## Box 1. On possible theoretical underpinnings for convergence hypothesis; Appendix 2 and Appendix 3 excerpts

## Source details

**Canonical URL:** [Box 1. On possible theoretical underpinnings for convergence hypothesis; Appendix 2 and Appendix 3 excerpts](https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023189-print-pdf.pdf)

## Other formats

- [Markdown version](/-/media/files/publications/wp/2023/english/wpiea2023189-print-pdf.pdf.md)
- [Structured JSON version](/-/media/files/publications/wp/2023/english/wpiea2023189-print-pdf.pdf.json)

---

### Theoretical motivations and mechanisms
- Deterministic output convergence for all countries follows naturally from neoclassical growth models (e.g., Ramsey and Solow), as discussed by Quah (2003).
- Quantitative models of female labor force participation (LFP) (Guner and others (2011); Bick and others (2017)) model female LFP as partly governed by the household tradeoff between:
  - the disutility (psychological or well-being) of female participation, and
  - the net salary of women.
- Bick and others (2017) characterize disutility as reflecting “the inconvenience of scheduling joint work, home production and leisure activities, or spending less family time with children.”
- Under a mechanism where disutility is invariant to economic changes outside the household, economic growth raising wages could reduce gender gaps via higher women’s net salaries.
- Malta and others (2019): increments in female education can increase women’s net salary, positively affecting female participation.
- Heathcote and others (2010): provide a symmetric labor supply model across genders to measure roles of preference shifts and consider contributions of gender wage gaps, income sharing rules, and marriage patterns to U.S. female participation trends.

### Caveats and prerequisites for endogenous long-run theories
- Modeling disutility or preference shifters explicitly is a pre-requisite for theories of endogenous gender LFP gaps in the long run.
- Theories should model mechanics of barriers to female LFP such as legal frameworks, social norms, and religion.

### Markov chain assessment of LFP gender gap dynamics — simple two-state example
- Data snapshot:
  - In 1991, 29 countries displayed LFP gender gaps below 10 percent (“low”), while 160 countries displayed gaps above 10 percent (“high”).
  - Out of the 29 low-gap countries in 1991, 21 remained low by 2021 and 8 moved to high by 2021.
  - Out of the 160 high-gap countries in 1991, 30 had moved to low by 2021 and 130 remained high.
- Empirical transition probabilities (from 1991–2021 data):
  - Probability of transition from low to high = 8/29 (i.e., 27.6 percent).
  - Probability of transition from high to low = 30/160 (i.e., 18.8 percent).
- Implications under a Markov chain with the above transition matrix:
  - Number of countries with low gaps would become 51 by 2051.
  - By around the year 2200, the distribution would settle with approximately 40 percent of countries in the low gap category and the remaining countries in the high gap category.
  - The dynamics implied by these transition probabilities do not point to a global closing of the LFP gender gap; a large share of countries would continue to display high gender gaps in the long run if recent dynamics continue.
- Mathematical insight:
  - A global closing of the gender gap in this framework would require a zero low-to-high transition probability (making the low state absorbing). This is the strong assumption underlying prevalent time-to-equality calculations.
  - The data indicate a transition probability from low to high close to 30 percent, contradicting the absorbing-low assumption.

### Limitations of the simple exercise and robustness directions
- Simplifications and limitations:
  - Only two years were used (1991 and 2021).
  - The range of possibilities was split into only two categories (“high” and “low”).
- Robustness and extensions explored:
  - Calculate transition matrices using the full sample (1991 to 2021) at annual, five-, and ten-year frequencies (one-, five-, and ten-year transition probabilities).
  - Reproduce results excluding years 2000 and 2021 to purge effects of COVID-19.
  - In Appendix 3, address measurement error concerns by adding population weights.
  - Use finer granularity with an eight-category distributional model (cutoffs at the 5th, 10th, 25th, 50th, 75th, 90th, and 95th percentiles).
  - Employ a nonlinear interpolation technique to plot evolution of several percentiles of the gap over time.
  - Analyze robustness to methodology variations, changes in dynamics over 1991–2021, and joint dynamics of the LFP gender gap and the wealth of nations using a bivariate Markov chain.

### Core conclusion from the Box 1 exercise
- Across all variations considered, the main conclusion is robust: the current dynamics of the LFP gender gap across countries imply elevated gaps in the long run for a large share of countries, absent stronger changes in transition dynamics.

### Appendix 2 — Data, methods, and main dynamic results
- Data and methods:
  - Compare first 15 years versus most recent 15 years to assess acceleration or deceleration of distributional movement.
  - Multi-year transition focus reduces observations; model stops at 10-year transitions to balance dynamic stability and precision.
  - Markovian models estimated at multiple measurement frequencies (including Annual Frequency) and with sample splits (first half vs second half; exclusion of 2020–2021).
  - Bivariate Markov chain jointly models income (real GDP per capita relative to global sample average) and LFP gender gaps. Cutoffs: LFP gap cutoffs at the 10th, 25th, and 75th percentiles; income cutoffs at the 25th and 75th percentiles, yielding 12 categories.
- Main dynamic results:
  - Transitionary period lasts until about the year 2150 for the methods compared; all percentiles decline gradually over time.
  - Notable narrowing of gender gaps over the next two centuries is possible if recent policies, trends, and patterns of shocks continue.
  - However, a substantial fraction of countries will continue to exhibit sizable LFP gender gaps over time.
- Key percentile outcomes (long run):
  - 25th percentile: exhibits the slowest convergence; is the last curve to fully flatten in all panels.
  - Median (50th percentile): converges to a value of approximately 10 percentage points in all exercises; consistent with introductory exercise result of 40 percent of the countries with gaps above 10 percentage points in the long run.
  - 90th percentile: located above roughly 15 to 20 percentage points in the long run across exercises.
  - Top 10 percent of countries (Table 3 summary): will have gaps above 14.6 to 20.7 percentage points in the long run (method-dependent).
- Convergence timing and speed of adjustment:
  - Gaps move within 1 percent of their long-run value after between 197 and 300 years (Table 3).
  - Gaps move within 0.1 percent of their long-run value after 295 to 430 years (Table 3).
  - Sample-split (one-year transition probabilities):
    - Later sample implies faster convergence mainly for the 25th percentile (flattens by year 2100) versus earlier sample (flattens after year 2200).
    - Faster convergence for 25th percentile in later sample accompanied by slightly higher long-run gaps for the more recent sample.
- Sensitivity to Covid-19 years (2020–2021 exclusion):
  - Excluding 2020 and 2021, for the Annual Frequency, Full Sample case:
    - All percentiles are pushed downward.
    - 25th percentile cut most: cut by 0.6 percentage points.
    - 90th percentile cut by 0.2 percentage points.
  - Percentiles from other methods are not impacted systematically.
  - Time to convergence roughly similar across samples, with longer convergence time for the Annual Frequency, Full Sample case when including 2020–2021.
  - Interpretation: pandemic had a devastating effect on status of women in countries with substantial progress; removing pandemic years lowers percentiles but can lengthen convergence where the 25th percentile is lowest.
- Role of large-population countries and measurement:
  - Results are not driven by a few mismeasured countries.
  - Long-run distribution exhibits higher percentiles when large countries receive more weight, suggesting some very large-population countries have gap dynamics elevating the long-run stationary distribution.
  - Observations with increasing gaps are distributed across all IMF regions and income levels: significant evidence of increasing gender gaps within all regions and income levels.
- Bivariate analysis (accounting for development):
  - Transitional dynamics are similar in bivariate and univariate cases; percentiles drop monotonically toward steady-state values in both.
  - Convergence speed contrast:
    - Bivariate case: long run distribution converges to a vicinity of 1 percent of its long run value in 25 years.
    - Univariate case (same specification): takes 280 years to reach vicinity of 1 percent.
  - Interpretation: taking development into account changes mixing and implied long-run gaps; overall, joint analysis leads to slightly larger gaps in the long-run distributions in the joint analysis.

### Appendix 2 — Deterministic-trend supplemental calculations
- Two trend regressions:
  - Log of the gender gap regressed on year: coefficient 0.0107 (which means that the gap falls by approximately 1.07 percent per year).
  - Gap in levels regressed on year: coefficient -0.24 (this means that the gap falls by 0.24 percentage points every year).
- Starting point: average gender LFP gap of 19.49 percent observed in year 2020.
- Extrapolated times to zero:
  - Logarithmic trend reaches zero in 425 years.
  - Levels trend reaches zero in 19.49/0.24 = 81.21 years.
- Goodness of fit:
  - R-squared statistics: 0.0214 (log trend) and 0.0134 (levels trend), reflecting vast heterogeneity and poor fit.

### Appendix 2 — Interpolation Method for Quantiles
- Interpolation uses a Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) implemented via interp1 in Matlab.
- Markov state cutoffs vector x = {x0,x1,x2,...,xN}:
  - Points 1 to N-1 match certain percentiles of the gender gaps in the dataset.
  - x0 and xN set to the minimum and the maximum in the dataset.
- Compute CDF in 2020 as {0,F1,F2,...,FN} with FN = 1 and PDF items F(n)-F(n-1).
- Use transition matrix to simulate future PDF paths and reconstruct CDFs; for any percentile p in [0,1] locate p in {0,F1,F2,...,FN} and interpolate to find corresponding x in {x0,x1,x2,...,xN}.

### Appendix 3 — Mitigating Measurement Error
- Assumption: measurement error decreases in population size; larger-population countries have more stable and accurate statistics.
- Method: recompute annual frequency full-sample transition matrix using population weights proportional to country population share of world population in that year; implemented with the “aweight” option in tabulate in Stata (assumes implicit variance of measurement error inversely proportional to country population).
- Key findings:
  - Certain countries with very large populations have dynamics that tend to increase the long-run gaps; smaller countries have dynamics that tend to decrease the gaps.
  - Main results are robust to population weighting.
  - Under the population-weighting assumption, measurement error tends to bias the percentiles downward; thus reported results can be interpreted as a lower bound for the long run gaps.
  - Changing the unit of observation from countries to individuals may affect the result.
- Acknowledgement: “We thank Jiajia Gu for suggesting this robustness check.” (footnote 11)

### Conclusions and policy implications (synthesized from Box 1 and Appendices)
- If episodes of increasing gaps persist in some countries, the global gap will not fully close; gender gaps will narrow but remain quite large for the foreseeable future absent substantial policy action.
- Dynamics of narrowing LFP gender gaps are very slow: stationarity within 1 percent of long-run steady state requires 197 to 430 years.
- Policy implications drawn from Markov-chain logic:
  - Interventions should focus on both reducing gender gaps and maintaining low gaps in countries that have achieved more progress.
  - Attention should be paid across the distribution, including where gaps are narrower but progress may be slowest (e.g., the 25th percentile).
  - Stronger policy interventions are needed to prioritize and achieve women’s economic empowerment and sustained utilization and improved allocation of human resources.
- Robustness notes:
  - Results are not exclusively caused by increasing gaps in emerging market and developing economies; increasing-gap cases are spread evenly across regions and income groups.
  - Joint dynamics of income and gender gaps do not support a deterministic route to universal gender equality; data do not provide evidence for deterministic U-shaped convergence to a single gender-equality point.

*Source: Box 1, Appendix 2, and Appendix 3 — wpiea2023189-print-pdf*

### Box 1. On possible theoretical underpinnings for convergence hypothesis

### Box 1. On possible theoretical underpinnings for convergence hypothesis

### Theoretical motivations and mechanisms
- For output dynamics, deterministic output convergence for all countries follows naturally from neoclassical growth models (e.g., Ramsey and Solow), as discussed by Quah (2003).
- For gender gaps, theoretical foundations are less definitive. Leading quantitative work (Guner and others (2011); Bick and others (2017)) models female LFP as partly governed by the household tradeoff between:
  - the disutility (psychological or well-being) of female participation, and
  - the net salary of women.
- Bick and others (2017) characterize disutility as reflecting “the inconvenience of scheduling joint work, home production and leisure activities, or spending less family time with children.”
- Under the mechanism where disutility is invariant to economic changes outside the household, economic growth raising wages could reduce gender gaps via higher women’s net salaries.
- Malta and others (2019): increments in female education can increase women’s net salary, positively affecting female participation.
- Heathcote and others (2010): provide a symmetric labor supply model across genders to measure roles of preference shifts and consider contributions of gender wage gaps, income sharing rules, and marriage patterns to U.S. female participation trends.

### Caveats and prerequisites for endogenous long-run theories
- Modeling disutility or preference shifters explicitly is a pre-requisite for theories of endogenous gender LFP gaps in the long run.
- Theories should model mechanics of barriers to female LFP such as legal frameworks, social norms, and religion.

### Markov chain assessment of LFP gender gap dynamics — simple two-state example
- Data snapshot:
  - In 1991, 29 countries displayed LFP gender gaps below 10 percent (“low”), while 160 countries displayed gaps above 10 percent (“high”).
  - Out of the 29 low-gap countries in 1991, 21 remained low by 2021 and 8 moved to high by 2021.
  - Out of the 160 high-gap countries in 1991, 30 had moved to low by 2021 and 130 remained high.
- Empirical transition probabilities (from 1991–2021 data):
  - Probability of transition from low to high = 8/29 (i.e., 27.6 percent).
  - Probability of transition from high to low = 30/160 (i.e., 18.8 percent).
- Implications under a Markov chain with the above transition matrix:
  - Number of countries with low gaps would become 51 by 2051.
  - By around the year 2200, the distribution would settle with approximately 40 percent of countries in the low gap category and the remaining countries in the high gap category.
  - The dynamics implied by these transition probabilities do not point to a global closing of the LFP gender gap; a large share of countries would continue to display high gender gaps in the long run if recent dynamics continue.
- Mathematical insight:
  - A global closing of the gender gap in this framework would require a zero low-to-high transition probability (making the low state absorbing). This is the strong assumption underlying prevalent time-to-equality calculations.
  - The data indicate a transition probability from low to high close to 30 percent, contradicting the absorbing-low assumption.

### Limitations of the simple exercise and robustness directions
- Simplifications and limitations of the two-state, two-year snapshot exercise:
  - Only two years were used (1991 and 2021).
  - The range of possibilities was split into only two categories (“high” and “low”).
- Robustness and extensions explored subsequently:
  - Calculate transition matrices using the full sample (1991 to 2021) at annual, five-, and ten-year frequencies (examining one-, five-, and ten-year transition probabilities).
  - Reproduce results excluding years 2000 and 2021 to purge effects of COVID-19.
  - In Appendix 3, address measurement error concerns by adding population weights.
  - Use finer granularity with an eight-category distributional model, where category cutoffs are the 5th, 10th, 25th, 50th, 75th, 90th, and 95th percentiles of the sample distribution.
  - Employ a nonlinear interpolation technique to plot evolution of several percentiles of the gap over time.
  - Analyze robustness to methodology variations, changes in dynamics over 1991–2021, and the joint dynamics of the LFP gender gap and the wealth of nations using a bivariate Markov chain.

### Core conclusion from the exercise
- Across all variations considered, the main conclusion is robust: the current dynamics of the LFP gender gap across countries imply elevated gaps in the long run for a large share of countries, absent stronger changes in transition dynamics.

*Source: Box 1. On possible theoretical underpinnings for convergence hypothesis — wpiea2023189-print-pdf*

### Appendix 2, which helps us to illustrate the evolution of the underlying distribution in a simple manner.

### Appendix 2, which helps us to illustrate the evolution of the underlying distribution in a simple manner.

### Data and methods
- Comparison of first 15 years versus most recent 15 years of data to assess acceleration or deceleration of distributional movement.
- Multi-year transition focus reduces observations; model stops at 10-year transitions to balance dynamic stability and precision.
- Markovian models estimated at multiple measurement frequencies (including Annual Frequency) and with sample splits (first half vs second half; exclusion of 2020–2021).
- Bivariate Markov chain extends univariate model by jointly modeling income (real GDP per capita relative to global sample average) and LFP gender gaps. Cutoffs: LFP gap cutoffs at the 10th, 25th, and 75th percentiles; income cutoffs at the 25th and 75th percentiles, yielding 12 categories.

### Main dynamic results
- Transitionary period lasts until about the year 2150 for the methods compared; all percentiles decline gradually over time.
- Notable narrowing of gender gaps over the next two centuries is possible if recent policies, trends, and patterns of shocks continue.
- However, a substantial fraction of countries will continue to exhibit sizable LFP gender gaps over time.

### Key percentile outcomes (long run)
- 25th percentile:
  - Exhibits the slowest convergence; is the last curve to fully flatten in all panels.
- Median (50th percentile):
  - Converges to a value of approximately 10 percentage points in all exercises.
  - Consistent with introductory exercise result of 40 percent of the countries with gaps above 10 percentage points in the long run.
- 90th percentile:
  - Located above roughly 15 to 20 percentage points in the long run across exercises.
- Top 10 percent of countries (Table 3 summary):
  - Will have gaps above 14.6 to 20.7 percentage points in the long run (method-dependent).

### Convergence timing and speed of adjustment
- Gaps move within 1 percent of their long-run value after between 197 and 300 years (Table 3).
- Gaps move within 0.1 percent of their long-run value after 295 to 430 years (Table 3).
- Sample-split (one-year transition probabilities):
  - Later sample implies faster convergence mainly for the 25th percentile (flattens by year 2100) versus earlier sample (flattens after year 2200).
  - Faster convergence for 25th percentile in later sample accompanied by slightly higher long-run gaps for the more recent sample (see last two rows of Table 3).

### Sensitivity to Covid-19 years (2020–2021 exclusion)
- Excluding 2020 and 2021 (Table 4), for the Annual Frequency, Full Sample case:
  - All percentiles are pushed downward.
  - 25th percentile cut most: cut by 0.6 percentage points.
  - 90th percentile cut by 0.2 percentage points.
- Percentiles from other methods are not impacted systematically.
- Time to convergence roughly similar across samples, with longer convergence time for the Annual Frequency, Full Sample case when including 2020–2021.
- Interpretation: pandemic had a devastating effect on status of women in countries with substantial progress; removing pandemic years lowers percentiles but can lengthen convergence where the 25th percentile is lowest.

### Role of large-population countries and measurement
- Results are not driven by a few mismeasured countries.
- Long-run distribution exhibits higher percentiles when large countries receive more weight, suggesting some very large-population countries have gap dynamics elevating the long-run stationary distribution.
- Observations with increasing gaps are distributed across all IMF regions and income levels (Table 5): significant evidence of increasing gender gaps within all regions and income levels.

### Bivariate analysis (accounting for development)
- Motivation: evaluate the U-shaped transition hypothesis linking income and LFP gender gaps.
- Method: bivariate Markov chain with 12 categories (income × gap categories); compare univariate versus bivariate transition paths and ergodic distributions.
- General finding: transitional dynamics are similar in bivariate and univariate cases; percentiles drop monotonically toward steady-state values in both.
- Convergence speed contrast:
  - Bivariate case: long run distribution converges to a vicinity of 1 percent of its long run value in 25 years.
  - Univariate case (same specification): takes 280 years to reach vicinity of 1 percent.
- Interpretation: mixing high-income countries in univariate estimation pushes upward movement probabilities higher, affecting convergence times; taking development into account can change mixing and implied long-run gaps.
- Overall: taking development into account leads to slightly larger gaps in the long-run distributions in the joint analysis.

### Deterministic-trend (Appendix 1) supplemental calculations
- Two trend regressions:
  - Log of the gender gap regressed on year: coefficient 0.0107 (which means that the gap falls by approximately 1.07 percent per year).
  - Gap in levels regressed on year: coefficient -0.24 (this means that the gap falls by 0.24 percentage points every year).
- Starting point: average gender LFP gap of 19.49 percent observed in year 2020.
- Extrapolated times to zero:
  - Logarithmic trend reaches zero in 425 years.
  - Levels trend reaches zero in 19.49/0.24 = 81.21 years.
- Goodness of fit:
  - R-squared statistics: 0.0214 (log trend) and 0.0134 (levels trend), reflecting vast heterogeneity and poor fit.

### Conclusions and policy implications
- If episodes of increasing gaps persist in some countries, the global gap will not fully close; gender gaps will narrow but remain quite large for the foreseeable future absent substantial policy action.
- Dynamics of narrowing LFP gender gaps are very slow: stationarity within 1 percent of long-run steady state requires 197 to 430 years.
- Policy implications drawn from Markov-chain logic:
  - Interventions should focus on both reducing gender gaps and maintaining low gaps in countries that have achieved more progress.
  - Attention should be paid across the distribution, including where gaps are narrower but progress may be slowest (e.g., the 25th percentile).
  - Stronger policy interventions are needed to prioritize and achieve women’s economic empowerment and sustained utilization and improved allocation of human resources.
- Robustness notes:
  - Results are not exclusively caused by increasing gaps in emerging market and developing economies; increasing-gap cases are spread evenly across regions and income groups.
  - Joint dynamics of income and gender gaps do not support a deterministic route to universal gender equality; data do not provide evidence for deterministic U-shaped convergence to a single gender-equality point.

*Source: Appendix 2 and related sections of the IMF working paper "When will Global Gaps Close?" (excerpt provided).*

### Appendix 2. Interpolation Method for Quantiles

### Appendix 2. Interpolation Method for Quantiles

### Interpolation method for the gender gap distribution
- The method interpolates the approximate cumulative distribution function (CDF) of the gender gap produced by the Markov chain model using a Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) available from interp1 in Matlab.
- Markov state cutoffs are fixed in a vector x = {x0,x1,x2,...,xN}:
  - Points 1 to N-1 are set to match certain percentiles of the gender gaps in the dataset.
  - x0 and xN are set to the minimum and the maximum in the dataset.
- Using this vector, the CDF of countries across categories in 2020 is calculated as {0,F1,F2,...,FN}, where FN = 1 and the items of the PDF are given by F(n)-F(n-1).
- The transition matrix is used to simulate the future path of the PDF and reconstruct the CDF for each simulation year.
- For any given percentile p in [0,1], locate p in the grid {0,F1,F2,...,FN} and use the interpolation technique to find the corresponding value x from the image array {x0,x1,x2,...,xN}.

### Appendix 3. Mitigating Measurement Error

### Assumptions and approach
- Measurement error is assumed to be decreasing in population size: countries with larger populations have more stable and accurate statistics.
- To gauge the potential effect of measurement error while keeping the country as the unit of observation, the annual frequency full-sample transition matrix is recomputed using population weights that increase the importance of each observation in proportion to the country population.
- The results of this exercise are depicted in Figure 6.

### Weighting implementation details
- The weight assigned to a country-year observation consists of the ratio of the country’s population to the world population in that given year.
- The weight is applied with the option “aweight” in the tabulate Stata command; this assumes that the implicit variance of the (measurement error of the) observation is inversely proportional to the country’s population.

### Key findings and interpretations
- The path for selected percentiles in Figure 6 indicates:
  - Certain countries with very large populations have dynamics that tend to increase the long-run gaps.
  - Smaller countries have dynamics that tend to decrease the gaps.
- Interpretations under the population-weighting assumption:
  - The main results are robust to population weighting.
  - Under the assumption that measurement error decreases with population size, measurement error tends to bias the percentiles downward.
  - Therefore, under these assumptions, the reported results can be interpreted as a lower bound for the long run gaps.
- Alternative interpretations include that changing the unit of observation from countries to individuals may affect the result.

- Acknowledgement: "We thank Jiajia Gu for suggesting this robustness check." (footnote 11)

*When will Global Gender Gaps Close? Working Paper No. WP/2023/189*

---


_Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023189-print-pdf.pdf_