Background

As a major structural shock on the NKPC explaining inflation variations, there is a typical supply shock called as the productivity shock. SW (2003, 2007), in addition to the productivity shock, introduced the so-called markup shock into the log-linearized NKPC, in a one-to-one correspondence with inflation. Considering the simple model in the previous section as an example, it means ε^m_t called the markup shock is added to the log-linearized NKPC as follows (see (1.52)):

ˆ

πt= β

1 +βEtπˆt+1+ 1

1 +βˆπt−1+κ

σL+ σ 1−h

ˆ

yt− σh

1−hyˆt−1−(1 +σL)ˆzt

+ε^m_t

The problem is the role of markup shock added to the NKPC in this ad hoc manner has played a tremendous role to explain inflation fluctuations. According to variance decompositions of SW (2007) using the U.S. data, most of inflation fluctuations are explained by the price markup shock in the short term and the wage markup shock in the long term (Over 80% of inflation fluctuations are explained by both markup shocks, whether in short or long term). And the contribution of the productivity shock ˆz_t, another key structural shock on the NKPC, to inflation fluctuations was extremely insignificant.

However, markup shocks on price and wage (from their names) are shocks related to parameters of the price elasticity of intermediate goods demand and the wage elasticity of labor demand, respectively. Normally, the markup rate will increase if the price elasticity of intermediate goods demand (wage elasticity of labor demand) decreases, in other words, if the necessity of intermediate goods (workers) increases and the market power of intermediate goods firms (workers) increases.

Therefore, SW (2007) tells that most of inflation fluctuations are caused by markup variations, that is, it is attributed to frequent fluctuations in the market powers of intermediate goods firms and workers (fluctuations in the necessity of intermediate goods and workers).

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The reason why the name is the markup shock can be understood from the simple model example. The optimal pricing condition of intermediate goods firms in the case of flexible price adjustment can be written as follows (see (1.14)):

Pt(i)^∗ Pt

= (1 +λ)Ψt

where λis the (net) price markup rate, Ψt is the real marginal cost, and the left-hand side is the relative price of the intermediate goods price P_t(i) against the general price P_t. Since the price adjustment is flexible, the aggregate supply curve becomes vertical. Defining the nominal marginal cost as M Ct ≡ PtΨt, the inflation after the log-linear approximation around the steady state is expressed as follows:

ˆ

π_t= ˆM C_t

In case of flexible price adjustment, if the nominal marginal cost (here, nominal wage) changes, it just means that the inflation will jump to exactly the same change as the nominal marginal cost and the price adjustment will be immediately completed. What SW did is equivalent to adding shockε^m_t to this right-hand side.

ˆ

π_t= ˆM C_t+ε^m_t

Now, the same result can be obtained by adding the following shock to the (gross) markup rate (1 +λ) before log-linear approximation:

P_t(i)^∗ Pt

= (1 +λ)e^εmt Ψ_t

From such a point of view, it certainly might be possible to capture it like a shock associated with the gross markup rate. However, when considering another shock called “relative price shock” that distorts the relative price of the intermediate goods price against the general price, the relative price shock could enter into the log-linearized NKPC in the same form as the markup shock (observational equivalent). It is criticized to add a structural interpretation by regarding ad hoc markup shocks as structural shocks (de Walque et al. 2006, Chari et al. 2009). For example, if the shock ofε^RP_t is added as “relative price shock” as follows, the resulting aggregate supply curve after the log-linear approximation is the same (only the sign of the shock is different):

P_t(i)^∗ Pt

e^εRPt = (1 +λ)Ψ_t

Therefore, for the question of what is the source of inflation variations in this model, it can be explained that the change in markup is the source and it can be also explained that fluctuation of a shock distorting the relative price of intermediate goods against general price. In other words, adding shocks easily leads to a problem of lowering the advantage of the empirical DSGE approach that it can provide structural interpretations in line with the theoretical model for output and inflation fluctuations.

Similarly, in the empirical aspect, there is also a contradiction in interpreting that the main source of inflation fluctuations is due to the SW type markup shocks. In the context of the busi-ness cycle in macroeconomics, there is a long debate whether output and markup are pro-cyclical or counter-cyclical. According to the current consensus based on empirical analysis, output and markup is counter-cyclical (Bils 1987, Warner and Barsky 1995, Chevalier and Scharfstein 1996).

2.1. INTRODUCTION 45 In this case, it is necessary for the markup to decline (against the nominal marginal cost) during the booming inflation phase.

Consider about the situation where the boom and inflation occur simultaneously due to the negative monetary policy shock (ε^R_t < 0: Monetary easing policy) in a simple model. To meet the empirical consensus that output and markup are counter-cyclical, the markup shock ε^m_t must respond negative. That is, a positive correlation is required between “structural shocks” (monetary policy shock and markup shock), which have to be independent from each other.

In summary, it is difficult to justify the SW type markup shock as a “structural shock” both theoretically and empirically. So, what kind of shock should we capture inflation fluctuations?

According to data, it is well known that inflation since the 1990s shows a very volatile behavior in both Japan and the U.S. How well can DSGE models or other time series models predict such volatile inflation?

Edge and Gurkaynak (2010) examined predictability for inflation in the U.S. using an official DSGE model of FED (EDO model; Estimated Dynamic Optimization Model). As a result, the prediction accuracy of the DSGE model for recent inflation was extremely low.

Unfortunately, however, the low prediction accuracy for inflation is not attributable to the DSGE model’s characteristics that there are a lot of cross-equation constraints and parameter restrictions.

According to VAR analysis by Cogley et al. (2010), inflation before the 1990s is smooth fluctuations with inertia, but since the 1990s it shows a noisy and volatile movement and therefore they concluded that it is difficult to predict inflation even in the VAR model. In addition, from the results using dynamic factor model (DFM), which is adopted to measure the diffusion index (DI) in the U.S., inflation after the 1990s is still volatile and it is difficult to predict (Stock and Watson 2007). In the end, even if we employ atheoretical time series models with little equation constraints or parameter restrictions, it is still difficult to capture inflation fluctuations.

Basically, it is the first best to further brush up the structural model expressing the pricing behavior of intermediate goods firms, to derive the aggregate supply curve that output and markup become counter-cyclical and can explain the volatile inflation. However, even reduced-form time series models such as VAR and DFM are difficult to predict inflation. It seems difficult to explain inflation fluctuations by building a structural DSGE model with many equation constraints and parameter restrictions. So, as a second best, we focused on the role of measurement errors.

Introducing measurement errors means to divide the data fluctuations into two components: a component explainable by the model and the other component unexplainable by the model. The state equation expresses any endogenous variable as a linear combination of all structural shocks (and initial values of endogenous variables). For example, fluctuations in the productivity shock of the NKPC systematically spread to various endogenous variables through E(θ) and G(θ) (see (1.64)). On the other hand, according to the measurement equation, any measurement error added to a certain data does not affect other data and other endogenous variables. In other words, the measurement error can be regarded as “idiosyncratic component” (or unique factor) that only the data has. Given that current inflation data shows volatile fluctuations and if it is not possible to predict or explain inflation well both in DSGE models and other time series models, we should separate the inflation data fluctuations into two components: The component that can be described by the model and the other component that cannot be explained by the model (measurement error=

idiosyncratic component of the data).

There are very few estimated DSGE models with measurement errors: Measurement errors are not introduced in SW (2003, 2007) for the Euro area and the U.S., in Sugo and Ueda (2008) for Japan, or in official estimated DSGE models of various central banks and governments. If

measurement errors are not introduced, it implies to capture inflation data by structural shocks.

However, according to SW (2007), the volatile fluctuations of inflation data is unfortunately forced to capture as fluctuations of ad hoc markup shocks which are difficult to interpret structurally.¹