TFPR: Dispersion and Cyclicality

Layer 1: Overview

This paper investigates what drives the countercyclical dispersion of TFPR — total factor productivity measured in revenue terms — a pattern that is well documented empirically but poorly understood theoretically. The central motivation is a gap between data measurement and model theory: empirical studies (Kehrig 2011; Bloom, Floetotto, Jaimovich, Eksten, and Terry 2018) document countercyclical dispersion of TFPR, yet the models that seek to explain it routinely conflate TFPR with TFPQ (quantity-based TFP) and treat the two as interchangeable. Cooper and Ozturk argue this conflation is misleading because the distribution of TFPR is endogenous — it depends both on the exogenous distribution of TFPQ and on the endogenous price-setting decisions of firms.

The paper builds an overlapping generations (OG) model with monopolistic competition and state-dependent pricing (menu costs). Young agents set prices ex ante, observe idiosyncratic productivity shocks, menu cost draws, and aggregate shocks, then decide whether to adjust prices ex post at a fixed cost. Old agents consume a CES bundle of goods produced by the young. The aggregate state includes shocks to the money supply, to the mean (µQ) and dispersion (dispQ) of TFPQ, and to the dispersion of idiosyncratic demand (dispD). The model is solved as a stationary rational expectations equilibrium (SREE) without linear approximations, allowing the nonlinear hazard of price adjustment to propagate to the aggregate.

The calibration matches three moments: the standard deviation of TFPR (dispR = 0.102 in data, 0.103 in model), the ratio of dispersion in TFPQ to TFPR (1.181 in both), and the monthly frequency of price adjustment (0.110 in data, 0.127 in model), using parameters from Vavra (2014) and Foster, Haltiwanger, and Syverson (2008). The model period is one month. A key structural feature is a U-shaped hazard of price adjustment: firms with very large or very small gaps between actual and desired prices are most and least likely to adjust, respectively.

The central empirical target is three jointly countercyclical moments: (i) dispersion of TFPR, (ii) dispersion of price changes, (iii) frequency of price adjustment. The paper’s first set of findings is negative. Taken individually, no single shock source reproduces all three patterns. Specifically, shocks to dispQ alone produce procyclical TFPR dispersion — output expands when dispersion rises because high-productivity firms can produce more, but TFPR dispersion rises with dispQ (and hence with output), contradicting the data. Money shocks produce procyclical TFPR dispersion and an inverse U-shaped relationship between dispR and the money shock: at extreme shock values, more firms adjust to the common nominal shock, compressing TFPR dispersion; at moderate values, idiosyncratic heterogeneity dominates and dispR is higher. Shocks to µQ alone leave TFPR dispersion nearly flat. Shocks to dispD produce slight countercyclical TFPR dispersion but counterfactually procyclical price adjustment moments.

Two combinations succeed. First, a joint shock to dispQ and µQ with perfect negative correlation (corr = -1, as in Vavra 2014) generates all three countercyclical moments: as dispQ rises, µQ falls, and output contracts while TFPR dispersion increases; from Table 5, dispR is 0.126 in contraction versus 0.020 in expansion, disp∆p is 0.208 in contraction versus 0.082 in expansion, and freq∆p is 0.328 in contraction versus 0.164 in expansion. Second, a monetary feedback rule where the central bank leans against the wind (ζ = -0.05) — tightening money when dispQ is above average — also replicates all three countercyclical moments (Table 5, leaning-against-the-wind rows).

Two additional findings emerge. The model generates state-dependent monetary policy effectiveness: the response of output to a monetary shock is larger in expansions (coefficient 0.644) than in contractions (0.578) when business cycle state is measured by output growth, consistent with Tenreyro and Thwaites (2016) only for the growth-based measure. The paper also finds no role for uncertainty distinct from realized dispersion: when Markov-switching uncertainty over TFPQ dispersion is introduced, the ex ante price is essentially unchanged, consistent with Berger, Dew-Becker, and Giglio (2020).

The theoretical contribution is a TFPR decomposition: Var(tfpr) = Var(tfpq) + Var(ln p) + 2·Cov(ln p, tfpq). In the FHS data, Var(tfpr) = 0.0484, Var(tfpq) = 0.0676, Var(ln p) = 0.0324, Cov(ln p, tfpq) = -0.0258. In recessions, Var(tfpr) rises to 0.0618, driven by an increase in Var(ln p) to 0.0506 while Var(tfpq) stays at 0.0676. This means countercyclical TFPR dispersion can be generated through endogenous price adjustment even holding TFPQ dispersion fixed — a mechanism entirely absent from models that equate TFPR with TFPQ.

Layer 2: Deep Dive

What is the fundamental measurement-theory gap the paper identifies?

Existing business cycle models (Bloom et al. 2018, Vavra 2014) are calibrated to observed countercyclical dispersion of TFPR but then build theoretical mechanisms around countercyclical dispersion of TFPQ, treating the two as equivalent. Cooper and Ozturk show this is incorrect: TFPR = TFPQ × (p/P), so the TFPR distribution is endogenous, shaped by both the exogenous TFPQ distribution and the endogenous price-setting decisions of firms. Changes in the distribution of prices — through extensive and intensive margins of price adjustment — can move TFPR dispersion independently of TFPQ dispersion.

Why does the OG framework give the model tractability advantages?

In the OG model, young sellers make price decisions within a single period, so the ex post price is independent of the ex ante price. This means the state space is simplified (no lagged own-price), individual choice problems are tractable, the ex post pricing problem is static, and the full SREE can be characterized without log-linear approximations. Crucially, this allows the nonlinear U-shaped price adjustment hazard to propagate to aggregate outcomes exactly, without the approximation errors that would arise in linearized dynamic models.

What are the main shocks in the model and how are they parameterized?

There are four aggregate shocks: (i) money supply shocks x, (ii) shocks to the mean of TFPQ (µQ), (iii) shocks to the dispersion of TFPQ (dispQ, implemented as a mean-preserving spread in z), and (iv) shocks to the dispersion of idiosyncratic demand (dispD). At the individual level, sellers face idiosyncratic productivity shocks z with standard deviation σz = 0.0378 and idiosyncratic demand shocks with σd = 0.0069. Menu costs follow the Dotsey and Wolman (2019) distribution with a fraction ψ = 0.053 of firms having zero adjustment costs.

Why does a dispQ shock alone produce procyclical, not countercyclical, TFPR dispersion?

An increase in dispQ expands the tails of the productivity distribution. High-productivity firms can produce more and expand output (reallocating labor to them raises aggregate output), so output rises with dispQ. Simultaneously, higher dispQ directly raises TFPR dispersion because TFPR = (p/P)×TFPQ and the increased heterogeneity in z carries through to TFPR. Since dispR rises when output rises, the cyclicality is procyclical — directly contradicting the empirical pattern. The pricing response (more adjustment for extreme z draws) magnifies rather than offsets this pattern.

How do monetary shocks affect TFPR dispersion, and why is the relationship non-monotone?

Money shocks cause a rightward shift in the price gap distribution rather than a spread. For moderate money shocks (near average), few firms adjust, so non-adjusters retain their ex ante prices and face heterogeneous gaps — TFPR dispersion is high. For extreme money shocks (very high or very low), many firms adjust to align their prices with the common nominal shock, compressing idiosyncratic price dispersion. Combined with U-shaped adjustment frequency, this creates an inverse U-shaped relationship between dispR and the money shock: TFPR dispersion is highest at moderate shocks and lowest at extreme shocks. Consequently, money shocks alone produce procyclical TFPR dispersion on average, but the model can produce countercyclical dispersion for extreme realizations.

How does the joint (dispQ, µQ) shock with perfect negative correlation work to match the data?

Following Vavra (2014), the paper assumes corr(dispQ, µQ) = -1: the highest dispQ state is paired with the lowest µQ state and so on. When dispQ rises, µQ falls. The mean productivity drop dominates in determining output (output contracts), while the dispersion increase drives up TFPR dispersion. This creates countercyclical dispR. From Table 5, in contractions: dispR = 0.126, disp∆p = 0.208, freq∆p = 0.328; in expansions: dispR = 0.020, disp∆p = 0.082, freq∆p = 0.164. All three moments are countercyclical, matching the data. The key mechanism is that the two shocks drive a wedge between the movements in mean output (dominated by µQ) and the movements in dispersion (dominated by dispQ).

How does the monetary ’leaning against the wind’ feedback rule generate countercyclical TFPR dispersion?

The central bank sets money growth as Mt+1 = Mt[Φ(st+1) + x̃t+1] where Φ(dispQ) = ζ × (dispQ − µdispQ) with ζ < 0 (specifically ζ = -0.05 in the main experiment). When dispQ is above average, the central bank contracts money supply. Since without this rule increased dispQ raises output (procyclical), the monetary contraction more than offsets this, turning the dispQ shock into a net recessionary force. Meanwhile TFPR dispersion still tracks dispQ and rises. Result: both dispR and recession coincide. Table 5 shows that with leaning against the wind on dispQ shocks, dispR = 0.093 in contraction versus 0.082 in expansion, and all three moments remain countercyclical. A second case (feedback to µQ shocks) also produces countercyclical dispR but fails to match the pricing-frequency moment (which becomes procyclical due to asymmetry in the U-shaped hazard).

What are the nonlinearities in the model and why does the paper avoid using correlations as summary statistics?

The U-shaped price adjustment hazard creates nonlinear aggregate responses: variables can be positively correlated with output in expansions and negatively correlated in contractions, or vice versa. For example, under money shocks the correlation of frequency of price adjustment with output is -0.648 in contractions and +0.977 in expansions (Table 7). The dispersion of TFPR under money shocks also switches sign across states. Standard unconditional correlations average over these sign switches and can give misleading or zero correlations, masking the underlying structure. The SREE is solved exactly without linearization so these nonlinearities are not averaged away in the solution.

What is the finding on the state-dependence of monetary policy effectiveness?

Table 8 reports regressions of log output on the log money shock separately in contractions and expansions. When recessions are defined by output below trend, the coefficient is 0.578 in contractions and 0.644 in expansions — monetary policy is less effective in recessions. When recessions are defined by three consecutive periods of negative output growth (as in Tenreyro and Thwaites 2016), coefficients are 0.589 in contractions and 0.611 in expansions — the same qualitative finding. However, this contrasts with Tenreyro and Thwaites (2016) in that the paper finds the asymmetry holds regardless of whether the cycle state is measured in levels or growth rates, whereas Tenreyro and Thwaites find the effect only for growth-based definitions. The mechanism is that recessions (high dispQ, low µQ) are associated with more frequent price adjustment, which attenuates the real effect of money shocks.

What is found regarding the effects of uncertainty versus realized dispersion?

The paper introduces Markov-switching uncertainty where firms do not know in advance which dispersion regime they are in (high or low dispQ). For the ex ante price setting problem, this amounts to taking an expectation over the future dispersion distribution. The quantitative finding is that the ex ante price is essentially unchanged when uncertainty over the dispersion regime is added versus the baseline without such uncertainty. This confirms that the effects on price adjustment and TFPR dispersion in the model come from the realized dispersion, not from ex ante uncertainty about which regime will prevail — consistent with Berger, Dew-Becker, and Giglio (2020) who find that uncertainty shocks have negligible real effects.

How does the variance decomposition of TFPR characterize the empirical patterns?

The paper uses the identity Var(tfpr) = Var(tfpq) + Var(ln p) + 2·Cov(ln p, tfpq). In the FHS data: Var(tfpr) = 0.0484, Var(tfpq) = 0.0676, Var(ln p) = 0.0324, Cov(ln p, tfpq) = -0.0258. The covariance is negative (prices are lower for high-productivity firms, consistent with markup compression), which is why Var(tfpr) < Var(tfpq). In recessions: Var(tfpr) rises to 0.0618, Var(tfpq) is held fixed at 0.0676 (by assumption in the thought experiment), Var(ln p) rises to 0.0506 (from Vavra 2014), and Cov(ln p, tfpq) becomes more negative at -0.0282. This decomposition shows that countercyclical TFPR dispersion can be generated by endogenous price changes — through both higher price variance and a larger (absolute) covariance between prices and productivity — even if TFPQ dispersion is fixed.

What is the role of the U-shaped adjustment hazard in the model?

The U-shaped hazard (probability of price adjustment as a function of the price gap or idiosyncratic shock z) is a key structural feature inherited from state-dependent pricing. Adjustment probability is near zero for small gaps (moderate z) and rises steeply for large gaps (extreme z). This creates nonlinear responses: a mean-preserving spread in z (dispQ shock) pushes more mass into the tails, sharply increasing adjustment frequency; a mean shift in z (µQ shock) shifts the gap distribution rightward, also raising adjustment but asymmetrically; a money shock shifts all gaps in one direction (rightward for a positive shock). The interaction between the shock type and the hazard shape determines whether the covariance of prices and productivity rises or falls, which in turn determines whether TFPR dispersion moves countercyclically.

How does price stickiness create a non-degenerate TFPR distribution without needing other frictions?

In the flexible-price monopolistic competition benchmark (used for comparison), if production is linear in labor (α=1), TFPR = ω/(1-η) and is independent of z — the TFPR distribution is degenerate. In the sticky-price model, non-adjusters set prices ex ante proportional to the money supply, while adjusters set prices that depend on both z and the money shock. The resulting cross-sectional distribution of prices is non-degenerate and generates a non-degenerate TFPR distribution. The coexistence of adjusters and non-adjusters — with prices reflecting both idiosyncratic productivity and aggregate conditions to different degrees — is sufficient to generate TFPR heterogeneity without additional distortions or wedges.

What are the robustness checks and how do they affect the main findings?

Table 6 reports robustness under money shocks alone across three parameter changes: (1) Higher elasticity of substitution ε = 4 (versus baseline 2.37): higher adjustment frequency, lower price change dispersion, but still procyclical TFPR dispersion. (2) Lower labor supply convexity φ = 1.5 (versus baseline 2): moments become nearly acyclical; TFPR dispersion is much higher than baseline. (3) Equal demand and productivity shock dispersion σd = σz: frequency of price adjustment is nearly four times the baseline, but the monetary shock model still fails to generate countercyclical TFPR dispersion. None of these alternatives bring the money-shock-only model into line with the data, confirming that the main positive results (joint dispQ-µQ shock, or monetary feedback) are not artifacts of baseline parameterization. The paper also notes its calibrated ε is lower than Vavra (2014) and Golosov-Lucas (2007), which use higher elasticities and linear labor disutility.

How does this paper relate to and differ from Vavra (2014) and Bloom et al. (2018)?

Vavra (2014) documents countercyclical dispersion of price changes and frequency, and argues this follows from countercyclical TFPQ dispersion driving volatility of firm-level productivity shocks. He calibrates to TFPR moments but treats TFPQ and TFPR as equivalent. Bloom et al. (2018) combine uncertainty and dispersion shocks to TFPQ to generate aggregate fluctuations, requiring both a rise in dispQ and a fall in mean TFPQ to avoid counterfactual negative correlation between consumption and investment. Cooper and Ozturk differ in three respects: (i) they explicitly model the TFPQ-to-TFPR mapping through state-dependent pricing; (ii) they show that dispQ shocks alone produce procyclical (not countercyclical) TFPR dispersion in their model; (iii) while they confirm that the joint (dispQ, µQ) combination matches data, they attribute the mechanism to the pricing wedge rather than uncertainty — uncertainty per se has no effect in their framework.

What are the limitations and directions for future work noted by the authors?

The OG model’s one-period price-setting horizon misses forward-looking dynamics in price adjustment — specifically, the distinction between permanent and temporary adjustment opportunities that matters in infinite-horizon models. However, the authors show the OG model’s policy functions and hazard shape closely replicate those from infinite-horizon state-dependent pricing models, so this limitation is argued to be minor. On the data side, the authors note the ideal structural estimation would use high-frequency joint data on prices and quantities at the firm level, which is not yet available. They suggest future work extending the model to incorporate real-options-style wait-and-see behavior (as in Bloom 2009) combined with state-dependent pricing, and point to the value of non-linear empirical methods (analogous to Tenreyro and Thwaites 2016) for studying price adjustment dynamics.

What is the relationship between idiosyncratic demand shocks and TFPR dispersion?

Idiosyncratic demand shocks (αi) directly affect a seller’s revenue without changing physical productivity z. Under flexible prices they would affect TFPR directly; under sticky prices the adjustment decision interacts with both the demand and productivity shocks. From Table 5, dispD shocks generate slightly countercyclical TFPR dispersion, but the pricing moments (dispersion of price changes and adjustment frequency) are procyclical — inconsistent with the data. Additionally, the dispersion of demand shocks (σd = 0.0069) is calibrated to be about 18% of productivity shock dispersion (σz = 0.0378), so demand shocks play a smaller quantitative role in the baseline. When σd = σz (equal dispersions), adjustment frequency is nearly four times the baseline but the model still fails to match all three target moments.

Key Concepts

TFPR (Revenue Total Factor Productivity): In this paper, TFPR = (p/P) × TFPQ, where p is a firm’s price and P is the aggregate price index. It is the revenue-based measure of productivity that is directly observed in plant-level data. Its distribution is endogenous because prices are set by sellers; unlike TFPQ, it is not a primitive of the model.

TFPQ (Quantity Total Factor Productivity): The physical or quantity-based measure of productivity, denoted z in the model. It is exogenous to the individual seller and drawn from a distribution that can shift in mean (µQ) or dispersion (dispQ). TFPQ is the primitive shock; TFPR is derived from TFPQ through the pricing decisions of sellers.

State-Dependent Pricing (SDP): A pricing framework in which firms adjust prices only when the gain from adjustment exceeds a menu cost. In this paper, sellers set prices ex ante and then decide ex post whether to pay a stochastic cost to reset. Price adjustment depends on the realized state (idiosyncratic z, money shock x), creating both extensive margin (who adjusts) and intensive margin (what price to set) decisions.

Stationary Rational Expectations Equilibrium (SREE): The equilibrium concept used in the paper. It is a set of ex ante prices, ex post prices, critical adjustment costs, and aggregate price levels that are mutually consistent across all aggregate and idiosyncratic states. The SREE is solved exactly without log-linear approximations, allowing the model’s nonlinearities to be preserved.

U-Shaped Adjustment Hazard: The probability of price adjustment as a function of the gap (difference between desired and actual log price) is U-shaped: near-zero for small gaps and sharply increasing for large gaps in either direction. This creates nonlinear aggregate responses to shocks — aggregate variables can comove differently in expansions versus contractions — and is a central driver of the model’s results on TFPR cyclicality.

Leaning Against the Wind (Monetary Feedback Rule): A monetary policy rule in the paper where the central bank contracts the money supply when the dispersion of TFPQ (dispQ) rises above its average (ζ < 0 in the feedback rule). By doing so, the authority converts what would otherwise be a procyclical dispQ shock into a recessionary one, generating countercyclical TFPR dispersion as a byproduct.

dispQ Shock: An aggregate mean-preserving spread in the distribution of idiosyncratic productivity z. It widens the cross-sectional distribution of TFPQ without changing its mean. Taken alone, it produces procyclical TFPR dispersion; combined with a negative shock to µQ (or with monetary tightening), it can produce countercyclical TFPR dispersion.

Price Gap: The difference between the log of the price a seller would optimally set if adjustment were free and the log of the seller’s current ex ante price. The gap is the sufficient statistic for the price adjustment decision: sellers with larger gaps (in absolute value) have larger gains to adjustment and hence higher adjustment probability. The distribution of gaps across sellers responds to aggregate shocks and shapes aggregate price dynamics.