A new approach for modelling and understanding optimal monetary policy Katarzyna Romaniuk1

PRISM, University of Paris 1 Panthéon-Sorbonne, UFR 06, 1, rue Victor Cousin, 75005 Paris, France Abstract: The coefficients of Taylor (1993)’s monetary policy rule can be seen as portfolio weights. Their optimal values are derived by adapting Merton (1971)’s asset allocation model. JEL classification: C61; E52; G11. Keywords: optimal monetary policy; portfolio choice; stochastic dynamic programming. In the standard literature dealing with optimal monetary policy, the central bank minimizes its intertemporal loss function, defined as a weighted sum of the squared deviations of inflation and output from its respective targets, subject to the state and dynamics of the economy. Solving this optimization program with the stochastic linear regulator results in a linear feedback interest rate rule, defined as a function of the current state of the economy (see, for example, Rudebusch and Svensson, 1998). 1

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E-mail address: [email protected].

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We propose an alternative approach for modelling and deriving optimal monetary policy, based on the standard financial management framework developed by Merton (1971). This continuous-time stochastic framework has been originally built with the objective of determining optimal portfolio rules. We emphasize that the interest rate rule governing monetary policy can be seen as a portfolio of two assets - the price index and output -, while the goal of monetary policy is to define the optimal portfolio weights to be attributed to these two variables. The original Merton’s model is first adapted to the needs of the optimal monetary policy framework. It is then solved with the use of the technique of stochastic dynamic programming. The result interpretation follows, discussed with respect to the optimal dynamic behavior of the central bank and to the compatibility of its logic with the one of the original optimal asset allocation model.

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The framework

The objective of the central bank is to maximize its expected utility over the monetary policy horizon. Its optimization program at date t takes the following form::

max Et




T t

 U (r(s), P (s), Y (s), s) ds

(1)

where Et denotes the expectation conditional on the information available at date t, T the horizon of the central bank’s monetary policy, U the utility function, r the 2

interest rate determined by the central bank, P the price index and Y output. The utility function is assumed to be strictly concave in r. It is increasing, and then decreasing with respect to r, while the utility maximum corresponds to the interest rate which allows to minimize the deviations of P and Y from their targeted values. The dynamics of the price index P and of output Y write:

dP (t) = µP (t, P (t), Y (t), r(t)) dt + σ P (t, P (t), Y (t), r(t)) dB P (t) P (t)

(2)

dY (t) = µY (t, Y (t), P (t), r(t)) dt + σ Y (t, Y (t), P (t), r(t)) dB Y (t) Y (t)

(3)

where µP and µY denote expectations of the instantaneous variation rates of P and Y respectively, σ P and σ Y their volatilities, B P and B Y standard Brownian motions, and the instantaneous correlation between the Wiener processes is defined by dB P (t)dB Y (t) = ρP Y dt, with −1 ≤ ρP Y ≤ 1. In the literature dealing with monetary policy, the following relations are frequently assumed for inflation and output: Inflation depends linearly on past values of inflation and on the output gap (between the actual output and the potential one) in the preceding period - which is a version of the standard Phillips curve, while the output gap is determined by past values of the output gap and, negatively, on the monetary policy interest rate in the preceding period - which represents an IS 3

curve (see, for example, Rudebusch and Svensson, 1998). As to the monetary policy mechanisms, a reduction (an increase) in the monetary policy instrument increases (decreases) output, while the monetary policy actions impact on inflation indirectly, in the following period only, via the effect on output. The modelling we propose here captures these basic mechanisms. The expectation, as well as the volatility, of the price index and of the output variation rates both depend on time, the price index, output and the interest rate. As to the dynamics of the interest rate r, Taylor (1993)’s contribution proposes the following formulation of the monetary policy rule:

    r(t) − r = γ P P (t) − P + γ Y Y (t) − Y

(4)

where r, P and Y denote respectively the equilibrium level of the interest rate and the targeted values of the price index and of output, and γ P and γ Y the weights attributed to the price index and output deviations.2 This equation, widely used in the literature, presents in a simple manner the mechanism of the interest rate setting by the central bank: The latter increases (decreases) the interest rate when the price index is higher (lower) than the targeted one and/or the output level is higher (lower) than its potential. 2

It is worthwhile to note that the equilibrium level of the interest rate differs from the targeted

one.

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We shall adapt this basic formulation to our needs and define r as follows:

r(t) = λP (t)P (t) + λY (t)Y (t)

(5)

with λP and λY the portions of P and Y respectively that will be taken into account when fixing the monetary policy interest rate. After differentiating and dividing by r, one obtains the interest rate dynamics:

dr(t) dP (t) dY (t) = δ P (t) + δ Y (t) r(t) P (t) Y (t)

(6)

(t) (t) , δ Y (t) ≡ λY (t) Yr(t) , the following condition being satisfied: with δ P (t) ≡ λP (t) Pr(t)

δ P (t) + δ Y (t) = 1. From now on, the ease of exposition objective will lead us to drop the dependence with respect to t, P , Y and r, except when a risk of confusion appears. Let us replace the dynamics of P and Y , given by the equations (2) and (3) respectively, and use the identity δ P + δ Y = 1. One obtains:

dr = [δ P (µP − µY ) + µY ] dt + δ P σ P dB P + (1 − δ P )σ Y dB Y r

(7)

The central bank thus maximizes the program defined by equation (1) with respect to δ P - the proportion of the r variation rate driven by the P variation rate - subject to the constraint of the r dynamics, as defined by equation (7).

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Solving the central bank’s optimization program

In solving the optimization program, the method of stochastic dynamic programming is used, following the lines of Merton (1971). The first step consists in defining the indirect utility function J:

J(r, P, Y, t) ≡ max Et δP




T

t

 U(r, P, Y, s)ds

(8)

The function J is assumed to be strictly concave in r and twice differentiable with respect to r, P and Y . The Bellman optimality conditions require:

0 = max [U (r, P, Y, t) + DJ(r, P, Y, t)] δP

(9)

where D denotes the Dynkin operator, the Dynkin of J being defined as:

DJ = Jt +



Ji iµi +

i

1  Jij ijσ ij 2 i j

(10)

with i = j = {r, P, Y }, where subscripts on J denote partial derivatives, σ ij the covariance between the variables i and j, and it is assumed that the general form of the r dynamics writes

dr r

= µr dt + σ r dB, with µr the drift, σ r the volatility,

vector of dimension (1 × 2), and B a bi-dimensional Brownian motion defined as

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



 dB P  . dB ≡    dB Y

In equation (10), let us replace the parameters of the r dynamics with their

formulations proposed in equation (7). After deriving the resulting DJ formulation with respect to δP , one obtains the first order condition for an optimum:

   0 = Jr r (µP − µY ) + Jrr r2 δ P (σ P − σ Y )2 + σ P Y − σ 2Y

(11)

    +JrP rP σ 2P − σ P Y + JrY rY σ P Y − σ 2Y

the condition Jrr < 0 (J strictly concave in r) being a sufficient one for a unique interior maximum. The optimal proportion δ P then takes the form:

Jr µP − µY Jrr r (σ P − σ Y )2 σ P Y − σ 2Y JrP P σ 2P − σ P Y JrY Y σ P Y − σ 2Y − − − Jrr r (σ P − σ Y )2 Jrr r (σ P − σ Y )2 (σ P − σ Y )2

δP = −

3

(12)

Result interpretation

Four elements form together the optimal proportion δ P : the usual Merton’s speculative fund, a preference-independent output-hedge fund and two preference-dependent hedge funds - against price index variations and against output variations. 7

The last three terms do not require long interpretations: These funds provide hedges against variations of the stochastic variables influencing the utility level. By taking account of them when determining the value of the δ P proportion, the central bank covers itself against variations of the concerned variables. More interesting, the first term reproduces the structure of the Merton’s speculative fund: Following the basic rules in portfolio theory, the individual invests more heavily in the asset providing the best reward relative to the risk taken (which is illustrative of the well-known arbitrage between µ and σ), while taking into account his degree of risk aversion (the term − JJrrr r defining the reciprocal of the relative risk aversion coefficient). Yet in our monetary policy framework, the interpretation of this “speculative fund” is different: The central bank is rather interested in defining a monetary policy rule which basically contains inflation in the case of inflationary pressures, and supports growth when the latter is threatened. The interest rate should thus increase in the first case, and decrease in the second one, which leads to the existence of an optimal - or targeted - interest rate level. The central bank will then mix the price index P and output Y in order to obtain an interest rate r which optimally decreases prices when they exceed the targeted level, and increases output when the economy stagnates or enters in a recession. Let us first focus on the case of the need of an interest rate increase. As to the signs

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of the involved variables, Jrr is always strictly negative, r and (σ P − σ Y )2 are always strictly positive. In the case under scrutiny, Jr is positive, as an interest rate increase increases the utility level. As a consequence, the first term in the optimal proportion δ P will be positive if µP > µY and negative if µP < µY . Thus, when the actual price index and/or the actual output level have to be decreased in order to reach their targeted values, a higher weighting will be given to the variable which records the higher expected variation rate, while the weighting level is set relative to the compared risk of the two variables because of the term (σ P − σ Y )2 in the denominator. A higher weighting is thus given to the variable whose evolution characteristics enable a more effective interest rate increase. The situation is opposite when an interest rate decrease is needed. Jr is then negative (utility decreases when the interest rate increases), so that the sign of the first term in the optimal proportion δ P is positive when µP < µY and negative when µP > µY . The central bank now gives a higher weighting to the variable with the lower expected variation rate. If an interest rate increase (decrease) is even more needed, Jr increases (decreases), which means that the former decisions become more pronounced (as the first term of the optimal proportion δ P increases in absolute value). The absolute value of this first term of the δ P proportion will vary positively with respect to µP − µY and negatively with respect to (σ P − σ Y )2 . In order to analyze

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the influence of the term µP − µY , let us take the example of a need for an interest rate increase, characterized by the situation µP > µY : If the spread between µP and µY increases, the δ P first term increases; If this spread decreases, the δ P first term follows; If µY increases so as to become higher than µP , the δ P first term changes its sign. As to the influence of the term (σ P − σ Y )2 , the decisions are more pronounced if the spread σ P − σ Y is lower, and less pronounced in the case of a higher spread. This relation of the δ P first term with respect to µP − µY and (σ P − σ Y )2 is illustrative of a sort of arbitrage between µ and σ. The central bank takes its decision when observing the relative expected variation rate of the concerned variables, yet by taking account of the compared risk of these variation rates. This reasoning is close to the famous arbitrage between risk and reward in the financial management framework. An important difference is however to note: The central bank does not aim at choosing the best investment opportunities (which are characterized by the highest relative expected returns), yet by obtaining the mix between the price index and output which best increases, or decreases, the interest rate, depending on the state of the economy.

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Conclusion

The monetary policy interest rate rule can be considered as a portfolio of two variables - the price index and output. The central bank’s optimization program solution shows

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that the higher (higher) the expected variation rate of the concerned variable, the higher (lower) the portfolio weight attributed to it when an interest rate increase (decrease) is needed, the decision being taken relative to the compared risk of the two variables.

References Merton, R.C., 1971, Optimum consumption and portfolio rules in a continuoustime model, Journal of Economic Theory 3, 373-413. Rudebusch, G. D. and L. E. O. Svensson, 1998, Policy rules for inflation targeting, CEPR Discussion Paper No 1999. Taylor, J. B., 1993, Discretion versus policy rules in practice, Carnegie Rochester Conference Series on Public Policy 39, 195-214.

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