## 2. Under some conditions on their correlation with R&D ## Source details **Canonical URL:** [2. Under some conditions on their correlation with R&D](https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025183-source-pdf.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2025/english/wpiea2025183-source-pdf.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2025/english/wpiea2025183-source-pdf.pdf.json) --- ### Amplification vs. dampening of R&D wedges by impact-value factors - Key mechanism: - Impact-value factors can amplify or dampen the growth cost of R&D wedges depending on their correlation with R&D wedges. - Positive correlation: constrained firms that also have high growth impact per private value amplify misallocation from R&D wedges. - (Weak) Negative correlation: constrained firms with lower growth impact per private value dampen the effect of R&D wedges. - Formal Proposition 2 (summary): - Define σ_∆,ζ as the ω_it–weighted covariance of log R&D wedges and impact-value factors and σ^2_∆ as the ω_it–weighted variance of log R&D wedges. - If σ_∆,ζ > −1/2 · σ^2_∆ and η = sqrt(1 + 2·σ_∆,ζ / σ^2_∆), then up to a 2nd-order approximation: - Ξ_t = [∫_0^1 ω_it·(1 + ∆_it)^(−γ/(1−γ))·η di] · [∫_0^1 ω_it·(1 + ∆_it)^(−1/(1−γ))·η di]^(−γ). - Illustrative examples: - Young, financially constrained firms: large R&D wedges and large impact-value factors ⇒ R&D wedges more costly for growth. - Large firms with market power: R&D wedges from monopsony but lower impact-value factors ⇒ market power shifts R&D away from low impact-value firms, less costly for growth. ### Link to optimal growth policy - Planner objective and policy implication: - Planner uses R&D subsidies to offset variation in ζ_it·(1 + ∆_it) across firms. - Frontier growth achieved by setting Λ_t·Ξ_t = 1; Λ_t·Ξ_t < 1 yields growth inside frontier. - Formal Proposition 3 (summary): - Let g^*_t be planner frontier growth; g^*_t = g^F_t. - Planner can achieve frontier by setting the R&D subsidy component of ∆_it to equalize ζ_it·(1 + ∆_it) across firms (i.e., set 1 + ∆_it = 1/ζ_it). ### Extensions (summary of Appendix E) - Entry/mass of firms: - Allowing endogenous entry amplifies cost of private frictions: R&D wedges reduce profits and lower entry; fewer firms → more inventors per firm → lower productivity because of decreasing returns. - Multi-research-line firms: - Formulae extend to multi-research-line frameworks when counterfactual holds distribution of research lines constant. Estimates are conservative if firms that expand absent frictions are more productive. - Labor market power and imperfect substitutability: - When firms are imperfect substitutes for workers, gains from reallocation are limited; formulae preserved with lower implied scale elasticity γ (specialization reduces R&D wedge cost). - Aggregate R&D input supply elasticity: - Fixed aggregate R&D input assumption can be relaxed; positive supply elasticity adds a supply adjustment term depending on average R&D wedge. Gains from reallocation coincide if average R&D wedge constant. - Multiple R&D production factors: - Extensions hold when multiple R&D factors have equally inelastic supply and frictions are firm-level common. ### Data, measurement, and empirical strategy (selection and measurement details) - Data scope and construction: - Sample: research-active firms listed on US stock exchanges, 1975–2018. - Firm-level annual R&D expenditure: WRDS Compustat (mandatory filings). - Patents: USPTO grants. - Patent valuations: Kogan et al. (2017) estimates from stock-return responses at patent grants. - Forward-citations: citations within first 5 years since grant, normalized by application year average. - Aggregation: forward-citations and valuations aggregated to firm-year using Kogan et al. (2017) mapping; patents recorded in application year. - Sample restrictions: drop firms with consistently low R&D expenditure (< 2.5m 2012 USD/yr), low patenting (< 2.5 patents/yr), or < 5 years in sample. - Coverage: > 80% of R&D expenditure in Compustat and patent valuations in Kogan et al. (2017); 40% of R&D recorded in BEA accounts. - Parameter and baseline choices: - R&D scale elasticity γ = 1/2 (consensus); implies R&D unit-cost elasticity of −1. - Alternative estimates: Terry (2023) γ = 0.4; Terry et al. (2023) γ = 0.8 (county-level); Dechezleprêtre et al. (2023) imply γ ≥ 2/3. Appendix C.4 reports sample estimates consistent with γ = 1/2. - Measurement of R&D wedges: - Identity: z_it·V_it / (W_t·ℓ_it) = 1/γ · (1 + ∆_it). - Measured over 5-year windows with a 1-year lag: - 1 + ∆_it = γ · (∑_{s=0}^4 Patent Valuations_{it+s}) / (∑_{s=0}^4 R&D Expenditure_{it−1+s}). - Measurement concerns and mitigations: - Heterogeneity in γ: residualize measured wedges by industry×year cells. - Need ex-ante expected returns: restrict to observations with ≥ 50 patents over 5-year window. - Non-patented inventions: alternative output measures use non-negative changes in Sales or Employment: - 1 + ∆_it = γ · (∑_{s=0}^4 max{X_{it+s} − X_{it+s−1}, 0}) / (∑_{s=0}^4 R&D Expenditure_{it−1+s}), X ∈ {Sales, Empl.}. - Measurement of R&D productivity θ_it: - From first-order conditions: - θ_it = (1 + ∆_it) × (W_t·ℓ_it)^{1−γ} · W_t^γ. - Scale-independent measure: - θ̂_it = (1 + ∆_it) · (∑_{s=0}^4 R&D Expenditure_{it−1+s})^{1−γ}. - Measurement approaches for impact-value factor ζ_it: - Approach 1: constant across firms. - Approach 2: infer from markups: ˆζ_it = ˆμ_it (Loecker et al. 2020) or ˆζ_it = (∑_{s=0}^4 Sales_{it+s}) / (∑_{s=0}^4 Sales_{it+s} − ∑_{s=0}^4 Profit_{it+s}). - Approach 3: forward-citations as direct measure (up to constant): ˆζ_it = (∑_{s=0}^4 Patent Citations_{it+s}) / (∑_{s=0}^4 Patent Valuations_{it−1+s}). - Mitigations for impact-value measurement concerns: - Residualize impact-value factors by industry-year fixed effects to address citation convention heterogeneity. - When relating ζ_it to R&D wedges, replace valuations with changes in sales to avoid mechanical correlation. ### Empirical exploration of R&D returns (micro facts) - Terminology: R&D wedges and R&D returns used interchangeably (measurement equivalence). - Large dispersion documented: - Percentiles: P25 = 1.12; P50 = 1.7; P75 = 2.27. - SD (log R&D returns) = .93 (reported as 0.9 and plotted as .93). - A firm at the 75th percentile has close to twice the median return in levels; similar gap between median and 25th percentile. - Rising dispersion over time: - R&D return dispersion increased by 32% comparing early and late samples (Table 1.B). - About 64% of industries have more dispersed R&D returns in the later sample. - R&D return dispersion is 0.93/0.64−1≈46% larger than return on capital dispersion in this sample. ### Robustness and persistence - Robustness checks examined four concerns: patent coverage, patent valuation procedure, realized vs expected returns, heterogeneity in γ. - Industry patenting conventions: within 6-digit industry-year variation accounts for 0.85/1.09≈78% of overall dispersion (Panel A, Table 1). - Alternative output measures: - Revenue growth yields dispersion 1.11/0.93−1≈19% larger than patent valuations (Panel C, Table 1). - Persistence: - 5-year autocorrelation for patent valuations = 0.697 (Std. err. (0.020), R2 45.7%, Observations 7,623). - Implied annual autocorrelation ≈ 0.697 1/5 ≈0.93 (text reported transformation). - ∆Sales 5-year autocorrelation = 0.564 (Std. err. (0.024), R2 29.6%, Observations 7,455). - Contemporaneous correlations: ∆Sales 0.597 (Std. err. (0.033), R2 25.1%, Observations 11,688); ∆Employment 0.551 (Std. err. (0.038), R2 14.1%, Observations 10,870). - Structural variance decomposition and winsorizing: - Transitory shocks contribute almost none of overall variation. - Winsorizing patent valuations at the 95th percentile does not materially reduce dispersion; excluding low-value patents does not reduce dispersion. - Heterogeneity in scale elasticity: - Estimated γ stable around 0.5 across deciles; heterogeneity in γ unlikely main driver of dispersion. ### Economic correlates of R&D returns (Table 3 summary) - Overall low explanatory power; correlates (each row a separate regression controlling for NAICS3×Year FE): - Frictions (Panel A): - Return on Capital: 0.043 (Std. err. (0.068), R2 0.1%, Obs 11,844). - Tobin’s Q: 0.202 ∗∗∗ (Std. err. (0.030), R2 6.3%, Obs 10,471). - Liquidity: −0.048 ∗∗ (Std. err. (0.022), R2 0.3%, Obs 10,568). - Dividend rate: 36.499 ∗∗∗ (Std. err. (7.142), R2 1.5%, Obs 11,499). - Risk (Panel B): - CAPMβ: 0.001 (Std. err. (0.065), R2 0.0%, Obs 6,799). - Valuation risk: 0.493 ∗∗∗ (Std. err. (0.132), R2 1.4%, Obs 10,961). - Taxation (Panel C): - R&D user cost 1−τ: −0.527 (Std. err. (0.617), R2 0.1%, Obs 11,247). - Public patent involvement: 1.392 (Std. err. (1.240), R2 0.2%, Obs 11,845). - Dynamics (Panel D): - Long-term R&D growth: 0.424 ∗∗∗ (Std. err. (0.054), R2 10.2%, Obs 6,525). - Long-term TFP growth: 0.451 ∗∗∗ (Std. err. (0.050), R2 3.7%, Obs 5,421). - Prior excess stock return: 0.260 ∗∗∗ (Std. err. (0.031), R2 1.1%, Obs 10,087). - Prior TFP growth: 0.316 ∗∗∗ (Std. err. (0.041), R2 1.1%, Obs 7,277). - Inventors and labor-market power (Panel E): - Inventors: 0.228 ∗∗∗ (Std. err. (0.032), R2 7.2%, Obs 11,845). - Firm dominance: 0.142 ∗∗∗ (Std. err. (0.045), R2 1.4%, Obs 10,477). - Inventor specialization: 0.233 ∗∗∗ (Std. err. (0.083), R2 0.4%, Obs 11,828). - Interpretation: - R&D returns uncorrelated with return on capital but strongly correlated with Tobin’s Q. - Higher liquidity associated with lower R&D returns; larger dividend rates associated with larger R&D returns. - Valuation risk positively associated with higher R&D returns (risk premium interpretation). - Long-term R&D growth accounts for ~10% of dispersion. - Evidence consistent with monopsony over inventors: firms hiring more inventors or displaying inventor-market dominance have larger R&D returns. ### Growth, wedges, and policy: combining data and model - Estimation target: - R&D Allocative Efficiency ˆΞt per Proposition 2 with weights ˆωit = ˆθ1/(1−γ)it normalized across firms. - Adjustment factor ˆηt scenarios: - Scenario 1: independence assumption: ˆηt = 1. - Scenario 2: estimate ˆηt = √(1 + 2·ˆβt) where ˆβt is regression coefficient of R&D returns on impact-value proxy (citations/sales growth) over centered rolling 10-year window. - Aggregation and inference: - Annual estimates collapsed via geometric averages (full sample 1975–2014 and subperiods 1975–90 and 2000–14). - Bootstrapping: 1000 draws per year with replacement; report SD and non-parametric 95% CIs. - Counterfactuals and growth scenarios: - Short-run: compare measured ˆΞt to hypothetical Ξt = 1. - Long-run endogenous growth: set L_t = L and φ = 0; calibrated frictionless gC = 1.5%·Ξ−1 to match long-run US growth; misallocation reduces long-run growth. - Semi-endogenous growth: gCt = A−φt · Lγt · gC with φ > 0 and Lt+1 = (1 + n) · Lt; population growth n = 1% and φ calibrated to achieve long-run growth rate 1.5%. - Implementation note: comparing 1975–90 vs 2000–14 to assess long-run changes and contribution to growth slowdown. - Inference caveat: estimates biased toward milder impact if large, established firms in sample are less impacted by frictions; more misallocation appears among smaller firms. ### Main quantitative findings: impact on growth and welfare (Table 4, Panels A–B) - Baseline (1975–2014): - Growth Impact Ξ−1 = -21.3% (0.41%) [ -21.9% -20.5% ]; Welfare Cost End. = 12.3%; Semi-End. = 11.7%. - Subsample 1975–1990 (Baseline): - Growth Impact Ξ−1 = -14.7% (0.49%) [ -15.4% -13.8% ]; Welfare Cost End. = 7.6%; Semi-End. = 7.4%. - Subsample 2000–2014 (Baseline): - Growth Impact Ξ−1 = -24.0% (0.68%) [ -25.0% -22.8% ]; Welfare Cost End. = 14.5%; Semi-End. = 13.7%. - Baseline change (∆Change): -10.9% growth-impact change; Welfare End. = 5.4%; Semi-End. = 5.3%. - Adjusted specification (Panel B) 1975–2014: - Growth Impact Ξ−1 = -17.9% (0.36%) [ -18.4% -17.3% ]; Welfare Cost End. = 9.8%; Semi-End. = 9.4%. - Adjusted ∆Change: -11.0% growth-impact change; Welfare End. = 5.5%; Semi-End. = 5.3%. - Interpretation: - Baseline -21.3% implies realized annual productivity growth of 1.5% corresponds to 1.9% absence of R&D wedges. - Welfare would be ~12% higher in absence of R&D wedges (baseline statement). - Comparative reference points: Berger et al. (2022) production monopsony reduces US output by 21% and welfare by 8%; Hsieh and Klenow (2009) find 30%–40% larger US output absent production factor misallocation. ### Time series pattern and contribution to growth slowdown - Decline in R&D Allocative Efficiency Ξt−1: - Declined from -15% in 1975–90 to -24% in 2000–14; associated welfare loss ≈ 5%. - Text decomposition: decline implies 24%−15% / (1−15%) ≈ 11% slower short-run growth rate, accounting for 11/40 ≈ 25% of observed growth slowdown. - Example TFP growth decline: 0.5% (1976–1995) to 0.3% (2005–2018) ⇒ 0.5%−0.3% / 0.5% ≈ 40% reduction (text note). ### Robustness exercises and alternative specifications (Table 5 and Section 4.3) - Baseline reported in robustness: Growth Impact (1975–2014) = -17.9%; ∆ = -11.0%; Welfare End. = 5.5%; Semi-E. = 5.3%. - Selected robustness results (highlights): - A. Value of Innovation - ∆Sales: Growth Impact 1975–2014 = -25.5%; 1975–90 = -19.6%; 2000–14 = -32.0%; ∆ = -15.4%; Welfare End. = 8.1%; Semi-E. = 7.8%. - ∆Employment: Growth Impact 1975–2014 = -40.3%; 1975–90 = -39.0%; 2000–14 = -43.9%; ∆ = -8.0%; Welfare End. = 3.8%; Semi-E. = 3.7%. - B. Impact-Value Adjustment (Profit-Based): Growth Impact 1975–2014 = -22.9%; ∆ = -11.5%; Welfare End. = 5.8%; Semi-E. = 5.6%. - C. Entry & Exit (Continuing Firms): Growth Impact 1975–2014 = -16.3%; ∆ = -8.9%; Welfare End. = 4.3%; Semi-E. = 4.2%. - D. Measurement Error - Direct adjustment: Growth Impact 1975–2014 = -18.0%; ∆ = -10.9%; Welfare End. = 5.4%; Semi-E. = 5.3%. - Bils et al. (2021) adjustment: Growth Impact 1975–2014 = -13.7%; ∆ = -5.1%; Welfare End. = 2.3%; Semi-E. = 2.3%. - E. Observation Weights - Unweighted: Growth Impact 1975–2014 = -21.6%; ∆ = -10.7%; Welfare End. = 5.3%; Semi-E. = 5.1%. - Sales growth weights: Growth Impact 1975–2014 = -23.9%; ∆ = -9.8%; Welfare End. = 4.8%; Semi-E. = 4.7%. - F. Firm Size - Small R&D: Growth Impact 1975–2014 = -25.2%; 1975–90 = -16.8%; 2000–14 = -30.7%; ∆ = -16.7%; Welfare End. = 9.0%; Semi-E. = 8.7%. - Larger R&D: Growth Impact 1975–2014 = -16.3%; ∆ = -10.7%; Welfare End. = 5.3%; Semi-E. = 5.1%. - Measurement notes: - Patent valuations yield largest estimates of R&D Allocative Efficiency; sales-based measures show more pronounced decline; employment-based less pronounced. - Entry & exit: continuing-firm analysis yields growth-impact -16% and time change predicting 9% reduction in growth. - Measurement error: bootstrapping and GMM show negligible contribution of patent-level uncertainty to dispersion; additive-noise adjustments (Bils et al. 2021) reduce estimated misallocation but still leave sizable welfare costs. ### Predictive power at industry level - Model prediction: R&D allocative efficiency negatively correlated with industry-level R&D expenditure and R&D return. - Empirical 10-year changes (Table C.4): - Panel A (∆R&D Expenditure): ∆R&D Efficiency coefficients range from −0.751*** (0.139) to −0.294** (0.130) across specifications. - Panel B (∆R&D Return): ∆R&D Efficiency coefficients range from −1.645*** (0.222) to −0.549*** (0.178). - Interpretation: industries with lower allocative efficiency spend more on R&D and have higher aggregate R&D returns — consistent with misallocation waste. ### Potential drivers of rising R&D misallocation (Section 4.4 and Figure 4) - Rising concentration: - Share of patents to top 5% of firms: 49% in 1975–90 vs 67% in 2000–14. - Similar rises in concentration of inventors and sales; slightly lower increases in patent valuations and R&D expenditure concentration. - Declining federal support: - Federal share of private R&D declined from 40% in 1970 to 10% in 2010 (author description). - Mechanisms: - Rising concentration can increase market power in input markets (inventor labor), raising R&D return dispersion and reducing allocative efficiency. - Declining federal funding may have removed a mechanism alleviating financial constraints or providing outside options for R&D workers, reducing allocative efficiency. ### Conclusion and policy-relevant implications - Core conclusions: - Frictions that increase dispersion in R&D returns (R&D wedges) materially contributed to the recent decline in U.S. productivity growth. - Aggregate estimates: frictions reduced US economic growth substantially and increasingly over time; reported reductions include statements of frictions reducing growth by 18% in one context and by 21.3% in another; welfare costs around 9%–12% depending on specification. - Rising frictions can account for an 11% lower growth rate for 2000–14 vs 1975–90, explaining ~25% of the observed productivity slowdown. - Policy and research avenues: - Better identify drivers of rising R&D wedges (adjustment costs, financial frictions, monopsony power over inventors). - Improve measurement of R&D wedges and impact-value factors to design potentially targeted policies to improve U.S. R&D productivity and growth. *Italic source attribution: wpiea2025183-source-pdf - 2. Under some conditions on their correlation with R&D* ### 2. Under some conditions on their correlation with R&D ### 2. Under some conditions on their correlation with R&D ### Amplification vs. dampening of R&D wedges by impact-value factors - Impact-value factors can amplify or dampen the growth cost of R&D wedges depending on their correlation with R&D wedges: - Positive correlation: if particularly constrained firms also achieve a high growth impact per private value created, then impact-value factors amplify the misallocation created from R&D wedges. - (Weak) Negative correlation: if constrained firms have lower growth impact per private value created, then impact-value factors dampen the effect of R&D wedges. - Proposition 2 (formal statement): - Let the ω_it–weighted covariance of log R&D wedges and impact-value factors, σ_∆,ζ, be greater than minus half the ω_it–weighted variance of log R&D wedges, σ^2_∆, and define η = sqrt(1 + 2·σ_∆,ζ / σ^2_∆). - Then, up to a 2nd-order approximation, Ξ_t = [∫_0^1 ω_it·(1 + ∆_it)^(−γ/(1−γ))·η di] · [∫_0^1 ω_it·(1 + ∆_it)^(−1/(1−γ))·η di]^(−γ). - Economic examples illustrating the mechanism: - Young firms that are financially constrained and less able to capture rents (large R&D wedges and large impact-value factors): R&D wedges are more costly for growth because these firms would have under-invested even without wedges and are pushed further from the optimum. - Large firms with market power that are better at capturing rents (R&D wedges from monopsony power but lower impact-value factors): market power shifts R&D away from low impact-value firms and is therefore less costly for growth. ### Link to optimal growth policy - R&D wedges relate directly to optimal growth policy: - To achieve maximal growth, the planner uses R&D subsidies to perfectly offset any variation in the product of impact-value factors and R&D wedges. - The frontier growth rate is achieved by setting Λ_t·Ξ_t = 1; any Λ_t·Ξ_t < 1 yields a growth rate within the growth possibility frontier. - Intuition: firms maximize private value, planner maximizes growth; misalignment implies inefficient allocation of R&D inputs from a growth-maximization perspective. - Proposition 3 (formal statement): - Let g^*_t be the growth frontier achieved in the Planner Growth Equilibrium, then g^*_t = g^F_t. - This allocation can be achieved by setting the R&D subsidy component of ∆_it to equalize ζ_it·(1 + ∆_it) across firms. ### Extensions (summary of Appendix E) - Entry/mass of firms: - Allowing for entry (rather than holding the mass of firms constant) can amplify the cost of private frictions. - R&D wedges reduce profits and lower entry; fewer firms imply more inventors per firm and, due to decreasing returns to scale, lower productivity. - Multi-research-line firms: - Formulae extend to frameworks with multi-research line firms (Klette and Kortum, 2004) when the counterfactual holds constant the distribution of research lines across firms. - Estimated growth impacts are conservative if firms that expand in absence of frictions also tend to be more productive. - Labor market power and imperfect substitutability: - When firms are imperfect substitutes for workers (amenities or specialization), gains from reallocation are limited. - Formulae are preserved with a lower implied scale elasticity γ reflecting specialization, making R&D wedges tend to be less costly. - Aggregate R&D input supply elasticity: - Main model assumes fixed aggregate R&D inputs (level of R&D wedges does not affect growth). - Formulae extend to positive input supply elasticity with an additional supply adjustment term depending on the average R&D wedge. - Gains from reallocation coincide as long as the average R&D wedge remains constant. - Multiple R&D production factors: - Formulae extend to multiple R&D production factors as long as their supply is equally inelastic and frictions are common at the firm-level. ### Data, measurement, and empirical strategy (selection and measurement details) - Data scope and sources: - Sample: research-active firms listed on US stock exchanges. - Firm-level annual R&D expenditure: WRDS Compustat (mandatory filings). - Patents: USPTO grants used to measure R&D outputs. - Patent valuations: Kogan et al. (2017) — estimated from firm stock returns around patent grant announcements to measure private value created from innovation. - Forward-citations: citations received by the patent within the first 5 years since grant date, normalized by average within application year. - Aggregation: forward-citations and valuations aggregated to firm-year using patent-to-firm mapping in Kogan et al. (2017); patents recorded in their application year. - Sample period and restrictions: 1975–2018; drop firms with consistently low R&D expenditure (less than 2.5m 2012 USD per year), low patenting (less than 2.5 patents per year), or less than 5 years in sample. - Final sample coverage: more than 80% of R&D expenditure in Compustat and patent valuations in Kogan et al. (2017), and 40% of the R&D recorded in BEA accounts. - Parameters and baseline choices: - R&D scale elasticity γ set to 1/2 (consensus in literature), implying an R&D unit-cost elasticity of -1. - Note: Terry (2023) estimates γ = 0.4; Terry et al. (2023) estimate γ = 0.8 at the county-level; Dechezleprêtre et al. (2023) imply γ of 2/3 or larger, with attribution to financial frictions. Appendix C.4 reports sample estimates consistent with γ = 1/2. - Measurement of R&D wedges: - Identity linking R&D wedges to average R&D return: z_it·V_it / (W_t·ℓ_it) = 1/γ · (1 + ∆_it). - Measured over 5-year windows with a 1-year lag between R&D expenditure and patent valuations: 1 + ∆_it = γ · (∑_{s=0}^4 Patent Valuations_{it+s}) / (∑_{s=0}^4 R&D Expenditure_{it−1+s}). - Measurement concerns and solutions: - Heterogeneity in γ across firms could confound wedges; residualize measured R&D wedges with respect to industry×year cells. - Need ex-ante expected R&D returns: restrict sample to observations with at least 50 patents over the 5-year window to leverage law of large numbers. - Not all inventions are patented and patenting rates differ across firms: alternative measures use non-negative changes in sales or employment as innovation output: 1 + ∆_it = γ · (∑_{s=0}^4 max{X_{it+s} − X_{it+s−1}, 0}) / (∑_{s=0}^4 R&D Expenditure_{it−1+s}), with X ∈ {Sales, Empl.}. - Measurement of R&D productivity θ_it: - From first-order conditions: θ_it = (1 + ∆_it) × (W_t·ℓ_it)^{1−γ} · W_t^γ. - Scale-independent measure used (dropping common wage intercept): θ̂_it = (1 + ∆_it) · (∑_{s=0}^4 R&D Expenditure_{it−1+s})^{1−γ}. - Measurement approaches for impact-value factor ζ_it: - Approach 1: assume constant factor across firms (workhorse growth models). - Approach 2: infer from markups (limit-pricing link); obtain markups ˆμ_it from Loecker et al. (2020) or via value implied by firms’ profit rates. Impact-value factor: ˆζ_it = ˆμ_it or ˆζ_it = (∑_{s=0}^4 Sales_{it+s}) / (∑_{s=0}^4 Sales_{it+s} − ∑_{s=0}^4 Profit_{it+s}). - Approach 3: use forward-citations as direct measure of growth-impact (up to constant factor): ˆζ_it = (∑_{s=0}^4 Patent Citations_{it+s}) / (∑_{s=0}^4 Patent Valuations_{it−1+s}). - Concerns with impact-value measurement and mitigations: - Heterogeneity in citation conventions across industries/time: residualize impact-value factors with respect to industry-year fixed effects. - Potential mechanical relation between impact-value factor and R&D return (due to patent valuations): when relating impact-value factor to R&D wedges, replace valuations with changes in sales. ### Empirical exploration of R&D returns (micro facts) - Terminology: R&D wedges and R&D returns used interchangeably due to measurement equivalence. - Frictionless benchmark implies equalized R&D wedges across firms; empirical finding: large dispersion in measured R&D returns (histogram referenced). *Italic source attribution: wpiea2025183-source-pdf - 2. Under some conditions on their correlation with R&D* ### 2. A firm at the 75th percentile of the ### 2. A firm at the 75th percentile of the ### R&D return dispersion: magnitudes and trends - A firm at the 75th percentile has close to twice the median return in levels, with a similar gap between the median and the 25th percentile. - P25 = 1.12; P50 = 1.7; P75 = 2.27. - Histogram summary: SD = .93 (standard deviation of log R&D returns reported as 0.9 and plotted as .93). - R&D return dispersion has increased throughout the sample: comparing early and late samples in Table 1.B, R&D return dispersion has risen by32%. - Rising dispersion is broad-based: about 64% of industries have more dispersed R&D returns in the later sample. - Comparing dispersion across factor returns (Table 1.A): R&D return dispersion is 0.93/0.64−1≈46% larger than return on capital dispersion in this sample. ### Robustness and measurement concerns - Four measurement concerns evaluated: patent coverage of innovation, patent valuation procedure, realized vs expected returns, and heterogeneity in R&D scale elasticity. - Industry patenting conventions: cross-industry differences contribute little relative to overall dispersion — within 6-digit industry-year variation accounts for 0.85/1.09≈78% of overall R&D return dispersion (Panel A of Table 1). - Alternative R&D output measures: using revenue growth yields dispersion 1.11/0.93−1≈19% larger than patent valuations (Panel C of Table 1). - Patent valuation measurement error: Kogan et al. (2017) approach may contain confounding events, but R&D returns are highly persistent, inconsistent with classical measurement error driving dispersion. - 5-year autocorrelation for patent valuations = 0.697 (Panel A, Table 2). - Implied annual autocorrelation coefficient reported as 0.697 1/5 ≈0.93. - All R&D return measures show high persistence; persistence remained stable over time. - Structural variance decomposition indicates transitory shocks, such as classical measurement error, contribute almost none of overall R&D return variation. - Winsorizing patent valuations at the 95th percentile does not materially reduce measured dispersion. - Excluding low-value patents does not reduce dispersion. - Differences in estimated R&D scale elasticity across deciles are stable around 0.5, suggesting scale-elasticity heterogeneity is not the main driver of measured R&D return differences. - Overall conclusion: robustness checks suggest neither patent/valuation measurement issues nor classical measurement error are major drivers of R&D return dispersion. ### Consistency across measures and time - Table 2 highlights: - 5-year autocorrelation: Patent valuations 0.697 (Std. err. (0.020), R2 45.7%, Observations 7,623). - 5-year autocorrelation: ∆Sales 0.564 (Std. err. (0.024), R2 29.6%, Observations 7,455). - Contemporaneous correlations with baseline R&D returns: ∆Sales 0.597 (Std. err. (0.033), R2 25.1%, Observations 11,688); ∆Employment 0.551 (Std. err. (0.038), R2 14.1%, Observations 10,870). - Persistence and cross-measure correlations reinforce that dispersion is not dominated by short-lived noise. ### Economic drivers of R&D return dispersion - Low explanatory power overall; mechanisms considered jointly explain a small share of dispersion, paralleling return-on-capital literature findings. - Correlations reported in Table 3 (each row a separate regression, controlling for NAICS3×Year fixed effects): - Frictions (Panel A): - Return on Capital: Estimate 0.043 (Std. err. (0.068), R2 0.1%, Observations 11,844) — essentially uncorrelated. - Tobin’s Q: Estimate 0.202 ∗∗∗ (Std. err. (0.030), R2 6.3%, Observations 10,471). - Liquidity: Estimate -0.048 ∗∗ (Std. err. (0.022), R2 0.3%, Observations 10,568). - Dividend rate: Estimate 36.499 ∗∗∗ (Std. err. (7.142), R2 1.5%, Observations 11,499). - Risk (Panel B): - CAPMβ: Estimate 0.001 (Std. err. (0.065), R2 0.0%, Observations 6,799). - Valuation risk: Estimate 0.493 ∗∗∗ (Std. err. (0.132), R2 1.4%, Observations 10,961). - Taxation (Panel C): - R&D user cost 1−τ: Estimate -0.527 (Std. err. (0.617), R2 0.1%, Observations 11,247). - Public patent involvement: Estimate 1.392 (Std. err. (1.240), R2 0.2%, Observations 11,845). - Dynamics (Panel D): - Long-term R&D growth: Estimate 0.424 ∗∗∗ (Std. err. (0.054), R2 10.2%, Observations 6,525). - Long-term TFP growth: Estimate 0.451 ∗∗∗ (Std. err. (0.050), R2 3.7%, Observations 5,421). - Prior excess stock return: Estimate 0.260 ∗∗∗ (Std. err. (0.031), R2 1.1%, Observations 10,087). - Prior TFP growth: Estimate 0.316 ∗∗∗ (Std. err. (0.041), R2 1.1%, Observations 7,277). - Inventors and labor-market power (Panel E): - Inventors: Estimate 0.228 ∗∗∗ (Std. err. (0.032), R2 7.2%, Observations 11,845). - Firm dominance: Estimate 0.142 ∗∗∗ (Std. err. (0.045), R2 1.4%, Observations 10,477). - Inventor specialization: Estimate 0.233 ∗∗∗ (Std. err. (0.083), R2 0.4%, Observations 11,828). - Interpretation highlights: - R&D returns are uncorrelated with return on capital but strongly correlated with Tobin’s Q. - Firms with greater liquidity tend to have lower R&D returns, while firms with larger dividend rates have larger R&D returns. - Valuation risk (patent-valuation volatility) is positively associated with higher R&D returns, consistent with a risk-premium interpretation. - Long-term R&D growth can account for around 10% of R&D return dispersion; long-term TFP growth also strongly correlated. - Evidence consistent with monopsony over inventors: firms hiring more inventors, dominating inventor labor markets, or hiring more specialized inventors tend to have larger R&D returns. ### Growth, wedges, and policy: combining data and model - Estimation target: R&D Allocative Efficiency ˆΞt per Proposition 2, with weights ˆωit defined as ˆθ1/(1−γ)it normalized over firms. - Two scenarios for adjustment factor ˆηt: - Scenario 1: assume R&D wedges and impact-value factors independent, set ˆηt = 1. - Scenario 2: estimate ˆηt using citations divided by sales growth as proxy for the impact-value factor, with ˆηt = √(1 + 2·ˆβt), where ˆβt is the regression coefficient of R&D returns on the impact-value factor over a centered rolling 10-year window. - Aggregation and inference: - Annual estimates collapsed using geometric averages to assess longer-run developments (full sample 1975–2014 and subperiods 1975–90 and 2000–14). - Bootstrapping: for each year, sample observations with replacement to true sample size, compute annual aggregates, repeat 1000 times, report standard deviation and non-parametric 95% confidence intervals. - Counterfactuals: - Short-run impact: compare growth rate under measured impact ˆΞt to hypothetical Ξt = 1 (assumes offsetting R&D wedges technologically feasible). - Long-run scenarios: - Endogenous growth scenario: set Lt = L and φ = 0; setting Ξt = 1 achieves frictionless growth rate gC, calibrated as gC = 1.5%·Ξ−1 to match long-run US growth rate. Misallocation reduces long-run growth in this scenario. - Semi-endogenous growth scenario: frictionless growth dynamics gCt = A−φt · Lγt · gC with φ > 0 and Lt+1 = (1 + n) · Lt; long-run growth pinned down by g = (1 + n)γ/φ − 1. In counterfactual, economy starts on long-run path before policy change and responses in productivity and consumption are traced. Population growth set to n = 1% and φ calibrated to achieve long-run growth rate of 1.5%. - Implementation note: comparing early and late periods (1975–90 vs 2000–14) provides insight into long-run changes in R&D Allocative Efficiency and potential contribution to declining economic growth. - Inference: estimates are biased toward milder impact if large, established firms in sample are less impacted by frictions; sample selection discussed in text, with more misallocation appearing among smaller firms in the sample. *Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025183-source-pdf.pdf* ### 4. The table also reports the welfare cost in consumption- ### wpiea2025183-source-pdf - 4. The table also reports the welfare cost in consumption- ### Main quantitative findings: Impact of R&D wedges on growth and welfare (Table 4, Panel A and B) - Baseline (1975–2014): Growth Impact Ξ−1 = -21.3% (0.41%) [ -21.9% -20.5% ]; Welfare Cost End. = 12.3%; Semi-End. = 11.7% - Subsample 1975–1990 (Baseline): Growth Impact Ξ−1 = -14.7% (0.49%) [ -15.4% -13.8% ]; Welfare Cost End. = 7.6%; Semi-End. = 7.4% - Subsample 2000–2014 (Baseline): Growth Impact Ξ−1 = -24.0% (0.68%) [ -25.0% -22.8% ]; Welfare Cost End. = 14.5%; Semi-End. = 13.7% - Baseline change (∆Change): -10.9% growth-impact change; Welfare End. = 5.4%; Semi-End. = 5.3% - Adjusted specification (Panel B) 1975–2014: Growth Impact Ξ−1 = -17.9% (0.36%) [ -18.4% -17.3% ]; Welfare Cost End. = 9.8%; Semi-End. = 9.4% - Adjusted 1975–1990: Growth Impact Ξ−1 = -12.0% (0.40%) [ -12.6% -11.2% ]; Welfare Cost End. = 6.0%; Semi-End. = 5.9% - Adjusted 2000–2014: Growth Impact Ξ−1 = -21.7% (0.62%) [ -22.6% -20.6% ]; Welfare Cost End. = 12.6%; Semi-End. = 12.0% - Adjusted ∆Change: -11.0% growth-impact change; Welfare End. = 5.5%; Semi-End. = 5.3% ### Interpretation and illustrative comparisons - Estimated average growth impact of R&D wedges for the full sample: -21.3%, implying a realized annual productivity growth rate of 1.5% corresponds to a growth rate of 1.9% in absence of R&D wedges. - Model-implied welfare effect: welfare would be 12% higher in absence of R&D wedges (baseline statement). - Comparative estimates cited in text: - Berger et al. (2022): monopsony in production sector reduces US output by 21% and welfare by 8%. - Hsieh and Klenow (2009): 30%–40% larger US output in absence of production factor misallocation. ### Time series pattern: Decline in R&D Allocative Efficiency (Figure 3 and discussion) - Estimated R&D Allocative Efficiency Ξt−1 declined throughout sample with: - Sharp downturn and partial recovery during the 1990s (potential Dot-Com dynamics). - Comparison 1975–90 vs 2000–14: R&D Allocative Efficiency declined from -15% to -24% with an associated welfare loss of around 5%. - Textual decomposition: decline implies a 24%−15% / (1−15%) ≈ 11% slower short-run growth rate, which accounts for 11/40 ≈ 25% of the growth slowdown (as stated). - Total factor productivity growth example: declined from 0.5% for 1976–1995 to 0.3% for 2005–2018, a 0.5%−0.3% / 0.5% ≈ 40% reduction (noted in text). ### Robustness exercises and alternative specifications (Table 5 and Section 4.3) - Baseline reported in robustness table: Growth Impact (1975–2014) = -17.9%; ∆ = -11.0%; Welfare End. = 5.5%; Semi-E. = 5.3% - A. Value of Innovation - ∆Sales: Growth Impact 1975–2014 = -25.5%; 1975–90 = -19.6%; 2000–14 = -32.0%; ∆ = -15.4%; Welfare End. = 8.1%; Semi-E. = 7.8% - ∆Employment: Growth Impact 1975–2014 = -40.3%; 1975–90 = -39.0%; 2000–14 = -43.9%; ∆ = -8.0%; Welfare End. = 3.8%; Semi-E. = 3.7% - B. Impact-Value Adjustment - Profit-Based: Growth Impact 1975–2014 = -22.9%; 1975–90 = -15.7%; 2000–14 = -25.4%; ∆ = -11.5%; Welfare End. = 5.8%; Semi-E. = 5.6% - C. Entry & Exit - Continuing Firms: Growth Impact 1975–2014 = -16.3%; 1975–90 = -11.5%; 2000–14 = -19.4%; ∆ = -8.9%; Welfare End. = 4.3%; Semi-E. = 4.2% - D. Measurement Error - Direct adjustment: Growth Impact 1975–2014 = -18.0%; 1975–90 = -12.1%; 2000–14 = -21.7%; ∆ = -10.9%; Welfare End. = 5.4%; Semi-E. = 5.3% - Bils et al. (2021) adjustment: Growth Impact 1975–2014 = -13.7%; 1975–90 = -10.5%; 2000–14 = -15.1%; ∆ = -5.1%; Welfare End. = 2.3%; Semi-E. = 2.3% - E. Observation Weights - Unweighted: Growth Impact 1975–2014 = -21.6%; 1975–90 = -16.1%; 2000–14 = -25.1%; ∆ = -10.7%; Welfare End. = 5.3%; Semi-E. = 5.1% - Sales growth weights: Growth Impact 1975–2014 = -23.9%; 1975–90 = -18.2%; 2000–14 = -26.3%; ∆ = -9.8%; Welfare End. = 4.8%; Semi-E. = 4.7% - F. Firm Size - Small R&D: Growth Impact 1975–2014 = -25.2%; 1975–90 = -16.8%; 2000–14 = -30.7%; ∆ = -16.7%; Welfare End. = 9.0%; Semi-E. = 8.7% - Larger R&D: Growth Impact 1975–2014 = -16.3%; 1975–90 = -10.3%; 2000–14 = -19.9%; ∆ = -10.7%; Welfare End. = 5.3%; Semi-E. = 5.1% ### Measurement issues examined - Patent valuations as R&D output: using patent valuations yields largest estimate of R&D Allocative Efficiency; decline is more (less) pronounced when using sales (employment) growth. - Impact-value factor measurement: preferred specification estimates adjustment factor η by regressing citations over sales growth on R&D returns; profit-based measure yields slightly larger estimates but time changes still ≈ -11% growth impact. - Entry & exit: focusing on continuing firms yields growth-impact -16% and time change predicting 9% reduction in growth. - Measurement error analysis: bootstrapping patent valuations and a GMM estimator for persistent vs non-persistent variation find almost no contribution of these two sources to main dispersion estimates (Panel D); persistence in R&D returns remained stable over time (Appendix Table C.3). - Alternative additive-in-levels measurement-error adjustment (Bils et al. 2021) improves estimated R&D Allocative Efficiency: adjusted measure declines by 5% vs 10% baseline, but associated welfare costs remain sizable. - Observation weights and joint measurement: equalizing weights or using sales growth yields smaller estimates of R&D Allocative Efficiency while holding the decline approximately equal. ### Predictive power at industry level - Model prediction: R&D allocative efficiency is negatively correlated with industry-level R&D expenditure and R&D return. - Empirical test: 10-year industry changes confirm a strong negative correlation of both aggregates with R&D allocative efficiency. - Note: R&D Allocative Efficiency is scale independent, i.e., HD(0), in R&D expenditure and patent valuations. ### Potential drivers of rising R&D misallocation (Section 4.4 and Figure 4) - Rising concentration in R&D sector (top 5% shares reported): - Share of patents granted to top 5% of firms: 49% in 1975–90 vs 67% in 2000–14. - Similar rises in concentration of inventors and sales; slightly lower but meaningful increases in concentration of patent valuations and R&D expenditure. - Declining federal support for R&D (NSF National Patterns, panel B text): - Federal share of private R&D declined from 40% in 1970 to 10% in 2010 (author’s description). - Mechanisms discussed: - Rising concentration could increase market power in input markets (including labor for inventors), raising dispersion in R&D returns and reducing allocative efficiency. - Decline in federal funding could have removed a mechanism that alleviated financial constraints, subsidized firms with market power over R&D inputs, or provided an outside option for R&D workers—each channel potentially reducing allocative efficiency. ### Conclusion and policy-relevant implications - Core conclusion: Frictions that increase dispersion in R&D returns (R&D wedges) materially contributed to the recent decline in U.S. productivity growth. - Aggregate estimates: frictions reduce US economic growth significantly and increasingly so; for full sample the paper reports frictions reduced growth by 18% in one statement and by 21.3% in another context; welfare cost estimates reported around 9%–12% depending on specification (consumption-equivalent terms). - Rising frictions can account for an 11% lower growth rate for 2000–14 compared to 1975–90, which accounts for 25% of the observed productivity slowdown. - Suggested research/policy avenues: - Better understanding drivers of rising R&D wedges (adjustment costs, financial frictions, monopsony power over inventors). - Improved measurement of R&D wedges and impact-value factors to enable potentially targeted policies to improve U.S. R&D productivity and economic growth. *Source: content from the provided PDF chapter/section (text and tables and figures excerpts as supplied).* ### References ### References ### Key literature cited - Acemoglu, Daron, “Training and Innovation in an Imperfect Labour Market,” Review of Economic Studies, 1997, 64, 445–464. - Acemoglu, Daron; Ufuk Akcigit; Harun Alp; Nicholas Bloom; William R. 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The In-State, Out-of-State, and Aggregate Effects of R&D Tax Credits,” The Review of Economics and Statistics, 2009, 91, 431–436. - Yeh, Chen; Claudia Macaluso; Brad Hershbein, “Monopsony in the U.S. Labor Market,” American Economic Review, 2022, 112, 2099–21399. ### Appendix (A): Proofs — analytical results and formulas - Definition: θ_it = φ_it · V_it; firm demand for R&D inputs: - ℓ_it = [θ_it · γ / ((1 + ∆_it) · W_t)]^(1/(1−γ)). - R&D input price solution (from R&D resource constraint): - W_t^γ = L_t^(−(1−γ)) · [∫_0^1 (θ_it /(1 + ∆_it))^(1/(1−γ)) di]^(1−γ). - Economic growth rate expression: - g_t = ∫_0^1 ζ_it · ℓ_it · W_t^γ · di. - Rearranged growth rate formula: - g_t = L_t^γ · [∫_0^1 ζ_it · θ_it^(1/(1−γ)) · (1 + ∆_it)^(−γ/(1−γ)) · di] · [∫_0^1 θ_it^(1/(1−γ)) · (1 + ∆_it)^(−1/(1−γ)) · di]^γ. - Corollary 1: Ξ_t ≤ 1 by Jensen’s inequality; equality if no dispersion in R&D wedges. Final statement follows from the second order approximation in Lemma 1. - Planner problem (Proposition 3): - max g_t = ∫_0^1 ζ_it · z_it · V_it · di - s.t. L_t = ∫_0^1 ℓ_it · di and z_it = φ_it · ℓ_it^γ. - First order condition yields R&D input demand: - ℓ_it = [ζ_it · θ_it · γ / (λ_W_t · W_t)]^(1/(1−γ)). (Equation A.1) - Competitive equilibrium allocation coincides with planner iff ζ_it · (1 + ∆_it) is constant. Planner can implement growth-maximizing allocation by setting 1 + ∆_it = 1/ζ_it. ### Appendix (A): Lemma and approximations - Lemma 1: Second-order approximation of Ξ_t around ζ_it = ζ and ∆_it = ∆: - Ξ_t ≈ exp[−1/2 · γ/(1−γ) · (σ_∆^2 + 2·σ_∆,ζ)], (Equation A.2) - where σ_∆^2 is the weighted variance of log R&D wedges and σ_∆,ζ is the ω_it-weighted covariance of log R&D and Impact-Value factors. - Approximation is precise if variables are jointly log-normal; then weights unnecessary. - Proposition 2: Second-order approximation of Ξ_t can be written as: - Ξ_t ≈ exp[−1/2 · γ/(1−γ) · σ_∆^2 · ̃β] with ̃β = 1 + 2·σ_∆,ζ/σ_∆^2. - A second-order approximation of Proposition 2’s formula yields the same expression. ### Data Appendix (B): variable definitions and construction - Inventor employment: - Let P_it→t+4 be set of successful patent applications for firm i between t and t+4. - I_it→t+4 be set of associated inventors. - P_ijt = number of patents assigned to firm i listing inventor j at time t. - P_jt = total number of patents listing j as inventor at time t. - Inventors_it→t+4 = Σ_{j∈I_it→t+4} (Σ_{s=0}^4 P_ijt+s) / (Σ_{s=0}^4 P_jt+s). (Equation B.1) - Return on Capital: - Follow David et al. (2016): return on capital = ratio of sales to beginning of period capital stock. - Constructed at 5-year level: - Return on Capital_it ≡ Σ_{s=0}^4 Sales_it+s / Σ_{s=0}^4 Capital_it+s. (Equation B.2) - Tobin’s Q: - Physical investment Q = (stock price × outstanding shares + debt net of cash holdings (prcc_f×csho + dltt + dlc − act)) / physical capital (ppeqgt). - Liquidity: - Liquidity = cash holdings / assets = ch/at. - Dividend rate: - Dividend rate = dividends / assets = dvt/at. - Public patent involvement: - Patents classified as connected to public actors if assigned to government entity, research lab, or university, or if they have a government interest statement. - Public involvement = share of patent valuations connected to public actors for the 5-year window. - Firm dominance (two-step construction): - Step 1: For each new firm patent within a 5-year window, calculate share of inventors working for the firm among those that worked on patents in the exactly same technology classification (complete CPC classification, >600 non-exclusive classes). - Step 2: Aggregate to firm-level by simple average over firm’s new patents. Measure ranges 0–1; 1 = maximal dominance. - Inventor specialization (two-step construction): - Step 1: For a 5-year window, inventor specialization = average cosine similarity between patent classifications in an inventor’s portfolio of new patents. For each patent, create indicator vector over available patent classifications weighted by inverse frequency; compute average cosine similarity across patents in portfolio; take simple average across patents. Measure ranges 0–1; 0 = completely different patents, 1 = identical technology classification across patents. - Step 2: Aggregate to firm-level by patent-weighted average across firm’s inventors. ### Empirical Appendix (C): robustness, dispersion metrics, and predictability ### C.1 Additional Robustness for R&D Return Dispersion - Table C.1: R&D Return Dispersion Across Specifications (Standard Deviation; Observations) - A. Aggregation horizon - 1-year: 1.00; 1,083 - 5-year: 0.93; 1,845 - 10-year: 0.92; 1,845 - 20-year: 0.91; 1,845 - B. Realization horizon - Same year: 0.86; 10,885 - 1-year: 0.93; 1,845 - 2-year: 0.97; 10,852 - 5-year: 1.08; 8,377 - C. Minimum Patents - 50 patents: 0.93; 1,845 - 100 patents: 0.84; 7,846 - 200 patents: 0.79; 4,859 - Note: All returns in logs and residualized with respect to NAICS3-Year fixed effects. Aggregation horizon is number of years over which valuations and R&D expenditure are summed. Realization horizon is difference between patent application year and year of R&D expenditure considered. Unless specified, R&D returns measured with a 5-year aggregation horizon, 1-year realization horizon, and 50 minimum patents. ### C.2 The Realization-Expectation Gap - Decomposition of realized R&D return: - ln(z_it · V_it / (ℓ_it · W_t)) = - ln(E[z_it · V_it] / (ℓ_it · W_t)) (Expected R&D Return) - + ln(z_it · V_it / E[z_it · V_it]) (Realization-Expectation Gap). (Equation C.1) - Theory: If firms optimize with rational expectations, the realization-expectation gap should be i.i.d.; it should not be predictable using information in firm’s information set at investment time. - Empirical test: Autoregressive model estimated: - ln R&D Return_it = α_{j(i)t} + β_h · ln R&D Return_{it−h} + ε_it. (Equation C.2) - α_{j(i)t} are industry-year fixed effects. - Under equalized expected R&D returns and rational expectations, expect β_h = 0 if ln R&D Return_{it−h} was in the firm’s information set at decision time. - Results summary: - Estimates in Figure C.1 indicate R&D returns are highly predictable at long horizons. - Autocorrelation coefficient at the 5-year horizon is 0.7 and declines to 0.3 for the 15-year horizon. - The author transforms coefficients using their h-th root to obtain implied annual autocorrelation coefficients; estimated values are consistently above (text truncated in source). ### C.2 Additional table: R&D Return Dispersion Across Industries (Table C.2) - Industry; Standard Deviation; Observations - All industries: 0.93; 1,844 - Life Science: 0.82; 1,630 - IT: 1.07; 4,732 - Manufacturing: 0.83; 5,108 - Other: 0.83; 374 - Note: R&D returns residualized with respect to NAICS3×Year fixed effects. Returns are measured in logs. Industries defined as in Mezzanotti and Simcoe (2023). *Source: References and appendices (Proofs, Data Appendix, Empirical Appendix) from the provided content unit.* ### 0.9 confirming that R&D returns are indeed highly persistent and, thus, predicable. We ### C.3 Model Prediction and Industry Trends ### R&D Return Persistence - Autocorrelation estimates show R&D returns are highly persistent and predictable; hypothesis that variation is driven primarily by stochastic realization vs expectation can be rejected. - Figure evidence (annualized implied AR(1) coefficients plotted with 95% confidence intervals) indicates strong persistence across horizons h = 1…15. - Table C.3 estimates of autocorrelation: - R&D Return (lag) coefficients: 0.697*** (0.020), 0.738*** (0.032), 0.752*** (0.042) across specifications (columns (1)-(3)). - Interaction terms: R&D Return × Year≥1995 = -0.064 (0.040); R&D Return × Year∈{1991−1999} = -0.093* (0.053); R&D Return × Year≥2000 = -0.052 (0.052). - R2 = 0.46 in all reported specifications. - Observations = 7,623. - Notes: All variables in logs. Regressions control for NAICS3×Year fixed effects and standard errors clustered at NAICS6 level. Significance levels: * 10% , ** 5%, *** 1%. ### Model prediction linking R&D allocative efficiency and industry-level performance - Model predicts positive correlation between R&D allocative efficiency and productivity growth (formula: lng_t = lng_F_t + lnΓ_t + lnΞ_t). - Practical concerns for industry-level tests: - Unclear time horizon for R&D translating into productivity (technology development vs deployment). - Formula need not hold at industry level if R&D workers move across industries; misallocation may not show up in industry R&D wages. - Under assumptions ((1 + ∆_{it}) and θ_{it} log-normal, ∆_{it} ⟂ ζ_{it}), derived expressions imply lnΞ_{it} = −1/2 · γ/(1−γ) · σ^2_{i,∆} and industry growth rate lng_{it} expression that can generate strong negative correlation between R&D efficiency and R&D measures unless covariances are large. ### Industry-level empirical tests (10-year differences, OLS) - Focus on total industry R&D expenditure and total industry R&D returns because these aggregate measures are informative about misallocation. - Table C.4 (10-year log-differences over 5-year aggregates; observations = 900; an observation is an industry-year): - Panel A: ∆R&D Expenditure regressions - ∆R&D Efficiency coefficients: -0.751*** (0.139), -0.660*** (0.122), -0.423*** (0.142), -0.294** (0.130) across columns (1)-(4). - Panel B: ∆R&D Return regressions - ∆R&D Efficiency coefficients: -1.553*** (0.214), -1.645*** (0.222), -0.549*** (0.178), -0.595*** (0.187) across columns (1)-(4). - Industry FEs and Year FEs included as noted. - Interpretation: Industries with lower allocative efficiency spend more on R&D and have higher aggregate R&D returns — consistent with waste from R&D misallocation. ### C.4 R&D Returns and the Scale Elasticity ### Equilibrium identity and identification strategy - Equilibrium R&D return identity: (z_{it}·V_{it})/(ℓ_{it}·W_t) = 1/γ_{it} · (1 + ∆_{it}). (Equation (C.3)) - Identification: assume common R&D scale elasticity γ within industry-year cells; recover R&D wedge by residualizing R&D return. This is conservative if assumption holds. - Concern: remaining variation in γ within industry-year cells could cause measurement error in wedges, biasing R&D Allocative Efficiency downward and estimated impact of misallocation upward. ### Empirical approach to test heterogeneity in scale elasticity - Dominant approach: relate R&D tax credits τ_{it} to R&D expenditure via ln(ℓ*_{it}·W_t) = (1/(1−γ_{it})) ln(γ_{it}·θ_{it}) − (γ_{it}/(1−γ_{it})) lnW_t − (1/(1−γ_{it})) ln(1−τ_{it}) − (1/(1−γ_{it})) ln(1−∆_{it}). - For γ_{it}=γ, estimate by regressing measures of (1−τ_{it}) on log R&D expenditure. For the US, use effective R&D tax credit measures from Lucking et al. (2019) linked by headquarter state. - To assess heterogeneity, estimate scale elasticity by deciles of R&D return distribution: - Specification (C.4): ln R&D Expenditure_{it} = α_i + γ_t + Σ_{j=1..10} β_j · {R&D Return_{it} ∈ D_{jt}} · ln(1−τ_{s(i)t}) + ε_{it}. - Implied scale elasticities computed as −(ˆβ_j+1)/ˆβ_j with standard errors via Delta-method. ### Findings - Panel A of Figure C.2: regression coefficients β_j (sensitivity of R&D expenditure to R&D tax credits) are not systematically different across deciles of R&D return. - Panel B of Figure C.2: implied estimates of γ_j centered around 0.5 across deciles. - Conclusion: no evidence that systematic differences in R&D scale elasticity across the R&D return distribution are a primary driver of R&D return dispersion; differences in γ are unlikely to be the main contributor to measured misallocation. ### Online Appendix D — Additional Empirical Results ### D.1 Measurement Robustness — summary of adjustments and effects on return dispersion - Table D.1: Return dispersion with adjustments (standard deviation of log R&D returns; all return measures residualized with respect to NAICS3×Year fixed effects) - Panel A (Acquisitions): - Unadjusted: 0.923 (Observations 11,829) - Adjusted (s= 6.3%): 0.910 (11,829) - Adjusted (s= 8.5%): 0.909 (11,829) - Adjusted (s= 100%): 0.982 (11,829) - Interpretation: adjusting for acquisitions marginally reduces dispersion; adjusting by 8.5% reduces measured dispersion by 1.5%; counting all acquisitions as R&D increases dispersion. - Panel B (Fixed-costs): - Unadjusted: 0.924 (11,807) - Adjusted: 0.937 (11,807) - Interpretation: fixed-cost adjustment increases dispersion marginally; fixed costs not a major source of measured dispersion. - Panel C (Knowledge capital): - R&D Expenditure: 0.925 (11,845) - Knowledge capital: 0.961 (11,845) - Organizational capital: 0.985 (11,845) - Interpretation: using knowledge or organizational capital as denominator yields strictly higher R&D return dispersion. - Outlier patents (Table D.2, Panel A): - Unadjusted valuations: 0.925 (11,845) - Winsorized at top 1%: 0.916 (11,845) - Winsorized at top 5%: 0.892 (11,845) - Interpretation: winsorizing reduces dispersion only modestly (1%–3.6%). - Low value patents (Table D.2, Panel B): - All valuations: 0.925 (11,845) - Valuations > 250k: 0.942 (11,845) - Valuations > 500k: 0.959 (11,842) - Interpretation: excluding low-value patents slightly increases measured dispersion. ### Patent valuation measurement and grant-probability adjustments - Kogan et al. (2017) valuation approach: V = ∆M/(1−π) where ∆M is the change in firm valuation at grant and π is ex-ante grant probability. - Concerns: grant probability differs by patent class and may depend on patent value. - Adjustments: - Class-specific grant probabilities (3.5-year grant share by CPC subclass) used to recalibrate valuations — this marginally increases dispersion (Panel C of Table D.2: Unadjusted 0.925 → Adjusted 0.933). - Value-dependent grant-rate specification 1−π_p = π_0 · V^{−η} and adjusted valuation ˜V_p = (V · (1− ̄π / π_0))^{1/(1−η)}: - η = 0 (Unadjusted): 0.925 (11,845) - η = .05: 0.968 (11,845) - η = .25: 1.163 (11,845) - η = .5: 1.740 (11,845) - η = 1.5: 2.584 (11,845) - η = 2: 1.648 (11,845) - Interpretation: assuming higher-value patents are more likely to be granted can substantially increase measured R&D return dispersion; class-based adjustments do not reduce dispersion. ### D.2 Relationship with Impact-Value Factors (Table D.3) - Regressions of various impact-value factor measures on R&D wedge (R&D wedge measured as patent valuations over R&D expenditure except where noted). All variables in logs; regressions control for NAICS3×Year fixed effects and standard errors clustered at NAICS6. - A. Markup-based measures: - Estimated Markup: Estimate 0.030*** (0.007), R2 = 2.7%, Observations = 10,615. - Profit-implied Markup: Estimate 0.066*** (0.016), R2 = 4.6%, Observations = 11,845. - Interpretation: positive correlation suggests firms with higher R&D returns tend to have larger markups. - B. Patent-based measures: - Citations/∆Sales: Estimate -0.077 (0.054), R2 = 0.3%, Observations = 11,688. - Citations/Valuations: Estimate -0.201*** (0.032), R2 = 3.8%, Observations = 11,688. - Text-Impact/∆Sales: Estimate -0.184*** (0.053), R2 = 1.9%, Observations = 7,481. - Interpretation: patent-based measures show negative correlations with R&D wedges; if these proxy the impact-value factor, they may partly offset misallocation due to R&D wedges. ### D.3 Constructing Estimates for Continuing Firms - Method: construct baseline measure in 1975 and annual changes in the impact of R&D wedges for subsequent years; accumulate changes to assess whether entry and exit contributed to evolution of R&D Allocative Efficiency. (Procedure described up to filtering for firms active in 1975.) *Source: wpiea2025183-source-pdf (selected sections C.3, C.4, Online Appendix D)* ### 1976. For these firms, I estimate of R&D efficiency for both 1975 and 1976. I then take the ### wpiea2025183-source-pdf - 1976. For these firms, I estimate of R&D efficiency for both 1975 and 1976. I then take the ### Continuing firms: construction of time series for Impact of R&D wedges - Definition and procedure: - Let ˆΞt be the baseline estimate for the Impact of R&D wedges for year t. - Let ˆΞt,t−1t be the estimate when using only firms that were active in both t and t−1 with ˆΞt,t−1t−1 being the respective value for t−1. - The time-series for the Impact of R&D wedges for continuing firms ˆΞCt is calculated as: - ˆΞCt = { ˆΞt if t = 1975; ˆΞC t−1 · [ ˆΞt,t−1t / ˆΞt,t−1t−1 ] if t = 1976,...,2014 } (Equation (D.4)) - Implementation: - For continuing firms, estimate R&D efficiency for both 1975 and 1976, take the ratio to get the rate of change, apply it to the original estimate for 1975, and calculate subsequent years accordingly. ### Robustness for aggregate measures (Table D.4) - Table D.4 reports estimates of R&D efficiency for alternative specifications and their implications for welfare. Changes in welfare are in consumption equivalent terms. - Summary of reported specifications and key numeric results (Growth Impact Ξ−1; Welfare 1975–2014, 1975–90, 2000–14; ∆End.; Semi-End.): - A. Fixed Effects - Year: Growth Impact Ξ−1 = -21.3% ; Welfare 1975–2014 = -15.4% ; Welfare 1975–90 = -25.2% ; Welfare 2000–14 = -11.7% ; ∆End. = 5.8% ; Semi-End. = 5.7% - NAICS3×Year: Growth Impact Ξ−1 = -17.9% ; Welfare 1975–2014 = -12.0% ; Welfare 1975–90 = -21.7% ; Welfare 2000–14 = -11.0% ; ∆End. = 5.5% ; Semi-End. = 5.3% - NAICS6×Year: Growth Impact Ξ−1 = -13.5% ; Welfare 1975–2014 = -8.1% ; Welfare 1975–90 = -17.6% ; Welfare 2000–14 = -10.3% ; ∆End. = 5.1% ; Semi-End. = 5.0% - B. Minimum Patents - 50 Patent: Growth Impact Ξ−1 = -17.9% ; Welfare 1975–2014 = -12.0% ; Welfare 1975–90 = -21.7% ; Welfare 2000–14 = -11.0% ; ∆End. = 5.5% ; Semi-End. = 5.3% - 100 Patents: Growth Impact Ξ−1 = -17.3% ; Welfare 1975–2014 = -10.7% ; Welfare 1975–90 = -21.7% ; Welfare 2000–14 = -12.3% ; ∆End. = 6.2% ; Semi-End. = 6.0% - 200 Patents: Growth Impact Ξ−1 = -16.5% ; Welfare 1975–2014 = -9.4% ; Welfare 1975–90 = -21.5% ; Welfare 2000–14 = -13.4% ; ∆End. = 6.8% ; Semi-End. = 6.7% - C. Time Horizon - 5-Year: Growth Impact Ξ−1 = -17.9% ; Welfare 1975–2014 = -12.0% ; Welfare 1975–90 = -21.7% ; Welfare 2000–14 = -11.0% ; ∆End. = 5.5% ; Semi-End. = 5.3% - 10–Year: Growth Impact Ξ−1 = -18.1% ; Welfare 1975–2014 = -16.3% ; Welfare 1975–90 = -19.8% ; Welfare 2000–14 = -4.2% ; ∆End. = 1.9% ; Semi-End. = 1.9% - 20–Year: Growth Impact Ξ−1 = -17.2% ; Welfare 1975–2014 = -17.9% ; Welfare 1975–90 = -18.6% ; Welfare 2000–14 = -0.9% ; ∆End. = 0.4% ; Semi-End. = 0.4% - Notes: - Table reports estimates of R&D efficiency across samples together with their implications for welfare. See text and Appendix for details. ### Unobserved firms and size heterogeneity - Aggregate decomposition: - Let total firm mass be Mt with MUt unobserved and MOt observed. - Overall R&D Allocative Efficiency decomposes as: - Ξt = ΩU t · ΞU t + ΩO t · ΞO t (Equation (D.5)) - Where - ΩX t = ( LX t / Lt ) · [ ∫R M X t ωX it · (1 + ∆it)−1/(1−γ) · di / ∫R M t ωit · (1 + ∆it)−1/(1−γ) · di ] −(1−γ) - and ωX it = θ1/(1−γ) it / ( ∫R M X t θ1/(1−γ) it · di ) for X ∈ {U,O}. - Implications and assumptions: - ΩX t equals the employment (and thus expenditure) share if the distribution of ∆it is equivalent across observed and unobserved groups. - If distributions differ, and assuming a common level of friction, the weight ΩX t is larger than relative employment when variation in R&D wedges is larger. - ΞO t is an unbiased estimator for Ξt if distributions are comparable across unobserved and observed firms. - Bias in evolution of estimated Ξt depends on whether unobserved firms follow similar trends; estimates are exaggerated if unobserved firms are constant or move in the opposite direction. - Selection and size heterogeneity in practice: - Firms in Compustat that patent heavily tend to be larger; unobserved firms are mostly smaller firms with, on average, lower R&D expenditure. - The author splits the sample at the median R&D expenditure within a given year to investigate size heterogeneity. - Empirical findings (from Figure D.3, described in text): - R&D Allocative Efficiency (hence misallocation) is worse among firms with low R&D expenditure. - R&D Allocative Efficiency declines faster for low R&D expenditure firms. - The gap between high and low R&D firms is modest in the early sample but opens up significantly post- (text ends here). *Source: wpiea2025183-source-pdf - 1976. For these firms, I estimate of R&D efficiency for both 1975 and 1976. I then take the (IMF working paper source PDF).* ### 2000. Thus, if we were to put more emphasis on smaller firms, aggregate R&D Allocative ### wpiea2025183-source-pdf - 2000. Thus, if we were to put more emphasis on smaller firms, aggregate R&D Allocative ### Model extensions and key analytical results - Specialization of R&D inputs: - Augmented R&D resource constraint: L_t = [∫_0^1 ℓ_it^(1+ξ) di]^(1/(1+ξ)), with ξ>0 capturing increasing marginal costs of R&D inputs to a given firm. - Firm wages: W_it = W_t · ℓ_it^ξ, with common factor W_t clearing the labor market. - First-order conditions at firm level: ∂z_it/∂ℓ_it · ℓ_it = ℓ*_it · V_it = (1 + ∆_it) · W_t · ℓ_it^ξ. (E.2) - Adjusted scale elasticity: ̃γ ≡ γ/(1+ξ). Frictions are less costly for larger ξ because reallocation benefits fall. - Proposition 4 (growth decomposition under (2),(E.1),(E.2),(5)): - g_t = L_t^γ A_t^φ · [∫_0^1 (θ_it·ζ_it)^(1/(1−̃γ)) di]^(1−̃γ) ≡ Frontier Growth Rate g^F_t · Policy Opportunity Λ_t · R&D Efficiency Ξ_t, with definitions in (E.3). - Unit cost elasticity of R&D identifies ̃γ; calibration of γ preserves quantitative implications when matching unit cost elasticity. - Multiple R&D lines per firm: - Line-level production: z_ij = φ_ij · ℓ_ij^γ. (E.4) - Line-level FOC: γ ℓ_ij^(1−γ) · θ_ij = (1 + ∆_ij) W. (E.5) - Aggregate growth rate recovers familiar form when aggregating product-line heterogeneity: see (E.7)–(E.9). - Abundant resources (elastic labor supply): - L_t = ̄L_t · (W_t/Y_t)^(ξ/(1−γ)). (E.10) - Proposition 5: Frontier growth g^F_t uses frictionless R&D input supply L*_t; R&D efficiency Ξ_t also reflects labor-supply effects as in (E.12). - Supply elasticity affects only the term depending on productivity-weighted average frictions; if average frictions unchanged, effect on growth same as ξ=0. - Dispersion implications under log-normal assumptions: - Proposition 6: If R&D returns, impact-value factors, and R&D productivity are jointly log-normal, and R&D returns and impact-value factors are nonnegative correlated, then: - R&D efficiency declines in dispersion of log-R&D wedges if 1/γ > ξ/(1−γ). - Holding average R&D wedges constant, Impact of R&D wedges declines in dispersion if γ ≥ ξ/(1+2ξ). - Empirical note: Aggregate estimates suggest ξ/(1−γ) ≈ 0.5 and γ = 0.5 (Chetty et al., 2012), satisfying the more stringent constraint. - Free entry and endogenous mass of firms: - Equilibrium wage: W_t^γ = [M_t L_t]^(1−γ) [∫_0^1 θ_it^(1/(1−γ))·(1+∆_it)^(-1/(1−γ)) di]^(1−γ). (E.13) - Entry/value conditions and equilibrium mass M_t derived under two entry-cost specifications; see (E.14) and (E.15). - Proposition 7: Frontier growth g^F_t reflects frictionless entry M^*_t; Ξ_t includes entry effects via M_t/M^*_t as in (E.16)–(E.17). - Private frictions worsen growth additionally through reduced number of innovative firms; entry-effect independent of impact-value factor. ### Mechanisms driving R&D wedges (R&D returns ∆ and impact-value ζ) - Baseline two-period model: - Final good: ln Y_t = ∫_0^1 ln y_jt dj; monopolists with marginal cost ψ/A_jt; competitive fringe with cost λ_jt·(ψ/A_jt); monopolist profits π_jt = Y_t·(1−λ_jt^−1). - Inventor production: z_i = φ_i·ℓ_i^γ; invention improves technology by factor λ_i (A_j1 = λ_i·A_j0). - Growth approximation: g ≈ ∫_0^1 (λ_i − 1)·z_i di (using ln λ_i ≈ λ_i − 1). - Planner problem equivalent to maximizing ∫ z_i·V_i·ζ_i di subject to L = ∫ ℓ_i di with ζ_i ≡ (λ_i − 1)/V_i. Planner and private allocation coincide iff ζ_i constant across firms. - Mechanisms generating dispersion in R&D returns (V_i·z_i / (W·ℓ_i)): - R&D subsidies/taxes τ_i: - V_i·z_i/(W·ℓ_i) = (1/γ)·(1−τ_i); ℓ_i = [V_i·φ_i/( (W/γ)·(1−τ_i) )]^(1/(1−γ)). - Capacity constraints ℓ_i ≤ ̄ℓ_i: - V_i·z_i/(W·ℓ_i) = (1/γ)·(1 + ̃λ_i) and ℓ_i = [V_i·φ_i/( (W/γ)·(1 + ̃λ_i) )]^(1/(1−γ)). - Heterogeneous discount rates R_i: - V_i measured = π_i/R with firm problem scaling by R/R_i; V_i·z_i/(W·ℓ_i) = (1/γ)·(R_i/R). - Adjustment costs φ·W·(ℓ_i − ̄ℓ_i)^2: - V_i·z_i/(W·ℓ_i) = (1/γ)·(1 + 2φ·(ℓ_i − ̄ℓ_i)); firms with high R&D relative to reference have higher returns. - Monopsony power / field-specialized R&D labor: - Labor clearing with fields: L = ∫ ℓ_i · [ (1/N_i) Σ_{j∈N_i} ℓ_j / L ]^ξ di. (F.2) - Field wage W_i = W·[ (1/N_i) Σ_{j∈N_i} ℓ_j / L ]^ξ. (F.3) - Under symmetry, R&D return = (1/γ)·[1 + (1/N_i)·ξ] with ∆_i = (1/N_i)·ξ; more competition in R&D labor lowers R&D returns. - Mechanisms generating dispersion in impact-value factors ζ_i: - Patent protection: monopolist profits realized with probability P_i ⇒ V_i = P_i·π_i/R while growth impact remains λ_i − 1 ⇒ variation in P_i induces ζ_i variation. - Exogenous or endogenous markup differences (ψ_i or λ_i): variation in ψ_i or λ_i yields variation in private value V_i without proportional growth effect, generating ζ_i dispersion. - Product-market frictions: frictions that reduce π_i but not λ_i − 1 create low-profit firms that under-provide innovation (ζ_i variation). - Knowledge externalities: if g = [∫ φ_i·z_i·(ζ_i·V_i) di]^φ · [∫ z_i·(ζ_i·V_i) di], perceived marginal benefits differ from social planner, generating ζ_i heterogeneity. (F.5) ### Measurement error in R&D returns and adjustment procedures - Two main sources: - Uncertainty across R&D projects within a firm (patent-level ex-post variation). - Firm-level uncertainty in R&D outcomes (correlated shocks to firm patent valuations or citations, misreporting). - Bootstrapping procedure for project-level uncertainty: - For each firm and 5-year window with ≥50 patents, draw with replacement alternative portfolios of same size 1000 times; compute log return-gap between alternative and true portfolio; compute within-period SD of return gaps. - Empirical result (Table G.1): - Standard deviation (overall period 1975–2014): 0.059 with 90% CI [0.051,0.069]. - Period subsamples: 1975–1990 SD = 0.056 [0.049,0.066]; 2000–2014 SD = 0.058 [0.051,0.069]. - Adjustment factor: 0.998 overall [0.997,0.998]; 0.997 [0.996,0.998] for 1975–1990; 0.998 [0.997,0.999] for 2000–2014. - Interpretation: average SD ≈ 0.06 -> contributes (0.06/0.93)^2 ≈ 0.4% of variance of R&D returns; small contribution from patent-level uncertainty. - GMM structural approach for firm-level transitory shocks (AR(1) with measurement noise): - Data-generating process: y_it = (1−ρ) μ_i + ρ y_it−1 + ε_it, ε_it iid N(0, σ_ε^2); μ_i ~ N(0, σ_μ^2). Observed with additive noise ̃y_it = y_it + ν_it, ν_it iid N(0, σ_ν^2). (G.1)–(G.2) - Moment covariance relationships m1..m4 defined in Lemma 2 and used to recover {ρ, σ_μ^2, σ_ε^2, σ_ν^2} (Proposition 8). - Empirical estimates (Table G.2) for two R&D-output measures (Valuations/∆Sales and R&D): - Valuations/∆Sales/R&D columns: - ρ = 0.626*** (0.083) and 0.787*** (0.149). - σ_ε^2 = 0.628*** (0.085) and 0.404*** (0.092). - σ_μ^2 = 0.197 (0.140) and −0.086 (0.726). - σ_ν^2 = 0.007 and 0.509*** (0.090). - Observations: 8,014 and 7,653. - Adjustment factor: 0.997 and 0.810. - Interpretation: For baseline R&D returns measure, negligible contribution of transitory shocks (adjustment factor 0.997; estimates imprecise). Using sales-growth measure yields larger transitory component (adjustment factor 0.810; transitory shocks contribute ~22%). - Bils et al. (2021) additive-noise adjustment implementation: - Two-step procedure: 1. Estimate ∆ln R&D Output_it = α_{k(i)} + Σ_{k=1}^{10} β_k · ∆ln R&D Expenditure_it · 1{i∈k} + ε_it, where α_{k(i)} decile fixed effects. 2. Create adjusted ln(R&D Return)_it = ln R&D Return_it + ln ˆβ_{k(i)} + σ_β · ε_it with σ_β^2 defined in (G.5). - Full-sample estimates for β_k (Table G.3) by decile: - Deciles 1–10 estimates: 1.477, 1.112, 1.154, 1.025, 1.088, 0.919, 0.861, 0.834, 0.803, 0.524. - Std. errors (respectively): (0.12), (0.14), (0.09), (0.08), (0.09), (0.10), (0.11), (0.08), (0.08), (0.07). - Implementation results: - Adjusted standard deviation of R&D returns: 0.80 versus unadjusted 0.93 for full sample. - Annual/rolling-window implementation: coefficients estimated over 15-year windows centered where possible; early/late observations use first/last available 15-year window. - Figure and visual notes: - Figure D.3 reports annual estimates for R&D Allocative Efficiency Ξ_{t−1}. Baseline assumes R&D wedges independent from impact-value factors; adjusted values estimate adjustment factor over a 10-year rolling window. Shaded area: 90% confidence interval via bootstrapping. (Figure spans 1975–2015 and shows separate series for All-firms, HighR&D, LowR&D; axis labeled Impact of R&D Wedges %.) *Italic: Source: Extracted from wpiea2025183-source-pdf (sections E–G and Figure D.3/Table G.1–G.3) provided in the content unit.* --- _Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025183-source-pdf.pdf_