Figure 1: Cost Growth is Associated with Expected and Realized Year-ahead Cost
Growth
Note: 50 bins, each contain an equal number of observations.
Data is residualized on the fixed effects then binned. We add back in
the unconditional mean for each variable.
This is a two-panel chart showing binned scatter plots with the x-axis displaying past cost growth ranging from -2 to 6 percent and the y-axis showing values from 1.0 to 3.5 percent. In the left panel with firm fixed effects only, two series are plotted on the y-axis: "Year-ahead expectations" (blue dots with slope 0.37) and "Year-ahead outcome" (orange dots with slope 0.16). The blue line shows a steeper positive relationship between past cost growth and expected future costs compared to the orange line showing realized future costs. The right panel shows the same relationship but controlling for both firm and sector-time fixed effects, with slopes of 0.3 and 0.07 respectively. Both panels demonstrate that firms' expectations about future costs are more responsive to past cost growth than actual future cost outcomes, suggesting firms overreact to their own past cost experiences. Note: 50 bins, each contain an equal number of observations. Data is residualized on the fixed effects then binned. We add back in the unconditional mean for each variable.
Figure 2: Heterogeneity analysis
(a) Comparison across sectors
(b) Comparison across firm size distribution
(c) Comparison across time
Note: The vertical bars in Figures (a) and (b) are 95%
confidence intervals based on standard errors clustered at the firm
level. The horizontal lines in all cases are the coefficients from the
full sample regression. We include PCA controls in (a) and (b), and
sector-time fixed effects in (c).
This is a three-panel chart showing regression coefficients across different dimensions. Panel (a) compares coefficients across 14 sectors, showing three regression coefficients with confidence intervals for each sector: "Year-ahead expectations" (blue), "Year-ahead outcome" (orange), and "Forecast error" (green). Most sectors show positive coefficients for expectations and outcomes, and negative coefficients for forecast errors. Panel (b) shows the same three coefficients across nine firm size categories based on employee count, from "1 to 4" employees to "1000 plus." The pattern of coefficients is relatively stable across size categories. Panel (c) compares the coefficients across years from 2012 to 2024, showing that the relationship between cost growth and expectations has remained fairly stable over time. Note: The vertical bars in Figures (a) and (b) are 95% confidence intervals based on standard errors clustered at the firm level. The horizontal lines in all cases are the coefficients from the full sample regression. PCA controls are included in (a) and (b), and sector-time fixed effects in (c).
Figure 3: Firm costs take time to drive beliefs
Note: Dotted lines are 95% confidence intervals, standard
errors clustered at firm level.
This figure contains two line graphs showing the coefficient β plotted against "Months ago, ℓ" on the x-axis (ranging from 0 to 11 months). The left panel shows results with firm fixed effects only, while the right panel includes both firm and sector-time fixed effects. Both graphs show coefficients declining as lag length increases, but remaining positive and significant (shown by dotted 95% confidence intervals) for several months. The left panel shows β values ranging from approximately 0.25 at lag 0 to 0.10 at lag 11, while the right panel shows smaller values ranging from about 0.2 to 0.05. This indicates that firms learn from their costs gradually over time, with the impact of past costs on beliefs declining but persisting for many months. Note: Dotted lines are 95% confidence intervals, standard errors clustered at firm level.
Figure 4: Underreaction and overreaction to Oil Prices
Note: 95% confidence intervals from a firm-level (cluster)
bootstrap with 2000 draws. In each draw, we resample firms with
replacement and re-compute the Fast/Slow split on the bootstrap
sample.
This figure presents two panels showing responses to oil price increases over time (0-11 months on the x-axis). Each panel plots three series: "Cumulative cost growth" (red), "Forecast" (blue), and "Forecast error" (green), with 95% confidence intervals. The left panel shows results for firms with "Fast exposure to oil," where costs rise quickly after an oil price increase, peak, and then decline. The blue forecast line rises initially but not enough (underreaction) and then doesn't fall as quickly as it should (delayed overreaction). The right panel shows firms with "Slow exposure to oil," where costs rise more gradually and continuously. For these firms, forecasts rise with cost increases, showing persistent underreaction throughout the period. The y-axis ranges from approximately -0.015 to 0.025, representing percentage point changes. Note: 95% confidence intervals from a firm-level (cluster) bootstrap with 2000 draws. In each draw, we resample firms with replacement and re-compute the Fast/Slow split on the bootstrap sample.
Figure 5: Inflation response to 1pp permanent change in nominal marginal costs
at date s
Note: Impulse response of inflation to a permanent rise in
nominal marginal costs under six models. Calibrated is our model with
\(\alpha^\textrm{FIRE}\), \(\tilde{\rho}\) and \(\tilde{\mathcal{K}}\) calibrated as in
Section 2.2
This figure contains two line charts showing inflation responses under different models. The x-axis shows time periods (0-20), and the y-axis shows inflation rates (0-0.18). Six different models are compared: FIRE (orange), Calibrated (blue), Mankiw Reis (red), Sticky Expectations (purple), Cognitive Discounting (brown), and Diagnostic (green). The left panel shows responses to an immediate shock (s = 0), where the Calibrated and Diagnostic models show faster initial pass-through than FIRE, while the other three models show slower pass-through. The right panel shows responses to a future shock (s = 10), where the Calibrated model shows the least anticipation before the shock occurs, followed by the strongest response once costs begin moving. This illustrates how different expectation formation models predict varying inflation dynamics in response to cost shocks. Note: Impulse response of inflation to a permanent rise in nominal marginal costs under six models. Calibrated is our model with α^(FIRE), ρ̃, and K̃ calibrated as in Section 2.2
Figure 6: Inflation response to supply and demand shocks
Note: The response of inflation to a TFP shock (left) and to a
monetary policy shock (right) under different models of firm beliefs.
Impulse responses normalized to 1 under FIRE. Our model, calibrated in
Section 2.2 above, is shown in
blue.
This figure consists of two line charts showing inflation responses to different shocks across multiple models. The x-axis shows time periods (0-15), and the y-axis shows normalized inflation responses (0-1.75). Six different models are compared: Calibrated (blue), FIRE (orange), Diagnostic (green), Cognitive Discounting (brown), Mankiw Reis (red), and Sticky Expectations (purple). The left panel shows responses to a TFP shock (supply shock), where the Calibrated model predicts a larger initial inflation response compared to FIRE. The right panel shows responses to a monetary policy shock (demand shock), where the Calibrated model predicts a smaller initial inflation response than FIRE. This demonstrates an asymmetric prediction unique to the Calibrated model: supply shocks propagate into inflation faster than under FIRE, while demand shocks propagate slower. Note: The response of inflation to a TFP shock (left) and to a monetary policy shock (right) under different models of firm beliefs. Impulse responses normalized to 1 under FIRE. Our model, calibrated in Section 2.2 above, is shown in blue.
Figure 7: Optimal inflation and output response to a persistent markup
shock
Note: Impulse responses for inflation, output, and the real
interest rate in response to a markup shock. The responses are
normalized so that inflation is identical and equal to an AR(1) with
persistence 0.9 in all models if the central bank holds the real
interest rate constant.
This is a three-panel figure showing optimal policy responses to a persistent markup shock over 30 time periods (x-axis). The left panel shows inflation (π_(t)) responses under four scenarios: FIRE with Commitment (orange solid line), Calibrated with Commitment (blue solid line), FIRE with Discretion (orange dashed line), and Calibrated with Discretion (blue dashed line). Inflation ranges from 0 to 0.8 on the y-axis. The middle panel shows output (Y_(t)) responses for the same four scenarios, with values ranging from approximately -1.4 to 0 on the y-axis. The right panel shows real interest rate (r_(t)) responses, with values ranging from approximately -0.20 to 0.10 on the y-axis. The figure demonstrates that under FIRE with commitment, the output response is much more delayed than under the Calibrated model. Under discretion, the FIRE response becomes more front-loaded, while the Calibrated model shows little difference between commitment and discretion approaches, suggesting limited benefits of commitment over discretion in the Calibrated model. Note: Impulse responses for inflation, output, and the real interest rate in response to a markup shock. The responses are normalized so that inflation is identical and equal to an AR(1) with persistence 0.9 in all models if the central bank holds the real interest rate constant.
Figure A.1: BIE question eliciting cost growth expectations over the
next year
This is an image of a survey question from the Business Inflation Expectations (BIE) survey. The question asks respondents to project and assign percentage likelihoods to different possible changes in unit costs over the next twelve months. The question format shows five possible unit cost change scenarios (-2%, 0%, 2%, 4%, and 6%) with empty fields next to each option where respondents can enter percentage values that should sum to 100%.
Figure A.2: Average BIE responses align well with macroeconomic data
Note: BIE is employment-weighted cross-sectional average of
\(\Delta_{12}\textrm{mc}_{i,t}\) (LHS)
and \(\mathbf{E}_{i,t}\left[\Delta_{12}\textrm{mc}_{i,t+12}\right]\)
(RHS). CPI and PGDP are the year on year percent change. Mich HH is the
median expected price change over next year from Michigan Surveys of
Consumers. Cleveland is 1 year expected inflation from the Federal
Reserve Bank of Cleveland. SPF is 1 year ahead CPI forecast.
This is a two-panel line chart tracking inflation metrics over time from 2012 to 2024. The left panel shows "Actual growth over last year" with y-axis ranging from 0 to 8 percent. Three lines are plotted: "BIE" (blue), "CPI" (orange), and "PGDP" (purple). All three metrics track closely together, particularly BIE and PGDP (outside of the COVID region where the BIE truncation is binding), showing a notable spike around 2022 where values reach 6-8%. The right panel shows "Forecasts year ahead growth" with y-axis ranging from 0 to 5 percent. Four lines are plotted: "BIE" (blue), "Mich HH" (orange), "Cleveland" (purple), and "SPF" (brown). These forecast measures show similar patterns but with less volatility than actual growth, all rising moderately during the 2021-2023 period. Note: BIE is employment-weighted cross-sectional average of year-on-year cost changes and expected future costs. CPI and PGDP are year-on-year percent changes. Mich HH is the median expected price change from Michigan Surveys of Consumers. Cleveland is 1-year expected inflation from the Federal Reserve Bank of Cleveland. SPF is 1-year ahead CPI forecast.
Figure A.3: Proportion of cost responses in each bin over time
Note: Left hand side breaks down past cost growth response into
five bins. Right hand side is average share of probability weight placed
on the given bin when forecasting costs.
This is a two-panel chart showing the distribution of cost responses in the BIE survey from 2012 to 2024. The left panel shows "Cost growth over last year, by bin" with the y-axis measuring percent (0-70%). Five different colored lines represent the percentage of firms reporting cost growth in each bin: -2% (blue), 0% (orange), 2% (green), 4% (red), and 6% (purple). Most notably, the 6% line (purple) spikes dramatically around 2022, reaching nearly 50% of responses. The right panel shows "Expected cost growth over next year, by bin" with the same structure. It shows a similar pattern but with the 6% expectations less pronounced than actual reported costs during the 2022 spike. Note: Left panel breaks down past cost growth responses into five bins. Right panel shows average share of probability weight placed on each bin when forecasting costs.
Figure A.4: Average beliefs and average costs over time for each firm
Note: Binscatter plot where each bin contains 5 firms. Only
firms with at least 30 observations are included.
This is a binned scatter plot showing the relationship between average costs and average beliefs over time for individual firms. The x-axis shows "Average costs over time" (0-6%), and the y-axis shows "Average beliefs over time" (0-6%). Each point represents a bin of 5 firms. A blue linear fit line and a 45-degree line (dashed) are shown. Most points cluster close to the 45-degree line, indicating that firms' average beliefs about costs align well with their actual average costs over time. The linear fit line has a slightly flatter slope than the 45-degree line. Note: Binscatter plot where each bin contains 5 firms. Only firms with at least 30 observations are included.
Figure A.5: Firms (over) react to their own costs: Binscatter robustness
Note: Binscatter following the methodology in
This is a two-panel binscatter plot similar to Figure 1 in the main text, but using an alternative statistical methodology. The left panel shows "Firm fixed effects" with past cost growth on the x-axis (-2% to 6%) and year-ahead measures on the y-axis (1.0 to 4.0%). Two series are plotted: "Year-ahead expectations" (blue dots) and "Year-ahead outcome" (orange dots). The blue line shows a steeper positive relationship between past costs and expectations than the orange line showing actual outcomes. The right panel shows "Firm and Sector-Time fixed effects" with similar axes but a narrower y-axis range (1.5 to 3.5%). The same pattern appears with expectations showing a stronger relationship to past costs than actual outcomes. Note: Binscatter following the methodology in Cattaneo et al. (2024).
Figure A.6: Costs drive firms’ long-run and CPI beliefs
Note: Black lines are 95% confidence intervals, standard errors
clustered at firm level.
This is a bar chart showing regression coefficients across different control specifications. The x-axis shows four control types: "None," "Time FE," "Sector Time FE," and "PCA factors." The y-axis ranges from 0.0 to 0.4. Three sets of bars represent coefficients for different dependent variables: "Short run own cost expectations" (blue), "Long run own cost expectations" (orange), and "CPI expectations" (green). Black vertical lines represent 95% confidence intervals. All coefficients are positive and significant across specifications, showing that firms' past costs influence not only their short-run cost expectations but also their long-run cost and CPI inflation expectations. Note: Black lines are 95% confidence intervals, standard errors clustered at firm level.
Figure A.7: Including sales and profits as controls
Note: Black lines are 95% confidence intervals, standard errors
clustered at firm level.
This is a coefficient bar plot showing regression results with additional controls. The x-axis shows four control types: "None," "Time FE," "Sector Time FE," and "PCA factors." The y-axis ranges from -0.3 to 0.4. Three series are plotted representing coefficients for different outcomes: "Year-ahead expectations" (blue bars), "Year-ahead outcomes" (orange bars), and "Forecast error" (green bars). Black vertical lines show 95% confidence intervals. The pattern matches the main results in Figure 2, with expectations showing higher coefficients than outcomes, and negative forecast error coefficients, even when controlling for sales and profits. Note: Black lines are 95% confidence intervals, standard errors clustered at firm level.
Figure A.8: Including CPI beliefs as a control
Note: Black lines are 95% confidence intervals, standard errors
clustered at firm level.
This is a coefficient bar plot similar to Figure A.7, showing regression results when controlling for CPI beliefs. The x-axis shows four control types: "None," "Time FE," "Sector Time FE," and "PCA factors." The y-axis ranges from -0.4 to 0.6. Three series are plotted: "Year-ahead expectations" (blue bars), "Year-ahead outcomes" (orange bars), and "Forecast error" (green bars). Black vertical lines show 95% confidence intervals. Despite controlling for CPI beliefs, firms still overreact to their own costs, as shown by the negative forecast error coefficients. Note: Black lines are 95% confidence intervals, standard errors clustered at firm level.
Figure A.9: Robustness to instrumenting costs
Note: Black lines are 95% confidence intervals, standard errors
clustered at firm level. F-stats all larger than 250.
This is a coefficient bar plot showing regression results when instrumenting for costs. The x-axis shows three control types: "None," "Time FE," and "PCA factors." The y-axis ranges from 0.0 to 0.5. Only one outcome is shown: "Year-ahead expectations" (blue bars). Black vertical lines show 95% confidence intervals. The coefficients are positive and significant across all specifications, confirming that the relationship between past costs and expectations is robust to instrumenting. Note: Black lines are 95% confidence intervals, standard errors clustered at firm level. F-stats all larger than 250.
Figure A.10: Robustness to truncation of the data
Note: Black lines are 95% confidence intervals, standard errors
clustered at firm level.
This is a coefficient bar plot showing regression results when removing boundary values. The x-axis shows four control types: "None," "Time FE," "Sector Time FE," and "PCA factors." The y-axis ranges from -0.3 to 0.4. Three series are plotted: "uc_exp" (year-ahead expectations, blue bars), "yoyuc_lead12" (year-ahead outcomes, orange bars), and "uc_FE12" (forecast error, green bars). Black vertical lines show 95% confidence intervals. The results remain consistent with the main findings even after removing boundary values (-2% and 6%). Note: Black lines are 95% confidence intervals, standard errors clustered at firm level.
Figure A.11: Heterogeneity by frequency of price adjustment
Note: Vertical lines are 95% confidence intervals, standard
errors clustered at firm level. Includes PCA controls.
This is a coefficient bar plot comparing results across different frequencies of price adjustment. The x-axis shows four categories: "At least monthly," "Quarterly or semiannually," "Annually or less," and "Whole economy." The y-axis ranges from -0.5 to 0.4. Three series are plotted: "Year-ahead expectations" (blue bars), "Year-ahead outcome" (orange bars), and "Forecast error" (green bars). Vertical lines show 95% confidence intervals. The pattern of overreaction to past costs is consistent across all price adjustment frequency groups. Note: Vertical lines are 95% confidence intervals, standard errors clustered at firm level. Includes PCA controls.
Figure A.12: Belief react too much to costs on average but too little to
individual cost persistence
Note: Binned scatter plot of \(\hat{\tilde{\rho}}_{i}\) and \(\hat{\rho}_{i}\), recovered from regression
equations (54) and (55) respectively.
We run these time series regressions only for firms with at least 30
observations. Our results are robust to other cutoffs. Each bin contains
five observations.
This is a scatter plot showing the relationship between "Actual cost persistence, ρ" (x-axis, -0.4 to 0.6) and "Perceived cost persistence, ρ̃ (y-axis, -0.6 to 0.8). A linear fit line (red) and a 45-degree line representing Rational Expectations (black) are shown. The linear fit has a slope of 0.39, much flatter than the 45-degree line, indicating that firms with highly persistent costs underestimate their persistence, while firms with less persistent costs overestimate their persistence. Note: Binned scatter plot of ρ̃_(i) and ρ_(i), recovered from regression equations. We run these time series regressions only for firms with at least 30 observations. Our results are robust to other cutoffs. Each bin contains five observations.
Figure B.13: Model predicted time series for sectoral cost beliefs
closely match the data
This is a multi-panel chart showing time series from 2014 to 2024 for cost beliefs across different sectors. Each panel represents a different sector (e.g., "Whole economy," "Construction," "Durable goods manufacturing," etc.), with the y-axis showing values from 0 to 5 percent. In each panel, two lines are plotted: "BIE data" (blue) showing actual survey responses, and "Our model" (orange) showing the model predictions. The model predictions generally track the BIE data closely across all sectors, with both showing notable increases around 2021-2022.
Figure B.14: Comparing model fits: Response of belief to an oil price
change
This is a two-panel chart showing the response of cost forecasts to oil price changes over time. The left panel shows "Fast exposure to oil" with months since oil price increase (0-10) on the x-axis and year-ahead cost growth forecast (-0.0025 to 0.0150) on the y-axis. Three lines are plotted: "Data" (blue with confidence intervals), "Calibrated model" (black dashed), and "FIRE" (orange dashed). For firms quickly exposed to oil prices, the data and calibrated model show an immediate jump followed by a slow decline, while FIRE predicts a larger jump and steeper decline. The right panel shows "Slow exposure to oil" with the same x-axis and y-axis (0.000 to 0.012). For firms slowly exposed to oil prices, the data and calibrated model show a gradual increase in forecasts, unlike the FIRE model predictions, which jump and gradually decline.
Figure D.15: Percent price level response to 1pp permanent change in nominal
marginal costs at date s
Note: Impulse response of prices to a permanent rise in nominal
marginal costs under six models. ‘Calibrated’ is our model with \(\alpha^\textrm{FIRE}\), \(\tilde{\rho}\) and \(\tilde{\mathcal{K}}\) calibrated as in
Section 2.2
This is a two-panel chart showing price level responses to cost shocks across different models. The left panel shows "Shock date: s = 0" (immediate shock) with time periods (0-100) on the x-axis and price level response (0.0-1.0) on the y-axis. Six different models are plotted: "FIRE" (orange), "Calibrated" (blue), "Mankiw Reis" (red), "Sticky Expectations" (purple), "Cog. Disc." (brown), and "Diagnostic" (green). All models eventually converge to the same long-run price level (1.0), but with different adjustment paths. The right panel shows "Shock date: s = 10" (future shock) with the same axes and models. The models show varying degrees of anticipation before the shock occurs at period 10, followed by different adjustment paths that again converge to the same long-run level. Note: Impulse response of prices to a permanent rise in nominal marginal costs under six models.
Figure D.16: Jacobian of the NKPC across belief models
This is a multi-panel chart showing the Jacobian (partial derivatives) of the New Keynesian Phillips Curve across nine different belief models: "FIRE," "Diagnostic," "Cognitive Discounting," "Mankiw Reis," "Sticky Expectations," "Angeletos Huo," "Hybrid of FIRE and Misspecified," "Hybrid with permanent component," and "Hybrid with static beliefs." Each panel shows how inflation at time t responds to marginal cost shocks at different times s (s=0, s=5, s=10, s=15). The x-axis shows time t (0-15), and the y-axis shows the magnitude of the response (scales vary by panel). The patterns differ substantially across models, illustrating how different belief formation mechanisms lead to different inflation dynamics.
Figure D.17: Inflation response to supply and demand shocks under Taylor Rule
Note: The response of inflation to a monetary policy shock
(left) and to a TFP shock (right) under different models of firm
beliefs. Impulse responses normalized to 1 under FIRE. Our model,
calibrated in Section 2.2
above, is shown in blue.
This is a two-panel chart showing inflation responses under a Taylor rule for monetary policy. The left panel shows "TFP shock = Supply shock" with time periods (0-15) on the x-axis and normalized inflation response (0.00-1.75) on the y-axis. Six different models are compared: "Calibrated" (blue), "FIRE" (orange), "Diagnostic" (green), "Cog Disc" (brown), "Mankiw Reis" (red), and "Sticky Exp." (purple). The calibrated model shows a stronger initial inflation response to the supply shock compared to FIRE. The right panel shows "MP shock = Demand shock" with the same axes and models. Here, the calibrated model shows a weaker initial inflation response to the demand shock compared to FIRE. Note: The responses are normalized to 1 under FIRE.
Figure D.18: Inflation response to supply and demand shocks
Note: The response of inflation to a monetary policy shock
(left) and to a TFP shock (right) under different models of firm
beliefs. Impulse responses normalized to 1 under FIRE. Our model,
calibrated in Section 2.2
above, is shown in blue.
This is a two-panel chart comparing three specific models' responses to macroeconomic shocks. The left panel shows "TFP shock = Supply shock" with time periods (0-15) on the x-axis and normalized inflation response (0.00-1.75) on the y-axis. Three models are compared: "Calibrated" (blue), "FIRE" (orange), and "LIRE" (light blue). Both the calibrated and LIRE models show stronger initial inflation responses to the supply shock compared to FIRE. The right panel shows "MP shock = Demand shock" with the same axes and models. Both the calibrated and LIRE models show weaker initial inflation responses to the demand shock compared to FIRE. Note: The responses are normalized to 1 under FIRE.
Figure D.19: Forward guidance is much less powerful at long horizons
Note: Impact inflation, \(\pi_{0}\), in response to date \(s\) monetary policy shock announced at date
\(0\).
This is a line chart showing the impact of forward guidance on current inflation. The x-axis shows "Date of announced interest rate change, s" (0-50), and the y-axis shows "Period 0 inflation response" (0.0-0.6). Two models are compared: "Calibrated" (blue) and "FIRE" (orange). For near-term forward guidance (s ≤ 3), the calibrated model shows a stronger inflation response than FIRE. However, as the horizon extends, the calibrated model's response flattens out, while the FIRE model maintains a much stronger response even at very long horizons.
Figure D.20: Inflation response to estimated TFP and Monetary Policy shocks
Note: The response of inflation to an estimated TFP shock
(left) and to an estimated monetary policy shock (right) under different
models of firm beliefs. Impulse responses normalized to 1 under FIRE.
Our model, calibrated in Section 2.2
above, is shown in blue.
This is a two-panel chart showing inflation responses to estimated shocks in a medium-scale model. The left panel shows "TFP shock = Supply shock" with quarters (0-15) on the x-axis and normalized inflation response (0.0-1.4) on the y-axis. Two models are compared: "Calibrated" (blue) and "FIRE" (orange). The calibrated model shows a stronger initial inflation response to the TFP shock. The right panel shows "MP shock = Demand shock" with the same axes and models. The calibrated model shows a weaker initial inflation response to the monetary policy shock. Note: The responses are normalized to 1 under FIRE.