Sectoral Heterogeneity in Nominal Rigidities in Korea: Implications for Monetary Policy Chung Gu Chee

Seul A Kimy

Jae Won Leez

April 2018

Abstract This paper documents how the frequency of price changes di¤ers across sectors in Korea and what implications such heterogeneity may have for monetary policy. We …nd that under heterogeneity: i) monetary policy has larger and more persistent real e¤ects; ii) it is welfare improving to stabilize an alternative (optimal) price index that places disproportionately larger weights on “stickier” sectors, rather than the Consumer Price Index; iii) the central bank, targeting such an alternative index, should move nominal interest rates more slowly, which may provide a justi…cation for the “gradualism” in monetary policy; and iv) the condition for equilibrium determinacy is di¤erent. Overall, our results suggest that it is potentially important for the monetary authority to take into account how …rms’pricing behaviors di¤er across sectors in the design of monetary policy. (JEL C51, E13, E31, E32, E44, J20 ) Keywords: heterogeneity, price stickiness, multiple sectors, DSGE model, monetary policy, economy of Korea

The Bank of Korea; [email protected] University of Maryland; [email protected] z University of Virginia; [email protected] y

1

1

Introduction

Earlier studies have shown that the frequency of …rms’price adjustments di¤er across sectors and such heterogeneity generally matters for monetary policy in several dimensions.1 Whether it matters signi…cantly or may safely be disregarded in practice, however, depends on the exact shape of the cross-sector distribution of the frequency of price adjustments. In this paper, we use a simple multi-sector New Keynesian framework as a laboratory to explore quantitative implications. We …rst infer the sectoral frequency of price adjustments along with the size of sectors, using time series data on sectoral prices and quantities. Given the estimates of the sector-speci…c measures of price stickiness, this paper then addresses four broad sets of questions. What are the e¤ects on output and in‡ation of an exogenous shift in monetary policy?; What measure of in‡ation should a central bank target?; How would nominal interest rates behave di¤erently if the central bank targeted alternative in‡ation indices?; What conditions an interest rate rule must satisfy to avoid sunspot-driven ‡uctuations in the in‡ation rate and output? These are all classic questions in the literature on monetary policy. What is new is that this paper revisits the questions, allowing for sectoral heterogeneity in nominal rigidities in the economy of Korea. We …nd that the degree of price stickiness does di¤er across sectors, which enables monetary policy to have larger and more persistent e¤ects on the economy. In addition, it is welfare improving to stabilize an alternative (optimal) price index that places disproportionately larger weights on “stickier”sectors, rather than the Consumer Price Index. It results in a moderate, yet non-negligible welfare gain of 0:18 percent of steady state consumption. Moreover, the central bank, targeting such an optimal index, adjusts nominal interest rates more gradually. This …nding may provide yet another justi…cation for the monetary authority’s preference for slow and cautious changes in the policy rate, which is often dubbed the “gradualism” in monetary policy. Finally, heterogeneity in price stickiness alters the condition for equilibrium determinacy –although not signi…cantly with our estimates. The paper is organized as follows. Section 2 works through a structural model on which our quantitative analysis will be based in the ensuing sections. Section 3 introduces the sectors we consider and presents sectoral characteristics, including the size and the price-adjustment frequency. Section 4 reports various implications for monetary policy. Section 5 provides concluding remarks with some caveats. 1

This research agenda is relatively young, but is growing rapidly. Some non-exhaustive recent contributions include Bils and Klenow (2004), Nakamura and Steinsson (2008), Dhyne et al. (2006), Aoki (2001), Benigno (2004), Mankiw and Reis (2003), Carvalho (2006), Carvalho and Lee (2011), Carvalho, Dam and Lee (2016), Barsky, House and Kimball (2006), Carvalho and Schwartzman (2006) Boukaez, Cardia and Ruge-Murcia (2005), Eusepi, Hobijin, and Tambalotti (2012), Lee (2016), and Carlstrom, Fuerst and Ghironi (2006).

2

2

Model

Our framework is a standard multi-sector New Keynesian model (Woodford, 2003; Benigno, 2004; Carvalho, 2006; Carvalho and Lee, 2011; Bhattarai, Lee and Park, 2014; and others.)2 The economy on the unit interval [0,1] is divided into J sectors. Sectors, indexed by j 2 f1; 2; ; Jg, produce di¤erentiated goods and are characterized by di¤erent degrees of price stickiness f j gJj=1 . We use Ij and nj =length(Ij ) to denote respectively the subinterval and the size for sector j. When the degree of price stickiness is identical across sectors (i.e. j = 8j), the model is reduced to a standard single-sector New Keynesian model.

2.1 2.1.1

Households Household problem

The representative household in sector j, who gains utility from consuming composite goods and disutility from working, maximizes the expected lifetime utility: E0

"

1 X t=0

t

(

1 nj

U (Cj;t )

Z

Ij

V (Nj;t (i)) di

)#

;

where the period (dis)utility functions have the following functional forms: 1 Cj;t U (Cj;t ) = 1

1

1+ Nj;t (i) and V (Nj;t (i)) = : 1+

The parameter is the coe¢ cient of relative risk aversion; 1 is the the Frisch elasticity of labor supply; Cj;t and Nj;t (i) denote respectively the household’s consumption of composite goods and labor supply to …rm i in sector j at time t. The household faces the ‡ow budget constraint: 1 Pt Cj;t + Et [Qt;t+1 Bj;t+1 ] = Bj;t + nj

Z

1 Wj;t (i)Nj;t (i)di + nj Ij

Z

j;t (i)di

Pt Tt ;

Ij

where Pt is the price of one unit of composite goods –i.e. the Consumer Price Index (CPI). The household’s disposable income at time t is given by the sum of labor income earned by R supplying labor hours to …rms in sector j, n1j Ij Wj;t (i)Nj;t (i)di, and pro…t income from the 2

The model is a simple extension of a textbook (three-equation) New Keynesian model. It abstract from many important features such as capital accumulation, dynamic price indexation, consumption habit formation, incomplete …nancial markets, sectoral heterogeneity other than price stickiness, and international trade, all of which may potentially a¤ect the quantitative results. Therefore, while our analysis may serve as a useful starting point for a future study that explores the implications of sectoral heterogeneity in Korea for monetary policy, our results should be taken with caution.

3

R ownership of …rms in sector j, n1j Ij j;t (i)di, net of lump-sum taxes, Pt Tt . We assume that a complete set of securities that completely spans all the states of nature is available: Bj;t is the nominal payo¤s and Qt;t+1 is the nominal stochastic discount factor. The …rst order optimality conditions are standard: Nj;t (i) Cj;t = Wj;t (i)=Pt and Rt 1 = Et Cj;t =Cj;t+1 (Pt =Pt+1 ) , where Et Qt;t+1 = Rt 1 . 2.1.2

Consumption aggregates and price indices

Following Obstfeld and Rogo¤ (1998, 2000), we assume that composite consumption goods are given by a Cobb-Douglas aggregate of sectoral goods: J Y

Cj;t

Cj;k;t nk

k=1

nk

where Cj;k;t is the sector j representative household’s consumption of sector k goods, which in turn is given by a CES aggregate of individual goods produced in that sector:

Cj;k;t

"

1

1 nk

Z

Cj;k;t (i)

1

di

Ik

#

1

:

The parameter measures the elasticity of substitution between goods. The solution for the standard expenditure minimization problem yields the price indices and the demand functions for sectoral and individual goods. The Consumer Price Index, Pt , and the sectoral price index, Pj;t , are given as: Pt =

J Y

(Pj;t )nj ;

j=1

Pj;t =

"

1 nj

Z

Pj;t (i)1

di

Ij

#11

:

The sector j representative household’s demand for sectoral and individual goods are then given as: Cj;k;t = nk Cj;k;t (i) =

Pk;t Pt

Pk;t (i) Pk;t

1

Cj;t ; Pk;t Pt

1

Cj;t :

Lastly, the aggregate consumption for the economy is obtained by taking the sum of con-

4

sumption of all representative households: J X

Ct

nj Cj;t

j=1

The total demand for sectoral and individual goods then can be written as: Pk;t Pt

Ck;t = nk

Pk;t (i) Pk;t

Ck;t (i) = where Ck;t =

2.2

PJ

j=1

nj Cj;k;t and Ck;t (i) =

PJ

j=1

1

Ct ; 1

Pk;t Pt

(1)

Ct ;

nj Cj;k;t (i).

Firms

A monopolistically competitive …rm produces a di¤erentiated product using a linear production function: Yj;t (i) = Aj;t Nj;t (i); where Aj;t is the level of sector-speci…c productivity and evolves exogenously as: log Aj;t =

A j

log Aj;t

1

+ "j;t ;

"j;t

i:i:d

N (0;

A2 j ):

(2)

Prices are sticky as in Calvo (1983) and Yun (1996): each producer in sector j resets his or her price with a …xed per-period probability (1 j ). A price-adjusting …rm therefore sets its price Pj;t (i) to maximize a discounted sum of the current and future expected pro…ts: max Et

Pj;t (i)

1 X

k j Qt;t+k

[Pj;t (i)Yj;t+k (i)

(1

s)Wj;t+k (i)Nj;t+k (i)] ;

k=0

subject to the demand function for its product: Yj;t (i) =

Pj;t (i) Pj;t

C

Pj;t Pt

1

Yt ;

j;t t where Qt;t+k = k PPt+k , obtained from the household intertemporal optimality Cj;t+k conditions, is the stochastic discount factor between t and t+k. At each time t, the government provides each …rm with an employment subsidy sWj;t (i)Nj;t (i), and the subsidy rate, s, is set to cancel out the mark-up charged by a …rm over its marginal cost. The demand function is

5

obtained from (1) imposing market clearing conditions.3 The …rst-order optimality condition is given by: Et

1 X

k j Qt;t+k

k=0

Pj;t (i) Pj;t+k

Wj;t+k (i) 1 Pj;t+k Aj;t+k

j Yt+k nj

Pj;t (i) Pj;t+k

!

= 0:

Since …rms within a sector that update their prices at the same time choose a common price Pj;t (i) = Pj;t , the law of motion for the sectoral price index is given by 1 j Pj;t 1

Pj;t = Notice that the Calvo parameter, rigidities.

2.3

j,

+ (1

j )Pj;t

1

1 1

:

is sector-speci…c, which induces heterogeneity in nominal

Government

The government collects lump-sum taxes and uses the tax revenues to …nance its purchases and the total employment subsidy given …rms. The government budget constraint is therefore written as: Z J X 1 Wj;t (i)Nj;t (i)di = Pt Tt : Pt Gt + s n j I j j=1 For simplicity, we assume Gt = 0 throughout the paper.

2.4

Market clearing conditions

We assume that one-period-ahead state-contingent assets have zero net-supply. The …nancial market clearing condition is: J X Bj;t+1 = 0; j=1

which should hold for every possible state that may occur at time t + 1. For each type of product, the quantity demanded must equal the quantity supplied: Ck;t (i) = Yk;t (i); 8i;

3

For simplicity, the model is abstract from government purchases and investment.

6

which implies that, at the aggregate level, consumption equals output: J X

nj Cj;t = Ct = Yt :

j=1

Furthermore, since asset markets are complete, one can easily show that Cj;t = Ct = Yt ; 8j; with an appropriate initial condition for wealth distribution.

2.5

Monetary policy

To close the model, two types of monetary policy are considered in turns. First, we consider a “strict in‡ation targeting” in which the central bank sets the growth rate of an aggregate price index to its target level (zero in our model). In other words, we assume the central bank commits itself to following a “targeting rule”of the form: target t

=

J X

j

j;t

= 0;

j=1

where j;t log Pj;t log Pj;t 1 is the growth rate of the sectoral price index (i.e. sectoral P in‡ation) and j is the relative weight with Jj=1 j = 1. An important special case arises when j = nj . In this case, target is simply the CPI in‡ation rate. It can be shown that t PJ the CPI in‡ation targeting ( j=1 nj j;t = 0) is optimal when the degree of price stickiness is identical across sectors.4 While the type of in‡ation targeting mentioned above has some good properties (at least in theory,) it is often not the best description of practical monetary policy as central banks around the world may have policy objectives other than in‡ation stabilization –such as real activity stabilization, interest rate smoothing and others. In empirical studies, therefore, researchers often consider various forms of “interest rate rules” (or Taylor rules.) In this paper, we consider a standard Taylor rule in which the central bank adjusts nominal interest rates when the in‡ation rate and output deviates from their respective target levels.

2.6

Equilibrium

Equilibrium is characterized by an allocation of quantities and prices that satisfy the households’optimality conditions and budget constraint, the …rms’optimality conditions, the mon4

It is suboptimal under heterogenous price stickiness.

7

etary policy rule, and the market-clearing conditions. We solve the model by log-linearizing the equilibrium conditions around a deterministic steady state. The Appendix provides the full set of log-linearized equations for the interested readers.

3

Sectoral frequency of price adjustments

Earlier studies have shown that heterogeneity in price stickiness generally matters for monetary policy in numerous ways, ranging from the size of real e¤ects of monetary policy changes to the appropriate in‡ation index a central bank should target to the design of optimal monetary policy. However, whether it matters signi…cantly or may safely be neglected in practice will depend on the exact shape of the cross-sector distribution of the degree of price stickiness. We therefore …rst estimate the sectoral infrequency of price adjustments f j g, along with the sector size fnj g, using the time series data on sectoral prices and quantities.

3.1

Sectoral frequency and size

For empirical exercises, we map the “model sectors” into the twelve consumption categories in household consumption. The twelve sectors are the second level disaggregation of the aggregate consumption and are reported in Table 1. We take a semi-structural approach to estimate the sectoral frequency of price adjustments. The Calvo pricing scheme implies that the dynamics of the in‡ation rate in sector j can be approximated by (3) j;t = j j;t 1 + "j;t ; where "j;t is a serially uncorrelated exogenous term.5 Equation (3) indicates that sectoral in‡ation is more persistent if …rms in that sector change prices less frequently. We take the quarterly data on sectoral price indices from the Bank of Korea and the Korean Statistical Information Service and estimate j by MLE methods. Given estimated j , we then derive the average duration of price spells by 1=(1 j ). Besides (3), we do not impose other restrictions implied by our structural model. This “semi-structural” approach gives ‡exibility relative to a fully-structural model and thus serves our purpose well.6 7 5

The underlying assumption is that …rms’marginal costs follow a random walk. We refer the interested readers to Bils and Klenow (2004) for details. 6 However, for other questions we address in the paper, a fully-structural model is necessary. 7 A better way to estimate j is perhaps looking at the micro-level data directly (with no economic models). Obvious costs of this approach are time and resources. Moreover, micro data on prices usually contain measurement errors and are often not publicly available. We leave this interesting endeavor for future research.

8

The estimation results are summarized in Table 1. The measure of price stickiness differs across sectors as expected. However, the overall magnitude of heterogeneity is not large. According to our estimates, prices of some expenditure categories – such as “Restaurants and Hotels” and “Clothing and Footwear” – are quite sticky. Firms, producing those goods and services, take about thirteen and eleven months, respectively, before they adjust their prices. On the other hand, prices are quite ‡exible in many other sectors –such as “Food and Non-alcoholic beverages,” “Alcoholic beverages and tobacco,” “Transportation,” “Communication,” and “Recreation and Culture.” Furthermore, the degree of price stickiness is quite similar among those sectors: our estimates indicate that …rms change their prices every 3-4 months on average. Table 1: Sectoral Frequency of Price Adjustments CPI expenditure categories nj (%) 1 Food and non-alcoholic beverages 14.15 2 Alcoholic beverages and tobacco 1.30 3 Clothing and footwear 6.44 4 Housing, water, electricity and other fuels 10.13 5 Furnishings, household equipment and routine household 3.73 maintenance 6 Health 6.36 7 Transportation 12.28 8 Communication 6.41 9 Recreation and culture 5.51 10 Education 11.77 11 Restaurants and hotels 13.27 12 Miscellaneous goods and services 8.65

0.1943 0.0435 0.7234 0.6180 0.4679

Duration 1.24 1.05 3.62 2.62 1.88

0.4690 0.3613 0.0583 0.2497 0.5578 0.7720 0.4231

1.88 1.57 1.06 1.33 2.26 4.39 1.73

j

We suspect the lack of signi…cant di¤erences in estimated j s among some sectors may have resulted from the fact that the sectors under consideration are still broad. Some idiosyncrasies that would be observed at a highly disaggregate level are likely to be averaged out through aggregation. It is therefore possible that our quantitative results in the ensuing sections understate the importance of heterogeneity.8 The table also reports the size of the sectors, fnj g. In the model, they are equal to the P Y steady-state value of Pj;tt Yj;t . We calibrate the parameters so that they match the average t expenditure share of each consumption category over our sample periods, 2003-2016. Interestingly, the stickiest consumption category, “Restaurants and Hotels,” happens to be one of the largest in size: about 13 percent of the total household consumption expenditure. 8

Investigating a lower level disaggregation of consumption data will be interesting. We leave that as a (near) future research project.

9

3.2

Other parameters

We mostly use the standard values for remaining parameters (Table 2.) We set the time discount factor to 0.99 so that the steady-state annual real interest rate is 4 percent. The risk aversion parameter, , and the inverse of Frisch elasticity of labor supply, are set to 1.3 and 1 respectively. Finally, the within-sector elasticity of substitution, is set to 6. We take the values from Bae (2013) which analyzes the economy of Korea using a DSGE model.9 We estimate the AR(1) process for the sectoral productivity, assuming the persistence A A 10 , and A :) and volatility parameters have common values across sectors ( A j = j = This assumption is made mainly because disaggregate labor hour data are unavailable for the sectors we consider. It also enables us to focus on the price stickiness heterogeneity in isolation, shutting down other sources of heterogeneity. We use the seasonally-adjusted quarterly data on real output and total hours worked to obtain the labor productivity and then estimate (2). Our estimate of A is 0.34, which is smaller than conventional values often used in other studies. Therefore we repeat the same analysis with larger values for robustness and …nd that our numerical results are largely una¤ected.11

A A

4

Table 2: Model Parameter Values 0.99 Time discount factor 1.3 Risk aversion 1 Inverse of Frisch elasticity 6 Within-sector elasticity of substitution 0.34 Persistence of the productivity shocks 0.0138 Standard deviation of the shock innovation

Implications for monetary policy

Given the estimates of f j g, we now use our structural model as a laboratory to explore the implications of heterogeneity in price stickiness for monetary policy in Korea. In particular, we address four broad sets of questions. First, what in‡ation index should an in‡ation targeting central bank stabilize? Will the CPI in‡ation targeting still be desired under heterogeneity in f j g ? If not, how will an alternative (and better) in‡ation index look like? Second, does the cross-sector heterogeneity 9 See Kim (2014) for another paper that uses a DSGE model to study Korea. Ahn and Kim.(2008) and Kim and Park (2006) and Chung, Jung, and Yang (2007) analyze monetary policy in Korea using a similar framework. 10 Realized shocks are still di¤erent across sectors. 11 For example, see Section C in the Appendix for the results when A = 0:84 and A = 0:0084, the values used in Bae (2013).

10

amplify the real e¤ects of monetary policy? If so, how much? Third, how does it a¤ect the equilibrium determinacy condition when monetary policy is characterized by an interest rate rule? Forth, how do the implied nominal interest rates look like if a monetary authority targets an alternative price index in‡ation (that takes the sectoral heterogeneity into account) instead of the standard CPI in‡ation? How do they compare to the data?

4.1 4.1.1

Implications for in‡ation targeting What measure of in‡ation should a central bank target?

It is an important question, and the answer depends on the cross-sector distribution of price stickiness summarized by f j ; nj g. To address the question, we consider an in‡ation targeting central bank that follows a strict targeting rule of the form: target t

=

J X

j

j;t

= 0;

j=1

P 2 [0; 1] and Jj=1 j = 1. An important special case is when j = nj , in which case, target is simply the growth rate of the CPI. It has been shown that it is desired to target the t CPI in‡ation under homogeneous price stickiness ( j = for all j) in our framework, as it maximizes the welfare of households. However, the CPI in‡ation targeting is generally suboptimal because it overlooks sectoral heterogeneity, thereby treating all sectors symmetrically. Since in‡ation in stickier sectors generates greater welfare losses, it is welfare improving to place disproportionately large weights on sticky sectors (Aoki, 2001; Benigno, 2004; Eusepi, Hobijn and Tambalotti, 2011; Lee and Sung, 2016.) To see this more clearly, we follow the literature and take a second order approximation of the household welfare. Notice that the period utility ‡ow of all households in the economy is given by Z J X 1 1 V (Nj;t (i))di ; Wt nj U (Cj;t ) nj 0 j=1 where

j

11

while the second order approximation is given as12 Wt W WC C

1 Lt 2

where Lt = ( + ) Y^t

Y~t

2

+ (1 + )

J X

R P^j;t

R P~j;t

2

+ (1 +

)

j=1

J X j=1

(1

nj j j )(1

j

)

2 j;t :

Then (disregarding third and higher order terms,) a welfare-maximizing central bank will minimize the following loss function: L=

E0

1 X t=0

t

Wt W WC C

1 X = E0 2 t=0 1

t

Lt :

The terms in the period loss function Lt clearly shows that nominal rigidities generate distortions at various levels of aggregation. As is well understood, at the aggregate level, output deviates from the natural level re‡ected by Y^t Y~t , while each sectoral in‡ation 2 j;t represents ine¢ cient …rm-level price (and production) dispersions within a sector. In addition, nominal rigidities prevent the relative prices from adjusting to allocate resources e¢ ciently across sectors in response to sectoral disturbances in productivity. In other words, the relative outputs deviate from their “e¢ cient” levels (that would prevail under ‡exible R R appear in the loss function. P~j;t prices,) and hence the relative price gaps P^j;t When the frequency of price adjustments di¤ers across sectors, targeting an in‡ation index that places a larger weight on stickier sectors is welfare improving over the simple CPI in‡ation targeting for two reasons. First, resource misallocations within sectors are more pronounced in a stickier sector as a larger fraction of …rms does not adjust prices in response to shocks. In the loss function, one can verify that the coe¢ cients on sectoral in‡ation are disproportionately large for sticky sectors (i.e. sectors with large j .) Second, targeting such a (non-symmetric) in‡ation index allows ‡exible-sector prices to move even more freely, which enables the relative prices to adjust not only faster but also with a smaller welfare cost.13 Given the parameter values, in particular the cross-sector distribution f j ; nj g, one can obtain the optimal set of weights j that minimizes the loss function. By construction, For a variable Xt , X^t denotes the log-deviation of Xt from its steady state value, log Xt log X, and X~t R denotes the same – but under fully ‡exible prices. In addition, X^j;t X^j;t X^t denotes the log-deviation R of sector j variable (Xj;t ) from its aggregate counterpart (Xt ), and X~j;t X~j;t X~t denotes the same – but under ‡exible prices. 13 This is because in‡ation ‡uctuations in ‡exible sectors do not result in large welfare losses, as mentioned above. 12

12

P targeting the “optimal price index” (OPI) in‡ation ( Jj=1 j j;t = 0) is welfare improving P over the CPI in‡ation targeting ( Jj=1 nj j;t = 0), except for the homogeneous frequency case ( j = .)14 Moreover, it has been shown that the OPI in‡ation targeting almost replicates (unconstrained) optimal monetary policy which does not have to follow any parametric rule, yet hard to implement in reality (Aoki, 2001 and Benigno, 2004.) 4.1.2

Optimal price index for the economy of Korea

In this subsection, we obtain numerically the optimal weights j for the twelve consumption categories and see how the weights are related to sectors’characteristics, such as stickiness of prices and size. We then compare the OPI to the CPI to see how the two have been behaving di¤erently over our sample periods. Table 3 reports the optimal weights along with the size and the price stickiness measure of the sectors. As discussed above, the optimal weights are positively correlated with the Calvo parameters. For example, our estimates indicate that “Restaurants and hotels” is the stickiest sector and thus should receive the highest attention by the central bank under the OPI in‡ation targeting: its weight (31.64 percent) in the OPI is more than doubled from that in the CPI. On the other hand, "Alcoholic beverages and tobacco," the most ‡exible sector, receives the lowest weight (0.15 percent,) which indicates that the central bank should let that sector’s prices to move freely. Table 3: Optimal weights CPI expenditure categories nj (%) 1 Food and non-alcoholic beverages 14.15 2 Alcoholic beverages and tobacco 1.30 3 Clothing and footwear 6.44 4 Housing, water, electricity and other fuels 10.13 5 Furnishings, household equipment and routine household 3.73 maintenance 6 Health 6.36 7 Transport 12.28 8 Communication 6.41 9 Recreation and culture 5.51 10 Education 11.77 11 Restaurants and hotels 13.27 12 Miscellaneous goods and services 8.65 Correlation with optimal weights 0.5164

j

j (%)

0.1943 0.0435 0.7234 0.6180 0.4679

3.46 0.15 15.91 15.90 2.50

0.4690 0.3613 0.0583 0.2497 0.5578 0.7720 0.4231 0.8403

5.78 6.04 0.47 1.69 11.25 31.64 5.20 1

The optimal weights are also positively correlated with the sector sizes. However, the e¤ect 14

In this case,

j

= nj .

13

of the size on the optimal weights is not as signi…cant as that of price stickiness. The correlation between the optimal weights ( j ) and the Calvo parameters ( j ) is 0.8403, while that between the optimal weights and the sector size (nj ) is 0.5164. Moreover, as can be inferred from the coe¢ cients in the loss function, the optimal weight increases at an accelerating rate as the degree of price stickiness increases. These results are illustrated in Figure 1, which shows scatter plots of the optimal weights and sector characteristics. Overall, our analysis suggests that a sector’s price stickiness is the most important determinant for the optimal price index. (b) Optimal weights and stickiness 35

30

30

Optimal weight (%)

Optimal weight (%)

(a) Optimal we ights and size 35

25 20 15 10 5

25 20 15 10 5

0

0 0

5

10

15

0

Size (%)

0.2

0.4

Stickiness (

0.6 j

0.8

)

Figure 1: Optimal Weights and Sector Characteristics Figure 2 shows the annualized OPI in‡ation rate ( opt;yr ) and the CPI in‡ation rates t cpi;yr ( t ) from 2003 to 2016. Not surprisingly, the two measures of in‡ation tend to move together. There are however noticeable di¤erences. The OPI in‡ation is more persistent and less volatile relative to the CPI in‡ation, as reported in Table 4.15 The OPI in‡ation is slowermoving because disproportionately larger weights are placed on slow-moving (sticky) sectors. As can be seen in the next section, this result has an important implication for the dynamics of nominal interest rates. Table 4: Volatility and persistence of in‡aton rates opt;yr

In‡ation rate index Standard deviation Autocorrelation

15

cpi;yr

1.6419 2.2215 0.3780 0.0626

We also report the statistics for the core-in‡ation for interested readers.

14

core;yr

1.5841 0.2756

10 cpi,yr t opt,yr t

Inflation rate(%)

8

6

4

2

0

-2 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016

Time

Figure 2: OPI in‡ation versus CPI in‡ation 4.1.3

Welfare analysis

Given small yet non-negligible di¤erences between the two measures of in‡ation, one may naturally ask the question of whether targeting the OPI in‡ation results in a signi…cant welfare improvement. To this end, we use the di¤erence between the (unconditionally) expected welfare losses associated with the two in‡ation targeting schemes as a measure of welfare improvement: (ELCP I

ELOP I )

100(%) =

E

P1

t=0

t

(WOP I;t WC C

WCP I;t )

100(%)

0;

where WOP I;t and WCP I;t denote the period utility ‡ow at time t under the two targeting regimes. Notice that the unit of the measure is WC C. Our numerical simulation shows that targeting the OPI in‡ation instead of the CPI in‡ation yields a welfare gain of 0:18 percent of steady state consumption. The welfare gain is not huge, yet should not be dismissed.

4.2

Implications for interest rate rules

While the (optimal) in‡ation targeting of the type described above is theoretically appealing, central banks often do not strictly commit themselves to such targeting rules in practice, perhaps to provide themselves with some ‡exibility. Instead it has been shown that a simple 15

Taylor rule – in which the central bank adjusts its policy instrument in response to changes in in‡ation and output – is a reasonably good characterization of monetary policy in many countries, including Korea. Given its empirical relevance, we address three questions associated with Taylor rules. Would a change in nominal interest rates have a di¤erent e¤ect on output and in‡ation under heterogenous price stickiness? Would the heterogeneity alter the condition for equilibrium determinacy? Last but not least, how would nominal interest rates behave di¤erently if the central bank responded to variations in the OPI in‡ation rather than the CPI in‡ation? 4.2.1

Implications for real e¤ects of monetary policy

It is well documented that the e¤ects of monetary policy depend crucially on the cross-sector distribution of price stickiness (Carvalho 2006, Nakamura and Steinsson 2008, Lee 2012 and others). In particular, a change in monetary policy has a bigger e¤ect on real output when the frequency of price adjustments di¤ers across sectors –due to strategic complementarity in price settings. The economic mechanism is straightforward. Firms in ‡exible sectors change their prices by a smaller amount than they would under homogeneous price stickiness because they do not wish to deviate from …rms in sticky sectors. The aggregate price level therefore adjusts more slowly. Motivated by the theoretical result, we investigate how heterogeneity in f j g we …nd in the previous section a¤ects the central bank’s ability to a¤ect real output in Korea. Figure 3 presents the impulse responses of output, in‡ation and the price level to an exogenous decrease in the nominal interest rate by 25 basis points.16 It shows that the response of in‡ation is more muted under heterogeneous price stickiness, and consequently monetary policy has larger and more persistent e¤ects on output. For example, two years after the shock, output is still above the steady state by about 0.2 percent under heterogeneity while the e¤ect of the shock nearly dies out under homogeneity, due to a faster adjustment of the price level. Panel (b) shows that at …rst the in‡ation rate under homogeneity is greater than that under heterogeneity to some point (6 quarters) and then becomes smaller. This implies that the price level increases and reaches the new steady state with a greater speed under homogeneity, which can be seen in panel (c). The results suggest that it may be important for the monetary authority to take into account how …rms’pricing behaviors di¤er across sectors when adjusting its policy instrument.

16

To produce the impulse responses, we have employed the conventional Taylor rule that will appear in the ensuing subsections.

16

(a) Re spons e of output

(b) Response of inflation

(c) Re sponse of the price level

0.7

Deviations from the steady state (%)

Deviations from the steady state (%)

1.5

1

0.5

0

1.8 homogeneous

0.6

1.6

0.5 1.4 0.4 1.2 0.3 1 0.2 0.8

0.1 0

0

5

10

heterogeneous

0.6 0

Periods after impact

5

10

0

Periods after impact

5

10

Periods after impact

Figure 3: Impulse responses to an expansionary monetary shock 4.2.2

Implications for determinacy condition

One of the most important guiding principles for practical monetary policy is that the central bank needs to adjust nominal interest rates more than one-for-one to a change in in‡ation in the long run in order to ensure price stability. Otherwise, an interest rate rule leads to equilibrium indeterminacy in which nonfundamental factors (often referred as sunspot shocks) may cause unintended ‡uctuations in the price level and output. To …x ideas, we consider a standard Taylor rule with an interest rate smoothing term: it =

i it 1

+ (1

i )f{t

+

(

t

t)

+

~t g yy

(4)

where it is the (net) nominal interest rate, it is the target (natural) rate of interest, t is the target rate of in‡ation, and y~t is the output gap. The parameter i measures the persistence of the interest rate, while and y measures how much the interest rate responses to the in‡ation gap and the output gap respectively. Given the Taylor rule, the determinacy condition in conventional New Keynesian models (with j = ) is given by 1 > 1: (5) + y The second part, y 1 , however, is insigni…cant as is close to one. Thus, under reasonable calibration, the coe¢ cient on in‡ation has to be greater than one to guarantee a unique equilibrium. The determinacy condition in theory depends on the cross-sector distribution of the fre-

17

2 Homogeneous Heterogeneous

1.8 1.6 Determinacy 1.4 1.2 1 0.8 0.6 0.4 Indeterminacy 0.2 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

y

Figure 4: Determinacy and indeterminacy regions quency of price adjustments, as shown in Lee and Park (2016), because the entire crosssectional distribution of …rms a¤ects equilibrium dynamics. This implies that the ranges of the policy coe¢ cients ; y that guarantee a unique equilibrium may well be substantially di¤erent from those given in (5). Our numerical simulations however show that the di¤erence is quantitatively insigni…cant for the sectors we consider here. Figure 4 presents the (in)determinacy regions for homogeneous (solid blue line) and heterogeneous frequency case (dashed red line.) It shows that for any given value of y , smaller values of can guarantee equilibrium determinacy when sectors di¤er in price stickiness. However, for the empirically relevant range of y (less than 0.5,) the di¤erence is small. Once again, we suspect that the small e¤ect is resulted from the lack of su¢ cient amount of heterogeneity across the sectors under consideration. If we instead considered more disaggregate sectors, the result might have been di¤erent. 4.2.3

Implications for policy instrument

Assuming the Taylor rule (4) approximates well the monetary policy of the Bank of Korea, how would the implied nominal interest rate look di¤erent if it targeted the OPI instead of the CPI? To address the question, we parameterize the policy coe¢ cients in (4) following Bae (2013).

18

We set the smoothing parameter i to 0.913, the coe¢ cient on the in‡ation gap to 1.66, and the coe¢ cient on the output gap y to 0.024. In addition, we use the previous-period uncollateralized call rates (on an annual basis) for it 1 , the low-frequency component (obtained from HP …lter) of uncollateralized call rates for {t , the target rate of the annualized in‡ation announced by the Bank of Korea for t , and the HP-…ltered log(GDP) as the measure of the output gap y~t . Finally, we use two alternative (annualized) in‡ation rates – the CPI in‡ation ( cpi;yr ) and the OPI in‡ation ( opt;yr ) –in place of t to compare the implied policy t t instrument.17 Figure 5 shows the interest rates (icpi;yr , iopt;yr ) implied by the Taylor rule with the two t t di¤erent target in‡ation rates, as well as the actual time series data (idata;yr ) on the uncollatt eralized call rate. Several …ndings are worth mentioning.

6 cpi,yr t iopt,yr t idata,yr t

i

5.5 5

Interest rate(%)

4.5 4 3.5 3 2.5 2 1.5 1 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016

Time

Figure 5: The nominal interest rate implied by the Taylor rule First, perhaps not surprisingly, the simple Taylor rule is in fact a good approximation of actual monetary policy. The actual policy rate is more persistent and less volatile than the arti…cial data generated by the rule. This may capture the Bank of Korea’s preference for smoothing out the movements in nominal rates. However, the discrepancy between the data and the model is relatively small. Second, the model implied interest rates (icpi;yr , iopt;yr ) are overall similar to each other. t t This implies that it would not be a signi…cant mistake if the monetary authority did not 17

For the interested readers, the Appendix provides the result also with the core in‡ation (

19

core;yr ). t

adopt the optimal index. However, in some time periods, non-negligible di¤erences do exist between icpi;yr and iopt;yr . For example, the di¤erence was about 40 basis points in 2014 and t t 2005. Overall we …nd that iopt;yr shows more persistence and less volatility compared to t cpi;yr it (as reported in Table 5,) precisely because the OPI in‡ation is more inertial than the CPI in‡ation as shown in Figure 2. This …nding naturally leads to our third (and perhaps somewhat surprising) point. Table 5: Volatility and persistence of nominal interest rates Interest rates from Taylor rule iopt;yr icpi;yr Standard deviation 1.1105 1.1399 Autocorrelation 0.9015 0.8610 Since both iopt;yr and idata;yr are more inertial than icpi;yr , the actual monetary policy t t t resembles the OPI in‡ation targeting more than the CPI in‡ation targeting. Table 6 reports the correlations between the model implied and the actual interest rates. Our analysis suggests that the Bank of Korea (perhaps with no intention) has been acting as if it is stabilizing the optimal index, which is welfare improving. This result is more pronounced in the recent periods (2013-2016,) in which the Bank of Korea did not lower the interest rate as much as what the CPI in‡ation targeting would dictate. Overall, our …nding provides another justi…cation for the “gradualism”in monetary policy. Table 6: Correlation between the model implied and the actual interest rates Interest rates from Taylor rule iopt;yr icpi;yr Correlation with idata;yr 0.9475 0.9365

5

Conclusion and caveats

This paper documents how the frequency of price changes di¤ers across sectors in Korea and what implications such heterogeneity may have for monetary policy. While our …ndings have potentially important implications for both policy and empirical analyses, they should be taken with a grain of salt – as the results are obviously model- as well as data-speci…c, and there are clearly some shortcomings in those regards. First, the structural framework employed in this paper is highly parsimonious. Moreover, it assumes a closed economy. While such simpli…cations are useful, they may have a¤ected the numerical results. Second, we consider only highly aggregated sectors in this paper, thereby assuming away much of heterogeneity that might be observed in more disaggregate sectors. Investigating 20

highly disaggregated consumption data is likely to lead us to a better understanding of this important topic. In sum, this paper can be a useful starting point for research on sectoral heterogeneity and its macroeconomic implications in the economy of Korea. At the same time, however, there remains much work to be done before we get more reliable answers.

21

References [1] Aoki, Kosuke, 2001. Optimal monetary policy responses to relative-price changes. Journal of Monetary Economics, 48(1): 55-80. [2] Barsky, R., C. House and M. Kimball, 2007. Sticky Price Models and Durable Goods. American Economic Review, 97(3): 984-998 [3] Bhattarai, Saroj, Jae Won Lee, and Woong Yong Park, 2015. Optimal Monetary Policy in a Currency Union with Interest Rate Spreads. Journal of International Economics, 96(2): 375-397. [4] Benigno, Pierpaolo, 2004. Optimal monetary policy in a currency area. Journal of International Economics, 63: 293-320. [5] Bils, Mark and Peter J. Klenow, 2004. Some Evidence on the Importance of Sticky prices. Journal of Political Economy, University of Chicago Press, vol.112(5): 947-985. [6] Bouakez, Hafedh, Emanuela Cardia, and Francisco J. Ruge-Murcia, 2009. The Transmission Of Monetary Policy in a Multisector Economy. International Economic Review, 50(4): 1243-1266. [7] Calvo, Guillermo A., 1983. Staggered prices in a utility-maximizing framework. Journal of Monetary Economics, vol.12(3): 383-398. [8] Carvalho, Carlos, Niels Arne Dam, and Jae Won Lee, 2016. The Cross-Sectional Distribution of Price Stickiness Implied by Aggregate Data. Unpublished working paper. [9] Carvalho, Carlos, and Jae Won Lee, 2011. Sectoral price facts in a sticky-price model, Sta¤ Reports 495, Federal Reserve Bank of New York. [10] Carvalho, Carlos, and Felipe Schwartzman, 2008. Heterogeneous Price Setting Behavior and Monetary Non-neutrality: Some General Results. Unpublished working paper. [11] Carlstrom, C., T. Fuerst and F. Ghironi, 2006. Does It Matter (for Equilibrium Determinacy) What Price Index the Central Bank Targets? Journal Economic Theory 128: 214-231. [12] Chung, Jaesik, Yongseung Jung, and Doo Yong Yang, 2007. Optimal monetary policy in a small open economy: The case of Korea. Journal of Asian Economics, 18: 125-143. [13] Eusepi, Stefano, Bart Hobijn, and Andrea Tambalotti, 2011. CONDI: A Cost-Of-NominalDistortions Index. American Economic Journal: Macroeconomics, 3(3): 53-91. 22

[14] Kim, Bae-Geun, and Byung Kwun Ahn, 2008. An Assessment of the New Keynesian Philips Curve in the Korean Economy. Economic analysis, vol.14, no.3. The Bank of Korea Economic Research Institute. [15] Kim, Soyoung, and Yung Chul Park, 2006. In‡ation targeting in Korea: a model of success?. BIS papers No.31 : 140-164. Bank for International Settlements. [16] Kim, Tae Bong, 2014. Analysis on Korean Economy with an Estimated DSGE Model after 2000. KDI Journal of Economic Policy, vol.36, no.2. Korea Development Institute. [17] Lee, Jae Won, 2014. Monetary Policy with Heterogeneous Households and Imperfect Risk-Sharing. Review of Economic Dynamics, 17(3): 505-522. [18] Lee, Jae Won, 2016. Heterogeneous Households, Real Rigidity, and Estimated Duration of Price Contracts in a Sticky Price DSGE Model. Unpublished working paper. [19] Lee, Jae Won, and Woong Yong Park 2016. The Cross-Sectional Distribution of Price Stickiness and In‡ation Stability. Unpublished working paper. [20] Lee, Jae Won, and Yeji Sung 2016. Optimal Index versus Simple Index for Monetary Policy. Unpublished working paper. [21] Mankiw, N. gregory, and Ricardo Reis, 2003. What measure of In‡ation Should a Central Bank Target?. Journal of the European Economic Association, MIT Press, vol 1(5): 10581086, 09. [22] Nakamura, Emi, and Jón Steinsson, 2008. Five Facts About Prices: A Reevaluation of Menu Cost Models. Quarterly Journal of Economics, 123(4): 1415-1464. [23] Obstfeld, M., K. Rogo¤, 1998. Risk and Exchange Rates. NBER Working Paper No.6694. [24] Obstfeld, M., K. Rogo¤, 2000. New directions for stochastic open economy models. Journal of International Economics, 60: 117-153. [25] Woodford, M., 2003. Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton NJ: Princeton University Press. [26] Yun, T., 1996. Nominal Price Rigidity, Money Supply Endogeneity, and Business Cycles. Journal of Monetary Economics 37, 345-370.

23

Appendix A

System of log-linearized equations For a variable Xt , we use the following notations: Xt

J X

nj Xj;t : Aggregate variable

j=1

R Xj;t

^t X ~t X

Xj;t

Xt : Relative variable

log Xt log XtN

log X : Percentage deviation from the steady state log X : Percentage deviation from the steady state under ‡exible prices.

Household’s optimality condition: 1 Y^t = Et Y^t+1 + (Et ^ t+1 ^j;t (i) + Y^t = W ^ j;t (i) N

P^t

De…nition of the relative price: R P^j;t = P^j;t R P~j;t = P~j;t

P^t P~t

Flexible price allocation: ( + )~ R P~j;t = Yt A^j;t ( + 1) J ( + 1) X ^ ~ Yt = nj Aj;t ( + ) j=1 Production function: ^j;t (i) Y^j;t (i) = A^j;t + N Demand function: Y^j;t =

(P^j;t

P^t ) + Y^t =

24

R P^j;t + Y^t

^t) R

Identities: ^ t;t+1 = Et Q j;t t R P^j;t

R P^j;t 1

^t R

= P^j;t P^j;t = P^t P^t 1 =

j;t

1

t

New Keynesian Phillips Curve (NKPC) from the …rm’s optimality condition: j;t

=

(1

j )(1 j (1

+

j

)

)

R [ ( + 1)(P^j;t

25

R ) + ( + )(Y^t P~j;t

Y~t )] + Et

j;t+1

B

Derivation of the welfare loss function

Following Woodford(2003), we derive the utility-based loss function. We take a second-order Taylor expansion of the utility function U (Cj;t ) around the steady state C. It is given by: U (Cj;t ) = U C + UC Cj;t

1 C + UCC Cj;t 2

C

2

+ o k k3

where o(k k3 ) denotes all the terms that are of third or higher order in the deviations of variables from their steady-state values. We also expand Cj;t with a second-order Taylor approximation and then it is obtained as: 1 2 Cj;t = C 1 + C^j;t + C^j;t + o k k3 2 where C^j;t = log (Cj;t ) be rewritten as this.

(6)

log C . Then, the deviation of Cj;t from its steady-state value C can

1 2 + o k k3 C = C C^j;t + C C^j;t 2 Substituting (7) into (5), we obtain

(7)

Cj;t

1 1 2 2 U (Cj;t ) = U C + UC C C^j;t + UC C C^j;t + UCC C 2 C^j;t + o k k3 2 2

(8)

Here, since U C is independent of monetary policy, we include it in t:i:p: which describes all the terms independent of monetary policy. Then, (8) becomes 1 1 2 2 + UCC C 2 C^j;t + t:i:p: + o k k3 ; U (Cj;t ) = UC C C^j;t + UC C C^j;t 2 2

(9)

which can be reorganized as 1 UCC 2 U (Cj;t ) = UC C C^j;t + UC C 1 + C C^j;t + t:i:p: + o k k3 2 UC Note that

UCC C UC

is de…ned from the utility function. Thus, we have

1 U (Cj;t ) = UC C C^j;t + (1 2

2 )C^j;t + t:i:p: + o k k3

(10)

Similarly, we take a second-order Taylor expansion of V (Nj;t (i)) around the steady state N . Then, we obtain V (Nj;t (i)) = V N + VN Nj;t (i)

1 N + VN N Nj;t (i) 2 26

N

2

+ o k k3

(11)

Also, the second order approximation of Nj;t (i) is derived as Nj;t (i) = N

^j;t (i) + 1 N ^j;t (i)2 + o k k3 1+N 2

(12)

The deviation of Nj;t (i) from its steady-state value N is written as: Nj;t (i)

^j;t (i) + 1 N N ^j;t (i)2 + o k k3 N = NN 2

(13)

By substituting (13) into (11), it is given that V (Nj;t (i)) = V N +VN

^j;t (i)2 + 1 VN N ^j;t (i) + 1 N N NN 2 2

^j;t (i) + 1 N N ^j;t (i)2 NN 2

2

+o k k3

By using the de…nition of o (k k3 ) and t:i:p:, we obtain ^j;t (i) + 1 VN N N ^j;t (i)2 + 1 VN N N 2 N ^j;t (i)2 + t:i:p: + o k k3 ; V (Nj;t (i)) = VN N N 2 2 which can be rewritten as ^j;t (i) + 1 VN N V (Nj;t (i)) = VN N N 2 Since

VN N N , VN

1+

VN N N VN

^j;t (i)2 + t:i:p: + o k k3 N

it becomes

^j;t (i) + 1 (1 + )N ^j;t (i)2 + t:i:p: + o k k3 V (Nj;t (i)) = VN N N 2

(14)

From the production function, we obtain ^j;t (i); Y^j;t (i) = A^j;t + N

(15)

^j;t (i) = Y^j;t (i) N

(16)

which derives A^j;t

Substituting (16) into (14), we have V (Nj;t (i)) = VN N (Y^j;t (i)

1 A^j;t ) + (1 + )(Y^j;t (i) 2

27

A^j;t )2 + t:i:p: + o(k k3 )

(17)

It can be rewritten as 1^ Yj;t (i)2 2

V (Nj;t (i)) = VN N Y^j;t (i) + (1 + )

A^j;t Y^j;t (i)

+ t:i:p: + o k k3

(18)

By integrating (18) over the range of Ij , we derive 1 nj

Z

V (Nj;t (i)) di =VN N

Ij

1 E Y^j;t (i) + (1 + )E Y^j;t (i)2 2

(1 + )A^j;t E Y^j;t (i)2

+ t:i:p: + o k k3

(19)

This implies that 1 nj

Z

V (Nj;t (i)) di

Ij

= VN N

1 1 E Y^j;t (i) + (1 + )V ar Y^j;t (i)2 + (1 + )E Y^j;t (i) 2 2

2

2 (1 + )A^j;t E Y^j;t

+ t:i:p: + o k k3

(20)

We take a second-order Taylor expansion of the aggregated obtaining 1 Y^j;t = E Y^j;t (i) + 2

1

V ar Y^j;t (i) + o k k3

1 2

1

V ar Y^j;t (i) + o k k3

It implies then, E Y^j;t (i) = Y^j;t and E Y^j;t (i)

2

2 = Y^j;t + +o k k3

By substituting (21) and (22) into (20), we derive 1 nj

Z

V (Nj;t (i)) di

Ij

= VN N

1 2 Y^j;t + (1 + )Y^j;t 2

1 1 (1 + )A^j;t Y^j;t + ( + )V ar Y^j;t (i) 2

+ t:i:p: + o k k3

28

(21)

(22)

Recall that we have N=

Y =Y Aj

Also, from the household’s intra-temporal optimality condition, we have W VN = = Aj = 1 UC P Thus, this implies VN = UC =)

VN N = UC Y = UC C

Combining these altogether, the household j’s utility function is given by wtj

= U (Cj;t )

1 nj

Z

V (Nj;t (i)) di

Ij

1 = UC C C^j;t + (1 2

1 2 UC C Y^j;t + (1 + )Y^j;t 2

2 )C^j;t

1 (1 + )A^j;t Y^j;t + 2

+

1

V ar Y^j;t (i)

+ t:i:p: + o k k3 Then, it is rearranged by 1 wtj = UC C C^j;t + (1 2

2 )C^j;t

Y^j;t

1 2 (1 + )Y^j;t + (1 + )A^j;t Y^j;t 2

+ t:i:p: + o k k3

1 2

+

1

V ar Y^j;t (i)

(23)

From (23), the weighted sum of the utility is J X j=1

nj wtj

=

J X j=1

= UC C

nj U (Cj;t )

Z

1

V (Nj;t (i)) di

0

" PJ

# P PJ PJ 1 2 ^2 ^ nj C^j;t + 21 (1 ) Jj=1 nj C^j;t (1 + ) n Y j j;t j=1 nj Yj;t j=1 2 P PJ ^ Y (1 + ) Jj=1 nj A^j;t Y^j;t 21 + 1 n V ar j;t (i) j=1 j j=1

+ t:i:p: + o k k3

(24)

Recall that the resource constraint is given by Yt =

J X j=1

29

nj Cj;t ;

which implies Y^t =

J X

nj C^j;t

(25)

j=1

Also, from the sectoral goods demand function, we have Y^j;t =

R P^j;t + Y^t ;

from which we get: J X

nj Y^j;t = Y^t

(26)

j=1

J X

2 nj Y^j;t =

j=1

J X

R 2 nj (P^j;t ) + (Y^t )2

(27)

j=1

P P R = Jj=1 nj (P^j;t P^t ) = 0. since Jj=1 nj P^j;t Meanwhile, from the ‡exible allocation, we obtain A^j;t =

+ ~ R )Yt P~j;t +( +1

Then, it is rewritten as: (1 + )A^j;t =

R (1 + )P~j;t + ( + )Y~t

30

(28)

Using (25), (26), (27), and (28) into (24), we have 2

2

PJ

+ t:i:p: + o k k3 2 1 ( + ) Y^t 2 4 = UC C +( + )Y~t Y^t

1 (1 2

R P^j;t

2

^ ) Y^t + ) Y^t + Y^t j=1 nj 6 Yt + 6 n o n o PJ R R nj wtj = UC C 6 (1 + )P~j;t + ( + )Y~t P^j;t + Y^t 6 + j=1 nj 4 j=1 PJ 1 ^ +1 j=1 nj V ar Yj;t (i) 2

J X

1 (1 2

7 7 7 7 5

3 2 P P R R ^R + ) Jj=1 nj P^j;t + (1 + ) Jj=1 nj P~j;t Pj;t 5 PJ 1 1 ^ ( + ) j=1 nj V ar Yj;t (i) 2

2

1 (1 2

+ t:i:p: + o k k3 2 2 2 2 1 ^t ~t ~t Y^t + Y~t Y Y ( + ) 2 Y 6 2 6 2 6 R R ~R R = UC C 6 1 (1 + ) PJ nj ^j;t P 2P^j;t Pj;t + P~j;t j=1 6 2 4 PJ 1 ^ +1 j=1 nj V ar Yj;t (i) 2

2

R P~j;t

3 7 7 7 7 7 5

2

+ t:i:p: + o k k3 Here, since Y~tW J X

2

nj wtj =

j=1

3

2

(29)

R and P~j;t

2

belong to t:i:p:, we can rewrite (29) by

2

2 P R ( + ) Y^t Y~t + (1 + ) Jj=1 nj P^j;t 1 UC C 4 PJ 2 ^ + +1 j=1 nj V ar Yj;t (i)

+ t:i:p: + o k k3

R P~j;t

2

3 5

(30)

Note that the demand for individual goods is given by Yj;t (i) =

Pj;t (i) Pj;t

Pj;t Pt

1

Yt

By log-linearization, it becomes Y^j;t (i) =

(P^j;t (i)

P^j;t )

(P^j;t

P^t ) + Y^t

Then, we have that V ar(Y^j;t (i)) =

2

V ar(P^j;t (i))

Here, V ar(P^j;t (i)) describes price dispersion within a sector. De…ning Pt

31

(31) Ei flog(Pj;t (i))g,

we obtain V ar(P^j;t (i)) = V ar(log(Pj;t (i))

Pt 1 )

= E((log(Pj;t (i)) Pt 1 )2 ) E(log(Pj;t (i)) Pt 1 )2 h i 2 2 ~ = j E((log(Pj;t 1 (i)) Pt 1 ) ) + (1 Pt 1 ) E(log(Pj;t (i)) j )(log(Pj;t (i)) h i ~ = j V ar(Pj;t 1 (i)) + (1 Pt 1 ) 2 E(log(Pj;t (i)) Pt 1 )2 j )(log(Pj;t (i)) h i 2 ~ = j V ar(Pj;t 1 (i)) + (1 )(log( P (i)) P ) (4Pt )2 j j;t t 1 (32) by de…ning 4Pt = Pt Pt 1 = log(Pj;t (i)) Pt 1 . Here, note that h i ~ 4Pt = j Pt 1 + (1 Pt 1 j )log(Pt (i)) h i ~ = (1 Pt 1 j ) log(Pt (i) Then, (32) is written

V ar(P^j;t (i)) =

jV

=

jV

1

ar(P^j;t 1 (i)) +

(4Pt )2

1

j j

ar(P^j;t 1 (i)) +

(4Pt )2

(4Pt )2

1

j

Note that Pt = log(Pj;t ) + o k k3 ; which implies 4Pt =

j;t

+ o k k3

Thus, we obtain V ar(P^j;t (i)) =

jV

ar(P^j;t 1 (i)) +

j

1

(

j;t )

2

j

+ o k k3 ;

and by backward iteration we have V ar(P^j;t (i)) =

t+1 j V

ar(Pj; 1 (i)) +

t X s=0

=

t X s=0

t s j

j

1

(

2 j;s )

j

32

t s j

j

1

(

2 j;s )

j

+ t:i:p: + o k k3

+ o(k k3 )

Pt 1 ) 2

since t+1 j V ar(Pj; 1 (i)) is independent of the policy conducted from t Then, we take the discounted value over time, we obtain 1 X

t

V ar(P^j;t (i)) =

t=0

j

(1

j )(1

)

j

1 X

t 2 j;t

t=0

0.

+ t:i:p: + o k k3

(33)

By substituting (33) into (31), 1 X

t

V ar(Y^j;t (i)) =

j

2

(1

t=0

j )(1

)

j

1 X

t 2 j;t

+ t:i:p: + o k k3

t=0

(34)

Now, we take the discounted sum of (30) with (34) 1 X t=0

t

J X

nj wtj =

j=1

1 UC C 2

2

2 ^t Y~t + (1 + ) PJ nj P^ R ( + ) Y j;t j=1 t4 P J ^ + +1 j=1 nj V ar Yj;t (i)

1 X t=0

+ t:i:p: + o k k3 =

X 1 UC C 2 t=0 1

1 UC C 2 =

1 UC C 2 1 UC C 2

=

1 UC C 2

+

t

1

1 UC C 2

( + ) Y^t

t

t=0

+

1

J X

nj

1 X

t

t=0

1 X

t

t=0

+ t:i:p: + o k k

3

Y~t

1 X

t

t=0

(

J X

4 2 4

+ (1 + )

J X

nj

R P^j;t

R P~j;t

2

j=1

nj

+

Y~t

J X

j

(1

1

j )(1

j

)

2

Y~t + (1 + ) P J j 2 j=1 nj (1 j )(1

( + ) Y^t + (1 +

+ (1 + )

2

( + ) Y^t +

2

nj

R P^j;t

j=1

(

2

3 5

)

V ar Y^j;t (i) + t:i:p: + o k k3

( + ) Y^t

2

2

j=1

j=1

1 X

+ t:i:p: + o k k3 =

(

R P~j;t

)

33

Y~t PJ

2

j=1

+ (1 + ) nj (1

j j )(1

1 X

t 2 j;t

t=0

PJ

j=1

nj

)

2 j;t

j

PJ j

j=1

nj

)

2 j;t

)

R P~j;t

2

)

+ t:i:p: + o k k3

R P^j;t

R P~j;t

R P^j;t

R P~j;t

2

3 5

2

3 5

By taking expectation on it, we obtain

E0

1 X t=0

t

J X

1 UC CE0 2

nj wtj =

j=1

1 X

t

t=0

+ t:i:p: + o k k3

2 4

( + ) Y^t +(1 +

)

Y~t PJ

2

j=1

+ (1 + ) j

nj (1

j )(1

PJ

j=1

j

)

nj

R P^j;t

2

R P~j;t

3 5

2 j;t

Thus, we derive Lt = ( + ) Y^t

Y~t

2

+ (1 + )

J X

nj

R P^j;t

R P~j;t

j=1

2

+ (1 +

)

J X j=1

34

nj

j

(1

j )(1

j

)

2 j;t

C

Numerical results when

A

= 0:84 and

A

= 0:0084

We obtain the OPI in‡ation and the implied nominal interest rates when A = 0:0084. We take these values fom Bae (2013).

Table 7: Optimal weights ( A = 0:84) CPI expenditure categories nj (%) 1 Food and non-alcoholic beverages 14.15 2 Alcoholic beverages and tobacco 1.30 3 Clothing and footwear 6.44 4 Housing, water, electricity and other fuels 10.13 5 Furnishings, household equipment and routine household 3.73 maintenance 6 Health 6.36 7 Transport 12.28 8 Communication 6.41 9 Recreation and culture 5.51 10 Education 11.77 11 Restaurants and hotels 13.27 12 Miscellaneous goods and services 8.65

35

A

= 0:84 and

j

j (%)

0.1943 0.0435 0.7234 0.6180 0.4679

2.95 0.04 15.03 13.42 2.52

0.4690 0.3613 0.0583 0.2497 0.5578 0.7720 0.4231

4.36 5.15 0.45 1.46 10.84 38.77 5.01

10 cpi,yr t opt,yr t

Inflation rate(%)

8

6

4

2

0

-2 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016

Time

Table 8: Volatility and persistence of in‡aton rates ( A = 0:84) opt;yr cpi;yr In‡ation rate index Standard deviation 1.6211 2.2215 Autocorrelation 0.4107 0.0626

36

core;yr

1.5841 0.2756

6 cpi,yr t iopt,yr t idata,yr t

i

5.5 5

Interest rate(%)

4.5 4 3.5 3 2.5 2 1.5 1 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016

Time

Table 9: Volatility and persistence of nominal interest rates ( A = 0:84) Interest rates iopt;yr icpi;yr icore;yr Standard deviation 1.1094 1.1399 1.1012 Autocorrelation 0.9031 0.8610 0.8886

Table 10: Correlation between the model implied and the actual interest rates ( A = 0:84) Correlation of idata;yr with iopt;yr icpi;yr icore;yr Correlation 0.9460 0.9365 0.9305

37

D

Figures with core in‡ation

10 cpi,yr t opt,yr t core,yr t

Inflation rate(%)

8

6

4

2

0

-2 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016

Time

Figure 6: In‡ation rates

Table 11: Volatility and Persistence of In‡aton Rates opt;yr cpi;yr In‡ation rate index Standard deviation 1.6419 2.2215 Autocorrelation 0.3780 0.0626

38

core;yr

1.5841 0.2756

6 cpi,yr t iopt,yr t icore,yr t data,yr it

i

5.5 5

Interest rate(%)

4.5 4 3.5 3 2.5 2 1.5 1

2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016

Time

Figure 7: Nominal interest rates

Table 12: Volatility and persistence of nominal interest rates Interest rates iopt;yr icpi;yr icore;yr Standard deviation 1.1105 1.1399 1.1012 Autocorrelation 0.9015 0.8610 0.8886

Table 13: Correlation between the model implied and the actual interest rates Correlation of idata;yr with iopt;yr icpi;yr icore;yr Correlation 0.9475 0.9365 0.9305

39