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### Key empirical facts on strategic complementarities and cost pass-through
- Firms' response to competitors’ price changes is about 1.5 times the response to an equivalent increase in marginal costs.
- Amiti et al. (2022) and Gagliardone et al. (2023) (Belgian manufacturing) find an elasticity of about 0.4 in response to competitors’ price changes and an elasticity of about 0.6 in response to own cost shocks.
- A 27 percent increase in competitors’ prices is sufficient to prompt the average firm to adjust the prices of all its products, whereas marginal costs would need to surge by 60 percent to achieve the same result.
- Intensive margin timing:
  - By the end of the year, firms have passed through nearly 70 percent of the competitor price increase.
  - Adjustment to marginal cost changes is more gradual: only 24 percent of the initial increase is reflected in sales prices by the end of the year.
- Sectoral heterogeneity:
  - Manufacturing pass-through of competitors’ prices ≈ 40 percent; retail ≈ 70 percent.
  - Cost pass-through rates: manufacturing 26 percent; retail 13 percent.
  - Retailers pass on a larger share of competitors’ price changes within the first couple of months; manufacturers take the entire year to pass on the same share.
- State dependence:
  - High-inflation environments: firms respond disproportionately to changes in marginal costs and competitors’ prices; firms stabilize markups relative to prices, prompting more aggressive responses.
  - Firms respond to cost changes mainly by adjusting prices more frequently; competitors’ price changes trigger both more frequent and larger price adjustments.
  - Monetary policy has less traction on real outcomes in these settings, but leads to faster price adjustment.
- Asymmetry:
  - Firms’ prices react more strongly when competitors raise prices than when they cut them; larger pass-through for competitors’ price increases driven by larger increase in frequency of price adjustments.
- Market structure:
  - Stronger strategic complementarities when firms have fewer competitors, larger market shares, and broader customer bases; more concentrated markets show more aggressive strategic pricing.

### Data, measurement, and sample
- Geographic and time coverage:
  - Chile, January 2014 to December 2023.
- Inflation environment (sample):
  - CPI inflation: declined to 1.4 percent in late 2017 after peaking at 5.9 percent at end-2014; rose from mid-2020 to peak of 14 percent in August 2022; declined to 3.8 percent by end-2023.
  - PPI inflation: turned negative in 2016, 2019, and 2023; trough −8.7 percent in July 2023; in 2014 approached 10 percent; surged to 27.9 percent in July 2022.
- Data source and construction:
  - Detailed administrative firm-to-firm electronic invoice records from Chile’s Internal Revenue Service (SII); transaction-level data aggregated at selling firm–product variety–month by median price per product variety per month.
- Cleaning and exclusions:
  - Month-to-month price changes smaller than 10 Chilean pesos treated as non-change.
  - One-month fully reversed changes replaced with prior month’s price.
  - Drop records with price changes outside bottom and top 5th percentile (price declines >50 percent or increases >100 percent).
  - Exclude firms with fewer than five employees on average; retain only firms in Chilean National Accounts registry; drop within-firm transactions and transactions subject to invoice corrections via credit notes.
- Sample size:
  - 21,722 firms.

### Firm-level inflation decomposition and measurement
- Firm-level inflation rate:
  - ∆p_{i,t} = Σ_{v∈i} ω_{i,v,t} ∆p_{i,v,t}, where ω_{i,v,t} is the one-year rolling average of sales share of variety v.
- Extensive and intensive margin decomposition:
  - ∆p_{i,t} = f_{i,t} s_{i,t}, where f_{i,t} is frequency and s_{i,t} is average size of non-zero price changes.
  - Frequency:
    - f_{i,t} = Σ_{v∈i} ω_{i,v,t} (Σ_{v∈i} D_{v,i,t} / Σ_{v∈i} O_{v,i,t}), with D_{v,i,t} = 1 if price changes in month t.
  - Average non-zero size:
    - s_{i,t} = Σ_{v∈i} ω^{,0}_{i,v,t} (Σ_{v∈i} ∆p^{,0}_{v,i,t} / Σ_{v∈i} O^{,0}_{v,i,t}).
  - Positive/negative split approximation:
    - ∆p_{i,t} ≈ f^{+}_{i,t} s^{+}_{i,t} + f^{-}_{i,t} s^{-}_{i,t}.
- Outlier handling:
  - Drop observations outside bottom and top 5 percentile interval (approximate monthly price declines and increases of over 20 percent).

### Frequency and size statistics (firm-level)
- Weighting schemes: sales-based and employment-based; report time averages.
- Frequency of observed price changes:
  - Sales-weighted median frequency: 40.5 percent.
  - Employment-weighted median frequency: 37.3 percent.
  - Interpretation: average firms adjust prices approximately every two months.
- Frequency of regular prices (Nakamura and Steinsson filter):
  - Sales-weighted median: 31.6 percent.
  - Employment-weighted median: 27.7 percent.
  - Interpretation: regular prices persist nearly three weeks longer than observed prices.
- Asymmetry in frequency:
  - Price increases occur more frequently than price decreases.
  - Observed prices: frequency difference ≈ 6 percentage points; regular prices: gap ≈ 4 percentage points.
- Intensive margin magnitudes:
  - Average magnitude of monthly price changes ranges between 0.7 and 0.8 percent over the sample period.

### Key empirical findings on price change frequency and size (table-style figures preserved)
- Price change frequency (observed prices, sales weights):
  - All changes: 40.5 percent (Median), 45.7 percent (Average)
  - Duration (months): 2.0 (Median), 1.6 (Average)
  - Increases: 20.2 percent (Median), 25.9 percent (Average)
  - Decreases: 14.8 percent (Median), 19.9 percent (Average)
- Price change frequency (observed prices, employment weights):
  - All changes: 37.3 percent (Median), 43.5 percent (Average)
  - Duration (months): 2.2 (Median), 1.8 (Average)
  - Increases: 17.0 percent (Median), 24.4 percent (Average)
  - Decreases: 11.8 percent (Median), 19.1 percent (Average)
- Price change frequency (regular prices, sales weights):
  - All changes: 31.6 percent (Median), 34.8 percent (Average)
  - Duration (months): 2.7 (Median), 2.4 (Average)
  - Increases: 15.7 percent (Median), 19.1 percent (Average)
  - Decreases: 11.5 percent (Median), 15.6 percent (Average)
- Price change frequency (regular prices, employment weights):
  - All changes: 27.7 percent (Median), 33.3 percent (Average)
  - Duration (months): 3.1 (Median), 2.5 (Average)
  - Increases: 13.4 percent (Median), 18.3 percent (Average)
  - Decreases: 8.9 percent (Median), 15.0 percent (Average)
- Size of price changes (observed prices, sales weights):
  - All changes: 0.8 (Median), 0.7 (Average)
  - Increases: 11.1 (Median), 12.3 (Average)
  - Decreases: -13.4 (Median), -14.6 (Average)
- Size of price changes (observed prices, employment weights):
  - All changes: 0.8 (Median), 0.7 (Average)
  - Increases: 12.0 (Median), 13.4 (Average)
  - Decreases: -14.4 (Median), -15.9 (Average)
- Size of price changes (regular prices, sales weights):
  - All changes: 0.7 (Median), 0.6 (Average)
  - Increases: 10.4 (Median), 11.6 (Average)
  - Decreases: -12.5 (Median), -13.8 (Average)
- Size of price changes (regular prices, employment weights):
  - All changes: 0.9 (Median), 0.7 (Average)
  - Increases: 11.1 (Median), 12.6 (Average)
  - Decreases: -13.5 (Median), -15.0 (Average)
- Additional observations:
  - Price decreases tend to be approximately 2.5 percentage points higher in magnitude than price increases.
  - Firms reduce prices less frequently but by a larger magnitude.
  - Regular-price increases and decreases are somewhat smaller than observed-price changes, suggesting temporary promotional sales drive larger observed adjustments.

### Empirical framework for strategic complementarities and pass-through
- Firms’ optimal pricing condition:
  - p_{i,t} = mc_{i,t} + M_i(p_{i,t}, p_{-i,t}, ξ_t), where mc_{i,t} is marginal cost and M_i is the markup function.
- Markup function:
  - M_i(p_{i,t}, p_{-i,t}, ξ_t) = log[ ε(p_{i,t}-p_{-i,t}, ξ_t) ε(p_{i,t}-p_{-i,t}, ξ_t)^{-1} ], with ε elasticity of demand.
- Reduced-form distributed lag estimating equation:
  - Δp_{i,t} = α_i + τ_t + Σ_{r=0}^R β_r Δmc_{i,t−r} + Σ_{r=0}^R γ_r Δp_{−i,t−r} + ε_{i,t}
- Interpretation:
  - β_r: pass-through of own marginal costs.
  - γ_r: strength of strategic complementarities (response to competitors’ prices).
  - Long-term (T = 12) reduced-form elasticities:
    - Σ_{r=0}^T β_{r}^i ≡ 1 / (1 + Γ_{i,T})
    - Σ_{r=0}^T γ_{r}^i ≡ Γ_{−i,T} / (1 + Γ_{i,T})
    - Γ_{i,T} ≡ −∂M_i(p_T; ξ_T) / ∂p_{i,T}; Γ_{−i,T} ≡ Σ_{j≠i} ∂M_i(p_T; ξ_T) / ∂p_{j,T}
- Mechanism:
  - Strategic complementarities cause positive firm responses to competitors’ price increases; they reduce pass-through of marginal cost changes because firms avoid being first to raise prices, generating coordination problems and price stickiness.

### Identification strategies and robustness
- Fixed effects:
  - Results robust to sector-time fixed effects.
- Instrumental variables:
  - Instrument marginal costs by excluding inputs from suppliers with few buyers (∆ip^90).
  - Instrument competitors’ prices using input-cost variation of competitors’ suppliers, excluding shared suppliers.
  - IV results corroborate dominance of strategic complementarities; OLS estimates are conservative relative to IV.
- Additional robustness:
  - Extended response horizons (24 months) and local projections confirm strength and consistency of findings, though some magnitudes vary (e.g., 24-month competitors’ prices pass-through ≈ 55 percent vs baseline ≈ 66 percent).

### Five facts about strategic complementarities (main quantitative conclusions)
- Fact 1: Strategic complementarities are overwhelmingly important.
  - Firms pass through 55 percent of competitors’ price changes within the same month and 66 percent over a one-year horizon.
  - About 10 percent of an increase in marginal costs is passed through on impact, reaching 18 percent after one year.
  - Pass-through of competitors’ prices is 3.7 times larger than pass-through of input prices.
  - Implied elasticities: competitors’ prices = 3.7; input prices = 4.7 (mapping to markup elasticities).
  - Extensive margin:
    - A one percentage point increase in marginal costs → one percentage point increase in share of products changing prices over 12 months.
    - A one percentage point increase in competitors’ prices → share of products adjusting rises by 2.2 percentage points after one year.
    - To adjust prices across all product varieties: marginal costs need to increase ≈ 60 percent; competitors’ prices need to increase ≈ 24 percent.
  - Intensive margin:
    - Marginal costs: on impact ≈ 16 percent pass-through; after one year ≈ 24 percent.
    - Competitors’ prices: on impact 64 percent; rises to 75 percent within first quarter; gradually declines to ≈ 67 percent thereafter.
- Fact 2: Strategic complementarities are first-order across sectors.
  - Sample: universe of manufacturing and retail firms; retail ≈ two-thirds of total.
  - Average market share (3-digit ISIC sector–municipality–product):
    - Retail: 6.1 percent.
    - Manufacturing: 8.9 percent.
  - Sectoral pass-through:
    - Marginal costs impact ≈ 10 percent across sectors; after one year: manufacturing 26 percent; retail nearly half that of manufacturing.
    - Competitors’ prices:
      - Manufacturing: <20 percent on impact; ≈ 40 percent one year after shock.
      - Retail: immediately pass through 65 percent; ≈ 75 percent by end of year.
  - Structural elasticities:
    - Retail markup elasticities: 4.7 (input prices); competitors’ prices elasticity 5.3 (vs 3.7 full sample).
    - Manufacturing elasticities: competitors’ prices 3; input prices 1.6.
- Fact 3: State-dependent strategic pricing (high vs low inflation).
  - Manufacturing marginal cost pass-through (on impact):
    - High inflation: ≈ 12 percent.
    - Low inflation: 7 percent.
  - Over time:
    - High inflation: almost 30 percent of cost changes passed through.
    - Low inflation: 17 percent passed through.
  - Frequency effect (marginal cost shock):
    - High-inflation: 2 percentage point increase in frequency per 1 percentage point cost shock.
    - Low-inflation: 0.6 percentage point increase.
  - Response to competitors’ price changes:
    - One percentage point increase in competitors’ prices → 0.5 percent increase in firm's sales prices when inflation is high, about half that when inflation is low.
    - Frequency (one year after):
      - High inflation: 2 percentage point increase.
      - Low inflation: transitory, peaks at 1 percentage point at 3 months, declines to 0.4 percentage points at one year.
  - Structural markup elasticity mapping:
    - Markup elasticity w.r.t own prices declines from 4.9 during low-inflation to 2.3 in high-inflation periods; this reduction accounts for ≈ 90 percent of the overall increase in reduced-form pass-through coefficients.
- Fact 4: Asymmetry — "rockets and feathers".
  - Marginal cost pass-through (manufacturing): higher for cost increases than decreases; immediate gap up to 7 percentage points that fades by year-end.
  - Competitors’ price pass-through (manufacturing):
    - On impact: increase pass-through 19 percent; price cut pass-through 9 percent (gap 10 percentage points).
    - By end of year: increases 34 percent; decreases 22 percent.
    - Frequency: a 1 percentage point competitors’ price increase → share of products changing prices up 2.8 percentage points after one year; competitors’ price cuts → frequency declines by ≈ 1.5 percentage points.
  - Size of price changes in response to competitors’ price increases vs decreases statistically indistinguishable.
- Fact 5: Market structure matters.
  - Stronger strategic complementarities when firms face less competition, have higher market shares, and serve larger customer bases.
  - Empirical examples:
    - One percentage point increase in competitors’ prices → 0.4 percentage point price increase for firms in less-competitive markets vs 0.3 percentage point for more-competitive markets (one-year horizon).
    - Firms with larger customer bases: 0.4 percentage point response vs 0.2 percentage point for smaller customer base.
  - No statistically significant differences in marginal cost pass-through across competition levels one year after shocks.

### Marginal costs and competitors’ prices measurement (Appendix summaries)
- Marginal costs model:
  - ∆mc_i,t = −∆a_i,t + (1−ρ_i)∆w_i,t + ρ_i ∆ip_i,t (equation 10).
  - Baseline simplification: ∆mc_i,t = −μ_i + ρ_i ∆ip_i,t + ε_i,t (equation 11).
  - Empirical marginal cost: ∆mc_i,t = ρ_i Σ_k ω_k,i,t ∆ip_k,i,t (equation 12).
- Data limitations:
  - Imported intermediate inputs excluded; ≈ 70 percent of Chilean firms source all intermediate inputs domestically.
  - No precise measure of variable labor costs; total labor costs ≈ 23 percent of combined value of intermediate inputs and labor costs on average.
- Competitors’ prices construction:
  - ∆p_{−i,t} constructed as sales-weighted average of firm-municipality-product price changes (equations 13–14); 290 product categories (CUP) via NLP; 54 manufacturing sectors, 5 wholesale, 11 retail; 345 municipalities and ≈ 1,700 products.
  - Distribution: leptokurtic with larger mass on positive side; nearly half observations have unchanged competitors’ prices; changes more likely increases.
  - Unconditional correlations:
    - Competitors’ prices vs firm prices ≈ 0.38.
    - Marginal costs vs competitors’ prices ≈ 0.15.
  - Competitors’ price changes range −40 to 40 percent (individual variety changes bounded −50 to 200 percent); trimming −20 to 20 percent yields virtually identical results.

### Decomposition of pass-through into frequency and magnitude (Appendix D)
- Cumulative R-period change:
  - ∆^R p_t = f_{t−R} ∆^R s_t + s_{t−R} ∆^R f_t + ∆^R s_t ∆^R f_t (equation (28)).
- Cumulative effect of marginal cost change:
  - ∂∆^R p_t / ∂∆mc_t = Σ_{r=0}^R β_r = f_{t−1} Σ_{r=0}^R β_{s,r} + s_{t−1} Σ_{r=0}^R β_{f,r} + (Σ_{r=0}^R β_{s,r})(Σ_{r=0}^R β_{f,r}) (equation (29)).
- Empirical implication: pass-through arises from changes in frequency, size, and their interaction.

### Robustness, IVs, and alternative estimators
- Identification challenges addressed:
  1. Sectoral demand shocks → sector-time fixed effects.
  2. Reverse causality in supplier prices → instrument marginal costs excluding suppliers with few buyers (∆ip^90).
  3. Simultaneity in firm–competitor pricing → instrument competitors’ prices using competitors’ suppliers’ input-cost variation excluding shared suppliers.
- IV results:
  - First-stage Kleibergen-Paap Wald F-statistic > 200.
  - IV one-year pass-through: competitors’ prices ≈ 85 percent; marginal costs ≈ 12 percent.
  - Sum of estimated elasticities statistically indistinguishable from one.
- Local projections (one-year-ahead):
  - Marginal cost pass-through = 17 percent (baseline 18 percent).
  - Competitors’ prices pass-through = 59 percent (baseline 66 percent).
  - Frequency and size responses differ modestly versus distributed-lag baseline.

### Policy implications and theoretical relevance
- Strategic complementarities are a source of real rigidities that are first-order in price setting and should be incorporated into macroeconomic models.
- Monetary policy effects:
  - Findings imply monetary policy can have relatively larger effects on real variables than standard models predict when strategic complementarities are present.
  - Magnitude of monetary policy effects depends on macro conditions (e.g., high inflation), direction of cost and competitor price changes, and market structure (concentration, market shares, customer base).
- Research avenues:
  - Precisely estimated elasticities of firms’ prices w.r.t. own costs and competitors’ prices and their variation across environments enable assessments of monetary policy effectiveness when nominal rigidities coexist with real rigidities.

*Source: wpiea2025164-source-pdf*

### 1.5 times the response to an equivalent increase in marginal costs. This implies that a 27

### wpiea2025164-source-pdf - 1.5 times the response to an equivalent increase in marginal costs. This implies that a 27

### Key empirical facts on strategic complementarities and cost pass-through
- Firms' response to competitors’ price changes is about 1.5 times the response to an equivalent increase in marginal costs.
- Using data on Belgian manufacturing firms, Amiti et al. (2022) and Gagliardone et al. (2023) find an elasticity of about 0.4 in response to its competitors’ price changes and an elasticity of about 0.6 in response to its own cost shocks.
- A 27 percent increase in competitors’ prices is sufficient to prompt the average firm to adjust the prices of all its products, whereas marginal costs would need to surge by 60 percent to achieve the same result.
- At the intensive margin:
  - By the end of the year, firms have passed through nearly 70 percent of the competitor price increase.
  - Adjustment to changes in marginal costs is more gradual: only 24 percent of the initial increase is reflected in sales prices by the end of the year.
- Sectoral heterogeneity:
  - Manufacturing firms pass through about 40 percent of changes in competitors’ prices.
  - Retail firms’ pass-through of competitors’ prices is about 70 percent.
  - Cost pass-through rates: 26 percent for manufacturing firms and 13 percent for retail firms.
  - Retailers pass on a much larger share of competitors’ price changes within the first couple of months; manufacturers take the entire year to pass on the same share.
  - Frequency of price adjustments to competitors’ price changes is significantly higher for retailers than manufacturers at the one-year horizon.
- State dependence:
  - In high inflation environments, firms disproportionately respond to changes in marginal costs and competitors’ prices.
  - Mapping pass-through estimates to structural markup elasticities shows firms tend to stabilize markups relative to prices, prompting more aggressive responses to input and competitors’ price variations.
  - Firms respond to cost changes mainly by adjusting prices more frequently; competitors’ price changes trigger both more frequent and larger price adjustments.
  - Policy implication: monetary policy has less traction on real outcomes in these settings, but leads to faster price adjustment.
- Asymmetry in strategic pricing:
  - Firms’ prices react more strongly when competitors raise prices than when they cut them.
  - Larger pass-through for competitors’ price increases is driven by a larger increase in the frequency of price adjustments; no statistically significant differences in the magnitude of price changes between upward and downward competitor price moves.
- Market structure effects:
  - Firms exhibit stronger strategic complementarities when they have fewer competitors, hold larger market shares, and serve broader customer bases.
  - Firms in more concentrated markets engage more aggressively in strategic pricing.

### Data, measurement, and sample
- Geographic and time coverage:
  - Empirical analysis focuses on Chile over the period January 2014 to December 2023.
- Inflation environment in sample:
  - CPI inflation: declined to 1.4 percent in late 2017 after peaking at 5.9 percent at the end of 2014; rose from mid-2020 and reached a peak of 14 percent in August 2022; declined to 3.8 percent by the end of 2023.
  - PPI inflation: turned negative in 2016, 2019, and 2023; trough of –8.7 percent in July 2023; in 2014 approached 10 percent; surged to 27.9 percent in July 2022.
- Data source and construction:
  - Detailed administrative firm-to-firm electronic invoice records from Chile’s Internal Revenue Service (SII) used to extract buyer, seller, sector, product variety (text descriptors), and transaction prices for the universe of domestic transactions since 2014.
  - Transaction-level data aggregated at the selling firm–product variety–month level by computing the median price per product variety per month.
- Cleaning and exclusions:
  - If month-to-month price change is smaller than 10 Chilean pesos (less than 2 U.S. cents), observation treated as non-change and previous month’s price carried forward.
  - If a price change in one month is fully reversed in the following month, it is replaced with the prior month’s price.
  - Drop records with price changes outside the bottom and top 5th percentile interval (price declines >50 percent or increases >100 percent).
  - Exclude firms with fewer than five employees on average over the year.
  - Retain only firms listed in the Chilean National Accounts registry.
  - Drop within-firm transactions and transactions subject to invoice corrections via credit notes.
- Sample size:
  - The sample consists of 21,722 firms.

### Firm-level inflation decomposition and measurement
- Firm-level inflation rate construction:
  - For firm i selling product variety v at time t:
    ∆p_{i,t} = Σ_{v∈i} ω_{i,v,t} ∆p_{i,v,t},
    where ω_{i,v,t} denotes the one-year rolling average of the sales share of product variety v in firm i’s total sales at time t.
- Decomposition into extensive and intensive margins:
  - ∆p_{i,t} = f_{i,t} s_{i,t},
    where f_{i,t} is the frequency of price changes and s_{i,t} is the average size of non-zero price changes.
  - Frequency measure:
    f_{i,t} = Σ_{v∈i} ω_{i,v,t} (Σ_{v∈i} D_{v,i,t} / Σ_{v∈i} O_{v,i,t}),
    where D_{v,i,t} = 1 if price of variety v sold by firm i changes in month t; O_{v,i,t} indicates price observed in that month.
  - Average size of non-zero changes:
    s_{i,t} = Σ_{v∈i} ω^{,0}_{i,v,t} (Σ_{v∈i} ∆p^{,0}_{v,i,t} / Σ_{v∈i} O^{,0}_{v,i,t}),
    where ∆p^{,0}_{v,i,t} denotes the price change when price differs from previous month; ω^{,0}_{i,v,t} are sales weights normalized over varieties with non-zero changes.
- Positive/negative split approximation:
  - ∆p_{i,t} ≈ f^{+}_{i,t} s^{+}_{i,t} + f^{-}_{i,t} s^{-}_{i,t},
    where + and − denote positive and negative price changes, respectively.
- Outlier handling:
  - Drop observations outside bottom and top 5 percentile interval (approximate monthly price declines and increases of over 20 percent).

### Frequency and size statistics (firm-level)
- Aggregation and weighting:
  - Two weighting schemes used: sales-based and employment-based; report time averages.
- Frequency of observed price changes:
  - Weighted median frequency using sales weights: 40.5 percent.
  - Weighted median frequency using employment weights: 37.3 percent.
  - Interpretation: on average, firms adjust prices approximately every two months.
- Frequency of regular prices (Nakamura and Steinsson methodology):
  - Weighted median using sales weights: 31.6 percent.
  - Weighted median using employment weights: 27.7 percent.
  - Interpretation: regular prices persist, on average, nearly three weeks longer than observed prices.
- Asymmetry in frequency:
  - Price increases occur more frequently than price decreases.
  - For observed prices, difference in frequency ~ 6 percentage points; for regular prices the gap narrows to ~ 4 percentage points.
  - Reduction in frequency from observed to regular prices largely driven by fewer price increases, consistent with filtering out temporary promotional sales.
- Intensive margin magnitudes:
  - Over the sample period, average magnitude of monthly price changes ranges between 0.7 and 0.8 percent (depending on median or average prices).

### Robustness checks and identification strategies
- Fixed effects:
  - Results robust to inclusion of sector-time fixed effects to account for sector-specific demand shocks and coordinated wage adjustments.
- Instrumental variables (IV) to address endogeneity:
  - To mitigate reverse causality from monopsonic power: exclude purchases from suppliers that have only a few buyers.
  - To tackle simultaneity in price setting between firms and competitors: instrument competitors’ prices using input cost changes faced by competitors’ suppliers, excluding inputs sold to both the firm and its competitors.
  - IV results corroborate dominance of strategic complementarities; OLS estimates are conservative relative to IV.
- Additional robustness:
  - Extended response horizons and alternative estimation methods confirm strength and consistency of findings.

### Policy implications and theoretical relevance
- Strategic complementarities constitute a source of real rigidities that are first-order in price setting and should be incorporated in macroeconomic models.
- Monetary policy effects:
  - Findings imply monetary policy can have relatively larger effects on real variables than standard models predict when strategic complementarities are present.
  - Magnitude of monetary policy effects depends on macroeconomic conditions (e.g., high inflation), direction of cost and competitor price changes, and market structural characteristics (market concentration, firm market shares, customer base).
- Research avenues:
  - Precisely estimated elasticities of firms’ prices w.r.t. own costs and competitors’ prices, and their variation across environments, open new avenues to assess monetary policy effectiveness when nominal rigidities are paired with real rigidities.

*Source: wpiea2025164-source-pdf*

### 11.1 to 13.4 percent,  depending on the weighting scheme and whether the median or

### 11.1 to 13.4 percent,  depending on the weighting scheme and whether the median or

### Key empirical findings on price change frequency and size
- Price change frequency (observed prices, sales weights):
  - All changes: 40.5 percent (Median), 45.7 percent (Average)
  - Duration (months): 2.0 (Median), 1.6 (Average)
  - Increases: 20.2 percent (Median), 25.9 percent (Average)
  - Decreases: 14.8 percent (Median), 19.9 percent (Average)
- Price change frequency (observed prices, employment weights):
  - All changes: 37.3 percent (Median), 43.5 percent (Average)
  - Duration (months): 2.2 (Median), 1.8 (Average)
  - Increases: 17.0 percent (Median), 24.4 percent (Average)
  - Decreases: 11.8 percent (Median), 19.1 percent (Average)
- Price change frequency (regular prices, sales weights):
  - All changes: 31.6 percent (Median), 34.8 percent (Average)
  - Duration (months): 2.7 (Median), 2.4 (Average)
  - Increases: 15.7 percent (Median), 19.1 percent (Average)
  - Decreases: 11.5 percent (Median), 15.6 percent (Average)
- Price change frequency (regular prices, employment weights):
  - All changes: 27.7 percent (Median), 33.3 percent (Average)
  - Duration (months): 3.1 (Median), 2.5 (Average)
  - Increases: 13.4 percent (Median), 18.3 percent (Average)
  - Decreases: 8.9 percent (Median), 15.0 percent (Average)
- Size of price changes (observed prices, sales weights):
  - All changes: 0.8 (Median), 0.7 (Average)
  - Increases: 11.1 (Median), 12.3 (Average)
  - Decreases: -13.4 (Median), -14.6 (Average)
- Size of price changes (observed prices, employment weights):
  - All changes: 0.8 (Median), 0.7 (Average)
  - Increases: 12.0 (Median), 13.4 (Average)
  - Decreases: -14.4 (Median), -15.9 (Average)
- Size of price changes (regular prices, sales weights):
  - All changes: 0.7 (Median), 0.6 (Average)
  - Increases: 10.4 (Median), 11.6 (Average)
  - Decreases: -12.5 (Median), -13.8 (Average)
- Size of price changes (regular prices, employment weights):
  - All changes: 0.9 (Median), 0.7 (Average)
  - Increases: 11.1 (Median), 12.6 (Average)
  - Decreases: -13.5 (Median), -15.0 (Average)
- Additional empirical observations:
  - Price decreases tend to be approximately 2.5 percentage points higher in magnitude than price increases.
  - Firms reduce prices less frequently but by a larger magnitude.
  - Regular-price increases and decreases are somewhat smaller than observed-price changes, suggesting price adjustments associated with temporary sales tend to be larger.

### Definitions and measurement notes
- Regular prices: identified using the filter of Nakamura and Steinsson (2008), which removes asymmetric V-shaped sales defined as temporary price reductions followed by a return to the regular price; the filter applies a return window of five months.
- Implied duration: expressed in months and calculated as the inverse of the frequency of price changes.
- The table reports both sales-weighted and employment-weighted statistics, and presents median and average measures for frequency, duration, and size.

### Relation to existing literature
- Most literature on price rigidity focuses on CPI or PPI products rather than firm-level measures.
- Appendix B presents analogous calculations at the product level for varieties included in the CPI and PPI indexes to relate firm-level estimates to the literature.
- While direct comparisons are challenging, available evidence for emerging markets suggests the firm-level estimates of price adjustment frequency fall broadly within the range reported for other countries.

### Empirical framework for strategic complementarities and pass-through
- Firms’ optimal price condition:
  - p_{i,t} = mc_{i,t} + M_i(p_{i,t}, p_{-i,t}, ξ_t)
  - mc_{i,t} is firm i’s marginal cost at time t; M_i is the markup function of the firm’s own price, competitors’ prices p_{-i,t}, and demand shifters ξ_t.
- Markup function:
  - M_i(p_{i,t}, p_{-i,t}, ξ_t) = log[ ε(p_{i,t}-p_{-i,t}, ξ_t) ε(p_{i,t}-p_{-i,t}, ξ_t)^{-1} ]
  - ε is the elasticity of demand; the formulation holds for any invertible demand system and competition structure.
- Reduced-form distributed lag estimating equation (derived):
  - Δp_{i,t} = α_i + τ_t + Σ_{r=0}^R β_r Δmc_{i,t−r} + Σ_{r=0}^R γ_r Δp_{−i,t−r} + ε_{i,t}
- Interpretation of coefficients:
  - β_r capture pass-through of changes in firms’ own marginal costs.
  - γ_r capture the strength of strategic complementarities via response to competitors’ prices.
  - Long-term reduced-form elasticities (for T = 12):
    - Σ_{r=0}^T β_{r}^i ≡ 1 / (1 + Γ_{i,T})
    - Σ_{r=0}^T γ_{r}^i ≡ Γ_{−i,T} / (1 + Γ_{i,T})
    - Γ_{i,T} ≡ −∂M_i(p_T; ξ_T) / ∂p_{i,T} (elasticity of the markup w.r.t. own price)
    - Γ_{−i,T} ≡ Σ_{j≠i} ∂M_i(p_T; ξ_T) / ∂p_{j,T} (elasticity of the markup w.r.t. competitors’ prices)
- Mechanism:
  - Strategic complementarities imply firms’ prices react positively to competitors’ price increases.
  - Strong strategic complementarities reduce pass-through of marginal cost changes because firms are reluctant to lose market share by being first to raise prices.
  - This can generate coordination problems and price stickiness.

### Extensions and robustness
- The distributed lag specification (baseline T = 12) captures dynamics in response to marginal cost and competitor price changes; the paper notes this is the first to characterize the dynamics of these coefficients.
- The paper re-estimates the specification replacing the dependent variable with:
  - Frequency of price changes (extensive margin) to gauge state dependence of nominal rigidities.
  - Size of price changes (intensive margin) to measure pass-through conditional on price changes.
- Local projections are considered as an alternative to obtain dynamics, but two downsides are highlighted:
  - Loss of the first year of data because independent variables have 12 lags.
  - Computational intensity: local projections require one regression per horizon versus a single distributed lag regression.
- Robustness section presents results using local projections.

*Source: wpiea2025164-source-pdf - 11.1 to 13.4 percent,  depending on the weighting scheme and whether the median or*

### Appendix D illustrates the decomposition of the pass-through into the frequency and the magnitude

### Appendix D illustrates the decomposition of the pass-through into the frequency and the magnitude of price adjustments

### 3.1 Marginal costs
- Model and measurement
  - Log change in marginal costs:
    - ∆mc_i,t = −∆a_i,t + (1−ρ_i)∆w_i,t + ρ_i ∆ip_i,t (equation 10).
    - Baseline assumption: wages are predetermined at the frequency of the firm’s pricing decisions.
    - Productivity evolves around a firm-specific trend plus idiosyncratic white noise, yielding:
      - ∆mc_i,t = −μ_i + ρ_i ∆ip_i,t + ε_i,t (equation 11).
  - Empirical marginal cost measure:
    - ∆mc_i,t = ρ_i Σ_k ω_k,i,t ∆ip_k,i,t (equation 12), where expenditure shares ω are one-year rolling averages.
- Data advantages and limitations
  - Granular electronic invoice data allow precise tracking of price changes in intermediate inputs purchased by each firm.
  - Limitation: price changes for imported intermediate inputs are excluded; in the Chilean context approximately 70 percent of firms source all their intermediate inputs domestically.
  - Limitation: no precise measure of variable labor costs from monthly Social Security records; on average total labor costs account for 23 percent of the combined value of intermediate inputs and labor costs.
- Descriptive evidence
  - Distribution of changes in marginal costs is skewed right (cost increases more frequent/larger than decreases).
  - Unconditional relationship between changes in marginal costs and changes in firms’ sales prices: slope = 0.17, statistically significant at the 1 percent level — prima facie evidence of positive pass-through from input cost changes to output prices.

### 3.2 Competitors’ prices
- Competitors definition and construction
  - Competitors’ price changes for firm i:
    - ∆p_{−i,t} = Σ_{r∈R_i} ω_{i,r,t} Σ_{d∈D_i} Σ_{j∈J_{s(i),d,r}} ω_{j,s,d,r,t} ∆p_{j,s,d,r,t} (equation 13).
    - Firm-municipality-product price change:
      - ∆p_{j,s,d,r,t} = Σ_{v∈V_{j,s,d,r,t}} ω_{j,v,t} ∆p_{j,v,t} (equation 14).
  - Municipality defined by buyer location; products classified into 290 standardized product categories (CUP) using NLP on text descriptors.
  - Scope: according to ISIC rev.4 3-digit sorting, 54 manufacturing sectors, 5 wholesale, 11 retail; 345 municipalities and about 1,700 products.
- Distributional and correlation facts
  - Distribution of competitors’ price changes is leptokurtic with larger mass on the positive side; nearly half of firm-time observations have unchanged competitors’ prices; when changes occur they are more likely price increases.
  - Unconditional correlation between changes in competitors’ prices and firms’ own sales prices ≈ 0.38.
  - Correlation between marginal costs and competitors’ prices ≈ 0.15 (helps identification of β_r and γ_r).
  - Competitors’ price changes range from −40 to 40 percent (individual product variety changes bounded between −50 and 200 percent); trimming between −20 and 20 percent yields virtually identical results.

### 3.3 Identification challenges and instruments
- Three key identification challenges:
  1. Omitted variable bias from sectoral demand shocks or coordinated wage adjustments → addressed by replacing time fixed effects with sector-time fixed effects.
  2. Reverse causality from firms with monopsony power influencing supplier prices → instrument for changes in marginal costs:
     - ∆mc*_i,t = ρ*_i Σ_k ω_k,i,t ∆ip^90_{k,i,t} (equation 15), where ∆ip^90 excludes inputs sold by suppliers with a small number of buyers (keeps suppliers whose number of buyers exceeds the 90th percentile).
  3. Simultaneity in price setting between a firm and its competitors → instrument for competitors’ price changes:
     - ∆p^*_{−i,t} = Σ_{r∈R_i} ω_{i,r,t} Σ_{d∈D_i} Σ_{j∈J_{s(i),d,r}} ω_{j,s,d,r,t} ρ_j Σ_{k∉S} ω_{k,j,t} ∆ip_{k,j,t} (equation 16), excluding inputs purchased from suppliers S that sell to both firm i and its competitors to isolate competitors’ input-cost-driven variation.

### 4 Five facts about strategic complementarities
- Overview: document five key facts about firms’ pricing behavior with respect to competitors’ prices (section introduces five facts; content below covers facts 1–2 and extensive/intensive margins).

#### 4.1 Fact 1: strategic complementarities are overwhelmingly important
- Main pass-through estimates (Equation 8; standard errors clustered at firm and month):
  - Firms pass through 55 percent of changes in competitors’ prices to their own sales prices within the same month and 66 percent over a one-year horizon.
  - About 10 percent of an increase in marginal costs is passed through to sales prices on impact, reaching 18 percent after one year.
  - Pass-through of competitors’ prices is 3.7 times larger than pass-through of input prices.
  - Implied elasticities from mapping to markup elasticities:
    - Implied elasticity to competitors’ prices = 3.7.
    - Implied elasticity to input prices = 4.7.
- Comparison and omitted-variable bias
  - Estimating without controlling for competitors’ prices yields a higher marginal cost pass-through of 37 percent.
- Robustness to competitor definition
  - Using a broader industry-only competitor definition (excluding geographic and product dimensions) biases estimates:
    - Estimated marginal cost pass-through would be 44 percent higher relative to baseline.
    - Pass-through to competitors’ prices would be 37 percent lower relative to baseline.

##### Extensive and intensive margins of price adjustments
- Extensive margin (frequency of price changes)
  - A one percentage point increase in a firm’s marginal costs → a one percentage point increase in the share of products changing prices over a 12-month horizon.
  - For the average firm (sales-weighted median frequency in Table 1), a 10 percentage point increase in marginal costs is needed to raise the share of products changing prices to nearly 50 percent.
  - In response to a one percentage point increase in competitors’ prices, the share of products adjusting prices rises by 2.2 percentage points after one year.
  - For the average firm to adjust prices across all product varieties:
    - Marginal costs would need to increase by approximately 60 percent.
    - Competitors’ prices would need to increase by 24 percent.
- Intensive margin (magnitude conditional on changing prices; excludes zero changes)
  - Marginal costs:
    - On impact, approximately 16 percent of a marginal cost increase is passed through to sales prices.
    - After one year, about 24 percent of the initial cost increase is reflected in sales prices.
  - Competitors’ prices:
    - On impact, 64 percent of the increase in competitors’ prices is reflected in firms’ sales prices.
    - Pass-through rises to 75 percent within the first quarter.
    - Pass-through then gradually declines to around 67 percent over subsequent months.
- Interpretation
  - Strong pass-through of competitors’ prices arises from both extensive (frequency) and intensive (size) adjustments.
  - Short-run rise in pass-through driven by larger size of adjustments; longer-run decline reflects smaller individual adjustments offset by steady increase in frequency — consistent with state-dependent pricing with strategic complementarities.

#### 4.2 Fact 2: strategic complementarities are of first-order importance across sectors
- Sample composition and heterogeneity strategy
  - Sample covers the universe of manufacturing and retail firms; retail firms account for about two-thirds of the total.
  - Estimate variant of Equation 8 allowing coefficients to vary between manufacturing and retail sectors (equation 17, M_i = 1 for manufacturing).
- Market power and market shares
  - Firms hold relatively large market shares within narrowly defined markets (3-digit ISIC sector–municipality–product); average market share for the disaggregated level:
    - Retail firms: 6.1 percent.
    - Manufacturing firms: 8.9 percent.
- Sectoral pass-through patterns
  - Marginal costs:
    - Impact pass-through similar across sectors ≈ 10 percent on impact.
    - After one year:
      - Manufacturing pass-through rises to 26 percent.
      - Retail pass-through nearly half that of manufacturing at one year (implied by statement: “nearly double that of retail firms”).
  - Competitors’ prices:
    - Manufacturing firms: less than 20 percent of competitors’ price changes passed through on impact; about 40 percent one year after the shock.
    - Retail firms: immediately pass through 65 percent of competitors’ price changes; about 75 percent by the end of the year.
- Interpretation
  - Retail markets (lower menu costs, more flexible pricing) exhibit stronger strategic complementarities than manufacturing (higher adjustment costs, longer production cycles).
  - Heterogeneous reduced-form coefficients reflect larger markup elasticities in retail relative to the full sample (retail elasticity for input prices reported as 6.7 in the text fragment).

*Source: wpiea2025164-source-pdf — Appendix D illustrates the decomposition of the pass-through into the frequency and the magnitude*

### 4.7 in the full sample; for competitors’ prices, the elasticity is 5.3 compared to 3.7, respec-

### wpiea2025164-source-pdf - 4.7 in the full sample; for competitors’ prices, the elasticity is 5.3 compared to 3.7, respec-

### Sectoral differences in structural elasticities and pass-through
- For retail firms, markup elasticities: 4.7 in the full sample; for competitors’ prices, the elasticity is 5.3 compared to 3.7, respectively.
- Manufacturing firms display lower elasticities: 3 with respect to competitors’ prices and 1.6 with respect to their own input prices.
- Interpretation: differences in structural elasticities across sectors help explain variation in estimated pass-through coefficients.
- Common conclusion across sectors: firms tend to stabilize markups with respect to competitors’ prices to a larger extent than markups with respect to input prices.

### Extensive and intensive margins of price adjustments (manufacturing vs retail)
- Frequency (extensive margin) response to a one percentage point increase in marginal costs:
  - Manufacturing: share of product varieties changing prices increases by 1.5 percentage points at the end of the year.
  - Retail: share increases by 0.7 percentage points at the end of the year.
- Intensive margin (size of price adjustment) response to a one percentage point increase in marginal costs:
  - Manufacturing: pass-through of 42 percent of the increase in marginal costs to sales prices.
  - Retail: pass-through of 15 percent.
- Response to a one percentage point increase in competitors’ prices (frequency):
  - Immediate increase in the share of product varieties changing prices is similar for manufacturing and retail on impact.
  - By the end of the year:
    - Manufacturing: increase of 1.5 percentage points.
    - Retail: increase of 2.5 percentage points.
- Dynamics:
  - Retail firms: frequency of price adjustments continues to rise steadily even 12 months after the shock.
  - Manufacturing firms: frequency peaks at 1.8 percentage points after six months and then marginally declines.
- Intensive margin dynamics for competitors’ price shocks:
  - Manufacturing: size of price adjustments rises from 28 percent on impact to 66 percent after 12 months.
  - Retail: pass through approximately 75 percent immediately, peaking at 82 percent two months after the shock; subsequently retrenches so that by 12 months the size is not statistically different from manufacturing.

### Industry heterogeneity
- Baseline specification estimated separately at the 2- or 3-digit ISIC Rev. 4 level for industries with at least 100 firms.
- For each industry, compute difference between one-year pass-through coefficient for competitors’ prices and corresponding one for marginal costs.
- Figure 8 finding: for most industries the positive difference between the two estimates is statistically significant; for a few industries where the difference is negative, such estimates are statistically indistinguishable from zero.
- Interpretation: evidence of the dominant role of strategic complementarities relative to firms’ own marginal costs in shaping pricing decisions.

### Fact 3 — State-dependent strategic pricing (high vs low inflation)
- Empirical specification conditions responses on periods when (monthly) inflation is above the median plus one standard deviation.
- Focus: manufacturing firms (nominal rigidities more important for manufacturing).
- Pass-through of marginal cost changes (manufacturing):
  - On impact:
    - High inflation: approximately 12 percent.
    - Low inflation: 7 percent.
  - Over time:
    - High inflation: almost 30 percent of cost changes passed through.
    - Low inflation: 17 percent passed through.
- Frequency effect (marginal cost shock):
  - One percentage point increase in marginal costs associated with:
    - High-inflation periods: two percentage point increase in frequency of price adjustments.
    - Low-inflation periods: 0.6 percentage point increase.
- Size effect: broadly similar across inflation regimes for marginal cost shocks.
- Response to competitors’ price changes:
  - One percentage point increase in competitors’ prices associated with a 0.5 percent increase in a firm’s sales prices when inflation is high, compared to approximately half of that when inflation is low.
  - Frequency (competitors’ price shock, one year after):
    - High inflation: two percentage point increase in frequency.
    - Low inflation: increase is transitory and modest, peaking at one percentage point three months after the shock and declining to 0.4 percentage points after one year.
  - Size (competitors’ price shock):
    - High inflation: firms fully reflect competitors’ price changes in the prices they adjust.
    - Low inflation: size effect is 0.4 percent.
- Structural markup elasticity mapping:
  - Markup elasticity with respect to the firm’s own prices declines from 4.9 during low-inflation periods to 2.3 in high-inflation periods.
  - Reduction in this elasticity accounts for approximately 90 percent of the overall increase in reduced-form pass-through coefficients; remaining 10 percent attributable to changes in the markup elasticity with respect to competitors’ prices.

### The 2021–2022 inflation surge: contributions of margins
- Decomposition: define inflation deviations and decompose into contributions from:
  - Changes in frequency of price adjustments.
  - Changes in size of price adjustments.
  - Residual term.
- Figure 10 finding: the frequency of price adjustment was the primary driver of the inflation surge that began in 2021 and the subsequent normalization that started in 2023.

### Fact 4 — Strategic complementarities as “rockets and feathers” (asymmetric responses)
- Specification uses indicators for positive versus negative changes in marginal costs and competitors’ prices to detect asymmetry.
- Marginal cost pass-through (manufacturing):
  - Higher for cost increases than for cost decreases.
  - Immediate gap up to 7 percentage points, narrowing over time and statistically insignificant by end of year.
  - Frequency: one year after a one percentage point increase in marginal costs, share of products changing prices rises by 1.7 percentage points versus 0.4 percentage points following a one percentage point decrease.
  - Initial size larger for cost increases than decreases, but difference fades within a quarter.
- Competitors’ price pass-through (manufacturing):
  - On impact:
    - Price increase pass-through: 19 percent.
    - Price cut pass-through: 9 percent.
    - Gap on impact: 10 percentage points.
  - By the end of the year:
    - Pass-through of competitors’ price increases: 34 percent.
    - Pass-through of competitors’ price decreases: 22 percent.
  - Frequency and magnitude dynamics:
    - A one percentage point increase in competitors’ prices leads to a gradual rise in share of products changing prices, reaching 2.8 percentage points higher after one year.
    - For competitors’ price cuts, frequency of price changes declines by about 1.5 percentage points.
    - The size of price changes in response to competitors’ price increases and decreases is statistically indistinguishable.
- Interpretation: evidence consistent with strong downward nominal rigidities and asymmetric “rockets and feathers” behavior for competitors’ prices.

### Fact 5 — Market structure and strategic complementarities
- Firms exhibit stronger strategic complementarities when they:
  - Face less competition.
  - Enjoy higher market shares.
  - Serve larger customer bases.
- Theoretical linkage: in oligopolistic frameworks, a larger number of competing firms reduces strategic complementarities; intensified competition shrinks market shares and increases demand elasticity, damping strategic complementarities.

*Source: wpiea2025164-source-pdf (PDF chapter/section).*

### Appendix Figure G.2 reports the corresponding estimates for retail firms. In contrast to manufacturing

### wpiea2025164-source-pdf - Appendix Figure G.2 reports the corresponding estimates for retail firms. In contrast to manufacturing

### Asymmetric pricing: retail vs manufacturing
- Appendix Figure G.2 reports the corresponding estimates for retail firms. In contrast to manufacturing firms, the marginal cost pass-through for retail firms is generally statistically indistinguishable between cost increases and decreases.
- For changes in competitors’ prices, the asymmetry is also limited; if anything, retail firms appear somewhat more sensitive to competitors’ price declines.

### Measurement of competition and empirical specification
- Competition proxy:
  - Ci,t = Σr∈Ri Σd∈Di,r ωi,r,d,t Fi,r,d,t
  - ωi,r,d,t is the share of firm i’s sales in municipality-product r,d.
  - Fi,r,d,t denotes the number of firms selling product d in municipality r.
- Market-share aggregation:
  - market sharei,t = Σr∈Ri Σd∈Di,r ωi,r,d,t salesi,d,r,t / Σj salesj,d,r,t
- Main estimating equation for manufacturing firms (distributed lag form):
  - ∆pi,t = αi + τt + Σr=0^12 ∆mc i,t−r (βr + βr CIC i,t−r) + Σr=0^12 ∆p−i,t−r (γr + γr CIC i,t−r) + Σr=0^12 βr CIC i,t−r + εi,t
  - CIC i,t = 1 if Ci,t is above the median at that time; otherwise 0.

### Role of market structure — key empirical findings
- Competition by number of competitors:
  - After one year, firms with a lower number of competitors respond more strongly to changes in competitors’ prices.
  - A one percentage point increase in competitors’ prices is associated with a 0.4 percentage point increase in prices for firms operating in less competitive markets, compared to a 0.3 percentage point increase for firms in more competitive environments.
  - No statistically significant difference in the pass-through of firms’ own input cost changes across competition levels.
- Competition by market share:
  - Firms with higher market shares respond more strongly to changes in competitors’ prices than those with lower market shares.
  - No statistically significant differences for marginal cost pass-through one year after the shock.
- Competition by number of buyers (customer base):
  - Firms with a large customer base: a one percentage point increase in competitors’ prices is associated with a 0.4 percentage point increase in prices.
  - Firms with a smaller customer base: the same shock is associated with a 0.2 percentage point increase in prices.
  - Firms with larger customer bases respond more strongly to competitors’ price changes and less to marginal cost changes relative to firms with fewer buyers, although differences in marginal cost pass-through are not statistically significant.
- Appendix Figure G.3 reports retail-firm evidence; except for the case of the number of competitors, patterns are broadly consistent with manufacturing-firm results.

### Robustness checks and identification strategies
- Sector-time fixed effects:
  - Adding sector-time fixed effects yields results "remarkably consistent" with baseline estimates (Figure 13), reducing concern about sectoral confounding factors.
- Instrumental variables (IV) for endogeneity:
  - Marginal costs instrument excludes purchases from suppliers with a limited number of buyers (Equation 15).
  - Competitors’ prices instrument based on costs of inputs used by competitors’ suppliers, excluding inputs from suppliers that transact with both the firm and its competitors (Equation 16).
  - First-stage Kleibergen-Paap Wald F-statistic consistently exceeds 200.
  - IV estimates:
    - Pass-through of competitors’ prices ≈ 85 percent over the one-year horizon.
    - Pass-through of marginal costs declines to 12 percent.
    - The sum of the two estimated elasticities is statistically indistinguishable from one.
  - Appendix H reports results when instrumenting each variable separately.
- Horizon extension (24-month distributed-lag):
  - Responses to marginal costs concentrated within the first 12 months; magnitude declines during the second year (estimates less precise).
  - Pass-through elasticity for competitors’ prices after 24 months ≈ 55 percent, about 10 percentage points lower than baseline; still considerably larger than cost pass-through.
  - Decline primarily due to continued reduction in size of price adjustments during second year, partially offset by modest increase in frequency of price adjustments.
- Local projections (horizon-specific elasticities):
  - Local projections estimated via: pi,t+h − pi,t−1 = αi + τt + βh (mci,t+h − mci,t−1) + γh (p−i,t+h − p−i,t−1) + εh i,t
  - One-year-ahead estimates from local projections:
    - Marginal cost pass-through = 17 percent (baseline: 18 percent).
    - Competitors’ prices pass-through = 59 percent (baseline: 66 percent).
  - Differences mainly due to variation in estimated responses of the frequency of price adjustments.
  - In response to an increase in marginal costs:
    - Frequency of price changes rises by 1.2 percentage points (baseline: 1 percentage point).
    - Size of individual price adjustments ≈ 0.2 percent.
  - Following a change in competitors’ prices:
    - Frequency of price adjustments increases by 2.8 percentage points (baseline: 2.2 percentage points).
    - Size of adjustments ≈ 0.7 percent.

### Main quantitative conclusions and stylized facts
- Five key facts documented:
  1. Firms set prices strategically: pass-through from competitors’ prices substantially exceeds that from their own costs.
     - Firms pass through approximately two-thirds of changes in competitors’ prices—about 3.7 times larger than the pass-through of marginal cost changes.
  2. Strength of strategic complementarities varies across sectors but is first-order important throughout.
     - Manufacturing firms: pass-through of competitors’ prices estimated at 40 percent.
     - Retail firms: pass-through above 70 percent.
     - Both sector estimates: pass-through of competitors’ prices considerably larger than cost pass-through.
  3. Strategic complementarities are state dependent.
     - During periods of high inflation:
       - Firms become more sensitive to changes in their own costs.
       - Firms are twice as likely to adjust prices in response to competitors’ price changes.
     - High inflation weakens nominal rigidities while strengthening strategic complementarities.
  4. Pass-through is asymmetric:
     - Larger for competitors’ price increases than for decreases (a "rockets and feathers" phenomenon) — this asymmetry extends to strategic pricing.
  5. Strategic complementarities are stronger when:
     - Firms face lower competition.
     - Firms have larger customer bases.
     - Firms enjoy higher market shares.
- Aggregate interpretation:
  - Strategic complementarities are a dominant source of real rigidities in pricing, leading firms to adjust their prices less than one-for-one with changes in marginal costs.
  - The pass-through of competitors’ prices — and the elasticity of markups to competitors’ prices — informs models of how rising market concentration affects general-equilibrium macroeconomic outcomes via firms’ strategic behavior and has implications for optimal monetary policy design.

*Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025164-source-pdf.pdf*

### References

### wpiea2025164-source-pdf - References

### References (bibliography)
- The document lists numerous academic and working papers relevant to price-setting, inflation dynamics, pass-through, and related micro and macro evidence. Authors and works cited include: Acevedo et al. (2025); Alvarez et al. (2019); Amiti et al. (2019, 2022); Amiti, Heise, Karahan, and Sahin (2022); Atkeson and Burstein (2008); Auer and Schoenle (2016); Bacon (1991); Ball and Romer (1990); Barros et al. (2009); Bils and Klenow (2004); Blanco et al. (2024); Boehm, Levchenko, and Pandalai-Nayar (2023); Burstein and Gopinath (2014); Campbell and Eden (2014); Campbell and Hopenhayn (2005); Cooper and John (1988); Creamer and Rankin (2008); Dedola et al. (2024); Eichenbaum, Jaimovich, and Rebelo (2011); Eichenbaum et al. (2014); Feenstra, Gagnon, and Knetter (1996); Fontagné, Martin, and Orefice (2018); Gagliardone et al. (2023); Gagnon (2009); Ganapati, Shapiro, and Walker (2020); Genakos and Pagliero (2022); Genesove and Mullin (1998); Gentzkow, Kelly, and Taddy (2019); Goldberg and Hellerstein (2007, 2013); Golosov and Lucas (2007); Gopinath and Itskhoki (2010, 2011); Gopinath, Itskhoki, and Rigobon (2010); Gouvea (2007); Joulin et al. (2016); Julio, Zárate, and Hernández (2009); Karadi and Reiff (2019); Karadi et al. (2024); Karrenbrock et al. (1991); Klenow and Willis (2016); Klenow and Kryvtsov (2008); Lafrogne-Joussier, Martin, and Mejean (2023); Medina, Rappoport, and Soto (2007); Nakamura and Zerom (2010); Nakamura and Steinsson (2008); Nakamura et al. (2018); Neumark and Sharpe (1992); Peltzman (2000); Pennings (2017); Rossi-Hansberg, Sarte, and Trachter (2021); Taylor (2000); Wang and Werning (2022).

### Appendix A — Regular prices: filter algorithm and parameters
- Purpose: compute regular prices r_t from observed prices p_t following Nakamura and Steinsson (2008) to filter out changes due to sales and stockouts.
- Algorithm steps (as listed):
  1. If p_t = r_{t−1}, then r_t = r_{t−1}
  2. If p_t > r_{t−1}, then r_t = p_t
  3. If r_{t−1} ∈ {p_{t+1}, ..., p_{t+J}} and p_t ≤ r_{t−1} before returning to r_{t−1}, then r_t = r_{t−1}
  4. If {p_t, p_{t+1}, ..., p_{t+L}} has K or more different elements, then r_t = p_t
  5. Define p_max = max{p_{t+1}, ..., p_{t+L}} and t_max = first-time max{p_{t+1}, ..., p_{t+L}}; if p_max ∈ {p_{t_max+1}, ..., p_{t_max+L}}, then r_t = p_max
  6. r_t = p_t
- Filter parameterization used in calculations: L = 3, K = 3, and J = 5.

### Appendix B — Price rigidity of CPI and PPI products: methods and key empirical findings
- Measurement definitions and aggregation:
  - Frequency of price changes at CPI or PPI product category level f_{c,t} and economy-level aggregation f_t = Σ_c ω_c f_{c,t} = Σ_c ω_c (Σ_{v∈c} D_{v,c,t} / Σ_{v∈c} O_{v,c,t}) (Equation 25), where ω_c denotes CPI/PPI weights, D_{j,c,t} is dummy for a price change, and O_{j,c,t} is dummy for price observation.
  - Implied price duration computed as −1 / ln(1 − f_t).
  - Size of price changes aggregated as s_t = Σ_c ω_c s_{c,t} = Σ_c ω_c (Σ_{v∈c} Δp_{,0 v,c,t} / Σ_{v∈c} O_{,0 v,c,t}) (Equation 26), with Δp_{,0 v,c,t} = p_{v,c,t} − p_{v,c,t−1,0} and O_{,0 j,c,t} indicating price differs from previous month.
  - Decomposition of frequency and size into positive and negative changes: Δp_t = f^+_t s^+_t + f^-_t s^-_t (Equation 27).

- Summary of time-average empirical statistics (Table B.1; Percent):
  - Frequency — Observed prices (Median / Average):
    - All changes: 30.3 / 36.4 (CPI); 30.9 / 37.7 (PPI)
    - Duration (months): 2.8 / 2.2 (CPI); 2.7 / 2.1 (PPI)
    - Increases: 17.3 / 21.2 (CPI); 16.6 / 22.0 (PPI)
    - Decreases: 11.5 / 15.1 (CPI); 11.6 / 15.8 (PPI)
  - Frequency — Regular prices (Median / Average):
    - All changes: 25.6 / 30.0 (CPI); 25.2 / 28.9 (PPI)
    - Duration (months): 3.4 / 2.9 (CPI); 3.5 / 3.0 (PPI)
    - Increases: 14.7 / 17.6 (CPI); 13.7 / 16.7 (PPI)
    - Decreases: 9.5 / 12.2 (CPI); 9.1 / 12.2 (PPI)
  - Size — Observed prices (Median / Average):
    - All changes: 1.3 / 1.3 (CPI); 1.3 / 1.3 (PPI)
    - Increases: 12.7 / 13.7 (CPI); 10.5 / 11.1 (PPI)
    - Decreases: -15.0 / -15.8 (CPI); -12.2 / -12.6 (PPI)
  - Size — Regular prices (Median / Average):
    - All changes: 1.2 / 1.1 (CPI); 1.5 / 1.4 (PPI)
    - Increases: 11.8 / 12.6 (CPI); 10.1 / 10.8 (PPI)
    - Decreases: -14.2 / -15.0 (CPI); -11.7 / -12.3 (PPI)

- Key empirical findings (bulleted):
  - Frequency of price changes (median, observed prices):
    - CPI: 30.3 percent
    - PPI: 30.9 percent
    - Implied median duration: about 2.8 months (CPI) and 2.7 months (PPI).
  - Regular prices (after excluding sales) exhibit lower frequency:
    - Median frequency: 25.6 percent (CPI) and 25.2 percent (PPI)
    - Implied median duration: about 3.4 months (CPI) and 3.5 months (PPI).
  - Sales are a notable feature: removing temporary sales reduces frequency estimates and increases implied duration by roughly half a month to one month.
  - Asymmetry in direction of changes:
    - Price increases are about 5 percentage points more frequent than price decreases for observed prices; similar pattern for regular prices.
    - Price decreases are less frequent but tend to be larger in magnitude than increases:
      - Observed median size for CPI decreases: -15.0 percent (2.3 percentage points larger than increases).
      - Observed median size for PPI decreases: -12.2 percent (1.7 percentage points larger than increases).
  - Size of price changes (median across full sample):
    - Observed prices: 1.3 percent (CPI and PPI).
    - Regular prices: slightly smaller, e.g., 1.2 percent (CPI median) and 1.5 percent (PPI median).
    - Removing temporary sales reduces the size of regular price changes by about 0.2 percentage points compared with observed price changes.
  - Heterogeneity across categories:
    - For CPI regular prices, category medians range from about 22 percent (restaurants and hotels; miscellaneous) to 48 percent (housing, water, electricity, etc.).
    - For PPI, 'tobacco' shows median frequency zero (most products did not change prices), while 'coke and refined petroleum' had about half of products changing prices.
    - Largest median regular price adjustment in CPI: 'food and non-alcoholic beverages' at 1.7 percent.
    - Smallest median regular price adjustment in CPI: 'clothing and footwear' at about 0.2 percent.
    - Largest PPI median price changes: 'tobacco' at about 4.5 percent; smallest: 'non-metallic minerals'.
  - Distributional features:
    - Regular prices display fatter left tails (lower frequency) than observed prices.
    - Modes: distribution mode for frequency around 0.25 (one-fourth of products change prices in a typical category-month).
    - Size distributions have modes around 1.5 percent for regular prices and fat right tails, consistent with price increases being more frequent than declines.
  - Comparisons with prior studies:
    - Estimates fall broadly within ranges reported for emerging markets, though cross-study comparisons are complicated by differing methodologies, sample periods, and data sources.
    - Examples cited: Gouvea (2007) 37.2 percent; Barros et al. (2009) 37 percent; Julio, Zárate and Hernández (2009) 21.1 percent average (median 11.9 percent); Gagnon (2009) range 22.1 percent to 61.9 percent depending on inflation; Creamer and Rankin (2008) 16 percent (CPI) and 19.5 percent (PPI); Medina et al. (2007) 44.7 percent average (median 28.9 percent) for Chile pre-2005.
  - Note on sample mapping:
    - For period January 2018–December 2023, product varieties are mapped to CPI and PPI categories using CPC and COICOP classifiers per the National Statistics Institute classification.
    - For periods before January 2018, mapping information is not available and the first available mapping is assumed to hold backwards.

### Appendix C — Classification of product varieties into product categories
- Approach: follows Acevedo et al. (2025) to classify product varieties into CPC-consistent product categories and an adapted COICOP (the CUP with 290 categories).
- The appendix states that the algorithm is described "in a nutshell" (algorithmic details follow in the source beyond the excerpt provided).

*Italic: Source: wpiea2025164-source-pdf - References (Appendices A–C and Table B.1 summaries from the provided content).*

### 1.  It builds a sample of text descriptors associated with each product category.

### 1.  It builds a sample of text descriptors associated with each product category.

### Methodology for mapping product descriptors to product categories
- It builds a sample of text descriptors associated with each product category.
- It then computes a score for each text descriptor within each economic sector of the seller based on the sales amount.
- For each product category, it selects up to 2,000 text descriptors with the highest score and manually classifies them into product categories.
- It cleans the text descriptors following Gentzkow, Kelly and Taddy (2019), as free text descriptors are unstructured and might contain uninformative parts.
- It employs the FastText classification algorithm proposed in Joulin et al. (2016) to predict the product category using the text of the training sample.
- It finally uses the trained model on the untrained sample and obtains the mapping between product varieties and product categories.
- Note: The algorithm also uses information about the economic sector of the selling firm and the price range of the product to improve the fit of the model, which can help distinguish between product categories (for example, distinguishing a computer sale from repair/maintenance when sector and price range are informative).

### Decomposing pass-through into frequency and magnitude of price adjustments
- The cumulative change in prices over an R-period horizon is expressed as:
  - ∆^R p_t = f_{t−R} ∆^R s_t + s_{t−R} ∆^R f_t + ∆^R s_t ∆^R f_t  (equation (28))
- Let β_{f,r} and β_{s,r} denote the estimated coefficients on marginal cost changes when the dependent variable is the frequency and the size of price changes, respectively.
- The cumulative effect of a marginal cost change on prices over horizon R is:
  - ∂∆^R p_t / ∂∆mc_t = Σ_{r=0}^R β_r = f_{t−1} Σ_{r=0}^R β_{s,r} + s_{t−1} Σ_{r=0}^R β_{f,r} + (Σ_{r=0}^R β_{s,r})(Σ_{r=0}^R β_{f,r})  (equation (29))
- The decomposition highlights that price adjustments result from:
  - changes in how often firms change prices (frequency),
  - changes in how much they change prices (size),
  - and the interaction between frequency and size.

### Alternative measures of high and low inflation (Figure E.1)
- The analysis shows pass-through estimates in high and low inflation periods using different inflation measures:
  - Panels (a) and (b): Change in marginal costs and change in competitors’ prices with monthly inflation used to identify high inflation periods (using the mean and standard deviation of monthly inflation).
  - Panels (c) and (d): Same responses using annual inflation to identify high inflation periods (using the mean and standard deviation of annual inflation).
- Notes on estimation:
  - The figures show the percent change in firms’ sales prices in response to a one percentage point change in marginal costs and a one percentage point change in competitors’ prices, differentiating between periods of low and high inflation, along with the 90 percent confidence intervals.
  - Specifications include firm and time fixed effects. Standard errors are clustered at the firm and time level.

### Statistical differences (Table F.1)
- The table reports one-year cumulated coefficients on interaction terms with changes in marginal costs (column 1) and changes in competitors’ prices (column 2) from regressions of pass-through, frequency of price adjustments, and size of price adjustments. All specifications include firm and time fixed effects. Standard errors are clustered at the firm and time level. Significance markers: ***p <0.01, **p <0.05, *p <0.1.
- Reported differences (exact values preserved):
  - Changes in marginal costs | Changes in competitors’ prices
  - Manufacturing firms - retail firms
    - Pass-through 0.121***    -0.331***
    - Frequency 0.737***    -1.071***
    - Size 0.279***    -0.014
  - High inflation - low inflation
    - Pass-through 0.109***    0.212***
    - Frequency 1.403***    1.618***
    - Size 0.149    0.612***
  - Positive changes - negative changes
    - Pass-through 0.027    0.124***
    - Frequency 2.077***    1.335***
    - Size -0.006    -0.094
  - Many competitors - few competitors
    - Pass-through 0.014    -0.076***
  - Large markets share - small market share
    - Pass-through -0.019    0.085***
  - Many buyers - few buyers
    - Pass-through -0.033    0.157***

### Evidence on retail firms (Figures G.1–G.3)
- Retail firms’ pass-through during high vs low inflation (Figure G.1):
  - Shows percent change in firms’ sales prices in response to a one percentage point change in marginal costs and a one percentage point change in competitors’ prices, with 90 percent confidence intervals; compares retail firm responses during high vs low inflation.
  - Specification includes firm and time fixed effects. Standard errors are clustered at the firm and time level.
- Asymmetric pass-through of retail firms (Figure G.2):
  - Shows percent change in firms’ sales prices in response to a one percentage point change in marginal costs and a one percentage point change in competitors’ prices, comparing responses to positive and negative changes; includes 90 percent confidence intervals.
  - Specification includes firm and time fixed effects. Standard errors are clustered at the firm and time level.
- Market structure of retail firms (Figure G.3):
  - Shows percent change in retail firms’ sales prices one year after a change in their marginal costs and competitors’ prices, conditional on the firm’s number of competitors, market share, and number of buyers, with 90 percent confidence intervals.
  - Specification includes firm and time fixed effects. Standard errors are clustered at the firm and time level.

### Other IV regressions (Figures H.1–H.2)
- Instrumenting changes in marginal costs (Figure H.1):
  - Reports percent change in firms’ sales prices, percent change in frequency of sales price adjustments, and percent change in size of non-zero sales price adjustments in response to a one percentage point change in marginal costs and a one percentage point change in competitors’ prices.
  - Results come from IV regressions where changes in marginal costs are instrumented. Figures include 90 percent confidence intervals.
  - Specification includes firm and time fixed effects. Standard errors are clustered at the firm and time level.
- Instrumenting changes in competitors’ prices (Figure H.2):
  - Reports analogous outcomes where changes in competitors’ prices are instrumented. Figures include 90 percent confidence intervals.
  - Specification includes firm and time fixed effects. Standard errors are clustered at the firm and time level.

*Source: wpiea2025164-source-pdf - 1.  It builds a sample of text descriptors associated with each product category.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025164-source-pdf.pdf_