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---

### Decline of the labor income share: context and research strategy
- The labor income share in the US economy has been declining in recent decades; the decline has been especially prominent in manufacturing, which represents about a third of GDP.
- Manufacturing in the dataset comprises 21 sectors; the aggregate is composed by 35 sectors. The series reflect 5-year moving averages. Source: 35-sector KLEMS (Jorgenson, 2008).
- Two leading narratives examined:
  - Capital accumulation narrative: central claim that capital deepening or increases in capital productivity shift income from labor to capital; crucial parameter is the elasticity of substitution σ, with capital deepening causing a decline in the labor share only if σ > 1.
  - Market power narrative: central claim that rising firm markups raise profit shares at the expense of labor and capital shares regardless of σ.
- Research objective: investigate roles of capital accumulation and market power in explaining the decline of the labor share in manufacturing via a two-step empirical approach:
  1. Estimate the production function for 21 manufacturing sectors to estimate σ along their time series.
  2. Use estimated production-function parameters to track long-run trend changes in labor shares and decompose changes into sectoral re-weights, capital-labor substitution (factor contributions), and market power effects.

### Theoretical/conceptual backbone (key mechanisms)
- Sector-level cost minimization with CRS CES production function F(·) and factor-efficiency terms Γ_j(t) for j ∈ (K,L).
- CES parameterization: capital weight π ∈ (0,1) and elasticity of substitution σ ∈ [0,+∞):
  - σ → 0 approximates Leontief; σ ≈ 1 yields Cobb-Douglas; σ < 1 implies gross complements; σ > 1 implies gross substitutes.
- Factor shares depend on markups μ(t), factor efficiencies Γ_j(t), factor quantities, π, and σ.
- Profit share under marginal cost = average cost: S_profit(t) = 1 − 1/μ(t).
- Implications:
  - Market power channel: rising markups shift income from labor and capital to profits, reducing both factor shares equally in magnitude; independent of σ.
  - Accumulation channel: capital deepening or capital-augmenting technical progress causes a decline in the labor share only if σ > 1; if σ < 1, labor share can increase.
  - Relative capital-to-labor share ratio Θ(t) = S_K(t)/S_L(t) equals π/(1−π) times [Γ_K/Γ_L · K/L]^{(σ−1)/σ}; markups cancel out in Θ(t), illustrating independence of accumulation versus market power in determining relative factor shares.

### Production function estimation: methodology and data
- Estimation approach:
  - System approach (normalized CRS production function and first-order conditions) extended to include time-varying markups.
  - Factor-augmenting technical progress Γ_jt = Γ_j0 · e^{α_j t} with α_j constant average annual growth rates.
  - Normalization sets C = Γ_j0 = 1; normalized variables divide by their averages; normalization constant ψ expected close to one.
  - System in logs (equations (13a)–(13c)) jointly identifies σ, α_L, α_K, and ψ using data on S_{L t}, S_{K t}, Y_t, L_t, K_t and μ_t.
- Estimation procedure:
  - Non-linear three-stage least squares (NL3SLS) with one-year-lagged instruments in stage 1 to address endogeneity; stage 2 non-linear least squares; stage 3 feasible GLS to account for contemporaneous correlation and heteroskedasticity.
  - Instruments: lagged normalized value added, capital, labor, labor and capital share, markup series, plus a time trend.
  - Non-linear least squares sensitivities: choose solutions with best fit (smallest log determinant of residual covariance matrix) where multiple convergence solutions exist.
- Data and markup assumptions:
  - Dataset: 35-sector US KLEM (Jorgenson, 2008); sectoral coverage at 2-digit SIC; period 1960–2005; includes gross output, intermediate inputs, physical capital, quality-adjusted labor, and rental costs.
  - Value added: P_t Y_t = gross output − intermediate inputs; with CRS 1 = S_{K t} μ_t + S_{L t} μ_t, and S_{K t} constructed as residual given observed S_{L t} and μ_t.
  - Baseline markup series: constant markup = 1 for 1960–1980; linearly growing markup for 1980–2005 reaching 1.15 by 2005 (same series for all sectors). Fixed-markup specifications set μ_t = 1 ∀t.

### Production-function estimation: key empirical results (21 manufacturing sectors)
- Elasticity of substitution σ:
  - Most sectors: σ < 1 under both fixed and time-varying markup specifications.
  - Including time-varying markups reduces downward bias; σ across sectors becomes less dispersed and generally lies within the 0.6−1 interval.
  - Median sector elasticity: 0.86.
  - Sectors close to Cobb-Douglas (σ ≈ 1): printing and publishing; tobacco manufactures.
  - Lumber and wood products: σ̂ = 1.88 (significantly larger than one).
  - Only one sector has σ > 1; that sector represents 2.5 percent of total value added in manufacturing.
- Labor-augmenting technical progress α_L:
  - α_L positive and varies between 0 and 0.10.
  - Interpretation: labor becomes between 0 to 10 percent more efficient each year in these sectors.
  - α_L and α_K are only identified when σ ≠ 1; when σ close to one estimates can be ill-behaved.
  - Inclusion of time-varying markups does not significantly change α_L point estimates for sectors with σ < 1; confidence intervals with fixed and flexible markups almost overlap.
- Capital-augmenting technical progress α_K:
  - No clear overall pattern: many sectors present negative and significant α_K; others close to zero; a few positive.
  - Point estimate of α_K is larger when including time-varying markups; sign changes observed only in a few sectors.
- Net technical progress (α_L − α_K):
  - Net technical progress is labor-augmenting for most sectors.
- Normalization constant ψ:
  - Estimated ψ are all close to one.
- Implication: since capital deepening causes a decline in the labor share only if σ > 1 and all sectors but one present σ ≤ 1 (the one sector accounts for 2.5 percent of manufacturing value added), the capital accumulation narrative cannot explain the overall decline of the labor share in manufacturing.

### Tracking the labor share in manufacturing: accounting framework
- Fundamental labor share for sector i:
  - ˆS^L_it = (1−π_i) μ_it ˆΓ^L_it (L_it / Y_it)^(ˆσ_i−1) ˆσ_i, with ˆΓ^L_it = e^{ˆα_{Li}·t}.
  - In Cobb-Douglas sectors (ˆσ_i = 1): ˆS^L_it = (1−π_i)/μ_it.
- Aggregate fundamental labor share:
  - ˆS^L_t = Σ_{i=1}^N θ_it ˆS^L_it, where θ_it is sector value added share of total manufacturing GDP.
- Decomposition components:
  1. Sectoral re-weight effects (changes in θ_it).
  2. Capital-labor income substitution effects (effective labor contribution; markups fixed).
  3. Market power effects (time-varying markups).

### Tracking results and quantitative accounting (1960–2005)
- Sectoral fit:
  - Fundamental labor shares built from estimated production parameters track observed sectoral labor shares well across sectors.
- Aggregate decomposition findings:
  - Sectoral re-weights (series (1)) played almost no role in the manufacturing labor share decline.
  - Series (1)+(2) (re-weights plus effective labor contribution) shows:
    - A constant fundamental labor share between 1960 and 1970.
    - Slight decline between 1970 and 1980.
    - Positive trend from 1980 to 2005.
    - No level changes between 1960 and 2005.
  - Adding market power (series (1)+(2)+(3)):
    - Time-varying markups begin affecting the series starting in 1980.
    - Rise in markups shifts income from labor to profits, driving labor share downwards.
- Key quantitative figures (1960–2005):
  - Number of manufacturing sectors analyzed: 21.
  - Observed manufacturing labor share decline: 17 percent.
  - Accounting-predicted decline using Barkai markups: 13 percent.
  - Share of observed decline explained by rise in market power (Barkai path): 76 percent.
  - Observed labor share in 2005: 0.62.
  - Accounting-predicted labor share in 2005 (baseline): 0.65.

### Sensitivity analyses: alternative markups and parameter estimates
- Transformation assumption for gross-output to value-added markups: share of intermediate inputs s = 0.5; μ_V = μ_G (1−s) / (1−μ_G s).
- Alternative markup series:
  - Hall (2018) transformed to value added: markups increase from 1.3 in 1980 to 1.7 in 2005.
    - Using Hall instead of Barkai: accounting predicts a much steeper decline of the labor share to about 0.5 in 2005 — i.e., 200 percent of the actual decline.
  - De Loecker et al. (2020) transformed to value added (per Basu (2019)): markups would reach a value of 4.
    - Predicted labor share would be no larger than 0.2 in 2005.
  - Implication: some influential markup estimates imply markup growth paths too large to be compatible with the actual labor share decline.
- Alternative estimated production-technology parameters (fixed-markup estimation):
  - Re-estimate production-technology parameters assuming μ_it = 1 ∀i,t; then build fundamental labor shares using those ˆσ and ˆα^L.
  - Result: series (1)+(2) predicts a slight labor share decline from 0.74 to 0.7 concentrated in the pre-1980 period, with a stable series thereafter.
  - Conclusion: results robust in the sense that capital-labor substitution alone does not account for the labor share decline in manufacturing; accounting sensitive to assumed markup growth path.

### Main conclusions from the analyzed content
- Production-technology estimates imply capital and labor are gross complements in production for most manufacturing sectors: estimated σ between 0.6 and 1 for most sectors.
- Absent any rise in markups, the labor share would have remained stable at the post-war level given estimated technology parameters; effective labor contribution predicts a constant labor share through the second half of the 20th century.
- Assuming a rise in markups similar to Barkai (2020), the rise in market power can explain up to 76 percent of the manufacturing labor share decline between 1960 and 2005.
- Accounting results are sensitive to the assumed markup growth path: some markup estimates (Hall; De Loecker et al.) imply declines larger than observed, motivating further exploration of cross-study differences in market power estimates.

*Source: wpiea2023032-print-pdf - https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023032-print-pdf.pdf*

### 1.  Introduction

### 1.  Introduction

### Decline of the labor income share: context and patterns
- The labor income share in the US economy has been declining in recent decades (Elsby et al.,2013).
- The decline has been especially prominent in manufacturing, which represents about a third of GDP.
- Measurement notes from the source figure: Manufacturing comprises 21 sectors and services 7 sectors. The aggregate is composed by 35 sectors. The series reflect 5-year moving averages. Source: 35-sector KLEMS (Jorgenson, 2008).

### Two leading narratives examined
- Capital accumulation narrative
  - Central claim: the rate of capital accumulation (capital deepening) or an increase in its productivity (e.g., automation and robotization) shifts income from labor to capital (Karabarbounis and Neiman,2014b; Rognlie,2015).
  - Crucial parameter: elasticity of substitution σ≥0. Capital deepening causes a decline of the labor share only if capital and labor are gross substitutes, i.e. σ >1.
- Market power narrative
  - Central claim: a rise in firms' market power allows higher markups above marginal costs, raising profit shares at the expense of labor and capital shares regardless of σ (De Loecker et al.,2020; Philippon,2019).

### Research objective and strategy
- Goal: investigate the roles of the capital accumulation and market power narratives in explaining the stark decline of the labor share in manufacturing.
- Two-step empirical approach:
  1. Estimate the production function for 21 manufacturing sectors to estimate σ along their time series.
  2. Use the estimated production function parameters to track long-run trend changes in labor shares and decompose changes into contributions from factor contributions and market power.

### Estimation methodology and data scope
- Production functions estimated using the system approach over the 1960-2005 period (León-Ledesma et al., 2010). The system approach jointly estimates a normalized (CRS) production function and its first order conditions along the time series.
- Methodology jointly estimates σ and the growth rates of labor and capital technical progress, where technical progress is assumed deterministic and grow at a constant rate for each factor.
- Extension: system specification includes time-varying markups (as in Jiang and León-Ledesma (2018)). Markups follow the path reported by Barkai (2020) and are assumed independent from production technology.

### Key empirical findings (summary)
- Estimated σ:
  - σ is within the 0.6−1 range for all but one of the 21 manufacturing sectors.
  - Evidence of σ ≤ 1 suggests capital deepening cannot explain declining labor shares in manufacturing.
- Decomposition and accounting exercise:
  - Built “fundamental” labor shares for each sector based on estimated production technology parameters; these capture effective labor contribution and markups.
  - Decomposed total labor share changes in manufacturing into:
    - sectoral re-weights (changes in sectors’ size),
    - capital-labor substitution (re-shifting of income between capital and labor due to relative factor efficiency changes and factor use),
    - market power effects (transfers from labor to profits triggered by rising markups).
- Two main quantitative results:
  - Capital-labor substitution effects remained stable between 1960 and 2005, despite most sectors having σ ≠ 1. Abstracting from changes in market power, the labor share in manufacturing would have remained constant in the post-war period.
  - Market power can account for a sizable portion of the labor share decline:
    - Using the reported markups from Barkai (2020), up to 76 percent of the labor share decline in this period can be accounted for.
    - Alternative markup measures (Hall (2018) and De Loecker et al. (2020)) overestimate the actual labor share decline by more than 200 percent.

### Relation to existing literature and contribution
- Places findings within multiple literatures: capital accumulation, market power, automation/robotization, measurement issues, and factor-bias of technical change.
- Contrasts with studies arguing σ >1 for the accumulation narrative; provides sectoral estimates for manufacturing sub-sectors and highlights heterogeneity in production technology across them.
- Aligns with other studies finding σ ≤ 1 in manufacturing (Herrendorf et al., 2015; Oberfield and Raval, 2019).
- Novelty: jointly estimates σ and analyzes capital accumulation and market power narratives with a unified methodology on the same data, tracking timing and heterogeneous decline patterns across sectors while allowing for within-sector factor substitution.

### Conceptual/theoretical backbone (key mechanisms)
- Setting: sector-level cost minimization with CRS production function F(·) and factor-efficiency terms Γj(t) for j ∈ (K,L).
- Production function assumed CES with capital share weight π∈(0,1) and elasticity of substitution σ∈[0,+∞):
  - σ → 0 approximates Leontief; σ close to 1 yields Cobb-Douglas; σ <1 implies gross complements; σ >1 implies gross substitutes.
- Factor shares (capital and labor) depend on markups μ(t) = P(t)/λ(t), factor efficiencies Γj(t), factor quantities, weights π, and σ (equations (5) and (6) in the source).
- Profit share under marginal cost = average cost: S_profit(t) = 1 − 1/μ(t).
- Implications:
  - Market power channel: rising markups shift income from labor and capital to profits, reducing both factor shares equally in magnitude; this mechanism is independent of σ.
  - Accumulation channel: changes in relative factor incomes depend on σ. An increase in capital per worker (capital deepening) or capital-augmenting technical progress causes a decline in the labor share only if σ >1 (gross substitutes). If σ <1 (gross complements), the labor share can increase when capital efficiency rises.
  - The relative capital-to-labor share ratio Θ(t) = S_K(t)/S_L(t) equals π/(1−π) times [Γ_K/Γ_L · K/L]^(σ−1)/σ, illustrating how net bias of technical change, capital deepening, and σ jointly determine shifts between factor shares.
- Note: markups cancel out in Θ(t), illustrating independence of accumulation vs. market power narratives in determining relative factor shares.

*Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023032-print-pdf.pdf*

### 3.  Production Function Estimation

### 3.  Production Function Estimation

### Methodology: system approach and production function
- Uses the system approach (Klump et al., 2007; León-Ledesma et al., 2010) extended to include time-varying markups (as in Jiang and León-Ledesma (2018)).
- Production function: CES
  - Main parameter of interest: σ ∈ [0,+∞).
  - Factor-augmenting technical progress: Γ_jt = Γ_j0 · e^{α_j t} for j ∈ (L, K), with α_j the average annual growth rate of factor j-augmenting technical progress.
- Normalization:
  - Normalization sets C = Γ_j0 = 1 and divides factors and efficiency levels by their averages (geometric for growing variables; arithmetic for factor income shares and time index).
  - Normalized production function (equation (10)) includes a normalization constant ψ (expected value close to one; included to check convergence but has no economic meaning).
- First-order conditions from cost minimization yield equations (11) and (12); combining with the normalized production function and adding multiplicative errors yields the system (13a)-(13c) in logs:
  - (13a) log(Y_t / \bar{Y}) = logψ + (σ/(σ−1)) log[ π ( e^{α_K (t−\bar{t})} K_t/\bar{K} )^{(σ−1)/σ} + (1−π) ( e^{α_L (t−\bar{t})} L_t/\bar{L} )^{(σ−1)/σ} ] + ε_{Yt}
  - (13b) log(S_{K t} μ_t) = log(π) + ((σ−1)/σ) α_K (t−\bar{t}) − ((σ−1)/σ) log( Y_t/\bar{Y} · \bar{K}/K_t ) + ε_{K t}
  - (13c) log(S_{L t} μ_t) = log(1−π) + ((σ−1)/σ) α_L (t−\bar{t}) − ((σ−1)/σ) log( Y_t/\bar{Y} · \bar{L}/L_t ) + ε_{L t}
- Objective: jointly estimate σ, α_L, α_K and ψ using data on S_{L t}, S_{K t}, Y_t, L_t, K_t and μ_t; estimate both time-varying and fixed markups (μ_t = 1 ∀t).

### Estimation procedure
- Estimation method: non-linear three-stage least squares (NL3SLS).
- Estimation details:
  - Parameters estimated freely (no point restrictions), with cross-equation restrictions for efficiency.
  - Addresses endogeneity/transmission bias (Ackerberg et al., 2015) via NL3SLS instrumenting right-hand side variables with their one-year lagged values in stage 1.
  - Stage 2: non-linear least squares with projected covariates equation-by-equation.
  - Stage 3: feasible generalized least squares to account for contemporaneous correlation and heteroskedasticity (Zellner and Theil, 1962).
- Instruments (first-stage) follow Fair (1970) and system literature:
  - logarithm of lagged normalized value added, capital, labor, labor and capital share, the markup series, plus a time trend.
- Practical notes:
  - Non-linear least squares is highly sensitive to starting values; for sectors with multiple convergence solutions, the set with the best fit (smallest log determinant of the estimated residual covariance matrix) is chosen.
  - Estimates are not necessarily consistent with a balanced growth path.

### Data and markup assumptions
- Dataset: 35 sector US KLEM dataset from Jorgenson (2008); sectoral coverage at 2-digit SIC; period 1960–2005; includes price and quantity of gross output, physical capital and quality-adjusted labor, and their rental cost.
- Value added construction: real value added P_t Y_t = gross output − intermediate inputs; with CRS production function 1 = S_{K t} μ_t + S_{L t} μ_t, and S_{K t} constructed as a residual given observed S_{L t} and μ_t.
- Markup series assumptions (based on De Loecker et al. (2020), Hall (2018), Barkai (2020)):
  - Constant markup = 1 (perfect competition) between 1960 and 1980.
  - Linearly growing markup for 1980–2005 reaching 1.15 by 2005 (same series for all sectors).
  - For fixed-markup specifications, μ_t = 1 ∀ sectors and years.

### Key estimation results (manufacturing, 21 sectors)
- Elasticity of substitution σ:
  - Most sectors: σ < 1 whether using fixed or time-varying markups.
  - Including time-varying markups reduces downward bias; σ across sectors becomes less spread and generally lies within the 0.6−1 interval.
  - Median sector elasticity: 0.86.
  - Sectors close to Cobb-Douglas (σ ≈ 1): printing and publishing; tobacco manufactures.
  - Lumber and wood products: σ̂ = 1.88 (significantly larger than one).
  - Only one sector has σ > 1; that sector represents 2.5 percent of total value added in manufacturing.
- Labor-augmenting technical progress α_L:
  - α_L positive and varies between 0 and 0.10.
  - Interpretation: labor becomes between 0 to 10 percent more efficient each year in these sectors.
  - α_L and α_K can only be identified when σ ≠ 1; when σ is close to one (e.g., tobacco manufactures, apparel and other textile products) estimates of α_L and α_K can be implausible/ill-behaved.
  - Inclusion of time-varying markups does not significantly change α_L point estimates for sectors with σ < 1; confidence intervals with fixed and flexible markups almost overlap.
- Capital-augmenting technical progress α_K:
  - No clear overall pattern in magnitude.
  - Many sectors present negative and significant α_K; others close to zero; a few positive.
  - Negative α_K estimates are not uncommon in system approach literature.
  - Point estimate of α_K is larger when including time-varying markups; change in sign (negative to positive) observed only in a few sectors in this study.
- Net technical progress (α_L − α_K):
  - Net technical progress is labor-augmenting for most sectors (magnitude presented in Appendix Figure 9).
- Normalization constant ψ:
  - Estimated ψ are all close to one (Appendix Figure 10).

### Implication / takeaways
- Section 2 (theoretical result) indicates capital deepening only causes a decline in the labor share if σ > 1.
- Empirical estimates: all sectors but one present σ ≤ 1; the one sector with σ > 1 accounts for 2.5 percent of total manufacturing value added.
- Therefore, the capital accumulation narrative cannot explain the overall decline of the labor share in manufacturing.

*Source: wpiea2023032-print-pdf - 3.  Production Function Estimation (https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023032-print-pdf.pdf)*

### 6.  Tracking the Labor Share in Manufacturing

### 6.  Tracking the Labor Share in Manufacturing

### Accounting framework and definitions
- Fundamental labor share for sector i:
  - ˆS^L_it = (1−π_i) μ_it ˆΓ^L_it (L_it / Y_it)^(ˆσ_i−1) ˆσ_i, with ˆΓ^L_it = e^{ˆα_{Li}·t}. (Equation (14))
  - Hat denotes estimated production parameters from prior estimation.
- Interpretation:
  - ˆS^L_it captures labor income share given sector market power (1/μ_it) and labor’s contribution in production.
  - In Cobb-Douglas sectors (ˆσ_i = 1): ˆS^L_it = (1−π_i)/μ_it.
  - For ˆσ_i ≠ 1 the term ˆΓ^L_it L_it / Y_it can either raise or lower ˆS^L_it depending on whether the effective labor contribution grows faster or slower than Y_it.

### Sector-level results
- Sample and construction:
  - Analysis covers 21 manufacturing sectors.
  - For sectors with ˆσ within the 0.97−1.03 interval, ˆσ is replaced with 1.
  - For ˆα^L_i, use point estimate when positive and statistically significant; otherwise replace with zero.
- Fit:
  - Fundamental labor shares track observed sectoral labor shares well across sectors (examples: decline in "food and kindred"; increase in "non-electrical machinery"; good fit even in lumber and wood products where capital and labor are gross complements).
- Implication:
  - The sectoral goodness of fit indicates the estimated production technology parameters capture sector labor-share dynamics.

### Aggregate decomposition and main accounting exercise
- Aggregate fundamental labor share:
  - ˆS^L_t = Σ_{i=1}^N θ_it ˆS^L_it, where θ_it is sector value added share of total manufacturing GDP. (Equation (15))
  - θ_it changes over time due to structural transformation.
- Decomposition components (as presented in Figure 6):
  - (1) Sectoral re-weight effects (change in labor share due to changes in sectors' weights).
  - (2) Capital-labor income substitution effects (adds contribution of effective labor in value added; markups fixed).
  - (3) Market power effects (adds time-varying markups).
- Findings from decomposition (1960–2005):
  - Sectoral re-weights (series (1)) played almost no role in explaining the decline of the labor share in manufacturing.
  - Series (1)+(2) (re-weights plus effective labor contribution) shows:
    - A constant fundamental labor share between 1960 and 1970.
    - Slight decline between 1970 and 1980.
    - Positive trend from 1980 to 2005.
    - No level changes between 1960 and 2005 — suggesting the weight of effective labor in manufacturing GDP did not decline over this period.
  - Adding market power (series (1)+(2)+(3)):
    - Time-varying markups begin affecting the series starting in 1980.
    - Rise in markups shifts income from labor to profits, driving labor share downwards.
    - Quantitative comparison (1960–2005):
      - Observed labor share fell by 17 percent in the data.
      - Accounting exercise predicts a decline of 13 percent.
      - Rise in market power accounts for 76 percent of the decline.
      - Observed labor share in 2005: 0.62.
      - Accounting-predicted labor share in 2005: 0.65.
    - Conclusion: The rise in market power (as in Barkai (2020)) can account for most of the observed decline in manufacturing labor share.

### Sensitivity analyses

- Overview:
  - Two exercises: (1) alternative markup series; (2) alternative estimated production-technology parameters (fixed markups in estimation).
  - Note: To transform gross-output markups into value-added markups, assume share of intermediate inputs s = 0.5 (standard in literature). Formula: μ_V = μ_G (1−s) / (1−μ_G s).

- 6.1.1 Alternative markup series
  - Hall (2018) markups (originally on gross output) transformed to value added:
    - Hall’s markups increase from 1.3 in 1980 to 1.7 in 2005.
  - Results using Hall instead of Barkai:
    - Accounting predicts a much steeper decline of the labor share to about 0.5 in 2005 — i.e., 200 percent of the actual decline.
  - Results using De Loecker et al. (2020) transformed to value added (per Basu (2019)):
    - Markups would reach a value of 4.
    - Predicted labor share would be no larger than 0.2 in 2005.
  - Implication:
    - While there is consensus that markups grew, some influential markup estimates (Hall; De Loecker et al.) imply markup growth paths that are too large to be compatible with the actual labor share decline.

- 6.1.2 Alternative estimated parameters of the production function
  - Exercise:
    - Re-estimate production-technology parameters using fixed markups (μ_it = 1 ∀i,t) in the econometric specification, then build fundamental labor shares using those ˆσ and ˆα^L.
  - Resulting aggregate behavior:
    - Sectoral tracking changes slightly but not substantively.
    - Series (1)+(2) predicts a slight labor share decline from 0.74 to 0.7 concentrated in the pre-1980 period, with a stable series thereafter.
    - Compared to baseline, differences in σ and α^L across sectors modify aggregate behavior, but the core finding remains: capital-labor substitution (factor contribution) alone does not account for the labor share decline in manufacturing.
  - Note: By assuming no markup variation in this exercise, μ_it = 1 for all i,t.

### Key quantitative and technical findings
- Number of manufacturing sectors analyzed: 21.
- Observed manufacturing labor share decline (1960–2005): 17 percent.
- Accounting-predicted decline using Barkai markups (1960–2005): 13 percent.
- Share of observed decline explained by rise in market power (Barkai path): 76 percent.
- Observed labor share in 2005: 0.62.
- Accounting-predicted labor share in 2005 (baseline): 0.65.
- Hall (1980→2005) markups (value-added): 1.3 → 1.7.
- De Loecker et al. transformed markups (value-added) per Basu (2019): markups would reach 4 → predicted labor share ≤ 0.2 in 2005.
- Assumed share of intermediate inputs for transformation: 0.5.
- Elasticity of substitution σ:
  - Estimated σ lies between 0.6 and 1 for most sectors.
  - For sectors with σ within 0.97−1.03, σ is set to 1 in the accounting exercise.

### Conclusions (section summary)
- Production-technology estimates imply capital and labor are gross complements in production; σ between 0.6 and 1 for most sectors eliminates capital accumulation as the main driver of falling labor shares.
- Absent any rise in markups, the labor share would have remained stable at the post-war level given estimated technology parameters; effective labor contribution predicts a constant labor share through the second half of the 20th century.
- Assuming a rise in markups similar to Barkai (2020), the rise in market power can explain up to 76 percent of the manufacturing labor share decline.
- The accounting results are sensitive to the assumed markup growth path; some markup estimates imply declines larger than observed, motivating further exploration of cross-study differences in market power estimates.

*Source: wpiea2023032-print-pdf - 6.  Tracking the Labor Share in Manufacturing*

### References

### References

### Major themes in the cited literature
- Labor- and capital-augmenting technical change and elasticity of substitution
  - Acemoglu, Daron, “Labor- and Capital-Augmenting Technical Change,” Journal of the European Economic Association, 2003, 1(1), 1–37.
  - Antras, Pol, “Is the US aggregate production function Cobb-Douglas? New estimates of the elasticity of substitution,” Contributions in Macroeconomics, 2004, 4(1).
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- Decline in labor share, automation, and market power
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### Appendix figures and tables (as reported)
- Figure 9: Net technical progress ˆα_L − ˆα_K
  - Note: Each point corresponds to net technical progress ˆα_L − ˆα_K in each sector. Red estimates include time-varying markups in the specification, while blue estimates assume them constant.

- Figure 10: Normalization constant ψ
  - Note: Each point estimate corresponds to the normalization constant ψ in each sector using the system approach. Red estimates include time-varying markups in the specification, while blue estimates assume them constant.

- Table 2: System estimation with time-varying markups (NL3SLS)
  - Caption notes:
    - This table shows the estimation of all parameters in the system and its respective standard errors using NL3SLS including time-varying markups.
    - Observations: 44 for each sector.
    - Standard errors in parentheses.
    - Sector refers to the sector ID (see Table 1 for the name of each sector).
    - ldrcov refers to log of the determinant of the residual covariance matrix.
  - Example parameter rows (sector ID and columns shown in the source): ID σα α_L α_K ψ ldrcov
    - 1 0.76 0.036 -0.016 1.003 -18.084 (0.015)(0.002)(0.003)(0.005)
    - 2 0.986 -0.348 0.187 1.103 -17.441 (0.006)(0.087)(0.055)(0.020)
    - 3 0.947 0.023 0.018 1.010 -21.213 (0.009)(0.005)(0.015)(0.004)
    - ... (full sector-by-sector estimates reported in the table in the source)

- Table 3: System estimation with fixed markups (NL3SLS)
  - Caption notes:
    - This table shows the estimation of all parameters in the system and its respective standard errors using NL3SLS assuming fixed markups.
    - Observations: 44 for each sector.
    - Standard errors in parentheses.
    - Sector refers to the sector ID (see Table 1 for the name of each sector).
    - ldrcov refers to log of the determinant of the residual covariance matrix.
  - Example parameter rows (sector ID and columns shown in the source): ID σα α_L α_K ψ ldrcov
    - 1 0.668 0.039 -0.021 1.003 -17.375 (0.018)(0.002)(0.003)(0.005)
    - 2 0.994 -0.572 0.327 1.099 -18.565 (0.004)(0.215)(0.136)(0.019)
    - 3 0.926 0.052 -0.073 1.007 -20.215 (0.018)(0.006)(0.018)(0.006)
    - ... (full sector-by-sector estimates reported in the table in the source)

*Content compiled from the "References" and Appendix (Figures 9–10; Tables 2–3) sections of the source PDF.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023032-print-pdf.pdf_