Mohamed Safouane

LAREQUAD Ben Aïssa

Fsegt

Olivier Musy
email: omusy@u-paris10.fr

The Dynamic Properties of Alternative Assumptions on Price Adjustment in New Keynesian Models

Keywords: E31; E52 New Keynesian Phillips Curves, Taylor Price Rule, Calvo Price Rule, Fixed Prices, Predetermined Prices, Disinflation policy

This paper presents a classification of the different new Phillips curves existing in the literature as a set of choices based on three assumptions: the choice of the structure of price adjustments (Calvo or Taylor), the presence of backward indexation, and the type of price contracts (fixed prices or predetermined prices). The paper suggests study of the dynamic properties of each specification, following different monetary shocks on the growth rate of the money stock. We develop the analytical form of the price dynamics, and we display graphics for the responses of prices, output, and inflation. We show that the choice made for each of the three assumptions has a strong influence on the dynamic properties. Notably, the choice of the price structure, while often considered as unimportant, is indeed the most influential choice concerning the dynamic responses of output and inflation.

Introduction

One of the central questions in the recent literature concerning inflation dynamics has been the ability of the New Keynesian Phillips curve (NKPC thereafter) to correctly reproduce the empirical impact of monetary shocks. This specification is attractive for several reasons: among others, its tractability and the existence of microfoundations. Moreover, as shown by [START_REF] Roberts | New Keynesian Economics and the Phillips Curve[END_REF], the different assumptions on price adjustment existing in the literature [START_REF] Taylor | Aggregate Dynamics an Staggered Contracts[END_REF], Rotemberg, 1982[START_REF] Calvo | Staggered Prices in a Utility Maximizing Framework[END_REF]) lead to the same structural relation between inflation and output. For these reasons, this specification, based on the assumption of fixed prices (FP thereafter), has been for some years something close to a "standard specification" (McCallum, 1997). It represents inflation as a pure forward looking variable. However, since the work of Ball (1994a) and [START_REF] Fuhrer | Inflation persistence[END_REF], the empirical plausibility of the NKPC has been strongly questioned. The main reason is that the forward-looking dynamics of the NKPC imply that inflation behaves like a "jump-variable". As a consequence, this model predicts an absence of inflation persistence, and a real neutrality of disinflation policies, while in reality, inflation is persistent and nearly all disinflations are costly1 .

One way to improve the empirical validation of the NKPC is to introduce lagged inflation in the dynamics, for example assuming the presence of backwardlooking agents (Gali and Gertler, 1999), or a backward-looking indexation mechanism (Woodford, 2003, Christiano, Eichenbaum and[START_REF] Eichenbaum | Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy[END_REF]. Recently, these "hybrid" Phillips curves, combining forward and backward inflation terms, have replaced the forward-looking NKPC as the new standard specification. Although the theoretical foundations of this lagged inflation term are unclear, it is often considered as a necessary condition for the reproduction of plausible inflation dynamics. Some authors, like [START_REF] Ball L | Monetary Policy for Inattentive Economies[END_REF] criticize all models based on FP, and prefer to give up this assumption. Deploring the "sorry state" of the literature, these authors say that forward-looking New Keynesian models are at odds with the facts, and that hybrid models are even worse. In response to the apparent failures, they propose to replace the FP assumption by the sticky information one [START_REF] Woodford | Sticky Information Versus Sticky Prices: A Proposal to Replace the New Keynesian Phillips Curve[END_REF], which formally is a resurgence of the [START_REF] Fischer | Long-term Contracts, Rational Expectations and the Optimal Money Supply Rule[END_REF] predetermined prices assumption (PP thereafter). They argue that this alternative outperforms the FP hypotheses. Notably, the predetermination of prices prevents inflation from jumping immediately after shocks.

However, the superior performance of PP models has been questioned by some recent papers. Analyzing the responses of FP and PP models built on the structure of price adjustment of Calvo, some authors conclude that the Mankiw and Reis PP model displays indeed a performance similar to the forward-looking NKPC [START_REF] Devereux | Predetermined Prices and the Persistent Effects of Money on Output[END_REF] and to the "hybrid" Phillips curve [START_REF] Trabandt | Sticky Information vs. Sticky Prices: A Horse Race in a DSGE Framework[END_REF]. [START_REF] Woodford | Interest and Prices: Foundations of a Theory of Monetary Policy[END_REF] also uses forward and "hybrid" versions of the NKPC and finds that these specifications correctly fit the facts. [START_REF] Collard | Sticky Information[END_REF], and [START_REF] Dupor | Sticky Information: The Impact of Different Updating Assumptions[END_REF] show that giving up the Calvo price structure significantly lowers the performance of the PP model.

Because the choice of the structural form of the economy is important for the recommendations of monetary policy, the current controversy and the presence of many contradicting declarations is embarrassing. Despite the apparent similarities of these models, their predictions in terms of responses to monetary shocks are quite different and there is no agreement on their relative performances. One reason that could explain these contrasted results is the absence in the literature of an exhaustive synthesis summarizing the dynamic properties of these different models. Some comparative works exist, but either they do not compare models within a common framework [START_REF]Sluggish Inflation and Optimizing Models of the Business Cycle[END_REF][START_REF] Walsh | Monetary Theory and Policy[END_REF], Jondeau and LeBihan, 2001), giving hardly comparable results, or the comparison only takes in account a limited set of alternatives2 (Jeanne, 1998, [START_REF] Pereau | Staggered Wage Setting in Normal and Real Terms: A Comparative Analysis[END_REF][START_REF] Woodford | Sticky Information Versus Sticky Prices: A Proposal to Replace the New Keynesian Phillips Curve[END_REF][START_REF] Devereux | Predetermined Prices and the Persistent Effects of Money on Output[END_REF][START_REF] Trabandt | Sticky Information vs. Sticky Prices: A Horse Race in a DSGE Framework[END_REF]. Another reason is the frequent use of simulation methods (for example [START_REF] Dupor | Sticky Information: The Impact of Different Updating Assumptions[END_REF]Tsuruga, 2005, or De Walque, Smets andWouters, 2005). The absence of analytical results does not clearly highlight the properties of each assumption on price adjustment, and conclusions remain uncertain.

In this paper, we will try to evaluate the relative performance of the most important new Phillips curves within a common framework, for different money shocks. We present both an analytical representation of the price dynamics, and some graphical illustrations. This allows for a better understanding of the dynamic properties of the different Phillips curves.

To do this comparative work, we adopt a classification of the different hypotheses presented in the literature based on three points: the first is the choice of the nature of the nominal rigidity (either PP or FP), the second is the choice of the price adjustment rule (either Calvo or Taylor), and the third is the presence or the absence of indexation. This permits to obtain comparable Phillips curves and to attribute the differences of response of each specification exclusively to the assumptions made on the three parameters presented. The other elements of the economy are voluntarily simplified in order to highlight the properties of the price equation.

In addition to the survey, our work shows the following elements:

a) The choice of the price structure (Calvo or Taylor) has very important implications for the price and output dynamic. The influential paper by [START_REF] Roberts | New Keynesian Economics and the Phillips Curve[END_REF] concluding to the unimportance of the specific type of price rigidity is misleading. For example, in the forward-looking FP models, the structure of Taylor implies structural expectation errors and a positive cost of disinflation absent from the structure of Calvo. b) Except for the disinflation case, the introduction of indexation does not by itself add significantly more persistence. The choice of the price structure (Calvo or Taylor) is more important. In other words, hybrid sticky-prices models have very different dynamic properties concerning persistence. Under the Taylor pricing rule, indexation does not add much persistence. Under the Calvo pricing rule, it can raise significantly the degree of persistence. c) Contrary to the affirmation of [START_REF] Ball L | Monetary Policy for Inattentive Economies[END_REF], the hybrid sticky price model built on the structure of Calvo produces plausible responses to all the shocks we consider. The PP assumption needs the presence of the Calvo pricing rule to generate sufficient persistence. This paper is organized as follows. Section 1 presents the derivation of the different Phillips curves. Section 2 presents the responses of the model to an auto-correlated shock on money growth. Section 3 is a study of the dynamic properties of each model consecutively to a disinflation policy. Section 4 suggests a comparison of our results with those of others important papers in the literature, and in the last section, we conclude.

PRESENTATION OF THE PHILLIPS CURVES

The common set-up

We follow a standard two-step procedure. In the first step, we present the level at which prices would be set if they were entirely flexible. We note p * t as the profit-maximizing price of a firm during the period t (all firms are identical). Following Romer (2001) or Kiley (2002a), in a monopolistic competition setup, absent wage rigidities, this price is given by:

p * t = p t + φy t (1)
where p t is the overall price level, y t is the output gap and φ is a measure of real rigidities. All variables are in logarithms.

In the second step we introduce nominal rigidities, with the standard assumption that firms can take new decisions on prices only when they receive signals of price changes. Each time a firm receives a signal, it sets an entire path of future prices. As signals of price changes are infrequent, we note λ j the expected probability of having a new signal of price change j periods after setting the price path. We note x t,t+j the price set at time t for the period t+j. The objective of the firm setting new prices is to minimize the sum of the differences between x t,t+j and p * t+j . We also assume that prices can be indexed to past inflation [START_REF] Woodford | Interest and Prices: Foundations of a Theory of Monetary Policy[END_REF]. We note γ the degree of indexation. Up to a linear approximation, a firm having a signal of price change during period t tries to minimize the following loss function3 :

Min x t,t , ... ,x t,t+j L t = (x t,t -p * t ) 2 + ∞ X j=1 (1 -λ j ) j E t ⎛ ⎝ x t,t+j -p * t+j + γ j-1 X i=0 π t+i ⎞ ⎠ 2 (2)
The price gap of period t + i is weighted by the probability of not having a new price signal until this date.

Most of the models presented in the literature can be seen as imposing some restrictions on three elements of equation (2): λ j , x t,t+i and γ.

1.2 The choice of the price adjustments structure (λ j )

The first distinction is about the choice of the rule governing price path adjustments. Each time a firm receives a new signal of price change, it can immediately set a new path of prices. We assume that the expected average duration between two signals is equal to N periods of the model (i.e. a firm changing its price path at the beginning of period t should have on average a new signal at the beginning of period t + N). Two assumptions have retained much attention in the literature. The first is based on the staggered prices structure of [START_REF] Taylor | Aggregate Dynamics an Staggered Contracts[END_REF]. In this specification, each price lasts exactly N periods, with an individual probability of price change of 1 every N periods and 0 otherwise. Firms in the economy are divided between N cohorts of equal size, each cohort being differentiated by the date of its price change. The alternative, which is more used, is based on the partial adjustment structure of [START_REF] Calvo | Staggered Prices in a Utility Maximizing Framework[END_REF]. A firm has a constant probability equal to 1/N to change its price each period. This probability is independent of the date of the last price change. The possible values of λ j are given in Table 1.

Structure

Probabilities of price signal

CALV O j ∈ [1; +∞[ ⇒ λ j = 1 N T AY LOR ⎧ ⎪ ⎨ ⎪ ⎩ j ∈ [1; N -1[ ⇒ λ j = 0 j = N ⇒ λ j = 1 ⎫ ⎪ ⎬ ⎪ ⎭
TAB. 1. Assumptions on the arrival of price signals

This presentation highlights the fact that the main difference between the Calvo and Taylor pricing rule is that the price signal probability in Calvo is time independent whereas under Taylor it is time dependent.

1.3 The nature of price rigidity (x t,t+j )

The second distinction concerns the restrictions imposed on the value of the prices set for each period of the price contract. We call a price contract the vector of prices set between two adjustments of the price path. In the literature, the prices specified in price contracts have taken two forms: fixity or predetermination. Following Blanchard and Fischer (1989), a price is predetermined from t to t + j if its path from t to t + j is predetermined as of time t. A price is fixed if it is predetermined and constant during that length of time. This is equivalent to imposing x t,t+j = x t during all the duration of the contract.

The indexation degree (γ)

Some authors have introduced an indexation mechanism for the price set (x t ) to lagged inflation. Each period, a fraction γ of the inflation rate observed during the last period can be added to the value of the price if there is no new signal of price change. Several γ values are considered in the literature. [START_REF] Eichenbaum | Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy[END_REF] assume that indexation is complete (γ = 1), and [START_REF] Woodford | Interest and Prices: Foundations of a Theory of Monetary Policy[END_REF] uses partial indexation (0 < γ < 1).

The motivation for indexation in FP models (apart from raising the degree of persistence) is that for a positive trend inflation, there is a growing gap between the optimal price and the price effectively set, which is inefficient. However, as under PP, all expected inflation is integrated in the price path.

This scheme of indexation is of little interest4 . Consequently, we shall only study the impact of indexation in FP models. We shall also restrict our analysis to the following cases: full indexation (noted I thereafter), and no indexation (noted FL, for forward-looking, thereafter).

Presentation of inflation dynamics

Combining the different assumptions, it is possible to derive the most important Phillips curves discussed in the recent literature.

Fixed prices in the Calvo model

First, we consider the most popular combinations: the FP version of the Calvo model (with and without indexation). The loss functions reduce to the following forms (respectively for the FL model, then the model with full indexation):

Min x t L t = (x t -p * t ) 2 + ∞ X j=1 µ N -1 N ¶ j E t ³ x t -p * t+j ´2 Min x t L t = (x t -p * t ) 2 + ∞ X j=1 µ N -1 N ¶ j E t ⎛ ⎝ x t -p * t+j + j-1 X i=0 π t+i ⎞ ⎠ 2
FOC give the following optimal prices (respectively with, then without indexation):

x t = µ 1 N ¶ ∞ X j=0 µ N -1 N ¶ j E t p * t+j (3) x t = µ 1 N ¶ ⎡ ⎣ ⎛ ⎝ ∞ X j=0 µ N -1 N ¶ j E t p * t+j ⎞ ⎠ -E t ⎛ ⎝ ∞ X j=0 µ N -1 N ¶ j+1 j X i=0 π t+i ⎞ ⎠ ⎤ ⎦ (4)
Up to an approximation, the aggregate price level is a weighted average of the prices coexisting at time t, given that a fraction (1/N ) of the prices is modified each period:

p t = µ 1 N ¶ x t + µ N -1 N ¶ p t-1 (5) p t = µ 1 N ¶ x t + µ N -1 N ¶ (p t-1 + π t-1 ) (6)
Given these equations and the value of p * t+i (equation 1), it is possible to derive a Phillips curve linking the current inflation to its expected value (and its lagged value in the case of indexation), and the output gap:

π t = E t π t+1 + φ N (N -1) y t (7) π t = 1 2 (π t-1 + E t π t+1 ) + φ 2N (N -1) y t (

Fixed prices in the Taylor model

We consider now the structure of the Taylor price adjustments, under the FP hypotheses. Given the probabilities of price adjustment, the loss functions are reduced to the following forms (respectively for the FL version, then with full indexation):

Min x t L t = (x t -p * t ) 2 + N -1 X j=1 E t ³ x t -p * t+j ´2 Min x t L t = (x t -p * t ) 2 + N -1 X j=1 E t ⎛ ⎝ x t -p * t+j + j-1 X i=0 π t+i ⎞ ⎠ 2
The FOC respectively give the following optimal prices:

x t = µ 1 N ¶ N-1 X j=0 E t p * t+j (9) x t = µ 1 N ¶ ⎡ ⎣ ⎛ ⎝ N-1 X j=0 E t p * t+j ⎞ ⎠ -E t ⎛ ⎝ N-2 X j=0 j X i=0 π t+i ⎞ ⎠ ⎤ ⎦ (10)
The price level is again approximated by an average of the existing prices:

p t = 1 N N-1 X j=0 (x t-j ) (11) p t = 1 N ⎡ ⎣ N -1 X i=0 x t-j + N-1 X j=1 N -j X i=1 π t-i ⎤ ⎦ (12)
Given these equations and the value of p * t+i , it is possible to obtain a Phillips curve. However, while with the Calvo structure, the form of the Phillips curve does not depend on the assumption made on N (the average length of contracts), this is not the case under the Taylor structure. In most of the literature, two-period contracts (N=2) are considered. We retain this assumption for numerical applications (in a semi-annual model, it represents a length of contracts equal to one year on average), which is standard (see [START_REF] Taylor | Staggered Price and Wage Setting in Macroeconomics[END_REF]. For this length of contracts, the differences between the Phillips curves resulting from the structure of Taylor and Calvo are minimized. This gives the following Phillips curves (respectively for the FL version, then with full indexation):

π t = 1 2 (E t-1 π t + E t π t+1 ) + φ 2 e y t (13)
π t = 1 3 (π t-1 + E t-1 π t + E t π t+1 ) + 2φ 3 e y t (14)
where e y t = y t + y t-1 + E t-1 y t + E t y t+1 .

Equation (13) corresponds to the Phillips curve of the Taylor model, and equation ( 14) to the Phillips curve of the Fuhrer and Moore model. This presentation of inflation dynamics is important because it differs from the well-known presentations of the same models made by [START_REF] Roberts | New Keynesian Economics and the Phillips Curve[END_REF], [START_REF] Fuhrer | Inflation persistence[END_REF] or [START_REF] Walsh | Monetary Theory and Policy[END_REF]. In those papers, equations ( 13) and ( 14) are presented with the same form as equations ( 7) and (8), representing their equivalent under the Calvo structure, adding a neglected expectation error. This error term is neglected in most papers and the choice concerning the Calvo or Taylor structure is often considered as equivalent [START_REF] Roberts | New Keynesian Economics and the Phillips Curve[END_REF][START_REF] Mankiw | The Inexorable and Mysterious Trade-off Between Inflation and Unemployment[END_REF]. However, as Ben Aïssa and [START_REF] Ben | Persistence with Staggered Price Setting in Nominal and Real Terms[END_REF] and [START_REF] Musy | Inflation Persistence and the Real Costs of Disinflation in Staggered Prices and Partial Adjustment Models[END_REF] have shown, this error term is crucial to properly understand the dynamic properties of the Taylor price structure. This point will be clearly illustrated in the following sections.

The origin of these terms indeed just reflects the interactions between the cohorts of price setters. The price level is an average of the prices set by cohorts in the past. As the prices are expected to be constant over several periods, price setters have to expect the future economic situation. The current price level (and then inflation dynamics) then contains these past expectations.

When the number of staggering price cohorts rises, the number of these lags also rises, because firms have to take into account a higher number of future prices. As an example, if we have 3 cohorts of firms in the model of Taylor, we obtain the following Phillips curve :

π t = 1 3 (E t π t+1 + E t-1 π t + E t-2 π t-1 ) + 1 6 (E t π t+2 + E t-1 π t+1 + E t-2 π t ) - 1 2 π t-1 + φ 6 ⎡ ⎢ ⎣ y t + y t-1 + y t-2 + E t (y t+1 + y t+2 ) +E t-1 (y t + y t+1 ) + E t-2 (y t-1 + y t ) ⎤ ⎥ ⎦
In a staggered framework, such terms appear because they resume the past fixations of prices. They do not appear only in cases such that all the past history of price decisions is resumed by the last index of the price level (p t-1 ). It is precisely the case if we follow the rule of Calvo. But is it of course a particular case, resulting from the use of a Poisson process, where the probability of changing the price is always the same for all the firms.

One should also note that the assumption leading to the presence of lagged inflation in the Fuhrer and Moore model is not based on the presence of indexation, but rather on an assumption of relative contracting. We show in the Appendix 1 that for two-period contracts, the Phillips curve resulting from the indexation assumption in the Taylor model and the Phillips curve derived by Fuhrer and Moore are exactly equivalent. However, we prefer to use indexation because the assumptions used by Fuhrer and Moore are hard to reconcile with microeconomic foundations [START_REF] Holden | Inflation Persistence and Relative Contracting[END_REF] 1.

Predetermined Prices

As indicated below, we only consider the case with zero indexation. The loss functions of a firm setting predetermined prices are the following (respectively for the Calvo and Taylor structures):

Min x t,t , ... ,x t,t+j L t = E t ∞ X j=0 µ N -1 N ¶ j ³ x t,t+j -p * t+j ´2 Min xt,t,...,x t,t+j L t = E t N -1 X j=0 ³ x t,t+j -p * t+j ´2
In both cases, the optimal price sequence is the same:

x t,t+j = E t p * t+j
The price level is an average of current prices predetermined at different dates, i.e., with different sets of information. We have (respectively under the Calvo and Taylor hypotheses):

p t = 1 N ∞ X j=0 µ N -1 N ¶ j x t-j,t p t = 1 N N-1 X j=0 x t-j,t
Even if the sequences of prices are identical in both cases, the different definitions of the price level lead to very different Phillips curves. To be in accordance with the previous section, we assume two-period contracts for the Taylor price structure :

π t = 1 N ∞ X j=0 µ N -1 N ¶ j E t-1-j (π t + φ∆y t ) + φ N -1 y t (15) π t = π t-1 + E t-1 π t -E t-2 π t-1 + b y t (16)
where ∆y t = y ty t-1 and b y t = φy tφy t-1 + E t-1 y t -E t-2 y t-1 . Equation (15) is equivalent to the Phillips curve derived by Mankiw and Reis, and equation (16) represents a Phillips curve closely related to the Fischer model (1977). Contrary to the models with FP, current inflation does not include any expectations of future variables. The expectation terms are related only to past expectations of current variables. The number of expectation lags is equal to maximum length of contracts, that is, N under the structure of Taylor, and infinity under the Calvo structure.

Summary of the Phillips Curves

In Table 2, we summarize the different Phillips curves presented. We now assume two-period contracts for all specifications. We note C for the Calvo hypotheses, T for Taylor, FL for the forward-looking version (i.e. zero indexation) and I for full indexation. For each combination, we indicate the main paper of the literature using these hypotheses, and the associated Phillips curve.

π t = 1 3 (π t-1 + E t-1 π t + E t π t+1 ) + ³ 2φ 3 ´e y t C/PP
Mankiw and Reis We assume a simple output function, which depends on the level of real balances:

π t = 1 2 P ∞ j=0 ³ 1 2 ´j E t-1-j (π t + φ∆y t ) +κy t T/PP Fischer π t = π t-1 +E t-1 π t -E t-2 π t-
y t = m t -p t (17)
The money path is exogenous and we study different money shocks in the following sections. As our aim is to understand the implications of each assumption for price rigidity, the simplified economy we study has the virtue to derive all dynamics from the nominal rigidity5 .

AN AUTO CORRELATED SHOCK ON MONEY GROWTH

Presentation

We assume that the growth of the money stock follows an AR(1) process:

∆m t = ρ∆m t-1 + ε t , where ∆m t ≡ m t -m t-1 .
The main criteria of evaluation in the literature is the presence of a delayed and gradual response of output and inflation to the monetary innovation [START_REF] Mankiw | The Inexorable and Mysterious Trade-off Between Inflation and Unemployment[END_REF][START_REF] Woodford | Sticky Information Versus Sticky Prices: A Proposal to Replace the New Keynesian Phillips Curve[END_REF]. As stated by [START_REF]The lead of output over inflation in sticky price models[END_REF], the inflation response should also be more delayed than the output response. Other evaluation elements, such as the reproduction of the "acceleration phenomenon", are also used to assess the relative performance of inflation dynamics models (Mankiw and[START_REF] Woodford | Sticky Information Versus Sticky Prices: A Proposal to Replace the New Keynesian Phillips Curve[END_REF][START_REF] Trabandt | Sticky Information vs. Sticky Prices: A Horse Race in a DSGE Framework[END_REF].

Price dynamics

Economy begins on its steady-state. The money shock occurs at the beginning of period t = 1. We assume that the initial value of m is equal to m 0 . The price dynamics6 consecutive to the shock are presented in Table 3.

Hypotheses Price Dynamics

Calvo/Fixed Prices Forward-looking

p t = θp t-1 + (1 -θ) m t + (1-θ)θρ 1-θρ ∆m t
Calvo/Fixed Prices Indexation

p t = (θ 1 + θ 2 ) p t-1 -θ 1 θ 2 p t-2 + B θ 3 -1 m t + Bρ (θ 3 -1)(θ 3 -ρ) ∆m t
Taylor/Fixed Prices Forward-looking

p t = θp t-1 + (1-θ) 2 (m t + m t-1 ) + (1-θ)(1+θ)ρ 4(1-θρ) (∆m t + ∆m t-1 )
Taylor/Fixed Prices Indexation

p t = (θ 1 + θ 2 ) p t-1 -θ 1 θ 2 p t-2 + A θ 3 -1 (m t + m t-1 ) + A(1+θ 3 )ρ 2(θ 3 -1)(θ 3 -ρ) (∆m t + ∆m t-1 ) Calvo/Predetermined Prices p t = m 0 + ∞ P j=0 • φ ( 1-λ j+1 )( 1-ρ j+1 ) (1-ρ) [ 1-( 1-λ j+1 ) (1-φ) ] ¸εt-j Taylor/Predetermined Prices p t = p t-1 + 1 1+φ [φ∆m t + (1 + ρ) ∆m t-1 -ρ∆m t-2 ]
TAB. 3. Price dynamics (auto correlated shock on money growth)

with A = [4φ/ (1 -2φ)] and B = φ/4. θ is the stable root of forward-looking FP models. In models with indexation, θ 1 and θ 2 are two complex roots, with a modulus below 1, and θ 3 > 1 is a real root.

We focus on price dynamics instead of inflation dynamics because it is helpful for a better understanding of the price adjustment mechanics. This will be more evident in the next section on disinflations.

Before seeing graphical illustrations, one can see the properties of each assumption on price adjustment.

• In FP models: -Indexation introduces a second price lag, with a negative influence;

-The Taylor price structure implies, even in its purely forward-looking form, the presence of one lag of the money stock value, and one lag of the money growth value 7 . This results from the predetermination at time t of one cohort of firms' expectations.

• In PP models, dynamics are very dependent on the choice of the price adjustment structure. A unique and permanent shock on ∆m has a very short impact in the PP model built on the Taylor structure. With the Calvo structure, dynamics include an infinity of predetermined expectations. Consequently, persistence is larger.

The response of output and inflation

For numerical applications, we arbitrarily assume that ρ = 0.5 8 . We also assume that φ = 0.1, which corresponds to an important degree of strategic interactions between firms. While there is a strong debate, [START_REF] Woodford | Interest and Prices: Foundations of a Theory of Monetary Policy[END_REF] shows that values around φ = 0.1 are possible. This allows to make easier comparisons with the literature 9 (see [START_REF] Taylor | Staggered Price and Wage Setting in Macroeconomics[END_REF].

In Figure 1, we present the reaction of inflation to the shock.

7 When N rises, the number of these lags also rises.

8 As this value is the same for all models, and as we focus essentially on qualitative aspects of the dynamics, this particular value is unimportant. 9 Kiley (2002a) and Dixon and Kara (2005b) study the impact of variations of φ on persistence. In Figure 2 we present the response of the output.

FIG. 2. Response of output (autocorrelated shock)

The three models built on the Calvo structure display a hump-shaped response of output. None of the models built on the Taylor structure can reproduce such output behavior11 . Since they reproduce a hump-shaped response of inflation, they all respect the condition of a more delayed response of inflation than output, a criterion advanced by [START_REF]The lead of output over inflation in sticky price models[END_REF]. The only model that reproduces a faster response of inflation (the peak of inflation occurs before the peak of output) is the Calvo forward-looking model.

The behavior of output is of interest because it can summarize the degree of persistence implied by nominal rigidities. If such rigidities were absent, we would have each period m = p (and then y = 0). The degree of persistence can be measured as the cumulative deviation between m and p, which is equal to y. If we adopt this measure of persistence, the strongest degree of persistence is displayed by the FL-FP model of Calvo (see Table 4), which is surprising given all the debate on the lack of persistence of this model. The forward-looking versions of the FP models imply a more important total output deviation than their alternatives with indexation. This results from the cyclical responses of the models with indexation. If we consider only the absolute value of deviations, this strongly raises the cumulative deviation of the Calvo FP model with indexation (from 0.03 to 0.057), but the impact is low with the Taylor model (from 0.008 to 0.013), which is still lower than with the forward-looking version.

One could note that the degree of persistence is lower with the Taylor structure. However, as shown by [START_REF] Dixon | How to compare Taylor and Calvo contracts: a comment on Michael Kiley[END_REF], the difference should be lower if we allow for a motivated change in the calibration of λ i . The value of this parameter should be higher in the models using the Calvo structure, which will lower the persistence of output and inflation. Despite the persuasive arguments of Dixon and Kara, we do not adopt this calibration here to facilitate the comparisons between our results and previous papers in the literature. We also note that our results are independent from this choice, and adopting the formulation by Dixon and Kara would reinforce our argument. Indeed, from a qualitative viewpoint, we show that the implications of the forward-looking Taylor contracts are in accordance with all the stylized facts, while their main weakness is to produce a lower degree of persistence than the models using the Calvo structure (the reason is shown by [START_REF]Partial Adjustment and Staggered Price Setting[END_REF]. As Dixon and Kara show, using their calibration simply reduces this quantitative difference. This would only reinforce the interest of Taylor contracts.

The acceleration phenomenon

As stated by [START_REF] Woodford | Sticky Information Versus Sticky Prices: A Proposal to Replace the New Keynesian Phillips Curve[END_REF], the acceleration phenomenon, which represents the correlation between the output gap y t (log from trend) and one-year inflation change (π t+1π t-1 ), focused on the observation date is a well-documented macroeconomic fact.

In Figure 3, we present the correlation between output gap y t and the annual change of inflation (π t+1π t-1 ) (the acceleration phenomenon), assuming that the output gap and the output coincide (real shocks are held constant). [START_REF] Woodford | Sticky Information Versus Sticky Prices: A Proposal to Replace the New Keynesian Phillips Curve[END_REF], a FP forward looking model can reproduce a positive correlation if we depart from the Calvo assumption. Models with backward looking components predict a higher correlation than the initial models, but the low value for the Taylor model is not necessarily a weakness since it is in line with the empirical values found using semi-annual US data 12 (Table 5). 12 For this calculation, output y t is the deviation of log real GDP from trend, where trend is calculated using the Hodrick-Prescott filter. The US inflation is based on the Consumer Price Index (CPI) from the FRED II Database of the Saint Louis Federal Reserve. As we use a semi-annual model, we consider corr(y t , π t+1π t-1 ), and not corr(y t , π t+2π t-2 ) as in Mankiw and Reis, who use a quarterly model.

A DISINFLATION POLICY

Presentation

A special case of the previous money process, where ρ = 1, is also often regarded as a criterion of evaluation. This particular value of ρ, while counterfactual, is of great interest because, with a negative monetary innovation, it corresponds to the case of a "cold-turkey" disinflation policy engineered by the central bank. This kind of disinflation policy has often been used as a test to dismiss the validity of the forward-looking New Keynesian inflation dynamics model (Ball, 1994a[START_REF] Fuhrer | Inflation persistence[END_REF][START_REF] Woodford | Sticky Information Versus Sticky Prices: A Proposal to Replace the New Keynesian Phillips Curve[END_REF]. This model is supposed to be unable to reproduce the presence of a positive output cost of the disinflation (Ball, 1994b), and the delayed response of inflation following the slowing of money growth [START_REF] Woodford | Sticky Information Versus Sticky Prices: A Proposal to Replace the New Keynesian Phillips Curve[END_REF].

The money process

The central bank sets the money growth rate ∆m t to a level compatible with its inflation target π * t , that is, each period ∆m t = π * t . Then, expectations are given by: E t ∆m t+i = π * t = ∆m t , ∀i ≥ 0.

The economy is initially at its steady-state, defined by ∆m t = π t = π * 1 and y t = 0. A sudden and unexpected disinflation policy is equivalent to a negative shock on the inflation target, e.g. from π * 1 to π * 2 with π * 2 < π * 1 . Let the disinflation begin in period 1. This policy is credible and permanent, but is not expected by the agents. For numerical applications, we assume that m 0 = 0, and the evolution of m t is given by m t = m t-1 + π * t , with the following path:

⎧ ⎪ ⎨ ⎪ ⎩ t ∈ [-∞, 0] , π * t = 0.025 t ∈ [1, +∞] , π * t = 0 ⎫ ⎪ ⎬ ⎪ ⎭

Price dynamics

Given the expectations of the money path, we can derive the analytic form of price dynamics, which correspond to the ones presented in Table 1, assuming ρ = 1 (the resolution of the Calvo model with PP is different, and we use the method presented by [START_REF] Woodford | Sticky Information Versus Sticky Prices: A Proposal to Replace the New Keynesian Phillips Curve[END_REF]. They are displayed in the Appendix 2.

Graphical illustrations

FIG. 4. Response of the price level (unexpected disinflation)

Although interest variables are mostly inflation and output, we also present price dynamics because they are important to understand clearly the origins of the real costs implied by disinflation. Given equation demand (17), in order to reproduce a positive disinflation cost, models have to display a mechanism of price over-shooting (i.e., the price level must continue to rise despite the stability of the money growth). The only model unable to reproduce this overshooting mechanism is the forward-looking Calvo FP model. This is not a general feature of forward-looking FP models, but only a characteristic of the very specific rule of Calvo adjustment. Even in its simplest form, the Taylor structure can reproduce this price over-shooting. We also could notice that when considering this kind of shock, the equivalence between PP and FP in the Calvo model when φ = 1 (showed by [START_REF] Devereux | Predetermined Prices and the Persistent Effects of Money on Output[END_REF] does not hold.

FIG. 5. Response of inflation (unexpected disinflation)

The behavior of prices implies that after reaching a peak, the price level has to fall. Consequently, inflation also displays an over-shooting response. For all models built on the structure of Taylor, this mechanism begins very quickly 13 (the second period consecutive to the shock), while for the models built on the Calvo structure (excepted the FL version), the peak response of inflation is more delayed.

Figure 6 represents the reaction of output. 13 The very quick response presented here is dependent on the very short length of contracts assumed. For the forward-looking version, the peak response of the price level occurs N periods after the shock, where N is the length of contracts.

21

FIG. 6. Response of output (unexpected disinflation)

The magnitude of the output response depends on the degree of over-shooting presented in Figure 1. The Christiano, Eichenbaum and Evans (Calvo hybrid) and Mankiw Reis (Calvo and PP) models display comparable responses. The two models with indexation (Fuhrer and Moore, and Christiano, Eichenbaum and Evans) reproduce oscillatory dynamics due to the presence of complex roots. It is important to note that the introduction of indexation in the Taylor structure does not significantly increase the output cost of disinflation. This can be more clearly illustrated by the calculation of sacrifice ratios14 (Figure 7).

FIG. 7. Sacrifice ratio (unexpected disinflation)

To have a comparison order, Ball (1994b) estimates sacrifice ratios ranging from 0.0 to 3.6, with quarterly data. Of course, the values obtained depend on the value chosen for parameters. Figure 7 has only some illustrative value.

To summarize the results, as it is well known, the forward-looking Calvo model does not produce any inflation persistence or output costs. However, contrary to a common view (among others [START_REF] Roberts | Inflation Expectations and the Transmission of Monetary Policy[END_REF][START_REF] Walsh | Monetary Theory and Policy[END_REF], this is a special feature of the probabilistic Calvo price structure and not a property shared by all FP forward-looking models. Even with the simplest deterministic structure (Taylor uniform contracts for two periods), inflation displays some persistence, and the disinflation output cost is positive. This result is general and does not depend on the specific value chosen for φ. The sacrifice ratio in the presented Taylor model is always positive and is equal to

³ 1/4 √ φ
´(see next section for a demonstration). While often neglected, the expectation errors present in the Taylor structure have very important implications for disinflations. When looking at the Phillips curves in the Taylor and Mankiw and Reis models, one can see that the sources of persistence are indeed similar: both models rely on the presence of predetermined expectations to reproduce a delay in the inflation adjustment. Even in the sticky prices model, past information sets of agents have an influence on current variables because they are included in the prices which are still in effect in the current period. Then, as in the sticky-information model, past money growth has an influence on current output. There is one exception to this case: the Calvo assumption of a constant probability of price changes among all firms (see next section).

The introduction of indexation in the Taylor structure does not significantly raise persistence. As shown by Ben Aïssa and [START_REF] Ben | Persistence with Staggered Price Setting in Nominal and Real Terms[END_REF], when taking into account the expectation errors, the Taylor, Fuhrer and Moore models have close dynamics properties, the latter produces only a little more persistence than the former. However, introducing indexation in the Calvo model significantly alters the dynamics. Inflation is very persistent and the output cost is the strongest of all the specifications considered. The corollary of the previous remark is that FP hybrid models have very different dynamic properties.

The properties of PP models are also sensitive to the choice of the pricing structure. As it is well known for this price rigidity, a monetary shock has real effects only as long as all contracts have not been modified (Blanchard and Fischer, 1989, p. 393). In the particular Calvo structure, some contracts have an infinite length, which produces a strong persistence and an asymptotic convergence. When the length of contracts is finite, persistence is much lower and there is strict convergence. Another presentation of this point is made by [START_REF] Collard | Sticky Information[END_REF], and [START_REF] Dupor | Sticky Information: The Impact of Different Updating Assumptions[END_REF]. The properties of Mankiw and Reis model, which depend on the values chosen for the parameters, are discussed extensively by [START_REF] Coibion | Inflation Inertia in Sticky Information Models[END_REF].

The sacrifice ratio in fixed price forward-looking models

The presence of a positive real cost following a disinflation is possible even with fully rational forward-looking agents facing sticky prices. This is a result often contested in the literature. We propose a general demonstration of this result for a "cold-turkey" disinflation, considering only the two Taylor and Calvo models, with fixed prices but no indexation. Let disinflation begin in period j. We derive the general form of inflation dynamics in the Calvo model15 , given the output equation (17):

π t = θ C π t-1 + ³ 1 -θ C ´∆m t + ³ 1 -θ C ´θC ∞ X i=0 ³ θ C ´i (E t ∆m t+i+1 -E t-1 ∆m t+i )
where θ C is the stable root of the dynamics. Inflation dynamics in the Taylor model are given by:

π t = θ T π t-1 + Ã 1 -θ T 2 ! (∆m t + ∆m t-1 ) + ³ 1 -θ T ´³1 + θ T 4 (E t ∆m t+1 -E t-2 ∆m t-1 ) + θ T ³ 1 -θ T ´³1 + θ T 4 " ∞ X i=0 ³ θ T ´i (E t ∆m t+i+2 -E t-2 ∆m t+i )
# where θ T is the stable root of the dynamics. Expected terms on money growth are different in the two structures. Given the money process, expectations are given by: E t ∆m t+i = π * t = ∆m t , ∀i ≥ 0. Inflation dynamics are reduced in both cases to the following forms:

π t = θ C (π t-1 -∆m t-1 ) + ∆m t ( 18
)
π t = θ T π t-1 + 1 -θ T 2 (∆m t + ∆m t-1 ) + 1 + θ T 4 (∆m t -∆m t-2 ) (19)
We note π * 1 the inflation rate before the disinflation. The initial state of the economy is given by: ∆m t = π t = π * 1 , and y t = 0. When the disinflation policy begins, the inflation target is changed permanently to π * 2 , with π * 2 < π * 1 . Let the disinflation begin during period j.

In the Calvo model,equation (18) implies that as soon as the economy begins on its steady-state (π j-1 = ∆m j-1 ), we have π j+i = ∆m j+i = π * 2 , and y j+i = 0, ∀i ≥ 0. In the Taylor model, due to the presence of lagged expectations, we have by contrast:

π j = 3 -θ T 4 π * 2 + 1 + θ T 4 π * 1 π j+1 = 5 -θ T 4 π * 2 - 1 - ³ θ T ´2 4 π * 1 π j+i = θ T π j+i-1 + ³ 1 -θ T ´π * 2 , ∀i º 2
The condition π j > π * 2 is always verified 16 . This means that there is some inflation persistence at the beginning of the disinflation. After, the inflation rate overshoots its long run value (π j+1 < π * 2 ). During subsequent periods, the inflation rate converges to this long-run value. Convergence is gradual and monotone if φ ≺ 1, immediate if φ = 0, and oscillatory if φ Â 1.

In the Taylor model, output is always negative during the process, and the 16 Because φ is always positive and θ T is given by: θ

T = ¡ 1 - √ φ ¢ / ¡ 1 + √ φ ¢ .
dynamics are given by:

y j+i = - ³ θ T ´i ³ 1 + θ T 4 [π * 1 -π * 2 ]
for i ≥ 0. The total disinflation cost (TDC), given by ∞ P i=0 y j+i , is equal to (respectively in the Calvo and in the Taylor models):

T DC C = 0 T DC T = - ³ 1 + θ T 4 ³ 1 -θ T ´[π * 1 -π * 2 ]
As θ T ∈ ]-1; 1[, we have T DC T < 0. In the forward-looking Taylor model , a cold-turkey disinflation always implies a positive real cost of disinflation. This result does not depend on the value of φ. This is an important point, as there are some debates on the plausible values of φ (see [START_REF] Chari | Sticky Prices Models of the Business Cycle: Can the Contract Multiplier Solve the Persistence Problem ?[END_REF][START_REF] Chari | Sticky Prices Models of the Business Cycle: Can the Contract Multiplier Solve the Persistence Problem ?[END_REF][START_REF] Woodford | Interest and Prices: Foundations of a Theory of Monetary Policy[END_REF]. However, the sacrifice ratio (SR) is decreasing in φ:

SR T = 1 4 √ φ A presentation of the mechanics underlying this positive cost is given by [START_REF] Musy | Inflation Persistence and the Real Costs of Disinflation in Staggered Prices and Partial Adjustment Models[END_REF]. The source of this cost is the presence of expectation errors. Consider first the price dynamics of the FL Calvo model (see Table 1). Assume for simplicity that when the disinflation takes place, ∆m j = 0 (i.e. m j = m j-1 ). It comes that p j = p j-1 and y j = 0. Thus, in the Calvo model, there is no inflation persistence and disinflation is costless. By contrast, in the price dynamics of the FL Taylor model, the forward looking component at time is driven by the expectations made both in j and j -1. Since half of the firms have set their prices without expecting the disinflation, one has E j-1 ∆m j > 0, which implies p j > p j-1 . Thus, in the Taylor model, there is a positive rate of inflation and a real cost of disinflation.

The difference in the models' properties comes from the difference in the distribution of contracts and in the timing of price changes. In the Calvo model, the probability of a price change is independent from the date of the last price modification. Thus, the price changes are equally distributed among the firms. During period j, the average unmodified price is equal to p j-1 . This implies that π j depends only on the price changes made by those firms that set their prices at time j. These firms fully respond to the monetary change. In the Taylor model, a cohort of firms charges the older prices and only this cohort changes its prices. For positive inflation rates, the average price of the firms keeping their prices fixed is greater than p j-1 . At the beginning of the disinflation, in period j, the only prices that are modified are those set in j -2. The index of the unmodified prices is then greater than p j-1 . Given the presence of real rigidities (equation 1 can be rewritten as p * t = (1φ) p t + φm t ), the new prices have to be close to the unmodified prices and will take a value between the index of the unchanged prices and the new equilibrium price level. In the following periods, the price level will continue to be higher than the money stock (see Figure 4) If the disinflation is announced in advance, these error terms disappear and the mechanisms presented do not play any role, and disinflation is not costly. But in this case, PP combined with the Taylor structure also implies a costless disinflation (see [START_REF] Taylor | Union Wage Settlements during a Disinflation[END_REF].

COMPARISON WITH THE LITERATURE

Our paper is close in spirit to the one by [START_REF]Sluggish Inflation and Optimizing Models of the Business Cycle[END_REF], who shows that many alternative sticky-price specifications may be written as special cases of a loglinear model of prices dynamics. Among others, Nelson (1998) concludes that the hybrid Fuhrer and Moore model is better suited to reproduce empirical inflation serial correlations than the forward-looking Taylor model. However, his conclusions are distorted for two reasons: the first one is the absence of expectations errors in the dynamics, while the results presented below show the importance of these errors. The second reason is the absence of a common framework to derive the implications of the alternative assumptions on price adjustment. Consequently, in his presentation, the numerical value of some parameters appears to artificially differs from one model to another. As an example, the value of φ is estimated when he considers the Fuhrer and Moore model, which gives a very small value of 0.008. This estimated value is common in empirical studies. But when he analyzes the Taylor model, he uses the calibration of φ proposed by [START_REF] Chari | Sticky Prices Models of the Business Cycle: Can the Contract Multiplier Solve the Persistence Problem ?[END_REF], who argue that φ cannot be smaller than 1.2 in a general equilibrium framework. As we know, the lower the value of φ, the greater the model persistence. The higher value of φ in his representation of the Taylor model can explain most of his contrasted conclusions about the relevance of the two models. As we have shown, Furher and Moore contracts do not add much more persistence.

Another relevant paper is Kiley (2002a). He considers a shock on the level of the money stock, and shows that the different price distributions between the two models have an impact in terms of output persistence, the Calvo model being more persistent. In this paper, we focus on shocks on the money growth rate. The different assumptions on price adjustment have other implications. Notably, in the Taylor model, price changes are made by the firms charging the oldest prices in the economy. As Kiley only considers shocks on the level of the money stock, the effects presented in the previous section do not play any role in the dynamics presented in his paper, as in this case, before the shock, all contracts are set at the same value.

The paper by [START_REF] Woodford | Sticky Information Versus Sticky Prices: A Proposal to Replace the New Keynesian Phillips Curve[END_REF] is also very relevant, because we adopt their criteria consisting in the reproduction of stylized facts. Their conclusion is that the sticky information model is consistent with "accepted views of how monetary policy works", while the forward-looking fixed price model fail on three points: 1-Disinflations are not contractional; 2-Money shocks have their maximum effect on inflation with a substantial delay (this concerns also the hybrid model, according to [START_REF] Ball L | Monetary Policy for Inattentive Economies[END_REF]; 3-The model cannot explain the acceleration phenomenon that vigorous economic activity is positively correlated with rising inflation (Mankiw andReis, 2002, p. 1318). They study only the Calvo version of the forward-looking fixed price model. Interestingly, we show that the Taylor model can successfully reproduce all these three points, even in its entire forward-looking version : the sacrifice ratio is positive, the correlation between the output gap and the annual change of inflation is positive (the acceleration phenomenon), and the maximum impact of the money shock on inflation does not occur immediately. The conclusion by Mankiw and Reis is biased toward the sticky information model because they only compare it to the forward-looking Calvo model, which implies very specific dynamic properties. Even simple departures from this special rule can greatly improve the fit of the macro implications of the sticky price assumption to the "accepted views of how monetary policy works".

Mankiw and Reis also use the "disinflation criterion"17 to discard the new Keynesian model. We show that indeed, all models, excepted the FL Calvo model, can fill this criterion. However, the analysis of price level dynamics we conducted in this paper shows that a clear comprehension of the costs of disinflations in those models, indeed leads to the following question: "why do firms continue to raise prices even if the money stock, and then optimal prices, are stable ?". Our opinion is that the most interesting response is given by the FL Taylor model : it results from the interactions between decisions of fully rational and informed firms which can change their prices, with firms which cannot. The sticky price mechanism is indeed a good mechanism to reproduce the effects of disinflation. All the alternatives rely on assumptions either of a lack of rationality from firms, or on a limited use of information. If we consider the new current standard model of inflation dynamics (the Calvo model with indexation), disinflation is costly only because some firms use an ad hoc rule of past inflation indexation. There is nothing to explain why and when firms should use indexation, and if they use it, what should be the appropriate degree of indexation (which is a free parameter, as in [START_REF] Woodford | Interest and Prices: Foundations of a Theory of Monetary Policy[END_REF]. If we consider that the New Neoclassical Synthesis emphasized the importance of optimization and rational expectations [START_REF] Goodfriend | MonterayPolicy in the New Neoclassical Synthesis: A Primer[END_REF], its current leading model provides all but a satisfactory explanation to the understanding of disinflations, and more generally on the phenomenon of inflation persistence18 .

CONCLUSION

Our main conclusion is that the most important element for dynamic properties is the choice of the price structure. In this paper, we have considered only the simplest forms of the Calvo and Taylor structures, but even with these restrictions, differences are important. This result is surprising because, since the influential paper by [START_REF] Roberts | New Keynesian Economics and the Phillips Curve[END_REF], this choice has often been considered as unimportant. As a consequence, the conclusion that there is a total absence of sources of inflation inertia in forward-looking FP models, as stated by [START_REF] Ball L | Monetary Policy for Inattentive Economies[END_REF], is not exact. The frequent negligence of expectation errors inherent to the Taylor structure explains this result. Indeed, the absence of disinflation costs is a very special characteristic attributable to the Calvo structure. For critics finding that the simplest Taylor structure can produce only a small degree of inflation persistence, the introduction of a small part of longer contracts seems able to significantly rise persistence [see [START_REF] Dixon | Persistence and Nominal Inertia in a Generalized Taylor Economy: How Longer Contracts Dominate Shorter Contracts[END_REF]. Then, when using the New Keynesian Phillips curve, it seems more important not to systematically use the Calvo model, rather than to apologize in a footnote, as suggested by [START_REF] Mankiw | Comment on Cecchetti and Kim[END_REF]. Another surprising result is the fact that the presence of lagged inflation in the dynamics improves only slightly the degree of inflation persistence produced by each price structure, except in the case of the Calvo structure submitted to a disinflation shock. Concerning the PP versus FP choice, we show that this choice by itself is not sufficient to determine dynamic inflation properties and output. Both hypotheses can produce an important degree of persistence. In PP models, the choice of the price structure is even more important than in FP models, because under the Taylor structure, the degree of persistence is very low.

An interesting extension would be to go deeper in the study of the foundations of these price rules. The assumption of exogenous time-dependent price rigidities is frequent in the literature, but for a comparative work, it would be interesting to associate output and inflation dynamics at a macro level with the microeconomic nature of the adjustment costs preventing firms from changing their prices (menu costs, contracting costs, rational inattentiveness, indexation decision etc. . . ). [START_REF] Yetman | Explaining Hump-shaped Inflation Responses to Monetary Policy Shocks[END_REF] provided an interesting paper in this way of research, and it seems to be a natural way to prolong our work.

APPENDIX 1: Equivalence between equation (14) and the Phillips curve of [START_REF] Fuhrer | Inflation persistence[END_REF] The general formulation of the Fuhrer and Moore model consists in introducing a relative term in the Taylor structure. Equations (1), ( 9) and (11) of the Taylor model can be re-written under the following form:

x t = 1 N 2 N -1 X i=0 N-1 X j=0 (x t+i-j ) + φ N N-1 X i=0 E t (y t+i ) (20)
There is no reference to the price level. Fuhrer and Moore (1995) criticize this point and argue that contracts should be written in relative terms. They propose the following rule for the evolution of x t :

x t -p t = µ 1 N ¶ N -1 X i=0 E t (v t+i + φy t+i ) (21)
where v t is the mean real price of other contracts coexisting during period t:

v t = µ 1 N ¶ N-1 X i=0 (x t-i -p t-i ) (22)
Then, (21) becomes:

x t -p t = N-1 X i=1 N -i N (N -1) (x t-i -p t-i )+ N-1 X i=1 N -i N (N -1) E t (x t+i -p t+i )+ φ N -1 E t y t+i (23)
For N = 2, this gives:

x tp t = 1 2 (x t-1p t-1 + E t x t+1 -E t p t+1 ) + φ(y t + E t y t+1 )

The corresponding Phillips curve is:

π t = 1 2 (π t-1 + E t π t+1 ) + φ (e y t ) + (1/2) η t ( 25
)
where e y t = y t +y t-1 +E t-1 y t +E t y t+1 and η π t = E t-1 π t -π t . When we take into account explicitly the expectation error, the last equation corresponds exactly to the equation ( 14) of the text: π t = (1/3) (π t-1 + E t-1 π t + E t π t+1 ) + (2φ/3) (e y t )

  8) Equation (7) corresponds to the widely used Calvo forward-looking Phillips curve, and equation (8) represents a simplified version of the hybrid Phillips curve presented by Christiano et al. (2005).

FIG. 1 .

 1 FIG. 1. Response of inflation (autocorrelated shock)

  deviation between m and p (over 25 periods)

Fischer

  FIG. 3. The acceleration phenomenon

Figure 4

 4 Figure 4 represents the reaction of the price level.

Figure

  Figure 5 represents the reaction of inflation.

  1 +b y t

	TAB. 2. The Phillips Curves
	where e y

t = y t-1 +E t-1 y t +y t +E t y t+1 and b y t = φy t -φy t-1 +E t-1 y t -E t-2 y t-1 .

Critics about the inflation dynamics implied by the NKPC are extensive in the literature (see among others[START_REF] Roberts | Inflation Expectations and the Transmission of Monetary Policy[END_REF], Walsh, 1998, Mankiw,

2001[START_REF] Woodford | Sticky Information Versus Sticky Prices: A Proposal to Replace the New Keynesian Phillips Curve[END_REF], Rudd and Wheelan, 2005).

Most of the time, these papersonly focus on one of the schemes of price rigidity(Calvo or Taylor). The justification for this limitation comes from a general feeling of homogeneity in the reduced form of the models using different assumptions on price rigidity[START_REF] Roberts | New Keynesian Economics and the Phillips Curve[END_REF].

The discount rate is equal to one. Its introduction is straightforward but unimportant for our analysis.

It is possible to imagine an indexation rule that would introduce a correction mechanism if realized inflation differs from expected inflation, but this issue is beyond the scope of our paper.

[START_REF] Trabandt | Sticky Information vs. Sticky Prices: A Horse Race in a DSGE Framework[END_REF] studies the implications of some alternative assumptions of price rigidity in dynamic stochastic general equilibrium models.

The PP model resolution under the Calvo structure is given by[START_REF] Woodford | Sticky Information Versus Sticky Prices: A Proposal to Replace the New Keynesian Phillips Curve[END_REF]. The Taylor model with PP is straightforward to resolve once the values of expectations are given. The FP models resolution is obtained by the method of factorization (seeRomer, 2006, or Mankiw and[START_REF] Woodford | Sticky Information Versus Sticky Prices: A Proposal to Replace the New Keynesian Phillips Curve[END_REF].

This result is dependent of the use of 2-periods contracts. With longer contracts, hump shaped dynamics can be reproduced.

This result is dependent of the use of 2-periods contracts. With longer contracts, hump shaped dynamics can be reproduced.

The sacrifice ratio is defined as the cumulative reduction in output required to achieve a one percentage point reduction in the inflation rate.

To make the comparison simpler, we index by C the variables relative to the Calvo model, and we index by T the variables relative to the model of Taylor.

The disinflation criterion represents the ability of the model to reproduce correctly the costs of disinflations.

Of course, the costs of disinflations can come from other equations than the inflation equation. As we use a simple model, we do not consider these effects.

APPENDIX 2: Price dynamics for the disinflation policy

Hypotheses

Price Dynamics

Calvo/Fixed Prices Forward-looking

Taylor/Fixed Prices Forward-looking

Taylor/Fixed Prices Indexation

TAB. 6. Price dynamics (Disinflation)


Olivier Musy
email: omusy@u-paris10.fr

Jean-Christophe Pereau

Disinflationary boom in a price-wage spiral model

Keywords: Disinflation policy, Shock timing, Sacrifice ratio, Price and wage staggered contracts. JEL classification: E31, E52

. When wages are updated every four quarters and prices every two quarters, we show that a "cold-turkey" disinflation is associated to an output boom when the policy is implemented during the last period of life of the wage contract and a recession the other quarters.

Introduction

There is a huge literature on the conduct of monetary policy in a framework with optimizing agents facing constraints when they want to adjust their price or wage decisions. Due to these nominal rigidities, monetary shocks have persistent real effects. While some divergences remain concerning the quantitative impact of monetary shocks in such models, a common feature of this literature is to consider that the impact of shocks on economic variables is independent from the date of their occurrence during the calendar year.

This paper shows that the timing of a disinflation policy may have a impact on the size of the sacrifice ratio, and, more importantly, on the sign of the output response. We determine the conditions under which an immediate and unexpected disinflation (often called "cold-turkey" disinflation) is associated to an output boom. Some papers already associate output booms to disinflations. However, it is always in the case of gradual or pre-announced disinflations, as in the sticky price models of Ball (1994a) or Mankiw and Reis (2002). In these models, the origin of the disinflationary boom is not due to a specific assumption about the kind of nominal rigidities. It comes from the combination of sticky prices with the assumption of an expected disinflation. Taking into account this future disinflation, sticky price setters begin to lower their prices immediately, leading to a rise of real balances and an output boom. However a cold-turkey disinflation in these models is at best associated with a real neutrality.

We derive our result from the 'wage-price' spiral model of [START_REF] Blanchard | The Wage-Price Spiral[END_REF]. It is based on staggered prices and wages contracts à la Taylor in a monopolistic competition setup. Our main assumption concerns the distinct duration of the price and wage contracts. We assume that prices are adjusted every two quarters while wages are set annually. This assumption has strong implications concerning the real impact of an unexpected disinflation. We show that the output boom only occurs when the disinflation begins in the last period of the wage contract. If the same policy is implemented in other periods, it implies recessions of various sizes. Our approach shares some common features with Olivei andTeneyro (2007, 2008). They show that the date of a monetary shock may have a different impact on output deviation depending on the percentage of wages which are changed during the quarter of the calendar year. But our work differs since we do not consider a non uniform distribution of staggering contracts à la [START_REF] Calvo | Staggered Prices in a Utility Maximizing Framework[END_REF].

The timing structure of our model is mainly guided by empirical considerations. As Ball (1994b) points out, "in many countries, virtually all wages are set for one year with no time-variation", and it concerns "more than 80 percent of US wages". Taylor (1999) and Dickens et al. (2006) also note that most of the wages are negotiated annually. Furthermore, Olivei andTenreyro (2007, 2008) support the idea that annual wage changes are mostly concentrated in the same quarter of the year in the United States (U.S.) and Japan (but not for major countries of the Euro area). Evidence for a faster adjustment of prices is also well documented. Among others, [START_REF] Bils | Some Evidence on the Importance of Sticky Prices[END_REF] find that half of goods' prices last 5.5 months or less in the U.S. 1 . Based on these empirical findings, we can plausibly assume that prices are more often adjusted than wages, leading to a higher duration for wage contracts than price contracts. Moreover this assumption is in line with the "old" Keynesian feature that wages are more sticky than most prices in the U.S. [START_REF] Friedman | Comment on King and Wolman[END_REF].

The structure of the paper is as follows. Section 2 introduces asymmetric price and wage contracts, and determines analytically the impact of monetary shocks on the dynamics of prices and wages. Section 3 shows that a "coldturkey" disinflation can have a different impact depending on the date of the shock. Finally, Section 4 presents a brief summary and conclusion.

The model

We consider the reduced form of [START_REF] Blanchard | The Wage-Price Spiral[END_REF] derived from the general equilibrium imperfect competition model of [START_REF] Blanchard | Monopolistic Competition and the effect of aggregate demand[END_REF]. Let p, w, m and y be the price level, the nominal wage, the stock of money and the output, respectively. 2 Let p * and w * be the price and wage that maximize the profit for a given period. The elasticities a and b, with 0 < a, b < 1, respectively measure the degree of rigidity of markups and real wage. 3 Following [START_REF] Blanchard | Monopolistic Competition and the effect of aggregate demand[END_REF], the optimal price and the optimal wage at time t are given by:

p * t = aw t + (1 -a)m t , (1)
w * t = bp t + (1 -b)m t . (2)
The output depends on the level of real balances:

y t = m t -p t . (3)
We introduce some nominal rigidities in the economy: prices set by one cohort of firms and wages set by one cohort of workers are not modified each period. The duration of each contract is known by advance and follows the adjustment rule of [START_REF] Taylor | Aggregate Dynamics an Staggered Contracts[END_REF]: for a contract x t (x t = p t , w t ) modified in period t, with a duration of N periods, the probability of changing the contract is equal to 0 during the N -1 subsequent periods, and equal to 1 in period t + N. Agents set their price or wage by minimizing the following loss function:

Min x t L t = N-1 X i=0 E t ¡ x t -x * t+i ¢ 2 , (4)
where E t stands for the mathematical expectation of future variables, given the set of available information at time t (this set of information contains the path of monetary policy announced at time t). The contract is then set as an average of the future optimal values:

x t = 1 N N-1 X i=0 x * t+i . (5)
Based on the empirical findings developed in the introduction, we assume a one-year duration of wage with wage changes mostly concentrated during the same period. Prices are assumed to be updated every two quarters leading a duration of one semester. It turns out that at time t, the wage cohort sets its nominal wage for four periods, from t to t + 4, while at time t + 1 the price cohort sets its price for two periods t + 1 and t + 2 and again at time t + 3 for two periods t + 3 and t + 4. Three different price contracts overlap with a wage contract. The first period of the wage contract coexists with the price contract set at time t -1, the second and third periods overlap with the price set at time t + 1 and the last period overlaps with the price set at time t + 3. We denote by p 1 and p 3 the prices set at the first and the third periods of the wage contract life. The timing of adjustment is summarized by the following Table :

4 Period t 0 1 2 3 4 5 6 7 8 ... prices -→ p 1 1 -→ p 3 3 -→ p 1 5 -→ p 3 7 -→ ... wage w 0 -→ -→ -→ w 4 -→ -→ -→ w 8 ... Table 1:
The timing of adjustment Given Eqs (1), ( 2) and (5), wages and prices are set according the follow-ing rules (∀t = 0, 4, 8...):

w t = 1 4 £ bp 3 t-1 + (1 -b) m t ¤ + 1 4 E t £ bp 1 t+1 + (1 -b) m t+1 ¤ + 1 4 E t £ bp 1 t+1 + (1 -b) m t+2 ¤ + 1 4 E t £ bp 3 t+3 + (1 -b) m t+3 ¤ , (6)
p 1 t+1 = 1 2 [aw t + (1 -a) m t+1 ] + 1 2 [aw t + (1 -a) E t+1 m t+2 ] , (7)
p 3 t+3 = 1 2 [aw t + (1 -a) m t+3 ] + 1 2 E t+3 [aw t+4 + (1 -a) m t+4 ] . (8)
The choice of an asymmetric structure has some implications on the weights of other contracts taken into account by price setters. As shown in Table 1, the choice of p 1 is made only with respect to the past value of wages (because the price contract will be renewed before the wage contract), while the determination of p 3 depends also on the future value of wages (because the contract will interact both with the current and following wage contracts). Eqs (6), ( 7) and (8) permit to obtain the following dynamic equation for the wage level (∀t = 0, 4, 8...):

w t = fw t-4 + fE t-1 w t + fE t w t+4 + g 1 m t-1 + g 1 E t-1 m t +g 2 m t + (2g 1 + g 2 ) E t m t+1 + (2g 1 + g 2 ) E t m t+2 + (g 1 + g 2 ) E t m +3 + g 1 E t m t+4 , (9)
where

f = ab 8 -5ab , g 1 = b (1 -a) 8 -5ab , g 2 = 2 (1 -b) 8 -5ab .
Eq [START_REF] Dickens | How wages change: micro evidence from the international wage flexibility project[END_REF] states that the dynamic of w t under rational expectations depends on both past and future expected values of w t through the terms E t-1 w t and E t w t+4 , as well as expected values of m t . Solving the model requires the elimination of the expected values of w t by the method of lag operators [START_REF] Romer | Advanced Macroeconomics[END_REF]. As in [START_REF] Blanchard | The Wage-Price Spiral[END_REF], we distinguish the initial impact of the shock from the subsequent dynamics. We obtain an equation which only depends on the past values of wages (the initial value of wages is known), and the expectations about the money path (which depends on the exogenous and publicly announced monetary policy). The dynamic of w t is then given by (∀t = 0, 4, 8...):

w t = λw t-4 + λ f ∞ X i=0 λ i E t µ g 2 m t+4i + (2g 1 + g 2 ) (m t+1+4i + m t+2+4i ) + (g 1 + g 2 ) m t+3+4i + g 1 m t+4+4i ¶ + λg 1 f ∞ X i=0 λ i E t-1 (m t-1+4i + m t+4i ) , ( 10
)
where λ is the stable eigenvalue:

λ = √ 2 -ab - p 2 (1 -ab) √ 2 -ab + p 2 (1 -ab) . (11)
To analyze the impact of a shock on the growth rate of the money, we have to determine both the growth path of the wage and the initial response (see Appendix). We denote by ∆w t = w tw t-4 the growth rate of wage (∀t = 0, 4, 8..), and ∆m t = m tm t-1 the growth rate of money (∀t = 0, 1, 2..). We have:

f (1 -λ) λ -1 ∆w t = f (1 -λ) ∆w t-4 -g 1 E t-5 (M 1 ) -E t-4 (M 2 ) +g 1 E t-1 (M 3 ) + E t (M 4 ) (12)
where

M 1 = ∆m t-4 + ∞ X i=1 λ i (∆m t-4+4i + ∆m t-5+4i ) , M 2 = (2g 1 + g 2 ) 3 X i=2 ∆m t-i + (2g 1 + g 2 ) ∆m t-3 + (g 1 + g 2 ) 3 X i=1 ∆m t-i +g 1 3 X i=0 ∆m t-i + ∞ X i=1 λ i µ g 2 ∆m t-4+4i + (2g 1 + g 2 ) (∆m t-3+4i + ∆m t-2+4i ) + (g 1 + g 2 ) ∆m t-1+4i + g 1 ∆m t+4i ¶ , M 3 = 4 X i=1 ∆m t-i + 4 X i=0 ∆m t-i + ∞ X i=1 λ i [∆m t-1+4i + ∆m t+4i ] , M 4 = g 2 3 X i=0 ∆m t-i + (2g 1 + g 2 ) " 4 X i=0 ∆m t+1-i + 5 X i=0 ∆m t+2-i # + (g 1 + g 2 ) 6 X i=0 ∆m t+3-i + g 1 7 X i=0 ∆m t+4-i + ∞ X i=1 λ i • g 2 ∆m t+4i + (2g 1 + g 2 ) (∆m t+1+4i + ∆m t+2+4i ) + (g 1 + g 2 ) ∆m t+3+4i + g 1 ∆m t+4+4i ¸,
The terms M 1 -M 4 contain the current and past values of money growth, as well as many past expected terms on money growth. Eq [START_REF] Ireland | Optimal Disinflationary Paths[END_REF] shows that the wage dynamic is driven by current and past expectations of the growth of the money stock, with different weights for each date of expectation (reflecting the position in the cycle of adjustment presented in Table 1). Let us remark that the lags of the expectation terms depends on the longer contract and go up to t -4 and t -5.

The initial impact of a change in the growth rate of money is given by:

(1 -f ) ∆w t = f (-E t-5 ∆w t-4 + 2∆w t-4 -E t-4 ∆w t + E t-1 ∆w t + E t ∆w t+4 ) -g 1 E t-5 ∆m t-4 -E t-4 [(2g 1 + g 2 ) (2∆m t-1 + 2∆m t-2 + 3∆m t-3 )] +2g 1 ∆m t-4 + 4 (2g 1 + g 2 ) [∆m t-1 + ∆m t-2 + ∆m t-3 ] + g 1 E t-1 ∆m t +2 (3g 1 + g 2 ) ∆m t + (2g 1 + g 2 ) E t [∆m t+3 + 2∆m t+2 + 3∆m t+1 ] +g 1 E t ∆m t+4 . (13)
The inflation rates ∆p 1 t+1 = p 1 t+1p 3 t-1 and ∆p 3 t+3 = p 3 t+3p 1 t+1 are given by (∀t = 0, 2, 4..):

∆p 1 t+1 = a 2 [2∆w t -E t-1 ∆w t ] (14)
+ µ 1 -a 2 ¶ [-E t-1 ∆m t + 2∆m t + 2∆m t+1 + E t+1 ∆m t+2 ] , ∆p 3 t+3 = a 2 [E t+3 ∆w t+4 ] (15)
+ µ 1 -a 2 ¶ [-E t+1 ∆m t+2 + 2∆m t+2 + 2∆m t+3 + E t+3 ∆m t+4 ] .
The inflation rate in Eq [START_REF] Olivei | The Timing of Monetary Policy Shocks[END_REF] contains past expectations of the current wage growth, while Eq (15) contains contemporary expectations of future wage growth. This feature is important when considering the impact of monetary shocks.

The impact of a disinflation policy

Many papers focuse on disinflation because it is often associated with high real costs that are difficult to reproduce in standard models (Ball, 1994a, Mankiw andReis, 2002). The estimated costs of disinflation policies are also quite heterogenous (Ball, 1994b). Explanations for the empirical variations of these costs often rely on the speed of disinflation, some labor market features (Ball, 1994b) or imperfect credibility [START_REF] Ball | Disinflation with Imperfect Credibility[END_REF][START_REF] Ireland | Optimal Disinflationary Paths[END_REF].

To illustrate the properties of our model, we consider a credible disinflation policy 5 occurring at time s. For t < s, we set ∆m t = µ and for t ≥ s, ∆m t = µ 0 , with µ > µ 0 . Since the shock is unique, permanent and totally unexpected, we have E t ∆m t+i = ∆m t . In our numerical application, we pass from an inflation rate of µ = 1% to a long term objective of price stability µ 0 = 0. The output cost obtained after the implementation of the disinflation policy is measured by the sacrifice ratio. This is defined as the output loss for one percent of inflation reduction. Before the shock, we assume that, at the regular state, prices and wage are set as an average of the expected levels of money during the contract life, such that x t = (1/N ) P n i=0 E t m t+i for x t = p t , w t . This implies that on average the output y equals its equilibrium value set to zero. The long run equilibrium values of wages and prices are equal to the value of the money stock.

Dynamics

The monetary shock can occur during each of the different phases of the price/wage cycle of adjustment. A shock in t = 0 coincides with a date of wage adjustment, while shocks in t = 1 and t = 3 coincide with periods of price revision. In t = 2, neither wages nor prices are adjusted. We show that, depending on the date of the shock, the disinflation can be associated either to an output loss or to an output boom. 6 Figure (1) displays the response of prices and wages when the disinflation policy occurs at time 0. At this time, the money growth is stopped (µ 0 = 0), implying a value of the money stock equal to its past value m 0 = m -1 = -µ. Since the shock was not expected, the price p 3 -1 set before the shock is based on the previous growth of the money stock (equal to µ) implying

p 3 -1 = (1/2) (m -1 + m 0 ) = -(1/2) µ.
When the disinflation occurs, the stop of the money growth yields a reduction of the real balances (p 0 > m 0 ) and a recession (y 0 = m 0p 0 < 0). To explain the slow convergence of the variables to their long run equilibrium values and the associated persistent output cost, note that the wage cohort sets optimally in t = 0 its wage using Eq (2) as a weighted average between the price level and the value of the money stock (the weight depends on the parameter b). It turns out that the wage level also overshoots the money stock. The same kind of adjustment holds for prices p 1 1 and p 3 3 according to the rule [START_REF] Altissimo | Inflation Persistence and Price-Setting Behaviour in the Euro Area: A Summary of the Evidence[END_REF]. Since the price level always overshoots the money stock, the disinflation policy is associated to output costs.

The same mechanisms are at stake when the disinflation policy is implemented in time 1 (Figure 2) and in time 2 (Figure 3). The main differences concern the magnitude of the nominal variables adjustment. When the shock occurs in t = 1, wages have been set in t = 0 with respect to the constant expected rate of the monetary growth. w 0 = (1/4) ¡P 4 t=0 m t ¢ , so we obtain w 0 = (1/4) (0 + µ + 2µ + 3µ) = (3/2) µ. This wage level remains unchanged until period 4. The price p 1 1 is updated at the date of the shock with respect to the new value of the money stock (m 1 = m 0 = 0) but also on the basis of the past value of the wage w 0 . Since wages are too high for the new monetary path, and because of the interactions between prices and wages, the price level continues to rise in order to catch the value of wages. It occurs despite of the stability of the money stock (which should optimally imply a stability of prices). In period t = 3, prices are revised for the second time after the implementation of the disinflation. p 3 3 is set relatively to the future value of wages according to Eq [START_REF] Olivei | Wage Setting Patterns and Monetary Policy: International Evidence[END_REF]. Hence, expecting an adjustment of wages in the following period, prices begin to fall and converge towards their long-run value. Since prices are higher than the money stock during all the process, there is a persistent recession in the economy. A similar mechanism is at work for recessions occurring when the disinflation shock is in t = 2. At this time we have

m 2 = m 1 = µ, w 0 = p 1 1 = (3/2) µ.
Figure 4 presents prices and wages dynamics for a disinflation implemented in period t = 3 at the last period of life of the wage contract. As indicated previously, before the shock, wages are set equal to the mean of the money stock expected to hold during the whole contract life (Eq 5). Then, wages at time t = 0 are fixed expecting a continuous growth of money at rate µ such that w 0 = (3/2) µ. When the disinflation occurs in period 3, the money stock is given by m 3 = m 2 = 2µ. It turns out that the money stock is higher than the wage level in period 3. Since prices are defined as a weighted average of the wage level and the money stock, it also implies that in period 3, they are set at a value located between these two variables (see Eq 1). The price level is then set below the money stock yielding an initial output boom. Subsequent adjustments of wages and prices are gradual to their equilibrium values but they remain below the money stock yielding a persistent output boom. Remark that this output boom does not depend on particular values m of a and b; these two parameters only affect the magnitude of the boom. 7In Figure 5, we present the comparative response of output for each scenario, for values a = b = 0.99.

Table 2 shows the robustness of our qualitative results. We display the sacrifice ratios for several values8 of a and b with respect to the dates of the disinflation policy. In absolute value, the size of the boom and the recessions decreases if a or b decreases, but even for lower values of a and b, our qualitative results remain valid. The highest sacrifice ratio occurs for a shock in t = 1 when prices are revised. Interestingly, it means that a date of price adjustment can imply a higher sacrifice ratio than a date which is not associated to any adjustment of nominal variables (this is the case for a shock occurring in t = 2). This result may seem to be paradoxical. However it comes from the asymmetric structure which implies that prices set during the first period are backward-looking with respect to the wages. Hence, prices continue to rise despite the stability of the money stock, increasing the output cost of the disinflation. The adjustment of prices indeed rises the initial output cost, and convergence towards the new path of money only begins in period 3. Then, for a shock in period 2, there is no initial rise of price. It results into a lower real cost.

Discussion about the price-wage structure

Our asymmetric temporal structure is highly stylized. In a more realistic framework, at least a fraction of prices should be modified each quarter, implying changes in the aggregate price index every quarter and not only in quarters 1 and 3 as in the previous structure. It does not alter the possibility of an output boom. Nevertheless, the boom will not be systematic as in our model. One can imagine a standard staggered prices structure à la Taylor (with two cohorts of equal size, setting 2 periods contracts, as in [START_REF] Romer | Advanced Macroeconomics[END_REF], combined with an annual wage modification (which still represents the U.S. case or the Japanese Shunto). Denoting the prices set by the first cohort of firms by p i and the prices set by the second cohort of firms by p j , the general price level is given by p = (1/2) (p i + p j ). In this case, the optimal wage and prices of the model of Blanchard and Kiyotaki (1987) are:

w * = a 1 2 (p i + p j ) + (1 -a 1 ) m p * i = b 1 w + b 2 p j + (1 -b 1 -b 2 ) m p * j = b 1 w + b 2 p i + (1 -b 1 -b 2 ) m with a 1 = 1 -α (β -1), b 1 = ³ 2 2+(α-1)(θ+1)
´, and b 2 =

³ (α-1)(θ+1) 2+(α-1)(θ+1)
´.

Assume that cohort i changes its price at even periods for two periods, cohort j at odd periods for two periods, and the wage cohort at even periods for four periods. Based on the timing procedure presented in section 2, we can derive a more complicated dynamic structure which is much less tractable. However, an output boom is still possible. However our result will depend on two contradictory effects. The first effect relies on the interactions between price setters in the staggered structure of Taylor. This generates output costs of disinflation [START_REF] Musy | Inflation persistence and the real costs of disinflation in staggered prices and partial adjustment models[END_REF]. The second effect is due to the duration of wage contracts. It can generate an output gain when the disinflation occurs just before its updating. When the two effects are active, the net impact depends on the relative weights of the parameters in the previous equations. Mechanisms are the same than those presented, excepted that, in addition, price setters have to take into account the prices set by their competitors. If the weight relative to competitors is not relatively high, the output boom is still present.

Conclusion

Most current models of output and inflation dynamics, despite their size and complexity, are built on very simple uniform staggering structures. They predict no relationship between the impact of a monetary shock and its date of occurrence. Olivei andTenreyro (2007, 2008) challenge this prediction, giving some evidence that shocks have asymmetric effects depending on their date of occurrence during the process of wage adjustment. This paper presents an original price and wage structure based on [START_REF] Blanchard | The Wage-Price Spiral[END_REF], assuming that wages have a longer duration than prices. We show that the strategic interactions between staggered price and wage give a combination of multiple expectations terms that differ for each period of the cycle of wage adjustments. Indeed, the date of the shock appears to be important because each date corresponds to a particular combination of price and wage expectations; and each combination determines a specific initial impact of the shock. Then this initial response is transmitted to the following periods due to the overlapping structure of the contracts. In our example, a disinflation causes a boom when the shock occurs during the last quarter of wage rigidity and recessions of different magnitudes occur during the other quarters. The highest sacrifice ratio occurs during a period of price change. It challenges the idea that the sacrifice ratio would always be lower when the nominal rigidity is also the lower (as it is the case in simple models building on [START_REF] Calvo | Staggered Prices in a Utility Maximizing Framework[END_REF].

One reader could object that our results are model specific. Indeed, the quantitative results obtained are model dependent, and are merely an illustration of the unexpected results that can be obtained when we combine price and wage rigidities using finite length contracts à la Taylor. However, the mechanisms presented rest only on the expected terms inherent to the structure of Taylor. Similar results could be obtained with alternative structures based on contracts à la Taylor. The Calvo structure is more used in the literature but its very specific nature erases all the expectations terms in the dynamics. This gives more tractable models but at the expense of less interesting dynamics as we have shown in this paper.

APPENDIX: Disinflation dynamics with asymmetric contracts

We present the general dynamics of the price and wage inflation for a specific money supply process and the initial responses of the monetary shock.

Wage and price inflation dynamics Eq (12) can be rewritten as for all t = 0, 4, 8...:

∆w t = λ∆w t-4 + λ f (1 -λ) 2 µ -h 5 ∆m t-5 -h 4 ∆m t-4 + h 3 ∆m t-3 +h 3 ∆m t-2 + h 2 ∆m t-1 + h 1 ∆m t ¶ with h 1 = (19 -13λ) g 1 + 2
(5 -3λ) g 2 ; h 2 = (9 -7λ) g 1 + 4(1λ)g 2 h 3 = 4(1 + λ)(2g 1 + g 2 ); h 4 = (11 -7λ) g 1 + 3 (2λ) g 2 h 5 = (1 + λ) g 1

Price dynamics are derived from eqs ( 14) and [START_REF] Olivei | Wage Setting Patterns and Monetary Policy: International Evidence[END_REF].

Initial Impact

The initial responses are given by eq [START_REF] Musy | Inflation persistence and the real costs of disinflation in staggered prices and partial adjustment models[END_REF]. They depend on the date of the shock.

Shock at time 0 When the monetary policy is implemented at time 0 during a period of wages revision. The wage growth path is given by (with z = (1f ) [1f (1 + λ)]-2f 2 > 0):

z∆w 0 = • [1 -f (1 + λ)] (4f + 13g 1 + 6g 2 ) -f (4f + g 1 ) ¸µ + • [1 -f (1 + λ)] (19g 1 + 10g 2 ) +f [4f (1 -λ) + 33g 1 + 16g 2 ] ¸µ0 z∆w 4 = [2f [4f + 13g 1 + 6g 2 ] -(4f + g 1 ) (1 -f )] µ + • (1 -f ) [4f (1 -λ) + 33g 1 + 16g 2 ] +2f (19g 1 + 10g 2 )

Shock at time 1

The shock occurs when prices p 1 are set, leading to a wage growth path:

∆w 0 = 4µ ∆w 4 = -6 • (2g 1 + g 2 ) 1 -f (2 + λ) ¸µ + • 4f (1 -λ) + 22 (2g 1 + g 2 ) 1 -f (2 + λ)

Shock at time 2

Since no prices are set at this period, the wage growth path is:

∆w 0 = 4µ ∆w 4 = -2 • 2g 1 + g 2 1 -f (2 + λ) ¸µ + • 4f (1 -λ) + 18 (2g 1 + g 2 ) 1 -f (2 + λ) ¸µ0
∆w t = λ∆w t-4 + 4 (1λ) µ 0 for for t = 8, 12

and the price inflation is given by:

Shock at time 3

The shock occurs when prices p 3 are set, we obtain the wage growth path:

∆w 0 = 4µ ∆w 4 = 2 • 2g 1 + g 2 1 -f (2 + λ) ¸µ + • 4f (1 -λ) + 14 (2g 1 + g 2 ) 1 -f (2 + λ)

Figure 1 :

 1 Figure 1: Shock in time 0 p, w, m 1 2 µ

Figure 2 :

 2 Figure 2: Shock in time 1 p, w, m 3 2 µ

Figure 3 :

 3 Figure 3: Shock in time 2

Figure 4 :

 4 Figure 4: Shock in time 3

¸µ0 ∆w t = λ∆w t- 4 + 4 ( 1 - 3 t = ∆p 1 t+2 = a 2

 44132 λ) µ 0 for t = 8, 12, ... Concerning the price inflation path, we obtain: ∆w t+1 + 2 (1a) µ 0 for t = 3, 7, 11...

¸µ0 ∆w t = λ∆w t- 4 + 4 ( 1 - 12 and 2

 441122 λ) µ 0 for t = 8, ∆w t+1 + 2 (1a) µ 0 for t = 3, 7, 11...

2

 2 ∆w t+1 + 2 (1a) µ 0 for t = 3, 7, 11...

¸µ0 ∆w t = λ∆w t- 4 + 4 ( 1 -λ) µ 0 for t = 8 ∆p 3 t = ∆p 1 t+2 = a 2

 441832 ∆w t+1 + 2 (1a) µ 0 for t = 7, 11, ..

Table 2 :

 2 Sacrifice ratios (in %)

	a	b	shock in 0 shock in 1 shock in 2 shock in 3
	0.99 0.99 4.8 0.7 0.99 1.11 0.7 0.85 0.9 0.99 0.7 1.12	16.47 4.1 3.8 5.82	4.95 1.17 1.07 1.44	-4.4 -0.67 -0.57 -0.94

[START_REF] Altissimo | Inflation Persistence and Price-Setting Behaviour in the Euro Area: A Summary of the Evidence[END_REF] highlight that the degree of price rigidity seems to be higher for the Euro area (around one year).

All variables are expressed in logarithm and are measured as deviations from their trend values normalized to zero. The steady state is defined as w = p = m and y = 0.

These elasticities depend on some structural parameters. Following[START_REF] Blanchard | The Wage-Price Spiral[END_REF], we have a = 1/α and b = 1α (β -1), where α ≥ 1 is the inverse of the degree of returns-to-scale, and β > 1 the marginal desutility of work. To ensure b > 0, we assume β < 1 + (1/α).

A more complex setup with two cohorts of price and one cohort of wage is discussed in section 3.2.

The derivation of the general price dynamics allows us to study other types of monetary shocks, since it just modifies the value of expectations on money growth.

The dynamics of both the price and wage inflation and the initial responses are displayed in Appendix.

In absolute value, the size of the boom and the recessions decreases if a or b decreases, but even for very small values of a and b, the qualitative features presented remain valid. Decreases in a imply more important reductions in the sacrifice ratio measured in absolute value.

These elasticities depend on the structural parameters α = 1/a and β = 1a(b -1) of the model (see footnote 3). We consider two values for α (1.01 and 1.42) and four values for β (1.007, 1.009, 1.105 and 1.297).