 denote the value functon for an agent wthout money. Wth cost heterogenety, the o er conssts of a par of probabltes L j ; j where j s the probablty goods are traded wth a type j seller whle j s the probablty money changes hands wth ths j seller. onjecture that, for a gven value of M, there s a cuto value 2 L ; H such that the barganng solutons yeld: 8 j > ; j = 1 and j = V 1 j < 1 8 j < ; j = j V 1 < 1 and j = 1 8 j = ; j = 1, j = 1 and V 1 = : Once agan, conjecture an equlbrum n whch buyers can choose approprate values of j and j such that both buyers and sellers are wllng to trade at the prce j = j. The value functon of a buyer s: Z H V 1 = (1 M) j U V 1 df () + (1 M) Z L U j V 1 df () Substtutng the lotteres nto ths expresson and rewrtng gves: V 1 = (1 M) V 1 Z H U= j 1 Z df () + (1 M) U j df () (2) L The rst term s the expected surplus receved when goods are acqured from hgh cost sellers, whch occurs wth probablty j = V 1 = j, whle the second term s the expected surplus from tradng wth low cost sellers. For the cuto seller t must be the case that V 1 = ; so solves: T " Z H 1 (1 M) U= j 1 # Z df () (1 M) U j df () = 0 (3) L We can now state the followng proposton: 5
Proposton 1 For M 2 1, = H and 1, = 1 for all sellers. For M 2 ^M1 ; ^M 2, there exsts a unque value 2 L ; H where for j > ; j = 1 and j < 1, whle for j <, j < 1 and j = 1. For M = ^M 2, = L and = 1, 1 for all sellers. For M > ^M 2, no monetary equlbrum exsts. The proof s n the appendx. Fgure 1 llustrates Proposton 1. Fgure 1: Equlbra wth lotteres over goods () and money (). For low values of the money stock (M < ^M 1 ), money s hghly valued so all sellers trade goods wth probablty one and receve money wth a probablty less than one. For ntermedate values of M, hgh cost sellers value money less than the cost of producng so they only want to gve up a fracton of the good for money. onsequently, buyers have to o er a lottery over goods for exchange to occur. On the other hand, low cost sellers value money more than the cost of producng so they agree to receve a fracton of the unt of money for ther unt of good; buyers o er a lottery over money. It then follows that there s a dstrbuton of expected prces wth p > 1 for hgh cost sellers ( j > ) and p < 1 for low cost sellers ( j < ). For hgh values of the money stock, no monetary equlbrum exsts. What s the optmal steady-state money stock n ths economy? In Kyotak and Wrght (1991, 1993), Rocheteau (2000) and Berentsen et al. (2002), the optmal value of the money stock s M = 1=2, whch maxmzes the extensve margn (number of matches). In these models, the ntensve margn s the expected producton n a match and all monetary equlbra have = 1 so the ntensve margn s constant. Hence, changes n M do not a ect the ntensve margn and so there s no tradeo between the extensve and ntensve margns. In our model, for M 2 ^M1 ; ^M 2 ; < 1 and, n ths range, ncreasng M has two negatve e ects. Frst, t lowers j for all sellers wth costs above thereby lowerng the expected quantty 6
of goods traded n those matches (t worsens the ntensve margn). Second, t lowers the cuto value meanng more sellers resort to lotteres over goods rather than lotteres over money (more sellers produce less). As s standard n the lterature, de ne steady-state welfare to be W (M) = MV 1 + (1 M) V 0. onsder the two possble ranges for M: M 2 1 wth H = and M 2 ^M1 ; ^M 2 wth 2 L ; H. Usng V 0 = 0, = V1 and (2) we have where 8 < M (1 M) (U e ) f M 2 1 W (M) = : M (1 M) Z f M 2 ^M1 ; ^M : 2 Z Z H U= j 1 Z df () + U j df () : (4) L The optmal steady-state money stock, M, sats es 8 < (1 2M ) (U e ) = 0 f M 2 W 0 (M 1 ) = : (1 2M ) Z + M (1 M ) @Z @ @ @M = 0 f M 2 ^M1 ; ^M : 2 From (4), @Z=@ > 0 and totally d erentatng (3) yelds @ =@M < 0 for M < ^M 2. It then follows that f 1=2 < ^M 1 ; M = 1=2 and the optmal steady-state money stock maxmzes the extensve margn snce the ntensve margn s una ected for = 1. On the other hand, f ^M1 < 1=2, then choosng M 2 ^M1 ; 1=2 s optmal snce W 0 (1=2) < 0. Therefore, f there are too few buyers for M < ^M 1, then t s optmal to ncease the number of trades on the extensve margn (by ncreasng M) even though t reduces the ntensve margn by lowerng. But t s not optmal to maxmze the number of trades that occur snce M < 1=2. 5 oncluson By extendng the Berentsen, Molco and Wrght (2002) monetary search model wth lotteres to the case where agents are heterogeneous ex post, we rst showed that there exst monetary equlbra wth lotteres over goods. Second, because goods may be exchanged wth probablty less than one, a tradeo between the ntensve and extensve margn emerges, modfyng the optmal monetary stock. 7
References Berentsen A., Molco M., Wrght R. (2002). Indvsbltes, lotteres, and monetary exchange, Journal of Economc Theory 107, pp. 70-94. amera G., Vesely F. (2006). Market Actvty and the Value of Money. An Example, Journal of Money, redt, and Bankng 38 (2), pp. 495-510. urts E., Wrght R. (2004). Prce settng, prce dsperson, and the value of money, or the law of two prces, Journal of Monetary Economcs 51, pp. 1599-1621. Johr A. (1999). Search, Money and Prces, Internatonal Economc Revew 40 (2), pp. 439-454. Kyotak N., Wrght R. (1991). A contrbuton to the pure theory of money, Journal of Economc Theory 53, pp. 215-235. Kyotak N., Wrght R. (1993). A search-theoretc approach to monetary economcs, Amercan Economc Revew 83, pp. 63-77. Lotz S., Shevchenko A., Waller. (2006). Heterogenety and lotteres n monetary search models, Journal of Money, redt, and Bankng, forthcomng. Rocheteau G. (2000). La quantté optmale de monnae dans un modèle avec apparements aléatores, Les Annales d Econome et Statstque 58, pp. 101-142. Shevchenko A., Wrght R. (2004). A smple search model of money wth heterogeneous agents and partal acceptablty, Economc Theory 24 (4), pp. 877-885. 8
Appendx Proof of Proposton 1: If = H, usng (2) gves us: V 1 = (1 M) (U e ) ; H = H = (1 M) (U e ) Assume H j M=0 < 1. Let ^M 1 2 (0; 1) denote the value of the money stock makng H = 1. Thus, for M 2 1, V 1 H and all sellers gve up the good wth probablty one and receve money wth probablty less than one. If = L, then j = V 1 = j for all j > L and j = 1, 8j: It follows that V 1 solves: So V 1 = 0, unless M = ^M 2 satsfyng: Z H V 1 = V 1 " (1 M) U= j 1 # df () : L 1 = 1 Z H ^M2 U= j 1 df () (5) L whch mples V 1 s ndetermnate and any value V 1 2 0; L s a monetary equlbrum. For M > ^M 2, no monetary equlbra exst. It s straghtforward to show that ^M 2 > ^M 1 s true 8. As a result for M 2 ^M1 ; ^M 2, t may be possble to have 2 L ; H. We now establsh exstence and unqueness of. Usng (5), (3) gves us T L = L M ^M 2 = 1 ^M2 < 0 T H = H M ^M 1 = 1 ^M1 > 0: Snce F s contnuous n by assumpton, there exsts a value such that T = 0, where 2 L ; H for M 2 ^M1 ; ^M 2. Gven that T L < 0 and T H > 0, there may be multple equlbra but f so there are an odd number of them. We now show that the equlbrum s unque. The proof s by contradcton. If there are multple equlbra then at least one of the equlbra must have T 0 < 0. Applyng Lebntz s rule, we obtan: T 0 = 1 (1 M) Z H U= j 1 df () : (6) Substtute ths nto (3) and rearrange to get whch requres T 0 T 0 = (1 M) Z L U j df () > 0 > 0 for all. Hence, the equlbrum s unque. 9