Keywords: Inflation persistence, intermediate goods, monetary policy
Abstract:
This paper studies the implications of inflation persistence (generated by backwardlooking price setters) for monetary policy in a New Keynesian "inputoutput" modela model with sticky prices in both intermediate and final goods sectors. Optimal policy under commitment depends on the degree of inflation persistence in both sectors. Under discretion, speedlimit targetingtargeting the change in the output gapoutperforms pricelevel and inflation targeting in the presence of inflation persistence. If inflation persistence is low in the intermediate goods sector, pricelevel targeting outperforms inflation targeting despite high inflation persistence in the final goods sector.
JEL codes: E50, E52, E58
How should monetary policy be conducted in a New Keynesian model in which prices are sticky at multiple stages of production? Huang and Liu (2005) and Strum (2009) examine this question using a forwardlooking New Keynesian "inputoutput" modela model that has sticky prices in both the intermediate and final goods sectors. They find that a central bank paying attention to price movements in both the final and intermediate goods sectors can more ably minimize household utility losses than if it considers only one sector. Furthermore, Strum (2009) finds that if the central bank acts under discretion, it performs better if it targets price levels rather than inflation rates.
One feature of the standard forwardlooking New Keynesian framework is that it generates Phillips curves that do not relate current inflation to lagged inflation. Yet studies such as Fuhrer (1997), Rudebusch (2002), and Roberts (2005) find an important empirical role for lagged inflation in the Phillips curve. Besides affecting the specification of the Phillips curve, inflation persistence can affect the evaluation of monetary policy in New Keynesian models. For example, when examining discretionary monetary policy regimes in onesector New Keynesian models, Walsh (2003) and Nessén and Vestin (2005) show that the type of regime that performs best depends on the degree of inflation persistence in the Phillips curve.
These findings suggest the question: How is monetary policy design in a New Keynesian inputoutput model affected by inflation persistence? To answer this question, the standard model must be extended so that the Phillips curve exhibits inflation persistence. As recent scholarship has found, inflation persistence can arise in New Keynesian models in a number of different ways.^{1} This paper builds inflation persistence into the inputoutput model developed by Huang and Liu (2005) and extended in Strum (2009) by following the approach pioneered in Galí and Gertler (1999), Amato and Laubach (2003), and Steinsson (2003): When resetting prices, some firms are modeled as employing a simple rule of thumb that uses information about past states of the world to set new prices. These firms may behave this way if, from time to time, they find it too costly to gather new information and calculate the optimal forwardlooking price. Firms using this rule of thumb generate inflation persistence in the sectoral Phillips curves. Furthermore, compared to models with sticky prices in one sector, this model can yield different degrees of inflation persistence in different sectors.
This paper examines four questions about monetary policy in New Keynesian models: How does inflation persistence at multiple stages of production affect the conduct of optimal policy under commitment? How does inflation persistence at multiple stages of production affect the type of loss function that should be assigned to a central bank acting under discretion? How well do different regimes perform when policies are set using incorrect assumptions about inflation persistence or the sources of shocks? How well do policies derived from onesector models perform when implemented in an inputoutput model?
I find that the timing and magnitude of the central bank's responses to shocks when implementing optimal monetary policy under commitment are affected not only by the presence of inflation persistence, but also by the relative degrees of sectoral inflation persistence. On the other hand, the nature of the central bank's responseswhether expansionary or contractionaryis not affected by the degrees of inflation persistence.
When studying monetary policy under discretion, I find that inflation persistence affects the type of loss function that best minimizes household losses. As in Strum (2009), pricelevel targeting performs best in a forwardlooking model. However, when inflation persistence is introduced, speedlimit targeting (a regime targeting the change in the output gap) performs best. Pricelevel targeting outperforms inflation targeting unless inflation persistence is high in both sectors.
When a regime is chosen and the loss function is crafted, the government may make an incorrect assessment of inflation persistence in the two sectors, the sources of shocks, or the need to use the inputoutput model. I find that, given the degrees of inflation persistence in the two sectors, the type of regime that performs best is not affected by the government's assumptions regarding inflation persistence or the sources of shocks (when crafting the loss function). However, the type of regime that performs best is affected by the incorrect use of a onesector model.
The remainder of the paper is organized as follows: Section 2 sets up the model and presents the linearized version used for later analysis. Section 3 discusses the calibration. Section 4 examines the qualitative characteristics of optimal policy under commitment for different degrees of inflation persistence in the two sectors. Section 5 compares the performances of simple loss functions under the more realistic case of discretionary optimization. Section 6 examines some robustness properties of the discretionary regimes. Section 7 concludes.
Huang and Liu (2005) develop a New Keynesian model with a vertical production chain consisting of two sectors. Firms in the first sector produce final (nondurable) goods using intermediate goods and labor. Final goods are consumed by households. In the second sector, intermediate (nondurable) goods are produced using only labor. Intermediate goods are used only by final goods firms in production. Each firm in each sector produces a unique differentiated good and engages in monopolistic competition within its sector. Prices in both sectors are sticky, and firms adjust their prices in a staggered manner in the spirit of Calvo (1983). There is one competitive market for homogenous labor that can be used by all firms. All firms are price takers in their input markets.
Strum (2009) extends the model by introducing costpush shocks and characterizing monetary policy as the minimization of an assigned loss function. This paper extends the model in Strum (2009) by assuming that in each sector and in each period, a fraction of firms find that solving for the optimal forwardlooking price is too costly. Following Galí and Gertler (1999), Amato and Laubach (2003), and Steinsson (2003), I assume that backwardlooking firms in each sector employ a rule of thumb that uses past information to set a new price. This section presents the basic elements of the model and its linearized version used for subsequent analysis.^{2}
The economy is populated by a large number of identical, infinitely lived households. Households derive utility from consumption and leisure. Given a fixed amount of time that households divide fully between leisure and labor each period, the household utility function can be written in terms of labor instead of leisure. Accordingly, households maximize expected lifetime utility, given by
(1) 
The period utility function for consumption is . The consumption good, , is a composite of a continuum of differentiated final goods in the spirit of Dixit and Stiglitz (1977), given by
,  (2) 
Households have equal ownership in all firms and divide all profits equally among themselves. Labor is homogeneous and supplied equally by households to all firms through one market with one wage rate, which households take as given. I assume complete financial markets. Finally, I assume standard budgetset and transversality conditions hold.
Each final goods firm has access to a constant returns to scale (CRS) CobbDouglas production function
,  (3) 
,  (4) 
Each firm minimizes costs to meet the demand for its good given its stated price. Final goods firms adjust their prices with probability each period, where . A random fraction of firms resetting prices find it worthwhile to determine the price that maximizes discounted expected profits over the time the price is expected to persist, where . This maximization problem is given by
The remaining fraction of firms setting a new price use a rule of thumb to determine a new backwardlooking price, , instead of solving an explicit optimization problem. Steinsson (2003) suggests a generalization of the Galí and Gertler (1999) approach by allowing firms to also react to an indicator of the previous period's output gap. Generalizing the rule of thumb in Steinsson (2003), I assume that the backwardlooking firms set prices according to
The aggregate price level can be written as
Each intermediate goods firm has access to a CRS production function given by
,  (10) 
The government serves two purposes in this model. First, it assigns a loss function to an independent central bank. The central bank acts to minimize its assigned loss function. I assume that the central bank can react to and affect state variables in the current period. Second, the government collects lumpsum taxes from households to provide subsidies to firms so that the steadystate equilibrium is not distorted from inefficiencies arising from monopolistic competition. Finally, I ignore the possible interactions between monetary and fiscal policy that would be present in a richer model.
I loglinearize the model using logdeviations from a hypothetical nondistorted efficient equilibrium (the equilibrium that would obtain if prices were flexible and there were no shocks to the elasticities of substitution).^{4} The natural rate of interest is the real interest rate that would obtain in the efficient equilibrium. I list the key variables and symbols from the model in Table 1.
The household intertemporal consumption equation is obtained from the household's firstorder conditions. Its loglinearized version is given by
The loglinearized pricing equations for backwardlooking firms can be combined with the loglinearized firstorder equations for forwardlooking firms to obtain Phillips curves for each sector, namely,
E  (12)  
, 
E  (13)  
, 
. 
The sectoral Phillips curves reveal how the percentage of backwardlooking firms affects the sectoral Phillips curves. An increase the percentage of backwardlooking firms increases the weight of lagged inflation relative to the forwardlooking component ( is decreasing and is increasing in ). The coefficient of the sectoral Phillips curves with respect to the sector's current real marginal cost gap (and, therefore, the output gap) decreases as the percentage of backwardlooking firms increases ( is decreasing in ).^{5} On the other hand, the coefficient of the sectoral Phillips curves with respect to the sector's lagged real marginal cost gap increases as the percentage of backwardlooking firms increases ( is increasing in ).
The weight on intermediate goods in the production function of final goods, represented by , affects the coefficients of the current and lagged output gaps in the final goods Phillips curve, given by and , respectively. As in the forwardlooking model in Strum (2009), higher values of correspond to lower values of the coefficient of the current output gap in the final goods Phillips curve. Moreover, higher values of also correspond to lower values of the coefficient of the lagged output gap in the final goods Phillips curve.^{6}
In standard hybrid New Keynesian Phillips curves, the costpush shocks are represented by terms like . In this model, the term arises from , variations in the elasticities of substitution of differentiated goods. The magnitude of a onestandarddeviation shock to does not depend on the percentage of backwardlooking firms. However, as the expression for shows, the effects of fluctuations in the elasticities of substitution are attenuated by higher percentages of backwardlooking firms. For the statespace representation of the model, I rewrite this term as
, ,  (14) 
As noted in Table 1, the "relative price" refers to the ratio of the price index for intermediate goods to the price index for final goods. The relative price gap evolves, by definition, according to
.  (15) 
,  (16) 
In this model, the nominal interest rate does not appear in the objective function of the central bank and is not a constraint in the central bank's maximization problem. Therefore, I simplify the setup by treating as the instrument of the central bank. Once the model is solved, I use (11) to determine the interest rates that are consistent with the desired equilibrium. I compute price levels using the identity for , where can be interpreted as the logdeviation in the price level from its initial value. I represent the structural equations of the economy as
where
The secondorder Taylor approximation to the household's utility function is given by
Et.i.p. ,  (18) 
(19)  
, 
, 
The period loss function is a natural extension of period loss functions in other models. Under the assumption of one sector with sticky prices, the first line of the period loss function is obtained. The assumption that both intermediate and final goods sectors have sticky prices adds the second line. The third line is obtained if backwardlooking price setters are included and (the formulation in Galí and Gertler, 1999). The last two lines of the loss function arise if backwardlooking firms react to past marginal cost gaps in their rule of thumb.
Inflation in each sector corresponds to lower household utility since the interaction of sticky prices and inflation produces a set of suboptimal relative prices of differentiated goods, which then leads to inefficient mixes of goods in each sector. In the loss function, the real marginal cost gap in the intermediate goods sector is connected to the relative price's role in the allocation of resources across sectors. Finally, as is standard in other models, deviations of consumption (final output) from the efficient level correspond to higher utility losses.
In order to understand the connection between the household loss function and the proportion of backwardlooking price setters, I present the loss function weights for a number of combinations of forwardlooking and backwardlooking price setters in Table 2. The calibration of the model determining these weights is explained in Section 3. The main effect of backwardlooking price setters is a dramatic increase in the importance of smoothing the change of inflation, represented by and

This section discusses the calibration used for the benchmark model. The assumption that implies that . I set the subjective time discount factor to , implying that the annual real interest rate in the steady state is about 4 percent, given that I interpret a time period as a quarter. The steadystate values of the elasticities of substitution for the differentiated goods, and , are set to 10, which implies a steadystate markup of about 11 percent. Consistent with earlier empirical work (e.g., Carlton, 1986; and Blinder et al., 1998) and following Huang and Liu (2005) and Strum (2009), I set the average price contract equal to one year, which means setting . I also follow Huang and Liu (2005) and Strum (2009) in setting .
Technology shocks are typically represented as small but persistent (see, for example, Cooley and Prescott, 1995, and Gomme and Rupert, 2007). I set the AR(1) coefficients for process governing the evolution of the technology factors to . I set the standard deviation of the innovations to the technology factor process in each sector, , to 0.02.
As noted earlier, I assume that the costpush shocks are white noise processes that do not depend on . I set so that the standard deviation of is in the purely forwardlooking case. I chose this value so that a negative twostandarddeviation shock does not cause the central bank to hit the zero bound on the nominal interest rate when implementing optimal policy with commitment.^{9} When , I use the expressions for derived earlier to adjust .
As I am not aware of any models with estimates of , I set these values so that the coefficient on lagged marginal cost in the backwardlooking rule of thumb equals that of the coefficient on current marginal cost in the Phillips curves in which there are no backwardlooking price setters (similar to the approach in Steinsson, 2003). This leads me to set .
A number of authors have fit empirical estimates to hybrid Phillips curves. Fuhrer (1997) finds that setting more relative weight on lagged inflation does better, whereas Galí and Gertler (1999) find that more weight should be put on the forwardlooking term. I take a middleoftheroad approach and set , implying and in (12), which is in line with the estimates in Roberts (2005). Since Clark (1999) finds that the prices of goods at earlier stages of processing are more responsive to monetary policy shocks, I assume that the intermediate goods sector does not have greater persistence than the final goods sector. Consequently, I set equal to or less than , with three possible values for : and 0.7.
In this section, I assume that the central bank is assigned the household loss function. Furthermore, I assume that the central bank can credibly commit to statecontingent future actions. The intertemporal loss function of the central bank is given by
E.  (20) 
(21) 
where
.  (22) 
E  (23)  
, 
The results in Section 2 show that inflation persistence affects the model in three ways that matter for policymakers: First, higher inflation persistence in a particular sector causes the household utility function to put more weight on smoothing inflation in that sector. Second, for the calibrations considered in this paper, a higher degree of sectoral inflation persistence leads to a flatter sectoral Phillips curve with respect to the output gap.^{12} Third, higher sectoral inflation persistence means that policymakers must work against a greater degree of momentum when attempting to affect sectoral inflation.
How do these additional complications arising from inflation persistence affect optimal policy under commitment? Figure 1 presents the optimal commitment responses to positive shocks under different combinations of sectoral inflation persistence. The standard forwardlooking case is obtained by setting . Two cases of high final goods inflation persistence and low intermediate goods inflation persistence are obtained by setting and or 0.2. Finally, high inflation persistence in both sectors is obtained by setting .
The first and second columns of Figure 1 show the central bank's responses to positive onestandarddeviation productivity shocks. The top row shows the (annualized) values of the nominal interest rate that are consistent with the central bank's policy. However, the real interest rate gap, shown in the second row, indicates the nature of monetary policy (whether policy is expansionary or contractionary).
Initially, a positive productivity shock in the final goods sector induces the central bank to enact contractionary policy (a positive real interest rate gap), whereas the central bank pursues expansionary policy (a negative real interest rate gap) in response to a positive productivity shock in the intermediate goods sector. The nature of the central bank's response does not depend on the degrees of inflation persistence in the two sectors.
However, the timing and magnitude of the central bank's responses to productivity shocks depend on the degrees of inflation persistence in the two sectors. When inflation persistence is high in the final goods sector but low in the intermediate goods sectors, the central bank delays its maximal response and increases the magnitude of its maximal response. When inflation persistence is high in both sectors, the central bank also delays its maximal response relative to the forwardlooking case; however, unlike the case of unequal inflation persistence, the maximal response is of lesser magnitude than in the forwardlooking case. Finally, the bottom two rows show that when a sector is populated by a large number of backwardlooking price setters, inflation (or deflation) in that sector has a delayed and muted maximal response to a productivity shock in either sector.
The third and fourth columns of Figure 1 show the dynamics following positive onestandarddeviation costpush shocks. The simulations show that, unlike the case with productivity shocks, the central bank engages in contractionary policy when responding to positive costpush shocks in either sector. On the other hand, just as is the case with productivity shocks, the nature of the response does not depend on the degrees of inflation persistence in the two sectors.
The timing of the central bank's maximal response to a costpush shock does not depend on the degrees of inflation persistence in the two sectors; however, the magnitude does. The maximal response comes in the initial period whether or not inflation is persistent. The magnitude of the central bank's response to a costpush shock is strongly affected by the attenuation of the effect of costpush shocks on sectoral inflation that occurs as the percentage of backwardlooking firms in a sector increases. Accordingly, the magnitude of the central bank's maximal response to a costpush shock decreases noticeably relative to the forwardlooking case if inflation persistence is high in the sector hit by the shock. Finally, as the bottom two rows show, when a sector is characterized by high inflation persistence, the magnitude of the initial jump in inflation (or deflation) in that sector following a costpush shock in either sector is lower than in the forwardlooking case.
Figures 2 and 3 show that the behavior of the price levels of final and intermediate goods following positive onestandarddeviation costpush shocks depends on the percentage of backwardlooking price setters in the sector hit by the shock. So, for example, the response of the price level of final goods to a costpush shock in the intermediate goods sector depends critically on the number of backwardlooking firms in the intermediate goods sector. Consistent with Steinsson (2003), the price levels of both sectors do not return to their preshock levels if there are backwardlooking price setters. However, if the intermediate goods sector has few backwardlooking price setters, the price levels of both sectors converge to levels very close to their preshock values following a costpush shock in the intermediate goods sector. Furthermore, even when inflation persistence is high in the intermediate goods sector, the magnitudes of the permanent effect of an intermediate goods costpush shock on price levels in both sectors are lower than if the shock occurred in the final goods sector (when persistence is high in the final goods sector). Given that previous studies have found that pricelevel targeting under discretion works well in the forwardlooking case, Figures 2 and 3 suggest that pricelevel targeting may perform well in an inputoutput model with inflation persistence, especially if the intermediate goods sector has fewer backwardlooking firms, is more likely to experience costpush shocks than the final goods sector, or has both of these characteristics.
Although an analysis of the optimal commitment policy is useful for understanding the characteristics of desirable policy, Clarida, Galí, and Gertler (1999) point out that central banks do not make binding commitments. Furthermore, simple loss functions are easier for the public to understand and monitor, facilitating transparency and communication. Accordingly, I examine household utility losses that arise under alternative simple loss functions assigned to central banks acting under discretion. I consider the same combinations of inflation persistence as in the previous section and examine three classes of loss functions: inflation targeting (IT), pricelevel targeting (PT), and speedlimit targeting (SL). I also report household losses under the optimal commitment policy (COM).
Under inflation targeting, the central bank acts to keep inflation and other variables (such as the output gap) close to a set of target values.^{13} Pricelevel targeting differs from inflation targeting in that the central bank reacts to deviations of the price level from a target value. Speedlimit targeting stabilizes fluctuations in the change in the output gap in addition to other target variables. In a forwardlooking model, Strum (2009) finds that targeting inflation or price levels in both sectors dominates singlesector targeting regimes; therefore, I consider only regimes that target inflation or price levels in both sectors.
I represent these regimes by loss functions given by
One disadvantage of the backwardlooking rule is that optimizing agents might not use the same backwardlooking rule in different regimes. Nevertheless, two factors suggest that the approach in this paper is useful. First, although different regimes lead to different shortrun dynamics of inflation, all of the regimes are consistent with the same steadystate inflation rate of zero. Second, firms do not adopt the rule to optimize or as a longterm rule, but rather at random times when information costs are high. Therefore, the backwardlooking rule can be seen as a simple rule employed by firms when they occasionally find the costs of information gathering and processing prohibitively expensive.
The central bank's problem can be put in a standard linearquadratic setup.^{15} I calculate and report the expected household losses under each loss function as
Table 3 reports the coefficients and expected household losses for each regime according to the percentages of backwardlooking firms (and, hence, degrees of inflation persistence) in the two sectors.^{17} Two observations jump out immediately. First, the absolute levels of losses are highest in the fully forwardlooking model and lowest in the model with high persistence in both sectors. When the percentage of backwardlooking price setters is high, the attenuation of the effects of costpush shocks in the sectoral Phillips becomes important. Second, even though there is always a clear ranking of regime performance, the difference between the best and worst regimes decreases as inflation persistence increases.


The qualitative results are summarized in Table 4. Consistent with Strum (2009), pricelevel targeting produces the lowest household losses in the purely forwardlooking model. When inflation persistence is high in the final goods sector but low in the intermediate goods sector, speedlimit targeting performs best. At the same time, pricelevel targeting outperforms inflation targeting. When inflation persistence is high in both sectors, speedlimit targeting performs best. However, in this instance, inflation targeting performs better than pricelevel targeting (and only slightly worse than speedlimit targeting). These results are broadly consistent with Walsh (2003), who finds that speedlimit targeting outperforms pricelevel targeting once moderate levels of inflation persistence are reached, and that inflation targeting performs well when inflation persistence is high. However, these results show that the degrees of inflation persistence in both intermediate and final goods sectors can be important when ranking regime performance.

In the previous section, the government crafted loss functions based on an accurate assessment of inflation persistence, the sources of shocks to the economy, and the model of the economy to be used. However, monetary policy may not be practiced in such favorable conditions. In this section, I examine how discretionary regimes perform when the central bank minimizes loss functions that were crafted based on assumptions that may not be true.
I begin by examining how policies crafted under incorrect assumptions about inflation persistence perform. Specifically, I follow a twostep process: First, I assume that the government chooses loss function coefficients based on assumptions regarding inflation persistence in each sector that may not be accurate. Second, I determine the household losses that occur when the central bank implements these policies in the true economy. Table 5 reports the results from this exercise. For example, in the block of columns under "," the table shows the household losses for policies crafted under four assumed combinations of degrees of inflation persistence, and implemented in a forwardlooking world.
Pricelevel targeting performs best in the forwardlooking world in three of the four scenarios. When inflation is persistent in the final goods sector but forwardlooking in the intermediate goods sector, pricelevel targeting performs best in two of the four scenarios, while speedlimit targeting performs best in the other two scenarios. When inflation persistence is high in the final goods sector and either low (but positive) or high in the intermediate goods sector, speedlimit targeting performs best. Although inflation targeting never performs best, it outperforms pricelevel targeting in three of the four scenarios when inflation persistence is high in both sectors. Finally, as the true levels of inflation persistence rise, the losses from making the wrong assumptions about the degrees of inflation persistence decrease.


Next, I examine regime performance when the loss function is based on the assumption that shocks hit either only the final goods sector or only the intermediate goods sector, when, in fact, shocks hit both sectors. I assume that the government correctly perceives the degrees of inflation persistence in both sectors when it sets the loss functions. I follow a twostep process similar to the one above: I calibrate loss function coefficients that are optimal under the assumption that shocks hit only one of the sectors, then I run the implied policies in economies that are subject to shocks in both sectors. Table 6 reports the results for the four possible combinations of inflation persistence. The columns labeled "F" show the results when the central bank is assigned a loss function based on the assumption that shocks arise only in the final goods sector. The columns labeled "M" show the results when the loss function is based on the assumption of shocks coming only from the intermediate goods sector. For example, in the twocolumn block under "," the column labeled "F" shows the household losses under regimes based on the assumption that shocks hit only the final goods sector, when, in fact, shocks hit both sectors.
If the government crafts the loss function based on the incorrect assumption that shocks hit only one sector, better results are usually obtained if the government assumes that shocks hit the intermediate goods sector. Regardless of which sector is assumed to be hit by shocks, pricelevel targeting performs best in the forwardlooking case. When inflation persistence is high in the final goods sector and either low or high in the intermediate goods sector, speedlimit targeting performs best. Although inflation targeting ranks last both when the economy is fully forwardlooking and when inflation persistence is high in the final goods sector but low in the intermediate goods sector, it outperforms pricelevel targeting when inflation persistence is high in both sectors.

Finally, suppose that the central bank is assigned a loss function based on a standard onesector model in which inflation is assumed to be final goods inflation. I consider two alternatives: a loss function based on a forwardlooking onesector model and a loss function based on a onesector model with many backwardlooking price setters. Table 7 reports the household losses when these policies are implemented in the four inputoutput economies examined in earlier sections. For example, in the twocolumn block under "," the column labeled "0" shows the household losses under regimes based on the assumption of a onesector model with no inflation persistence, when the true economy has an inputoutput structure with high inflation persistence in the final goods sector and no inflation persistence in the intermediate goods sector.
When the onesector regimes are incorrectly implemented in inputoutput economies, regime rankings from earlier exercises do not hold as tightly. In the forwardlooking inputoutput economy, onesector inflation targeting performs best if a forwardlooking model is assumed, whereas onesector pricelevel targeting performs best if inflation persistence is assumed when crafting the loss functions.^{18} If inflation is persistent in at least one sector in the true inputoutput economy, onesector inflation targeting always performs best.

The three exercises in this section point to a few qualitative results about the robustness of loss functions that are formed under assumptions that may not be true of the economy in which they are implemented. Table 8 reports the bestperforming regime for each characterization of the economy under the possibly incorrect assumptions studied in the previous three exercises and the bestperforming regime when correct assumptions are used. The type of regime that performs best under different combinations of inflation persistence is not affected by the government's assumptions regarding inflation persistence or the sources of shocks (when crafting the loss function). In particular, if the economy is fully forwardlooking, pricelevel targeting usually performs best. When inflation is persistent in one or both sectors, speedlimit targeting usually performs best. However, the type of regime that performs best is affected by the incorrect use of a onesector model. In this case, inflation targeting performs best in one of the two forwardlooking cases and in every case when inflation in the true economy is persistent in one or both sectors.

Adding inflation persistence (through backwardlooking price setters) to a New Keynesian model in which prices are sticky in both intermediate and final goods sectors alters the household loss function and the sectoral Phillips curves. Consequently, the degrees of inflation persistence in both sectors can affect the implementation and design of monetary policy in New Keynesian models.
When conducting the optimal commitment policy, the nature of the central bank's responses to shockswhether expansionary or contractionaryis not affected by the degrees of inflation persistence. However, the timing and magnitude of the central bank's responses shocks can be affected by the degrees of inflation persistence in the two sectors. When inflation is persistent, the maximal response to productivity shocks by the central bank is delayed relative to the forwardlooking case. When inflation persistence is high in the final goods sector but low in the intermediate goods sector, the central bank's maximal response is greater in magnitude than in the forwardlooking case. On the other hand, the magnitude of the central bank's maximal response is lower than in the forwardlooking case when inflation persistence is high in both sectors. The timing of the central bank's response to costpush shocks does not depend on inflation persistence. However, the magnitude of the central bank's response to a costpush shock decreases as the percentage of backwardlooking firms and the degree of inflation persistence in the sector hit by the shock increase.
When the central bank acts under discretion, the type of regime that performs best depends on the degrees of inflation persistence in both sectors. As in Strum (2009), pricelevel targeting performs best when both sectors are fully forwardlooking. Speedlimit targeting performs best when inflation persistence is high in the final goods sector but low in the intermediate goods sector. In this case, both speedlimit targeting and pricelevel targeting outperform inflation targeting. When inflation persistence is high in both sectors, speedlimit targeting still performs best; however, in this case, inflation targeting outperforms pricelevel targeting.
When crafting the loss function to assign to the central bank, incorrect assumptions can be made about the degree of inflation persistence, the sources of shocks, or whether to use a onesector model. Under the calibration considered here, the type of regime that performs best under different combinations of inflation persistence is not affected by the government's assumptions regarding inflation persistence or the sources of shocks (when crafting the loss function). However, the type of regime that performs best is affected by the incorrect use of a onesector model.
Finally, in assessing these results, it is important to remember that the mechanism generating inflation persistence in the model may be important. Further research into the sources of inflation persistence would enable clearer connections between the design of policy in a model and in the real world. Nevertheless, this paper has shown that accounting for both sticky prices and inflation persistence at different stages of production can be important for stickyprice models used to study monetary policy. Inflation dynamics and policy tradeoffs can depend on the degrees of inflation persistence at multiple stages of production.
I represent the structural equations of the economy as
The matrices that describe the exact statespace structural relations for the economy are given by(24) 
where