monetary aggregate versus interest rate rules

BE-I-I-Y C. DANIEL State University of New York, Albany

Monetary Aggregate versus Interest Rate Rules*

This paper analyzes the issue regarding whether the central bank should stabilize the interest rate, a monetary aggregate, or some combination thereof, in the context of a model with rational expectations and with permanent and temporary aggregate supply, IS, and LM disturbances. Monetary policy has a role due to the assumption that the monetary authority uses information on the current interest rate while wage setters do not. The analysis confirms Poole (1970) for the case of temporary IS and LM shocks, and provides several additional results when disturbances are permanent and when they affect supply.

1. Introduction The question whether the central bank should stabilize a mon-

etary aggregate, an interest rate, or some combination of both con- tinues to be a prominent issue in the formulation of monetary policy. Poole’s (1970) seminal contribution to the problem, using the Hicks IS-LM framework, demonstrates that an interest rate rule is preferable if shocks are to the LM curve whereas a money supply rule dominates when shocks are to IS. With uncertainty regarding the source of the disturbance, the optimal combination policy depends on the parameters of the model. The greater the variance of LM shocks relative to IS shocks, the more weight should be placed on the interest rate rule and vice versa.

Recently, authors have reexamined these results in models with rational expectations. Woglom (1979) and Canzoneri, Henderson and Rogoff (1983) confirm Poole’s fundamental results in models in which wage setters do not use the interest rate to forecast prices. Parkin (1978) and Turnovsky (1980) show that a policy of real or nominal interest rate stabilization, depending upon the assumed information structure, reduces the variance of output around a fixed target when stochastic disturbances are to supply.

The purpose of this paper is to further reexamine the Poole

*The author would like to thank John Boschen, Richard Froyen, and two anon- ymous referees for helpful comments on an earlier draft.

Journal of Macroeconomics, Winter 1986, Vol. 8, No. 1, pp. 75-86 75 Copyright 0 1986 by Wayne State University Press.

Betty C. Daniel

(1970) results, allowing temporary and permanent disturbances to IS, LM, and aggregate supply. The policy rule considered provides for a long-run growth path for money from which temporary deviations are made to offset interest rate forecast errors. For temporary IS and LM shocks, the analysis is essentially the same as that in Canzoneri, Henderson and Rogoff (1983), and it thereby confirms Poole (1970). The analysis also demonstrates that permanent IS shocks require the same policy as temporary IS shocks, but that permanent LM shocks require pegging the real interest rate instead of the nominal rate. Supply disturbances, however, do not necessarily require real or nominal interest rate stabilization, as suggested by Parkin (1978) and Turnovsky (1980). If the output goal is stabilization of output around its full information level rather than around a fixed target, the productivity disturbances affect target output. Thus, the real interest rate must be allowed to adjust such that demand adjusts with full information supply.

2. The Model

A. Equations of the Model The model contains conventional IS and LM curves and an

aggregate supply curve. It is specified as follows:

YP = h - Nit - E(P,+I - PJI + ult + vt ; (1)

mt - pt = -4 + P yt + uzt + 02r ; (2)

ys = a, + X(p, - &IPJ + w3t-l f ht + ugt , 0 < aI < 1 ; (3)

The variables, yt, pt, and m,, are the logarithms of real output, the price level, and the money supply, respectively. The variable, i,, is the nominal interest rate. nit, i = 1, 2, 3, are random walk disturbances with normally-distributed, finite variance, zero mean in- crements, which are denoted by iii,. The uil, i = 1, 2, 3 are temporary disturbances with normal distributions, zero means, and finite variances. Superscripts d and s denote demand and supply respectively; E is expectational notation.

Equation (l), the IS curve, states that aggregate demand depends negatively on the real interest rate and that it shifts with permanent and temporary disturbances. Following Canzoneri, Hen- derson and Rogoff (1983), the expectation of inflation, used to de-

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Monetary Aggregate vs. Interest Rate Rules

termine the real interest rate, is assumed to be conditional on the knowledge of values of all past variables and on knowledge of the current interest rate. The unsubscripted E denotes the expectation conditional on this information set. This assumption is justified because interest rates are readily observable while variables like the price level, the level of output, the money supply, etc. are observable only with a lag. Thus, economic agents have some information on the nature of current disturbances, which is provided by the interest rate, but not full information since they cannot observe current values of other variables.

Equation (2) for the LM curve is based on the assumption that the money supply is exogenously determined by the central bank and that real money demand depends negatively on the nominal interest rate and positively on real income. Further, real money demand shifts with permanent and temporary disturbances.

Equation (3), the aggregate supply curve, states that output depends positively on price surprises and on stochastic disturbances. Price surprises appear, due to the assumption, following Fischer (1977) and Gray (1976), that the labor market meets in ad- vance of other markets and determines a wage to prevail over the period. Therefore, when the period t wage is negotiated, neither firms nor workers know the value of period t variables. E,-1 rep- resents the expectations operator conditional on knowledge of variables dated t - 1 or earlier. Other assumptions used in the derivation of Equation (3) include: 1) production is Cobb-Douglas and is shifted by random permanent and temporary disturbances; 2) firms maximize profits; 3) labor supply is an increasing function of the real wage; 4) the predetermined wage for period t equates expected labor supply and demand for the coming period; and 5) actual employment is determined by demand.

Note that a permanent disturbance has a larger direct effect (ignoring price surprises) this period than next. This is because, in next period’s wage negotiations, a disturbance which increases (de- creases) productivity and labor demand will be partially offset by an increase (decrease) in the wage.’

Finally, for the purpose of designing stabilization policy, it is necessary to determine the optimal value of output. If economic

‘The assumption that wages are set based on identical information structures for both firms and workers is not crucial to the analysis. Qualitatively similar results emerge if it is assumed, following Friedman (1968) and Phelps (19f37), that firms have current information when wages are set, while workers have only lagged information. In this case, a, = 1.

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agents are rational, then, in the absence of externalities, they will make optimal decisions, given full information. Following Barro (1976), it is assumed that optimal output is full information output. The expression for full information output is obtained by letting the wage equate full information labor supply and demand to yield:

Yf = a0 + alhf + u3J - (4)

Thus, if the policy maker’s goal is to stabilize output around its full information level, then in the event of IS or LM curve disturbances, he should seek to eliminate price surprises, while in the event of disturbances to the production function, he should want a price surprise to offset 1 - al of the supply disturbance. It is never desirable to completely offset the supply disturbance because a fraction, al, does contribute to full information output.’

B. Policy Rule The policy rule analyzed is that of a long-run money supply

path from which temporary deviations are made to offset (fully or partially) interest rate forecast errors. The short-run choice faced by the monetary authority is the extent to which it wishes to de- viate from the long-run money supply path to offset a current interest rate forecast error. Whatever deviation occurs, the money supply is expected to return to its pre-set growth path next period. Note that this rule differs from the fixed interest rate rule analyzed by Poole (1970) and by Sargent and Wallace (1975). Regardless of the extent to which the monetary authority seeks to offset current interest rate forecast errors, the interest rate must adjust to permanent real disturbances with a one period lag. The rule therefore provides a determinate price level by establishing an anchor for expected future money supplies.

It is assumed that the policy instrument available to the central bank is the money supply. Therefore, letting rnf be the long-

“Barr0 (1981) suggests that full information output should be increasing in the real interest rate due to the inter-temporal substitution of labor based on returns to current work effort versus future work effort. However, Blinder and Fischer (1981) show that in a model with storable output, firms reduce labor demand thereby reducing current production as the real interest rate rises. Thus, if both firms and workers are allowed to set up intertemporal maximization problems, the effect of the real interest rate on full information output is ambiguous. This issue is beyond the scope of this paper, and the interest rate does not enter the determination of actual or of full information output.

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run money supply target and At be that level of the money supply necessary to eliminate current interest rate forecast errors, the money supply can be expressed as:

m, = (1 - p)m,* + p?fz, (5)

where p is the policy choice variable. To solve for &, solve Equa- tion (2) for the interest rate using Equation (3) to substitute for output to yield:

liz, is the value of m, which implies II,-,i, - i, = 0. Thus, 7jzf is expressed as:

At = mt* + (1 + PQ(p, - Et-d4 + Q(fh + ~3) + &T.~ + 02 . (7) Substituting into the expression for m, yields the monetary rule:

mf = mf + PLO + PWp, - Et-,?4 + P&t + 4 + k?t + %I . (8)

The policy problem is to choose p to stabilize output around its full information level. p = 0 implies a pure money rule while p = 1 implies a pure interest rule. Notice, however, that the choice of p is irrelevant for the current expectation of the future stock of money. Taking the current expectation of future money using equation (8) yields I&m,+, = rnr+{. The long-run money supply path is independent of temporary deviations, which are made to offset interest rate forecast errors.

C. Solution The model is solved using the familiar technique of undeter-

mined coefficients. Equilibrium price can be expressed as:

(9)

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Betty C. Daniel

where g is the long-term trend rate of growth of money and where

7F = --[a + fib, (1 - p) + oH]/[b,(l + Ph)(l - p) + a(h + b,$)] ;

+r, $2, and +a are respectively the proportion of one period interest rate variance accounted by IS shocks, by LM shocks, and by aggregate supply shocks.3 + = $r + (b2 + +a and is equal to 1 unless p = 1. When p = 1, the interest rate provides no information on the disturbances, and the expected value of all current disturbances is zero. This case is modeled by setting all 4’s equal to zero.4 The 4’s, j = 1, 2, 3 represent the fraction of the variance of each type of disturbance attributable to its permanent component.’ The solution for the real interest rate can be expressed as:

it - VP,+, - Pt) = (b” - a0 + Ull+l - %-d/b,

+ [Cl - P) - 4hfflh - +)I[(1 + PUfiu + Ou)

3. Optimal Policy Monetary policy has a role in this model due to the assump-

tion that the monetary authority uses the current interest rate in

3Letting UT, I$, and cri represent respectively, the variance of IS, LM and aggregate supply disturbances, and UT the variance of the interest rate, then:

u: = [f3 t n(1 + PA)]’ [(1 + PAY4 + A%$ + o-‘$ ,

4, = (1 + PA)%:

(1 + PA)%; + Aso; + u: ’

42 = A%;

(1 t PA)%: + A%; t u; ’

43 = d

(1 + pA)“of + A%; + u: ’

4That this is legitimate can be proved by requiring expected inflation to be conditional on information available at t - 1 and resolving the model.

“0, = ugu; j = 1, 2, 3.

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determining the money supply, but that wage setters do not use the interest rate.6 The goal of the monetary authority is to choose p to minimize the expected squared deviation of output from its full information level. An expression for the latter as a function of p can be obtained by using equations (1) and (10). Assuming that the disturbances are not mutually correlated, minimization with re- spect to p yields:

p* = { -)ca(l + PX)[ 1 - pb,+ + (1 + ph)H]a; + X2& + (Y(b&

- AH)]ue” + [b, (1 - 40 + 44 - Aal 0% + 4

- ahH]u~}/{h2blcr~+ [b,(l - al) - ha@bJ ai},

where of, I$ and c$ represent respectively the one period variances of IS, LM and aggregate supply disturbances.

To compare p* with Poole’s policy recommendations, it is use- ful to consider special cases. Assume first that all disturbances are to aggregate demand, implying that ai = (~32 = 0. In this case, as ui + 0 and u,” * 0, p* 3 --TV. Demand disturbances affect output only by causing price surprises and as p + -*, the effect on the real interest rate of a demand disturbance * l/b,, from Equation (lo), implying perfect stabilization of demand. The result is therefore qualitatively the same as those of Poole (1970) and of Canzo- neri, Henderson and Rogoff (1983). The smaller p is, the better, irrespective of whether the disturbance is temporary or permanent. If the policy choice is constrained to a pure policy, then a pure high-powered money rule is superior to pure interest rate rule.

Next consider the special case in which all disturbances are to the LM curve crt = crz = 0. In this case,

P*lu; = 0; = 0 = 1 + cy+e, . (11)

Optimal policy can be described as a pure real interest rate rule. With p set as above the expression for the real interest rate, Equa- tion (lo), is unchanged by an LM disturbance. If all disturbances were temporary, (0, = 0), then optimal p would be unity, as in Poole and Canzoneri, Henderson and Rogoff. With optimal policy,

‘See Canzoneri, Henderson and Rogoff (1983) f or a discllssion of this isslle. If wage-setters are allowed to condition on the current interest rate in predicting p,. G,, and Q. then monetary policy is redundant.

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the monetary disturbance is initially offset by a change in the money supply to maintain the interest rate at its forecast value. In the next period, the disturbance disappears, the money supply returns to its growth path, and there are no remaining effects. Since there is no inflation, stabilizing the real and nominal interest rates are equivalent.

For the case in which all LM disturbances are permanent (0, = l), optimal policy explicitly requires stabilization of the real interest rate. Stabilization of the nominal interest rate is sufficient, but not necessary. To understand this, suppose that a permanent random disturbance reduces excess money demand. Since the disturbance reducing money demand is permanent, and since money is expected to return to its preannounced path next period, price is expected to have risen by next period. Using Equation (9), and the fact that Eii,, = @I&, expected inflation increases by +h[a + pb, (1 - p)]/[b,(l + ph)(l - p) + a(h + bl+)]. If p = 1, then the current interest rate provides no information on the disturbance, + = 0, and expected inflation does not change. If, however, p # 1, expected inflation does change. For example, assuming 1 - b# > 0,’ and letting p take its other optimal value of 1 + a, expected inflation increases. The nominal interest rate must be allowed to rise such that the real interest rate remains unchanged, stabilizing demand and thereby output. The appropriate policy is therefore to “overstabilize” the interest rate, i.e. requiring it to rise when the disturbance alone says it should fall. This implication is absent from Woglom’s (1979) analysis since his analysis assumes aggregate demand depends on the nominal interest rate instead of on the real interest rate.

This result is important in considering optimal combination policies. If optimal p differs from one, then its optimal value ap- proaches 1 + a, not 1, as the variance of permanent LM disturbances becomes relatively more important. It should be noted that this result arises solely because investors condition on the interest rate when forecasting expected inflation. The result is also specific to permanent disturbances because, unlike temporary disturbances, they do require price level adjustment once the money supply has returned to its long-run growth path.

‘This is necessary for an increase in the money supply to reduce the interest rate. In the absence of this assumption, optimal policy would allow the nominal interest rate to fall (remain constant if b, = 1) in order to peg the real interest rate.

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Finally, consider the case in which all disturbances are to aggregate supply. The optimal p becomes:

p*j-; = of = rJ = b, (1 - aJ(1 + +a) - haI (fib, + a)(1 + +a%)

h (1 - 4 - W% (12)

Whether disturbances are permanent or temporary, a policy of real interest rate stabilization would be desirable only if al were equal to zero. To demonstrate this, substitute optimal p into the expression for the real interest rate, Equation (10). With optimal p, a positive supply shock reduces the real interest rate by a,/bl, cre- ating the desired increase in aggregate demand of al. Thus, since al > 0, the real interest rate must be allowed to fall sufficiently to stimulate demand to its higher full information level. This result differs from those of Parkin (1978) and Turnovsky (1980) due to the policy objective. In those analyses, the goal for output is stabilization around a fixed target, instead of around a full information target.

4. Summary and Conclusion This paper analyzes the issue regarding whether the central

bank should stabilize the interest rate, a monetary aggregate, or some combination thereof, in the context of a model with rational expectations and with permanent and temporary aggregate supply, IS, and LM disturbances. Monetary policy has a role due to the assumption that the monetary authority uses information on the current interest rate while wage setters do not. The policy rule considered is that of allowing temporary deviations from the long-run money supply path to offset interest rate forecast errors. Regardless of the combination of money supply and interest rate target chosen today, the money supply is expected to return to its pre-set growth path next period.

The analysis confirms Poole (1970) and presents two new results. When economic agents use the current interest rate to pre- dict inflation, real interest rate stabilization instead of nominal interest rate stabilization is optimal in the presence of LM shocks. Also, when disturbances are to aggregate supply, real interest rate stabilization fixes output and is not optimal when the goal is to stabilize output around its stochastic full information level, Consid- eration of supply disturbances does not add weight to a real interest

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rate rule in an optimal combination policy designed without current knowledge of the source of disturbances.

References Barro, R. J., “Rational Expectations and the Role of Monetary Pol-

icy.” journal of Monetary Economics 2 (January 1976): l-32. -. “A Capital Market in an Equilibrium Business Cycle Model.”

Econometrica, 48 (September 1980): 1393-1417. Blinder, A.S. and S. Fischer. “Inventories, Rational Expectations,

and the Business Cycle.” Journal of Monetary Economics 8 (No- vember 1981): 277-304.

Canzoneri, M., D. Henderson, and K. Rogoff, “The Information Content of the Interest Rate and Optimal Monetary Policy.” Quarterly Journal of Economics 48 (November 1983): 545-566.

Fischer, S. “Long-Term Contracts, Rational Expectations and the Optimal Policy Rule.” Journal of Political Economy 85 (February 1977): 191-296.

Friedman, M. “The Role of Monetary Policy.” American Economic Review 58 (March 1968): 1-17.

Gray, J. “Wage Indexation: A Macroeconomic Approach.” Journal of Monetary Economics 2 (April 1976): 221-35.

McCallum, B. “Rational Expectations and Macroeconomic Stabili- zation Policy: An Overview.” Journal of Money, Credit and Banking 12 (N ovember 1980): 716-47.

-. “Price Level Determinacy with an Interest Rate Policy Rule and Rational Expectations. ” Journal of Monetary Economics 8 (November 1981): 319-29.

Parkin, M. “A Comparison of Alternative Techniques of Monetary Control Under Rational Expectations.” The Manchester School 46 (September 1978): 252-87.

Phelps, E.S. “Phillips Curves, Expectations of Inflation, and Op- timal Unemployment over Time.” Economica 34 (August 1967): 254-81.

Poole, W. “Optimal Choice of Monetary Policy Instruments in a Simple Stochastic Macro Model.” Quarterly Journal of Econom- ics 84 (May 1970): 197-216.

Sargent, T.J. and N. Wallace. “Rational Expectations, the Optimal Monetary Instrument, and the Optimal Money Supply Rule.” Journal of Political Economy 83 (April 1975): 241-54.

Turnovsky, S. J. “The Choice of Monetary Instruments under Al-

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Monetary Aggregate vs. interest Rate Rules

temative Forms of Price Expectations.” The Manchester School 48 (March 1980): 39-62.

Woglom, G. “Rational Expectations and Monetary Policy in a Sim- ple Macroeconomic Model.” Quarterly Journal of Economics 93 (February 1979): 91-105.

Appendix: Derivation of Aggregate Supply Let output be Cobb-Douglas which can be expressed in log-

arithms as:

y, = a& + f&k + q, + Ef (Al. 1)

where y is the log of real output, C is the log of labor supply, k is the log of the capital stock, which is assumed fixed, and Q and E, represent, respectively, permanent and temporary well-behaved stochastic disturbances. o1 and o2 are elasticities. Assuming profit maximization, the log of expected labor demand can be written as:

E,-& = [l/(1 - c.ur)][lnar + cL2k

+ s-1 + Et-,pt -4 . (A1.2)

Labor supply is assumed to depend positively on the real wage, allowing the log of expected labor supply (t?) to be expressed as:

E,-,4’s = 0~~ + CC& - E,-,p,) . (A1.3)

Equating E,-ref and E,-,8f, to determine the wage, substituting the expression for the wage into labor demand to determine employment, and substituting the expression for employment into Equa- tion (Al. 1) yields Equation (3) in the text for aggregate supply;

where a0 = [ha&l - a,) + (a,k + lnoLI)crIoI]/[l + a,(1 - or)] ,

al = (1 - or)(l + crJ/[l + a,(1 - or)] < 1,

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Betty C. Daniel

ii31 = (1 + Ahit ,

ugt = (1 + A)$ .

Full information aggregate supply is found by replacing E,-ip, with p, in the equations for labor supply and labor demand and repeating the solution procedure to yield Equation (4) in the text.

If, following Friedman (1968) and Phelps (1967), it is assumed that firms have current information while workers have only lagged information, the supply equation can be written as Equation (3) in the text where

a, = A(a3 + ha, + a&)

A= ala4

1 + a4(1 - aJ

a, = 1.

Production function disturbances, which increase productivity and labor demand, are known to firms when they occur, and therefore have equivalent current and lagged effects on aggregate supply, given price surprises.

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monetary aggregate versus interest rate rules

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