Monetary policy and multiple equilibria in a cash-in-advance economy

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1 Economics Leers 74 (2002) locae/ econbase Moneary policy and muliple equilibria in a cash-in-advance economy Qinglai Meng* The Chinese Universiy of Hong Kong, Deparmen of Economics, Shain, N.T., Hong Kong, China Received 5 Ocober 2000; acceped 5 May 200 Absrac This aricle shows ha wheher acive ineres rae rules can generae equilibrium uniqueness or indeerminacy depends upon: () he magniude of he ineremporal elasiciy of subsiuion, and (2) he value of he seady sae inflaion rae. In paricular, if he ineremporal elasiciy is smaller han one, hen an acive ineres rae rule renders equilibrium indeerminae when he seady sae inflaion rae is sufficienly high, and ensures uniqueness of equilibrium when i is sufficienly low Elsevier Science B.V. All righs reserved. Keywords: Ineres rae rules; Indeerminacy JEL classificaion: E3; E52. Inroducion A well-known resul in he lieraure on ineres rae policy is ha acive ineres rae feedback rules, ha is, rules ha respond o increases in inflaion wih a more han one-for-one increase in he nominal ineres rae, are sabilizing by ensuring uniqueness of equilibrium. Some auhors have recenly sared o challenge he early resuls. For example, Benhabib e al. (200) demonsrae ha depending on how money eners preference or producion, acive ineres rae rules may desabilize he economy by generaing equilibrium indeerminacy. Resuls obained by all prior auhors sugges wheher or no a paricular ineres rae rule generaes equilibrium uniqueness or indeerminacy is independen of he value of he seady sae inflaion rae. Therefore, all hese resuls imply ha regardless of he magniude of he seady sae inflaion rae, *Tel.: ; fax: address: meng2000@cuhk.edu.hk (Q. Meng). See, for examples, Eric Leeper (99), Chrisopher Sims (994), Michael Woodford (994), Richard Clarida e al. (997), and Sephanie Schmi-Grohe and Marın Uribe (2000) / 02/ $ see fron maer 2002 Elsevier Science B.V. All righs reserved. PII: S (0)

2 66 Q. Meng / Economics Leers 74 (2002) cerain policy eiher sabilizes or desabilizes he economy. The presen paper shows ha, however, wheher acive ineres rae rules can generae uniqueness of equilibrium depends upon he magniude of he seady sae inflaion rae. In paricular, in a sandard cash-in-advance economy wih non-separable uiliy in consumpion and leisure, if he ineremporal elasiciy is smaller han one, hen acive ineres rae rules render equilibrium indeerminae if he seady sae inflaion rae is sufficienly high, and ensure uniqueness of equilibrium if he seady sae inflaion rae is sufficienly low. 2. A cash-in-advance economy 2.. The model The represenaive household s lifeime uiliy is given by: ` 2r U 5E e 0 uc, s l dd () where r. 0 denoes he rae of ime preference, c consumpion, l leisure. Assume ha cash has o be held in advance of purchasing goods. The liquidiy consrain ha he household faces can be formalized as: M $ apc (2) where M are nominal money balances held for purchases, P is he nominal price level, and a is a posiive parameer. The household also holds nominal bonds, B, which pay he nominal ineres rae R. 0. The household is subjec o he liquidiy and budge consrains: m $ ac (3) ~a 5sR 2 p da 2 Rm y 2 c 2 (4) where m are real cash balances, p he inflaion rae, lump axes, a5sm B d P, and y he oupu flow. Subsiue (3) ino (4), we have: ~a 5 R 2 p a 2 c ar y 2 (5) s d s d The household is also subjec o he following no-ponzi-game condiion: ` 2Efsd Rs2psdg s ds ` 0 lim e a > 0 (6) The household maximizes () subjec o (5) and (6). The opimaliy condiions are: u sc, l d5 l s ar d (7)

3 Q. Meng / Economics Leers 74 (2002) u (c, l ) 5 l w 2 ~l 5 lsr p 2 R d (8) (9) ` 2Efsd Rs2psdg s ds ` 0 lim e a 5 0 (0) where erm s ar d is he usual moneary wedge generaed in cash-in-advance models, due o he fac ha any form of wealh (excep cash iself) has o be made liquid before goods can be consumed. This increases he effecive price of consumpion bu no ha of leisure. w here is he real wage rae. We assume he following sandard uiliy funcion used exensively in he RBC lieraure: s b 2b 2s d d uc, s l 5 c l s 2 s d, 0, b, () Assume ha he household is endowed wih uni of ime and choose endogenously o have leisure 2 and o work. To simplify he analysis, we assume he linear producion funcion: y5 h where h5 2 l is he labor supply Ineres rae rules Following Leeper (99) and Benhabib e al. (200), we assume ha he moneary policy akes he form of an ineres rae feedback rule whereby he nominal ineres rae is se as an increasing funcion of inflaion rae. Specifically: R 5 cp s d where c s? d is coninuous, non-decreasing, and here exiss a leas one p *.2r such ha cp s * d5 r p *. We say ha he moneary auhoriy implemens acive moneary policy if c9sp * d., and passive moneary policy if c9sp * d,. Governmen purchases are assumed o be zero a all imes. The budge consrain of he governmen is given by B~ 5 RB 2 M~ 2 P, which can be wrien as: ~a 5sR 2 p da 2 Rm 2 We assume he fiscal policy is Ricardian in he sense ha (0) is always saisfied boh in and off equilibrium Equilibrium and local dynamics By subsiuion, he firs-order condiions are: b 2b 2s b2 2b s d s s dd c l bc l 5 l ac p (4) (2) (3) The major resuls obained below do no change if we insead use he producion funcion y 5 h (0,h, ). 2 h

4 68 Q. Meng / Economics Leers 74 (2002) s d s d b 2b 2s 2b b c l 2 b l c 5 l ~l 5 lsr p 2 cp s dd In equilibrium, he goods marke clear: c 5 h 5 2 l (5) (6) (7) From (4), (5) and (7), he following equaion can be derived: b l ]]]]]] 5 acsp s 2 bd s 2 l d d Since csp d is an increasing funcion of p, from (8), l is also an increasing funcion of p. Therefore: l5 lsp d, l9sp d. 0 (9) From (7) c is a decreasing funcion of inflaion rae p c5 csp d, c9sp d, 0 (20) Inuiion for boh (9) and (20) is sraighforward. When inflaion rae is high, people have less incenive o work and hence consume more leisure. They also consume less of regular goods when he cos of purchasing is high. From (5) and (7), b s 2sd ss2bdb s ds d l 5 2 b 2 l l Since l is a funcion of p, from (2) l is also a funcion of p. (6) can be wrien as: lp s dsr p2 cp s dd ~p 5]]]]]]] l9sp d Linearize (2) a he seady sae p *, l(p *)( 2 c9(p *) ~p 5 []]]]]]](p2 p *) 5D(p2 p *) (23) l9(p *) The dynamics of he above differenial equaion hinges on he sign of l(p *)( 2 c9(p *) D5 F]]]]]] G. l9(p *) In paricular, if D.0 he unique soluion is he seady sae p 5 p *.IfD,0, on he oher hand, here are a coninuum of equilibrium rajecories ha converge o he seady sae, and hence equilibrium indeerminacy arises. In order o see he connecion beween he sance of ineres rae policies and equilibrium, we consider nex wo cases separaely. Case. s < (c and l are separable when s 5 ), from (2), he righ-hand side is a decreasing (8) (2) (22)

5 Q. Meng / Economics Leers 74 (2002) funcion of l, and hence a decreasing funcion of p. Therefore l9sp d, 0. We have he following proposiion. Proposiion. If he ineremporal elasiciy of subsiuion (he inverse of s) is no less han one, hen for acive ineres rae rules he only equilibrium is he seady sae. Case 2. s., (2) can be wrien as: bs2 s d ss2bdb s d l l 5 ( 2 b ) 2 l (24) The RHS of (24) is a U-shaped funcion of l (or p ), which reaches is minimum value when: s 2 l 5 2 bs]]d s Solve for he corresponding p by subsiuing (25) ino (8): (25) ss 2 bd b ]]]]] 5 acsp ss 2 2 b ds d d (26) Therefore l5 lp s d has he propery such ha for p, p, l9sp d, 0, and for p. p, l9sp d. 0. One can see ha he connecion beween he ineres rae policy and equilibrium now depends upon he relaive values of p and he seady sae inflaion rae p *. In paricular, we can obain he following proposiion. Proposiion 2. If he ineremporal elasiciy of subsiuion is less han one, hen for acive ineres rae rules he only equilibrium is he seady sae for sufficienly low seady sae inflaion rae, and equilibrium is indeerminae for sufficienly high seady sae inflaion rae. The inuiion behind he indeerminacy resul is as follows. Suppose he consumpion is reduced below is seady sae level, and a he same ime leisure is increased above is seady sae level, i follows ha he nominal ineres rae has o be above is seady sae level. Acive ineres rae rules imply ha he increase in he nominal ineres rae is associaed wih an increase in he real ineres rae. The rae of he marginal uiliy of consumpion increases in response due o low ineremporal elasiciy and high inflaion. The inflaion rae and hence he nominal ineres rae will decrease. The consumpion level will increase and reurn o is seady sae level and his rajecory is consisen wih an equilibrium. I is well known ha here is no consensus on he esimae of he parameer s, and mos auhors believe i is greaer han one. In his sense, he resul in Proposiion 2 may be more plausible han ha in Proposiion. In he cash-in-advance and endowmen economy, one can show ha Proposiion holds, which is similar o he resuls obained in Leeper (99) or Benhabib e al. (200) in money in uiliy funcion models. Schmi-Grohe and Uribe (2000) sudy a cash-in-advance model wih endogenous labor similar o he presen paper, bu hey assume a uiliy funcion separable in consumpion and leisure (s 5 ). In all hese papers, connecion beween he ineres rae policies and he magniude of seady sae inflaion rae can no be esablished.

6 70 Q. Meng / Economics Leers 74 (2002) Acknowledgemens I would like o hank Jess Benhabib, Mark Gerler, Yong Wang and Andres Velasco for useful commens. References Benhabib, J., Schmi-Grohe, S., Uribe, M., 200. Moneary policy and muliple equilibria. American Economic Review 9, Clarida, R., Gali, J., Gerler, M., 997. Moneary policy rules and macroeconomic sabiliy: evidence and some heory. Working Paper, New York Universiy. Leeper, E., 99. Equilibria under acive and passive moneary and fiscal policies. Journal of Moneary Economics 27, Sims, C., 994. A simple model for he sudy of he deerminaion of he price level and he ineracion of moneary and fiscal policy. Economic Theory 4, Woodford, M., 994. Moneary policy and price level deerminacy in a cash-in-advance economy. Economic Theory 4, Schmi-Grohe, S., Uribe, M., Price-level deerminacy and he moneary policy under a balanced-budge requiremen. Journal of Moneary Economics 45,