Tight money and inflation

Journal of Monetary Exonom~cs 13 (1984114) S- 5. lu‘urth-Holland

TIGHT

MONEY

AND INFLATION

issan LIVIATAN

In a recent paper Wpent and Wallace present an example where tlghtenq of moneta~ pohc\ ovx a sgecified time Interval, in cqunctlon with a constant deficit. leads to an IV( rtptlsein the rate of inflation not only beyond this interval but even from the outset In the preseqlr paper I shot\ how rhc latter result can be derivsd from a standard monetary-growth model of the ‘Sicrauhki- BrKk type, This enables B relatively simple interpretation of the apparent paradox

1. hwhtction

In a recent paper Sargent and Wallace (SW) (1981) investigate the effeil:ton inflation of tightening of monetary policy over a specified time interval under the condition of a given government deficit (net of interest on the public d.&t). They find indications that restrictive monetary policy under these conditions may not be very eflective in reducing inflation. They even presenr a numerical example, which seems to them ‘spectacular’, which shows that tightening monetary policy leads to a uniformly higher rate of inflation than in the absence of such a policy. The example provided by SW is based on an overlapping generations model with various arbitrary assumptions, such as no bequest motive, a minimum scale of investment in assets, legal restrictions on intermediation. a speciFc variable time path of the government’s deficit, and so on. This creates the impression that the ‘spectacular‘ results can hold only under very special circumstances. In the present paper I shall show that the paradoxical reu& holds for a standard rn~net~~ rowth model wirh long-run perfect foresight of the Sidruuski -- ock’ type. The main assumption which limits the generalit> of linear form of the utility function. ‘Plrefact that thr: radaxicul result is obtained in a fairIF standard moctel of ary-growth may indicate that it is of a more general nature than mi.l>he 1 in view of the SW study. 1 also find that the much simpler snal?;tiul ure enables a clearer economic interpretation of the apparent parad80s. Finally I show that the results depend crucially o the concept of government deficit in question. In the final part of the paper prove hat the previous

0304-3923/R4/$3.00

.cL\ 1984, Elsevier Science Puhhshers B.V. (North- Holland)

N. Yviatapr, Tighs monnqvand inflalion

6

results are reversed wlhen we use alternatively a more conventional definition of a deficit name& the one incluske of interest payments on government debt. t’l%is is not m$eant to imply that the latter concept of deticit is the more relevant one.], m;odel

2,

The model alssumes a representative fautily with an infinitely long horizon which rtuaximizes a dkcounted utility integral, based on per capita variables -.rCX -I I

,: - “Iu[

~‘(,t ), m(t)]

dr,

subje ,i to

(1)

“‘0

ii(f) = (1 - :r)r(r)b(t) -[n

a,

+-(1

-

(2)

+T(f)]m(i)-c(t),

and

a(r)=m(t)-t-b(t),

,9(O)=

-nb(r)

=

a given initial pondition.

(3)

The notation is as follows: u is a standard utility function, c consumption, m where M is nsmirral balances, p price level and 1, real balances ( m = M/PI,, population size), 6 government indexed2 bonds held by the public, y a constant endmvmenr of #he cmmodity, n rate of peculation growth, r gross real interest rate o:n bonds, T prop0 nal income tax, 6 is a positive constant subjective discount f;Wor and k = dx/dr, where t denotes time. I7 = p/y is the inflation rate and s denotes real transfer payments to the public. The variables ilre

per

cagisa.

Assuming a separatble utility function U(C, ns) s &logc + 18,logan.

(4

we obtain from the IEuler conditions (omitting for simplicity th,e time index) &+=c)--6,) ( P&4,

N a/‘n

(5) ) = i,

(6)

hen p 5%(I - I’)S - n: and i is (1 - ~)r + 31= p + n + Il. p represents theposiI& Get real in t,erest rate (.neNof population growth) while i is the net nominal ~~~~r~~trate.

5 !Srhc aaumplion

u.xd by SW, but it is in no way basic to the analysis.

Let USassume that p is constant, as it will in fact turn out to be later. Then we may define consumers’ wealth at any I = 0 as w-a(r)+(l

-T)/@=e-@gy+s)dr’. 0

(7)

We maythen derive from the o timization the demand functions for consumption and real balances c=

g*w*

m=(S,/i)w,

(8) iS,=S--6,.

(9)

On the macro level we have the national income identity

where g denotes real government expenditures per capita. The latter is a component of the government budget erjuation D-(1

-7)sS(g-7y)+(l

-T)rh=~I+(r?f71)M+i)+nb.

The right-hand side indicates that the deficit by new ( and A equilibrium characterized I+I b 0 hence p 6. latter p c independent p = The is by system

assumed bonds the that

market

market

be

(11) is

properties: steady macroeconomic

(13

(13)

-.r)s~(,~-T)‘)~(p+n)b=m(n+Pz)~nh.

It can be shown using the rePationships given earlier, that any one of these

N. Liviotan,

8

Tight money and inflation

equations is determined by the other two. The two independent equations can demmine the values of I7 and m if we treat 6 a given parameter, or can determine rn and Ir if dZ is given (note that iI can be determined by the rate of mc~netary expansion, p, since in steady states h -= n + II).

‘The effect of a reduction in the rate of monetary ‘expansion, ~1,on Kf is very clear when we dieal with steady s’l:ates.In thlis case we simply have An= 4~. The problem becomes more complicated when we consider a reduction of p for a limited time interval, say between t = 0 and c = IY There is the question of how to define the framework within wh:.;h we perform the ‘ tight money’ eqperiment. After defining the .Prameworkone may consider the: effect of tight money on PJ before ancl after L!Y SW use the following framework. Throughout the experiment they hold fixed some time path of ‘net govemmcnt deficit’ which is the usual concept of government deficit but exclusive of anything that has to do with government bonds. In terms of our notation this concept is represented by

D,N=g--qy+(l

-T)S=P-(l--7)f$,

(15)

where g, s and 11’are assumed to be constants. It turns out that tne precise: dletinition of the deficit is crucial Ito the results of the analysis. ‘Tight monetary policy means that within thle interval [O,T] ,Uis kept below, its onginal path while after T the value of gofpernment bonds is frozen at the: level it attained at ‘p: Note that in the first stage (t s T) p is determined exogenously while b( t ) is endogenous whereas in the second stage (t > T) b( I 111 is fixed exogenotAy by b( f ) = 6( 7’) while p is determined endogenously. Given this concept of ‘tight Qnc>ney’ we can investigate its’effect on the time paths ol II,, ~1~b and m in i;he two stages. The dynamic s,ystem for our analysis is derived as follows. Using the budget constraints (2) and (3) we can write

b=pb-((n+,~)mt-~]+(y+s)(l that

path c ence Q, is constant.

-+-c.

(16)

can be written

be constar

since c = .V- g whi:h

9

From rir = (p - II -

ra)m

and (6) we obtain

ril=(p+=p)m*Q,.

Q?= - 4 PJP, L

(18)

where again Q2 is constant. Since c must be held constant it follows from (5) that p = cf= consta a constant c implies that wealth must be constant an Suppose we start at po. At time I = 0 we reduce p to a lower (constant) value jr’ over the interval ]Q,7’1. The only variable which can change discontinuously at t = 0 is m(O), since the price level is flexible. However this will not happer, since m(O) is determined through (8) by the condition that c must remain constant and by the fact that b(O) is momentarily fixed. It then follows from (18) that a reduction in p makes h(O) < 0 so that m(t) will be declinin . Solving (18) we obtain

where m, < H( p’). Thus m(t) will be declimng at #anincreasing absolute rate over the interval [O,T]. The fact that m(r) is decreasing during the first stage. i.e.. over [O.T]. implies that the rate of inflation (I7 I is increasing, si ace by 46) (Bz/P,)Ss/m)=i=p--n-n.

(19’:

As n7(t) declines continuously over [0, 7’1, III T) is ver=vsimple. Note that throughout the first stage b(t) has been increasing since tit + 6 = 0 and ri3 tive. At T we have to freeze tS(r ) at b(T) for all t > T, Since riz+ i>= C must continue to hold we can o this by making rit -0 0 for I Z-T. The latter can be achieved by than p, discontinuously ,~t T so as to satisfy in eq. (18).

Recall that as we reached T in the first stage rizwas negative hence (23) implies th,at iui has to be raised above p’. Moreover, since m(O)> m( T) it follows that IA!> cr, > P’. Note that since rLs= 0 by (20) and consequently i>= 0 by (8) we have been put by p, in a steady-state solution. Hence if we maintain p1 for t > 7’ h(l) will remain constant at b(T) and the SW requirement for the second stage will be :satisfied.Throughout the second stage we then have 17(f)- II(T) :’ n(o),

t > T,

CL0ll= P,

t > T,

> P,

’

I-&‘*

(21)

which shows that tight money oper [0, T; raises the inflation rate in,horh stages. It also shows that the reduction of ~1in the first stage below its original level muss be followed by an increase in ,L(in the second stage above the original It should be noted that the present formulation of tight money cannot be carriled ou! 3ver an arbitrarily large interval since nt( t) will run eventually into negative values, which is infeasible. Another way of looking at this problem is by uloting that a steady state wit.ha given D, and ~1’< cl0 requires an i~rcrease in rleal balances while a policy of keeping /L’conslant during the dynamic process involves, as we have scan, a reduction in real balances>.We may .>f o~urse reach thsenew steady state, with the lower p, if we do not maintain p c0nstanr s: ~1’. 4, An inWive wqhuWion Our

model enables an intuitive explanation of the paradoxical results along t!he fcrllow9.ng lines. In a world without bonds the government’s deficit is given pm. If Dv is held constant and p is reduced this must lead to an increiuc in m and oonsequentlr to a reduction in n. Hence, under rational expectations, a reduction in p is deflationary in the short run, If the public”sportfolio includes government bonds as well then D, & +-b. Therefore a reduction in p at t = 0 does not necessarily require an ~l‘rvrcasrin nr, since there is the alterrlative of raising the additional funds by ~:~~~s of nfw bonds as expressed by h. Thus suppose that at t = 0 the reduction of p is acL:smp’aniedby an increase in h while m(O) and b(O) remain constant. Since la(C) will also remain constant, along with m(O), this implies, by rh =I~ (p-n -~:?)m, that & < 0, and hence that m(r) will be declining, This involves ~~e~~~le in “BI and an additional uction in m(t). t; tbc existence of bonds sna a reduction in p over the first sta he ott”$ctby an increase in bond finance __and a reduction in nr( r), gsrc cr3urselimited in tim,e by the non-ne ativity constraint on m(l) 1~~~~ an ~~~~r bound on 6( 1’), us wa ointed out by SW.

N. Lwloron. pi&t

ntonev ond

rnfiarron

11

The remaining question is whether it is possible that the introducaic>n of a light monetary policy will leave m(0) unchanged. In our model this follows from the goods market equilibrium v - R = S,( m + b + (_v - s j/p). Since all the variables in this equation except for m are momentarily fixed this must also be true for m(O). In general however it is not necessarily true that’nt(0) should s we shaII see later. money leads to more inflation in the second stage can be follows. If we consider the starting point and the second stage as steady state positions then the government budget equation can be expressed as D, =t pb = m( n c VT 1. If tight money leads to an increase in b then, since p > 6, this must involve a larger injection of real money balances pm [ = m(n + a)] which must involve a hi er level of n as long as the to (n + n) is less than unitary. elasticity of the demand for money with resp o put it differently, the reliance on more bond finance raises in steady states the burden of ,nterest payments which is inflationary if financed by the inflation tax. 5. Remarks on generalizations

A crucial point in the foregoing argument is that nt(0) remains up#lffected by the introduction of tight money. In order to show that this result may not hold under more general conditions let us drop the separability assumption in the utility function and assume that the cross derivative II,nris (say) positive. rather than zero. The Euler condition (5) can then be wrrtten as

where we used the fact that along the quilihrrum path of the ccontlnay 2 must constant). It follows that in general p fill vary over be zero (since c tnry to our earlier conclusion, fact let us assume. ASbefore. that the implications of the fore the first stage and that the second stage corresponds to a steady e B 2 S while during of (22) this means thal durrn ec!uals ,v - .s. which the Second one p 6. Sincrt in the 0 policy. w:: see that remains constant after the introd substitution etkct the itrcreusc in p over the lirst st TCWGS the demand for current consumption. This will reduce the e level and increase the rquilihrium value el ilritial real balances M(0). ion of n~~~-s~~~~? aMit]; is t0 It seems theref The increax in nr(t.N

ttea~~~iat~d

with a decroa:;e in (n(b))so that the i

may be d~l~at,i~~~~ry, However, since we havclassum PITmay fall bylaw its 0 inal vnlraeat some point ult6 rcmwin os3entiall sts however that n~nms~~arabilitymay also I, if consunqtian is ralativ insensitive to tbon the el&ct of ti t money leads to heavier thtit a crucial question L whether bond flnanc , IIf this is the cwsd n in the second st i~c~‘~a interebitpayments will increase inflation since, by assumpti lbrc ~r~anc~~by the inflation tax. By considerations of continuity the rate of ~~~ati~~n will :rise(compared with the initial position) at some point during the

tt urns

~~~~~th~r feature of the more general case is that the mere requirement that b ~,~~uldremain constant during the second stage [at a Icvel equal to b(T >] does MN n~~ssarily imply that the secand stage corresponds to a steady state f6tuation for ths economy. However since our starting paint is a steady state it ijsnatural to reqxriruthat the same apply to the second stage. Finally we must point out that the specific concept of the government ~3ud~~tpcrlicqarssumcdby SW iq crucial to the results. In principle one may fhink of various (conceptsof govu-mmentdeficits and government fiscal policies lvhich have difl’eremtimplications for monetary policy, In order to stress the importance al’ Ihe governmeant’sbudgetary policy in the context of tight money we shall define a fairly ’ traciition~al’kind of policy which will lead to a reversal OFthle results &rived in our previous example in section 2. However, as noted in the introduction, it is not claimed that the latter concept of deficit is in any MYMmore apprlraprititethen the one used in section 2, kationn td a

amstwt

kowmtianal’ deficit (D)

The results of section 2 depend crucially on the concept of the deficit which d constant duting the experiment. In particular if it is D in (11) which is teld cQn~tanttlhen one can show ,that the basic results are reversed, Since D is a more conventionul concept of budget deficit, it is of interest to demonstrate sratennent formally, er to keep D constant in (I 1) while h(t) varies it is necessary to some compensating chang,esi other components of the bud mc that these compensatin changes are made through (err taxes) s. It is therefore uired to maintain rb -+s =ck constant. From (2) iand (3) WC the:11 obtain

13

where we u,sed &I= (p - I7 - n jm. We al50 have, at; in (18). El.

Z,

=

-6-f

&//3,

) =

constant.

(24)

ven ~(23) and (24) form a dynamic system in nt and h where the uil~b~um is a saddle point. The n;iture of the system is de1 in term of d~~~ti~ns from the; steady state. The stable insides witt the horizontal axis, where ction in the constant level of ~1then ‘II,, equilibrium value and stay 1y b( t ) will change. The reason a jur q irb the price level) is that nge SOas to enable wealth, and not possible in the preceeding y derive ahernative steady-state values of na and h. Alternatively, if we consider h as a par ‘meter and w and 1~ as endogenous we may derive the steady-state relation between rn and h. This is given by (W

(V‘@P - (2, t- zrLQ* and is represented by the SS curve in fig. 2. m

N. Limatan, Tight money and iqjfotion

In fig. 2 A represents the original steady state with p i= po. Now let p be rleduced to a constant value p’ over the interval [0, T], as required by the tight money experiment. The trajectories corresponding to ~1’are derived from (23) and (24) with p = ,u’. The steady-state solution for p = p’ is given by A’ which must of course be located on SS. If T were infinitely large then the dynamic solution ,would be along the stablt: trajectory BA’. If T < oo then the solution may be derived as follows. .Po each value of T, 0 c T <: 00, there corresponds a trajectory RR and a vallue of KU,say rn& which will bring the system to SS curve after exactly T periods. The point which corresponds to rni is represented by N on the Trajectory RR. Starting at N we shall land on SS at AT exactly at time 7“. Note that A’ is a steady-state solution for some p, which we denote ‘by pl, Iwhich must be larger than p’. Suppose that at the instant we arrive at .4-I‘we increase ~1from c’ to &. Then this will leave the system in a steadv-state equilibrium at A? In particular 6(1) will remain constant at b( t ) = h( ) for I > T as required by the second stage of the tight money experiment. IUithe foregoing, solution the second stage (where b is frozen) takes the form of a steady state. Is there another possibility where the second stage is not a

fr

steady state? e answer is negative. For m ntaining D constant requires that rb + s = constant, so that if b is constant then s must also be constant. However the demand function c implies that if c is constant so must bc N (as deftned in (?)I. Since s has to constant then this is also true of cx= m + 6, which implies that if constant so must be m. This establishes the fact that the solution for the s nd stage is a steady state. Since the point N 2 must be between A and B it follows that the f the i~tr~uction of tight monetary policy is to raise m,, which tion bclth of p(0) (the price level) and of H(0). The latter result follows from (6) and i = p + n + Z?. On the dynamic path of the first stage along NAT, m decreases and hence I7 increases. However at A“ m is still above R(F,) so that II will remain permanently below its level before tighter money was introduced. Thus the introduction of tighter money reduces the rate of inflation below its 0 nal level for all c following the change in policy. The economic int tation of this result can be seen most clearly when we compare the steady state points A and A? Tight money causes b to increase during the first stage. However since in steady states D = m( n + II) + nb. it follows that m(n -t l7) must decrease. In other words the increase in nb increases the share of bond finance (b/L = nb) at the expense of the inflation tax m(n c I?). A reduction in the reliance on the inflation tax must be deflationary if the demand for money with respect to (II+ II> is less than unitary, as it is in our model. The foregoing remarks explain the difference in the effects of tight money under regimes of constant D, and D. In first case the increase in rt>(due to the increase in b during the first stage) is financed by the inflation tax and is therefore inflationary. In the second case, where D is held ccnstant. the increased interest on government bonds is financed by adoitional Faxes which leads to a smaller reliance on the inflation tax.