Wealth Effects in the New Neoclassical Models

International Finance Discussion Papers Number 158

April 1980

WEALTH EFFECTS IN THE NEW NEOCLASSICAL MODELS

by

Matthew B. Canzoneri

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment by a writer that he has had access to unpublished material) should be cleared with the author or authors,

Wealth Effects in. the New Neoclassical Models

by

Matthew B. Canzoneri*

In a model incorporating "rational" expectations and the "natural rate" hypothesis, Sargent and Wallace (1975) demonstrated three familiar propositions: (I) There is no stabilization role for monetary policy since fluctuations in output are not affected by systematic policy rules. (IT) There is no growth role for Monetary policy since the real rate of interest is not affected by systematic policy rules either, (III) Pegging the interest rate is destabilizing in the sense that it results in an indeterminant price level. So Poole's (1970) results do not carry over to this new generation of models. The implication of these three propositions would seem to be that the monetary authorities should only be concerned with controlling inflation,

These propositions are not new. They were much discussed within the framework of the old fashioned non-stochastic neoclassical models, and they were all valid as long as wealth effects on consumption were ignored. As is well known, wealth effects invalidate propositions (II) and (III), and "sticky" nominal wages invalidate all three,

In the new stochastic framework, proposition (I) has been questioned by some because of the existence of long-term labor contracts. Fischer (1977) and others have argued that these contracts imply a temporary stickiness in wages that allows monetary policy to work in the familiar way. But propositions (II) and (III) seem to be have been generally

accepted, even by proponents of the contracting models.

*I would like to thank Dale Henderson and Ken Rogoff for useful discussions of the material presented here, However, the views expressed are solely

those of the author and do not necessarily represent the views of the Federal Reserve Board or other members of the staff,

1

Taylor (1979), for example, reports that "interest rate targeting generally leads to instability in rational expectations models, whether prices are flexible or temporarily rigid.”

The purpose of the present paper is to show that propositions (IT) and (III) are not valid if there are wealth effects in consumption, + and to demonstrate that Poole's results carry over to this new class of models in a very robust manner. Interest rate policies are better than money supply policies if monetary disturbances are "large" relative to real disturbances, and this is true in contracting models, where there is scope for lagged feedback policies, and in the Sargent-Wallace model, where there

fs not,

It is interesting to note that, as Taylor (1979) has asserted, the very existence of long-term contracts is not sufficient to invalidate proposition (III). The wage stickiness these contracts imply allows monetary policy to influence real wages and employment, but it does not result in a determinant price level if the monetary authorities peg the rate of interest.’ In this respect the new stochastic models are at odds with their non-stochastic counterparts,

Contracting models will be discussed in an appendix. The main focus of the paper will be upon wealth effects in a model quite similar

to the one analyzed by Sargent and Wallace.

lsargent and Wallace themselves noted (in footnote 5) that proposition (TIT) resulted from their exclusion of wealth from the aggregate demand schedule.

I. The Model and Its Solution

The model is, for the most part, a familiar one:

GQ) yryt s(p, = Pelee (2) a = log 1, + BY) = 0

(3) oy, =~ yey Prarie 7 Pp] + 4,@ - PL) ta,

@) om - P= -M/e) r+ 4,0,-) + -2)@-p) ty,

where y is the log of aggregate output, p is the log of the price level, M + B is the total nominal indebtedness of the government to the private sector, r is the nominal rate of interest, and m is the log of the money

supply. u and v are stochastic disturbances, and p and p

t|t-1 tt1|t-1 27°

predictions of Py and p based upon information available at the end of

tt1 period t-1; these predictions are assumed to be "rational" in the sense of Muth (1961).

Equation (1) is an aggregate supply schedule incorporating the "natural rate" hypothesis; price prediction errors cause output to fluctuate about its natural rate, ye This hypothesis has been motivated in several ways. Under the "signal extraction" interpretation, individual

suppliers of goods and labor observe price changes in their own markets,

but they are unable to immediately decompose them into relative price

changes and movements in a general price index. Consequently, an unexpected increase in the price index will be confused at the local level with an increase in relative prices, and suppliers will provide more goods and services than they would have had they known the true relative prices. Lucas (1973) used this argument to explain the observed correlation between rates of economic activity and rates of inflation. Under the "long-term contracts" interpretation, suppliers of, say, labor lock them selves into nominal wage contracts for a specified length of time. The nominal wage settings are based upon cost of living predictions; the goal is to set a real wage consistent with a desired or "natural" rate of employment. If prices turn out to be higher than anticipated, the real wage will be lower than intended, and employment and production will ex-~ ceed their natural rates. Equation (1) can be explained in this way if the length of a period is defined by the length of labor contracts.

Equation (2) gives the nominal wealth of the private sector. Since the capital stock has been suppressed (for algebraic simplicity), this consists of money and government bonds, Monetary policy consists of "open market" exchanges of money and bonds. These exchanges leave the nominal value of wealth unchanged; units are chosen so that a, the nominal value of wealth, equals zero,

Equation (3) is an aggregate demand schedule. Demand for output varies inversly with the (expected) real rate of interest and directly

Ieee Fischer (1977) or Canzoneri (1980).

with the real value of wealth.- Equation (4) is the equilibrium condition for the asset sector. The fraction of wealth held in real money balances depends upon the nominal rate of interest and the ratio of income to wealth’; Los the income elasticity of demand for money, is assumed to be less than unity.

The first step in solving this model is to find a reduced form for the price level, The supply and demand equations ((1) and (3)) can be

solved for Gr) P, = hy Petife-1 ~ By, - Pelee “Ar, thy uy where

hy = dh, hy = sh, h, = 1/(d, + d,)

The expected value of u is zero, and in the main body of the paper, it

eee Some explanation of the dating of expected inflation may be in order. Two other dating schemes have been used in the literature: (a) Peatft-17

Pelee1 and (b) Prtile ~ P_s Sargent and Wallace (1975) used (a), and

indeed (a) is the most popular specification in the closed economy literature, It can be argued, however, that purchasers should at least know the price of what they are buying. Specification (b), which is popular in the international finance literature, assumes full knowledge of all current information. The choice among these specifications does note affect the qualitative conclusions of the present paper; some minor differences will be pointed out in footnotes.

2 equation (4) is a log linearization of

MesyA = aa D a D where a = f(r, Alp )

will be assumed that u ts serially uncorrelated. Equation (5r) determines the path of prices if the nominal rate of interest is the instrument of

monetary policy. Using (4) to eliminate vr in (5r), one obtains

= - - + (sm) P, = fy Peetle-1 ~ £2, Peleea? + £3 ™ +,

where

Fh I

= h,/(1 + hy 2,25) f.= (h, + h, 2, 2,8)/(1 + hy 2,25)

Fh t

37 4,2, /0 + hy 2,25) w. = (hgu, - heyy /(1 + hy 2,25)

which determines the path of prices if the money supply is the instrument of monetary policy. w is a combination of real and monetary disturbances; it too will be assumed to be serially uncorrelated and to have a zero mean,

Suppose first that the interest rate is the instrument of monetary policy. Forwarding (5r) j periods and taking the expectation based upon

information available at the end of period t-l,

=h h

Petgfe-2 7 [1Petjea|e-2 7 PuTe+y | e-1

This difference equation can be solved forward to obtain an expression

1 for Peti[t-1} that is, 1 eér) is obtained by repeated forward substitutions:

Pettfe-2 ~ yPeeofe-a + Oy Tosa }ena

1 Pe+3|e-1 t hy (r

t+1]t-1 * Byte 4+2/t-1)

= lim ht ~hr bye me} Peer|e-a 7 %y 1 “ttHit1|t-1

(6r) Pear lena io

To obtain a unique solution, one must specify a terminal condition

for the first term in (6r). Since hy <1, it is plausible to assume that

T-1

1 Petr|e-1 = °

(7r) lim h

To

If, for example, the price level is not expected to blow up (that is, if

Peet |t-1 is bounded for all T), then (7r) will be satisfied. Sargent (1973) calls this a "no speculative bubbles" assumption. Suppose the interest rate is simply pegged at Ts then substituting (7r) into (6r), and (6r)

into (5r), one obtains a reduced form for the price level under a fixed

interest rate policy:

-1 — (85) Pp = why Cohy) r= Bye ~ Pe fend) + Byte

The reasoning used to derive (8r) does require the presence of

wealth in the aggregate demand schedule, (3). If d, = 0, then hy =1

and (7r) is not the only plausible terminal condition. In particular,

T-1

t 42 = (7r)" Lim by Pear fend Prteo|t—1

To0

and simply asserting that the price level is not expected to blow up is

not sufficient to pin down the terminal condition. Any finite Prto|t-1

yields a particular solution for p in (6r) and a new solution for

t+1|t-1

Pye Unless some cogent argument can be given for a particular value of

1y¢ dating scheme (a) in footnote is used, one also needs d, > d, to ensure that the equivalanet of hy is less than unity.

1 Pr teo|t-1? the price level is indeterminent.

The same procedure can be used to solve (5m) for a reduced form “under a fixed money supply policy:

- a (8m) Pp, = £, (1 f,) mn - £,@, Pelee tw

Here wealth effects are not necessary for the argument; f, <1, even if

1

d, = 0. These reduced forms can be used to calculate price prediction errors

under the two regimes, and then substitution into (1) gives reduced forms

for output. With a fixed interest rate policy,

_= “1 (9r) y7y + s(1 + h,) hau,

and with a fixed money supply policy,

-> -1 (9m) y= +s(1 + h, + 9) (hau, hy%,v,)

where ¢ = 44%,%,0 +s) >0O

The values of rf and m do not appear in these reduced forms, but the choice of an instrument clearly makes a difference in the stochastic structure of

the fluctuations in output.

=_OOOCOOOOO

And th n e problems do not end here. Unless the Tetit1 | t-1 converge quickly

to zero, the infinite sum is (6r) will not be finite. If the interest

rate is pegged at zero, Taylor's (1977) condition can be imposed to achieve uniqueness,

II. The Real Rate of Interest and Monetary Policy

The values of r and m do affect the real rate of interest in this

model, It can be shown that

(lor) E[r, - Pet fend -pJle=r

under a fixed interest rate policy, and

-1 — (10m) Elr, - Prater - P)] =- &, (1-h,) (i-h, + hy 2425) m

under a fixed money supply policy. (Here, E[*] is the unconditioned expectation operator.) In either case, monetary policy is not neutral; the choice of r or m affects the real rate of interest.

This non-neutrality depends crucially upon the presence of wealth in the aggregate demand schedules. (If d, = 0, then the interest rate

policy is infeasible and h, = 1, som dissapears in (10m).) This result

1 is essentially Metzler's (1951).

III. The Yr Policy versus the m Policy

If the goal of monetary policy is to minimize the fluctuations in output, then (9r) and (9m) can be used to choose between pegging the

interest rate at r and setting the money supply at mM. With a fixed

1 It is clear from (8r) and (8m) that ETP rea |e-a - P,] = 0, no matter

which instrument is chosen. (10r) follows immediately. Using (4) to calculate E(r,) and (8m) to evaluate E(p,)> one obtains (10m).

10

interest rate policy,

. 2 2.2.2 Oo, rss (th) hy o,

correlated),

2 2 ~2

2 2 2, 2 2 3 a, + hy hy o., )

The ratio of these variances

2 2 : 2 (11) Os ny > >= ofl + (hy 2, /h,) R]} where 2 2 _ 2 -2 R= C / oy Q@ = (1 + hy) (1 +h, + 9) < 1

depends upon the relative sizes of real and monetary disturbances.

Equation (11) is graphed in figure 1. If monetary disturbances are "small" in comparison with real disturbances (that is, if R <R*), then the m policy is better than the r policy. If monetary disturbances are "large" (R > R*), the © policy is better.

The reason for this is clear. The interest rate policy does not allow a monetary disturbance to affect the goods market. On the other hand, a flexible interest rate allows financial markets to absorb some of the effects of a real disturbance. This result is essentially Poole's

(1970).

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Iv. A Better Interest Rate Policy

The instrument selection problem, as posed in the last section,

probably presents too stark a contrast. The FED has rarely, if ever,

pegged an interest rate, Instead, we have witnessed a variety of interest

rate policies, each characterized by a degree of interest rate flexibility.

2 This continuum of policies might be represented by (12) m= m+ g(r, - 1) g>0

Larger values of g correspond to more vigorous stabilization efforts. (In fact, it will be seen below that g = corresponds to an T policy,

while g = 0 corresponds to an m policy.)

ee It is not even clear that the goal of these policies was to fix an

interest rate per se. The idea may have been to create a demand for money consistent with a money supply target.

The reader might wonder why policy rules of the form

(12)' me =m ~g (y, - ¥)

are not discussed, since the ultimate goal is to stabilize output. The motivation for the present discussion is that "current" information on interest rates is available to the monetary authorities while information about real variables comes with a lag. It is well known that rules of the form

(12)" m =m ~g(y,) - y)

have no effect on 7 y in the present model.

12

Clearly, r and m can not be chosen independently.~ The choice of r and m will affect the real rate of interest, but it will not affect the fluctuation of output about its natural rate. So without loss of generality, r and m can be set equal to zero; this combination works, and it results in

simple algebraic expressions.

Substituting the policy rule (12) into (4), one obtains

=~ _ -1 _ (13) r,-rer, (1+2, 8) £4 [2 opts (Pp, Pelee +viJ

As g>~, the interest rate is pegged at r. As g is decreased, interest

rate fluctuations become larger. Substituting into (5r), one obtains

(14) P= Pett [tat ~ ho, - Pe|e-1)

-1 ~ (142, 8) h, £, [25 Pp. + 298 (Pe-Pe lea) + v,1 + hau,

which reduces to (5r) as g+~ and (5m) as g>0. Equation (14) can be

2 solved in the same way for the reduced

~l- G5) pps [L+h, + G+ ayg)

61 Thu, - G+ ty9)? nye.)

~—-__------

Taking the (unconditional) expected value of (3) and (4), m= -g7l rt

gp and p=-dd. tt, som=-(2. +244 he * 2 P 1%2. «iT me 7G) 2%1%9 Ts

A auicker wav of derivine (15) is to note that €44) is tdenticai to (5m) with Ly replaced by (142, 8)-12,5 (15) can then be inferred from (8m).

13

So with policy rule (12),

2 y's

2 2 -1,,-2 2 2 -2 2 2 2 (16) o = 8 [1+h, + (1+2,8) 6] fh, ot (+2, 8) hy 25 ov]

The best policy can be found by differentiating (16) with respect

to g; it turns out that the variance minimizing g is (17) g* = @2,)* (Re)

where

2 8 = h, &, (+s) /h, 2, Ath)

As shown in figure 2, the m policy (or g= 0) is best if monetary disturbances are "small" relative to real disturbances (that is, if R < Bg).

The "larger" are monetary disturbances relative to real disturbances, the

less flexible should be the interest rate policy. However, it should be

noted that the r policy (g = *) is never best. Pegging the interest rate is a feasible policy, but it is always dominated by a more flexible policy.

The reason for this should be clear. The interest rate provides some information about the unobserved disturbances if it is allowed to

fluctuate in response to them; pegging the interest rate is tantemount

to throwing this information away.”

lonis discussion assumes that interest rate destabilizing policies (g < 0)

are not even considered.

2pinally, it may be interesting to note that R*> 8. The analysis of section II leads one to turn away from interest rate policies at too low a value for R. The reason for this is again the r policy is not the best interest rate policy.

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V. Concluston

Even in models without long-term labor contracts, there are policy choices facing the monetary authority that affect real variables. The choice of a monetary instrument (or a "combination" policy) has implica~ tions for stability in the goods market, and these implications are basically those identified by Poole (1970) in a much simpler context. In addition, monetary policy can affect the real rate of interest and thus capital formation. These conclusions, which differ substantially from those of Sargent and Wallace (1975), are directly attributable to wealth effects in

consumption.

’ References

Canzoneri,.M..."Labor.Contracts and Monetary Policy," Journal of Monetary

‘Economtes, forthcoming. —

Fischer, A. "Long Term Contracts, Rational.Expectations, and the Optimal

Money Supply Rule," Journal of Political Economy, Feb. 1977, pp. 191-206.

Lucas, R. "Some International. Evidence on Output-Inflation Tradeoffs," American Economic Review, June 1973, pp. 326-334.

Metzler, L. "Wealth, Savings and the Rate of Interest," Journal of Political ‘Economy, Vol. LIX, No. 2, pp. 93-116.

Muth, J. "Rational Expectations and the Theory of Price Movements," ‘Econometrica, July 1961, pp. 315-335.

Poole, W. "Optimal Choice of Monetary Policy Instruments in a Simple Stochastic Macro Model," Quarterly Journal of Economics, May 1970, pp. 197-216.

Sargent, T, "Rational Expectations, the Real Rate of Interest and the

Natural Rate of Unemployment, Brookings Papers on Economic Activity,

1973, pp. 429-472,

Sargent, T. and N. Wallace, "Rational Expectations, the Optimal Money Supply Rule," "Journal of Political Economy, April 1975, pp. 241-254,

Taylor, J. "Conditions for Unique Solutions in Stochastic Macroeconomic Models with Rational Expectations," Econometrica, Sept 1977,

Taylor, J. "Recent Developments in the Theory of Stabilization Policy," Columbia working paper number 41, Nov. 1979,

FIGURE 1

oA /02 y,m y,r

m policy better r policy better

FIGURE 2

r policies best

-(/4)

Appendix: Wealth Effects in Models with Long-Term Labor Contracts

This purpose of this appendix is two-fold. First, it will be shown that the very existence of such contracts does not allow the FED to peg interest rates; if wealth effects are absent, the price level is once again indeterminent under an Yr policy. Second, it will be shown that the choice of a monetary instrument (or more generally, an interest rate policy) will depend upon much the same considerations as were discussed in sections ITI and IV; in particular, it depends upon the relative sizes of real and monetary disturbances. The long-term contracts do imply a role for lagged feedback policies, but the instrument selection problem is largely in~ dependent of this fact.

If labor contracts span two periods, the supply schedule (1) becomes! 1) y,=y tsp, - Pele-a) * s@, - Pr e-2 the rest of the model remains unchanged. Equations (I) and (3) imply = - - - - - + (TIr) p, = hy Preafe-1 ~ 2), Pelt-1? ~ hy, Peleeg? ~ Ayr, + Agu,

and using (4) to eliminate Ths

(tim) p, = fy Peaa|t-1 7 Fo, - Peitea - £,0@, - Pel e-o) + ism ty, ma

Ieee Fischer (1977) or Canzoneri (1980) for a derivation of (I). The length of a period is determined by the length of the lag in policy feedback rules, and this in turn is usually identified with the data lag for real variables. (In section IV, it was assumed that there is no lag for financial data.)

The present author usually thinks in terms of quarters, It is clear that most labor contracts span more than two quarters; however, the results presented here are easily generalized to models with n-period contracts.

-~A2-

These equations correspond to (5r) and (5m) in the main text, and they

can be solved in exactly the same way for the reduced forms

ys - - (TIr) py = hy ~ by) By, Pep eeg) — BL (Py Peyeen) + hyd,

and -1 —- (ttm) p, = £,01 - £)) “ m- £) PL - Pei pm F,@, Pelee2? +

which corresvond to (8r) and (8m). ‘The reduced form (IIIr) is only valid

if d, #0 (so that hy #1). The diseussion in the main text carrys over word for word. The price level is indeterminent without wealth effects in the

aggregate demand schedule.

Now if the disturbances are seriallv correlated the reduced forms will

be more complicated ,+ However, it is well known that in thie ease the nriae pre-

diction error P, - Peppeo will depend upon both current and lagged inno-

vations in the disturbances.“ This will cause output to "cycle" about its

natural rate. It is also well known that lagged feedback rules for monetary '

policy can completely offset this cyclical component in output?

1 the expressions for p will include u Ww t+j|t-1 ttj|t-1 °F “ety |t-1°

2500, for example, Fischer (1977) or Canzoneri (1980).

3yb4a.

- A3-

In the present context, those rules take the form

(IIIr) re=3rt (hg/hy Uy py

or

(IIIm) m, =m- (1/£,)w

t t|t-1

depending upon which instrument is selected. It should be clear what

these rules do. Noting that h,/h, = 1/d, ,(1IIr) simply moves the interest rate to offset the predictable part of the demand disturbance in (3); (II Im) has to offset the predictable effects of both monetary and real disturbances.

With (IIIr) and (IIIm), (IIr) and (IIm) become

(Ivr) p, = MPa feed - ho, - Pele-D - ho, - Pe} to) -hyr + h(u, - Me tI

(Ivm) Pp, = FPeaz jena ~ £2 _ ~ Pele-1? ~ £2_ - Peje_p? + f3m + Cw, - )

We |t-1 Now (IVr) and (IVm) are exactly analygeous to (5r) and (5m) in the main

text (since it will turn out that P.- Pele =P. - Pele-2)° So in models with long-term contracts, one should add the feedback terms described in (IIIr) and (IIIm), but then the analysis proceeds as before. It is elear

from (9r) and (9m) that the cyclical component has been eliminated.