Anatomy of the Phillips Curve: Micro Evidence and Macro Implications

Layer 1 — Overview

Research Question

This paper addresses a fundamental puzzle in macroeconomics: why do estimates of the New Keynesian Phillips curve (NKPC) slope differ sharply depending on whether real marginal cost or the output gap is used as the real activity variable? The conventional, output gap-based NKPC yields very flat slope estimates (e.g., 0.006 to 0.024 in Hazell et al. 2022 and Rotemberg and Woodford 1997), which has led to the widespread view that the Phillips curve is “flat,” at least during the pre-pandemic period. The authors argue that this view conflates two distinct structural relationships: the elasticity of inflation with respect to real marginal cost, and the elasticity of marginal cost with respect to the output gap.

Data and Methodology

The authors assemble a unique quarterly micro-level dataset covering 4,598 manufacturing firms in Belgium over 84 quarters (1999:Q1–2019:Q4), totaling 132,915 observations. The dataset combines product-level domestic prices and quantities from the PRODCOM administrative database, customs data on foreign competitors’ prices, and firms’ variable production costs (labor costs from social security declarations plus intermediate input costs from VAT declarations). Intermediate inputs account for approximately 75 percent of total variable costs on average and are the most volatile cost component (within-firm coefficient of variation 1.77, versus 0.77 for labor costs).

Their estimation strategy follows a “bottom-up” approach. Starting from a theoretical framework with heterogeneous firms subject to Calvo (1983) nominal rigidities and strategic complementarities in price setting (imperfect competition including dynamic oligopoly and Kimball demand), they derive a forward-looking dynamic pass-through regression linking a firm’s current price to discounted present values of its own marginal costs and competitors’ prices, plus a lagged price level that serves as an error-correction term. This is Model A; robustness variants include Model B (absorbing competitor prices via industry-by-time fixed effects), Model C (imposing an AR(1) process for marginal cost), and Model A-U (unrestricted lagged-price coefficient).

The structural parameters governing the NKPC slope — the degree of nominal rigidity (θ) and the strength of strategic complementarities (Ω) — are estimated jointly via GMM. Instruments for marginal cost are four-quarter-lagged firm-level total factor productivity (TFPQ), and instruments for competitors’ prices exploit variation in EU-area export prices to third-country destinations and bilateral exchange rates between non-EU competitor currencies and the Euro. Sector-by-time fixed effects and firm fixed effects absorb confounding trends, shifting trend inflation, and permanent markups.

Main Findings with Quantitative Magnitudes

The baseline estimate (Model A) yields θ = 0.711 (SE 0.014), implying that prices remain fixed for approximately three to four quarters on average, consistent with Nakamura and Steinsson (2008) Belgian PPI data (0.72). The strategic complementarity parameter is Ω = 0.570 (SE 0.059), indicating that competitor price dynamics reduce the pass-through of own marginal cost shocks by approximately half relative to the no-complementarities benchmark.

These structural estimates imply a slope of the marginal cost-based NKPC of λ = 0.052 (SE 0.007), tightly estimated and robust across specifications: λ = 0.077 in Model B, λ = 0.069 in Model C, and λ = 0.056 in the unrestricted Model A-U. This slope is two to ten times larger than existing estimates of the conventional output gap-based NKPC slope (κ ≈ 0.024, Rotemberg and Woodford 1997; κ ≈ 0.006, Hazell et al. 2022).

Reconciling the High Cost-Based Slope with the Flat Output-Based Slope

The paper shows that the output-based slope κ equals the product of the cost-based slope λ and the output elasticity of marginal cost σ_y: κ = λ · σ_y. Using Bartik-style instruments based on high-frequency ECB monetary policy surprises interacted with industry-level sensitivities, the authors estimate σ_y using two models. Model D yields σ_y = 0.406 and κ = 0.021; Model E (directly regressing changes in marginal cost on changes in output) yields σ_y = 0.112 and κ = 0.006. These estimates are consistent with, and overlap with, Rotemberg and Woodford (1997) and Hazell et al. (2022) during the pre-pandemic sample period. The low elasticity of marginal cost to output is attributed to near-constant short-run returns to scale at the firm level and wage rigidity that mutes general equilibrium effects.

Aggregate Inflation Dynamics

Feeding an aggregate marginal cost index (constructed as a Törnqvist-weighted average of firm-level marginal costs) into the model-implied inflation expression produces a series that tracks Belgian manufacturing PPI inflation well: marginal cost fluctuations alone account for approximately 70 percent of inflation variation (R² = 0.68, correlation 0.8), without appealing to unobservable cost-push shocks or inflation lags.

Model Validation via Supply Shocks

A validation exercise using identified oil shocks (Känzig 2021 — measured as unexpected OPEC-day movements in oil futures prices) confirms the model. A one-standard-deviation shock to oil prices (a 15.7 percent increase in Brent crude) raises firms’ real marginal costs by approximately 1.5 to 3 percent within the first three quarters, before reverting. The price response peaks at approximately 3 percent after six quarters, consistent with nominal rigidities generating a delayed but persistent response. Impulse-response matching yields λ_IRF = 0.042 (SE 0.005), within the confidence bands of the micro-level estimate λ = 0.052, validating the bottom-up approach.

Scope Conditions

All estimates are drawn from Belgian manufacturing firms over 1999–2019, a period of moderate inflation during which Calvo pricing provides a good approximation of firm behavior. The authors note that the elasticity of marginal cost to output may be time-varying and nonlinear, and that during large aggregate shocks (such as the post-pandemic inflation surge), both the frequency of price adjustment and the sensitivity of marginal cost to output can rise substantially, requiring state-dependent pricing models (addressed in a companion paper, Gagliardone et al. 2025).

Layer 2 — Q&A

Q1: What is the primitive formulation of the NKPC, and how does it differ from the conventional formulation?

A1: The primitive NKPC features real marginal cost (in log-deviation from its steady state) as the real activity variable: π_t = λ·mc_t + β·E_t{π_{t+1}} + u_t, where λ is the slope depending on nominal rigidities and strategic complementarities. The conventional formulation uses the output gap (or unemployment gap) as a proxy for marginal cost, which is valid only under specific conditions including perfectly flexible wages. When those conditions fail, the output gap is a poor proxy for marginal cost, typically leading to downward bias in slope estimates. Even when a proportionality holds, the output-based slope κ equals λ multiplied by σ_y (the output elasticity of marginal cost), so the two slopes carry different economic content.

Q2: What structural parameters govern the slope of the cost-based NKPC, and what is the formula?

A2: The slope is λ = (1−θ)(1−βθ)/θ, where θ is the Calvo probability of price non-adjustment (capturing nominal rigidity) and Ω = Γ/(1+Γ) is the strategic complementarities parameter derived from the markup elasticity Γ with respect to relative prices. High nominal rigidity (high θ) flattens the slope by making individual price adjustments less frequent; strong strategic complementarities (high Ω) flatten it further because firms mute their price response to marginal cost in order to avoid deviating from competitors. The discount factor β is calibrated at 0.99 for quarterly data.

Q3: How does the dynamic pass-through regression differ from the static (long-run) pass-through regressions used in prior literature?

A3: The dynamic pass-through regression (Model A) includes the firm’s lagged price as a regressor, which functions as an error-correction term controlling for persistent deviations between the price and the optimal reset price. Failing to include this term with quarterly data leads to omitted variable bias of magnitude −θ·Var(Δp_ft), since the cointegration error is autocorrelated with coefficient θ. Static pass-through regressions (as in Amiti, Itskhoki and Konings 2019 using annual data) are appropriate only when nominal rigidities can be ignored (θ ≈ 0); with quarterly data and θ ≈ 0.711, the orthogonality condition of the static model fails and the dynamic framework is necessary.

Q4: What are the baseline estimates of the structural parameters, and how robust are they?

A4: The baseline Model A yields θ = 0.711 (SE 0.014) and Ω = 0.570 (SE 0.059), implying prices fixed for approximately three to four quarters and competitor-price influence roughly equal to own marginal cost influence. The implied NKPC slope is λ = 0.052 (SE 0.007). Robustness checks across six specifications (Models B, C, A-U, variable SR-RTS controls, Translog TFPQ, eight-quarter-lagged instrument) yield λ in the range 0.044 to 0.077, with all estimates statistically significant and within each other’s confidence bands. The unrestricted model (A-U) cannot reject the restriction Ϛ = θ on the lagged-price coefficient (p-value 0.90).

Q5: What is the short-run elasticity of a firm’s own price to a permanent marginal cost shock, and how do nominal rigidities and strategic complementarities each contribute?

A5: The short-run pass-through elasticity is (1−Ω)(1−θ) ≈ (1−0.570)(1−0.711) ≈ 0.125. This is substantially below one because both forces dampen price adjustment: nominal rigidity (1−θ ≈ 0.289) means most firms cannot adjust in any given quarter, and strategic complementarities (1−Ω ≈ 0.430) mean that adjusting firms reduce their pass-through to avoid deviating from competitors’ prices. Without strategic complementarities (Ω = 0), the elasticity would be roughly 0.289; without nominal rigidities (θ = 0), it would be roughly 0.430; both together produce the observed 0.125.

Q6: How is marginal cost measured in the data, and why is the inclusion of intermediate input costs important?

A6: Marginal cost is proxied by average variable cost per unit of output: the log-nominal marginal cost equals ln(TVC_ft/Y_ft) + ln(1+ν_ft), where TVC is the sum of intermediate input costs (from VAT declarations) and labor costs (wage bill from social security declarations), and Y_ft is a quantity index. Intermediate inputs account for approximately 75 percent of total variable costs on average and are the most volatile component (within-firm coefficient of variation 1.77 vs 0.77 for labor). The authors note that DSGE models typically feature only labor as a variable input, but accounting for intermediates is pivotal because intermediate goods price shocks were among the most important drivers of the post-pandemic inflation surge.

Q7: What instruments are used for marginal cost and competitors’ prices, and what are the identifying assumptions?

A7: The instrument for marginal cost is the four-quarter lagged firm-level TFPQ (physical total factor productivity), estimated as the residual from a gross-output production function. Its relevance depends on TFP persistence (confirmed); the exclusion restriction requires that persistent TFP variation is orthogonal to current and future demand shocks after removing permanent demand components (via firm fixed effects) and industry trends (via sector-by-time fixed effects). Two instruments for competitors’ prices exploit international trade variation: (i) sales-weighted average export prices of EU-area competitors to non-Belgium, non-EU destinations (orthogonal to Belgian demand shocks by construction), and (ii) bilateral exchange rate movements between non-EU competitor currencies and the Euro. All instruments pass the Cragg-Donald and Kleibergen-Paap F-statistics (strongly rejecting weak instruments) and Hansen-Sargan over-identification tests (failing to reject validity).

Q8: What evidence supports the validity of the TFPQ instrument against capacity utilization concerns?

A8: The authors run two empirical tests. First, regressing marginal cost on four-quarter-lagged capacity utilization yields a small, statistically insignificant elasticity (0.011, SE 0.052), suggesting the TFPQ instrument’s predictive power does not reflect capacity utilization variation. Second, re-estimating with “purified” TFPQ instruments adjusted for capital utilization (Column 4) and for both capital and labor utilization (Column 5) produces parameter estimates and NKPC slopes essentially unchanged from baseline. Additionally, regression residuals show only weak and short-lived autocorrelation (−0.09 at one-quarter lag, p=0.09; −0.01 at two-quarter lag, p=0.69), indicating demand shocks are highly transitory after conditioning on fixed effects.

Q9: How does the model track aggregate Belgian manufacturing PPI inflation, and what does this imply for cost-push shocks?

A9: Using the reduced-form expression π_t = λ̃(mc_t^n − p_{t-1}) + α + θu_t, where the reduced-form slope λ̃ = 0.22 is evaluated at baseline structural estimates, the model produces a model-implied inflation series that accounts for approximately 70 percent of variation in manufacturing PPI inflation (R² = 0.68, correlation 0.8), without including inflation lags or cost-push shocks. The model captures the inflation drop during the 2008 financial crisis, the run-up in 2016, and the subsequent decline. This contrasts with the quantitative DSGE literature in which cost-push shocks (variation in desired price and wage markups) account for approximately 70 percent of inflation volatility (e.g., Primiceri, Schaumburg and Tambalotti 2006).

Q10: How do the authors estimate the output elasticity of marginal cost σ_y, and what do they find?

A10: They use two approaches. Model D is a pricing equation directly relating firm-level prices and nominal output (value added), estimated via GMM, instrumented with Bartik-style shifters based on high-frequency ECB monetary policy surprises (Altavilla et al. 2019) interacted with industry-level sensitivities. Model E directly regresses changes in nominal marginal cost on changes in nominal output, also instrumented. Model D yields σ_y = 0.406 (SE 0.099) and implied κ = 0.021 (SE 0.005); Model E yields σ_y = 0.112 (SE 0.026) and κ = 0.006 (SE 0.001). The low σ_y is consistent with near-constant short-run returns to scale at the firm level and wage rigidity muting general equilibrium labor-market feedback, at least during the moderate-inflation pre-pandemic period.

Q11: How does the oil shock validation exercise confirm the cost-based NKPC slope estimate?

A11: Following Känzig (2021), the authors identify oil shocks as unexpected movements in Brent crude oil futures around OPEC meeting days, normalizing to a one-standard-deviation shock (15.7 percent Brent increase). Local linear projection IRFs show that firms’ real marginal costs rise 1.5 to 3 percent within three quarters and then revert, while prices peak at approximately 3 percent increase after six quarters (consistent with nominal rigidity delaying the price response). Impulse-response matching — minimizing the weighted distance between empirical and model-implied price IRFs — yields λ_IRF = 0.042 (SE 0.005), which is close to and within the confidence bands of the micro-level estimate λ = 0.052, validating the bottom-up estimation approach.

Q12: What do the estimates imply about why the conventional NKPC appears flat in normal times?

A12: The flat conventional NKPC slope (κ ≈ 0.006–0.024) does not reflect limited transmission of marginal cost fluctuations to inflation — that transmission is high (λ ≈ 0.052–0.077). Rather, flatness reflects a weak link between the output gap and marginal cost during the pre-pandemic period (σ_y ≈ 0.112–0.406), attributable to near-constant short-run returns to scale in production and wage rigidity. This decomposition matters for policy: supply shocks that directly raise marginal cost will pass through strongly to inflation even when output does not move much, whereas demand shocks that operate through the output-cost channel face attenuated transmission.

Q13: Under what conditions does the cost-based Phillips curve decompose cleanly into a product of the two elasticities?

A13: The decomposition κ = λ · σ_y requires assuming that real wages are flexible and determined in general equilibrium at the industry level, with real wages increasing in industry output with elasticity σ_w; that the natural level of output is defined as the equilibrium under flexible prices and constant desired markups; and that the firm’s marginal product of labor depends on productivity and output with a common short-run returns-to-scale parameter ν (homogeneous across firms and time-invariant). Under these assumptions (which parallel those used to derive the conventional NKPC in the standard NK model), the output elasticity of marginal cost is σ_y = σ_w + ν, and the theoretical restriction κ = λ · σ_y holds exactly.

Q14: How do macroeconomic complementarities from aggregate decreasing returns to scale affect the NKPC slope?

A14: If aggregate SR-RTS fall below unity, the NKPC slope formula gains an additional term Θ = 1/(1+γν(1−Ω)) < 1, where ν is inversely related to average SR-RTS and γ is the within-industry elasticity of substitution. However, empirical estimates of sectoral SR-RTS range from 0.93 to 0.98, with an aggregate estimate of approximately 0.965 (implying ν ≈ 0.036). Given this and calibrating γ = 4, Θ ≈ 0.941, so macroeconomic complementarities would reduce the NKPC slope by only about 6 percent — well within the confidence bounds of the baseline estimates. The authors conclude that the constant-returns assumption in their main framework is a good approximation.

Key Concepts

Primitive (cost-based) NKPC slope (λ): The coefficient linking inflation to real marginal cost in the underlying New Keynesian pricing equation, defined as λ = (1−θ)(1−βθ)/θ. It captures how strongly firms’ aggregate price setting responds to movements in real marginal cost per unit of output, holding the discount factor, nominal rigidity, and strategic complementarities fixed. Estimated at 0.052 (tightly, range 0.044–0.077 across specifications) for Belgian manufacturing.

Calvo probability of price non-adjustment (θ): The parameter from Calvo (1983) staggered price setting capturing the share of firms that cannot change their price in a given period, equal to one minus the per-period probability of price adjustment. In this paper, θ is estimated directly from the dynamic pass-through regression coefficient on lagged prices, yielding θ ≈ 0.711, implying prices fixed approximately three to four quarters on average.

Strategic complementarities parameter (Ω): Defined as Ω = Γ/(1+Γ), where Γ is the elasticity of a firm’s desired markup with respect to its own relative price. Captures the extent to which a firm weights competitors’ prices (rather than its own marginal cost) when resetting its price. High Ω means firms strongly mute price responses to own cost changes to avoid relative price deviations from competitors. Estimated at Ω ≈ 0.570, implying competitor prices and own marginal cost enter the reset price with roughly equal weight.

Dynamic pass-through regression: A forward-looking pricing equation (Model A) relating observed firm prices to the discounted present values of own marginal costs and competitors’ prices, plus lagged own price as an error-correction term. The structural parameters θ and Ω are identified jointly from the regression coefficients, using GMM with instruments for the present values. The dynamic specification is necessary at quarterly frequency because the error-correction term (omitted in static pass-through models) is non-negligible when θ > 0.

Output elasticity of marginal cost (σ_y): The elasticity of firm-level real marginal cost with respect to the firm-level output gap, defined under the assumptions that real wages are flexible and industry-level, equal to σ_y = σ_w + ν (wage elasticity with respect to industry output plus the short-run returns-to-scale parameter). This parameter bridges the cost-based and output-based Phillips curve slopes via κ = λ · σ_y. Estimated from micro data using monetary policy shock instruments at σ_y ≈ 0.112–0.406 in the pre-pandemic period.

Short-run returns to scale (SR-RTS): The extent to which a firm’s marginal cost rises with output scale in the short run, parameterized by ν in the cost function MC^n_ft = C_{it} · A_{ft} · Y_ft^ν. If ν = 0, marginal cost is independent of output scale (constant returns), which the authors assume in their baseline. Firm- and sector-level estimates from Translog production functions yield SR-RTS ≈ 0.93–0.98 across sectors (aggregate ≈ 0.965), broadly consistent with the constant-returns assumption and implying modest macroeconomic complementarities.

Reduced-form aggregate pass-through slope (λ̃): A composite parameter capturing the contemporaneous pass-through of aggregate real marginal cost (defined as nominal marginal cost relative to the lagged price level) into quarterly inflation under the assumption that nominal marginal cost follows a random walk. Evaluated at θ ≈ 0.70 and Ω ≈ 0.52 (median across models), λ̃ = 0.22. This is distinct from the structural NKPC slope λ because it also captures the persistence of cost shocks.