Money/monetary policy issues an enduring fascination in macroeconomics. How can/should central bank control the economy? Should it/can it at all?

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1 SIMPLE DSGE MODELS OF MONEY PART I SEPTEMBER 22, 211 Inroducion BASIC ISSUES Money/moneary policy issues an enduring fascinaion in macroeconomics How can/should cenral bank conrol he economy? Should i/can i a all? Roles of money Medium of exchange (ransacions role) Uni of accoun (numeraire role) Sore of value (asse role) Highlighed in CIA, MIU, and money-search approaches Highlighed in New Keynesian approach How o model money in DSGE environmen? Which role o model? Which role is racable o model? Which role is mos relevan for conduc of moneary policy? Sepember 22,

2 HOUSEHOLDS s.. P he nominal price of c equivalenly, he nominal price level Household opimizaion max E β uc (, n) c, n, M, B = Pc + M + B = Pw n + M + (1 + i ) B + T Flow budge consrain Lump-sum moneary injecion/conracion Nominal consumpion spending Nominal money Nominal bond holdings holdings Pc M Nominal labor income Nominal ineres rae (on previously-accumulaed nominal bonds) Cash-in-advance (CIA) consrain - forces consumers o hold money - ariculaes a ransacions moive Sepember 22, HOUSEHOLDS s.. P he nominal price of c equivalenly, he nominal price level Household opimizaion max E β uc (, n) c, n, M, B = Pc + M + B = Pw n + M + (1 + i ) B + T Flow budge consrain Lump-sum moneary injecion/conracion Nominal consumpion spending Nominal money Nominal bond holdings holdings Pc M Nominal labor income CIA consrain a fricion on economy Pareo-opimal allocaions do no require i Nominal ineres rae (on previously-accumulaed nominal bonds) Cash-in-advance (CIA) consrain - forces consumers o hold money - ariculaes a ransacions moive Money no essenial as in models of Kiyoaki and Wrigh (1993), Lagos and Wrigh (25) Does no ENDOGENOUSLY EXPAND consumers se of feasible rades. Because underlying DSGE model feaures full se (including over all sae-dae pairs) of Arrow-Debreu securiies complee markes! Trade does no require money Sepember 22,

3 Removing moneary fricion.requires an allocaion ha feaures a zero muliplier on CIA consrain implies zero nominal ineres rae Friedman Rule Benchmark resul in moneary heory Compleely relaxing moneary fricion requires eliminaing any (opporuniy) cos of holding money Sepember 22, Removing moneary fricion.requires an allocaion ha feaures a zero muliplier on CIA consrain implies zero nominal ineres rae Friedman Rule Benchmark resul in moneary heory Compleely relaxing moneary fricion requires eliminaing any (opporuniy) cos of holding money Really he same hing Oher Inerpreaions Eliminae he wedge beween alernaive nominal asses: i = makes money and nominal bonds equivalen asses (in erms of heir cos and benefi properies) Eliminae he wedge in he consumpion-leisure opimaliy condiion Are moneary fricions empirically imporan?...and hus, is he Friedman Rule of pracical use for advising moneary policy? Sepember 22,

4 Household opimaliy condiions hh muliplier on CIA consrain hh muliplier on budge consrain i φ = λ i No-arbirage beween money and nominal bonds (Assumpion: i in he informaion se of ime ) Sepember 22, Household opimaliy condiions hh muliplier on CIA consrain hh muliplier on budge consrain i φ = λ i No-arbirage beween money and nominal bonds (Assumpion: i in he informaion se of ime ) un( c, n ) i = w + uc( c, n) 1+ i Consumpion-leisure opimaliy condiion - relaive price depends on w AND i Efficiency requires C-L opimaliy depends on real wage. bu no on moneary aspecs of economy (nonechnology) Friedman Rule achieves Pareo efficiency along his margin Sepember 22,

5 Household opimaliy condiions hh muliplier on CIA consrain hh muliplier on budge consrain Noe disuiliy of labor appears in ineremporal MRS If moneary fricion were shu down, would have u c here as usual. i φ = λ i un( c, n ) i = w + uc( c, n) 1+ i Eiher hrough Friedman Rule or hrough cashless New Keynesian environmen (laer ) un( c, n ) un( c 1, n 1) P + + = (1 + i) β E w w+ 1 P+ 1 No-arbirage beween money and nominal bonds (Assumpion: i in he informaion se of ime ) Sepember 22, Consumpion-leisure opimaliy condiion - relaive price depends on w AND i Efficiency requires C-L opimaliy depends on real wage. bu no on moneary aspecs of economy (nonechnology) Friedman Rule achieves Pareo efficiency along his margin Consumpion-savings opimaliy condiion (aka bond Euler equaion) (aka Fisher equaion!) Household opimaliy condiions (coninued) φ i = λ i un( c, n) i = w + uc( c, n) 1+ i un( c, n ) un( c 1, n 1) P + + = (1 + i) β E w w+ 1 P+ 1 M c = P No-arbirage beween money and nominal bonds Consumpion-leisure opimaliy condiion Consumpion-savings opimaliy condiion (aka bond Euler equaion) (aka Fisher equaion) Binding CIA consrain Obvious if i > (why hold excess money?) Also assume i even in saes where i = : pins down a moneary equilibrium level of M, hence is an equilibrium selecion device Sepember 22,

6 Household opimaliy condiions (coninued) φ i = λ i un( c, n) i = w + uc( c, n) 1+ i un( c, n ) un( c 1, n 1) P + + = (1 + i) β E w w+ 1 P+ 1 M c = P Res of he environmen T = M M = (1 + μ ) M No-arbirage beween money and nominal bonds Consumpion-leisure opimaliy condiion Consumpion-savings opimaliy condiion (aka bond Euler equaion) (aka Fisher equaion) Binding CIA consrain Obvious if i > (why hold excess money?) Also assume i even in saes where i = : pins down a moneary equilibrium level of M, hence is an equilibrium selecion device w = marginal produc of labor (linear producion + compeiive facor marke) Gov budge: Resource consrain: c = zn Sepember 22, Household opimaliy condiions (coninued) φ i = λ i Define un( c, n) i π +1 = P +1 / P 1 = w + uc( c, n) 1+ i μ +1 = M +1 / M -1 un( c, n) un( c+ 1, n+ 1) 1 = (1 + i) β E w w π + 1 Combine and -1 (binding) c 1+ μ CIA consrains c = inflaion 1 + π Ariculaes a quaniy-heoreic channel No-arbirage beween money and nominal bonds Consumpion-leisure opimaliy condiion Consumpion-savings opimaliy condiion (aka bond Euler equaion) (aka Fisher equaion) Equilibrium link beween money growh and Res of he environmen w = marginal produc of labor (linear producion + compeiive facor marke) Gov budge: T = M M = (1 + μ ) M Resource consrain: c = zn Examine seady-sae equilibrium Sepember 22,

7 Household opimaliy condiions in deerminisic seady sae un (, c n) i = w 1 u (, c n) + 1+ i c φ i = λ i No-arbirage beween money and nominal bonds Consumpion-leisure opimaliy condiion Friedman Rule: i = π = β -1 BUT ONLY IN STEADY STATE! NOT (necessarily) dynamically. and opimal policy calls for μ = β -1 (i.e., SHRINK nominal money supply!) 1 + π = β (1 + i) Consumpion-savings opimaliy condiion (aka bond Euler equaion) (aka Fisher equaion) 1 1 = + μ 1 + π Ariculaes a quaniy-heoreic channel Equilibrium link beween money growh and inflaion Res of he environmen w = marginal produc of labor (linear producion + compeiive facor marke) Gov budge: T / P= (1 + μ)( M / P)(1/(1 + π )) Resource consrain: c= n Sepember 22, OTHER ANALYSIS Imply φ <, i.e., money NOT valued for exchange Oher aspecs of equilibrium μ < β - 1 (in seady-sae!) inconsisen wih moneary equilibrium Dynamic analog: i < inconsisen wih moneary equilibrium Zero-lower-bound consrain Model s policy rae ypically idenified wih a (shor-run Euler equaion) marke ineres rae Wheher CIA models, MIU models, New Keynesian models, money search models Model mechanism: change in policy rae (poenially) affecs ineremporal incenives (i.e., he real ineres rae) A valid empirical idenificaion? Term-srucure issues? Oher issues? See Canzoneri, Cumby, and Diba (27 JME) Sepember 22,

8 CIA Models OTHER VARIANTS OF CIA Cash good/credi good model Lucas and Sokey (1983) Foundaion for Ramsey models of opimal fiscal and moneary policy see Chari and Kehoe (1999 Macro Handbook chaper) Subse of goods (c 1 ) require cash in advance Subse of goods (c 2 ) do no require cash in advance uc 1 1 i u = + MRS cash/credi = gross nominal ineres rae c2 Moneary policy creaes a STATIC wedge!... Invesmen in CIA consrain Sockman (1981): long-run inflaion lowers long-run capial sock M c + k+ 1 (1 δ ) k P Basic Idea: Posiive nominal ineres rae axes whaever is in he CIA consrain Sepember 22, Oher Approaches ALTERNATIVE MONETARY MODELS Alernaives o CIA Money in he uiliy funcion (MIU) models E = M β u c, P Shopping-ime & ransacions coss models Nominal money holdings reduce cos of acquiring goods Feensra (1986 JME) shows condiions under which CIA, MIU, shopping-ime are equivalen Can hink of as Friedman Rule running in he background Go cashless New Keynesian models don model money demand a all (or, a bes, as an appendage separae from he main equilibrium) Go for deep micro-foundaions Kiyoaki and Wrigh (1989, 1993) Lagos and Wrigh (25), Aruoba, Waller and Wrigh (26) Sepember 22,