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OxCarre Research Paper 63

Inflation Persistence and Exchange Rate Regime: Implications for dynamic adjustment to

shocks in a small open economy

Karlygash Kuralbayeva

OxCarre

Inflation persistence and exchange rate regime: Implications for

dynamic adjustment to shocks in a small open economy

Karlygash Kuralbayeva

Abstract

The paper examines implications of inflation persistence for business cycle dynamics

following terms of trade shock in a small oil producing economy, under inflation targeting

and exchange rate targeting regimes. It is shown that due to the ‘Walters critique’ effect, the

country’s adjustment paths are slow and cyclical if there is a significant backward-looking

element in the inflation dynamics and the exchange rate is fixed. It is also shown that such

cyclical adjustment paths are moderated if there is a high proportion of forward-looking

price setters in the economy, so that when the Phillips curve becomes completely forward-

looking cyclicality in adjustment paths disappears and the response of the real exchange rate

becomes hump-shaped. In contrast, with an independent monetary policy, irrespective of the

degree of inflation persistence, flexible exchange rate allows to escape severe cycles, which

results in a smooth response of the real exchange rate.

JEL Classifications: E32, F40, F41

Keywords: inflation inertia, inflation targeting, exchange rate targeting, Phillips curve,

oil shocks, small open economy

1 Introduction

New Keynesian macroeconomics1, the modern approach featuring rational expectations and

some form of microfoundations has become the main workhorse framework in modern monetary

policy analysis. The essential macroeconomic feature of the New Keynesian macroeconomics

is the absence of instantaneous market clearing, which is rationalized by the assumption of

price rigidity. In the New Keynesian model, prices fail to adjust rapidly to clear markets within1See Gordon (1990) for discussion of New Keynesian economics.

1

short period of time, so there can be periods when aggregate output is different from its so-called

potential level, and monetary policy can have short-run effects on economic activity. Thus, price

stickiness and so price-setting behavior is the essence of New Keynesian macroeconomics, which

have been modeled in many different ways. Seminal papers by Fisher (1977), Taylor (1980), and

Calvo (1983) present some well-known examples of modeling price inertia from the price-setting

behavior of firms. Price setting behavior approach within a general equilibrium framework has

resulted in a derivation of the New Keynesian Phillips Curve (NKPC), which describes the

process of inflation and relates it to a level of economic activity within New Keynesian models.

However, New Keynesian Phillips Curve has been severely criticized, mostly on empirical

grounds. The most relevant to the discussions in this paper critique of the NKPC is persistence

problem, that is failure of the NKPC generate the degree of inflation persistence observed in the

data.2 The origin of this problem is forward-looking nature of the NKPC where inflation today

depends on the sequence of expected future output gaps. This implies that there is no structural

relationship between current and lagged inflation, and all persistence in inflation comes from

the persistence in the output-gap term.

These observations about the NKPC have motivated many researches to develop theoretical

models which introduce inflation persistence. In the micro-founded way the literature suggests

two main mechanisms. First is an introduction of a rule-of-thumb price setting behavior by some

firms (Galı and Gertler (1999), Steinsson (2003)). The second mechanism is a backward-looking

price indexation (Christiano et al. (2001), Smets and Wouters (2002)).

As a result, there is a large number of papers that incorporate inflation persistence into

stochastic general equilibrium models to study different issues (e.g., Christiano et al. (2001),

Leitemo (2002), Benigno and Thoenissen (2003), Smets and Wouters (2002)). Within this

literature, closest to the spirit of this paper, are papers by Leitemo (2002) and Leitemo and

Røisland (2002), which have studied the implications of inflation persistence for a design of

optimal monetary policy rules for open economies. Results of both of these papers suggest that

exchange rate targeting regime leads to the highest social loss of all regimes considered in the

papers. The main reason behind this result is that targeting the nominal exchange rate gives2See for example, Fuhrer and Moore (1995), Galı and Gertler (1999), Benigno and Lopez-Salido (2002), and

Mehra (2004).

2

rise to the ‘Walters Critique’ effect, mechanism coming from the real interest rate under a fixed

exchange rate regime, named after Sir Alan Walters, who drew attention to the potentially

destabilizing real interest rate response at the time when the UK entered the ERM in 1990.

It is important to note that the main limitation of the methodology employed in Leitemo

and Leitemo and Røisland (2002) remains to be reduced-form structure of their macromodels

used for policy evaluation. As a result, the non-microfounded structure of their models make

them subject to the Lucas critique. The reduced form macromodels also lacks an important

feature of the microfounded models those coefficients are directly related to the deep structural

parameters of the model.

In contrast, this paper employees a micro-founded optimization framework and introduces

inflation persistence via rule-of-thumb price setting behavior. The model is then used to study

how inflation persistence shapes the dynamics of adjustment to the terms of trade shock in a

oil producing country, under two alternative monetary policy regimes: inflation targeting and

exchange rate targeting regimes. The aim of the paper is threefold. First, I analyze in details

the dynamic adjustment to the terms of trade shock in oil-rich economy. Specifically, I draw

attention to the possibility of the ‘Walters Critique’ effect under exchange rate targeting regime

in resource-rich economies subject to terms of trade shock. Second, I show that the extent of

‘cyclicality’ depends on the number of forward-looking price setters in the economy. The more

forward-looking price setters in the economy, the less pronounced the Walters critique effect.

Third, the microfounded structure of the model of the paper allows to derive the degree of

inflation persistence as an explicit function of the structural parameters of the model and to

analyze sensitivity of the results to the choice of those parameters. In particular, the paper

studies how the emergence of oscillatory behavior in series is affected by the value of the Calvo

parameter and it also provides simulated estimate of the threshold value of degree of inflation

inertia which generates the Walters effect. This will suggest an important guidelines for a design

of monetary policy in resource-rich economies, given the uncertainty of inflation persistence and

price stickiness in those countries.

It is also necessary to note that this study does not attempt to evaluate whether one ex-

change rate regime would be potentially beneficial than the other, when the economy exhibits

significant degree of inflation inertia. Neither does this study examine the question of optimal

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monetary policy (optimal interest rate rule) and no attempt is made to derive ‘optimal’ feedback

parameters in the interest rate rules, conditional on the degree of inflation persistence in the

Phillips curve.

The reminder of the paper is structured as follows. Section 2 describes the model and defines

a competitive equilibrium for the economy. Section 3 discusses calibration of the model. The

simulation results are presented in section 4. Section 5 concludes.

2 Model of a Small Open Economy

I consider a two-sector dynamic stochastic general equilibrium model of oil producing economy.

The domestic economy is open and small and comprises three sectors: traded, non-traded goods,

and the oil sector. It is assumed that oil production requires no domestic factor inputs, and

all of its production is exported. The exogenous price of oil is subject to stochastic shocks.

The model assumes flexible price traded goods, which are traded in competitive markets, and a

continuum of monopolistically produced non-traded goods. I assume Calvo-type price stickiness

in the non-traded sector. I also allow for some additional inflation inertia, using a rule-of-

thumb price setting mechanism outlined in Steinsson (2003). Monetary authority is assumed

to implement one of two alternative monetary policy regimes: inflation targeting and exchange

rate targeting regimes. In both cases the monetary authority uses the nominal interest rate as

a policy instrument and monetary policy is modeled through an interest rate rule.

2.1 Consumers

The economy is inhabited by a representative household who seeks to maximize:

U = E0

∞∑t=0

βtu(Ct, Ht) (2.1)

where Ht is labor supply and Ct is a composite consumption index defined as a CES function of

consumption of non-traded and traded goods, Ct = [a1ρC

(ρ−1)/ρN,t +(1−a)

1ρC

(ρ−1)/ρT,t ]ρ/(1−ρ), where

ρ > 0, The implied consumer price index is then Pt = [aP 1−ρN,t + (1−a)P 1−ρ

T,t ]1

1−ρ , with PN,t(PT,t)

is the price of non-traded (traded) good. I assume imperfect competition in the non-traded

sector, so that consumption of non-traded goods is differentiated, with elasticity of substitution

4

across different categories equal to λ: CNt = (∫ 10 CN,t(j)

λ−1λ dj)

λλ−1 ,with λ > 1.

Households may borrow and lend in the form of non state-contingent bonds that are denom-

inated in units of the traded goods. If households borrow an amount Dt+1, then the portfolio

adjustment costs are κ2 (Dt+1 − D)2, which are denominated in the units of the traded goods.

Households can also obtain loans from domestic capital markets, where Bt denotes the stock of

domestic currency debt.

Households own all home production firms and thus receive the profits on these firms (profits

on traded goods firms are zero). Households also accumulate capital and rent it out to the firms.

Capital stocks in non-traded and traded sectors are assumed to evolve according to the following:

KN,t+1 = IN,t + (1− δ)KN,t, KT,t+1 = IT,t + (1− δ)KT,t, (2.2)

where investment in both sectors is traded good. Installation of capital in both sectors requires

adjustment costs, which represent a basket of goods composed of non-traded goods and traded

goods in the same mix as the household’s consumption basket. Capital adjustment costs are

defined as: φi,t(Ii,tKi,t

)Ki,t = ψI2 ( Ii,tKi,t

− δ)2Ki,t, where i = N,T , and so φ′ > 0, and φ′′ > 0.

The household’s budget constraint in nominal terms is:

PtCt + Pt(φNtKNt + φTtKTt) + PTt(INt + ITt) = WtHt + StDt+1 − (1 + iFt )StDt +Bt+1 − (1 + it)Bt+

+RNtKNt +RTtKTt + PTtOt+

+ ΠNt − PTtκ

2(Dt+1 −D)2 (2.3)

where Wt is the wage rate; RNt (RTt) is the nominal rate of return of capital in the non-traded

(traded) sector; Dt is the outstanding amount of foreign debt, denominated in foreign currency,

Bt is the stock of domestic debt, denominated in domestic currency, and St is the nominal

exchange rate expressed as units of domestic currency needed for one unit of foreign currency.

The household receives profits from the firms producing the non-traded goods, ΠNt. Households

take as given the money coming from export of oil, those price is determined exogenously at the

world markets:

log(εt) = ρ log(εt−1) + νt (2.4)

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and thus, Ot = Oεt. Since oil revenues are denominated in the units of the traded goods, and

because PT,t = St following the assumption of a small open economy, εt can be described as the

world price of oil, denominated in foreign currency. It implies that νt shocks are international

oil price shocks. At the same time, εt represents the price of the oil relative to the price of the

traded goods (which are goods imported in the model), that is εt are terms of trade and shocks

νt are terms of trade shocks. Thus in this model, terms of trade (or oil price) shocks may be

described as the oil income shocks and they will be used interchangeably in the paper.

The household chooses non-traded and traded goods to minimize expenditure conditional on

total composite demand. Demand for non-traded and traded goods is then given by:

CNt = a(PN,tPt

)−ρCt; CTt = (1− a)(PT,tPt

)−ρCt (2.5)

In what follows I assume that the period utility function takes the form u(Ct, Ht) = C1−σt

1−σ −

ηH1+ψt

1+ψ , where the parameter σ measures the (inverse) intertemporal elasticity of substitution.

The household optimum is characterized by the following equations:

Wt = ηHψt PtC

σt (2.6)

1 = βEt[Cσt Pt

Cσt+1Pt+1(1 + it+1)] (2.7)

1− κ(Dt+1 −D) = βEt[Cσt Pt

Cσt+1Pt+1

St+1

St(1 + iFt+1)] (2.8)

qNt = βEtCσt Pt

Cσt+1Pt+1{RNt+1 + Pt+1(φ′N,t+1

INt+1

KNt+1− φN,t+1) + qNt+1(1− δ)} (2.9)

qNt = PTt + Ptφ′N,t (2.10)

qTt = βEtCσt Pt

Cσt+1Pt+1{RT,t+1 + Pt+1(φ′T,t+1

IT,t+1

KT,t+1− φT,t+1) + qTt+1(1− δ)} (2.11)

qTt = PTt + Ptφ′T,t (2.12)

along with the capital accumulation, (2.2), and budget constraint, (2.3), equations.

Equation (2.6) equates the marginal disutility of the labor effort to the utility value of the

wage rate, and defines the households labor supply curve. Equation (2.7) is a Euler equation that

determines intertemporal allocation: it equates the intertemporal marginal rate of substitution

in consumption to the real rate of return on domestic bonds. Equation (2.9) is the counterpart

of equation (2.8) for foreign bonds. Equation (2.9) is the pricing condition for physical capital in

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the non-traded sector. It equates the revenue from selling one unit of capital today (qNt ), to the

discounted value of renting the unit of capital for one period, and then selling it, RN,t+1 + qNt+1,

net of depreciation and adjustment costs3. Equation (2.10) relates the cost of producing a unit

of capital in the non-traded sector to the shadow price of installed capital, or Tobin’s Q, qNt .

Equations (2.11), and (2.12) are the traded sector counterparts of (2.9) and (2.10).

2.2 Production by Firms

There is a continuum of firms of measure unity in each of two sectors. The production technology

for a firm in non-traded and traded goods sector is given by:

YN,t(j) = ANKN,t(j)αHN,t(j)1−α; YT,t(j) = ATKT,t(j)γHT,t(j)1−γ (2.13)

respectively. Productivity parameter AN (AT ) is the same across the firms in the non-traded

(traded) sector. Firms are monopolistically competitive in the non-traded sector, while firms in

the traded sector operate under perfect competition. We assume that all firms rent capital and

labor in perfectly competitive factor markets. Cost minimization implies equations:

Wt = MCN,t(1− α)YN,tHN,t

; RN,t = MCN,tαYN,tKN,t

(2.14)

Wt = PT,t(1− γ)YT,tHT,t

; RT,t = PT,tγYT,tKT,t

(2.15)

where YN,t = ANKN,tαHN,t

1−α and YT,t = ATKT,tγHT,t

1−γ are aggregate supply functions of

non-traded and traded goods4. Demand for labor and capital in the non-traded goods sector

is described by equations (2.14), where MCN,t represents the (nominal) marginal costs in that

sector. It is noteworthy that the marginal costs in the non-traded sector are identical across firms

as long as their production functions exhibit constant returns to scale and prices of inputs are3Adjustments are costs stemming from decreasing the capital stock. The installation function φNtKNt

as a function of INt shifts upwards as KNt decreases, which is represented by ∂/∂KNt+1(φNt+1KNt+1) =

−φ′Nt+1IN,t+1/KNt+1 + φNt+1 in (2.9).4With respect to aggregation in the non-traded sector, in the technical appendix I show that the non-traded

market equilibrium equation has an additional term that deals with the distribution of prices in the non-traded

sector. However, as shown in the appendix (see also Erceg et. al. (2000) and Christiano et. al. (2001)), this term

does not appear in the log-linear approximation of the resource constraint in the non-traded sector.

7

fully flexible in perfectly competitive markets. Producers of the traded goods are price-takers,

so that equations (2.15) describe the demand for labor and capital inputs in the traded sector,

with PT,t representing the unit cost of production.

2.3 Price setting in the non-traded sector

In order to describe the price setting decisions we split firms into two groups according to their

pricing behavior, following Steinsson (2003). In each period, each firm changes its price with

probability 1 − θN , and otherwise, with probability θN , its price will rise at the steady state

rate of inflation ΠN,t = PN,t/PN,t−1. Among those firms which reset their price, a proportion of

1−ω are forward-looking and set prices optimally, while a fraction ω are backward-looking and

set their prices according to a rule of thumb.

Forward-looking firms are profit-maximizing and reset prices (PFN,t) optimally, which in terms

of log-deviations from the steady state implies:

pFN,t = mcN,t + Et

∞∑i=1

(θNβ)i[mcN,t+i − mcN,t+i−1] + Et

∞∑i=1

(θNβ)iπN,t+i (2.16)

where πN,t is inflation in the non-traded sector and mcN,t stands for the deviation of real

marginal costs from its steady state. As discussed in Christiano et al. (2001), relation (2.16)

shows several important features of the behavior of the forward-looking firms. When firms

expect real marginal costs to be higher in the future and/or expect future increases in the price

level, then the firms set pFN,t higher than mcN,t. Christiano et al. (2001) describe this behavior

as ‘front loading ’. Firms understand that they might not be allowed to change their price when

higher real marginal costs or higher prices materialize. So, anticipating this, forward-looking

firms set prices to maximize their current and future profits, taking into account the future

evolution of real marginal costs and prices. Further, for derivation of the Phillips curve below,

it is convenient to re-write relation (2.16) as:

pFN,t = (1− βθN )mcN,t + βθN pFN,t+1 + βθNπN,t+1 (2.17)

Backward-looking firms set their prices according to the following rule:

PBN,t = P rN,t−1ΠN,t−1(YN,t−1

Y nN,t−1

)ϑ (2.18)

8

where ΠN,t−1 = PN,t−1/PN,t−2 is the past period growth rate of prices in the non-traded sector,

YN,t−1/YnN,t−1 is output relative to the flexible-price equilibrium, P rN,t−1 is an index of prices set

at date t−1, given in terms of log-deviations from the steady state (see Technical appendix) by:

prN,t−1 = (1− ω)pFN,t−1 + ωpNN,t−1 (2.19)

The rule of thumb (2.18) shows that backward-looking firms set their prices equal to the

average of the newly set prices in the previous period updated by the previous period inflation

rate of the non-traded goods price level and by the deviation of the non-traded goods output

relative to the flexible price equilibrium non-traded output. This assumption, as discussed in

Galı and Gertler (1999) has the following appealing properties: first, the rule of thumb behavior

converges to the optimal behavior over time; and second, PBN,t depends only on information up

to the period t − 1, but implicitly incorporates past expectations about the future, since the

price index P rN,t−1 is partly determined by forward-looking price setters.

For the whole non-traded sector, the price index in the non-traded sector is given by:

PN,t = [(1− θN )(1− ω)(PFN,t)1−λ + (1− θN )ω(PBN,t)

1−λ + θN (ΠNPN,t−1)1−λ]1

1−λ (2.20)

Following Steinsson (2003), we can derive the following Phillips curve for the non-traded

sector, written in terms of log-deviations from the steady state5:

πN,t = χfβEtπN,t+1 + χbπN,t−1 + κ1yN,t−1 + κ2yN,t + κmcmcN,t (2.21)

where the coefficients are:

χf =θN

θN + ω(1 + θNβ − θN ), χb =

ω

θN + ω(1 + θNβ − θN ),

κ1 =ωϑ(1− θN )

θN + ω(1 + θNβ − θN ), κ2 = − βθNωϑ(1− θN )

θN + ω(1 + θNβ − θN ),

κmc =(1− βθN )(1− ω)(1− θN )θN + ω(1 + θNβ − θN )

. (2.22)

and where yN,t is the output gap in the non-traded sector, defined as the deviation from the

flexible price output of the non-traded sector (Y nN,t), and mcN,t are real marginal costs in the

non-traded sector.5A detailed derivation is given in the technical appendix.

9

All coefficients are explicit functions of three model parameters: θN which measures the

degree of price stickiness in the non-traded sector; ω which measures the degree of ‘backwardness’

in price setting, and the discount factor β. Coefficients on the output gaps, κ1 and κ2, also depend

on the elasticity ϑ. In this model, the degree of inflation inertia can be measured by the fraction

of backward-looking firms, as a larger fraction of backward-looking firms implies a higher value

of the coefficient of the lagged inflation, χb,as one would anticipate. Less intuitive is however the

fact that high degrees of price stickiness yields lower levels of inflation persistence in the current

specification, configuration consistent with structural estimates in Galı and Gertler (1999). As

one would anticipate, a rise in ω leads to a fall in the coefficients of current variables, κ2 and

κmc, and to a rise in the coefficient of the predetermined variable, κ1. Moreover, a higher degree

of backwardness in the model implies a lower weight on the currently expected future inflation,

χf . This is because only forward-looking firms react immediately to changes in current market

conditions.

Note that when ω = 0 the Phillips curve collapses to the standard forward-looking specifi-

cation:

πN,t =(1− βθN )(1− θN )

θNmcN,t + βEtπN,t+1 = λNmcN,t + βEtπN,t+1 (2.23)

When ω = 1 the Phillips curve takes the specification:

πN,t =θNβ

1 + θNβEtπN,t+1 +

11 + θNβ

πN,t−1 −1− θN

1 + θNβ[βθNϑyN,t − ϑyN,t−1] (2.24)

As noted in Steinsson (2003), (2.24) has a unique bounded solution:

πt = πt−1 + (1− θN )ϑyN,t−1 (2.25)

which is a form of the accelerationist Phillips curve and has no forward-looking component.

2.4 Local Currency Pricing

We assume that the price of the traded good is flexible and determined by the law of one price,

PT,t = StP∗T,t where P ∗T,t is the foreign currency price of the traded good, and St is the nominal

exchange rate. The economy is small also in the respect that the economy’s export share is

negligible in the foreign aggregate price index, implying that the foreign price of traded goods

10

is equal to the foreign aggregate price level, and we assume that it is equal to unity, so P ∗T,t =

P ∗t = 1, and PT,t = St. The real exchange rate in the model is defined as et = StP∗t /Pt, so that

the real exchange rate depreciates (appreciates) when et rises (decreases).

2.5 Monetary policy regimes

In this paper I consider two alternative monetary policy regimes: inflation targeting and ex-

change rate targeting regimes. As may be argued, they are two groups of two alternative

exchange rate regimes for a small open economies, which comprise a continuum of variants of

the regimes within each group. I have chosen to consider one variant from each group: strict

exchange rate targeting, which corresponds to the traditional fixed exchange rate regime; and

inflation targeting regime, characterized by simple Taylor rule. So, both regimes are assumed to

be implemented via interest rate rules. The inflation targeting regime is defined through Taylor

rule:

log(1 + it1 + i

) = ρY log(YNtY nN,t

) + ρπ log(1 + πN,t1 + πN

) (2.26)

where πN is the target for the annual inflation in the non-traded sector, i is the stationary value

of the interest rate, and YN,t/Y nN,t is output of the non-traded sector relative to its flexible-price

equilibrium6. Log-linearization of the feedback rule yields:

it = ρY yN,t + ρππt (2.27)

where ρY > 0, and ρπ > 1 are the reaction coefficients on non-traded goods inflation and yN,t is

the output gap in the non-traded sector.

Alternatively, monetary policy is characterized by the following interest rate rule that delivers

a fixed exchange rate:

1 + it = (1 + iFt )(St

S)ωS (2.28)

where ωS > 0, and St = S ∀t. Under this rule, the monetary authority pegs the nominal

exchange rate at a target level S in all periods by varying the nominal interest rate in reaction6The model collapses to the flexible price equilibrium in the case when firms change their price with probability

one, that is θN = 0. In flexible price equilibrium there is no distinction between backward-looking and forward-

looking firms, product markets are monopolistically competitive and firms set prices as a mark-up over marginal

costs.

11

to movements in the foreign interest rate and deviations of the nominal exchange rate from the

target7.

2.6 Equilibrium

In each period, the non-traded goods market must clear:∫ 1

0Y SNt(j)di = ∆tY

DNt (2.29)

where ∆t is a measure of relative price dispersion in the non-traded goods sector8 and Y DNt is

the aggregate demand of non-traded goods in the economy, defined as:

Y DNt ≡ a(

PNtPt

)−ρ[Ct + φN,tKNt + φT,tKTt] (2.30)

The aggregate balance of payments condition of the economy is given by:

ATt + (1 + iFt )Dt +κ

2(Dt+1 −D)2 = Dt+1 + YTt +Ot (2.31)

In addition, labor market, capital market, domestic loan market conditions must be satisfied:

HXt +HNt = Ht (2.32)

KSN,t = KD

N,t KSX,t = KD

X,t (2.33)

Bt = 0 (2.34)

3 Calibration

In calibrating the model one period is meant to be one quarter. The parameter choices of the

model are described in Table 1. I set the following parameters of the utility function: σ, the

inverse intertemporal elasticity of substitution in consumption, equal to unity9; the value of7Benigno et. al. (2007) show that such kind interest rate rule ensures determinacy of the exchange rate and

the real economy. Such rules produce equality between the domestic and foreign interest rate endogenously in all

periods as a feature of the rational expectations equilibrium.8See technical appendix for more details on this.9Ostry and Reinhart (1992) provides an estimate of σ for a group of Asian countries at 0.8. Gali and Monacelli

(2005) assume log-utility of consumption, which also implies a unit intertemporal elasticity of substitution.

12

coefficient on labor η = 1, and ψ = 0.4510, so that elasticity of the labor supply is 2.22. The

elasticity of substitution between non-traded and traded goods (ρ) is set to 1.11

The value of a, the share of non-traded goods in CPI, is set to equal to 0.65, which implies

the steady state share of non-traded goods in GDP is 45 percent. The depreciation rate is

set at 10 percent per annum, a standard value in the business cycle literature. The value of

the adjustment cost parameter, ψI is set at 0.1. This is consistent with empirical estimates

of the adjustment cost parameter in the literature, although these estimates are for developed

countries12.

The steady-state real interest rate faced by the small economy in international markets is

set at 11 percent per annum, with a world interest rate r∗ of 4 percent and a country premium

of 7 percent. These parameters yield a value of the subjective discount factor, β, of 0.973. The

steady state value of oil income, O, was chosen such that oil transfers constitute 25 percent of

GDP. I also set the steady state value of foreign borrowing equal to 60 percent of GDP. The

elasticity of substitution between differentiated goods is set to equal 11, which implies a steady

state mark-up of 10%. This is within the range suggested by the literature.13

In simulations, perhaps the most important variable is ω, the proportion of rule-of-thumb

price setters. As 0 6 ω 6 1, our model nests the purely forward-looking new Keynesian model

(ω = 0), the purely backward-looking acceleration model (ω = 1), as well as other models such as

in Fuhrer and Moore (1995) (0 < ω < 1). Empirical studies conclude that an empirical Phillips

curve has a statistically significant backward-looking component. The estimates of coefficients

on ‘backwardness’ and ‘forwardness’, however, vary widely among studies. Backward-looking

behavior is of limited quantitative importance in the estimated specifications of Galı and Gertler

(1999) and Benigno and Lopez-Salido (2006). Mehra (2004) finds an extremely backward-looking

specification of the Phillips curve, while Fuhrer and Moore (1995) claim that an equal weight on10Uribe and Yue (2006) also set ψ = 0.4511Ostry and Reinhart’s estimates of the parameter for Asian and Latin American countries equal 0.655 and

0.76 respectively, using one set of instruments, and 1.15 and 1.1 for a different set of instruments respectively.12Hall (2002) estimates a quadratic adjustment cost for capital and finds a slightly higher value of 0.91 for ψI ,

on average, across industries. A much closer value of 0.096 is found recently by Groth (2005) for estimates of

capital adjustment costs for UK manufacturing covering the period 1970-2000.13Gali (2003) sets λ = 11 as well, while in Gali and Monacelli (2005) the value of this parameter is equal to 6.

The empirically plausible range of 10% - 40% for markups, as Gali et. al. (2001) discuss, yields similar results.

13

forward and backward inflation terms matches the pattern of US data much better than either

a purely backward-looking or purely forward-looking model.

To analyze how backward-looking behavior affects adjustment dynamics in response to the

shock compared with forward-looking behavior, I consider two limiting cases. In the first case,

inflation exhibits very little persistence and price setters are almost completely forward-looking,

assuming ω = 0.01. In the second case, inflation is almost completely persistent and I set

ω = 0.9. These values of ω, with the appropriate choice of elasticity ϑ, imply the values of

structural parameters of the Phillips curve, reported in Table 2. I discuss the sensitivity of the

main results of the paper to the choice of the parameter ω in Section 4.3.

To calibrate the parameter ϑ, I follow Steinsson’s procedure, which is as follows. The value

of ω varies between zero and one. So, when ω → 0, it collapses to the familiar forward-looking

specification πN,t = λNmcN,t + βEtπN,t+1, whilst when ω → 1, it collapses to πt = πt−1 + (1−

θN )ϑyN,t−1, which is an accelerationist Phillips curve. In calibrating ϑ, Steinsson assumes that

demand pressure is the same across these two extreme cases, i.e. λN = (1 − θN )ϑ. Chosen in

this way, ϑ = (1− βθN )/θN = 0.36.

I follow the literature in setting θN = 0.75, which implies that, on average, prices last for one

year. The micro analysis on price stickiness, however, gives mixed results. Golosov and Lucas

(2007) suggest that firms change prices on average about 2 quarters. Baudry’s et al. (2004)

analysis suggests 3 quarters. The sensitivity of the simulation exercises to the choice of the

parameter θN is analyzed after I present the results.

4 Simulations

This section presents simulations results of the model for a temporary 1 percentage improvement

in the price of oil, under forward-looking and backward-looking Phillips curves, and under two

different exchange rate regimes).

4.1 Inflation targeting regime

Figure 1 displays impulse responses to the shock for ω = 0.01 and ω = 0.9 under inflation

targeting regime.

14

The economic rationale behind transmission of the shock to the economy is similar in many

ways to the conventional Dutch disease story of the response to the resource boom. Specifically,

a rise in the real income, stemming from the increased price of oil, leads to an increase in demand

for all goods, including non-traded and traded goods. As a result, consumption rises on impact

of the shock. In response to positive terms of trade shock, real exchange rate appreciates. Trade

balance responds positively to an improvement in the terms of trade, as well as the current

account.

Increased demand for non-traded goods pushes demand for labor up. As no imperfections

exist in the labor market, the nominal wage must increase. Due to the price stickiness, this

generates an increase in the real wage. As real marginal costs increase on the impact of the

shock, forward-looking firms – those able to reset their prices – find it profitable to set a price

above the average price of the previous period, which causes an increase in the price of non-traded

goods and inflation in that sector. The higher proportion of forward-looking firms in an economy,

the more initial rise in inflation, as indicated by the figure. This is because backward-looking

firms do not react contemporaneously to unexpected shocks. As a result, when there is high

proportion of backward-looking firms in the economy, inflation rises only slightly initially and

starts picking up in subsequent periods following the shock, as more and more backward-looking

firms adjust their prices. This results in hump-shaped behavior of inflation.

An increase in the price of the non-traded goods relative to the prices of traded goods

causes producers to reallocate production from non-oil traded goods to non-traded goods, so

that output in the non-traded sector rises, and output of the traded sector contracts on the

impact to the shock.

Under both cases the central bank responses to the shock by increasing the (nominal) interest

rate. The real interest rate also rises, given the Taylor principle. This monetary tightening helps

to suppress demand, directly via the real interest rate effect and indirectly via its effect on the

real exchange rate. Both of these effects curb inflation. There is a corresponding increase in the

output gap, which causes additional monetary tightening and reinforces the interest rate effect.

One further point about the behavior of the real interest rate and inflation is worth noting.

Despite very different dynamic responses of inflation and the (nominal) interest rate in the two

cases, the real interest rate and other real variables responses are very similar. This is because

15

nominal interest rates closely follows the dynamics of inflation, and hence the real interest rate

reaction in case of the accelerationist Phillips curve has the same pattern of behavior as in the

case of the forward-looking Phillips curve. So, both real interest rates rise on the impact of the

shock and then gradually converge to zero in the long-run. Given similar variations in the real

interest rate, it is not surprising that the reaction of real variables are very similar too.

This is because, as discussed in Woodford (2003), such a Taylor rule policy, with character-

istics above, would succeed in stabilizing inflation. As a result, the flexible price equilibrium

is replicated here in the sense that real variables respond to the oil shock in a similar way to

reactions of variables in the flexible-price model. As shown further in Leitemo and Røisland

(2002), the simple Taylor rule also performs well in comparison with more complex targeting

rules.14 Within two-sector estimated model of a small open economy, they find that the Taylor

rule outperforms all targeting regimes (and comes close in matching the performance of flexible

inflation targeting regime) in terms of expected social loss.

4.2 Exchange rate targeting regime

In this section I report results from similar simulations with a exchange rate targeting regime.

The capital-mobility condition is then simply it = iFt . Now a country must give up an inde-

pendent monetary policy to keep the exchange rate fixed. The relevant impulse responses are

summarized in Figure 2 below.

There are two important features of impulse response functions in the exchange rate targeting

regime, compared to inflation targeting regime, which are worth noting. First, the behavior of

the real exchange rate is hump-shaped when ω = 0.01; Second, when there is high proportion of

backward-looking firms, the dynamic adjustment of series is slow and cyclical.15

The hump-shaped behavior of the real exchange rate is a result of the country’s commitment

to the fixed exchange rate regime. In the case of inflation targeting regime, for instance, the14They consider strict and flexible versions of both inflation targeting and exchange rate targeting regimes. In

the strict version, the objective is to minimize the variability of the target variable, while in the flexible version,

output and the change in the interest rate are targeted in addition to the primary target variable.15Leitemo and Røisland (2002) also find that both strict and flexible versions of exchange rate targeting regimes

generate oscillations and slow convergence to equilibrium, with flexible exchange rate targeting generating less

pronounced oscillations.

16

nominal exchange rate carries out the burden of adjustment to the shock, resulting in a smooth

response of the real exchange rate. In contrast, with a fixed exchange rate, prices do all the

adjustment, which causes equivalent changes in the real exchange rate. But because prices are

sticky, an equivalent change in the real exchange rate would not be quick and immediate. This

results in the hump-shaped behavior of the real exchange rate.

The adjustment dynamics of the series, including the real exchange rate, changes considerably

and become oscillatory when there is high proportion of backward-looking firms in the economy.

The mechanism behind this phenomenon is the following. As the output gap is positive on the

impact of the shock, inflation starts rising gradually (because of inertia) and the real interest

rate starts falling for some time. This falling real interest rate would cause further output gains

which could push prices further up and in turn cause the real interest rate to fall further, and

so on. This could be destabilizing. This destabilizing real interest rate mechanism has become

known as ‘Walters critique’ (Walters, 1998).16

However, the destabilization does not occur in the economy, because of the stabilizing effect

of the real exchange rate channel. Since nominal exchange rate is fixed, the gradual increase in

inflation leads to a gradual real exchange rate appreciation or an increase in the price of non-

traded goods. A gradual rise in the price of non-traded goods eventually reaches a level high

enough to reduce demand and to cause the output gap return to zero. At this point the cycle

in inflation dynamics turns, because the dampening effect of the real exchange rate more than

offsets the expansionary effects of a lower real interest rate. However, at this point prices are still

rising (and real exchange rate continues to depreciate) because of the rule-of-thumb behavior

of backward-looking firms. This will lead to a further decline in the demand for non-traded

goods, which will cause a further fall in the output gap, which is now below its potential level.

The output gap then falls substantially causing a fall in the price of non-traded goods, and thus

leading to a decline in inflation so that inflation reaches its long-run equilibrium level. However,

at this point the output gap is below its potential level, so with an accelerationist Phillips

curve17 it is necessary for the inflation rate to fall further, and inflation will start falling below16The destabilizing real interest rate effect first has been noticed by Wicksell (1907). Walter’s mechanism is

similar and it happens when an inflationary economy entering a fixed exchange rate regime is forced to reduce

nominal interest rates, which in turn will reduce real interest rates, stimulate the economy and exacerbate inflation.17Under accelerationist Phillips curve, πt = πt−1 + ςgapt inflation tends to rise when output is above potential

17

the steady state level. As inflation starts falling below zero, real interest rate starts increasing

and real exchange rate starts depreciating. Eventually when the expansionary effect of a real

exchange rate depreciation more than offsets the dampening effect of a high real interest rate,

the cycle in inflation dynamics turns again. At this point output gap returns to zero. However,

at this point prices of the non-traded goods are still falling and exchange rate continues to

depreciate, and so on. This explains the oscillatory response of the economy to an oil shock in

the case of a fixed exchange rate regime. As in the case of the forward-looking Phillips curve

discussed earlier, in the absence of a ‘shock absorbing’ role of the nominal exchange rate, the

real exchange rate facilitates adjustment to the shock, resulting in an oscillatory reaction.

In addition, the presence of forward-looking firms (though a small proportion) can reinforce

the stabilizing real exchange rate effect on the economy. This is because forward-looking con-

sumers know that, as prices of non-traded goods are anchored by the price level outside the

country, and if prices rise they would need to fall again, which would in due course cause the

real interest rate to increase again. That would lead them to reduce their expenditure and thus

demand. Thus, forward-looking nature of the price setting behavior helps to prevent instability

coming from the real interest rate channel. The higher proportion of forward-looking firms, the

lower consequences of the destructive Walters effect and a more smooth adjustment path. That

is why, in most earlier papers on monetary policy in small open economies the ‘Walters critique’

problems are absent, because these papers assume forward-looking wage and price setters.

Oscillatory responses of the variables to the shock are not desirable because they imply high

welfare losses for the economy. In this paper I do not perform welfare analysis. However, within

the context of the reduced-form structure of the macromodels, Leitemo (2002) and Leitemo

and Røisland (2002) analyze the performance of alternative monetary policy regimes and the

role of inflation persistence for the ranking of different regimes. As they found, the presence

of the Walters effect is the main reason why exchange rate targeting regime underperforms

compared with flexible inflation targeting and Taylor rule with respect to providing stability of

the real exchange rate, and why it is associated with the highest social loss among all considered

growth- and level-targeting exchange rate regimes when inflation is persistent. Exchange rate

targeting regime, on the other hand, achieves the greatest improvement in terms of social loss

and fall when it is below.

18

(Leitemo (2002)), with a more forward-looking Phillips curve as the Walters effect becomes less

pronounced in that case (as has been also discussed above).

4.3 Sensitivity analysis

A key finding so far is that oil price shocks may produce movements in real interest rates

that generate cycles and instability when there is significant backward-looking element in the

inflation dynamics and exchange rate is fixed. Such cyclicality in series is moderated if there

is high proportion of forward-looking price setters, so that when the Phillips curve becomes

completely forward-looking cyclicality in adjustment paths disappears and response of the real

exchange rate becomes hump-shaped.

The elements of the model that are crucial for determining the degree of ’backwardness’ in

inflation dynamics are the fraction of backward-looking firms (ω) and the degree of stickiness

in the price level (θN ), as can be seen from (2.22). So, now I analyze how the value of these

parameters affect the key findings of the paper. First, I want to quantify the effect of the degree

of inflation inertia in the Phillips curve on the ‘cyclicality’ result of the paper. As high degree of

inflation persistence generates cycles, which disappear in case of forward-looking Phillips curve,

the obvious question is to ask: what is the critical value of ω that generates cycles? Numerical

simulations of the model reveal that, under the baseline calibration, the series start oscillating

for ω ≥ 0.3.

Now I consider changes in the degree of stickiness in the price level (θN ). As noted before,

empirical literature on price stickiness provides mixed evidence. As Golosov and Lucas (2007)

suggest that firms change prices on average about 2 quarters, Baudry’s et al. (2004) analysis

suggests 3 quarters, and Aucremanne and Dhyne (2004) find a value of 5, I compare simulation

results for the values of θN : 0.3, 0.5, and 0.75 (benchmark specification), which are summarized

in Figure 3.

As the figure illustrates, for fixed ω, decreasing the price stickiness generates larger fluctua-

tions in series and faster adjustment to the shock. The intuition is clear: the more flexible are

prices, the more prices adjust to the shock initially and inflation rises more. And because of

the higher proportion of the firms, which can change prices to the new optimal level, the faster

disappearance of the cycles and so as the adjustment to the shock.

19

Finally, I should discuss how the result of the paper are robust to the choice of the shock that

hits the economy. Although in this paper I study the dynamic adjustment of the economy to

the oil price shock, the cyclicality behavior of the series emerge if to consider other shocks, such

as productivity shocks, or preference shocks. The key elements of the model which generate this

result are strict exchange rate targeting regime and high degree of inflation persistence. The

focus on oil price shocks was motivated by two reasons. First, the purpose was to contribute to

the literature on study of dynamic behavior of a small open economy subject to terms of trade

shocks, by drawing attention to the possibility of the ‘Walters critique’ effect under exchange rate

targeting regime. Second, given uncertainty on the actual degrees of inflation persistence and

price stickiness, the model provides some simulated estimates of the critical values of degree of

inflation persistence depending on the Calvo parameter, which can be used as general guidelines

in setting monetary policy in oil producing countries.

5 Conclusion

This paper studies implications of inflation persistence on the dynamic adjustment to the terms

of trade shock in a small oil producing economy with endogenous inflation persistence. It is

shown that the country’s adjustment paths are slow and cyclical if there is a significant backward-

looking element in the inflation dynamics with a strict exchange rate targeting regime. Such

adjustment dynamics are moderated if there is a higher proportion of forward-looking price

setters. In the case of an almost entirely forward-looking Phillips curve, the response of the real

exchange rate becomes hump-shaped. The reason is that with a fixed exchange rate, prices do all

the adjustment, which causes equivalent changes in the real exchange rate. But because prices are

sticky, an equivalent change in the real exchange rate would not be quick and immediate, which

results in the hump-shaped behavior of the real exchange rate. In contrast, with an independent

monetary policy, a freely floating exchange rate carries out the burden of adjustment, which

results in a smooth response of the real exchange rate.

Obviously, the objective of this paper has been to provide detailed analysis of the effects of

inflation inertia on dynamic adjustment to the shocks under alternative exchange rate regimes:

inflation targeting and exchange rate targeting regimes. Although the results suggest that fixed

20

exchange rates can be potentially disadvantageous in face of the real shocks in a small open

economy, the merits of alternative exchange rate regimes should be evaluated by using utility-

derived welfare function. Such welfare-based analysis will allow study the benefits of different

exchange rate regimes under different assumptions about price-setting behavior. This could be

an interesting extension of the paper which I leave for future research.

Acknowledgements I would like to thank David Vines, Florin Bilbiie, Chris Adam, Rick van

der Ploeg and anonymous referees for providing helpful comments and suggestions to previous

versions of the paper. I have greatly benefited from presenting the paper at XV International

”Tor Vergata” conference on Banking and Finance, 11th International Conference on Macroeco-

nomic Analysis and International Finance and seminars at the University of Oxford. The paper

has been awarded as the best young economist paper presented at the ”Tor Vergata” conference.

All errors and omissions are my own.

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6 Appendix

Table 1 Calibration of the model

23

Parameter Value Description

σ 1 Inverse of elasticity of substitution in consumption

β 0.973 Discount factor

ρ 1 Elasticity of substitution between non-traded and traded goods in Ct

η 1 Coefficient on labor in utility

ψ 0.45 Inverse of elasticity of labor supply (1/ψ = 2.22)

γ 0.65 Share of capital in traded sector

α 0.25 Share of capital in non-traded sector

δ 0.025 Quarterly rate of capital depreciation (same across sectors)

a 0.65 Share on non-traded goods in CPI

ψI 0.1 Investment adjustment cost (same across sectors)

κ 0.0001 Portfolio adjustment costs

θN 0.75 Probability of fixed price

ϑ 0.36 Elasticity of deviation of the output gap in the rule-of-thumb

λ 11 Elasticity of substitution between differentiated goods

ω 0.01 and 0.9 Proportion of backward-looking firms

ρY 0.5 coefficient on output gap in Taylor rule

ρπ 1.5 coefficient on inflation in Taylor rule

24

Table 2 Values for the structural parameters of the Phillips curve for different values of ω

ω χf χb κ1 κ2 κmc

0.01 0.98 0.013 0 0 0.087

0.9 0.46 0.55 0.05 -0.036 0.004

25

7 Technical appendix (not to be published)

7.1 Production

In this subsection we focus on the structural equations arising from the representative firm‘s

decision problem that concern production, labor and capital demand. In the next subsection

we focus on pricing implications. So, we think about breaking the firm into two separate parts

for planning purposes: production unit and pricing unit. First, the production unit takes, given

the output level of the firm and the rental price of capital and wage rate (each firm operates

under perfect competition in the inputs markets). Thus, each firm j in the non-traded sector

determines its labor and capital demand so as to minimize its total costs. Second, the pricing

unit determines the price of the firm‘s output, taking into account costs of production and

demand conditions for the output.

minKN,t(j),HN,t(j)

WtHN,t(j) +RN,tKN,t(j)

subject to YN,t(j) = ANKαN,t(j)L

1−αN,t (j).To derive the first order conditions for firms optimiza-

tion problem we write Lagrangian for the firm j in the non-traded sector as:

L = (WtHN,t(j) +RN,tKN,t(j)) + λft (YN,t(j)−ANKαN,t(j)H

1−αN,t (j))

so the first order conditions are:

∂L

∂Ht(j)= Wt − λftAN (1− α)Kα

N,t(j)H−αN,t(j) = 0 (7.1)

∂L

∂Kt(j)= RN,t − λftANαK

α−1N,t (j)H1−α

N,t (j) = 0 (7.2)

∂L

∂λft= YN,t −ANKα

N,t(j)H1−αN,t (j) = 0 (7.3)

The lagrangian multiplier on the constraint is interpretable as nominal marginal cost, so

λft = MCN,t(j), and (7.1)-(7.3) correspond to the following:

Wt = MCN,t(j)(1− α)YN,t(j)HN,t(j)

(7.4)

26

RN,t = MCN,t(j)αYN,t(j)KN,t(j)

(7.5)

YN,t(j) = ANKαN,t(j)H

1−αN,t (j) (7.6)

In the traded sector we assume perfect competition and flexible prices, so the cost of unit

of production in that sector MCT,t = PT,t, such as the counterpart of equations (7.4)-(7.6) in

traded sector is:

Wt = PT,t(1− γ)YT,tHT,t

(7.7)

RT,t = PT,tγYT,tKT,t

(7.8)

YT,t = ATKγT,tH

1−γT,t (7.9)

7.2 Price setting by non-traded goods firms

Firms in the non-traded sector set their prices as monopolistic competitors. Pricing behavior is

taken as in Rotemberg and Woodford (1997) and Steinsson (2003). We use Calvo (1983) sticky

price specification and assume that the firm j changes its price with probability (1− θN ).That

is, each period there is a constant probability (1 − θN ) that the firm will be able to change its

price, independent of past history18. We also assume that if prices are not reset, the old price

is adjusted by a steady state inflation factor:

ΠN,t = PN,t/PN,t−1 (7.10)

Hence, even if the firm is not allowed to change its price, the latter grows at the same rate as

trend inflation.18Hence, the average time over which a price is fixed is given by (1−θN )1+θN (1−θN )2+...+θi−1

N (1−θN )i+... =

= (1− θN )∞∑i=0

iθi−1N = 1/(1− θN ). Thus, for example, with θN = 0.75 in a quarterly model, prices are fixed on

average for a year.

27

Those who re-set a new price (with probability 1− θN ), are split into backward-looking and

forward-looking firms, in proportion ω, such that the aggregate index of prices set by the firms

is

P rN,t = [(1− ω)(PFN,t)1−λ + ω(PBN,t)

1−λ]1

1−λ (7.11)

Backward-looking firms set their prices according to the rule of thumb:

PBN,t = P rN,t−1ΠN,t−1(YN,t−1

Y nN,t−1

)ϑ (7.12)

As in Galı and Gertler, this rule of thumb has the following features: (1) in a steady state

equilibrium the rule is consistent with optimal behavior, that is PBN = PFN , (2) the price set in

period t depends only on information dated t− 1 or earlier.

7.2.1 Forward-looking price-setters

The problem of the firm j changing price at time t consists of choosing price PnewN,t (j) to maximize:

Et

∞∑i=0

θiNΦt+i[PnewN,t (j)ΠiNYN,t+i(j)− TCN,t+i(j)] (7.13)

subject to the total demand it faces:

YN,t+i(j) = (PnewN,t (j)PN,t+i

ΠiN )−λYN,t+i (7.14)

and where Φt+i is an appropriate stochastic discount factor, θiN is the probability that the price

PnewN,t (j) set for good j still holds i periods ahead, and TCN,t+i(j) represents total (nominal)

costs. The discount factor relates to the way how the households value their future consumption

relative to the current consumption, and we define the discount factor as:

Φt+i = βiΛt+iΛt

= βiP−1t+iC

−σt+i

P−1t C−σt

Cost minimizing behavior of the firm in non-traded sector yields the following expression

for the total costs: TCN,t+i(j) = PN,t+imcN,t+iYN,t+i(j), where mcN,t+i = MCNt+i/PN,t+i

represents real marginal costs.

The FOC of this maximization problem yields the following optimal price:

PnewN,t (j) =λ

λ− 1Et

∑∞i=0(θNβ)iΛt+i(PnewN,t (j)Πi

N/PN,t+i)λPN,t+imcN,t+iYN,t+i

Et∑∞

i=0(θNβΠ)iΛt+i(PnewN,t (j)ΠiN/PN,t+i)λYN,t+i

(7.15)

28

From (7.15) it is clear that all firms that reset their prices in period t, set it at the same level, so

PnewN,t (j) = PnewN,t , for all j ∈ [0, 1], and we could omit subscript j. If we define two new variables

P 1N,t = Et

∞∑i=0

(θNβ)iΛt+i(PnewN,t Πi

N

PN,t+i)λPN,t+imcN,t+iYN,t+i (7.16)

and

P 2N,t = Et

∞∑i=0

(θNβΠn)iΛt+i(PnewN,t Πi

N

PN,t+i)λYN,t+i (7.17)

then (7.15) can be rewritten as:

PnewN,t =λ

λ− 1P 1N,t

P 2N,t

(7.18)

Both P 1N,t and P 2

N,t can be expressed recursively that does away the use of infinite sums, such

that:

P 1N,t = Λt(

PnewN,t

PN,t)−λPN,tmcN,tYN,t + βθNEtP

1N,t+1 (7.19)

P 2N,t = Λt(

PnewN,t

PN,t)−λYN,t + βθNΠNEtP

2N,t+1 (7.20)

Finally, it is necessary to note that the optimal price set by forward-looking firms PFN,t ≡

PnewN,t .

To obtain formula for the price set by forward-looking firms, we log-linearize expressions

(7.16)-(7.18):

p1N,t = (1− βθN )(−σCt − λpnewN,t + pN,t + mcN,t + YN,t) + βθNEtp

1N,t+1 (7.21)

p1N,t = (1− βθN )(−σCt − λpnewN,t + pN,t + mcN,t + YN,t) + βθNEtp

1N,t+1 (7.22)

pnewN,t = p1N,t − p2

N,t (7.23)

where pnewN,t = PnewN,t /PN,t; p1N,t = P 1

N,t; p2N,t = P 2

N,tPN,t. Substituting (7.21) and (7.22) into

(7.23), we obtain the following formula for the forward-looking firms:

29

pFN,t = (1− βθN )mcN,t + βθN pFN,t+1 + βθNπN,t+1 (7.24)

where we turn to the notation pnewN,t ≡ pFN,t.

7.2.2 Aggregate price in the non-traded sector

The price index in the non-traded sector is given by:

PN,t = (∫ 1

0PN,t(j)1−λdj)1/(1−λ) (7.25)

which can be expressed as the average of all prices set i periods ago (in period t − i) that still

hold in period t:

PN,t = (∞∑i=0

(1− θN )θiN (ΠiNP

rN,t−i)

1−λ)1/(1−λ) (7.26)

where (1− θN )θiN is the fraction of firms that last adjusted price (P rN,t−i) i periods ago.

Using (7.11), expression in (7.26) can be rewritten recursively as:

PN,t = [(1− θN )(P rN,t)1−λ + θN (ΠNPN,t−1)1−λ]

11−λ =

= [(1− θN )(1− ω)(PFN,t)1−λ + (1− θN )ω(PBN,t)

1−λ + θN (ΠNPN,t−1)1−λ]1

1−λ (7.27)

Linearization of equation above yields:

πN,t =1− θNθN

[(1− ω)pFN,t + ωpBN,t] (7.28)

where, as before, pFN,t and pBN,t denote percent deviations of PFN,t/PN,t and PBN,t/PN,t respectively,

from their steady-state values of one.

7.2.3 Rule of thumb price-setters and Phillips curve

The rule of thumb price-setters use formula (7.12) to set the new price. The log-linearization of

this equation straightforwardly yields:

pBN,t = prN,t−1 + πN,t−1 − πN,t + δyN,t−1 (7.29)

30

where the log-linearized version of the index of prices set at date t − 1 (prN,t−1) result of log-

linearization of formula (7.11), and given by

prN,t−1 = (1− ω)pFN,t−1 + ωpNN,t−1 (7.30)

and yN,t denotes percent deviation of YN,t/Y nN,t from steady state value of one.

So, now we have equation (7.24) that determines the price set by forward-looking firms,

equation (7.29) that determines the price set by backward-looking firms and equation (7.28),

thus, by doing manipulations similar to Steinsson (2003) (A.1)-(A.6), we eliminate pFN,t−1 and

pNN,t−1and obtain the following specification of the Phillips curve:

πN,t =θNβ

θN + ω(1 + θNβ − θN )EtπN,t+1 +

ω

θN + ω(1 + θNβ − θN )πN,t−1 + (7.31)

+ωϑ(1− θN )

θN + ω(1 + θNβ − θN )yN,t−1 −

βθNωϑ(1− θN )θN + ω(1 + θNβ − θN )

yN,t +

+(1− βθN )(1− ω)(1− θN )θN + ω(1 + θNβ − θN )

mcN,t

Note that when ω = 0 then the Phillips curve collapses to the standard forward-looking

specification:

πN,t =(1− βθN )(1− θN )

θNmcN,t + βEtπN,t+1 (7.32)

When ω = 1 then the Phillips curve takes the specification:

πN,t =θNβ

1 + θNβEtπN,t+1 +

11 + θNβ

πN,t−1 −1− θN

1 + θNβ[βθNϑyN,t − ϑyN,t−1] (7.33)

7.3 Aggregation issue

In this subsection we focus on issues of aggregation in the non-traded sector. From cost mini-

mization problem considered in the previous subsection we get:

Wt

RN,t=

1− αα

KN,t(j)HN,t(j)

⇒ 1− αα

KN,t(j) = HN,t(j)Wt

RN,t(7.34)

Integrating second expression in (7.34) over all firms, and defining

KN,t =

1∫0

KN,t(j)dj; HN,t =

1∫0

HN,t(j)dj; (7.35)

31

we getKN,t(j)HN,t(j)

=KN,t

HN,t(7.36)

The production function for a given good j therefore becomes19

YN,t(j) = ANHN,t(j)(KN,t

HN,t)α (7.37)

which implies that aggregate supply in the non-traded sector is

1∫0

YN,t(j)dj = ANKαN,tH

1−αN,t (7.38)

The demand for each differentiated good j is given by:

Y DN,t(j) = (

PN,t(j)PN,t

)−λ{CN,t + a(PN,tPt

)−ρ[φN,tKN,t + φT,tKT,t]} ≡ (PN,t(j)PN,t

)−λY DN,t (7.39)

so aggregate demand in the non-traded sector is given by:

1∫0

Y DN,t(j)dj = Y D

N,t

1∫0

(PN,t(j)PN,t

)−λdj (7.40)

where

∆t ≡1∫

0

(PN,t(j)PN,t

)−λdj (7.41)

is a measure of relative price dispersion in the non-traded good sector. The steady state value of

the dispersion is unity, while off-steady state it always ≥ 1, and rises with the variance of non-

traded prices. As the equation (7.40) shows the higher variability of prices, given aggregate non-

traded goods output, there will less aggregate consumption of the non-traded goods. Combining

identities (7.38) and (7.40), the overall non-traded goods market equilibrium equation is:

ANKαN,tH

1−αN,t = ∆tY

DN,t (7.42)

where Y DN,t stands for aggregate demand of non-traded goods in the economy. Note that the

aggregation introduces an additional term that deal with distribution of prices of non-traded

goods. However, as shown by Yun (1996), Erceg et. al. (2000) and Christiano et. al. (2001) that19Note that (7.37) in conjuncture with (7.4) can be used to show that nominal marginal costs are identical

across firms in the non-traded sector.

32

term does not appear in the log-linear approximation of the aggregate constraint. We briefly

summarize their argument here for the non-traded sector. For that, we define

PN,t ≡ (

1∫0

P−λN,t(j)dj)−1/λ (7.43)

so that dispersion can be re-written as:

∆t = (PN,tPN,t

)−λ (7.44)

As for the aggregate price level in the non-traded sector, this price aggregate admits a

recursive representation

PN,t = [(1− θN )(P rN,t)−λ + θN (ΠNPN,t−1]−1/λ (7.45)

It can be easily seen that dispersion does not appear in the log-linearized version of the

model. Log-linearization of aggregate price level of the non-traded sector, and the aggregate

price index given by (7.45) yields the following expressions:

PN,t = (1− θN )P rN,t + θN PN,t−1 (7.46)

PN,t = (1− θN )P rN,t + θN PN,t−1 (7.47)

while the log-linearized version of the price dispersion ∆t is

∆t = λ(PN,t − PN,t) (7.48)

So, subtracting expression (7.46) from (7.47), and using expression in (7.48), we get that

∆t = θN∆t−1 (7.49)

implying that if the economy is started from its steady state level, ∆t = 0 for all t, which we

will consider hereafter.

33

0 10 20 30 40 50 600

0.01

0.02

YN

0 10 20 30 40 50 60−0.2

−0.1

0

YT

0 10 20 30 40 50 600.01

0.02

0.03

C

0 10 20 30 40 50 60−0.5

0

0.5

CA

/Y

0 10 20 30 40 50 60−0.01

0

0.01

i

0 10 20 30 40 50 60−0.04

−0.02

0

RE

R

0 10 20 30 40 50 60−5

0

5x 10

−3

Yn g

ap

0 10 20 30 40 50 60−5

0

5x 10

−3

π n

0 10 20 30 40 50 60−5

0

5x 10

−3

real

rat

e

ω =0.01

ω=0.9

Figure 1: Inflation targeting regime: responses to oil price shock

34

0 10 20 30 40 50 60−0.1

−0.05

0

0.05

0.1

YN

0 10 20 30 40 50 60−0.1

−0.05

0

0.05

0.1

YT

0 10 20 30 40 50 600

0.02

0.04

0.06

0.08

0.1

C

0 10 20 30 40 50 60−0.2

−0.1

0

0.1

0.2

CA

/Y

0 10 20 30 40 50 60−0.1

−0.05

0

0.05

0.1

Yn g

ap

0 10 20 30 40 50 60−0.02

−0.01

0

0.01

0.02

π n

0 10 20 30 40 50 60−0.1

−0.08

−0.06

−0.04

−0.02

0

RE

R

ω=0.01

ω=0.9

Figure 2: Exchange rate targeting regime: responses to oil price shock

35

0 10 20 30 40 50 60−0.1

−0.05

0

0.05

0.1

YN

0 10 20 30 40 50 60−0.1

−0.05

0

0.05

0.1

YT

0 10 20 30 40 50 600

0.02

0.04

0.06

0.08

0.1

C

0 10 20 30 40 50 60−0.2

−0.1

0

0.1

0.2

CA

/Y

0 10 20 30 40 50 60−0.1

−0.05

0

0.05

0.1

Yn g

ap

0 10 20 30 40 50 60−0.05

0

0.05

π n

0 10 20 30 40 50 60−0.1

−0.08

−0.06

−0.04

−0.02

0

RE

R

θN=0.75,ω=0.9

θN=0.5,ω=0.9

θN=0.3,ω=0.9

Figure 3: Effect on ’cyclicality’ of varying price stickiness

36