A Positive Theory of Monetary Policy in a Natural Rate Model

7/29/2019 A Positive Theory of Monetary Policy in a Natural Rate Model

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A Positive Theory of Monetary Policy in a Natural Rate ModelAuthor(s): Robert J. Barro and David B. GordonSource: Journal of Political Economy, Vol. 91, No. 4 (Aug., 1983), pp. 589-610Published by: The University of Chicago PressStable URL: http://www.jstor.org/stable/1831069 .

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A Positive Theory of MonetaryPolicy in a Natural Rate Model

RobertJ. BarroUniversityof Chicago and National Bureau of Economic Research

David B. GordonUniversityof Rochester

A discretionary policymaker can create surprise inflation, which mayreduce unemployment and raise government revenue. But whenpeople understand the policymaker's objectives, these surprises can-not occur systematically. In equilibrium people form expectationsrationally and the policymaker optimizes in each period, subject tothe way that people form expectations. Then, we find that (1) therates of monetary growth and inflation are excessive; (2) these ratesdepend on the slope of the Phillips curve, the natural unemploy-ment rate, and other variables that affect the benefits and costs frominflation; (3) the monetary authority behaves countercyclically; and(4) unemployment is independent of monetary policy. Outcomesimprove if rules commit future policy choices in the appropriatemanner. The value of these commitments-which amount to long-term contracts between the government and the private sector-underlies the argument for rules over discretion.

The primary purpose of this paper is to develop a positive theory ofmonetary policy and inflation. On the one hand, the theory turns outto accord with two perceptions about the world in recent years:

Some useful suggestions were provided on earlier drafts by Ken Arrow, Gary Beck-er, Bob Brito, Ben Eden, Donald Gordon, Herschel Grossman, Bob Hall, Bob Lucas,Bill Oakland, Alan Stockman, and Larry Weiss. This research is supported in part bythe National Science Foundation.UouJral of Political EconlomY,983, vol. 91, no. 4]?) 19833 'Ihe UniVersity of Chicago. All rights reserved. 0022-3808/83/9104-0001$01.50

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590 JOURNAL OF POLITICAL ECONOMY1. Average rates of inflation and monetary growth are excessiverelative to an efficiency criterion.2. There is a tendency to pursue activist, countercyclical monetary

policies.Yet the model exhibits three other properties:3. The unemployment rate-our proxy for real economic activ-ity-is invariant with monetary policy (neglecting the familiardeadweight-loss aspect of inflation).4. The policymaker and the public all act rationally, subject to theirenvironments.5. The policymaker's objectives reflect the public's preferences.'Natural rate models with rational expectations-such as Sargentand Wallace (1975)-suggest that the systematic parts of monetarypolicy are irrelevant for real economic activity. Some empirical evi-dence on the real effects of monetary disturbances in the post-WorldWar II United States (e.g., Barro 1977, 1981) is consistent with thisresult-in particular, there is some support for the proposition thatanticipated monetary changes are neutral with respect to output, un-employment, and so on. On the other hand, these empirical studiesand others indicated the presence of countercyclical monetary policyat least for the post-World War II United States-rises in the unem-ployment rate appear to generate subsequent expansions in monetarygrowth. Within the natural rate framework, it is difficult to reconcilethis countercyclical monetary behavior with rationality of the policy-maker.2 A principal object of our analysis is to achieve this reconcilia-tion.The natural rate models that have appeared in the macroeconomicsliterature of the last decade share the characteristic that policy choice

is over a class of prespecified monetary rules. With the policy rulepredetermined, there is no scope for ongoing policymaking; discre-tionary policy choice is excluded a priori. If private agents can deducethe characteristics of the monetary process once it is implemented, itdefines their expectations. Thus, the policy decision is made subject tothe constraint that agents' expectations of future monetary policy will' The model that we consider is sufficiently simple to allow for unanimity aboutdesirable governmental actions.2 Many people respond with a willingness to view public policy as irrational. Despitethe obvious attractions of this viewpoint, it does leave us without a theory of systematicgovernmental behavior. An earlier attempted reconciliation with rationality (Barro1977, p. 104) relied on public finance considerations associated with cyclical changes inthe revenue obtained from printing money. This avenue appears to be quantitativelyinsufficient to explain the facts about countercyclical monetary response. However, therevenue motive for money creation is important in some extreme cases. See, e.g.,Hercowitz (1981) for an analysis of monetary behavior and government spendingduring the German hyperinflation.



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THEORY OF MONETARY POLICY 591equal the realization. This framework allows the analysis to be re-duced to a pair of single-agent decision problems, which can be con-sidered independently. But, this approach cannot deal with the game-theoretic situation that arises when policy decisions are made on anongoing basis.In our framework an equilibrium will include the followingfeatures:a) a decision rule for private agents, which determines their actionsas a function of their current information,b) an expectations function, which determines the expectations ofprivate agents as a function of their current information, and

c) a policy rule, which specifies the behavior of policy instrumentsas a function of the policymaker's current information set.The outcome is said to be a rational expectations equilibrium if, first,the decision rule specified in a is optimal for agents given their expec-tations as calculated under b; and second, it is optimal for the policy-maker, whose actions are described by c, to perform in accordancewith agents' expectations b, given that the policymaker recognizes theform of the private decision rules under a. Faced by a maximizingpolicymaker, it would be unreasonable for agents to maintain expec-tations from which they know it will be in the policymaker's interest todeviate.If policy is precommitted, the only reasonable expectations thatagents can hold are those defined by the rule. But, if policy is sequen-tially chosen, the equality of policy expectations and realizations is acharacteristic of equilibrium-not a prior constraint. We have to de-termine which expectations agents can reasonably expect to berealized.We view the policymaker as attempting to maximize an objectivethat reflects "society's" preferences on inflation and unemployment.(Additional arguments for the preference function are mentionedlater.) Although the equilibrium involves a path of unemploymentthat is invariant with policy, the rational policymaker adopts an activ-ist rule. The extent of countercyclical response depends, among otherthings, on society's relative dislikes for inflation and unemployment.There is an apparent contradiction because the policymaker pursues

an activist policy that ends up having no desirable effects-in fact,unemployment is unaltered but inflation ends up being excessive.This outcome reflects the assumed inability of the policymaker-thatis, of the institutional apparatus that is set up to manage monetaryaffairs-to commit its course of future actions. This feature has beenstressed in an important paper by Kydland and Prescott (1977). Ifcommitment were feasible through legal arrangements or other pro-cedures, the countercyclical aspect of monetary policy would disap-



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592 JOURNAL OF POLITICAL ECONOMYpear (and, abstracting from costs of erecting and maintaining institu-tions, everyone would be better off). When this type of advancerestriction is precluded, so that the policymaker sets instruments ateach date subject only to the initial conditions prevailing for that date(which do not include restraints on policy choices), the equilibriummay involve an activist form of policy. This solution conforms tooptimal behavior of private agents subject to a rationally anticipatedpolicy rule. It corresponds also to optimality for the policymaker eachperiod, subject to agents' decision rules. Although an equilibriumobtains, the results are suboptimal, relative to outcomes where com-mitment is permitted. Given an environment where this type of policycommitment is absent-as appears to characterize the United Statesand other countries in recent years-the results constitute a positivetheory of monetary growth and inflation.We illustrate the results with a simple model, which comes from anexample in Kydland and Prescott (1977, pp. 477-80). We augmenttheir example along the lines detailed in Gordon (1980) to include atheory of expectations formation. People form their expectations byeffectively solving the problem that the optimizing policymaker willface. The policymaker's problem is then conditioned on the expecta-tions function of private agents. Ultimately, there are no systematicdifferences between expected and realized inflation. But this prop-erty emerges as part of the equilibrium rather than as a constraint onthe policy problem.

I. The Model of Unemployment and InflationThe unemployment rate U1, which is a convenient proxy for theoverall state of real activity, equals a "natural rate," U/, plus a termthat depends negatively on contemporaneous unexpected inflation,7Tt _ Tt 9

Ut == Un _ ot(,t _t e), at > 0.(l)For convenience, we treat the "Phillips curve slope" parameter, a, as aconstant.3 Given the relevant inflationary expectations, Srt, equation(1) is assumed to reflect the maximizing behavior of private agents ondecentralized markets. The formulation of St is detailed below. Equa-

3 The prior expectation of inflation for period t could be distinguished from theexpectation that is conditioned on partial information about current prices. This dis-tinction arises in models (e.g., Lucas 1972, 1973; Barro 1976) in which peopleoperate in localized markets with incomplete information about contemporaneousnominal aggregates. In this setting the Phillips curve slope coefficient, Ox,urns outto depend on the relative variances for general and market-specific shocks.



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THEORY OF MONETARY POLICY 593tion (1) could be reformulated without changing the main conclusionsby expressing U, as a reduced-form function of monetary shocks.The natural unemployment rate can shift over time due to autono-mous real shocks, Et. A single real disturbance is allowed to have apersisting influence on unemployment, output, etc. This behavior ismodeled as

Un = X;UnL+ (1 - X)Utl + et, 0 A 1, (2)where t iS independently, identically distributed with zero mean. If 0< X < 1 applies, then the realization for the shock t affects futurenatural unemployment rates in the same direction. However, the ef-fect dissipates gradually over time-equation (2) implies that thelong-run mean of the natural unemployment rate is ULP, constant.For convenience, we assume that Ut in equation (1) depends only oncontemporaneous unexpected inflation, St - St, and not on laggedvalues. These additional terms could be introduced without changingthe main results (see below). Thus, the main thrust of our analysis iscompatible with either monetary or real theories of business cycles.The policymaker's (and society's) objective for each period is sum-marized by a cost, Zt, which depends on that period's values for theunemployment rate and inflation. We assume a simple quadraticform,

Zt = a(Ut - kU')2 + b(rrt)2; a, b > 0 , 0 k S 1. (3)We do not consider any divergence across individuals in their assess-ments of relative costs for unemployment and inflation.The first term in equation (3) indicates that costs rise with thedeparture of the unemployment rate from a target value, kUW,whichdepends positively on the contemporaneous natural rate. In the ab-sence of external effects, k = 1 would correspond to an efficiencycriterion-that is, departures of Ut from Ut'in either direction wouldbe penalized. In the presence of unemployment compensation, in-come taxation, and the like, the natural unemployment rate will tendto exceed the efficient level-that is, privately chosen quantities ofmarketable output and employment will tend to be too low. Theinequality k < 1 captures this possibility.4 Not surprisingly, we shallneed some existing distortion in the economy-that is, k < 1-inorder to generate activist policy in our model. This result conformswith those stressed by Calvo (1978a).Governmental decisions on taxes and transfers will generally in-

The target unemployment rate is U* = kU' < U". The formulation implies alsothat aUl*aU7 < 1. The last condition, which we use for some conclusions, is moredifficult tojustify.



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594 JOURNAL OF POLITICAL ECONOMYfluence the value of k. However, given that some government expen-ditures are to be carried out, it will generally be infeasible to select afiscal policy that avoids all distortions and yields k = 1. We assumethat the government's optimization on the fiscal side-which we donot analyze explicitly-results in a value of k that satisfies 0 < k < 1.The choice of monetary policy is then carried out conditional on thisvalue of k.Equation (3) regards departures of St from zero as generating costs.Economists have not come up with convincing arguments to explainwhy inflation is very costly. However, direct costs of changing priceswould fit most easily into our model. More generally, the form of thecost function could be changed to include a term in (-r, - St)2, whereSt might involve the optimal rate of taxation on cash balances. A latersection expands the analysis to consider the revenue from moneycreation.We assume that the policymaker controls an instrument-say,monetary growth, pa-which has a direct connection to inflation, Wt. ineach period. This specification neglects any dynamic relation betweeninflation and monetary growth or a correlation between (mt - pt) andthe real disturbances, Et, Et- 1,... In effect, we pretend that thepolicymaker chooses St directly in each period. We discuss later whathappens when we allow a separation between inflation and monetarygrowth.The choice of 7Tt at each date is designed to minimize the expectedpresent value of costs, as calculated at some starting date zero. That is,the objective is to minimizeELL t I'o (4)where I( represents the initial state of information and r is a constant,exogenous real discount rate. It should be stressed that the policy-maker's objective conforms with society's preferences.The determination of inflation and unemployment can be charac-terized as a game between the policymaker and a large number ofprivate-sector agents. The structure of this game is as follows. Thepolicymaker enters period t with the information set, It 1. The in-flation rate, St, is set based on It I in order to be consistent with thecost-minimization objective that we set out in expression (4). Simulta-neously, each individual formulates expectations, St, for the policy-maker's choice of inflation for period t. These expectations are basedon the same information set, It -, as that available to the policymaker.Most important, in forming inflationary expectations, people incor-porate the knowledge that a, will emerge from the policymaker's cost-minimization problem that we specify in equation (4). Finally, the



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THEORY OF MONETARY POLICY 595choices for St and -rr, together with the random disturbance, Et, deter-mine Ut and the cost, Zt, in accordance with equations (1)-(3).TheExpectationsMechanismIn order to determine a', agents must consider the policymaker'soptimization problem, which determines the choice of St. Suppose forthe moment that the policymaker, when selecting St, treats 7


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596 JOURNAL OF POLITICAL ECONOMYable at the start of period t. Suppose that people perceive this processas described by the reaction function, he(It-)f5 Therefore, infla-tionary expectations formed on the basis of It 1- are given by6

T'= he(I1) (5)A solution to the model involves finding a function he(e),such thatsetting -rt = he(It 1) is a solution to the policymaker's cost-minimi-zation problem, given that m' = he(It 1). Expecting inflation asspecified by he(e) must not contradict the policymaker's minimizationof expected costs, as set out in equation (3). The previous discussionsuggests that lagged values of inflation will not appear as parts of the

solution, he(e). That is, we are looking for an equilibrium where am


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THEORY OF MONETARY POLICY 597rationality entails using equation (8) in order to calculate he(I- 1 inequation (5). Consistency requires he(I1 l) = ft, The unexpected in-flation term, frt - he(I l), then cancels out in equation (8), whichleads to the formula for the expectations function,

rT = he(It- 1) = ab (1 - k)[XUtL1 + (1 - X)Ut]- aba(1 - k)Et1Ut. (

EquilibriumPolicyBy the construction of the problem, a policymaker who faces theexpectations given in equation (9) will be motivated from the first-order condition of equation (8) to choose an inflation rate, -ft, thatcoincides with -rr2.That is, the equilibrium involves

ITt = aa(1 - IU' = 7T' (10)bt bSince frt = w', Ut = Un applies also as part of the equilibrium.

Equation (10) provides an equilibrium (Nash equilibrium) in thefollowing sense. Given the public's equilibrium perceptions, TTtrhe(e), minimization of Et 1Zt (for a given value of 7re) induces thepolicymaker to choose rt = he(e) in each period.7 Expectations arerational and individuals optimize subject to these expectations (assummarized in eqq. [1] and [2]).In order to provide perspective on the present framework, con-sider an alternative manner in which the policymaker's choice prob-lem could have been formulated. Policy could have been viewed as theonce-and-for-all choice of reaction function, h(.), so that 7Te = he() =h(.) holds automatically in every period for all choices of h(-). Thisperspective applies, for example, to the analysis of macro policy inSargent and Wallace (1975). In their setting the choice of the func-tion, h(.), affects not only 7Tt but also 7Tt in each period. The indepen-dence of 7Tte from 7t is broken in the context of a once-and-for-allselection of policy functions. The condition -Tt - ITTe= t - he(1t1 ) =0 is then a constraint on the policy problem, which can be substitutedinto equation (7). In particular, with erreguaranteed to move one-to-one with changes in rt, the policymaker must regard unemployment,Ut = U', as invariant with h(.). Given the simple objective from equa-tion (3), which penalizes departures of Tt from zero, the choice of h()

7 Note that no equilibrium exists if the policymaker gives no direct weight to infla-tion-i.e., if b = 0. More generally, we require that the marginal cost of inflation, 'rT,,epositive at a point where 7rr= -rt.



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598 JOURNAL OF POLITICAL ECONOMYthat minimizes EZ, for all periods is a variant of the constant growthrate rule,8

At h(It - ) = ? l.Note that U1 = U' obtains again as part of this solution.Given the public's perceptions, rT' = he(It-1), Ut depends on theterm, wt - -rT = Wt- he(It 1). People have observed (Taylor 1975;Friedman 1979) that the policymaker can fool the public and reduceunemployment ("temporarily") by setting rrt -te = he(It 1) in periodt. This possibility is ruled out in the case where policy amounts to aonce-and-for-all binding choice of h(.). However, there may be nomechanism in place to constrain the policymaker to stick to the rule,h(It1 ), as time evolves. This consideration leads to the setup for pol-icy choice that we assumed before-namely, for given initial condi-tions in each period, including the expectations mechanism, rt =he(It I) set -rtin order to minimize Et -Zt. The policymaker is notrequired to select an inflation rate that equals the given expectedinflation rate. However, people also realize that the policymaker hasthe power to fool them at each date. Since the formation of expecta-tions takes this potential for deception into account, a full equilibriumwill ultimately involve -at = iTe'.The crucial point is that-unlike for aonce-and-for-all choice of policy rules-the policymaker does not re-gard St = ITrr s occurring automatically for all possible choices of wt.For this reason the (noncooperative) equilibrium does not correspondto equation (1 1).Compare the equilibrium solution, 7frtrom equation (10), with thechoice, St* = 0, which arises from a once-and-for-all selection of pol-icy rules. The equilibrium solution delivers the same unemploymentrate and a higher rate of inflation at each date. Therefore, the equilib-rium cost, Zt, exceeds that, Zt4 which would arise under the rule.(Note that, with Ut the same in both cases, costs end up dependingonly on the path of the inflation rate.) Of course, this conclusionneglects any costs of setting up or operating the different institutionalenvironments. Notably, the costs of enforcing commitments are ex-cluded. With this cost neglected, the present type of result provides anormative argument (and positive theory?) for policy rules -.that is,for commitment on future choices of wt.We highlight these aspects ofour results in a later section.It may be useful to demonstrate directly that 7Tt = 0 is not anequilibrium for the case where the policymaker optimizes subject togiven expectations in each period. Conjecture that re = he(11 1) = 0holds. In this case the choice of rTt> 0 would reduce unemployment

8 If Z, in eq. (3) depended on (7r, - W,)2, Trt*= -t would emerge.



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THEORY OF MONETARY POLICY 599for period t. A trade-off arises between reduced costs of unemploy-ment and increased costs from inflation. The balancing of these costsdetermines the chosen inflation rate, as shown in equation (8). Underthe assumed conditions (marginal cost of inflation is zero at Str = 0and marginal benefit from reduced unemployment is positive whenU, = U;'), he selected inflation rate will be positive. However, sincepeople understand this policy choice, the result 7, > 0 is inconsistentwith the conjecture that ri = 0. Zero inflation is not a reasonableexpectation for individuals to hold.An analogous argument can be used to find the positive rate ofinflation that does provide an equilibrium. If a small positive value for-7' had been conjectured, the policymaker would still have beenmotivated to select ,t> -rr2,which would be inconsistent with equilib-rium. The equilibrium obtains when Srt is sufficiently high, so that 7T,= St is the policymaker'sbest choice, given this value of nT".At thispoint the policymaker retains the option of choosing7t > 7Tr (or -rr

A Positive Theory of Monetary Policy in a Natural Rate Model

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