Cash In Advance Models Econ602. Spring Lutz Hendricks

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1 Cash In Avance Moels Econ602. Spring Luz Henricks In his secion we suy a secon moel of money. Recall he cenral quesions of moneary heory: 1. Why o people hol money, an asse ha oes no pay ineres (rae of reurn ominance)? 2. Why is money value in equilibrium? 3. Wha are he effecs of moneary policy: one ime increases in he money supply or changes in he money growh rae? The OLG moels we suie provies a saisfacory answer why money is value: i is a bubble. However, moels of his kin fall shor because money is never hel if is rae of reurn is ominae by anoher asse. The iea of CIA moels is o explicily moel he fricions ha make money valuable, bu he moeling has very much he characer of a reuce form: i is simply assume ha cerain goos can only be purchase wih bis of green paper. Oher variaions of moneary moels (ha we won cover) have much cleaner micro founaion. In search moels a househol canno ea his own proucion an mus herefore rae. Bu fining a raing parner is cosly. Money is value because of he expecaion ha ohers will value i in he fuure as well. If I mee an iniviual who woul like o consume my goo, I may sell i for money, because I on like o ea his goo. Bu I may expec o mee someone in he fuure who has somehing eible o offer in exchange for my money. Limie paricipaion moels are similar o CIA moels, bu spell ou he insiuional srucure ha gives rise o marke fricions. 1. Timing The overall moel srucure resembles ha of he neoclassical growh moel. Bu i is now assume ha some goos can only be purchase for money, no for oher goos. The iming of he moel is crucial. 1. The househol eners he perio wih capial k an a sock of money m He hen receives a ransfer of money from he governmen so ha his perio money holings are m m = 1 + τ. Cash In Avance Moels 1

2 3. Nex he househol prouces an sells his oupu for money o be receive a he en of he perio. A he same ime he uses m o buy goos from oher househols (c an k +1 ). 4. The househol is pai for he goos he sol in sep 3 so ha his en of perio money sock is m. Noe ha money is in limbo for a while. The househol nees money o pay ohers, bu oes no acually receive paymens for his own sales unil he en of he perio. One way of hinking abou his is ha he money is in he mail or in escrow unil he en of he perio. I canno be use for anyhing uring ha ime. As a consequence, househols mus hol money a if hey wan o buy consumpion goos a Househol Problem While his souns like a complicae seup, he moificaion o he househol problem is acually quie minor. There is simply an aiional consrain (he CIA consrain) which says ha expeniures on goos ha nee o be pai for wih cash canno excee m. The househol solves (BC) (CIA) max = β 1 u ( c ) subjec o c + m / p = f ( k ) k + k m / p m / p 1 (1 δ) k c + k+ m + 1 = m + τ+1 The buge consrain is fairly sanar. Afer proucion he househol receives goos f( an unepreciae capial. In aiion, he househol carries money balances m ino he perio, which mus be ivie by he price level because hey are in nominal erms. There are hree uses for he househol s resources: consumpion, invesmen in capial, an money eman, m. The law of moion for money saes ha m +1 equals oay s money eman plus a money ransfer from he governmen, τ +1. Exacly wha kins of goos have o be bough wih cash is arbirary. Some moels have cash an crei goos. Here we assume ha all goos (incluing replacemen invesmen) are bough wih cash. I will urn ou ha he CIA consrain hols wih equaliy if he rae of reurn on money is less han ha on capial (he nominal ineres rae is posiive). Cash In Avance Moels 2

3 2.1 DP Formulaion Noe ha he househol now has wo (iniviual) sae variables (m,. The Bellman equaion is herefore: V ( m, = max u( c) + βv ( m, ) + λ ( BC) + γ ( CIA). To wrie his properly in recursive form, we nee o impose as a choice variable (his woul no work uner uncerainy). FOC: V ( m, = max u( c) + βv ( m, ) m m + λ[ f ( k + m / p c ( m τ ) / p] + γ[ m / p c k] u (c) = λ + γ, β V m ( ) = λ / p, β V ( ) = λ + γ KT: γ[ m / p c k] = 0, γ 0 k Thus: u ( c) = βv ( ) k = 1 + τ. We can hen use m +1 Envelope coniions: V m = ( λ + γ) / p, = λ[ f ( + 1 δ] + γ[1 δ] V k Eliminae erivaives of V: β [ λ + γ ]/ p = λ / p, ( λ + γ) / β = λ f ( ) + [1 δ][ λ + γ ] (1) βu ( c ) p / p = λ an obain he Euler equaion for c: 2 (2) u ( c) = β u ( c )( p / p ) f ( ) βu ( c ) To summarize, he soluion of he househol problem akes prices (p) as given an consiss of a value funcion an policy funcions [c(k, m), k (k, m), m (k, m)] ha saisfy he Bellman equaion an he policy funcions solve he max problem an saisfy he buge consrain. In sequence language a soluion consiss of sequences (c, k, m ) ha saisfy he Euler equaion an he buge consrain. If he CIA consrain bins, i provies he hir equaion. Wha happens if he CIA consrain oes no bin is iscusse below. We also nee wo ransversaliy coniions ha preven he presen value of wealh from blowing up. Noe ha we nee four bounary coniions because we have wo firs-orer an one secon-orer ifference equaion. The inerpreaion of he Euler equaion is as follows. Suppose he househol gives up a uni of c an buys more capial insea (k rises). He gains wo benefis: Nex perio, he can consume he unepreciae capial. Plus he prouces aiional oupu ( f ( )), which he can inves in money ha yiels ( p / p ) unil perio +2, when i can be consume. The complicaion is ha any reucion in consumpion oay can only be consume in +2. Of course, i coul be Cash In Avance Moels 3

4 consume in +1, if he househol hel money insea of aiional capial, bu ha woul no be opimal as long as money pays a lower ineres rae han k. 2.2 When oes he CIA consrain bin? We nex show ha he CIA consrain bins unless he reurn on money equals ha on capial, i.e. he nominal ineres rae is zero. If he CIA consrain oes no bin, hen γ = 0 an herefore λ / β = λ [ f ( ) + 1 δ], so we ge a sanar Euler equaion he same as in a non-moneary economy. Wha happens is ha holing money is free; i has no opporuniy cos. Money hen oes no isor he real allocaion in any way. Wha oes moneary policy have o o in orer o achieve oucome? Wih γ = 0, we have β λ / p = λ / p an hus λ /( βλ ) = p / p = [ f ( + 1 δ]. The real reurn on money is he negaive of he inflaion rae an he RHS of he equaion is he real reurn on capial. Wha happens o he soluion of he househol problem in ha case? We sill nee o eermine c, k, an m sequences using he Euler equaion, he buge consrain an a hir coniion ha is no longer he CIA consrain. Now he capial sock is eermine by p / p = f ( + 1 δ, while money eman follows from he buge consrain. The inuiion is illusrae in he nex figure. The househol can hol wo asses, k an m. The rae of reurn o m is given (p / p ). The rae of reurn o k is a ecreasing funcion. If he CIA consrain oes no bin, he househol accumulaes k unil is reurn falls o ha of money. Any aiional wealh is accumulae in he form of money. Rae of reurn p / p' CIA bins CIA oes no bin f'( + 1 δ Asse eman Cash In Avance Moels 4

5 3. Equilibrium Governmen: The governmen s only role is o han ou lump-sum ransfers of money. The money growh rule is = g m 1 τ so ha money holings in perio are m m 1 + τ = ( 1+ g) m 1 =. Marke clearing: Goos marke clearing requires (he same as feasibiliy): c + = f ( k. Money marke clearing is implici in he noaion. Definiion: An equilibrium is a sequence k, m, c, τ, p ) ha saisfies ( 1. he money growh rule an efiniion of τ (sor of a governmen buge consrain); 2. he househol opimaliy coniions (see above) (3 equaions) 3. he goos marke clearing coniion. 4. Seay Sae In seay sae, all real variables are consan ( c, k, m / p). This requires π = g o hol real money balances consan. Since u (c) is consan, he Euler equaion (2) implies 1 = β 2 (1 + π) 1 f ( ) β. Using 1 + π = 1+ g his can be solve for he capial sock: f ( = (1 + g)[1 β(1 δ)]/ β 2 Assuming ha he CIA consrain bins: implies c = f ( δ k. f ( = m / p. Goos marke clearing wih consan k 4.1 Properies of he Seay Sae 1. Money is no superneural. Higher inflaion (g ) implies a lower capial sock an lower consumpion. The reason is ha inflaion increases he cos of holing money, which is require for invesmen. There is an inflaion ax on any aciviy ha requires money holings. Moels of his kin are, in fac, commonly use o suy he welfare coss of inflaion (e.g. Abel 1997). Definiion: Money is calle neural if changing he level of M oes no affec he real allocaion. I is calle superneural if changing he growh rae of M oes no affec he real allocaion. Exercise: Show ha super-neuraliy woul be resore, if he CIA consrain applie only o consumpion ( m / p c). Wha is he inuiion for his fining? Cash In Avance Moels 5

6 2. The velociy of money is one. Higher inflaion reuces money eman only be reucing oupu. This is a irec consequence of he rigi CIA consrain an probably an unesirable resul. Obviously, his woul no be a goo moel of hyperinflaion. 3. The Frieman rule ensures ha he CIA consrain oes no bin. I can be shown ha his maximizes welfare. The Frieman rule requires he nominal ineres rae o be zero. The iea is o eliminae he opporuniy cos of holing money, which inuces agens o inefficienly economize on holing a goo ha can be prouce coslessly. The opporuniy cos of money is he gap beween is rae of reurn an ha of oher asses (here. Seing he real reurn on money (he negaive inflaion rae) equal o ha of capial, in which case he nominal ineres rae is zero, eliminaes his inefficiency. This policy rule urns ou o be opimal (in seay sae) in a number of moels. To see ha he Frieman rule is opimal here, i is easy o show ha he Frieman rule inuces he firs-bes allocaion. The planner maximizes iscoune uiliy subjec o he feasibiliy consrain only. There is no money in he planner s problem. Even if he planner faces somehing akin o a ransacions echnology ha requires pieces of paper as inpu, he fac ha paper can be prouce coslessly means ha such a echnology canno affec he allocaion. The planner s problem hus reuces o a sanar non-moneary growh moel. The Pareo opimal allocaion is characerize by an Euler equaion an a feasibiliy consrain. The Euler equaion is he same as he househol Euler equaion when he CIA consrain oes no bin. The feasibiliy consrain is he same as he goos marke clearing coniion. Hence, boh allocaions mus be he same an he Frieman rule is firs bes. Is his a goo heory of money? For many purposes he answer is yes (alhough Neil Wallace isagrees). The moel is consisen wih rae of reurn ominance an explains why money is value: i provies liquiiy services. Of course, his resul is essenially assume. Why oes one nee money insea of -bills o pay bills? Clearly, one shoul hink of he CIA consrain as a reuce form for a fully specifie moel wih raing fricions (a search moel?). Sill, he moel oes allow o hink abou quesions for which enogeneiy of he insiuional srucure (ha gives rise o he CIA consrain) is no essenial. There is one cavea, hough: he moel s properies epen on he exac naure of he CIA consrain (see he homewor, which is arbirary. 5. Reaing BF chaper 4 (CIA moel in coninuous ime). Cash In Avance Moels 6

7 Abel, Anrew B. (1985). Dynamic Behavior of Capial Accumulaion in a Cash in Avance Moel. Journal of Moneary Economics. [Discusses he ynamics of he CIA moel.] Abel, Anrew B. (1997). Commen on The Coss an Benefis of Going from Low Inflaion o Price Sabiliy by Marin Felsein. In: C. Romer an D. Romer, es., Reucing Inflaion: Moivaion an Sraegy. Chicago: Universiy of Chicago Press. [Uses a CIA moel o quanify he welfare coss of inflaion. They urn ou o be iny.] Lucas, Rober E. (1996). Nobel Lecure: Moneary Neuraliy. Journal of Poliical Economy 104(4): McCanless, George T.; Warren E. Weber (1995). Some Moneary Facs. Feeral Reserve Bank of Minneapolis Quarerly Review 19(3): Wallace, Neil (1998). A icum for moneary heory. Feeral Reserve Bank of Minneapolis Quarerly Review, Winer: [A criique of reuce form moels of money (such as CIA or money in he uiliy funcion).] The exension o a shopping ime moel in which he CIA consrain is less rigi is iscusse in Ljunquis an Sargen (2000), ch. 17. They also show ha he implicaions of moneary policy can be very ifferen, if money is use o finance governmen spening. Cash In Avance Moels 7