JOURNAL

OF ECONOMIC

23, 261-266

THEORY

(1980)

Adaptive Monetary Policy and Rational Expectations JESS

Department

BENHABIB

University of Southern of Economics, University Park, Received

May

16, 1979;

1.

revised

California, Los Angeles, November

California

9ooO7

26, 1979

INTR~DUCTI~N

Models of money and growth under the assumption of rational expectations were initiated by Brock [4, 51 and Sargent and Wallace [9]. In such a context, Fisher Black [2] proposed that adaptive monetary rules, where the rate of growth of the money supply is tied to the level of inflation of the previous period, can generate dynamic price paths that are convergent for any initial level of money balances.’ This implies that the rational expectation path of prices may not be unique. Black also claimed that no matter what the parameters of the money demand function are, the prices are convergent for any adaptive monetary policy as described above. This contradicts the results of Sargent and Wallace [9] who show that a necessary condition for the convergence of prices is that the money supply grows slower than a certain rate which depends on the parameters of the money demand function. We will first show the mathematical error of Black’s analysis and give the correct condition for convergence in terms of the inflation elasticities of the demand for money function and the adaptive money supply policy. Then we will show that non-convergent, structurally robust and bounded price paths are always possible if any minute amount of non-linearity is allowed into the model. These price paths which remain bounded are also rational expectations paths since they would satisfy the

I More recently Fisher Black has argued that the nominal money supply cannot be changed by the monetary authority since any attempt to alter the nominal supply would be offset by the actions of the agents in the economic. (See 131.)

261 0022-053 All

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Copyright C 1980 by Academic Press. Inc. rights of reproduction in any form reserved.

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JESS BENHABIB

transversality conditions for an explicit intertemporal optimization problem faced by the representative consumer.’ Before considering the mathematics, let us consider an intuitive explanation of the role of adaptive monetary policy. First consider the case when the supply of money is fixed and the demand for money depends negativeIy (and linearly) on the level of inflation. As shown by Sargent and Wallace [9] there will be a certain level of initial real money balances that will be held willingly with a zero level of inflation. If initial balances are below (above) this level, equilibrium in the money market will require inflation (deflation). This inflation (deflation) will reduce (increase) money balances for the following period, requiring further inflation (deflation) for equilibrium, and so on.3 If, however, we adopt a rule whereby the supply of money is positively related to the level of previous inflation, we may reduce the burden born by prices to establish equilibrium in the money market. If inflation is accompanied by a sufficient increase in money, real balances may not decline too much in the subsequent period, easing the level of inflation required to bring about equilibrium in that period. This is the mechanism by which adaptive policy allows the possibility of convergence of prices, no matter what the level of initial real balances.4 However, the adaptive policy may be overly strong relative to the parameters of the money demand function. Indeed, it is possible to change the money supply by too large a multiple of the inflation level of the previous period, inducing an increasing in real balances in the subsequent period and push them beyond the equilibrium level. Equilibrium will then require deflation, and an oscillatory pattern wil develop. Depending on the relative magnitudes of the parameters of the money demand function and the adaptive policy rule, these oscillations may be convergent or divergent. The mathematics that follow show exactly this.

“‘Ad hoc” demand for money functions which relate the demand for money inversely to the rate of inflation, as in 191 or 121, can also be shown to arise from an explicit intertemporal optimization problem for the individual. Consider, for example, Brock’s [4] model and assume for simplicity that current utility varies linearly with current consumption. Then a necessary first order condition (assuming interiority) appearing in Eq. (2) in Brock’s paper 14, p. 7581 can be written as a’(M,/P,) = 1 -/3/I,. where u’(M,/P,) is the marginal utility of real balances, /I is the discount factor and J,(= P,, ,/P,) is the inflation rate. It is easily seen from inspection that the demand for real balances varies inversely with L when the utility of real balances, @%4,/P,), is strictly concave. 3 In the context of explicit optimization for a representative individual and a fixed money supply, only the stationary path is optimal since the other parts would violate transversality conditions (see 141). 4 For an early discussion along similar lines, see [IO].

263

DYNAMIC MONETARY MODELS

2. THE MODEL

AND THE DYNAMIC

TRAJECTORIES

Following Black [2] we define the money demand function log(?)

=-/!Hog(J+),

where M, is the money supply and P, is the price level in the ith period and /3 is a positive parameter. The adaptive rule is given by log(%)

=klog(+-))

(2)

where k is positive. (Black [2] assumesk > 1 but this is not essential.) Let X, = log(M,+ r/M,), y1= log(P,+ JPt). Rewriting Eq. (1) by substituting t + 1 for t, and subtracting (1) from it, then substituting (2) and using the definitions of x1 and y, we obtain, as does Black [2], &Jr+,-(l+P).v,+b-,=(A

(3)

where we used xt = ky,-, . This is a second order linear difference equation in y, the logarithm of the inflation rate. This inflation rate will converge to its stationary value, y= 0, if the roots of Eq. (3) are within the unit circle, whether real or complex. Black [2] claimed that convergence is assuredif the real roots are within the unit circle and the complex roots have real parts between zero and one. This is erroneous; the modulus, and not the real parts of the complex roots must be within the unit circle. Below is a diagram showing regions of stability for combinations of the parameters k and /I. (For

1

FIG. 1. convergent, divergent.

Parameter values M.D. = monotonic

k

giving stability or instability divergent, O.C. = oscillatory

for Eq. (3): convergent,

M.C. O.D.

= monotonic = oscillatory

264

JESS BENHABIB

necessary and sufficient conditions for the stability of a second order system see Gandolfo [6, p. 561. As is easily seen from the diagram, whether Eq. (3) has a convergent solution or not depends on whether both k//l and k are less than unity. (Note from Eq. (3) that k//3 is the product of the roots.) Expressed in economic terms, this means that the necessary and sufficient conditions for convergence are that the inflation elasticity of money demand, ,f3,be greater than the inflation elasticity of the adaptive money supply function, k, and that the elasticity of the adaptive money supply function be greater than unity.

3. DISCUSSION OF RESULTS WHEN SMALL NON-LINEARITIES ARE INTRODUCED For the model above we observe that for values of p > 1, a choice of k, where k = p gives rise to periodic solutions of Eq. (3) that have constant amplitude. In linear models this situation is considered degenerate, since the slightest disturbance in the parameters k or b results in either convergence or divergence. Such a description is highly deceptive, however, since linear systems themselves are very special. Consider introducing any small nonlinearity into the system described by Eqs. (1) and (2). If certain nondegeneracy conditions are satisfied (see below), we can show that for any given p = PO, there will be at least a right or left neighborhood of k = /I,, (for /3 > 1) for which locally unique, non-trivial periodic solutions exist. As a very simple example, consider adding the term -E(P~/P~-~)~ to the righthand side of (2), where E is any non-zero number. We can transform this new second order equation to a first order system Y ItI=

1 +P TYt-xt-EXf3

k x,+ 1 = -Y, P

(4)

Evaluated at equilibrium values, x = y = 0, the roots of the Jacobian of (4) are simply the roots of the linear system of Eqs. (1), (2). The non-linear term does not affect these roots at equilibrium. Since the Jacobian does not vanish, the mapping represented by the right-hand side of (4) is a local diffeomorphism. We can now apply the Hopf Bifurcation Theorem for we vary k diffeomorphisms (see [8, p. 23]), where for given p =& parametrically (Varying p with k fixed will also do). If PO > 1 as assumed by Black [2], the two roots of the Jacobian will be complex and will pass through the unit circle with non-zero speed as k passes through k, =&.

DYNAMIC

MONETARY

MODELS

265

Then in general, if certain technical conditions which rule out a few degeneracies are satisfied,’ there will be a continuous one parameter family of invariant circles as the solution of (4), one for each k/P, in either a right or left neighborhood of k,,/& (see footnote below). Let us also point out that there are, for certain other non-linear systems, possibilities of almost periodic trajectories and other erratic behavior (see [7], Appendix D, and the cited references).

4. FINAL REMARKS

Section 5 shows that rational expectation price paths discussed by Black [2] may exhibit a wide variety of behavior, including cyclical fluctations, if some non-linearity is introduced into the model. They are not necessarily unique, and those that do not converge to an equilibrium may nevertheless remain bounded. Brock, in unpublished notes, has pointed out that the arbitrary money demand function of Sargent and Wallace [9] and Black [Z] should be derived from microeconomic behavior. In the context of a model where real balances enter the utility function, Brock argues that nonuniqueness of paths may occur only for very special money supply policies. Benhabib and Nishimura [l] have also studied rational expectations equilibria in a real, multi-capital economy and have shown that limit cycles in stocks, outputs and relative prices can occur endogenously, without outside shocks, and without further possibilities of arbitrage. This is possible since agents equate rates of return on assets which consist of yields plus capital gains. If yields endogenously fluctuate against one another due to the capital accumulation process, so will capital gains, yielding a rational expectations equilibrium with cyclical relative prices. A similar argument should explain the cyclical pattern of prices in models where money yields liquidity services. Along the perfect foresight rational expectation path the net yield of a unit of money, consisting of its liquidity services plus (discounted) utility from potential future consumption, must equal the additional utility obtainable from the current consumption value of that unit of money. A ’ Let the roots going through the unit circle be a f bi. or in polar form reei, r = a’ + b* = 1. H= tan b/a. We require that emei # 1 for m = 1,2, 3,4, 5. In other words, we need tan(mb/a) # 0 for m = 1.2. 3.4, 5. This rules out certain isolated ratios b/a (like b/a = 2n, n, n/2. n/3. etc.). If this is satisfied, there is a k-dependent change of parameters bringing (4) (or any alternative second order system) into polar coordinate form F(r, 0) = (( 1 + k) r-f,(k) r’. 0 +f,(k) +f,(k) r*) + terms of order i. The invariant circles will be stable (unstable) and will exist in right (left) neighborhood of k,/& if J, > 0 (f, < 0). (See [8, p. 23; Sects. 6, 7. pp. 206224)). The case where f, = 0 is very special and includes the case where the difference equations are linear. General results for this case are not available. It is in this sense that linear systems are exceptional and misleading. See also (71 especially Appendix D, for further discussion.

266

JESS BENHABIB

variable and foreseeable future inflation path does not result in speculative movements into or out of money holdings since, along that path, the inflation loss (or deflation gain) has to be added to the fluctuating yield of money in terms of its liquidity services. Thus a foreseeable fluctuating price level is quite consistent with the optimality condition cited in the sentencebefore the last one.

ACKNOWLEDGMENTS I would like to thank useful discussions.

Professor

W. A. Brock

for showing

me his unpublished

notes and for

REFERENCES 1. J. BENHABIB AND K. NISHIMLJRA. The Hopf bifurcation and the existence and stability of closed orbits in multisector models of optimal economic growth, J. Econ. Theory. in press. 2. F. BLACK, Uniqueness of the price level in monetary growth model with rational expectations, J. Econ. Theory 7 (January 1974), 53-65. 3. F. BLACK, On markets. unpublished, 1976. 4. W. A. BROCK, Money and growth: The case of long-run perfect foresight. Internut. Econ. Rev. 15 (October 1977), 75b-774. 5. W. A. BROCK, A simple perfect foresight monetary model, J. Monelar?: Econ. I (April 1975), 133-150. 6. G. GANDOLFO, “Mathematical Methods and Models in Economic Dynamics.” NorthHolland, Amsterdam/London, 197 1. 7. J. GUCKENHEIMER, G. OSTER, AND A. IPAKTCHI. The dynamics of density dependent population models, J. Math. Biol. 4 (1977), 101-147. 8. J. E. MARSDEN AND M. MCCRAKEN. “The Hopf Bifurcation and its Applications.” Applied Mathematical Sciences Series No. 19, Springer-Verlag. New York. 1976. 9. T. J. SARGENT AND N. WALLACE, The stability of Models of money and growth with perfect foresight, Econometrica 41 (November 1973). 1043-1048. 10. W. S. VICKREY, Stability through inflation, in “Post-Keynesian Economics” (K. K. Kurihara. Ed.), pp. 89-122, Allen and Unwin, Winchester, Mass., 1955.