2.2.1 Sectoral inflation

We first define a proxy for sectoral inflation. Again, the proxy for the inflation of the goods sold by firm $i$ in sector $s$ is denoted by $\pi _{t}(i;\,s)$. The proxy for the sectoral inflation in sector $s$ is then defined as the average of each firm’s inflation, i.e.,

\begin{align*} \pi _{t}(s)\equiv 100\frac {1}{I}\sum \limits _{i=1}^{I}\pi _{t}(i;\,s) . \end{align*}

2.2.2 Changes in sectoral costs

Next, we define a proxy of the changes in sectoral costs. The change in a firm’s cost in sector $s$ is denoted by $\Delta Cost_{t}(i;\,s)$, Then, the change in sectoral costs is defined as follows:

\begin{align*} \Delta Cost_{t}(s)\equiv 100\frac {1}{I}\sum \limits _{i=1}^{I}\Delta Cost_{t}(i;\,s). \end{align*}

2.2.3 Degree of dispersion

We define three indicators to measure the degree of dispersion regarding the changes in sectoral costs. While the first measure is our baseline measure, we also examine with the second and third measures for robustness.

First, we define the following indicator based on the distribution of the answers by firms in each sector ($s$).

\begin{eqnarray*} d_{s,t} & \equiv &\frac{1}{I}\sum \limits _{i=1}^{I}\left (\Delta Cost_{t}(i;\,s)-\frac{\Delta Cost_{t}(s)}{100}\right )^2 \end{eqnarray*}

This indicator represents the degree of dispersion in changes in firms’ own costs within each industry ($s$) in period $t$. Because firms’ own costs are noisy signals about sectoral costs for them, this indicator is expected to capture the degree of fragmentation of information on changes in sectoral costs across firms in the sector.Footnote ¹⁰ This measure is intuitive and straightforward, but there could be systematic correlation between the change of the costs ($\Delta Cost_{t}(s)$) and the dispersion ($d_{s,t}$). Because firms’ answers are qualitative and unable to fully capture the magnitude of the changes in their costs, their answers tend to be more concentrated when sectoral costs more substantially change.Footnote ¹¹

Second, we define an alternative measure of dispersion by conditioning with the magnitude of the change in sectoral cost. This is given by:

\begin{align*} {\hat {\kern-2pt d}}_{s,t}\equiv d_{s,t}- \bar {\kern-2pt d}(\Delta Cost_{t}(s)), \end{align*}

which can be calculated by the following algorithm:

1. Step 1. We split the samples (i.e., all industries in each period) into ten groups based on the percentile rank of their $\Delta Cost_{t}(s)$. For example, the sample (sector $s$ in period $t$) is assigned to group 1 if $\Delta Cost_{t}(s)$ is smaller than the 10th percentile, to group 2 if $\Delta Cost_{t}(s)$ is between the 10th and 20th percentiles, and so on.

2. Step 2. We then calculate the median dispersion of changes in input prices in each group ($\, \bar{\kern-2pt d}(\Delta Cost_{t}(s))$) for all groups.

3. Step 3. Finally, we calculate relative dispersion ($\,\hat{\kern-2pt d}_{s,t}$) by subtracting the median dispersion from the dispersion as defined earlier ($d_{s,t}$).

The advantage of this indicator is that it controls the potential systematic relationship between $\Delta Cost_{t}(s)$ and $d_{s,t}$ by focusing on the relative dispersion within similar values of $\Delta Cost_{t}(s)$.Footnote ¹² In fact, while correlation between $\Delta Cost_{t}(s)$ and $d_{s,t}$ across all periods and industries is moderately positive (0.24), the correlation between $\Delta Cost_{t}(s)$ and $\ \hat{\kern-2pt d}_{s,t}$ is close to zero (−0.09), implying that this measure successfully avoids the potential influence of systematic relationship between $\Delta Cost_{t}(s)$ and $d_{s,t}$.

Finally, for further robustness, we develop the dummy variable for measuring relative dispersion as follows:

\begin{align*} D_{s,t}\equiv 1\text { if }d_{s,t}\geq\, \bar{\kern-2pt d}(\Delta Cost_{t}(s)),\text { otherwise 0} \end{align*}

where the dummy variable is calculated as follows:

1. Step 1. Following the calculation of relative dispersion, we split all sample (i.e., all industries in each period) into ten groups based on the percentile rank of their $\Delta Cost_{t}(s)$.

2. Step 2. We then calculate the median dispersion of changes in input prices in each group ($\, \bar{\kern-2pt d}(\Delta Cost_{t}(s))$) for all groups.

3. Step 3. Finally, if the dispersion of changes in input prices of a sample ($d_{s,t}$) is higher than the median in the same group ($\, \bar{\kern-2pt d}(\Delta Cost_{t}(s))$), we assign one to its dummy variable ($D_{s,t}=1$), and zero otherwise ($D_{s,t}=0$).

Similar to the previous measure, this measure succeeds in avoiding the potential influence of systematic relationship between $\Delta Cost_{t}(s)$ and $d_{s,t}$ as the correlation between $\Delta Cost_{t}(s)$ and $\hat{\kern-2pt d}_{s,t}$ is nearly zero (−0.02).

2.3.1 Dispersion in changes in firms’ own costs

We examine how the current and past dispersion in changes in firms’ own costs ($d_{s,t},d_{s,t-1}$) affects the response of sectoral inflation ($\pi _{t}(s)$) to changes in sectoral costs ($\Delta Cost_{t}(s)$). Specifically, we estimate the following fixed- and period-effects model by ordinary least squares (OLS).

(1) \begin{align} \pi _{t}(s)={\beta }^{A}(s)+{\beta }_{t}^{A}+\left ({\beta }_{1}^{A}(s)+{\beta }_{2}^{A}d_{s,t}+{\beta }_{3}^{A}d_{s,t-1}\right ) \Delta Cost_{t}(s)+{\beta }_{4}^{A}(s)\Delta Demand_{t-1}(s)+\epsilon _{t}^{A}(s), \end{align}

where ${\beta }^{A}(s)$ represents sector-level fixed-effects, ${\beta }_{t}^{A}$ indicates time-effects, and $\epsilon _{t}^{A}(s)$ is an error term. $\Delta Demand_{t-1}(s)$ controls the lagged changes in demand surveyed in Tankan survey,Footnote ¹⁴ where the lag is taken to mitigate endogeneity. Our interest in this regression is on the signs and significance of $\hat{\beta }_{2}^{A}$ and $\hat{\beta }_{3}^{A}$. Specifically, if $\hat{\beta }_{2}^{A}$ ($\hat{\beta }_{3}^{A}$) is negative, it means that the response of sectoral inflation to changes in sectoral costs decreases in accordance with the current (past) dispersion of firms’ own costs on sectoral costs.

As shown in Table 1, the estimates of coefficients ($\hat{\beta }_{2}^{A}$ and $\hat{\beta }_{3}^{A}$) are all negative and statistically significant at least at five percent significance level.Footnote ¹⁵ These indicate that an increase in fragmentation of information on sectoral costs, either in the current and previous periods, lowers the sensitivity of sectoral inflation to sectoral costs.

Table 1.

Regression results: ordinary least squares with dispersion

Note: ***, ** and * indicate statistically significant at 1%, 5%, and 10% levels, respectively.

Standard errors are cross-section cluster robust standard errors.

For robustness, we estimate the same equation by replacing the dispersion of firms’ own costs ($d_{s,t},d_{s,t-1}$) with relative dispersion ($\, \hat{\kern-2pt d}_{s,t},\,\hat{\kern-2pt d}_{s,t-1}$).

(2) \begin{align} \pi _{t}(s)={\beta }^{B}(s)+{\beta }_{t}^{B}+\left ({\beta }_{1}^{B}(s)+{\beta }_{2}^{B}\,\hat{\kern-2pt d}_{s,t}+{\beta }_{3}^{B}\,\hat{\kern-2pt d}_{s,t-1}\right ) \Delta Cost_{t}(s)+{\beta }_{4}^{B}(s)\Delta Demand_{t-1}(s)+\epsilon _{t}^{B}(s), \end{align}

where ${\beta }^{B}(s)$ represents sector-level fixed-effects, ${\beta }_{t}^{B}$ indicates time-effects, and $\epsilon _{t}^{B}(s)$ is an error term. As before, if the signs of $\hat{\beta }_{2}^{B}$ ($\hat{\beta }_{3}^{B}$) is negative, it indicates that the response of sectoral inflation to changes in sectoral costs decreases in accordance with the current (past) dispersion of firms’ own costs on sectoral costs. Table 2 shows the estimation results and the estimated $\hat{\beta }_{2}^{B}$ and $\hat{\beta }_{3}^{B}$ are negative and statistically significant at least at ten percent significance level, confirming the results in Table 1.

Table 2.

Regression results: ordinary least squares with relative dispersion

Note: ***, ** and * indicate statistically significant at 1%, 5%, and 10% levels, respectively.

Standard errors are cross-section cluster robust standard errors.

Finally, we estimate the same equation by replacing the dispersion of firms’ own costs ($d_{s,t},d_{s,t-1}$) by dummy variable ($D_{s,t},D_{s,t-1}$). Specifically, we estimate the following fixed- and period-effects model by OLS.

(3) \begin{align} \pi _{t}(s) & = {\beta }^{C}(s)+{\beta }_{t}^{C}+\left ({\beta }_{1}^{C}(s)+{\beta }_{2}^{C}D_{s,t}+{\beta }_{3}^{C}D_{s,t-1}\right ) \Delta Cost_{t}(s)+{\beta }_{4}^{C}(s)\Delta Demand_{t-1}(s) \nonumber \\ & \qquad + \epsilon _{t}^{C}(s), \end{align}

where ${\beta }^{C}(s)$ represents sector-level fixed-effects, ${\beta }_{t}^{C}$ indicates time-effects, and $\epsilon _{t}^{C}(s)$ is an error term. As previously mentioned, if the signs of $\hat{\beta }_{2}^{C}$ ($\hat{\beta }_{3}^{C}$) are negative, it suggests that the sectoral inflation response to changes in sectoral costs diminishes in line with the current (past) relative dispersion of firms’ own costs. Table 3 presents the estimation outcomes, and the estimated $\hat{\beta }_{2}^{C}$ and $\hat{\beta }_{3}^{C}$ are negative and statistically significant at least at five percent significance level, corroborating the findings in Table 1.

Table 3.

Regression results: ordinary least squares with dummy

Note: ***, ** and * indicate statistically significant at 1%, 5%, and 10% levels, respectively.

Standard errors are cross-section cluster robust standard errors.

To sum up, the estimation results in Tables 1, 2, and 3 indicate that the response of sectoral inflation to changes in sectoral costs is decreasing in the degree of fragmentation of (i) current and (ii) past information.Footnote ¹⁶ In what follows, we reconcile these findings with a theoretical model which generalizes the standard dispersed information model by introducing the concept of fragmentation of information, defined as the situation in which changes in firms’ costs are shared only with a subset of their competitors. We then show that changes in the dispersion of firm-specific costs, the key parameter in standard dispersed information model, can replicate only (i), while changes in the degree of fragmentation of information can account for both (i) and (ii).

3.1.1 Households

Preferences of the representative household are described as a function of consumption, $C_{t}$, and labor, $L_{t}$, by the utility function:

\begin{align*} \sum _{t=0}^{\infty }\beta ^{t}\left ( \log C_{t}-L_{t}\right ), \end{align*}

where $\beta \in (0,1)$ is the discount rate. The household’s consumption is described by the CES consumption aggregator:

\begin{eqnarray*} C_{t} &=&\left [ \int _{0}^{1}\left ( C_{t}(s)\right ) ^{\frac{\tilde{\eta }-1}{\tilde{\eta }}}ds\right ] ^{\frac{\tilde{\eta }}{\tilde{\eta }-1}}, \\ C_{t}(s) &=&\left [ \int _{0}^{1}\left ( C_{t}(i;\,s)\right ) ^{\frac{\eta -1}{\eta }}di\right ] ^{\frac{\eta }{\eta -1}}, \end{eqnarray*}

where $\tilde{\eta }\gt 0$ is the elasticity of the substitution across sectors, $\eta \gt 0$ is the elasticity of the substitution across goods in the same sector, and $C_{t}(i;\,s)$ is the consumption of each good in sector $s$.

3.1.2 Firms

Preferences of the representative household imply the following demand for firms $i$ in sector $s$:

\begin{align*} C_{t}(i;\,s)=\left (\frac {P_{t}(i;\,s)}{P_{t}(s)}\right )^{-\eta }\left (\frac {P_{t}(s)}{P_{t}}\right )^{-\widetilde {\eta }}C_{t}, \end{align*}

where $P_{t}\equiv \left [\int _{0}^{1}P_{t}^{1-\tilde{\eta }}(s)ds\right ]^{\frac{1}{1-\tilde{\eta }}}$ and $P_{t}(i;\,s)\equiv \left [\int _{0}^{1}P_{t}^{1-\eta }(i;\,s)di\right ]^{\frac{1}{1-\eta }}$ are the aggregate price index and the sectoral price index, respectively.Footnote ¹⁷

Each firm produces a single goods using the following production technology:

\begin{align*} Y_{t}(i;\,s)=A_{t}(i;\,s)L_{t}^{\epsilon }(i;\,s), \end{align*}

where $A_{t}(i;\,s)$ is a firm-specific productivity and $\epsilon \in (0,1)$ is the degree of diminishing returns.

3.1.3 Market clearing

Market clearing implies $Y_{t}(i;\,s)=C_{t}(i;\,s)$ for each firm, and $Y_{t}=C_{t}$ in the economy. Aggregate nominal demand, $Q_{t}$, is given by the following cash-in-advance constraint:

\begin{align*} Q_{t}=P_{t}C_{t}. \end{align*}

In the subsequent analysis, we use lower-case variables to indicate logarithms of the corresponding upper-case variables (i.e., $p_{t}\equiv \log P_{t}$).

3.1.4 Optimal price setting

We first derive the optimal price-setting rule with flexible prices, assuming perfect information about the current nominal variables. We then describe the change in the environment under imperfect information. During each period $t$, firm $i$ in sector $s$ sets the optimal price as follows:

(4) \begin{align} p_{t}(i;\,s)=\mu +mc_{t}(i;\,s) \end{align}

where $\mu \equiv \eta /(\eta -1)\gt 0$ is the markup and $mc_{t}(i;\,s)$ is the firm $i$’s nominal marginal cost. The nominal marginal cost is defined as nominal wage $w_{t}$ net of the marginal product of labor:

\begin{align*} mc_{t}(i;\,s)=w_{t}+\left (1-\epsilon \right )l_{t}(i;\,s)-a_{t}(i;\,s)-\log (\epsilon ). \end{align*}

Using the production technology defined above, we express labor input as: $l_{t}(i;\,s)=[y_{t}(i;\,s)-a_{t}(i;\,s)]/\epsilon$. Then the nominal marginal cost is rewritten as:

\begin{align*} mc_{t}(i;\,s)=w_{t}+\frac {1-\epsilon }{\epsilon }y_{t}(i;\,s)-\frac {1}{\epsilon }a(i;\,s)-\log (\epsilon ). \end{align*}

The optimal labor supply condition for the representative household is:

\begin{align*} w_{t}-p_{t}=c_{t}, \end{align*}

and the consumer demand is:

(5) \begin{align} c_{t}(i;\,s)=-\eta \left (p_{t}(i;\,s)-p_{t}(s)\right )-\tilde{\eta }\left (p_{t}(s)-p_{t}\right )+c_{t}. \end{align}

We derive the optimal price-setting rule for firm $i$ in sector $s$ by using Equations (4), (5), the market clearing conditions ($y_{t}(i;\,s)=c_{t}(i;\,s)$ for all $i$ and $s$ and $y_{t}=c_{t}$), and the cash-in-advance constraint, $y_{t}=q_{t}-p_{t}$, as follows:Footnote ¹⁸

(6) \begin{align} p_{t}(i;\,s)=r_{1}p_{t}(s)+r_{2}p_{t}+\left ( 1-r_{1}-r_{2}\right ) x_{t}(i;\,s)+\xi, \end{align}

where

\begin{eqnarray*} x_{t}(i;\,s) &=&q_{t}-a_{t}(i;\,s), \\ \xi &=&\frac{\epsilon }{\epsilon +{\eta }\left ( 1-\epsilon \right ) }(\mu -\log (\epsilon )), \\ r_{1} &=&\frac{\left ({\eta }-\tilde{\eta }\right ) \left ( 1-\epsilon \right ) }{\epsilon +{\eta }\left ( 1-\epsilon \right ) }, \\ r_{2} &=&\frac{\left ( \tilde{\eta }-1\right ) \left ( 1-\epsilon \right ) }{\epsilon +{\eta }\left ( 1-\epsilon \right ) }, \end{eqnarray*}

and $p_{t}(s)=\int _{0}^{1}p{_{t}(i;\,s)di}$. Equation (6) shows that the optimal pricing rule for firm $i$ in sector $s$ is a weighted average of sectoral prices ($p_{t}(s)$), aggregate prices ($p_{t}$) and (the component of) their cost ($x_{t}(i;\,s)$). The weight between them is determined by the parameters $(r_{1},r_{2})$, which reflect the degree of strategic complementarity among firms in the same sector and across sectors, respectively. In what follows, for the sake of simplicity, without loss of generalizability, we normalize $\xi$ as zero, and assume $\eta \gt \tilde{\eta }=1$ which yields $r_{2}=0$. The optimal price for firm $i$ in sector $s$ is then simplified as:

(7) \begin{align} p_{t}(i;\,s)=rp_{t}(s)+\left ( 1-r\right ) x_{t}(i;\,s), \end{align}

where

\begin{align*} r=\frac {\left ( {\eta }-1\right ) \left ( 1-\epsilon \right ) }{\epsilon +{\ \eta }\left ( 1-\epsilon \right ) }. \end{align*}

Equation (7) shows that the optimal pricing rule for firm $i$ is a weighted average of sectoral prices ($p_{t}(s)$) and its own costs ($x_{t}(i;\,s)$). The weights for sectoral prices and firms’ own costs are determined by the parameter $r$, which reflects the degree of strategic complementarity between firms within sectors.

3.1.5 Cost structures

In what follows, we focus on price dynamics in a sector and take the notation $s$ away for the sake of simplicity. We assume that $a_{t}(i)(=a_{t}(i;\,s))$ comprises sectoral component ($a_{t}$) and firm-specific component ($a_{t}(i)-a_{t}$), which is shared by $1/N_{t}$ proportion of competing firms in the sector. We then denote firms’ prices which receive a shock of type $n\in \{1,2,\ldots, N_{t}-1,N_{t}\}$ by $p_{t}(i;\,n)$. Firms’ costs that receive the shock of type $n$ at time $t$ is denoted as $x_{t}(n)$, and is decomposed as $x_{t}(n)=\widetilde{mc}_{t}+v_{t}(n)$ where $\widetilde{mc}_{t}\equiv q_{t}-a_{t}$ represents sectoral costs, and $v_{t}(n)\equiv a_{t}-a_{t}(n)$ represents firm-specific costs where $v_{t}(n)\sim \mathcal{N}(0,\tau _{t}^{2})$. The parameter $\tau _{t}^{2}$ indicates the variance of firm-specific costs at time $t$, which is randomly and exogenously given and common knowledge for households and firms.Footnote ¹⁹

We assume that sectoral costs follow the random-walk process:

(8) \begin{align} \widetilde{mc}_{t}=\widetilde{mc}_{t-1}+\epsilon _{t}, \end{align}

where $\epsilon _{t}\sim \mathcal{N}(0,\sigma _{t}^{2})$.

Note that there are a variety of drivers affecting $N_t$, but one possible driver is the supply chain relationship. For example, the firms that share the same supplier may receive common shocks from the common supplier, and changes in the supply chain relationship could result in time-varying $N_t.$Footnote ²⁰ To avoid further complexity, the model considers the shock structure as given.

3.1.6 Information structures

We introduce imperfect information for firms as follows. Regarding the current variables, firms observe their own cost ($x_{t}(n)$) but they cannot disentangle sectoral component ($\widetilde{mc}_{t}$) and firm-specific component ($v_{t}(n)$) from the observed cost ($x_{t}(n)$). Hence, $x_{t}(n)$ serves as noisy signals for $\widetilde{mc_t}$ where $v_{t}(n)$ is the noise whose variance is $\tau _{t}^{2}$. $N_{t}$ represents the degree of fragmentation of information and this generalizes the information structures of standard dispersed information models. Namely, if $N_{t}=1$, the signals correspond to the public signals and if $N_{t}\rightarrow \infty$, the signals correspond to private signals.

Firms observe their own past costs and their competitors’ prices while they cannot observe past sectoral costs. Moreover, as shown later, firms can form endogenous signal ($s_{t-1}(n)$) on $\widetilde{mc}_{t-1}$ after all firms set their prices in the previous period. Then the type $n$ firms’ information set in period $t$ when they set their prices is then defined as:

\begin{align*} \mathcal {I}_{t}(n)\equiv \left\{x_{t}(n), \widehat {\mathcal {I}}_{t}(n) \right\}, \end{align*}

where the firms’ information set before observing the signal, $x_{t}(n)$, is given as:

\begin{align*} \widehat {\mathcal {I}}_{t}(n)\equiv \left\{{\mathcal {I}}_{t-1}(n),\left \{ p_{t-1}(j;\,n)\right \} _{j\in \lbrack 0,1]}, s_{t-1}(n) \right\}. \end{align*}

Under these settings, firms’ log-linearized best response functions about their pricing decisions (Equation (7)) under this imperfect information are given by:

(9) \begin{eqnarray} p_{t}(i;\,n) &=&r\mathbb{E} \left[p_{t}|\mathcal{I}_{t}(n) \right]+(1-r)\mathbb{E}\left[\widetilde{mc}_{t}+v_{t}(n)|\mathcal{I}_{t}(n)\right] \notag \\ &=&r\mathbb{E}\left[p_{t}|\mathcal{I}_{t}(n)\right]+(1-r)x_{t}(n) \end{eqnarray}

where $p_{t}=\int \nolimits _{i\in \lbrack 0,1]}p_{t}(i;\,n)di$ represents the sectoral price. Note that the sectoral price, $p_{t}$, can be expressed as follows:

(10) \begin{align} p_{t}=r\overline{\mathbb{E}}\left[p_{t}|\mathcal{I}_{t}(n)\right]+(1-r)\left [ \widetilde{mc}_{t}+\frac{1}{N_{t}}\sum \limits _{n=1}^{N_t}v_{t}(n)\right ] = \frac{1}{N_{t}}\sum \limits _{n=1}^{N_{t}}p_{t}(n), \end{align}

where $\overline{\mathbb{E}}[{\cdot}]\equiv \int \nolimits _{i\in \lbrack 0,1]}\mathbb{E}[{\cdot}]di$ represents an operator of the average of firms’ expectations and $p_{t}(n)$ represents the common price for the firms receiving a common shock of type $n$. Hence, unlike standard imperfect information models where firms receive private signals and cannot know any of their competitors’ prices, in this model, firms can know a proportion of $1/N_{t}$ of their competitors’ prices. In the following, the dynamics of sectoral prices are to be derived.

3.2.1 Prices with exogenous prior beliefs

Denote the mean and the imprecision of the type $n$ firms’ prior beliefs about sectoral costs by $\mathbb{E}[\widetilde{mc}_{t}|\widehat{\mathcal{I}}_{t}(n)]$ and $\mathbb{V}[\widetilde{mc}_{t}|\widehat{\mathcal{I}}_{t}(n)]$, respectively. As shown in the following, firms’ prior beliefs are common across all firms in the sector. Then, by taking this prior beliefs as given, we can calculate the equilibrium sectoral price as follows:

Lemma 1. Given $\mathbb{E}[\widetilde{mc}_{t}|\widehat{\mathcal{I}}_{t}(n)]$ and $\mathbb{V}[\widetilde{mc}_{t}|\widehat{\mathcal{I}}_{t}(n)]$, the firm $i$’s price and average price in the equilibrium are given by:

(11) \begin{eqnarray} p_{t}(i;\,n) &=& \alpha _t x_{t}(n)+\left ( 1-\alpha _t\right ) \mathbb{E}\left[\widetilde{mc}_{t}|\widehat{\mathcal{I}}_{t}(n)\right], \\[-25pt] \nonumber \end{eqnarray}

(12) \begin{eqnarray} p_{t} &=&\alpha _t \left ( \widetilde{mc}_{t}+\frac{1}{N_{t}}\sum \limits _{n=1}^{N_{t}}v_{t}(n)\right ) +\left ( 1-\alpha _t\right ) \mathbb{E}\left[\widetilde{mc}_{t}|\widehat{\mathcal{I}}_{t}(n)\right], \end{eqnarray}

where

(13) \begin{align} \alpha _t \equiv \frac{1-r }{1-r\left ( \frac{N_{t}-1}{N_{t}}\widehat{\lambda }_{t}+\frac{1}{N_{t}}\right ) } \end{align}

and

(14) \begin{align} \widehat{\lambda }_{t}\equiv \frac{\mathbb{V}\left[\widetilde{mc}_{t}|\widehat{\mathcal{I}}_{t}(n)\right]}{\mathbb{V}\left[\widetilde{mc}_{t}|\widehat{\mathcal{I}}_{t}(n)\right]+\tau _{t}^{2}}. \end{align}

Equation (11) in Lemma (1) indicates that each firm’s equilibrium price ($p_{t}(i;\,n)$) is a linear combination of its own cost ($x_{t}(n)$) and its prior beliefs about the sectoral cost ($\mathbb{E}[\widetilde{mc}_{t}|\widehat{\mathcal{I}}_{t}(n)]$). The latter is included because the firm infers competitors’ prices based on its beliefs about sectoral price. Similarly, Equation (12) indicates that the sectoral price is affected by prior beliefs while it also depends on the sectoral costs ($\widetilde{mc}_{t}$) and the averages of the firm-specific costs $ \left(\frac{1}{N_{t}}\sum \limits _{n=1}^{N_{t}}v_{t}(n) \right)$. Note that because $N_{t}$ is finite, aggregation process does not eliminate the idiosyncratic variables.

Moreover, these equations show an important observation: the weights for the variables in period $t$ $\left( x_{t}(n),\, \widetilde{mc}_{t}+\frac{1}{N_{t}}\sum \limits _{n=1}^{N_{t}}v_{t}(n) \right)$ are monotonically decreasing in $N_{t}$ as the following inequality holds:Footnote ²¹

\begin{align*} \frac {\partial \alpha _t}{\partial N_{t}} =-\frac {\left ( 1-\widehat {\lambda }_{t}\right ) \left ( 1-r\right ) r}{\left [ r\left ( 1-\widehat {\lambda }_{t}\right ) -N_{t}\left ( 1-r\widehat {\lambda }_{t}\right ) \right ] ^{2}}\lt 0. \end{align*}

The intuition behind this relationship is as follows: If $N_{t}$ is small, then each firm shares the same costs and the information with many of its competitors. In this situation, the firm does not hesitate to incorporate the signal ($x_{t}(n)$) into its price because it expects that many others will set the same prices. By contrast, if $N_{t}$ is large, then each firm shares the same costs and the information with only a few of its competitors. In this situation, the firm does hesitate to set its price in accordance with the signal ($x_{t}(n)$) as it is afraid that the price may be very different from the others. In summary, the primary mechanism is that firms experiencing common shocks in their costs can effectively coordinate their prices with each other, and the degree of commonality in these shocks varies depending on $N_t$.

3.2.2. Endogenizing prior beliefs

Next, we endogenize prior belifs $\mathbb{E}[\widetilde{mc}_{t}|\widehat{\mathcal{I}}_{t}(n)]$ and $\mathbb{V}[\widetilde{mc}_{t}|\widehat{\mathcal{I}}_{t}(n)]$. Specifically, we calculate how firms generate endogenous signal on the sectoral costs based on competitors’ prices in the previous period, and then how they update thier beliefs based on the endogenous signal. To this end, we model the process of firms’ learning form their competitors’ prices as follows. Suppose a firm’s cost is type $n$. Then, by aggregating Equation (11) across firms whose cost type is not $n$, endogenous signal of $\widetilde{mc}_{t-1}$ is formed in period $t$ as follows. The information included in the aggregated price $\left(\frac{1}{N_{t-1}-1}\sum \limits _{n^{\prime}=1, n^{\prime}\neq n}^{N_{t-1}}p_{t-1}(n^{\prime}) \right)$ is given as:

\begin{align*} & \frac{1}{N_{t-1}-1}\sum \limits _{n^{\prime}=1,n^{\prime}\neq n}^{N_{t-1}}p_{t-1}(n^{\prime}) =\alpha _{t-1} \left ( \widetilde{mc}_{t-1}+\frac{1}{N_{t-1}-1}\sum \limits _{n^{\prime}=1, n^{\prime}\neq n}^{N_{t-1}}v_{t-1}(n^{\prime})\right ) \\ & \qquad + \left ( 1-\alpha _{t-1}\right ) \mathbb{E}\left[\widetilde{mc}_{t-1}|\widehat{\mathcal{I}}_{t-1}(n)\right]. \end{align*}

Note that in this model, the prior beliefs become identical across firms, which is to be shown later. Hence, the following signal ($s_{t-1}(n)$), consisting of observed variables, is a noisy signal on $\widetilde{mc}_{t-1}$:

\begin{eqnarray*} s_{t-1}(n) &\equiv &\frac{\frac{1}{N_{t-1}-1}}{\alpha _{t-1}}\sum \limits _{n^{\prime}=1, n^{\prime} \neq n}^{N_{t-1}}p_{t-1}(n^{\prime})-\frac{1-\alpha _{t-1}}{\alpha _{t-1}}\mathbb{E}\left[\widetilde{mc}_{t-1}|\widehat{\mathcal{I}}_{t-1}(n)\right] \\ &=&\widetilde{mc}_{t-1}+\frac{1}{N_{t-1}-1}\sum \limits _{n^{\prime}=1, n^{\prime} \neq n}^{N_{t-1}}v_{t-1}(n^{\prime})\sim \mathcal{N}\left ( \widetilde{mc}_{t-1},\frac{1}{N_{t-1}-1}\tau _{t-1}^{2}\right ). \end{eqnarray*}

This signal is composed of the true past sectoral cost ($\widetilde{mc}_{t-1}$) and noises ($v_{t-1}(n)$) due to finite types of firm-specific costs within the industry. Combining this signal, and its own cost $x_{t-1}(n)$, each firm updates its beliefs about sectoral prices as below.

Lemma 2. $\mathbb{E}[\widetilde{mc}_{t}|\widehat{\mathcal{I}}_{t}(n)]$ and $\mathbb{V}[\widetilde{mc}_{t}|\widehat{\mathcal{I}}_{t}(n)]$ are identical across firms and expressed as follows:

(15) \begin{align} \mathbb{E}\left[\widetilde{mc}_{t}|\widehat{\mathcal{I}}_{t}(n)\right]=\sum \limits _{s=1}^{\infty }\prod \limits _{u=1}^{s}\frac{1-\lambda _{t-u}}{1-\lambda _{t-s}}\lambda _{t-s}\left ( \widetilde{mc}_{t-s}+\frac{1}{N_{t-s}}\sum \limits _{n=1}^{N_{t-s}}v_{t-s}(n)\right ), \\[-25pt] \nonumber \end{align}

(16) \begin{align} \mathbb{V}\left[\widetilde{mc}_{t}|\widehat{\mathcal{I}}_{t}(n)\right]=\frac{\mathbb{V}\left[\widetilde{mc}_{t-1}|\widehat{\mathcal{I}}_{t-1}(n)\right]\frac{1}{N_{t-1}}\tau _{t-1}^{2}}{\mathbb{V}\left[\widetilde{mc}_{t-1}|\widehat{\mathcal{I}}_{t-1}(n)\right]+\frac{1}{N_{t-1}}\tau _{t-1}^{2}}+\sigma _{t}^{2}, \end{align}

where

(17) \begin{align} \lambda _{t-s}=\frac{\mathbb{V}\left[\widetilde{mc}_{t-s}|\widehat{\mathcal{I}}_{t-s}(n)\right]}{\mathbb{V}\left[\widetilde{mc}_{t-s}|\widehat{\mathcal{I}}_{t-s}(n)\right]+\frac{1}{N_{t-s}}\tau _{t-s}^{2}}, \end{align}

and

(18) \begin{align} \mathbb{V}\left[\widetilde{mc}_{t-s}|\widehat{\mathcal{I}}_{t-s}(n)\right]=\frac{\mathbb{V}\left[\widetilde{mc}_{t-1-s}|\widehat{\mathcal{I}}_{t-1-s}(n)\right]\frac{1}{N_{t-1-s}}\tau _{t-1-s}^{2}}{\mathbb{V} \left[\widetilde{mc}_{t-1-s}|\widehat{\mathcal{I}}_{t-1-s}(n)\right]+\frac{1}{N_{t-1-s}}\tau _{t-1-s}^{2}}+\sigma _{t-s}^{2}, \end{align}

for $s\geq 1$.

Equations (15) and (16) in Lemma2 indicate that firms’ prior beliefs about sectoral costs are fully endogenized variables in this model, and the beliefs rely not only on the past sectoral costs, but also on the firm-specific costs due to imperfect information. Furthermore, both the mean and imprecision of the prior beliefs depend on the degree of fragmentation of information in the past ($N_{t-1}$, $N_{t-2}, ...$), which is an important channel through which past information structures affect the firms’ current beliefs.

3.2.3 Prices with learning

Combining the equilibrium prices (11) and (12) with Equations (15) and (16), equilibrium prices are derived as follows:

Proposition 1. The firm $i$’s price and sectoral price in the equilibrium are given by:

(19) \begin{eqnarray} p_{t}(i;\,n)=\alpha _t x_{t}(n) +\left ( 1-\alpha _t \right ) \left [ \sum \limits _{s=1}^{\infty }\prod \limits _{u=1}^{s}\frac{1-\lambda _{t-u}}{1-\lambda _{t-s}}\lambda _{t-s}\left ( \widetilde{mc}_{t-s}+\frac{1}{N_{t-s}}\sum \limits _{n=1}^{N_{t-s}}v_{t-s}(n)\right ) \right ] \\[-25pt] \nonumber \end{eqnarray}

(20) \begin{align} p_{t} & = \alpha _t \left ( \widetilde{mc}_{t}+\frac{1}{N_{t}}\sum \limits _{n=1}^{N_{t-1}}v_{t}(n)\right ) \nonumber \\[3pt] & \quad + (1-\alpha _t) \left [ \sum \limits _{s=1}^{\infty }\prod \limits _{u=1}^{s}\frac{1-\lambda _{t-u}}{1-\lambda _{t-s}}\lambda _{t-s}\left ( \widetilde{mc}_{t-s}+\frac{1}{N_{t-s}}\sum \limits _{n=1}^{N_{t-s}}v_{t-s}(n)\right ) \right ] \end{align}

where $\alpha _t$, $\widehat{\lambda }_{t}$, $\lambda _{t-s}$, and $\mathbb{V}[\widetilde{mc}_{t-s}|\widehat{\mathcal{I}}_{t-s}(n)]$ for $s\geq 0$ are, respectively, given by Equations ( 13), ( 14), ( 17), and ( 18).

Proposition 1 shows the full-blown equilibrium prices, which depend on the current and an infinite number of past variables. The weight for the current variables in Equations (19) and (20) are the same and it depends not only on the degree of fragmentation in the current period ($N_{t}$), but also the degrees of fragmentation in the previous periods ($N_{t-1},N_{t-2},\ldots$) via $\widehat{\lambda }_{t}$. Similarly, the weight also depends not only on the variance of idiosyncratic costs in the current period ($\tau _{t}^{2}$), but also on those in past periods ($\tau _{t-1}^2,\tau _{t-2}^2,\ldots$). The next section examines the theoretical relationship between the sensitivity of sectoral prices to the sectoral average of the firms costs and the degree of fragmentation of information, which we examined empirically in Section 2. Although Section 2 examines the sensitivity of sectoral inflation to changes in sectoral average costs, for the sake of simplicy, we first show the property of the equilibrium prices and then connect it to that of sectoral inflation later.

3.3.1 Standard imperfect information environments

In standard imperfect information environments, $N_{t-s}$ is infinite in all periods ($s\geq 0$). In this extreme case, Proposition1 is transformed as follows.

Corollary 1. Suppose $N_{t-s}\rightarrow \infty$ for all $s\geq 0$. Then, the firm $i$’s price and sectoral price in the equilibrium are given by:

(21) \begin{align} p_{t}(i;\,n)=\left ( \frac{1-r}{1-r\widehat{\lambda }_{t}}\right ) x_{t}(n)+\left ( 1-\frac{1-r}{1-r\widehat{\lambda }_{t}}\right ) \widetilde{mc}_{t-1}, \end{align}

(22) \begin{align} p_{t}=\left ( \frac{1-r}{1-r\widehat{\lambda }_{t}}\right ) \widetilde{mc}_{t}+\left ( 1-\frac{1-r}{1-r\widehat{\lambda }_{t}}\right ) \widetilde{mc}_{t-1} \end{align}

where

(23) \begin{align} \widehat{\lambda }_{t}\equiv \frac{\sigma _{t}^{2}}{\sigma _{t}^{2}+\tau _{t}^{2}}, \end{align}

and $\alpha _{t}\equiv \frac{1-r}{1-r\hat{\lambda }_{t}}$ is decreasing in $\tau _{t}^{2}$ while invariant to $\tau _{t-1}^{2}$, $\tau _{t-2}^{2}$, ….

Corollary1 indicates that under standard imperfect information models, prices depend on the variables only in the current and previous periods, and do not depend on the variables for more than two periods ago. $\tau _{t}^{2}$ reduces the sensitivity of sectoral prices to sectoral costs. More importantly, unlike Proposition2, $\tau _{t-s}^{2}$ for $s\geq 1$ has no relationship with the sensitivity. When $N_{t-s}\rightarrow \infty$, the endogenous signals become perfectly accurate as the idiosynscratic noises are fully canceled out. Hence, the variance of the past firm-specific costs does not affect the efficiency of the learning. Compared with Proposition2, the takeaway from this result is that $\alpha _{t}$ never decreases in $\tau _{t-s}^{2}$ for $s\geq 1$.