Amonetary authority (MA) can avoid the inflationary bias associated with the time-consistency problem in monetary policy by credibly committing to a zero-inflation policy rule. If monetary policy plays a role in stabilizing employment and output, the ex ante optimal policy rule will be a state-contingent rule. In a recent article, Flood and Isard (1989a) argue that commitment to this rule is infeasible if the probability distributions of relevant disturbances are not well-defined. Given this constraint, society might improve on the outcomes achieved under a discretionary regime or a regime of rigid adherence to a simple rule, by motivating the MA to follow a hybrid policy: in “normal” times, the MA adheres to a simple rule, while it responds to “unusual” circumstances at its discretion.^{1} In addition, the authors examine the optimal parameter setting for the simple rule, given that the private sector understands that the rule will not be adhered to under all circumstances.

Flood and Isard’s article is a valuable contribution to our understanding of central bank behavior. In particular, it offers some insight as to why central banks occasionally deviate from the policy rules to which they claim to be committed.

The purpose of this note is to show that there exist some inconsistencies in the derivation of the optimal degree of commitment to and parameter setting for the simple rule; and to outline a procedure by which these can be derived.

## I. Flood and Isard’s Procedure and Results

The goal of monetary policy is to minimize the social loss function:

where the domestic base-money growth rate *b* is the monetary policy instrument; v is a zero-mean money-demand shock; *E* is an expectations operator based on information available before v and *b* are observed; *k* is the difference between the socially desired and natural level of output; and *a* is the relative weight on the output and inflation goals, *a* > 0.

If the MA is motivated to minimize (1) and sets base-money growth *b* at its discretion, it will offset the money demand shock v to avoid destabilizing output and inflation. In addition, the MA has an incentive to stimulate output above the natural level by setting an inflationary base-money growth rate. In anticipation of this incentive, the private sector forms inflationary expectations that are sufficiently high to give the inflation-averse MA a disincentive to systematically stimulate output. As a result, society achieves the natural level of output on average, while base-money growth exhibits an inflationary bias.

In this model, the ex ante optimal policy rule sets zero inflation on average, but allows for a flexible response to the shock v. Flood and Isard’s article is driven by the assumption that the probability density of v is not well-defined. As a consequence, the commitment to the optimal state-contingent rule is infeasible.

Flood and Isard assume that the MA’s political principal^{2} sets the parameter τ_{0} in a simple rule of the form

where the subscript *R* stands for rule. Moreover, the MA is motivated to minimize the loss function

which is identical to the social loss function (1) except for an additive cost term. The MA’s political principal sets the cost c, which is incurred by the MA if it deviates from the simple rule given in equation (2); *d* is a dummy variable, which takes on the value of unity if the MA deviates from the rule, and zero otherwise.

The time sequence in the model is as follows. After the MA’s political principal has set τ_{0} and c, the wage setters form inflationary expectations *Eb*. Then v is realized. Finally, the MA sets *b*. The MA either follows the rule (2) or deviates from it (at cost *c*) to set *b* at its discretion:

where the subscript *D* stands for discretion.

Flood and Isard’s objective is to calculate the optimal τ_{0} and to characterize the optimal *c* in terms of whether this setting of *c* will give rise to a partial commitment to the optimal simple rule. The optimal setting of τ_{0} and *c* minimizes the expected social loss

where *q* is the probability that the MA adheres to the rule; the MA follows the rule for shock realizations in set *R* and deviates for shock realizations in set *D*. The wage setters form inflationary expectations:^{3}

At this point, the authors make some “simplifying” assumptions. First, *q* is assumed to be an exogeneously given parameter. Second, the MA is assumed to deviate from the rule in a symmetric way, so that *E*(*v* | *v*∈*D*) = 0. Implicitly, *E*(*v* | *v* ∈*D*) is assumed to be invariant to changes in τ_{0}. Equations (2), (4), and (6) are substituted into (5), and the derivative of the resulting expression with respect to τ_{0} is taken. As a result, the authors assert that the optimal parameter setting for the simple rule, the discretionary base-money growth rate set in the event of a deviation from the rule, and the private sector’s inflationary expectations are given by

Assuming that *v* is uniformly distributed with finite support, *q* is asserted to be equal to prob {| *v* | ≤ θ}, where θ is a parameter, θ ≥ 0 Substituting (7), (8), and (9) into (5) and taking the derivative of the resulting expression with respect to *q* leads to the result that the optimal *q* is strictly positive and, for some parameters, strictly smaller than unity. The authors conclude that the hybrid policy of mixing a simple zero-inflation rule with discretion dominates the discretionary regime and, for some parameters, the regime of rigid adherence to the simple rule.

## II. Discussion

I now show that the assumptions employed in the derivation of equations (7), (8), and (9) are inconsistent with the constraints implied by the MA’s incentive to deviate from the simple rule (7).

After τ_{0}, *c*, and *Eb* are set, and *v* is realized, the MA’s incentive to renege on the simple rule (2) is given by

Equations (7), (8), and (9) are substituted into (10), resulting in

The MA deviates from the rule if the incentive to deviate (11) is positive. This expression is symmetric around a strictly positive constant:

It follows from *Ev* = 0 and (12) that *E*(*v* | *v* ∈*D*) ≠ 0, which is inconsistent with the assumption that *E*(*v* | *v* ∈*D*) = 0. Furthermore,

This result is inconsistent with the assertion that *q* = prob {| *v* | ≤ θ}.

Taking the derivative of *I* with respect to τ_{0} and substituting equations (7), (8), and (9) into the resulting expression leads to

It follows from equations (12)–(14) that *E(∂q/∂*τ_{0}) ≠ 0, and *E[∂E(v |v ∈D)/∂τ _{0}]*≠0. These implications are inconsistent with the assumption that

*q*and

*E(v|v ∈ D)*can be taken as exogeneously fixed parameters in the derivation of the optimal τ

_{0}.

As a consequence of these inconsistencies, the derivation of the optimal τ_{0} presented in the previous section is invalid. The same holds for the proof that the optimal *q* lies strictly between zero and (for some parameters) unity, since this proof makes use of equations (7), (8), and (9).

The following procedure can be utilized to derive the optimal setting of *τ*_{0} and *c*:

Minimize equation (3) with respect to

*b*and solve for*b*(τ_{D}_{0}, c, Eb (τ_{0}, c), v).Substitute

*b*(τ_{R}_{0}) and*b*into equation (6) and solve for_{D}(τ_{0}, c, Eb(τ_{0}, c), v)*E*._{b}(τ_{0}, c)Substitute

*Eb(τ*into_{0}, c)*b*and solve for_{D}(τ_{0}, c, Eb(τ_{0}, c), v)*b*._{D}(τ_{0}, c, v)Substitute

*b*,_{D}(τ_{0}, c, v)*b*, and_{R}= τ_{0}*Eb (τ*into equation (5) to get_{0}, c)*EL(τ*._{0}, c)Minimize

*EL (τ*with respect to_{0}, c)*τ*, taking into account that the quantities_{0}and c*q(τ*, and_{0}, c),**R**(τ_{0}, c)change as τ**D**(τ_{0}, c)_{0}and*c*vary, in a way implicitly defined by the MA’s incentive to deviate equation (10) from the simple rule.^{4}

The optimal degree of commitment to and parameter setting for the simple rule will depend crucially on the probability distribution of the shock *v*.

FloodRobert P. andPeterIsard “Monetary Policy Strategies,” NBER Working Paper 2770 (Cambridge, Massachusetts: National Bureau of Economic Research1988).

FloodRobert P. and PeterIsard “Monetary Policy Strategies,” NBER Working Paper 2770 (Cambridge, Massachusetts: National Bureau of Economic Research(1989a) “Monetary Policy Strategies,” Staff PapersInternational Monetary Fund (Washington) Vol. 36 (September) pp. 612-32.

FloodRobertP. and PeterIsard “Monetary Policy Strategies,” NBER Working Paper 2770 (Cambridge, Massachusetts: National Bureau of Economic Research (1989b) “Simple Rules, Discretion, and Monetary Policy,” NBER Working Paper 2924 (Cambridge, Massachusetts: National Bureau of Economic Research).

Susanne Lohmann is a doctoral candidate in economics and political economy at the Graduate School of Industrial Administration, Carnegie Mellon University. The author thanks John Londregan for insightful comments.

Flood and Isard (1989a, 1989b) initially assume that the policymaker can commit to a partially state-contingent rule. In later sections of these versions and in Flood and Isard (1988), they examine the case where the policymaker is restricted to a simple (non-state-contingent) rule. For brevity’s sake, I concentrate on this case, too.

The political principal is the set of politicians who have the power to bring about changes in the institutional arrangements underlying the making of monetary policy.

The corresponding equation (26) in Flood and Isard (1989a) contains a typographical error.

If the cost *c* imposed on the MA for deviating from the rule is a deadweight loss for society, then *EL*(τ_{0}, *c*) will contain an additional term *q c*, which will affect the size of the optimal *c* (compare footnote 13 in Flood and Isard, 1989a).