Optimal Inflation Stabilization in a Medium-Scale Macroeconomic Model

8/11/2019 Optimal Inflation Stabilization in a Medium-Scale Macroeconomic Model

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Optimal Ination Stabilization in a

Medium-Scale Macroeconomic Model

Stephanie Schmitt-Grohe Martn Uribe

October 17, 2005Incomplete

Abstract

This paper characterizes Ramsey-optimal monetary policy in a medium-scale macro-economic model that has been estimated to t well postwar U.S. business cycles. Wend that mild deation is Ramsey optimal in the long run. However, the optimalination rate appears to be highly sensitive to the assumed degree of price stickiness.Within the window of available estimates of price stickiness (between 2 and 5 quarters)the optimal rate of ination ranges from -4.2 percent per year (close to the Friedmanrule) to -0.4 (close to price stability). This sensitivity disappears when one assumes thatlump-sum taxes are unavailable and scal instruments take the form of distortionaryincome taxes. In this case, mild deation emerges as a robust Ramsey prediction. Thepaper characterizes simple interest-rate feedback rules that mimic Ramsey-optimal sta-bilization policy. We nd that the optimal interest-rate rule is active in price and wageination, mute in output growth, and moderately inertial. JEL Classication: E52,E61, E63.

Keywords: Ramsey Policy, Interest-Rate Rules, Nominal Rigidities, Real Rigidities.

This paper was prepared for the Central Bank of Chile Annual Conference to be held on October 20-21,2005 in Santiago, Chile.

Duke University, CEPR, and NBER. Phone: 919 660-1889. E-mail: [email protected] University and NBER. Phone: 919 660-1888. E-mail: [email protected].


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Contents

1 Introduction 2

2 The Model 4

2.1 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 The Government . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4 Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.1 Market Clearing in the Final Goods Market . . . . . . . . . . . . . . 172.4.2 Market Clearing in the Labor Market . . . . . . . . . . . . . . . . . . 19

2.5 Functional Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.6 Inducing Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.7 Competitive Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.8 Ramsey Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Calibration 23

4 Ramsey Steady States 25

4.1 Additional Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 Optimal Operational Interest-Rate Rules 33

6 Conclusion 35

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1 Introduction

Two fundamental but separate questions in the theory of monetary stabilization policy arewhat is the optimal monetary policy and how can the central bank implement it. Bothquestions have been extensively studied in the existing related literature, but always in thecontext of simple theoretical structures, which by design are limited in their ability to accountfor actual observed business-cycle uctuations. Our contribution is to characterize optimalmonetary policy and its implementation using a medium-scale, empirically plausible modelof the U.S. business cycle.

The model we consider is the one developed in Altig et al. (2005). This model has beenestimated econometrically and shown to account fairly well for business-cycle uctuationsin the postwar United States. The theoretical framework emphasizes the importance of combining nominal as well as real rigidities in explaining the propagation of macroeconomic

shocks. Specically, the model features four nominal frictions, sticky prices, sticky wages,a transactional demand for money by households, and a cash-in-advance constraint on thewage bill of rms, and four sources of real rigidities, investment adjustment costs, vari-able capacity utilization, habit formation, and imperfect competition in product and factormarkets. Aggregate uctuations are driven by three shocks: a permanent neutral technol-ogy shock, a permanent investment-specic technology shock, and temporary variations ingovernment spending. Altig et al. (2005) and Christiano, Eichenbaum, and Evans (2005)argue that the model economy for which we seek to design optimal monetary policy canindeed explain the observed responses of ination, real wages, nominal interest rates, moneygrowth, output, investment, consumption, labor productivity, and real prots to neutral andinvestment-specic productivity shocks and monetary shocks in the postwar United States.

We address rst question posed above, namely, what business-cycle uctuations shouldlook like under optimal monetary policy by characterizing the Ramsey equilibrium associatedwith our model. The central policy problem faced by the monetary authority is, on the onehand, the need to stabilize prices so as to minimize price dispersion stemming from nominalrigidities and, on the other hand, the need to minimize and stabilize the opportunity cost of holding money to avoid transactional frictions. The task of characterizing Ramsey-optimal

policy is challenging because the model is large and highly distorted. A methodologicalcontribution of this paper is the development of a numerical procedure to characterize exactlythe steady-state of the Ramsey equilibrium. 1 We nd that the policy tradeoff faced by theRamsey planner is resolved in favor of price stability. In effect, the Ramsey optimal ination

1 Matlab code to replicate the quantitative results reported in this paper is available on the authorswebsites.

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rate is -0.4 percent per annum. This prediction, however, is highly sensitive to the assumeddegree of price stickiness. Available estimates of the degree of price stickiness vary between2 and 5 quarters. Within this range, the optimal rate of ination increases from a deationof about 4 when prices are fairly exible to a mild deation of about half a percent. So,

depending on depending on what available estimate of price rigidity one chooses to pick, theRamsey-optimal policy can range from close to the Friedman rule, to close to price stability.

We address the question of implementation of optimal monetary policy by characterizingoptimal, simple, and implementable interest-rate feedback rules. We restrict attention towhat we call operational interest rate rules. By an operational interest-rate rule we meanan interest-rate rule that satises three requirements. First, it prescribes that the nominalinterest rate is set as a function of a few readily observable macroeconomic variables. In thetradition of Taylor (1993), we focus on rules whereby the nominal interest rate depends onmeasures of ination, aggregate activity, and possibly its own lag. Second, the operationalrule must induce an equilibrium satisfying the zero lower bound on nominal interest rates.And third, operational rules must render the rational expectations equilibrium unique. Thislast restriction closes the door to expectations driven aggregate uctuations.

The object that monetary policy aims to maximize in our study is the expectation of life-time utility of the representative household. Here, one can distinguish between a measure of welfare conditional on a particular initial state of the economy or on an unconditional mea-sure of welfare. The key difference between conditional and unconditional welfare measuresis that the latter ignores the welfare effects of transitioning from a particular initial state

to the stochastic steady state induced by the policy under consideration. We nd that thecoefficients of the optimal operational interest-rate rule are quite similar under both typesof welfare measures.

In our welfare evaluations, we depart from the widespread practice in the neo-Keynesianliterature on optimal monetary policy of limiting attention to models in which the nonsto-chastic steady state is undistorted. Most often, this approach involves assuming the existenceof a battery of subsidies to production and employment aimed at eliminating the long-rundistortions originating from monopolistic competition in factor and product markets. Theefficiency of the deterministic steady-state allocation is assumed for purely computationalreasons. For it allows the use of rst-order approximation techniques to evaluate welfareaccurately up to second order (see, e.g., Rotemberg and Woodford, 1997). This practice hastwo potential shortcomings. First, the instruments necessary to bring about an undistortedsteady state (e.g., labor and output subsidies nanced by lump-sum taxation) are empiri-cally uncompelling. Second, it is ex ante not clear whether a policy that is optimal for aneconomy with an efficient steady state will also be so for an economy where the instruments

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necessary to engineer the nondistorted steady state are unavailable. For these reasons, werefrain from making the efficient-steady-state assumption and instead work with a modelwhose steady state is distorted.

Departing from a model whose steady state is Pareto efficient has a number of important

ramications. One is that to obtain a second-order accurate measure of welfare it no longersuffices to approximate the equilibrium of the model up to rst order. Instead, we obtain asecond-order accurate approximation to welfare by solving the equilibrium of the model up tosecond order. Specically, we use the methodology and computer code developed in Schmitt-Grohe and Uribe (2004c) to compute higher-order approximations to policy functions of dynamic, stochastic models. One advantage of this numerical strategy is that because it isbased on perturbation arguments, it is particularly well suited to handle economies with alarge number of state variables like the one studied in this paper.

The results from our numerical work suggest that in the model economy we study, theoptimal operational interest-rate rule responds aggressively to deviations of price and wageination from target. The price-ination coefficient is about 5 and the wage-ination co-efficient is about 2. In addition, the optimal interest-rate rule prescribes a mute responseto deviations of output growth from target. In this sense, the implementation of optimalpolicy calls for following a regime of ination targeting. The parameters of the optimizedrule appear to be robust to using a conditional or unconditional measure of welfare. Theimpulse responses of all variables of the model to all three shocks are remarkably similarunder the Ramsey policy and under the optimal operational rule.

The remainder of the paper is organized in ve sections. Section 2 presents the theoreticaleconomy and derives nonlinear recursive representations for the price and wage Phillipscurves as well as for the state variables summarizing the degree of wage and price dispersion.Section 3 describes the calibration of the model and discusses the solution method. Section 4characterizes the steady state of the Ramsey equilibrium. Section 5 computes the optimaloperational interest-rate rule. Section 6 provides concluding remarks.

2 The Model

2.1 Households

The economy is assumed to be populated by a large representative family with a continuumof members. Consumption and hours worked are identical across family members. Thehouseholds preferences are dened over per capita consumption, ct , and per capita labor

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effort, h t , and are described by the utility function

E 0

t=0

t U (ct bct 1, h t), (1)

where E t denotes the mathematical expectations operator conditional on information avail-able at time t, (0, 1) represents a subjective discount factor, and U is a period utilityindex assumed to be strictly increasing in its rst argument, strictly decreasing in its secondargument, and strictly concave. Preferences display internal habit formation, measured bythe parameter b [0, 1). The consumption good is assumed to be a composite made of acontinuum of differentiated goods cit indexed by i [0, 1] via the aggregator

ct =

1

0cit 1

1/ di1/ (1 1/ )

, (2)

where the parameter > 1 denotes the intratemporal elasticity of substitution across differ-ent varieties of consumption goods.

For any given level of consumption of the composite good, purchases of each individualvariety of goods i [0, 1] in period t must solve the dual problem of minimizing totalexpenditure,

10 P it cit di, subject to the aggregation constraint (2), where P it denotes the

nominal price of a good of variety i at time t. The demand for goods of variety i is thengiven by

cit =

P itP t

ct , (3)where P t is a nominal price index dened as

P t 1

0P 1 it di

11

. (4)

This price index has the property that the minimum cost of a bundle of intermediate goodsyielding ct units of the composite good is given by P tct .

Labor decisions are made by a central authority within the household, a union, which

supplies labor monopolistically to a continuum of labor markets of measure 1 indexed by j [0, 1]. In each labor market j , the union faces a demand for labor given by W jt /W t

hdt .

Here W jt denotes the nominal wage charged by the union in labor market j at time t, W t isan index of nominal wages prevailing in the economy, and hdt is a measure of aggregate labordemand by rms. We postpone a formal derivation of this labor demand function until weconsider the rms problem. In each particular labor market, the union takes W t and hdt as

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exogenous. 2 Given the wage it charges in each labor market j [0, 1], the union is assumedto supply enough labor, h jt , to satisfy demand. That is,

h jt =w jt

wt

hdt , (5)

where w jt W jt /P t and wt W t /P t . In addition, the total number of hours allocated to

the different labor markets must satisfy the resource constraint ht = 1

0 h jt dj. Combining this

restriction with equation (5), we obtain

h t = hdt 1

0

w jtwt

dj. (6)

Our setup of imperfectly competitive labor markets departs from most existing exposi-tions of models with nominal wage inertia (e.g., Erceg, et al., 2000). For in these models, it isassumed that each household supplies a differentiated type of labor input. This assumptionintroduces equilibrium heterogeneity across households in the number of hours worked. Toavoid this heterogeneity from spilling over into consumption heterogeneity, it is typically as-sumed that preferences are separable in consumption and hours and that nancial marketsexist that allow agents to fully insure against employment risk. Our formulation has theadvantage that it avoids the need to assume both separability of preferences in leisure andconsumption and the existence of such insurance markets. As we will explain later in more

detail, our specication gives rise to a wage-ination Phillips curve with a larger coefficienton the wage-markup gap than the model with employment heterogeneity across households.

The household is assumed to own physical capital, kt , which accumulates according tothe following law of motion

kt+1 = (1 )kt + it 1 S itit 1

, (7)

where i t denotes gross investment and is a parameter denoting the rate of depreciation of physical capital. The function S introduces investment adjustment costs. It is assumed thatin the steady state, the function S satises S = S = 0 and S > 0. These assumptionsimply the absence of adjustment costs up to rst-order in the vicinity of the deterministicsteady state.

2 The case in which the union takes aggregate labor variables as endogenous can be interpreted as anenvironment with highly centralized labor unions. Higher-level labor organizations play an important rolein some European and Latin American countries, but are less prominent in the United States.

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As in Fisher (2005) and Altig et al. (2004), it is assumed that investment is subjectto permanent investment-specic technology shocks. Fisher argues that this type of shockis needed to explain the observed secular decline in the relative price of investment goodsin terms of consumption goods. More importantly, Fisher argues that investment-specic

technology shocks account for about 50 percent of aggregate uctuations at business-cyclefrequencies in the postwar U.S. economy. (As we will discuss below, Altig et al., 2005, ndsignicantly smaller numbers in the context of the model studied in our paper.)

We assume that investment goods are produced from consumption goods by means of alinear technology whereby t units of consumption goods yield one unit of investment goods,where t denotes an exogenous, permanent technology shock in period t. The growth rateof t is assumed to follow an AR(1) process of the form:

,t = ,t 1 + ,t ,

where ,t ln(,t / ) denotes the percentage deviation of the gross growth rate of in-vestment specic technological change and denotes the steady-state growth rate of t .

Owners of physical capital can control the intensity at which this factor is utilized. For-mally, we let u t measure capacity utilization in period t. We assume that using the stock of capital with intensity ut entails a cost of

1t a(ut )kt units of the composite nal good. The

function a is assumed to satisfy a(1) = 0, and a (1), a (1) > 0. Both the specication of cap-ital adjustment costs and capacity utilization costs are somewhat peculiar. More standardformulations assume that adjustment costs depend on the level of investment rather thanon its growth rate, as is assumed here. Also, costs of capacity utilization typically take theform of a higher rate of depreciation of physical capital. The modeling choice here is guidedby the need to t the response of investment and capacity utilization to a monetary shockin the U.S. economy. For further discussion of this issue, see Christiano, Eichenbaum, andEvans (2005) and Altig et al. (2004).

Households rent the capital stock to rms at the real rental rate rkt per unit of capital.Total income stemming from the rental of capital is given by r kt u tkt . The investment good isassumed to be a composite good made with the aggregator function shown in equation (2).

Thus, the demand for each intermediate good i [0, 1] for investment purposes, i it , is givenby i it =

1t it (P it /P t )

.As in our earlier related work (Schmitt-Grohe and Uribe, 2004), we motivate a demand

for money by households by assuming that purchases of consumption goods are subjectto a proportional transaction cost that is increasing in consumption-based money velocity.

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Formally, the purchase of each unit of consumption entails a cost given by (vt). Here,

vt ctmht

(8)

is the ratio of consumption to real money balances held by the household, which we denoteby mht . The transaction cost function satises the following assumptions: (a) (v) isnonnegative and twice continuously differentiable; (b) There exists a level of velocity v > 0, towhich we refer as the satiation level of money, such that (v) = (v) = 0; (c) ( v v) (v) > 0for v = v; and (d) 2 (v) + v (v) > 0 for all v v. Assumption (a) implies that thetransaction process does not generate resources. Assumption (b) ensures that the Friedmanrule, i.e., a zero nominal interest rate, need not be associated with an innite demand formoney. It also implies that both the transaction cost and the associated distortions inthe intra and intertemporal allocation of consumption and leisure vanish when the nominalinterest rate is zero. Assumption (c) guarantees that in equilibrium money velocity is alwaysgreater than or equal to the satiation level v. As will become clear shortly, assumption (d)ensures that the demand for money is decreasing in the nominal interest rate. Assumption (d)is weaker than the more common assumption of strict convexity of the transaction costfunction.

Households are assumed to have access to a complete set of nominal state-contingentassets. Specically, each period t 0, consumers can purchase any desired state-contingentnominal payment X ht+1 in period t + 1 at the dollar cost E tr t,t +1 X ht+1 . The variable r t,t +1

denotes a stochastic nominal discount factor between periods t and t + 1. Households payreal lump-sum taxes in the amount t per period. The households period-by-period budgetconstraint is given by:

E t r t,t +1 xht+1 + ct [1 + (vt )] + 1t [i t + a(ut )kt ] + m

ht + t =

xht + mht 1t

+ rkt u tkt (9)

+ 1

0w jt

w jtwt

hdt dj + t .

The variable xht / t X h

t /P t denotes the real payoff in period t of nominal state-contingent

assets purchased in period t 1. The variable t denotes dividends received from the own-ership of rms and t P t /P t 1 denotes the gross rate of consumer-price ination.

We introduce wage stickiness in the model by assuming that each period the household(or unions) cannot set the nominal wage optimally in a fraction [0, 1) of randomly chosenlabor markets. In these markets, the wage rate is indexed to average real wage growth and tothe previous periods consumer-price ination according to the rule W jt = W

jt 1(z t 1) ,

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where [0, 1] is a parameter measuring the degree of wage indexation. When equals 0,there is no wage indexation. When equals 1, there is full wage indexation to long-run realwage growth and to past consumer price ination.

The household chooses processes for ct , ht , xht+1 , w jt , kt+1 , it , ut , and mht so as to maximize

the utility function (1) subject to (6)-(9), the wage stickiness friction, and a no-Ponzi-gameconstraint, taking as given the processes wt , rkt , hdt , r t,t +1 , t , t , and t and the initialconditions xh0 , k0, and mh 1. The households optimal plan must satisfy constraints (6)-(9).In addition, letting t t wt t , t t q t , and t t denote Lagrange multipliers associated withconstraints (6), (7), and (9), respectively, the Lagrangian associated with the householdsoptimization problem is

L = E 0

t=0

t {U (ct bct 1, h t )

+ t hdt 1

0wit

witwt

di + rkt u t kt + t t

ct 1 + ctmht

1t [it + a(ut )kt] r t,t +1 xht+1 m

ht +

mht 1 + xhtt

+ twtt

ht hdt 1

0

witwt

di

+ tq t (1 )kt + it 1 S itit 1

kt+1 .

The rst-order conditions with respect to ct , xht+1 , h t , kt+1 , i t , mht , ut , and wit , in that order,are given by

U c(ct bct 1, h t) bE t U c(ct+1 bct , h t+1 ) = t [1 + (vt ) + vt (vt )], (10)

t r t,t +1 = t+1P t

P t+1(11)

U h (ct bct 1, h t) = t wt

t, (12)

t q t = E t t+1 r kt+1 ut+1 1t+1 a(u t+1 ) + q t+1 (1 ) , (13)

1t t = tq t 1 S itit 1

itit 1

S itit 1

+ E t t+1 q t+1it+1it

2

S it+1it

(14)

v2t (vt ) = 1 E t t+1

t t+1. (15)

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r kt = 1t a (ut ) (16)

wit = wt if wit is set optimally in t

wit 1(z t 1) / t otherwise,

where wt denotes the real wage prevailing in the 1 labor markets in which the union

can set wages optimally in period t. Let h t denote the level of labor effort supplied to thosemarkets. Because the labor demand curve faced by the union is identical across all labormarkets, and because the cost of supplying labor is the same for all markets, one can assumethat wage rates, wt , and employment, ht , are identical across all labor markets updatingwages in a given period. By equation (5), we have that wt h t = whdt . It is of use to track theevolution of real wages in a particular labor market. In any labor market j where the wageis set optimally in period t, the real wage in that period is wt . If in period t +1 wages are notreoptimized in that market, the real wage is wt (z t ) / t+1 . This is because the nominal

wage is indexed by percent of the sum of past price ination and long-run real wage growth.In general, s periods after the last reoptimization, the real wage is wt sk=1

(z t + k 1 )

t + k . Toderive the households rst-order condition with respect to the wage rate in those marketswhere the wage rate is set optimally in the current period, it is convenient to reproduce theparts of the Lagrangian given above that are relevant for this purpose,

Lw = E t

s=0

( )s t+ s hdt+ s wt+ s

s

k=1

t+ k(z t+ k 1)

w1 ts

k=1

t+ k(z t+ k 1)

1

wt+ st+ s

w t .

The rst-order condition with respect to wt is

0 = E t

s=0

( )s t+ s wt+ s hdt+ ss

k=1

t+ k(z t+ k 1)

1

wtsk=1

t + k(z t + k 1 )

wt+ st+ s

.

Using equation (12) to eliminate t+ s , we obtain that the real wage wt must satisfy

0 = E t

s=0

( )s t+ s wtwt+ s

hdt+ ss

k=1

t+ k(z t+ k 1)

1

wtsk=1

t + k(

z t + k 1 )

U ht + s

t+ s.

This expression states that in labor markets in which the wage rate is reoptimized in periodt, the real wage is set so as to equate the unions future expected average marginal revenueto the average marginal cost of supplying labor. The unions marginal revenue s periodsafter its last wage reoptimization is given by 1 wt

sk=1

(z t + k 1 )

t + k . Here, / ( 1)represents the markup of wages over marginal cost of labor that would prevail in the absence

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of wage stickiness. The factor sk=1(z t + k 1 )

t + k in the expression for marginal revenuereects the fact that as time goes by without a chance to reoptimize, the real wage declinesas the price level increases when wages are imperfectly indexed. In turn, the marginal costof supplying labor is given by the marginal rate of substitution between consumption and

leisure, or

U ht + s t + s = wt + st + s . The variable t is a wedge between the disutility of labor andthe average real wage prevailing in the economy. Thus, t can be interpreted as the averagemarkup that unions impose on the labor market. The weights used to compute the averagedifference between marginal revenue and marginal cost are decreasing in time and increasingin the amount of labor supplied to the market.

We wish to write the wage-setting equation in recursive form. To this end, dene

f 1t = 1

wt E t

s=0

( )s t+ swt+ swt

hdt+ ss

k=1

t+ k(z t+ k 1)

1

and

f 2t = w t E t

s=0

( )swt+ shdt+ s U ht + ss

k=1

t+ k(z t+ k 1)

.

One can express f 1t and f 2t recursively as

f 1t = 1

wt t

wtwt

hdt + E t t+1

(z t ) 1 wt+1

wt

1

f 1t+1 , (17)

f 2t = U ht wtwt

hdt + E t t+1(z t )

wt+1wt

f 2t+1 . (18)

With these denitions at hand, the wage-setting equation becomes

f 1t = f 2t . (19)

The households optimality conditions imply a liquidity preference function featuring anegative relation between real balances and the short-term nominal interest rate. To see this,we rst note that the absence of arbitrage opportunities in nancial markets requires that

the gross risk-free nominal interest rate, which we denote by Rt , be equal to the reciprocalof the price in period t of a nominal security that pays one unit of currency in every stateof period t + 1. Formally, Rt = 1/E t r t,t +1 . This relation together with the householdsoptimality condition (11) implies that

t = R tE t t+1t+1

, (20)

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which is a standard Euler equation for pricing nominally risk-free assets. Combining thisexpression with equations (10) and (15), we obtain

v2t (vt) = 1 1R t

.

The right-hand side of this expression represents the opportunity cost of holding money,which is an increasing function of the nominal interest rate. Given the assumptions regardingthe form of the transactions cost function , the left-hand side is increasing in money velocity.Thus, this expression denes a liquidity preference function that is decreasing in the nominalinterest rate and unit elastic in consumption.

2.2 Firms

Each variety of nal goods is produced by a single rm in a monopolistically competitiveenvironment. Each rm i [0, 1] produces output using as factor inputs capital services, kit ,and labor services, hit . The production technology is given by

F (kit , z th it ) z

t ,

where the function F is assumed to be homogenous of degree one, concave, and strictly in-creasing in both arguments. The variable z t denotes an aggregate, exogenous, and stochasticneutral productivity shock. The parameter > 0 introduces xed costs of operating a rm

in each period. In turn, the presence of xed costs implies that the production function ex-hibits increasing returns to scale. We model xed costs to ensure a realistic prot-to-outputratio in steady state. Finally, we follow Altig et al. (2005) and assume that xed costs aresubject to permanent shocks, z t , with

z tz t

=

1 t .

This formulation of xed costs ensures that along the balanced-growth path xed costs donot vanish. Let z,t z t /z t 1 denote the gross growth rate of the neutral technology shock.By assumption, in the non-stochastic steady state z,t is constant and equal to z . Also, letz,t = ln( z,t / z) denote the percentage deviation of the growth rate of neutral technologyshocks. Then, the evolution of z,t is assumed to be given by:

z,t = z z,t 1 + z ,t ,

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with z ,t (0, 2z ).Aggregate demand for good i, which we denote by yit , is given by

yit = ( P it /P t ) yt ,

whereyt ct [1 + (vt )] + gt +

1t [it + a(ut )kt ], (21)

denotes aggregate absorption. The variable gt denotes government consumption of the com-posite good in period t.

We rationalize a demand for money by rms by imposing that wage payments be subject to a working-capital requirement that takes the form of a cash-in-advance constraint.Formally, we impose

mf it

= wt hit , (22)

where mf it denotes the demand for real money balances by rm i in period t and 0 is aparameter indicating the fraction of the wage bill that must be backed with monetary assets.

Firms incur nancial costs in the amount (1 R 1t )mf it stemming from the need to

hold money to satisfy the working-capital constraint. Letting the variable it denote realdistributed prots, the period-by-period budget constraint of rm i can then be written as

E t r t,t +1 xf it +1 + mf it

xf it + mf it 1

t=

P itP t

1

yt r kt kit wt hit it ,

where E t r t,t +1 xf it +1 denotes the total real cost of one-period state-contingent assets that therm purchases in period t in terms of the composite good. 3 We assume that the rm mustsatisfy demand at the posted price. Formally, we impose

F (kit , z t h it ) z

t P itP t

yt . (23)

The objective of the rm is to choose contingent plans for P it , hit , kit , xf it +1 , and mf it so as

3

Implicit in this specication of the rms budget constraint is the assumption that rms rent capitalservices from a centralized market. This is a common assumption in the related literature (e.g., Christianoet al., 2003; Kollmann, 2003; Carlstrom and Fuerst, 2003; and Rotemberg and Woodford, 1992). A polarassumption is that capital is rm specic, as in Woodford (2003, chapter 5.3) and Sveen and Weinke (2003).Both assumptions are clearly extreme. A more realistic treatment of investment dynamics would incorporatea mix of rm-specic and homogeneous capital.

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to maximize the present discounted value of dividend payments, given by

E t

s=0

r t,t + s P t+ s it + s ,

where r t,t + s sk=1 r t+ k 1,t + k , for s 1, denotes the stochastic nominal discount factor

between t and t + s, and r t,t 1. Firms are assumed to be subject to a borrowing constraintthat prevents them from engaging in Ponzi games.

Clearly, because r t,t + s represents both the rms stochastic discount factor and the marketpricing kernel for nancial assets, and because the rms objective function is linear in assetholdings, it follows that any asset accumulation plan of the rm satisfying the no-Ponziconstraint is optimal. Suppose, without loss of generality, that the rm manages its portfolioso that its nancial position at the beginning of each period is nil. Formally, assume that

xf it +1 + m

f it = 0 at all dates and states. Note that this nancial strategy makes x

f it +1 state

noncontingent. In this case, distributed dividends take the form

it =P itP t

1

yt r kt kit wt h it (1 R 1t )m

f it . (24)

For this expression to hold in period zero, we impose the initial condition xf i0 + mf i 1 = 0.

The last term on the right-hand side of the above expression for dividends represents therms nancial costs associated with the cash-in-advance constraint on wages. This nancialcost is increasing in the opportunity cost of holding money, 1 R 1t , which in turn is anincreasing function of the short-term nominal interest rate R t .

Letting r t,t + sP t+ s mcit + s denote the Lagrange multiplier associated with constraint (23),the rst-order conditions of the rms maximization problem with respect to capital andlabor services are, respectively,

mcit z t F 2(kit , z t hit ) = wt 1 + R t 1

R t (25)

and

mcit F 1(kit , z t h it ) = rkt . (26)

It is clear from these optimality conditions that the presence of a working-capital requirementintroduces a nancial cost of labor that is increasing in the nominal interest rate. We notealso that because all rms face the same factor prices and because they all have access tothe same production technology with the function F being linearly homogeneous, marginalcosts, mc it , are identical across rms. Indeed, because the above rst-order conditions hold

14


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for all rms independently of whether they are allowed to reset prices optimally, marginalcosts are identical across all rms in the economy.

Prices are assumed to be sticky `a la Calvo (1983) and Yun (1996). Specically, eachperiod t 0 a fraction [0, 1) of randomly picked rms is not allowed to optimally set

the nominal price of the good they produce. Instead, these rms index their prices to pastination according to the rule P it = P it 1t 1. The interpretation of the parameter is thesimilar to that of its wage counterpart . The remaining 1 rms choose prices optimally.Consider the price-setting problem faced by a rm that has the opportunity to reoptimizethe price in period t. This price, which we denote by P t , is set so as to maximize the expectedpresent discounted value of prots. That is, P t maximizes the following Lagrangian:

L = E t

s=0

r t,t + s P t+ s s P tP t

1 s

k=1

t+ k 1t+ k

1

yt+ s r kt+ s kit + s wt+ s h it + s [1 + (1 R 1t+ s )]

+mc it + s F (kit + s , z t+ sh it + s ) z

t+ s P tP t

s

k=1

t+ k 1t+ k

yt+ s .

The rst-order condition with respect to P t is

E t

s=0

r t,t + s P t+ s s P tP t

s

k=1

t+ k 1t+ k

yt+ s 1

P tP t

s

k=1

t+ k 1t+ k

mcit + s = 0.

(27)

According to this expression, optimizing rms set nominal prices so as to equate averagefuture expected marginal revenues to average future expected marginal costs. The weightsused in calculating these averages are decreasing with time and increasing in the size of the demand for the good produced by the rm. Under exible prices ( = 0), the aboveoptimality condition reduces to a static relation equating marginal costs to marginal revenuesperiod by period.

It will prove useful to express this rst-order condition recursively. To that end, let

x1t

E t

s=0

rt,t + s

syt+ s

mcit + s

P tP t

1 s

k=1

t+ k 1 (1+ )/t+ k

and

x2t E t

s=0

r t,t + s syt+ s P tP t

s

k=1

t+ k 1/ (

1)t+ k

1

.

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Express x1t and x2t recursively as

x1t = ytmct p 1t + E t

t+1 t

( pt / pt+1 ) 1

t

t+1

x1t+1 , (28)

x2t = yt p t + E t

t+1 t

tt+1

1 pt pt+1

x2t+1 . (29)

Then we can write the rst-order condition with respect to P t as

x1t = ( 1)x2t . (30)

The labor input used by rm i [0, 1], denoted hit , is assumed to be a composite madeof a continuum of differentiated labor services, h jit indexed by j [0, 1]. Formally,

h it = 1

0h jit

1 1/ dj

1/ (1 1/ ), (31)

where the parameter > 1 denotes the intratemporal elasticity of substitution across dif-ferent types of activities. For any given level of hit , the demand for each variety of labor j [0, 1] in period t must solve the dual problem of minimizing total labor cost,

10 W

jt h

jit dj ,

subject to the aggregation constraint (31), where W jt denotes the nominal wage rate paid tolabor of variety j at time t. The optimal demand for labor of type j is then given by

h jit = W jt

W t

h it , (32)

where W t is a nominal wage index given by

W t 1

0W jt

1 dj

11

. (33)

This wage index has the property that the minimum cost of a bundle of intermediate laborinputs yielding h it units of the composite labor is given by W t h it .

2.3 The Government

Each period, the government consumes gt units of the composite good. We assume that thegovernment minimizes the cost of producing gt . As a result, public demand for each varietyi [0, 1] of differentiated goods git is given by git = ( P it /P t ) gt .

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We assume that along the balanced-growth path the share of government spending invalue added is constant, that is, we impose lim j E t gt+ j /y t+ j = sg, where sg is a constantindicating the share of government consumption in value added. To this end we impose:

gt = z

t gt ,

where gt is an exogenous stationary stochastic process. This assumption ensures that gov-ernment purchases and output are cointegrated. We impose the following law of motion forgt :

lngtg

= g lngt 1

g+ g,t .

The government issues money given in real terms by m t mht + 1

0 mf it di. For simplicity, we

assume that government debt is zero at time zero and that the scal authority levies lump-

sum taxes, t to bridge any gap between seignorage income and government expenditures,that is, t = gt (m t m t 1/ t ). As a consequence, government debt is nil at all times.

We postpone the presentation of the monetary policy regime until after we characterizea competitive equilibrium.

2.4 Aggregation

We limit attention to a symmetric equilibrium in which all rms that have the opportunity tochange their price optimally at a given time choose the same price. It then follows from (4)

that the aggregate price index can be written as P 1

t = (P t 1 t 1)1 + (1 ) P 1

t .Dividing this expression through by P 1 t one obtains

1 = 1t (1 )t 1 + (1 ) p

1 t . (34)

2.4.1 Market Clearing in the Final Goods Market

Naturally, the set of equilibrium conditions includes a resource constraint. Such a restrictionis typically of the type F (kt , z t h t ) z t = ct [1 + (vt)] + gt +

1t [it + a(ut )kt]. In the present

model, however, this restriction is not valid. This is because the model implies relative pricedispersion across varieties. This price dispersion, which is induced by the assumed natureof price stickiness, is inefficient and entails output loss. To see this, consider the followingexpression stating that supply must equal demand at the rm level:

F (kit , z t h it ) z

t = [1 + (vt )]ct + gt + 1t [i t + a(ut )kt ]

P itP t

.

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Integrating over all rms and taking into account that (a) the capital-labor ratio is commonacross rms, (b) that the aggregate demand for the composite labor input, hdt , satises

hdt =

1

0h it di,

and that (c) the aggregate effective level of capital, ut kt satises

u tkt = 1

0kit di,

we obtain

z t hdt F ut ktz t hdt

, 1 z t = [1 + (vt )]ct + gt + 1t [it + a(ut )kt ]

1

0

P itP t

di.

Let s t 1

0P itP t

di. Then we have

s t = 1

0

P itP t

di

= (1 ) P tP t

+ (1 ) P t 1t 1

P t

+ (1 ) 2 P t 2t 1

t 2

P t

+ . . .

= (1 )

j =0

j P t j js=1

t j 1+ s

P t

= (1 ) p t + tt 1

s t 1.

Summarizing, the resource constraint in the present model is given by the following twoexpressions

F (ut kt , z t hdt ) z

t = [1 + (vt )]ct + gt + 1t [i t + a(ut )kt ] s t (35)

ands t = (1 ) p

t + t t 1

s t 1, (36)

with s 1 given. The state variable st summarizes the resource costs induced by the inefficientprice dispersion featured in the Calvo model in equilibrium. Three observations are in orderabout the price dispersion measure st . First, st is bounded below by 1. That is, pricedispersion is always a costly distortion in this model. To see that s t is bounded below by 1,

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let vit (P it /P t )1 . It follows from the denition of the price index given in equation (4) that

1

0 vit/ ( 1)

= 1. Also, by denition we have st = 1

0 v/ ( 1)it . Then, taking into account

that / ( 1) > 1, Jensens inequality implies that 1 =

10 vit

/ ( 1)

10 v

/ ( 1)it = st .

Second, in an economy where the non-stochastic level of ination is nil (i.e., when = 1)or where prices are fully indexed to any variable t with the property that its deterministicsteady-state level equals the deterministic steady-state value of ination (i.e., = ), thenthe variable st follows, up to rst order, the univariate autoregressive process s t = s t 1.In these cases, the price dispersion measure st has no rst-order real consequences for thestationary distribution of any endogenous variable of the model. This means that studies thatrestrict attention to linear approximations to the equilibrium conditions are justied to ignorethe variable st if the model features no price dispersion in the deterministic steady state.But st matters up to rst order when the deterministic steady state features movements in

relative prices across goods varieties. More importantly, the price dispersion variable s t mustbe taken into account if one is interested in higher-order approximations to the equilibriumconditions even if relative prices are stable in the deterministic steady state. Omitting stin higher-order expansions would amount to leaving out certain higher-order terms whileincluding others. Finally, when prices are fully exible, = 0, we have that pt = 1 andthus st = 1. (Obviously, in a exible-price equilibrium there is no price dispersion acrossvarieties.)

As discussed above, equilibrium marginal costs and capital-labor ratios are identicalacross rms. Therefore, one can aggregate the rms optimality conditions with respect tolabor and capital, equations (25) and (26), as

mct z t F 2(ut kt , z t hdt ) = wt 1 + R t 1

R t (37)

andmct F 1(u tkt , z thdt ) = r

kt . (38)

2.4.2 Market Clearing in the Labor Market

It follows from equation (32) that the aggregate demand for labor of type j [0, 1], whichwe denote by h jt

10 h

jit di, is given by

h jt =W jtW t

hdt , (39)

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where hdt 1

0 hit di denotes the aggregate demand for the composite labor input. Takinginto account that at any point in time the nominal wage rate is identical across all labormarkets at which wages are allowed to change optimally, we have that labor demand in eachof those markets is

h t = wtwt

hdt .

Combining this expression with equation (39), describing the demand for labor of type j [0, 1], and with the time constraint (6), which must hold with equality, we can write

h t = (1 )hdt

s=0

s W t s sk=1 (z t+ k s 1)

W t

.

Let s t (1 )

s=0 s W t s

sk =1 (z t + k s 1 )

W t

. The variable s t measures the degree of

wage dispersion across different types of labor. The above expression can be written as

ht = s t hdt . (40)

The state variable s t evolves over time according to

s t = (1 ) wtwt

+ wt 1wt

t(z t 1)

s t 1. (41)

We note that because all job varieties are ex-ante identical, any wage dispersion is inefficient.This is reected in the fact that s t is bounded below by 1. The proof of this statement isidentical to that offered earlier for the fact that st is bounded below by unity. To see this, notethat s t can be written as s t =

10

W itW t

di. This inefficiency introduces a wedge that makes

the number of hours supplied to the market, h t , larger than the number of productive unitsof labor input, hdt . In an environment without long-run wage dispersion, the dead-weightloss created by wage dispersion is nil up to rst order. Formally, a rst-order approximationof the law of motion of s t yields a univariate autoregressive process of the form s t = s t 1,as long as there is no wage dispersion in the deterministic steady state. When wages are

fully exible, = 0, wage dispersion disappears, and thus s t equals 1.It follows from our denition of the wage index given in equation (33) that in equilibrium

the real wage rate must satisfy

w1 t = (1 ) w1 t + w

1 t 1

(z t 1)

t

1

. (42)

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Aggregating the expression for rms prots given in equation (24) yields

t = yt r kt u tkt wt hdt (1 R 1t )wthdt . (43)

In equilibrium, real money holdings can be expressed as

m t = mht + wt hdt , (44)

and the government budget constraint is given by

t = gt (m t m t 1/ t ). (45)

2.5 Functional Forms

We use the following standard functional forms for utility and technology:

U =(ct bct 1)1

4 (1 ht )41 3

1

1 3

andF (k, h) = kh1 .

The functional form for the investment adjustment cost function is taken from Christiano,

Eichenbaum, and Evans (2005):

S iti t 1

= 2

itit 1

I 2

,

where I is the steady-state growth rate of investment.Following Schmitt-Grohe and Uribe (2004) we assume that the transaction cost technol-

ogy takes the form(v) = 1v + 2/v 2 12. (46)

The money demand function implied by the above transaction technology is of the form

v2t = 21

+ 11

R t 1R t

.

Note the existence of a satiation point for consumption-based money velocity, v, equal to

2/ 1. Also, the implied money demand is unit elastic with respect to consumption expen-ditures. This feature is a consequence of the assumption that transaction costs, c (c/m ), are21


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homogenous of degree one in consumption and real balances and is independent of the par-ticular functional form assumed for (). Further, as the parameter 2 approaches zero, thetransaction cost function () becomes linear in velocity and the demand for money adoptsthe Baumol-Tobin square root form with respect to the opportunity cost of holding money,

(R 1)/R . That is, the log-log elasticity of money demand with respect to the opportunitycost of holding money converges to 1/2, as 2 vanishes.

The costs of higher capacity utilization are parameterized as follows:

a(u) = 1(u 1) + 22

(u 1)2.

2.6 Inducing Stationarity

This economy features two types of permanent shocks. As a result, a number of variables,

such as output and the real wage, will not be stationary along the balanced-growth path.We therefore perform a change of variables so as to obtain a set of equilibrium conditionsthat involve only stationary variables. To this end we note that the variables ct , mht , mt ,wt , wt , yt , gt , t , x1t , x2t , and t are cointegrated with z

t . Similarly, the variables kt+1 and i tare cointegrated with t z t , the variable t is cointegrated with z

t(1 3 )(1 4 ) 1, the variables

q t and rkt are cointegrated with 1 / t , and the variables f 1t and f 2t are cointegrated withz t

(1 3 )(1 4 ) . We therefore divide these variables by the appropriate cointegrating factorand denote the corresponding stationary variables with capital letters.

2.7 Competitive Equilibrium

A stationary competitive equilibrium is a set of stationary processes ut , C t , ht , I t , K t+1 ,vt , M ht , M t , t , t , W t , t , Qt , Rkt , t , F 1t , F 2t , W t , hdt , Y t , mct , X 1t , X 2t , pt , st , s t , andT t satisfying (7), (8), (10), (12)-(21), (28)-(30), (34)-(38), and (40)-(45) written in terms of the stationary variables, given exogenous stochastic processes ,t , z,t , and gt , the policyprocess, R t , and initial conditions c 1, w 1, s 1, s 1, 1, i 1, and k0. A complete list of thecompetitive equilibrium conditions in terms of stationary variables is given in the technicalappendix to this paper (Schmitt-Grohe and Uribe, 2005b).

2.8 Ramsey Equilibrium

We assume that at t = 0 the benevolent government has been operating for an innite numberof periods. In choosing optimal policy, the government is assumed to honor commitmentsmade in the past. This form of policy commitment has been referred to as optimal fromthe timeless perspective (Woodford, 2003).

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Formally, we dene a Ramsey equilibrium as a set of stationary processes ut , C t , ht , I t ,K t+1 , vt , M ht , M t , t , t , W t , t , Q t , Rkt , t , F 1t , F 2t , W t , hdt , Y t , mct , X 1t , X 2t , pt , s t , s t , T t ,and R t for t 0 that maximize

E 0

t=0

tz

0 ts=1 z ,s(1 4 )(1 3 )

C t bC t

1z ,t

1 4

(1 h t)41 3

1

1 3

subject to the competitive equilibrium conditions (7), (8), (10), (12)-(21), (28)-(30), (34)-(38), and (40)-(45) written in stationary variables, and Rt 1, for t > , given exogenousstochastic processes z,t , ,t , and gt , values of the variables listed above dated t < 0, andvalues of the Lagrange multipliers associated with the constraints listed above dated t < 0.

Technically, the difference between the usual Ramsey equilibrium concept and the oneemployed here is that here the structure of the optimality conditions associated with theRamsey equilibrium is time invariant. By contrast, under the standard Ramsey equilibriumdenition, the equilibrium conditions in the initial periods are different from those applyingto later periods.

Our results concerning the business-cycle properties of Ramsey-optimal policy are com-parable to those obtained in the existing literature under the standard denition of Ramseyoptimality (e.g., Chari, Christiano, and Kehoe, 1995). The reason is that existing studies of business cycles under the standard Ramsey policy focus on the behavior of the economy inthe stochastic steady state (i.e., they limit attention to the properties of equilibrium time

series excluding the initial transition).

3 Calibration

The time unit is meant to be one quarter. For most of the calibration we draw on the papersby Altig et al. (2005) and Christiano et al. (2005) (hereafter ACEL and CEE, respectively).We assign most of the parameter values from he high-markup case of the ACEL estimationresults. In this case, the steady-state markup in product markets is 20 percent (or = 6).

ACEL estimate the parameters of the exogenous stochastic processes for the investment-specic and neutral technology shocks ,t and z,t to be, respectively, ( , , ) =(1.0042, 0.0031, 0.20) and (z , z , z ) = (1 .00213, 0.0007, 0.89)

Following ACEL we assume that in the deterministic steady state of the competitiveequilibrium the rate of capacity utilization equals one ( u = 1) and prots are zero ( = 0).ACEL calibrate the discount factor, , to be 1.03 1/ 4, the depreciation rate, , to be 0.025,the capital share, , to be 0.36, and the steady-state share of money held by households,

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mh /m , to be 0.44. ACEL assume that preferences are separable in consumption and leisureand logarithmic in habit-adjusted consumption ( 3 = 1). Their functional for the periodutility function implies a unit Frisch elasticity of labor supply. ACEL assume full indexationof wages ( = 1) and a steady-state markup of wages over the marginal rate of substitution

between leisure and consumption of 5 percent (or = 21).ACEL estimate the degree of nominal wage stickiness to be slightly above 3 quarters

( = .69). They also estimate the degree of habit formation measured by the parameter bto be 0.69, the elasticity of the marginal capital adjustment cost, , to be 2.79, the elasticityof the marginal cost of capacity utilization, 2/ 1, to be 1.46, and the annualized interestsemielasticity of money demand by households, (1 / 4) ln(mht )/ (R t ), to be -0.81.

CEE estimate the degree of price stickiness in product markets to be 2.5 quarters ( =0.6). ACEL estimate the degree of price stickiness to be 5 quarters (or = 0.8) when capitalis not rm specic, which is the assumption maintained in this paper. We set equal to 0.6in our baseline calibration, but will report results for economies with higher degrees of pricestickiness.

We do not draw from the work of CEE or ACEL to calibrate the degree of indexationin product prices and wages. The reason is that in their study the parameters governingthe degree of indexation are not estimated. They simply assume full indexation of all pricesto past product price ination. Instead, we draw from the econometric work of Cogley andSbordone (2005) and Levin et al. (2005) who nd no evidence of indexation in product prices.We therefore set = 0. At the same time, Levin et al. estimate a high degree of indexation

in nominal wages. We therefore assume that = 1, which happens to be the value assumedin ACEL and CEE.

Using postwar U.S. data, we measure the average money-to-output ratio as the ratio of M1 to GDP, and set it equal to 17 percent per year. Neither ACEL nor CEE impose thiscalibration restriction. Instead, they assume that all of the wage bill is subject to a cash-in-advance constrainti.e., they impose = 1. By contrast, our calibration implies that only60 percent of wage payments must be held in money (or = 0 .6).

In calibrating the model we assume that in deterministic steady state of the competitiveequilibrium the rate of ination equals 4.2 percent per year. This value coincides with theaverage growth rate of the U.S. postwar GDP deator.

The process for government purchases is taken from Christiano and Eichenbaum (1992).According to their estimates, g = 0 .96 and g = 0.020. Finally, we impose that the steady-state share of government consumption in value added is 17 percent, which equals the averagevalue observed in the U.S. over the postwar period.

Table 1 presents the values of the deep structural parameters implied by our calibration

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strategy.

4 Ramsey Steady States

Consider the long-run state of the Ramsey equilibrium in an economy without uncertainty.We refer to this state as the Ramsey steady state. Note that the Ramsey steady state is ingeneral different from the allocation/policy that maximizes welfare in the steady state of acompetitive equilibrium.

We nd that the most striking characteristic of the Ramsey steady state is the highsensitivity of the optimal rate of ination with respect to the parameter governing the degreeof price stickiness, , for the range of values of this parameter that is empirically relevant.Available empirical estimates of the degree of price rigidity using macroeconomic data varyfrom about 2 quarters to 5 quarters, or [0.5, 0.8]. For example, CEE (2005) in thecontext of a model similar to ours estimate to be 0.6. By contrast, ACEL (2005), using amodel identical to the present one, estimate an output-gap coefficient in the Phillips curvethat is consistent with a value of of around 0.8 when the market for capital is assumed tobe centralized, as is maintained in our formulation. 4 Both CEE and ACEL use an impulse-response matching technique to estimate . Bayesian estimates of this parameter includeDel Negro et al. (2004) and Levin et al. (2005) who report posterior means of 0.67 and 0.83,respectively, and 90-percent probability intervals of (0.51,0.83) and (0.81,0.86), respectively.Figure ?? displays the relationship between the degree of price stickiness, , and the optimal

rate of ination in percent per year, . The optimal rate of ination equals -4 percent(virtually equal to the level called for by the Friedman rule) when equals 0.5 and rises to-0.4 percent (close to price stability) when takes the value 0.8.

Evidence on price stickiness based on microeconomic data suggest a much higher fre-quency of price changes than the one based on macro data. The evidence reported in Bilsand Klenow (2004) and Golosov and Lucas (2003), for example, suggest values of of around1/3, or a degree of price stickiness of about 1.5 quarters. As is evident from gure ?? suchvalues of would imply that the Friedman rule is Ramsey optimal in the long run in thecontext of our model.

The above analysis suggests that it is of outmost importance to devote further researchinto rening the available estimates of the degree of price stickiness. This research shouldaim not only at narrowing the range of values that stem from macro evidence but also atreconciling the apparent disconnect between estimates emerging from macro and micro data.

4 If, instead, capital accumulation is assumed to be rm-specic, then ACELs estimate of the Phillipscurve is consistent with a value of of about 0.6xxx.

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Table 1: Structural Parameters

Parameter Value Description 1.031/ 4 Subjective discount factor (quarterly) 0.36 Share of capital in value added 0.254 Fixed cost parameter 0.025 Depreciation rate (quarterly) 0.6011 Fraction of wage bill subject to a CIA constraint 6 Price-elasticity of demand for a specic good variety 21 Wage-elasticity of demand for a specic labor variety 0.6 Fraction of rms not setting prices optimally each quarter 0.69 Fraction of labor markets not setting wages optimally each quarterb 0.69 Degree of habit persistence1 0.0459 Transaction cost parameter2 0.1257 Transaction cost parameter3 1 Preference parameter4 0.5301 Preference parameter 2.79 Parameter governing investment adjustment costs 1 0.0412 Parameter of capacity-utilization cost function

2 0.0601 Parameter of capacity-utilization cost function 0 Degree of price indexation 1 Degree of wage indexation 1.0042 Quarterly growth rate of investment-specic technological change 0.0031 Std. dev. of the innovation to the investment-specic technology shock 0.20 Serial correlation of the log of the investment-specic technology shockz 1.0021 Quarterly growth rate of neutral technology shockz 0.0007 Std. dev. of the innovation to the neutral technology shockz 0.89 Serial correlation of the log of the neutral technology shockg 0.2161 Steady-state value of government consumption (quarterly) g 0.02 Std. dev. of the innovation to log of gov. consumptiong 0.96 Serial correlation of the log of government spending

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Figure 1: Degree of Price Stickiness and the Optimal Rate of Ination

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 15

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

ACEL

CEE

* Benchmark Parameter Value

Note: CEE and ACEL indicate, respectively, the parameter values used by Chris-tiano, Eichenbaum, and Evans (2005) and Altig, Christiano, Eichenbaum, andLinde (2005). All parameters other than take their baseline values, given intable 1.

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Besides the uncertainty surrounding the estimation of the degree of price stickiness anaspect of the apparent difficulty in establishing reliably the long-run level of ination hasto do with the shape of the relationship linking the degree of price stickiness to the optimallevel of ination. The problem resides in the fact that this relationship becomes signicantly

steep precisely for that range of values of that is empirically most compelling. The prob-lem would not arise if the steep portion of the relationship would take place at values of below 1/3 or above 0.8, say. It turns out that an important factor determining the shape of the function relating to is the underlying scal policy regime. In this paper, we followthe widespread practice in the literature on optimal monetary policy in the Neo Keynesianframework of ignoring scal considerations by implicitly or explicitly assuming the existenceof lump-sum, nondistorting taxes that balance the government budget at all times and underall circumstances. This assumption is clearly unrealistic and usually maintained on the solebasis of simplicity. We wish to argue that taking explicitly into account the scal side of op-timal policy has crucial consequences for establishing the optimal long-run level of ination.Fiscal considerations fundamentally change the long-run tradeoff between price stability andthe Friedman rule. To see this, we now consider an economy where lump-sum taxes areunavailable. Instead, the scal authority must nance government purchases by means of proportional capital and labor income taxes. The social planner sets jointly monetary andscal policy in a Ramsey-optimal fashion. The details of this environment are contained inSchmitt-Grohe and Uribe (2005a). Figure 2 displays the relationship between the degree of price stickiness, , and the optimal rate of ination, . The solid line corresponds to the

baseline case considered in this paper (featuring lump-sum taxes) and is reproduced fromgure 1. The dash-circled line corresponds to the economy with optimally chosen incometaxes analyzed in Schmitt-Grohe and Uribe (2005). Note that in stark contrast to whathappens under lump-sum taxation, under distortionary taxation the function linking and is at and very close to zero for the entire range of macro-data-based empirically plausiblevalues of , namely 0.5 to 0.8. In other words, when taxes are distortionary and optimallydetermined, price stability emerges as a result that is robust to the existing uncertaintyabout the exact degree of price stickiness. Even if one focuses on the evidence of price stick-iness stemming from micro data, the model with distortionary Ramsey taxation predicts anoptimal long-run level of ination that is much closer to zero than to the level predicted bythe Friedman rule.

Our intuition for why price stability arises as a robust policy recommendation in theeconomy with optimally set distortionary taxation runs as follows. Consider the economywith lump-sum taxation. Deviating from the Friedman rule (by raising the ination rate) hasthe benet of reducing the price dispersion that originates in the presence of price stickiness.

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Figure 2: Price Stickiness, Fiscal Policy, and Optimal Ination

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 15

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

ACEL

CEE

ACEL CEE

i l i iLump-Sum Taxes -o-o- Optimal Distortionary Taxes

Note: CEE and ACEL indicate, respectively, the values for the parameter usedby Christiano, Eichenbaum, and Evans (2005) and Altig, Christiano, Eichen-baum, and Linde (2005).

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Consider next the economy with Ramsey-optimal income taxation and no lump-sum taxes.In this economy, deviating from the Friedman rule still provides the benet of reducingprice dispersion. However, in this economy increasing ination has the additional benetof increasing seignorage revenue thereby allowing the social planner to lower regular income

tax rates. Therefore, the Friedman-rule versus price-stability tradeoff is tilted in favor of price stability.

Price Indexation and the Optimal Ination Rate

The parameter measuring the degree of price indexation is crucial in determining the opti-mal level of long-run ination. The reason is that when prices are fully indexed ( = 1), pricedispersion disappears in the deterministic steady state. As a result the social planner facesno longer a tradeoff between minimizing price dispersion and minimizing the opportunity

cost of holding money. In such an environment the Friedman rule is Ramsey optimal. Inthe absence of perfect indexation ( < 1), any deviation from zero ination will entail pricedispersion, and the lower the degree of indexation, the higher will be the price dispersionassociated with a given level of ination. Consequently, the Ramsey optimal deation rateis increasing in the degree of price indexation. Figure 3 shows that the Ramsey optimal in-ation rate is a decreasing function of the indexation parameter . CEE and ACEL assumethat prices are perfectly indexed to lagged ination, that is, they calibrate the parameter to be unity. Under this assumption, the Friedman rule is optimal in the deterministicRamsey steady state. However, the few existing studies that attempt to econometricallyestimate the indexation parameter nd little empirical support for price indexation. Forexample, Levin et al. (2005) using Bayesian methods report a tight estimate of of 0.08.Similarly, Cogley and Sbordone (2005) using a different empirical strategy than Levin et al.also nd virtually no evidence of price indexation in U.S. data. As we argued above, thesetwo empirical studies motivate our setting = 0. But more importantly, this evidence givessupport to near-zero ination rates being Ramsey optimal.

4.1 Additional Sensitivity Analysis

Figure 4 displays the sensitivity of the optimal rate of ination to variations in four structuralparameters. One of these parameters is , the intratempral elasticity of substitution acrossthe different varieties of intermediate goods. The remaining three parameters, , 1, and 2,dene the demand for money by rms and households.

The upper-left panel of the gure plots the Ramsey ination rate against / ( 1), whichis roughly equal to the gross product markup in steady state. The lower is the elasticity of

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Figure 3: Degree of Price Indexation and the Optimal Rate of Ination

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 15

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0

CEE&ACEL


Note: CEE and ACEL indicate, respectively, the value of used by Christiano,Eichenbaum, and Evans (2005) and Altig, Christiano, Eichenbaum, and Linde(2005). All parameters other than take their baseline values, given in table 1.

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Figure 4: Sensitivity of the Optimal Rate of Ination

1 1.1 1.2 1.3 1.45

4.5

4

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/( 1)

0 0.5 1 1.50.7

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0 0.05 0.1 0.15 0.2 0.250.75

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0 0.05 0.1 0.15 0.21.1

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2


Note: CEE and ACEL indicate, respectively, the parameter values used by Chris-tiano, Eichenbaum, and Evans (2005) and Altig, Christiano, Eichenbaum, andLinde (2005). In each panel, all parameters other than the one shown take theirbaseline values, given in table 1.

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substitution across varieties the higher is the markup. The gure shows that the optimalrate of ination decreases as the markup increases. Optimal ination is close to zero andquite insensitive to changes in the markup for markup values below 40 percent. At aroundthis level the optimal ination rate falls vertically to the value consistent with the Friedman

rule. The intuition for the negative relationship between the ination rate and the markup isthat for a given ination rate the resulting dispersion in output across different intermediategoods varieties decreases with the markup because a higher markup is associated with a lowerelasticity of substitution and hence price differences lead to less dispersion in production of different varieties. Price dispersion is the most costly, when the elasticity of substitution isthe highest. Most empirical studies would support markups that are less than 30 percent,and for that range of markups, the optimal ination rate is rather insensitive to the level of the markup.

Clearly, the tradeoff between setting ination so as to minimize the distortions introducedby money demand and setting it so as to minimize the distortions stemming from the stickyprice friction, should depend on the exact specication of money demand. In the threepanels in the right column of gure ?? , we consider how robust our nding of low inationis to changes in the money demand specication. The top right panel considers variationsin working capital requirement on the wage bill, measured by the parameter . When = 0money demand by rms vanishes and all demand for money stems from transaction demandby household. Clearly, in this case the optimal ination rate rise, but quantitatively theeffect is relatively minor, ination rises from -0.4 to -0.24. Similarly, the panel shows that

requiring rms to hold the entire wage bill in cash, that is, increasing from its baselinevalue of 0.6 to unity, would move the economy closer to the Friedman rule by only about 10basis points.

The middle right panel considers increases in the transactions costs that have relativelyminor effects on the money demand elasticity. Specically, the panels shows that increasing1 beyond its baseline value of 0.04 by a factor of 5, does not move the economy much closerto the Friedman rule. Ination continues to be above -0.75 percent per year.

5 Optimal Operational Interest-Rate RulesRamsey outcomes are mute on the issue of what policy regimes can implement them. Theinformation on policy one can extract from the solution to the Ramsey problem is limitedto the equilibrium behavior of policy variables such as the nominal interest rate. But thisinformation is in general of little use for central banks seeking to implement the Ramsey equi-librium. Specically, the equilibrium process of policy variables in the Ramsey equilibrium

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is a function of all of the states of the Ramsey equilibrium. These state variables include allof the exogenous driving forces and all of the endogenous predetermined variables. Amongthis second set of variables are past values of the Lagrange multipliers associated with theconstraints of the Ramsey problem. Even if the policymaker could observe the state of all of

these variables, using the equilibrium process of the policy variables to dene a policy regimewould not guarantee the Ramsey outcome as the competitive equilibrium. The problem isthat such a policy regime could give rise to multiple equilibria.

In this section, we do not attempt to resolve the issue of what policy implements theRamsey equilibrium in the medium-scale model under study. Rather, we focus on ndingparameterizations of interest-rate rules that satisfy the following 3 conditions: (a) Theyare simple, in the sense that they involve only a few observable macroeconomic variables;(b) They guarantee local uniqueness of the rational expectations equilibrium; and (c) Theymaximize the expected lifetime utility of the representative household conditional on theinitial state of the economy being the deterministic steady state of the Ramsey economy.We refer to rules that satisfy criteria (a) and (b) as implementable. We refer to implementablerules that satisfy criterion (c) as optimized rules. 5

The family of rules that we consider here consists of an interest-rule whereby the nominalinterest rate depends linearly on its own lag, the rates of price and wage ination, and thegrowth rate of output. Formally, the interest-rate rule is given by

ln(R t /R) = ln(t /

) + W ln(W t /) + y ln(yt /y t 1/

y) + R ln(R t 1/R).

The target values R, , and y are assumed to be the Ramsey steady-state values of theirassociated endogenous variables. (The steady-state growth of output is indeed exogenousand given by z .) The variable W t denotes nominal wage ination. It follows that in oursearch for the optimized policy rule, we pick 4 parameters so as to maximize conditionalwelfare.

The optimized operational interest rate rule is

ln(R t /R) = 5 .5ln(t /

) + 1 .7ln(W t /) + 0 .06ln(yt /y t 1/

y) + 0 .5ln(R t 1/R).

The results from our numerical work suggest that in the model economy we study, the optimaloperational interest-rate rule is active in both price and wage ination, as both coefficientsare greater than unity. In additions, the rule prescribes virtually no response to output

5 A further criterion one could impose is that the nominal interest rate not violate the zero bound. InSchmitt-Grohe and Uribe (2004c), we approximate this constraint by requiring that in the competitiveequilibrium the standard deviation be less than a fraction of the steady-state value of the nominal interestrate. WE WILL ADD THIS ANALYSIS IN FUTURE DRAFTS OF THIS PAPER

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growth. In this sense the optimized interest-rate rule can indeed be interpreted as a pureination targeting rule. According to the above rule, the policymaker reacts positively tolagged nominal interest rates. Because the interest-rate coefficient is less than unity, the ruleis inertial but not superinertial. Thus, the policymaker is backward looking in its response

to ination deviations from target.Figures 5, 6, and 7 compare the impulse responses of all variables of the model to the

three shocks driving aggregate uctuations under the Ramsey-optimal policy (solid lines)and under the optimized operational interest-rate rule (broken lines). In the gures, inationand the nominal interest rate are in percent per quarter deviations from their steady-statevalues. All other variables are expressed in percent deviations from their deterministic steadystate. As described earlier in section 2.6 variables in capital letters are stationarity-inducingtransformations of the corresponding variables in lowercase letters.

The gures suggest a remarkable match between the Ramsey responses and the impulseresponses associated with the optimized operational interest-rate rule.

The optimized operational interest rate rule is quite robust to changing the welfare mea-sure the unconditional expectation of lifetime utility. We the unconditional measure of welfare is used the optimal operational interest-rate rule becomes

ln(R t /R) = 5 .7ln(t /

) + 1 .6ln(W t /) 0.01ln(yt /y t 1/

y) + 0 .5ln(R t 1/R).

TO BE CONTINUED

6 Conclusion

To be added.

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Figure 5: Ramsey And Optimized Responses To An Investment-Specic Productivity Shock

0 20 400.4

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Figure 6: Ramsey And Optimized Responses To A Neutral Productivity Shock

0 20 401

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Figure 7: Ramsey And Optimized Responses To A Government Purchases Shock

0 20 401

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