## wp18179

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### 3.1 Firm size distribution (data & empirical findings)
- Data sources (2014 reference year):
  - ENE firm survey: sample size 19,204 firms; coverage: formal firms with annual sales larger than 76,000 Peruvian soles; reports number of workers disaggregated between salaried and non-salaried workers.
  - SUNAT administrative dataset: sample size 313,810 firms with a unique taxpayer number; coverage: near universe of firms affected by the profit sharing regulation; excludes the public sector and micro firms with sales less than 76,000 soles; employment variable mixes salaried and some non-salaried workers.
  - Additional figures: Total registered firms in 2014 according to SUNAT: 1,647,529; Number of firms reporting annual sales of less than 76,000 soles: 1,309,399.
- Key empirical patterns:
  - Discontinuity around the 20-employee cut-off (N = 20):
    - More firms than expected just under 20 employees; fewer firms than expected with 21 employees or more.
    - Discontinuity sharper in ENE (which separates salaried vs. non-salaried employment).
    - SUNAT’s mixed employment measure produces a bulge in firms with fewer than 20 employees rather than a single-point drop at 20.
  - On a log-log scale of firms by employment brackets:
    - Firm size distribution follows a power law overall.
    - Distribution flattens approaching the 20-employee cut-off (consistent with bunching).
    - After 20 employees, an intercept shift occurs and the distribution resumes an approximate power law for larger firms.
- Interpretation:
  - Bunching below 20 salaried employees is consistent with mandatory profit sharing and related regulations creating hiring disincentives for firms near the threshold.
  - Measurement differences across datasets explain differences in the sharpness/form of the discontinuity.
  - Patterns imply the profit sharing requirement materially affects hiring behavior near the regulatory threshold, with implications for labor allocation and aggregate productivity.

### 3.2 Salaried and non-salaried employment (composition, productivity, model motivation)
- Institutional fact and empirical regularities:
  - Profit-sharing legal requirement applies to all firms with more than 20 salaried employees (threshold N = 20), incentivizing expansion via informal, non-salaried workers.
  - ENE excludes micro firms, explaining slope differences in the 0–5 employee range compared to SUNAT.
  - ENE (2014): firm-size measured as salaried employment (annual average of monthly employment, rounded).
    - Firms with < 20 salaried workers tend not to hire non-salaried workers.
    - At N = 20, median non-salaried employment jumps discontinuously from zero to about one-fifth of firm employment.
    - For firms with > 20 salaried employees, non-salaried employment decreases as salaried employment rises.
- Productivity and contractual differences:
  - Salaried workers: subject to schedules/directives enabling production organization, monitoring, quality control, inventory control, task coordination.
  - Non-salaried contracts (fee-based, consultants, family members): less subordination, harder to monitor, anecdotal higher turnover, less productive contributions, narrower task scope.
  - Paper interprets productivity gap as arising from position nature rather than worker characteristics.
- Sales, profits, wages around N = 20 (SUNAT, 2014; variables in thousands of Peruvian soles; employment = annual average monthly employment, rounded):
  - Clear bunching in sales per worker and profits per worker around 20: firms with < 20 salaried employees exhibit much larger sales and profits per worker than expected.
  - No similar discontinuity in wages per worker across the threshold.
    - Implies hours worked or worker quality unlikely to fully explain sales/profits discontinuity.
    - Flatter wage profile for firms > 20 salaried employees consistent with part of total compensation occurring via shared profits.
  - Potential explanations considered:
    - Measurement: SUNAT salaried employment excludes most informal workers, potentially inflating sales/profits per salaried worker; missing workers unlikely to be the non-salaried workers showing discontinuity at N = 20.
    - Firm splitting: possible splitting into sub-units < 20 salaried employees; no direct evidence due to absent owner identifiers.
    - Hours/worker quality: absence of wage discontinuity argues against sole explanation.
    - Higher TFP/capital intensity in 10–20 salaried employees range could contribute but datasets lack capital measures; observed near 40 percent difference in sales per worker unlikely to be fully explained by manager productivity or capital alone.
- Theoretical motivation:
  - Model extension (based on Garicano et al., 2016 and Lucas (1978)) allows firms to choose salaried (`l`) and non-salaried (`n`) labor.
  - Key modeling assumptions:
    - Output of a non-salaried worker = fraction φ of salaried worker output; assume φ(1 + τ_w) < 1 so non-salaried workers are not chosen absent size-dependent policy.
    - Workers identical and paid same wage w regardless of managerial productivity; supported by no observed wage-bill discontinuity and appendix evidence.
    - Size-dependent policy features:
      - Profit tax τ_π applies only to firms hiring more than threshold N (= 20) salaried workers.
      - Salaried labor tax τ_p and pension/social cost τ_w enter through T = (1 + τ_w + τ_p) ̄k with Tw = (1 + τ_w + τ_p) c(τπ, τp) in model derivations.
  - Managerial-productivity thresholds partition behavior (definitions preserved in modeling section):
    - α_c = w(1 + τ_w)/g′(N).
    - α_a = w/(φ g′(N)) = α_c/(φ(1 + τ_w)).
    - α_r implicitly defined by an indifference condition equating regulated and unregulated profit outcomes.
  - Labor demand schedule as function of α (piecewise forms preserved in source) and market clearing integral condition also specified.

### 5.1 Firm splitting (assumptions, estimation, counterfactuals)
- Firm splitting assumptions:
  - Assumption 1: Each firm has probability δ ∈ (0,1) of splitting; δ independent of firm characteristics.
  - Assumption 2: If splitting occurs, firm divides into smallest possible number of sub-firms (all sub-firms < N salaried employees) due to fixed costs.
  - Assumption 3: Splits yield equal-sized units; managers keep same production function/span of control; integer number of units required.
- Partially rigid wages:
  - Wage adjustment: w = ρ w* + (1−ρ) w0.
    - w0: wage before regulatory change; w*: target full-employment wage; ρ calibrates partial rigidity.
  - If ρ < 1, unemployment u > 0 arises; labor market clearing and sorting conditions modified accordingly.
- Estimation and calibration inputs:
  - Calibration: set τ_w = 0.13 (cost of pension contributions as share of wage in Peru).
  - Estimation steps:
    1. Estimate θ and φ using production function.
    2. Estimate γ, `r, T and `max using firm size distribution.
    3. Use model restrictions to find remaining parameters.
  - Outliers: sample a little over 19,200 observations; 170 flagged/removed (local discontinuities criterion).
  - Production function specification: ln(y_j) = θ ln(`_j + φ n_j) + ln(α_j); regression ln(sales_j) = β0 + β1 ln(salaried emp_j) + β2 ln(non-salaried emp_j) + γ X_j + ε_j with controls including physical capital, education indexes, firm type, 4-digit industry FE, and 3rd-degree polynomial on manager education and capital purchases.
  - Empirical estimates used in counterfactuals: θ = 0.7; φ = 0.8.
- Firm size distribution estimation (measurement error and MLE):
  - Assume ψ(α) = C α^{−β}; define γ = β(1−θ) + θ.
  - Observed firm size has multiplicative log-normal measurement error: `(α, ε) = `(α) e^{ε}, ε ∼ N(0, σ).
  - Constrained maximum likelihood with constraint that firms hiring `r salaried employees are not worse off than hiring N salaried plus non-salaried workers; use τ^min_π = 0.05 in constraint.
  - Maximum likelihood estimates (Table 3):
    - γ = 1.99 [1.98, 2.00]
    - σ = 0.182 [0.180, 0.184]
    - `r = 35.5 [35.1, 35.9]
    - T = 1.143 [1.142, 1.144]
    - `max = 33451 [29255, 37647]
- Model with firm splitting: estimation and results
  - Splitting redistributes mass of large firms to smaller sizes; fraction δ split implies ζ_s(`) = (1−δ) ζ(`) for ` > `r and bulging density in (N/2, N].
  - Combinatorial splitting rule yields ζ_s(`) expressions (source provides full piecewise forms).
  - GMM estimator used to fit (γ, σ, `r, T, δ) numerically; initialization from non-splitting ML solution.
  - GMM estimates (Table A.3):
    - γ = 1.98 [1.95, 2.01]
    - σ = 0.15 [0.11, 0.20]
    - `r = 36.0 [35.3, 42.5]
    - T = 1.143 [1.140, 1.147]
    - δ = 0.12 [0.04, 0.13]
  - Note: `max not identified in GMM; `max set to 23179 for numerical scaling in this estimation exercise.

### Counterfactuals, aggregate impacts, and distributional effects
- Counterfactual scenarios considered:
  1. Remove size-dependent regulations entirely (threshold → ∞).
  2. Remove adjustment margin via non-salaried labor (firms cannot use non-salaried workers to circumvent regulations).
  3. Vary the regulatory threshold N between 2 and 100 (and conceptually 0 or very large values).
- Who is most affected:
  - Medium-sized firms near 20 employees: about 3 percent of firms and employ about 10 percent of formal workers.
  - Managers with ability α ∈ (α_a, α_r) experience an output drop of almost 20% when regulations are introduced.
  - Small firms (< 20 employees): profits increase by about 2% when regulations are introduced.
  - Large firms: after-tax profits decrease by a little over 5% due to profit sharing.
- Aggregate impacts of introducing size-dependent regulations (Table 4, percent changes):
  - Baseline Model (no firm splitting):
    - Flexible wages: Aggregate Wages −1.034 ; Aggregate Profits −2.884 ; Aggregate Output −0.096 ; Share of Managers 3.504
    - Rigid Wages: Aggregate Wages −0.644 ; Aggregate Profits −3.774 ; Aggregate Output −1.006 ; Share of Managers 3.501
  - With firm splitting:
    - Flexible wages: Aggregate Wages −0.775 ; Aggregate Profits −2.928 ; Aggregate Output −0.084 ; Share of Managers 2.576
    - Rigid wages: Aggregate Wages −0.384 ; Aggregate Profits −3.809 ; Aggregate Output −0.995 ; Share of Managers 2.573
- Interpretation of aggregate effects:
  - Aggregate wages fall because reduced labor demand pushes wages down; decline larger with fully flexible wages and smaller with downward wage rigidity and firm splitting.
    - Example magnitudes: wage declines of −1.034 percent (flexible, no splitting) vs −0.384 percent (rigid, with splitting).
  - Aggregate profits decline by about 3 to 4 percent as losses by larger firms outweigh gains by smaller firms.
  - Aggregate output effect depends on wage flexibility:
    - Flexible wages: output decreases marginally (about −0.1 percent).
    - Partially rigid wages: output declines more substantially (about −1.0 percent).
  - Composition shifts: share of managers increases (e.g., 3.504 in baseline flexible case), reflecting increased entrepreneurship attractiveness for some agents.
- Unemployment and wage rigidity:
  - Wage rigidity calibration: ρ ≈ 0.49 (wage adjusts about half way to full-employment wage).
  - Calibration result: unemployment increases by 1.3 percent when regulations are implemented and wages are not fully flexible.
  - For context: unemployment rate in Peru in 2014 was 5.5 percent, implying size-dependent regulations may account for about one fourth of structural unemployment.
- Removing non-salaried labor counterfactual:
  - Directly affected firms are those in “active, unregulated and constrained” regions.
  - Two opposing effects on labor demand for these firms nearly cancel in aggregate, producing little change in aggregate labor demand.
  - Local firm effects: some firms produce up to 15% less or 25% more if non-salaried labor is removed.
  - Unemployment change: increases by only 0.005 percent in the fully flexible case.
  - Interpretation: removal of non-salaried workers causes large effects on few firms but small aggregate change; presence of informal labor does not mitigate overall effects of size-dependent policies.
- Effects of changing the threshold N:
  - Example: raising threshold from 20 to 50 salaried employees → profits increase by about 0.5 percent and average firm size increases by about 0.2 percent.
  - Unemployment decreases with higher threshold (no unemployment when wages fully flexible).
  - Share of informal, non-salaried workers increases with threshold because constrained firms become bigger and substitute salaried for non-salaried workers.
  - Output response non-monotonic in baseline flexible case: for small thresholds “lower-misallocation” effect dominates (output rises); for large thresholds fewer firms affected, output increases; intermediate thresholds can lower output relative to current levels.
  - With firm splitting and rigid wages: output monotonically increases with the threshold.
  - Shape parameter γ crucially affects responses (controls mass above/below threshold); range of affected firms increases with threshold value (α_r − α_c grows with N).
  - Informality non-monotonic at extreme thresholds; model-calibrated match: about 12 percent of Peruvian workforce is non-salaried, 0.3 percent hired by firms with 20 salaried employees; model equivalent share = 0.29.
- Distributional conclusions and policy implications:
  - As designed, size-dependent policies (profit sharing and other firm-size thresholds) have an overall negative effect on aggregate output.
  - Distributional winners: small firms and entrants benefit from lower wages and expand employment/profits.
  - Distributional losers: larger firms lose profitability; workers suffer lower wages and higher unemployment.
  - Size-dependent policies increase labor informality as firms hire informal workers to avoid regulation.
  - If objective is raising workers’ incomes, results suggest these policies make labor worse off via lower wages, higher informality, and higher unemployment.
  - Policy approaches:
    - First-best: remove distortionary size-dependent regulations.
    - Second-best: adjust threshold value while accounting for non-monotonic effects and firm-size distribution sensitivity.
    - Consider enforcement and firm-splitting responses when designing thresholds/enforcement strategies.
- Suggested future research:
  - Obtain direct evidence of firm splitting by linking legally separate firms to same owner to assess costs/trade-offs/prevalence.
  - Study other size-dependent policies (simplified tax regimes, minimum thresholds) common in developing economies to improve tax collection and formalization design.

*Source: wp18179 (IMF Working Paper content units: sections 3.1, 3.2, 5.1 and related estimation/counterfactual results as provided).*

### 3.1  Firm size distribution  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  11

### 3.1  Firm size distribution

### Data and measurement
- ENE firm survey (2014 reference year)
  - Sample size: 19,204 firms.
  - Coverage: formal firms with annual sales larger than 76,000 Peruvian soles (roughly $ US 20,000).
  - Key feature: reports number of workers disaggregated between salaried and non-salaried workers.
- SUNAT administrative dataset (2014 reference year)
  - Sample size: 313,810 firms with a unique taxpayer number.
  - Coverage: near universe of firms affected by the profit sharing regulation; excludes the public sector and micro firms with sales less than 76,000 soles.
  - Limitation: does not provide information on non-salaried employment by firm; its employment variable includes both salaried workers and some types of non-salaried workers such as trainees.
- Additional contextual figures from the source:
  - Total registered firms in 2014 according to SUNAT: 1,647,529.
  - Number of firms reporting annual sales of less than 76,000 soles: 1,309,399.

### Empirical findings on firm size distribution
- Both ENE and SUNAT datasets show a discontinuity around the 20-employee cut-off:
  - More firms than expected with just under 20 employees.
  - Fewer firms than expected with 21 employees or more.
- The discontinuity is sharper in the ENE survey where salaried vs. non-salaried employment is observed.
- In the SUNAT data the employment measure includes some non-salaried workers (e.g., trainees), which produces a bulge in the number of firms with fewer than 20 employees rather than a clear single-point discontinuity at 20.
- On a log-log scale of firms by employment brackets, the firm size distribution:
  - Follows a power law overall, consistent with prior findings for other countries.
  - Appears to flatten as it approaches the 20 employee cut-off, consistent with firm bunching below that cut-off.
  - Exhibits an intercept shift after 20 employees, after which the distribution resumes an approximate power law for larger firms.

### Interpretation and implications
- The observed bunching below 20 employees is consistent with the hypothesis that the mandatory profit sharing and related regulations that apply to firms with more than 20 salaried employees create hiring disincentives for firms close to the threshold.
- Measurement differences across datasets (ability to separate salaried and non-salaried workers in ENE vs. mixed employment counts in SUNAT) explain differences in the sharpness and form of the discontinuity.
- The distributional patterns suggest the profit sharing requirement may materially affect hiring behavior, particularly for firms near the regulatory threshold, with potential implications for allocation of labor and aggregate productivity.

*Source: wp18179 - 3.1  Firm size distribution (excerpt).*

### 3.2  Salaried and non-salaried employment

### 3.2  Salaried and non-salaried employment

### Key empirical findings on employment composition
- The profit-sharing legal requirement applies to all firms with more than 20 salaried employees (threshold N = 20), creating incentives for firms to expand using informal, non-salaried workers.
- Micro firms are excluded from the ENE, explaining the different slope of the firm size distribution in the 0-5 employee range compared to SUNAT data.
- Figure (2) (ENE, year 2014) shows:
  - Firm size on the horizontal axis is measured as salaried employment (annual average of monthly employment, rounded to the nearest integer).
  - The median and interquartile range of non-salaried employment are plotted by firm sized in consecutive four salaried worker bins.
  - Firms with fewer than 20 salaried workers tend not to hire non-salaried workers.
  - At the 20 salaried worker cutoff, the median number of non-salaried workers increases discontinuously from zero to about one-fifth of firm employment.
  - As firms hire more salaried employees beyond 20, the number of non-salaried workers decreases.
- Explanations for non-salaried hiring patterns:
  - Firms bunch at the N = 20 threshold and choose to expand their workforce using informal labor to avoid profit-sharing obligations.
  - Remaining non-salaried hires in larger firms may be outside consultants, lawyers, accountants, or due to frictions in adjusting employment composition.

### Productivity and contractual differences between salaried and non-salaried workers
- Salaried workers are subject to specific schedules and directives allowing firms to: organize production, monitor quality, control inventories, and coordinate tasks.
- Non-salaried contracts (fee-based, service provision, consultants, family members) lack the same level of subordination, leading to:
  - More difficult monitoring and control.
  - Anecdotally higher turnover.
  - Less productive contributions and a more limited range of tasks performed.
- The paper interprets the productivity difference as arising from the nature of the position rather than underlying worker characteristics (cites Busso et al., 2012).

### Empirical patterns in sales, profits and wages around N = 20
- Data sources and units:
  - Annual sales: value added tax returns or simplified tax regime for smaller firms.
  - Profits: corporate tax returns, measured as profits before tax, dividends and profit sharing distributions.
  - All SUNAT variables expressed in thousands of Peruvian soles; employment defined as the annual average of monthly employment, rounded to the nearest integer. Data refer to year 2014.
- Figure (3) findings:
  - Clear bunching in both sales per worker and profits per worker around the 20 employee cut-off: firms with under 20 salaried employees exhibit much larger sales and profits per worker than expected given workforce size.
  - No similar discontinuity is observed in wages per worker across the threshold (bottom left panel of Figure (3)).
    - This suggests changes in hours worked or worker quality are unlikely to fully explain the sales/profits discontinuity.
    - A flatter wage profile for firms with more than 20 salaried employees is consistent with part of total compensation taking place via shared profits for larger firms.
- Potential explanations for the discontinuities in sales and profits per worker:
  - Measurement: SUNAT salaried employment excludes most informal workers, potentially underestimating total firm employment and inflating measured sales and profits per salaried worker. However, missing workers are unlikely to be the non-salaried workers that show a discontinuity at N = 20 (per Figure (2)).
  - Firm splitting: firms may split into sub-units of less than 20 salaried employees with separate legal standing under the same management to circumvent the profit sharing rule. The authors have no direct evidence of splitting in their data (owner identifiers absent).
  - Hours or worker quality: smaller firms may increase average hours per worker or hire more qualified workers, but absence of wage discontinuity argues against this as the sole explanation.
  - Higher TFP or capital intensity in firms with between 10 and 20 salaried employees could contribute, but the datasets lack adequate firm capital measures; a near 40 percent difference in sales per worker seems unlikely to be fully explained by manager productivity or capital intensity alone.

### Theoretical interpretation: incentives and hiring margin
- The empirical patterns motivate a model extension (based on Garicano et al., 2016 and Lucas (1978)) that allows firms to choose between salaried (`l`) and non-salaried (`n`) labor.
- Key modeling assumptions and implications:
  - Output generated by a non-salaried worker equals a fraction φ of the output of a salaried worker; assume φ(1 + τ_w) < 1 so firms will not hire non-salaried workers unless constrained by size-dependent policy.
  - Workers are identical and paid the same wage w regardless of managerial productivity; empirically supported by no observed discontinuity in the total wage bill and Figure A.1 in the appendix showing no significant average wage difference for employees in firms with 16-25 salaried employees.
  - First-order condition for salaried labor in absence of size-dependent regulation:
    - α g′(l) = w(1 + τ_w), yielding l(α) = [g′]^{-1}(w(1 + τ_w)/α) and n(α) = 0.
  - Size-dependent policies modeled as:
    - A profit tax τ_π that applies only to firms hiring more than threshold N (= 20) salaried workers.
    - A salaried labor tax τ_p reflecting higher labor costs (health and safety committee, potential unionization), allowing (1 + τ_w + τ_p) c(τ_π, τ_p) = T w with T as a re-scaled labor tax.
    - Assumption T φ < 1 ensures large firms actively hiring salaried workers will not hire non-salaried workers.
  - Under multiplicative separability of worker indirect utility, compensation c(τ_π, τ_p) is proportional to w, justifying the re-scaling to T.

### Model-derived firm behavior around the threshold
- Definitions of managerial-productivity thresholds (α) partition firm behavior:
  - α_c = w(1 + τ_w)/g′(N): managerial productivity of a manager who hires exactly N salaried workers regardless of size-dependent policies; managers with α ≤ α_c are unconstrained.
  - α_a = w/(φ g′(N)) = α_c/(φ(1 + τ_w)): lower bound where hiring non-salaried workers becomes profitable; managers with α ∈ [α_c, α_a) are constrained but inactive (do not hire non-salaried workers).
  - For α ∈ [α_c, α_a) firms choose l = N and n = 0; for α ≥ α_a some constrained managers become active and hire non-salaried workers solving:
    - n(α) = (1/φ)[ [g′]^{-1}(w/(α φ)) − N ] with FOC α g′(N + φ n(α)) φ = w.
  - α_r: managerial ability where a manager is indifferent between coping with regulation costs (hiring more salaried workers) and hiring only N salaried workers and supplementing with non-salaried workers. Implicitly defined by:
    - (1 − τ_π)[α_r g(l(α_r)) − T w l(α_r)] = α_r g(N + φ n(α_r)) − w(1 + τ_w) N − w n(α_r).
  - For α ≥ α_r, managers will not hire non-salaried workers; labor demand reverts to l(α) = [g′]^{-1}(w T/α), n(α) = 0.
- Complete labor demand schedule as a function of α:
  - l(α) =
    - 0, if α ∈ [α_min, α)
    - [g′]^{-1}(w(1 + τ_w)/α), if α ∈ [α, α_c]
    - N, if α ∈ (α_c, α_r)
    - [g′]^{-1}(w T/α), if α ∈ [α_r, α_max]
  - n(α) =
    - 0, if α ∈ [α_min, α_a]
    - (1/φ) [ [g′]^{-1}(w/(α φ)) − N ], if α ∈ (α_a, α_r)
    - 0, if α ∈ [α_r, α_max]
- Market clearing in labor:
  - ∫_{α_min}^{α} ψ(α) dα = ∫_{α_min}^{α_max} (l(α) + n(α)) ψ(α) dα.

### Conceptual implications
- Size-dependent regulations (profit-sharing τ_π and extra labor costs T) induce:
  - Bunching at the N = 20 salaried worker threshold.
  - Use of non-salaried labor as an adjustment margin for firms constrained by the threshold.
  - Discontinuities in measured sales and profits per salaried worker around N = 20 that are not matched by discontinuities in wages per worker.
- The model rules out, under its assumptions, persistent wage gains for workers in larger firms because firms can adjust wages and workers are identical; potential non-wage benefits (health/safety committees, unionization gains) are excluded from the model due to measurement and modeling friction considerations.

*Source: IMF Working Paper chapter section "3.2  Salaried and non-salaried employment" (SUNAT and ENE data, 2014) contained in the supplied content unit.*

### 5.1  Firm splitting

### wp18179 - 5.1  Firm splitting

### Firm splitting: assumptions and modeling
- Assumption 1: Each firm has a probability δ ∈ (0,1) of splitting. Furthermore, δ is independent of the firm’s characteristics.
  - The model assumes firms are unable to split with probability 1−δ to match the empirical fact that not all firms split.
  - The independence of δ across firms implies the split decision is unrelated to firm size (it may still depend on omitted factors such as sector, local regulations, manager preferences).
- Assumption 2: If a firm splits, it will divide itself in the smallest possible number of sub-firms (as long as all sub-firms have less than N salaried employees).
  - Splitting entails fixed costs per new unit, motivating firms to split into the minimal feasible number of subunits.
  - Example: a firm with 64 employees would split into 4 sub-firms of 16 employees each (not 8 sub-firms of 8) under Assumption 2.
- Assumption 3: Firms split into units of equal size.
  - A manager who splits retains the same production function and span of control: e.g., one manager with 30 employees produces the same as one manager who manages two firms with 15 employees each.
  - Firms can hire non-integer employees; the number of units a firm splits into is restricted to an integer.

### Partially rigid wages: setup and implications
- Wage adjustment specification (Garicano et al. (2016) formulation):
  - w = ρ w* + (1−ρ) w0. (Equation (5))
    - w0: wage before the regulatory change.
    - w*: target wage that would equilibrate markets with no unemployment.
    - ρ calibrates partial wage rigidity (discussed in Appendix F).
- Labor-market consequences when ρ < 1:
  - The economy features an unemployment rate u > 0.
  - Labor market clearing equation becomes:
    - (1−u) ∫_{α_min}^{α_max} min ψ(α) dα = ∫_{α_min}^{α_max} (`(α) + n(α)) ψ(α) dα. (Equation (4’))
  - Sorting condition becomes:
    - α g(`(α)) − w(1 + τ_w)`(α) = w(1−u). (Equation (1’))
  - Worker’s wage outcome: employed worker gets w, unemployed gets 0.

### Estimation: inputs, steps, and calibration choices
- Calibration: set τ_w = 0.13 (cost of pension contributions as a share of the wage in Peru).
- Estimation procedure (3 steps):
  1. Estimate θ and φ using the production function specification.
  2. Estimate γ, `r, T and `max using the firm size distribution.
  3. Use model restrictions to find remaining parameters.
- Outlier treatment:
  - Sample size: a little over 19,200 observations; 170 flagged as outliers and removed.
  - Outliers defined by local discontinuities in the share distribution of firms by salaried employees.
- Production function and identification:
  - Production function specification: ln(y_j) = θ ln(`_j + φ n_j) + ln(α_j).
  - Regression specification used for identification (Equation (7)):
    - ln(sales_j) = β0 + β1 ln(salaried emp_j) + β2 ln(non-salaried emp_j) + γ X_j + ε_j.
  - Controls X_j include physical capital, worker and manager education indexes, firm type fixed effects, 4-digit industry fixed effects, and a 3rd-degree polynomial on manager education and capital purchases as a proxy for managerial productivity.
  - Robust mappings and cross-checks:
    - Mapping θ to labor cost share gives ˆθ ≈ 0.7.
    - Semi-elasticities identification yields implied φ via levels regression.
- Empirical estimates used for counterfactuals:
  - θ = 0.7
  - φ = 0.8

### Firm size distribution: theory, measurement error, and MLE estimates
- Theoretical form:
  - Assume ψ(α) = C α^{−β}. Define γ = β(1−θ) + θ.
  - The distribution of salaried employment ζ(`) is given by the piecewise expression in equation (6) with parameters γ, `min, `r, `max, T, τ_w, w, etc. (see equation (6) for full functional form).
- Measurement error on observed firm size:
  - Observed `(`α`, ε) = `* (α) e^{ε}, where ε ∼ N(0, σ).
  - Measurement error transforms the mass point at N into a bulge and smooths the discontinuous drop between N and `r.
  - Observed density ω(x) computed via integrating over ε and differentiating P(`(α,ε) < x).
- Estimation by constrained maximum likelihood:
  - Constraint: firms hiring `r salaried employees are not worse off than hiring N salaried and more non-salaried workers, enforced using the weakest profit-sharing level τ^min_π = 0.05 and the indifference condition:
    - (1−τ^min_π) [α_r g(`(α_r)) − T w ` (α_r)] ≥ α_r g(N + φ n(α_r)) − w(1 + τ_w) N − w n(α_r).
- Maximum likelihood parameter estimates (Table 3):
  - γ = 1.99 [1.98, 2.00]
  - σ = 0.182 [0.180, 0.184]
  - `r = 35.5 [35.1, 35.9]
  - T = 1.143 [1.142, 1.144]
  - `max = 33451 [29255, 37647]

### Counterfactuals and quantitative results
- Counterfactual scenarios considered:
  1. Remove regulations entirely (no size-dependent regulations).
  2. Remove the margin of adjustment via non-salaried labor (i.e., firms cannot use non-salaried workers to circumvent regulations).
  3. Vary the value of the regulatory threshold instead of removing regulations.
- Key empirical observations about affected firms:
  - Firms most affected are medium-sized firms in the vicinity of 20 employees.
    - These correspond to about 3 percent of firms in the Peruvian economy and employ about 10 percent of formal workers.
  - Firms whose manager’s ability lies in (α_a, α_r) experience a drop of almost 20% in their output when regulations are put in place.
  - Small firms (less than 20 employees) see slight increases in employment and profits:
    - Small firms’ profits increase by about 2% when regulations are introduced.
  - Large firms’ (after-tax) profits decrease by a little over 5% due to the profit sharing rule.
- Aggregate effects of introducing size-dependent regulations (Table 4: aggregate impacts expressed in percent)
  - Baseline Model
    - Flexible wages: Aggregate Wages −1.034 ; Aggregate Profits −2.884 ; Aggregate Output −0.096 ; Share of Managers 3.504
    - Rigid Wages: Aggregate Wages −0.644 ; Aggregate Profits −3.774 ; Aggregate Output −1.006 ; Share of Managers 3.501
  - With firm splitting
    - Flexible wages: Aggregate Wages −0.775 ; Aggregate Profits −2.928 ; Aggregate Output −0.084 ; Share of Managers 2.576
    - Rigid wages: Aggregate Wages −0.384 ; Aggregate Profits −3.809 ; Aggregate Output −0.995 ; Share of Managers 2.573
- Interpretation of aggregate impacts:
  - Wages fall in the aggregate because reduced labor demand pushes wages down; the decline is larger when wages are fully flexible and smaller with downward wage rigidity and firm splitting.
    - Example magnitudes: wage declines of −1.034 percent (flexible, no splitting) versus −0.384 percent (rigid, with splitting).
  - Aggregate profits decline by about 3 to 4 percent as losses by larger firms outweigh gains by smaller firms.
  - Aggregate output impact depends on wage flexibility:
    - Flexible wages: output decreases marginally (about −0.1 percent).
    - Partially rigid wages: output declines more substantially (about −1 percent of GDP) because wage rigidity amplifies employment/output effects.
  - Regulatory effects shift the composition of the economy:
    - The share of managers increases (e.g., 3.504 in baseline flexible wages), reflecting increased attractiveness of entrepreneurship for some agents.
- Notes on firm-splitting in counterfactuals:
  - All counterfactual figures reported in the paper text are computed using the model without firm splitting; figures with firm splitting are noted to be similar.
  - Inclusion of firm splitting moderates wage and output effects (see Table 4 rows “With firm splitting”).

*Italic: Source — wp18179 - 5.1  Firm splitting (PDF chapter/section)*

### introduction of the size-dependent regulations lowers labor demand, lowering wages. In the

### wp18179 - introduction of the size-dependent regulations lowers labor demand, lowering wages. In the

### Effect of size-dependent regulations on labor markets and unemployment
- Introduction of size-dependent regulations lowers labor demand, lowering wages.
- In the presence of wage rigidities, wages cannot fully adjust, leading to a larger decline in employment in equilibrium.
- Wage rigidity generates unemployment, as wages remain too high for markets to clear.
- Calibration result: unemployment increases by 1.3 percent when the regulations are implemented and wages are not fully flexible.
- For comparison: the unemployment rate in Peru in 2014 was 5.5 percent, suggesting that these size-dependent regulations may account for about one fourth of total structural unemployment in the country.

### Distributional impacts: winners and losers
- Winners:
  - Small firms and entrants (agents who start small firms once the regulations are introduced) benefit because they face a lower overall wage, expand employment, and increase profitability.
- Losers:
  - Larger firms suffer a severe reduction in profitability.
  - Workers face lower wages and higher unemployment.
- When firms can split:
  - Wages fall by less because firm splitting moderates the decrease in labor demand.
  - Aggregate profits and output fall by roughly the same amount as when firm splitting is not allowed, since firms that split avoid profit sharing but face a higher cost of labor.
  - Smaller increase in the share of firms and smaller decrease in average firm productivity because some larger firms split and continue growing.

### Counterfactual: removing non-salaried labor
- Directly affected firms: only firms in the “active, unregulated and constrained” regions.
- Two counteracting effects on labor demand for these firms:
  - Downward: constrained firms hire less labor because they can no longer hire non-salaried employees, hiring only Nsalaried employees.
  - Upward: more productive constrained firms now have a stronger incentive to comply and resume hiring salaried employees, increasing labor demand.
- Aggregate outcome: the two effects almost cancel, resulting in little change in aggregate demand for labor.
- Local effects: some firms produce up to 15% less or 25% more when non-salaried labor is removed.
- Unemployment change: increases by only 0.005 percent in the fully flexible case.
- Interpretation: removal of non-salaried workers causes large effects on a few firms, but most firms are unaffected, diluting aggregate quantities. Presence of informal labor does not mitigate effects of size-dependent policies on the economy.

### Aggregate impacts reported (as in Table 5)
- Table 5 text (preserved format):
  - Aggregate
  - Wages
  - Aggregate
  - Profits
  - Aggregate
  - Output
  - Share of
  - Managers
  - Baseline model
  - Flexible wages-0.0050.0020.0400.017
  - Rigid Wages-0.003-0.0020.0350.017
  - With firm splitting
  - Flexible wages-0.003-0.0020.0360.010
  - Rigid wages-0.001-0.0060.0320.010

### Effects of changing the policy threshold (size cutoff)
- Conceptual extremes:
  - Removing all size-dependent regulations = threshold set to infinity.
  - Setting threshold to 0 removes the “size-dependent” part of the policies, affecting all firms.
- Range considered: threshold varied between 2 (size of the smallest firm in the model) and 100.
- Example quantitative effects:
  - Setting the threshold at 50 salaried employees instead of 20 salaried employees would lead to an increase in profits of about 0.5 percent, and average firm size would increase by about 0.2 percent.
- Informality and unemployment:
  - Unemployment rate decreases with the size of the threshold (panel 11d; no unemployment when wages are flexible).
  - Share of informal, non-salaried workers increases with the threshold because constrained firms become bigger and substitute salaried employees for non-salaried ones.
- Non-monotonic and threshold-dependent output responses:
  - Panel 11e (flexible-wage baseline) shows non-monotonic output variations:
    - For small threshold values the “lower-misallocation” effect dominates (output rises).
    - For large values (i.e., from 60 salaried employees onwards) the effect that fewer firms are affected dominates (output increases).
    - In between, increasing the threshold can cause output to fall below current levels.
  - Panel 11f (with firm splitting and rigid wages): output monotonically increases with the threshold because firms that can split are unaffected by the threshold and wage rigidities reduce indirect effects.
- Sensitivities and drivers:
  - Shape parameter γ of the firm-size distribution critically affects the curve in panel 11e, since it controls how many firms are above/below any threshold.
  - Range of most affected firms increases with the threshold value (difference αr − αc increases with N), aggravating the impact of regulations.
- Non-monotonic behavior of informality:
  - For very high thresholds, it is possible that αr > αmax and even the largest firms are constrained, causing the share of informal workers to decrease; eventually with thresholds so high that αc > αmax all firms are smaller than the threshold and share of informal workers is zero.
  - Note: the share of informal workers only starts to decrease for unrealistically high values of the threshold (around many thousands of workers).
- Calibration note on informality in Peru:
  - About 12 percent of the Peruvian workforce is non-salaried, of which 0.3 percent are hired by firms with 20 salaried employees.
  - This suggests that 0.3 percent of the workforce is employed in a non-salaried position because of the size-dependent policies.
  - Model equivalent share is 0.29.

### Policy implications and conclusions
- As currently designed, the size-dependent policies (profit sharing and other policies applicable only to firms with more than 20 formal workers) have an overall negative effect on aggregate output.
- Distributional effects: owners of small firms win; larger firms and workers lose.
- Size-dependent policies lead to an increase in labor informality, as some firms hire informal workers to avoid regulation.
- If the purpose of these policies is to raise workers' income, the results suggest they make labor worse off through lower wages and higher informality and unemployment.
- Possible policy approaches:
  - First-best: remove distortionary size-dependent regulations.
  - Second-best: adjust the value of the threshold, acknowledging non-monotonic effects and sensitivity to firm-size distribution.
  - Consider enforcement and responses such as firm splitting when designing thresholds and enforcement strategies.

### Suggested avenues for future research (as noted)
- Obtain direct evidence of firm splitting by linking legally separate firms to the same owner to better understand costs, trade-offs and prevalence.
- Study other types of size-dependent policies, including simplified tax regimes and minimum thresholds, common in developing economies, to improve design for tax collection and formalization.

*wp18179 - introduction of the size-dependent regulations lowers labor demand, lowering wages. In the*

### References

### wp18179 - References

### References
- Citations included (authors and titles preserved exactly as in source): Ackerberg, Daniel A., Kevin Caves, and Garth Frazer, “Identification properties of recent production function estimators,” Econometrica, 2015, 83(6), 2411–2451.
- Other cited works (selection as listed): Alaimo, Verónica et al., “Measuring the Cost of Salaried Labor in Latin America and the Caribbean,” June 2017; Amirapu, Amrit and Michael Gechter, “Labor Regulations and the Cost of Corruption: Evidence from the Indian Firm Size Distribution,” 2017; Gandhi, Amit, Salvador Navarro, and David Rivers, “On the Identification of Gross Output Production Functions,” 2017; Hsieh, Chang-Tai and Benjamin A. Olken, “The Missing “Missing Middle”,” Journal of Economic Perspectives, 2014, 28(3), 89–108; Hsieh and Peter J. Klenow, “The Life Cycle of Plants in India and Mexico,” The Quarterly Journal of Economics, 2014, 129(August), 1035–1084; Levy, Santiago, Good Intentions, Bad Outcomes: Social Policy, Informality, and Economic Growth in Mexico, Brookings Institution Press, 2008; Lucas, Robert E., “On the Size Distribution of Business Firms,” The Bell Journal of Economics, 1978, 9(2), 508–523; Olley, G. Steven and Ariel Pakes, “The Dynamics of Productivity in the Telecommunications Equipment Industry,” Econometrica, 1996, 64(6), 1263–1297; Schivardi, Fabiano and Roberto Torrini, “Identifying the effects of firing restrictions through size-contingent differences in regulation,” Labour Economics, 2008, 15, 482–511; Ulyssea, Gabriel, “Firms , Informality and Development: Theory and evidence from Brazil,” American Economic Review, 2018; Viollaz, Mariana, “Are Labor Inspections Protecting Workers’ Rights? Adding the Evidence from Size-based Labor Regulations and Fines in Peru,” International Labour Review, 2018.
- Note: Full reference list preserved in source; above captures representative entries as provided.

### Other tables and figures
- Figure A.1: Wages around the policy threshold - ENE.
  - Note: Average wages computed as total personnel expenses divided by the number of workers (both salaried and non-salaried).
  - Data source: ENE 2015 firm survey, refer to the year 2014.
  - Boxplot construction: each box depicts the 25th, 50th and 75th percentiles; whiskers represent adjacent values (upper/lower quartile ± 1.5×interquartile range).

### Computing salaried employees’ compensations
- Model assumptions and setup:
  - Worker indirect utility v(·) multiplicatively separable: v(xy) = v(x)v(y).
  - Firm profit shock ξ is i.i.d. with E[ξ] = 1 and realized profit ξπ(α).
  - Indifference condition: E[v(c(τπ, τp) + τπ ξ π(α) `(α))] = v(w).
- Firm profit maximization and derivations:
  - E[ξπ(α)] = π(α) = max` (1−τπ)[α g(`) − (1 + τw + τp) c(τπ, τp)`].
  - With g(x) = xθ, π(α)/`(α) = (1−θ)/θ (1 + τw + τp) c(τπ, τp).
  - Define ̄k by 1/v(1/̄k) = E[v(1 + ξ τπ (1−θ)/θ (1 + τw + τp))].
  - Implied c(τπ, τp) = w/̄k and define T = (1 + τw + τp) ̄k.
  - Result: Tw = (1 + τw + τp) c(τπ, τp).
- Comparative statements:
  - v(1/̄k) ≤ 1 implies c(τπ, τp) ≤ w.
  - Observation: Small firms often avoid profit sharing, suggesting T ≥ 1 + τw.

### Estimating the elasticity of labor
- Estimating θ:
  - From first-order conditions: α θ (`+ φ n)^{θ−1} = w(1 + τ).
  - Algebra leads to θ = [w(1 + τ)(`+ φ n)] / [α(`+ φ n)^{θ}].
  - For firms not hiring non-salaried labor: θ = [w(1 + τ)`] / [α`] ≡ wage bill / sales.
  - Empirical approach: use ENE dataset; labor expenditure = own personnel + services from third-party firms; prefer total cost of firm as denominator to avoid extreme labor cost share values.
- Distribution of labor cost share (Table A.1):
  - Mean: 0.67
  - Std. Dev.: 0.25
  - p10: 0.31
  - p25: 0.47
  - p50: 0.71
  - p75: 0.91
  - p90: 0.93
  - N: 1,233
  - Note: mean and median close to values estimated in section 6.2.1.
- Estimating φ:
  - Three methods: (i) method described in section 6.2.1; (ii) non-linear regression of log(sales) on θ log(` + φ n) and controls; (iii) linearized approximation regressing log(sales) on log(`) and labor ratio/`.
  - Approximation: θ log(` + φ n) ≈ θ log(`) + θ φ n/`.
- Regression results (Table A.2 summary):
  - Column (1) non-linear: ˆθ = 0.745*** (0.08); ˆφ = 0.846 (0.533); Implied φ = 0.846.
  - Column (2): logsalaried coefficient 0.729*** (0.064); labor ratio coefficient 0.499** (0.235); Implied φ = 0.685.
  - Column (3): # Salaried 0.074*** (0.019); # Non-Salaried 0.062*** (0.005); Implied φ = 0.831*** (0.107).
  - Controls: average worker education, 4-digit industry sector, firm type; standard errors robust to heteroskedasticity.
  - Samples and observations:
    - Column (1): Sample Full, Observations 2214, R-squared 0.99.
    - Column (2): Sample labor ratio ≤ 2, Observations 1472, R-squared 0.69.
    - Column (3): Sample # Salaried < 20, Observations 2013, R-squared 0.66.
  - Interpretation: estimated ˆφ roughly consistent across methods; larger standard errors in first two columns due to measurement error in non-salaried labor data.

### Estimating the firm size distribution
- Derivation highlights:
  - Salaried labor demand uses g(`) = `θ and ψ(α) = C α^{−β}.
  - Change-of-variables yields ζ(`(α)) = C α(`)^{−β} / P α′(`).
  - Definitions and regimes for ζ(`) depend on thresholds `min, N, `r, `max and parameters T, τw, θ, γ where γ = β(1−θ) + θ.
  - Constant K defined: K = C/P (θ/w)^{(γ−1)/(1−θ)} (1−θ)/(γ−1).
  - Density ζ(`) expressions piecewise given, with normalization ∫`max`min ζ(`) d` = 1 imposing restriction:
    - 1/K = (1 + τw)^{(1−γ)/(1−θ)} `min^{1−γ} − T^{(1−γ)/(1−θ)} `max^{1−γ}.
- Estimation procedure (Garicano et al. (2016) adaptation):
  - Measurement error in observed labor: `(α, ε) = e^{ε} `(α) with ε ∼ N(0, σ).
  - Conditional distribution P(`(α,ε) < x | ε) computed piecewise across regions; integrate over ε using normal density φ and distribution Φ.
  - Closed-form expressions obtained for P(`(α,ε) < x) and density ω(x) = ∂P/∂x with multiple terms; simplifications reduce ω(x) to a tractable form.
- Maximum likelihood estimation:
  - Parameters to maximize: γ, σ, `r, T, `max via max logL = Σ_j log ω(`_j) subject to constraint that firms hiring salaried employees find it optimal to do so.
  - Constraint derived (equation 8): `r ≥ N (1/φ − (1 + τw))^{(1−θ)/θ} [T (1−τ_min^π) − T^{1/(1−θ)} φ^{θ/(1−θ)}] (rewritten in source algebraic form).
  - Profit-sharing lower bound: τ_min^π = 0.05.
- Remaining parameter determination:
  - Normalization α_max = 1.
  - Wage implied: `max^{1−θ} = α_max^{θ} Tw ⇒ w = α_max^{θ} T `max^{1−θ}/θ (as in source; exact algebra in text).
  - Thresholds computed: α_c, α_a, α_r, α via given functional relations (preserved algebraic forms).
  - τπ found via restriction: (1−τπ)[α_r g(`(α_r)) − Tw`(α_r)] = α_r g(N + φ n(α_r)) − w(1 + τw) N − w n(α_r); solution gives τπ = 5.6%.
  - Market clearing condition used to determine α_min; C computed from ψ(α) normalization: C = [ (1−γ)/(1−θ) (α_max^{(1−γ)/(1−θ)} − α_min^{(1−γ)/(1−θ)}) ]^{−1}.
  - P computed: P = ∫_{α}^{α_max} C α^{−β} dα = C (1−θ)/(γ−1) [ (w(1 + τw)/θ)^{(1−γ)/(1−θ)} − α_max^{(1−γ)/(1−θ)} ].

### Model with firm splitting
- Conceptual effect:
  - Splitting redistributes mass from firms with > `r salaried employees to smaller firms.
  - A fraction δ of large firms split: ζ_s(`) = (1−δ) ζ(`) for ` > `r.
  - No firms with ` ∈ (N, `r) and no split yields sub-units < N/2, so ζ_s(`) = ζ(`) for ` ∈ [ `min, N/2 ] ∪ (N, `r).
  - Increased density for ` ∈ (N/2, N], asymmetric bulging below N.
- Combinatorial splitting rule:
  - For each ` ∈ [N/2, N), largest firm splitting into units of size ` does so into J_` pieces where J_` = max{ ⌈`/(N−`)⌉, 1 } and J_N = ⌊`max / N⌋.
  - ζ_s(n) for integer n in [`min, N) given by:
    - ζ_s(n) = K(γ−1)(1 + τw)^{(1−γ)/(1−θ)} n^{−γ} [ (1−δ) + δ Σ_{j=1}^{J_n} j^{1−γ} ].
  - Special case ` = N: ζ_s(N) = (1−δ)K[(1 + τw)^{(1−γ)/(1−θ)} N^{−γ} − T^{(1−γ)/(1−θ)} `r^{−γ}] + δ K(γ−1)(1 + τw)^{(1−γ)/(1−θ)} N^{−γ} Σ_{j=1}^{J_N} j^{1−γ}.
  - Full piecewise ζ_s(`) specified in source.
- Estimation approach:
  - Measurement error model retained: `(α, ε) = e^{ε} `(α).
  - ζ_s(`) lacks closed-form integral and has discontinuities; use numerical computation and GMM estimator.
  - Construct conditional distribution matrix P_{i,j} = P(` < x(i) | ε(j)) on grid for x and ε, integrate over ε using standard normal weights, obtain density ω_s by finite differences.
  - GMM objective: choose (γ, σ, `r, T, δ) to minimize Σ_{x=1}^{L} [ω_s(x) − s(x)]^2 subject to constraint equivalent to equation (8); L = 150.
  - Initialization: use ML solution from non-splitting model as starting guess for (γ, σ, `r, T) and multiple initial δ ∈ [0.01, 0.25].
- Estimation results (Table A.3):
  - γ = 1.98 with 95th CI [1.95, 2.01].
  - σ = 0.15 with 95th CI [0.11, 0.20].
  - `r = 36.0 with 95th CI [35.3, 42.5].
  - T = 1.143 with 95th CI [1.140, 1.147].
  - δ = 0.12 with 95th CI [0.04, 0.13].
  - Note: `max not identified in GMM; `max set to 23179 (largest firm in data) for numerical scaling in results.
- Comparative implications:
  - Adding firm splitting reduces mass of large firms by proportion δ; main empirical effect is bulging of distribution between N/2 and N, asymmetric below N.
  - Estimated σ lower with splitting (less measurement error needed); T lower but similar because δ and T both reduce mass of large firms and compete in fitting data.

### Wage rigidity
- Estimating ρ:
  - Definition: ρ = (w − w0) / (w* − w0) (rearranged from Equation (5)).
  - Empirical identification: compare unemployment rates of Peru and Mexico to attribute share due to size-dependent regulations.
  - Average unemployment 2005–2015: Peru 5.5%, Mexico 4.2% → difference 1.3 percentage points attributed to regulation effect.
  - Model-implied estimate: ρ ≈ 0.49 (wage adjusts about half way to full-employment wage).
- Counterfactual mechanics under partial wage rigidity:
  - For a small change in threshold (e.g., from 20 to 21 employees), direct wage comparison can be sensitive due to large initial wage differences.
  - Thought experiment to remove “artificial” variation: allow wages to vary one time without underlying primitive changes.
  - Final wage normalization used in counterfactuals:
    - w = w0 + ρ (w* − w0) − ρ (w*_20 − w0) = w0 + ρ (w* − w*_20),
    - where w*_20 is the wage that clears the market of the 20-employee threshold economy with no unemployment.
  - This normalization subtracts the artificial variation ρ (w*_20 − w0) so reported wage changes reflect structural changes in equilibrium rather than one-time adjustment artifacts.

*Italic: Content derived from source "wp18179 - References" (source PDF content provided).*

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