TY - JOUR
AU - Rossi,, Lorenza
AB - Abstract We compare the Calvo and Rotemberg price‐setting mechanisms in a New Keynesian model with trend inflation. We show that: the long‐run relationship between inflation and output is positive in Rotemberg and negative in Calvo; the dynamics of the two models differ even to a first‐order approximation; positive trend inflation enlarges the determinacy region in the Rotemberg model, whereas it shrinks it in the Calvo model; the responses of output and inflation to technology shocks are amplified by trend inflation in Calvo, whereas they are dampened in Rotemberg; the two models imply differing non‐linear adjustments after a disinflation. We consider the two most commonly used approaches to model firms’ price‐setting behaviour within the standard New Keynesian (NK henceforth) framework of monopolistically competitive firms: the Rotemberg (1982) quadratic cost of price adjustment and the Calvo (1983) random price adjustment signal. The Calvo price‐setting mechanism produces relative‐price dispersion among firms, whereas the Rotemberg model is consistent with a symmetric equilibrium. Despite the economic difference between these two pricing specifications, the literature has pointed out that to a first‐order approximation the implied dynamics are equivalent. As shown by Rotemberg (1987) and Roberts (1995), both approaches imply the same reduced form New Keynesian Phillips Curve (NKPC henceforth). They therefore lead to observationally equivalent dynamics for inflation and output. In particular, both models deliver the well‐known result of immediate adjustment of the economy to the new steady state following a disinflation, despite nominal rigidities in price‐setting (Ball, 1994; Mankiw, 2001). Furthermore, Nisticò (2007) shows that up to a second‐order approximation, provided that the steady state is efficient, both models imply the same welfare costs of inflation. Thus, they imply the same prescriptions for welfare‐maximising Central Banks.1 Therefore, to the best of our knowledge, but with some exceptions, there is widespread agreement in the literature that the two models are almost equivalent and that up to a first order they imply the same dynamics. To date, comparison of the two approaches has used models in which the steady‐state level of the inflation rate is equal to zero. In this article, we show that the dynamics of the two pricing models differ both qualitatively and quantitatively when the steady‐state inflation rate is not zero and not fully indexed. Hence, the way in which trend inflation affects the dynamics of a log‐linearised NK model is particularly sensitive to the choice of the price‐setting mechanisms.2 This discrepancy derives specifically from differences between the nominal rigidities that underlie the two models. The Calvo mechanism creates a price dispersion term in the model. The price dispersion term generates a wedge between output and hours and, for its backward‐looking behaviour introduces an inertial mechanism in the model. The Rotemberg model, by contrast, assumes a quadratic cost of changing prices, which generates a wedge between output and consumption without introducing any inertial mechanism. If trend inflation is zero, these two wedges vanish and the two models are equivalent up to first order. Both these wedges, however, are quite sensitive to inflation and they accordingly induce a difference in the two models whenever trend inflation is positive. Further, trend inflation generates an opposite effect on the average markup in the two models. The average markup generally increases with trend inflation in the Calvo model, whereas it decreases with steady‐state inflation in the Rotemberg model. Trend inflation, hence, naturally brings to light the different implications of the two types of nominal rigidity. Five main results follow. First, trend inflation has an opposite effect on the long‐run relationship between inflation and output in the two models. While the long‐run NKPC is negatively sloped in the Calvo model, it is positively sloped in the Rotemberg model. Second, the log‐linear NKPCs implied by the two models are radically different once the model is log‐linearised around a generic steady‐state inflation level. On the one hand, the price dispersion term in the Calvo model generates a backward‐looking variable that is absent in the Rotemberg model. On the other hand, in the Rotemberg model the marginal costs depend on inflation because of the price‐adjustment term. Third, trend inflation has opposite effects on the determinacy conditions of the two models. In contrast to the Calvo model, where an increase in trend inflation shrinks the determinacy region, positive trend inflation enlarges the determinacy area in the Rotemberg model. This means that when we look for the optimal and implementable rules, e.g. Schmitt‐Grohé and Uribe (2007b), the set of the possible rules will depend on the pricing assumption. Rules that can be optimal and implementable under Rotemberg pricing, thus, could prove to be not implementable under Calvo. Fourth, trend inflation has opposite effects on the responses of output and inflation to a positive technology shock. In the Calvo model, the higher trend inflation is, the higher both the decrease in inflation and the increase in output are following a positive technology shock. In the Rotemberg model, the higher trend inflation is, the lower both the decrease in inflation and the increase in output are. Fifth, the two pricing assumptions also imply different dynamics after a disinflation. As some papers have recently shown (Ascari, 2004; Yun, 2005; Ascari and Merkl, 2009) non‐linear simulations are important because the interplay between long‐run effects and short‐run dynamics is crucial in the adjustment path after a disinflation. Contrary to the common view, this interaction leads to differences in results between the respective implied non‐linear dynamics of the Rotemberg and the Calvo model in response to a Central Bank disinflation experiment. Ascari and Merkl (2009) show that in the Calvo model a credible disinflation implies an inertial adjustment (due to the backward‐looking price dispersion term) and leads to a permanently higher level of output in the non‐linear model. The non‐linear dynamics of the Rotemberg model implies that output immediately adjusts to a permanently lower level. To date, both the theoretical and the empirical literature on trend inflation has concentrated on the NK model with Calvo pricing. Ascari (2004) shows that both the long‐run and the short‐run properties of DSGE‐NK models based on the Calvo staggered price model change dramatically in the presence of a trend inflation term. Yun (2005) shows that optimal inflation targets respond to changes in the level of relative price distortion in the presence of initial price dispersion due to trend inflation. Schmitt‐Grohé and Uribe (2007a) find a positive relationship between trend inflation and price dispersion. Amano et al. (2007) numerically study the macroeconomic effects of trend inflation and compare three common time‐dependent pricing schemes: Calvo, truncated‐Calvo and Taylor. They show that, regardless of the price‐setting mechanism, as trend inflation increases, the stochastic means of output, consumption and employment decrease, while that of inflation increases. Moreover, they show that the variability of most aggregate variables increases with trend inflation. Damjanovic and Nolan (2010) show that a contractionary monetary shock has a persistent, negative hump‐shaped impact on inflation and a positive hump‐shaped impact on output. They quantify the utility cost of price dispersion and its impact on optimal monetary policy. Overall, the theoretical literature shows the importance of trend inflation, demonstrating that the results obtained when the model is log‐linearised around a zero inflation steady state can be quite misleading. Moreover, the assumption of non‐zero trend inflation is supported by the empirical evidence. First, a low and positive trend inflation seems to be much more realistic, as the post‐war economic history of industrialised countries shows. Furthermore, the practice of many Central Banks suggests that a zero inflation steady state is not an actual target (Primiceri, 2006; Sargent et al., 2006). Ireland (2007) estimates an NK model characterised by Rotemberg pricing and draws inferences about the behaviour of the Federal Reserve's unobserved inflation target. The paper shows that trend inflation rose from 1.25% in 1959 to over 8% in the mid‐to‐late 1970s before falling back below 2.5% in 2004. Trend inflation is also important in explaining the persistence of the inflation process, as shown in Benati (2008). Using data from various countries, Benati (2008) shows that persistence has fallen whenever countries have adopted an explicit inflation target and thus reduced the average level of inflation. Cogley and Sbordone (2008) estimate a purely forward‐looking Phillips Curve, based on Calvo pricing, that allows for shifts in trend inflation. They find that it successfully describes US inflation dynamics. Benati (2009) finds similar results for other industrialised countries. Cogley et al. (2010) show that the persistence in the inflation gap increased during the Great Inflation and declined after the Volcker disinflation. The main reason behind this shift in inflation volatility and persistence is the stability of the Fed's long‐run inflation target. Thus, the empirical evidence suggests that the inflation gap is a purely forward looking variable, while the main source of inflation persistence is related to trend inflation. Finally, from a normative perspective, some recent papers (Kimbrough, 2006; Damjanovic et al., 2008; Levine et al., 2008) show that the optimal trend inflation may become positive in a NK model with Calvo pricing.3 Similarly, Schmitt‐Grohé and Uribe (2004) and Adam and Billi (2008, 2010) find positive values for the optimal trend inflation with Rotemberg (1982) pricing. In particular, Schmitt‐Grohé and Uribe (2004) study optimal Ramsey monetary and fiscal policy in an NK model with money and sticky prices à laRotemberg (1982). In this setup, the long‐run inflation target from highly negative approaches zero and turns positive in a model with high monopolistic distortions. More recently, Adam and Billi (2008, 2010) show that when the Central Bank plays Nash with the fiscal authority the optimal trend inflation is positive in an NK model with Rotemberg pricing, even with small monopolistic distortions. Therefore, the NK literature cannot disregard the role of trend inflation. The literature on trend inflation, however, has so far focused mainly on staggered price models (Calvo or Taylor). Despite the fact that the Rotemberg (1982) model of price rigidity is widely employed in the NK literature, the said literature has neither assessed the effects of trend inflation in such a framework, nor compared the steady‐state properties as well as the dynamics of the Calvo and the Rotemberg model once trend inflation is taken into account. This is what we undertake in this article. The article is organised as follows. Section 1 describes the basic NK model under the two‐pricing assumptions. Section 2 presents the log‐linear approximation of the models around the special case of a zero inflation steady state. Section 3 compares the long‐run properties and the dynamics of the two pricing‐models under a generic value of trend inflation. Section 4 concludes. 1. The Basic Model In this Section, we briefly present a very simple and standard cashless NK model in the two versions of Rotemberg and the Calvo price‐setting scheme. The model economy is composed of a continuum of infinitely‐lived consumers, producers of final and intermediate goods. 1.1. Households and Technology Consider an economy with a representative household which maximises the following intertemporal separable utility function (1) subject to the period‐by‐period budget constraint (2) where Ct is consumption, it is the nominal interest rate, Bt are one‐period bond holdings, Wt is the nominal wage rate, Nt is the labour input and Πt is the profit income. The following first‐order conditions hold (3) (4) The final good market is competitive and the production function is given by (5) Final good producers’ demand for intermediate inputs is therefore equal to (6) Intermediate inputs Yi,t are produced by a continuum of firms indexed by i ∈ [0, 1] with the following simple linear technology (7) where labour is the only input and  ln At = at is an exogenous productivity shock, which follows an AR(1) process (8) where . The labour demand and the real marginal cost of firm i are therefore (9) and (10) Given our simple linear production function, the real marginal cost is the same across firms and simply equal to the productivity‐adjusted real wage. 1.2. Price Setting: Rotemberg (1982) and Calvo (1983) The intermediate‐good sector is monopolistically competitive and the intermediate‐good producer therefore has market power. We now present the Rotemberg (1982) and the Calvo (1983) price‐setting mechanisms. 1.2.1. The Rotemberg model The Rotemberg model assumes that a monopolistic firm faces a quadratic cost of adjusting nominal prices, which can be measured in terms of the final good and given by (11) where ϕ > 0 determines the degree of nominal price rigidity. As stressed in Rotemberg (1982), the adjustment cost accounts for the negative effects of price changes on the customer–firm relationship. These negative effects increase in magnitude with the size of the price change and with the overall scale of economic activity, Yt. The problem for the firm is then (12) where is the stochastic discount factor, is the real marginal cost function. Firms can change their price in each period, subject to the payment of the adjustment cost. Hence, all the firms face the same problem, and thus will choose the same price, and produce the same quantity. In other words: Pi,t = Pt,Yi,t = Yt and ∀i. Therefore, from the first‐order condition, after imposing the symmetric equilibrium, we get (13) where πt = Pt/Pt−1. As all the firms will employ the same amount of labour, the aggregate production function is simply given by (14) The aggregate resource constraint should take the adjustment cost into account, that is (15) In the following, it is important to note that the Rotemberg adjustment cost model creates an inefficiency wedge, Ψt, between output and consumption4 (16) Note that the inefficiency wedge increases with inflation: the higher inflation, the higher the size of the price change and hence the higher the adjustment costs that firms have to pay. 1.2.2. The Calvo model The Calvo model assumes that for each period there is a fixed probability 1 − θ that a firm can re‐optimise its nominal price, i.e. The price‐setting problem becomes (17) subject to (6), which specialises to (18) The equation for the optimal price is (19) while the aggregate price dynamics is given by (20) In the Calvo price‐setting framework, firms charging prices in different periods will generally have different prices. Thus, the model features a distribution of different prices, that is, there will be price dispersion. Price dispersion results in an inefficiency loss in aggregate production. In fact (21) Schmitt‐Grohé and Uribe (2007a) show that st is bounded below 1, so that price dispersion is always costly in terms of aggregate output: the higher st, the more labour is needed to produce a given level of output. The intuition is that price dispersion causes firms, irrespectively of their symmetry, to charge different prices and thus to produce different levels of output. This in turn decreases the level of aggregate output by Jensen inequality, because the elasticity of substitution among goods is greater than one (Ascari, 1998; King and Wolman, 1999; Graham and Snower, 2004). This is the reason why higher price dispersion acts as a negative productivity shift in the aggregate production function: Yt = (At/st)Nt. Moreover, as we will see, price dispersion is a backward‐looking variable, and introduces an inertial component into the model. To close the model, the aggregate resource constraint is simply given by (22) In the Rotemberg model, the cost of nominal rigidities, i.e. the adjustment cost, creates a wedge between aggregate consumption and aggregate output, because part of the output goes into the price adjustment cost. In the Calvo model, the cost of nominal rigidities, i.e. price dispersion, contrastingly creates a wedge between aggregate hours and aggregate output, thus making aggregate production less efficient and introducing an inertial component into the model. Note that both of these wedges in (16) and (21) are non‐linear functions of inflation. Moreover, they behave very similarly in steady state. Both wedges are minimised at one when steady‐state inflation equals zero, and both wedges increase as trend inflation moves away from zero. It is very important to stress that these wedges can vanish in specific cases. In the Rotemberg model, the wedge Ψt in (16) equals one when inflation is zero, because firms are not changing their prices and thus there is no adjustment cost to pay. In the Calvo model, the wedge st in (21) equals one, when there is no price dispersion, i.e. when all the firms have the same price. There is one special case in which both these conditions hold: the zero inflation steady‐state case 2. A Special Case: Zero Steady‐State Inflation It is well known5 that the two models deliver equivalent dynamics when log‐linearised around a zero inflation steady state. In this case, Calvo‐pricing yields the following NKPC: (23) where lower case hatted letters denote log‐deviations of the variable with respect to its steady‐state value. Similarly, under Rotemberg‐pricing to a first‐order approximation, the NKPC is (24) Therefore, up to a first‐order approximation the two models are identical, except as regards the coefficient of the slope of the NKPC. Note that, by imposing (25) and therefore by setting ϕ = [(ɛ − 1)θ]/[(1 − θ)(1 − βθ)], the two models imply the same first‐order dynamics. Zero trend inflation, however, is a special case. When the steady‐state level of inflation is equal to zero, the difference between the two models cancels out. The reason is that, in this case, the two wedges in (16) and (21) disappear, because in steady state π = s = 1. This is not very surprising, as the zero inflation steady state of both models is equivalent to the steady state of the flexible price version of the model. In all the other, more interesting and realistic cases, the two models entail a different dynamics. The next Section investigates these differences thoroughly. 3. Rotemberg and Calvo under Trend Inflation This Section investigates how the two models differ regarding: the long‐run relationship between output and inflation; the NKPC; the dynamic response to shocks; the determinacy properties; and the dynamic response to a disinflation. Many results will be analytical, whereas some will be visualised through numerical simulations. Calibration: In the figures below, the calibration considers the following standard parameter specification: σ = 1, β = 0.99, ɛ = 10, φ = 1, θ = 0.75, ϕ = [(ɛ − 1)θ]/[(1 − θ)(1 − βθ)], unless explicitly stated otherwise. The persistence of the technology shock (8) is set equal to ρa = 0.95. None of the figures qualitatively depends on the parameter values. 3.1. The Long‐Run Phillips Curve This subsection investigates the non‐linear long‐run Phillips curve implied by the two price‐setting mechanisms. To understand the differences in the dynamics of the two models, it is necessary to analyse their steady‐state properties first. At the root of the different effects of trend inflation on the dynamics of the two price‐setting models lies the fact that trend inflation affects the steady‐state properties of the two models in two differing ways. As we will see, in the Rotemberg model the higher trend inflation is the higher the steady‐state level of output will be, whereas in the Calvo model the opposite holds. 3.1.1. The Rotemberg model The online Appendix A.2 shows that the long‐run Phillips Curve in the Rotemberg model is equal to (26) Online Appendix A.2 proves that (if β < 1) Note that this implies that for i.e. the higher trend inflation is, the more output is produced. Minimum output occurs at a negative rate of steady‐state inflation, unless β = 1, in which case for . The intuition is straightforward if the steady‐state output level is rewritten as (27) where P/MC is the average markup. Equation (27) shows that there are two effects at work as trend inflation changes: the average markup effect, due to time discounting and the wedge effect. Both these effects increase the steady‐state output. First of all, consider the average markup effect: in changing their price, firms weight today's adjustment cost of moving away from yesterday's price, relatively more than tomorrow's adjustment cost of fixing a new price away from today's one, because of discounting. Trend inflation thus reduces the average mark‐up. Indeed, the steady‐state mark‐up is given by (28) which monotonically decreases with (for economically relevant values of Given that the model defines a monopolistic competitive economy, if the mark‐up decreases with trend inflation, it follows that output increases with trend inflation. Secondly, the ‘wedge effect’ relates directly to the adjustment costs term, which is equal to 1/(1 − ADC), where ADC is the steady‐state adjustment cost. The latter increases with trend inflation, because in steady state prices grow at the same rate as that of trend inflation. Thus the larger trend inflation is, the larger the size of the price adjustment, and the greater the wedge Ψ will be. The price adjustment cost, however, has to be paid in terms of aggregate output by the firms, so it has a positive effect on aggregate production. However, a fraction of this production is not consumed, but it is eaten up by the adjustment cost. Given that the wedge between output and consumption (16) increases with trend inflation, consumption decreases with trend inflation in steady state. Thus, output and hours increase with trend inflation, but consumption and welfare decrease with trend inflation. (1) [ Steady State and the Long‐Run Phillips Curve in the Rotemberg Model ] 3.1.2. The Calvo model Figure 2 shows the long‐run relationship between inflation and output in the standard Calvo model. Fig. 2. Open in new tabDownload slide Steady State and the Long‐Run Phillips Curve in the Calvo Model Fig. 2. Open in new tabDownload slide Steady State and the Long‐Run Phillips Curve in the Calvo Model The long‐run Phillips Curve is negatively sloped. Thus, the steady output declines with trend inflation and so does consumption, while aggregate hours increase. As a consequence, steady‐state welfare decreases. To grasp the intuition, it is useful to rewrite steady‐state output level as6 (29) The symmetry between (29) and (27) in the two models makes the comparison clear. In the Calvo model as well there are two effects at work as trend inflation changes: the average markup effect and the wedge effect. However, both these effects decrease steady‐state output. First, in contrast to the Rotemberg model, the average markup increases with trend inflation in the Calvo model. King and Wolman (1996) nicely explain the intuition for this result by decomposing the average markup into two terms: . The authors call the first term, , the price adjustment gap which decreases with trend inflation, because trend inflation mechanically erodes the markups and the relative prices that were set by non‐adjusting firms in previous periods. By contrast, the second term, , is the ratio of the newly adjusted price to marginal cost and is called the marginal markup by King and Wolman (1996). This term is influenced by the forward‐looking behaviour of the price resetting firms. Firms know that they can change their price in the future only with a given probability. Thus, price resetting forward‐looking firms will set higher prices relative to their current marginal costs, exactly to offset the erosion of markups and relative prices that trend inflation creates, if they are not allowed to change their price in the future. The marginal markup term dominates the price adjustment gap one,7 such that higher trend inflation yields a larger average markup, hence a larger monopolistic distortion in the economy and a larger negative effect on the steady‐state output. Second, the wedge effect in (29) is due to price dispersion, s. Price dispersion increases rapidly with trend inflation, so that the negative effect of trend inflation on output through this channel is quite powerful. Both the average markup and price dispersion therefore increase with trend inflation and generate a negative steady‐state relationship between inflation and output.8 Thus, for positive trend inflation, the slope of the long‐run Phillips Curve is positive in the Rotemberg model and negative in the Calvo model. This discrepancy between the two models highlights the difference in the microfoundations of the two pricing frameworks. The nominal rigidity in the Rotemberg model arises from an adjustment cost in the price, and all the firms charge the same price. The core of the Calvo model is the staggered structure of price decisions that generates different prices and thus price dispersion. Therefore, the average markup goes in opposite directions as a function of trend inflation: negative in the case of Rotemberg, positive in the case of Calvo. In Rotemberg, the average markup decreases because of the discounting effect of a standard adjustment cost problem (and this effect is small because discounting is small). In Calvo, the forward‐looking behaviour of the price‐setters and a constant probability of resetting induce the price setters to increase the marginal markup with trend inflation and this increases the average markup. Finally, trend inflation amplifies the two wedges created by the microfoundations of nominal rigidities in both models but the wedges have an opposite effect on output. The adjustment cost increases output because it is a real cost, price dispersion decreases output because it creates an inefficiency in aggregate production. As is shown in Section 2, the fundamental difference in the microfoundations of the two models is wiped out when the model is log‐linearised around a zero inflation steady state, because the steady state in both models is then exactly the same as the flexible price steady state. By contrast, trend inflation makes this difference evident and transparent. Finally, it is worth pointing out that there is obviously another way to remove the effects of trend inflation and thus the difference between the two approaches. As often occurs in the literature, one may assume an ad hoc system of subsidies to make the steady state of the model coincide with the efficient one. As is evident from (16) and (21), the standard subsidy à la Woodford, which is used to remove the steady‐state distortion that derives from the average markup in the basic NK model with zero steady‐state inflation, would not be sufficient to remove the effects of trend inflation. Indeed, such a subsidy would remove the distortion that derives from the average markup effect, so that Another subsidy, however, would be necessary to take care of the wedge effect, so that too. Further, diversely from the standard NK model, which assumes zero steady‐state inflation, the two subsidies would be contingent on trend inflation. Additionally, the system of subsidy would differ in the two pricing approaches, and would be very difficult to justify on any grounds.9 The next subsections show how the opposite slope of the long‐run Phillips Curve between the two models determines their different dynamic properties. 3.2. The Generalised NKPC As we saw above, the relationship between trend inflation and the steady‐state values of the variables is generally non‐linear. We now show how the effect of trend inflation on the coefficients of the log‐linearised equations depends on the specific pricing assumption. 3.2.1. The Rotemberg model The log‐linearisation of (13) yields the following generalised NKPC under Rotemberg pricing (30) where (31) are the log‐linearised real marginal costs and γf, γdy, γmc and ςc are complicated convolution parameters that depend on trend inflation, Equation (30) encompasses the standard NKPC, because, under a zero steady‐state inflation (i.e. ςc = γdy = 0, γf = 1, and γmc = (ɛ − 1)/ϕ, so that (30) becomes (24). Moreover, log‐linearising (3), (4), (14), (16) and combining them together delivers the following log‐linearised IS curve, (32) 3.2.2. The Calvo model As shown by Ascari and Ropele (2009), the log‐linearisation of the Calvo model is described by the following first‐order difference equations:10 (33) (34) (35) (36) where, as explained in Damjanovic and Nolan (2010), is the expected discounted marginal revenue for the firms updating their price in the current period. λ, η, κ and ξ are complicated convolution parameters that depend on trend inflation, Notice that trend inflation alters the inflation dynamics implied by the usual Calvo model in three ways. First, trend inflation enriches the dynamic structure by adding two new endogenous variables: a forward looking variable, and a predetermined variable, which represents price dispersion. Second, trend inflation directly affects the NKPC coefficients. Higher trend inflation makes the NKPC more ‘forward‐looking’, and leads to a smaller coefficient on current output and a larger coefficient on future expected inflation. The short‐run NKPC, hence, flattens when drawn in the plane . Third, trend inflation increases the inertia of the equation of the relative price dispersion This means that, ceteris paribus, higher trend inflation yields a more persistent adjustment of the inflation rate. The two log‐linearised systems present three main differences. First of all, in the Calvo model the presence of a price dispersion wedge creates an endogenous predetermined variable in the NKPC, which is absent in the Rotemberg model. Secondly, in the Rotemberg model, the presence of price adjustment costs causes the real marginal cost to depend additionally on actual inflation (see the additional term in (31)). Finally, the price adjustment cost generates a wedge between output and consumption in the resource constraint, one that appears in the IS curve as the additional term (see (32)). Not surprisingly these differences in the log‐linear model will deliver different dynamic responses and determinacy properties. 3.3. The Dynamics In this subsection, we compare the dynamics of the two price‐setting models. We assume that the Central Bank follows a Taylor‐type feedback rule and we study the responses of output and inflation to a technology shock. It is well known that the dynamics of the two models will be equivalent under zero trend inflation. We here investigate the extent to which the dynamics will differ between the two models as trend inflation varies. We simply assume that the Central Bank sets the short‐run nominal interest rate in accordance with the following standard Taylor‐type rule (37) and we set α = 1.5 and αy = 0.5/4, in the simulation. Figure 3 compares the impulse response functions (IRFs henceforth) of output and inflation to a positive technology shock, for different values of trend inflation, under Rotemberg (panel a) and under Calvo pricing (panel b). Fig. 3. Open in new tabDownload slide Impulse Response Functions to a Positive Technology Shock: Rotemberg Model (Panel a) and Calvo Model (Panel b) Fig. 3. Open in new tabDownload slide Impulse Response Functions to a Positive Technology Shock: Rotemberg Model (Panel a) and Calvo Model (Panel b) As expected, in the Rotemberg model output increases on impact while inflation decreases in response to a positive technology shock. Subsequently, they return to their initial level. Note that, the higher trend inflation is, the lower both the decrease in inflation and the increase in output will be. The effects of varying trend inflation, however, are quantitatively minor. Moreover, the persistence of output and inflation is likewise basically unaffected by the level of trend inflation. As in the Rotemberg model, in the Calvo model output increases and inflation decreases in response to a positive productivity shock. They then return to their initial levels. Actually, the IRFs coincide when the model is log‐linearised around zero inflation. In contrast to what occurs in the Rotemberg model, however, the IRFs are very sensitive to varying trend inflation in the Calvo model. As trend inflation increases, the responses of output and inflation amplify and become more persistent. As shown in Ascari (2004), this happens because of the strong effects that trend inflation has on the coefficient of the NKPC in the Calvo model. Moreover, trend inflation increases the inertia in the dynamic equation of the relative price dispersion which is a predetermined variable.11 This means that, ceteris paribus, higher trend inflation yields a more persistent adjustment of the inflation rate. As a consequence, the response of output becomes more persistent too. In the Rotemberg model, instead, there is no price dispersion and the model is completely forward‐looking. Overall these results show that, if moderate levels of trend inflation are considered, the two models exhibit different dynamics in response to a productivity shock, even to a first‐order approximation. Trend inflation has opposite effects on the adjustment dynamics of output and inflation in the two models. 3.4. Determinacy and the Taylor Principle To assess the determinacy of the rational expectations equilibrium (REE henceforth), we first substitute the Taylor rule (37) into the IS curve and then we write the structural equations in the following matrix format (38) where vector includes the endogenous variables of the model, while at is the technology shock. The determinacy of REE obtains if the standard Blanchard and Kahn (1980) conditions are satisfied. Next, we analyse how trend inflation affects the determinacy of REE. 3.4.1. The Rotemberg model We first present the analytical derivation of our main results under Rotemberg pricing. We then compare our results with those obtained by Ascari and Ropele (2009) for the Calvo model. To derive simple analytical results, in this subsection we will assume that φ = 0, σ = 1, α ∈ [0, ∞) and αy ∈ [0, ∞). In particular, we are able to state the following Proposition:12 Proposition 1. Necessary and sufficient conditions for the determinacy of REE. Let φ = 0, σ = 1, α ∈ [0, ∞), αy ∈ [0, ∞) and α or αy different from zero. Determinacy of REE under positive trend inflation obtains if and only if (39)where (1 + ςcγcm − βγf)/γcm is the long‐run elasticity of output to inflation (see online Appendix A.4). With zero steady‐state inflation, i.e. with condition (39), becomes: (40) where κ = (ɛ − 1)/ϕ is the slope of the NKPC. We also know that in this special case, by imposing condition (25) so that the Rotemberg and the Calvo model coincides up to first order, then the conditions to ensure determinacy of REE are identical under the two pricing models. As stressed by Woodford (2001, 2003, see ch. 4.2.2) among others, condition (40) is a generalisation of the standard Taylor principle. Note that the coefficient (1 − β)/κ represents the long‐run elasticity of output to inflation in a standard NKPC log‐linearised around the zero‐inflation steady state (see (24)). Hence, given a standard Taylor rule as (37), Woodford (2003) shows that the generalised Taylor principle could be written as (41) where the sub‐index LR stands for Long Run. In other words, to ensure the determinacy of REE the nominal interest rate should rise by more than the increase in inflation in the long run. The generalised Taylor principle in its formulation (41) is still a crucial condition for the determinacy of REE in the Rotemberg model with trend inflation. Indeed, the long‐run elasticity of output to inflation in the generalised Rotemberg model with trend inflation, i.e. coincides with the coefficient (1 + ςcγcm − βγf)/γcm in (39) (see online Appendix A.4). Hence (39) corresponds exactly to (41) in the general case of a Rotemberg model with trend inflation. What then are the effects of trend inflation on the determinacy region in the Rotemberg model? Proposition 2. The effects of trend inflation on the determinacy region. Let φ = 0, σ = 1, α ∈ [0, ∞), αy ∈ [0, ∞) and α or αy different from zero. Then (42) which is positive for β sufficiently close to 1. (see online Appendix A.4.3) Corollary. Let φ = 0, σ = 1, α ∈ [0, ∞), αy ∈ [0, ∞) and α or αy different from zero, and β sufficiently close to 1. Then, the determinacy region widens in the parameter space (α, αy). The derivative in (42) reveals the effects of trend inflation on condition (39). Recall that (39) is equivalent to (41) in the case of the Rotemberg model. Hence (42) demonstrates that increases with trend inflation around the point .13 As indicated in the corollary, if increases, then the region in the parameter space (α, αy) that guarantees the determinacy of the REE gets larger. In fact, for a given αy, the condition (39) is satisfied by lower α values. Figures 4a and b visualise the content of Proposition 2. Figure 4a shows the usual graph of the Taylor principle in the space (α, αy) in the case which is identical to the one we get under Calvo pricing with zero trend inflation. In the case where , and condition (41) implies . As trend inflation increases, Proposition 2 shows that increases and the line rotates anti‐clockwise (see Figure 4b). Fig. 4. Open in new tabDownload slide The Effect of Trend Inflation on the Taylor Principle. (a) Zero Trend Inflation (b) Rotemberg Pricing (c) Calvo Pricing Fig. 4. Open in new tabDownload slide The Effect of Trend Inflation on the Taylor Principle. (a) Zero Trend Inflation (b) Rotemberg Pricing (c) Calvo Pricing Moreover, from a quantitative perspective, Figure 5 depicts the determinacy regions for the values α ∈ [0, 5] and αY ∈ [−1, 5] for differing levels of trend inflation, i.e. from 0% to 4% (for the calibration, see the beginning of Section 3). The determinacy frontier rotates anti‐clockwise and thus enlarges the determinacy region and remains negatively sloped, as suggested by Proposition 2. Fig. 5. Open in new tabDownload slide The Effect of Trend Inflation on the Determinacy Region in the Rotemberg Model Fig. 5. Open in new tabDownload slide The Effect of Trend Inflation on the Determinacy Region in the Rotemberg Model 3.4.2. The Calvo model In a recent paper, Ascari and Ropele (2009) show that trend inflation shrinks the determinacy region in the Calvo model. The two approaches therefore cause trend inflation to affect the determinacy region in the opposite way. In particular, Ascari and Ropele (2009) show that the generalised Taylor principle, (41), is still a necessary condition in the Calvo model. However, as trend inflation increases, decreases, and then very rapidly switches sign from positive to negative, such that the determinacy frontier rotates clockwise (Figure 4c shows the equivalent of Proposition 2 in the Calvo model). So trend inflation strongly shrinks the determinacy region in the space (α, αy) in the Calvo model, whereas it does the opposite in the Rotemberg model. Moreover, the two authors, show that the generalised Taylor principle is a necessary, but not sufficient condition for the local determinacy of the REE in the positive orthant of the parameter space (α, αy). This happens because, generally, there is a second determinacy frontier that needs to be satisfied. This frontier lies entirely below the positive orthant when such that it is usually disregarded in the literature (see Figure 4a). Trend inflation, however, moves this second determinacy frontier upwards, causing it to cross the positive orthant for moderate rates of trend inflation. Hence, this condition becomes necessary, even exclusively considering positive values of α and αy. Figure 5 above shows that, in the Rotemberg model too, this second determinacy frontier is relevant and that it lies entirely below the positive orthant when (as the Rotemberg model is equivalent to the Calvo model in this case). However, the simulation shows that trend inflation shifts this frontier upwards as in the Calvo model, but the effects are very minor and the frontier never crosses the positive orthant, given our calibration. Therefore, in contrast to the Calvo model, in the Rotemberg model the generalised Taylor principle remains not only a necessary but also a sufficient condition for the determinacy of the REE in the positive orthant of the space (α, αy). To sum up, the determinacy conditions in the two models are equivalent when the model is log‐linearised around zero trend inflation, i.e. but they are different in the presence of moderate levels of inflation. In particular, trend inflation has opposite effects on the condition that defines the generalised Taylor principle in the two models. Moderate inflation enlarges the determinacy region in the Rotemberg model, whereas it shrinks it in the Calvo model. Moreover, from a quantitative perspective, these effects are small in the former case and large in the latter. 3.5. Disinflation Dynamics In this subsection, we look at an unanticipated and permanent reduction in the inflation target of the Central Bank. The Central Bank follows the standard Taylor rule (37). In particular, we employ a non‐linear simulation method by means of the DYNARE package. We plot the path for output, inflation, nominal interest rate, real wages, consumption and hours in response to given changes in the Central Bank policy regime. We consider three cases: a disinflation from 4%, 6% and 8% trend inflation to zero. 3.5.1. The Rotemberg model When prices are set à la Rotemberg, the economy will immediately adjust to the new steady state without any transitional dynamics (see Figure 6). Thus, the non‐linear version of the simple NK model above with Rotemberg pricing is completely forward‐looking. Note that this is the same result as that obtainable from the log‐linear model. Fig. 6. Open in new tabDownload slide Disinflation in the Rotemberg Model Fig. 6. Open in new tabDownload slide Disinflation in the Rotemberg Model Consideration of the role of trend inflation, however, reveals the long‐run effects of such a policy. A disinflation policy permanently decreases output together with the real wage and hours worked but it increases consumption. As explained in Section 3.1, a disinflation causes an increase in firms’ markup, and a fall in output (and hence in hours). Moreover, a disinflation reduces the size of the adjustment costs, so it reduces the wedge between output and consumption, as shown by (16). Consumption increases because the decrease in the fraction of output wasted by price adjustments more than compensates for the decrease in output. Thus, a disinflation would cause output and consumption to move in opposite directions. So two main results stem from this analysis of the effects of a disinflation policy in the Rotemberg model. First, there are no transitional dynamics, and the economy immediately adjusts to the new steady‐state level, because the non‐linear model is completely forward‐looking. Second, there are, however, long‐run effects of such a policy: output and hours decrease, whereas consumption increases. 3.5.2. The Calvo model As with Figure 6 above, Figure 7 plots the responses of the main economic variables to disinflation policies from 4%, 6% and 8% trend inflation to zero in the case of the Calvo model (Ascari and Merkl, 2009). When non‐linear simulations are employed, the adjustment path of the Calvo model is completely different from that described above for the Rotemberg model. Indeed, the two main results above are inverted. Fig. 7. Open in new tabDownload slide Disinflation in the Calvo Model Fig. 7. Open in new tabDownload slide Disinflation in the Calvo Model First, the dynamic adjustment of the non‐linear Calvo model after a disinflation is inertial. The Calvo model implies price dispersion, i.e. st, which is a backward‐looking variable that adjusts sluggishly after a disinflation, as evident from (38). Thus, the non‐linear solution of the model features a new endogenous state variable and the model dynamic is inertial. By contrast, the Rotemberg model does not feature any inertial component and, as shown in Figure 6, each variable immediately adjusts to the new steady state. Second, output and consumption increase, whereas hours decrease. Output increases sluggishly to the new higher steady‐state level (see Section 3.1). As output is entirely consumed, consumption and output show the same adjustment path. By contrast, the adjustment dynamics in hours worked is different. Hours soar on impact, because output increases but then they decrease throughout the adjustment path, following the dynamics of price dispersion and regardless of the increasing output path. As explained in Section 1.2, inflation in the Calvo model creates a wedge between aggregate hours and aggregate output, through price dispersion, since Y = AN/s (see (21)). The lower price dispersion is, the fewer the hours needed to produce a given level of output will be. For all the cases considered, price dispersion decreases monotonically to the new lower steady‐state level. This is why hours thus peak on impact (when output increases, while s is predetermined), and then start decreasing (as s decreases along the adjustment path). Indeed, throughout the adjustment, output increases, whereas price dispersion decreases. From period 2 onwards, the latter effect dominates, thus, making aggregate production more efficient and in turn saving hours worked, despite the rise in output. Note that the differing behaviour of hours worked, as caused by the presence of price dispersion, implies a different dynamics for real wages. Unlike the Rotemberg model, disinflation leads to a short‐run overshooting of the real wages above their new higher steady‐state value.14 We therefore show that, when the economy is hit by a permanent and unanticipated inflation target shock, the two non‐linear models, as based on the two different price‐setting mechanisms, show very different and opposite dynamics. The Calvo model implies that output and consumption move closely together, whereas output and hours move in opposite directions during the adjustment, after the impact period. The opposite is true for the Rotemberg model. Moreover, while in the non‐linear Calvo model the adjustment is inertial, in the non‐linear Rotemberg model the adjustment is immediate. The intuition for these differences is straightforward and lies in the differences between the wedges that nominal rigidities create in the two models. Both wedges decrease after a disinflation. In the Rotemberg model, however, a disinflation reduces the wedge between output and consumption, so that they move in opposite directions, whereas in the Calvo model a disinflation reduces the wedge between output and hours, so that they move in opposite directions. Finally, the results in the Rotemberg model are qualitatively similar to those of the standard linear model. The standard NK model log‐linearised around zero steady‐state inflation would imply an immediate adjustment after a disinflation. Indeed, if log‐linearised around a zero inflation steady state, price dispersion would not matter for the model dynamics up to first order. So nothing prevents the model from jumping to the new steady state.15 In other words, the results in the Rotemberg model are qualitatively robust with regard to trend inflation and non‐linear analysis, whereas this is not the case for the Calvo model. We will conclude this main Section of the article with a few words on indexation. Quite obviously, allowing for price indexation counteracts the effect of trend inflation, because it reduces price dispersion by allowing the non‐price‐resetting firms to keep up with the pace of inflation. By dampening the effects of trend inflation, partial indexation tends to mitigate the differences between the two pricing models, as discussed above.16 However, our results are qualitatively very robust, because they vanish only upon full indexation. In fact, with full indexation, the two models are again equivalent; and in turn, they are both equivalent to the standard NK model that assumes no trend inflation (zero steady‐state inflation), because the two wedges in (16) and (21) disappear. This is not very surprising, since with full indexation the steady state of both models is equivalent to the steady state of the flexible price version of the model. This is exactly the reason why the dynamics of Rotemberg and Calvo models are identical under full indexation and under zero trend inflation. 4. Conclusion This article analyses the dynamics of an NK model with two firms’ price‐setting mechanisms: the Rotemberg (1982) quadratic cost of price adjustment and the staggered price setting introduced by Calvo (1983). The conventional wisdom is to consider these two models as observationally equivalent, because they deliver the same log‐linear NKPC. We show, however, that the two models differ substantially, once trend inflation is considered. Indeed, the two different nominal rigidity assumptions generate two different wedges in the two models. First, price dispersion in the Calvo model generates a wedge between output and hours and introduces an inertial component in the model, whereas the adjustment cost in the Rotemberg model generates a wedge between output and consumption and the model remains a pure forward‐looking model. However, these two wedges vanish under the particular case of zero steady‐state inflation, simply because in these specific conditions there is no cost of price rigidities in steady state. Further, trend inflation has an opposite effect on the steady‐state average markup. The latter generally increases with trend inflation in the Calvo model, whereas the average markup decreases with trend inflation in the Rotemberg model. Overall, the average markup and the wedge effects distinguish the Calvo from the Rotemberg model when steady‐state inflation is not zero. Indeed, by altering the cost of the nominal rigidities, trend inflation reveals the difference between the two pricing rigidity assumptions in the two models. In particular: the long‐run NKPC is negatively sloped in the Calvo model and positively sloped in the Rotemberg model; the log‐linear NKPC in the two models is qualitatively very different, implying two different dynamic systems; positive trend inflation shrinks the determinacy region in the Calvo model, while it enlarges the determinacy region in the Rotemberg model; positive trend inflation amplifies the IRFs to a technology shock in the Calvo model, while it dampens them in the Rotemberg model; a permanent and credible disinflation implies inertial adjustment and output gains in the Calvo model, whereas it implies immediate adjustment and output losses in the Rotemberg model. Throughout the article, we assume that the Rotemberg adjustment cost represents a pure waste for the economy, and that it therefore goes into the resource constraint. This is the most common assumption in the NK literature. Nevertheless, to understand the robustness of our results on the difference between the Rotemberg and the Calvo model, we also considered the case in which the adjustment costs are rebated to consumers.17 In this case, what we have defined as the ‘wedge effect’ is shut down, and the aggregate resource constraint becomes Ct = Yt. The results, however, remain qualitatively unaffected, even if quantitatively mitigated.18 Moreover, consumption and output move in the same direction after a disinflation in this case. Summing up, as a general point, this article stresses the importance of the interplay between long‐run effects and short‐run dynamics. The two models are non‐linear in trend inflation. Accordingly, the two price‐setting mechanisms imply a very different dynamics even to a first‐order approximation, once trend inflation is considered. Log‐linearising the model around a zero inflation steady state, by contrast, removes these interesting and intrinsic differences between the two models. Additional Supporting information may be found in the online version of this article: Appendix A. The Rotemberg Model. Please note: The RES and Wiley-Blackwell are not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the authors of the article. Footnotes 1 " Lombardo and Vestin (2008) and Damjanovic and Nolan (2011) show that with a distorted steady state the two models might imply different welfare costs at a second order of approximation. Damjanovic and Nolan (2008) also show that the performances of the two models differ when the Government maximises seigniorage. 2 " We treat ‘trend inflation’ and ‘steady‐state inflation’ as synonyms. 3 " Kimbrough (2006) shows that it can happen when the government maximises seignorage. Damjanovic et al. (2008) show that the optimal inflation rate becomes positive when the government discount factor differs from the private one. Levine et al. (2008) find that the optimal trend inflation rate becomes positive in an NK model with habit persistence. Schmitt‐Grohé and Uribe (2007b)), among others, also find non‐zero optimal steady‐state inflation rate with Calvo pricing. 4 " Note that this expression implicitly defines the condition 1 > (ϕ/2)(πt − 1)2 for the model to be well‐defined, that is: 5 " See for example Rotemberg (1987), Roberts (1995) and more recently Nisticò (2007) and Lombardo and Vestin (2008). 6 " See Ascari and Merkl (2009) for a derivation. 7 " Actually, the price adjustment gap is stronger for extremely low values of trend inflation, such that average markup first slightly decreases and then increases with trend inflation (King and Wolman, 1996). 8 " To be more precise, the derivative of the long‐run Phillips Curve evaluated at zero inflation, i.e. the tangent at zero inflation of the curve depicted in Figure 1, is positive. Only a feeble discounting effect is present in this case. Indeed, this positive slope equals the positive long‐run relationship between inflation and output implied by the standard log‐linear NKPC (23) popularised by Woodford (2003) among others. See also King and Wolman (1996) and Graham and Snower (2004) . 9 " See for example Schmitt‐Grohé and Uribe (2007b) for a discussion on the problems related to the Woodfordian subsidies. 10 " For a detailed derivation and description of the reduced form solution of the Calvo model under trend inflation see Ascari and Ropele (2009). See also Cogley and Sbordone (2008). 11 " In a recent paper, Damjanovic and Nolan (2010) show that, in a model with low trend inflation, a negative monetary shock can have a persistent and hump‐shaped impact on output and a positive impact on inflation. 12 " In the Rotemberg model vector in the representation (38) includes two non‐predetermined variables, i.e. . Hence, the determinacy of REE obtains if and only if all eigenvalues of A lie inside the unit circle. 13 " In general, the derivative in (42) yields a very cumbersome expression that would not allow any analytical insights. We were able, however, to derive the condition in (42) by evaluating the derivative at to understand how trend inflation affects the Taylor principle when slightly moves from one. By continuity, one may argue that the result holds for the values of very close to one, such as the ones we consider (recall that is the gross quarterly inflation rate). The simulations later in the article, indeed, confirm such conjecture. 14 " Ascari and Rossi (2011) consider the effects of disinflation in an economy characterised by real wage rigidities. They show that in the Calvo model real wage rigidities generate a long‐lasting boom in output, whereas in the Rotemberg model real wage rigidities cause a moderate output slump throughout the adjustment path after a disinflation experiment. 15 " Moreover, output would decrease, as implied by the non‐linear Rotemberg model. The NKPC, indeed, is positively sloped when the Calvo and Rotemberg model is log‐linearised around a zero inflation steady state. 16 " For more details, see Ascari and Rossi (2009) who study the effects of price indexation, both on the long‐run properties and on the dynamics of the two pricing models. 17 " This means that, as in the standard model, (ϕp/2)[(Pt/Pt−1) − 1]2Yt is a cost for the intermediate goods‐producing firm and therefore it lowers firms profits Πt. However, we now assume that the cost of adjusting prices is paid to the representative consumer. Then, (ϕp/2)[(Pt/Pt−1) − 1]2Yt enters the household budget constraint increasing his revenues. 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TI - Trend Inflation and Firms Price‐Setting: Rotemberg Versus Calvo
JO - Economic Journal
DO - 10.1111/j.1468-0297.2012.02517.x
DA - 2012-09-01
UR - https://www.deepdyve.com/lp/oxford-university-press/trend-inflation-and-firms-price-setting-rotemberg-versus-calvo-CI3Hm6vDoe
SP - 1115
VL - 122
IS - 563
DP - DeepDyve
ER -