Measurement with Minimal Theory Ellen R. McGrattan. Asset Prices, Liquidity, and Monetary Policy in the Search Theory of Money Ricardo Lagos

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1 QUARTERLY REVIEW July 2010 Meauremen wih Minimal Theory Ellen R. McGraan Ae Price, Liquidiy, and Moneary Policy in he Search Theory of Money Ricardo Lago

2 Quarerly Review vol.33, No. 1 ISSN Thi publicaion primarily preen economic reearch aimed a improving policymaking by he Federal Reerve Syem and oher governmenal auhoriie. Any view expreed herein are hoe of he auhor and no necearily hoe of he Federal Reer\>e Bank of Minneapoli or he Federal Reerve Syem. EDITOR: Arhur J. Rolnick ASSOCIATE EDITORS: Parick J. Kehoe, Warren E. Weber MANAGING EDITOR: Warren E. Weber ARTICLE EDITOR: Joan M. Gieeke PRODUCTION EDITOR: Joan M. Gieeke DESIGNER: Phil Swenon TYPESETTING: Mary E. Anomalay CIRCULATION ASSISTANTS: Mary E. Anomalay, Barbara Drucker The Quarerly Review i publihed by he Reearch Deparmen of he Federal Reerve Bank of Minneapoli. Subcripion are available free of charge. Thi ha become an occaional publicaion; however, i coninue o be known a he Quarerly Review for ciaion purpoe. Quarerly Review aricle ha are reprin or reviion of paper publihed elewhere may no be reprined wihou he wrien permiion of he original publiher. All oher Quarerly Review aricle may be reprined wihou charge. If you reprin an aricle, pleae fully credi he ource-he Minneapoli Federal Reerve Bank a well a he Quarerly Review-and include wih he reprin a verion of he andard Federal Reerve diclaimer (ialicized lef). Alo, pleae end one copy of any publicaion ha include a reprin o he Minneapoli Fed Reearch Deparmen. Elecronic file of Quarerly Review aricle are available hrough he Minneapoli Fed' home page on he World Wide Web: hp:// Commen and queion abou he Quarerly Review may be en o: Quarerly Review Reearch Deparmen Federal Reerve Bank of Minneapoli P.O. Box 291 Minneapoli, MN (Phone /Fax ) Subcripion reque may be en o he circulaion aian a mea@re.mpl.frb.fed.u.

3 QR Ae Price, Liquidiy, and Moneary Policy in he Search Theory of Money* Ricardo Lago Aociae Profeor of Economic New York Univeriy value of he ream of conumpion ha hey yield, bu alo for heir uefulne in faciliaing exchange. Conider a buyer who canno commi or be forced o honor deb, and who wihe o make a purchae from a eller. eller (e.g., an equiy hare, a bond, money) helpful in carrying ou he ranacion. For example, he buyer could ele he ranacion on he po by uing he ae direcly a a mean of paymen. In ome modern ae o ener a repurchae agreemen wih he eller, or a collaeral o borrow he fund needed o pay he eller. Once ripped from he ubidiary conracual complexiie, he eence of hee ranacion i ha he ae help he unruworhy buyer o obain wha ae are rouinely employed in he exchange proce and play a role akin o a medium of exchange. Tha i, hey provide liquidiy he erm ha moneary heori ue o refer o he uefulne of an ae in faciliaing ranacion. reuling from aggregae hock, o o he exen ha hee ae erve a a ource of liquidiy, hock o heir price will ranlae ino aggregae liquidiy hock ha dirup he mechanim of exchange and he enuing allo- in ae price can dirup he exchange proce in ome key marke, and hrough hi channel, propagae o he macroeconomy. Much of he policy advice offered o cenral bank i framed in erm of imple inere-rae feedback rule looely moivaed by a paricular cla of model where nominal rigidiy. Such policy recommendaion are baed on he premie ha he primary goal of moneary policy i o miigae he effec of hee rigidiie. Wih no room or role for a noion of liquidiy (and ypically even no meaningful role for money), hi convenional view ha dominae policy circle ha failed o offer nancial crii. I inerpre hi failure a an indicaion ha he conenu ance oward moneary policy, wih i heoreical focu on icky-price fricion and i imple- *Thi aricle wa prepared for he Proceeding of he Thiry-fourh Annual Economic Policy Conference of he Federal Reerve Bank of S. Loui. Financial uppor from he C.V. Sarr Cener for Applied Economic a New York Univeriy i graefully acknowledged. 14

4 Ae Price, Liquidiy, and Moneary Policy Ricardo Lago menaion emphai on ad hoc feedback inere-rae rule, i oo narrow in ha i neglec he fundamenal fricion ha give rie o a demand for liquidiy. In hi aricle I preen a dynamic equilibrium microfounded moneary ae-pricing framework wih muliple ae and aggregae uncerainy regarding liquidiy need, and dicu he main normaive and poiive policy implicaion of he heory. The broad view ha emerge from explicily modeling he role of money and oher liquid ae in he exchange proce i ha of a moneary auhoriy ha eek o provide he privae ecor wih he liquidiy needed o conduc hree propoiion ha anwer he following queion: How hould moneary policy be conduced in order o miigae he advere effec of hock o he valuaion vae ecor? Wha are he implicaion for ae price of deviaing from he opimal moneary policy? Are uch deviaion capable of cauing real ae price o be above heir fundamenal value for exended period of ime? Model In order o addre he queion ju poed, in hi ecion I ouline a bare-bone model ha encompae he key economic mechanim. 1 The model combine elemen of he ae-pricing model of Luca (1978) wih elemen of he model of moneary exchange of Lago and Wrigh (2005). Time i dicree and he horizon [,] 01 agen. Each ime period i divided ino wo ubperiod where differen aciviie ake place. There are hree nonorable and perfecly diviible conumpion good a each dae: frui, general good, and pecial good. Frui and general good are homogeneou good, wherea pecial good come in many varieie. The only durable commodiy in he economy i a e of Luca ree. The agen. Tree yield a random quaniy x of frui in he econd ubperiod of every period. Each ree yield he ame amoun of frui a every oher ree, o x i an aggregae hock. The realizaion of x become known a he beginning of period ubperiod). Producion of frui i enirely exogenou: no reource are uilized, and i i no poible o affec he oupu a any ime. The moion of x will be aken o fol- F( x, x) Pr( x1 x x x). x, F(, x) i a diribuion funcion wih uppor (, 0 ). In each ubperiod, every agen i endowed wih n uni of ime which can be employed a labor ervice. In he econd ubperiod, each agen ha acce o a linear producion echnology ha ranform labor er- agen ha acce o a linear producion echnology ha ranform hi own labor inpu ino a paricular variey of he pecial good ha he himelf doe no conume. Thi pecializaion i modeled a follow. Given wo agen i and j drawn a random, here are hree poible even. The probabiliy ha i conume he variey of pecial good ha j produce bu no vice vera (a ingle coincidence) i denoed. Symmerically, he probabiliy ha j conume he pecial good ha i produce bu no vice vera i alo. In a ingle-coincidence meeing, he agen who wihe o conume i he buyer, and he agen who produce, he eller. The probabiliy ha neiher wan wha he oher can produce i 1 2, wih 12 /. Frui and general good are homogeneou and hence conumed (and, in he cae of general good, alo produced) by all agen. decenralized marke where rade i bilaeral (each meeing i a random draw from he e of pairwie meeing), and he erm of rade are deermined by bargaining (a ake-i-or-leave-i offer by he buyer, for impliciy). The pecializaion of agen over conumpion and producion of he pecial good, combined wih bilaeral rade, give rie o a double-coincidence-of-wan problem rade in a cenralized marke. Agen canno make binding commimen, and rading hiorie are privae in a way ha preclude any borrowing and lending beween people, o all rade in boh he cenralized and decenralized marke mu be quid pro quo. Each ree ha ouanding one durable and perfecly diviible equiy hare ha repreen he bearer ownerhip and confer him he righ o collec he frui which i inrinically uele (i i no an argumen of any uiliy or producion funcion), and unlike equiy, ownerhip of money doe no coniue a righ o collec any reource. Money i iued by a governmen ha 1 The analyi ha follow i baed on Lago (2009, 2010). See alo Lago (2006). 15

5 QR a = 0 commi o a moneary policy repreened by a equence of poiive real-valued funcion, { } 0. Given an iniial ock of money, M 0 > 0, a moneary policy induce a money upply proce, { M } 0, via M + 1 = ( x ) M, where x denoe a hiory of realizaion of frui dividend hrough period, i.e., x ( x, x1,..., x0). The governmen injec or wihdraw money via lumpum ranfer or axe in he econd ubperiod of every period, i.e., along every ample pah, M+ 1 = M + T, where T i he lump-um ranfer (or ax, if negaive). All ae are perfecly recognizable, canno be forged, and can be raded among agen in boh he cenralized and decenralized marke. A = 0 each agen i endowed wih a 0 equiy hare and a 0 m money. Le he uiliy funcion for pecial good, u :, and he uiliy funcion for frui, U :, be coninuouly differeniable, bounded by B on, increaing, and ricly concave, wih u( 0) = U( 0) = 0. Le n be he uiliy from working nriod. Alo, uppoe here exi q * (, 0 ) u( q* ) 1, wih q* n. Le he uiliy for general good, and he diuiliy from working in he econd ubperiod, be linear. The agen prefer a conumpion and labor equence { q, n, c, y, h} 0 over anoher equence { q, n, c, y, h } 0 if T liminf E [ u( q) n U( c) y h] T 0 0 [ uq ( ) n Uc ( ) y h ] 0, where ( 01,), q, and n are he quaniie of pecial good conumed and produced in he decenralized marke, c denoe conumpion of frui, y conumpion of general good, and h he hour worked in he econd ubperiod. Here, E i an expecaion operaor condiional on he informaion available o he agen a ime and he probabiliy meaure induced by F. In he remainder of hi ecion, I decribe he individual opimizaion problem ha each agen face, convenien o analyze he deciion ha an agen ha o make in any given period by working backward from he he cenralized marke, I hen explain he deerminaion I provide an expreion for he agen value funcion when he ener he decenralized marke before he round of bilaeral rade. Le a = ( a, a m ) denoe he m porfolio of an agen who hold a hare and a uni of money. Le W ( a ) and V ( a ) be he maximum aainable expeced dicouned uiliy of an agen when he ener he cenralized and decenralized marke, repecively, a ime wih porfolio a. Then, (1) W ( a ) = max { Uc ( ) y h EV ( a )} c, y, h, a1.. c + w y + a + a + 1 m m = ( + x ) a + ( a + T ) + wh m m 0c, 0h n, 0 a 1. Tha i, when he ener he cenralized marke of period wih porfolio a, he agen chooe conumpion of frui ( c ), conumpion of general good ( y ), labor upply ( h ), and an end-of-period porfolio ( a +1 ) in order o maximize hi expeced dicouned uiliy. Frui i ued a he numéraire: w i he relaive price of he general m good, i he (ex-dividend) price of a hare, and 1/ i he dollar price of frui. Nex, conider a meeing in he decenralized marke of period beween a buyer wih porfolio a and a eller wih porfolio a. Le [ q,( aa, ), p ( aa, )] denoe he erm a which a buyer who own porfolio a rade wih a eller who own porfolio a, where q ( aa, ) i he quaniy of pecial good raded, and p ( a, a ) = [ p ( aa, ), p m ( aa, )] i he ranfer of ae ranfer of equiy). A menioned earlier, he erm of rade, ( q, p ), are deermined by he buyer, who make a ake-i-or-leave-i offer o he eller. In order o chooe hi offer, he buyer olve max [ uq ( ) W ( a p ) W( a )] q, p a.. W ( a p ) q W ( a ). The conrain p a pecify ha he buyer canno pend more ae han he own, and he oher conrain enure ha he buyer offer i accepable o he 16

6 Ae Price, Liquidiy, and Moneary Policy Ricardo Lago eller. Le = (, ), wih ( 1/ w)( x ) and m ( / w) m 1. Inuiively, i he period real m cum-dividend value of equiy, and i he period real value of a uni of money, boh expreed in erm of period general good. The bargaining oucome i a follow. If a q *, he buyer buy q = q * in exchange for a vecor p of ae wih real value p q * a. Oherwie, he buyer pay he eller p = a in exchange for q = a. Hence, he quaniy of pecial good exchanged i min(, * a q ) q( a), and he real value (expreed in erm of general good) of he porfolio ued a paymen i p( a, a ) = q( a). Wih he bargaining oluion, he value of earch o an agen who ener he decenralized marke of period wih porfolio a can be wrien a m where i he expeced gain from rade in he decenralized marke. Subiue he budge conrain (1) and V ( a ) ino he righ ide of W ( a ) o arrive a c (2) W( a) a max U( c) c 0 w max { a / w a10 1 where m = T and = (, m ). Thi expreion provide a imple recurive repreenaion of he maximum aainable expeced dicouned uiliy of an agen who ener he cenralized marke of period wih porfolio a. Given a proce { M } 0, a (ymmeric) equilibrium i a plan { c, a 1} 0, pricing funcion { w, } 0, and bilaeral erm of rade { q, p } 0 uch ha: (i) given price and he bargaining proocol, { c, a 1} 0 olve he agen opimizaion problem on he righ ide of (2); (ii) he bilaeral erm of rade are q = min( a, q* ) and p = q ; and (iii) he cenralized marke clear, i.e., c = x, and a + 1 = 1. The equilibrium i moneary if m > 0 for all, and in hi cae he money-markeclearing condiion i a m + 1 = M+ 1. The marke-clearing condiion imply { c, a 1, a m 1} 0 { x, 1, M1} 0, w 1/ U( x), and once { } 0 ha been found, { q } 0 { p } 0 {min(, * 1 q )} 0, where i he real value of he equilibrium porfolio ha each agen bring ino he decenralized marke of period +1. Therefore, given a money upply proce { M } 0, and leing L( 1 ) [ 1 ( 1)], a moneary equilibrium can be ummarized by a equence { } 0 - m + 1M + 1 U( x ) E [ L( ) U( x )( x )] U( x ) E [ L( ) U( x ) ] m m lim E [ U( x ) ] 0 0 lim E [ U( x ) M ]. 0 m 1 0 The la wo condiion are he ranveraliy condiion are he Euler equaion for equiy and money, repecively. The fac ha boh ae can be ued a a medium of exchange in bilaeral rade implie ha he uual ochaic dicoun facor i augmened by he liquidiy facor L( 1 ), which capure he liquidiy value of he ae, i.e., he degree o which he ae i ueful a a medium of exchange, a he margin. Normaive Reul: Opimal Policy and Implemenaion The Pareo opimal allocaion in hi environmen can be found by olving he problem of a ocial planner who wihe o maximize average (equally weighed acro agen) expeced uiliy. The planner chooe a plan { c, q, n, y, h} 0 ubjec o he feaibiliy conrain, i.e., 0 c d, y h, and 0 q n for hoe agen and q = n =0 for hoe agen who are no. Under hee feaible plan { c, q} 0 uch ha for all feaible plan { c, q }, 0 T liminf E { [ u( q) q] U( c)} T 0 0 { [ uq ( ) q] Uc ( )} 0. Here, E 0 denoe he expecaion wih repec o he 17

7 QR probabiliy meaure over equence of dividend realizaion induced by F. The oluion o he planner problem i { c, q} { x, q * 0 } 0. In equilibrium, marke clearing implie c = x, i.e., However, he equilibrium allocaion ha q q *, which may hold wih ric inequaliy in ome ae. Tha i, equilibrium conumpion and producion in he decen- hi conex, he role of moneary policy i o enure ha he value of money i large enough o allow for q = q * wih probabiliy one, for all. To ae he reul ha follow, i will be convenien o inroduce he following noion of nominal inere rae. Imagine ha here exied an addiional ae in hi economy, an illiquid nominal bond, i.e., a one-period rik-free governmen bond ha pay a uni of money in he cenralized marke and which canno be ued in decenralized exchange. Le n denoe he price of hi n ae. In equilibrium, hi price mu aify U( x ) E[ U( x1) m 1 ]. Since n / m i he money price of a nominal bond, he (ne) nominal inere rae in a moneary equilibrium i i / n 1, or m equivalenly, (3) i E [ L( 1 ) 1 ] 1. E ( ) m m 1 PROPOSITION 1. Equilibrium quaniie in a moneary equilibrium are Pareo opimal if and only if i = 0 wih probabiliy one for all, in he moneary equilibrium. Propoiion 1 eablihe he opimaliy of he Friedman rule Milon Friedman (1969) precripion ha moneary policy hould induce a zero nominal inere rae in order o lead o an opimal allocaion of reource. The proof i a follow. The equilibrium allocaion i ef- q( ) q*, and hi equaliy hold if and only if q *, i.e., if and only if he real value of he equilibrium porfolio,, i a lea a large a he real liquidiy need, repreened by q *.The nominal inere rae, i, i zero if and only if L( 1) 1, and hi equaliy hold if and only if q *. Hence, q( ) q* if and only if i = 0. Inuiively, he co of producing real balance i zero o he governmen, o he opimum quaniy of real balance hould be uch ha marginal co, i i zero o he economic agen. I nex urn o he queion of implemenaion: Which moneary policie are conien wih a moneary equilibrium in which he nominal inere rae i a i opimal arge level of zero? The following reul addree he iue of (weak) implemenaion by characerizing a family of moneary policie ha are conien wih an equilibrium wih i = 0 for all. PROPOSITION 2. Le * j E j 1 [ U( xj)/ U( x)] xj, * = U( x)( * x ), and be he e of dae for which q * * 0 hold wih probabiliy > 0. Aume ha inf 0. A moneary equilibrium wih i = 0 wih probabiliy one for all exi under a deerminiic money upply proce { M } 0 if and only if he following wo condiion hold: M (4) lim (5) inf M 0 0 if. Condiion (4) and (5) are raher unrericive aympoic ply converge o zero. The econd condiion require ha aympoically, on average over he e of dae rae of he money upply mu be a lea a large a he dicoun facor. Verion of hi reul have been proven by Wilon (1979) and Cole and Kocherlakoa (1998) for deerminiic compeiive economie wih cah-in-advance conrain ha are impoed on agen every period wih probabiliy one. Propoiion 2 ha everal implicaion. Fir, even hough liquidiy need are ochaic in hi environmen (becaue equiy, whoe value i ochaic, can be ued alongide money a a mean of paymen), a deermin- a zero nominal rae in every ae of he world. Second, even wihin he cla of deerminiic moneary policie, here i a large family of policie ha can implemen he Pareo opimal equilibrium. Finally, i would be im- for he pah of he money upply o deermine wheher an opimal moneary policy i being followed. On he oher hand, a ingle obervaion of a poiive nominal an opimal moneary policy. 18

8 Ae Price, Liquidiy, and Moneary Policy Ricardo Lago Poiive Reul: Inere-Rae Targe and Ae Price In hi ecion I conider perurbaion of he opimal moneary policy ha coni of argeing a conan poiive nominal inere rae. I alo dicu ome of he poiive implicaion of change in he nominal and equiy reurn. To hi end, i i convenien o focu on a recurive formulaion in which price are invarian funcion of he aggregae ae = ( x, M ), i.e., = ( ), m = m ( ), m = ( ( ), ( )), where ( ) = U( x )[ ( ) x] and m m ( ) = U( x ) ( ), m and ( ) ( ) M. Alo, reric aenion o aionary moneary policie, i.e., :, o ha M+ 1 = ( x) M. To illurae he main idea a imply a poible, he following propoiion pecialize he analyi o he cae of independenly and idenically diribued dividend and liquidiy conrain ha would bind wih probabiliy one a every dae in he abence of money. PROPOSITION 3. Aume df( x, x) df( x). Le l( ) = 1 u( q * ) and be defined by l( ) = 1 /. Le (, ) be given, and uppoe ha B [1 0 1 * l( 0)] 0q. Then for any [ 0, 1], here exi a recurive moneary equilibrium under he moneary policy (6) ( x ; ) = l( ) q * 1 q* xu ( x) df( x) 1 l( ). l( ) xu ( x) df( ) ( ) 1 l( ) x xu x The equilibrium price of equiy and money are l( ) (7) ( x; )= 1 l( ) (8) m ( ; ) = xu ( x) df( x) U( x) l( ) q* xu ( x) df( x) xu( x) 1 l( ). U( x) M Togeher wih (3), he ae price (7) and (8) imply ha he moneary policy (6) induce an equilibrium gro nominal inere rae ha i conan (independen of ) and equal o l( ) 1 (wih equaliy only if = 1). The funcion (; ) dexed by he parameer, which effecively deermine he level of he conan nominal inere rae implemened by he policy. According o (8), real money balance and he value of money are decreaing in he nominal inere-rae arge (increaing in ). According o (7), under he propoed policy, he real price of equiy i increaing in he nominal inere-rae arge (decreaing in ). A 1, l ( ) 1, and herefore (according o Propoiion 1) he policy ( x ; ) approache an opimal policy under which he recurive moneary equilibrium decenralize he Pareo opimal allocaion. Noice ha ( x; 1 ) i he equilibrium equiy price ha would reul in a Luca (1978) economy wih no liquidiy need. Therefore, he fac ha ( x; 1 ) < ( x; ) for all x and any [ 0, 1 ) implie ha deviaion from he opimal policy raie real ae price above he value peced ream of dividend dicouned by he ochaic dicoun facor of Luca model, U( x1)/ U( x). On average, liquidiy conideraion generae a negaive relaionhip beween he nominal inere rae nominal rae, l( ) 1, rae i higher, real money balance are lower, and he liquidiy reurn on equiy rie, which caue i price o rie and i meaured real rae of reurn o fall. Inuiively, a higher nominal inere-rae arge implie ha buyer are on average hor of liquidiy, o equiy become more valuable a i i ued by buyer o relax heir rading conrain. Thi addiional liquidiy value average, a a higher inere rae. Propoiion 3 alo how explicily how moneary policy mu be conduced in order o uppor a recurive moneary equilibrium wih a conan nominal inere rae (wih he Pareo opimal equilibrium in which he nominal rae i zero a a pecial cae): he growh rae of he money upply mu be relaively low following ae in which he real value of he equilibrium equiy holding i below average. Equivalenly, he implied x and a nex-period ae x, if he realized real value of he equilibrium equiy holding in ae x i below he ae-x condiional expecaion of i value nex period. 19

9 QR Concluion I have preened a imple verion of a prooypical earch-baed moneary model in which money coexi hi formulaion, money i no aumed o be he only ae ha mu, nor he only ae ha can, play he role of a medium of exchange: nohing in he environmen preven agen from uing equiy along wih money, or inead of money, a a mean of paymen. Since he equiy hare i a claim o a riky aggregae endowmen, implie ha hey face aggregae liquidiy rik, in he ene ha in ome ae of he world, he value of equiy holding may urn ou o be oo low relaive o wha would be needed o carry ou he ranacion ha require a medium of exchange. Thi eem like a naural aring poin o udy he role of money and moneary policy in providing liquidiy o lubricae he mechanim of exchange in modern economie. In hi conex, I characerized a large family of opimal moneary policie. Every policy in hi family implemen Friedman precripion of zero nominal inere rae. Under an opimal policy, equiy price and reurn are independen of moneary conideraion. I have alo udied a cla of moneary policie ha arge a conan bu nonzero nominal inere rae. For hi perurbaion of he family of opimal policie, I found ha he model ariculae he idea ha, o he exen ranacion, i real rae of reurn will include a liquidiy reurn ha depend on moneary conideraion. A a reul of hi liquidiy channel, perien deviaion from he opimal moneary policy will caue he real price of ae ha can be ued o relax borrowing or oher rading conrain o exhibi perien deviaion from heir fundamenal value. Reference Cole, Harold L., and Narayana Kocherlakoa Zero nominal inere rae: Why hey re good and how o ge hem. Federal Reerve Bank of Minneapoli Quarerly Review 22 (Spring): Friedman, Milon The opimum quaniy of money. In The opimum quaniy of money and oher eay. Chicago: Aldine. Lago, Ricardo Ae price and liquidiy in an exchange economy. Reearch Deparmen Saff Repor 373. Federal Reerve Bank of Minneapoli Ae price, liquidiy, and moneary policy in an exchange economy. Working paper. New York Univeriy Some reul on he opimaliy and implemenaion of he Friedman rule in he Search Theory of Money. Journal of Economic Theory 145 (July): and policy analyi. Journal of Poliical Economy 113 (June): Luca, Rober E., Jr Ae price in an exchange economy. Economerica 46 (November): General equilibrium, growh, and rade: Eay in honor of Lionel McKenzie, ed. Jerry R. Green and Joé Alexandre Scheinkman, pp New York: Academic Pre. 20