Keiichi Tanaka Graduate School of Economics, Osaka University. Abstract

Transcription

1 Indeerminacy of equilibrium price of money, marke price of risk and ineres raes Keiichi Tanaka Graduae School of Economics, Osaka Universiy Absrac This paper shows ha a marke price of nominal risk plays an imporan role in he deerminacy of he price of money under a sochasic coninuous ime moneary economy. I is presened ha a sufficien condiion for he deerminacy of he price of money is eiher an exogenously given nominal shor rae or an exogenously given marke price of nominal risk, which implies ha a differen marke price of nominal risk may yield a differen price of money. Thus he nominal pricing kernel is no endogenously deermined under he original assumpions of he model. If cenral banks can deermine no only he money supply bu also he nominal shor rae process, he policy leads o he deerminacy of a process of he price of money and he pricing kernels. Explici soluions for real quaniies and nominal quaniies are provided for wo cases of uiliy funcional forms. Ciaion: Tanaka, Keiichi, (23 "Indeerminacy of equilibrium price of money, marke price of risk and ineres raes." Economics Bullein, Vol. 7, No. 4 pp. Submied: April 3, 23. Acceped: May 28, 23. URL: hp:// 3G2A.pdf

2 . Inroducion The purpose of his paper is (i o show ha in general i is no possible o derive he price of money endogenously in a sochasic moneary economy wih money-inhe-uiliies, and (ii o presen a way o overcome he issue of he indeerminacy of he price of money, real ineres raes and nominal ineres raes from financial economics poin of view. We follow he sochasic coninuous-ime seing of Basak and Gallmeyer (999 and exend heir resuls on he price of money o more general cases in he closed moneary economy. Wih paying more aenion o he imporan role of he marke prices of he risks we sudy he endogenous relaionship among real quaniies and nominal quaniies. Then we show ha a sufficien condiion for he deerminacy of he price of money is eiher (bu no boh an exogenously given nominal shor rae or an exogenously given marke price of nominal risk in addiion o he nominal money supply and he consumpion commodiy endowmen. The equaion for he price of money should be solved backwards from he ransversaliy condiion. Our idea is ha if he diffusion erm is given, hen he price of money can be obained accordingly, which implies ha a differen marke price of nominal risk may yield a differen price of money. I leads o he indeerminacy in he model. There are wo implicaions from our resuls ha are no observed in he deerminisic cases. Firs, he nominal pricing kernel is no endogenously deermined under he original assumpions of he model. Secondly, if cenral banks can deermine no only he money supply bu also he nominal shor rae process, he policy leads o he deerminacy of he price of money and he pricing kernels. Explici soluions for real quaniies and nominal quaniies are provided for wo cases of uiliy funcional forms. 2. The Model 2. Basic Seing and Pricing Kernels In his subsecion we seup he model by following Basak and Gallmeyer (999 and hen briefly review he basic resuls of he real pricing kernel and he nominal pricing kernel. 2 We consider a closed moneary economy where a represenaive agen is endowed wih a perishable consumpion commodiy δ and he nominal money supply M and maximizes his lifeime uiliy under cerain budge consrains in an infinie ime horizon. The uncerainiies in he economy are caused by an n-dimensional Brownian moion W on a complee filered probabiliy space (Ω, F, P,{F } where P is some objecive probabiliy measure. The filraion {F } is he augmened filraion generaed by W. Bakshi and Chen (996 solves he price of money wih a logarihm uiliy funcion. Basak and Gallmeyer (999 considers a wo-counry model wih more general uiliy funcions. I is easily checked ha heir discussion can be applied o our model. 2 The erms sae-price densiy or sochasic discoun facor are ofen used synonymously wih pricing kernel. The real pricing kernel is used for he cash flow in real erm while he nominal pricing kernel is used for he cash flow in nominal erm.

3 The endowmen δ in he commodiy is given by an Io process dδ( =δ( µ δ (d + σ δ (dw (, δ( = δ, where µ δ and σ δ are respecively R-valued and R n -valued processes. (Hereafer we assume ha hese coefficien processes of d and dw( have appropriae dimensions and condiions. The nominal money supply M is also exogenously given by dm( =M( µ M (d + σ M (dw (, M( = M. The price of money q is supposed o follow dq( =q( µ q (d + σ q (dw (, which will be deermined laer. We assume ha he represenaive agen can rade n securiies wihou any fricion: (i a real money accoun which pays he real shor rae r coninuously as dividend in he uni of he commodiy, (ii a nominal money accoun which pays he nominal shor rae R coninuously as dividend in he uni of money, (iii a sock which pays he commodiy δ and (iv zero-coupon bonds wih disinc n 3 mauriies. Ne supplies of hese securiies in his economy are assumed o be zero. R is assumed o follow an Io process dr( =R(µ R (d + σ R (dw (. The price of he sock a ime is denoed by S(. Given he price of money q in unis of he commodiy, he represenaive agen chooses he consumpion c and he nominal money balance M D o maximize a life uiliy E e ρ u(c(,q(m D (d subjec o he budge consrain E z((c(+r(q(m D (d S( + E z(r(q(m(d, where z is he real pricing kernel. u(c, m is assumed o be increasing, sricly concave and hree imes coninuously differeniable in boh argumens. Noice ha we assume and impose he ransversaliy condiion lim z(t q(t M D (T F = T ( in he budge consrain of he maximizaion problem, and we can show ha he saic budge consrain above is equivalen o he dynamic consrain ha he consumpion a each insan is financed by he endowmen, he money supply and a porfolio of securiies as argued in Basak and Gallmeyer (999 Proposiion 2.. By he firs order condiions of he maximizaion problem and he general equilibrium condiions c = δ and M D = M, i is well-known ha he real pricing kernel z is proporional o he marginal uiliy of consumpion and he nominal 2

4 shor rae is he marginal rae of subsiuion beween he commodiy and he real money balance, z( = e ρ u c(δ(,q(m( u c (δ(,q(m(, (2 R( = u m(δ(,q(m( u c (δ(,q(m(. (3 We can define he nominal pricing kernel Z as Z( =z(q(/q(. (4 We call λ and Λ respecively he marke price of real risk and he marke price of nominal risk. Then he pricing kernels are characerised by he pair of he shor rae and he marke price of he risk, dz( = z( r(d + λ(dw (, z( =, (5 dz( = Z( R(d +Λ(dW (, Z( =. (6 If he process of he real money m = qm, which is no ye deermined, is assumed o follow dm( =m( µ m (d + σ m (dw (, we have he following general expressions by applying Io s formula o equaions (2 and (4 and comparing wih (5 and (6, r( = ρ + A cc (µ δ (δ(+a cm (µ m (m( ( A ccc ( σ δ (δ( A ccm (σ δ (σ m (δ(m(+a cmm ( σ m (m( 2, (7 R( = r( µ q (+σ q (λ(, (8 λ( = A cc (σ δ (+A cm (σ m (, (9 Λ( = λ( σ q (, ( where A ij ( = u ij(δ(,m( u c (δ(,m(, A ijk( = u ijk(δ(,m( u c (δ(,m(, and represens an n-dimensional Euclidean norm. Noice ha equaion (7 represens he real ineres rae, as being well-known, as a sum of (i he ime-preference, (ii he risk aversion coefficien imes he expeced growh rae and (iii he effec of precauionary savings. Equaion (8 is he modified Fisher equaion since he expeced inflaion rae is µ q ( and he covariance erm appears on he righ-hand side. 2.2 Indeerminacy due o a volailiy Our concern is wheher he price of money q can be expressed wih exogenous variables δ and M. By expressing equaion (6 in he inegral form and aking he expecaion we see ha q( = z( E z(sq(sr(sds F + z( lim E z(t q(t F, ( T 3

5 which saes ha he money is priced as a securiy paying he dividend sream q(sr(s as he liquidiy service. Equaion ( should be solved backwards from he erminal value since he process q appears on he boh sides. The firs erm of he righ-hand side of ( can be recognised as he forward-looking (or fundamenal erm and he second erm as he bubble erm. Regarding o he bubble erm, if p(, T denoes he ime price of he nominal zero coupon bond wih he mauriy dae T, he bubble erm is proporional o lim T p(, T which we assume is zero, z( lim E z(t q(t F = q( T Z( lim E Z(T F = q( lim p(, T =.(2 T T The assumpion is economically desirable and reasonable in he sense ha (i deflaionary pahs are ruled ou and (ii he nominal shor raes are always posiive from he assumpions on he uiliy funcion and equaion (3. 3 I is imporan o noe ha he forward-looking erm may depend on he volailiies of he price of money. 4 Our basic idea is ha if he volailiy of zq ha is he marke price of nominal risk is given, hen he equaion ( should be solvable for zq. In oher words a differen volailiy process will give a differen value of he forward-looking erm. This indeerminacy is no observed in he deerminisic cases. Inuiively he indeerminacy of he price of money is similar o he fac ha in Black-Scholes model he volailiies should be exogenously given o calculae an opion premium. Thus we will look for a soluion by assuming he marke price of nominal risk is exogenously given somehow. In addiion o he sanding assumpions ( (for he opimal real money balance and (2 (for he posiive nominal ineres raes, for he simpliciy of our discussion we focus on he cases wih he furher ransversaliy condiion of (z(t lim E k q(t F (T F = for all, (3 T ( where F ( = exp Λ(u 2 du + Λ(udW (u 2 for some posiive consan k. 5 Wih his condiion we can solve ( for uiliy funcions wih some funcional forms as shown in he subsequen secion. The following auxiliary lemma is useful o ge he soluion. The essence of he proof of Lemma is ha he process XG does no have a diffusion erm so ha he 3 The assumpion makes he nominal pricing kernel a poenial. A poenial X is a righconinuous nonnegaive supermaringale which saisfies lim E(X =. See Jin and Glasserman (2. 4 An excepion is a case of a logarihm uiliy funcion u(c, m =β ln c +( βlnm in which case q does no appear in he forward-looking erm. See Bakshi and Chen ( Condiion (3 is purely for echnical convenience o obain a simple closed-form soluion. The relaionship wih (2 is no clear due o he erm of F (T. In principle i is possible o solve wih oher forms of he ransversaliy condiion hough he resuls will be differen accordingly. 4

6 equaion becomes an ordinary differenial equaion wih respec o ime which can be easily solved. Lemma. Le X =(X(; be a sochasic process saisfying a sochasic differenial equaion 6 dx( = f(x( k d + X(σ(dW (, (X(T k wih lim T E G(T F =for all,, where f and σ are known inegrable sochasic processes, k> is a consan, and ( G( = exp σ(u 2 du 2 Then he soluion is given by ( ( G(s X( = ke f(s G( Proof. See Appendix A. σ(udw (u. kds F /k. 2.3 Equilibrium in he case of a separable uiliy funcion In his subsecion we seek soluions by assuming he separable uiliy funcion u(c, m =β c γ γ +( βm α α, (4 where α, β and γ are consans wih α>,β (,,γ >. In he case of he separable uiliy, from equaions (2, (7 and (9, real quaniies are deermined independenly of he price of money as z( = e ρ δ( γ /δ( γ, λ( =γσ δ (, γ( + γ r( = ρ + γµ δ ( σ δ ( 2. 2 Equaion (3 is reduced o R( = β β d(zq( = β β δ( γ herefore equaion (6 implies q( α M( α δ( γ e ρ z( α M( α ( z(q( αd z(q(λ(dw (. (5 6 A soluion of he equaion means a process X saisfying X( = f(sx(s k ds X(sσ(sdW(s,,. This soluion is a special case of a soluion of a backward sochasic differenial equaion in he lieraure such as Yong and Zhou (999 because he diffusion erm σ is already known in his problem. 5

7 Since z is known as above, he price of money q can be obained if a soluion zq of equaion (5 exiss. Lemma ensures ha his is he case if he marke price of nominal risk Λ is specified. Furhermore if he nominal shor rae R, insead of Λ, is specified, hen q is also deermined as in Proposiion. Two equaions (6 and (7 below link hree variables, q, R and Λ. Proposiion. If he marke price of nominal risk Λ is exogenously given, hen he price of money and he nominal shor rae saisfying (3 wih k = α is deermined by β δ( γ /α q( = β R( M(, (6 ( s = αe K(, s exp b (udu ds F, (7 R( where K(, s = exp s s σ R (u 2 du + σ R (udw (u, 2 σ R (u = ( αλ(u+αλ(u ασ M (u, b (u = ρ ( αr(u+αµ M (u ( ( α λ(u 2 + α Λ(u α σ M (u 2 + ( αλ(u+αλ(u ασ M (u 2, if he soluions saisfy ( and (2. Conversely, if he nominal shor rae process R, ha is sricly posiive, is exogenously given, hen he price of money q is deermined by equaion (6 and he marke price of nominal risk Λ mus saisfy equaion (7. Proof. See Appendix B. There are a couple of implicaions. Noice ha he firs wo observaions below are no seen in he deerminisic economies ha do no involve he marke prices of risks. Firs, Proposiion saes ha he nominal pricing kernel is no endogenously deermined wihou a furher specificaion, and once eiher variable of hree variables q, R and Λ is specified hen he remaining wo variables can be deermined. The marke price of nominal risk plays a criical role in he deerminacy of he price of money ha bridges he gap beween real quaniies and nominal ones. Secondly, some policy rules can be embeded ino he model in order o obain he price of money or he marke price of nominal risk. The rules may include Taylor rule or a direc conrol of he nominal shor rae. If cenral banks can deermine no only he money supply bu also he nominal shor rae process, he policy leads o he deerminacy of a process of he price of money (an inflaion process and he pricing kernels. Thirdly, he resul is a generalizaion of Basak and Gallmeyer (999 and Bakshi and Chen (996. If one wans o see deerminisic R (σ R ( and b ( 6

8 is assumed o be a posiive consan b>, hen R( =bα ha is he corresponding resul o one obained in Basak and Gallmeyer (999 Proposiion 4.2. If coefficiens in he processes of δ and M are assumed o be consan and one ses α = γ = and Λ( =σ M, hen q( = β δ( as is shown in β (ρ + µ M σm 2 M( Bakshi and Chen (996 Theorem 3. Remark. Lemma is so useful ha, insead of he separable uiliy (4, he similar discussion can be applied o he following non-separable uiliy funcion u(c, m = γ (c β m β, γ where β and γ are consans wih <β<, <γ<(2 β/( β. Since his uiliy funcion has non-zero cross derivaives, real quaniies are affeced by nominal quaniies and he real and nominal pricing kernels should be solved simulaneously as opposed o he separable uiliy. Even hough his problem looks more complicaed, i is possible o solve equaions simulaneously for boh real quaniies and nominal quaniies in he similar way as Proposiion by using Lemma. The corresponding formula are obained as follows wih κ =( β( γ. q( = β β = R( δ( R(M(, +κ E ( K(, s exp where K(, s = exp s s σ R (u 2 du + 2 σ R (u = ( γσ δ (u+λ(u σ M (u, +κ b 2 (u = +κ s σ R (udw (u, b 2 (udu ds F, ρ ( γ ( µ δ (u+ 2 σ δ(u 2 2 Λ(u 2 + µ M (u+ 2 σ M(u 2 2( + κ ( γσ δ(u+λ(u σ M (u Conclusion We demonrae he very close relaionship beween he price of money, he nominal shor rae and he marke price of nominal risk. When a soluion is buil from he ransversaliy condiion backwards, he volailiy erm is required. However he marke price of nominal risk is no endogenously deermined wih endowmen processes. I suggess ha he marke price of nominal risk is an equivalenly imporan facor as he oher wo variables, he price of money and he nominal shor rae, in he conex of moneary and fiscal policy. The deerminacy of he yield curve of nominal ineres raes depends on he specificaion of hese policies, which is lef for fuure research. 7

9 References Bakshi, G.S., and Z. Chen (996 Inflaion, Asse Prices, and he Term Srucure of Ineres Raes in Moneary Economies Review of Financial Sudies 9, Basak, S., and M. Gallmeyer (999 Currency Prices, he Nominal Exchange Rae, and Securiy Prices in a Two-counry Dynamic Moneary Equilibrium Mahemaical Finance 9, Jin, Y., and P. Glasserman (2 Equilibrium Posiive Ineres Raes: A Unified View Review of Financial Sudies 4, Yong, J., and X. Y. Zhou (999 Sochasic Conrols: Hamilonian Sysems and HJB Equaions, Springer-Verlag: New York 8

10 A Proof of Lemma Appendix Since G saisfies dg( =G( σ( 2 d G(σ(dW (, i can be easily checked ha he diffusion erm of he process XF disappears and he evoluion of XF can be wrien as an ordinary differenial equaion The soluion is d(xg( d = f(g( k (XG( k. T X(T k G(T k X( k G( k = k f(sg(s k ds. Taking he condiional expecaion E F and making T yields he resul. B Proof of Proposiion By seing X( =z(q(, f( = β β δ( γ e ρ z( α M( α, k = α, and applying Lemma wih equaion (4 we have β δ( γ e ρs ( F (s αds /α, z(q( = αe F β z(s α M(s α F ( ( where F ( = exp Λ(u 2 du + Λ(udW (u. Thus 2 β q( = β δ( γ e ρ z(m( ααe e ρ(s ( z( z(s α ( M( M(s α ( F (s αds /α. F F ( Recalling he soluion Y = {Y (} of he sochasic differenial equaion dy ( = Y (µ Y (d + σ Y (dw ( saisfies ( s( Y (s =Y ( exp µy (u 2 σ Y (u 2 s du + σ Y (udw (u, he erms in he inegrand can be wrien as e ρ(s F (s α z(s α M(s = α F ( α ( z( α M( K(, s exp α 9 s b (udu.

11 For he laer par of he proposiion, when R and z are known, he price of money is given by equaion (6 from equaion (3. Then Z( =z(q(/q( = e ρ δ( δ( γ( /α R( /α M( R( M( By comparing he diffusion erms on boh sides, we have Λ( = ( ( αγσ δ (+σ R (+ασ M (. α Applying Lemma for X( = Z( wih equaion (6 gives he equaion (7.