interest-rate policy and rational expectations

TORBEN M. ANDERSEN CORE

Belgium

University of Aarhus

Denmark

Interest-Rate Policy and Rational Expectations*

The choice of monetary instrument under rational expectations is discussed in a general equilibrium model for the financial sector. It is shown that a supply rule leads to indeterminate asset prices, whereas the prices are determinate under an appropriately formulated interest-rate policy.

1. introduction The defeat of Keynesian theory by the New Classical Macro-

economic theory seems to be complete. Not only has systematic economic policy been shown to be ineffective, but the favored Keynesian monetary instrument, viz., interest-rate policy, has been shown to imply an indeterminate price level.

Sargent and Wallace (1975)’ argue that an interest-rate rule under rational expectations leads to an indeterminate price level because agents (rightly) expect that an increase in the price level will be met by an increase in the money supply. Thus, there is nothing to anchor the price level.”

Usually simple, ad hoc macromodels are used to discuss the choice of monetary instrument under rational expectations. The financial part of these models is given by a liquidity preference schedule (LM curve). However, since the Lucas-Sargent-Wallace (hereinafter LSW) proposition makes systematic monetary policy ineffective with respect to real variables, it is of interest to expand the financial side of the model to see how the choice of monetary instrument affects the financial system.

For the purpose of this paper we shall thus take the LSW proposition to be valid, and the aim is to set up a general equilib-

*Comments and suggestions from au anonymous referee, S. Hylleberg, and C. Vastrup are gratefully acknowledged. Thanks are due to the Danish Social Science Research Council for financial support of the research underlying this paper.

‘Also see Sargent (1979). “McCallum (1981) shows that the special form of the interest-rate rule is critical

for this result.

Journal of Macroeconomics, Summer 198.4, Vol. 6, No. 3, pp. 311-322 311 Copyright 0 1985 by Wayne State University Press.

Torben M. Andersen

rium model for the financial sector, and to trace in this the effects of monetary control under rational expectations. 3

The simplified model which is used to this effect is similar to the general equilibrium approach introduced by Tobin and Brainard (1963). This approach assumes that the financial and the real sectors can be separated. The neglect of the interaction between these sectors is very restrictive. However, given the LSW proposition this separation is acceptable and allows a fairly simple way of modeling the financial system. 4

Furthermore, the general equilibrium model set up is shown under rational expectations to be a model for efficient capital markets. The question is, therefore, really how to conduct monetary policy in efficient capital markets.

The expansion of the financial side of the model proposed in this paper has important consequences for the choice of monetary policy instruments, since it turns out that a pure supply rule for the quantities of the (nonrisky) financial assets imply that the asset prices become indeterminate,5 whereas in general there exists an interest policy for the (nonrisky) financial assets which make asset prices determinate.” In this way we obtain the opposite result to that of Sargent and Wallace (1975), although in respect to asset prices rather than the general price level.

This paper has nothing to say about the optimal feedback rule for the policy instruments. It is a purely positive analysis of the implications of different policy rules for the working of the financial sector under rational expectations.

The plan of the paper is as follows. In Section 2 the general equilibrium model for the financial sector is set up. In Sections 3 and 4, we shall analyze the model under a supply rule and an interest-rate rule, respectively. Section 5 contains the summary and conclusion.

3Notice that the LSW proposition does not make the choice of monetary instrument trivial [see Parkin (19’78), Woglom (1979), Turnovsky (1980), and Minford and Peel (1981).

*A similar assumption is made in McCallum (1981), where output is set equal to the constant natural-rate output. This assumption, of course, makes it impossible to analyze the effects of unanticipated changes in the monetary instruments on real variables.

‘See Andersen (1983-84) for a discussion of indeterminate asset prices in relation to the efficient capital market hypothesis.

‘Notice that we do not consider the implications of an interest-rate rule for the information disseminated by interest rates [see Dotsey and King (1983)].

312

Interest-Rate Policy and Rational Expectations

2. The Model Consider an economy with K -I- 1 financial assets. Asset 0 re-

fers to money. There are N riskless assets with return r:, i = 1, . . . , N. There are M risky assets, i = N + 1, . . . , K. Let pf denote the price at time t of asset i. Money is chosen as numeraire, i.e., p: = 1. Since asset 1 to N are riskless the prices p;, i = 1, . . . , N, are constant and set equal to one.

Monetary policy is taken to mean open market operations by the monetary authorities in the N riskless assets either by regulating the rate of interest or the supply of these assets.7 By a supply rule will be understood an active choice vver the quantities of the assets to sell/purchase in the market, letting the rates of interest be whatever they must be to equilibrate the system. By an interest- rate policy is understood an active choice over the interest rates at which the monetary authority is willing to make transactions, letting the quantities be whatever they must be to achieve portfolio bal- ance.

Private Investors A representative investor is assumed to maximize the value of

period t + 1 wealth:

A’ K

W t+1 = x; + 2 (1 + r”l) x’l + c p:+$j ; (1) i=l i=N+l

where xi is the number of asset i bought at time t. Investment at time t must satisfy the budget constraint given

by the initial wealth

w, = i: pfxj . i=O

(2)

Combining (1) and (2) gives*

W t+1 = wt+ i r:x; + i (pf, 1 - p$7! t i=l i=iV+l

‘Restricting policy to be operations in the N riskless assets simplifies the exposition, but does not affect any qualitative results.

‘For expositional purposes, the cash income associated with holding each risky asset has been stated as a capital gain.

313

However, the period 1 -t 1 prices are not known with certainty when the investlnent decision is to be made at time 1. Let 1~: 1 be the expectation at time t of the price of asset i at time t -t 1. We will now, as has bccomc standard practice, postulate demand func- tions of the form

where the q’s are coeficients relating the demand for asset i to the return on asset j.

Wealth is assumed to be constant and income is constant due to the LSW proposition, and fbr these reasons wealth and income have 1~w1i ldi out of tln2 demand fiinctioil. ?‘110 assets will be as- sllliicd to be gross sut)stitrites

uzj = 1 > 0 for i = j ; 5OforiZj. (5)

For this demand system to be consistent we nlust impose the fol- lowing adding-up constraint on the coctficicnts:

K c ay=o;j= I, . . . ,K. i=l

Finally, we shall assume expectations to be rational conditional on the available information set, Z,-,, at date 1. Ttiis information set is assumed to consist of data on current and past values of all endog- enous and exogenous variables oliscrved as of the end of period t - 1.

Thus,

pp+, - E(pj, ,/Z,-,) , i = N + 1, . . _ , K . (7)

An implication of this assumption is

K(pf., , - $+,lZ, 1) = 0 ; i = N + 1, . , K Pi

Equation (8) ilnplics that the market is eflicient with respect to the information set I,-,, and that it is impossible to make economic profiti by trading on the basis of this information set.

Interest-linte Policy unci Hutio~d Erpectutiot~s

SUPPkl Let the supply of asset i be denoted S:. The supply of asset

N -I- 1 to K is given by

s; = K: : i = iv + 1, . . ) K ; 60

where9

Kt~N(O,&, i=N+ 1, . . . , K. (W

The supply of the assets 1 to N is determined by the monetary authorities’ open market operations. Either by a supply rule stating the quantities to be sold/purchased (see Section 3), or an infinitely elastic demand/supply br the assets at the policy determined rates of interest (see Section 4).

Equibihriutn The equilibrium condition fi)r each asset is given by

Si = xi , i = 0, . . . , K ; (11)

Any one of the equilibrium conditions in (11) can be dropped, and naturally the one for the numeraire, viz., money, is dropped.

Inserting in (11) we can write the equilibrium condition as

h(t) = A,,r(t) -I- A,,{~b(t + 1)1L11 - p(t)1 ; (12)

(13)

where S,(t) and S,(t) arc (N X 1) and (A4 X 1) vectors for the supply of riskless and risky assets respectively; r(t) and {E[p(i + l)lf,- 13 - p(t)} are (‘V x 1) ilIl<l (114 X 1) vectors for 1.:lks of in tcresl and cx- petted price increases respoc~tivc!l~; and A,,, A,,, A,,, all(j A.,., ar(’ N x N, N x M, M x N, and M X M matrices with elements give12

.lre CIloSI111 to simplify tllc exposition. risky assets is not cssserttial, altd (9) al,d (10)

Torhen 31. Andersen

by the demand coetficicmts (all matrices are assumed to be nonsin- gular).

3. Supply Rule Assume that the monetary authorities subscribe to the view

that monetary policy should aim at controlling monetary aggregates. In the present model this is taken to mean that the central bank sets targets for the aggregate amount of the controllable assets 1 to N with the purpose of regulating monetary aggregates.

111 this case the policy rule will be a function of the monetary aggregate or equivalently of the linear combination of the stock of the assets defining the monetary aggregate. A general form of feedback ride for the supply of asset I to N is in this situation given by

S,,(t) = Q,S,(t - 1) + Q2 S,(t - 1) + u(t) ; (14)

whcrc u(t) is an (N x 1) vector of u:, i = 1, . , N, which are serial incleyendent white noise random disturbance terms.

11: - N(C), vii) , i = l., . . , LX

Q, and Qz are (N X M) and (IV X N) matrices of policy dctcrmined a Justmcnt cocfYicients. d’

From (13) we have

E[p(t 4 l)lZ,-,I - p(t) = &iS,(1) - A,,‘A,,r(t) . (15)

lnscrting in (12) gives

FlY~lll (15) W1' Ilntl

E[p(l f 1)/f,+,] -- p(l) = I-‘,S,(/) - I’&(/) ; (17)

We shall now turn to the task of deternlining expectations. Once expectations are determined the stochastic process for the asset prices can be determined from (17). Taking expectations conditional on all available information 1,-l, and using (14), WC find from (17)

zc[p(t)lZ,. 11 = E[p(t + 1)11,-j I + IY2V,&(t -- 1)

+ IT’rQzS,,r(t - 1) .

Equation (18) is in the form of a first-order diflerence ecluation in the expected asset prices, and this equation tnust be solved to find an expression for the expected prices.”

It is, however, seen that it is impossible to solve: the diff’er- ence Equation (1.8) in the forward direction due to the one-to-one relation between the expected period t and t + 1 prices.” That is, the expected prices cannot be determined from (18) unless we impose a terminal condition in the form of exogcnously given expected prices as of some future date, i.e., the solution cannot be found solely from the initial conditions of the model.

The economics behind the result that a supply rule in the present portfolio model leads to indeterminate expected prices is quite obvious. Given the supply rule it follows from (18) that any increase in the expected period t + 1 prices will be met by an equal increase in the expected period t prices. Hence, any expeo- tations of the period t prices are as rational as any other, and by implication the process for the expected prices is indetermirlate. I2 Given a supply rule WC can thus conclude that the asset prices become indeterminate.

‘“As is well known, linear rational t!xpectations models have iu gener-al an infinity of solution [see. e.g., Burrneister (198(I)]. As has h ecome standal-cl practice we sh‘all, however, only search for a convergent (forward-looking) particular solution to the diKerence equation determining expectations.

“Notice that it is an implication of (18) what the expectations of the lxire of asset j (j 2 N + 1) at any date only depends on exogenous variables and expectations of the price of asset j at tilture dates. Howcvcr, Imder an interest-rate rule expectations of the other asset prices at future dates will intluencc: the expwtations of the price of asset j at any date [cf. (Zl)].

“Equation (17) shows that there is a one-to-one relation between the current spot prices and the expected period t + 1 prices. This corresponds to what Ilicks [(1946), Chapter 201 in a discussion of the implications of introducing securities in a temporal equilibrium model denoted the critical case between stability and insta- bility.

317

To&en M. Andersetl

Notice that the rates of interest on the N riskless asset are determinate [cf. (16)], even though the prices of the M risky assets are indeterminate.

For analytical convenience we have somewhat artificially par- titioned assets into a group of riskless and a group of risky assets; but, in reality a large class of assets are of an intermediary kind, being risky assets over a certain horizon, but eventually riskless at the date of maturity. Hence, for each single asset of this type there is an obvious and well-defined terminal condition for its price, namely, the face value or principal at maturity.13 This does not, however, constitute the sort of terminal condition which is needed in (18) to make expectations and hence asset prices determinate. The structure of (18) follows from the fact that in efficient capital markets investors are only concerned with the capital gains to be made over a short time-horizon. This implies under rational expectations that the price of an asset of type j (j z N + 1) at time t depends on the expected price of an asset of type j at time t + I, which again depends on the expected price of an asset of type j at time t + 2, etc., etc. Let an asset of type j be, e.g., a bond with say 20 years to maturity, in one year’s time this will be a bond with I9 years to maturity,i4 and hence no longer an asset of type j but of type k (k # j). The fact that assets as the date of maturity approaches become riskless in the sense that their price at maturity is given does not, therefore, impose a terminal condition on (18).

4. Interest-Rate Rule Assume instead that the central bank subscribes to the view

that the structure of interest rates or asset yields rather than some monetary aggregates is the proper target for monetary policy. In this case the central bank stipulates an interest-rate rule for the assets 1 to N with the intention of regulating the interest level and the asset prices. At the policy determined interest rate the central bank is willing to sell/purchase whatever amount the public demand/supply of the assets.

The interest policy will be determined by the rates of interest and asset prices which by the monetary authorities are seen as essential for monetary control.

I31 am grateful to an anonymous referee for raising this problem. 14The specification of the model presupposes that the time period is small and

significantly less than a year.

318

r(t) = Q:, r(t -. 1) -C Q4 p(t - 1) +- e(t) ; (1%

where e(t) is a (iv x 1) vector of e’(t), i = 1, . . , N, which are serial independent white noise disturbance terms.

Qn and Q4 are N x A: and N x M matrices of policy-determined adjustment coefficients.

From (13) we find that

kJ p(t i- l)lI,-,] = p(t) + AG1 S,(t) - A,-,’ &r(t) . W)

Given a solution to the cxpcclations process the asset prices are determined from (20). The analysis of the expectations process is in this case slightly more complicated because r(t) at any date appears in (20) as a variable predetermined from (19). Leading (19) one pc- riod and taking expectations conditional on the information set I, r we can combine (19) and (20) as follows:

where

Equation (21) gives the difhcrencc equation which determines cx- petted asset prices, and it can be solved using the method devel- oped in Rtanchard and Kahn (1980).

There exists a unique solution to (21), viz., the forward solution, if @r has the strict saddle-point property, i.e., tire number (T) of characteristic roots to @r outside the unit circle is equal to the number of norrprcdetermined variables (&I).‘” If this condition

ISA more general form of (19) would be to let r(l) depend on [p(t - I) - p(t - l)]. i.e., the past return on tho risky asset, rather than only p(t - 1). It can easily 1~ verified that this change does not affect any qualitative results.

‘“See Bllrrncister (1980) and Blancharcl and Kahn (1980). If 7’ > M, there doc:s not w.ist a nonexplosive solution (21); and if’ 7‘ < M, (21) has an infinity of’ solutions.

31Y

is MfIl~tl WC can ~:o~~cl~~d~~ that expectations, a11t1 herw asset prices, are iiniqucly dctcrniiried III&~ an interest rate rule. Since @, is a (M + AI) x (N + M) matrix WC face a (A’ + M)th order polynomial to calculate the characteristic roots. Hence, the condition for de- termii~atc cxpcctations and asset prices is quite complex and it is difficult to state any gerieral cmiditions for liniquencss.

As can be seen a1 depends OII the parmleters of’ the monetary ftiedback rule, viz.. Q, and Q.,. ‘I’hc cboicc of feedback rule is thcre- fime essential for the existence of’ a uniqtw solution to the cxpec- tations process under ark intwcst rate rule. AS noted there arc IM I- N characteristic roots to Qr, ad M rionpretleterlninc:cl variables,

mnd by choice of the [(N x A’) + (A’ x M)]-elements in Q.) and Q4 the monetary authority has some possibility of ensuring existence of a unique soltltion to (21). It should be noted that the f&t that the characteristic equation is nonlinear implies that we cannot con- cludc that the strict sadtll~~-point corldition generally can bc ~nade to be f~~lfilletl by a proper choice of the clcmcnts of Qn and Q,,. In gcucral there should, bowcvcr, be son1e scope for- rrmuetary policy to achieve urlicl[m~css and thils dctcrniinate asset prices. “.‘”

WC can tllrls conclude that, althollgh uniqueness is not in gen- cral ensured, the condition of the strict saddle-point property in tlir case of an intc~rctst-rilte poliq is not as strict as the terrnirial conditiott ucccmary for deternlinntc~ asset prices in the case of a wl)ld~ rtlltt [cl’. Silrg!c.wt (1979). 1). X2]. I”

5. Conclusion The iclea that an illtercst-rate policy Icads to ind~ttcrlnillan~y,

of noulinal niagnitudcs in the long run is a familiar argunlcnt Of

“To suhstantintc! this claim WC shall show that there exists a choice of feedback

pxamctcrs which mukrs Ihc strict sudtlle-point cmdition to hold for a three asset’ model oftell used tn Iltacro(‘cotlolnic analysis [cl.. , e.g., 13urnieistcr and Tllrnovsky

(J%l)]. AsW 0 is rnonry, 9sW I is n capitid certain bond. aid asscl 2 is a capital risky atsnt. I.ct qI = 0 \vc find Ihe (rwl) roots to be A = ‘/2 [ 1 ? (1 - 4h)“*],

where b = q1011/f~S2 (n2, <. 0). It is easily chcckcd that [AlI > 1 ant1 IA21 < 1 if (nJ4+,) < qr < (-2r1,~/n,,), nntl helice the strict saddle-point condition holds.

“If’ tllc fcctlhack rule for the interest rates dots not involve ally feedback IO the

past asset prices, i.e , Q, = 0. the asset prices become indctcrminatc as in the case of a slpply rule. ‘I’llis can he seen 1Lorn (20) since this cqlration has the salnc.

mathcrnatic.tl stlucturr as (Ii) for Q, 7 0.

320

classical ccononlics [see Olivora (19X)]. Wicksell is normally given credit for this result, and Sargent and Wallace (1975) also see their result as an illustration of the Wicksellian indeterminaucy result. In the present paper wc have analyzed the implications of the choice of monetary instrument for the dctctrmiiiation of nominal mngni- tudes under rational expectations in a gcrleral e(luilibriunl ntodel for tlie financial sector Isc*c also (;lirloy anti Sllaw ( IWiO), C:haptc:r 31. ‘l‘liis analysis illIlIOSl r(:\‘crses tllc results dcrivcrl witllill a11 IS Lhl fAnlc!work IIII~CI. rational (~xi)c(:t;ltiolls 1s~~: Sargt‘nt ;111rl \\‘;I]- lace (Iwg].

That is, within the present 11rocle1 it has been shown that if the nwnetary authorities pursue a policy of controlling rnonctary aggregates, the prices of the fina1lcial assets becomr~ indetcrnlinate. It is, however, possible by an appropriately fbrmulated interest-rnle policy to establish determinate asset prices.

Given the assumption of rational expectation formation it was shown that the model for tht: financial sector set up is ;I rrlodcl for eficient capital markets. It can thus be co1~11decl that asset prices will IX indcterminatc in eff‘icient capital markets unless the monetary authorities intervene irt the market with an appropriately formulated interest rate policy.

Finally, it should be remarket1 that both tllc IS-L\,1 model and the gcIicral equilibrium model for the finalIcia sector 11sed in this paper are ad hoc models, and tllc: highly divergent results show that policy canclllsioris and rec~oInmcndations baseci on such inodels ~l~ou1d he interpreted with cart.

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