## _wp10101

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### I. Introduction and research objective
- Development accounting: explains cross-country differences in income level by decomposing income variance into inputs (physical capital K, human capital h) and total factor productivity A.
- Main research question: How has the contribution of TFP differences to explaining cross-country income level differences evolved over time (annual series 1970–2000), departing from the single-year practice?
- Key claim: The share of TFP differences in explaining income inequality across countries increases steadily over 1970–2000.

### Data, sample, and parameter choices
- Sample: up to 94 countries; annual series constructed for 1970–2000. Sample with all three series available: 88 countries in 1970 decreasing to 82 countries in 2000; maximum in 1996.
- Data sources:
  - Penn World Tables version 6.1 for output, stock of physical capital, and number of workers.
  - Barro and Lee (2001) for educational attainment (average years of schooling for population over 25).
- Capital stock construction:
  - Perpetual inventory with depreciation rate ߜ = 6 percent.
  - Initial K0 imputed as I0 /(g+ߜ ), where I0 is first-year investment and g is geometric average growth rate of investment to 1970.
- Human capital h:
  - h = Φ(s) with Φ(s) piecewise linear:
    - Φ(s)=0.134*s if s≤4,
    - Φ(s)=0.134*4 +0.101*(s-4) if 4<s≤8,
    - Φ(s)=0.134.4+0.101*4+0.068*(s-8) if 8<s.
- Technology and parameters:
  - Benchmark Cobb-Douglas with α = 1/3 for all countries.
- Note: schooling data available every five years; linear interpolation used between observations.

### Measuring “ignorance” (TFP residual) and success indicators
- Factor-only model: y explained by k and h; A (TFP) computed as residual.
- Success indicators (Caselli-style):
  - success1 = variance(log(factor-only model)) / variance(log(y)).
  - success2 = (90th-to-10th percentile ratio of income distribution from factor-only model) / (90th-to-10th percentile ratio of observed income distribution).
  - success1* and success2* computed using Hall and Jones (1999) capital-output representation.
- Interpretation: success indicators equal 1 if technology identical across countries; lower values indicate larger role of TFP (ignorance).

### Main empirical results (1970–2000)
- Secular trend:
  - Explanatory power of factor-only model decreases steadily over time.
- Key numeric findings (Caselli-style indicators, entire sample):
  - 1970: success1 = 52 percent.
  - 2000: success1 = 39 percent.
  - 1996 (benchmark year in Caselli 2005): success1 = 0.38 and success2 = 0.35.
  - Using Laspeyres investment and schooling imputation as in Caselli (2005): success1 = 0.39 and success2 = 0.34.
- Hall and Jones style indicators (success1*, success2*):
  - Show similar decreasing trends but at systematically lower levels than success1 and success2.
- Decomposition and dispersion:
  - Variance of log(y) increases substantially over time.
  - Variance of log(factor-only model) relatively flat (Caselli representation) and decreasing (Hall and Jones representation).
  - Income divergence not matched by divergence in factor accumulation.
- Relative contributions (geometric averages, five richest vs five poorest):
  - Five richest were on average 25 times richer than five poorest in 1970; ratio reaches 40 in the 1990s.
  - Caselli representation: human capital ≈ 16 percent of total income difference; physical capital ≈ 50 percent.
  - Hall and Jones representation: human capital ≈ 20 percent; capital-income ratio ≈ 18 percent.
- Variance dynamics of inputs:
  - Variance of log(physical capital per capita) and variance of log(human capital per capita) relatively constant over 1970–2000.
  - Variance of capital-output ratio declines substantially to reach 50 percent of its 1970 value by end of period.

### Robustness checks and subgroup analysis
- Changing-sample robustness:
  - Restricting to same sample of 87 countries with data 1970–1996 yields virtually unchanged results for success1 and success1*; success2 and success2* trends unchanged.
  - Results qualitatively similar using PWT63 dataset.
- OECD vs non-OECD:
  - Both groups show declining explanatory power of factor-only model over time.
  - Decrease less pronounced for OECD; success indicators for OECD greater than for non-OECD but very close overall.
  - For OECD, smaller covariance between log(TFP) and log(factor model) drives closer success1 vs success1*.
  - Small OECD sample size makes success2 less appropriate.

### Measurement error in physical capital: analysis and findings
- Three approaches to assess measurement error concerns:
  1. Restrict sample to countries with data starting in 1950 (51 countries, 1950–2000). Result: success1 decreases from about 50 percent in 1970 to about 40 percent in 2000; pattern similar.
  2. Compute magnitude of measurement error needed to offset decreasing trend in success1 and show required measurement error is unrealistically large.
  3. Replace capital-output ratios by investment rates (proportional along balanced growth path); success1* and success1** and success2* and success2** are close, indicating investment-series-based measures produce similar indicators.
- Specific quantitative statements:
  - An over-estimation of the true variance by 20 percent would not fully offset the decreasing trend of the success indicator.
  - Computed var(k*) is about 40 percent of the observed value of var(k) in 1996.
  - The required var(k*) in 1970 that would produce a constant success1 equal to observed 1996 value is less than half of the measured variance in 1970.
- Conclusion: It is unlikely that measurement error in physical capital stock alone accounts for the secular decrease in success indicators.

### Robustness to non-factor-neutral technologies (CES)
- Alternative production specification: CES production function parameterized by ߪ in [-0.5,+0.5]; elasticity of substitution ߟ = 1/(1 - ߪ).
- Method: infer ܣ௞ and ܣ௛ from U.S. and impose for all countries; compute success indicators across ߪ grid.
- Empirical result:
  - The negative trend in success1 persists for ߪ in [-0.5,+0.5]; levels and rate of change lower for ߪ close to 0.5.
  - Assuming ߪ ≥ 0.5 would explain most of the result, but:
    - ߪ ≥ 0.5 implies elasticity of substitution ߟ greater than 2, above literature-typical range (between 0 and 1).
    - For σ close to 0.5, explanatory power of factor-only model would be only 20 percent in 2000.
- Conclusion: For reasonable elasticity values, factor-neutral assumption is not essential to main finding of secular decrease in success indicators.

### Institutions, TFP, and consistency with rising role of TFP
- Observed facts on institutions:
  - Rule of law variance (Political Risk Services) displays a decrease over the past two decades of available data, indicating convergence in institutional arrangements.
  - Polity2 score rescaled to range 0 to 20 (from original -10 to 10); from 1985 onwards Polity2 variance decreases similarly to rule of law; prior to 1985 Polity2 variance relatively stable.
- Implication:
  - A secular rise in TFP’s explanatory role is inconsistent with evolution of cross-country institutional quality if the institution–TFP relationship is assumed stable.
  - Possible reconciliation: the relationship between institutions (and other determinants) and TFP may have changed over time; increasing complementarities as production processes become more complex.
- Research priority: test whether the nature of the relationship between institutions (and other potential determinants) and output has changed over time and whether such change can be sufficiently swift to be consistent with rising TFP role given observed institutional convergence.

### Conclusion and implications
- Core evidence: The contribution of differences in total factor productivity (TFP) to income differences across countries steadily increased between 1970 and 2000.
- Robustness: This finding is robust to plausible measurement-error adjustments in input factors and to relaxing factor-neutral technology assumptions for reasonable elasticity values.
- Theoretical implication: Challenges theories based on time-invariant forces (geography, ethnic fractionalization, legal origin) as sole determinants of cross-country income differences.
- Suggested research directions:
  - Explain why the unexplained share (TFP residual) appears to increase over time.
  - Investigate whether and how the relationship between institutions (and other determinants) and output has changed over time.

### Key preserved numerical facts and parameters
- Sample years: 1970 to 2000; also referenced 1950–2000.
- Sample counts: 88 countries in 1970; 82 countries in 2000; 51 countries for 1950–2000 restricted sample.
- Success indicators:
  - 1970: success1 = 52 percent.
  - 1996: success1 = 0.38; success2 = 0.35.
  - 2000: success1 = 39 percent.
  - Laspeyres imputation: success1 = 0.39; success2 = 0.34.
- Over-estimation example: 20 percent.
- var(k*) ≈ 40 percent of observed var(k) in 1996.
- Variance of capital-output ratio reaches 50 percent of its 1970 value by end of period.
- Relative income: five richest / five poorest = 25 in 1970; = 40 in 1990s.
- CES parameter grid: ߪ in [-0.5,+0.5]; ߟ = 1/(1 - ߪ).
- Literature-typical elasticity of substitution range: between 0 and 1.
- ߪ ≥ 0.5 implies elasticity of substitution greater than 2.
- For σ close to 0.5: factor-only explanatory power ≈ 20 percent in 2000.
- Polity2 rescaled range: 0 to 20 (from original -10 to 10).
- Piecewise Φ(s) coefficients preserved exactly (0.134, 0.101, 0.068).
- Depreciation rate: ߜ = 6 percent.
- α = 1/3.

*Extracted from the PDF chapter titled "_wp10101 - introduction of non-factor neutral difference in technology. Section V studies the evolution" (content provided).*

### References .............................................................................................................

### _wp10101 - References .............................................................................................................

### I. Introduction
- Development accounting aims to explain cross-country differences in income level by calculating the relative contribution of input factors and total factor productivity (TFP) differences.
- Main finding in the literature: TFP differences explain a large fraction of observed cross-country income differences; Caselli (2005) finds that as much as 60 percent of the observed variance in cross-country income in the year 1996 is associated with TFP differences.
- Rationale for levels focus: Measuring differences in levels captures better long-run differences in welfare, while growth rate differences are considered transitory (Hall and Jones (1999)). Standard practice is to focus on a single year when explaining long-run level differences.
- Contribution of the paper: Breaks with single-year practice by investigating how the contribution of TFP differences to explaining cross-country income level differences has evolved through time.

### Data and empirical scope
- Sample: up to 94 countries.
- Period studied: 1970 to 2000.
- Approach: Standard development accounting exercise applied through time rather than at a single year.

### Key findings
- The share of TFP differences in explaining income inequality across countries increases steadily over the period studied (1970 to 2000).
- Measurement errors in input factors, in particular in physical capital, are unlikely to be responsible for the observed increase in the share of TFP differences.
- Departing from the assumption of Cobb-Douglas technology and factor neutrality in efficiency does not alter the main finding.

### Interpretation and implications
- The paper interprets the increase in the TFP share as an increase in the measure of "ignorance," i.e., the share of the variance in income explained by the variance in TFP.
- Although no theoretical explanation is provided, the authors argue the finding may challenge understanding of cross-country income level differences and suggests re-examining main theories through this lens.
- The paper studies the evolution of differences in the quality of institutions and assesses consistency with a narrative where most TFP variance is explained by differences in the quality of institutions and the rise of TFP inequality.

### Paper structure (as described)
- Section II: Brief description of methodology and data used to conduct development accounting.
- Section III: Description of results.
- Section IV: Discussion of sensitivity of results to measurement errors in input factors and the ... (text ends here).

*Source: _wp10101 - References .............................................................................................................*

### introduction of non-factor neutral difference in technology. Section V studies the evolution

### _wp10101 - introduction of non-factor neutral difference in technology. Section V studies the evolution

### Benchmark: production function and objective
- Adopt a benchmark Cobb-Douglas technology as in Hall and Jones (1999) with A as TFP, K as aggregate capital stock, L as number of workers, h as average human capital (Lh is quality-adjusted workforce), and α a constant in [0,1].
- Production in per-worker terms: y as GDP per worker, k as capital-labor ratio.
- Objective: determine how much of the variation in income is driven by variation in k and h on an annual basis between 1970 and 2000.

### Data and parameter choices
- Data sources:
  - Penn World Tables version 6.1 [PWT61-Heston, Summers and Aten (2002)] for output, stock of physical capital, and number of workers.
  - Barro and Lee (2001) for educational attainment (average years of schooling for population over 25).
- Construction of K:
  - Perpetual inventory equation with depreciation rate ߜ set to 6 percent.
  - Initial capital stock K0 imputed as I0 /(g+ߜ ), where I0 is first-year investment available and g is geometric average growth rate of investment between first available year and 1970.
- Human capital measure h:
  - h computed via h = Φ(s) where s is average years of schooling and Φ(s) is piecewise linear:
    - Φ(s)=0.134*s if s≤4,
    - Φ(s)=0.134*4 +0.101*(s-4) if 4<s≤8,
    - Φ(s)=0.134.4+0.101*4+0.068*(s-8) if 8<s.
- Parameter α assumed to be 1/3 for all countries (consistent with U.S. time-series data).
- Sample coverage:
  - Construct annual series 1970–2000.
  - Sample with all three series available: 88 countries in 1970 decreasing to 82 countries in 2000; maximum in 1996.
- Note: schooling data available only every five years; linear interpolation used between observations.

### Measuring “ignorance” (TFP residual) and success indicators
- Factor-only model defines the portion of y explained by k and h; A (TFP) computed as the residual.
- Two indicators of "success" (Caselli 2005):
  - success1 = variance(log(factor-only model)) / variance(log(y)).
    - Interprets fraction of cross-country income variance explained by factor accumulation; equals 1 if technology identical across countries.
  - success2 = (90th-to-10th percentile ratio of income distribution from factor-only model) / (90th-to-10th percentile ratio of observed income distribution).
    - Also equals 1 if common technology across countries.
- Alternative indicators success1* and success2* computed using the Hall and Jones (1999) representation (capital-output ratio formulation).

### Main results from development accounting (1970–2000)
- Main finding: explanatory power of factor-only model decreases steadily over time.
- Key numeric findings (Caselli-style indicators, entire sample):
  - In 1970, success1 = 52 percent (combining human and physical capital explains 52 percent of total variance of output).
  - In 2000, success1 = 39 percent.
  - For 1996 (benchmark year in Caselli 2005): success1 is 0.38 and success2 is 0.35 (authors’ measures very close to Caselli’s).
  - Using Laspeyres investment and schooling imputation as in Caselli (2005) yields identical results: success1 = 0.39 and success2 = 0.34.
- Hall and Jones (1999) style indicators (success1*, success2*) show similar decreasing trends but at systematically lower levels than success1 and success2.
- Decomposition evidence:
  - Variance of log(y) (income dispersion) has substantially increased over time.
  - Variance of log(factor-only model) has been relatively flat for the Caselli representation and decreasing for the Hall and Jones representation.
  - Income divergence not matched by divergence in factor accumulation.
- Relative contributions (geometric averages for five richest vs five poorest):
  - Five richest were on average 25 times richer than five poorest in 1970; ratio reaches 40 in the 1990s.
  - Using Caselli representation: human capital contributes about 16 percent of total income difference; physical capital about 50 percent.
  - Using Hall and Jones representation: human capital about 20 percent; capital-income ratio about 18 percent.
- Variance dynamics:
  - Variance of log(physical capital per capita) and variance of log(human capital per capita) relatively constant over 1970–2000.
  - Variance of capital-output ratio declines substantially to reach 50 percent of its 1970 value by end of period.

### Robustness checks and subgroup analysis
- Changing-sample robustness:
  - Restricting to same sample of 87 countries with data 1970–1996 yields virtually unchanged results for success1 and success1*; trends for success2 and success2* unchanged.
  - Authors use approach that maximizes number of countries at each point to extend through 2000.
  - Results qualitatively similar when using PWT63 dataset [PWT63-Heston, Summers, and Aten (2009)].
- OECD vs non-OECD comparison:
  - Both groups show declining explanatory power of factor-only model over time.
  - Observations:
    - Decrease in success indicators is less pronounced for OECD group.
    - Success indicators for OECD group are greater than for non-OECD group.
    - Success indicators for OECD and non-OECD are nevertheless very close.
  - For OECD countries, smaller covariance between log(TFP) and log(factor model) drives closer success1 vs success1*.
  - Note: small OECD sample size makes success2 less appropriate.

### Is the trend driven by measurement error or factor neutrality?
- Two potential explanations investigated: measurement error in capital stock (especially in earlier years) and violation of factor neutrality (Cobb-Douglas assumption).
- Measurement error analysis and evidence:
  - Three approaches to address measurement error concerns (focus on physical capital):
    1. Restrict sample to countries with data starting in 1950 (best-measured K in 1970).
       - 51 countries available for 1950–2000; using this restricted sample, success1 decreases from about 50 percent in 1970 to about 40 percent in 2000 (Figure 8), pattern remains similar.
    2. Compute magnitude of measurement error needed to fully offset the decreasing trend in success1 and show it is unrealistically large.
    3. Replace capital-output ratios by investment rates (proportional along balanced growth path); success1* not materially affected, especially in 1970s — less concern about measurement error in investment series versus capital stock.
  - Specific finding on required variance:
    - Compute var(k*) = required variance of log(physical capital) that would produce a constant success1 equal to its observed 1996 value, assuming covariance between physical and human capital constant at its 1996 value.
    - Result: required var(k*) in 1970 that would achieve constant success1 is less than half of the measured variance in 1970.
- Conclusion from measurement-error analysis:
  - It is unlikely that measurement error in physical capital stock alone accounts for the secular decrease in the success indicators.
  - Decreasing explanatory power of factor accumulation is robust to measurement-error focused checks.

*Italic source attribution: Extracted from the PDF chapter titled "_wp10101 - introduction of non-factor neutral difference in technology. Section V studies the evolution" (content provided).*

### 1970. The magnitude of the measurement error required to offset the decreasing trend

### _wp10101 - 1970. The magnitude of the measurement error required to offset the decreasing trend

### Measurement error and alternative capital measures
- The magnitude of the measurement error required to offset the decreasing trend observed is described as huge.
- Even a substantial measurement error leading to an over-estimation of the true variance by 20 percent would not fully offset the decreasing trend of the success indicator.
- var(k*) is about 40 percent of the observed value of the observed variance var(k) in 1996.
- It is stated that it is difficult to believe that cross-country differences in physical capital per capita were halved between 1970 and 2000 for such a large sample.
- Rationale for using representation of income per capita in terms of capital-output and investment:
  - The capital-output ratio is proportional to the investment rate along the balanced growth path.
  - Replacing the capital-output (k/y) by the investment rate should, in theory, produce the same value of the success indicators success1* and success2*.
- Computation performed:
  - success1** and success2** are computed based on the factor-only model using investment-rate based representation.
- Empirical finding:
  - Figures 10 and 11 show that success1* and success1** and success2* and success2** are close.
  - The proximity reflects that the proportionality of capital-output and investment rate is a good approximation in the sample.
  - Therefore, the secular decrease in the success indicators is robust to the use of investment rates instead of capital-output ratios, while investment data are more reliable compared with the estimated stock of physical capital.
- Author conclusion for this theme:
  - Three different approaches were used; none is an unchallengeable proof, but together they lead the authors to believe that obtaining the true values of the stock of physical capital will not affect the decreasing trend in the success indicators.

### Factor neutrality and robustness to non-neutral technologies (CES)
- Question addressed: Do main findings depend on the assumption of factor neutral differences in technology?
- Method:
  - Replace Cobb-Douglas by a CES production function allowing non-factor neutrality, parameterized by ߪ (sigma) in [-0.5,+0.5].
  - Elasticity of substitution denoted ߟ, with ߟ = 1/(1 - ߪ).
  - Assume factor markets are competitive as in Caselli and Coleman (2005); derive relationships for returns to physical capital and quality-adjusted labor and infer ܣ௞ and ܣ௛.
  - Infer ܣ௞ and ܣ௛ for the U.S. and impose those values for all countries to compute success indicators under non-neutral technologies.
- Empirical finding:
  - Figure 12 displays the evolution of success1 for the class of production functions parameterized by ߪ in [-0.5,+0.5].
  - The negative trend in the success indicator seems always present, although the level and the rate of change are substantially lower for ߪ close to 0.5.
  - Assuming ߪ greater or equal to 0.5 would explain most of the result; however:
    - Such an assumption would lead to an extremely high elasticity of substitution (greater than 2) compared to the usual literature range (between 0 and 1).
    - For σ close to 0.5, the explanatory power of the only-factor model would be only 20 percent in 2000.
- Conclusion for this theme:
  - For reasonable values of the elasticity of substitution, the assumption of factor neutral differences in technology is not essential to the main finding of a secular decrease in the success indicators.

### Institutions and consistency with rising role of TFP
- Main empirical context:
  - The paper presents evidence that the role of TFP in explaining income differences rose over the period 1950-2000.
- General argument:
  - Any theory relying on predetermined, time-invariant factors (geography, ethnic fragmentation, legal origin) cannot be consistent with a secular rise in the explanatory power of TFP over recent decades.
- Illustrative exercise on institutions:
  - Build on North (1990) and subsequent empirical literature linking political/institutional measures to macroeconomic outcomes.
  - Use cross-country indicators: rule of law from Political Risk Services (2009) and Polity2 score from Polity IV (Marshall and Jaggers, 2005).
- Empirical findings on institutional measures:
  - Figure 13 shows that the cross-country variance of the indicator of rule of law displays a decrease over the past two decades for which data are available; the observed decrease indicates convergence in cross-country qualities of institutional arrangements.
  - The main finding (rising role of TFP) is inconsistent with the evolution of cross-country differences in the quality of institutions if one assumes a stable relationship between institutions and TFP over time.
  - Polity2 score was rescaled so that it ranges from 0 to 20 instead of from -10 to 10; a higher score indicates a more democratic system.
  - From 1985 onwards the decreasing pattern of Polity2 score’s variance is similar to that of rule of law; prior to 1985, Polity2 variance is relatively stable.
- Interpretation and research implication:
  - Developments related to cross-country differences in political institutions appear inconsistent with a secular rise in the role of TFP if the institution–TFP relationship is assumed stable.
  - A plausible reconciliation is that the nature of the relationship between institutions (and other potential determinants) and output may have changed over time; as production processes are more complex, complementarities between quality of institutions and output could have increased.
  - Important direction for research: test whether the relationship between potential factors shaping cross-country TFP differences, including institutions, and output has changed over time; any change would have to be swift enough to remain consistent with the rise of TFP given observed convergence in institutional quality.

### Key statistics, parameters, and numerical facts preserved from the analysis
- Measurement error over-estimation example: 20 percent.
- var(k*) is about 40 percent of var(k) in 1996.
- Time periods referenced: 1950-2000; 1970 to 2000; 1970, 1975, 1980, 1985, 1990, 1995, 2000.
- CES parameterization: ߪ in [-0.5,+0.5].
- Elasticity of substitution ߟ related by ߟ = 1/(1 - ߪ).
- Literature-typical elasticity of substitution range: between 0 and 1.
- Elasticity implied by ߪ ≥ 0.5 would be greater than 2.
- For σ close to 0.5, explanatory power of factor-only model would be only 20 percent in 2000.
- Polity2 rescaled range: 0 to 20 (from original -10 to 10).

### Conclusion and implications
- The paper’s core evidence: the contribution of differences in total factor productivity (TFP) to income differences across countries steadily increased between 1970 and 2000.
- Robustness:
  - The finding neither stems from plausible measurement errors in input factors nor hinges on the assumption of factor neutral technology differences for reasonable elasticity values.
- Implications for theory:
  - The rise of TFP’s explanatory role challenges theories based on time-invariant forces (geography, ethnic fractionalization, legal origin).
  - The observed convergence in institutional quality is inconsistent with a rise in TFP under a stable institution–TFP relationship; a changing relationship or increasing complementarities is a possible reconciliation.
- Research priorities:
  - Test whether the nature of the relationship between institutions (and other potential determinants) and output has changed over time.
  - Explain why what is unexplained by input factors appears to be increasing over time.

*Italic: Content derived from the provided extract of the chapter "1970. The magnitude of the measurement error required to offset the decreasing trend" from the supplied PDF.*

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