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---

### Context and datasets
- Two firm-level datasets:
  - Compustat: quarterly United States data covering 1990-2016; focuses on US public firms.
  - Bureau Van Dijk’s Orbis: annual panel of 14 countries covering 2000-2015; includes private firms and better represents younger firms.
- US Compustat sample details:
  - Sample period: 1986 to 2016; cleaned sample: 1990-2016.
  - Final cleaned sample: 424,949 observations for 13,691 firms.
  - Exclusions: finance, insurance and real estate (FIRE) industries; privately held firms excluded.
- Orbis cross-country sample details:
  - Countries: Czech Republic, Germany, Spain, France, the UK, the US, Hungary, Italy, Japan, Korea, the Netherlands, Poland, Slovakia and Turkey.
  - Sector coverage: non-farm, non-financial business sector (NACE Rev.2 two-digit codes 5-82).
  - Sample period: 2001-2015.
  - Final sample after cleaning and merging with shocks: 14 countries, 339,296 firms, and more than 2 million observations.
  - Cleaning exclusions: firms with less than 3 employees and firms not reporting at least four consecutive periods.
- Markup computation:
  - Definition: markup μ_{i,t} = P_{i,t} / MC_{i,t}.
  - Estimation: De Loecker and Warzynski (2012) expression μ_{i,t} = β^{v}_{i,t} / α^{v}_{i,t}; Cobb-Douglas production function; control function approach of Ackerberg et al. (2015).
  - Flexible input: cost of goods sold.
  - Annual markups estimated using same code as Dı́ez et al. (2021).
  - Robustness: alternative markup definitions (Lerner index) reported in Appendix A.2.
- Monetary policy shock series:
  - US (Compustat): high-frequency identification implemented as in Albrizio et al. (2020); instrument: changes in Federal funds futures around FOMC announcements; policy rate instrumented: one-year government bond rate; VAR estimated over 1973m1-2016m8 at monthly frequency; VAR variables: log(IP_t), log(CPI_t), i_t, EBP_t; baseline first-stage F-statistic: 12 (on average); allow unconventional monetary policy coefficient to differ post 2008m8.
  - Cross-country (Orbis): Furceri et al. (2018) approach using Consensus Economics forecast errors; shocks are residuals ̂ε_{i,t} from FE^r_{i,t} regression on FE^{Δy}_{i,t}, FE^{π}_{i,t}, and lags; further purification by projecting residuals on current and lagged GDP growth and inflation; uses 3-month rates as proxy for policy rates.

### Empirical approach (summary and framework)
- Outcome variables:
  - Main: firms’ real sales = total revenue deflated by (2-digit NACE Rev. 2) industry-level price indices.
  - Extension: real fixed assets deflated using economy-wide capital stock prices.
- Estimation strategy:
  - Local projection method (Jordà, 2005) to estimate dynamic responses.
  - Control for unobserved confounders via fixed effects: industry-time (US) or country-industry-time (cross-country) to absorb average impact of monetary shocks.
  - Control for observed firm-level confounders: firm age, size, leverage, asset liquidity, tangibility, Tobin’s q; include two lags of dependent variable and monetary shocks.
- Main specifications (equations referenced in the source):
  - Unconditional effect (Equation 6): ỹ_{i,t+h} = ln(y_{i,t+h}) − ln(y_{i,t−1}) = α^h_i + β^h ε^m_t + ρ^h X_{i,t} + ε_{i,t+h}.
  - Heterogeneous effects by markup bins (Equation 7): firms binned into bottom 25 percentile (low), middle (25-75), top 25 percentile (high) of time-invariant average markup per firm.
  - Triple-interaction heterogeneity (Equation 8): interaction of markup bins with firm characteristic bins f ∈ {age, size, leverage, asset liquidity, tangibility, Tobin’s q}.
- Inference:
  - US analysis period: 1990-2016; OLS with standard errors clustered by time.
  - Cross-country analysis period: 2001-2015; OLS with standard errors clustered at country-sector and time levels.
- Identification note: heterogeneous effects are deviations from average effect absorbed by time or sector-time fixed effects.

### Key empirical findings — US (Compustat) and cross-country (Orbis)
- Unconditional impulse responses to a 100-basis point contractionary monetary policy shock:
  - US: real sales decline by about 1½ percent 12 quarters after the shock; real total assets show similar shape and magnitude.
  - Panel of advanced economies: real sales decline by about 1½ percent 2 years after the shock (peak response similar to US).
- Differential response by firm markup:
  - High-markup firms’ real sales (and fixed assets) respond less to monetary policy shocks than low-markup firms, both within-industry (Compustat) and within country-industry (Orbis), after controlling for observed and unobserved confounders.
  - Quantitative differences in the one-year-ahead response of real sales to a 100 basis points monetary policy shock between top quartile and bottom quartile of the markup distribution:
    - about 2 percentage points in Compustat (US).
    - 0.8 percentage point in Orbis (14 advanced economies).
- Interaction with financial-friction-related characteristics:
  - The role of markups for monetary policy transmission is amplified for firms with characteristics associated with greater financial frictions (small size, young age).
  - Small low-markup and young low-markup firms are most responsive to monetary policy shocks.
  - Difference between large high-markup and small low-markup firms in US is about 2 percentage points (larger than baseline markup-only difference).
- Robustness and falsification checks:
  - Results robust to sample variations, alternative markup definitions (Lerner index), alternative bins for low- and high-markup firms, and removing any particular country from the Orbis sample.
  - Evidence against relative-price interpretation: following a monetary policy shock, no significant change in the relative markup of high-markup firms vis-à-vis low-markup ones within the same industry.

### Theoretical rationale — stylized partial equilibrium model (Section 5 and Appendix)
- Objective: rationalize empirical findings with a simple two-period partial equilibrium model highlighting markups and financial constraints.
- Key elements of the model:
  - Production: Cobb-Douglas Y = k^α l^{1−α}, 0 < α < 1.
  - Borrowing constraint: k ≤ φ(R) + k_0, where R = r + δ and φ(R) is decreasing in R.
  - Variable cost VC(Q) solves min_{k,l} Rk + wl s.t. Q = k^α l^{1−α}, k ≤ φ(R) + k_0.
  - Marginal cost MC(Q) has unconstrained and constrained regimes with cutoff Q^*(R) defined explicitly.
  - Demand: q = κ^{ξ} p^{-ξ}; profits Π = max_q pq − VC(q) − FC; MR(q) = κ^{ξ−1}/ξ q^{-1/ξ} = MC(q); p = ξ/(ξ−1) MC(q).
  - A higher-markup firm (smaller ξ) has steeper MR and thus responds less to MC shifts.
- Mechanisms for larger responsiveness of low-markup firms:
  - Interest-rate cut lowers marginal cost for all firms and relaxes borrowing constraints (increasing Q^*).
  - Low-markup firms have flatter MR curves and therefore expand output more in response to the same MC shift.
  - When constraints bind, monetary easing benefits constrained firms disproportionately because it both lowers cost of capital and relaxes the borrowing constraint.
- Formal lemmas and comparative statics (selected statements preserved verbatim):
  - Lemma 1: Marginal cost MC(Q) has unconstrained and constrained expressions; cutoff Q^*(R) given by equation (11).
  - Lemma 2 & Assumption 1: conditions under which ∂Q^*/∂R ≤ 0.
  - Lemma 3: MR(q) = κ^{ξ−1}/ξ q^{-1/ξ} = MC(q); p = ξ/(ξ−1) MC(q).
  - Lemma 4: constrained portion of MC is more sensitive to R changes than unconstrained portion.
  - Lemma 6: If both unconstrained ex ante and same q^∗, and only one becomes constrained after r decline, it will be the low-markup firm.
  - Lemma 7 and Lemma 8: statements on relative responses when both firms are constrained or one becomes unconstrained post shock.
  - Lemma 9–16 (Appendix): detailed characterizations of MC(Q), cutoff Q^*(R), factor demands, constrained and unconstrained regimes, and comparative statics showing low-markup firms respond more to interest-rate declines under stated sufficient conditions (explicit functional forms and sufficiency conditions preserved in the source).
- Stylized-model summary findings:
  - i) low-markup firms are more responsive to interest rate shifts than high-markup ones, all else equal; and
  - ii) the role of markups for the response of output is greater when firms face tighter financial constraints.

### Empirical conclusions (paper-level summary)
- Datasets and identification:
  - Uses two alternative firm-level datasets for the United States and 14 advanced economies, recent markup measures, and exogenous monetary policy shocks.
- Main empirical conclusions:
  - High-markup firms’ real sales (and real fixed assets) respond significantly less to monetary policy shocks than low-markup firms.
  - The difference in one-year-ahead response of real sales to a 100 basis points monetary policy shock between top and bottom quartiles of the markup distribution is about 2 percentage points in Compustat (US) and 0.8 percentage point in Orbis (14 advanced economies).
  - A firm’s markup has a material impact on its responsiveness to monetary policy shocks even after controlling for size, age and financial characteristics—implying the role of markup is distinct from other channels.
  - Suggestive evidence that financial frictions amplify the impact of markups on firms’ responsiveness to monetary policy.
  - Low-markup firms are more responsive regardless of size or age, but smaller or younger firms tend to be most responsive.

*Source: Extract from the paper’s Introduction, Sections 2, 3, 4, 5, Appendix B and related figures in the provided PDF.*

### 2.1    Firm-level data.  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 

### 2.1    Firm-level data.

### Context and datasets
- The paper uses two alternative firm-level datasets to study how firms’ markups shape their real sales (and investment) response to monetary policy shocks:
  - Compustat: quarterly data for the United States covering 1990-2016; focuses on US public firms.
  - Bureau Van Dijk’s Orbis: annual data for a panel of 14 countries covering 2000-2015; includes private firms and better represents younger firms.

- Markup computation:
  - Firm-level markups are computed following De Loecker and Warzynski (2012).
  - Data cleaning procedures and markup calculations follow De Loecker et al. (2020) for Compustat and D́ıez et al. (2021) for Orbis.

- Monetary policy shock series:
  - For the United States (Compustat-based analysis): use the series put together by Albrizio et al. (2020), who follow Gertler and Karadi (2015) relying on high-frequency identification of the impact of monetary policy surprises on interest rates.
  - For each of the 14 countries in Orbis (Orbis-based analysis): construct the monetary policy shock series as the unexpected change in policy rates that is orthogonal to growth and inflation surprises following Duval and Furceri (2018) and Furceri et al. (2018).

### Empirical approach (summary)
- Outcome variables:
  - Main variable: firms’ real sales, computed as total revenue deflated by (2-digit NACE Rev. 2) industry-level price indices.
  - Additional outcome in extensions: real fixed assets (deflated using economy-wide capital stock prices).

- Estimation strategy:
  - Estimate the impact of monetary policy shocks on firms’ sales growth conditioning on their markup levels.
  - Control for unobserved confounding factors via fixed effects that absorb the average impact of monetary policy shocks on firms’ sales growth in each industry (US analysis) or country-industry (cross-country analysis).
  - Control for observed firm-level confounding factors including firm age, size and financials.
  - Use the local projection method (Jordà, 2005) to estimate the dynamic response of firms’ real sales to monetary policy shocks.

### Key empirical findings
- Differential response by markup:
  - High-markup firms’ real sales (and fixed assets) respond less to monetary policy shocks than low-markup firms’, both within the same industry (Compustat) and within the same country-industry (Orbis), after controlling for observed and unobserved confounders.
  - The difference in the one-year-ahead response of real sales to a 100 basis points monetary policy shock between the top quartile and the bottom quartile of the firm markup distribution is:
    - about 2 percentage points in Compustat
    - 0.8 percentage point in Orbis

- Robustness and falsification checks:
  - Results are robust to a battery of sensitivity checks including sample variations, other markup definitions (such as a Lerner index), and alternative bins for low- and high-markup firms.
  - The authors show results are unlikely to reflect changes in relative prices rather than real output: following a monetary policy shock, there is no significant change in the relative markup of high-markup firms vis-à-vis low-markup ones within the same industry.

- Interaction with financial-friction-related characteristics:
  - The role of markups for monetary policy transmission is amplified for firms with characteristics typically associated with greater financial frictions, such as small size and young age.
  - Small low-markup and young low-markup firms are found to be most responsive to monetary policy shocks.

### Theoretical rationale (preview)
- The paper offers a rationalization via a simple partial equilibrium model highlighting how differences in markups and the tightness of financial constraints across firms shape relative responses to monetary policy shocks.
- The model emphasizes financial frictions as one potential factor behind the observed role of market power for monetary policy transmission at the firm level.
- The parsimonious partial equilibrium framework is chosen because the empirical differences-in-differences strategy identifies relative, but not aggregate, impacts of markups.

### Contribution to literature
- Adds direct empirical evidence on how firm-level markups affect monetary policy transmission, complementing literature on:
  - Firm-level monetary policy transmission and the role of financial frictions (e.g., Gertler and Gilchrist (1994); Kashyap et al. (1994); Cloyne et al. (2018); Anderson and Cesa-Bianchi (2020); Jeenas (2019); Ottonello and Winberry (2020)).
  - Heterogeneous firm responses to financial shocks (e.g., Siemer (2016); Giroud and Mueller (2017); Benmelech et al. (2011); Chodorow-Reich (2014); Huber (2017); Duval et al. (2020)).
  - Rising corporate market power and macroeconomic consequences (e.g., Autor and Katz (2017); De Loecker et al. (2020); D́ıez et al. (2021); Gutiérrez and Philippon (2016); Baqaee et al. (2021); Wang and Werning (2020)).

*Source: Extract from the paper’s Introduction and Section 2.1 (firm-level data, markup computation, and monetary policy shocks) in the provided PDF.*

### Section 5 presents a simple model and uses it to rationalize our key findings. Section 6 concludes.

### wpiea2021184-print-pdf - Section 5 presents a simple model and uses it to rationalize our key findings. Section 6 concludes.

### Data (Section 2)
- US firm-level data:
  - Source: quarterly Compustat (publicly listed U.S. firms).
  - Sample period: 1986 to 2016; cleaned sample: 1990-2016.
  - Final sample after cleaning: 424,949 observations for 13,691 firms.
  - Exclusions: finance, insurance and real estate (FIRE) industries; privately held firms excluded by dataset.
  - Cleaning follows De Loecker et al. (2020); drop firms with extreme or unreliable data per references.
- Cross-country firm-level data:
  - Source: annual ORBIS (listed and non-listed firms) for 14 advanced economies.
  - Countries: Czech Republic, Germany, Spain, France, the UK, the US, Hungary, Italy, Japan, Korea, the Netherlands, Poland, Slovakia and Turkey.
  - Sector coverage: non-farm, non-financial business sector (NACE Rev.2 two-digit codes 5-82).
  - Sample period: 2001-2015.
  - Final sample after cleaning and merging with shocks: 14 countries, 339,296 firms, and more than 2 million observations.
  - Cleaning follows Kalemli-Ozcan et al. (2015), Boz et al. (2017), Gal (2013); exclude firms with less than 3 employees and firms not reporting at least four consecutive periods.

### Markups (Section 2.2)
- Definition: markup μ_{i,t} = P_{i,t} / MC_{i,t}.
- Estimation approach: De Loecker and Warzynski (2012) expression — markup estimated as ratio of output elasticity of a flexible input (β^{v}_{i,t}) to its expenditure share (α^{v}_{i,t}): μ_{i,t} = β^{v}_{i,t} / α^{v}_{i,t}.
- Production function specification: Cobb-Douglas production function estimated using the control function approach of Ackerberg et al. (2015).
- Variable input used: cost of goods sold as the flexible input.
- Markups are estimated at the annual level; same code as Dı́ez et al. (2021) applying De Loecker and Warzynski (2012) methodology.
- Robustness: alternative markup definitions based on a Lerner index are reported in Appendix A.2.

### Monetary policy shocks (Section 2.3)
- Identification requirement: shocks must be orthogonal to current and lagged conditions, uncorrelated with other exogenous shocks, and unanticipated.
- US shocks (Compustat-based analysis, Section 2.3.1):
  - Methodology: high-frequency identification following Gürkaynak et al. (2005), Gertler and Karadi (2015), Nakamura and Steinsson (2018), Cloyne et al. (2018); implemented as in Albrizio et al. (2020).
  - Instrument: changes in Federal funds futures in a narrow window around FOMC announcements used as instruments in a proxy-SVAR / TSLS framework.
  - Policy rate instrumented: one-year government bond rate (to capture unconventional policy effects when short-term rates at the zero lower bound).
  - VAR estimated over: 1973m1-2016m8 at monthly frequency.
  - VAR variables: log(IP_t), log(CPI_t), i_t (government bond yield), EBP_t (Gilchrist and Zakrajsek (2012) credit spread).
  - Baseline instrument-policy combination: one-year government bond rate instrumented by one-month ahead Fed funds future.
  - First-stage F-statistic for baseline combination: 12 (on average across regressions).
  - Structural break: allow unconventional monetary policy coefficient to differ post 2008m8.
- Cross-country shocks (Orbis-based analysis, Section 2.3.2):
  - Methodology: Furceri et al. (2018) approach using Consensus Economics forecast errors, following Auerbach and Gorodnichenko (2013).
  - Two-step identification:
    1. Compute forecast errors of policy rates (proxied by 3-month rates), GDP growth and inflation (FE^r_{i,t}, FE^{Δy}_{i,t}, FE^{π}_{i,t}) as actual minus Consensus forecast.
    2. Regress FE^r_{i,t} on FE^{Δy}_{i,t} and FE^{π}_{i,t} and lags of realized Δy and π; shocks are residuals ̂ε_{i,t} from:
       FE^r_{i,t} = α_i + β FE^{Δy}_{i,t} + γ FE^{π}_{i,t} + Σ_{j=0}^4 δ_j Δy_{i,t−j} + Σ_{j=0}^4 θ_j π_{i,t−j} + ε_{i,t}.
  - Further purification: project residuals on current and lagged GDP growth and inflation to remove predictable components.
  - Advantage: addresses “policy foresight” and mitigates endogeneity; for euro-area countries shocks capture relative exposure to ECB actions (robustness checks drop euro-area countries other than Germany).

### Empirical framework (Section 3)
- Estimation method: local projection method of Jorda (2005) on firm-level panel data.
- Three specifications:
  1. Unconditional effect of monetary policy shocks on firm real sales/investment:
     - Equation (6): ỹ_{i,t+h} = ln(y_{i,t+h}) − ln(y_{i,t−1}) = α^h_i + β^h ε^m_t + ρ^h X_{i,t} + ε_{i,t+h}.
     - Controls X include two lags of dependent variable and monetary shocks, size, age, leverage, asset liquidity, tangibility, Tobin’s q.
     - Firm fixed effects α^h_i included.
  2. Heterogeneous effects by markup bins:
     - Equation (7): non-parametric heterogeneity with firms binned into bottom 25 percentile (low), middle (25-75), top 25 percentile (high) of markup distribution; use time-invariant average markup per firm to mitigate endogeneity.
     - Include sector-time fixed effects to absorb average sectoral response.
  3. Heterogeneity interaction with other firm characteristics:
     - Equation (8): triple interactions between markup bins and bins of firm characteristics f ∈ F (age, size, leverage, asset liquidity, tangibility, Tobin’s q).
- Estimation samples and inference:
  - US analysis period: 1990-2016; OLS with standard errors clustered by time.
  - Cross-country analysis period: 2001-2015; OLS with standard errors clustered at country-sector and time levels.
- Interpretation note: estimates of heterogeneous effects are deviations from the average effect (absorbed by time or sector-time fixed effects).

### Results — US analysis (Section 4.1)
- Unconditional responses (Equation 6):
  - Impulse response to a 100-basis point contractionary monetary policy shock:
    - Real sales decline by about 1½ percent 12 quarters after the shock.
    - Real total assets show similar shape and magnitude.
  - Magnitude and dynamics broadly similar to Albrizio et al. (2020) GDP estimates.
- High- vs. low-markup firms (Equation 7 and 8):
  - Comparison: bottom 25th percentile (low-markup) vs top 25th percentile (high-markup).
  - Key quantitative result: difference in the one-year-ahead response of real sales to a 100 basis points monetary policy shock between top quartile and bottom quartile of the markup distribution is about 2 percentage points.
  - Heterogeneity robust to controls for size, age, leverage, asset liquidity, asset tangibility, and Tobin’s q — i.e., markup effect is distinct from financial-friction-related channels.
  - Triple interactions (markup × size) findings:
    - Small low-markup firms are most responsive to monetary policy shocks.
    - Difference between large high-markup and small low-markup firms is about 2 percentage points (larger than baseline markup-only difference).
    - Difference in response between low- and high-markup firms is statistically significant for any level of size.
  - Similar patterns hold when conditioning on age and other firm characteristics (Figures 7, 8, A.3–A.6 referenced).
  - Robustness: results hold with alternative markup definitions (Lerner index) and alternative bin choices (Appendix A.2).

### Results — Cross-country analysis (Section 4.2)
- Unconditional responses (Equation 6'):
  - For the panel of advanced economies, response of real sales to a 100-basis point contractionary monetary policy shock:
    - Real sales decline by about 1½ percent 2 years after the shock (peak response similar to US).
- Heterogeneity by markup (Equation 7'):
  - Low-markup firms: negative and statistically significant response relative to average.
  - High-markup firms: response not statistically significant relative to average.
  - Quantitative difference: peaks at about 0.8 percentage point one year after a 100 basis points contractionary shock (difference between low- and high-markup firms).
  - Robust to controls for firm characteristics related to financial frictions (size, age, leverage).
- Detailed heterogeneity:
  - Small low-markup firms are most responsive; difference between low-markup-small and high-markup-large firms is larger than baseline markup-only difference.
  - Small firms are more responsive than large ones, particularly among low-markup firms.
  - Young low-markup firms most responsive; difference between old high-markup and young low-markup firms larger than baseline difference.
- Robustness: results hold when using alternative markup measures (Lerner index), alternative bins, and removing any particular country from the sample (Appendix A.3).

### Theoretical rationale for the empirical results (Section 5)
- Objective: simple two-period partial equilibrium model illustrating:
  - (i) low-markup firms are generally more responsive than high-markup firms to monetary-policy-driven shifts in the real interest rate;
  - (ii) this difference is larger when monetary policy action removes a binding financial constraint;
  - (iii) difference is larger when firms are initially constrained;
  - (iv) if a smaller/younger firm is constrained while a larger/older firm is unconstrained, monetary easing benefits the constrained (via an easier financial constraint and lower cost of capital) more than the unconstrained (only via lower cost of capital).
- Model setup:
  - Production: Cobb-Douglas Y = k^α l^{1−α}, 0 < α < 1.
  - Cost & borrowing constraint:
    - Firm has initial capital k_0 and can borrow to increase capital subject to k ≤ φ(R) + k_0, where R = r + δ and φ(R) is a decreasing function of R.
    - Variable cost: VC(Q) = min_{k,l} Rk + wl s.t. Q = k^α l^{1−α}, k ≤ φ(R) + k_0, k,l ≥ 0.
    - Lemma 1: Marginal cost MC(Q) has two parts: unconstrained region and constrained region; explicit expressions given and cutoff Q^*(R) defined by equation (11).
    - Lemma 2 & Assumption 1: conditions under which ∂Q^*/∂R ≤ 0 (i.e., lower R relaxes constraint and raises cutoff).
  - Demand & markup:
    - Demand q = κ^{ξ} p^{-ξ}; profits Π = max_q pq − VC(q) − FC.
    - Lemma 3: MR(q) = κ^{ξ−1}/ξ q^{-1/ξ} = MC(q); p = ξ/(ξ−1) MC(q). A higher-markup firm (smaller ξ) has steeper MR and thus responds less to MC shifts.
- Effects of an interest-rate cut:
  - Two effects on MC: interest-rate effect (downward shift) and borrowing-constraint effect (flattening for q > Q^*); constrained portion affected more (Lemma 4).
  - Case analyses:
    - Case 1 (both firms unconstrained): low-markup firm increases output more after interest rate cut because it has flatter MR curve (Lemma 5). Sufficient condition for larger response is provided by κ inequality.
    - Additional cases (briefly motivated in text): firms may become constrained after rate cut; low-markup firm is first to become constrained due to flatter MR, and even if constrained its response remains larger than the high-markup firm (further formal results follow in Appendix B).
- Main theoretical intuition:
  - Lowering the interest rate reduces marginal cost for all firms but relaxes borrowing constraints; low-markup firms face flatter marginal revenue and therefore expand output more in response to the same MC shift.
  - When financial constraints bind, monetary easing has an amplified effect for constrained low-markup firms because it both lowers the cost of capital and relaxes the constraint.

*Italic source: wpiea2021184-print-pdf - Section 5 presents a simple model and uses it to rationalize our key findings. Section 6 concludes. (canonical URL provided in the source metadata)*

### Appendix B).

### Appendix B)

### Lemmas and case distinctions on markups and financial constraints
- Lemma 6. Consider high- and low-markup firms that are both financially unconstrained and have the same output levelq
∗
ex ante. If only one firm becomes financially constrained following a decline in the interest rate, it will be the low-markup firm.
- Case 2. Financially constrained firms
  - If the two firms are financially constrainedex anteandex post, the low-markup firm will respond more to the interest rate shock because the constrained portion of the marginal cost curve is increasing in the quantity producedqand theMRcurve of the low-markup firm is flatter.
- Lemma 7.If high- and low-markup firms are both financially constrainedex anteandex post, and they initially have the same output levelq
∗
, then a reduction in the interest rate triggers a larger output response of the low-markup firm (
q
L
> q
H
)
for a sufficient large initial capital stock initial capital stockk
0
and constantκ.
- If both firms are financially constrainedex antebut one firm becomes financially unconstrainedex post, it is straightforward to show that the latter will be the low-markup firm.
- Lemma 8. Consider high- and low-markup firms that are both financially constrained and have the same output level
q
∗
ex ante. If only one firm becomes financially unconstrained following a decline in the interest rate, it will be the low-markup firm.
- If the two firms are financially constrained ex ante and the interest rate decline lifts the constraint for both, the difference in response between the low- and high-markup firms is larger than in the unconstrained case.
  - The interest rate shock shifts theMCcurve fromMC
1
toMC
2
, resulting in both the low- and high-markup firms becoming unconstrained and achieving their optimal scales (q
L
andq
H
).
  - The interest rate decline is more powerful in this case, because it relaxes the borrowing constraint (raising output by the difference betweenq
∗
andq
I
(region A)) in addition to reducing the cost of capital (which raises output by the difference betweenq
I
andq
H
(region B)).
  - This effect is disproportionately more powerful for the low-markup firm (as can be seen by comparing Panels A and B).
- Formal sufficiency condition (as stated): Formally, it is sufficientκ >
w
1−α
ξ−1
ξ
andk
0
>1−φ
(
R
)
.

### Interplay between financial constraints and markups
- Departing from the assumption of identical cost structures, assume one firm unconstrained and the other constrained.
- An interest rate decline benefits the constrained firm more because it obtains not only a lower cost of capital but also a looser borrowing constraint.
- The difference in output responses between the constrained and unconstrained firms is larger the lower the markup of the constrained firm—that is, the farther it operates from its optimal scale before the interest rate cut.
- This result follows directly from Lemma 4 and aligns with the paper's empirical analysis.

### Stylized-model summary findings
- Two key results from the stylized partial equilibrium model:
  i) low-markup firms are more responsive to interest rate shifts than high-markup ones, all else equal; and
  ii) the role of markups for the response of output is greater when firms face tighter financial constraints.

### Empirical conclusions (paper-level)
- The paper uses two alternative firm-level datasets for the United States and 14 advanced economies, with recent measures of markups and exogenous monetary policy shocks.
- Main empirical findings:
  - High-markup firms’ real sales (and real fixed assets) respond significantly less to monetary policy shocks than low-markup firms.
  - The difference in the one-year-ahead response of real sales to a 100 basis points monetary policy shock between the top quartile and the bottom quartile of the firm markup distribution is about 2 percentage points in Compustat (US) and 0.8 percentage point in Orbis (14 advanced economies).
  - Even after controlling for firm size, age and financial characteristics, a firm’s markup has a material impact on its responsiveness to monetary policy shocks—implying the role of markup is independent from other channels.
  - Suggestive evidence that financial frictions amplify the impact of markups on firms’ responsiveness to monetary policy.
  - Results confirm previous findings that firms' age and size affect transmission: while low-markup firms are more responsive regardless of size or age, smaller or younger firms tend to be most responsive to monetary policy.

*Source: Appendix B) of the provided PDF content.*

### Part I. The Wealth Residual. NBER Working Papers 21189, National Bureau of Economic Research, Inc.

### Part I. The Wealth Residual.

### Empirical responses to monetary policy shocks (firm-level and cross-country)
- Figures document impulse responses of real sales and real total assets to a 100-basis point monetary policy shock.
- Confidence bands:
  - Vertical lines denote 90 percent confidence bands.
- Timing:
  - The x-axis denotes time; t=0 is the quarter of the shock for US-based estimates and the year of the shock for the panel of advanced countries.
- Estimation equations referenced:
  - Estimates in US firm-level analysis are based on equation 6 and equation 7.
  - Heterogeneity by firm characteristics uses equation 7 and equation 8.
  - Panel of advanced countries uses equation 6’, equation 7’, and equation 8’.
- Heterogeneity analyses shown:
  - High- vs. low-markup firms: comparisons for real sales and real total assets (Figures 5, 9).
  - Interactions controlled for: Size, Age, Leverage, Liquidity, Tangibility, Tobin’s Q (Figures 6, 6 continued, 10).
  - Subgroup analyses by Size and Age for high- vs. low-markup firms (Figures 7, 8, 11, 12).
- Robustness and restricted samples:
  - US restricted sample responses (Figure A.1).
  - Responses for high vs. low values of selected control variables: Size, Age, Leverage (Figure A.2).
  - Additional subgroup breakdowns by Leverage, Liquidity, Asset Tangibility, Tobin’s Q (Figures A.3–A.6).
- Alternative markup measures and binning:
  - De Loecker–Warzynski markup measure — different bins based on by-industry median over the whole period (Figure A.7).
  - Lerner index markup measure and by-age/size breakdowns (Figures A.8, B.3-1, B.3-2).
  - Orbis-based panel of advanced countries: alternative bins and Lerner index variants (Figures A.9–A.11).

### Theoretical Appendix — key lemmas and comparative static results
- Marginal cost characterization (Lemma 9):
  - The marginal cost curve MC(Q) implied by (1) is:
    - MC(Q) = (R/α)^α (w/1−α)^{1−α}, Q ≤ Q^*
    - MC(Q) = 1/(1−α) w Q^{α/(1−α)} (φ(R)+k_0)^{−α/(1−α)}, otherwise
  - The cutoff Q^*(R) is given by Q^*(R) = (R / [w^{1−α} α])^{1−α} (φ(R)+k_0)
- Monotonicity of cutoff Q^* (Lemma 10 and Assumption 3):
  - dQ^*/dR ≤ 0 if and only if −φ′(R)/(φ(R)+k_0) ≥ (1−α)/R.
  - Assumption 3: φ(·) is such that −φ′(R)/(φ(R)+k_0) ≥ (1−α)/R, hence Q^* in (3) is strictly decreasing in R.
- Firm first-order conditions when unconstrained:
  - R = α k^{α−1} l^{1−α}
  - w = (1−α) k^{α} l^{−α}
  - Implied factor demands (equations 14, 15):
    - l = (R / [w (1−α)/α])^{α} Q
    - k = (w / [R α (1−α)])^{1−α}
  - Unconstrained marginal cost simplifies to MC(Q) = (R/α)^α (w/1−α)^{1−α} (equation 16).
- Constrained regime:
  - Borrowing constraint implies k = φ(R) + k_0 (equation 17).
  - Constrained marginal cost given by MC(Q) = 1/(1−α) w Q^{α/(1−α)} (φ(R)+k_0)^{−α/(1−α)} (equation 18).
- Marginal revenue and markup effects (Lemma 11):
  - MR(q) := κ (ξ−1)/ξ q^{−1/ξ} = MC(q), solution p = ξ/(ξ−1) MC(q).
  - MR curve is decreasing in ξ; for given output, a high-markup firm (lower ξ) has a steeper MR and responds less to shifts in MC.
- Comparative statics of interest rate changes on MC (Lemma 12):
  - For given Q, a decline in r leads to a larger fall in marginal cost on the constrained portion of the MC curve than on the unconstrained portion.
  - Formal comparisons use MC_u and MC_c at different R levels and note:
    - ∂MC_c/∂Q > 0, ∂MC_u/∂Q = 0.
    - Under Assumption 1 and φ decreasing, Q^*_2 > Q^*_1 and constrained MC differences increase in Q.
- Output responsiveness to interest rate and markups (Lemmas 13–16):
  - Lemma 13: If both firms are unconstrained ex ante and ex post and have same initial q^*, a reduction in r triggers a larger output response for the low-markup firm (q_L > q_H).
    - Unconstrained solution: q = κ^ξ ( (ξ−1)/ξ )^ξ [ (α/r)^α ( (1−α)/w )^{1−α} ]^ξ.
    - ∂q/∂ξ > 0 under sufficient κ condition: κ > (r/α)^α (w/1−α)^{(1−α)}^{ξ/(ξ−1)} (as stated).
    - ∂q/∂r < 0; cross-derivative ∂^2 q/(∂ξ ∂ r) < 0 under the same sufficient condition.
  - Lemma 14: If initially unconstrained and same q^*, then if only one firm becomes constrained after r decline, it will be the low-markup firm (follows from Lemma 13 and negative cross-derivative).
  - Lemma 15: If both firms are constrained both ex ante and ex post and have same q^*, a reduction in r triggers a larger output response for the low-markup firm (q_L > q_H).
    - Constrained solution: q = [ κ/(1−α) w^{(ξ−1)/ξ} ]^{ξ(1−α)/(αξ+(1−α))} (φ(R)+k_0)^{αξ/(αξ+(1−α))}.
    - ∂q/∂ξ > 0 under sufficient conditions: κ > w^{1−α} ξ^{ξ/(ξ−1)} and k_0 > 1−φ(R).
    - ∂q/∂r = (… ) φ′(R) < 0 since φ(·) is decreasing in R.
    - Cross-derivative ∂^2 q/(∂ξ ∂r) < 0 under stated sufficient conditions (equation 21).
  - Lemma 16 (restatement): When both firms are constrained, the impact of a decline in r is stronger for firms with larger ξ (lower markup); thus low-markup firms have larger output responses and are more likely to become constrained.

### Figures and robustness structure (inventory of empirical visualizations)
- US firm-level impulse responses and heterogeneity: Figures 4–8.
- Panel of advanced countries impulse responses and heterogeneity: Figures 9–12.
- Appendix empirical figures: Figures A.1–A.11, B.3-1, B.3-2.
- Robustness checks split by markup construction (De Loecker–Warzynski, Lerner index) and alternative bin definitions.

*Part I. The Wealth Residual. NBER Working Papers 21189, National Bureau of Economic Research, Inc.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021184-print-pdf.pdf_